Properties

Label 45.14.a.g.1.4
Level 4545
Weight 1414
Character 45.1
Self dual yes
Analytic conductor 48.25448.254
Analytic rank 00
Dimension 44
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] = [45,14,Mod(1,45)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 14, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("45.1"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Level: N N == 45=325 45 = 3^{2} \cdot 5
Weight: k k == 14 14
Character orbit: [χ][\chi] == 45.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 48.253918028448.2539180284
Analytic rank: 00
Dimension: 44
Coefficient field: Q[x]/(x4)\mathbb{Q}[x]/(x^{4} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4x324774x286616x+52534656 x^{4} - x^{3} - 24774x^{2} - 86616x + 52534656 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 233453 2^{3}\cdot 3^{4}\cdot 5^{3}
Twist minimal: yes
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.4
Root 152.425152.425 of defining polynomial
Character χ\chi == 45.1

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+148.425q2+13838.1q4+15625.0q5+215324.q7+838027.q8+2.31915e6q10+3.50194e6q111.22382e6q13+3.19596e7q14+1.10227e7q16+1.14596e8q17+1.26403e8q19+2.16221e8q20+5.19777e8q221.54080e8q23+2.44141e8q251.81646e8q26+2.97968e9q283.50622e9q29+6.23402e9q315.22907e9q32+1.70089e10q34+3.36444e9q35+2.11673e10q37+1.87614e10q38+1.30942e10q40+5.91552e10q411.84126e10q43+4.84603e10q442.28694e10q463.33828e10q475.05244e10q49+3.62367e10q501.69354e10q527.18621e10q53+5.47178e10q55+1.80448e11q565.20412e11q58+2.91405e10q595.52018e11q61+9.25288e11q628.66425e11q641.91222e10q65+9.87816e11q67+1.58579e12q68+4.99369e11q70+1.33203e11q71+1.18101e10q73+3.14177e12q74+1.74918e12q76+7.54053e11q773.39311e12q79+1.72230e11q80+8.78014e12q821.44794e12q83+1.79056e12q852.73290e12q86+2.93472e12q881.62879e11q892.63518e11q912.13218e12q924.95485e12q94+1.97504e12q95+3.58718e12q977.49911e12q98+O(q100)q+148.425 q^{2} +13838.1 q^{4} +15625.0 q^{5} +215324. q^{7} +838027. q^{8} +2.31915e6 q^{10} +3.50194e6 q^{11} -1.22382e6 q^{13} +3.19596e7 q^{14} +1.10227e7 q^{16} +1.14596e8 q^{17} +1.26403e8 q^{19} +2.16221e8 q^{20} +5.19777e8 q^{22} -1.54080e8 q^{23} +2.44141e8 q^{25} -1.81646e8 q^{26} +2.97968e9 q^{28} -3.50622e9 q^{29} +6.23402e9 q^{31} -5.22907e9 q^{32} +1.70089e10 q^{34} +3.36444e9 q^{35} +2.11673e10 q^{37} +1.87614e10 q^{38} +1.30942e10 q^{40} +5.91552e10 q^{41} -1.84126e10 q^{43} +4.84603e10 q^{44} -2.28694e10 q^{46} -3.33828e10 q^{47} -5.05244e10 q^{49} +3.62367e10 q^{50} -1.69354e10 q^{52} -7.18621e10 q^{53} +5.47178e10 q^{55} +1.80448e11 q^{56} -5.20412e11 q^{58} +2.91405e10 q^{59} -5.52018e11 q^{61} +9.25288e11 q^{62} -8.66425e11 q^{64} -1.91222e10 q^{65} +9.87816e11 q^{67} +1.58579e12 q^{68} +4.99369e11 q^{70} +1.33203e11 q^{71} +1.18101e10 q^{73} +3.14177e12 q^{74} +1.74918e12 q^{76} +7.54053e11 q^{77} -3.39311e12 q^{79} +1.72230e11 q^{80} +8.78014e12 q^{82} -1.44794e12 q^{83} +1.79056e12 q^{85} -2.73290e12 q^{86} +2.93472e12 q^{88} -1.62879e11 q^{89} -2.63518e11 q^{91} -2.13218e12 q^{92} -4.95485e12 q^{94} +1.97504e12 q^{95} +3.58718e12 q^{97} -7.49911e12 q^{98} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q15q2+16837q4+62500q5+343040q714865q8234375q10+12697800q11+34336040q13+26944650q14+66562801q16+84377280q17131821144q19+263078125q20+17650752985395q98+O(q100) 4 q - 15 q^{2} + 16837 q^{4} + 62500 q^{5} + 343040 q^{7} - 14865 q^{8} - 234375 q^{10} + 12697800 q^{11} + 34336040 q^{13} + 26944650 q^{14} + 66562801 q^{16} + 84377280 q^{17} - 131821144 q^{19} + 263078125 q^{20}+ \cdots - 17650752985395 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
22 148.425 1.63989 0.819943 0.572446i 0.194005π-0.194005\pi
0.819943 + 0.572446i 0.194005π0.194005\pi
33 0 0
44 13838.1 1.68922
55 15625.0 0.447214
66 0 0
77 215324. 0.691761 0.345880 0.938279i 0.387580π-0.387580\pi
0.345880 + 0.938279i 0.387580π0.387580\pi
88 838027. 1.13025
99 0 0
1010 2.31915e6 0.733379
1111 3.50194e6 0.596014 0.298007 0.954564i 0.403678π-0.403678\pi
0.298007 + 0.954564i 0.403678π0.403678\pi
1212 0 0
1313 −1.22382e6 −0.0703211 −0.0351605 0.999382i 0.511194π-0.511194\pi
−0.0351605 + 0.999382i 0.511194π0.511194\pi
1414 3.19596e7 1.13441
1515 0 0
1616 1.10227e7 0.164252
1717 1.14596e8 1.15146 0.575731 0.817639i 0.304717π-0.304717\pi
0.575731 + 0.817639i 0.304717π0.304717\pi
1818 0 0
1919 1.26403e8 0.616393 0.308197 0.951323i 0.400275π-0.400275\pi
0.308197 + 0.951323i 0.400275π0.400275\pi
2020 2.16221e8 0.755444
2121 0 0
2222 5.19777e8 0.977394
2323 −1.54080e8 −0.217028 −0.108514 0.994095i 0.534609π-0.534609\pi
−0.108514 + 0.994095i 0.534609π0.534609\pi
2424 0 0
2525 2.44141e8 0.200000
2626 −1.81646e8 −0.115318
2727 0 0
2828 2.97968e9 1.16854
2929 −3.50622e9 −1.09459 −0.547295 0.836940i 0.684343π-0.684343\pi
−0.547295 + 0.836940i 0.684343π0.684343\pi
3030 0 0
3131 6.23402e9 1.26159 0.630794 0.775950i 0.282729π-0.282729\pi
0.630794 + 0.775950i 0.282729π0.282729\pi
3232 −5.22907e9 −0.860893
3333 0 0
