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

Newform orbit 769.2.a.b

Trace form

The 36 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$36q + 9q^{2} + 5q^{3} + 41q^{4} + 9q^{5} + q^{6} + 4q^{7} + 24q^{8} + 47q^{9} - 6q^{10} + 36q^{11} + 3q^{12} - 4q^{13} + 15q^{14} + 42q^{15} + 51q^{16} + 4q^{17} + 13q^{18} + 5q^{19} + 20q^{20} + 8q^{21} + 7q^{22} + 53q^{23} - 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 36. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

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

Atkin–Lehner signs

$p$	$w_p$
769	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	9	5	9	4	36	−4	4	5	53	40	29	7	17	−5	55	21	71	−21	9	159	−37	32	13	33	−45

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

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)[1]   # 769.2.a.b

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

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