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Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 949 β€Ί 949.2.a β€Ί 949.2.a.c

Newform orbit 949.2.a.c

Trace form

The 16 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

16qβˆ’3q2βˆ’11q3+11q4βˆ’6q5βˆ’4q6+q7βˆ’9q8+9q9βˆ’8q10βˆ’23q11βˆ’23q12+16q13βˆ’16q14βˆ’q15+9q16βˆ’18q17+14q18βˆ’18q19βˆ’9q20βˆ’12q21βˆ’5q22βˆ’38q23βˆ’19q24+O(q25)16q - 3q^{2} - 11q^{3} + 11q^{4} - 6q^{5} - 4q^{6} + q^{7} - 9q^{8} + 9q^{9} - 8q^{10} - 23q^{11} - 23q^{12} + 16q^{13} - 16q^{14} - q^{15} + 9q^{16} - 18q^{17} + 14q^{18} - 18q^{19} - 9q^{20} - 12q^{21} - 5q^{22} - 38q^{23} - 19q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 16. It is generated by the eigenvalue Ξ±\alpha of T=T2+3T3T = T_{2} + 3T_{3}, whose minimal polynomial is

x16+36x15+382x14βˆ’767x13βˆ’38262x12βˆ’147275x11+1110613x10+7825688x9βˆ’9490000x8βˆ’150093005x7βˆ’49926604x6+1333714352x5+1048154170x4βˆ’5488731330x3βˆ’4473869485x2+7661253473x+6993036273x^{16} + 36x^{15} + 382x^{14} - 767x^{13} - 38262x^{12} - 147275x^{11} + 1110613x^{10} + 7825688x^{9} - 9490000x^{8} - 150093005x^{7} - 49926604x^{6} + 1333714352x^{5} + 1048154170x^{4} - 5488731330x^{3} - 4473869485x^{2} + 7661253473x + 6993036273

Atkin–Lehner signs

ppwpw_p
13βˆ’1-1
73βˆ’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’11βˆ’61βˆ’2316βˆ’18βˆ’18βˆ’38βˆ’30βˆ’17βˆ’5βˆ’11βˆ’2βˆ’3βˆ’9βˆ’35βˆ’23βˆ’3116βˆ’35βˆ’60βˆ’3938

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 0.291 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, 13, 73, 949]: [0, 1, 6, 86] (certified at ell = 2147482873)
  • T = [(2, 1), (3, 3)]; h of degree 73 = 1 x 73 (phi(ord eps) x dim S_k^new); 10 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(949, 2)[2]   # 949.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 949:2 1000