3434 1.70089e10 1.88827
3535 3.36444e9 0.309365
3636 0 0
3737 2.11673e10 1.35630 0.678148 0.734925i 0.262783π-0.262783\pi
0.678148 + 0.734925i 0.262783π0.262783\pi
3838 1.87614e10 1.01081
3939 0 0
4040 1.30942e10 0.505462
4141 5.91552e10 1.94490 0.972451 0.233107i 0.0748892π-0.0748892\pi
0.972451 + 0.233107i 0.0748892π0.0748892\pi
4242 0 0
4343 −1.84126e10 −0.444192 −0.222096 0.975025i 0.571290π-0.571290\pi
−0.222096 + 0.975025i 0.571290π0.571290\pi
4444 4.84603e10 1.00680
4545 0 0
4646 −2.28694e10 −0.355901
4747 −3.33828e10 −0.451737 −0.225869 0.974158i 0.572522π-0.572522\pi
−0.225869 + 0.974158i 0.572522π0.572522\pi
4848 0 0
4949 −5.05244e10 −0.521467
5050 3.62367e10 0.327977
5151 0 0
5252 −1.69354e10 −0.118788
5353 −7.18621e10 −0.445356 −0.222678 0.974892i 0.571480π-0.571480\pi
−0.222678 + 0.974892i 0.571480π0.571480\pi
5454 0 0
5555 5.47178e10 0.266546
5656 1.80448e11 0.781860
5757 0 0
5858 −5.20412e11 −1.79500
5959 2.91405e10 0.0899412 0.0449706 0.998988i 0.485681π-0.485681\pi
0.0449706 + 0.998988i 0.485681π0.485681\pi
6060 0 0
6161 −5.52018e11 −1.37186 −0.685929 0.727669i 0.740604π-0.740604\pi
−0.685929 + 0.727669i 0.740604π0.740604\pi
6262 9.25288e11 2.06886
6363 0 0
6464 −8.66425e11 −1.57602
6565 −1.91222e10 −0.0314485
6666 0 0
6767 9.87816e11 1.33410 0.667052 0.745011i 0.267556π-0.267556\pi
0.667052 + 0.745011i 0.267556π0.267556\pi
6868 1.58579e12 1.94508
6969 0 0
7070 4.99369e11 0.507323
7171 1.33203e11 0.123406 0.0617030 0.998095i 0.480347π-0.480347\pi
0.0617030 + 0.998095i 0.480347π0.480347\pi
7272 0 0
7373 1.18101e10 0.00913385 0.00456693 0.999990i 0.498546π-0.498546\pi
0.00456693 + 0.999990i 0.498546π0.498546\pi
7474 3.14177e12 2.22417
7575 0 0
7676 1.74918e12 1.04123
7777 7.54053e11 0.412299
7878 0 0
7979 −3.39311e12 −1.57044 −0.785222 0.619215i 0.787451π-0.787451\pi
−0.785222 + 0.619215i 0.787451π0.787451\pi
8080 1.72230e11 0.0734555
8181 0 0
8282 8.78014e12 3.18942
8383 −1.44794e12 −0.486118 −0.243059 0.970011i 0.578151π-0.578151\pi
−0.243059 + 0.970011i 0.578151π0.578151\pi
8484 0 0
8585 1.79056e12 0.514950
8686 −2.73290e12 −0.728424
8787 0 0
8888 2.93472e12 0.673643
8989 −1.62879e11 −0.0347400 −0.0173700 0.999849i 0.505529π-0.505529\pi
−0.0173700 + 0.999849i 0.505529π0.505529\pi
9090 0 0
9191 −2.63518e11 −0.0486454
9292 −2.13218e12 −0.366609
9393 0 0
9494 −4.95485e12 −0.740798
9595 1.97504e12 0.275659
9696 0 0
9797 3.58718e12 0.437258 0.218629 0.975808i 0.429842π-0.429842\pi
0.218629 + 0.975808i 0.429842π0.429842\pi
9898 −7.49911e12 −0.855146
9999 0 0
100100 3.37845e12 0.337845
101101 1.32945e13 1.24619 0.623095 0.782146i 0.285875π-0.285875\pi
0.623095 + 0.782146i 0.285875π0.285875\pi
102102 0 0
103103 −2.14656e13 −1.77134 −0.885669 0.464317i 0.846300π-0.846300\pi
−0.885669 + 0.464317i 0.846300π0.846300\pi
104104 −1.02559e12 −0.0794801
105105 0 0
106106 −1.06662e13 −0.730332
107107 −4.53570e12 −0.292180 −0.146090 0.989271i 0.546669π-0.546669\pi
−0.146090 + 0.989271i 0.546669π0.546669\pi
108108 0 0
109109 −2.75975e13 −1.57615 −0.788074 0.615580i 0.788922π-0.788922\pi
−0.788074 + 0.615580i 0.788922π0.788922\pi
110110 8.12152e12 0.437104
111111 0 0
112112 2.37346e12 0.113623
113113 −3.59293e13 −1.62345 −0.811725 0.584040i 0.801471π-0.801471\pi
−0.811725 + 0.584040i 0.801471π0.801471\pi
114114 0 0
115115 −2.40751e12 −0.0970580
116116 −4.85194e13 −1.84901
117117 0 0
118118 4.32519e12 0.147493
119119 2.46752e13 0.796537
120120 0 0
121121 −2.22591e13 −0.644767
122122 −8.19335e13 −2.24969
123123 0 0
124124 8.62671e13 2.13110
125125 3.81470e12 0.0894427
126126 0 0
127127 −8.24154e13 −1.74295 −0.871473 0.490443i 0.836835π-0.836835\pi
−0.871473 + 0.490443i 0.836835π0.836835\pi
128128 −8.57630e13 −1.72359
129129 0 0
130130 −2.83822e12 −0.0515720
131131 −9.61500e13 −1.66221 −0.831105 0.556115i 0.812291π-0.812291\pi
−0.831105 + 0.556115i 0.812291π0.812291\pi
132132 0 0
133133 2.72176e13 0.426397
134134 1.46617e14 2.18778
135135 0 0
136136 9.60342e13 1.30144
137137 1.15756e14 1.49575 0.747877 0.663837i 0.231073π-0.231073\pi
0.747877 + 0.663837i 0.231073π0.231073\pi
138138 0 0
139139 8.85611e13 1.04147 0.520735 0.853718i 0.325658π-0.325658\pi
0.520735 + 0.853718i 0.325658π0.325658\pi
140140 4.65576e13 0.522586
141141 0 0
142142 1.97708e13 0.202372
143143 −4.28574e12 −0.0419123
144144 0 0
145145 −5.47846e13 −0.489516
146146 1.75292e12 0.0149785
147147 0 0
148148 2.92916e14 2.29109
149149 1.45878e14 1.09215 0.546073 0.837738i 0.316122π-0.316122\pi
0.546073 + 0.837738i 0.316122π0.316122\pi
150150 0 0
151151 1.41584e13 0.0971994 0.0485997 0.998818i 0.484524π-0.484524\pi
0.0485997 + 0.998818i 0.484524π0.484524\pi
152152 1.05929e14 0.696676
153153 0 0
154154 1.11921e14 0.676123
155155 9.74066e13 0.564199
156156 0 0
157157 −3.21755e14 −1.71466 −0.857329 0.514768i 0.827878π-0.827878\pi
−0.857329 + 0.514768i 0.827878π0.827878\pi
158158 −5.03624e14 −2.57535
159159 0 0
160160 −8.17041e13 −0.385003
161161 −3.31773e13 −0.150132
162162 0 0
163163 −1.93760e14 −0.809181 −0.404591 0.914498i 0.632586π-0.632586\pi
