Properties

Label 3775.2.a.i.1.2
Level 37753775
Weight 22
Character 3775.1
Self dual yes
Analytic conductor 30.14430.144
Analytic rank 00
Dimension 22
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3775,2,Mod(1,3775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3775, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3775.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 3775=52151 3775 = 5^{2} \cdot 151
Weight: k k == 2 2
Character orbit: [χ][\chi] == 3775.a (trivial)

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: yes
Analytic conductor: 30.143526763030.1435267630
Analytic rank: 00
Dimension: 22
Coefficient field: Q(ζ8)+\Q(\zeta_{8})^+
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x22 x^{2} - 2 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 755)
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.2
Root 1.414211.41421 of defining polynomial
Character χ\chi == 3775.1

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+1.00000q2+2.41421q31.00000q4+2.41421q6+2.00000q73.00000q8+2.82843q9+1.00000q112.41421q12+0.414214q13+2.00000q141.00000q16+4.82843q17+2.82843q18+8.65685q19+4.82843q21+1.00000q226.07107q237.24264q24+0.414214q260.414214q272.00000q282.17157q29+0.171573q31+5.00000q32+2.41421q33+4.82843q342.82843q36+1.17157q37+8.65685q38+1.00000q398.82843q41+4.82843q42+10.4853q431.00000q446.07107q46+11.6569q472.41421q483.00000q49+11.6569q510.414214q52+1.17157q530.414214q546.00000q56+20.8995q572.17157q58+11.8284q5912.0000q61+0.171573q62+5.65685q63+7.00000q64+2.41421q66+4.75736q674.82843q6814.6569q691.65685q718.48528q7214.5563q73+1.17157q748.65685q76+2.00000q77+1.00000q78+4.82843q799.48528q818.82843q82+2.75736q834.82843q84+10.4853q865.24264q873.00000q88+4.82843q89+0.828427q91+6.07107q92+0.414214q93+11.6569q94+12.0711q96+18.4853q973.00000q98+2.82843q99+O(q100)q+1.00000 q^{2} +2.41421 q^{3} -1.00000 q^{4} +2.41421 q^{6} +2.00000 q^{7} -3.00000 q^{8} +2.82843 q^{9} +1.00000 q^{11} -2.41421 q^{12} +0.414214 q^{13} +2.00000 q^{14} -1.00000 q^{16} +4.82843 q^{17} +2.82843 q^{18} +8.65685 q^{19} +4.82843 q^{21} +1.00000 q^{22} -6.07107 q^{23} -7.24264 q^{24} +0.414214 q^{26} -0.414214 q^{27} -2.00000 q^{28} -2.17157 q^{29} +0.171573 q^{31} +5.00000 q^{32} +2.41421 q^{33} +4.82843 q^{34} -2.82843 q^{36} +1.17157 q^{37} +8.65685 q^{38} +1.00000 q^{39} -8.82843 q^{41} +4.82843 q^{42} +10.4853 q^{43} -1.00000 q^{44} -6.07107 q^{46} +11.6569 q^{47} -2.41421 q^{48} -3.00000 q^{49} +11.6569 q^{51} -0.414214 q^{52} +1.17157 q^{53} -0.414214 q^{54} -6.00000 q^{56} +20.8995 q^{57} -2.17157 q^{58} +11.8284 q^{59} -12.0000 q^{61} +0.171573 q^{62} +5.65685 q^{63} +7.00000 q^{64} +2.41421 q^{66} +4.75736 q^{67} -4.82843 q^{68} -14.6569 q^{69} -1.65685 q^{71} -8.48528 q^{72} -14.5563 q^{73} +1.17157 q^{74} -8.65685 q^{76} +2.00000 q^{77} +1.00000 q^{78} +4.82843 q^{79} -9.48528 q^{81} -8.82843 q^{82} +2.75736 q^{83} -4.82843 q^{84} +10.4853 q^{86} -5.24264 q^{87} -3.00000 q^{88} +4.82843 q^{89} +0.828427 q^{91} +6.07107 q^{92} +0.414214 q^{93} +11.6569 q^{94} +12.0711 q^{96} +18.4853 q^{97} -3.00000 q^{98} +2.82843 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+2q2+2q32q4+2q6+4q76q8+2q112q122q13+4q142q16+4q17+6q19+4q21+2q22+2q236q242q26+2q27+6q98+O(q100) 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} + 4 q^{7} - 6 q^{8} + 2 q^{11} - 2 q^{12} - 2 q^{13} + 4 q^{14} - 2 q^{16} + 4 q^{17} + 6 q^{19} + 4 q^{21} + 2 q^{22} + 2 q^{23} - 6 q^{24} - 2 q^{26} + 2 q^{27}+ \cdots - 6 q^{98}+O(q^{100}) Copy content Toggle raw display

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 0.707107 0.353553 0.935414i 0.384973π-0.384973\pi
0.353553 + 0.935414i 0.384973π0.384973\pi
33 2.41421 1.39385 0.696923 0.717146i 0.254552π-0.254552\pi
0.696923 + 0.717146i 0.254552π0.254552\pi
44 −1.00000 −0.500000
55 0 0
66 2.41421 0.985599
77 2.00000 0.755929 0.377964 0.925820i 0.376624π-0.376624\pi
0.377964 + 0.925820i 0.376624π0.376624\pi
88 −3.00000 −1.06066
99 2.82843 0.942809
1010 0 0
1111 1.00000 0.301511 0.150756 0.988571i 0.451829π-0.451829\pi
0.150756 + 0.988571i 0.451829π0.451829\pi
1212 −2.41421 −0.696923
1313 0.414214 0.114882 0.0574411 0.998349i 0.481706π-0.481706\pi
0.0574411 + 0.998349i 0.481706π0.481706\pi
1414 2.00000 0.534522
1515 0 0
1616 −1.00000 −0.250000
1717 4.82843 1.17107 0.585533 0.810649i 0.300885π-0.300885\pi
0.585533 + 0.810649i 0.300885π0.300885\pi
1818 2.82843 0.666667
1919 8.65685 1.98602 0.993009 0.118036i 0.0376599π-0.0376599\pi
0.993009 + 0.118036i 0.0376599π0.0376599\pi
2020 0 0
2121 4.82843 1.05365
2222 1.00000 0.213201
2323 −6.07107 −1.26591 −0.632953 0.774191i 0.718157π-0.718157\pi
−0.632953 + 0.774191i 0.718157π0.718157\pi
2424 −7.24264 −1.47840
2525 0 0
2626 0.414214 0.0812340
2727 −0.414214 −0.0797154
2828 −2.00000 −0.377964
2929 −2.17157 −0.403251 −0.201625 0.979463i 0.564622π-0.564622\pi
−0.201625 + 0.979463i 0.564622π0.564622\pi
3030 0 0
3131 0.171573 0.0308154 0.0154077 0.999881i 0.495095π-0.495095\pi
0.0154077 + 0.999881i 0.495095π0.495095\pi
