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Properties

Label 164.2.o.b.15.10
Level $164$
Weight $2$
Character 164.15
Analytic conductor $1.310$
Analytic rank $0$
Dimension $288$
Inner twists $4$

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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] = [164,2,Mod(7,164)] 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("164.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(164, base_ring=CyclotomicField(40)) chi = DirichletCharacter(H, H._module([20, 39])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 164.o (of order \(40\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [288] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.30954659315\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(18\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label 15.10
Character \(\chi\) \(=\) 164.15
Dual form 164.2.o.b.11.10

$q$-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+(0.0433699 - 1.41355i) q^{2} +(0.739847 + 1.78615i) q^{3} +(-1.99624 - 0.122611i) q^{4} +(1.08360 + 2.12669i) q^{5} +(2.55689 - 0.968344i) q^{6} +(-0.938602 + 3.90956i) q^{7} +(-0.259893 + 2.81646i) q^{8} +(-0.521631 + 0.521631i) q^{9} +(3.05317 - 1.43949i) q^{10} +(-2.35255 - 2.75448i) q^{11} +(-1.25791 - 3.65629i) q^{12} +(2.54584 - 4.15443i) q^{13} +(5.48565 + 1.49632i) q^{14} +(-2.99688 + 3.50890i) q^{15} +(3.96993 + 0.489521i) q^{16} +(0.0520733 - 0.661654i) q^{17} +(0.714728 + 0.759974i) q^{18} +(0.534152 - 0.327329i) q^{19} +(-1.90237 - 4.37824i) q^{20} +(-7.67748 + 1.21599i) q^{21} +(-3.99563 + 3.20598i) q^{22} +(3.84791 - 2.79567i) q^{23} +(-5.22290 + 1.61954i) q^{24} +(-0.409682 + 0.563879i) q^{25} +(-5.76207 - 3.77884i) q^{26} +(4.04081 + 1.67376i) q^{27} +(2.35303 - 7.68933i) q^{28} +(-0.0823757 - 1.04668i) q^{29} +(4.83002 + 4.38842i) q^{30} +(-1.06140 - 3.26666i) q^{31} +(0.864137 - 5.59046i) q^{32} +(3.17939 - 6.23990i) q^{33} +(-0.933021 - 0.102304i) q^{34} +(-9.33149 + 2.24029i) q^{35} +(1.10526 - 0.977342i) q^{36} +(-3.25281 + 10.0111i) q^{37} +(-0.439529 - 0.769246i) q^{38} +(9.30395 + 1.47360i) q^{39} +(-6.27135 + 2.49921i) q^{40} +(-5.78985 - 2.73452i) q^{41} +(1.38589 + 10.9052i) q^{42} +(-1.20002 + 7.57663i) q^{43} +(4.35852 + 5.78705i) q^{44} +(-1.67459 - 0.544106i) q^{45} +(-3.78493 - 5.56046i) q^{46} +(-0.561062 - 2.33699i) q^{47} +(2.06279 + 7.45306i) q^{48} +(-8.16664 - 4.16111i) q^{49} +(0.779303 + 0.603561i) q^{50} +(1.22034 - 0.396512i) q^{51} +(-5.59147 + 7.98108i) q^{52} +(10.8014 - 0.850086i) q^{53} +(2.54119 - 5.63929i) q^{54} +(3.30870 - 7.98790i) q^{55} +(-10.7672 - 3.65960i) q^{56} +(0.979848 + 0.711901i) q^{57} +(-1.48311 + 0.0710475i) q^{58} +(-3.02089 - 4.15790i) q^{59} +(6.41271 - 6.63714i) q^{60} +(1.97137 + 12.4467i) q^{61} +(-4.66362 + 1.35867i) q^{62} +(-1.54974 - 2.52895i) q^{63} +(-7.86491 - 1.46396i) q^{64} +(11.5938 + 0.912455i) q^{65} +(-8.68251 - 4.76484i) q^{66} +(-9.59192 - 8.19228i) q^{67} +(-0.185077 + 1.31443i) q^{68} +(7.84035 + 4.80457i) q^{69} +(2.76206 + 13.2877i) q^{70} +(7.99816 - 6.83108i) q^{71} +(-1.33359 - 1.60472i) q^{72} +(-7.30490 - 7.30490i) q^{73} +(14.0101 + 5.03218i) q^{74} +(-1.31027 - 0.314569i) q^{75} +(-1.10643 + 0.587933i) q^{76} +(12.9769 - 6.61208i) q^{77} +(2.48652 - 13.0877i) q^{78} +(1.92843 - 0.798783i) q^{79} +(3.26077 + 8.97325i) q^{80} +10.6689i q^{81} +(-4.11649 + 8.06564i) q^{82} -9.71576i q^{83} +(15.4752 - 1.48607i) q^{84} +(1.46356 - 0.606225i) q^{85} +(10.6579 + 2.02488i) q^{86} +(1.80858 - 0.921519i) q^{87} +(8.36931 - 5.91000i) q^{88} +(-4.79593 - 1.15140i) q^{89} +(-0.841747 + 2.34351i) q^{90} +(13.8525 + 13.8525i) q^{91} +(-8.02413 + 5.10903i) q^{92} +(5.04947 - 4.31265i) q^{93} +(-3.32778 + 0.691733i) q^{94} +(1.27493 + 0.781280i) q^{95} +(10.6247 - 2.59261i) q^{96} +(1.98311 + 1.69374i) q^{97} +(-6.23612 + 11.3635i) q^{98} +(2.66399 + 0.209660i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9} - 12 q^{10} - 8 q^{12} - 24 q^{13} - 4 q^{14} - 28 q^{16} - 12 q^{18} - 44 q^{20} - 32 q^{21} - 40 q^{22} + 8 q^{24} - 40 q^{25} + 4 q^{26}+ \cdots + 104 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(83\) \(129\)
\(\chi(n)\) \(-1\) \(e\left(\frac{37}{40}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0433699 − 1.41355i 0.0306671 − 0.999530i
\(3\) 0.739847 + 1.78615i 0.427151 + 1.03123i 0.980187 + 0.198076i \(0.0634692\pi\)
−0.553036 + 0.833157i \(0.686531\pi\)
\(4\) −1.99624 − 0.122611i −0.998119 − 0.0613054i
\(5\) 1.08360 + 2.12669i 0.484601 + 0.951084i 0.995794 + 0.0916177i \(0.0292038\pi\)
−0.511193 + 0.859466i \(0.670796\pi\)
\(6\) 2.55689 − 0.968344i 1.04385 − 0.395325i
\(7\) −0.938602 + 3.90956i −0.354758 + 1.47767i 0.454778 + 0.890605i \(0.349719\pi\)
−0.809536 + 0.587070i \(0.800281\pi\)
\(8\) −0.259893 + 2.81646i −0.0918860 + 0.995770i
\(9\) −0.521631 + 0.521631i −0.173877 + 0.173877i
\(10\) 3.05317 − 1.43949i 0.965498 − 0.455206i
\(11\) −2.35255 − 2.75448i −0.709321 − 0.830508i 0.282349 − 0.959312i \(-0.408886\pi\)
−0.991670 + 0.128804i \(0.958886\pi\)
\(12\) −1.25791 − 3.65629i −0.363127 − 1.05548i
\(13\) 2.54584 − 4.15443i 0.706088 − 1.15223i −0.275515 − 0.961297i \(-0.588848\pi\)
0.981603 − 0.190934i \(-0.0611516\pi\)
\(14\) 5.48565 + 1.49632i 1.46610 + 0.399908i
\(15\) −2.99688 + 3.50890i −0.773791 + 0.905993i
\(16\) 3.96993 + 0.489521i 0.992483 + 0.122380i
\(17\) 0.0520733 − 0.661654i 0.0126296 − 0.160475i −0.987368 − 0.158442i \(-0.949353\pi\)
0.999998 − 0.00203275i \(-0.000647046\pi\)
\(18\) 0.714728 + 0.759974i 0.168463 + 0.179128i
