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

Newform orbit 497.2.a.d

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{9} + 23x^{8} + 108x^{7} - 871x^{6} - 7234x^{5} + 786x^{4} + 97130x^{3} + 166112x^{2} - 39944x - 153392$

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 odd 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	−7	−4	−9	1	−15	−8	−14	−9	−1	−22	−7	−1	−24	−28	16	10	−25	−12	−9	−36	−14	−13	12	−15

$\operatorname{tr} a_n$ 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)[3]   # 497.2.a.d

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

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