Atlas › Modular forms › Weight 4 › Level 123 › 123.4.a › 123.4.a.c

Newform orbit 123.4.a.c

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 2q^{2} - 18q^{3} + 30q^{4} + 21q^{5} - 6q^{6} + 28q^{7} + 18q^{8} + 54q^{9} - 29q^{10} - 54q^{11} - 90q^{12} + 247q^{13} + 192q^{14} - 63q^{15} + 378q^{16} + 133q^{17} + 18q^{18} + 121q^{19} + 405q^{20} - 84q^{21} + 286q^{22} - 146q^{23} - 54q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{6} - 23x^{5} - 308x^{4} + 9514x^{3} - 25280x^{2} - 353516x + 661608$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
41	$+1$

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
$\operatorname{tr} a_p$	2	−18	21	28	−54	247	133	121	−146	76	135	240	−246	−96	102	874	367	1044	1591	−79	1091	1062	−525	1521	4304

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.112 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, 3, 41, 123]: [1, 2, 12, 44] (certified at ell = 2147482921)
T = [(2, 1), (5, 1)]; h of degree 20 = 1 x 20 (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(123, 4)[2]   # 123.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 123:4 1000