Properties

Label 150.8.a.s.1.2
Level 150150
Weight 88
Character 150.1
Self dual yes
Analytic conductor 46.85846.858
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] = [150,8,Mod(1,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: N N == 150=2352 150 = 2 \cdot 3 \cdot 5^{2}
Weight: k k == 8 8
Character orbit: [χ][\chi] == 150.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: 46.857753822646.8577538226
Analytic rank: 00
Dimension: 22
Coefficient field: Q(2641)\Q(\sqrt{2641})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x660 x^{2} - x - 660 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 225 2^{2}\cdot 5
Twist minimal: no (minimal twist has level 30)
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.2
Root 25.1953-25.1953 of defining polynomial
Character χ\chi == 150.1

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+8.00000q227.0000q3+64.0000q4216.000q6+1309.81q7+512.000q8+729.000q95457.35q111728.00q12+5237.72q13+10478.5q14+4096.00q16+4870.92q17+5832.00q18+32781.2q1935365.0q2143658.8q2283486.0q2313824.0q24+41901.8q2619683.0q27+83828.0q2873409.7q29+244719.q31+32768.0q32+147348.q33+38967.3q34+46656.0q36+107537.q37+262250.q38141418.q39+396774.q41282920.q42+725839.q43349270.q44667888.q46+801478.q47110592.q48+892068.q49131515.q51+335214.q521.22395e6q53157464.q54+670624.q56885092.q57587278.q58+2.32931e6q59+2.63553e6q61+1.95775e6q62+954854.q63+262144.q64+1.17879e6q66209887.q67+311739.q68+2.25412e6q69+3.76578e6q71+373248.q72+2.81943e6q73+860298.q74+2.09800e6q767.14810e6q771.13135e6q788.68583e6q79+531441.q81+3.17419e6q825.77317e6q832.26336e6q84+5.80672e6q86+1.98206e6q872.79416e6q88+1.02714e7q89+6.86043e6q915.34310e6q926.60742e6q93+6.41183e6q94884736.q965.19683e6q97+7.13654e6q983.97841e6q99+O(q100)q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -216.000 q^{6} +1309.81 q^{7} +512.000 q^{8} +729.000 q^{9} -5457.35 q^{11} -1728.00 q^{12} +5237.72 q^{13} +10478.5 q^{14} +4096.00 q^{16} +4870.92 q^{17} +5832.00 q^{18} +32781.2 q^{19} -35365.0 q^{21} -43658.8 q^{22} -83486.0 q^{23} -13824.0 q^{24} +41901.8 q^{26} -19683.0 q^{27} +83828.0 q^{28} -73409.7 q^{29} +244719. q^{31} +32768.0 q^{32} +147348. q^{33} +38967.3 q^{34} +46656.0 q^{36} +107537. q^{37} +262250. q^{38} -141418. q^{39} +396774. q^{41} -282920. q^{42} +725839. q^{43} -349270. q^{44} -667888. q^{46} +801478. q^{47} -110592. q^{48} +892068. q^{49} -131515. q^{51} +335214. q^{52} -1.22395e6 q^{53} -157464. q^{54} +670624. q^{56} -885092. q^{57} -587278. q^{58} +2.32931e6 q^{59} +2.63553e6 q^{61} +1.95775e6 q^{62} +954854. q^{63} +262144. q^{64} +1.17879e6 q^{66} -209887. q^{67} +311739. q^{68} +2.25412e6 q^{69} +3.76578e6 q^{71} +373248. q^{72} +2.81943e6 q^{73} +860298. q^{74} +2.09800e6 q^{76} -7.14810e6 q^{77} -1.13135e6 q^{78} -8.68583e6 q^{79} +531441. q^{81} +3.17419e6 q^{82} -5.77317e6 q^{83} -2.26336e6 q^{84} +5.80672e6 q^{86} +1.98206e6 q^{87} -2.79416e6 q^{88} +1.02714e7 q^{89} +6.86043e6 q^{91} -5.34310e6 q^{92} -6.60742e6 q^{93} +6.41183e6 q^{94} -884736. q^{96} -5.19683e6 q^{97} +7.13654e6 q^{98} -3.97841e6 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+16q254q3+128q4432q6+564q7+1024q8+1458q93720q113456q12+7392q13+4512q14+8192q1624176q17+11664q18+34728q19+2711880q99+O(q100) 2 q + 16 q^{2} - 54 q^{3} + 128 q^{4} - 432 q^{6} + 564 q^{7} + 1024 q^{8} + 1458 q^{9} - 3720 q^{11} - 3456 q^{12} + 7392 q^{13} + 4512 q^{14} + 8192 q^{16} - 24176 q^{17} + 11664 q^{18} + 34728 q^{19}+ \cdots - 2711880 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 8.00000 0.707107
