Properties

Label 525.2.a.k.1.3
Level 525525
Weight 22
Character 525.1
Self dual yes
Analytic conductor 4.1924.192
Analytic rank 00
Dimension 33
CM no
Inner twists 11

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [525,2,Mod(1,525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("525.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 525=3527 525 = 3 \cdot 5^{2} \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 525.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,1,-3,5,0,-1,-3,9,3,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 4.192146106124.19214610612
Analytic rank: 00
Dimension: 33
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x3x23x+1 x^{3} - x^{2} - 3x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 105)
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.3
Root 2.170092.17009 of defining polynomial
Character χ\chi == 525.1

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
f(q)f(q) == q+2.70928q21.00000q3+5.34017q42.70928q61.00000q7+9.04945q8+1.00000q9+2.00000q115.34017q120.921622q132.70928q14+13.8371q161.07838q17+2.70928q18+3.07838q19+1.00000q21+5.41855q222.34017q239.04945q242.49693q261.00000q275.34017q286.68035q297.75872q31+19.3896q322.00000q332.92162q34+5.34017q3610.8371q37+8.34017q38+0.921622q39+6.49693q41+2.70928q42+6.52359q43+10.6803q446.34017q464.68035q4713.8371q48+1.00000q49+1.07838q514.92162q523.75872q532.70928q549.04945q563.07838q5718.0989q58+10.5236q594.15676q6121.0205q621.00000q63+24.8576q645.41855q664.68035q675.75872q68+2.34017q69+2.00000q71+9.04945q727.07838q7329.3607q74+16.4391q762.00000q77+2.49693q78+6.15676q79+1.00000q81+17.6020q826.83710q83+5.34017q84+17.6742q86+6.68035q87+18.0989q88+8.34017q89+0.921622q9112.4969q92+7.75872q9312.6803q9419.3896q96+8.43907q97+2.70928q98+2.00000q99+O(q100)q+2.70928 q^{2} -1.00000 q^{3} +5.34017 q^{4} -2.70928 q^{6} -1.00000 q^{7} +9.04945 q^{8} +1.00000 q^{9} +2.00000 q^{11} -5.34017 q^{12} -0.921622 q^{13} -2.70928 q^{14} +13.8371 q^{16} -1.07838 q^{17} +2.70928 q^{18} +3.07838 q^{19} +1.00000 q^{21} +5.41855 q^{22} -2.34017 q^{23} -9.04945 q^{24} -2.49693 q^{26} -1.00000 q^{27} -5.34017 q^{28} -6.68035 q^{29} -7.75872 q^{31} +19.3896 q^{32} -2.00000 q^{33} -2.92162 q^{34} +5.34017 q^{36} -10.8371 q^{37} +8.34017 q^{38} +0.921622 q^{39} +6.49693 q^{41} +2.70928 q^{42} +6.52359 q^{43} +10.6803 q^{44} -6.34017 q^{46} -4.68035 q^{47} -13.8371 q^{48} +1.00000 q^{49} +1.07838 q^{51} -4.92162 q^{52} -3.75872 q^{53} -2.70928 q^{54} -9.04945 q^{56} -3.07838 q^{57} -18.0989 q^{58} +10.5236 q^{59} -4.15676 q^{61} -21.0205 q^{62} -1.00000 q^{63} +24.8576 q^{64} -5.41855 q^{66} -4.68035 q^{67} -5.75872 q^{68} +2.34017 q^{69} +2.00000 q^{71} +9.04945 q^{72} -7.07838 q^{73} -29.3607 q^{74} +16.4391 q^{76} -2.00000 q^{77} +2.49693 q^{78} +6.15676 q^{79} +1.00000 q^{81} +17.6020 q^{82} -6.83710 q^{83} +5.34017 q^{84} +17.6742 q^{86} +6.68035 q^{87} +18.0989 q^{88} +8.34017 q^{89} +0.921622 q^{91} -12.4969 q^{92} +7.75872 q^{93} -12.6803 q^{94} -19.3896 q^{96} +8.43907 q^{97} +2.70928 q^{98} +2.00000 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 3q+q23q3+5q4q63q7+9q8+3q9+6q115q126q13q14+13q16+q18+6q19+3q21+2q22+4q239q24+10q26++6q99+O(q100) 3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} - 3 q^{7} + 9 q^{8} + 3 q^{9} + 6 q^{11} - 5 q^{12} - 6 q^{13} - q^{14} + 13 q^{16} + q^{18} + 6 q^{19} + 3 q^{21} + 2 q^{22} + 4 q^{23} - 9 q^{24} + 10 q^{26}+ \cdots + 6 q^{99}+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 2.70928 1.91575 0.957873 0.287190i 0.0927213π-0.0927213\pi
0.957873 + 0.287190i 0.0927213π0.0927213\pi
33 −1.00000 −0.577350
44 5.34017 2.67009
55 0 0
66 −2.70928 −1.10606
77 −1.00000 −0.377964
88 9.04945 3.19946
99 1.00000 0.333333
1010 0 0
1111 2.00000 0.603023 0.301511 0.953463i 0.402509π-0.402509\pi
0.301511 + 0.953463i 0.402509π0.402509\pi
1212 −5.34017 −1.54158
1313 −0.921622 −0.255612 −0.127806 0.991799i 0.540793π-0.540793\pi
−0.127806 + 0.991799i 0.540793π0.540793\pi
1414 −2.70928 −0.724084
1515 0 0
1616 13.8371 3.45928
1717 −1.07838 −0.261545 −0.130773 0.991412i 0.541746π-0.541746\pi
−0.130773 + 0.991412i 0.541746π0.541746\pi
1818 2.70928 0.638582
1919 3.07838 0.706228 0.353114 0.935580i 0.385123π-0.385123\pi
0.353114 + 0.935580i 0.385123π0.385123\pi
2020 0 0
2121 1.00000 0.218218
2222 5.41855 1.15524
2323 −2.34017 −0.487960 −0.243980 0.969780i 0.578453π-0.578453\pi
−0.243980 + 0.969780i 0.578453π0.578453\pi
2424 −9.04945 −1.84721
2525 0 0
2626 −2.49693 −0.489688
2727 −1.00000 −0.192450
2828 −5.34017 −1.00920
2929 −6.68035 −1.24051 −0.620255 0.784401i 0.712971π-0.712971\pi
−0.620255 + 0.784401i 0.712971π0.712971\pi
3030 0 0
3131 −7.75872 −1.39351 −0.696754 0.717310i 0.745373π-0.745373\pi
−0.696754 + 0.717310i 0.745373π0.745373\pi
3232 19.3896 3.42763
3333 −2.00000 −0.348155
3434 −2.92162 −0.501054
3535 0 0
3636 5.34017 0.890029
3737 −10.8371 −1.78161 −0.890804 0.454387i 0.849858π-0.849858\pi
−0.890804 + 0.454387i 0.849858π0.849858\pi
3838 8.34017 1.35295
3939 0.921622 0.147578
4040 0 0
4141 6.49693 1.01465 0.507325 0.861755i 0.330634π-0.330634\pi
0.507325 + 0.861755i 0.330634π0.330634\pi
