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

Newform orbit 307.2.a.g

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{10} + 19x^{9} + 34x^{8} - 1102x^{7} - 5887x^{6} + 2605x^{5} + 55115x^{4} + 40175x^{3} - 66063x^{2} - 26982x + 17348$

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 odd 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
$\operatorname{tr} a_p$	−7	−4	−15	−3	−10	−11	−17	1	−20	−22	1	−28	7	−4	−21	−59	−3	2	8	7	1	12	−19	−16	−17

$\operatorname{tr} a_n$ 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)[6]   # 307.2.a.g

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

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