Properties

Label 162.4.c.i.109.1
Level 162162
Weight 44
Character 162.109
Analytic conductor 9.5589.558
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,4,Mod(55,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.55");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 162=234 162 = 2 \cdot 3^{4}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 162.c (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: 9.558309420939.55830942093
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q(ζ12)\Q(\zeta_{12})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x2+1 x^{4} - x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 33 3^{3}
Twist minimal: yes
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

Embedding invariants

Embedding label 109.1
Root 0.866025+0.500000i-0.866025 + 0.500000i of defining polynomial
Character χ\chi == 162.109
Dual form 162.4.c.i.55.1

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+(1.000001.73205i)q2+(2.00000+3.46410i)q4+(5.59808+9.69615i)q5+(9.1961515.9282i)q7+8.00000q8+22.3923q10+(11.7846+20.4115i)q11+(33.873158.6699i)q13+(18.3923+31.8564i)q14+(8.0000013.8564i)q16+117.158q17+110.315q19+(22.392338.7846i)q20+(23.569240.8231i)q22+(34.6077+59.9423i)q23+(0.1769150.306425i)q25135.492q26+73.5692q28+(99.1865171.796i)q29+(155.531269.387i)q31+(16.0000+27.7128i)q32+(117.158202.923i)q34+205.923q35206.608q37+(110.315191.072i)q38+(44.7846+77.5692i)q40+(66.3154114.862i)q41+(167.588+290.272i)q4394.2769q44+138.431q46+(189.531+328.277i)q47+(2.361564.09034i)q49+(0.353829+0.612850i)q50+(135.492+234.679i)q52+190.908q53263.885q55+(73.5692127.426i)q56+(198.373+343.592i)q58+(168.862292.477i)q59+(138.735240.295i)q61622.123q62+64.0000q64+(379.248+656.877i)q65+(332.535+575.967i)q67+(234.315+405.846i)q68+(205.923356.669i)q70528.431q7173.8306q73+(206.608+357.855i)q74+(220.631+382.144i)q76+(216.746375.415i)q77+(239.904+415.526i)q79+179.138q80265.261q82+(89.8846+155.685i)q83+(655.858+1135.98i)q85+(335.177580.543i)q86+(94.2769+163.292i)q88+846.458q891246.01q91+(138.431239.769i)q92+(379.061656.554i)q94+(617.554+1069.63i)q95+(336.492+582.822i)q979.44624q98+O(q100)q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-5.59808 + 9.69615i) q^{5} +(-9.19615 - 15.9282i) q^{7} +8.00000 q^{8} +22.3923 q^{10} +(11.7846 + 20.4115i) q^{11} +(33.8731 - 58.6699i) q^{13} +(-18.3923 + 31.8564i) q^{14} +(-8.00000 - 13.8564i) q^{16} +117.158 q^{17} +110.315 q^{19} +(-22.3923 - 38.7846i) q^{20} +(23.5692 - 40.8231i) q^{22} +(-34.6077 + 59.9423i) q^{23} +(-0.176915 - 0.306425i) q^{25} -135.492 q^{26} +73.5692 q^{28} +(-99.1865 - 171.796i) q^{29} +(155.531 - 269.387i) q^{31} +(-16.0000 + 27.7128i) q^{32} +(-117.158 - 202.923i) q^{34} +205.923 q^{35} -206.608 q^{37} +(-110.315 - 191.072i) q^{38} +(-44.7846 + 77.5692i) q^{40} +(66.3154 - 114.862i) q^{41} +(167.588 + 290.272i) q^{43} -94.2769 q^{44} +138.431 q^{46} +(189.531 + 328.277i) q^{47} +(2.36156 - 4.09034i) q^{49} +(-0.353829 + 0.612850i) q^{50} +(135.492 + 234.679i) q^{52} +190.908 q^{53} -263.885 q^{55} +(-73.5692 - 127.426i) q^{56} +(-198.373 + 343.592i) q^{58} +(168.862 - 292.477i) q^{59} +(-138.735 - 240.295i) q^{61} -622.123 q^{62} +64.0000 q^{64} +(379.248 + 656.877i) q^{65} +(-332.535 + 575.967i) q^{67} +(-234.315 + 405.846i) q^{68} +(-205.923 - 356.669i) q^{70} -528.431 q^{71} -73.8306 q^{73} +(206.608 + 357.855i) q^{74} +(-220.631 + 382.144i) q^{76} +(216.746 - 375.415i) q^{77} +(239.904 + 415.526i) q^{79} +179.138 q^{80} -265.261 q^{82} +(89.8846 + 155.685i) q^{83} +(-655.858 + 1135.98i) q^{85} +(335.177 - 580.543i) q^{86} +(94.2769 + 163.292i) q^{88} +846.458 q^{89} -1246.01 q^{91} +(-138.431 - 239.769i) q^{92} +(379.061 - 656.554i) q^{94} +(-617.554 + 1069.63i) q^{95} +(336.492 + 582.822i) q^{97} -9.44624 q^{98} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q4q28q412q516q7+32q8+48q1036q1110q1332q1432q16+240q1716q1948q2072q22180q23+124q25+40q26+1368q98+O(q100) 4 q - 4 q^{2} - 8 q^{4} - 12 q^{5} - 16 q^{7} + 32 q^{8} + 48 q^{10} - 36 q^{11} - 10 q^{13} - 32 q^{14} - 32 q^{16} + 240 q^{17} - 16 q^{19} - 48 q^{20} - 72 q^{22} - 180 q^{23} + 124 q^{25} + 40 q^{26}+ \cdots - 1368 q^{98}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/162Z)×\left(\mathbb{Z}/162\mathbb{Z}\right)^\times.

nn 8383
χ(n)\chi(n) e(13)e\left(\frac{1}{3}\right)

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 −1.00000 1.73205i −0.353553 0.612372i
33 0 0
44 −2.00000 + 3.46410i −0.250000 + 0.433013i
55 −5.59808 + 9.69615i −0.500707 + 0.867250i 0.499293 + 0.866433i 0.333593π0.333593\pi
−1.00000 0.000816748i 0.999740π0.999740\pi
66 0 0
77 −9.19615 15.9282i −0.496546 0.860042i 0.503447 0.864026i 0.332065π-0.332065\pi
−0.999992 + 0.00398426i 0.998732π0.998732\pi
88 8.00000 0.353553
99 0 0
1010 22.3923 0.708107
1111 11.7846 + 20.4115i 0.323018 + 0.559483i 0.981109 0.193455i 0.0619694π-0.0619694\pi
−0.658092 + 0.752938i 0.728636π0.728636\pi
1212 0 0
1313 33.8731 58.6699i 0.722669 1.25170i −0.237257 0.971447i 0.576248π-0.576248\pi
0.959926 0.280253i 0.0904183π-0.0904183\pi
1414 −18.3923 + 31.8564i −0.351111 + 0.608142i
1515 0 0
1616 −8.00000 13.8564i −0.125000 0.216506i
1717 117.158 1.67147 0.835733 0.549137i 0.185043π-0.185043\pi
0.835733 + 0.549137i 0.185043π0.185043\pi
1818 0 0
1919 110.315 1.33200 0.666002 0.745950i 0.268004π-0.268004\pi
0.666002 + 0.745950i 0.268004π0.268004\pi
2020 −22.3923 38.7846i −0.250354 0.433625i
2121 0 0
2222 23.5692 40.8231i 0.228408 0.395614i
2323 −34.6077 + 59.9423i −0.313748 + 0.543427i −0.979171 0.203039i 0.934918π-0.934918\pi
0.665423 + 0.746467i 0.268251π0.268251\pi
2424 0 0
2525 −0.176915 0.306425i −0.00141532 0.00245140i
2626 −135.492 −1.02201
2727 0 0
2828 73.5692 0.496546
2929 −99.1865 171.796i −0.635120 1.10006i −0.986490 0.163823i 0.947617π-0.947617\pi
0.351370 0.936237i 0.385716π-0.385716\pi
3030 0 0
3131 155.531 269.387i 0.901101 1.56075i 0.0750350 0.997181i 0.476093π-0.476093\pi
0.826066 0.563573i 0.190574π-0.190574\pi
3232 −16.0000 + 27.7128i −0.0883883 + 0.153093i
3333 0 0
3434 −117.158 202.923i −0.590952 1.02356i
3535 205.923 0.994496
3636 0 0
