Atlas › Modular forms › Weight 2 › Level 757 › 757.2.a › 757.2.a.a

Newform orbit 757.2.a.a

Trace form

The 29 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$29q - 12q^{2} - 21q^{3} + 26q^{4} - 11q^{5} - 2q^{6} - 10q^{7} - 30q^{8} + 26q^{9} - 8q^{10} - 22q^{11} - 33q^{12} - 21q^{13} - 8q^{14} - 10q^{15} + 24q^{16} - 41q^{17} - 16q^{18} - 24q^{19} - 19q^{20} - 6q^{21} - 3q^{22} - 27q^{23} + 17q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 29. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{29} + 33x^{28} + 448x^{27} + 3023x^{26} + 7887x^{25} - 24935x^{24} - 238720x^{23} - 472466x^{22} + 1256333x^{21} + 7049546x^{20} + 4844554x^{19} - 33273301x^{18} - 70584748x^{17} + 47014821x^{16} + 281419743x^{15} + 123543980x^{14} - 538092861x^{13} - 589007699x^{12} + 476830936x^{11} + 975273994x^{10} - 43093729x^{9} - 803818465x^{8} - 256683136x^{7} + 314482475x^{6} + 184529903x^{5} - 36685867x^{4} - 41821800x^{3} - 4317219x^{2} + 1651455x + 259002$

Atkin–Lehner signs

$p$	$w_p$
757	$+1$

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 odd 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
$\operatorname{tr} a_p$	−12	−21	−11	−10	−22	−21	−41	−24	−27	−17	−4	−13	−25	−39	−54	−43	−69	−3	−27	−9	0	1	−135	−28	8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.244 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, 757]: [0, 63] (certified at ell = 2147478481)
T = [(2, 1), (3, 1)]; h of degree 62 = 1 x 62 (phi(ord eps) x dim S_k^new); 6 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(757, 2)[0]   # 757.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 757:2 1000