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

Newform orbit 35.6.a.d

Trace form

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

$4q + 7q^{2} + 14q^{3} + 49q^{4} + 100q^{5} + 136q^{6} - 196q^{7} + 489q^{8} + 774q^{9} + 175q^{10} + 770q^{11} + 840q^{12} + 58q^{13} - 343q^{14} + 350q^{15} - 615q^{16} + 2006q^{17} - 1409q^{18} + 564q^{19} + 1225q^{20} - 686q^{21} - 1736q^{22} - 6340q^{23} - 6244q^{24} + O(q^{25})$

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

$x^{4} - 21x^{3} - 877x^{2} + 19823x - 99526$

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
$\operatorname{tr} a_p$	7	14	100	−196	770	58	2006	564	−6340	8066	−5856	29544	13156	−5692	39926	20300	8432	30540	32792	−83920	−75424	129486	−187520	−30324	180270

$\operatorname{tr} a_n$ 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)[3]   # 35.6.a.d

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

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