3737 −206.608 −0.918003 −0.459001 0.888436i 0.651793π-0.651793\pi
−0.459001 + 0.888436i 0.651793π0.651793\pi
3838 −110.315 191.072i −0.470935 0.815683i
3939 0 0
4040 −44.7846 + 77.5692i −0.177027 + 0.306619i
4141 66.3154 114.862i 0.252603 0.437521i −0.711639 0.702546i 0.752047π-0.752047\pi
0.964242 + 0.265025i 0.0853800π0.0853800\pi
4242 0 0
4343 167.588 + 290.272i 0.594349 + 1.02944i 0.993638 + 0.112618i 0.0359235π0.0359235\pi
−0.399290 + 0.916825i 0.630743π0.630743\pi
4444 −94.2769 −0.323018
4545 0 0
4646 138.431 0.443707
4747 189.531 + 328.277i 0.588211 + 1.01881i 0.994467 + 0.105051i 0.0335007π0.0335007\pi
−0.406256 + 0.913759i 0.633166π0.633166\pi
4848 0 0
4949 2.36156 4.09034i 0.00688502 0.0119252i
5050 −0.353829 + 0.612850i −0.00100078 + 0.00173340i
5151 0 0
5252 135.492 + 234.679i 0.361335 + 0.625850i
5353 190.908 0.494777 0.247388 0.968916i 0.420428π-0.420428\pi
0.247388 + 0.968916i 0.420428π0.420428\pi
5454 0 0
5555 −263.885 −0.646949
5656 −73.5692 127.426i −0.175555 0.304071i
5757 0 0
5858 −198.373 + 343.592i −0.449098 + 0.777860i
5959 168.862 292.477i 0.372609 0.645377i −0.617357 0.786683i 0.711797π-0.711797\pi
0.989966 + 0.141306i 0.0451301π0.0451301\pi
6060 0 0
6161 −138.735 240.295i −0.291199 0.504372i 0.682894 0.730517i 0.260721π-0.260721\pi
−0.974094 + 0.226145i 0.927388π0.927388\pi
6262 −622.123 −1.27435
6363 0 0
6464 64.0000 0.125000
6565 379.248 + 656.877i 0.723691 + 1.25347i
6666 0 0
6767 −332.535 + 575.967i −0.606352 + 1.05023i 0.385485 + 0.922714i 0.374034π0.374034\pi
−0.991836 + 0.127518i 0.959299π0.959299\pi
6868 −234.315 + 405.846i −0.417866 + 0.723766i
6969 0 0
7070 −205.923 356.669i −0.351607 0.609002i
7171 −528.431 −0.883284 −0.441642 0.897191i 0.645604π-0.645604\pi
−0.441642 + 0.897191i 0.645604π0.645604\pi
7272 0 0
7373 −73.8306 −0.118373 −0.0591865 0.998247i 0.518851π-0.518851\pi
−0.0591865 + 0.998247i 0.518851π0.518851\pi
7474 206.608 + 357.855i 0.324563 + 0.562159i
7575 0 0
7676 −220.631 + 382.144i −0.333001 + 0.576775i
7777 216.746 375.415i 0.320786 0.555617i
7878 0 0
7979 239.904 + 415.526i 0.341662 + 0.591776i 0.984741 0.174024i 0.0556769π-0.0556769\pi
−0.643080 + 0.765799i 0.722344π0.722344\pi
8080 179.138 0.250354
8181 0 0
8282 −265.261 −0.357234
8383 89.8846 + 155.685i 0.118869 + 0.205887i 0.919320 0.393512i 0.128740π-0.128740\pi
−0.800451 + 0.599398i 0.795407π0.795407\pi
8484 0 0
8585 −655.858 + 1135.98i −0.836915 + 1.44958i
8686 335.177 580.543i 0.420268 0.727926i
8787 0 0
8888 94.2769 + 163.292i 0.114204 + 0.197807i
8989 846.458 1.00814 0.504069 0.863663i 0.331836π-0.331836\pi
0.504069 + 0.863663i 0.331836π0.331836\pi
9090 0 0
9191 −1246.01 −1.43535
9292 −138.431 239.769i −0.156874 0.271714i
9393 0 0
9494 379.061 656.554i 0.415928 0.720408i
9595 −617.554 + 1069.63i −0.666944 + 1.15518i
9696 0 0
9797 336.492 + 582.822i 0.352223 + 0.610068i 0.986639 0.162924i 0.0520926π-0.0520926\pi
−0.634416 + 0.772992i 0.718759π0.718759\pi
9898 −9.44624 −0.00973689
9999 0 0
100100 1.41532 0.00141532
101101 −103.261 178.854i −0.101732 0.176204i 0.810667 0.585508i 0.199105π-0.199105\pi
−0.912398 + 0.409304i 0.865772π0.865772\pi
102102 0 0
103103 685.885 1187.99i 0.656138 1.13646i −0.325469 0.945553i 0.605522π-0.605522\pi
0.981607 0.190912i 0.0611444π-0.0611444\pi
104104 270.985 469.359i 0.255502 0.442543i
105105 0 0
106106 −190.908 330.662i −0.174930 0.302988i
107107 −1267.00 −1.14472 −0.572362 0.820001i 0.693973π-0.693973\pi
−0.572362 + 0.820001i 0.693973π0.693973\pi
108108 0 0
109109 1725.13 1.51594 0.757970 0.652289i 0.226191π-0.226191\pi
0.757970 + 0.652289i 0.226191π0.226191\pi
110110 263.885 + 457.061i 0.228731 + 0.396174i
111111 0 0
112112 −147.138 + 254.851i −0.124136 + 0.215011i
113113 870.098 1507.05i 0.724353 1.25462i −0.234886 0.972023i 0.575472π-0.575472\pi
0.959240 0.282594i 0.0911949π-0.0911949\pi
114114 0 0
115115 −387.473 671.123i −0.314192 0.544196i
116116 793.492 0.635120
117117 0 0
118118 −675.446 −0.526948
119119 −1077.40 1866.11i −0.829959 1.43753i
120120 0 0
121121 387.746 671.596i 0.291319 0.504580i
122122 −277.469 + 480.591i −0.205909 + 0.356645i
123123 0 0
124124 622.123 + 1077.55i 0.450551 + 0.780377i
125125 −1395.56 −0.998580
126126 0 0
127127 492.131 0.343855 0.171927 0.985110i 0.445001π-0.445001\pi
0.171927 + 0.985110i 0.445001π0.445001\pi
128128 −64.0000 110.851i −0.0441942 0.0765466i
129129 0 0
130130 758.496 1313.75i 0.511727 0.886337i
131131 −959.638 + 1662.14i −0.640030 + 1.10857i 0.345395 + 0.938457i 0.387745π0.387745\pi
−0.985425 + 0.170108i 0.945588π0.945588\pi
132132 0 0
133133 −1014.48 1757.13i −0.661401 1.14558i
134134 1330.14 0.857511
135135 0 0
136136 937.261 0.590952
137137 −1311.43 2271.46i −0.817832 1.41653i −0.907277 0.420534i 0.861843π-0.861843\pi
0.0894453 0.995992i 0.471491π-0.471491\pi
138138 0 0
139139 314.284 544.357i 0.191779 0.332171i −0.754061 0.656804i 0.771908π-0.771908\pi
0.945840 + 0.324634i 0.105241π0.105241\pi
140140 −411.846 + 713.338i −0.248624 + 0.430629i
141141 0 0
142142 528.431 + 915.269i 0.312288 + 0.540899i
143143 1596.72 0.933739
144144 0 0
145145 2221.02 1.27204
146146 73.8306 + 127.878i 0.0418511 + 0.0724883i
147147 0 0
148148 413.215 715.710i 0.229501 0.397507i
149149 −284.156 + 492.173i −0.156235 + 0.270606i −0.933508 0.358557i 0.883269π-0.883269\pi
0.777273 + 0.629163i 0.216602π0.216602\pi
150150 0 0
151151 −178.715 309.544i −0.0963155 0.166823i 0.813841 0.581087i 0.197372π-0.197372\pi
−0.910157 + 0.414264i 0.864039π0.864039\pi
152152 882.523 0.470935
153153 0 0
154154 −866.985 −0.453660
155155 1741.35 + 3016.10i 0.902376 + 1.56296i
156156 0 0
157157 363.627 629.820i 0.184844 0.320160i −0.758680 0.651464i 0.774155π-0.774155\pi
0.943524 + 0.331304i 0.107489π0.107489\pi
158158 479.808 831.051i 0.241591 0.418449i
159159 0 0
160160 −179.138 310.277i −0.0885134 0.153310i
161161 1273.03 0.623161
162162 0 0
163163 −396.554 −0.190555 −0.0952776 0.995451i 0.530374π-0.530374\pi
−0.0952776 + 0.995451i 0.530374π0.530374\pi
164164 265.261 + 459.446i 0.126301 + 0.218761i
165165 0 0
166166 179.769 311.369i 0.0840530 0.145584i
