Atlas › Modular forms › Weight 2 › Level 303 › 303.2.a › 303.2.a.e

Newform orbit 303.2.a.e

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{7} - 18x^{6} - 171x^{5} + 4302x^{4} - 5210x^{3} - 231371x^{2} + 1102212x - 738193$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
101	$-1$

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
$\operatorname{tr} a_p$	0	−7	6	6	−10	10	20	2	6	−10	10	8	4	−4	0	0	−16	12	2	−20	8	14	4	10	52

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.187 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, 101, 303]: [0, 1, 9, 36] (certified at ell = 2147482801)
T = [(2, 1), (5, 3)]; h of degree 17 = 1 x 17 (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(303, 2)[4]   # 303.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 303:2 1000