Atlas › Modular forms › Weight 4 › Level 121 › 121.4.a › 121.4.a.b

Newform orbit 121.4.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 2q^{2} - 8q^{3} + 10q^{4} - 10q^{5} + 32q^{6} - 8q^{7} - 42q^{8} + 2q^{9} + 34q^{10} - 88q^{12} + 130q^{13} - 160q^{14} + 64q^{15} - 62q^{16} - 14q^{17} - 194q^{18} - 48q^{19} - 98q^{20} - 136q^{21} - 128q^{23} + 144q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} + 26x - 23$

Atkin–Lehner signs

$p$	$w_p$
11	— ( $11^2 \mid 121$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	−8	−10	−8	0	130	−14	−48	−128	−30	−184	126	370	−264	256	−162	−1304	−300	−656	−1176	−668	416	−960	−1074	−338

$\operatorname{tr} a_n$ 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, 11, 121]: [1, 4, 34] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 23 = 1 x 23 (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(121, 4)[1]   # 121.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 121:4 1000