Properties

Label 1323.2.f.g.442.5
Level 13231323
Weight 22
Character 1323.442
Analytic conductor 10.56410.564
Analytic rank 00
Dimension 1212
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,2,Mod(442,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1323.442");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1323=3372 1323 = 3^{3} \cdot 7^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1323.f (of order 33, degree 22, not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 10.564208187410.5642081874
Analytic rank: 00
Dimension: 1212
Relative dimension: 66 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q[x]/(x12)\mathbb{Q}[x]/(x^{12} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x127x10+37x878x6+123x436x2+9 x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 35 3^{5}
Twist minimal: no (minimal twist has level 441)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

Embedding invariants

Embedding label 442.5
Root 1.829041.05600i-1.82904 - 1.05600i of defining polynomial
Character χ\chi == 1323.442
Dual form 1323.2.f.g.883.5

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q+(1.23025+2.13086i)q2+(2.02704+3.51094i)q4+(1.82904+3.16799i)q55.05408q89.00071q10+(0.203210+0.351971i)q11+(0.2433980.421578i)q13+(2.163723.74766i)q164.85584q171.97351q19+(7.4150712.8433i)q20+(0.500000+0.866025i)q22+(2.323834.02499i)q23+(4.190767.25860i)q25+1.19777q26+(3.82383+6.62307i)q29+(3.51360+6.08573i)q31+(0.2697480.467216i)q32+(5.9739110.3471i)q34+2.32743q37+(2.427924.20528i)q38+(9.2441116.0113i)q40+(3.757006.50731i)q41+(1.16372+2.01561i)q431.64766q44+11.4356q46+(3.158115.47002i)q47+(10.311417.8598i)q50+(0.986757+1.70911i)q52+3.56867q531.48672q55+(9.40856+16.2961i)q58+(3.059195.29868i)q59+(4.01356+6.95169i)q6117.2905q627.32743q64+(0.890369+1.54216i)q65+(1.80039+3.11836i)q67+(9.8429917.0486i)q688.46050q71+1.97351q73+(2.86333+4.95943i)q74+(4.000406.92889i)q76+(4.081137.06872i)q79+15.8301q80+18.4882q82+(6.08600+10.5413i)q83+(8.8815115.3832i)q85+(2.86333+4.95943i)q86+(1.027041.77889i)q8814.8301q89+(9.42101+16.3177i)q92+(7.7705613.4590i)q94+(3.609636.25206i)q95+(4.74375+8.21642i)q97+O(q100)q+(1.23025 + 2.13086i) q^{2} +(-2.02704 + 3.51094i) q^{4} +(-1.82904 + 3.16799i) q^{5} -5.05408 q^{8} -9.00071 q^{10} +(0.203210 + 0.351971i) q^{11} +(0.243398 - 0.421578i) q^{13} +(-2.16372 - 3.74766i) q^{16} -4.85584 q^{17} -1.97351 q^{19} +(-7.41507 - 12.8433i) q^{20} +(-0.500000 + 0.866025i) q^{22} +(2.32383 - 4.02499i) q^{23} +(-4.19076 - 7.25860i) q^{25} +1.19777 q^{26} +(3.82383 + 6.62307i) q^{29} +(-3.51360 + 6.08573i) q^{31} +(0.269748 - 0.467216i) q^{32} +(-5.97391 - 10.3471i) q^{34} +2.32743 q^{37} +(-2.42792 - 4.20528i) q^{38} +(9.24411 - 16.0113i) q^{40} +(3.75700 - 6.50731i) q^{41} +(1.16372 + 2.01561i) q^{43} -1.64766 q^{44} +11.4356 q^{46} +(-3.15811 - 5.47002i) q^{47} +(10.3114 - 17.8598i) q^{50} +(0.986757 + 1.70911i) q^{52} +3.56867 q^{53} -1.48672 q^{55} +(-9.40856 + 16.2961i) q^{58} +(3.05919 - 5.29868i) q^{59} +(4.01356 + 6.95169i) q^{61} -17.2905 q^{62} -7.32743 q^{64} +(0.890369 + 1.54216i) q^{65} +(-1.80039 + 3.11836i) q^{67} +(9.84299 - 17.0486i) q^{68} -8.46050 q^{71} +1.97351 q^{73} +(2.86333 + 4.95943i) q^{74} +(4.00040 - 6.92889i) q^{76} +(-4.08113 - 7.06872i) q^{79} +15.8301 q^{80} +18.4882 q^{82} +(6.08600 + 10.5413i) q^{83} +(8.88151 - 15.3832i) q^{85} +(-2.86333 + 4.95943i) q^{86} +(-1.02704 - 1.77889i) q^{88} -14.8301 q^{89} +(9.42101 + 16.3177i) q^{92} +(7.77056 - 13.4590i) q^{94} +(3.60963 - 6.25206i) q^{95} +(4.74375 + 8.21642i) q^{97} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 12q+2q26q424q8+8q116q166q22+4q2312q25+22q29+16q3212q376q43+28q44+24q46+56q5056q5318q5848q64++60q95+O(q100) 12 q + 2 q^{2} - 6 q^{4} - 24 q^{8} + 8 q^{11} - 6 q^{16} - 6 q^{22} + 4 q^{23} - 12 q^{25} + 22 q^{29} + 16 q^{32} - 12 q^{37} - 6 q^{43} + 28 q^{44} + 24 q^{46} + 56 q^{50} - 56 q^{53} - 18 q^{58} - 48 q^{64}+ \cdots + 60 q^{95}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1323Z)×\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times.

nn 785785 10811081
χ(n)\chi(n) e(13)e\left(\frac{1}{3}\right) 11

Coefficient data

For each nn we display the coefficients of the qq-expansion ana_n, the Satake parameters αp\alpha_p, and the Satake angles θp=Arg(αp)\theta_p = \textrm{Arg}(\alpha_p).



Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)
Significant digits:
nn ana_n an/n(k1)/2a_n / n^{(k-1)/2} αn \alpha_n θn \theta_n
pp apa_p ap/p(k1)/2a_p / p^{(k-1)/2} αp \alpha_p θp \theta_p
22 1.23025 + 2.13086i 0.869920 + 1.50675i 0.862078 + 0.506776i 0.169163π0.169163\pi
0.00784213 + 0.999969i 0.497504π0.497504\pi
33 0 0
44 −2.02704 + 3.51094i −1.01352 + 1.75547i
55 −1.82904 + 3.16799i −0.817970 + 1.41677i 0.0892047 + 0.996013i 0.471567π0.471567\pi
−0.907175 + 0.420753i 0.861766π0.861766\pi
66 0 0
77 0 0
88 −5.05408 −1.78689
99 0 0
1010 −9.00071 −2.84628
1111 0.203210 + 0.351971i 0.0612702 + 0.106123i 0.895033 0.445999i 0.147152π-0.147152\pi
−0.833763 + 0.552122i 0.813818π0.813818\pi
1212 0 0
1313 0.243398 0.421578i 0.0675065 0.116925i −0.830297 0.557322i 0.811829π-0.811829\pi
0.897803 + 0.440397i 0.145162π0.145162\pi
1414 0 0
1515 0 0
1616 −2.16372 3.74766i −0.540929 0.936916i
1717 −4.85584 −1.17771 −0.588857 0.808237i 0.700422π-0.700422\pi
−0.588857 + 0.808237i 0.700422π0.700422\pi
1818 0 0
1919 −1.97351 −0.452755 −0.226378 0.974040i 0.572688π-0.572688\pi
−0.226378 + 0.974040i 0.572688π0.572688\pi
2020 −7.41507 12.8433i −1.65806 2.87185i
2121 0 0
2222 −0.500000 + 0.866025i −0.106600 + 0.184637i
2323 2.32383 4.02499i 0.484552 0.839269i −0.515290 0.857016i 0.672316π-0.672316\pi
0.999843 + 0.0177464i 0.00564915π0.00564915\pi
2424 0 0
2525 −4.19076 7.25860i −0.838151 1.45172i
2626 1.19777 0.234901
2727 0 0
2828 0 0
2929 3.82383 + 6.62307i 0.710068 + 1.22987i 0.964831 + 0.262870i 0.0846690π0.0846690\pi
−0.254764 + 0.967003i 0.581998π0.581998\pi
3030 0 0