4242 2.70928 0.418050
4343 6.52359 0.994838 0.497419 0.867510i 0.334281π-0.334281\pi
0.497419 + 0.867510i 0.334281π0.334281\pi
4444 10.6803 1.61012
4545 0 0
4646 −6.34017 −0.934808
4747 −4.68035 −0.682699 −0.341349 0.939937i 0.610884π-0.610884\pi
−0.341349 + 0.939937i 0.610884π0.610884\pi
4848 −13.8371 −1.99721
4949 1.00000 0.142857
5050 0 0
5151 1.07838 0.151003
5252 −4.92162 −0.682506
5353 −3.75872 −0.516300 −0.258150 0.966105i 0.583113π-0.583113\pi
−0.258150 + 0.966105i 0.583113π0.583113\pi
5454 −2.70928 −0.368686
5555 0 0
5656 −9.04945 −1.20928
5757 −3.07838 −0.407741
5858 −18.0989 −2.37650
5959 10.5236 1.37005 0.685027 0.728517i 0.259790π-0.259790\pi
0.685027 + 0.728517i 0.259790π0.259790\pi
6060 0 0
6161 −4.15676 −0.532218 −0.266109 0.963943i 0.585738π-0.585738\pi
−0.266109 + 0.963943i 0.585738π0.585738\pi
6262 −21.0205 −2.66961
6363 −1.00000 −0.125988
6464 24.8576 3.10720
6565 0 0
6666 −5.41855 −0.666977
6767 −4.68035 −0.571795 −0.285898 0.958260i 0.592292π-0.592292\pi
−0.285898 + 0.958260i 0.592292π0.592292\pi
6868 −5.75872 −0.698348
6969 2.34017 0.281724
7070 0 0
7171 2.00000 0.237356 0.118678 0.992933i 0.462134π-0.462134\pi
0.118678 + 0.992933i 0.462134π0.462134\pi
7272 9.04945 1.06649
7373 −7.07838 −0.828461 −0.414231 0.910172i 0.635949π-0.635949\pi
−0.414231 + 0.910172i 0.635949π0.635949\pi
7474 −29.3607 −3.41311
7575 0 0
7676 16.4391 1.88569
7777 −2.00000 −0.227921
7878 2.49693 0.282721
7979 6.15676 0.692689 0.346345 0.938107i 0.387423π-0.387423\pi
0.346345 + 0.938107i 0.387423π0.387423\pi
8080 0 0
8181 1.00000 0.111111
8282 17.6020 1.94381
8383 −6.83710 −0.750469 −0.375235 0.926930i 0.622438π-0.622438\pi
−0.375235 + 0.926930i 0.622438π0.622438\pi
8484 5.34017 0.582661
8585 0 0
8686 17.6742 1.90586
8787 6.68035 0.716208
8888 18.0989 1.92935
8989 8.34017 0.884057 0.442028 0.897001i 0.354259π-0.354259\pi
0.442028 + 0.897001i 0.354259π0.354259\pi
9090 0 0
9191 0.921622 0.0966123
9292 −12.4969 −1.30289
9393 7.75872 0.804542
9494 −12.6803 −1.30788
9595 0 0
9696 −19.3896 −1.97894
9797 8.43907 0.856858 0.428429 0.903576i 0.359067π-0.359067\pi
0.428429 + 0.903576i 0.359067π0.359067\pi
9898 2.70928 0.273678
9999 2.00000 0.201008
100100 0 0
101101 −5.81658 −0.578772 −0.289386 0.957213i 0.593451π-0.593451\pi
−0.289386 + 0.957213i 0.593451π0.593451\pi
102102 2.92162 0.289284
103103 2.15676 0.212511 0.106256 0.994339i 0.466114π-0.466114\pi
0.106256 + 0.994339i 0.466114π0.466114\pi
104104 −8.34017 −0.817821
105105 0 0
106106 −10.1834 −0.989101
107107 16.4969 1.59482 0.797409 0.603439i 0.206203π-0.206203\pi
0.797409 + 0.603439i 0.206203π0.206203\pi
108108 −5.34017 −0.513858
109109 −12.8371 −1.22957 −0.614786 0.788694i 0.710757π-0.710757\pi
−0.614786 + 0.788694i 0.710757π0.710757\pi
110110 0 0
111111 10.8371 1.02861
112112 −13.8371 −1.30748
113113 5.23513 0.492480 0.246240 0.969209i 0.420805π-0.420805\pi
0.246240 + 0.969209i 0.420805π0.420805\pi
114114 −8.34017 −0.781129
115115 0 0
116116 −35.6742 −3.31227
117117 −0.921622 −0.0852040
118118 28.5113 2.62468
119119 1.07838 0.0988547
120120 0 0
121121 −7.00000 −0.636364
122122 −11.2618 −1.01960
123123 −6.49693 −0.585808
124124 −41.4329 −3.72079
125125 0 0
126126 −2.70928 −0.241361
127127 −1.84324 −0.163562 −0.0817808 0.996650i 0.526061π-0.526061\pi
−0.0817808 + 0.996650i 0.526061π0.526061\pi
128128 28.5669 2.52498
129129 −6.52359 −0.574370
130130 0 0
131131 1.47641 0.128995 0.0644973 0.997918i 0.479456π-0.479456\pi
0.0644973 + 0.997918i 0.479456π0.479456\pi
132132 −10.6803 −0.929605
133133 −3.07838 −0.266929
134134 −12.6803 −1.09542
135135 0 0
136136 −9.75872 −0.836804
137137 −4.43907 −0.379255 −0.189628 0.981856i 0.560728π-0.560728\pi
−0.189628 + 0.981856i 0.560728π0.560728\pi
138138 6.34017 0.539711
139139 13.6020 1.15370 0.576852 0.816849i 0.304281π-0.304281\pi
0.576852 + 0.816849i 0.304281π0.304281\pi
140140 0 0
141141 4.68035 0.394156
142142 5.41855 0.454715
143143 −1.84324 −0.154140
144144 13.8371 1.15309
145145 0 0
146146 −19.1773 −1.58712
147147 −1.00000 −0.0824786
148148 −57.8720 −4.75705
149149 15.6742 1.28408 0.642040 0.766671i 0.278088π-0.278088\pi
0.642040 + 0.766671i 0.278088π0.278088\pi
150150 0 0
151151 5.84324 0.475516 0.237758 0.971324i 0.423587π-0.423587\pi
0.237758 + 0.971324i 0.423587π0.423587\pi
152152 27.8576 2.25955
153153 −1.07838 −0.0871817
154154 −5.41855 −0.436639
155155 0 0
156156 4.92162 0.394045
157157 −4.92162 −0.392788 −0.196394 0.980525i 0.562923π-0.562923\pi
−0.196394 + 0.980525i 0.562923π0.562923\pi
158158 16.6803 1.32702
159159 3.75872 0.298086
160160 0 0
161161 2.34017 0.184431
162162 2.70928 0.212861
163163 9.84324 0.770982 0.385491 0.922712i 0.374032π-0.374032\pi
0.385491 + 0.922712i 0.374032π0.374032\pi
164164 34.6947 2.70920
165165 0 0
166166 −18.5236 −1.43771
167167 19.2039 1.48605 0.743023 0.669266i 0.233391π-0.233391\pi
0.743023 + 0.669266i 0.233391π0.233391\pi
168168 9.04945 0.698180
169169 −12.1506 −0.934662
170170 0 0
171171 3.07838 0.235409
172172 34.8371 2.65630
173173 22.4391 1.70601 0.853005 0.521902i 0.174777π-0.174777\pi
0.853005 + 0.521902i 0.174777π0.174777\pi
174174 18.0989 1.37207
175175 0 0
176176 27.6742 2.08602
