Atlas › Modular forms › Weight 2 › Level 235 › 235.2.a › 235.2.a.e

Newform orbit 235.2.a.e

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + q^{2} + q^{3} + 7q^{4} + 7q^{5} + q^{6} - 3q^{7} + 3q^{8} + 10q^{9} + q^{10} + q^{11} - 4q^{12} + 2q^{13} + 10q^{14} + q^{15} - q^{16} + 12q^{17} - 17q^{18} + 3q^{19} + 7q^{20} + 7q^{21} - 5q^{22} + q^{23} + 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{7} - 131x^{6} + 4906x^{5} + 12310x^{4} - 3898854x^{3} + 57876471x^{2} + 29352837x - 3289336256$

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
$\operatorname{tr} a_p$	1	1	7	−3	1	2	12	3	1	26	−5	0	12	−33	−7	4	13	20	−32	11	−8	17	19	13	−12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 5, 47, 235]: [0, 1, 5, 26] (certified at ell = 2147483137)
T = [(2, 1), (3, 3), (11, 5), (13, 7), (17, 9)]; h of degree 15 = 1 x 15 (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(235, 2)[4]   # 235.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 235:2 1000