3131 −3.51360 + 6.08573i −0.631061 + 1.09303i 0.356274 + 0.934381i 0.384047π0.384047\pi
−0.987335 + 0.158648i 0.949286π0.949286\pi
3232 0.269748 0.467216i 0.0476851 0.0825930i
3333 0 0
3434 −5.97391 10.3471i −1.02452 1.77452i
3535 0 0
3636 0 0
3737 2.32743 0.382627 0.191314 0.981529i 0.438725π-0.438725\pi
0.191314 + 0.981529i 0.438725π0.438725\pi
3838 −2.42792 4.20528i −0.393861 0.682187i
3939 0 0
4040 9.24411 16.0113i 1.46162 2.53160i
4141 3.75700 6.50731i 0.586744 1.01627i −0.407911 0.913022i 0.633743π-0.633743\pi
0.994655 0.103249i 0.0329240π-0.0329240\pi
4242 0 0
4343 1.16372 + 2.01561i 0.177465 + 0.307378i 0.941012 0.338374i 0.109877π-0.109877\pi
−0.763547 + 0.645753i 0.776544π0.776544\pi
4444 −1.64766 −0.248395
4545 0 0
4646 11.4356 1.68609
4747 −3.15811 5.47002i −0.460658 0.797884i 0.538335 0.842731i 0.319053π-0.319053\pi
−0.998994 + 0.0448469i 0.985720π0.985720\pi
4848 0 0
4949 0 0
5050 10.3114 17.8598i 1.45825 2.52576i
5151 0 0
5252 0.986757 + 1.70911i 0.136839 + 0.237011i
5353 3.56867 0.490195 0.245097 0.969498i 0.421180π-0.421180\pi
0.245097 + 0.969498i 0.421180π0.421180\pi
5454 0 0
5555 −1.48672 −0.200469
5656 0 0
5757 0 0
5858 −9.40856 + 16.2961i −1.23540 + 2.13978i
5959 3.05919 5.29868i 0.398273 0.689829i −0.595240 0.803548i 0.702943π-0.702943\pi
0.993513 + 0.113719i 0.0362763π0.0362763\pi
6060 0 0
6161 4.01356 + 6.95169i 0.513884 + 0.890073i 0.999870 + 0.0161063i 0.00512703π0.00512703\pi
−0.485987 + 0.873966i 0.661540π0.661540\pi
6262 −17.2905 −2.19589
6363 0 0
6464 −7.32743 −0.915929
6565 0.890369 + 1.54216i 0.110437 + 0.191282i
6666 0 0
6767 −1.80039 + 3.11836i −0.219952 + 0.380969i −0.954793 0.297271i 0.903924π-0.903924\pi
0.734841 + 0.678240i 0.237257π0.237257\pi
6868 9.84299 17.0486i 1.19364 2.06744i
6969 0 0
7070 0 0
7171 −8.46050 −1.00408 −0.502039 0.864845i 0.667416π-0.667416\pi
−0.502039 + 0.864845i 0.667416π0.667416\pi
7272 0 0
7373 1.97351 0.230982 0.115491 0.993309i 0.463156π-0.463156\pi
0.115491 + 0.993309i 0.463156π0.463156\pi
7474 2.86333 + 4.95943i 0.332855 + 0.576522i
7575 0 0
7676 4.00040 6.92889i 0.458877 0.794798i
7777 0 0
7878 0 0
7979 −4.08113 7.06872i −0.459163 0.795293i 0.539754 0.841823i 0.318517π-0.318517\pi
−0.998917 + 0.0465297i 0.985184π0.985184\pi
8080 15.8301 1.76986
8181 0 0
8282 18.4882 2.04168
8383 6.08600 + 10.5413i 0.668025 + 1.15705i 0.978456 + 0.206457i 0.0661933π0.0661933\pi
−0.310431 + 0.950596i 0.600473π0.600473\pi
8484 0 0
8585 8.88151 15.3832i 0.963336 1.66855i
8686 −2.86333 + 4.95943i −0.308760 + 0.534789i
8787 0 0
8888 −1.02704 1.77889i −0.109483 0.189630i
8989 −14.8301 −1.57199 −0.785996 0.618231i 0.787849π-0.787849\pi
−0.785996 + 0.618231i 0.787849π0.787849\pi
9090 0 0
9191 0 0
9292 9.42101 + 16.3177i 0.982208 + 1.70123i
9393 0 0
9494 7.77056 13.4590i 0.801472 1.38819i
9595 3.60963 6.25206i 0.370340 0.641448i
9696 0 0
9797 4.74375 + 8.21642i 0.481655 + 0.834251i 0.999778 0.0210547i 0.00670241π-0.00670241\pi
−0.518123 + 0.855306i 0.673369π0.673369\pi
9898 0 0
9999 0 0
100100 33.9794 3.39794
101101 4.35588 + 7.54461i 0.433426 + 0.750716i 0.997166 0.0752364i 0.0239711π-0.0239711\pi
−0.563739 + 0.825953i 0.690638π0.690638\pi
102102 0 0
103103 −4.01356 + 6.95169i −0.395468 + 0.684970i −0.993161 0.116755i 0.962751π-0.962751\pi
0.597693 + 0.801725i 0.296084π0.296084\pi
104104 −1.23016 + 2.13069i −0.120627 + 0.208931i
105105 0 0
106106 4.39037 + 7.60434i 0.426430 + 0.738599i
107107 −12.8420 −1.24148 −0.620742 0.784015i 0.713169π-0.713169\pi
−0.620742 + 0.784015i 0.713169π0.713169\pi
108108 0 0
109109 2.60078 0.249109 0.124555 0.992213i 0.460250π-0.460250\pi
0.124555 + 0.992213i 0.460250π0.460250\pi
110110 −1.82904 3.16799i −0.174392 0.302056i
111111 0 0
112112 0 0
113113 −6.97509 + 12.0812i −0.656162 + 1.13651i 0.325440 + 0.945563i 0.394488π0.394488\pi
−0.981601 + 0.190942i 0.938846π0.938846\pi
114114 0 0
115115 8.50075 + 14.7237i 0.792699 + 1.37300i
116116 −31.0043 −2.87867
117117 0 0
118118 15.0543 1.38586
119119 0 0
120120 0 0
121121 5.41741 9.38323i 0.492492 0.853021i
122122 −9.87538 + 17.1047i −0.894075 + 1.54858i
123123 0 0
124124 −14.2444 24.6721i −1.27919 2.21562i
125125 12.3698 1.10639
126126 0 0
127127 −15.5438 −1.37929 −0.689643 0.724149i 0.742233π-0.742233\pi
−0.689643 + 0.724149i 0.742233π0.742233\pi
128128 −9.55408 16.5482i −0.844470 1.46266i
129129 0 0
130130 −2.19076 + 3.79450i −0.192142 + 0.332800i
131131 −4.25696 + 7.37327i −0.371932 + 0.644205i −0.989863 0.142027i 0.954638π-0.954638\pi
0.617931 + 0.786233i 0.287971π0.287971\pi
132132 0 0
133133 0 0
134134 −8.85973 −0.765364
135135 0 0
136136 24.5418 2.10444
137137 0.188621 + 0.326702i 0.0161150 + 0.0279120i 0.873970 0.485979i 0.161537π-0.161537\pi
−0.857855 + 0.513891i 0.828204π0.828204\pi
138138 0 0
139139 9.50067 16.4556i 0.805837 1.39575i −0.109888 0.993944i 0.535049π-0.535049\pi
0.915725 0.401806i 0.131617π-0.131617\pi
140140 0 0
141141 0 0
142142 −10.4086 18.0281i −0.873467 1.51289i
143143 0.197844 0.0165446
144144 0 0
145145 −27.9757 −2.32326
146146 2.42792 + 4.20528i 0.200936 + 0.348032i
147147 0 0
148148 −4.71780 + 8.17147i −0.387801 + 0.671691i
149149 −4.85087 + 8.40196i −0.397399 + 0.688315i −0.993404 0.114665i 0.963420π-0.963420\pi
0.596005 + 0.802981i 0.296754π0.296754\pi
150150 0 0
151151 6.41741 + 11.1153i 0.522242 + 0.904549i 0.999665 + 0.0258756i 0.00823738π0.00823738\pi
−0.477424 + 0.878673i 0.658429π0.658429\pi
152152 9.97430 0.809023
153153 0 0
154154 0 0
155155 −12.8530 22.2621i −1.03238 1.78813i
156156 0 0
157157 −10.4743 + 18.1420i −0.835937 + 1.44789i 0.0573276 + 0.998355i 0.481742π0.481742\pi
−0.893265 + 0.449531i 0.851591π0.851591\pi
158158 10.0416 17.3926i 0.798869 1.38368i
159159 0 0
160160 0.986757 + 1.70911i 0.0780100 + 0.135117i
161161 0 0
162162 0 0
163163 −11.1623 −0.874295 −0.437148 0.899390i 0.644011π-0.644011\pi
−0.437148 + 0.899390i 0.644011π0.644011\pi
164164 15.2312 + 26.3812i 1.18936 + 2.06002i
165165 0 0
