Atlas › Modular forms › Weight 4 › Level 127 › 127.4.a › 127.4.a.c

Newform orbit 127.4.a.c

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q + 9q^{2} + 22q^{3} + 81q^{4} + 69q^{5} + 26q^{6} + 55q^{7} + 60q^{8} + 191q^{9} + 60q^{10} + 130q^{11} + 49q^{12} + 131q^{13} + 103q^{14} - 2q^{15} + 361q^{16} + 510q^{17} + 177q^{18} + 142q^{19} + 362q^{20} - 38q^{21} - 194q^{22} + 509q^{23} + 660q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{17} - 31x^{16} + 21x^{15} + 7971x^{14} - 55846x^{13} - 752735x^{12} + 8286587x^{11} + 28837526x^{10} - 537189110x^{9} - 71778389x^{8} + 17686772875x^{7} - 25528604560x^{6} - 287781173457x^{5} + 644721219507x^{4} + 1904757844993x^{3} - 4811436431847x^{2} - 1606766330658x + 1622846153868$

Atkin–Lehner signs

$p$	$w_p$
127	$+1$

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
$\operatorname{tr} a_p$	9	22	69	55	130	131	510	142	509	416	55	345	1010	−964	719	729	1358	−339	1768	−897	1167	743	2046	4300	3910

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.109 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 engine's checks

dim M^+ at levels [1, 127]: [1, 33] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 31 = 1 x 31 (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(127, 4)[2]   # 127.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 127:4 1000