Atlas › Modular forms › Weight 4 › Level 35 › 35.4.a › 35.4.a.a

Newform orbit 35.4.a.a

$q$ -expansion

$q + q^{2} - 8q^{3} - 7q^{4} - 5q^{5} - 8q^{6} + 7q^{7} - 15q^{8} + 37q^{9} - 5q^{10} + 12q^{11} + 56q^{12} - 78q^{13} + 7q^{14} + 40q^{15} + 41q^{16} - 94q^{17} + 37q^{18} + 40q^{19} + 35q^{20} - 56q^{21} + 12q^{22} + 32q^{23} + 120q^{24} + O(q^{25})$

Coefficient field

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

$x + 7$

Atkin–Lehner signs

$p$	$w_p$
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 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
$\operatorname{tr} a_p$	1	−8	−5	7	12	−78	−94	40	32	−50	−248	−434	402	−68	536	22	−560	−278	−164	672	82	−1000	−448	−870	1026

$\operatorname{tr} a_n$ 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 35, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.096 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, 5, 7, 35]: [1, 3, 3, 14] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 6 = 1 x 6 (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(35, 4)[0]   # 35.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 35:4 1000