3232 5.00000 0.883883
3333 2.41421 0.420261
3434 4.82843 0.828068
3535 0 0
3636 −2.82843 −0.471405
3737 1.17157 0.192605 0.0963027 0.995352i 0.469298π-0.469298\pi
0.0963027 + 0.995352i 0.469298π0.469298\pi
3838 8.65685 1.40433
3939 1.00000 0.160128
4040 0 0
4141 −8.82843 −1.37877 −0.689384 0.724396i 0.742119π-0.742119\pi
−0.689384 + 0.724396i 0.742119π0.742119\pi
4242 4.82843 0.745042
4343 10.4853 1.59899 0.799495 0.600672i 0.205100π-0.205100\pi
0.799495 + 0.600672i 0.205100π0.205100\pi
4444 −1.00000 −0.150756
4545 0 0
4646 −6.07107 −0.895130
4747 11.6569 1.70033 0.850163 0.526519i 0.176503π-0.176503\pi
0.850163 + 0.526519i 0.176503π0.176503\pi
4848 −2.41421 −0.348462
4949 −3.00000 −0.428571
5050 0 0
5151 11.6569 1.63229
5252 −0.414214 −0.0574411
5353 1.17157 0.160928 0.0804640 0.996758i 0.474360π-0.474360\pi
0.0804640 + 0.996758i 0.474360π0.474360\pi
5454 −0.414214 −0.0563673
5555 0 0
5656 −6.00000 −0.801784
5757 20.8995 2.76821
5858 −2.17157 −0.285141
5959 11.8284 1.53993 0.769965 0.638086i 0.220274π-0.220274\pi
0.769965 + 0.638086i 0.220274π0.220274\pi
6060 0 0
6161 −12.0000 −1.53644 −0.768221 0.640184i 0.778858π-0.778858\pi
−0.768221 + 0.640184i 0.778858π0.778858\pi
6262 0.171573 0.0217898
6363 5.65685 0.712697
6464 7.00000 0.875000
6565 0 0
6666 2.41421 0.297169
6767 4.75736 0.581204 0.290602 0.956844i 0.406144π-0.406144\pi
0.290602 + 0.956844i 0.406144π0.406144\pi
6868 −4.82843 −0.585533
6969 −14.6569 −1.76448
7070 0 0
7171 −1.65685 −0.196632 −0.0983162 0.995155i 0.531346π-0.531346\pi
−0.0983162 + 0.995155i 0.531346π0.531346\pi
7272 −8.48528 −1.00000
7373 −14.5563 −1.70369 −0.851846 0.523792i 0.824517π-0.824517\pi
−0.851846 + 0.523792i 0.824517π0.824517\pi
7474 1.17157 0.136193
7575 0 0
7676 −8.65685 −0.993009
7777 2.00000 0.227921
7878 1.00000 0.113228
7979 4.82843 0.543240 0.271620 0.962405i 0.412441π-0.412441\pi
0.271620 + 0.962405i 0.412441π0.412441\pi
8080 0 0
8181 −9.48528 −1.05392
8282 −8.82843 −0.974937
8383 2.75736 0.302660 0.151330 0.988483i 0.451644π-0.451644\pi
0.151330 + 0.988483i 0.451644π0.451644\pi
8484 −4.82843 −0.526825
8585 0 0
8686 10.4853 1.13066
8787 −5.24264 −0.562070
8888 −3.00000 −0.319801
8989 4.82843 0.511812 0.255906 0.966702i 0.417626π-0.417626\pi
0.255906 + 0.966702i 0.417626π0.417626\pi
9090 0 0
9191 0.828427 0.0868428
9292 6.07107 0.632953
9393 0.414214 0.0429519
9494 11.6569 1.20231
9595 0 0
9696 12.0711 1.23200
9797 18.4853 1.87690 0.938448 0.345421i 0.112264π-0.112264\pi
0.938448 + 0.345421i 0.112264π0.112264\pi
9898 −3.00000 −0.303046
9999 2.82843 0.284268
100100 0 0
101101 15.6569 1.55792 0.778958 0.627077i 0.215749π-0.215749\pi
0.778958 + 0.627077i 0.215749π0.215749\pi
102102 11.6569 1.15420
103103 9.17157 0.903702 0.451851 0.892093i 0.350764π-0.350764\pi
0.451851 + 0.892093i 0.350764π0.350764\pi
104104 −1.24264 −0.121851
105105 0 0
106106 1.17157 0.113793
107107 9.72792 0.940434 0.470217 0.882551i 0.344176π-0.344176\pi
0.470217 + 0.882551i 0.344176π0.344176\pi
108108 0.414214 0.0398577
109109 −14.1421 −1.35457 −0.677285 0.735720i 0.736844π-0.736844\pi
−0.677285 + 0.735720i 0.736844π0.736844\pi
110110 0 0
111111 2.82843 0.268462
112112 −2.00000 −0.188982
113113 10.1421 0.954092 0.477046 0.878878i 0.341708π-0.341708\pi
0.477046 + 0.878878i 0.341708π0.341708\pi
114114 20.8995 1.95742
115115 0 0
116116 2.17157 0.201625
117117 1.17157 0.108312
118118 11.8284 1.08889
119119 9.65685 0.885242
120120 0 0
121121 −10.0000 −0.909091
122122 −12.0000 −1.08643
123123 −21.3137 −1.92179
124124 −0.171573 −0.0154077
125125 0 0
126126 5.65685 0.503953
127127 −11.3137 −1.00393 −0.501965 0.864888i 0.667389π-0.667389\pi
−0.501965 + 0.864888i 0.667389π0.667389\pi
128128 −3.00000 −0.265165
129129 25.3137 2.22875
130130 0 0
131131 12.8284 1.12082 0.560412 0.828214i 0.310643π-0.310643\pi
0.560412 + 0.828214i 0.310643π0.310643\pi
132132 −2.41421 −0.210130
133133 17.3137 1.50129
134134 4.75736 0.410973
135135 0 0
136136 −14.4853 −1.24210
137137 −8.48528 −0.724947 −0.362473 0.931994i 0.618068π-0.618068\pi
−0.362473 + 0.931994i 0.618068π0.618068\pi
138138 −14.6569 −1.24767
139139 2.34315 0.198743 0.0993715 0.995050i 0.468317π-0.468317\pi
0.0993715 + 0.995050i 0.468317π0.468317\pi
140140 0 0
141141 28.1421 2.37000
142142 −1.65685 −0.139040
143143 0.414214 0.0346383
144144 −2.82843 −0.235702
145145 0 0
146146 −14.5563 −1.20469
147147 −7.24264 −0.597363
148148 −1.17157 −0.0963027
149149 −1.17157 −0.0959790 −0.0479895 0.998848i 0.515281π-0.515281\pi
−0.0479895 + 0.998848i 0.515281π0.515281\pi
150150 0 0
151151 1.00000 0.0813788
152152 −25.9706 −2.10649
153153 13.6569 1.10409
154154 2.00000 0.161165
155155 0 0
156156 −1.00000 −0.0800641
157157 −1.10051 −0.0878299 −0.0439149 0.999035i 0.513983π-0.513983\pi
−0.0439149 + 0.999035i 0.513983π0.513983\pi
158158 4.82843 0.384129
159159 2.82843 0.224309
160160 0 0
161161 −12.1421 −0.956934
162162 −9.48528 −0.745234
163163 −3.58579 −0.280860 −0.140430 0.990091i 0.544849π-0.544849\pi
