Atlas › Modular forms › Weight 6 › Level 70 › 70.6.a › 70.6.a.h

Newform orbit 70.6.a.h

Trace form

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

$2q + 8q^{2} + 5q^{3} + 32q^{4} + 50q^{5} + 20q^{6} + 98q^{7} + 128q^{8} + 91q^{9} + 200q^{10} + 415q^{11} + 80q^{12} + 429q^{13} + 392q^{14} + 125q^{15} + 512q^{16} + 1319q^{17} + 364q^{18} + 1918q^{19} + 800q^{20} + 245q^{21} + 1660q^{22} - 1334q^{23} + 320q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{1129})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{11}$ , whose minimal polynomial is

$x^{2} - 420x + 3456$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$-1$
7	$-1$

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	5	50	98	415	429	1319	1918	−1334	−1131	−5472	−9156	−7822	2078	−8507	−34242	6952	−45138	−54556	−62272	40896	62069	189600	64190	155547

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.287 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, 5, 7, 10, 14, 35, 70]: [1, 2, 3, 5, 9, 12, 22, 64] (certified at ell = 2147483629)
T = [(3, 1), (11, 1)]; 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(70, 6)[7]   # 70.6.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 70:6 1000