33 −27.0000 −0.577350
44 64.0000 0.500000
55 0 0
66 −216.000 −0.408248
77 1309.81 1.44333 0.721666 0.692241i 0.243377π-0.243377\pi
0.721666 + 0.692241i 0.243377π0.243377\pi
88 512.000 0.353553
99 729.000 0.333333
1010 0 0
1111 −5457.35 −1.23625 −0.618126 0.786079i 0.712108π-0.712108\pi
−0.618126 + 0.786079i 0.712108π0.712108\pi
1212 −1728.00 −0.288675
1313 5237.72 0.661212 0.330606 0.943769i 0.392747π-0.392747\pi
0.330606 + 0.943769i 0.392747π0.392747\pi
1414 10478.5 1.02059
1515 0 0
1616 4096.00 0.250000
1717 4870.92 0.240458 0.120229 0.992746i 0.461637π-0.461637\pi
0.120229 + 0.992746i 0.461637π0.461637\pi
1818 5832.00 0.235702
1919 32781.2 1.09645 0.548223 0.836332i 0.315305π-0.315305\pi
0.548223 + 0.836332i 0.315305π0.315305\pi
2020 0 0
2121 −35365.0 −0.833308
2222 −43658.8 −0.874162
2323 −83486.0 −1.43076 −0.715379 0.698737i 0.753746π-0.753746\pi
−0.715379 + 0.698737i 0.753746π0.753746\pi
2424 −13824.0 −0.204124
2525 0 0
2626 41901.8 0.467547
2727 −19683.0 −0.192450
2828 83828.0 0.721666
2929 −73409.7 −0.558934 −0.279467 0.960155i 0.590158π-0.590158\pi
−0.279467 + 0.960155i 0.590158π0.590158\pi
3030 0 0
3131 244719. 1.47537 0.737687 0.675142i 0.235918π-0.235918\pi
0.737687 + 0.675142i 0.235918π0.235918\pi
3232 32768.0 0.176777
3333 147348. 0.713751
3434 38967.3 0.170030
3535 0 0
3636 46656.0 0.166667
3737 107537. 0.349022 0.174511 0.984655i 0.444166π-0.444166\pi
0.174511 + 0.984655i 0.444166π0.444166\pi
3838 262250. 0.775304
3939 −141418. −0.381751
4040 0 0
4141 396774. 0.899082 0.449541 0.893260i 0.351588π-0.351588\pi
0.449541 + 0.893260i 0.351588π0.351588\pi
4242 −282920. −0.589238
4343 725839. 1.39220 0.696099 0.717946i 0.254918π-0.254918\pi
0.696099 + 0.717946i 0.254918π0.254918\pi
4444 −349270. −0.618126
4545 0 0
4646 −667888. −1.01170
4747 801478. 1.12603 0.563014 0.826447i 0.309642π-0.309642\pi
0.563014 + 0.826447i 0.309642π0.309642\pi
4848 −110592. −0.144338
4949 892068. 1.08321
5050 0 0
5151 −131515. −0.138829
5252 335214. 0.330606
5353 −1.22395e6 −1.12927 −0.564636 0.825340i 0.690983π-0.690983\pi
−0.564636 + 0.825340i 0.690983π0.690983\pi
5454 −157464. −0.136083
5555 0 0
5656 670624. 0.510295
5757 −885092. −0.633033
5858 −587278. −0.395226
5959 2.32931e6 1.47654 0.738270 0.674506i 0.235643π-0.235643\pi
0.738270 + 0.674506i 0.235643π0.235643\pi
6060 0 0
6161 2.63553e6 1.48667 0.743334 0.668921i 0.233244π-0.233244\pi
0.743334 + 0.668921i 0.233244π0.233244\pi
6262 1.95775e6 1.04325
6363 954854. 0.481111
6464 262144. 0.125000
6565 0 0
6666 1.17879e6 0.504698
6767 −209887. −0.0852557 −0.0426279 0.999091i 0.513573π-0.513573\pi
−0.0426279 + 0.999091i 0.513573π0.513573\pi
6868 311739. 0.120229
6969 2.25412e6 0.826049
7070 0 0
7171 3.76578e6 1.24868 0.624339 0.781154i 0.285368π-0.285368\pi
0.624339 + 0.781154i 0.285368π0.285368\pi
7272 373248. 0.117851
7373 2.81943e6 0.848265 0.424132 0.905600i 0.360579π-0.360579\pi
0.424132 + 0.905600i 0.360579π0.360579\pi
7474 860298. 0.246796
7575 0 0
7676 2.09800e6 0.548223
7777 −7.14810e6 −1.78432
7878 −1.13135e6 −0.269939
7979 −8.68583e6 −1.98206 −0.991030 0.133642i 0.957333π-0.957333\pi
−0.991030 + 0.133642i 0.957333π0.957333\pi
8080 0 0
8181 531441. 0.111111