−0.140430 + 0.990091i 0.544849π0.544849\pi
164164 8.82843 0.689384
165165 0 0
166166 2.75736 0.214013
167167 3.51472 0.271977 0.135989 0.990710i 0.456579π-0.456579\pi
0.135989 + 0.990710i 0.456579π0.456579\pi
168168 −14.4853 −1.11756
169169 −12.8284 −0.986802
170170 0 0
171171 24.4853 1.87244
172172 −10.4853 −0.799495
173173 7.31371 0.556051 0.278025 0.960574i 0.410320π-0.410320\pi
0.278025 + 0.960574i 0.410320π0.410320\pi
174174 −5.24264 −0.397444
175175 0 0
176176 −1.00000 −0.0753778
177177 28.5563 2.14643
178178 4.82843 0.361906
179179 −15.7990 −1.18087 −0.590436 0.807084i 0.701044π-0.701044\pi
−0.590436 + 0.807084i 0.701044π0.701044\pi
180180 0 0
181181 −10.0000 −0.743294 −0.371647 0.928374i 0.621207π-0.621207\pi
−0.371647 + 0.928374i 0.621207π0.621207\pi
182182 0.828427 0.0614071
183183 −28.9706 −2.14157
184184 18.2132 1.34270
185185 0 0
186186 0.414214 0.0303716
187187 4.82843 0.353090
188188 −11.6569 −0.850163
189189 −0.828427 −0.0602592
190190 0 0
191191 4.00000 0.289430 0.144715 0.989473i 0.453773π-0.453773\pi
0.144715 + 0.989473i 0.453773π0.453773\pi
192192 16.8995 1.21962
193193 10.4853 0.754747 0.377374 0.926061i 0.376827π-0.376827\pi
0.377374 + 0.926061i 0.376827π0.376827\pi
194194 18.4853 1.32717
195195 0 0
196196 3.00000 0.214286
197197 5.17157 0.368459 0.184230 0.982883i 0.441021π-0.441021\pi
0.184230 + 0.982883i 0.441021π0.441021\pi
198198 2.82843 0.201008
199199 −24.6274 −1.74579 −0.872896 0.487907i 0.837761π-0.837761\pi
−0.872896 + 0.487907i 0.837761π0.837761\pi
200200 0 0
201201 11.4853 0.810109
202202 15.6569 1.10161
203203 −4.34315 −0.304829
204204 −11.6569 −0.816143
205205 0 0
206206 9.17157 0.639014
207207 −17.1716 −1.19351
208208 −0.414214 −0.0287205
209209 8.65685 0.598807
210210 0 0
211211 −5.65685 −0.389434 −0.194717 0.980859i 0.562379π-0.562379\pi
−0.194717 + 0.980859i 0.562379π0.562379\pi
212212 −1.17157 −0.0804640
213213 −4.00000 −0.274075
214214 9.72792 0.664987
215215 0 0
216216 1.24264 0.0845510
217217 0.343146 0.0232943
218218 −14.1421 −0.957826
219219 −35.1421 −2.37469
220220 0 0
221221 2.00000 0.134535
222222 2.82843 0.189832
223223 −17.3137 −1.15941 −0.579706 0.814826i 0.696833π-0.696833\pi
−0.579706 + 0.814826i 0.696833π0.696833\pi
224224 10.0000 0.668153
225225 0 0
226226 10.1421 0.674645
227227 −6.00000 −0.398234 −0.199117 0.979976i 0.563807π-0.563807\pi
−0.199117 + 0.979976i 0.563807π0.563807\pi
228228 −20.8995 −1.38410
229229 −15.6274 −1.03269 −0.516344 0.856381i 0.672708π-0.672708\pi
−0.516344 + 0.856381i 0.672708π0.672708\pi
230230 0 0
231231 4.82843 0.317687
232232 6.51472 0.427712
233233 −26.8995 −1.76224 −0.881122 0.472889i 0.843211π-0.843211\pi
−0.881122 + 0.472889i 0.843211π0.843211\pi
234234 1.17157 0.0765881
235235 0 0
236236 −11.8284 −0.769965
237237 11.6569 0.757194
238238 9.65685 0.625961
239239 −4.65685 −0.301227 −0.150613 0.988593i 0.548125π-0.548125\pi
−0.150613 + 0.988593i 0.548125π0.548125\pi
240240 0 0
241241 −2.17157 −0.139883 −0.0699417 0.997551i 0.522281π-0.522281\pi
−0.0699417 + 0.997551i 0.522281π0.522281\pi
242242 −10.0000 −0.642824
243243 −21.6569 −1.38929
244244 12.0000 0.768221
245245 0 0
246246 −21.3137 −1.35891
247247 3.58579 0.228158
248248 −0.514719 −0.0326847
249249 6.65685 0.421861
250250 0 0
251251 11.4853 0.724945 0.362472 0.931995i 0.381933π-0.381933\pi
0.362472 + 0.931995i 0.381933π0.381933\pi
252252 −5.65685 −0.356348
253253 −6.07107 −0.381685
254254 −11.3137 −0.709885
255255 0 0
256256 −17.0000 −1.06250
257257 −4.07107 −0.253946 −0.126973 0.991906i 0.540526π-0.540526\pi
−0.126973 + 0.991906i 0.540526π0.540526\pi
258258 25.3137 1.57596
259259 2.34315 0.145596
260260 0 0
261261 −6.14214 −0.380189
262262 12.8284 0.792543
263263 17.5858 1.08439 0.542193 0.840254i 0.317594π-0.317594\pi
0.542193 + 0.840254i 0.317594π0.317594\pi
264264 −7.24264 −0.445754
265265 0 0
266266 17.3137 1.06157
267267 11.6569 0.713388
268268 −4.75736 −0.290602
269269 −17.0000 −1.03651 −0.518254 0.855227i 0.673418π-0.673418\pi
−0.518254 + 0.855227i 0.673418π0.673418\pi
270270 0 0
271271 −22.6274 −1.37452 −0.687259 0.726413i 0.741186π-0.741186\pi
−0.687259 + 0.726413i 0.741186π0.741186\pi
272272 −4.82843 −0.292766
273273 2.00000 0.121046
274274 −8.48528 −0.512615
275275 0 0
276276 14.6569 0.882239
277277 26.1421 1.57073 0.785364 0.619034i 0.212476π-0.212476\pi
0.785364 + 0.619034i 0.212476π0.212476\pi
278278 2.34315 0.140533
279279 0.485281 0.0290530
280280 0 0
281281 10.0000 0.596550 0.298275 0.954480i 0.403589π-0.403589\pi
0.298275 + 0.954480i 0.403589π0.403589\pi
282282 28.1421 1.67584
283283 −3.72792 −0.221602 −0.110801 0.993843i 0.535342π-0.535342\pi
−0.110801 + 0.993843i 0.535342π0.535342\pi
284284 1.65685 0.0983162
285285 0 0
286286 0.414214 0.0244930
287287 −17.6569 −1.04225
288288 14.1421 0.833333
289289 6.31371 0.371395
290290 0 0
291291 44.6274 2.61611
292292 14.5563 0.851846
293293 −16.4853 −0.963080 −0.481540 0.876424i 0.659922π-0.659922\pi
−0.481540 + 0.876424i 0.659922π0.659922\pi
294294 −7.24264 −0.422399
295295 0 0
