Atlas › Modular forms › Weight 12 › Level 17 › 17.12.a › 17.12.a.a

Newform orbit 17.12.a.a

Trace form

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

$6q - 9q^{2} - 476q^{3} + 4613q^{4} - 12884q^{5} + 552q^{6} - 23436q^{7} + 74805q^{8} + 296398q^{9} - 676038q^{10} - 962060q^{11} - 2352344q^{12} - 435268q^{13} - 7990948q^{14} - 9450288q^{15} - 12496671q^{16} + 8519142q^{17} - 20195421q^{18} - 26398480q^{19} - 47202914q^{20} - 56428792q^{21} - 51774200q^{22} - 99172772q^{23} - 110557128q^{24} + O(q^{25})$

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

$x^{6} + 485x^{5} - 571030x^{4} - 268719892x^{3} + 80932345936x^{2} + 36890201127040x + 101392495911936$

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
$\operatorname{tr} a_p$	−9	−476	−12884	−23436	−962060	−435268	8519142	−26398480	−99172772	165683964	−199133468	−785778644	166444428	−1110947880	−5828211928	−9889898636	204095112	−15864546948	17196640232	8751653884	−13704156916	−89923384436	−26042106648	53269579420	−106272517044

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.504 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]: [2, 18] (certified at ell = 2147483489)
T = [(2, 1), (3, 1)]; h of degree 14 = 1 x 14 (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(17, 12)[0]   # 17.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 17:12 1000