Atlas › Modular forms › Weight 2 › Level 313 › 313.2.a › 313.2.a.b

Newform orbit 313.2.a.b

Trace form

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

$11q - 8q^{2} - 8q^{3} + 10q^{4} - 6q^{5} + q^{6} - 14q^{7} - 18q^{8} + 5q^{9} - q^{10} - 11q^{11} - 12q^{12} - 7q^{13} + 3q^{14} - 11q^{15} + 16q^{16} - 15q^{17} - 6q^{18} + q^{19} - 11q^{20} + 7q^{21} - 8q^{22} - 44q^{23} + 13q^{24} + O(q^{25})$

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

$x^{11} + 32x^{10} + 322x^{9} + 391x^{8} - 11989x^{7} - 61376x^{6} - 6518x^{5} + 449731x^{4} + 325772x^{3} - 1097423x^{2} - 189982x + 264164$

Atkin–Lehner signs

$p$	$w_p$
313	$+1$

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
$\operatorname{tr} a_p$	−8	−8	−6	−14	−11	−7	−15	1	−44	−17	−3	−3	−9	−8	−26	−17	7	14	9	−38	5	−22	−2	18	−19

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.175 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, 313]: [0, 26] (certified at ell = 2147482273)
T = [(2, 1), (3, 3)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 4 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(313, 2)[1]   # 313.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 313:2 1000