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

Newform orbit 235.4.a.c

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q - 3q^{2} - 10q^{3} + 53q^{4} - 65q^{5} - 26q^{6} - 19q^{7} - 57q^{8} + 133q^{9} + 15q^{10} - 91q^{11} - 241q^{12} - 162q^{13} - 266q^{14} + 50q^{15} + 437q^{16} - 353q^{17} - 252q^{18} - 119q^{19} - 265q^{20} - 22q^{21} - 296q^{22} - 355q^{23} - 859q^{24} + O(q^{25})$

Coefficient field

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

$x^{13} + 13x^{12} - 210x^{11} - 2400x^{10} + 19679x^{9} + 147838x^{8} - 995119x^{7} - 2943679x^{6} + 22534962x^{5} - 1051126x^{4} - 153632113x^{3} + 255419122x^{2} - 146304104x + 26398040$

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	−65	−19	−91	−162	−353	−119	−355	−495	116	−482	−426	−808	611	−2162	−181	−725	−200	−1416	−281	1694	−3217	108	1499

$\operatorname{tr} a_n$ 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)[2]   # 235.4.a.c

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

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