Atlas › Modular forms › Weight 8 › Level 42 › 42.8.a › 42.8.a.b

Newform orbit 42.8.a.b

$q$ -expansion

$q - 8q^{2} - 27q^{3} + 64q^{4} - 18q^{5} + 216q^{6} + 343q^{7} - 512q^{8} + 729q^{9} + 144q^{10} + 8172q^{11} - 1728q^{12} - 14242q^{13} - 2744q^{14} + 486q^{15} + 4096q^{16} - 21462q^{17} - 5832q^{18} - 5884q^{19} - 1152q^{20} - 9261q^{21} - 65376q^{22} - 98784q^{23} + 13824q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{11}$ is a root of

$x - 24498$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$+1$
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 odd 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	−27	−18	343	8172	−14242	−21462	−5884	−98784	165174	−241312	185438	59682	−809308	942096	226398	−2205732	−1156690	−3740404	−2593296	−1038742	2280032	−283404	−5227230	6168770

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

Sato–Tate

The normalized $a_p / 2p^{7/2}$ for the primes $p < 1000$ not dividing 42, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.326 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, 7, 14, 21, 42]: [1, 3, 3, 9, 5, 16, 20, 60] (certified at ell = 2147483647)
T = [(5, 1), (11, 3)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 3 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(42, 8)[1]   # 42.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 42:8 1000