167167 −1589.26 + 2752.68i −0.736411 + 1.27550i 0.217690 + 0.976018i 0.430148π0.430148\pi
−0.954101 + 0.299484i 0.903185π0.903185\pi
168168 0 0
169169 −1196.27 2072.00i −0.544501 0.943104i
170170 2623.43 1.18358
171171 0 0
172172 −1340.71 −0.594349
173173 −1076.32 1864.25i −0.473014 0.819285i 0.526509 0.850170i 0.323501π-0.323501\pi
−0.999523 + 0.0308850i 0.990167π0.990167\pi
174174 0 0
175175 −3.25387 + 5.63586i −0.00140554 + 0.00243446i
176176 188.554 326.585i 0.0807544 0.139871i
177177 0 0
178178 −846.458 1466.11i −0.356431 0.617356i
179179 −4490.29 −1.87497 −0.937487 0.348022i 0.886854π-0.886854\pi
−0.937487 + 0.348022i 0.886854π0.886854\pi
180180 0 0
181181 1407.32 0.577931 0.288966 0.957340i 0.406689π-0.406689\pi
0.288966 + 0.957340i 0.406689π0.406689\pi
182182 1246.01 + 2158.15i 0.507474 + 0.878970i
183183 0 0
184184 −276.862 + 479.538i −0.110927 + 0.192131i
185185 1156.61 2003.30i 0.459650 0.796138i
186186 0 0
187187 1380.66 + 2391.37i 0.539913 + 0.935156i
188188 −1516.25 −0.588211
189189 0 0
190190 2470.22 0.943201
191191 −386.138 668.811i −0.146283 0.253369i 0.783568 0.621306i 0.213398π-0.213398\pi
−0.929851 + 0.367937i 0.880064π0.880064\pi
192192 0 0
193193 −1826.34 + 3163.31i −0.681154 + 1.17979i 0.293475 + 0.955967i 0.405188π0.405188\pi
−0.974629 + 0.223826i 0.928145π0.928145\pi
194194 672.985 1165.64i 0.249059 0.431383i
195195 0 0
196196 9.44624 + 16.3614i 0.00344251 + 0.00596260i
197197 2647.40 0.957460 0.478730 0.877962i 0.341097π-0.341097\pi
0.478730 + 0.877962i 0.341097π0.341097\pi
198198 0 0
199199 1470.22 0.523723 0.261861 0.965106i 0.415664π-0.415664\pi
0.261861 + 0.965106i 0.415664π0.415664\pi
200200 −1.41532 2.45140i −0.000500390 0.000866701i
201201 0 0
202202 −206.523 + 357.708i −0.0719351 + 0.124595i
203203 −1824.27 + 3159.73i −0.630732 + 1.09246i
204204 0 0
205205 742.477 + 1286.01i 0.252960 + 0.438140i
206206 −2743.54 −0.927919
207207 0 0
208208 −1083.94 −0.361335
209209 1300.02 + 2251.71i 0.430261 + 0.745233i
210210 0 0
211211 −768.003 + 1330.22i −0.250576 + 0.434010i −0.963685 0.267043i 0.913953π-0.913953\pi
0.713109 + 0.701054i 0.247287π0.247287\pi
212212 −381.815 + 661.323i −0.123694 + 0.214245i
213213 0 0
214214 1267.00 + 2194.51i 0.404721 + 0.700997i
215215 −3752.69 −1.19038
216216 0 0
217217 −5721.14 −1.78975
218218 −1725.13 2988.01i −0.535966 0.928320i
219219 0 0
220220 527.769 914.123i 0.161737 0.280137i
221221 3968.49 6873.63i 1.20792 2.09217i
222222 0 0
223223 828.765 + 1435.46i 0.248871 + 0.431057i 0.963213 0.268740i 0.0866071π-0.0866071\pi
−0.714342 + 0.699797i 0.753274π0.753274\pi
224224 588.554 0.175555
225225 0 0
226226 −3480.39 −1.02439
227227 757.131 + 1311.39i 0.221377 + 0.383436i 0.955226 0.295876i 0.0956116π-0.0956116\pi
−0.733850 + 0.679312i 0.762278π0.762278\pi
228228 0 0
229229 −2149.52 + 3723.08i −0.620280 + 1.07436i 0.369153 + 0.929369i 0.379648π0.379648\pi
−0.989433 + 0.144988i 0.953685π0.953685\pi
230230 −774.946 + 1342.25i −0.222167 + 0.384805i
231231 0 0
232232 −793.492 1374.37i −0.224549 0.388930i
233233 1336.78 0.375860 0.187930 0.982182i 0.439822π-0.439822\pi
0.187930 + 0.982182i 0.439822π0.439822\pi
234234 0 0
235235 −4244.03 −1.17809
236236 675.446 + 1169.91i 0.186304 + 0.322688i
237237 0 0
238238 −2154.80 + 3732.22i −0.586869 + 1.01649i
239239 3439.31 5957.07i 0.930840 1.61226i 0.148951 0.988845i 0.452410π-0.452410\pi
0.781889 0.623418i 0.214256π-0.214256\pi
240240 0 0
241241 −765.646 1326.14i −0.204646 0.354457i 0.745374 0.666646i 0.232271π-0.232271\pi
−0.950020 + 0.312190i 0.898938π0.898938\pi
242242 −1550.98 −0.411988
243243 0 0
244244 1109.88 0.291199
245245 26.4404 + 45.7961i 0.00689476 + 0.0119421i
246246 0 0
247247 3736.72 6472.19i 0.962598 1.66727i
248248 1244.25 2155.10i 0.318587 0.551810i
249249 0 0
250250 1395.56 + 2417.18i 0.353051 + 0.611503i
251251 1181.60 0.297139 0.148570 0.988902i 0.452533π-0.452533\pi
0.148570 + 0.988902i 0.452533π0.452533\pi
252252 0 0
253253 −1631.35 −0.405384
254254 −492.131 852.395i −0.121571 0.210567i
255255 0 0
256256 −128.000 + 221.703i −0.0312500 + 0.0541266i
257257 −2190.68 + 3794.36i −0.531714 + 0.920956i 0.467600 + 0.883940i 0.345119π0.345119\pi
−0.999315 + 0.0370161i 0.988215π0.988215\pi
258258 0 0
259259 1900.00 + 3290.89i 0.455830 + 0.789521i
260260 −3033.98 −0.723691
261261 0 0
262262 3838.55 0.905140
263263 1400.04 + 2424.94i 0.328251 + 0.568548i 0.982165 0.188021i 0.0602074π-0.0602074\pi
−0.653914 + 0.756569i 0.726874π0.726874\pi
264264 0 0
265265 −1068.72 + 1851.07i −0.247738 + 0.429095i
266266 −2028.95 + 3514.25i −0.467681 + 0.810047i
267267 0 0
268268 −1330.14 2303.87i −0.303176 0.525116i
269269 2803.73 0.635489 0.317745 0.948176i 0.397075π-0.397075\pi
0.317745 + 0.948176i 0.397075π0.397075\pi
270270 0 0
271271 −6332.36 −1.41942 −0.709711 0.704493i 0.751175π-0.751175\pi
−0.709711 + 0.704493i 0.751175π0.751175\pi
272272 −937.261 1623.38i −0.208933 0.361883i
273273 0 0
274274 −2622.86 + 4542.92i −0.578294 + 1.00163i
275275 4.16974 7.22220i 0.000914344 0.00158369i
276276 0 0
277277 −463.831 803.378i −0.100610 0.174261i 0.811326 0.584594i 0.198746π-0.198746\pi
−0.911936 + 0.410332i 0.865413π0.865413\pi
278278 −1257.14 −0.271216
279279 0 0
280280 1647.38 0.351607
281281 −570.764 988.592i −0.121170 0.209873i 0.799059 0.601253i 0.205331π-0.205331\pi
−0.920230 + 0.391379i 0.871998π0.871998\pi
282282 0 0
283283 −1805.52 + 3127.26i −0.379248 + 0.656877i −0.990953 0.134209i 0.957151π-0.957151\pi
0.611705 + 0.791086i 0.290484π0.290484\pi
284284 1056.86 1830.54i 0.220821 0.382473i
285285 0 0
286286 −1596.72 2765.61i −0.330127 0.571796i
287287 −2439.38 −0.501715
288288 0 0
289289 8812.92 1.79380
290290 −2221.02 3846.91i −0.449733 0.778960i
291291 0 0
292292 147.661 255.757i 0.0295932 0.0512570i
293293 −1162.11 + 2012.83i −0.231710 + 0.401333i −0.958311 0.285726i 0.907765π-0.907765\pi
0.726602 + 0.687059i 0.241099π0.241099\pi
294294 0 0
295295 1890.60 + 3274.61i 0.373136 + 0.646290i
296296 −1652.86 −0.324563
297297 0 0
298298 1136.62 0.220949
299299 2344.54 + 4060.86i 0.453472 + 0.785436i
300300 0 0
301301 3082.34 5338.77i 0.590243 1.02233i
302302 −357.430 + 619.088i −0.0681053 + 0.117962i
