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

Newform orbit 215.4.a.c

Trace form

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

$13q + 5q^{2} + 5q^{3} + 63q^{4} - 65q^{5} + 8q^{6} + 34q^{7} + 33q^{8} + 152q^{9} - 25q^{10} + 29q^{11} + 213q^{12} + 221q^{13} - 39q^{14} - 25q^{15} + 627q^{16} + 122q^{17} + 451q^{18} - 29q^{19} - 315q^{20} + 232q^{21} + 611q^{22} + 400q^{23} + O(q^{25})$

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

$x^{13} - 10x^{12} - 293x^{11} + 2120x^{10} + 34051x^{9} - 134852x^{8} - 1933343x^{7} + 2137715x^{6} + 50832604x^{5} + 43788047x^{4} - 465530684x^{3} - 834417867x^{2} + 899266560x + 2049772335$

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 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
$\operatorname{tr} a_p$	5	5	−65	34	29	221	122	−29	400	−311	154	527	66	−559	1650	1310	−409	−21	2349	−322	2192	4074	−502	−3312	3930

$\operatorname{tr} a_n$ 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)[2]   # 215.4.a.c

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

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