−0.404591 + 0.914498i 0.632586π0.632586\pi
164164 8.18596e14 3.28537
165165 0 0
166166 −2.14911e14 −0.797178
167167 −3.35780e14 −1.19784 −0.598919 0.800809i 0.704403π-0.704403\pi
−0.598919 + 0.800809i 0.704403π0.704403\pi
168168 0 0
169169 −3.01377e14 −0.995055
170170 2.65764e14 0.844458
171171 0 0
172172 −2.54796e14 −0.750340
173173 −1.10123e14 −0.312306 −0.156153 0.987733i 0.549909π-0.549909\pi
−0.156153 + 0.987733i 0.549909π0.549909\pi
174174 0 0
175175 5.25694e13 0.138352
176176 3.86010e13 0.0978962
177177 0 0
178178 −2.41754e13 −0.0569697
179179 7.53350e14 1.71180 0.855898 0.517145i 0.173005π-0.173005\pi
0.855898 + 0.517145i 0.173005π0.173005\pi
180180 0 0
181181 −1.93468e14 −0.408977 −0.204488 0.978869i 0.565553π-0.565553\pi
−0.204488 + 0.978869i 0.565553π0.565553\pi
182182 −3.91128e13 −0.0797728
183183 0 0
184184 −1.29124e14 −0.245296
185185 3.30739e14 0.606554
186186 0 0
187187 4.01307e14 0.686288
188188 −4.61954e14 −0.763085
189189 0 0
190190 2.93147e14 0.452050
191191 3.06017e14 0.456067 0.228033 0.973653i 0.426770π-0.426770\pi
0.228033 + 0.973653i 0.426770π0.426770\pi
192192 0 0
193193 2.32828e14 0.324274 0.162137 0.986768i 0.448161π-0.448161\pi
0.162137 + 0.986768i 0.448161π0.448161\pi
194194 5.32429e14 0.717052
195195 0 0
196196 −6.99163e14 −0.880874
197197 3.64429e14 0.444204 0.222102 0.975023i 0.428708π-0.428708\pi
0.222102 + 0.975023i 0.428708π0.428708\pi
198198 0 0
199199 −2.43854e14 −0.278346 −0.139173 0.990268i 0.544444π-0.544444\pi
−0.139173 + 0.990268i 0.544444π0.544444\pi
200200 2.04596e14 0.226049
201201 0 0
202202 1.97325e15 2.04361
203203 −7.54974e14 −0.757195
204204 0 0
205205 9.24300e14 0.869787
206206 −3.18604e15 −2.90479
207207 0 0
208208 −1.34898e13 −0.0115503
209209 4.42655e14 0.367379
210210 0 0
211211 1.86277e15 1.45319 0.726596 0.687065i 0.241101π-0.241101\pi
0.726596 + 0.687065i 0.241101π0.241101\pi
212212 −9.94436e14 −0.752305
213213 0 0
214214 −6.73213e14 −0.479141
215215 −2.87697e14 −0.198649
216216 0 0
217217 1.34234e15 0.872717
218218 −4.09617e15 −2.58470
219219 0 0
220220 7.57192e14 0.450255
221221 −1.40244e14 −0.0809721
222222 0 0
223223 8.44142e14 0.459657 0.229829 0.973231i 0.426183π-0.426183\pi
0.229829 + 0.973231i 0.426183π0.426183\pi
224224 −1.12595e15 −0.595532
225225 0 0
226226 −5.33282e15 −2.66227
227227 7.84425e14 0.380525 0.190263 0.981733i 0.439066π-0.439066\pi
0.190263 + 0.981733i 0.439066π0.439066\pi
228228 0 0
229229 2.50019e14 0.114563 0.0572813 0.998358i 0.481757π-0.481757\pi
0.0572813 + 0.998358i 0.481757π0.481757\pi
230230 −3.57335e14 −0.159164
231231 0 0
232232 −2.93830e15 −1.23716
233233 2.82252e15 1.15564 0.577822 0.816163i 0.303903π-0.303903\pi
0.577822 + 0.816163i 0.303903π0.303903\pi
234234 0 0
235235 −5.21606e14 −0.202023
236236 4.03249e14 0.151931
237237 0 0
238238 3.66243e15 1.30623
239239 3.19985e15 1.11056 0.555281 0.831663i 0.312611π-0.312611\pi
0.555281 + 0.831663i 0.312611π0.312611\pi
240240 0 0
241241 1.14929e15 0.377848 0.188924 0.981992i 0.439500π-0.439500\pi
0.188924 + 0.981992i 0.439500π0.439500\pi
242242 −3.30382e15 −1.05734
243243 0 0
244244 −7.63888e15 −2.31737
245245 −7.89444e14 −0.233207
246246 0 0
247247 −1.54694e14 −0.0433454
248248 5.22428e15 1.42591
249249 0 0
250250 5.66198e14 0.146676
251251 4.56021e15 1.15108 0.575540 0.817773i 0.304792π-0.304792\pi
0.575540 + 0.817773i 0.304792π0.304792\pi
252252 0 0
253253 −5.39580e14 −0.129352
254254 −1.22325e16 −2.85823
255255 0 0
256256 −5.63166e15 −1.25048
257257 2.41893e15 0.523671 0.261835 0.965113i 0.415672π-0.415672\pi
0.261835 + 0.965113i 0.415672π0.415672\pi
258258 0 0
259259 4.55784e15 0.938233
260260 −2.64615e14 −0.0531236
261261 0 0
262262 −1.42711e16 −2.72583
263263 1.39889e15 0.260657 0.130329 0.991471i 0.458397π-0.458397\pi
0.130329 + 0.991471i 0.458397π0.458397\pi
264264 0 0
265265 −1.12285e15 −0.199169
266266 4.03978e15 0.699242
267267 0 0
268268 1.36695e16 2.25360
269269 −7.53557e15 −1.21262 −0.606312 0.795227i 0.707352π-0.707352\pi
−0.606312 + 0.795227i 0.707352π0.707352\pi
270270 0 0
271271 −3.68276e15 −0.564772 −0.282386 0.959301i 0.591126π-0.591126\pi
−0.282386 + 0.959301i 0.591126π0.591126\pi
272272 1.26316e15 0.189130
273273 0 0
274274 1.71812e16 2.45287
275275 8.54966e14 0.119203
276276 0 0
277277 −2.90631e15 −0.386566 −0.193283 0.981143i 0.561914π-0.561914\pi
−0.193283 + 0.981143i 0.561914π0.561914\pi
278278 1.31447e16 1.70789
279279 0 0
280280 2.81950e15 0.349659
281281 4.98905e15 0.604542 0.302271 0.953222i 0.402255π-0.402255\pi
0.302271 + 0.953222i 0.402255π0.402255\pi
282282 0 0
283283 1.44585e16 1.67306 0.836529 0.547922i 0.184581π-0.184581\pi
0.836529 + 0.547922i 0.184581π0.184581\pi
284284 1.84329e15 0.208460
285285 0 0
286286 −6.36113e14 −0.0687314
287287 1.27376e16 1.34541
288288 0 0
289289 3.22757e15 0.325866
290290 −8.13143e15 −0.802750
291291 0 0
292292 1.63429e14 0.0154291
293293 1.80867e16 1.67001 0.835007 0.550239i 0.185463π-0.185463\pi
0.835007 + 0.550239i 0.185463π0.185463\pi
294294 0 0
