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

Newform orbit 122.4.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 12q^{2} + 5q^{3} + 24q^{4} + 13q^{5} + 10q^{6} + 44q^{7} + 48q^{8} + 113q^{9} + 26q^{10} + 43q^{11} + 20q^{12} + 103q^{13} + 88q^{14} + 122q^{15} + 96q^{16} + 106q^{17} + 226q^{18} - 57q^{19} + 52q^{20} - 288q^{21} + 86q^{22} - 164q^{23} + 40q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{6} - 44x^{5} - 4067x^{4} + 190717x^{3} - 1242320x^{2} - 10570229x + 25666806$

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 even 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
$\operatorname{tr} a_p$	12	5	13	44	43	103	106	−57	−164	231	−101	37	−148	−510	144	−321	−971	366	−371	−1735	1806	185	1019	1880	589

$\operatorname{tr} a_n$ 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)[3]   # 122.4.a.d

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

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