🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 6 β€Ί Level 103 β€Ί 103.6.a β€Ί 103.6.a.a

Newform orbit 103.6.a.a

Trace form

The 19 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

19qβˆ’21q2βˆ’38q3+271q4βˆ’147q5βˆ’137q6βˆ’206q7βˆ’1116q8+777q9βˆ’202q10βˆ’999q11βˆ’2217q12βˆ’1589q13+463q14βˆ’113q15+4571q16βˆ’6815q17βˆ’5028q18βˆ’1455q19βˆ’5571q20βˆ’5162q21βˆ’7647q22βˆ’10540q23βˆ’3438q24+O(q25)19q - 21q^{2} - 38q^{3} + 271q^{4} - 147q^{5} - 137q^{6} - 206q^{7} - 1116q^{8} + 777q^{9} - 202q^{10} - 999q^{11} - 2217q^{12} - 1589q^{13} + 463q^{14} - 113q^{15} + 4571q^{16} - 6815q^{17} - 5028q^{18} - 1455q^{19} - 5571q^{20} - 5162q^{21} - 7647q^{22} - 10540q^{23} - 3438q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 19. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x19+59x18βˆ’1259x17βˆ’115634x16+297028x15+87916042x14+111694134x13βˆ’34453824454x12βˆ’32107600436x11+7632468898320x10βˆ’7400851566048x9βˆ’946362610021818x8+3132607904056225x7+58403721711916643x6βˆ’344036949173262441x5βˆ’1071586133148306950x4+12434520790741172708x3βˆ’26114786961301924872x2βˆ’8120478403539106960x+46404360183877805248x^{19} + 59x^{18} - 1259x^{17} - 115634x^{16} + 297028x^{15} + 87916042x^{14} + 111694134x^{13} - 34453824454x^{12} - 32107600436x^{11} + 7632468898320x^{10} - 7400851566048x^{9} - 946362610021818x^{8} + 3132607904056225x^{7} + 58403721711916643x^{6} - 344036949173262441x^{5} - 1071586133148306950x^{4} + 12434520790741172708x^{3} - 26114786961301924872x^{2} - 8120478403539106960x + 46404360183877805248

Atkin–Lehner signs

ppwpw_p
103+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’21βˆ’38βˆ’147βˆ’206βˆ’999βˆ’1589βˆ’6815βˆ’1455βˆ’10540βˆ’22006βˆ’12968βˆ’13052βˆ’48836βˆ’18596βˆ’24571βˆ’71859βˆ’4613βˆ’308389566βˆ’669118773358712152723120βˆ’89198

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 0.273 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, 45] (certified at ell = 2147482501)
  • T = [(2, 1), (3, 1)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 8 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, 6)[0]   # 103.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 103:6 1000