Atlas › Modular forms › Weight 2 › Level 305 › 305.2.a › 305.2.a.b

Newform orbit 305.2.a.b

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{4} + 21x^{3} + 122x^{2} + 96x - 568$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
61	$-1$

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
$\operatorname{tr} a_p$	−3	−6	4	−10	−2	−1	−4	−7	−13	5	−21	−15	6	−1	−11	3	2	4	−29	3	−2	−21	−16	0	−8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.185 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, 5, 61, 305]: [0, 1, 5, 32] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 3 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(305, 2)[1]   # 305.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 305:2 1000