Atlas › Modular forms › Weight 10 › Level 35 › 35.10.a › 35.10.a.a

Newform orbit 35.10.a.a

$q$ -expansion

$q + 28q^{2} - 116q^{3} + 272q^{4} + 625q^{5} - 3248q^{6} + 2401q^{7} - 6720q^{8} - 6227q^{9} + 17500q^{10} - 25548q^{11} - 31552q^{12} - 42306q^{13} + 67228q^{14} - 72500q^{15} - 327424q^{16} - 526342q^{17} - 174356q^{18} - 350060q^{19} + 170000q^{20} - 278516q^{21} - 715344q^{22} - 621976q^{23} + 779520q^{24} + O(q^{25})$

Coefficient field

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

$x + 88$

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
$\operatorname{tr} a_p$	28	−116	625	2401	−25548	−42306	−526342	−350060	−621976	6720430	−6412208	−2317682	−10224678	30114004	−23644912	57292654	84934780	14677822	−244557812	61901952	−283763726	276107480	−72995956	−896368470	1205809578

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

Sato–Tate

The normalized $a_p / 2p^{9/2}$ 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 0.324 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, 5, 7, 38] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 5 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, 10)[0]   # 35.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 35:10 1000