296296 −3.51472 −0.204289
297297 −0.414214 −0.0240351
298298 −1.17157 −0.0678674
299299 −2.51472 −0.145430
300300 0 0
301301 20.9706 1.20872
302302 1.00000 0.0575435
303303 37.7990 2.17150
304304 −8.65685 −0.496505
305305 0 0
306306 13.6569 0.780710
307307 18.4853 1.05501 0.527505 0.849552i 0.323127π-0.323127\pi
0.527505 + 0.849552i 0.323127π0.323127\pi
308308 −2.00000 −0.113961
309309 22.1421 1.25962
310310 0 0
311311 −6.31371 −0.358018 −0.179009 0.983847i 0.557289π-0.557289\pi
−0.179009 + 0.983847i 0.557289π0.557289\pi
312312 −3.00000 −0.169842
313313 −22.6274 −1.27898 −0.639489 0.768801i 0.720854π-0.720854\pi
−0.639489 + 0.768801i 0.720854π0.720854\pi
314314 −1.10051 −0.0621051
315315 0 0
316316 −4.82843 −0.271620
317317 −26.5563 −1.49155 −0.745777 0.666196i 0.767921π-0.767921\pi
−0.745777 + 0.666196i 0.767921π0.767921\pi
318318 2.82843 0.158610
319319 −2.17157 −0.121585
320320 0 0
321321 23.4853 1.31082
322322 −12.1421 −0.676655
323323 41.7990 2.32576
324324 9.48528 0.526960
325325 0 0
326326 −3.58579 −0.198598
327327 −34.1421 −1.88806
328328 26.4853 1.46241
329329 23.3137 1.28533
330330 0 0
331331 −28.0000 −1.53902 −0.769510 0.638635i 0.779499π-0.779499\pi
−0.769510 + 0.638635i 0.779499π0.779499\pi
332332 −2.75736 −0.151330
333333 3.31371 0.181590
334334 3.51472 0.192317
335335 0 0
336336 −4.82843 −0.263412
337337 −21.7279 −1.18360 −0.591798 0.806087i 0.701582π-0.701582\pi
−0.591798 + 0.806087i 0.701582π0.701582\pi
338338 −12.8284 −0.697774
339339 24.4853 1.32986
340340 0 0
341341 0.171573 0.00929119
342342 24.4853 1.32401
343343 −20.0000 −1.07990
344344 −31.4558 −1.69599
345345 0 0
346346 7.31371 0.393187
347347 25.4558 1.36654 0.683271 0.730165i 0.260557π-0.260557\pi
0.683271 + 0.730165i 0.260557π0.260557\pi
348348 5.24264 0.281035
349349 −14.5147 −0.776955 −0.388478 0.921458i 0.626999π-0.626999\pi
−0.388478 + 0.921458i 0.626999π0.626999\pi
350350 0 0
351351 −0.171573 −0.00915788
352352 5.00000 0.266501
353353 4.75736 0.253209 0.126604 0.991953i 0.459592π-0.459592\pi
0.126604 + 0.991953i 0.459592π0.459592\pi
354354 28.5563 1.51775
355355 0 0
356356 −4.82843 −0.255906
357357 23.3137 1.23389
358358 −15.7990 −0.835003
359359 −12.1421 −0.640837 −0.320419 0.947276i 0.603824π-0.603824\pi
−0.320419 + 0.947276i 0.603824π0.603824\pi
360360 0 0
361361 55.9411 2.94427
362362 −10.0000 −0.525588
363363 −24.1421 −1.26713
364364 −0.828427 −0.0434214
365365 0 0
366366 −28.9706 −1.51432
367367 −13.3137 −0.694970 −0.347485 0.937686i 0.612964π-0.612964\pi
−0.347485 + 0.937686i 0.612964π0.612964\pi
368368 6.07107 0.316476
369369 −24.9706 −1.29992
370370 0 0
371371 2.34315 0.121650
372372 −0.414214 −0.0214760
373373 −25.8701 −1.33950 −0.669750 0.742586i 0.733599π-0.733599\pi
−0.669750 + 0.742586i 0.733599π0.733599\pi
374374 4.82843 0.249672
375375 0 0
376376 −34.9706 −1.80347
377377 −0.899495 −0.0463263
378378 −0.828427 −0.0426097
379379 6.00000 0.308199 0.154100 0.988055i 0.450752π-0.450752\pi
0.154100 + 0.988055i 0.450752π0.450752\pi
380380 0 0
381381 −27.3137 −1.39932
382382 4.00000 0.204658
383383 −4.34315 −0.221924 −0.110962 0.993825i 0.535393π-0.535393\pi
−0.110962 + 0.993825i 0.535393π0.535393\pi
384384 −7.24264 −0.369599
385385 0 0
386386 10.4853 0.533687
387387 29.6569 1.50754
388388 −18.4853 −0.938448
389389 −31.9411 −1.61948 −0.809740 0.586789i 0.800392π-0.800392\pi
−0.809740 + 0.586789i 0.800392π0.800392\pi
390390 0 0
391391 −29.3137 −1.48246
392392 9.00000 0.454569
393393 30.9706 1.56226
394394 5.17157 0.260540
395395 0 0
396396 −2.82843 −0.142134
397397 29.6569 1.48843 0.744217 0.667937i 0.232823π-0.232823\pi
0.744217 + 0.667937i 0.232823π0.232823\pi
398398 −24.6274 −1.23446
399399 41.7990 2.09257
400400 0 0
401401 −34.9706 −1.74635 −0.873173 0.487410i 0.837942π-0.837942\pi
−0.873173 + 0.487410i 0.837942π0.837942\pi
402402 11.4853 0.572834
403403 0.0710678 0.00354014
404404 −15.6569 −0.778958
405405 0 0
406406 −4.34315 −0.215547
407407 1.17157 0.0580727
408408 −34.9706 −1.73130
409409 −14.0000 −0.692255 −0.346128 0.938187i 0.612504π-0.612504\pi
−0.346128 + 0.938187i 0.612504π0.612504\pi
410410 0 0
411411 −20.4853 −1.01046
412412 −9.17157 −0.451851
413413 23.6569 1.16408
414414 −17.1716 −0.843937
415415 0 0
416416 2.07107 0.101542
417417 5.65685 0.277017
418418 8.65685 0.423421
419419 9.79899 0.478712 0.239356 0.970932i 0.423064π-0.423064\pi
0.239356 + 0.970932i 0.423064π0.423064\pi
420420 0 0
421421 −30.8284 −1.50249 −0.751243 0.660026i 0.770545π-0.770545\pi
−0.751243 + 0.660026i 0.770545π0.770545\pi
422422 −5.65685 −0.275371
423423 32.9706 1.60308
424424 −3.51472 −0.170690
425425 0 0
426426 −4.00000 −0.193801
427427 −24.0000 −1.16144
428428 −9.72792 −0.470217
429429 1.00000 0.0482805
430430 0 0
431431 8.82843 0.425250 0.212625 0.977134i 0.431799π-0.431799\pi
0.212625 + 0.977134i 0.431799π0.431799\pi
432432 0.414214 0.0199289
433433 −5.17157 −0.248530 −0.124265 0.992249i 0.539657π-0.539657\pi
−0.124265 + 0.992249i 0.539657π0.539657\pi
434434 0.343146 0.0164715
435435 0 0
