↑

Properties

Label 164.3.l.a.23.1
Level $164$
Weight $3$
Character 164.23
Analytic conductor $4.469$
Analytic rank $0$
Dimension $160$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [164,3,Mod(23,164)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("164.23"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(164, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 9])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 164.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.46867633551\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 23.1
Character \(\chi\) \(=\) 164.23
Dual form 164.3.l.a.107.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.99597 - 0.126903i) q^{2} +2.39245 q^{3} +(3.96779 + 0.506589i) q^{4} +(-2.73540 + 1.98738i) q^{5} +(-4.77525 - 0.303609i) q^{6} +(-2.96576 + 9.12767i) q^{7} +(-7.85530 - 1.51466i) q^{8} -3.27620 q^{9} +(5.71198 - 3.61963i) q^{10} +(2.71615 + 1.97340i) q^{11} +(9.49273 + 1.21199i) q^{12} +(-17.2359 + 5.60027i) q^{13} +(7.07790 - 17.8422i) q^{14} +(-6.54430 + 4.75471i) q^{15} +(15.4867 + 4.02008i) q^{16} +(12.0144 - 16.5364i) q^{17} +(6.53919 + 0.415759i) q^{18} +(-4.37350 + 13.4602i) q^{19} +(-11.8603 + 6.49980i) q^{20} +(-7.09543 + 21.8375i) q^{21} +(-5.17091 - 4.28352i) q^{22} +(1.61948 - 0.526201i) q^{23} +(-18.7934 - 3.62375i) q^{24} +(-4.19271 + 12.9038i) q^{25} +(35.1129 - 8.99069i) q^{26} -29.3702 q^{27} +(-16.3915 + 34.7143i) q^{28} +(23.0834 + 31.7716i) q^{29} +(13.6656 - 8.65977i) q^{30} +(-31.7277 + 43.6695i) q^{31} +(-30.4009 - 9.98928i) q^{32} +(6.49824 + 4.72124i) q^{33} +(-26.0789 + 31.4815i) q^{34} +(-10.0277 - 30.8619i) q^{35} +(-12.9993 - 1.65969i) q^{36} +(36.0109 - 26.1634i) q^{37} +(10.4375 - 26.3112i) q^{38} +(-41.2359 + 13.3984i) q^{39} +(24.4976 - 11.4683i) q^{40} +(37.4054 - 16.7879i) q^{41} +(16.9335 - 42.6865i) q^{42} +(61.7639 - 20.0683i) q^{43} +(9.77740 + 9.20599i) q^{44} +(8.96170 - 6.51106i) q^{45} +(-3.29921 + 0.844764i) q^{46} +(0.586520 + 1.80512i) q^{47} +(37.0512 + 9.61784i) q^{48} +(-34.8768 - 25.3395i) q^{49} +(10.0061 - 25.2236i) q^{50} +(28.7439 - 39.5625i) q^{51} +(-71.2253 + 13.4892i) q^{52} +(-17.3436 - 23.8714i) q^{53} +(58.6219 + 3.72716i) q^{54} -11.3516 q^{55} +(37.1223 - 67.2085i) q^{56} +(-10.4634 + 32.2029i) q^{57} +(-42.0418 - 66.3444i) q^{58} +(-19.4885 + 6.33220i) q^{59} +(-28.3751 + 15.5504i) q^{60} +(4.21742 - 12.9799i) q^{61} +(68.8694 - 83.1366i) q^{62} +(9.71641 - 29.9040i) q^{63} +(59.4116 + 23.7963i) q^{64} +(36.0171 - 49.5732i) q^{65} +(-12.3711 - 10.2481i) q^{66} +(-15.9786 + 11.6091i) q^{67} +(56.0479 - 59.5267i) q^{68} +(3.87452 - 1.25891i) q^{69} +(16.0984 + 62.8720i) q^{70} +(-41.1928 - 29.9283i) q^{71} +(25.7355 + 4.96233i) q^{72} -30.9183 q^{73} +(-75.1969 + 47.6516i) q^{74} +(-10.0308 + 30.8717i) q^{75} +(-24.1720 + 51.1919i) q^{76} +(-26.0679 + 18.9395i) q^{77} +(84.0059 - 21.5097i) q^{78} +145.964 q^{79} +(-50.3519 + 19.7816i) q^{80} -40.7808 q^{81} +(-76.7905 + 28.7614i) q^{82} +74.5638i q^{83} +(-39.2158 + 83.0521i) q^{84} +69.1110i q^{85} +(-125.826 + 32.2177i) q^{86} +(55.2258 + 76.0118i) q^{87} +(-18.3471 - 19.6157i) q^{88} +(112.870 + 36.6735i) q^{89} +(-18.7136 + 11.8586i) q^{90} -173.932i q^{91} +(6.69232 - 1.26744i) q^{92} +(-75.9069 + 104.477i) q^{93} +(-0.941601 - 3.67740i) q^{94} +(-14.7874 - 45.5110i) q^{95} +(-72.7325 - 23.8988i) q^{96} +(-34.9867 - 48.1551i) q^{97} +(66.3974 + 55.0029i) q^{98} +(-8.89862 - 6.46523i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q - 3 q^{2} - 3 q^{4} - 2 q^{5} - 25 q^{6} - 36 q^{8} + 424 q^{9} + 44 q^{10} - 50 q^{12} - 10 q^{13} - 43 q^{16} - 10 q^{17} - 84 q^{18} + 92 q^{20} - 44 q^{21} - 5 q^{22} - 50 q^{24} - 226 q^{25}+ \cdots - 50 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(129\)
\(\chi(n)\) \(-1\) \(e\left(\frac{9}{10}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99597 − 0.126903i −0.997985 − 0.0634515i
\(3\) 2.39245 0.797482 0.398741 − 0.917063i \(-0.369447\pi\)
0.398741 + 0.917063i \(0.369447\pi\)
\(4\) 3.96779 + 0.506589i 0.991948 + 0.126647i
\(5\) −2.73540 + 1.98738i −0.547080 + 0.397477i −0.826708 − 0.562632i \(-0.809789\pi\)
0.279628 + 0.960108i \(0.409789\pi\)
\(6\) −4.77525 − 0.303609i −0.795875 − 0.0506015i
\(7\) −2.96576 + 9.12767i −0.423680 + 1.30395i 0.480572 + 0.876955i \(0.340429\pi\)
−0.904253 + 0.426998i \(0.859571\pi\)
\(8\) −7.85530 − 1.51466i −0.981913 − 0.189333i
\(9\) −3.27620 −0.364022
\(10\) 5.71198 − 3.61963i 0.571198 − 0.361963i
\(11\) 2.71615 + 1.97340i 0.246922 + 0.179400i 0.704362 − 0.709841i \(-0.251233\pi\)
−0.457439 + 0.889241i \(0.651233\pi\)
\(12\) 9.49273 + 1.21199i 0.791061 + 0.100999i
\(13\) −17.2359 + 5.60027i −1.32584 + 0.430790i −0.884495 − 0.466550i \(-0.845497\pi\)
−0.441340 + 0.897340i \(0.645497\pi\)
\(14\) 7.07790 − 17.8422i 0.505564 − 1.27444i
\(15\) −6.54430 + 4.75471i −0.436287 + 0.316981i
\(16\) 15.4867 + 4.02008i 0.967921 + 0.251255i
\(17\) 12.0144 − 16.5364i 0.706730 − 0.972731i −0.293131 − 0.956072i \(-0.594697\pi\)
0.999861 − 0.0166586i \(-0.00530283\pi\)
\(18\) 6.53919 + 0.415759i 0.363288 + 0.0230977i
\(19\) −4.37350 + 13.4602i −0.230184 + 0.708434i 0.767540 + 0.641001i \(0.221481\pi\)
−0.997724 + 0.0674326i \(0.978519\pi\)
\(20\) −11.8603 + 6.49980i −0.593014 + 0.324990i
