🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 753 β€Ί 753.2.a β€Ί 753.2.a.h

Newform orbit 753.2.a.h

Trace form

The 9 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

9qβˆ’9q3+6q4+q5βˆ’14q7βˆ’3q8+9q9βˆ’12q10βˆ’q11βˆ’6q12βˆ’15q13βˆ’3q14βˆ’q15βˆ’4q16βˆ’9q17βˆ’7q19βˆ’9q20+14q21βˆ’16q22βˆ’16q23+3q24+O(q25)9q - 9q^{3} + 6q^{4} + q^{5} - 14q^{7} - 3q^{8} + 9q^{9} - 12q^{10} - q^{11} - 6q^{12} - 15q^{13} - 3q^{14} - q^{15} - 4q^{16} - 9q^{17} - 7q^{19} - 9q^{20} + 14q^{21} - 16q^{22} - 16q^{23} + 3q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 9. It is generated by the eigenvalue Ξ±\alpha of T=T2+T5T = T_{2} + T_{5}, whose minimal polynomial is

x9βˆ’x8βˆ’23x7βˆ’2x6+75x5βˆ’7x4βˆ’64x3+31x2βˆ’xβˆ’1x^{9} - x^{8} - 23x^{7} - 2x^{6} + 75x^{5} - 7x^{4} - 64x^{3} + 31x^{2} - x - 1

Atkin–Lehner signs

ppwpw_p
3+1+1
251+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_p0βˆ’91βˆ’14βˆ’1βˆ’15βˆ’9βˆ’7βˆ’1617βˆ’22βˆ’167βˆ’29βˆ’25βˆ’2βˆ’1βˆ’21βˆ’259βˆ’27βˆ’12βˆ’318βˆ’7

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 0.244 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, 3, 251, 753]: [0, 1, 22, 86] (certified at ell = 2147482501)
  • T = [(2, 1), (5, 1)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 5 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(753, 2)[7]   # 753.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 753:2 1000