166166 −14.9746 + 25.9368i −1.16226 + 2.01309i
167167 1.73012 2.99665i 0.133880 0.231888i −0.791289 0.611443i 0.790590π-0.790590\pi
0.925169 + 0.379555i 0.123923π0.123923\pi
168168 0 0
169169 6.38151 + 11.0531i 0.490886 + 0.850239i
170170 43.7060 3.35210
171171 0 0
172172 −9.43560 −0.719458
173173 −3.02680 5.24258i −0.230124 0.398586i 0.727721 0.685874i 0.240580π-0.240580\pi
−0.957844 + 0.287288i 0.907246π0.907246\pi
174174 0 0
175175 0 0
176176 0.879379 1.52313i 0.0662857 0.114810i
177177 0 0
178178 −18.2448 31.6010i −1.36751 2.36859i
179179 −9.13307 −0.682638 −0.341319 0.939948i 0.610874π-0.610874\pi
−0.341319 + 0.939948i 0.610874π0.610874\pi
180180 0 0
181181 11.9478 0.888074 0.444037 0.896008i 0.353546π-0.353546\pi
0.444037 + 0.896008i 0.353546π0.353546\pi
182182 0 0
183183 0 0
184184 −11.7448 + 20.3427i −0.865841 + 1.49968i
185185 −4.25696 + 7.37327i −0.312978 + 0.542093i
186186 0 0
187187 −0.986757 1.70911i −0.0721588 0.124983i
188188 25.6065 1.86755
189189 0 0
190190 17.7630 1.28867
191191 4.57014 + 7.91571i 0.330683 + 0.572760i 0.982646 0.185491i 0.0593874π-0.0593874\pi
−0.651963 + 0.758251i 0.726054π0.726054\pi
192192 0 0
193193 −8.47150 + 14.6731i −0.609792 + 1.05619i 0.381483 + 0.924376i 0.375414π0.375414\pi
−0.991274 + 0.131814i 0.957920π0.957920\pi
194194 −11.6720 + 20.2165i −0.838003 + 1.45146i
195195 0 0
196196 0 0
197197 21.3173 1.51880 0.759398 0.650627i 0.225494π-0.225494\pi
0.759398 + 0.650627i 0.225494π0.225494\pi
198198 0 0
199199 9.97430 0.707060 0.353530 0.935423i 0.384981π-0.384981\pi
0.353530 + 0.935423i 0.384981π0.384981\pi
200200 21.1804 + 36.6856i 1.49768 + 2.59406i
201201 0 0
202202 −10.7177 + 18.5635i −0.754092 + 1.30613i
203203 0 0
204204 0 0
205205 13.7434 + 23.8042i 0.959879 + 1.66256i
206206 −19.7508 −1.37610
207207 0 0
208208 −2.10658 −0.146065
209209 −0.401038 0.694619i −0.0277404 0.0480478i
210210 0 0
211211 −2.44592 + 4.23645i −0.168384 + 0.291649i −0.937852 0.347036i 0.887188π-0.887188\pi
0.769468 + 0.638685i 0.220521π0.220521\pi
212212 −7.23385 + 12.5294i −0.496823 + 0.860523i
213213 0 0
214214 −15.7989 27.3645i −1.07999 1.87060i
215215 −8.51392 −0.580644
216216 0 0
217217 0 0
218218 3.19961 + 5.54189i 0.216705 + 0.375344i
219219 0 0
220220 3.01364 5.21978i 0.203179 0.351917i
221221 −1.18190 + 2.04712i −0.0795034 + 0.137704i
222222 0 0
223223 11.7044 + 20.2727i 0.783786 + 1.35756i 0.929722 + 0.368263i 0.120047π0.120047\pi
−0.145936 + 0.989294i 0.546619π0.546619\pi
224224 0 0
225225 0 0
226226 −34.3245 −2.28323
227227 −3.05919 5.29868i −0.203046 0.351686i 0.746463 0.665427i 0.231751π-0.231751\pi
−0.949508 + 0.313742i 0.898417π0.898417\pi
228228 0 0
229229 −0.730195 + 1.26473i −0.0482526 + 0.0835760i −0.889143 0.457630i 0.848699π-0.848699\pi
0.840890 + 0.541206i 0.182032π0.182032\pi
230230 −20.9161 + 36.2278i −1.37917 + 2.38879i
231231 0 0
232232 −19.3260 33.4736i −1.26881 2.19765i
233233 13.2484 0.867934 0.433967 0.900929i 0.357113π-0.357113\pi
0.433967 + 0.900929i 0.357113π0.357113\pi
234234 0 0
235235 23.1052 1.50722
236236 12.4022 + 21.4813i 0.807316 + 1.39831i
237237 0 0
238238 0 0
239239 9.69436 16.7911i 0.627076 1.08613i −0.361060 0.932543i 0.617585π-0.617585\pi
0.988136 0.153584i 0.0490817π-0.0490817\pi
240240 0 0
241241 2.52684 + 4.37662i 0.162768 + 0.281923i 0.935860 0.352371i 0.114624π-0.114624\pi
−0.773092 + 0.634294i 0.781291π0.781291\pi
242242 26.6591 1.71371
243243 0 0
244244 −32.5426 −2.08333
245245 0 0
246246 0 0
247247 −0.480350 + 0.831990i −0.0305639 + 0.0529383i
248248 17.7580 30.7578i 1.12764 1.95312i
249249 0 0
250250 15.2180 + 26.3584i 0.962472 + 1.66705i
251251 15.0928 0.952647 0.476324 0.879270i 0.341969π-0.341969\pi
0.476324 + 0.879270i 0.341969π0.341969\pi
252252 0 0
253253 1.88891 0.118755
254254 −19.1228 33.1216i −1.19987 2.07823i
255255 0 0
256256 16.1804 28.0253i 1.01128 1.75158i
257257 −3.85592 + 6.67865i −0.240526 + 0.416603i −0.960864 0.277020i 0.910653π-0.910653\pi
0.720338 + 0.693623i 0.243986π0.243986\pi
258258 0 0
259259 0 0
260260 −7.21926 −0.447720
261261 0 0
262262 −20.9485 −1.29420
263263 2.10603 + 3.64776i 0.129864 + 0.224930i 0.923624 0.383301i 0.125213π-0.125213\pi
−0.793760 + 0.608231i 0.791879π0.791879\pi
264264 0 0
265265 −6.52724 + 11.3055i −0.400965 + 0.694492i
266266 0 0
267267 0 0
268268 −7.29893 12.6421i −0.445853 0.772240i
269269 20.7507 1.26519 0.632596 0.774482i 0.281989π-0.281989\pi
0.632596 + 0.774482i 0.281989π0.281989\pi
270270 0 0
271271 28.4889 1.73057 0.865287 0.501276i 0.167136π-0.167136\pi
0.865287 + 0.501276i 0.167136π0.167136\pi
272272 10.5067 + 18.1981i 0.637060 + 1.10342i
273273 0 0
274274 −0.464103 + 0.803851i −0.0280375 + 0.0485624i
275275 1.70321 2.95005i 0.102707 0.177895i
276276 0 0
277277 −8.58113 14.8629i −0.515590 0.893028i −0.999836 0.0180962i 0.994239π-0.994239\pi
0.484246 0.874932i 0.339094π-0.339094\pi
278278 46.7529 2.80405
279279 0 0
280280 0 0
281281 4.72140 + 8.17770i 0.281655 + 0.487841i 0.971793 0.235837i 0.0757833π-0.0757833\pi
−0.690138 + 0.723678i 0.742450π0.742450\pi
282282 0 0
283283 8.43422 14.6085i 0.501362 0.868385i −0.498636 0.866811i 0.666166π-0.666166\pi
0.999999 0.00157378i 0.000500949π-0.000500949\pi
284284 17.1498 29.7043i 1.01765 1.76263i
285285 0 0
286286 0.243398 + 0.421578i 0.0143924 + 0.0249284i
287287 0 0
288288 0 0
289289 6.57918 0.387011
290290 −34.4172 59.6124i −2.02105 3.50056i
291291 0 0
292292 −4.00040 + 6.92889i −0.234105 + 0.405483i
293293 1.86143 3.22409i 0.108746 0.188353i −0.806517 0.591211i 0.798650π-0.798650\pi
0.915262 + 0.402858i 0.131983π0.131983\pi
294294 0 0
295295 11.1908 + 19.3830i 0.651551 + 1.12852i
296296 −11.7630 −0.683712
297297 0 0
298298 −23.8712 −1.38282
299299 −1.13123 1.95935i −0.0654209 0.113312i
300300 0 0
301301 0 0
302302 −15.7901 + 27.3492i −0.908617 + 1.57377i
303303 0 0
304304 4.27012 + 7.39607i 0.244908 + 0.424194i
305305 −29.3638 −1.68137
306306 0 0
307307 −30.5691 −1.74467 −0.872335 0.488908i 0.837395π-0.837395\pi
