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

Newform orbit 613.2.a.a

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{5} + 9x^{4} + 25x^{3} + 13x^{2} - 34x - 31$

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 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
$\operatorname{tr} a_p$	−4	−5	0	−2	1	6	−2	−8	−3	−16	−10	9	10	−22	−29	−12	−38	−5	−20	−26	−25	5	−35	15	14

$\operatorname{tr} a_n$ 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)[0]   # 613.2.a.a

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

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