Atlas › Modular forms › Weight 2 › Level 907 › 907.2.a › 907.2.a.a

Newform orbit 907.2.a.a

Trace form

The 35 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$35q - 11q^{2} - 13q^{3} + 33q^{4} - 32q^{5} - 8q^{6} - 9q^{7} - 30q^{8} + 26q^{9} - 9q^{10} - 14q^{11} - 36q^{12} - 33q^{13} - 10q^{14} + 33q^{16} - 63q^{17} - 18q^{18} - 3q^{19} - 50q^{20} - 28q^{21} - 3q^{22} - 30q^{23} - 14q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 35. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{35} + 24x^{34} + 179x^{33} - 134x^{32} - 8203x^{31} - 28049x^{30} + 112787x^{29} + 827652x^{28} - 65266x^{27} - 10819454x^{26} - 14916130x^{25} + 78685703x^{24} + 192230325x^{23} - 330606103x^{22} - 1253585393x^{21} + 735285896x^{20} + 5033610931x^{19} - 426025111x^{18} - 13287254443x^{17} - 1716065843x^{16} + 23722929293x^{15} + 3997397786x^{14} - 28599392441x^{13} - 2766315067x^{12} + 22270776836x^{11} - 474862348x^{10} - 10190568684x^{9} + 1160328828x^{8} + 2530922795x^{7} - 411769952x^{6} - 303457548x^{5} + 49285123x^{4} + 13467097x^{3} - 1124818x^{2} - 161172x + 4792$

Atkin–Lehner signs

$p$	$w_p$
907	$+1$

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
$\operatorname{tr} a_p$	−11	−13	−32	−9	−14	−33	−63	−3	−30	−66	−10	−26	−36	−25	−50	−71	−1	−39	−1	8	−47	2	−61	−34	−3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.292 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, 907]: [0, 76] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 75 = 1 x 75 (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(907, 2)[0]   # 907.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 907:2 1000