Atlas › Modular forms › Weight 8 › Level 59 › 59.8.a › 59.8.a.a

Newform orbit 59.8.a.a

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q - 9q^{2} - 54q^{3} + 575q^{4} - 430q^{5} - 346q^{6} - 2390q^{7} - 2463q^{8} + 4504q^{9} - 5362q^{10} - 5030q^{11} - 10368q^{12} - 24364q^{13} - 12717q^{14} + 5828q^{15} + 33859q^{16} + 14504q^{17} - 47889q^{18} - 80234q^{19} - 190220q^{20} - 240088q^{21} - 266687q^{22} - 113272q^{23} - 458658q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} + 63x^{13} - 16414x^{12} - 778234x^{11} + 108969268x^{10} + 2953857592x^{9} - 372659036057x^{8} - 1931712077059x^{7} + 641697130751251x^{6} - 9049902065411305x^{5} - 365543974817926331x^{4} + 12758132007812346643x^{3} - 137286780125869890294x^{2} + 365779297591004785340x + 1516024177011042689408$

Atkin–Lehner signs

$p$	$w_p$
59	$-1$

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	−54	−430	−2390	−5030	−24364	14504	−80234	−113272	−490250	−379844	−1203748	−681860	−967090	287456	−1227618	2875306	−4711840	7298936	5657790	4750028	−17385506	−15067470	−15451868	2400932

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.293 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, 59]: [1, 36] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 34 = 1 x 34 (phi(ord eps) x dim S_k^new); 8 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(59, 8)[0]   # 59.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 59:8 1000