🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 863 β€Ί 863.2.a β€Ί 863.2.a.c

Newform orbit 863.2.a.c

Trace form

The 46 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

46q+4q2+6q3+52q4+11q5+2q6+28q7+9q8+62q9+17q10+2q11+2q12+52q13βˆ’9q14+3q15+56q16+40q17+7q18+30q19+16q20+11q21+4q22βˆ’3q23βˆ’5q24+O(q25)46q + 4q^{2} + 6q^{3} + 52q^{4} + 11q^{5} + 2q^{6} + 28q^{7} + 9q^{8} + 62q^{9} + 17q^{10} + 2q^{11} + 2q^{12} + 52q^{13} - 9q^{14} + 3q^{15} + 56q^{16} + 40q^{17} + 7q^{18} + 30q^{19} + 16q^{20} + 11q^{21} + 4q^{22} - 3q^{23} - 5q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 46. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x46βˆ’10x45βˆ’124x44+1518x43+5974x42βˆ’102374x41βˆ’116011x40+4069871x39βˆ’1045480x38βˆ’106795767x37+115727513x36+1964724210x35βˆ’3197611595x34βˆ’26307233865x33+52729459109x32+263094973657x31βˆ’595996027435x30βˆ’2003851775771x29+4868481282296x28+11802759340169x27βˆ’29490933472232x26βˆ’54389704222550x25+134173571270776x24+197482820670140x23βˆ’460596021108031x22βˆ’565390694459455x21+1191114920767312x20+1267220721736176x19βˆ’2304707919704650x18βˆ’2188859618624828x17+3299038798178660x16+2846073259055039x15βˆ’3440488844053013x14βˆ’2703559144016765x13+2564462362757947x12+1810273594276592x11βˆ’1333163444531387x10βˆ’818132279475743x9+466521564724692x8+235716757607846x7βˆ’103277330159144x6βˆ’39723058511038x5+12744786693544x4+3367373989531x3βˆ’630482044248x2βˆ’102022043123xβˆ’3274265657x^{46} - 10x^{45} - 124x^{44} + 1518x^{43} + 5974x^{42} - 102374x^{41} - 116011x^{40} + 4069871x^{39} - 1045480x^{38} - 106795767x^{37} + 115727513x^{36} + 1964724210x^{35} - 3197611595x^{34} - 26307233865x^{33} + 52729459109x^{32} + 263094973657x^{31} - 595996027435x^{30} - 2003851775771x^{29} + 4868481282296x^{28} + 11802759340169x^{27} - 29490933472232x^{26} - 54389704222550x^{25} + 134173571270776x^{24} + 197482820670140x^{23} - 460596021108031x^{22} - 565390694459455x^{21} + 1191114920767312x^{20} + 1267220721736176x^{19} - 2304707919704650x^{18} - 2188859618624828x^{17} + 3299038798178660x^{16} + 2846073259055039x^{15} - 3440488844053013x^{14} - 2703559144016765x^{13} + 2564462362757947x^{12} + 1810273594276592x^{11} - 1333163444531387x^{10} - 818132279475743x^{9} + 466521564724692x^{8} + 235716757607846x^{7} - 103277330159144x^{6} - 39723058511038x^{5} + 12744786693544x^{4} + 3367373989531x^{3} - 630482044248x^{2} - 102022043123x - 3274265657

Atkin–Lehner signs

ppwpw_p
863βˆ’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_p4611282524030βˆ’321632314511114327βˆ’8168428βˆ’10160

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 0.27 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, 863]: [0, 73] (certified at ell = 2147466121)
  • T = [(2, 1), (3, 1)]; h of degree 72 = 1 x 72 (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(863, 2)[2]   # 863.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 863:2 1000