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

Newform orbit 17.10.a.a

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 33q^{2} - 236q^{3} + 853q^{4} + 1480q^{5} + 7578q^{6} - 13202q^{7} - 42423q^{8} + 10981q^{9} - 89328q^{10} - 68036q^{11} - 406010q^{12} - 158862q^{13} - 84700q^{14} - 687324q^{15} + 350225q^{16} - 417605q^{17} - 1911585q^{18} - 370992q^{19} + 1632640q^{20} + 1783880q^{21} + 122290q^{22} + 1645870q^{23} + 9678702q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 269x^{4} - 27802x^{3} - 6757522x^{2} - 351003836x - 5442576560$

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 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
$\operatorname{tr} a_p$	−33	−236	1480	−13202	−68036	−158862	−417605	−370992	1645870	3668616	−7262362	−31420708	−7996938	−56908268	−16903336	−83362982	−37946604	−77685452	−304503600	−476602922	−289980486	−828240610	194681148	376848106	692035246

$\operatorname{tr} a_n$ 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)[0]   # 17.10.a.a

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

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