177177 −10.5236 −0.791001
178178 22.5958 1.69363
179179 10.0000 0.747435 0.373718 0.927543i 0.378083π-0.378083\pi
0.373718 + 0.927543i 0.378083π0.378083\pi
180180 0 0
181181 −8.52359 −0.633553 −0.316777 0.948500i 0.602601π-0.602601\pi
−0.316777 + 0.948500i 0.602601π0.602601\pi
182182 2.49693 0.185085
183183 4.15676 0.307276
184184 −21.1773 −1.56121
185185 0 0
186186 21.0205 1.54130
187187 −2.15676 −0.157718
188188 −24.9939 −1.82286
189189 1.00000 0.0727393
190190 0 0
191191 15.3607 1.11146 0.555730 0.831363i 0.312439π-0.312439\pi
0.555730 + 0.831363i 0.312439π0.312439\pi
192192 −24.8576 −1.79394
193193 −8.36683 −0.602258 −0.301129 0.953583i 0.597363π-0.597363\pi
−0.301129 + 0.953583i 0.597363π0.597363\pi
194194 22.8638 1.64152
195195 0 0
196196 5.34017 0.381441
197197 11.7587 0.837774 0.418887 0.908038i 0.362420π-0.362420\pi
0.418887 + 0.908038i 0.362420π0.362420\pi
198198 5.41855 0.385080
199199 −22.5958 −1.60178 −0.800888 0.598814i 0.795639π-0.795639\pi
−0.800888 + 0.598814i 0.795639π0.795639\pi
200200 0 0
201201 4.68035 0.330126
202202 −15.7587 −1.10878
203203 6.68035 0.468868
204204 5.75872 0.403191
205205 0 0
206206 5.84324 0.407118
207207 −2.34017 −0.162653
208208 −12.7526 −0.884232
209209 6.15676 0.425872
210210 0 0
211211 −13.6742 −0.941371 −0.470685 0.882301i 0.655993π-0.655993\pi
−0.470685 + 0.882301i 0.655993π0.655993\pi
212212 −20.0722 −1.37857
213213 −2.00000 −0.137038
214214 44.6947 3.05527
215215 0 0
216216 −9.04945 −0.615737
217217 7.75872 0.526696
218218 −34.7792 −2.35555
219219 7.07838 0.478312
220220 0 0
221221 0.993857 0.0668541
222222 29.3607 1.97056
223223 21.6742 1.45141 0.725706 0.688005i 0.241513π-0.241513\pi
0.725706 + 0.688005i 0.241513π0.241513\pi
224224 −19.3896 −1.29552
225225 0 0
226226 14.1834 0.943467
227227 11.5174 0.764440 0.382220 0.924071i 0.375160π-0.375160\pi
0.382220 + 0.924071i 0.375160π0.375160\pi
228228 −16.4391 −1.08870
229229 12.8371 0.848300 0.424150 0.905592i 0.360573π-0.360573\pi
0.424150 + 0.905592i 0.360573π0.360573\pi
230230 0 0
231231 2.00000 0.131590
232232 −60.4534 −3.96896
233233 6.76487 0.443181 0.221591 0.975140i 0.428875π-0.428875\pi
0.221591 + 0.975140i 0.428875π0.428875\pi
234234 −2.49693 −0.163229
235235 0 0
236236 56.1978 3.65816
237237 −6.15676 −0.399924
238238 2.92162 0.189381
239239 −23.3607 −1.51108 −0.755539 0.655104i 0.772625π-0.772625\pi
−0.755539 + 0.655104i 0.772625π0.772625\pi
240240 0 0
241241 −14.6803 −0.945644 −0.472822 0.881158i 0.656765π-0.656765\pi
−0.472822 + 0.881158i 0.656765π0.656765\pi
242242 −18.9649 −1.21911
243243 −1.00000 −0.0641500
244244 −22.1978 −1.42107
245245 0 0
246246 −17.6020 −1.12226
247247 −2.83710 −0.180520
248248 −70.2122 −4.45848
249249 6.83710 0.433284
250250 0 0
251251 9.16290 0.578357 0.289179 0.957275i 0.406618π-0.406618\pi
0.289179 + 0.957275i 0.406618π0.406618\pi
252252 −5.34017 −0.336399
253253 −4.68035 −0.294251
254254 −4.99386 −0.313342
255255 0 0
256256 27.6803 1.73002
257257 5.07838 0.316781 0.158390 0.987377i 0.449370π-0.449370\pi
0.158390 + 0.987377i 0.449370π0.449370\pi
258258 −17.6742 −1.10035
259259 10.8371 0.673385
260260 0 0
261261 −6.68035 −0.413503
262262 4.00000 0.247121
263263 −5.65983 −0.349000 −0.174500 0.984657i 0.555831π-0.555831\pi
−0.174500 + 0.984657i 0.555831π0.555831\pi
264264 −18.0989 −1.11391
265265 0 0
266266 −8.34017 −0.511369
267267 −8.34017 −0.510410
268268 −24.9939 −1.52674
269269 −27.8576 −1.69851 −0.849255 0.527984i 0.822948π-0.822948\pi
−0.849255 + 0.527984i 0.822948π0.822948\pi
270270 0 0
271271 25.1194 1.52590 0.762948 0.646460i 0.223751π-0.223751\pi
0.762948 + 0.646460i 0.223751π0.223751\pi
272272 −14.9216 −0.904756
273273 −0.921622 −0.0557791
274274 −12.0267 −0.726557
275275 0 0
276276 12.4969 0.752227
277277 28.1978 1.69424 0.847121 0.531401i 0.178334π-0.178334\pi
0.847121 + 0.531401i 0.178334π0.178334\pi
278278 36.8515 2.21020
279279 −7.75872 −0.464503
280280 0 0
281281 −20.3545 −1.21425 −0.607125 0.794606i 0.707677π-0.707677\pi
−0.607125 + 0.794606i 0.707677π0.707677\pi
282282 12.6803 0.755104
283283 −23.5174 −1.39797 −0.698984 0.715138i 0.746364π-0.746364\pi
−0.698984 + 0.715138i 0.746364π0.746364\pi
284284 10.6803 0.633762
285285 0 0
286286 −4.99386 −0.295293
287287 −6.49693 −0.383502
288288 19.3896 1.14254
289289 −15.8371 −0.931594
290290 0 0
291291 −8.43907 −0.494707
292292 −37.7998 −2.21206
293293 2.92162 0.170683 0.0853415 0.996352i 0.472802π-0.472802\pi
0.0853415 + 0.996352i 0.472802π0.472802\pi
294294 −2.70928 −0.158008
295295 0 0
296296 −98.0698 −5.70019
297297 −2.00000 −0.116052
298298 42.4657 2.45997
299299 2.15676 0.124728
300300 0 0
301301 −6.52359 −0.376014
302302 15.8310 0.910969
303303 5.81658 0.334154
304304 42.5958 2.44304
305305 0 0
306306 −2.92162 −0.167018
307307 10.4703 0.597570 0.298785 0.954321i 0.403419π-0.403419\pi
0.298785 + 0.954321i 0.403419π0.403419\pi
308308 −10.6803 −0.608569
309309 −2.15676 −0.122694
310310 0 0
311311 23.8310 1.35133 0.675665 0.737209i 0.263857π-0.263857\pi
0.675665 + 0.737209i 0.263857π0.263857\pi
312312 8.34017 0.472169
313313 −32.7526 −1.85129 −0.925643 0.378399i 0.876475π-0.876475\pi
−0.925643 + 0.378399i 0.876475π0.876475\pi
