🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 701 β€Ί 701.2.a β€Ί 701.2.a.b

Newform orbit 701.2.a.b

Trace form

The 21 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

21qβˆ’3q2βˆ’14q3+7q4βˆ’7q5βˆ’7q6βˆ’9q7βˆ’6q8+9q9βˆ’14q10βˆ’12q11βˆ’12q12βˆ’17q13βˆ’3q14βˆ’4q15βˆ’13q16βˆ’5q17+4q18βˆ’53q19βˆ’4q20βˆ’13q21βˆ’5q22βˆ’9q23βˆ’6q24+O(q25)21q - 3q^{2} - 14q^{3} + 7q^{4} - 7q^{5} - 7q^{6} - 9q^{7} - 6q^{8} + 9q^{9} - 14q^{10} - 12q^{11} - 12q^{12} - 17q^{13} - 3q^{14} - 4q^{15} - 13q^{16} - 5q^{17} + 4q^{18} - 53q^{19} - 4q^{20} - 13q^{21} - 5q^{22} - 9q^{23} - 6q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 21. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x21+17x20+91x19βˆ’1593x17βˆ’3869x16+7626x15+37382x14βˆ’883x13βˆ’153625x12βˆ’93442x11+328433x10+290780x9βˆ’385183x8βˆ’376011x7+250361x6+226782x5βˆ’87079x4βˆ’58567x3+14627x2+4799xβˆ’798x^{21} + 17x^{20} + 91x^{19} - 1593x^{17} - 3869x^{16} + 7626x^{15} + 37382x^{14} - 883x^{13} - 153625x^{12} - 93442x^{11} + 328433x^{10} + 290780x^{9} - 385183x^{8} - 376011x^{7} + 250361x^{6} + 226782x^{5} - 87079x^{4} - 58567x^{3} + 14627x^{2} + 4799x - 798

Atkin–Lehner signs

ppwpw_p
701+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’14βˆ’7βˆ’9βˆ’12βˆ’17βˆ’5βˆ’53βˆ’9βˆ’14βˆ’53βˆ’12βˆ’3βˆ’330βˆ’2βˆ’34βˆ’31βˆ’371βˆ’6βˆ’48βˆ’2βˆ’14βˆ’11

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 0.238 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, 701]: [0, 59] (certified at ell = 2147479601)
  • T = [(2, 1), (3, 1)]; h of degree 58 = 1 x 58 (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(701, 2)[1]   # 701.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 701:2 1000