🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 953 β€Ί 953.2.a β€Ί 953.2.a.b

Newform orbit 953.2.a.b

Trace form

The 47 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

47q+2q2+16q3+58q4+6q5+5q6+42q7+6q8+57q9+17q10+6q11+36q12+32q13+17q15+72q16+4q17βˆ’6q18+21q19βˆ’q20+10q22+34q23βˆ’q24+O(q25)47q + 2q^{2} + 16q^{3} + 58q^{4} + 6q^{5} + 5q^{6} + 42q^{7} + 6q^{8} + 57q^{9} + 17q^{10} + 6q^{11} + 36q^{12} + 32q^{13} + 17q^{15} + 72q^{16} + 4q^{17} - 6q^{18} + 21q^{19} - q^{20} + 10q^{22} + 34q^{23} - q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 47. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x47βˆ’18x46βˆ’18x45+2146x44βˆ’7913x43βˆ’104427x42+678502x41+2497019x40βˆ’27171719x39βˆ’20264017x38+661517002x37βˆ’498976637x36βˆ’10717271086x35+18871884979x34+120346616553x33βˆ’314504747578x32βˆ’951460918787x31+3357637138438x30+5265326195264x29βˆ’25209116486996x28βˆ’19590241759870x27+138459050715994x26+42240744061941x25βˆ’567339505543125x24βˆ’9481995185660x23+1749904815099402x22βˆ’265204466297502x21βˆ’4066389382943764x20+954266383041472x19+7066993881563239x18βˆ’1810989461789825x17βˆ’9030432179800878x16+2136263861580523x15+8245494187098665x14βˆ’1609524352119399x13βˆ’5158697860670231x12+771944398025902x11+2086539528036078x10βˆ’231502306795514x9βˆ’500878805032725x8+41178267666528x7+62059869679044x6βˆ’3672563919373x5βˆ’3120841055222x4+112160729169x3+19062661431x2βˆ’16630723xβˆ’10339x^{47} - 18x^{46} - 18x^{45} + 2146x^{44} - 7913x^{43} - 104427x^{42} + 678502x^{41} + 2497019x^{40} - 27171719x^{39} - 20264017x^{38} + 661517002x^{37} - 498976637x^{36} - 10717271086x^{35} + 18871884979x^{34} + 120346616553x^{33} - 314504747578x^{32} - 951460918787x^{31} + 3357637138438x^{30} + 5265326195264x^{29} - 25209116486996x^{28} - 19590241759870x^{27} + 138459050715994x^{26} + 42240744061941x^{25} - 567339505543125x^{24} - 9481995185660x^{23} + 1749904815099402x^{22} - 265204466297502x^{21} - 4066389382943764x^{20} + 954266383041472x^{19} + 7066993881563239x^{18} - 1810989461789825x^{17} - 9030432179800878x^{16} + 2136263861580523x^{15} + 8245494187098665x^{14} - 1609524352119399x^{13} - 5158697860670231x^{12} + 771944398025902x^{11} + 2086539528036078x^{10} - 231502306795514x^{9} - 500878805032725x^{8} + 41178267666528x^{7} + 62059869679044x^{6} - 3672563919373x^{5} - 3120841055222x^{4} + 112160729169x^{3} + 19062661431x^{2} - 16630723x - 10339

Atkin–Lehner signs

ppwpw_p
953βˆ’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=+1(-1)^{k/2} w_N = +1: each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

pp2357111317192329313741434753596167717379838997
tr⁑ap\operatorname{tr} a_p21664263242134βˆ’76623βˆ’96287βˆ’44166811263220βˆ’663

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 0.303 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, 953]: [0, 80] (certified at ell = 2147477809)
  • T = [(2, 1), (3, 1)]; h of degree 79 = 1 x 79 (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(953, 2)[1]   # 953.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 953:2 1000