314314 −13.3340 −0.752483
315315 0 0
316316 32.8781 1.84954
317317 17.9155 1.00623 0.503117 0.864218i 0.332187π-0.332187\pi
0.503117 + 0.864218i 0.332187π0.332187\pi
318318 10.1834 0.571058
319319 −13.3607 −0.748055
320320 0 0
321321 −16.4969 −0.920769
322322 6.34017 0.353324
323323 −3.31965 −0.184710
324324 5.34017 0.296676
325325 0 0
326326 26.6681 1.47701
327327 12.8371 0.709893
328328 58.7936 3.24633
329329 4.68035 0.258036
330330 0 0
331331 −1.36069 −0.0747904 −0.0373952 0.999301i 0.511906π-0.511906\pi
−0.0373952 + 0.999301i 0.511906π0.511906\pi
332332 −36.5113 −2.00382
333333 −10.8371 −0.593870
334334 52.0288 2.84689
335335 0 0
336336 13.8371 0.754876
337337 25.3607 1.38148 0.690742 0.723101i 0.257284π-0.257284\pi
0.690742 + 0.723101i 0.257284π0.257284\pi
338338 −32.9194 −1.79058
339339 −5.23513 −0.284333
340340 0 0
341341 −15.5174 −0.840317
342342 8.34017 0.450985
343343 −1.00000 −0.0539949
344344 59.0349 3.18295
345345 0 0
346346 60.7936 3.26829
347347 −16.8638 −0.905294 −0.452647 0.891690i 0.649520π-0.649520\pi
−0.452647 + 0.891690i 0.649520π0.649520\pi
348348 35.6742 1.91234
349349 9.51745 0.509457 0.254729 0.967013i 0.418014π-0.418014\pi
0.254729 + 0.967013i 0.418014π0.418014\pi
350350 0 0
351351 0.921622 0.0491926
352352 38.7792 2.06694
353353 35.7998 1.90543 0.952715 0.303867i 0.0982778π-0.0982778\pi
0.952715 + 0.303867i 0.0982778π0.0982778\pi
354354 −28.5113 −1.51536
355355 0 0
356356 44.5380 2.36051
357357 −1.07838 −0.0570738
358358 27.0928 1.43190
359359 −22.3135 −1.17766 −0.588831 0.808256i 0.700412π-0.700412\pi
−0.588831 + 0.808256i 0.700412π0.700412\pi
360360 0 0
361361 −9.52359 −0.501242
362362 −23.0928 −1.21373
363363 7.00000 0.367405
364364 4.92162 0.257963
365365 0 0
366366 11.2618 0.588663
367367 −20.3135 −1.06036 −0.530178 0.847886i 0.677875π-0.677875\pi
−0.530178 + 0.847886i 0.677875π0.677875\pi
368368 −32.3812 −1.68799
369369 6.49693 0.338217
370370 0 0
371371 3.75872 0.195143
372372 41.4329 2.14820
373373 16.0000 0.828449 0.414224 0.910175i 0.364053π-0.364053\pi
0.414224 + 0.910175i 0.364053π0.364053\pi
374374 −5.84324 −0.302147
375375 0 0
376376 −42.3545 −2.18427
377377 6.15676 0.317089
378378 2.70928 0.139350
379379 6.15676 0.316251 0.158126 0.987419i 0.449455π-0.449455\pi
0.158126 + 0.987419i 0.449455π0.449455\pi
380380 0 0
381381 1.84324 0.0944323
382382 41.6163 2.12928
383383 −26.8371 −1.37131 −0.685656 0.727926i 0.740485π-0.740485\pi
−0.685656 + 0.727926i 0.740485π0.740485\pi
384384 −28.5669 −1.45780
385385 0 0
386386 −22.6681 −1.15377
387387 6.52359 0.331613
388388 45.0661 2.28788
389389 −5.63317 −0.285613 −0.142806 0.989751i 0.545613π-0.545613\pi
−0.142806 + 0.989751i 0.545613π0.545613\pi
390390 0 0
391391 2.52359 0.127623
392392 9.04945 0.457066
393393 −1.47641 −0.0744750
394394 31.8576 1.60496
395395 0 0
396396 10.6803 0.536708
397397 −37.7998 −1.89712 −0.948558 0.316604i 0.897457π-0.897457\pi
−0.948558 + 0.316604i 0.897457π0.897457\pi
398398 −61.2183 −3.06860
399399 3.07838 0.154112
400400 0 0
401401 −13.6332 −0.680808 −0.340404 0.940279i 0.610564π-0.610564\pi
−0.340404 + 0.940279i 0.610564π0.610564\pi
402402 12.6803 0.632438
403403 7.15061 0.356197
404404 −31.0616 −1.54537
405405 0 0
406406 18.0989 0.898233
407407 −21.6742 −1.07435
408408 9.75872 0.483129
409409 12.3545 0.610893 0.305447 0.952209i 0.401194π-0.401194\pi
0.305447 + 0.952209i 0.401194π0.401194\pi
410410 0 0
411411 4.43907 0.218963
412412 11.5174 0.567424
413413 −10.5236 −0.517832
414414 −6.34017 −0.311603
415415 0 0
416416 −17.8699 −0.876144
417417 −13.6020 −0.666091
418418 16.6803 0.815862
419419 28.9939 1.41644 0.708221 0.705991i 0.249498π-0.249498\pi
0.708221 + 0.705991i 0.249498π0.249498\pi
420420 0 0
421421 −15.1629 −0.738994 −0.369497 0.929232i 0.620470π-0.620470\pi
−0.369497 + 0.929232i 0.620470π0.620470\pi
422422 −37.0472 −1.80343
423423 −4.68035 −0.227566
424424 −34.0144 −1.65188
425425 0 0
426426 −5.41855 −0.262530
427427 4.15676 0.201159
428428 88.0965 4.25830
429429 1.84324 0.0889927
430430 0 0
431431 −10.3135 −0.496784 −0.248392 0.968660i 0.579902π-0.579902\pi
−0.248392 + 0.968660i 0.579902π0.579902\pi
432432 −13.8371 −0.665738
433433 −20.4391 −0.982239 −0.491120 0.871092i 0.663412π-0.663412\pi
−0.491120 + 0.871092i 0.663412π0.663412\pi
434434 21.0205 1.00902
435435 0 0
436436 −68.5523 −3.28306
437437 −7.20394 −0.344611
438438 19.1773 0.916326
439439 −16.9216 −0.807625 −0.403812 0.914842i 0.632315π-0.632315\pi
−0.403812 + 0.914842i 0.632315π0.632315\pi
440440 0 0
441441 1.00000 0.0476190
442442 2.69263 0.128075
443443 12.8104 0.608642 0.304321 0.952569i 0.401570π-0.401570\pi
0.304321 + 0.952569i 0.401570π0.401570\pi
444444 57.8720 2.74648
445445 0 0
446446 58.7214 2.78054
447447 −15.6742 −0.741364
448448 −24.8576 −1.17441
449449 −14.6270 −0.690292 −0.345146 0.938549i 0.612171π-0.612171\pi
−0.345146 + 0.938549i 0.612171π0.612171\pi
450450 0 0
451451 12.9939 0.611857
452452 27.9565 1.31496
453453 −5.84324 −0.274540
454454 31.2039 1.46447
455455 0 0
456456 −27.8576 −1.30455
457457 −14.1568 −0.662225 −0.331113 0.943591i 0.607424π-0.607424\pi
−0.331113 + 0.943591i 0.607424π0.607424\pi
458458 34.7792 1.62513
