🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 739 β€Ί 739.2.a β€Ί 739.2.a.c

Newform orbit 739.2.a.c

Trace form

The 23 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

23qβˆ’6q2βˆ’6q3+20q4βˆ’24q5βˆ’9q6βˆ’7q7βˆ’18q8+5q9βˆ’6q10βˆ’18q11βˆ’20q12βˆ’17q13βˆ’8q14βˆ’13q15+18q16βˆ’9q17βˆ’8q18βˆ’24q19βˆ’53q20βˆ’29q21βˆ’2q22βˆ’16q23βˆ’14q24+O(q25)23q - 6q^{2} - 6q^{3} + 20q^{4} - 24q^{5} - 9q^{6} - 7q^{7} - 18q^{8} + 5q^{9} - 6q^{10} - 18q^{11} - 20q^{12} - 17q^{13} - 8q^{14} - 13q^{15} + 18q^{16} - 9q^{17} - 8q^{18} - 24q^{19} - 53q^{20} - 29q^{21} - 2q^{22} - 16q^{23} - 14q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 23. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x23+12x22+11x21βˆ’364x20βˆ’1104x19+3806x18+16763x17βˆ’19576x16βˆ’119790x15+63599x14+485987x13βˆ’186959x12βˆ’1162965x11+516131x10+1548112x9βˆ’881034x8βˆ’977019x7+638945x6+285503x5βˆ’198185x4βˆ’37859x3+25031x2+2024xβˆ’1003x^{23} + 12x^{22} + 11x^{21} - 364x^{20} - 1104x^{19} + 3806x^{18} + 16763x^{17} - 19576x^{16} - 119790x^{15} + 63599x^{14} + 485987x^{13} - 186959x^{12} - 1162965x^{11} + 516131x^{10} + 1548112x^{9} - 881034x^{8} - 977019x^{7} + 638945x^{6} + 285503x^{5} - 198185x^{4} - 37859x^{3} + 25031x^{2} + 2024x - 1003

Atkin–Lehner signs

ppwpw_p
739+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’6βˆ’24βˆ’7βˆ’18βˆ’17βˆ’9βˆ’24βˆ’16βˆ’48βˆ’17βˆ’23βˆ’6122βˆ’28βˆ’42βˆ’60βˆ’120βˆ’30146βˆ’34βˆ’1017

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, 739]: [0, 62] (certified at ell = 2147483323)
  • T = [(2, 1), (3, 1)]; h of degree 61 = 1 x 61 (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(739, 2)[2]   # 739.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 739:2 1000