303303 0 0
304304 −882.523 1528.57i −0.166501 0.288387i
305305 3106.59 0.583222
306306 0 0
307307 6968.51 1.29548 0.647742 0.761860i 0.275713π-0.275713\pi
0.647742 + 0.761860i 0.275713π0.275713\pi
308308 866.985 + 1501.66i 0.160393 + 0.277809i
309309 0 0
310310 3482.69 6032.20i 0.638076 1.10518i
311311 −3170.15 + 5490.87i −0.578016 + 1.00115i 0.417691 + 0.908589i 0.362840π0.362840\pi
−0.995707 + 0.0925637i 0.970494π0.970494\pi
312312 0 0
313313 −1194.42 2068.79i −0.215694 0.373594i 0.737793 0.675027i 0.235868π-0.235868\pi
−0.953487 + 0.301434i 0.902535π0.902535\pi
314314 −1454.51 −0.261409
315315 0 0
316316 −1919.23 −0.341662
317317 2930.63 + 5076.00i 0.519245 + 0.899359i 0.999750 + 0.0223668i 0.00712015π0.00712015\pi
−0.480505 + 0.876992i 0.659547π0.659547\pi
318318 0 0
319319 2337.75 4049.10i 0.410310 0.710677i
320320 −358.277 + 620.554i −0.0625884 + 0.108406i
321321 0 0
322322 −1273.03 2204.95i −0.220321 0.381606i
323323 12924.3 2.22640
324324 0 0
325325 −23.9706 −0.00409122
326326 396.554 + 686.851i 0.0673714 + 0.116691i
327327 0 0
328328 530.523 918.892i 0.0893086 0.154687i
329329 3485.91 6037.77i 0.584147 1.01177i
330330 0 0
331331 −2482.48 4299.78i −0.412234 0.714010i 0.582900 0.812544i 0.301918π-0.301918\pi
−0.995134 + 0.0985339i 0.968585π0.968585\pi
332332 −719.077 −0.118869
333333 0 0
334334 6357.04 1.04144
335335 −3723.11 6448.61i −0.607209 1.05172i
336336 0 0
337337 1548.83 2682.65i 0.250357 0.433630i −0.713267 0.700892i 0.752785π-0.752785\pi
0.963624 + 0.267262i 0.0861188π0.0861188\pi
338338 −2392.54 + 4144.00i −0.385021 + 0.666875i
339339 0 0
340340 −2623.43 4543.91i −0.418457 0.724789i
341341 7331.48 1.16429
342342 0 0
343343 −6395.43 −1.00677
344344 1340.71 + 2322.17i 0.210134 + 0.363963i
345345 0 0
346346 −2152.65 + 3728.50i −0.334472 + 0.579322i
347347 −4022.10 + 6966.48i −0.622241 + 1.07775i 0.366827 + 0.930289i 0.380444π0.380444\pi
−0.989067 + 0.147464i 0.952889π0.952889\pi
348348 0 0
349349 572.353 + 991.345i 0.0877862 + 0.152050i 0.906575 0.422045i 0.138687π-0.138687\pi
−0.818789 + 0.574095i 0.805354π0.805354\pi
350350 13.0155 0.00198773
351351 0 0
352352 −754.215 −0.114204
353353 −3900.63 6756.09i −0.588129 1.01867i −0.994477 0.104951i 0.966531π-0.966531\pi
0.406348 0.913718i 0.366802π-0.366802\pi
354354 0 0
355355 2958.20 5123.75i 0.442267 0.766029i
356356 −1692.92 + 2932.22i −0.252035 + 0.436537i
357357 0 0
358358 4490.29 + 7777.41i 0.662903 + 1.14818i
359359 341.307 0.0501769 0.0250885 0.999685i 0.492013π-0.492013\pi
0.0250885 + 0.999685i 0.492013π0.492013\pi
360360 0 0
361361 5310.48 0.774235
362362 −1407.32 2437.56i −0.204329 0.353909i
363363 0 0
364364 2492.02 4316.30i 0.358838 0.621526i
365365 413.310 715.873i 0.0592702 0.102659i
366366 0 0
367367 −59.6839 103.376i −0.00848903 0.0147034i 0.861750 0.507334i 0.169369π-0.169369\pi
−0.870239 + 0.492630i 0.836036π0.836036\pi
368368 1107.45 0.156874
369369 0 0
370370 −4626.42 −0.650044
371371 −1755.62 3040.81i −0.245679 0.425529i
372372 0 0
373373 2187.22 3788.37i 0.303619 0.525883i −0.673334 0.739338i 0.735139π-0.735139\pi
0.976953 + 0.213456i 0.0684718π0.0684718\pi
374374 2761.31 4782.74i 0.381776 0.661255i
375375 0 0
376376 1516.25 + 2626.22i 0.207964 + 0.360204i
377377 −13439.0 −1.83593
378378 0 0
379379 8949.46 1.21294 0.606468 0.795108i 0.292586π-0.292586\pi
0.606468 + 0.795108i 0.292586π0.292586\pi
380380 −2470.22 4278.54i −0.333472 0.577590i
381381 0 0
382382 −772.277 + 1337.62i −0.103437 + 0.179159i
383383 −103.232 + 178.802i −0.0137726 + 0.0238548i −0.872830 0.488025i 0.837717π-0.837717\pi
0.859057 + 0.511880i 0.171051π0.171051\pi
384384 0 0
385385 2426.72 + 4203.21i 0.321240 + 0.556403i
386386 7305.35 0.963297
387387 0 0
388388 −2691.94 −0.352223
389389 1014.47 + 1757.11i 0.132225 + 0.229021i 0.924534 0.381099i 0.124454π-0.124454\pi
−0.792309 + 0.610120i 0.791121π0.791121\pi
390390 0 0
391391 −4054.56 + 7022.70i −0.524419 + 0.908320i
392392 18.8925 32.7228i 0.00243422 0.00421620i
393393 0 0
394394 −2647.40 4585.44i −0.338513 0.586322i
395395 −5372.00 −0.684290
396396 0 0
397397 −6646.07 −0.840193 −0.420097 0.907479i 0.638004π-0.638004\pi
−0.420097 + 0.907479i 0.638004π0.638004\pi
398398 −1470.22 2546.49i −0.185164 0.320713i
399399 0 0
400400 −2.83063 + 4.90280i −0.000353829 + 0.000612850i
401401 0.817240 1.41550i 0.000101773 0.000176276i −0.865975 0.500088i 0.833301π-0.833301\pi
0.866076 + 0.499912i 0.166634π0.166634\pi
402402 0 0
403403 −10536.6 18249.9i −1.30240 2.25582i
404404 826.091 0.101732
405405 0 0
406406 7297.08 0.891990
407407 −2434.79 4217.18i −0.296531 0.513607i
408408 0 0
409409 3101.68 5372.26i 0.374983 0.649490i −0.615342 0.788261i 0.710982π-0.710982\pi
0.990324 + 0.138771i 0.0443152π0.0443152\pi
410410 1484.95 2572.02i 0.178870 0.309812i
411411 0 0
412412 2743.54 + 4751.95i 0.328069 + 0.568232i
413413 −6211.51 −0.740068
414414 0 0
415415 −2012.72 −0.238074
416416 1083.94 + 1877.44i 0.127751 + 0.221271i
417417 0 0
418418 2600.05 4503.41i 0.304240 0.526960i
419419 4875.18 8444.07i 0.568421 0.984534i −0.428302 0.903636i 0.640888π-0.640888\pi
0.996722 0.0808979i 0.0257788π-0.0257788\pi
420420 0 0
421421 5030.69 + 8713.41i 0.582377 + 1.00871i 0.995197 + 0.0978939i 0.0312106π0.0312106\pi
−0.412820 + 0.910813i 0.635456π0.635456\pi
422422 3072.01 0.354368
423423 0 0
424424 1527.26 0.174930
425425 −20.7269 35.9000i −0.00236565 0.00409743i
426426 0 0
427427 −2551.65 + 4419.59i −0.289187 + 0.500887i
428428 2534.00 4389.02i 0.286181 0.495680i
429429 0 0
430430 3752.69 + 6499.85i 0.420862 + 0.728955i
431431 −6763.21 −0.755853 −0.377926 0.925836i 0.623363π-0.623363\pi
−0.377926 + 0.925836i 0.623363π0.623363\pi
432432 0 0
433433 −10601.4 −1.17660 −0.588302 0.808641i 0.700204π-0.700204\pi
−0.588302 + 0.808641i 0.700204π0.700204\pi
434434 5721.14 + 9909.30i 0.632773 + 1.09599i
435435 0 0
436436 −3450.26 + 5976.03i −0.378985 + 0.656422i
437437 −3817.76 + 6612.55i −0.417914 + 0.723848i
438438 0 0
439439 6284.46 + 10885.0i 0.683237 + 1.18340i 0.973987 + 0.226602i 0.0727616π0.0727616\pi
−0.290751 + 0.956799i 0.593905π0.593905\pi
440440 −2111.08 −0.228731
