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

Newform orbit 122.4.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 6q^{2} - 2q^{3} + 12q^{4} - q^{5} + 4q^{6} - 31q^{7} - 24q^{8} + 27q^{9} + 2q^{10} - 15q^{11} - 8q^{12} - 21q^{13} + 62q^{14} - 106q^{15} + 48q^{16} - 30q^{17} - 54q^{18} - 272q^{19} - 4q^{20} - 220q^{21} + 30q^{22} - 139q^{23} + 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{3} + 5x^{2} - 205x - 1241$

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
$\operatorname{tr} a_p$	−6	−2	−1	−31	−15	−21	−30	−272	−139	−56	−50	−82	771	−370	−312	534	725	183	−1411	1166	−773	−1091	−1344	1452	−1910

$\operatorname{tr} a_n$ 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)[1]   # 122.4.a.b

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

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