Atlas › Modular forms › Weight 6 › Level 35 › 35.6.a › 35.6.a.b

Newform orbit 35.6.a.b

Trace form

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

$2q + q^{2} + 3q^{3} - 31q^{4} - 50q^{5} - 96q^{6} - 98q^{7} - 15q^{8} - 189q^{9} - 25q^{10} - 601q^{11} - 144q^{12} - 577q^{13} - 49q^{14} - 75q^{15} - 543q^{16} + 41q^{17} - 387q^{18} + 630q^{19} + 775q^{20} - 147q^{21} + 2852q^{22} - 442q^{23} + 4560q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{65})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} - 4x - 61$

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	3	−50	−98	−601	−577	41	630	−442	5885	−396	−8904	1774	−27122	−21289	−55582	59600	−51846	−45344	80744	−13532	−51795	109828	−37650	−96339

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.228 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, 5, 22] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 2 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, 6)[1]   # 35.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 35:6 1000