🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 643 β€Ί 643.2.a β€Ί 643.2.a.b

Newform orbit 643.2.a.b

Trace form

The 24 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

24qβˆ’13q2βˆ’7q3+23q4βˆ’23q5βˆ’4q6βˆ’10q7βˆ’33q8+13q9+6q10βˆ’17q11βˆ’14q12βˆ’23q13βˆ’7q14βˆ’10q15+33q16βˆ’30q17βˆ’33q18βˆ’q19βˆ’38q20βˆ’29q21βˆ’9q22βˆ’27q23βˆ’7q24+O(q25)24q - 13q^{2} - 7q^{3} + 23q^{4} - 23q^{5} - 4q^{6} - 10q^{7} - 33q^{8} + 13q^{9} + 6q^{10} - 17q^{11} - 14q^{12} - 23q^{13} - 7q^{14} - 10q^{15} + 33q^{16} - 30q^{17} - 33q^{18} - q^{19} - 38q^{20} - 29q^{21} - 9q^{22} - 27q^{23} - 7q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 24. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x24+20x23+126x22βˆ’3086x20βˆ’9131x19+18858x18+119288x17+20994x16βˆ’635063x15βˆ’563739x14+1727241x13+1966296x12βˆ’2827535x11βˆ’2991236x10+3062908x9+2122572x8βˆ’2083077x7βˆ’522854x6+703588x5βˆ’57096x4βˆ’58334x3+10425x2+807xβˆ’193x^{24} + 20x^{23} + 126x^{22} - 3086x^{20} - 9131x^{19} + 18858x^{18} + 119288x^{17} + 20994x^{16} - 635063x^{15} - 563739x^{14} + 1727241x^{13} + 1966296x^{12} - 2827535x^{11} - 2991236x^{10} + 3062908x^{9} + 2122572x^{8} - 2083077x^{7} - 522854x^{6} + 703588x^{5} - 57096x^{4} - 58334x^{3} + 10425x^{2} + 807x - 193

Atkin–Lehner signs

ppwpw_p
643+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’13βˆ’7βˆ’23βˆ’10βˆ’17βˆ’23βˆ’30βˆ’1βˆ’27βˆ’65βˆ’2βˆ’34βˆ’31βˆ’4βˆ’24βˆ’80βˆ’120βˆ’5βˆ’37βˆ’173βˆ’657

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 0.218 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, 643]: [0, 54] (certified at ell = 2147476519)
  • T = [(2, 1), (3, 1)]; h of degree 53 = 1 x 53 (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(643, 2)[1]   # 643.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 643:2 1000