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Properties

Label 80.11.r.a.11.2
Level $80$
Weight $11$
Character 80.11
Analytic conductor $50.829$
Analytic rank $0$
Dimension $160$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,11,Mod(11,80)] 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("80.11"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.r (of order \(4\), degree \(2\), 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: \(50.8285802139\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(80\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 11.2
Character \(\chi\) \(=\) 80.11
Dual form 80.11.r.a.51.2

$q$-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+(-31.9268 + 2.16369i) q^{2} +(228.134 + 228.134i) q^{3} +(1014.64 - 138.159i) q^{4} +(-988.212 - 988.212i) q^{5} +(-7777.20 - 6789.97i) q^{6} -19077.2 q^{7} +(-32095.1 + 6606.33i) q^{8} +45041.4i q^{9} +(33688.6 + 29412.2i) q^{10} +(-216752. + 216752. i) q^{11} +(262992. + 199955. i) q^{12} +(-88873.7 + 88873.7i) q^{13} +(609075. - 41277.2i) q^{14} -450890. i q^{15} +(1.01040e6 - 280363. i) q^{16} -33441.9 q^{17} +(-97455.5 - 1.43803e6i) q^{18} +(-267243. - 267243. i) q^{19} +(-1.13921e6 - 866146. i) q^{20} +(-4.35217e6 - 4.35217e6i) q^{21} +(6.45121e6 - 7.38917e6i) q^{22} +4.45535e6 q^{23} +(-8.82913e6 - 5.81487e6i) q^{24} +1.95312e6i q^{25} +(2.64516e6 - 3.02975e6i) q^{26} +(3.19562e6 - 3.19562e6i) q^{27} +(-1.93565e7 + 2.63570e6i) q^{28} +(1.70802e7 - 1.70802e7i) q^{29} +(975585. + 1.43955e7i) q^{30} +2.24213e7i q^{31} +(-3.16522e7 + 1.11373e7i) q^{32} -9.88971e7 q^{33} +(1.06769e6 - 72357.8i) q^{34} +(1.88524e7 + 1.88524e7i) q^{35} +(6.22288e6 + 4.57007e7i) q^{36} +(3.72249e7 + 3.72249e7i) q^{37} +(9.11045e6 + 7.95399e6i) q^{38} -4.05503e7 q^{39} +(3.82453e7 + 2.51883e7i) q^{40} -1.11994e8i q^{41} +(1.48368e8 + 1.29534e8i) q^{42} +(-5.41572e7 + 5.41572e7i) q^{43} +(-1.89978e8 + 2.49871e8i) q^{44} +(4.45104e7 - 4.45104e7i) q^{45} +(-1.42245e8 + 9.64000e6i) q^{46} -4.23715e8i q^{47} +(2.94467e8 + 1.66546e8i) q^{48} +8.14661e7 q^{49} +(-4.22595e6 - 6.23570e7i) q^{50} +(-7.62923e6 - 7.62923e6i) q^{51} +(-7.78959e7 + 1.02453e8i) q^{52} +(-1.62720e8 - 1.62720e8i) q^{53} +(-9.51114e7 + 1.08940e8i) q^{54} +4.28394e8 q^{55} +(6.12287e8 - 1.26031e8i) q^{56} -1.21935e8i q^{57} +(-5.08361e8 + 5.82273e8i) q^{58} +(-4.58422e7 + 4.58422e7i) q^{59} +(-6.22945e7 - 4.57489e8i) q^{60} +(1.03911e9 - 1.03911e9i) q^{61} +(-4.85128e7 - 7.15841e8i) q^{62} -8.59266e8i q^{63} +(9.86455e8 - 4.24062e8i) q^{64} +1.75652e8 q^{65} +(3.15746e9 - 2.13982e8i) q^{66} +(-1.39365e9 - 1.39365e9i) q^{67} +(-3.39314e7 + 4.62030e6i) q^{68} +(1.01642e9 + 1.01642e9i) q^{69} +(-6.42686e8 - 5.61104e8i) q^{70} -1.22728e9 q^{71} +(-2.97558e8 - 1.44561e9i) q^{72} -3.08051e9i q^{73} +(-1.26901e9 - 1.10793e9i) q^{74} +(-4.45575e8 + 4.45575e8i) q^{75} +(-3.08077e8 - 2.34233e8i) q^{76} +(4.13503e9 - 4.13503e9i) q^{77} +(1.29464e9 - 8.77381e7i) q^{78} +3.73141e9i q^{79} +(-1.27555e9 - 7.21432e8i) q^{80} +4.11771e9 q^{81} +(2.42319e8 + 3.57559e9i) q^{82} +(-2.29209e9 - 2.29209e9i) q^{83} +(-5.01717e9 - 3.81458e9i) q^{84} +(3.30477e7 + 3.30477e7i) q^{85} +(1.61189e9 - 1.84624e9i) q^{86} +7.79317e9 q^{87} +(5.52475e9 - 8.38862e9i) q^{88} -1.88788e9i q^{89} +(-1.32477e9 + 1.51738e9i) q^{90} +(1.69547e9 - 1.69547e9i) q^{91} +(4.52057e9 - 6.15548e8i) q^{92} +(-5.11507e9 + 5.11507e9i) q^{93} +(9.16786e8 + 1.35278e10i) q^{94} +5.28186e8i q^{95} +(-9.76174e9 - 4.68016e9i) q^{96} +1.39168e10 q^{97} +(-2.60095e9 + 1.76267e8i) q^{98} +(-9.76281e9 - 9.76281e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q + 2508 q^{4} + 34428 q^{6} - 62500 q^{10} + 91808 q^{11} - 1218300 q^{12} - 390132 q^{14} + 2979120 q^{16} - 4129760 q^{18} + 5107040 q^{19} - 2375000 q^{20} + 18896772 q^{22} + 16559488 q^{23} - 81623160 q^{24}+ \cdots + 37107476960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.9268 + 2.16369i −0.997711 + 0.0676152i
\(3\) 228.134 + 228.134i 0.938824 + 0.938824i 0.998234 − 0.0594100i \(-0.0189219\pi\)
−0.0594100 + 0.998234i \(0.518922\pi\)
\(4\) 1014.64 − 138.159i 0.990856 − 0.134921i
\(5\) −988.212 − 988.212i −0.316228 − 0.316228i
\(6\) −7777.20 − 6789.97i −1.00015 − 0.873196i
\(7\) −19077.2 −1.13508 −0.567539 − 0.823347i \(-0.692104\pi\)
−0.567539 + 0.823347i \(0.692104\pi\)
\(8\) −32095.1 + 6606.33i −0.979466 + 0.201609i
\(9\) 45041.4i 0.762780i
\(10\) 33688.6 + 29412.2i 0.336886 + 0.294122i
\(11\) −216752. + 216752.i −1.34586 + 1.34586i −0.455753 + 0.890106i \(0.650630\pi\)
−0.890106 + 0.455753i \(0.849370\pi\)
\(12\) 262992. + 199955.i 1.05691 + 0.803572i
\(13\) −88873.7 + 88873.7i −0.239363 + 0.239363i −0.816586 − 0.577223i \(-0.804136\pi\)
0.577223 + 0.816586i \(0.304136\pi\)
\(14\) 609075. − 41277.2i 1.13248 − 0.0767485i
\(15\) − 450890.i − 0.593764i
\(16\) 1.01040e6 − 280363.i 0.963593 − 0.267375i
\(17\) −33441.9 −0.0235530 −0.0117765 − 0.999931i \(-0.503749\pi\)
−0.0117765 + 0.999931i \(0.503749\pi\)
\(18\) −97455.5 − 1.43803e6i −0.0515755 − 0.761034i
\(19\) −267243. − 267243.i −0.107929 − 0.107929i 0.651080 − 0.759009i \(-0.274316\pi\)
−0.759009 + 0.651080i \(0.774316\pi\)
\(20\) −1.13921e6 − 866146.i −0.356002 − 0.270671i
