Atlas › Modular forms › Weight 4 › Level 102 › 102.4.a › 102.4.a.a

Newform orbit 102.4.a.a

$q$ -expansion

$q - 2q^{2} - 3q^{3} + 4q^{4} - 3q^{5} + 6q^{6} + 20q^{7} - 8q^{8} + 9q^{9} + 6q^{10} - 51q^{11} - 12q^{12} - 61q^{13} - 40q^{14} + 9q^{15} + 16q^{16} + 17q^{17} - 18q^{18} - 43q^{19} - 12q^{20} - 60q^{21} + 102q^{22} - 219q^{23} + 24q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13}$ is a root of

$x + 625$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$+1$
17	$-1$

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
$\operatorname{tr} a_p$	−2	−3	−3	20	−51	−61	17	−43	−219	−150	290	56	15	83	426	−378	−210	−448	−124	900	−1078	722	−78	−144	−268

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 102, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.143 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, 17, 34, 51, 102]: [1, 2, 2, 5, 6, 16, 20, 58] (certified at ell = 2147483489)
T = [(5, 1), (7, 3), (11, 5), (13, 7)]; h of degree 8 = 1 x 8 (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(102, 4)[0]   # 102.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 102:4 1000