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

Newform orbit 935.2.a.i

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{9} - 321x^{8} + 37662x^{7} - 1673175x^{6} - 18029415x^{5} + 4064262457x^{4} - 110483585239x^{3} - 18830643442x^{2} + 33594374545822x - 272608762193991$

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
$\operatorname{tr} a_p$	3	3	−9	8	9	23	−9	12	−1	−19	14	15	−7	33	−10	18	−7	13	8	−16	30	−21	33	−11	27

$\operatorname{tr} a_n$ 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)[8]   # 935.2.a.i

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

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