🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 653 β€Ί 653.2.a β€Ί 653.2.a.b

Newform orbit 653.2.a.b

Trace form

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

17qβˆ’4q2βˆ’10q3+16q4βˆ’10q5βˆ’12q6βˆ’9q7βˆ’12q8+21q9βˆ’20q10βˆ’3q11βˆ’33q12βˆ’37q13+4q14+30q16βˆ’21q17+7q18βˆ’36q19βˆ’2q20βˆ’18q21βˆ’30q22βˆ’21q23βˆ’2q24+O(q25)17q - 4q^{2} - 10q^{3} + 16q^{4} - 10q^{5} - 12q^{6} - 9q^{7} - 12q^{8} + 21q^{9} - 20q^{10} - 3q^{11} - 33q^{12} - 37q^{13} + 4q^{14} + 30q^{16} - 21q^{17} + 7q^{18} - 36q^{19} - 2q^{20} - 18q^{21} - 30q^{22} - 21q^{23} - 2q^{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+14x16+49x15βˆ’123x14βˆ’954x13βˆ’461x12+5305x11+6997x10βˆ’10252x9βˆ’21889x8βˆ’194x7+21714x6+14832x5+808x4βˆ’2612x3βˆ’1046x2βˆ’151xβˆ’7x^{17} + 14x^{16} + 49x^{15} - 123x^{14} - 954x^{13} - 461x^{12} + 5305x^{11} + 6997x^{10} - 10252x^{9} - 21889x^{8} - 194x^{7} + 21714x^{6} + 14832x^{5} + 808x^{4} - 2612x^{3} - 1046x^{2} - 151x - 7

Atkin–Lehner signs

ppwpw_p
653+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’10βˆ’10βˆ’9βˆ’3βˆ’37βˆ’21βˆ’36βˆ’214βˆ’32βˆ’4226βˆ’69299βˆ’11βˆ’3βˆ’38420βˆ’47βˆ’53βˆ’2βˆ’3

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 0.219 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, 653]: [0, 55] (certified at ell = 2147482621)
  • T = [(2, 1), (3, 1)]; h of degree 54 = 1 x 54 (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(653, 2)[1]   # 653.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 653:2 1000