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

Newform orbit 123.4.a.a

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 4q^{2} + 12q^{3} + 6q^{4} - 33q^{5} - 12q^{6} - 28q^{7} - 84q^{8} + 36q^{9} - 13q^{10} - 96q^{11} + 18q^{12} - 39q^{13} + 40q^{14} - 99q^{15} + 170q^{16} - 205q^{17} - 36q^{18} + 47q^{19} + 11q^{20} - 84q^{21} - 34q^{22} - 150q^{23} - 252q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} + 37x^{3} + 420x^{2} + 1462x + 268$

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 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	12	−33	−28	−96	−39	−205	47	−150	−262	−259	−430	−164	258	−904	230	−239	1226	857	−1513	−271	−906	−1039	321	148

$\operatorname{tr} a_n$ 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)[0]   # 123.4.a.a

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

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