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

Newform orbit 313.2.a.c

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 6q^{2} + q^{3} + 16q^{4} - q^{5} + 2q^{6} + 6q^{7} + 21q^{8} + 11q^{9} - 9q^{10} + 9q^{11} + 4q^{12} - 8q^{13} + 5q^{14} + 7q^{15} + 16q^{16} + 7q^{17} + 16q^{18} - q^{19} - 12q^{20} - 3q^{21} - 3q^{22} + 41q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{12} - 9x^{11} - 197x^{10} + 1614x^{9} + 14584x^{8} - 102562x^{7} - 502992x^{6} + 2816990x^{5} + 7873681x^{4} - 30640686x^{3} - 41742920x^{2} + 68223533x - 19671289$

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 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
$\operatorname{tr} a_p$	6	1	−1	6	9	−8	7	−1	41	29	−14	5	−15	21	34	19	13	−28	−9	45	−16	4	12	−3	−31

$\operatorname{tr} a_n$ 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)[2]   # 313.2.a.c

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

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