Atlas › Modular forms › Weight 8 › Level 35 › 35.8.a › 35.8.a.b

Newform orbit 35.8.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 23q^{2} - 50q^{3} + 233q^{4} - 375q^{5} + 1144q^{6} + 1029q^{7} - 2247q^{8} + 2449q^{9} + 2875q^{10} + 2858q^{11} - 16624q^{12} - 15852q^{13} - 7889q^{14} + 6250q^{15} - 27711q^{16} + 1464q^{17} - 110051q^{18} - 90364q^{19} - 29125q^{20} - 17150q^{21} - 128516q^{22} + 63372q^{23} + 169224q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 73x^{2} - 3293x + 26131$

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
$\operatorname{tr} a_p$	−23	−50	−375	1029	2858	−15852	1464	−90364	63372	105644	−71408	−82702	−258086	−359312	−373146	−1174498	1091968	−2628822	−2996428	−7975232	−7466346	5040874	−2965128	15507394	−27897424

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.273 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, 5, 30] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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, 8)[1]   # 35.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 35:8 1000