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

Newform orbit 907.2.a.b

Trace form

The 40 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$40q + 9q^{2} + 9q^{3} + 41q^{4} + 32q^{5} + 8q^{6} + 7q^{7} + 24q^{8} + 43q^{9} + 7q^{10} + 12q^{11} + 14q^{12} + 33q^{13} + 6q^{14} + 35q^{16} + 59q^{17} + 12q^{18} + 3q^{19} + 48q^{20} + 4q^{21} - 3q^{22} + 32q^{23} + 10q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 40. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{40} - 18x^{39} + 12x^{38} + 1612x^{37} - 7798x^{36} - 53489x^{35} + 452550x^{34} + 613879x^{33} - 12784203x^{32} + 8867260x^{31} + 213748620x^{30} - 409588525x^{29} - 2254714071x^{28} + 6676332596x^{27} + 15112581324x^{26} - 64496631229x^{25} - 60385119142x^{24} + 411638293761x^{23} + 99176513746x^{22} - 1808372829236x^{21} + 287955436780x^{20} + 5556032573757x^{19} - 2167330400564x^{18} - 11946076943105x^{17} + 6076557526133x^{16} + 17727993474801x^{15} - 9694031691242x^{14} - 17571556804232x^{13} + 9321497496307x^{12} + 10929268310246x^{11} - 5361732747689x^{10} - 3813362781411x^{9} + 1779012935785x^{8} + 608533323533x^{7} - 299200012603x^{6} - 24967191090x^{5} + 17231424608x^{4} - 22937076x^{3} - 308878180x^{2} + 9586136x - 58328$

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 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	9	32	7	12	33	59	3	32	58	−6	26	42	5	26	67	17	49	−27	14	35	−18	37	28	7

$\operatorname{tr} a_n$ 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)[1]   # 907.2.a.b

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

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