Atlas › Modular forms › Weight 2 › Level 977 › 977.2.a

Newspace 977.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(977))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
977.2.a.a	36	degree 36	$36q - 7q^{2} - 13q^{3} + 27q^{4} - 9q^{5} - 9q^{6} - 48q^{7} + O(q^{8})$	+	$-1$
977.2.a.b	45	degree 45	$45q + 4q^{2} + 11q^{3} + 50q^{4} + 7q^{5} + 9q^{6} + 52q^{7} + O(q^{8})$	−	$+1$

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 $T$
977.2.a.a	$x^{36} + 20x^{35} + 90x^{34} - 705x^{33} - 6884x^{32} + 1234x^{31} + 177266x^{30} + 322792x^{29} - 2398729x^{28} - 7502773x^{27} + 18986388x^{26} + 88015076x^{25} - 86895248x^{24} - 650469144x^{23} + 176370496x^{22} + 3279167627x^{21} + 326756487x^{20} - 11738715979x^{19} - 3300949531x^{18} + 30503830560x^{17} + 10376051957x^{16} - 58062118793x^{15} - 18234977617x^{14} + 80627719655x^{13} + 18718336313x^{12} - 80115706472x^{11} - 9758383867x^{10} + 54845297239x^{9} + 449046195x^{8} - 24351491779x^{7} + 2137774075x^{6} + 6430377324x^{5} - 975455369x^{4} - 899852510x^{3} + 165152072x^{2} + 51142851x - 9604225$
977.2.a.b	$x^{45} - 15x^{44} - 58x^{43} + 1907x^{42} - 2628x^{41} - 103193x^{40} + 371508x^{39} + 3065795x^{38} - 16825905x^{37} - 52434343x^{36} + 436438933x^{35} + 436546265x^{34} - 7389588468x^{33} + 1263743756x^{32} + 86180371017x^{31} - 81363164913x^{30} - 709398132393x^{29} + 1091470949780x^{28} + 4149257894417x^{27} - 8639311845724x^{26} - 17050218115339x^{25} + 46058594083602x^{24} + 47150484091226x^{23} - 171912943577296x^{22} - 76231107861469x^{21} + 451841175868129x^{20} + 22312041908433x^{19} - 822447675184403x^{18} + 200989550928649x^{17} + 992411712342427x^{16} - 487105253783233x^{15} - 721858335993547x^{14} + 540011736928757x^{13} + 244972307752156x^{12} - 298680371746719x^{11} + 5154010068510x^{10} + 67735604052263x^{9} - 16538634321426x^{8} - 5048282622886x^{7} + 2573114273316x^{6} - 149394360749x^{5} - 98563983248x^{4} + 22384380766x^{3} - 1641080679x^{2} + 3712900x + 3202207$

Orbit

Characteristic polynomial of

T

977.2.a.a

x^{36} + 20x^{35} + 90x^{34} - 705x^{33} - 6884x^{32} + 1234x^{31} + 177266x^{30} + 322792x^{29} - 2398729x^{28} - 7502773x^{27} + 18986388x^{26} + 88015076x^{25} - 86895248x^{24} - 650469144x^{23} + 176370496x^{22} + 3279167627x^{21} + 326756487x^{20} - 11738715979x^{19} - 3300949531x^{18} + 30503830560x^{17} + 10376051957x^{16} - 58062118793x^{15} - 18234977617x^{14} + 80627719655x^{13} + 18718336313x^{12} - 80115706472x^{11} - 9758383867x^{10} + 54845297239x^{9} + 449046195x^{8} - 24351491779x^{7} + 2137774075x^{6} + 6430377324x^{5} - 975455369x^{4} - 899852510x^{3} + 165152072x^{2} + 51142851x - 9604225

977.2.a.b

x^{45} - 15x^{44} - 58x^{43} + 1907x^{42} - 2628x^{41} - 103193x^{40} + 371508x^{39} + 3065795x^{38} - 16825905x^{37} - 52434343x^{36} + 436438933x^{35} + 436546265x^{34} - 7389588468x^{33} + 1263743756x^{32} + 86180371017x^{31} - 81363164913x^{30} - 709398132393x^{29} + 1091470949780x^{28} + 4149257894417x^{27} - 8639311845724x^{26} - 17050218115339x^{25} + 46058594083602x^{24} + 47150484091226x^{23} - 171912943577296x^{22} - 76231107861469x^{21} + 451841175868129x^{20} + 22312041908433x^{19} - 822447675184403x^{18} + 200989550928649x^{17} + 992411712342427x^{16} - 487105253783233x^{15} - 721858335993547x^{14} + 540011736928757x^{13} + 244972307752156x^{12} - 298680371746719x^{11} + 5154010068510x^{10} + 67735604052263x^{9} - 16538634321426x^{8} - 5048282622886x^{7} + 2573114273316x^{6} - 149394360749x^{5} - 98563983248x^{4} + 22384380766x^{3} - 1641080679x^{2} + 3712900x + 3202207

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(977))$	81
Old forms	0
Cusp forms $S_{2}(\Gamma_0(977))$	81
Eisenstein series $E_{2}(\Gamma_0(977))$	1
Modular forms $M_{2}(\Gamma_0(977))$	82

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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, 977]: [0, 82] (certified at ell = 2147475233)
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(977, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 977:2 1000

← 976.2.a · 978.2.a →