Atlas › Modular forms › Weight 10 › Level 17 › 17.10.a › 17.10.a.b

Newform orbit 17.10.a.b

Trace form

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

$7q - q^{2} + 88q^{3} + 2389q^{4} + 1362q^{5} - 11720q^{6} + 9388q^{7} + 16821q^{8} + 81419q^{9} + 154226q^{10} + 135536q^{11} + 198160q^{12} + 166122q^{13} + 447252q^{14} + 159048q^{15} + 1463585q^{16} + 584647q^{17} + 149027q^{18} + 777172q^{19} - 917162q^{20} - 3412104q^{21} - 1222520q^{22} + 1357764q^{23} - 8487360q^{24} + O(q^{25})$

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

$x^{7} - 87x^{6} - 97082x^{5} + 8639428x^{4} + 2573036208x^{3} - 199559273264x^{2} - 19853344053984x + 1414782466180416$

Atkin–Lehner signs

$p$	$w_p$
17	$-1$

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	88	1362	9388	135536	166122	584647	777172	1357764	967002	3546740	18296498	10285686	21913204	56639800	121813562	29222388	−49915846	301863420	652473940	306656342	959147884	−1512945268	−1971327114	2006526254

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.29 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 engine's checks

dim M^+ at levels [1, 17]: [1, 14] (certified at ell = 2147483489)
T = [(2, 1), (3, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 4 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(17, 10)[1]   # 17.10.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 17:10 1000