\(19\) 0.534152 − 0.327329i 0.122543 − 0.0750944i −0.459876 − 0.887983i \(-0.652106\pi\)
0.582419 + 0.812889i \(0.302106\pi\)
\(20\) −1.90237 − 4.37824i −0.425383 − 0.979003i
\(21\) −7.67748 + 1.21599i −1.67536 + 0.265351i
\(22\) −3.99563 + 3.20598i −0.851870 + 0.683518i
\(23\) 3.84791 − 2.79567i 0.802345 − 0.582938i −0.109256 − 0.994014i \(-0.534847\pi\)
0.911601 + 0.411076i \(0.134847\pi\)
\(24\) −5.22290 + 1.61954i −1.06612 + 0.330588i
\(25\) −0.409682 + 0.563879i −0.0819365 + 0.112776i
\(26\) −5.76207 − 3.77884i −1.13004 − 0.741091i
\(27\) 4.04081 + 1.67376i 0.777654 + 0.322115i
\(28\) 2.35303 − 7.68933i 0.444681 − 1.45315i
\(29\) −0.0823757 − 1.04668i −0.0152968 − 0.194364i −0.999818 − 0.0190966i \(-0.993921\pi\)
0.984521 − 0.175267i \(-0.0560790\pi\)
\(30\) 4.83002 + 4.38842i 0.881837 + 0.801211i
\(31\) −1.06140 − 3.26666i −0.190634 − 0.586710i 0.809366 − 0.587304i \(-0.199811\pi\)
−1.00000 0.000594276i \(0.999811\pi\)
\(32\) 0.864137 − 5.59046i 0.152759 − 0.988263i
\(33\) 3.17939 − 6.23990i 0.553460 − 1.08623i
\(34\) −0.933021 − 0.102304i −0.160012 − 0.0175450i
\(35\) −9.33149 + 2.24029i −1.57731 + 0.378678i
\(36\) 1.10526 − 0.977342i 0.184210 − 0.162890i
\(37\) −3.25281 + 10.0111i −0.534758 + 1.64582i 0.209414 + 0.977827i \(0.432845\pi\)
−0.744172 + 0.667989i \(0.767155\pi\)
\(38\) −0.439529 − 0.769246i −0.0713010 − 0.124788i
\(39\) 9.30395 + 1.47360i 1.48982 + 0.235965i
\(40\) −6.27135 + 2.49921i −0.991588 + 0.395160i
\(41\) −5.78985 − 2.73452i −0.904223 − 0.427061i
\(42\) 1.38589 + 10.9052i 0.213848 + 1.68271i
\(43\) −1.20002 + 7.57663i −0.183001 + 1.15543i 0.709609 + 0.704596i \(0.248872\pi\)
−0.892610 + 0.450829i \(0.851128\pi\)
\(44\) 4.35852 + 5.78705i 0.657072 + 0.872431i
\(45\) −1.67459 − 0.544106i −0.249633 − 0.0811106i
\(46\) −3.78493 − 5.56046i −0.558058 − 0.819845i
\(47\) −0.561062 − 2.33699i −0.0818393 − 0.340885i 0.916396 − 0.400274i \(-0.131085\pi\)
−0.998235 + 0.0593886i \(0.981085\pi\)
\(48\) 2.06279 + 7.45306i 0.297737 + 1.07576i
\(49\) −8.16664 − 4.16111i −1.16666 − 0.594445i
\(50\) 0.779303 + 0.603561i 0.110210 + 0.0853564i
\(51\) 1.22034 − 0.396512i 0.170881 − 0.0555228i
\(52\) −5.59147 + 7.98108i −0.775398 + 1.10678i
\(53\) 10.8014 − 0.850086i 1.48368 − 0.116768i 0.689428 − 0.724354i \(-0.257862\pi\)
0.794254 + 0.607586i \(0.207862\pi\)
\(54\) 2.54119 − 5.63929i 0.345812 − 0.767410i
\(55\) 3.30870 − 7.98790i 0.446145 − 1.07709i
\(56\) −10.7672 − 3.65960i −1.43883 − 0.489035i
\(57\) 0.979848 + 0.711901i 0.129784 + 0.0942936i
\(58\) −1.48311 + 0.0710475i −0.194742 + 0.00932900i
\(59\) −3.02089 − 4.15790i −0.393286 − 0.541312i 0.565757 − 0.824572i \(-0.308584\pi\)
−0.959043 + 0.283260i \(0.908584\pi\)
\(60\) 6.41271 − 6.63714i 0.827878 − 0.856851i
\(61\) 1.97137 + 12.4467i 0.252408 + 1.59364i 0.709817 + 0.704386i \(0.248778\pi\)