441441 0 0
442442 −15874.0 −1.70825
443443 5127.71 + 8881.46i 0.549944 + 0.952530i 0.998278 + 0.0586643i 0.0186842π0.0186842\pi
−0.448334 + 0.893866i 0.647983π0.647983\pi
444444 0 0
445445 −4738.53 + 8207.38i −0.504782 + 0.874308i
446446 1657.53 2870.93i 0.175978 0.304803i
447447 0 0
448448 −588.554 1019.41i −0.0620682 0.107505i
449449 −4080.23 −0.428860 −0.214430 0.976739i 0.568789π-0.568789\pi
−0.214430 + 0.976739i 0.568789π0.568789\pi
450450 0 0
451451 3126.00 0.326381
452452 3480.39 + 6028.22i 0.362177 + 0.627308i
453453 0 0
454454 1514.26 2622.78i 0.156537 0.271130i
455455 6975.25 12081.5i 0.718691 1.24481i
456456 0 0
457457 1091.60 + 1890.71i 0.111735 + 0.193531i 0.916470 0.400104i 0.131026π-0.131026\pi
−0.804735 + 0.593634i 0.797693π0.797693\pi
458458 8598.08 0.877209
459459 0 0
460460 3099.78 0.314192
461461 −1625.13 2814.81i −0.164186 0.284379i 0.772180 0.635404i 0.219166π-0.219166\pi
−0.936366 + 0.351025i 0.885833π0.885833\pi
462462 0 0
463463 −9495.57 + 16446.8i −0.953124 + 1.65086i −0.214520 + 0.976720i 0.568819π0.568819\pi
−0.738604 + 0.674140i 0.764515π0.764515\pi
464464 −1586.98 + 2748.74i −0.158780 + 0.275015i
465465 0 0
466466 −1336.78 2315.37i −0.132887 0.230166i
467467 6906.52 0.684359 0.342180 0.939635i 0.388835π-0.388835\pi
0.342180 + 0.939635i 0.388835π0.388835\pi
468468 0 0
469469 12232.2 1.20432
470470 4244.03 + 7350.88i 0.416516 + 0.721427i
471471 0 0
472472 1350.89 2339.81i 0.131737 0.228175i
473473 −3949.93 + 6841.48i −0.383970 + 0.665056i
474474 0 0
475475 −19.5164 33.8034i −0.00188521 0.00326527i
476476 8619.20 0.829959
477477 0 0
478478 −13757.3 −1.31641
479479 −3690.14 6391.50i −0.351997 0.609677i 0.634602 0.772839i 0.281164π-0.281164\pi
−0.986599 + 0.163162i 0.947831π0.947831\pi
480480 0 0
481481 −6998.44 + 12121.6i −0.663412 + 1.14906i
482482 −1531.29 + 2652.28i −0.144706 + 0.250639i
483483 0 0
484484 1550.98 + 2686.38i 0.145660 + 0.252290i
485485 −7534.84 −0.705442
486486 0 0
487487 −8756.51 −0.814774 −0.407387 0.913256i 0.633560π-0.633560\pi
−0.407387 + 0.913256i 0.633560π0.633560\pi
488488 −1109.88 1922.36i −0.102954 0.178322i
489489 0 0
490490 52.8808 91.5922i 0.00487533 0.00844432i
491491 −5918.97 + 10252.0i −0.544031 + 0.942290i 0.454636 + 0.890677i 0.349769π0.349769\pi
−0.998667 + 0.0516124i 0.983564π0.983564\pi
492492 0 0
493493 −11620.5 20127.2i −1.06158 1.83871i
494494 −14946.9 −1.36132
495495 0 0
496496 −4976.98 −0.450551
497497 4859.53 + 8416.95i 0.438591 + 0.759662i
498498 0 0
499499 −4289.19 + 7429.09i −0.384791 + 0.666477i −0.991740 0.128264i 0.959060π-0.959060\pi
0.606950 + 0.794740i 0.292393π0.292393\pi
500500 2791.12 4834.35i 0.249645 0.432398i
501501 0 0
502502 −1181.60 2046.59i −0.105055 0.181960i
503503 −3611.17 −0.320107 −0.160054 0.987108i 0.551167π-0.551167\pi
−0.160054 + 0.987108i 0.551167π0.551167\pi
504504 0 0
505505 2312.26 0.203751
506506 1631.35 + 2825.59i 0.143325 + 0.248246i
507507 0 0
508508 −984.261 + 1704.79i −0.0859636 + 0.148893i
509509 −4529.87 + 7845.96i −0.394465 + 0.683234i −0.993033 0.117838i 0.962404π-0.962404\pi
0.598567 + 0.801072i 0.295737π0.295737\pi
510510 0 0
511511 678.958 + 1175.99i 0.0587775 + 0.101806i
512512 512.000 0.0441942
513513 0 0
514514 8762.70 0.751957
515515 7679.27 + 13300.9i 0.657066 + 1.13807i
516516 0 0
517517 −4467.09 + 7737.23i −0.380005 + 0.658188i
518518 3799.99 6581.78i 0.322321 0.558276i
519519 0 0
520520 3033.98 + 5255.01i 0.255863 + 0.443169i
521521 −12834.1 −1.07922 −0.539609 0.841916i 0.681428π-0.681428\pi
−0.539609 + 0.841916i 0.681428π0.681428\pi
522522 0 0
523523 7061.21 0.590373 0.295187 0.955440i 0.404618π-0.404618\pi
0.295187 + 0.955440i 0.404618π0.404618\pi
524524 −3838.55 6648.57i −0.320015 0.554283i
525525 0 0
526526 2800.08 4849.88i 0.232109 0.402024i
527527 18221.6 31560.8i 1.50616 2.60875i
528528 0 0
529529 3688.11 + 6388.00i 0.303124 + 0.525027i
530530 4274.86 0.350355
531531 0 0
532532 8115.81 0.661401
533533 −4492.61 7781.43i −0.365097 0.632366i
534534 0 0
535535 7092.76 12285.0i 0.573172 0.992762i
536536 −2660.28 + 4607.73i −0.214378 + 0.371313i
537537 0 0
538538 −2803.73 4856.21i −0.224679 0.389156i
539539 111.320 0.00889593
540540 0 0
541541 −21645.6 −1.72018 −0.860091 0.510141i 0.829593π-0.829593\pi
−0.860091 + 0.510141i 0.829593π0.829593\pi
542542 6332.36 + 10968.0i 0.501842 + 0.869215i
543543 0 0
544544 −1874.52 + 3246.77i −0.147738 + 0.255890i
545545 −9657.41 + 16727.1i −0.759042 + 1.31470i
546546 0 0
547547 −4253.52 7367.32i −0.332482 0.575875i 0.650516 0.759492i 0.274553π-0.274553\pi
−0.982998 + 0.183617i 0.941219π0.941219\pi
548548 10491.4 0.817832
549549 0 0
550550 −16.6790 −0.00129308
551551 −10941.8 18951.7i −0.845982 1.46528i
552552 0 0
553553 4412.38 7642.47i 0.339301 0.587687i
554554 −927.661 + 1606.76i −0.0711418 + 0.123221i
555555 0 0
556556 1257.14 + 2177.43i 0.0958894 + 0.166085i
557557 −636.486 −0.0484179 −0.0242090 0.999707i 0.507707π-0.507707\pi
−0.0242090 + 0.999707i 0.507707π0.507707\pi
558558 0 0
559559 22706.9 1.71807
560560 −1647.38 2853.35i −0.124312 0.215315i
561561 0 0
562562 −1141.53 + 1977.18i −0.0856805 + 0.148403i
563563 −6781.31 + 11745.6i −0.507635 + 0.879249i 0.492326 + 0.870411i 0.336147π0.336147\pi
−0.999961 + 0.00883838i 0.997187π0.997187\pi
564564 0 0
565565 9741.75 + 16873.2i 0.725378 + 1.25639i
566566 7222.09 0.536338
567567 0 0
568568 −4227.45 −0.312288
569569 10558.4 + 18287.6i 0.777909 + 1.34738i 0.933145 + 0.359500i 0.117053π0.117053\pi
−0.155236 + 0.987877i 0.549614π0.549614\pi
570570 0 0
571571 7263.55 12580.8i 0.532347 0.922052i −0.466940 0.884289i 0.654643π-0.654643\pi
0.999287 0.0377630i 0.0120232π-0.0120232\pi
572572 −3193.45 + 5531.21i −0.233435 + 0.404321i
573573 0 0
574574 2439.38 + 4225.14i 0.177383 + 0.307237i
575575 24.4904 0.00177621
576576 0 0
577577 −14590.9 −1.05273 −0.526366 0.850258i 0.676446π-0.676446\pi
−0.526366 + 0.850258i 0.676446π0.676446\pi
578578 −8812.92 15264.4i −0.634203 1.09847i
579579 0 0
580580 −4442.03 + 7693.82i −0.318009 + 0.550808i
581581 1653.18 2863.40i 0.118048 0.204464i
582582 0 0
583583 2249.77 + 3896.72i 0.159822 + 0.276819i
