🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 103 β€Ί 103.4.a β€Ί 103.4.a.a

Newform orbit 103.4.a.a

Trace form

The 10 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

10qβˆ’11q2βˆ’8q3+27q4βˆ’30q5βˆ’17q6βˆ’33q7βˆ’108q8βˆ’24q9βˆ’2q10βˆ’28q11βˆ’57q12βˆ’75q13βˆ’149q14βˆ’206q15βˆ’29q16βˆ’401q17βˆ’222q18βˆ’171q19βˆ’387q20βˆ’236q21βˆ’179q22βˆ’463q23βˆ’54q24+O(q25)10q - 11q^{2} - 8q^{3} + 27q^{4} - 30q^{5} - 17q^{6} - 33q^{7} - 108q^{8} - 24q^{9} - 2q^{10} - 28q^{11} - 57q^{12} - 75q^{13} - 149q^{14} - 206q^{15} - 29q^{16} - 401q^{17} - 222q^{18} - 171q^{19} - 387q^{20} - 236q^{21} - 179q^{22} - 463q^{23} - 54q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 10. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x10+19x9+21x8βˆ’936x7βˆ’2024x6+15022x5+33595x4βˆ’93071x3βˆ’205527x2+186260x+406092x^{10} + 19x^{9} + 21x^{8} - 936x^{7} - 2024x^{6} + 15022x^{5} + 33595x^{4} - 93071x^{3} - 205527x^{2} + 186260x + 406092

Atkin–Lehner signs

ppwpw_p
103βˆ’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=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’11βˆ’8βˆ’30βˆ’33βˆ’28βˆ’75βˆ’401βˆ’171βˆ’463βˆ’395188βˆ’142βˆ’561βˆ’108βˆ’412βˆ’1050βˆ’503βˆ’629βˆ’1054168βˆ’1642629βˆ’2013βˆ’926βˆ’291

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 0.102 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 engine's checks
  • dim M^+ at levels [1, 103]: [1, 27] (certified at ell = 2147482501)
  • T = [(2, 1), (3, 1)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 4 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(103, 4)[0]   # 103.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 103:4 1000