436436 14.1421 0.677285
437437 −52.5563 −2.51411
438438 −35.1421 −1.67916
439439 19.2843 0.920388 0.460194 0.887818i 0.347780π-0.347780\pi
0.460194 + 0.887818i 0.347780π0.347780\pi
440440 0 0
441441 −8.48528 −0.404061
442442 2.00000 0.0951303
443443 30.0000 1.42534 0.712672 0.701498i 0.247485π-0.247485\pi
0.712672 + 0.701498i 0.247485π0.247485\pi
444444 −2.82843 −0.134231
445445 0 0
446446 −17.3137 −0.819828
447447 −2.82843 −0.133780
448448 14.0000 0.661438
449449 −14.4853 −0.683603 −0.341801 0.939772i 0.611037π-0.611037\pi
−0.341801 + 0.939772i 0.611037π0.611037\pi
450450 0 0
451451 −8.82843 −0.415714
452452 −10.1421 −0.477046
453453 2.41421 0.113430
454454 −6.00000 −0.281594
455455 0 0
456456 −62.6985 −2.93613
457457 −5.51472 −0.257968 −0.128984 0.991647i 0.541172π-0.541172\pi
−0.128984 + 0.991647i 0.541172π0.541172\pi
458458 −15.6274 −0.730221
459459 −2.00000 −0.0933520
460460 0 0
461461 −0.343146 −0.0159819 −0.00799095 0.999968i 0.502544π-0.502544\pi
−0.00799095 + 0.999968i 0.502544π0.502544\pi
462462 4.82843 0.224639
463463 22.1421 1.02903 0.514516 0.857481i 0.327972π-0.327972\pi
0.514516 + 0.857481i 0.327972π0.327972\pi
464464 2.17157 0.100813
465465 0 0
466466 −26.8995 −1.24610
467467 5.31371 0.245889 0.122945 0.992414i 0.460766π-0.460766\pi
0.122945 + 0.992414i 0.460766π0.460766\pi
468468 −1.17157 −0.0541560
469469 9.51472 0.439349
470470 0 0
471471 −2.65685 −0.122421
472472 −35.4853 −1.63334
473473 10.4853 0.482114
474474 11.6569 0.535417
475475 0 0
476476 −9.65685 −0.442621
477477 3.31371 0.151724
478478 −4.65685 −0.213000
479479 −1.85786 −0.0848880 −0.0424440 0.999099i 0.513514π-0.513514\pi
−0.0424440 + 0.999099i 0.513514π0.513514\pi
480480 0 0
481481 0.485281 0.0221269
482482 −2.17157 −0.0989124
483483 −29.3137 −1.33382
484484 10.0000 0.454545
485485 0 0
486486 −21.6569 −0.982375
487487 −3.17157 −0.143718 −0.0718588 0.997415i 0.522893π-0.522893\pi
−0.0718588 + 0.997415i 0.522893π0.522893\pi
488488 36.0000 1.62964
489489 −8.65685 −0.391476
490490 0 0
491491 37.6569 1.69943 0.849715 0.527242i 0.176774π-0.176774\pi
0.849715 + 0.527242i 0.176774π0.176774\pi
492492 21.3137 0.960896
493493 −10.4853 −0.472233
494494 3.58579 0.161332
495495 0 0
496496 −0.171573 −0.00770385
497497 −3.31371 −0.148640
498498 6.65685 0.298301
499499 18.4853 0.827515 0.413757 0.910387i 0.364216π-0.364216\pi
0.413757 + 0.910387i 0.364216π0.364216\pi
500500 0 0
501501 8.48528 0.379094
502502 11.4853 0.512613
503503 −10.1421 −0.452215 −0.226108 0.974102i 0.572600π-0.572600\pi
−0.226108 + 0.974102i 0.572600π0.572600\pi
504504 −16.9706 −0.755929
505505 0 0
506506 −6.07107 −0.269892
507507 −30.9706 −1.37545
508508 11.3137 0.501965
509509 21.6569 0.959923 0.479962 0.877290i 0.340651π-0.340651\pi
0.479962 + 0.877290i 0.340651π0.340651\pi
510510 0 0
511511 −29.1127 −1.28787
512512 −11.0000 −0.486136
513513 −3.58579 −0.158316
514514 −4.07107 −0.179567
515515 0 0
516516 −25.3137 −1.11437
517517 11.6569 0.512668
518518 2.34315 0.102952
519519 17.6569 0.775050
520520 0 0
521521 25.9706 1.13779 0.568896 0.822410i 0.307371π-0.307371\pi
0.568896 + 0.822410i 0.307371π0.307371\pi
522522 −6.14214 −0.268834
523523 16.5563 0.723959 0.361979 0.932186i 0.382101π-0.382101\pi
0.361979 + 0.932186i 0.382101π0.382101\pi
524524 −12.8284 −0.560412
525525 0 0
526526 17.5858 0.766777
527527 0.828427 0.0360869
528528 −2.41421 −0.105065
529529 13.8579 0.602516
530530 0 0
531531 33.4558 1.45186
532532 −17.3137 −0.750644
533533 −3.65685 −0.158396
534534 11.6569 0.504441
535535 0 0
536536 −14.2721 −0.616460
537537 −38.1421 −1.64595
538538 −17.0000 −0.732922
539539 −3.00000 −0.129219
540540 0 0
541541 18.1127 0.778726 0.389363 0.921084i 0.372695π-0.372695\pi
0.389363 + 0.921084i 0.372695π0.372695\pi
542542 −22.6274 −0.971931
543543 −24.1421 −1.03604
544544 24.1421 1.03509
545545 0 0
546546 2.00000 0.0855921
547547 28.2843 1.20935 0.604674 0.796473i 0.293303π-0.293303\pi
0.604674 + 0.796473i 0.293303π0.293303\pi
548548 8.48528 0.362473
549549 −33.9411 −1.44857
550550 0 0
551551 −18.7990 −0.800864
552552 43.9706 1.87151
553553 9.65685 0.410651
554554 26.1421 1.11067
555555 0 0
556556 −2.34315 −0.0993715
557557 −23.2426 −0.984822 −0.492411 0.870363i 0.663884π-0.663884\pi
−0.492411 + 0.870363i 0.663884π0.663884\pi
558558 0.485281 0.0205436
559559 4.34315 0.183695
560560 0 0
561561 11.6569 0.492153
562562 10.0000 0.421825
563563 9.45584 0.398516 0.199258 0.979947i 0.436147π-0.436147\pi
0.199258 + 0.979947i 0.436147π0.436147\pi
564564 −28.1421 −1.18500
565565 0 0
566566 −3.72792 −0.156696
567567 −18.9706 −0.796689
568568 4.97056 0.208560
569569 −14.9706 −0.627599 −0.313799 0.949489i 0.601602π-0.601602\pi
−0.313799 + 0.949489i 0.601602π0.601602\pi
570570 0 0
571571 10.1716 0.425667 0.212834 0.977088i 0.431731π-0.431731\pi
0.212834 + 0.977088i 0.431731π0.431731\pi
572572 −0.414214 −0.0173191
573573 9.65685 0.403421
574574 −17.6569 −0.736983
575575 0 0
576576 19.7990 0.824958
577577 −35.3137 −1.47013 −0.735064 0.677997i 0.762848π-0.762848\pi
