Atlas › Modular forms › Weight 8 › Level 17 › 17.8.a › 17.8.a.c

Newform orbit 17.8.a.c

Trace form

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

$6q + 15q^{2} + 40q^{3} + 485q^{4} - 184q^{5} + 1882q^{6} + 2064q^{7} + 6729q^{8} + 5846q^{9} + 5592q^{10} + 2000q^{11} + 3646q^{12} + 8708q^{13} - 19844q^{14} + 12552q^{15} + 5841q^{16} - 29478q^{17} - 108697q^{18} - 45400q^{19} - 202744q^{20} - 37896q^{21} + 1378q^{22} - 27208q^{23} + 21366q^{24} + O(q^{25})$

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

$x^{6} - 55x^{5} - 10480x^{4} + 594968x^{3} + 21446476x^{2} - 1401799020x + 10072288656$

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
$\operatorname{tr} a_p$	15	40	−184	2064	2000	8708	−29478	−45400	−27208	404808	532984	437968	−441660	1152240	−90296	−137764	−2050080	89808	4686632	−5553232	−1436452	12387160	−1877808	−19324324	−7630812

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.257 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, 12] (certified at ell = 2147483489)
T = [(2, 1), (3, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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, 8)[2]   # 17.8.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 17:8 1000