Atlas › Modular forms › Weight 2 › Level 295 › 295.2.a › 295.2.a.d

Newform orbit 295.2.a.d

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{7} - 4x^{6} - 29x^{5} + 119x^{4} + 179x^{3} - 764x^{2} - 137x + 541$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
59	$-1$

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	3	−7	−4	3	−9	24	3	3	4	−1	−24	45	−19	16	−7	7	9	−16	15	5	1	23	7	−4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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, 59, 295]: [0, 1, 6, 32] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 19 = 1 x 19 (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(295, 2)[3]   # 295.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 295:2 1000