🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 719 β€Ί 719.2.a β€Ί 719.2.a.c

Newform orbit 719.2.a.c

Trace form

The 45 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

45q+3q2+4q3+59q4+5q5+4q6+16q7+6q8+73q9+21q10+10q11+20q13+6q14+5q15+83q16+15q17βˆ’5q18+27q19βˆ’11q20+24q21+30q22+q23βˆ’6q24+O(q25)45q + 3q^{2} + 4q^{3} + 59q^{4} + 5q^{5} + 4q^{6} + 16q^{7} + 6q^{8} + 73q^{9} + 21q^{10} + 10q^{11} + 20q^{13} + 6q^{14} + 5q^{15} + 83q^{16} + 15q^{17} - 5q^{18} + 27q^{19} - 11q^{20} + 24q^{21} + 30q^{22} + q^{23} - 6q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 45. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x45βˆ’7x44βˆ’158x43+1192x42+10865x41βˆ’90277x40βˆ’427631x39+4036666x38+10628598x37βˆ’119364816x36βˆ’172455321x35+2479191644x34+1788398460x33βˆ’37515582867x32βˆ’10175224299x31+423571172312x30βˆ’272606805x29βˆ’3625112151947x28+535461862599x27+23750400601017x26βˆ’4859051239860x25βˆ’119653214491497x24+23644525228574x23+463122789342508x22βˆ’69270334887326x21βˆ’1368546079216980x20+111141800742874x19+3052980129206047x18βˆ’32802858717122x17βˆ’5063091918094853x16βˆ’239786001458841x15+6126472997505910x14+511666893415950x13βˆ’5286826476722800x12βˆ’494165077784072x11+3154626716420761x10+248289121961980x9βˆ’1239962798789582x8βˆ’59215485672406x7+294883072902390x6+5635820653270x5βˆ’36406604023323x4βˆ’923010494805x3+1893323660667x2+110886594211xβˆ’11852305957x^{45} - 7x^{44} - 158x^{43} + 1192x^{42} + 10865x^{41} - 90277x^{40} - 427631x^{39} + 4036666x^{38} + 10628598x^{37} - 119364816x^{36} - 172455321x^{35} + 2479191644x^{34} + 1788398460x^{33} - 37515582867x^{32} - 10175224299x^{31} + 423571172312x^{30} - 272606805x^{29} - 3625112151947x^{28} + 535461862599x^{27} + 23750400601017x^{26} - 4859051239860x^{25} - 119653214491497x^{24} + 23644525228574x^{23} + 463122789342508x^{22} - 69270334887326x^{21} - 1368546079216980x^{20} + 111141800742874x^{19} + 3052980129206047x^{18} - 32802858717122x^{17} - 5063091918094853x^{16} - 239786001458841x^{15} + 6126472997505910x^{14} + 511666893415950x^{13} - 5286826476722800x^{12} - 494165077784072x^{11} + 3154626716420761x^{10} + 248289121961980x^{9} - 1239962798789582x^{8} - 59215485672406x^{7} + 294883072902390x^{6} + 5635820653270x^{5} - 36406604023323x^{4} - 923010494805x^{3} + 1893323660667x^{2} + 110886594211x - 11852305957

Atkin–Lehner signs

ppwpw_p
719βˆ’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=βˆ’1w_N = -1, so the sign of the functional equation is (βˆ’1)k/2wN=+1(-1)^{k/2} w_N = +1: each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

pp2357111317192329313741434753596167717379838997
tr⁑ap\operatorname{tr} a_p345161020152714021304423βˆ’2410βˆ’5742034683βˆ’28βˆ’757

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 0.242 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, 719]: [0, 61] (certified at ell = 2147481997)
  • T = [(2, 1), (3, 1)]; h of degree 60 = 1 x 60 (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(719, 2)[2]   # 719.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 719:2 1000