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

Newform orbit 35.10.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 24q^{2} - 174q^{3} - 720q^{4} + 1250q^{5} + 2952q^{6} + 4802q^{7} + 20544q^{8} + 22428q^{9} - 15000q^{10} + 18566q^{11} + 41904q^{12} - 51090q^{13} - 57624q^{14} - 108750q^{15} + 112768q^{16} - 373910q^{17} - 419472q^{18} - 143276q^{19} - 450000q^{20} - 417774q^{21} - 76808q^{22} - 498908q^{23} - 2291904q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 198x - 14399$

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
$\operatorname{tr} a_p$	−24	−174	1250	4802	18566	−51090	−373910	−143276	−498908	−11577554	−3953760	−3205412	1058992	15948180	65501290	−25114688	−116159208	−44688544	118092496	−294165824	−57419332	−692852854	−540679928	−779043704	−2673039406

$\operatorname{tr} a_n$ 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)[1]   # 35.10.a.b

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

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