\(21\) −7.09543 + 21.8375i −0.337877 + 1.03988i
\(22\) −5.17091 − 4.28352i −0.235042 − 0.194706i
\(23\) 1.61948 − 0.526201i 0.0704121 − 0.0228783i −0.273599 − 0.961844i \(-0.588214\pi\)
0.344011 + 0.938966i \(0.388214\pi\)
\(24\) −18.7934 − 3.62375i −0.783058 − 0.150990i
\(25\) −4.19271 + 12.9038i −0.167708 + 0.516153i
\(26\) 35.1129 − 8.99069i 1.35050 − 0.345796i
\(27\) −29.3702 −1.08778
\(28\) −16.3915 + 34.7143i −0.585411 + 1.23980i
\(29\) 23.0834 + 31.7716i 0.795979 + 1.09557i 0.993338 + 0.115240i \(0.0367636\pi\)
−0.197359 + 0.980331i \(0.563236\pi\)
\(30\) 13.6656 − 8.65977i 0.455520 − 0.288659i
\(31\) −31.7277 + 43.6695i −1.02347 + 1.40869i −0.113736 + 0.993511i \(0.536282\pi\)
−0.909739 + 0.415181i \(0.863718\pi\)
\(32\) −30.4009 − 9.98928i −0.950028 − 0.312165i
\(33\) 6.49824 + 4.72124i 0.196916 + 0.143068i
\(34\) −26.0789 + 31.4815i −0.767028 + 0.925928i
\(35\) −10.0277 − 30.8619i −0.286504 − 0.881770i
\(36\) −12.9993 − 1.65969i −0.361091 − 0.0461024i
\(37\) 36.0109 − 26.1634i 0.973267 − 0.707120i 0.0170736 − 0.999854i \(-0.494565\pi\)
0.956194 + 0.292734i \(0.0945651\pi\)
\(38\) 10.4375 − 26.3112i 0.274672 − 0.692401i
\(39\) −41.2359 + 13.3984i −1.05733 + 0.343547i
\(40\) 24.4976 − 11.4683i 0.612440 − 0.286707i
\(41\) 37.4054 − 16.7879i 0.912327 − 0.409462i
\(42\) 16.9335 − 42.6865i 0.403179 − 1.01635i
\(43\) 61.7639 − 20.0683i 1.43637 − 0.466705i 0.515606 − 0.856826i \(-0.327567\pi\)
0.920763 + 0.390121i \(0.127567\pi\)
\(44\) 9.77740 + 9.20599i 0.222214 + 0.209227i
\(45\) 8.96170 − 6.51106i 0.199149 − 0.144690i
\(46\) −3.29921 + 0.844764i −0.0717219 + 0.0183644i
\(47\) 0.586520 + 1.80512i 0.0124791 + 0.0384069i 0.957102 − 0.289751i \(-0.0935723\pi\)
−0.944623 + 0.328157i \(0.893572\pi\)
\(48\) 37.0512 + 9.61784i 0.771900 + 0.200372i
\(49\) −34.8768 − 25.3395i −0.711772 − 0.517133i
\(50\) 10.0061 − 25.2236i 0.200121 − 0.504472i
\(51\) 28.7439 − 39.5625i 0.563605 − 0.775736i
\(52\) −71.2253 + 13.4892i −1.36972 + 0.259408i
\(53\) −17.3436 − 23.8714i −0.327238 − 0.450404i 0.613422 − 0.789755i \(-0.289792\pi\)
−0.940660 + 0.339351i \(0.889792\pi\)
\(54\) 58.6219 + 3.72716i 1.08559 + 0.0690215i
\(55\) −11.3516 −0.206393
\(56\) 37.1223 − 67.2085i 0.662898 − 1.20015i
\(57\) −10.4634 + 32.2029i −0.183568 + 0.564964i
\(58\) −42.0418 − 66.3444i −0.724859 − 1.14387i
\(59\) −19.4885 + 6.33220i −0.330314 + 0.107325i −0.469479 − 0.882944i \(-0.655558\pi\)
0.139165 + 0.990269i \(0.455558\pi\)
\(60\) −28.3751 + 15.5504i −0.472918 + 0.259174i
\(61\) 4.21742 − 12.9799i 0.0691381 − 0.212785i −0.910518 − 0.413470i \(-0.864317\pi\)
0.979656 + 0.200685i \(0.0643167\pi\)
\(62\) 68.8694 − 83.1366i 1.11080 − 1.34091i
