Atlas › Modular forms › Weight 2 › Level 335 › 335.2.a › 335.2.a.e

Newform orbit 335.2.a.e

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{11} - 273x^{9} + 239x^{8} + 25323x^{7} - 47964x^{6} - 950391x^{5} + 2548553x^{4} + 12225995x^{3} - 35463805x^{2} - 41689188x + 109010584$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
67	$+1$

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
$\operatorname{tr} a_p$	0	0	11	4	6	4	2	10	−10	26	−8	12	14	14	−16	−2	2	20	−11	−24	16	8	−36	14	20

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.187 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, 67, 335]: [0, 1, 6, 36] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 3 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(335, 2)[4]   # 335.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 335:2 1000