Atlas › Modular forms › Weight 2 › Level 613 › 613.2.a › 613.2.a.c

Newform orbit 613.2.a.c

Trace form

The 27 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$27q + 8q^{2} + 18q^{3} + 28q^{4} + 14q^{5} + 4q^{6} + 6q^{7} + 24q^{8} + 33q^{9} + 2q^{10} + 20q^{11} + 26q^{12} + 3q^{13} + q^{14} + 14q^{15} + 22q^{16} + 14q^{17} + 2q^{18} + 13q^{19} + 27q^{20} - 5q^{22} + 27q^{23} - 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 27. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{27} - 26x^{26} + 236x^{25} - 442x^{24} - 6760x^{23} + 44458x^{22} - 16365x^{21} - 723928x^{20} + 2134990x^{19} + 3184967x^{18} - 25162334x^{17} + 19399408x^{16} + 117802397x^{15} - 239428979x^{14} - 186131255x^{13} + 907940240x^{12} - 284200411x^{11} - 1560486555x^{10} + 1352634548x^{9} + 1144790962x^{8} - 1624872049x^{7} - 232710040x^{6} + 765871749x^{5} - 31404965x^{4} - 129083990x^{3} - 3369997x^{2} + 4054947x + 3577$

Atkin–Lehner signs

$p$	$w_p$
613	$-1$

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
$\operatorname{tr} a_p$	8	18	14	6	20	3	14	13	27	3	6	−11	13	8	70	27	90	−19	28	14	−4	−32	129	23	−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 0.212 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, 613]: [0, 51] (certified at ell = 2147478013)
T = [(2, 1), (3, 1)]; h of degree 50 = 1 x 50 (phi(ord eps) x dim S_k^new); 5 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(613, 2)[2]   # 613.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 613:2 1000