584584 −590.645 −0.0418511
585585 0 0
586586 4648.42 0.327687
587587 −9170.74 15884.2i −0.644833 1.11688i −0.984340 0.176280i 0.943594π-0.943594\pi
0.339507 0.940603i 0.389740π-0.389740\pi
588588 0 0
589589 17157.4 29717.5i 1.20027 2.07893i
590590 3781.20 6549.23i 0.263847 0.456996i
591591 0 0
592592 1652.86 + 2862.84i 0.114750 + 0.198753i
593593 28380.4 1.96534 0.982668 0.185376i 0.0593504π-0.0593504\pi
0.982668 + 0.185376i 0.0593504π0.0593504\pi
594594 0 0
595595 24125.5 1.66227
596596 −1136.62 1968.69i −0.0781173 0.135303i
597597 0 0
598598 4689.08 8121.72i 0.320653 0.555387i
599599 11940.6 20681.7i 0.814487 1.41073i −0.0952079 0.995457i 0.530352π-0.530352\pi
0.909695 0.415276i 0.136315π-0.136315\pi
600600 0 0
601601 −8310.38 14394.0i −0.564039 0.976944i −0.997138 0.0755978i 0.975914π-0.975914\pi
0.433100 0.901346i 0.357420π-0.357420\pi
602602 −12329.4 −0.834729
603603 0 0
604604 1429.72 0.0963155
605605 4341.26 + 7519.29i 0.291731 + 0.505293i
606606 0 0
607607 785.389 1360.33i 0.0525172 0.0909624i −0.838572 0.544791i 0.816609π-0.816609\pi
0.891089 + 0.453829i 0.149942π0.149942\pi
608608 −1765.05 + 3057.15i −0.117734 + 0.203921i
609609 0 0
610610 −3106.59 5380.77i −0.206200 0.357149i
611611 25680.0 1.70033
612612 0 0
613613 −5766.40 −0.379939 −0.189969 0.981790i 0.560839π-0.560839\pi
−0.189969 + 0.981790i 0.560839π0.560839\pi
614614 −6968.51 12069.8i −0.458023 0.793319i
615615 0 0
616616 1733.97 3003.32i 0.113415 0.196440i
617617 −3481.28 + 6029.76i −0.227149 + 0.393434i −0.956962 0.290213i 0.906274π-0.906274\pi
0.729813 + 0.683647i 0.239607π0.239607\pi
618618 0 0
619619 −1330.21 2303.99i −0.0863741 0.149604i 0.819602 0.572934i 0.194195π-0.194195\pi
−0.905976 + 0.423329i 0.860861π0.860861\pi
620620 −13930.8 −0.902376
621621 0 0
622622 12680.6 0.817438
623623 −7784.15 13482.5i −0.500587 0.867042i
624624 0 0
625625 7834.55 13569.8i 0.501411 0.868470i
626626 −2388.83 + 4137.58i −0.152519 + 0.264171i
627627 0 0
628628 1454.51 + 2519.28i 0.0924222 + 0.160080i
629629 −24205.7 −1.53441
630630 0 0
631631 −20432.5 −1.28908 −0.644538 0.764573i 0.722950π-0.722950\pi
−0.644538 + 0.764573i 0.722950π0.722950\pi
632632 1919.23 + 3324.20i 0.120796 + 0.209224i
633633 0 0
634634 5861.26 10152.0i 0.367162 0.635943i
635635 −2754.98 + 4771.77i −0.172170 + 0.298208i
636636 0 0
637637 −159.987 277.105i −0.00995118 0.0172359i
638638 −9351.00 −0.580266
639639 0 0
640640 1433.11 0.0885134
641641 −13034.2 22575.8i −0.803149 1.39109i −0.917534 0.397658i 0.869823π-0.869823\pi
0.114385 0.993436i 0.463510π-0.463510\pi
642642 0 0
643643 −10585.2 + 18334.1i −0.649207 + 1.12446i 0.334106 + 0.942535i 0.391566π0.391566\pi
−0.983313 + 0.181923i 0.941768π0.941768\pi
644644 −2546.06 + 4409.91i −0.155790 + 0.269836i
645645 0 0
646646 −12924.3 22385.5i −0.787151 1.36339i
647647 8291.45 0.503818 0.251909 0.967751i 0.418942π-0.418942\pi
0.251909 + 0.967751i 0.418942π0.418942\pi
648648 0 0
649649 7959.87 0.481436
650650 23.9706 + 41.5182i 0.00144647 + 0.00250535i
651651 0 0
652652 793.108 1373.70i 0.0476388 0.0825128i
653653 −12342.5 + 21377.8i −0.739662 + 1.28113i 0.212985 + 0.977055i 0.431681π0.431681\pi
−0.952648 + 0.304077i 0.901652π0.901652\pi
654654 0 0
655655 −10744.3 18609.6i −0.640936 1.11013i
656656 −2122.09 −0.126301
657657 0 0
658658 −13943.6 −0.826108
659659 14977.2 + 25941.3i 0.885325 + 1.53343i 0.845341 + 0.534227i 0.179397π0.179397\pi
0.0399839 + 0.999200i 0.487269π0.487269\pi
660660 0 0
661661 2063.38 3573.88i 0.121416 0.210299i −0.798910 0.601450i 0.794590π-0.794590\pi
0.920326 + 0.391151i 0.127923π0.127923\pi
662662 −4964.96 + 8599.56i −0.291493 + 0.504881i
663663 0 0
664664 719.077 + 1245.48i 0.0420265 + 0.0727920i
665665 22716.5 1.32467
666666 0 0
667667 13730.5 0.797070
668668 −6357.04 11010.7i −0.368206 0.637751i
669669 0 0
670670 −7446.21 + 12897.2i −0.429362 + 0.743676i
671671 3269.87 5663.58i 0.188125 0.325842i
672672 0 0
673673 13164.9 + 22802.2i 0.754039 + 1.30603i 0.945851 + 0.324602i 0.105230π0.105230\pi
−0.191812 + 0.981432i 0.561436π0.561436\pi
674674 −6195.32 −0.354058
675675 0 0
676676 9570.15 0.544501
677677 −3301.93 5719.11i −0.187450 0.324673i 0.756949 0.653473i 0.226689π-0.226689\pi
−0.944399 + 0.328801i 0.893356π0.893356\pi
678678 0 0
679679 6188.87 10719.4i 0.349789 0.605853i
680680 −5246.86 + 9087.83i −0.295894 + 0.512503i
681681 0 0
682682 −7331.48 12698.5i −0.411637 0.712977i
683683 12706.4 0.711854 0.355927 0.934514i 0.384165π-0.384165\pi
0.355927 + 0.934514i 0.384165π0.384165\pi
684684 0 0
685685 29365.9 1.63798
686686 6395.43 + 11077.2i 0.355946 + 0.616516i
687687 0 0
688688 2681.42 4644.35i 0.148587 0.257361i
689689 6466.63 11200.5i 0.357560 0.619312i
690690 0 0
691691 7865.10 + 13622.8i 0.432999 + 0.749977i 0.997130 0.0757086i 0.0241219π-0.0241219\pi
−0.564131 + 0.825686i 0.690789π0.690789\pi
692692 8610.60 0.473014
693693 0 0
694694 16088.4 0.879982
695695 3518.78 + 6094.70i 0.192050 + 0.332640i
696696 0 0
697697 7769.35 13456.9i 0.422217 0.731301i
698698 1144.71 1982.69i 0.0620742 0.107516i
699699 0 0
700700 −13.0155 22.5434i −0.000702769 0.00121723i
701701 −652.959 −0.0351811 −0.0175905 0.999845i 0.505600π-0.505600\pi
−0.0175905 + 0.999845i 0.505600π0.505600\pi
702702 0 0
703703 −22792.0 −1.22278
704704 754.215 + 1306.34i 0.0403772 + 0.0699354i
705705 0 0
706706 −7801.26 + 13512.2i −0.415870 + 0.720308i
707707 −1899.22 + 3289.54i −0.101029 + 0.174987i
708708 0 0
709709 12442.2 + 21550.6i 0.659065 + 1.14153i 0.980858 + 0.194724i 0.0623812π0.0623812\pi
−0.321793 + 0.946810i 0.604285π0.604285\pi
710710 −11832.8 −0.625460
711711 0 0
712712 6771.66 0.356431
713713 10765.1 + 18645.7i 0.565438 + 0.979367i
714714 0 0
715715 −8938.58 + 15482.1i −0.467530 + 0.809786i
716716 8980.58 15554.8i 0.468743 0.811887i
717717 0 0
718718 −341.307 591.162i −0.0177402 0.0307270i
719719 21323.8 1.10604 0.553020 0.833168i 0.313475π-0.313475\pi
0.553020 + 0.833168i 0.313475π0.313475\pi
720720 0 0
721721 −25230.0 −1.30321
722722 −5310.48 9198.02i −0.273733 0.474120i
723723 0 0
724724 −2814.65 + 4875.11i −0.144483 + 0.250251i
