Atlas › Modular forms › Weight 4 › Level 246 › 246.4.a › 246.4.a.h

Newform orbit 246.4.a.h

Trace form

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

$4q + 8q^{2} + 12q^{3} + 16q^{4} + 11q^{5} + 24q^{6} + 42q^{7} + 32q^{8} + 36q^{9} + 22q^{10} + 70q^{11} + 48q^{12} - 35q^{13} + 84q^{14} + 33q^{15} + 64q^{16} + 73q^{17} + 72q^{18} + 97q^{19} + 44q^{20} + 126q^{21} + 140q^{22} - 192q^{23} + 96q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{4} - 137x^{3} + 1156x^{2} + 238168x - 5212224$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$-1$
41	$+1$

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	12	11	42	70	−35	73	97	−192	180	−17	−110	−164	−358	102	−1030	101	458	−71	−1181	−319	1374	−565	−435	−790

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.227 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, 3, 6, 41, 82, 123, 246]: [1, 2, 2, 5, 12, 34, 44, 130] (certified at ell = 2147482921)
T = [(5, 1), (7, 3)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 5 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(246, 4)[7]   # 246.4.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 246:4 1000