Atlas › Modular forms › Weight 2 › Level 761 › 761.2.a › 761.2.a.c

Newform orbit 761.2.a.c

Trace form

The 41 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$41q + 3q^{2} + 4q^{3} + 59q^{4} + 3q^{5} + 27q^{7} + 6q^{8} + 53q^{9} + 7q^{10} + 17q^{11} - 6q^{12} + 12q^{13} + 4q^{14} + 43q^{15} + 91q^{16} - 7q^{17} + 2q^{18} + 35q^{19} + 9q^{21} + 8q^{22} + 19q^{23} - 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 41. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{41} - 7x^{40} - 134x^{39} + 1024x^{38} + 7615x^{37} - 65518x^{36} - 239484x^{35} + 2432883x^{34} + 4526946x^{33} - 58730097x^{32} - 51016422x^{31} + 979718856x^{30} + 280110902x^{29} - 11730892179x^{28} + 649741894x^{27} + 103374052167x^{26} - 24745829475x^{25} - 681132004250x^{24} + 213977973926x^{23} + 3382996941624x^{22} - 1053334721297x^{21} - 12673756488247x^{20} + 3234428292052x^{19} + 35566081848218x^{18} - 5908617440484x^{17} - 73656837641666x^{16} + 4545570935290x^{15} + 109936746229324x^{14} + 4805548616920x^{13} - 114308337205673x^{12} - 15936996253454x^{11} + 79017323572689x^{10} + 17216394676817x^{9} - 34120280319948x^{8} - 9338976785016x^{7} + 8514106590742x^{6} + 2580814418699x^{5} - 1113489866850x^{4} - 337776327640x^{3} + 63879820267x^{2} + 16502153740x - 848766755$

Atkin–Lehner signs

$p$	$w_p$
761	$-1$

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
$\operatorname{tr} a_p$	3	4	3	27	17	12	−7	35	19	−4	90	24	−9	23	9	14	19	11	80	76	−1	122	−30	−16	20

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.253 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, 761]: [0, 64] (certified at ell = 2147466641)
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(761, 2)[2]   # 761.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 761:2 1000