−0.457409 + 0.889257i \(0.651222\pi\)
\(62\) −4.66362 + 1.35867i −0.592280 + 0.172551i
\(63\) −1.54974 − 2.52895i −0.195249 − 0.318618i
\(64\) −7.86491 − 1.46396i −0.983114 − 0.182995i
\(65\) 11.5938 + 0.912455i 1.43804 + 0.113176i
\(66\) −8.68251 − 4.76484i −1.06874 − 0.586511i
\(67\) −9.59192 − 8.19228i −1.17184 − 1.00085i −0.999876 − 0.0157427i \(-0.994989\pi\)
−0.171964 − 0.985103i \(-0.555011\pi\)
\(68\) −0.185077 + 1.31443i −0.0224438 + 0.159399i
\(69\) 7.84035 + 4.80457i 0.943867 + 0.578402i
\(70\) 2.76206 + 13.2877i 0.330129 + 1.58818i
\(71\) 7.99816 − 6.83108i 0.949207 − 0.810700i −0.0328156 − 0.999461i \(-0.510447\pi\)
0.982023 + 0.188762i \(0.0604474\pi\)
\(72\) −1.33359 − 1.60472i −0.157165 − 0.189118i
\(73\) −7.30490 − 7.30490i −0.854974 − 0.854974i 0.135767 − 0.990741i \(-0.456650\pi\)
−0.990741 + 0.135767i \(0.956650\pi\)
\(74\) 14.0101 + 5.03218i 1.62864 + 0.584979i
\(75\) −1.31027 − 0.314569i −0.151297 − 0.0363233i
\(76\) −1.10643 + 0.587933i −0.126916 + 0.0674406i
\(77\) 12.9769 − 6.61208i 1.47886 − 0.753516i
\(78\) 2.48652 − 13.0877i 0.281543 − 1.48189i
\(79\) 1.92843 − 0.798783i 0.216966 − 0.0898701i −0.271553 − 0.962423i \(-0.587537\pi\)
0.488519 + 0.872553i \(0.337537\pi\)
\(80\) 3.26077 + 8.97325i 0.364565 + 1.00324i
\(81\) 10.6689i 1.18543i
\(82\) −4.11649 + 8.06564i −0.454590 + 0.890701i
\(83\) − 9.71576i − 1.06644i −0.845976 − 0.533222i \(-0.820981\pi\)
0.845976 − 0.533222i \(-0.179019\pi\)
\(84\) 15.4752 − 1.48607i 1.68848 − 0.162143i
\(85\) 1.46356 − 0.606225i 0.158745 − 0.0657544i
\(86\) 10.6579 + 2.02488i 1.14927 + 0.218349i
\(87\) 1.80858 − 0.921519i 0.193901 − 0.0987972i
\(88\) 8.36931 − 5.91000i 0.892171 − 0.630008i
\(89\) −4.79593 − 1.15140i −0.508367 − 0.122048i −0.0288555 − 0.999584i \(-0.509186\pi\)
−0.479512 + 0.877535i \(0.659186\pi\)
\(90\) −0.841747 + 2.34351i −0.0887279 + 0.247028i
\(91\) 13.8525 + 13.8525i 1.45213 + 1.45213i
\(92\) −8.02413 + 5.10903i −0.836573 + 0.532653i
\(93\) 5.04947 − 4.31265i 0.523606 − 0.447201i
\(94\) −3.32778 + 0.691733i −0.343234 + 0.0713468i
\(95\) 1.27493 + 0.781280i 0.130805 + 0.0801577i
\(96\) 10.6247 − 2.59261i 1.08438 − 0.264607i
\(97\) 1.98311 + 1.69374i 0.201354 + 0.171973i 0.744408 − 0.667725i \(-0.232732\pi\)
−0.543053 + 0.839698i \(0.682732\pi\)
\(98\) −6.23612 + 11.3635i −0.629943 + 1.14788i
\(99\) 2.66399 + 0.209660i 0.267741 + 0.0210717i
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 164.2.o.b.15.10 yes 288
4.3 odd 2 inner 164.2.o.b.15.7 yes 288
41.11 odd 40 inner 164.2.o.b.11.7 ✓ 288
164.11 even 40 inner 164.2.o.b.11.10 yes 288
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.2.o.b.11.7 ✓ 288 41.11 odd 40 inner
164.2.o.b.11.10 yes 288 164.11 even 40 inner
164.2.o.b.15.7 yes 288 4.3 odd 2 inner
164.2.o.b.15.10 yes 288 1.1 even 1 trivial