459459 1.07838 0.0503344
460460 0 0
461461 0.340173 0.0158434 0.00792172 0.999969i 0.497478π-0.497478\pi
0.00792172 + 0.999969i 0.497478π0.497478\pi
462462 5.41855 0.252094
463463 9.84324 0.457454 0.228727 0.973491i 0.426544π-0.426544\pi
0.228727 + 0.973491i 0.426544π0.426544\pi
464464 −92.4366 −4.29126
465465 0 0
466466 18.3279 0.849023
467467 11.5174 0.532964 0.266482 0.963840i 0.414139π-0.414139\pi
0.266482 + 0.963840i 0.414139π0.414139\pi
468468 −4.92162 −0.227502
469469 4.68035 0.216118
470470 0 0
471471 4.92162 0.226776
472472 95.2327 4.38344
473473 13.0472 0.599910
474474 −16.6803 −0.766154
475475 0 0
476476 5.75872 0.263951
477477 −3.75872 −0.172100
478478 −63.2905 −2.89484
479479 −19.5174 −0.891775 −0.445887 0.895089i 0.647112π-0.647112\pi
−0.445887 + 0.895089i 0.647112π0.647112\pi
480480 0 0
481481 9.98771 0.455401
482482 −39.7731 −1.81162
483483 −2.34017 −0.106482
484484 −37.3812 −1.69915
485485 0 0
486486 −2.70928 −0.122895
487487 −23.1506 −1.04905 −0.524527 0.851394i 0.675758π-0.675758\pi
−0.524527 + 0.851394i 0.675758π0.675758\pi
488488 −37.6163 −1.70281
489489 −9.84324 −0.445127
490490 0 0
491491 2.00000 0.0902587 0.0451294 0.998981i 0.485630π-0.485630\pi
0.0451294 + 0.998981i 0.485630π0.485630\pi
492492 −34.6947 −1.56416
493493 7.20394 0.324449
494494 −7.68649 −0.345831
495495 0 0
496496 −107.358 −4.82053
497497 −2.00000 −0.0897123
498498 18.5236 0.830062
499499 27.2039 1.21782 0.608908 0.793241i 0.291608π-0.291608\pi
0.608908 + 0.793241i 0.291608π0.291608\pi
500500 0 0
501501 −19.2039 −0.857969
502502 24.8248 1.10799
503503 18.8371 0.839905 0.419952 0.907546i 0.362047π-0.362047\pi
0.419952 + 0.907546i 0.362047π0.362047\pi
504504 −9.04945 −0.403094
505505 0 0
506506 −12.6803 −0.563710
507507 12.1506 0.539628
508508 −9.84324 −0.436723
509509 6.81044 0.301867 0.150934 0.988544i 0.451772π-0.451772\pi
0.150934 + 0.988544i 0.451772π0.451772\pi
510510 0 0
511511 7.07838 0.313129
512512 17.8599 0.789303
513513 −3.07838 −0.135914
514514 13.7587 0.606871
515515 0 0
516516 −34.8371 −1.53362
517517 −9.36069 −0.411683
518518 29.3607 1.29003
519519 −22.4391 −0.984966
520520 0 0
521521 −25.8166 −1.13105 −0.565523 0.824733i 0.691325π-0.691325\pi
−0.565523 + 0.824733i 0.691325π0.691325\pi
522522 −18.0989 −0.792167
523523 −4.00000 −0.174908 −0.0874539 0.996169i 0.527873π-0.527873\pi
−0.0874539 + 0.996169i 0.527873π0.527873\pi
524524 7.88428 0.344426
525525 0 0
526526 −15.3340 −0.668595
527527 8.36683 0.364465
528528 −27.6742 −1.20437
529529 −17.5236 −0.761895
530530 0 0
531531 10.5236 0.456685
532532 −16.4391 −0.712724
533533 −5.98771 −0.259357
534534 −22.5958 −0.977817
535535 0 0
536536 −42.3545 −1.82944
537537 −10.0000 −0.431532
538538 −75.4740 −3.25391
539539 2.00000 0.0861461
540540 0 0
541541 25.8843 1.11285 0.556426 0.830897i 0.312172π-0.312172\pi
0.556426 + 0.830897i 0.312172π0.312172\pi
542542 68.0554 2.92323
543543 8.52359 0.365782
544544 −20.9093 −0.896480
545545 0 0
546546 −2.49693 −0.106859
547547 −11.3197 −0.483993 −0.241997 0.970277i 0.577802π-0.577802\pi
−0.241997 + 0.970277i 0.577802π0.577802\pi
548548 −23.7054 −1.01264
549549 −4.15676 −0.177406
550550 0 0
551551 −20.5646 −0.876083
552552 21.1773 0.901365
553553 −6.15676 −0.261812
554554 76.3956 3.24574
555555 0 0
556556 72.6369 3.08049
557557 26.6491 1.12916 0.564580 0.825378i 0.309038π-0.309038\pi
0.564580 + 0.825378i 0.309038π0.309038\pi
558558 −21.0205 −0.889870
559559 −6.01229 −0.254293
560560 0 0
561561 2.15676 0.0910583
562562 −55.1461 −2.32620
563563 −46.3545 −1.95361 −0.976806 0.214128i 0.931309π-0.931309\pi
−0.976806 + 0.214128i 0.931309π0.931309\pi
564564 24.9939 1.05243
565565 0 0
566566 −63.7152 −2.67815
567567 −1.00000 −0.0419961
568568 18.0989 0.759413
569569 −14.3668 −0.602289 −0.301145 0.953579i 0.597369π-0.597369\pi
−0.301145 + 0.953579i 0.597369π0.597369\pi
570570 0 0
571571 38.7214 1.62044 0.810220 0.586126i 0.199348π-0.199348\pi
0.810220 + 0.586126i 0.199348π0.199348\pi
572572 −9.84324 −0.411567
573573 −15.3607 −0.641702
574574 −17.6020 −0.734692
575575 0 0
576576 24.8576 1.03573
577577 −43.4740 −1.80984 −0.904922 0.425577i 0.860071π-0.860071\pi
−0.904922 + 0.425577i 0.860071π0.860071\pi
578578 −42.9071 −1.78470
579579 8.36683 0.347714
580580 0 0
581581 6.83710 0.283651
582582 −22.8638 −0.947733
583583 −7.51745 −0.311341
584584 −64.0554 −2.65063
585585 0 0
586586 7.91548 0.326985
587587 −36.0288 −1.48707 −0.743533 0.668699i 0.766851π-0.766851\pi
−0.743533 + 0.668699i 0.766851π0.766851\pi
588588 −5.34017 −0.220225
589589 −23.8843 −0.984135
590590 0 0
591591 −11.7587 −0.483689
592592 −149.954 −6.16307
593593 −31.4863 −1.29299 −0.646493 0.762920i 0.723765π-0.723765\pi
−0.646493 + 0.762920i 0.723765π0.723765\pi
594594 −5.41855 −0.222326
595595 0 0
596596 83.7030 3.42861
597597 22.5958 0.924786
598598 5.84324 0.238948
599599 29.0349 1.18633 0.593167 0.805080i 0.297877π-0.297877\pi
0.593167 + 0.805080i 0.297877π0.297877\pi
600600 0 0
601601 15.3607 0.626576 0.313288 0.949658i 0.398570π-0.398570\pi
0.313288 + 0.949658i 0.398570π0.398570\pi
602602 −17.6742 −0.720347
603603 −4.68035 −0.190598
604604 31.2039 1.26967
605605 0 0
