🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 743 β€Ί 743.2.a β€Ί 743.2.a.a

Newform orbit 743.2.a.a

Trace form

The 21 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

21qβˆ’4q2βˆ’9q3+12q4βˆ’10q5βˆ’7q6βˆ’18q7βˆ’6q8+8q9βˆ’15q10βˆ’3q11βˆ’18q12βˆ’28q13βˆ’2q14βˆ’5q15βˆ’6q16βˆ’26q17βˆ’5q18βˆ’12q19βˆ’12q20βˆ’9q21βˆ’16q22βˆ’2q23βˆ’12q24+O(q25)21q - 4q^{2} - 9q^{3} + 12q^{4} - 10q^{5} - 7q^{6} - 18q^{7} - 6q^{8} + 8q^{9} - 15q^{10} - 3q^{11} - 18q^{12} - 28q^{13} - 2q^{14} - 5q^{15} - 6q^{16} - 26q^{17} - 5q^{18} - 12q^{19} - 12q^{20} - 9q^{21} - 16q^{22} - 2q^{23} - 12q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 21. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x21+13x20+29x19βˆ’269x18βˆ’1196x17+1678x16+14297x15+738x14βˆ’83873x13βˆ’51957x12+271112x11+236280x10βˆ’495394x9βˆ’463919x8+509773x7+421036x6βˆ’298445x5βˆ’159852x4+100402x3+17053x2βˆ’14743x+1727x^{21} + 13x^{20} + 29x^{19} - 269x^{18} - 1196x^{17} + 1678x^{16} + 14297x^{15} + 738x^{14} - 83873x^{13} - 51957x^{12} + 271112x^{11} + 236280x^{10} - 495394x^{9} - 463919x^{8} + 509773x^{7} + 421036x^{6} - 298445x^{5} - 159852x^{4} + 100402x^{3} + 17053x^{2} - 14743x + 1727

Atkin–Lehner signs

ppwpw_p
743+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’4βˆ’9βˆ’10βˆ’18βˆ’3βˆ’28βˆ’26βˆ’12βˆ’2βˆ’6βˆ’27βˆ’44βˆ’20βˆ’20βˆ’98βˆ’8βˆ’31βˆ’2819βˆ’10990βˆ’22βˆ’144

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 0.248 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, 743]: [0, 63] (certified at ell = 2147477851)
  • T = [(2, 1), (3, 1)]; h of degree 62 = 1 x 62 (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(743, 2)[0]   # 743.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 743:2 1000