Atlas › Modular forms › Weight 2 › Level 307 › 307.2.a › 307.2.a.f

Newform orbit 307.2.a.f

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 3q^{2} + q^{3} + 13q^{4} + 5q^{5} - 6q^{6} - 5q^{7} + 24q^{8} + 16q^{9} - 2q^{10} - 9q^{12} - 3q^{13} - 4q^{14} + 10q^{15} + 25q^{16} + 3q^{17} - 9q^{18} - q^{19} + 16q^{20} + 10q^{22} + 16q^{23} - 29q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{9} - 6x^{8} - 173x^{7} + 840x^{6} + 10807x^{5} - 34968x^{4} - 299362x^{3} + 304682x^{2} + 3415922x + 4051801$

Atkin–Lehner signs

$p$	$w_p$
307	$-1$

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
$\operatorname{tr} a_p$	3	1	5	−5	0	−3	3	−1	16	21	−4	23	10	−11	29	55	−7	−12	−4	−7	−7	−32	4	4	−26

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.176 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, 307]: [0, 26] (certified at ell = 2147478013)
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(307, 2)[5]   # 307.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 307:2 1000