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

Newform orbit 235.4.a.a

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - 3q^{2} - 10q^{3} + 9q^{4} + 40q^{5} - 42q^{6} - 15q^{7} - 57q^{8} + 14q^{9} - 15q^{10} - 89q^{11} + 32q^{12} - 68q^{13} + 49q^{14} - 50q^{15} - 147q^{16} - 169q^{17} + 111q^{18} - 169q^{19} + 45q^{20} - 454q^{21} - 36q^{22} - 93q^{23} + 102q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} + 13x^{7} - 25x^{6} - 739x^{5} - 105x^{4} + 15193x^{3} + 186x^{2} - 115960x + 94728$

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
$\operatorname{tr} a_p$	−3	−10	40	−15	−89	−68	−169	−169	−93	−583	−356	−578	−1336	−280	−376	−1106	−941	−1197	728	608	1383	−922	2979	−3544	383

$\operatorname{tr} a_n$ 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)[0]   # 235.4.a.a

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

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