Atlas › Modular forms › Weight 6 › Level 78 › 78.6.a › 78.6.a.h

Newform orbit 78.6.a.h

Trace form

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

$2q + 8q^{2} + 18q^{3} + 32q^{4} + 10q^{5} + 72q^{6} + 138q^{7} + 128q^{8} + 162q^{9} + 40q^{10} + 544q^{11} + 288q^{12} + 338q^{13} + 552q^{14} + 90q^{15} + 512q^{16} + 1228q^{17} + 648q^{18} - 522q^{19} + 160q^{20} + 1242q^{21} + 2176q^{22} - 1632q^{23} + 1152q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3241})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{2} - 424x + 31980$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$-1$
13	$-1$

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	18	10	138	544	338	1228	−522	−1632	2208	874	−2160	−12638	−17760	4300	−12020	−27532	−38016	−90974	−4384	−95312	−62168	137892	142534	4896

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.301 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, 13, 26, 39, 78]: [1, 2, 3, 7, 7, 19, 26, 74] (certified at ell = 2147483629)
T = [(5, 1), (7, 3)]; h of degree 10 = 1 x 10 (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(78, 6)[7]   # 78.6.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 78:6 1000