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

Newform orbit 35.10.a.e

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 15q^{2} - 124q^{3} + 3009q^{4} + 3750q^{5} + 4888q^{6} - 14406q^{7} + 22041q^{8} + 111090q^{9} + 9375q^{10} - 47796q^{11} - 541656q^{12} + 102168q^{13} - 36015q^{14} - 77500q^{15} + 2371065q^{16} - 38472q^{17} + 1837015q^{18} + 361056q^{19} + 1880625q^{20} + 297724q^{21} + 2068680q^{22} + 697032q^{23} + 4828076q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + 109x^{5} - 116582x^{4} - 13602632x^{3} + 3145846451x^{2} + 412032473931x + 8912066589538$

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 even 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
$\operatorname{tr} a_p$	15	−124	3750	−14406	−47796	102168	−38472	361056	697032	16028184	1362912	−3912924	22452756	−29998992	−121271508	−20308596	120280392	−87693540	495050664	253762512	187195308	831079500	−767650536	582579684	−1184506872

$\operatorname{tr} a_n$ 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)[4]   # 35.10.a.e

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

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