−0.872335 + 0.488908i 0.837395π0.837395\pi
308308 0 0
309309 0 0
310310 31.6249 54.7759i 1.79617 3.11106i
311311 5.21739 9.03678i 0.295851 0.512429i −0.679332 0.733831i 0.737730π-0.737730\pi
0.975182 + 0.221403i 0.0710635π0.0710635\pi
312312 0 0
313313 −0.309930 0.536815i −0.0175183 0.0303426i 0.857133 0.515095i 0.172243π-0.172243\pi
−0.874652 + 0.484752i 0.838910π0.838910\pi
314314 −51.5440 −2.90879
315315 0 0
316316 33.0905 1.86148
317317 5.12422 + 8.87541i 0.287805 + 0.498493i 0.973285 0.229598i 0.0737412π-0.0737412\pi
−0.685481 + 0.728091i 0.740408π0.740408\pi
318318 0 0
319319 −1.55408 + 2.69175i −0.0870120 + 0.150709i
320320 13.4021 23.2132i 0.749203 1.29766i
321321 0 0
322322 0 0
323323 9.58307 0.533216
324324 0 0
325325 −4.08009 −0.226323
326326 −13.7324 23.7852i −0.760567 1.31734i
327327 0 0
328328 −18.9882 + 32.8885i −1.04845 + 1.81596i
329329 0 0
330330 0 0
331331 10.1819 + 17.6356i 0.559648 + 0.969339i 0.997526 + 0.0703042i 0.0223970π0.0223970\pi
−0.437878 + 0.899035i 0.644270π0.644270\pi
332332 −49.3463 −2.70823
333333 0 0
334334 8.51392 0.465861
335335 −6.58596 11.4072i −0.359829 0.623242i
336336 0 0
337337 2.85594 4.94662i 0.155573 0.269460i −0.777695 0.628642i 0.783611π-0.783611\pi
0.933267 + 0.359182i 0.116944π0.116944\pi
338338 −15.7017 + 27.1962i −0.854062 + 1.47928i
339339 0 0
340340 36.0064 + 62.3649i 1.95272 + 3.38221i
341341 −2.85600 −0.154661
342342 0 0
343343 0 0
344344 −5.88151 10.1871i −0.317110 0.549251i
345345 0 0
346346 7.44746 12.8994i 0.400378 0.693475i
347347 4.44066 7.69145i 0.238387 0.412899i −0.721865 0.692034i 0.756715π-0.756715\pi
0.960252 + 0.279136i 0.0900480π0.0900480\pi
348348 0 0
349349 −10.4874 18.1648i −0.561379 0.972337i −0.997376 0.0723893i 0.976938π-0.976938\pi
0.435997 0.899948i 0.356396π-0.356396\pi
350350 0 0
351351 0 0
352352 0.219262 0.0116867
353353 −7.38268 12.7872i −0.392941 0.680593i 0.599895 0.800078i 0.295209π-0.295209\pi
−0.992836 + 0.119485i 0.961876π0.961876\pi
354354 0 0
355355 15.4746 26.8028i 0.821306 1.42254i
356356 30.0613 52.0677i 1.59325 2.75959i
357357 0 0
358358 −11.2360 19.4613i −0.593840 1.02856i
359359 −7.21206 −0.380638 −0.190319 0.981722i 0.560952π-0.560952\pi
−0.190319 + 0.981722i 0.560952π0.560952\pi
360360 0 0
361361 −15.1052 −0.795013
362362 14.6988 + 25.4591i 0.772554 + 1.33810i
363363 0 0
364364 0 0
365365 −3.60963 + 6.25206i −0.188937 + 0.327248i
366366 0 0
367367 5.48711 + 9.50396i 0.286425 + 0.496103i 0.972954 0.231000i 0.0741998π-0.0741998\pi
−0.686529 + 0.727103i 0.740866π0.740866\pi
368368 −20.1124 −1.04843
369369 0 0
370370 −20.9485 −1.08906
371371 0 0
372372 0 0
373373 0.271884 0.470916i 0.0140776 0.0243831i −0.858901 0.512142i 0.828852π-0.828852\pi
0.872978 + 0.487759i 0.162185π0.162185\pi
374374 2.42792 4.20528i 0.125545 0.217450i
375375 0 0
376376 15.9614 + 27.6459i 0.823145 + 1.42573i
377377 3.72286 0.191737
378378 0 0
379379 −22.6912 −1.16557 −0.582785 0.812626i 0.698037π-0.698037\pi
−0.582785 + 0.812626i 0.698037π0.698037\pi
380380 14.6337 + 25.3464i 0.750695 + 1.30024i
381381 0 0
382382 −11.2448 + 19.4766i −0.575336 + 0.996511i
383383 17.8569 30.9291i 0.912447 1.58041i 0.101851 0.994800i 0.467523π-0.467523\pi
0.810596 0.585606i 0.199143π-0.199143\pi
384384 0 0
385385 0 0
386386 −41.6883 −2.12188
387387 0 0
388388 −38.4632 −1.95267
389389 19.3296 + 33.4798i 0.980048 + 1.69749i 0.662156 + 0.749366i 0.269641π0.269641\pi
0.317892 + 0.948127i 0.397025π0.397025\pi
390390 0 0
391391 −11.2842 + 19.5447i −0.570664 + 0.988420i
392392 0 0
393393 0 0
394394 26.2257 + 45.4242i 1.32123 + 2.28844i
395395 29.8581 1.50233
396396 0 0
397397 −11.9478 −0.599644 −0.299822 0.953995i 0.596927π-0.596927\pi
−0.299822 + 0.953995i 0.596927π0.596927\pi
398398 12.2709 + 21.2538i 0.615085 + 1.06536i
399399 0 0
400400 −18.1352 + 31.4111i −0.906761 + 1.57056i
401401 16.1783 28.0216i 0.807906 1.39933i −0.106406 0.994323i 0.533934π-0.533934\pi
0.914312 0.405011i 0.132732π-0.132732\pi
402402 0 0
403403 1.71041 + 2.96251i 0.0852015 + 0.147573i
404404 −35.3182 −1.75715
405405 0 0
406406 0 0
407407 0.472958 + 0.819187i 0.0234437 + 0.0406056i
408408 0 0
409409 −9.48751 + 16.4328i −0.469127 + 0.812552i −0.999377 0.0352893i 0.988765π-0.988765\pi
0.530250 + 0.847841i 0.322098π0.322098\pi
410410 −33.8157 + 58.5704i −1.67004 + 2.89259i
411411 0 0
412412 −16.2713 28.1827i −0.801630 1.38846i
413413 0 0
414414 0 0
415415 −44.5261 −2.18570
416416 −0.131312 0.227439i −0.00643811 0.0111511i
417417 0 0
418418 0.986757 1.70911i 0.0482639 0.0835955i
419419 −8.64523 + 14.9740i −0.422347 + 0.731526i −0.996169 0.0874539i 0.972127π-0.972127\pi
0.573822 + 0.818980i 0.305460π0.305460\pi
420420 0 0
421421 −9.30039 16.1087i −0.453273 0.785092i 0.545314 0.838232i 0.316410π-0.316410\pi
−0.998587 + 0.0531397i 0.983077π0.983077\pi
422422 −12.0364 −0.585922
423423 0 0
424424 −18.0364 −0.875924
425425 20.3496 + 35.2466i 0.987103 + 1.70971i
426426 0 0
427427 0 0
428428 26.0313 45.0876i 1.25827 2.17939i
429429 0 0
430430 −10.4743 18.1420i −0.505114 0.874883i
431431 15.8784 0.764835 0.382418 0.923990i 0.375092π-0.375092\pi
0.382418 + 0.923990i 0.375092π0.375092\pi
432432 0 0
433433 40.4367 1.94326 0.971631 0.236501i 0.0760007π-0.0760007\pi
0.971631 + 0.236501i 0.0760007π0.0760007\pi
434434 0 0
435435 0 0
436436 −5.27188 + 9.13117i −0.252477 + 0.437304i
437437 −4.58611 + 7.94338i −0.219384 + 0.379984i
438438 0 0
439439 −6.23047 10.7915i −0.297364 0.515050i 0.678168 0.734907i 0.262774π-0.262774\pi
−0.975532 + 0.219857i 0.929441π0.929441\pi
440440 7.51399 0.358216
441441 0 0
442442 −5.81616 −0.276646
443443 −4.11537 7.12802i −0.195527 0.338663i 0.751546 0.659680i 0.229308π-0.229308\pi
−0.947073 + 0.321018i 0.895975π0.895975\pi
444444 0 0
445445 27.1249 46.9817i 1.28584 2.22715i
446446 −28.7988 + 49.8810i −1.36366 + 2.36193i
447447 0 0
448448 0 0
449449 −5.64474 −0.266392 −0.133196 0.991090i 0.542524π-0.542524\pi
