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

Newform orbit 215.4.a.d

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + q^{2} + 7q^{3} + 83q^{4} + 75q^{5} + 40q^{6} + 36q^{7} + 33q^{8} + 242q^{9} + 5q^{10} + 87q^{11} - 37q^{12} - 35q^{13} + 237q^{14} + 35q^{15} + 139q^{16} + 144q^{17} - 325q^{18} + 293q^{19} + 415q^{20} + 208q^{21} + 25q^{22} + 22q^{23} + 480q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} - 8x^{14} - 433x^{13} + 3704x^{12} + 64573x^{11} - 575938x^{10} - 4127695x^{9} + 38095121x^{8} + 117858376x^{7} - 1176500319x^{6} - 1097050040x^{5} + 16796536397x^{4} - 6138229198x^{3} - 88835608271x^{2} + 89878337536x + 53831237154$

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
$\operatorname{tr} a_p$	1	7	75	36	87	−35	144	293	22	495	408	1013	820	645	−622	276	−1405	1969	−439	494	958	−1846	−1524	1380	3240

$\operatorname{tr} a_n$ 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)[3]   # 215.4.a.d

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

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