−0.735064 + 0.677997i 0.762848π0.762848\pi
578578 6.31371 0.262616
579579 25.3137 1.05200
580580 0 0
581581 5.51472 0.228789
582582 44.6274 1.84987
583583 1.17157 0.0485216
584584 43.6690 1.80704
585585 0 0
586586 −16.4853 −0.681001
587587 −6.68629 −0.275973 −0.137986 0.990434i 0.544063π-0.544063\pi
−0.137986 + 0.990434i 0.544063π0.544063\pi
588588 7.24264 0.298681
589589 1.48528 0.0612000
590590 0 0
591591 12.4853 0.513576
592592 −1.17157 −0.0481513
593593 −36.0711 −1.48126 −0.740631 0.671912i 0.765473π-0.765473\pi
−0.740631 + 0.671912i 0.765473π0.765473\pi
594594 −0.414214 −0.0169954
595595 0 0
596596 1.17157 0.0479895
597597 −59.4558 −2.43337
598598 −2.51472 −0.102834
599599 38.2843 1.56425 0.782126 0.623120i 0.214135π-0.214135\pi
0.782126 + 0.623120i 0.214135π0.214135\pi
600600 0 0
601601 −39.9706 −1.63043 −0.815217 0.579156i 0.803382π-0.803382\pi
−0.815217 + 0.579156i 0.803382π0.803382\pi
602602 20.9706 0.854696
603603 13.4558 0.547964
604604 −1.00000 −0.0406894
605605 0 0
606606 37.7990 1.53548
607607 −24.6274 −0.999596 −0.499798 0.866142i 0.666592π-0.666592\pi
−0.499798 + 0.866142i 0.666592π0.666592\pi
608608 43.2843 1.75541
609609 −10.4853 −0.424885
610610 0 0
611611 4.82843 0.195337
612612 −13.6569 −0.552046
613613 31.1127 1.25663 0.628315 0.777959i 0.283745π-0.283745\pi
0.628315 + 0.777959i 0.283745π0.283745\pi
614614 18.4853 0.746005
615615 0 0
616616 −6.00000 −0.241747
617617 23.0416 0.927621 0.463811 0.885934i 0.346482π-0.346482\pi
0.463811 + 0.885934i 0.346482π0.346482\pi
618618 22.1421 0.890687
619619 −31.7990 −1.27811 −0.639055 0.769161i 0.720674π-0.720674\pi
−0.639055 + 0.769161i 0.720674π0.720674\pi
620620 0 0
621621 2.51472 0.100912
622622 −6.31371 −0.253157
623623 9.65685 0.386894
624624 −1.00000 −0.0400320
625625 0 0
626626 −22.6274 −0.904373
627627 20.8995 0.834645
628628 1.10051 0.0439149
629629 5.65685 0.225554
630630 0 0
631631 13.5147 0.538012 0.269006 0.963138i 0.413305π-0.413305\pi
0.269006 + 0.963138i 0.413305π0.413305\pi
632632 −14.4853 −0.576194
633633 −13.6569 −0.542811
634634 −26.5563 −1.05469
635635 0 0
636636 −2.82843 −0.112154
637637 −1.24264 −0.0492352
638638 −2.17157 −0.0859734
639639 −4.68629 −0.185387
640640 0 0
641641 37.2843 1.47264 0.736320 0.676633i 0.236562π-0.236562\pi
0.736320 + 0.676633i 0.236562π0.236562\pi
642642 23.4853 0.926890
643643 18.0000 0.709851 0.354925 0.934895i 0.384506π-0.384506\pi
0.354925 + 0.934895i 0.384506π0.384506\pi
644644 12.1421 0.478467
645645 0 0
646646 41.7990 1.64456
647647 −35.3137 −1.38833 −0.694163 0.719818i 0.744225π-0.744225\pi
−0.694163 + 0.719818i 0.744225π0.744225\pi
648648 28.4558 1.11785
649649 11.8284 0.464306
650650 0 0
651651 0.828427 0.0324686
652652 3.58579 0.140430
653653 19.6569 0.769232 0.384616 0.923077i 0.374334π-0.374334\pi
0.384616 + 0.923077i 0.374334π0.374334\pi
654654 −34.1421 −1.33506
655655 0 0
656656 8.82843 0.344692
657657 −41.1716 −1.60626
658658 23.3137 0.908863
659659 5.97056 0.232580 0.116290 0.993215i 0.462900π-0.462900\pi
0.116290 + 0.993215i 0.462900π0.462900\pi
660660 0 0
661661 3.79899 0.147764 0.0738818 0.997267i 0.476461π-0.476461\pi
0.0738818 + 0.997267i 0.476461π0.476461\pi
662662 −28.0000 −1.08825
663663 4.82843 0.187521
664664 −8.27208 −0.321019
665665 0 0
666666 3.31371 0.128404
667667 13.1838 0.510477
668668 −3.51472 −0.135989
669669 −41.7990 −1.61604
670670 0 0
671671 −12.0000 −0.463255
672672 24.1421 0.931303
673673 19.3137 0.744489 0.372244 0.928135i 0.378588π-0.378588\pi
0.372244 + 0.928135i 0.378588π0.378588\pi
674674 −21.7279 −0.836928
675675 0 0
676676 12.8284 0.493401
677677 −29.4558 −1.13208 −0.566040 0.824378i 0.691525π-0.691525\pi
−0.566040 + 0.824378i 0.691525π0.691525\pi
678678 24.4853 0.940352
679679 36.9706 1.41880
680680 0 0
681681 −14.4853 −0.555077
682682 0.171573 0.00656986
683683 25.2426 0.965883 0.482941 0.875653i 0.339568π-0.339568\pi
0.482941 + 0.875653i 0.339568π0.339568\pi
684684 −24.4853 −0.936218
685685 0 0
686686 −20.0000 −0.763604
687687 −37.7279 −1.43941
688688 −10.4853 −0.399748
689689 0.485281 0.0184877
690690 0 0
691691 −7.17157 −0.272819 −0.136410 0.990653i 0.543556π-0.543556\pi
−0.136410 + 0.990653i 0.543556π0.543556\pi
692692 −7.31371 −0.278025
693693 5.65685 0.214886
694694 25.4558 0.966291
695695 0 0
696696 15.7279 0.596165
697697 −42.6274 −1.61463
698698 −14.5147 −0.549390
699699 −64.9411 −2.45630
700700 0 0
701701 2.68629 0.101460 0.0507299 0.998712i 0.483845π-0.483845\pi
0.0507299 + 0.998712i 0.483845π0.483845\pi
702702 −0.171573 −0.00647560
703703 10.1421 0.382518
704704 7.00000 0.263822
705705 0 0
706706 4.75736 0.179046
707707 31.3137 1.17767
708708 −28.5563 −1.07321
709709 −36.6569 −1.37668 −0.688339 0.725390i 0.741660π-0.741660\pi
−0.688339 + 0.725390i 0.741660π0.741660\pi
710710 0 0
711711 13.6569 0.512172
712712 −14.4853 −0.542859
713713 −1.04163 −0.0390094
714714 23.3137 0.872494
715715 0 0
716716 15.7990 0.590436
717717 −11.2426 −0.419864
718718 −12.1421 −0.453140
719719 16.1421 0.602000 0.301000 0.953624i 0.402680π-0.402680\pi
