Atlas › Modular forms › Weight 2 › Level 215 › 215.2.a › 215.2.a.c

Newform orbit 215.2.a.c

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 2q^{2} - q^{3} + 8q^{4} + 5q^{5} - 12q^{6} + 5q^{7} + 3q^{8} + 18q^{9} + 2q^{10} - 6q^{11} + 5q^{13} + q^{14} - q^{15} + 14q^{16} - 17q^{17} - 5q^{18} - 6q^{19} + 8q^{20} + 20q^{21} - 8q^{22} + q^{23} - 45q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{5} + x^{4} - 121x^{3} + 124x^{2} + 3641x - 9088$

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
$\operatorname{tr} a_p$	2	−1	5	5	−6	5	−17	−6	1	6	6	5	2	−5	0	−23	−1	20	21	4	5	41	−7	20	37

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.178 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 5, 43, 215]: [0, 1, 4, 24] (certified at ell = 2147483353)
T = [(2, 1), (3, 3)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 2 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, 2)[2]   # 215.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 215:2 1000