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

Newform orbit 235.2.a.d

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{5} + 227x^{4} + 18995x^{3} + 724177x^{2} + 12663129x + 81429688$

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 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
$\operatorname{tr} a_p$	−4	−5	−5	−5	−1	−11	−14	5	−6	−16	3	−16	−24	0	−5	−14	10	−9	4	4	−27	−18	17	4	−30

$\operatorname{tr} a_n$ 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)[3]   # 235.2.a.d

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

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