\(21\) −4.35217e6 − 4.35217e6i −1.06564 − 1.06564i
\(22\) 6.45121e6 − 7.38917e6i 1.25178 − 1.43378i
\(23\) 4.45535e6 0.692218 0.346109 − 0.938194i \(-0.387503\pi\)
0.346109 + 0.938194i \(0.387503\pi\)
\(24\) −8.82913e6 − 5.81487e6i −1.10882 − 0.730270i
\(25\) 1.95312e6i 0.200000i
\(26\) 2.64516e6 − 3.02975e6i 0.222630 − 0.255000i
\(27\) 3.19562e6 − 3.19562e6i 0.222708 − 0.222708i
\(28\) −1.93565e7 + 2.63570e6i −1.12470 + 0.153146i
\(29\) 1.70802e7 − 1.70802e7i 0.832730 − 0.832730i −0.155160 − 0.987889i \(-0.549589\pi\)
0.987889 + 0.155160i \(0.0495893\pi\)
\(30\) 975585. + 1.43955e7i 0.0401475 + 0.592405i
\(31\) 2.24213e7i 0.783164i 0.920143 + 0.391582i \(0.128072\pi\)
−0.920143 + 0.391582i \(0.871928\pi\)
\(32\) −3.16522e7 + 1.11373e7i −0.943309 + 0.331916i
\(33\) −9.88971e7 −2.52705
\(34\) 1.06769e6 − 72357.8i 0.0234991 − 0.00159254i
\(35\) 1.88524e7 + 1.88524e7i 0.358943 + 0.358943i
\(36\) 6.22288e6 + 4.57007e7i 0.102915 + 0.755805i
\(37\) 3.72249e7 + 3.72249e7i 0.536816 + 0.536816i 0.922592 − 0.385777i \(-0.126066\pi\)
−0.385777 + 0.922592i \(0.626066\pi\)
\(38\) 9.11045e6 + 7.95399e6i 0.114980 + 0.100385i
\(39\) −4.05503e7 −0.449439
\(40\) 3.82453e7 + 2.51883e7i 0.373489 + 0.245980i
\(41\) − 1.11994e8i − 0.966660i −0.875438 − 0.483330i \(-0.839427\pi\)
0.875438 − 0.483330i \(-0.160573\pi\)
\(42\) 1.48368e8 + 1.29534e8i 1.13525 + 0.991146i
\(43\) −5.41572e7 + 5.41572e7i −0.368395 + 0.368395i −0.866892 − 0.498496i \(-0.833886\pi\)
0.498496 + 0.866892i \(0.333886\pi\)
\(44\) −1.89978e8 + 2.49871e8i −1.15197 + 1.51514i
\(45\) 4.45104e7 − 4.45104e7i 0.241212 − 0.241212i
\(46\) −1.42245e8 + 9.64000e6i −0.690634 + 0.0468045i
\(47\) − 4.23715e8i − 1.84750i −0.382998 − 0.923749i \(-0.625108\pi\)
0.382998 − 0.923749i \(-0.374892\pi\)
\(48\) 2.94467e8 + 1.66546e8i 1.15566 + 0.653626i
\(49\) 8.14661e7 0.288401
\(50\) −4.22595e6 − 6.23570e7i −0.0135230 − 0.199542i
\(51\) −7.62923e6 − 7.62923e6i −0.0221121 − 0.0221121i
\(52\) −7.78959e7 + 1.02453e8i −0.204879 + 0.269469i
\(53\) −1.62720e8 − 1.62720e8i −0.389101 − 0.389101i 0.485266 − 0.874367i \(-0.338723\pi\)
−0.874367 + 0.485266i \(0.838723\pi\)
\(54\) −9.51114e7 + 1.08940e8i −0.207140 + 0.237257i
\(55\) 4.28394e8 0.851196
\(56\) 6.12287e8 − 1.26031e8i 1.11177 − 0.228842i
\(57\) − 1.21935e8i − 0.202653i
\(58\) −5.08361e8 + 5.82273e8i −0.774519 + 0.887129i
\(59\) −4.58422e7 + 4.58422e7i −0.0641218 + 0.0641218i −0.738440 − 0.674319i \(-0.764437\pi\)
0.674319 + 0.738440i \(0.264437\pi\)
\(60\) −6.22945e7 − 4.57489e8i −0.0801113 − 0.588335i
\(61\) 1.03911e9 − 1.03911e9i 1.23030 − 1.23030i 0.266454 − 0.963848i \(-0.414148\pi\)
