Atlas › Modular forms › Weight 2 › Level 769 › 769.2.a

Newspace 769.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(769))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
769.2.a.a	27	degree 27	$27q - 10q^{2} - 7q^{3} + 18q^{4} - 13q^{5} - 7q^{6} - 10q^{7} + O(q^{8})$	+	$-1$
769.2.a.b	36	degree 36	$36q + 9q^{2} + 5q^{3} + 41q^{4} + 9q^{5} + q^{6} + 4q^{7} + O(q^{8})$	−	$+1$

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 $T$
769.2.a.a	$x^{27} + 17x^{26} + 70x^{25} - 343x^{24} - 3200x^{23} - 1471x^{22} + 47032x^{21} + 95570x^{20} - 321142x^{19} - 1068362x^{18} + 1000967x^{17} + 5832778x^{16} - 590104x^{15} - 18026634x^{14} - 4441456x^{13} + 33288006x^{12} + 12128778x^{11} - 37043998x^{10} - 13108237x^{9} + 23968887x^{8} + 6592224x^{7} - 8217590x^{6} - 1444034x^{5} + 1261965x^{4} + 112644x^{3} - 63198x^{2} + 283x + 467$
769.2.a.b	$x^{36} - 14x^{35} - 37x^{34} + 1343x^{33} - 2353x^{32} - 52707x^{31} + 192183x^{30} + 1092332x^{29} - 5838901x^{28} - 12571208x^{27} + 101409946x^{26} + 67859468x^{25} - 1133850197x^{24} + 130386267x^{23} + 8599706377x^{22} - 4894058306x^{21} - 45371818476x^{20} + 37602076656x^{19} + 168215212210x^{18} - 159539933577x^{17} - 438202534075x^{16} + 416000106904x^{15} + 794480967220x^{14} - 670623682943x^{13} - 979437255473x^{12} + 644088425794x^{11} + 780763898102x^{10} - 342502796990x^{9} - 363231499724x^{8} + 92345891290x^{7} + 81680869742x^{6} - 14947645301x^{5} - 7358069085x^{4} + 1522330714x^{3} + 126999713x^{2} - 40068650x + 1935563$

Orbit

Characteristic polynomial of

T

769.2.a.a

x^{27} + 17x^{26} + 70x^{25} - 343x^{24} - 3200x^{23} - 1471x^{22} + 47032x^{21} + 95570x^{20} - 321142x^{19} - 1068362x^{18} + 1000967x^{17} + 5832778x^{16} - 590104x^{15} - 18026634x^{14} - 4441456x^{13} + 33288006x^{12} + 12128778x^{11} - 37043998x^{10} - 13108237x^{9} + 23968887x^{8} + 6592224x^{7} - 8217590x^{6} - 1444034x^{5} + 1261965x^{4} + 112644x^{3} - 63198x^{2} + 283x + 467

769.2.a.b

x^{36} - 14x^{35} - 37x^{34} + 1343x^{33} - 2353x^{32} - 52707x^{31} + 192183x^{30} + 1092332x^{29} - 5838901x^{28} - 12571208x^{27} + 101409946x^{26} + 67859468x^{25} - 1133850197x^{24} + 130386267x^{23} + 8599706377x^{22} - 4894058306x^{21} - 45371818476x^{20} + 37602076656x^{19} + 168215212210x^{18} - 159539933577x^{17} - 438202534075x^{16} + 416000106904x^{15} + 794480967220x^{14} - 670623682943x^{13} - 979437255473x^{12} + 644088425794x^{11} + 780763898102x^{10} - 342502796990x^{9} - 363231499724x^{8} + 92345891290x^{7} + 81680869742x^{6} - 14947645301x^{5} - 7358069085x^{4} + 1522330714x^{3} + 126999713x^{2} - 40068650x + 1935563

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(769))$	63
Old forms	0
Cusp forms $S_{2}(\Gamma_0(769))$	63
Eisenstein series $E_{2}(\Gamma_0(769))$	1
Modular forms $M_{2}(\Gamma_0(769))$	64

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 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, 769]: [0, 64] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 63 = 1 x 63 (phi(ord eps) x dim S_k^new); 7 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(769, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 769:2 1000

← 768.2.a · 770.2.a →