295295 4.55320e14 0.0402229
296296 1.77388e16 1.53295
297297 0 0
298298 2.16521e16 1.79099
299299 1.88567e14 0.0152617
300300 0 0
301301 −3.96469e15 −0.307275
302302 2.10147e15 0.159396
303303 0 0
304304 1.39330e15 0.101244
305305 −8.62528e15 −0.613513
306306 0 0
307307 −2.08974e16 −1.42460 −0.712301 0.701874i 0.752347π-0.752347\pi
−0.712301 + 0.701874i 0.752347π0.752347\pi
308308 1.04347e16 0.696465
309309 0 0
310310 1.44576e16 0.925222
311311 −7.80809e15 −0.489330 −0.244665 0.969608i 0.578678π-0.578678\pi
−0.244665 + 0.969608i 0.578678π0.578678\pi
312312 0 0
313313 −1.63797e16 −0.984620 −0.492310 0.870420i 0.663847π-0.663847\pi
−0.492310 + 0.870420i 0.663847π0.663847\pi
314314 −4.77566e16 −2.81184
315315 0 0
316316 −4.69543e16 −2.65283
317317 3.59584e15 0.199028 0.0995141 0.995036i 0.468271π-0.468271\pi
0.0995141 + 0.995036i 0.468271π0.468271\pi
318318 0 0
319319 −1.22786e16 −0.652391
320320 −1.35379e16 −0.704816
321321 0 0
322322 −4.92435e15 −0.246199
323323 1.44852e16 0.709754
324324 0 0
325325 −2.98784e14 −0.0140642
326326 −2.87590e16 −1.32696
327327 0 0
328328 4.95737e16 2.19822
329329 −7.18812e15 −0.312494
330330 0 0
331331 −2.21791e16 −0.926961 −0.463480 0.886107i 0.653400π-0.653400\pi
−0.463480 + 0.886107i 0.653400π0.653400\pi
332332 −2.00367e16 −0.821162
333333 0 0
334334 −4.98384e16 −1.96432
335335 1.54346e16 0.596629
336336 0 0
337337 3.17420e16 1.18043 0.590215 0.807246i 0.299043π-0.299043\pi
0.590215 + 0.807246i 0.299043π0.299043\pi
338338 −4.47321e16 −1.63178
339339 0 0
340340 2.47779e16 0.869865
341341 2.18312e16 0.751924
342342 0 0
343343 −3.17417e16 −1.05249
344344 −1.54303e16 −0.502047
345345 0 0
346346 −1.63451e16 −0.512146
347347 −1.06948e16 −0.328876 −0.164438 0.986387i 0.552581π-0.552581\pi
−0.164438 + 0.986387i 0.552581π0.552581\pi
348348 0 0
349349 −4.43687e16 −1.31435 −0.657176 0.753737i 0.728249π-0.728249\pi
−0.657176 + 0.753737i 0.728249π0.728249\pi
350350 7.80264e15 0.226882
351351 0 0
352352 −1.83119e16 −0.513104
353353 1.05902e16 0.291319 0.145660 0.989335i 0.453470π-0.453470\pi
0.145660 + 0.989335i 0.453470π0.453470\pi
354354 0 0
355355 2.08130e15 0.0551888
356356 −2.25394e15 −0.0586837
357357 0 0
358358 1.11816e17 2.80715
359359 5.10726e15 0.125914 0.0629570 0.998016i 0.479947π-0.479947\pi
0.0629570 + 0.998016i 0.479947π0.479947\pi
360360 0 0
361361 −2.60753e16 −0.620059
362362 −2.87156e16 −0.670675
363363 0 0
364364 −3.64659e15 −0.0821729
365365 1.84532e14 0.00408478
366366 0 0
367367 5.49713e16 1.17437 0.587187 0.809452i 0.300235π-0.300235\pi
0.587187 + 0.809452i 0.300235π0.300235\pi
368368 −1.69839e15 −0.0356472
369369 0 0
370370 4.90902e16 0.994679
371371 −1.54737e16 −0.308080
372372 0 0
373373 4.60597e16 0.885551 0.442775 0.896633i 0.353994π-0.353994\pi
0.442775 + 0.896633i 0.353994π0.353994\pi
374374 5.95642e16 1.12543
375375 0 0
376376 −2.79757e16 −0.510575
377377 4.29097e15 0.0769728
378378 0 0
379379 1.36123e16 0.235926 0.117963 0.993018i 0.462364π-0.462364\pi
0.117963 + 0.993018i 0.462364π0.462364\pi
380380 2.73309e16 0.465650
381381 0 0
382382 4.54207e16 0.747897
383383 −1.14040e17 −1.84615 −0.923074 0.384623i 0.874331π-0.874331\pi
−0.923074 + 0.384623i 0.874331π0.874331\pi
384384 0 0
385385 1.17821e16 0.184386
386386 3.45576e16 0.531772
387387 0 0
388388 4.96399e16 0.738626
389389 5.53953e16 0.810589 0.405295 0.914186i 0.367169π-0.367169\pi
0.405295 + 0.914186i 0.367169π0.367169\pi
390390 0 0
391391 −1.76569e16 −0.249900
392392 −4.23408e16 −0.589386
393393 0 0
394394 5.40905e16 0.728443
395395 −5.30174e16 −0.702324
396396 0 0
397397 −7.12560e16 −0.913447 −0.456723 0.889609i 0.650977π-0.650977\pi
−0.456723 + 0.889609i 0.650977π0.650977\pi
398398 −3.61941e16 −0.456455
399399 0 0
400400 2.69110e15 0.0328503
401401 −1.25835e17 −1.51135 −0.755674 0.654948i 0.772691π-0.772691\pi
−0.755674 + 0.654948i 0.772691π0.772691\pi
402402 0 0
403403 −7.62932e15 −0.0887162
404404 1.83971e17 2.10509
405405 0 0
406406 −1.12057e17 −1.24171
407407 7.41267e16 0.808372
408408 0 0
409409 9.42268e16 0.995343 0.497672 0.867365i 0.334188π-0.334188\pi
0.497672 + 0.867365i 0.334188π0.334188\pi
410410 1.37190e17 1.42635
411411 0 0
412412 −2.97044e17 −2.99219
413413 6.27466e15 0.0622178
414414 0 0
415415 −2.26240e16 −0.217399
416416 6.39943e15 0.0605389
417417 0 0
418418 6.57012e16 0.602459
419419 2.06207e17 1.86171 0.930857 0.365383i 0.119062π-0.119062\pi
0.930857 + 0.365383i 0.119062π0.119062\pi
420420 0 0
421421 8.51583e16 0.745406 0.372703 0.927951i 0.378431π-0.378431\pi
0.372703 + 0.927951i 0.378431π0.378431\pi
422422 2.76482e17 2.38307
423423 0 0
424424 −6.02224e16 −0.503362
425425 2.79774e16 0.230293
426426 0 0
427427 −1.18863e17 −0.948997
428428 −6.27656e16 −0.493556
429429 0 0
430430 −4.27016e16 −0.325761
431431 −8.69860e16 −0.653653 −0.326826 0.945084i 0.605979π-0.605979\pi
−0.326826 + 0.945084i 0.605979π0.605979\pi
432432 0 0
433433 1.26824e17 0.924759 0.462380 0.886682i 0.346996π-0.346996\pi
0.462380 + 0.886682i 0.346996π0.346996\pi