−0.133196 + 0.991090i 0.542524π0.542524\pi
450450 0 0
451451 3.05384 0.143800
452452 −28.2776 48.9783i −1.33007 2.30374i
453453 0 0
454454 7.52716 13.0374i 0.353267 0.611876i
455455 0 0
456456 0 0
457457 −2.53443 4.38977i −0.118556 0.205345i 0.800640 0.599146i 0.204493π-0.204493\pi
−0.919196 + 0.393801i 0.871160π0.871160\pi
458458 −3.59330 −0.167904
459459 0 0
460460 −68.9255 −3.21367
461461 −3.88831 6.73475i −0.181097 0.313669i 0.761158 0.648567i 0.224631π-0.224631\pi
−0.942254 + 0.334898i 0.891298π0.891298\pi
462462 0 0
463463 4.58998 7.95008i 0.213314 0.369472i −0.739435 0.673228i 0.764907π-0.764907\pi
0.952750 + 0.303756i 0.0982408π0.0982408\pi
464464 16.5474 28.6609i 0.768192 1.33055i
465465 0 0
466466 16.2989 + 28.2306i 0.755033 + 1.30776i
467467 13.7654 0.636989 0.318494 0.947925i 0.396823π-0.396823\pi
0.318494 + 0.947925i 0.396823π0.396823\pi
468468 0 0
469469 0 0
470470 28.4253 + 49.2340i 1.31116 + 2.27100i
471471 0 0
472472 −15.4614 + 26.7800i −0.711669 + 1.23265i
473473 −0.472958 + 0.819187i −0.0217466 + 0.0376663i
474474 0 0
475475 8.27052 + 14.3250i 0.379477 + 0.657274i
476476 0 0
477477 0 0
478478 47.7060 2.18202
479479 4.35588 + 7.54461i 0.199025 + 0.344722i 0.948213 0.317636i 0.102889π-0.102889\pi
−0.749187 + 0.662358i 0.769556π0.769556\pi
480480 0 0
481481 0.566492 0.981194i 0.0258298 0.0447386i
482482 −6.21731 + 10.7687i −0.283191 + 0.490501i
483483 0 0
484484 21.9626 + 38.0404i 0.998302 + 1.72911i
485485 −34.7060 −1.57592
486486 0 0
487487 −18.0364 −0.817306 −0.408653 0.912690i 0.634001π-0.634001\pi
−0.408653 + 0.912690i 0.634001π0.634001\pi
488488 −20.2849 35.1344i −0.918253 1.59046i
489489 0 0
490490 0 0
491491 1.02344 1.77266i 0.0461874 0.0799989i −0.842007 0.539466i 0.818626π-0.818626\pi
0.888195 + 0.459467i 0.151960π0.151960\pi
492492 0 0
493493 −18.5679 32.1606i −0.836257 1.44844i
494494 −2.36381 −0.106353
495495 0 0
496496 30.4097 1.36544
497497 0 0
498498 0 0
499499 19.5438 33.8508i 0.874899 1.51537i 0.0180291 0.999837i 0.494261π-0.494261\pi
0.856870 0.515532i 0.172406π-0.172406\pi
500500 −25.0742 + 43.4297i −1.12135 + 1.94224i
501501 0 0
502502 18.5679 + 32.1606i 0.828727 + 1.43540i
503503 −5.11846 −0.228221 −0.114111 0.993468i 0.536402π-0.536402\pi
−0.114111 + 0.993468i 0.536402π0.536402\pi
504504 0 0
505505 −31.8683 −1.41812
506506 2.32383 + 4.02499i 0.103307 + 0.178933i
507507 0 0
508508 31.5079 54.5732i 1.39794 2.42130i
509509 −14.7636 + 25.5713i −0.654386 + 1.13343i 0.327662 + 0.944795i 0.393739π0.393739\pi
−0.982047 + 0.188634i 0.939594π0.939594\pi
510510 0 0
511511 0 0
512512 41.4078 1.82998
513513 0 0
514514 −18.9750 −0.836952
515515 −14.6819 25.4298i −0.646962 1.12057i
516516 0 0
517517 1.28352 2.22313i 0.0564493 0.0977730i
518518 0 0
519519 0 0
520520 −4.50000 7.79423i −0.197338 0.341800i
521521 −1.06470 −0.0466454 −0.0233227 0.999728i 0.507425π-0.507425\pi
−0.0233227 + 0.999728i 0.507425π0.507425\pi
522522 0 0
523523 13.3819 0.585149 0.292574 0.956243i 0.405488π-0.405488\pi
0.292574 + 0.956243i 0.405488π0.405488\pi
524524 −17.2581 29.8918i −0.753922 1.30583i
525525 0 0
526526 −5.18190 + 8.97532i −0.225942 + 0.391343i
527527 17.0615 29.5513i 0.743210 1.28728i
528528 0 0
529529 0.699612 + 1.21176i 0.0304179 + 0.0526853i
530530 −32.1206 −1.39523
531531 0 0
532532 0 0
533533 −1.82889 3.16774i −0.0792181 0.137210i
534534 0 0
535535 23.4885 40.6833i 1.01550 1.75889i
536536 9.09931 15.7605i 0.393031 0.680749i
537537 0 0
538538 25.5286 + 44.2168i 1.10062 + 1.90632i
539539 0 0
540540 0 0
541541 −34.0875 −1.46554 −0.732769 0.680478i 0.761772π-0.761772\pi
−0.732769 + 0.680478i 0.761772π0.761772\pi
542542 35.0485 + 60.7058i 1.50546 + 2.60754i
543543 0 0
544544 −1.30985 + 2.26873i −0.0561594 + 0.0972709i
545545 −4.75692 + 8.23922i −0.203764 + 0.352930i
546546 0 0
547547 2.97150 + 5.14678i 0.127052 + 0.220060i 0.922533 0.385918i 0.126115π-0.126115\pi
−0.795481 + 0.605978i 0.792782π0.792782\pi
548548 −1.52937 −0.0653316
549549 0 0
550550 8.38151 0.357389
551551 −7.54638 13.0707i −0.321487 0.556831i
552552 0 0
553553 0 0
554554 21.1139 36.5704i 0.897044 1.55373i
555555 0 0
556556 38.5165 + 66.7126i 1.63346 + 2.82924i
557557 30.0803 1.27454 0.637272 0.770639i 0.280063π-0.280063\pi
0.637272 + 0.770639i 0.280063π0.280063\pi
558558 0 0
559559 1.13298 0.0479202
560560 0 0
561561 0 0
562562 −11.6170 + 20.1213i −0.490035 + 0.848765i
563563 9.81060 16.9925i 0.413468 0.716147i −0.581799 0.813333i 0.697651π-0.697651\pi
0.995266 + 0.0971860i 0.0309842π0.0309842\pi
564564 0 0
565565 −25.5154 44.1940i −1.07344 1.85926i
566566 41.5049 1.74458
567567 0 0
568568 42.7601 1.79417
569569 −0.687159 1.19019i −0.0288072 0.0498955i 0.851262 0.524740i 0.175838π-0.175838\pi
−0.880070 + 0.474845i 0.842504π0.842504\pi
570570 0 0
571571 −8.69076 + 15.0528i −0.363697 + 0.629941i −0.988566 0.150788i 0.951819π-0.951819\pi
0.624869 + 0.780729i 0.285152π0.285152\pi
572572 −0.401038 + 0.694619i −0.0167683 + 0.0290435i
573573 0 0
574574 0 0
575575 −38.9545 −1.62451
576576 0 0
577577 27.0548 1.12631 0.563153 0.826353i 0.309588π-0.309588\pi
0.563153 + 0.826353i 0.309588π0.309588\pi
578578 8.09406 + 14.0193i 0.336668 + 0.583127i
579579 0 0
580580 56.7080 98.2211i 2.35467 4.07841i
581581 0 0
582582 0 0
583583 0.725191 + 1.25607i 0.0300344 + 0.0520210i
584584 −9.97430 −0.412740
585585 0 0
586586 9.16010 0.378400
587587 −3.75700 6.50731i −0.155068 0.268585i 0.778016 0.628245i 0.216226π-0.216226\pi
−0.933084 + 0.359659i 0.882893π0.882893\pi
588588 0 0
589589 6.93414 12.0103i 0.285716 0.494875i
590590 −27.5349 + 47.6919i −1.13359 + 1.96344i
591591 0 0
592592 −5.03590 8.72243i −0.206974 0.358490i
593593 35.5808 1.46113 0.730565 0.682843i 0.239257π-0.239257\pi
0.730565 + 0.682843i 0.239257π0.239257\pi
594594 0 0
595595 0 0
596596 −19.6659 34.0623i −0.805545 1.39524i
597597 0 0
598598 2.78340 4.82100i 0.113822 0.197145i
599599 −5.74105 + 9.94379i −0.234573 + 0.406292i −0.959148 0.282903i 0.908702π-0.908702\pi
