Properties

Label 187.2.g.f
Level 187
Weight 2
Character orbit 187.g
Analytic conductor 1.493
Analytic rank 0
Dimension 36
CM No

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Newspace parameters

Level: \( N \) = \( 187 = 11 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 187.g (of order \(5\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(1.4932025178\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(9\) over \(\Q(\zeta_{5})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36q - 4q^{2} - q^{3} - 14q^{4} + q^{5} - 5q^{6} + q^{7} + 17q^{8} - 22q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q - 4q^{2} - q^{3} - 14q^{4} + q^{5} - 5q^{6} + q^{7} + 17q^{8} - 22q^{9} - 10q^{10} + 3q^{11} + 28q^{12} - 13q^{13} + 14q^{14} - 24q^{15} + 16q^{16} - 9q^{17} + 2q^{18} + 10q^{19} + 19q^{20} - 50q^{21} - 25q^{22} + 38q^{23} - 17q^{24} - 28q^{25} + 20q^{26} - 16q^{27} + 31q^{28} - 45q^{29} + 68q^{30} - 13q^{31} - 40q^{32} - 29q^{33} - 4q^{34} + 13q^{35} - 25q^{36} + q^{37} + 65q^{38} - 34q^{39} - 54q^{40} + 37q^{41} + 28q^{42} - 8q^{43} - 2q^{44} + 42q^{45} + 22q^{46} - 35q^{47} + 48q^{48} - 2q^{49} - 49q^{50} - q^{51} + 56q^{52} + 58q^{53} - 58q^{54} - 19q^{55} - 28q^{56} + 9q^{57} - 52q^{58} + 16q^{59} + 97q^{60} - 14q^{61} - 64q^{62} + 34q^{63} - 33q^{64} - 42q^{65} - 28q^{66} + 54q^{67} - 14q^{68} + 19q^{69} + 4q^{70} + 25q^{71} - 72q^{72} + 8q^{73} + 84q^{74} + 30q^{75} - 140q^{76} - 31q^{77} - 48q^{78} + 19q^{79} - 19q^{80} + 56q^{81} + 48q^{82} + 42q^{83} - 91q^{84} - 9q^{85} + 30q^{86} - 32q^{87} + 126q^{88} + 12q^{89} + 160q^{90} - 59q^{91} + 69q^{92} - 40q^{93} - 77q^{94} - 11q^{95} + 192q^{96} - 49q^{97} - 212q^{98} + 38q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1 −2.14992 + 1.56201i 0.943884 + 2.90498i 1.56424 4.81425i 1.58671 + 1.15281i −6.56686 4.77110i −0.547865 + 1.68616i 2.51450 + 7.73883i −5.12092 + 3.72057i −5.21199
69.2 −1.73247 + 1.25872i 0.331943 + 1.02161i 0.799065 2.45927i −2.02945 1.47448i −1.86100 1.35210i 1.12029 3.44788i 0.387671 + 1.19313i 1.49354 1.08512i 5.37193
69.3 −1.36392 + 0.990946i −0.936920 2.88354i 0.260271 0.801032i 2.28514 + 1.66025i 4.13532 + 3.00449i 0.207430 0.638405i −0.603152 1.85631i −5.00995 + 3.63994i −4.76196
69.4 −1.28769 + 0.935561i 0.00348887 + 0.0107376i 0.164835 0.507309i 2.53950 + 1.84506i −0.0145383 0.0105627i −0.112651 + 0.346704i −0.721344 2.22007i 2.42695 1.76328i −4.99625
69.5 −0.415893 + 0.302164i 0.814643 + 2.50721i −0.536370 + 1.65078i −3.29479 2.39381i −1.09640 0.796578i −0.581623 + 1.79005i −0.593447 1.82644i −3.19543 + 2.32161i 2.09360