434434 1.99237e17 1.43116
435435 0 0
436436 −3.81897e17 −2.66247
437437 −1.94762e16 −0.133775
438438 0 0
439439 4.81345e16 0.320950 0.160475 0.987040i 0.448697π-0.448697\pi
0.160475 + 0.987040i 0.448697π0.448697\pi
440440 4.58550e16 0.301262
441441 0 0
442442 −2.08158e16 −0.132785
443443 −1.59797e17 −1.00449 −0.502244 0.864726i 0.667492π-0.667492\pi
−0.502244 + 0.864726i 0.667492π0.667492\pi
444444 0 0
445445 −2.54499e15 −0.0155362
446446 1.25292e17 0.753785
447447 0 0
448448 −1.86562e17 −1.09023
449449 −8.88441e16 −0.511714 −0.255857 0.966715i 0.582358π-0.582358\pi
−0.255857 + 0.966715i 0.582358π0.582358\pi
450450 0 0
451451 2.07158e17 1.15919
452452 −4.97194e17 −2.74237
453453 0 0
454454 1.16429e17 0.624017
455455 −4.11747e15 −0.0217549
456456 0 0
457457 1.18218e16 0.0607056 0.0303528 0.999539i 0.490337π-0.490337\pi
0.0303528 + 0.999539i 0.490337π0.490337\pi
458458 3.71092e16 0.187869
459459 0 0
460460 −3.33153e16 −0.163953
461461 −2.74119e17 −1.33010 −0.665049 0.746799i 0.731590π-0.731590\pi
−0.665049 + 0.746799i 0.731590π0.731590\pi
462462 0 0
463463 1.93108e17 0.911011 0.455506 0.890233i 0.349458π-0.349458\pi
0.455506 + 0.890233i 0.349458π0.349458\pi
464464 −3.86481e16 −0.179788
465465 0 0
466466 4.18934e17 1.89512
467467 −1.32213e17 −0.589812 −0.294906 0.955526i 0.595288π-0.595288\pi
−0.294906 + 0.955526i 0.595288π0.595288\pi
468468 0 0
469469 2.12701e17 0.922881
470470 −7.74195e16 −0.331295
471471 0 0
472472 2.44205e16 0.101656
473473 −6.44800e16 −0.264745
474474 0 0
475475 3.08600e16 0.123279
476476 3.41459e17 1.34553
477477 0 0
478478 4.74939e17 1.82119
479479 −2.43680e17 −0.921804 −0.460902 0.887451i 0.652474π-0.652474\pi
−0.460902 + 0.887451i 0.652474π0.652474\pi
480480 0 0
481481 −2.59050e16 −0.0953762
482482 1.70583e17 0.619628
483483 0 0
484484 −3.08024e17 −1.08916
485485 5.60497e16 0.195548
486486 0 0
487487 3.83909e17 1.30404 0.652018 0.758203i 0.273922π-0.273922\pi
0.652018 + 0.758203i 0.273922π0.273922\pi
488488 −4.62606e17 −1.55054
489489 0 0
490490 −1.17174e17 −0.382433
491491 4.57681e15 0.0147412 0.00737060 0.999973i 0.497654π-0.497654\pi
0.00737060 + 0.999973i 0.497654π0.497654\pi
492492 0 0
493493 −4.01797e17 −1.26038
494494 −2.29605e16 −0.0710815
495495 0 0
496496 6.87160e16 0.207218
497497 2.86820e16 0.0853674
498498 0 0
499499 −5.27519e17 −1.52962 −0.764812 0.644254i 0.777168π-0.777168\pi
−0.764812 + 0.644254i 0.777168π0.777168\pi
500500 5.27882e16 0.151089
501501 0 0
502502 6.76851e17 1.88764
503503 3.47045e17 0.955418 0.477709 0.878518i 0.341467π-0.341467\pi
0.477709 + 0.878518i 0.341467π0.341467\pi
504504 0 0
505505 2.07727e17 0.557313
506506 −8.00875e16 −0.212122
507507 0 0
508508 −1.14047e18 −2.94423
509509 −1.22973e17 −0.313433 −0.156717 0.987644i 0.550091π-0.550091\pi
−0.156717 + 0.987644i 0.550091π0.550091\pi
510510 0 0
511511 2.54300e15 0.00631844
512512 −1.33311e17 −0.327048
513513 0 0
514514 3.59031e17 0.858760
515515 −3.35400e17 −0.792167
516516 0 0
517517 −1.16904e17 −0.269242
518518 6.76500e17 1.53859
519519 0 0
520520 −1.60249e16 −0.0355446
521521 −6.98024e17 −1.52906 −0.764531 0.644587i 0.777029π-0.777029\pi
−0.764531 + 0.644587i 0.777029π0.777029\pi
522522 0 0
523523 −1.17031e17 −0.250057 −0.125028 0.992153i 0.539902π-0.539902\pi
−0.125028 + 0.992153i 0.539902π0.539902\pi
524524 −1.33053e18 −2.80784
525525 0 0
526526 2.07630e17 0.427448
527527 7.14392e17 1.45267
528528 0 0
529529 −4.80296e17 −0.952899
530530 −1.66659e17 −0.326614
531531 0 0
532532 3.76640e17 0.720279
533533 −7.23953e16 −0.136768
534534 0 0
535535 −7.08703e16 −0.130667
536536 8.27816e17 1.50787
537537 0 0
538538 −1.11847e18 −1.98857
539539 −1.76933e17 −0.310801
540540 0 0
541541 −4.80930e17 −0.824707 −0.412353 0.911024i 0.635293π-0.635293\pi
−0.412353 + 0.911024i 0.635293π0.635293\pi
542542 −5.46616e17 −0.926162
543543 0 0
544544 −5.99228e17 −0.991286
545545 −4.31210e17 −0.704875
546546 0 0
547547 −1.58434e16 −0.0252889 −0.0126445 0.999920i 0.504025π-0.504025\pi
−0.0126445 + 0.999920i 0.504025π0.504025\pi
548548 1.60185e18 2.52666
549549 0 0
550550 1.26899e17 0.195479
551551 −4.43195e17 −0.674698
552552 0 0
553553 −7.30620e17 −1.08637
554554 −4.31370e17 −0.633923
555555 0 0
556556 1.22552e18 1.75928
557557 7.17753e17 1.01840 0.509198 0.860650i 0.329942π-0.329942\pi
0.509198 + 0.860650i 0.329942π0.329942\pi
558558 0 0
559559 2.25337e16 0.0312361
560560 3.70854e16 0.0508137
561561 0 0
562562 7.40502e17 0.991380
563563 8.56529e17 1.13354 0.566771 0.823876i 0.308193π-0.308193\pi
0.566771 + 0.823876i 0.308193π0.308193\pi
564564 0 0
565565 −5.61395e17 −0.726029
566566 2.14601e18 2.74362
567567 0 0
568568 1.11628e17 0.139479
569569 4.50162e17 0.556083 0.278041 0.960569i 0.410315π-0.410315\pi
0.278041 + 0.960569i 0.410315π0.410315\pi
570570 0 0
571571 −8.83282e16 −0.106651 −0.0533255 0.998577i 0.516982π-0.516982\pi
−0.0533255 + 0.998577i 0.516982π0.516982\pi
572572 −5.93066e16 −0.0707993
573573 0 0
574574 1.89058e18 2.20631
575575 −3.76173e16 −0.0434057
576576 0 0
