Atlas › Modular forms › Weight 2 › Level 309 › 309.2.a › 309.2.a.d

Newform orbit 309.2.a.d

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{8} + 2x^{7} - 42x^{6} - 8x^{5} + 430x^{4} - 271x^{3} - 948x^{2} + 688x - 64$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
103	$+1$

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	8	−1	6	6	9	−4	16	−11	0	17	−6	12	9	−6	−16	11	5	5	−10	14	14	−23	−14	3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.182 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, 3, 103, 309]: [0, 1, 9, 36] (certified at ell = 2147482501)
T = [(2, 1), (5, 1)]; h of degree 17 = 1 x 17 (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(309, 2)[3]   # 309.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 309:2 1000