🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 993 β€Ί 993.2.a β€Ί 993.2.a.e

Newform orbit 993.2.a.e

Trace form

The 15 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

15qβˆ’6q2βˆ’15q3+18q4βˆ’8q5+6q6βˆ’15q7βˆ’18q8+15q9+3q10βˆ’9q11βˆ’18q12+7q13βˆ’4q14+8q15+28q16βˆ’15q17βˆ’6q18+8q19βˆ’32q20+15q21βˆ’20q22βˆ’38q23+18q24+O(q25)15q - 6q^{2} - 15q^{3} + 18q^{4} - 8q^{5} + 6q^{6} - 15q^{7} - 18q^{8} + 15q^{9} + 3q^{10} - 9q^{11} - 18q^{12} + 7q^{13} - 4q^{14} + 8q^{15} + 28q^{16} - 15q^{17} - 6q^{18} + 8q^{19} - 32q^{20} + 15q^{21} - 20q^{22} - 38q^{23} + 18q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 15. It is generated by the eigenvalue Ξ±\alpha of T=T2+T5T = T_{2} + T_{5}, whose minimal polynomial is

x15+14x14+27x13βˆ’369x12βˆ’1394x11+2960x10+16287x9βˆ’4834x8βˆ’71128x7βˆ’28464x6+104696x5+88496x4βˆ’15536x3βˆ’34135x2βˆ’9500xβˆ’562x^{15} + 14x^{14} + 27x^{13} - 369x^{12} - 1394x^{11} + 2960x^{10} + 16287x^{9} - 4834x^{8} - 71128x^{7} - 28464x^{6} + 104696x^{5} + 88496x^{4} - 15536x^{3} - 34135x^{2} - 9500x - 562

Atkin–Lehner signs

ppwpw_p
3+1+1
331+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’6βˆ’15βˆ’8βˆ’15βˆ’97βˆ’158βˆ’38βˆ’18βˆ’9βˆ’9βˆ’8βˆ’24βˆ’33βˆ’14βˆ’1139βˆ’32βˆ’44βˆ’6βˆ’43βˆ’51βˆ’20βˆ’8

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 0.284 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 label, dimension and tr⁑ap\operatorname{tr} a_p for every p<1000p < 1000 equal LMFDB's.

The engine's checks
  • dim M^+ at levels [1, 3, 331, 993]: [0, 1, 28, 112] (certified at ell = 2147482921)
  • T = [(2, 1), (5, 1)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 6 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(993, 2)[4]   # 993.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 993:2 1000