69.6 0.905302 0.657740i −0.440186 1.35475i −0.231085 + 0.711206i 1.95004 + 1.41679i −1.28958 0.936932i 1.26946 3.90701i 0.950176 + 2.92434i 0.785461 0.570671i 2.69726
69.7 1.31285 0.953841i −0.637980 1.96350i 0.195728 0.602389i −2.95777 2.14895i −2.71044 1.96925i −0.195368 + 0.601281i 0.685306 + 2.10916i −1.02126 + 0.741992i −5.93286
69.8 1.61769 1.17532i 1.01397 + 3.12067i 0.617507 1.90049i −1.06499 0.773760i 5.30807 + 3.85654i 1.03762 3.19345i 0.00105288 + 0.00324042i −6.28340 + 4.56516i −2.63224
69.9 2.11405 1.53595i 0.334210 + 1.02859i 1.49204 4.59203i −0.441442 0.320727i 2.28641 + 1.66117i −1.38827 + 4.27266i −2.28388 7.02906i 1.48074 1.07582i −1.42585
86.1 −0.841398 + 2.58956i −1.62645 + 1.18168i −4.37981 3.18212i 0.802755 + 2.47063i −1.69155 5.20605i −2.56930 1.86671i 7.51983 5.46348i 0.321909 0.990733i −7.07326
86.2 −0.768807 + 2.36614i 1.93066 1.40271i −3.38954 2.46265i −1.37389 4.22840i 1.83471 + 5.64664i −2.07663 1.50876i 4.40735 3.20213i 0.832818 2.56315i 11.0613
86.3 −0.709792 + 2.18452i 0.676145 0.491248i −2.65027 1.92553i 0.327553 + 1.00811i 0.593217 + 1.82573i 2.54751 + 1.85087i 2.37098 1.72262i −0.711203 + 2.18886i −2.43472
86.4 −0.343162 + 1.05614i −2.58082 + 1.87507i 0.620355 + 0.450715i 0.625693 + 1.92569i −1.09471 3.36916i 2.66865 + 1.93889i −2.48572 + 1.80598i 2.21766 6.82526i −2.24851
86.5 −0.135862 + 0.418139i −0.0312072 + 0.0226734i 1.46165 + 1.06195i 0.919117 + 2.82875i −0.00524076 0.0161294i −2.52440 1.83408i −1.35401 + 0.983744i −0.926591 + 2.85175i −1.30768
86.6 0.0881614 0.271333i −1.31211 + 0.953301i 1.55218 + 1.12773i −1.06412 3.27502i 0.142985 + 0.440062i 3.21023 + 2.33237i 0.904452 0.657123i −0.114211 + 0.351506i −0.982436
86.7 0.282168 0.868424i 1.40032 1.01739i 0.943492 + 0.685487i −0.0971410 0.298969i −0.488401 1.50314i −2.74134 1.99170i 2.33897 1.69936i −0.00124648 + 0.00383627i −0.287042
86.8 0.648313 1.99530i −2.15638 + 1.56670i −1.94289 1.41159i 1.10893 + 3.41293i 1.72804 + 5.31835i 0.243832 + 0.177154i −0.681533 + 0.495163i 1.26837 3.90365i 7.52877
86.9 0.780378 2.40176i 1.77278 1.28800i −3.54141 2.57299i 0.678152 + 2.08714i −1.71003 5.26293i 0.932422 + 0.677444i −4.85721 + 3.52897i 0.556762 1.71354i 5.54201
103.1 −2.14992 1.56201i 0.943884 2.90498i 1.56424 + 4.81425i 1.58671 1.15281i −6.56686 + 4.77110i −0.547865 1.68616i 2.51450 7.73883i −5.12092 3.72057i −5.21199
103.2 −1.73247 1.25872i 0.331943 1.02161i 0.799065 + 2.45927i −2.02945 + 1.47448i −1.86100 + 1.35210i 1.12029 + 3.44788i 0.387671 1.19313i 1.49354 + 1.08512i 5.37193
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.9
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(187, [\chi])\):

\(T_{2}^{36} + \cdots\)
\(T_{3}^{36} + \cdots\)