0.724576 + 0.689195i 0.242036π0.242036\pi
600600 0 0
601601 −0.190030 0.329142i −0.00775150 0.0134260i 0.862124 0.506698i 0.169134π-0.169134\pi
−0.869875 + 0.493272i 0.835801π0.835801\pi
602602 0 0
603603 0 0
604604 −52.0335 −2.11721
605605 19.8173 + 34.3246i 0.805688 + 1.39549i
606606 0 0
607607 −9.27044 + 16.0569i −0.376275 + 0.651728i −0.990517 0.137390i 0.956129π-0.956129\pi
0.614242 + 0.789118i 0.289462π0.289462\pi
608608 −0.532351 + 0.922058i −0.0215897 + 0.0373944i
609609 0 0
610610 −36.1249 62.5702i −1.46265 2.53339i
611611 −3.07472 −0.124390
612612 0 0
613613 7.32451 0.295834 0.147917 0.989000i 0.452743π-0.452743\pi
0.147917 + 0.989000i 0.452743π0.452743\pi
614614 −37.6077 65.1385i −1.51772 2.62877i
615615 0 0
616616 0 0
617617 −12.7427 + 22.0710i −0.513002 + 0.888546i 0.486884 + 0.873466i 0.338133π0.338133\pi
−0.999886 + 0.0150791i 0.995200π0.995200\pi
618618 0 0
619619 16.4482 + 28.4891i 0.661108 + 1.14507i 0.980325 + 0.197391i 0.0632468π0.0632468\pi
−0.319217 + 0.947682i 0.603420π0.603420\pi
620620 104.214 4.18535
621621 0 0
622622 25.6748 1.02947
623623 0 0
624624 0 0
625625 −1.67111 + 2.89444i −0.0668443 + 0.115778i
626626 0.762585 1.32084i 0.0304790 0.0527912i
627627 0 0
628628 −42.4636 73.5490i −1.69448 2.93493i
629629 −11.3016 −0.450626
630630 0 0
631631 29.8683 1.18904 0.594519 0.804082i 0.297343π-0.297343\pi
0.594519 + 0.804082i 0.297343π0.297343\pi
632632 20.6264 + 35.7259i 0.820472 + 1.42110i
633633 0 0
634634 −12.6082 + 21.8380i −0.500734 + 0.867297i
635635 28.4301 49.2424i 1.12822 1.95413i
636636 0 0
637637 0 0
638638 −7.64766 −0.302774
639639 0 0
640640 69.8991 2.76301
641641 5.73025 + 9.92509i 0.226331 + 0.392017i 0.956718 0.291016i 0.0939934π-0.0939934\pi
−0.730387 + 0.683034i 0.760660π0.760660\pi
642642 0 0
643643 −8.69078 + 15.0529i −0.342731 + 0.593627i −0.984939 0.172903i 0.944685π-0.944685\pi
0.642208 + 0.766531i 0.278019π0.278019\pi
644644 0 0
645645 0 0
646646 11.7896 + 20.4202i 0.463855 + 0.803421i
647647 25.3439 0.996372 0.498186 0.867070i 0.334000π-0.334000\pi
0.498186 + 0.867070i 0.334000π0.334000\pi
648648 0 0
649649 2.48664 0.0976091
650650 −5.01954 8.69410i −0.196883 0.341011i
651651 0 0
652652 22.6264 39.1900i 0.886116 1.53480i
653653 7.04163 12.1965i 0.275560 0.477284i −0.694716 0.719284i 0.744470π-0.744470\pi
0.970276 + 0.242000i 0.0778033π0.0778033\pi
654654 0 0
655655 −15.5723 26.9720i −0.608459 1.05388i
656656 −32.5163 −1.26955
657657 0 0
658658 0 0
659659 19.0854 + 33.0569i 0.743462 + 1.28771i 0.950910 + 0.309467i 0.100151π0.100151\pi
−0.207449 + 0.978246i 0.566516π0.566516\pi
660660 0 0
661661 −0.176866 + 0.306341i −0.00687930 + 0.0119153i −0.869445 0.494031i 0.835523π-0.835523\pi
0.862565 + 0.505946i 0.168856π0.168856\pi
662662 −25.0526 + 43.3924i −0.973698 + 1.68649i
663663 0 0
664664 −30.7591 53.2764i −1.19369 2.06752i
665665 0 0
666666 0 0
667667 35.5438 1.37626
668668 7.01403 + 12.1487i 0.271381 + 0.470046i
669669 0 0
670670 16.2048 28.0675i 0.626045 1.08434i
671671 −1.63119 + 2.82531i −0.0629715 + 0.109070i
672672 0 0
673673 10.5555 + 18.2827i 0.406886 + 0.704748i 0.994539 0.104365i 0.0332811π-0.0332811\pi
−0.587653 + 0.809113i 0.699948π0.699948\pi
674674 14.0541 0.541343
675675 0 0
676676 −51.7424 −1.99009
677677 −10.5732 18.3133i −0.406361 0.703837i 0.588118 0.808775i 0.299869π-0.299869\pi
−0.994479 + 0.104938i 0.966536π0.966536\pi
678678 0 0
679679 0 0
680680 −44.8879 + 77.7482i −1.72137 + 2.98151i
681681 0 0
682682 −3.51360 6.08573i −0.134543 0.233035i
683683 −34.7716 −1.33050 −0.665249 0.746622i 0.731674π-0.731674\pi
−0.665249 + 0.746622i 0.731674π0.731674\pi
684684 0 0
685685 −1.37998 −0.0527264
686686 0 0
687687 0 0
688688 5.03590 8.72243i 0.191992 0.332539i
689689 0.868609 1.50447i 0.0330914 0.0573159i
690690 0 0
691691 17.3246 + 30.0071i 0.659059 + 1.14152i 0.980860 + 0.194716i 0.0623785π0.0623785\pi
−0.321801 + 0.946807i 0.604288π0.604288\pi
692692 24.5418 0.932940
693693 0 0
694694 21.8525 0.829511
695695 34.7542 + 60.1960i 1.31830 + 2.28336i
696696 0 0
697697 −18.2434 + 31.5985i −0.691017 + 1.19688i
698698 25.8044 44.6945i 0.976710 1.69171i
699699 0 0
700700 0 0
701701 48.6050 1.83579 0.917894 0.396826i 0.129889π-0.129889\pi
0.917894 + 0.396826i 0.129889π0.129889\pi
702702 0 0
703703 −4.59322 −0.173236
704704 −1.48901 2.57904i −0.0561192 0.0972012i
705705 0 0
706706 18.1651 31.4629i 0.683654 1.18412i
707707 0 0
708708 0 0
709709 −2.05408 3.55778i −0.0771428 0.133615i 0.824873 0.565318i 0.191246π-0.191246\pi
−0.902016 + 0.431702i 0.857913π0.857913\pi
710710 76.1506 2.85788
711711 0 0
712712 74.9528 2.80898
713713 16.3300 + 28.2844i 0.611564 + 1.05926i
714714 0 0
715715 −0.361864 + 0.626767i −0.0135330 + 0.0234398i
716716 18.5131 32.0657i 0.691868 1.19835i
717717 0 0
718718 −8.87266 15.3679i −0.331125 0.573525i
719719 48.2816 1.80060 0.900299 0.435271i 0.143347π-0.143347\pi
0.900299 + 0.435271i 0.143347π0.143347\pi
720720 0 0
721721 0 0
722722 −18.5833 32.1872i −0.691597 1.19788i
723723 0 0
724724 −24.2187 + 41.9481i −0.900082 + 1.55899i
725725 32.0495 55.5114i 1.19029 2.06164i
726726 0 0
727727 −20.5151 35.5332i −0.760863 1.31785i −0.942406 0.334470i 0.891443π-0.891443\pi
0.181543 0.983383i 0.441891π-0.441891\pi
728728 0 0
729729 0 0
730730 −17.7630 −0.657439
731731 −5.65082 9.78750i −0.209003 0.362004i
732732 0 0
733733 15.2714 26.4508i 0.564062 0.976983i −0.433075 0.901358i 0.642571π-0.642571\pi
0.997136 0.0756253i 0.0240953π-0.0240953\pi
734734 −13.5011 + 23.3845i −0.498334 + 0.863139i
735735 0 0
736736 −1.25370 2.17147i −0.0462118 0.0800413i
737737 −1.46343 −0.0539061
738738 0 0
739739 23.8200 0.876234 0.438117 0.898918i 0.355646π-0.355646\pi
0.438117 + 0.898918i 0.355646π0.355646\pi
740740 −17.2581 29.8918i −0.634419 1.09885i
741741 0 0
742742 0 0
743743 5.26089 9.11213i 0.193003 0.334292i −0.753241 0.657745i 0.771510π-0.771510\pi
0.946244 + 0.323453i 0.104844π0.104844\pi
744744 0 0
