Atlas › Modular forms › Weight 2 › Level 599 › 599.2.a › 599.2.a.c

Newform orbit 599.2.a.c

Trace form

The 37 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$37q + q^{2} + 4q^{3} + 47q^{4} + 6q^{5} + 8q^{6} + 12q^{7} - 3q^{8} + 57q^{9} + 20q^{10} + 3q^{11} + 5q^{12} + 26q^{13} - q^{14} - 6q^{15} + 59q^{16} + 27q^{17} - 3q^{18} + 39q^{19} - 3q^{20} + 25q^{21} - 8q^{22} - 6q^{23} + 20q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 37. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{37} - 5x^{36} - 140x^{35} + 720x^{34} + 8503x^{33} - 44991x^{32} - 297030x^{31} + 1613223x^{30} + 6704963x^{29} - 37073173x^{28} - 104844093x^{27} + 578847565x^{26} + 1196385071x^{25} - 6356681407x^{24} - 10390984350x^{23} + 50086690359x^{22} + 70731555877x^{21} - 285484119961x^{20} - 381023743595x^{19} + 1172385899121x^{18} + 1604123335160x^{17} - 3400075844553x^{16} - 5121156887506x^{15} + 6638797681253x^{14} + 11885849152391x^{13} - 7758284305057x^{12} - 18983435215631x^{11} + 3299396094546x^{10} + 19270135316239x^{9} + 3457053578571x^{8} - 10809350342055x^{7} - 4938041777852x^{6} + 2355386131976x^{5} + 1848416194658x^{4} + 77052534928x^{3} - 133981873631x^{2} - 11985208190x + 2578689028$

Atkin–Lehner signs

$p$	$w_p$
599	$-1$

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
$\operatorname{tr} a_p$	1	4	6	12	3	26	27	39	−6	9	22	20	40	32	−8	−5	−7	106	−21	−31	88	26	14	47	40

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.218 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, 599]: [0, 51] (certified at ell = 2147480791)
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(599, 2)[2]   # 599.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 599:2 1000