577577 −2.23416e17 −0.252042 −0.126021 0.992028i 0.540221π-0.540221\pi
−0.126021 + 0.992028i 0.540221π0.540221\pi
578578 4.79054e17 0.534384
579579 0 0
580580 −7.58116e17 −0.826901
581581 −3.11776e17 −0.336278
582582 0 0
583583 −2.51657e17 −0.265438
584584 9.89717e15 0.0103235
585585 0 0
586586 2.68453e18 2.73863
587587 9.27969e17 0.936237 0.468119 0.883666i 0.344932π-0.344932\pi
0.468119 + 0.883666i 0.344932π0.344932\pi
588588 0 0
589589 7.87997e17 0.777634
590590 6.75811e16 0.0659610
591591 0 0
592592 2.33322e17 0.222774
593593 −7.24111e16 −0.0683832 −0.0341916 0.999415i 0.510886π-0.510886\pi
−0.0341916 + 0.999415i 0.510886π0.510886\pi
594594 0 0
595595 3.85550e17 0.356222
596596 2.01868e18 1.84488
597597 0 0
598598 2.79881e16 0.0250274
599599 −7.61377e17 −0.673481 −0.336740 0.941597i 0.609325π-0.609325\pi
−0.336740 + 0.941597i 0.609325π0.609325\pi
600600 0 0
601601 8.87128e17 0.767896 0.383948 0.923355i 0.374564π-0.374564\pi
0.383948 + 0.923355i 0.374564π0.374564\pi
602602 −5.88461e17 −0.503895
603603 0 0
604604 1.95925e17 0.164192
605605 −3.47799e17 −0.288349
606606 0 0
607607 −7.33626e17 −0.595317 −0.297658 0.954672i 0.596206π-0.596206\pi
−0.297658 + 0.954672i 0.596206π0.596206\pi
608608 −6.60968e17 −0.530649
609609 0 0
610610 −1.28021e18 −1.00609
611611 4.08545e16 0.0317667
612612 0 0
613613 1.49204e18 1.13576 0.567880 0.823111i 0.307764π-0.307764\pi
0.567880 + 0.823111i 0.307764π0.307764\pi
614614 −3.10171e18 −2.33618
615615 0 0
616616 6.31917e17 0.466000
617617 1.83926e17 0.134211 0.0671057 0.997746i 0.478624π-0.478624\pi
0.0671057 + 0.997746i 0.478624π0.478624\pi
618618 0 0
619619 1.24283e16 0.00888019 0.00444009 0.999990i 0.498587π-0.498587\pi
0.00444009 + 0.999990i 0.498587π0.498587\pi
620620 1.34792e18 0.953058
621621 0 0
622622 −1.15892e18 −0.802445
623623 −3.50719e16 −0.0240318
624624 0 0
625625 5.96046e16 0.0400000
626626 −2.43117e18 −1.61466
627627 0 0
628628 −4.45248e18 −2.89644
629629 2.42568e18 1.56173
630630 0 0
631631 −1.68478e18 −1.06256 −0.531279 0.847197i 0.678289π-0.678289\pi
−0.531279 + 0.847197i 0.678289π0.678289\pi
632632 −2.84352e18 −1.77499
633633 0 0
634634 5.33714e17 0.326383
635635 −1.28774e18 −0.779469
636636 0 0
637637 6.18327e16 0.0366701
638638 −1.82245e18 −1.06985
639639 0 0
640640 −1.34005e18 −0.770815
641641 −2.25951e18 −1.28658 −0.643290 0.765623i 0.722431π-0.722431\pi
−0.643290 + 0.765623i 0.722431π0.722431\pi
642642 0 0
643643 1.14956e18 0.641449 0.320725 0.947173i 0.396074π-0.396074\pi
0.320725 + 0.947173i 0.396074π0.396074\pi
644644 −4.59111e17 −0.253606
645645 0 0
646646 2.14997e18 1.16391
647647 2.91569e18 1.56266 0.781328 0.624121i 0.214543π-0.214543\pi
0.781328 + 0.624121i 0.214543π0.214543\pi
648648 0 0
649649 1.02048e17 0.0536062
650650 −4.43471e16 −0.0230637
651651 0 0
652652 −2.68128e18 −1.36689
653653 2.16676e18 1.09364 0.546821 0.837249i 0.315838π-0.315838\pi
0.546821 + 0.837249i 0.315838π0.315838\pi
654654 0 0
655655 −1.50234e18 −0.743363
656656 6.52052e17 0.319453
657657 0 0
658658 −1.06690e18 −0.512455
659659 4.11523e18 1.95722 0.978608 0.205732i 0.0659574π-0.0659574\pi
0.978608 + 0.205732i 0.0659574π0.0659574\pi
660660 0 0
661661 1.51002e17 0.0704162 0.0352081 0.999380i 0.488791π-0.488791\pi
0.0352081 + 0.999380i 0.488791π0.488791\pi
662662 −3.29194e18 −1.52011
663663 0 0
664664 −1.21341e18 −0.549434
665665 4.25275e17 0.190690
666666 0 0
667667 5.40239e17 0.237557
668668 −4.64657e18 −2.02342
669669 0 0
670670 2.29089e18 0.978404
671671 −1.93313e18 −0.817646
672672 0 0
673673 −4.17178e18 −1.73071 −0.865354 0.501161i 0.832906π-0.832906\pi
−0.865354 + 0.501161i 0.832906π0.832906\pi
674674 4.71133e18 1.93577
675675 0 0
676676 −4.17049e18 −1.68087
677677 1.27061e18 0.507206 0.253603 0.967308i 0.418384π-0.418384\pi
0.253603 + 0.967308i 0.418384π0.418384\pi
678678 0 0
679679 7.72408e17 0.302478
680680 1.50053e18 0.582020
681681 0 0
682682 3.24030e18 1.23307
683683 −2.64932e18 −0.998619 −0.499310 0.866424i 0.666413π-0.666413\pi
−0.499310 + 0.866424i 0.666413π0.666413\pi
684684 0 0
685685 1.80869e18 0.668922
686686 −4.71128e18 −1.72596
687687 0 0
688688 −2.02958e17 −0.0729593
689689 8.79463e16 0.0313179
690690 0 0
691691 4.37627e18 1.52931 0.764657 0.644438i 0.222909π-0.222909\pi
0.764657 + 0.644438i 0.222909π0.222909\pi
692692 −1.52390e18 −0.527554
693693 0 0
694694 −1.58738e18 −0.539319
695695 1.38377e18 0.465760
696696 0 0
697697 6.77892e18 2.23948
698698 −6.58545e18 −2.15539
699699 0 0
700700 7.27462e17 0.233708
701701 −3.76541e18 −1.19852 −0.599260 0.800555i 0.704538π-0.704538\pi
−0.599260 + 0.800555i 0.704538π0.704538\pi
702702 0 0
703703 2.67561e18 0.836012
704704 −3.03417e18 −0.939328
705705 0 0
706706 1.57186e18 0.477730
707707 2.86264e18 0.862066
708708 0 0
709709 1.74388e18 0.515604 0.257802 0.966198i 0.417002π-0.417002\pi
0.257802 + 0.966198i 0.417002π0.417002\pi
710710 3.08919e17 0.0905034
711711 0 0
712712 −1.36497e17 −0.0392648
713713 −9.60541e17 −0.273800
714714 0 0
715715 −6.69647e16 −0.0187438
716716 1.04249e19 2.89160