725725 −35.0951 + 60.7865i −0.00179779 + 0.00311387i
726726 0 0
727727 −4016.00 6955.92i −0.204877 0.354857i 0.745217 0.666822i 0.232346π-0.232346\pi
−0.950093 + 0.311965i 0.899013π0.899013\pi
728728 −9968.06 −0.507474
729729 0 0
730730 −1653.24 −0.0838207
731731 19634.3 + 34007.6i 0.993433 + 1.72068i
732732 0 0
733733 −14228.7 + 24644.8i −0.716984 + 1.24185i 0.245205 + 0.969471i 0.421145π0.421145\pi
−0.962189 + 0.272381i 0.912189π0.912189\pi
734734 −119.368 + 206.751i −0.00600265 + 0.0103969i
735735 0 0
736736 −1107.45 1918.15i −0.0554633 0.0960653i
737737 −15675.2 −0.783449
738738 0 0
739739 −11006.3 −0.547868 −0.273934 0.961748i 0.588325π-0.588325\pi
−0.273934 + 0.961748i 0.588325π0.588325\pi
740740 4626.42 + 8013.20i 0.229825 + 0.398069i
741741 0 0
742742 −3511.23 + 6081.63i −0.173721 + 0.300894i
743743 −2326.45 + 4029.54i −0.114871 + 0.198963i −0.917728 0.397209i 0.869979π-0.869979\pi
0.802857 + 0.596172i 0.203312π0.203312\pi
744744 0 0
745745 −3181.45 5510.44i −0.156456 0.270989i
746746 −8748.86 −0.429381
747747 0 0
748748 −11045.3 −0.539913
749749 11651.5 + 20181.0i 0.568408 + 0.984511i
750750 0 0
751751 8678.87 15032.3i 0.421700 0.730406i −0.574406 0.818571i 0.694767π-0.694767\pi
0.996106 + 0.0881649i 0.0281003π0.0281003\pi
752752 3032.49 5252.43i 0.147053 0.254703i
753753 0 0
754754 13439.0 + 23277.0i 0.649098 + 1.12427i
755755 4001.85 0.192903
756756 0 0
757757 119.139 0.00572019 0.00286010 0.999996i 0.499090π-0.499090\pi
0.00286010 + 0.999996i 0.499090π0.499090\pi
758758 −8949.46 15500.9i −0.428838 0.742769i
759759 0 0
760760 −4940.43 + 8557.08i −0.235800 + 0.408418i
761761 4421.51 7658.28i 0.210617 0.364799i −0.741291 0.671184i 0.765786π-0.765786\pi
0.951908 + 0.306385i 0.0991194π0.0991194\pi
762762 0 0
763763 −15864.6 27478.2i −0.752734 1.30377i
764764 3089.11 0.146283
765765 0 0
766766 412.926 0.0194773
767767 −11439.7 19814.2i −0.538545 0.932788i
768768 0 0
769769 −1346.55 + 2332.30i −0.0631442 + 0.109369i −0.895869 0.444318i 0.853446π-0.853446\pi
0.832725 + 0.553687i 0.186779π0.186779\pi
770770 4853.45 8406.41i 0.227151 0.393437i
771771 0 0
772772 −7305.35 12653.2i −0.340577 0.589897i
773773 −18116.5 −0.842958 −0.421479 0.906838i 0.638489π-0.638489\pi
−0.421479 + 0.906838i 0.638489π0.638489\pi
774774 0 0
775775 −110.063 −0.00510137
776776 2691.94 + 4662.57i 0.124530 + 0.215692i
777777 0 0
778778 2028.94 3514.23i 0.0934975 0.161942i
779779 7315.60 12671.0i 0.336468 0.582780i
780780 0 0
781781 −6227.35 10786.1i −0.285316 0.494183i
782782 16218.2 0.741640
783783 0 0
784784 −75.5700 −0.00344251
785785 4071.22 + 7051.56i 0.185106 + 0.320613i
786786 0 0
787787 −15873.7 + 27494.1i −0.718980 + 1.24531i 0.242424 + 0.970170i 0.422057π0.422057\pi
−0.961404 + 0.275140i 0.911276π0.911276\pi
788788 −5294.81 + 9170.88i −0.239365 + 0.414593i
789789 0 0
790790 5372.00 + 9304.58i 0.241933 + 0.419040i
791791 −32006.2 −1.43870
792792 0 0
793793 −18797.5 −0.841763
794794 6646.07 + 11511.3i 0.297053 + 0.514511i
795795 0 0
796796 −2940.43 + 5092.97i −0.130931 + 0.226778i
797797 14132.9 24478.9i 0.628122 1.08794i −0.359806 0.933027i 0.617157π-0.617157\pi
0.987928 0.154912i 0.0495095π-0.0495095\pi
798798 0 0
799799 22205.0 + 38460.2i 0.983174 + 1.70291i
800800 11.3225 0.000500390
801801 0 0
802802 −3.26896 −0.000143929
803803 −870.065 1507.00i −0.0382365 0.0662276i
804804 0 0
805805 −7126.52 + 12343.5i −0.312021 + 0.540436i
806806 −21073.2 + 36499.9i −0.920933 + 1.59510i
807807 0 0
808808 −826.091 1430.83i −0.0359676 0.0622976i
809809 −42553.4 −1.84932 −0.924659 0.380795i 0.875650π-0.875650\pi
−0.924659 + 0.380795i 0.875650π0.875650\pi
810810 0 0
811811 −6900.03 −0.298758 −0.149379 0.988780i 0.547727π-0.547727\pi
−0.149379 + 0.988780i 0.547727π0.547727\pi
812812 −7297.08 12638.9i −0.315366 0.546230i
813813 0 0
814814 −4869.58 + 8434.36i −0.209679 + 0.363175i
815815 2219.94 3845.05i 0.0954123 0.165259i
816816 0 0
817817 18487.6 + 32021.4i 0.791675 + 1.37122i
818818 −12406.7 −0.530306
819819 0 0
820820 −5939.81 −0.252960
821821 −11179.6 19363.6i −0.475238 0.823136i 0.524360 0.851497i 0.324305π-0.324305\pi
−0.999598 + 0.0283606i 0.990971π0.990971\pi
822822 0 0
823823 −395.785 + 685.520i −0.0167633 + 0.0290349i −0.874285 0.485412i 0.838670π-0.838670\pi
0.857522 + 0.514447i 0.172003π0.172003\pi
824824 5487.08 9503.90i 0.231980 0.401801i
825825 0 0
826826 6211.51 + 10758.6i 0.261654 + 0.453198i
827827 23005.9 0.967343 0.483671 0.875250i 0.339303π-0.339303\pi
0.483671 + 0.875250i 0.339303π0.339303\pi
828828 0 0
829829 −15420.2 −0.646040 −0.323020 0.946392i 0.604698π-0.604698\pi
−0.323020 + 0.946392i 0.604698π0.604698\pi
830830 2012.72 + 3486.14i 0.0841718 + 0.145790i
831831 0 0
832832 2167.88 3754.87i 0.0903336 0.156462i
833833 276.675 479.215i 0.0115081 0.0199326i
834834 0 0
835835 −17793.6 30819.4i −0.737453 1.27731i
836836 −10400.2 −0.430261
837837 0 0
838838 −19500.7 −0.803869
839839 −23145.1 40088.4i −0.952391 1.64959i −0.740228 0.672356i 0.765282π-0.765282\pi
−0.212163 0.977234i 0.568051π-0.568051\pi
840840 0 0
841841 −7481.44 + 12958.2i −0.306755 + 0.531314i
842842 10061.4 17426.8i 0.411803 0.713263i
843843 0 0
844844 −3072.01 5320.88i −0.125288 0.217005i
845845 26787.2 1.09054
846846 0 0
847847 −14263.1 −0.578613
848848 −1527.26 2645.29i −0.0618471 0.107122i
849849 0 0
850850 −41.4538 + 71.8001i −0.00167277 + 0.00289732i
851851 7150.22 12384.5i 0.288021 0.498868i
852852 0 0
853853 −13951.2 24164.2i −0.559999 0.969947i −0.997496 0.0707272i 0.977468π-0.977468\pi
0.437496 0.899220i 0.355865π-0.355865\pi
854854 10206.6 0.408973
855855 0 0
856856 −10136.0 −0.404721
857857 −3276.46 5675.00i −0.130597 0.226201i 0.793310 0.608818i 0.208356π-0.208356\pi
−0.923907 + 0.382617i 0.875023π0.875023\pi
858858 0 0
859859 −17428.8 + 30187.6i −0.692275 + 1.19906i 0.278815 + 0.960345i 0.410058π0.410058\pi
−0.971091 + 0.238711i 0.923275π0.923275\pi
860860 7505.38 12999.7i 0.297595 0.515449i
861861 0 0
862862 6763.21 + 11714.2i 0.267234 + 0.462863i
863863 −10885.2 −0.429359 −0.214680 0.976685i 0.568871π-0.568871\pi
−0.214680 + 0.976685i 0.568871π0.568871\pi
864864 0 0
865865 24101.4 0.947367
