Atlas › Modular forms › Weight 2 › Level 935 › 935.2.a › 935.2.a.h

Newform orbit 935.2.a.h

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{13} + 9T_{19}$ , whose minimal polynomial is

$x^{6} - 92x^{5} - 8752x^{4} + 1449588x^{3} - 68605567x^{2} + 1358399674x - 9522159621$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
11	$+1$
17	$-1$

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
$\operatorname{tr} a_p$	−2	4	−6	6	−6	10	6	−2	14	−4	4	18	12	−10	20	10	14	2	−34	−18	10	8	−30	−30	58

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.48 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, 5, 11, 17, 55, 85, 187, 935]: [0, 1, 2, 2, 8, 10, 20, 112] (certified at ell = 2147482801)
T = [(2, 1), (3, 3), (7, 5), (13, 7), (19, 9)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 14 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(935, 2)[7]   # 935.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 935:2 1000