0.301000 + 0.953624i 0.402680π0.402680\pi
720720 0 0
721721 18.3431 0.683134
722722 55.9411 2.08191
723723 −5.24264 −0.194976
724724 10.0000 0.371647
725725 0 0
726726 −24.1421 −0.895999
727727 −5.65685 −0.209801 −0.104901 0.994483i 0.533452π-0.533452\pi
−0.104901 + 0.994483i 0.533452π0.533452\pi
728728 −2.48528 −0.0921107
729729 −23.8284 −0.882534
730730 0 0
731731 50.6274 1.87252
732732 28.9706 1.07078
733733 −45.1838 −1.66890 −0.834450 0.551083i 0.814215π-0.814215\pi
−0.834450 + 0.551083i 0.814215π0.814215\pi
734734 −13.3137 −0.491418
735735 0 0
736736 −30.3553 −1.11891
737737 4.75736 0.175240
738738 −24.9706 −0.919179
739739 12.6274 0.464507 0.232254 0.972655i 0.425390π-0.425390\pi
0.232254 + 0.972655i 0.425390π0.425390\pi
740740 0 0
741741 8.65685 0.318017
742742 2.34315 0.0860196
743743 −20.8284 −0.764121 −0.382060 0.924137i 0.624785π-0.624785\pi
−0.382060 + 0.924137i 0.624785π0.624785\pi
744744 −1.24264 −0.0455574
745745 0 0
746746 −25.8701 −0.947170
747747 7.79899 0.285350
748748 −4.82843 −0.176545
749749 19.4558 0.710901
750750 0 0
751751 11.8579 0.432700 0.216350 0.976316i 0.430585π-0.430585\pi
0.216350 + 0.976316i 0.430585π0.430585\pi
752752 −11.6569 −0.425082
753753 27.7279 1.01046
754754 −0.899495 −0.0327577
755755 0 0
756756 0.828427 0.0301296
757757 11.5147 0.418510 0.209255 0.977861i 0.432896π-0.432896\pi
0.209255 + 0.977861i 0.432896π0.432896\pi
758758 6.00000 0.217930
759759 −14.6569 −0.532010
760760 0 0
761761 −11.4558 −0.415274 −0.207637 0.978206i 0.566577π-0.566577\pi
−0.207637 + 0.978206i 0.566577π0.566577\pi
762762 −27.3137 −0.989471
763763 −28.2843 −1.02396
764764 −4.00000 −0.144715
765765 0 0
766766 −4.34315 −0.156924
767767 4.89949 0.176911
768768 −41.0416 −1.48096
769769 40.7696 1.47019 0.735094 0.677965i 0.237138π-0.237138\pi
0.735094 + 0.677965i 0.237138π0.237138\pi
770770 0 0
771771 −9.82843 −0.353962
772772 −10.4853 −0.377374
773773 25.8579 0.930043 0.465021 0.885299i 0.346047π-0.346047\pi
0.465021 + 0.885299i 0.346047π0.346047\pi
774774 29.6569 1.06599
775775 0 0
776776 −55.4558 −1.99075
777777 5.65685 0.202939
778778 −31.9411 −1.14514
779779 −76.4264 −2.73826
780780 0 0
781781 −1.65685 −0.0592869
782782 −29.3137 −1.04826
783783 0.899495 0.0321453
784784 3.00000 0.107143
785785 0 0
786786 30.9706 1.10468
787787 −14.3431 −0.511278 −0.255639 0.966772i 0.582286π-0.582286\pi
−0.255639 + 0.966772i 0.582286π0.582286\pi
788788 −5.17157 −0.184230
789789 42.4558 1.51147
790790 0 0
791791 20.2843 0.721226
792792 −8.48528 −0.301511
793793 −4.97056 −0.176510
794794 29.6569 1.05248
795795 0 0
796796 24.6274 0.872896
797797 2.34315 0.0829985 0.0414992 0.999139i 0.486787π-0.486787\pi
0.0414992 + 0.999139i 0.486787π0.486787\pi
798798 41.7990 1.47967
799799 56.2843 1.99119
800800 0 0
801801 13.6569 0.482541
802802 −34.9706 −1.23485
803803 −14.5563 −0.513682
804804 −11.4853 −0.405055
805805 0 0
806806 0.0710678 0.00250326
807807 −41.0416 −1.44473
808808 −46.9706 −1.65242
809809 −50.4264 −1.77290 −0.886449 0.462826i 0.846835π-0.846835\pi
−0.886449 + 0.462826i 0.846835π0.846835\pi
810810 0 0
811811 −28.6274 −1.00524 −0.502622 0.864506i 0.667631π-0.667631\pi
−0.502622 + 0.864506i 0.667631π0.667631\pi
812812 4.34315 0.152415
813813 −54.6274 −1.91587
814814 1.17157 0.0410636
815815 0 0
816816 −11.6569 −0.408072
817817 90.7696 3.17562
818818 −14.0000 −0.489499
819819 2.34315 0.0818761
820820 0 0
821821 20.3431 0.709981 0.354990 0.934870i 0.384484π-0.384484\pi
0.354990 + 0.934870i 0.384484π0.384484\pi
822822 −20.4853 −0.714506
823823 8.34315 0.290824 0.145412 0.989371i 0.453549π-0.453549\pi
0.145412 + 0.989371i 0.453549π0.453549\pi
824824 −27.5147 −0.958521
825825 0 0
826826 23.6569 0.823127
827827 31.1127 1.08189 0.540947 0.841057i 0.318066π-0.318066\pi
0.540947 + 0.841057i 0.318066π0.318066\pi
828828 17.1716 0.596753
829829 3.34315 0.116112 0.0580561 0.998313i 0.481510π-0.481510\pi
0.0580561 + 0.998313i 0.481510π0.481510\pi
830830 0 0
831831 63.1127 2.18936
832832 2.89949 0.100522
833833 −14.4853 −0.501885
834834 5.65685 0.195881
835835 0 0
836836 −8.65685 −0.299404
837837 −0.0710678 −0.00245646
838838 9.79899 0.338500
839839 12.6863 0.437979 0.218990 0.975727i 0.429724π-0.429724\pi
0.218990 + 0.975727i 0.429724π0.429724\pi
840840 0 0
841841 −24.2843 −0.837389
842842 −30.8284 −1.06242
843843 24.1421 0.831499
844844 5.65685 0.194717
845845 0 0
846846 32.9706 1.13355
847847 −20.0000 −0.687208
848848 −1.17157 −0.0402320
849849 −9.00000 −0.308879
850850 0 0
851851 −7.11270 −0.243820
852852 4.00000 0.137038
853853 47.7990 1.63661 0.818303 0.574787i 0.194915π-0.194915\pi
0.818303 + 0.574787i 0.194915π0.194915\pi
854854 −24.0000 −0.821263
855855 0 0
856856 −29.1838 −0.997481
857857 −2.07107 −0.0707463 −0.0353732 0.999374i 0.511262π-0.511262\pi
−0.0353732 + 0.999374i 0.511262π0.511262\pi
858858 1.00000 0.0341394
859859 15.3137 0.522497 0.261248 0.965272i 0.415866π-0.415866\pi
0.261248 + 0.965272i 0.415866π0.415866\pi
860860 0 0
