🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 593 β€Ί 593.2.a β€Ί 593.2.a.d

Newform orbit 593.2.a.d

Trace form

The 18 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

18qβˆ’6q2βˆ’9q3+10q4βˆ’q5βˆ’5q6βˆ’30q7βˆ’12q8+13q9βˆ’8q10βˆ’3q11βˆ’12q12βˆ’22q13+5q14βˆ’31q15βˆ’6q16βˆ’9q17βˆ’14q18βˆ’6q19βˆ’10q20βˆ’3q21βˆ’18q22βˆ’23q23+10q24+O(q25)18q - 6q^{2} - 9q^{3} + 10q^{4} - q^{5} - 5q^{6} - 30q^{7} - 12q^{8} + 13q^{9} - 8q^{10} - 3q^{11} - 12q^{12} - 22q^{13} + 5q^{14} - 31q^{15} - 6q^{16} - 9q^{17} - 14q^{18} - 6q^{19} - 10q^{20} - 3q^{21} - 18q^{22} - 23q^{23} + 10q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 18. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x18+15x17+61x16βˆ’113x15βˆ’1300x14βˆ’1495x13+7397x12+15445x11βˆ’18320x10βˆ’54011x9+23983x8+91671x7βˆ’21656x6βˆ’77181x5+15088x4+27942x3βˆ’4625x2βˆ’3006x+67x^{18} + 15x^{17} + 61x^{16} - 113x^{15} - 1300x^{14} - 1495x^{13} + 7397x^{12} + 15445x^{11} - 18320x^{10} - 54011x^{9} + 23983x^{8} + 91671x^{7} - 21656x^{6} - 77181x^{5} + 15088x^{4} + 27942x^{3} - 4625x^{2} - 3006x + 67

Atkin–Lehner signs

ppwpw_p
593+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’9βˆ’1βˆ’30βˆ’3βˆ’22βˆ’9βˆ’6βˆ’23βˆ’4βˆ’28βˆ’46βˆ’1βˆ’53βˆ’40βˆ’19018βˆ’88βˆ’40βˆ’9βˆ’39βˆ’1632βˆ’18

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 0.222 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, 593]: [0, 50] (certified at ell = 2147470529)
  • T = [(2, 1), (3, 1)]; h of degree 49 = 1 x 49 (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(593, 2)[3]   # 593.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 593:2 1000