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

Newform orbit 59.8.a.b

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + 15q^{2} + 1535q^{4} + 570q^{5} + 1382q^{6} + 1040q^{7} + 2145q^{8} + 17626q^{9} + 4638q^{10} + 8280q^{11} + 23970q^{13} + 3747q^{14} - 15409q^{15} + 61611q^{16} - 8175q^{17} + 18625q^{18} + 88188q^{19} + 229652q^{20} + 255359q^{21} + 375285q^{22} + 142760q^{23} + 808080q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{20} - 15x^{19} - 34000x^{18} + 396025x^{17} + 472052408x^{16} - 2911059070x^{15} - 3552754121672x^{14} - 1374826293785x^{13} + 15991781079996949x^{12} + 115705562436722720x^{11} - 44164935677746449009x^{10} - 615808226138426530080x^{9} + 72749535295874045358451x^{8} + 1493636838466489814921375x^{7} - 64171656327934873892797505x^{6} - 1795155010460998798297911080x^{5} + 20861775449565486940802034739x^{4} + 908048904433515345397319811290x^{3} + 3058406798694838602198105689478x^{2} - 68021086190713764162698972034980x - 119796481022401687978995917552304$

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 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	0	570	1040	8280	23970	−8175	88188	142760	609298	501252	988540	134044	1098090	−192100	−223030	−4107580	268196	18460	−7557879	11309150	15100684	17914560	25286376	21354480

$\operatorname{tr} a_n$ 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)[1]   # 59.8.a.b

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

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