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

Newform orbit 122.4.a.c

Trace form

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

$4q - 8q^{2} + 7q^{3} + 16q^{4} - 6q^{5} - 14q^{6} + 39q^{7} - 32q^{8} + 27q^{9} + 12q^{10} + 40q^{11} + 28q^{12} + 96q^{13} - 78q^{14} - 76q^{15} + 64q^{16} + 58q^{17} - 54q^{18} + 235q^{19} - 24q^{20} + 340q^{21} - 80q^{22} + 287q^{23} - 56q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{4} + 11x^{3} - 4540x^{2} - 23565x + 5132477$

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
$\operatorname{tr} a_p$	−8	7	−6	39	40	96	58	235	287	−61	163	495	−207	−192	−500	−1243	158	−244	682	−963	7	1054	−89	−862	765

$\operatorname{tr} a_n$ 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)[2]   # 122.4.a.c

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

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