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

Newform orbit 70.6.a.g

Trace form

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

$2q + 8q^{2} + 3q^{3} + 32q^{4} - 50q^{5} + 12q^{6} - 98q^{7} + 128q^{8} + 1203q^{9} - 200q^{10} + 959q^{11} + 48q^{12} - 393q^{13} - 392q^{14} - 75q^{15} + 512q^{16} - 2231q^{17} + 4812q^{18} + 3342q^{19} - 800q^{20} - 147q^{21} + 3836q^{22} - 450q^{23} + 192q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3369})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{11}$ , whose minimal polynomial is

$x^{2} - 962x + 227992$

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	3	−50	−98	959	−393	−2231	3342	−450	−4515	−2036	−1928	19318	−14146	−19745	26378	−11104	3554	65208	17104	−36740	72245	99676	82454	−104627

$\operatorname{tr} a_n$ 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)[6]   # 70.6.a.g

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

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