606606 15.7587 0.640154
607607 13.0472 0.529569 0.264784 0.964308i 0.414699π-0.414699\pi
0.264784 + 0.964308i 0.414699π0.414699\pi
608608 59.6886 2.42069
609609 −6.68035 −0.270701
610610 0 0
611611 4.31351 0.174506
612612 −5.75872 −0.232783
613613 15.5174 0.626744 0.313372 0.949630i 0.398541π-0.398541\pi
0.313372 + 0.949630i 0.398541π0.398541\pi
614614 28.3668 1.14479
615615 0 0
616616 −18.0989 −0.729225
617617 22.7649 0.916479 0.458240 0.888829i 0.348480π-0.348480\pi
0.458240 + 0.888829i 0.348480π0.348480\pi
618618 −5.84324 −0.235050
619619 7.92777 0.318644 0.159322 0.987227i 0.449069π-0.449069\pi
0.159322 + 0.987227i 0.449069π0.449069\pi
620620 0 0
621621 2.34017 0.0939079
622622 64.5646 2.58881
623623 −8.34017 −0.334142
624624 12.7526 0.510512
625625 0 0
626626 −88.7358 −3.54659
627627 −6.15676 −0.245877
628628 −26.2823 −1.04878
629629 11.6865 0.465971
630630 0 0
631631 19.2039 0.764497 0.382248 0.924060i 0.375150π-0.375150\pi
0.382248 + 0.924060i 0.375150π0.375150\pi
632632 55.7152 2.21623
633633 13.6742 0.543501
634634 48.5380 1.92769
635635 0 0
636636 20.0722 0.795916
637637 −0.921622 −0.0365160
638638 −36.1978 −1.43308
639639 2.00000 0.0791188
640640 0 0
641641 −5.94668 −0.234880 −0.117440 0.993080i 0.537469π-0.537469\pi
−0.117440 + 0.993080i 0.537469π0.537469\pi
642642 −44.6947 −1.76396
643643 30.8904 1.21820 0.609100 0.793094i 0.291531π-0.291531\pi
0.609100 + 0.793094i 0.291531π0.291531\pi
644644 12.4969 0.492448
645645 0 0
646646 −8.99386 −0.353859
647647 19.2039 0.754985 0.377492 0.926013i 0.376786π-0.376786\pi
0.377492 + 0.926013i 0.376786π0.376786\pi
648648 9.04945 0.355496
649649 21.0472 0.826174
650650 0 0
651651 −7.75872 −0.304088
652652 52.5646 2.05859
653653 28.5548 1.11744 0.558718 0.829358i 0.311294π-0.311294\pi
0.558718 + 0.829358i 0.311294π0.311294\pi
654654 34.7792 1.35998
655655 0 0
656656 89.8987 3.50995
657657 −7.07838 −0.276154
658658 12.6803 0.494331
659659 −27.9877 −1.09025 −0.545123 0.838356i 0.683517π-0.683517\pi
−0.545123 + 0.838356i 0.683517π0.683517\pi
660660 0 0
661661 −22.1445 −0.861320 −0.430660 0.902514i 0.641719π-0.641719\pi
−0.430660 + 0.902514i 0.641719π0.641719\pi
662662 −3.68649 −0.143279
663663 −0.993857 −0.0385982
664664 −61.8720 −2.40110
665665 0 0
666666 −29.3607 −1.13770
667667 15.6332 0.605319
668668 102.552 3.96787
669669 −21.6742 −0.837973
670670 0 0
671671 −8.31351 −0.320940
672672 19.3896 0.747971
673673 −2.21008 −0.0851923 −0.0425962 0.999092i 0.513563π-0.513563\pi
−0.0425962 + 0.999092i 0.513563π0.513563\pi
674674 68.7091 2.64658
675675 0 0
676676 −64.8864 −2.49563
677677 −19.5486 −0.751315 −0.375658 0.926758i 0.622583π-0.622583\pi
−0.375658 + 0.926758i 0.622583π0.622583\pi
678678 −14.1834 −0.544711
679679 −8.43907 −0.323862
680680 0 0
681681 −11.5174 −0.441350
682682 −42.0410 −1.60983
683683 −11.8166 −0.452149 −0.226074 0.974110i 0.572589π-0.572589\pi
−0.226074 + 0.974110i 0.572589π0.572589\pi
684684 16.4391 0.628564
685685 0 0
686686 −2.70928 −0.103441
687687 −12.8371 −0.489766
688688 90.2676 3.44142
689689 3.46412 0.131973
690690 0 0
691691 11.7587 0.447323 0.223661 0.974667i 0.428199π-0.428199\pi
0.223661 + 0.974667i 0.428199π0.428199\pi
692692 119.829 4.55520
693693 −2.00000 −0.0759737
694694 −45.6886 −1.73431
695695 0 0
696696 60.4534 2.29148
697697 −7.00614 −0.265377
698698 25.7854 0.975991
699699 −6.76487 −0.255871
700700 0 0
701701 9.94668 0.375681 0.187840 0.982200i 0.439851π-0.439851\pi
0.187840 + 0.982200i 0.439851π0.439851\pi
702702 2.49693 0.0942405
703703 −33.3607 −1.25822
704704 49.7152 1.87371
705705 0 0
706706 96.9914 3.65032
707707 5.81658 0.218755
708708 −56.1978 −2.11204
709709 −11.0472 −0.414886 −0.207443 0.978247i 0.566514π-0.566514\pi
−0.207443 + 0.978247i 0.566514π0.566514\pi
710710 0 0
711711 6.15676 0.230896
712712 75.4740 2.82851
713713 18.1568 0.679976
714714 −2.92162 −0.109339
715715 0 0
716716 53.4017 1.99572
717717 23.3607 0.872421
718718 −60.4534 −2.25610
719719 −6.15676 −0.229608 −0.114804 0.993388i 0.536624π-0.536624\pi
−0.114804 + 0.993388i 0.536624π0.536624\pi
720720 0 0
721721 −2.15676 −0.0803218
722722 −25.8020 −0.960252
723723 14.6803 0.545968
724724 −45.5174 −1.69164
725725 0 0
726726 18.9649 0.703854
727727 −2.89043 −0.107200 −0.0536000 0.998562i 0.517070π-0.517070\pi
−0.0536000 + 0.998562i 0.517070π0.517070\pi
728728 8.34017 0.309107
729729 1.00000 0.0370370
730730 0 0
731731 −7.03489 −0.260195
732732 22.1978 0.820454
733733 25.7998 0.952936 0.476468 0.879192i 0.341917π-0.341917\pi
0.476468 + 0.879192i 0.341917π0.341917\pi
734734 −55.0349 −2.03138
735735 0 0
736736 −45.3751 −1.67255
737737 −9.36069 −0.344806
738738 17.6020 0.647937
739739 −1.04718 −0.0385212 −0.0192606 0.999814i 0.506131π-0.506131\pi
−0.0192606 + 0.999814i 0.506131π0.506131\pi
740740 0 0
741741 2.83710 0.104224
742742 10.1834 0.373845
743743 9.97334 0.365886 0.182943 0.983123i 0.441438π-0.441438\pi
0.182943 + 0.983123i 0.441438π0.441438\pi
744744 70.2122 2.57410
745745 0 0
746746 43.3484 1.58710
747747 −6.83710 −0.250156
748748 −11.5174 −0.421120
749749 −16.4969 −0.602785
750750 0 0
751751 3.26633 0.119190 0.0595950 0.998223i 0.481019π-0.481019\pi
