🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 773 β€Ί 773.2.a β€Ί 773.2.a.b

Newform orbit 773.2.a.b

Trace form

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

24qβˆ’3q2βˆ’15q3+11q4βˆ’3q5+q6βˆ’26q7βˆ’15q8+23q9βˆ’16q10βˆ’21q11βˆ’25q12βˆ’24q13βˆ’3q14βˆ’22q15+9q16βˆ’6q17βˆ’33q18βˆ’45q19+5q20+2q21βˆ’19q22+2q23βˆ’5q24+O(q25)24q - 3q^{2} - 15q^{3} + 11q^{4} - 3q^{5} + q^{6} - 26q^{7} - 15q^{8} + 23q^{9} - 16q^{10} - 21q^{11} - 25q^{12} - 24q^{13} - 3q^{14} - 22q^{15} + 9q^{16} - 6q^{17} - 33q^{18} - 45q^{19} + 5q^{20} + 2q^{21} - 19q^{22} + 2q^{23} - 5q^{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+3T3+5T5+7T7T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}, whose minimal polynomial is

x24+245x23+24737x22+1241011x21+23112114x20βˆ’747328611x19βˆ’55499973884x18βˆ’1295148898021x17βˆ’5706038649147x16+363951979642581x15+7718769500352685x14+24404654075696169x13βˆ’1095782759499365803x12βˆ’13916740324078385976x11+11987736286624284089x10+1316048672488956685099x9+6423312557746343874084x8βˆ’42974702677054936690657x7βˆ’436577669395101105586993x6+33005353018738192029961x5+10516619806018666060299789x4+20367700931150119202703024x3βˆ’86228980579801024488026788x2βˆ’215842876222358007504207959x+144766832133517776639104591x^{24} + 245x^{23} + 24737x^{22} + 1241011x^{21} + 23112114x^{20} - 747328611x^{19} - 55499973884x^{18} - 1295148898021x^{17} - 5706038649147x^{16} + 363951979642581x^{15} + 7718769500352685x^{14} + 24404654075696169x^{13} - 1095782759499365803x^{12} - 13916740324078385976x^{11} + 11987736286624284089x^{10} + 1316048672488956685099x^{9} + 6423312557746343874084x^{8} - 42974702677054936690657x^{7} - 436577669395101105586993x^{6} + 33005353018738192029961x^{5} + 10516619806018666060299789x^{4} + 20367700931150119202703024x^{3} - 86228980579801024488026788x^{2} - 215842876222358007504207959x + 144766832133517776639104591

Atkin–Lehner signs

ppwpw_p
773+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’3βˆ’15βˆ’3βˆ’26βˆ’21βˆ’24βˆ’6βˆ’452βˆ’6βˆ’5βˆ’31βˆ’1βˆ’90βˆ’29βˆ’14βˆ’21βˆ’11513βˆ’22βˆ’6βˆ’4635βˆ’2

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 0.297 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, 773]: [0, 65] (certified at ell = 2147480893)
  • T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 64 = 1 x 64 (phi(ord eps) x dim S_k^new); 14 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(773, 2)[1]   # 773.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 773:2 1000