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

Newform orbit 295.2.a.c

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} - 3x^{5} - 14x^{4} + 25x^{3} + 58x^{2} - 50x - 71$

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
$\operatorname{tr} a_p$	2	1	6	2	3	11	16	−5	11	6	−11	10	15	5	−2	7	−6	−21	−8	−29	9	−27	13	25	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)[2]   # 295.2.a.c

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

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