🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 6 β€Ί Level 101 β€Ί 101.6.a β€Ί 101.6.a.a

Newform orbit 101.6.a.a

Trace form

The 17 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

17qβˆ’5q2βˆ’46q3+143q4βˆ’79q5βˆ’194q6βˆ’198q7βˆ’135q8+837q9βˆ’366q10βˆ’1342q11βˆ’1868q12βˆ’677q13βˆ’653q14βˆ’2990q15βˆ’1825q16βˆ’1937q17βˆ’1385q18βˆ’8665q19βˆ’6156q20βˆ’6532q21βˆ’9642q22βˆ’1995q23βˆ’5262q24+O(q25)17q - 5q^{2} - 46q^{3} + 143q^{4} - 79q^{5} - 194q^{6} - 198q^{7} - 135q^{8} + 837q^{9} - 366q^{10} - 1342q^{11} - 1868q^{12} - 677q^{13} - 653q^{14} - 2990q^{15} - 1825q^{16} - 1937q^{17} - 1385q^{18} - 8665q^{19} - 6156q^{20} - 6532q^{21} - 9642q^{22} - 1995q^{23} - 5262q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 17. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x17+51x16βˆ’1333x15βˆ’90939x14+550059x13+61816357x12βˆ’91966089x11βˆ’20606819783x10+23465027639x9+3487669621405x8βˆ’8222153409293x7βˆ’263261707513947x6+975087544280059x5+6373616069106241x4βˆ’31146672210033994x3βˆ’9314680464241778x2+175803399082912552xβˆ’168122683175015208x^{17} + 51x^{16} - 1333x^{15} - 90939x^{14} + 550059x^{13} + 61816357x^{12} - 91966089x^{11} - 20606819783x^{10} + 23465027639x^{9} + 3487669621405x^{8} - 8222153409293x^{7} - 263261707513947x^{6} + 975087544280059x^{5} + 6373616069106241x^{4} - 31146672210033994x^{3} - 9314680464241778x^{2} + 175803399082912552x - 168122683175015208

Atkin–Lehner signs

ppwpw_p
101+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’5βˆ’46βˆ’79βˆ’198βˆ’1342βˆ’677βˆ’1937βˆ’8665βˆ’1995βˆ’19110βˆ’34877βˆ’26106βˆ’9012βˆ’57540βˆ’2565βˆ’244885290215818βˆ’135166βˆ’49655103364βˆ’161485115848βˆ’6102βˆ’100524

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 0.264 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, 101]: [1, 43] (certified at ell = 2147482801)
  • T = [(2, 1), (3, 1)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 8 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(101, 6)[0]   # 101.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 101:6 1000