Atlas › Modular forms › Weight 2 › Level 915 › 915.2.a › 915.2.a.e

Newform orbit 915.2.a.e

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{3} + 7x^{2} + 9x - 1$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
5	$-1$
61	$+1$

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	−3	3	−5	12	0	0	7	10	7	−4	3	18	−7	−1	18	−19	−3	8	1	−20	8	−7	2	8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.423 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, 3, 5, 15, 61, 183, 305, 915]: [0, 1, 1, 4, 5, 22, 32, 128] (certified at ell = 2147482921)
T = [(2, 1), (7, 1)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 5 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(915, 2)[4]   # 915.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 915:2 1000