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

Newform orbit 78.6.a.g

Trace form

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

$2q - 8q^{2} - 18q^{3} + 32q^{4} + 2q^{5} + 72q^{6} - 150q^{7} - 128q^{8} + 162q^{9} - 8q^{10} - 820q^{11} - 288q^{12} + 338q^{13} + 600q^{14} - 18q^{15} + 512q^{16} + 1556q^{17} - 648q^{18} + 1470q^{19} + 32q^{20} + 1350q^{21} + 3280q^{22} + 3240q^{23} + 1152q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{433})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{2} + 448x + 48444$

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	2	−150	−820	338	1556	1470	3240	−3912	10978	2448	11666	17328	−3376	−3556	−34376	38832	15250	−113420	6448	11656	173184	−12250	133488

$\operatorname{tr} a_n$ 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)[6]   # 78.6.a.g

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

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