866866 10601.4 + 18362.1i 0.415993 + 0.720520i
867867 0 0
868868 11442.3 19818.6i 0.447438 0.774985i
869869 −5654.35 + 9793.61i −0.220726 + 0.382308i
870870 0 0
871871 22527.9 + 39019.5i 0.876383 + 1.51794i
872872 13801.0 0.535966
873873 0 0
874874 15271.0 0.591019
875875 12833.8 + 22228.7i 0.495840 + 0.858821i
876876 0 0
877877 13956.9 24174.1i 0.537390 0.930787i −0.461653 0.887061i 0.652744π-0.652744\pi
0.999044 0.0437269i 0.0139231π-0.0139231\pi
878878 12568.9 21770.0i 0.483121 0.836791i
879879 0 0
880880 2111.08 + 3656.49i 0.0808686 + 0.140069i
881881 10694.5 0.408975 0.204488 0.978869i 0.434447π-0.434447\pi
0.204488 + 0.978869i 0.434447π0.434447\pi
882882 0 0
883883 3265.74 0.124463 0.0622315 0.998062i 0.480178π-0.480178\pi
0.0622315 + 0.998062i 0.480178π0.480178\pi
884884 15874.0 + 27494.5i 0.603958 + 1.04609i
885885 0 0
886886 10255.4 17762.9i 0.388869 0.673541i
887887 −4696.44 + 8134.47i −0.177780 + 0.307924i −0.941120 0.338073i 0.890225π-0.890225\pi
0.763340 + 0.645997i 0.223558π0.223558\pi
888888 0 0
889889 −4525.71 7838.75i −0.170739 0.295729i
890890 18954.1 0.713870
891891 0 0
892892 −6630.12 −0.248871
893893 20908.2 + 36214.0i 0.783499 + 1.35706i
894894 0 0
895895 25137.0 43538.6i 0.938812 1.62607i
896896 −1177.11 + 2038.81i −0.0438888 + 0.0760177i
897897 0 0
898898 4080.23 + 7067.17i 0.151625 + 0.262622i
899899 −61706.2 −2.28923
900900 0 0
901901 22366.3 0.827002
902902 −3126.00 5414.39i −0.115393 0.199867i
903903 0 0
904904 6960.78 12056.4i 0.256098 0.443574i
905905 −7878.30 + 13645.6i −0.289374 + 0.501211i
906906 0 0
907907 −6769.24 11724.7i −0.247816 0.429229i 0.715104 0.699018i 0.246379π-0.246379\pi
−0.962919 + 0.269789i 0.913046π0.913046\pi
908908 −6057.04 −0.221377
909909 0 0
910910 −27901.0 −1.01638
911911 −18903.2 32741.3i −0.687476 1.19074i −0.972652 0.232269i 0.925385π-0.925385\pi
0.285175 0.958475i 0.407948π-0.407948\pi
912912 0 0
913913 −2118.51 + 3669.37i −0.0767935 + 0.133010i
914914 2183.20 3781.41i 0.0790086 0.136847i
915915 0 0
916916 −8598.08 14892.3i −0.310140 0.537179i
917917 35299.9 1.27122
918918 0 0
919919 30674.2 1.10103 0.550515 0.834825i 0.314431π-0.314431\pi
0.550515 + 0.834825i 0.314431π0.314431\pi
920920 −3099.78 5368.98i −0.111084 0.192402i
921921 0 0
922922 −3250.26 + 5629.62i −0.116097 + 0.201086i
923923 −17899.6 + 31003.0i −0.638322 + 1.10561i
924924 0 0
925925 36.5519 + 63.3098i 0.00129926 + 0.00225039i
926926 37982.3 1.34792
927927 0 0
928928 6347.94 0.224549
929929 −14058.7 24350.4i −0.496502 0.859967i 0.503490 0.864001i 0.332049π-0.332049\pi
−0.999992 + 0.00403418i 0.998716π0.998716\pi
930930 0 0
931931 260.516 451.228i 0.00917087 0.0158844i
932932 −2673.56 + 4630.74i −0.0939650 + 0.162752i
933933 0 0
934934 −6906.52 11962.4i −0.241957 0.419083i
935935 −30916.1 −1.08135
936936 0 0
937937 31859.0 1.11077 0.555384 0.831594i 0.312571π-0.312571\pi
0.555384 + 0.831594i 0.312571π0.312571\pi
938938 −12232.2 21186.7i −0.425793 0.737495i
939939 0 0
940940 8488.06 14701.8i 0.294521 0.510126i
941941 1131.57 1959.93i 0.0392009 0.0678980i −0.845759 0.533565i 0.820852π-0.820852\pi
0.884960 + 0.465667i 0.154185π0.154185\pi
942942 0 0
943943 4590.04 + 7950.19i 0.158507 + 0.274543i
944944 −5403.57 −0.186304
945945 0 0
946946 15799.7 0.543016
947947 3492.67 + 6049.48i 0.119848 + 0.207583i 0.919707 0.392604i 0.128426π-0.128426\pi
−0.799859 + 0.600188i 0.795092π0.795092\pi
948948 0 0
949949 −2500.87 + 4331.63i −0.0855444 + 0.148167i
950950 −39.0328 + 67.6068i −0.00133304 + 0.00230890i
951951 0 0
952952 −8619.20 14928.9i −0.293435 0.508244i
953953 26436.8 0.898608 0.449304 0.893379i 0.351672π-0.351672\pi
0.449304 + 0.893379i 0.351672π0.351672\pi
954954 0 0
955955 8646.53 0.292979
956956 13757.3 + 23828.3i 0.465420 + 0.806131i
957957 0 0
958958 −7380.27 + 12783.0i −0.248900 + 0.431107i
959959 −24120.2 + 41777.4i −0.812181 + 1.40674i
960960 0 0
961961 −33484.1 57996.2i −1.12397 1.94677i
962962 27993.7 0.938206
963963 0 0
964964 6125.17 0.204646
965965 −20448.0 35416.9i −0.682117 1.18146i
966966 0 0
967967 50.0781 86.7379i 0.00166536 0.00288449i −0.865192 0.501442i 0.832803π-0.832803\pi
0.866857 + 0.498557i 0.166137π0.166137\pi
968968 3101.97 5372.77i 0.102997 0.178396i
969969 0 0
970970 7534.84 + 13050.7i 0.249411 + 0.431993i
971971 678.145 0.0224127 0.0112063 0.999937i 0.496433π-0.496433\pi
0.0112063 + 0.999937i 0.496433π0.496433\pi
972972 0 0
973973 −11560.8 −0.380908
974974 8756.51 + 15166.7i 0.288066 + 0.498945i
975975 0 0
976976 −2219.75 + 3844.73i −0.0727998 + 0.126093i
977977 4874.13 8442.24i 0.159608 0.276449i −0.775119 0.631815i 0.782310π-0.782310\pi
0.934727 + 0.355366i 0.115644π0.115644\pi
978978 0 0
979979 9975.17 + 17277.5i 0.325646 + 0.564036i
980980 −211.523 −0.00689476
981981 0 0
982982 23675.9 0.769376
983983 1757.84 + 3044.67i 0.0570361 + 0.0987894i 0.893134 0.449791i 0.148502π-0.148502\pi
−0.836098 + 0.548581i 0.815168π0.815168\pi
984984 0 0
985985 −14820.4 + 25669.6i −0.479407 + 0.830358i
986986 −23240.9 + 40254.5i −0.750651 + 1.30017i
987987 0 0
988988 14946.9 + 25888.8i 0.481299 + 0.833635i
989989 −23199.4 −0.745903
990990 0 0
991991 −612.517 −0.0196339 −0.00981697 0.999952i 0.503125π-0.503125\pi
−0.00981697 + 0.999952i 0.503125π0.503125\pi
992992 4976.98 + 8620.39i 0.159294 + 0.275905i
993993 0 0
994994 9719.06 16833.9i 0.310131 0.537162i
995995 −8230.38 + 14255.4i −0.262232 + 0.454198i
996996 0 0
997997 −11478.2 19880.8i −0.364611 0.631525i 0.624102 0.781343i 0.285465π-0.285465\pi
−0.988714 + 0.149817i 0.952131π0.952131\pi
998998 17156.8 0.544176
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 162.4.c.i.109.1 4
3.2 odd 2 162.4.c.j.109.2 4
9.2 odd 6 162.4.c.j.55.2 4
9.4 even 3 162.4.a.h.1.2 yes 2
9.5 odd 6 162.4.a.e.1.1 2
9.7 even 3 inner 162.4.c.i.55.1 4
36.23 even 6 1296.4.a.j.1.1 2
36.31 odd 6 1296.4.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.4.a.e.1.1 2 9.5 odd 6
162.4.a.h.1.2 yes 2 9.4 even 3
162.4.c.i.55.1 4 9.7 even 3 inner
162.4.c.i.109.1 4 1.1 even 1 trivial
162.4.c.j.55.2 4 9.2 odd 6
162.4.c.j.109.2 4 3.2 odd 2
1296.4.a.j.1.1 2 36.23 even 6
1296.4.a.s.1.2 2 36.31 odd 6