Atlas › Modular forms › Weight 2 › Level 301 › 301.2.a › 301.2.a.d

Newform orbit 301.2.a.d

Trace form

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

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

Coefficient field

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

$x^{7} - 5x^{6} - 13x^{5} + 83x^{4} + 11x^{3} - 334x^{2} + 201x + 4$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
43	$+1$

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
$\operatorname{tr} a_p$	4	1	0	7	16	−2	4	0	6	12	8	−7	14	−7	1	15	45	−20	7	13	−19	12	−6	−17	−15

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.181 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, 7, 43, 301]: [0, 1, 4, 30] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; 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(301, 2)[3]   # 301.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 301:2 1000