0.0595950 + 0.998223i 0.481019π0.481019\pi
752752 −64.7624 −2.36164
753753 −9.16290 −0.333915
754754 16.6803 0.607462
755755 0 0
756756 5.34017 0.194220
757757 49.9877 1.81683 0.908417 0.418065i 0.137292π-0.137292\pi
0.908417 + 0.418065i 0.137292π0.137292\pi
758758 16.6803 0.605857
759759 4.68035 0.169886
760760 0 0
761761 2.61265 0.0947083 0.0473542 0.998878i 0.484921π-0.484921\pi
0.0473542 + 0.998878i 0.484921π0.484921\pi
762762 4.99386 0.180908
763763 12.8371 0.464734
764764 82.0288 2.96770
765765 0 0
766766 −72.7091 −2.62709
767767 −9.69878 −0.350202
768768 −27.6803 −0.998828
769769 −15.6742 −0.565226 −0.282613 0.959234i 0.591201π-0.591201\pi
−0.282613 + 0.959234i 0.591201π0.591201\pi
770770 0 0
771771 −5.07838 −0.182893
772772 −44.6803 −1.60808
773773 −5.81205 −0.209045 −0.104522 0.994523i 0.533331π-0.533331\pi
−0.104522 + 0.994523i 0.533331π0.533331\pi
774774 17.6742 0.635286
775775 0 0
776776 76.3689 2.74148
777777 −10.8371 −0.388779
778778 −15.2618 −0.547162
779779 20.0000 0.716574
780780 0 0
781781 4.00000 0.143131
782782 6.83710 0.244494
783783 6.68035 0.238736
784784 13.8371 0.494182
785785 0 0
786786 −4.00000 −0.142675
787787 −39.3484 −1.40262 −0.701310 0.712857i 0.747401π-0.747401\pi
−0.701310 + 0.712857i 0.747401π0.747401\pi
788788 62.7936 2.23693
789789 5.65983 0.201495
790790 0 0
791791 −5.23513 −0.186140
792792 18.0989 0.643116
793793 3.83096 0.136041
794794 −102.410 −3.63439
795795 0 0
796796 −120.666 −4.27688
797797 −28.2823 −1.00181 −0.500905 0.865502i 0.667000π-0.667000\pi
−0.500905 + 0.865502i 0.667000π0.667000\pi
798798 8.34017 0.295239
799799 5.04718 0.178556
800800 0 0
801801 8.34017 0.294686
802802 −36.9360 −1.30426
803803 −14.1568 −0.499581
804804 24.9939 0.881465
805805 0 0
806806 19.3730 0.682384
807807 27.8576 0.980635
808808 −52.6369 −1.85176
809809 −15.6742 −0.551076 −0.275538 0.961290i 0.588856π-0.588856\pi
−0.275538 + 0.961290i 0.588856π0.588856\pi
810810 0 0
811811 −42.1666 −1.48067 −0.740335 0.672238i 0.765333π-0.765333\pi
−0.740335 + 0.672238i 0.765333π0.765333\pi
812812 35.6742 1.25192
813813 −25.1194 −0.880976
814814 −58.7214 −2.05818
815815 0 0
816816 14.9216 0.522361
817817 20.0821 0.702583
818818 33.4719 1.17032
819819 0.921622 0.0322041
820820 0 0
821821 −39.0472 −1.36276 −0.681378 0.731932i 0.738619π-0.738619\pi
−0.681378 + 0.731932i 0.738619π0.738619\pi
822822 12.0267 0.419478
823823 36.5646 1.27456 0.637281 0.770631i 0.280059π-0.280059\pi
0.637281 + 0.770631i 0.280059π0.280059\pi
824824 19.5174 0.679922
825825 0 0
826826 −28.5113 −0.992035
827827 −50.2245 −1.74648 −0.873238 0.487294i 0.837984π-0.837984\pi
−0.873238 + 0.487294i 0.837984π0.837984\pi
828828 −12.4969 −0.434298
829829 32.8371 1.14048 0.570240 0.821478i 0.306850π-0.306850\pi
0.570240 + 0.821478i 0.306850π0.306850\pi
830830 0 0
831831 −28.1978 −0.978171
832832 −22.9093 −0.794238
833833 −1.07838 −0.0373636
834834 −36.8515 −1.27606
835835 0 0
836836 32.8781 1.13711
837837 7.75872 0.268181
838838 78.5523 2.71355
839839 −13.3607 −0.461262 −0.230631 0.973041i 0.574079π-0.574079\pi
−0.230631 + 0.973041i 0.574079π0.574079\pi
840840 0 0
841841 15.6270 0.538863
842842 −41.0805 −1.41573
843843 20.3545 0.701048
844844 −73.0226 −2.51354
845845 0 0
846846 −12.6803 −0.435959
847847 7.00000 0.240523
848848 −52.0098 −1.78603
849849 23.5174 0.807117
850850 0 0
851851 25.3607 0.869353
852852 −10.6803 −0.365903
853853 39.6430 1.35735 0.678675 0.734438i 0.262554π-0.262554\pi
0.678675 + 0.734438i 0.262554π0.262554\pi
854854 11.2618 0.385371
855855 0 0
856856 149.288 5.10256
857857 29.7054 1.01472 0.507359 0.861735i 0.330622π-0.330622\pi
0.507359 + 0.861735i 0.330622π0.330622\pi
858858 4.99386 0.170487
859859 −3.07838 −0.105033 −0.0525164 0.998620i 0.516724π-0.516724\pi
−0.0525164 + 0.998620i 0.516724π0.516724\pi
860860 0 0
861861 6.49693 0.221415
862862 −27.9421 −0.951713
863863 6.39350 0.217637 0.108819 0.994062i 0.465293π-0.465293\pi
0.108819 + 0.994062i 0.465293π0.465293\pi
864864 −19.3896 −0.659648
865865 0 0
866866 −55.3751 −1.88172
867867 15.8371 0.537856
868868 41.4329 1.40633
869869 12.3135 0.417707
870870 0 0
871871 4.31351 0.146158
872872 −116.169 −3.93397
873873 8.43907 0.285619
874874 −19.5174 −0.660188
875875 0 0
876876 37.7998 1.27714
877877 1.21622 0.0410689 0.0205345 0.999789i 0.493463π-0.493463\pi
0.0205345 + 0.999789i 0.493463π0.493463\pi
878878 −45.8453 −1.54721
879879 −2.92162 −0.0985439
880880 0 0
881881 15.9733 0.538155 0.269078 0.963118i 0.413281π-0.413281\pi
0.269078 + 0.963118i 0.413281π0.413281\pi
882882 2.70928 0.0912260
883883 −11.6865 −0.393282 −0.196641 0.980476i 0.563003π-0.563003\pi
−0.196641 + 0.980476i 0.563003π0.563003\pi
884884 5.30737 0.178506
885885 0 0
886886 34.7070 1.16600
887887 25.6209 0.860265 0.430132 0.902766i 0.358467π-0.358467\pi
0.430132 + 0.902766i 0.358467π0.358467\pi
888888 98.0698 3.29101
889889 1.84324 0.0618204
890890 0 0
891891 2.00000 0.0670025
892892 115.744 3.87540
893893 −14.4079 −0.482141
894894 −42.4657 −1.42027
895895 0 0
896896 −28.5669 −0.954353
897897 −2.15676 −0.0720120
898898 −39.6286 −1.32242
899899 51.8310 1.72866
900900 0 0
901901 4.05332 0.135036
902902 35.2039 1.17216
