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

Newform orbit 42.8.a.c

$q$ -expansion

$q - 8q^{2} + 27q^{3} + 64q^{4} - 122q^{5} - 216q^{6} - 343q^{7} - 512q^{8} + 729q^{9} + 976q^{10} - 1012q^{11} + 1728q^{12} + 3126q^{13} + 2744q^{14} - 3294q^{15} + 4096q^{16} - 28294q^{17} - 5832q^{18} - 22228q^{19} - 7808q^{20} - 9261q^{21} + 8096q^{22} - 108640q^{23} - 13824q^{24} + O(q^{25})$

Coefficient field

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

$x + 3158$

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	−122	−343	−1012	3126	−28294	−22228	−108640	−41354	46656	−85714	−155694	926804	−529152	−294066	−667292	833430	1153996	3842336	1483690	−3763824	−5393092	2502690	−4597550

$\operatorname{tr} a_n$ 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)[2]   # 42.8.a.c

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

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