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

Newform orbit 123.4.a.b

Trace form

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

$4q - 12q^{3} + 10q^{4} - 9q^{5} - 14q^{7} - 6q^{8} + 36q^{9} - 89q^{10} + 4q^{11} - 30q^{12} - 155q^{13} - 50q^{14} + 27q^{15} - 198q^{16} - 27q^{17} - 119q^{19} - 133q^{20} + 42q^{21} - 232q^{22} - 70q^{23} + 18q^{24} + O(q^{25})$

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

$x^{4} + 9x^{3} - 90x^{2} + 50x + 300$

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
$\operatorname{tr} a_p$	0	−12	−9	−14	4	−155	−27	−119	−70	−8	−163	−226	164	−266	−126	−528	869	−1726	−577	1447	−1819	1008	−923	−173	−1548

$\operatorname{tr} a_n$ 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)[1]   # 123.4.a.b

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

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