745745 −17.7449 30.7350i −0.650121 1.12604i
746746 1.33794 0.0489855
747747 0 0
748748 8.00079 0.292538
749749 0 0
750750 0 0
751751 5.13521 8.89445i 0.187386 0.324563i −0.756992 0.653425i 0.773332π-0.773332\pi
0.944378 + 0.328862i 0.106665π0.106665\pi
752752 −13.6665 + 23.6711i −0.498367 + 0.863197i
753753 0 0
754754 4.58005 + 7.93288i 0.166796 + 0.288899i
755755 −46.9507 −1.70871
756756 0 0
757757 8.03930 0.292193 0.146097 0.989270i 0.453329π-0.453329\pi
0.146097 + 0.989270i 0.453329π0.453329\pi
758758 −27.9159 48.3518i −1.01395 1.75622i
759759 0 0
760760 −18.2434 + 31.5985i −0.661757 + 1.14620i
761761 −13.8302 + 23.9547i −0.501345 + 0.868355i 0.498654 + 0.866801i 0.333828π0.333828\pi
−0.999999 + 0.00155404i 0.999505π0.999505\pi
762762 0 0
763763 0 0
764764 −37.0554 −1.34062
765765 0 0
766766 87.8742 3.17502
767767 −1.48920 2.57938i −0.0537720 0.0931359i
768768 0 0
769769 −16.9613 + 29.3778i −0.611640 + 1.05939i 0.379324 + 0.925264i 0.376157π0.376157\pi
−0.990964 + 0.134128i 0.957177π0.957177\pi
770770 0 0
771771 0 0
772772 −34.3442 59.4858i −1.23607 2.14094i
773773 −28.5956 −1.02851 −0.514256 0.857637i 0.671932π-0.671932\pi
−0.514256 + 0.857637i 0.671932π0.671932\pi
774774 0 0
775775 58.8986 2.11570
776776 −23.9753 41.5265i −0.860664 1.49071i
777777 0 0
778778 −47.5605 + 82.3772i −1.70513 + 2.95337i
779779 −7.41449 + 12.8423i −0.265652 + 0.460122i
780780 0 0
781781 −1.71926 2.97785i −0.0615200 0.106556i
782782 −55.5294 −1.98573
783783 0 0
784784 0 0
785785 −38.3157 66.3647i −1.36754 2.36866i
786786 0 0
787787 23.0017 39.8402i 0.819923 1.42015i −0.0858145 0.996311i 0.527349π-0.527349\pi
0.905738 0.423838i 0.139317π-0.139317\pi
788788 −43.2111 + 74.8438i −1.53933 + 2.66620i
789789 0 0
790790 36.7330 + 63.6235i 1.30690 + 2.26362i
791791 0 0
792792 0 0
793793 3.90757 0.138762
794794 −14.6988 25.4591i −0.521642 0.903511i
795795 0 0
796796 −20.2183 + 35.0192i −0.716620 + 1.24122i
797797 18.4558 31.9664i 0.653739 1.13231i −0.328469 0.944515i 0.606533π-0.606533\pi
0.982208 0.187795i 0.0601339π-0.0601339\pi
798798 0 0
799799 15.3353 + 26.5615i 0.542524 + 0.939679i
800800 −4.52179 −0.159869
801801 0 0
802802 79.6136 2.81125
803803 0.401038 + 0.694619i 0.0141523 + 0.0245126i
804804 0 0
805805 0 0
806806 −4.20847 + 7.28928i −0.148237 + 0.256754i
807807 0 0
808808 −22.0150 38.1311i −0.774484 1.34145i
809809 −47.6887 −1.67665 −0.838323 0.545174i 0.816463π-0.816463\pi
−0.838323 + 0.545174i 0.816463π0.816463\pi
810810 0 0
811811 6.02728 0.211646 0.105823 0.994385i 0.466252π-0.466252\pi
0.105823 + 0.994385i 0.466252π0.466252\pi
812812 0 0
813813 0 0
814814 −1.16372 + 2.01561i −0.0407882 + 0.0706472i
815815 20.4162 35.3619i 0.715148 1.23867i
816816 0 0
817817 −2.29661 3.97784i −0.0803481 0.139167i
818818 −46.6881 −1.63241
819819 0 0
820820 −111.434 −3.89143
821821 −19.6321 34.0038i −0.685165 1.18674i −0.973385 0.229176i 0.926397π-0.926397\pi
0.288220 0.957564i 0.406936π-0.406936\pi
822822 0 0
823823 −11.3815 + 19.7134i −0.396735 + 0.687165i −0.993321 0.115384i 0.963190π-0.963190\pi
0.596586 + 0.802549i 0.296523π0.296523\pi
824824 20.2849 35.1344i 0.706657 1.22397i
825825 0 0
826826 0 0
827827 28.9286 1.00595 0.502973 0.864302i 0.332240π-0.332240\pi
0.502973 + 0.864302i 0.332240π0.332240\pi
828828 0 0
829829 −33.9767 −1.18006 −0.590029 0.807382i 0.700884π-0.700884\pi
−0.590029 + 0.807382i 0.700884π0.700884\pi
830830 −54.7783 94.8788i −1.90138 3.29329i
831831 0 0
832832 −1.78348 + 3.08908i −0.0618312 + 0.107095i
833833 0 0
834834 0 0
835835 6.32889 + 10.9620i 0.219020 + 0.379354i
836836 3.25169 0.112462
837837 0 0
838838 −42.5432 −1.46963
839839 19.0206 + 32.9446i 0.656663 + 1.13737i 0.981474 + 0.191595i 0.0613659π0.0613659\pi
−0.324811 + 0.945779i 0.605301π0.605301\pi
840840 0 0
841841 −14.7434 + 25.5363i −0.508392 + 0.880561i
842842 22.8837 39.6356i 0.788623 1.36593i
843843 0 0
844844 −9.91595 17.1749i −0.341321 0.591185i
845845 −46.6881 −1.60612
846846 0 0
847847 0 0
848848 −7.72159 13.3742i −0.265161 0.459272i
849849 0 0
850850 −50.0704 + 86.7245i −1.71740 + 2.97463i
851851 5.40856 9.36790i 0.185403 0.321127i
852852 0 0
853853 −4.90746 8.49996i −0.168028 0.291033i 0.769698 0.638408i 0.220407π-0.220407\pi
−0.937726 + 0.347374i 0.887073π0.887073\pi
854854 0 0
855855 0 0
856856 64.9046 2.21840
857857 13.3680 + 23.1541i 0.456642 + 0.790928i 0.998781 0.0493613i 0.0157186π-0.0157186\pi
−0.542139 + 0.840289i 0.682385π0.682385\pi
858858 0 0
859859 9.24411 16.0113i 0.315405 0.546297i −0.664119 0.747627i 0.731193π-0.731193\pi
0.979523 + 0.201330i 0.0645263π0.0645263\pi
860860 17.2581 29.8918i 0.588495 1.01930i
861861 0 0
862862 19.5344 + 33.8346i 0.665345 + 1.15241i
863863 9.14786 0.311397 0.155698 0.987805i 0.450237π-0.450237\pi
0.155698 + 0.987805i 0.450237π0.450237\pi
864864 0 0
865865 22.1445 0.752937
866866 49.7473 + 86.1649i 1.69048 + 2.92800i
867867 0 0
868868 0 0
869869 1.65865 2.87287i 0.0562660 0.0974555i
870870 0 0
871871 0.876423 + 1.51801i 0.0296964 + 0.0514358i
872872 −13.1445 −0.445130
873873 0 0
874874 −22.5683 −0.763385
875875 0 0
876876 0 0
877877 −11.3442 + 19.6487i −0.383065 + 0.663488i −0.991499 0.130117i 0.958465π-0.958465\pi
0.608434 + 0.793605i 0.291798π0.291798\pi
878878 15.3301 26.5525i 0.517366 0.896104i
879879 0 0
880880 3.21683 + 5.57172i 0.108439 + 0.187823i
881881 −15.3696 −0.517815 −0.258907 0.965902i 0.583362π-0.583362\pi
−0.258907 + 0.965902i 0.583362π0.583362\pi
882882 0 0
883883 29.6372 0.997370 0.498685 0.866783i 0.333817π-0.333817\pi
0.498685 + 0.866783i 0.333817π0.333817\pi
884884 −4.79153 8.29918i −0.161157 0.279132i
885885 0 0
886886 10.1259 17.5385i 0.340185 0.589219i
887887 −9.38252 + 16.2510i −0.315034 + 0.545655i −0.979445 0.201712i 0.935349π-0.935349\pi
0.664410 + 0.747368i 0.268683π0.268683\pi
888888 0 0
889889 0 0
890890 133.482 4.47432
891891 0 0
892892 −94.9014 −3.17753
893893 6.23258 + 10.7952i 0.208565 + 0.361246i
894894 0 0
895895 16.7047 28.9335i 0.558378 0.967139i