717717 0 0
718718 7.58048e17 0.206485
719719 4.92987e18 1.33075 0.665377 0.746507i 0.268271π-0.268271\pi
0.665377 + 0.746507i 0.268271π0.268271\pi
720720 0 0
721721 −4.62207e18 −1.22534
722722 −3.87024e18 −1.01683
723723 0 0
724724 −2.67724e18 −0.690853
725725 −8.56010e17 −0.218918
726726 0 0
727727 5.68333e18 1.42767 0.713837 0.700312i 0.246956π-0.246956\pi
0.713837 + 0.700312i 0.246956π0.246956\pi
728728 −2.20835e17 −0.0549813
729729 0 0
730730 2.73893e16 0.00669858
731731 −2.11001e18 −0.511471
732732 0 0
733733 2.88616e18 0.687298 0.343649 0.939098i 0.388337π-0.388337\pi
0.343649 + 0.939098i 0.388337π0.388337\pi
734734 8.15913e18 1.92584
735735 0 0
736736 8.05696e17 0.186838
737737 3.45927e18 0.795145
738738 0 0
739739 −7.62132e18 −1.72124 −0.860620 0.509248i 0.829924π-0.829924\pi
−0.860620 + 0.509248i 0.829924π0.829924\pi
740740 4.57681e18 1.02461
741741 0 0
742742 −2.29669e18 −0.505215
743743 6.95681e18 1.51699 0.758495 0.651679i 0.225935π-0.225935\pi
0.758495 + 0.651679i 0.225935π0.225935\pi
744744 0 0
745745 2.27935e18 0.488422
746746 6.83643e18 1.45220
747747 0 0
748748 5.55333e18 1.15929
749749 −9.76647e17 −0.202118
750750 0 0
751751 3.81292e18 0.775528 0.387764 0.921759i 0.373248π-0.373248\pi
0.387764 + 0.921759i 0.373248π0.373248\pi
752752 −3.67969e17 −0.0741986
753753 0 0
754754 6.36890e17 0.126227
755755 2.21225e17 0.0434689
756756 0 0
757757 −7.07019e18 −1.36555 −0.682775 0.730629i 0.739227π-0.739227\pi
−0.682775 + 0.730629i 0.739227π0.739227\pi
758758 2.02041e18 0.386892
759759 0 0
760760 1.65514e18 0.311563
761761 −4.26743e18 −0.796465 −0.398232 0.917285i 0.630376π-0.630376\pi
−0.398232 + 0.917285i 0.630376π0.630376\pi
762762 0 0
763763 −5.94241e18 −1.09032
764764 4.23469e18 0.770398
765765 0 0
766766 −1.69265e19 −3.02747
767767 −3.56627e16 −0.00632476
768768 0 0
769769 −5.55109e16 −0.00967959 −0.00483980 0.999988i 0.501541π-0.501541\pi
−0.00483980 + 0.999988i 0.501541π0.501541\pi
770770 1.74876e18 0.302371
771771 0 0
772772 3.22190e18 0.547771
773773 −6.47032e18 −1.09083 −0.545417 0.838165i 0.683629π-0.683629\pi
−0.545417 + 0.838165i 0.683629π0.683629\pi
774774 0 0
775775 1.52198e18 0.252318
776776 3.00616e18 0.494209
777777 0 0
778778 8.22208e18 1.32927
779779 7.47738e18 1.19882
780780 0 0
781781 4.66471e17 0.0735517
782782 −2.62074e18 −0.409807
783783 0 0
784784 −5.56917e17 −0.0856517
785785 −5.02742e18 −0.766819
786786 0 0
787787 −8.41309e18 −1.26218 −0.631088 0.775712i 0.717391π-0.717391\pi
−0.631088 + 0.775712i 0.717391π0.717391\pi
788788 5.04301e18 0.750360
789789 0 0
790790 −7.86913e18 −1.15173
791791 −7.73645e18 −1.12304
792792 0 0
793793 6.75570e17 0.0964705
794794 −1.05762e19 −1.49795
795795 0 0
796796 −3.37448e18 −0.470188
797797 7.81826e18 1.08052 0.540258 0.841499i 0.318327π-0.318327\pi
0.540258 + 0.841499i 0.318327π0.318327\pi
798798 0 0
799799 −3.82552e18 −0.520159
800800 −1.27663e18 −0.172179
801801 0 0
802802 −1.86772e19 −2.47844
803803 4.13582e16 0.00544390
804804 0 0
805805 −5.18395e17 −0.0671409
806806 −1.13238e18 −0.145484
807807 0 0
808808 1.11412e19 1.40850
809809 −1.29209e19 −1.62041 −0.810207 0.586144i 0.800645π-0.800645\pi
−0.810207 + 0.586144i 0.800645π0.800645\pi
810810 0 0
811811 −7.85766e18 −0.969745 −0.484872 0.874585i 0.661134π-0.661134\pi
−0.484872 + 0.874585i 0.661134π0.661134\pi
812812 −1.04474e19 −1.27907
813813 0 0
814814 1.10023e19 1.32564
815815 −3.02751e18 −0.361877
816816 0 0
817817 −2.32741e18 −0.273797
818818 1.39857e19 1.63225
819819 0 0
820820 1.27906e19 1.46926
821821 −3.45810e17 −0.0394101 −0.0197051 0.999806i 0.506273π-0.506273\pi
−0.0197051 + 0.999806i 0.506273π0.506273\pi
822822 0 0
823823 8.61836e18 0.966776 0.483388 0.875406i 0.339406π-0.339406\pi
0.483388 + 0.875406i 0.339406π0.339406\pi
824824 −1.79888e19 −2.00205
825825 0 0
826826 9.31319e17 0.102030
827827 1.58749e19 1.72554 0.862768 0.505600i 0.168729π-0.168729\pi
0.862768 + 0.505600i 0.168729π0.168729\pi
828828 0 0
829829 6.99647e18 0.748643 0.374321 0.927299i 0.377876π-0.377876\pi
0.374321 + 0.927299i 0.377876π0.377876\pi
830830 −3.35798e18 −0.356509
831831 0 0
832832 1.06035e18 0.110827
833833 −5.78987e18 −0.600450
834834 0 0
835835 −5.24657e18 −0.535690
836836 6.12551e18 0.620585
837837 0 0
838838 3.06064e19 3.05300
839839 1.67940e18 0.166227 0.0831134 0.996540i 0.473514π-0.473514\pi
0.0831134 + 0.996540i 0.473514π0.473514\pi
840840 0 0
841841 2.03292e18 0.198128
842842 1.26397e19 1.22238
843843 0 0
844844 2.57772e19 2.45476
845845 −4.70902e18 −0.445002
846846 0 0
847847 −4.79293e18 −0.446025
848848 −7.92117e17 −0.0731503
849849 0 0
850850 4.15256e18 0.377653
851851 −3.26147e18 −0.294355
852852 0 0
853853 −1.88404e19 −1.67464 −0.837322 0.546711i 0.815880π-0.815880\pi
−0.837322 + 0.546711i 0.815880π0.815880\pi
854854 −1.76423e19 −1.55625
855855 0 0
856856 −3.80104e18 −0.330235
857857 −1.46107e19 −1.25978 −0.629892 0.776683i 0.716901π-0.716901\pi
−0.629892 + 0.776683i 0.716901π0.716901\pi
858858 0 0
859859 8.75747e17 0.0743743 0.0371872 0.999308i 0.488160π-0.488160\pi
