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

Newform orbit 35.10.a.c

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 19q^{2} - 18q^{3} + 1729q^{4} - 2500q^{5} - 144q^{6} - 9604q^{7} - 30495q^{8} + 5382q^{9} + 11875q^{10} + 82438q^{11} + 41328q^{12} - 72962q^{13} + 45619q^{14} + 11250q^{15} + 64257q^{16} - 357542q^{17} - 965367q^{18} + 300732q^{19} - 1080625q^{20} + 43218q^{21} - 5595068q^{22} - 2340836q^{23} - 7636320q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 37x^{3} - 43117x^{2} + 1361315x + 119627428$

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
$\operatorname{tr} a_p$	−19	−18	−2500	−9604	82438	−72962	−357542	300732	−2340836	−1372022	−4590888	−38868456	57287084	−43403452	91822222	−61086884	72569680	−225681036	−4429720	−470468984	−168326464	−598805646	−1159074304	−1380153012	754874082

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

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)[2]   # 35.10.a.c

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

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