🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 991 β€Ί 991.2.a β€Ί 991.2.a.b

Newform orbit 991.2.a.b

Trace form

The 49 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

49q+4q2+4q3+52q4+26q5+12q6+3q7+9q8+77q9+7q10+22q11βˆ’4q12+14q13+25q14βˆ’3q15+46q16+72q17βˆ’9q18+18q19+55q20+13q21βˆ’5q22+12q23+27q24+O(q25)49q + 4q^{2} + 4q^{3} + 52q^{4} + 26q^{5} + 12q^{6} + 3q^{7} + 9q^{8} + 77q^{9} + 7q^{10} + 22q^{11} - 4q^{12} + 14q^{13} + 25q^{14} - 3q^{15} + 46q^{16} + 72q^{17} - 9q^{18} + 18q^{19} + 55q^{20} + 13q^{21} - 5q^{22} + 12q^{23} + 27q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 49. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x49βˆ’8x48βˆ’167x47+1482x46+11997x45βˆ’122937x44βˆ’477998x43+6056504x42+11043597x41βˆ’198270366x40βˆ’121924200x39+4575077640x38βˆ’756739998x37βˆ’77072586900x36+53938951963x35+969424731636x34βˆ’1044527079995x33βˆ’9242450464891x32+12428086356136x31+67504574903519x30βˆ’102934477744534x29βˆ’380714040736493x28+619450781800602x27+1668300728077565x26βˆ’2754374825166251x25βˆ’5705922388317379x24+9082863609046682x23+15258349253671926x22βˆ’22085144978989170x21βˆ’31785816214736454x20+38997867429696570x19+50930163611967636x18βˆ’48674159708753594x17βˆ’61201797026987042x16+41075950918389835x15+53024221054160421x14βˆ’21734025378112711x13βˆ’31417383233256855x12+6149170960198181x11+11938152478108160x10βˆ’408954290650682x9βˆ’2673825936081040x8βˆ’217073115631265x7+306781729923469x6+49257009369793x5βˆ’13024272957857x4βˆ’2739782193853x3+63502712325x2+24302281308x+137189455x^{49} - 8x^{48} - 167x^{47} + 1482x^{46} + 11997x^{45} - 122937x^{44} - 477998x^{43} + 6056504x^{42} + 11043597x^{41} - 198270366x^{40} - 121924200x^{39} + 4575077640x^{38} - 756739998x^{37} - 77072586900x^{36} + 53938951963x^{35} + 969424731636x^{34} - 1044527079995x^{33} - 9242450464891x^{32} + 12428086356136x^{31} + 67504574903519x^{30} - 102934477744534x^{29} - 380714040736493x^{28} + 619450781800602x^{27} + 1668300728077565x^{26} - 2754374825166251x^{25} - 5705922388317379x^{24} + 9082863609046682x^{23} + 15258349253671926x^{22} - 22085144978989170x^{21} - 31785816214736454x^{20} + 38997867429696570x^{19} + 50930163611967636x^{18} - 48674159708753594x^{17} - 61201797026987042x^{16} + 41075950918389835x^{15} + 53024221054160421x^{14} - 21734025378112711x^{13} - 31417383233256855x^{12} + 6149170960198181x^{11} + 11938152478108160x^{10} - 408954290650682x^{9} - 2673825936081040x^{8} - 217073115631265x^{7} + 306781729923469x^{6} + 49257009369793x^{5} - 13024272957857x^{4} - 2739782193853x^{3} + 63502712325x^{2} + 24302281308x + 137189455

Atkin–Lehner signs

ppwpw_p
991βˆ’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_p44263221472181261βˆ’22127βˆ’530164039βˆ’41634βˆ’213011915

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, 991]: [0, 83] (certified at ell = 2147476321)
  • T = [(2, 1), (3, 1)]; h of degree 82 = 1 x 82 (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(991, 2)[1]   # 991.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 991:2 1000