Atlas › Modular forms › Weight 2 › Level 314 › 314.2.a › 314.2.a.b

Newform orbit 314.2.a.b

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{6} - 4x^{5} - 39x^{4} + 184x^{3} + 104x^{2} - 710x - 607$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
157	$-1$

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	3	1	3	9	4	−7	17	−3	−5	18	14	−12	14	−8	−25	−6	−17	23	−24	−20	1	0	9	−2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.183 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, 2, 157, 314]: [0, 1, 13, 41] (certified at ell = 2147483053)
T = [(3, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 2 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(314, 2)[1]   # 314.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 314:2 1000