903903 6.52359 0.217091
904904 47.3751 1.57567
905905 0 0
906906 −15.8310 −0.525948
907907 57.7563 1.91777 0.958883 0.283802i 0.0915959π-0.0915959\pi
0.958883 + 0.283802i 0.0915959π0.0915959\pi
908908 61.5052 2.04112
909909 −5.81658 −0.192924
910910 0 0
911911 −35.9877 −1.19233 −0.596163 0.802863i 0.703309π-0.703309\pi
−0.596163 + 0.802863i 0.703309π0.703309\pi
912912 −42.5958 −1.41049
913913 −13.6742 −0.452550
914914 −38.3545 −1.26866
915915 0 0
916916 68.5523 2.26503
917917 −1.47641 −0.0487553
918918 2.92162 0.0964279
919919 46.7214 1.54120 0.770598 0.637321i 0.219958π-0.219958\pi
0.770598 + 0.637321i 0.219958π0.219958\pi
920920 0 0
921921 −10.4703 −0.345007
922922 0.921622 0.0303520
923923 −1.84324 −0.0606711
924924 10.6803 0.351358
925925 0 0
926926 26.6681 0.876367
927927 2.15676 0.0708371
928928 −129.529 −4.25201
929929 −53.0493 −1.74049 −0.870245 0.492619i 0.836040π-0.836040\pi
−0.870245 + 0.492619i 0.836040π0.836040\pi
930930 0 0
931931 3.07838 0.100890
932932 36.1256 1.18333
933933 −23.8310 −0.780191
934934 31.2039 1.02102
935935 0 0
936936 −8.34017 −0.272607
937937 16.1256 0.526799 0.263400 0.964687i 0.415156π-0.415156\pi
0.263400 + 0.964687i 0.415156π0.415156\pi
938938 12.6803 0.414028
939939 32.7526 1.06884
940940 0 0
941941 24.7070 0.805425 0.402713 0.915326i 0.368067π-0.368067\pi
0.402713 + 0.915326i 0.368067π0.368067\pi
942942 13.3340 0.434446
943943 −15.2039 −0.495108
944944 145.616 4.73940
945945 0 0
946946 35.3484 1.14928
947947 6.53797 0.212455 0.106228 0.994342i 0.466123π-0.466123\pi
0.106228 + 0.994342i 0.466123π0.466123\pi
948948 −32.8781 −1.06783
949949 6.52359 0.211765
950950 0 0
951951 −17.9155 −0.580949
952952 9.75872 0.316282
953953 −6.11327 −0.198028 −0.0990142 0.995086i 0.531569π-0.531569\pi
−0.0990142 + 0.995086i 0.531569π0.531569\pi
954954 −10.1834 −0.329700
955955 0 0
956956 −124.750 −4.03471
957957 13.3607 0.431890
958958 −52.8781 −1.70842
959959 4.43907 0.143345
960960 0 0
961961 29.1978 0.941864
962962 27.0595 0.872432
963963 16.4969 0.531606
964964 −78.3956 −2.52495
965965 0 0
966966 −6.34017 −0.203992
967967 25.6209 0.823912 0.411956 0.911204i 0.364846π-0.364846\pi
0.411956 + 0.911204i 0.364846π0.364846\pi
968968 −63.3461 −2.03602
969969 3.31965 0.106643
970970 0 0
971971 4.05332 0.130077 0.0650387 0.997883i 0.479283π-0.479283\pi
0.0650387 + 0.997883i 0.479283π0.479283\pi
972972 −5.34017 −0.171286
973973 −13.6020 −0.436059
974974 −62.7214 −2.00972
975975 0 0
976976 −57.5174 −1.84109
977977 3.81205 0.121958 0.0609791 0.998139i 0.480578π-0.480578\pi
0.0609791 + 0.998139i 0.480578π0.480578\pi
978978 −26.6681 −0.852751
979979 16.6803 0.533106
980980 0 0
981981 −12.8371 −0.409857
982982 5.41855 0.172913
983983 −24.0000 −0.765481 −0.382741 0.923856i 0.625020π-0.625020\pi
−0.382741 + 0.923856i 0.625020π0.625020\pi
984984 −58.7936 −1.87427
985985 0 0
986986 19.5174 0.621562
987987 −4.68035 −0.148977
988988 −15.1506 −0.482005
989989 −15.2663 −0.485441
990990 0 0
991991 −42.4079 −1.34713 −0.673565 0.739128i 0.735238π-0.735238\pi
−0.673565 + 0.739128i 0.735238π0.735238\pi
992992 −150.439 −4.77643
993993 1.36069 0.0431803
994994 −5.41855 −0.171866
995995 0 0
996996 36.5113 1.15690
997997 −43.4740 −1.37683 −0.688417 0.725315i 0.741694π-0.741694\pi
−0.688417 + 0.725315i 0.741694π0.741694\pi
998998 73.7030 2.33303
999999 10.8371 0.342871
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 525.2.a.k.1.3 3
3.2 odd 2 1575.2.a.w.1.1 3
4.3 odd 2 8400.2.a.dj.1.2 3
5.2 odd 4 105.2.d.b.64.6 yes 6
5.3 odd 4 105.2.d.b.64.1 6
5.4 even 2 525.2.a.j.1.1 3
7.6 odd 2 3675.2.a.bj.1.3 3
15.2 even 4 315.2.d.e.64.1 6
15.8 even 4 315.2.d.e.64.6 6
15.14 odd 2 1575.2.a.x.1.3 3
20.3 even 4 1680.2.t.k.1009.6 6
20.7 even 4 1680.2.t.k.1009.3 6
20.19 odd 2 8400.2.a.dg.1.2 3
35.2 odd 12 735.2.q.e.214.6 12
35.3 even 12 735.2.q.f.79.6 12
35.12 even 12 735.2.q.f.214.6 12
35.13 even 4 735.2.d.b.589.1 6
35.17 even 12 735.2.q.f.79.1 12
35.18 odd 12 735.2.q.e.79.6 12
35.23 odd 12 735.2.q.e.214.1 12
35.27 even 4 735.2.d.b.589.6 6
35.32 odd 12 735.2.q.e.79.1 12
35.33 even 12 735.2.q.f.214.1 12
35.34 odd 2 3675.2.a.bi.1.1 3
60.23 odd 4 5040.2.t.v.1009.2 6
60.47 odd 4 5040.2.t.v.1009.1 6
105.62 odd 4 2205.2.d.l.1324.1 6
105.83 odd 4 2205.2.d.l.1324.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.1 6 5.3 odd 4
105.2.d.b.64.6 yes 6 5.2 odd 4
315.2.d.e.64.1 6 15.2 even 4
315.2.d.e.64.6 6 15.8 even 4
525.2.a.j.1.1 3 5.4 even 2
525.2.a.k.1.3 3 1.1 even 1 trivial
735.2.d.b.589.1 6 35.13 even 4
735.2.d.b.589.6 6 35.27 even 4
735.2.q.e.79.1 12 35.32 odd 12
735.2.q.e.79.6 12 35.18 odd 12
735.2.q.e.214.1 12 35.23 odd 12
735.2.q.e.214.6 12 35.2 odd 12
735.2.q.f.79.1 12 35.17 even 12
735.2.q.f.79.6 12 35.3 even 12
735.2.q.f.214.1 12 35.33 even 12
735.2.q.f.214.6 12 35.12 even 12
1575.2.a.w.1.1 3 3.2 odd 2
1575.2.a.x.1.3 3 15.14 odd 2
1680.2.t.k.1009.3 6 20.7 even 4
1680.2.t.k.1009.6 6 20.3 even 4
2205.2.d.l.1324.1 6 105.62 odd 4
2205.2.d.l.1324.6 6 105.83 odd 4
3675.2.a.bi.1.1 3 35.34 odd 2
3675.2.a.bj.1.3 3 7.6 odd 2
5040.2.t.v.1009.1 6 60.47 odd 4
5040.2.t.v.1009.2 6 60.23 odd 4
8400.2.a.dg.1.2 3 20.19 odd 2
8400.2.a.dj.1.2 3 4.3 odd 2