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

Newform orbit 313.2.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} - 10x + 20$

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
$\operatorname{tr} a_p$	1	3	3	2	0	7	4	−2	5	−16	13	−16	12	−21	−2	4	2	6	10	−7	−17	−2	6	1	−6

$\operatorname{tr} a_n$ 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)[0]   # 313.2.a.a

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

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