Atlas › Modular forms › Weight 2 › Level 971 › 971.2.a

Newspace 971.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(971))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
971.2.a.a	26	degree 26	$26q - 4q^{2} - 6q^{3} + 8q^{4} - 6q^{5} - 7q^{6} - 12q^{7} + O(q^{8})$	+	$-1$
971.2.a.b	55	degree 55	$55q + 4q^{2} + 4q^{3} + 76q^{4} + 8q^{5} + 9q^{6} + 14q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
971.2.a.a	$x^{26} + 10x^{25} - 13x^{24} - 401x^{23} - 469x^{22} + 6540x^{21} + 13101x^{20} - 58595x^{19} - 141229x^{18} + 326346x^{17} + 838628x^{16} - 1210783x^{15} - 2996269x^{14} + 3123471x^{13} + 6539308x^{12} - 5686530x^{11} - 8443126x^{10} + 7085586x^{9} + 5859193x^{8} - 5502018x^{7} - 1627542x^{6} + 2226749x^{5} - 109029x^{4} - 321213x^{3} + 69112x^{2} + 5755x - 1825$
971.2.a.b	$x^{55} - 8x^{54} - 192x^{53} + 1653x^{52} + 16467x^{51} - 155552x^{50} - 832490x^{49} + 8859951x^{48} + 27513533x^{47} - 342552565x^{46} - 620255808x^{45} + 9560336810x^{44} + 9542568603x^{43} - 200039004039x^{42} - 93479542287x^{41} + 3218125643891x^{40} + 381456721201x^{39} - 40508476201694x^{38} + 4188178170046x^{37} + 403903807198794x^{36} - 93725186018064x^{35} - 3216694666905702x^{34} + 932832887759030x^{33} + 20563759089771175x^{32} - 6049516113460207x^{31} - 105729683545596067x^{30} + 27359377150002232x^{29} + 436802720088623628x^{28} - 85803059029424205x^{27} - 1444722223339655247x^{26} + 171553326681309108x^{25} + 3802312480220463621x^{24} - 136637826187435892x^{23} - 7898183377210371285x^{22} - 343242990581223620x^{21} + 12821856658593940981x^{20} + 1447058216365319705x^{19} - 16086245699443563870x^{18} - 2696063266367872796x^{17} + 15401865488221484443x^{16} + 3151020663404426835x^{15} - 11091187643226348311x^{14} - 2459346459369150402x^{13} + 5899575793486099503x^{12} + 1282287035949473613x^{11} - 2262199897609493302x^{10} - 428839964626186720x^{9} + 603464765180898600x^{8} + 82225463481453372x^{7} - 105741192233558918x^{6} - 5918747055723043x^{5} + 10908247964279863x^{4} - 532820999319964x^{3} - 489878290830202x^{2} + 83074878499544x - 3818667782183$

Orbit

Characteristic polynomial of

T

971.2.a.a

x^{26} + 10x^{25} - 13x^{24} - 401x^{23} - 469x^{22} + 6540x^{21} + 13101x^{20} - 58595x^{19} - 141229x^{18} + 326346x^{17} + 838628x^{16} - 1210783x^{15} - 2996269x^{14} + 3123471x^{13} + 6539308x^{12} - 5686530x^{11} - 8443126x^{10} + 7085586x^{9} + 5859193x^{8} - 5502018x^{7} - 1627542x^{6} + 2226749x^{5} - 109029x^{4} - 321213x^{3} + 69112x^{2} + 5755x - 1825

971.2.a.b

x^{55} - 8x^{54} - 192x^{53} + 1653x^{52} + 16467x^{51} - 155552x^{50} - 832490x^{49} + 8859951x^{48} + 27513533x^{47} - 342552565x^{46} - 620255808x^{45} + 9560336810x^{44} + 9542568603x^{43} - 200039004039x^{42} - 93479542287x^{41} + 3218125643891x^{40} + 381456721201x^{39} - 40508476201694x^{38} + 4188178170046x^{37} + 403903807198794x^{36} - 93725186018064x^{35} - 3216694666905702x^{34} + 932832887759030x^{33} + 20563759089771175x^{32} - 6049516113460207x^{31} - 105729683545596067x^{30} + 27359377150002232x^{29} + 436802720088623628x^{28} - 85803059029424205x^{27} - 1444722223339655247x^{26} + 171553326681309108x^{25} + 3802312480220463621x^{24} - 136637826187435892x^{23} - 7898183377210371285x^{22} - 343242990581223620x^{21} + 12821856658593940981x^{20} + 1447058216365319705x^{19} - 16086245699443563870x^{18} - 2696063266367872796x^{17} + 15401865488221484443x^{16} + 3151020663404426835x^{15} - 11091187643226348311x^{14} - 2459346459369150402x^{13} + 5899575793486099503x^{12} + 1282287035949473613x^{11} - 2262199897609493302x^{10} - 428839964626186720x^{9} + 603464765180898600x^{8} + 82225463481453372x^{7} - 105741192233558918x^{6} - 5918747055723043x^{5} + 10908247964279863x^{4} - 532820999319964x^{3} - 489878290830202x^{2} + 83074878499544x - 3818667782183

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(971))$	81
Old forms	0
Cusp forms $S_{2}(\Gamma_0(971))$	81
Eisenstein series $E_{2}(\Gamma_0(971))$	1
Modular forms $M_{2}(\Gamma_0(971))$	82

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.315 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 971]: [0, 82] (certified at ell = 2147479121)
T = [(2, 1), (3, 1)]; h of degree 81 = 1 x 81 (phi(ord eps) x dim S_k^new); 8 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(971, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 971:2 1000

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