Atlas › Modular forms › Weight 2 › Level 497 › 497.2.a › 497.2.a.c

Newform orbit 497.2.a.c

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{8} - 15x^{7} - 36x^{6} + 1146x^{5} - 1716x^{4} - 20126x^{3} + 51004x^{2} + 12718x - 25898$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
71	$-1$

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
$\operatorname{tr} a_p$	0	5	2	−8	−1	9	4	6	17	−5	20	1	−1	20	24	8	18	11	16	8	0	−10	−9	12	11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.201 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, 7, 71, 497]: [0, 1, 7, 50] (certified at ell = 2147482681)
T = [(2, 1), (3, 3)]; h of degree 35 = 1 x 35 (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(497, 2)[2]   # 497.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 497:2 1000