Atlas › Modular forms › Weight 4 › Level 235 › 235.4.a › 235.4.a.d

Newform orbit 235.4.a.d

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + 7q^{2} + 8q^{3} + 69q^{4} + 75q^{5} + 30q^{6} + 13q^{7} + 63q^{8} + 239q^{9} + 35q^{10} + 87q^{11} + 73q^{12} + 132q^{13} - 166q^{14} + 40q^{15} + 349q^{16} + 67q^{17} + 236q^{18} + 119q^{19} + 345q^{20} + 386q^{21} - 266q^{22} + 155q^{23} + 161q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} - 15x^{14} - 334x^{13} + 5320x^{12} + 35389x^{11} - 617554x^{10} - 1737293x^{9} + 31289583x^{8} + 48110844x^{7} - 691894342x^{6} - 807265985x^{5} + 5327365336x^{4} + 5288905670x^{3} - 12081352468x^{2} - 6162319520x + 6932606800$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
47	$-1$

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	8	75	13	87	132	67	119	155	955	388	8	1206	146	705	300	1273	821	−836	−978	337	1076	−2195	2330	−1897

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.147 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, 47, 235]: [1, 3, 13, 74] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 46 = 1 x 46 (phi(ord eps) x dim S_k^new); 7 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(235, 4)[3]   # 235.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 235:4 1000