🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 821 β€Ί 821.2.a β€Ί 821.2.a.b

Newform orbit 821.2.a.b

Trace form

The 41 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

41q+q2+16q3+53q4+7q5+6q6+19q7+49q9+13q10+20q11+21q12+10q13+12q14+10q15+69q16+3q17βˆ’6q18+76q19+5q20+14q21+4q22+2q23+5q24+O(q25)41q + q^{2} + 16q^{3} + 53q^{4} + 7q^{5} + 6q^{6} + 19q^{7} + 49q^{9} + 13q^{10} + 20q^{11} + 21q^{12} + 10q^{13} + 12q^{14} + 10q^{15} + 69q^{16} + 3q^{17} - 6q^{18} + 76q^{19} + 5q^{20} + 14q^{21} + 4q^{22} + 2q^{23} + 5q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 41. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x41βˆ’17x40βˆ’15x39+1792x38βˆ’6510x37βˆ’73696x36+473507x35+1374667x34βˆ’15676623x33βˆ’4757655x32+307007377x31βˆ’316966444x30βˆ’3896849142x29+7632171026x28+33287182350x27βˆ’93421951977x26βˆ’191842890173x25+739624729725x24+712088602843x23βˆ’4060245695066x22βˆ’1375503117137x21+15865009357500x20βˆ’863975228698x19βˆ’44428010182783x18+14010730165654x17+88684193718613x16βˆ’43734098339404x15βˆ’124180565845559x14+76620274163186x13+118555826394918x12βˆ’84086264728619x11βˆ’73514634499016x10+58484791837377x9+26985008121460x8βˆ’25012125813896x7βˆ’4577140461159x6+6063887625624x5βˆ’79515545788x4βˆ’678206673220x3+100630356200x2+14863371272xβˆ’2928198464x^{41} - 17x^{40} - 15x^{39} + 1792x^{38} - 6510x^{37} - 73696x^{36} + 473507x^{35} + 1374667x^{34} - 15676623x^{33} - 4757655x^{32} + 307007377x^{31} - 316966444x^{30} - 3896849142x^{29} + 7632171026x^{28} + 33287182350x^{27} - 93421951977x^{26} - 191842890173x^{25} + 739624729725x^{24} + 712088602843x^{23} - 4060245695066x^{22} - 1375503117137x^{21} + 15865009357500x^{20} - 863975228698x^{19} - 44428010182783x^{18} + 14010730165654x^{17} + 88684193718613x^{16} - 43734098339404x^{15} - 124180565845559x^{14} + 76620274163186x^{13} + 118555826394918x^{12} - 84086264728619x^{11} - 73514634499016x^{10} + 58484791837377x^{9} + 26985008121460x^{8} - 25012125813896x^{7} - 4577140461159x^{6} + 6063887625624x^{5} - 79515545788x^{4} - 678206673220x^{3} + 100630356200x^{2} + 14863371272x - 2928198464

Atkin–Lehner signs

ppwpw_p
821βˆ’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_p11671920103762βˆ’77825171318βˆ’968497621143820βˆ’840

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 0.257 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, 821]: [0, 69] (certified at ell = 2147455361)
  • T = [(2, 1), (3, 1)]; h of degree 68 = 1 x 68 (phi(ord eps) x dim S_k^new); 7 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(821, 2)[1]   # 821.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 821:2 1000