Atlas › Modular forms › Weight 8 › Level 7 › 7.8.a › 7.8.a.a

Newform orbit 7.8.a.a

$q$ -expansion

$q - 6q^{2} - 42q^{3} - 92q^{4} - 84q^{5} + 252q^{6} + 343q^{7} + 1320q^{8} - 423q^{9} + 504q^{10} - 5568q^{11} + 3864q^{12} - 5152q^{13} - 2058q^{14} + 3528q^{15} + 3856q^{16} - 13986q^{17} + 2538q^{18} + 55370q^{19} + 7728q^{20} - 14406q^{21} + 33408q^{22} - 91272q^{23} - 55440q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x + 48$

Atkin–Lehner signs

$p$	$w_p$
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
$\operatorname{tr} a_p$	−6	−42	−84	343	−5568	−5152	−13986	55370	−91272	41610	150332	−136366	−510258	−172072	−519036	−59202	1979250	−2988748	2409404	1504512	−1821022	−1669240	696738	5558490	9876734

$\operatorname{tr} a_n$ 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 7, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.234 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, 7]: [1, 5] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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(7, 8)[0]   # 7.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 7:8 1000