0.963848 − 0.266454i \(-0.0858522\pi\)
\(62\) −4.85128e7 − 7.15841e8i −0.0529538 − 0.781372i
\(63\) − 8.59266e8i − 0.865814i
\(64\) 9.86455e8 − 4.24062e8i 0.918707 − 0.394939i
\(65\) 1.75652e8 0.151386
\(66\) 3.15746e9 − 2.13982e8i 2.52127 − 0.170867i
\(67\) −1.39365e9 − 1.39365e9i −1.03224 − 1.03224i −0.999463 − 0.0327782i \(-0.989565\pi\)
−0.0327782 − 0.999463i \(-0.510435\pi\)
\(68\) −3.39314e7 + 4.62030e6i −0.0233376 + 0.00317779i
\(69\) 1.01642e9 + 1.01642e9i 0.649871 + 0.649871i
\(70\) −6.42686e8 − 5.61104e8i −0.382392 − 0.333852i
\(71\) −1.22728e9 −0.680221 −0.340111 − 0.940385i \(-0.610465\pi\)
−0.340111 + 0.940385i \(0.610465\pi\)
\(72\) −2.97558e8 − 1.44561e9i −0.153783 − 0.747117i
\(73\) − 3.08051e9i − 1.48596i −0.669311 − 0.742982i \(-0.733411\pi\)
0.669311 − 0.742982i \(-0.266589\pi\)
\(74\) −1.26901e9 − 1.10793e9i −0.571884 − 0.499290i
\(75\) −4.45575e8 + 4.45575e8i −0.187765 + 0.187765i
\(76\) −3.08077e8 − 2.34233e8i −0.121504 − 0.0923804i
\(77\) 4.13503e9 − 4.13503e9i 1.52765 − 1.52765i
\(78\) 1.29464e9 − 8.77381e7i 0.448410 − 0.0303889i
\(79\) 3.73141e9i 1.21266i 0.795214 + 0.606329i \(0.207358\pi\)
−0.795214 + 0.606329i \(0.792642\pi\)
\(80\) −1.27555e9 − 7.21432e8i −0.389266 − 0.220163i
\(81\) 4.11771e9 1.18095
\(82\) 2.42319e8 + 3.57559e9i 0.0653610 + 0.964448i
\(83\) −2.29209e9 − 2.29209e9i −0.581890 − 0.581890i 0.353532 − 0.935422i \(-0.384981\pi\)
−0.935422 + 0.353532i \(0.884981\pi\)
\(84\) −5.01717e9 − 3.81458e9i −1.19967 − 0.912117i
\(85\) 3.30477e7 + 3.30477e7i 0.00744811 + 0.00744811i
\(86\) 1.61189e9 − 1.84624e9i 0.342643 − 0.392462i
\(87\) 7.79317e9 1.56357
\(88\) 5.52475e9 − 8.38862e9i 1.04689 − 1.58956i
\(89\) − 1.88788e9i − 0.338084i −0.985609 − 0.169042i \(-0.945933\pi\)
0.985609 − 0.169042i \(-0.0540674\pi\)
\(90\) −1.32477e9 + 1.51738e9i −0.224351 + 0.256970i
\(91\) 1.69547e9 − 1.69547e9i 0.271695 − 0.271695i
\(92\) 4.52057e9 − 6.15548e8i 0.685889 − 0.0933948i
\(93\) −5.11507e9 + 5.11507e9i −0.735253 + 0.735253i
\(94\) 9.16786e8 + 1.35278e10i 0.124919 + 1.84327i
\(95\) 5.28186e8i 0.0682604i
\(96\) −9.76174e9 − 4.68016e9i −1.19721 − 0.573990i
\(97\) 1.39168e10 1.62062 0.810312 − 0.585999i \(-0.199298\pi\)
0.810312 + 0.585999i \(0.199298\pi\)
\(98\) −2.60095e9 + 1.76267e8i −0.287741 + 0.0195003i
\(99\) −9.76281e9 − 9.76281e9i −1.02659 − 1.02659i
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 80.11.r.a.11.2 ✓ 160
16.3 odd 4 inner 80.11.r.a.51.2 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.r.a.11.2 ✓ 160 1.1 even 1 trivial
80.11.r.a.51.2 yes 160 16.3 odd 4 inner