896896 0 0
897897 0 0
898898 −6.94445 12.0281i −0.231739 0.401384i
899899 −53.7416 −1.79238
900900 0 0
901901 −17.3289 −0.577310
902902 3.75700 + 6.50731i 0.125094 + 0.216670i
903903 0 0
904904 35.2527 61.0595i 1.17249 2.03081i
905905 −21.8530 + 37.8505i −0.726419 + 1.25819i
906906 0 0
907907 15.5541 + 26.9405i 0.516465 + 0.894543i 0.999817 + 0.0191175i 0.00608567π0.00608567\pi
−0.483352 + 0.875426i 0.660581π0.660581\pi
908908 24.8045 0.823165
909909 0 0
910910 0 0
911911 8.73764 + 15.1340i 0.289491 + 0.501413i 0.973688 0.227884i 0.0731806π-0.0731806\pi
−0.684197 + 0.729297i 0.739847π0.739847\pi
912912 0 0
913913 −2.47347 + 4.28418i −0.0818600 + 0.141786i
914914 6.23599 10.8010i 0.206268 0.357267i
915915 0 0
916916 −2.96027 5.12734i −0.0978101 0.169412i
917917 0 0
918918 0 0
919919 42.1986 1.39200 0.696002 0.718040i 0.254960π-0.254960\pi
0.696002 + 0.718040i 0.254960π0.254960\pi
920920 −42.9635 74.4150i −1.41647 2.45339i
921921 0 0
922922 9.56720 16.5709i 0.315079 0.545733i
923923 −2.05927 + 3.56676i −0.0677818 + 0.117401i
924924 0 0
925925 −9.75370 16.8939i −0.320700 0.555468i
926926 22.5873 0.742266
927927 0 0
928928 4.12588 0.135439
929929 24.2056 + 41.9253i 0.794159 + 1.37552i 0.923372 + 0.383907i 0.125422π0.125422\pi
−0.129213 + 0.991617i 0.541245π0.541245\pi
930930 0 0
931931 0 0
932932 −26.8551 + 46.5145i −0.879670 + 1.52363i
933933 0 0
934934 16.9350 + 29.3322i 0.554129 + 0.959780i
935935 7.21926 0.236095
936936 0 0
937937 −22.6750 −0.740762 −0.370381 0.928880i 0.620773π-0.620773\pi
−0.370381 + 0.928880i 0.620773π0.620773\pi
938938 0 0
939939 0 0
940940 −46.8353 + 81.1211i −1.52760 + 2.64588i
941941 −24.9753 + 43.2584i −0.814170 + 1.41018i 0.0957523 + 0.995405i 0.469474π0.469474\pi
−0.909922 + 0.414779i 0.863859π0.863859\pi
942942 0 0
943943 −17.4613 30.2438i −0.568617 0.984873i
944944 −26.4769 −0.861749
945945 0 0
946946 −2.32743 −0.0756713
947947 8.85613 + 15.3393i 0.287786 + 0.498459i 0.973281 0.229618i 0.0737476π-0.0737476\pi
−0.685495 + 0.728077i 0.740414π0.740414\pi
948948 0 0
949949 0.480350 0.831990i 0.0155928 0.0270075i
950950 −20.3496 + 35.2466i −0.660230 + 1.14355i
951951 0 0
952952 0 0
953953 −38.0229 −1.23168 −0.615842 0.787870i 0.711184π-0.711184\pi
−0.615842 + 0.787870i 0.711184π0.711184\pi
954954 0 0
955955 −33.4358 −1.08196
956956 39.3017 + 68.0726i 1.27111 + 2.20163i
957957 0 0
958958 −10.7177 + 18.5635i −0.346272 + 0.599761i
959959 0 0
960960 0 0
961961 −9.19076 15.9189i −0.296476 0.513512i
962962 2.78771 0.0898795
963963 0 0
964964 −20.4881 −0.659876
965965 −30.9894 53.6752i −0.997583 1.72786i
966966 0 0
967967 15.5064 26.8579i 0.498652 0.863691i −0.501346 0.865247i 0.667162π-0.667162\pi
0.999999 + 0.00155529i 0.000495066π0.000495066\pi
968968 −27.3801 + 47.4236i −0.880028 + 1.52425i
969969 0 0
970970 −42.6972 73.9537i −1.37092 2.37451i
971971 14.5675 0.467494 0.233747 0.972297i 0.424901π-0.424901\pi
0.233747 + 0.972297i 0.424901π0.424901\pi
972972 0 0
973973 0 0
974974 −22.1893 38.4330i −0.710991 1.23147i
975975 0 0
976976 17.3684 30.0830i 0.555949 0.962932i
977977 −23.9399 + 41.4651i −0.765904 + 1.32659i 0.173863 + 0.984770i 0.444375π0.444375\pi
−0.939767 + 0.341815i 0.888958π0.888958\pi
978978 0 0
979979 −3.01364 5.21978i −0.0963163 0.166825i
980980 0 0
981981 0 0
982982 5.03638 0.160717
983983 −22.2128 38.4737i −0.708479 1.22712i −0.965421 0.260695i 0.916048π-0.916048\pi
0.256942 0.966427i 0.417285π-0.417285\pi
984984 0 0
985985 −38.9902 + 67.5329i −1.24233 + 2.15178i
986986 45.6864 79.1313i 1.45495 2.52005i
987987 0 0
988988 −1.94738 3.37296i −0.0619544 0.107308i
989989 10.8171 0.343964
990990 0 0
991991 −41.6156 −1.32196 −0.660981 0.750403i 0.729860π-0.729860\pi
−0.660981 + 0.750403i 0.729860π0.729860\pi
992992 1.89557 + 3.28322i 0.0601844 + 0.104242i
993993 0 0
994994 0 0
995995 −18.2434 + 31.5985i −0.578354 + 1.00174i
996996 0 0
997997 −6.51407 11.2827i −0.206303 0.357327i 0.744244 0.667908i 0.232810π-0.232810\pi
−0.950547 + 0.310581i 0.899477π0.899477\pi
998998 96.1751 3.04437
999999 0 0
Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1323.2.f.g.442.5 12
3.2 odd 2 441.2.f.g.148.2 yes 12
7.2 even 3 1323.2.g.g.361.6 12
7.3 odd 6 1323.2.h.g.226.2 12
7.4 even 3 1323.2.h.g.226.1 12
7.5 odd 6 1323.2.g.g.361.5 12
7.6 odd 2 inner 1323.2.f.g.442.6 12
9.2 odd 6 441.2.f.g.295.2 yes 12
9.4 even 3 3969.2.a.bd.1.2 6
9.5 odd 6 3969.2.a.be.1.5 6
9.7 even 3 inner 1323.2.f.g.883.5 12
21.2 odd 6 441.2.g.g.67.1 12
21.5 even 6 441.2.g.g.67.2 12
21.11 odd 6 441.2.h.g.373.5 12
21.17 even 6 441.2.h.g.373.6 12
21.20 even 2 441.2.f.g.148.1 12
63.2 odd 6 441.2.h.g.214.5 12
63.11 odd 6 441.2.g.g.79.1 12
63.13 odd 6 3969.2.a.bd.1.1 6
63.16 even 3 1323.2.h.g.802.1 12
63.20 even 6 441.2.f.g.295.1 yes 12
63.25 even 3 1323.2.g.g.667.6 12
63.34 odd 6 inner 1323.2.f.g.883.6 12
63.38 even 6 441.2.g.g.79.2 12
63.41 even 6 3969.2.a.be.1.6 6
63.47 even 6 441.2.h.g.214.6 12
63.52 odd 6 1323.2.g.g.667.5 12
63.61 odd 6 1323.2.h.g.802.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.f.g.148.1 12 21.20 even 2
441.2.f.g.148.2 yes 12 3.2 odd 2
441.2.f.g.295.1 yes 12 63.20 even 6
441.2.f.g.295.2 yes 12 9.2 odd 6
441.2.g.g.67.1 12 21.2 odd 6
441.2.g.g.67.2 12 21.5 even 6
441.2.g.g.79.1 12 63.11 odd 6
441.2.g.g.79.2 12 63.38 even 6
441.2.h.g.214.5 12 63.2 odd 6
441.2.h.g.214.6 12 63.47 even 6
441.2.h.g.373.5 12 21.11 odd 6
441.2.h.g.373.6 12 21.17 even 6
1323.2.f.g.442.5 12 1.1 even 1 trivial
1323.2.f.g.442.6 12 7.6 odd 2 inner
1323.2.f.g.883.5 12 9.7 even 3 inner
1323.2.f.g.883.6 12 63.34 odd 6 inner
1323.2.g.g.361.5 12 7.5 odd 6
1323.2.g.g.361.6 12 7.2 even 3
1323.2.g.g.667.5 12 63.52 odd 6
1323.2.g.g.667.6 12 63.25 even 3
1323.2.h.g.226.1 12 7.4 even 3
1323.2.h.g.226.2 12 7.3 odd 6
1323.2.h.g.802.1 12 63.16 even 3
1323.2.h.g.802.2 12 63.61 odd 6
3969.2.a.bd.1.1 6 63.13 odd 6
3969.2.a.bd.1.2 6 9.4 even 3
3969.2.a.be.1.5 6 9.5 odd 6
3969.2.a.be.1.6 6 63.41 even 6