861861 −42.6274 −1.45274
862862 8.82843 0.300697
863863 −39.2548 −1.33625 −0.668125 0.744049i 0.732903π-0.732903\pi
−0.668125 + 0.744049i 0.732903π0.732903\pi
864864 −2.07107 −0.0704592
865865 0 0
866866 −5.17157 −0.175737
867867 15.2426 0.517667
868868 −0.343146 −0.0116471
869869 4.82843 0.163793
870870 0 0
871871 1.97056 0.0667700
872872 42.4264 1.43674
873873 52.2843 1.76955
874874 −52.5563 −1.77775
875875 0 0
876876 35.1421 1.18734
877877 21.1005 0.712513 0.356257 0.934388i 0.384053π-0.384053\pi
0.356257 + 0.934388i 0.384053π0.384053\pi
878878 19.2843 0.650813
879879 −39.7990 −1.34239
880880 0 0
881881 33.1716 1.11758 0.558789 0.829310i 0.311266π-0.311266\pi
0.558789 + 0.829310i 0.311266π0.311266\pi
882882 −8.48528 −0.285714
883883 −0.485281 −0.0163310 −0.00816551 0.999967i 0.502599π-0.502599\pi
−0.00816551 + 0.999967i 0.502599π0.502599\pi
884884 −2.00000 −0.0672673
885885 0 0
886886 30.0000 1.00787
887887 23.8701 0.801478 0.400739 0.916192i 0.368753π-0.368753\pi
0.400739 + 0.916192i 0.368753π0.368753\pi
888888 −8.48528 −0.284747
889889 −22.6274 −0.758899
890890 0 0
891891 −9.48528 −0.317769
892892 17.3137 0.579706
893893 100.912 3.37688
894894 −2.82843 −0.0945968
895895 0 0
896896 −6.00000 −0.200446
897897 −6.07107 −0.202707
898898 −14.4853 −0.483380
899899 −0.372583 −0.0124263
900900 0 0
901901 5.65685 0.188457
902902 −8.82843 −0.293954
903903 50.6274 1.68477
904904 −30.4264 −1.01197
905905 0 0
906906 2.41421 0.0802069
907907 −53.5980 −1.77969 −0.889846 0.456261i 0.849188π-0.849188\pi
−0.889846 + 0.456261i 0.849188π0.849188\pi
908908 6.00000 0.199117
909909 44.2843 1.46882
910910 0 0
911911 −23.8284 −0.789471 −0.394736 0.918795i 0.629164π-0.629164\pi
−0.394736 + 0.918795i 0.629164π0.629164\pi
912912 −20.8995 −0.692051
913913 2.75736 0.0912553
914914 −5.51472 −0.182411
915915 0 0
916916 15.6274 0.516344
917917 25.6569 0.847264
918918 −2.00000 −0.0660098
919919 20.0000 0.659739 0.329870 0.944027i 0.392995π-0.392995\pi
0.329870 + 0.944027i 0.392995π0.392995\pi
920920 0 0
921921 44.6274 1.47052
922922 −0.343146 −0.0113009
923923 −0.686292 −0.0225896
924924 −4.82843 −0.158844
925925 0 0
926926 22.1421 0.727636
927927 25.9411 0.852018
928928 −10.8579 −0.356427
929929 −10.0000 −0.328089 −0.164045 0.986453i 0.552454π-0.552454\pi
−0.164045 + 0.986453i 0.552454π0.552454\pi
930930 0 0
931931 −25.9706 −0.851151
932932 26.8995 0.881122
933933 −15.2426 −0.499022
934934 5.31371 0.173870
935935 0 0
936936 −3.51472 −0.114882
937937 −22.3431 −0.729919 −0.364959 0.931023i 0.618917π-0.618917\pi
−0.364959 + 0.931023i 0.618917π0.618917\pi
938938 9.51472 0.310667
939939 −54.6274 −1.78270
940940 0 0
941941 21.1716 0.690174 0.345087 0.938571i 0.387849π-0.387849\pi
0.345087 + 0.938571i 0.387849π0.387849\pi
942942 −2.65685 −0.0865650
943943 53.5980 1.74539
944944 −11.8284 −0.384983
945945 0 0
946946 10.4853 0.340906
947947 −16.0711 −0.522240 −0.261120 0.965306i 0.584092π-0.584092\pi
−0.261120 + 0.965306i 0.584092π0.584092\pi
948948 −11.6569 −0.378597
949949 −6.02944 −0.195724
950950 0 0
951951 −64.1127 −2.07900
952952 −28.9706 −0.938941
953953 −49.5980 −1.60664 −0.803318 0.595550i 0.796934π-0.796934\pi
−0.803318 + 0.595550i 0.796934π0.796934\pi
954954 3.31371 0.107285
955955 0 0
956956 4.65685 0.150613
957957 −5.24264 −0.169471
958958 −1.85786 −0.0600249
959959 −16.9706 −0.548008
960960 0 0
961961 −30.9706 −0.999050
962962 0.485281 0.0156461
963963 27.5147 0.886649
964964 2.17157 0.0699417
965965 0 0
966966 −29.3137 −0.943153
967967 28.4142 0.913740 0.456870 0.889534i 0.348971π-0.348971\pi
0.456870 + 0.889534i 0.348971π0.348971\pi
968968 30.0000 0.964237
969969 100.912 3.24175
970970 0 0
971971 −40.4853 −1.29923 −0.649617 0.760261i 0.725071π-0.725071\pi
−0.649617 + 0.760261i 0.725071π0.725071\pi
972972 21.6569 0.694644
973973 4.68629 0.150236
974974 −3.17157 −0.101624
975975 0 0
976976 12.0000 0.384111
977977 −5.17157 −0.165453 −0.0827266 0.996572i 0.526363π-0.526363\pi
−0.0827266 + 0.996572i 0.526363π0.526363\pi
978978 −8.65685 −0.276816
979979 4.82843 0.154317
980980 0 0
981981 −40.0000 −1.27710
982982 37.6569 1.20168
983983 −35.9411 −1.14634 −0.573172 0.819435i 0.694287π-0.694287\pi
−0.573172 + 0.819435i 0.694287π0.694287\pi
984984 63.9411 2.03837
985985 0 0
986986 −10.4853 −0.333919
987987 56.2843 1.79155
988988 −3.58579 −0.114079
989989 −63.6569 −2.02417
990990 0 0
991991 −12.6569 −0.402058 −0.201029 0.979585i 0.564429π-0.564429\pi
−0.201029 + 0.979585i 0.564429π0.564429\pi
992992 0.857864 0.0272372
993993 −67.5980 −2.14516
994994 −3.31371 −0.105104
995995 0 0
996996 −6.65685 −0.210930
997997 42.9706 1.36089 0.680446 0.732799i 0.261786π-0.261786\pi
0.680446 + 0.732799i 0.261786π0.261786\pi
998998 18.4853 0.585141
999999 −0.485281 −0.0153536
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 3775.2.a.i.1.2 2
5.4 even 2 755.2.a.g.1.1 2
15.14 odd 2 6795.2.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
755.2.a.g.1.1 2 5.4 even 2
3775.2.a.i.1.2 2 1.1 even 1 trivial
6795.2.a.t.1.1 2 15.14 odd 2