🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 6 β€Ί Level 59 β€Ί 59.6.a β€Ί 59.6.a.a

Newform orbit 59.6.a.a

Trace form

The 9 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

9qβˆ’9q2+63q4βˆ’72q5βˆ’274q6βˆ’208q7βˆ’327q8+163q9βˆ’402q10βˆ’1176q11βˆ’1928q13βˆ’89q14βˆ’2953q15βˆ’3533q16βˆ’3031q17βˆ’8061q18βˆ’4516q19βˆ’5540q20βˆ’9211q21βˆ’2075q22βˆ’2192q23βˆ’5190q24+O(q25)9q - 9q^{2} + 63q^{4} - 72q^{5} - 274q^{6} - 208q^{7} - 327q^{8} + 163q^{9} - 402q^{10} - 1176q^{11} - 1928q^{13} - 89q^{14} - 2953q^{15} - 3533q^{16} - 3031q^{17} - 8061q^{18} - 4516q^{19} - 5540q^{20} - 9211q^{21} - 2075q^{22} - 2192q^{23} - 5190q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 9. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x9+9x8βˆ’1036x7βˆ’2081x6+270650x5βˆ’14210x4βˆ’20441539x3+10268630x2+327603856xβˆ’270245344x^{9} + 9x^{8} - 1036x^{7} - 2081x^{6} + 270650x^{5} - 14210x^{4} - 20441539x^{3} + 10268630x^{2} + 327603856x - 270245344

Atkin–Lehner signs

ppwpw_p
59+1+1

For pβˆ₯Np \parallel N every form in the orbit has ap=βˆ’pk/2βˆ’1wpa_p = -p^{k/2-1} w_p. The Fricke sign is wN=+1w_N = +1, so the sign of the functional equation is (βˆ’1)k/2wN=βˆ’1(-1)^{k/2} w_N = -1: each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

pp2357111317192329313741434753596167717379838997
tr⁑ap\operatorname{tr} a_pβˆ’90βˆ’72βˆ’208βˆ’1176βˆ’1928βˆ’3031βˆ’4516βˆ’2192βˆ’3096βˆ’10148βˆ’1155411294βˆ’125261171616552βˆ’31329βˆ’143662βˆ’119488βˆ’38295βˆ’197332βˆ’75556βˆ’49800βˆ’85950βˆ’23462

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of TT over Z\mathbb{Z} is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks
  • dim M^+ at levels [1, 59]: [1, 26] (certified at ell = 2147482481)
  • T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 5 primes, 0 rejected
Open in the notebook Β· JSON Β· stored data
Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(59, 6)[0]   # 59.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 59:6 1000