Atlas › Modular forms › Weight 4 › Level 122 › 122.4.a › 122.4.a.a

Newform orbit 122.4.a.a

Trace form

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

$2q + 4q^{2} - 4q^{3} + 8q^{4} - 22q^{5} - 8q^{6} - 40q^{7} + 16q^{8} - 22q^{9} - 44q^{10} - 12q^{11} - 16q^{12} - 66q^{13} - 80q^{14} - 28q^{15} + 32q^{16} + 24q^{17} - 44q^{18} + 108q^{19} - 88q^{20} + 20q^{21} - 24q^{22} - 152q^{23} - 32q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{2} + 70x + 457$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
61	$+1$

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	−4	−22	−40	−12	−66	24	108	−152	−180	−24	−408	162	−112	216	260	−348	−122	−844	−224	−810	704	2280	−360	−380

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.115 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, 2, 61, 122]: [1, 2, 17, 49] (certified at ell = 2147482921)
T = [(3, 1), (5, 3)]; h of degree 15 = 1 x 15 (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(122, 4)[0]   # 122.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 122:4 1000