Atlas › Modular forms › Weight 4 › Level 215 › 215.4.a › 215.4.a.b

Newform orbit 215.4.a.b

Trace form

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

$8q - q^{2} - 7q^{3} + 27q^{4} - 40q^{5} + 8q^{6} - 36q^{7} - 39q^{8} + 53q^{9} + 5q^{10} - 25q^{11} - 148q^{12} - 85q^{13} + 17q^{14} + 35q^{15} - 77q^{16} - 74q^{17} - 469q^{18} - 181q^{19} - 135q^{20} - 32q^{21} - 270q^{22} - 10q^{23} + O(q^{25})$

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

$x^{8} + 8x^{7} - 156x^{6} - 575x^{5} + 8914x^{4} - 5701x^{3} - 93275x^{2} + 80158x + 283886$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
43	$-1$

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
$\operatorname{tr} a_p$	−1	−7	−40	−36	−25	−85	−74	−181	−10	95	−484	−1149	44	344	−46	−334	−585	825	−1519	1514	−1046	−1542	−44	110	−4414

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.138 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, 43, 215]: [1, 3, 12, 68] (certified at ell = 2147483353)
T = [(2, 1), (3, 1)]; h of degree 42 = 1 x 42 (phi(ord eps) x dim S_k^new); 6 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(215, 4)[1]   # 215.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 215:4 1000