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

Newform orbit 215.2.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{3} - x^{2} - 69x - 3$

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	3	−3	9	−2	10	6	−6	2	13	9	15	−3	−22	8	13	−10	−6	−6	3	−17	−12	8	6

$\operatorname{tr} a_n$ 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)[1]   # 215.2.a.b

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

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