Atlas › Modular forms › Weight 2 › Level 591 › 591.2.a › 591.2.a.d

Newform orbit 591.2.a.d

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{13} - 17x^{12} + 86x^{11} + 59x^{10} - 1937x^{9} + 5844x^{8} + 79x^{7} - 37198x^{6} + 94464x^{5} - 115669x^{4} + 77153x^{3} - 25982x^{2} + 2816x + 300$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
197	$+1$

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
$\operatorname{tr} a_p$	5	13	12	−1	20	−2	12	−1	13	7	−12	−9	17	−9	19	10	26	1	−14	20	−8	−12	23	54	−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 0.211 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, 197, 591]: [0, 1, 17, 68] (certified at ell = 2147479489)
T = [(2, 1), (5, 1)]; h of degree 33 = 1 x 33 (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(591, 2)[3]   # 591.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 591:2 1000