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

Newform orbit 309.2.a.c

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{5} + 7x^{4} + x^{3} - 67x^{2} - 117x - 36$

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 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
$\operatorname{tr} a_p$	−2	−5	−5	−2	−12	1	−10	−16	−5	−16	−13	4	−20	7	8	−8	−19	−19	11	−10	20	−14	−11	22	7

$\operatorname{tr} a_n$ 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)[2]   # 309.2.a.c

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

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