\(63\) 9.71641 − 29.9040i 0.154229 − 0.474667i
\(64\) 59.4116 + 23.7963i 0.928306 + 0.371817i
\(65\) 36.0171 − 49.5732i 0.554109 − 0.762665i
\(66\) −12.3711 − 10.2481i −0.187442 − 0.155274i
\(67\) −15.9786 + 11.6091i −0.238486 + 0.173270i −0.700609 − 0.713546i \(-0.747088\pi\)
0.462122 + 0.886816i \(0.347088\pi\)
\(68\) 56.0479 − 59.5267i 0.824233 − 0.875393i
\(69\) 3.87452 − 1.25891i 0.0561525 − 0.0182450i
\(70\) 16.0984 + 62.8720i 0.229977 + 0.898172i
\(71\) −41.1928 − 29.9283i −0.580180 − 0.421525i 0.258609 − 0.965982i \(-0.416736\pi\)
−0.838789 + 0.544457i \(0.816736\pi\)
\(72\) 25.7355 + 4.96233i 0.357438 + 0.0689212i
\(73\) −30.9183 −0.423539 −0.211770 − 0.977320i \(-0.567923\pi\)
−0.211770 + 0.977320i \(0.567923\pi\)
\(74\) −75.1969 + 47.6516i −1.01617 + 0.643940i
\(75\) −10.0308 + 30.8717i −0.133745 + 0.411623i
\(76\) −24.1720 + 51.1919i −0.318052 + 0.673577i
\(77\) −26.0679 + 18.9395i −0.338545 + 0.245967i
\(78\) 84.0059 − 21.5097i 1.07700 − 0.275766i
\(79\) 145.964 1.84765 0.923824 − 0.382818i \(-0.125046\pi\)
0.923824 + 0.382818i \(0.125046\pi\)
\(80\) −50.3519 + 19.7816i −0.629398 + 0.247269i
\(81\) −40.7808 −0.503467
\(82\) −76.7905 + 28.7614i −0.936470 + 0.350748i
\(83\) 74.5638i 0.898359i 0.893442 + 0.449179i \(0.148284\pi\)
−0.893442 + 0.449179i \(0.851716\pi\)
\(84\) −39.2158 + 83.0521i −0.466855 + 0.988715i
\(85\) 69.1110i 0.813070i
\(86\) −125.826 + 32.2177i −1.46309 + 0.374624i
\(87\) 55.2258 + 76.0118i 0.634779 + 0.873699i
\(88\) −18.3471 − 19.6157i −0.208490 − 0.222905i
\(89\) 112.870 + 36.6735i 1.26820 + 0.412062i 0.864409 − 0.502790i \(-0.167693\pi\)
0.403789 + 0.914852i \(0.367693\pi\)
\(90\) −18.7136 + 11.8586i −0.207928 + 0.131762i
\(91\) − 173.932i − 1.91134i
\(92\) 6.69232 − 1.26744i 0.0727426 − 0.0137766i
\(93\) −75.9069 + 104.477i −0.816203 + 1.12341i
\(94\) −0.941601 − 3.67740i −0.0100170 − 0.0391213i
\(95\) −14.7874 − 45.5110i −0.155657 − 0.479063i
\(96\) −72.7325 − 23.8988i −0.757631 − 0.248946i
\(97\) −34.9867 − 48.1551i −0.360688 − 0.496445i 0.589652 − 0.807657i \(-0.299265\pi\)
−0.950340 + 0.311213i \(0.899265\pi\)
\(98\) 66.3974 + 55.0029i 0.677525 + 0.561254i
\(99\) −8.89862 − 6.46523i −0.0898851 − 0.0653053i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 164.3.l.a.23.1 ✓ 160
4.3 odd 2 inner 164.3.l.a.23.33 yes 160
41.25 even 10 inner 164.3.l.a.107.33 yes 160
164.107 odd 10 inner 164.3.l.a.107.1 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.3.l.a.23.1 ✓ 160 1.1 even 1 trivial
164.3.l.a.23.33 yes 160 4.3 odd 2 inner
164.3.l.a.107.1 yes 160 164.107 odd 10 inner
164.3.l.a.107.33 yes 160 41.25 even 10 inner