0.0371872 + 0.999308i 0.488160π0.488160\pi
860860 −3.98119e18 −0.335562
861861 0 0
862862 −1.29109e19 −1.07192
863863 1.80636e19 1.48845 0.744226 0.667928i 0.232819π-0.232819\pi
0.744226 + 0.667928i 0.232819π0.232819\pi
864864 0 0
865865 −1.72068e18 −0.139667
866866 1.88238e19 1.51650
867867 0 0
868868 1.85754e19 1.47421
869869 −1.18825e19 −0.936006
870870 0 0
871871 −1.20891e18 −0.0938156
872872 −2.31274e19 −1.78144
873873 0 0
874874 −2.89076e18 −0.219375
875875 8.21397e17 0.0618730
876876 0 0
877877 3.65611e18 0.271345 0.135673 0.990754i 0.456680π-0.456680\pi
0.135673 + 0.990754i 0.456680π0.456680\pi
878878 7.14439e18 0.526321
879879 0 0
880880 6.03140e17 0.0437805
881881 1.10129e18 0.0793521 0.0396761 0.999213i 0.487367π-0.487367\pi
0.0396761 + 0.999213i 0.487367π0.487367\pi
882882 0 0
883883 8.49972e18 0.603476 0.301738 0.953391i 0.402433π-0.402433\pi
0.301738 + 0.953391i 0.402433π0.402433\pi
884884 −1.94072e18 −0.136780
885885 0 0
886886 −2.37180e19 −1.64725
887887 −2.42544e19 −1.67219 −0.836096 0.548583i 0.815168π-0.815168\pi
−0.836096 + 0.548583i 0.815168π0.815168\pi
888888 0 0
889889 −1.77460e19 −1.20570
890890 −3.77741e17 −0.0254776
891891 0 0
892892 1.16813e19 0.776463
893893 −4.21967e18 −0.278448
894894 0 0
895895 1.17711e19 0.765538
896896 −1.84669e19 −1.19231
897897 0 0
898898 −1.31867e19 −0.839152
899899 −2.18578e19 −1.38092
900900 0 0
901901 −8.23508e18 −0.512810
902902 3.07475e19 1.90094
903903 0 0
904904 −3.01097e19 −1.83490
905905 −3.02294e18 −0.182900
906906 0 0
907907 2.28334e19 1.36183 0.680915 0.732362i 0.261582π-0.261582\pi
0.680915 + 0.732362i 0.261582π0.261582\pi
908908 1.08550e19 0.642792
909909 0 0
910910 −6.11137e17 −0.0356755
911911 1.92508e19 1.11578 0.557890 0.829915i 0.311611π-0.311611\pi
0.557890 + 0.829915i 0.311611π0.311611\pi
912912 0 0
913913 −5.07059e18 −0.289733
914914 1.75466e18 0.0995502
915915 0 0
916916 3.45979e18 0.193522
917917 −2.07034e19 −1.14985
918918 0 0
919919 −2.09489e19 −1.14712 −0.573561 0.819163i 0.694438π-0.694438\pi
−0.573561 + 0.819163i 0.694438π0.694438\pi
920920 −2.01756e18 −0.109699
921921 0 0
922922 −4.06863e19 −2.18121
923923 −1.63017e17 −0.00867804
924924 0 0
925925 5.16780e18 0.271259
926926 2.86621e19 1.49395
927927 0 0
928928 1.83342e19 0.942325
929929 3.06906e18 0.156640 0.0783201 0.996928i 0.475044π-0.475044\pi
0.0783201 + 0.996928i 0.475044π0.475044\pi
930930 0 0
931931 −6.38642e18 −0.321429
932932 3.90584e19 1.95214
933933 0 0
934934 −1.96237e19 −0.967224
935935 6.27042e18 0.306917
936936 0 0
937937 1.98628e19 0.958809 0.479405 0.877594i 0.340853π-0.340853\pi
0.479405 + 0.877594i 0.340853π0.340853\pi
938938 3.15702e19 1.51342
939939 0 0
940940 −7.21804e18 −0.341262
941941 −3.52700e19 −1.65605 −0.828023 0.560694i 0.810534π-0.810534\pi
−0.828023 + 0.560694i 0.810534π0.810534\pi
942942 0 0
943943 −9.11465e18 −0.422099
944944 3.21208e17 0.0147730
945945 0 0
946946 −9.57047e18 −0.434151
947947 4.03654e19 1.81859 0.909295 0.416153i 0.136622π-0.136622\pi
0.909295 + 0.416153i 0.136622π0.136622\pi
948948 0 0
949949 −1.44534e16 −0.000642302 0
950950 4.58041e18 0.202163
951951 0 0
952952 2.06785e19 0.900283
953953 6.85871e18 0.296578 0.148289 0.988944i 0.452623π-0.452623\pi
0.148289 + 0.988944i 0.452623π0.452623\pi
954954 0 0
955955 4.78151e18 0.203959
956956 4.42799e19 1.87599
957957 0 0
958958 −3.61683e19 −1.51165
959959 2.49251e19 1.03470
960960 0 0
961961 1.44455e19 0.591603
962962 −3.84496e18 −0.156406
963963 0 0
964964 1.59040e19 0.638270
965965 3.63793e18 0.145020
966966 0 0
967967 4.77789e19 1.87916 0.939581 0.342328i 0.111215π-0.111215\pi
0.939581 + 0.342328i 0.111215π0.111215\pi
968968 −1.86538e19 −0.728746
969969 0 0
970970 8.31921e18 0.320675
971971 −5.51002e18 −0.210974 −0.105487 0.994421i 0.533640π-0.533640\pi
−0.105487 + 0.994421i 0.533640π0.533640\pi
972972 0 0
973973 1.90694e19 0.720448
974974 5.69818e19 2.13847
975975 0 0
976976 −6.08474e18 −0.225330
977977 3.66274e18 0.134738 0.0673691 0.997728i 0.478539π-0.478539\pi
0.0673691 + 0.997728i 0.478539π0.478539\pi
978978 0 0
979979 −5.70393e17 −0.0207055
980980 −1.09244e19 −0.393939
981981 0 0
982982 6.79315e17 0.0241739
983983 −1.33604e19 −0.472303 −0.236151 0.971716i 0.575886π-0.575886\pi
−0.236151 + 0.971716i 0.575886π0.575886\pi
984984 0 0
985985 5.69420e18 0.198654
986986 −5.96369e19 −2.06688
987987 0 0
988988 −2.14067e18 −0.0732201
989989 2.83703e18 0.0964023
990990 0 0
991991 3.12719e19 1.04876 0.524379 0.851485i 0.324297π-0.324297\pi
0.524379 + 0.851485i 0.324297π0.324297\pi
992992 −3.25981e19 −1.08609
993993 0 0
994994 4.25713e18 0.139993
995995 −3.81022e18 −0.124480
996996 0 0
997997 5.94349e18 0.191656 0.0958282 0.995398i 0.469450π-0.469450\pi
0.0958282 + 0.995398i 0.469450π0.469450\pi
998998 −7.82973e19 −2.50841
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 45.14.a.g.1.4 4
3.2 odd 2 45.14.a.h.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.14.a.g.1.4 4 1.1 even 1 trivial
45.14.a.h.1.1 yes 4 3.2 odd 2