8282 3.17419e6 0.635747
8383 −5.77317e6 −1.10826 −0.554129 0.832431i 0.686949π-0.686949\pi
−0.554129 + 0.832431i 0.686949π0.686949\pi
8484 −2.26336e6 −0.416654
8585 0 0
8686 5.80672e6 0.984432
8787 1.98206e6 0.322701
8888 −2.79416e6 −0.437081
8989 1.02714e7 1.54441 0.772206 0.635373i 0.219154π-0.219154\pi
0.772206 + 0.635373i 0.219154π0.219154\pi
9090 0 0
9191 6.86043e6 0.954348
9292 −5.34310e6 −0.715379
9393 −6.60742e6 −0.851808
9494 6.41183e6 0.796222
9595 0 0
9696 −884736. −0.102062
9797 −5.19683e6 −0.578146 −0.289073 0.957307i 0.593347π-0.593347\pi
−0.289073 + 0.957307i 0.593347π0.593347\pi
9898 7.13654e6 0.765943
9999 −3.97841e6 −0.412084
100100 0 0
101101 2.84866e6 0.275116 0.137558 0.990494i 0.456075π-0.456075\pi
0.137558 + 0.990494i 0.456075π0.456075\pi
102102 −1.05212e6 −0.0981666
103103 −1.49365e7 −1.34685 −0.673423 0.739257i 0.735177π-0.735177\pi
−0.673423 + 0.739257i 0.735177π0.735177\pi
104104 2.68171e6 0.233774
105105 0 0
106106 −9.79161e6 −0.798516
107107 −1.04317e6 −0.0823213 −0.0411606 0.999153i 0.513106π-0.513106\pi
−0.0411606 + 0.999153i 0.513106π0.513106\pi
108108 −1.25971e6 −0.0962250
109109 5.23084e6 0.386882 0.193441 0.981112i 0.438035π-0.438035\pi
0.193441 + 0.981112i 0.438035π0.438035\pi
110110 0 0
111111 −2.90351e6 −0.201508
112112 5.36499e6 0.360833
113113 1.58182e7 1.03130 0.515649 0.856800i 0.327551π-0.327551\pi
0.515649 + 0.856800i 0.327551π0.327551\pi
114114 −7.08074e6 −0.447622
115115 0 0
116116 −4.69822e6 −0.279467
117117 3.81830e6 0.220404
118118 1.86345e7 1.04407
119119 6.37999e6 0.347061
120120 0 0
121121 1.02955e7 0.528320
122122 2.10842e7 1.05123
123123 −1.07129e7 −0.519085
124124 1.56620e7 0.737687
125125 0 0
126126 7.63883e6 0.340197
127127 4.24970e6 0.184096 0.0920481 0.995755i 0.470659π-0.470659\pi
0.0920481 + 0.995755i 0.470659π0.470659\pi
128128 2.09715e6 0.0883883
129129 −1.95977e7 −0.803786
130130 0 0
131131 5.59155e6 0.217312 0.108656 0.994079i 0.465345π-0.465345\pi
0.108656 + 0.994079i 0.465345π0.465345\pi
132132 9.43029e6 0.356875
133133 4.29372e7 1.58254
134134 −1.67910e6 −0.0602849
135135 0 0
136136 2.49391e6 0.0850148
137137 −2.21174e7 −0.734874 −0.367437 0.930048i 0.619765π-0.619765\pi
−0.367437 + 0.930048i 0.619765π0.619765\pi
138138 1.80330e7 0.584105
139139 −2.55140e7 −0.805799 −0.402900 0.915244i 0.631998π-0.631998\pi
−0.402900 + 0.915244i 0.631998π0.631998\pi
140140 0 0
141141 −2.16399e7 −0.650113
142142 3.01262e7 0.882948
143143 −2.85841e7 −0.817424
144144 2.98598e6 0.0833333
145145 0 0
146146 2.25554e7 0.599814
147147 −2.40858e7 −0.625390
148148 6.88238e6 0.174511
149149 −1.64001e6 −0.0406158 −0.0203079 0.999794i 0.506465π-0.506465\pi
−0.0203079 + 0.999794i 0.506465π0.506465\pi
150150 0 0
151151 −4.65795e7 −1.10097 −0.550485 0.834845i 0.685557π-0.685557\pi
−0.550485 + 0.834845i 0.685557π0.685557\pi
152152 1.67840e7 0.387652
153153 3.55090e6 0.0801527
154154 −5.71848e7 −1.26171
155155 0 0
156156 −9.05078e6 −0.190875
157157 −6.21860e7 −1.28246 −0.641230 0.767349i 0.721575π-0.721575\pi
−0.641230 + 0.767349i 0.721575π0.721575\pi
158158 −6.94867e7 −1.40153
159159 3.30467e7 0.651985
160160 0 0
161161 −1.09351e8 −2.06506
162162 4.25153e6 0.0785674
163163 5.86864e7 1.06140 0.530702 0.847558i 0.321928π-0.321928\pi
0.530702 + 0.847558i 0.321928π0.321928\pi
164164 2.53935e7 0.449541
165165 0 0
166166 −4.61854e7 −0.783657
167167 2.74875e7 0.456696 0.228348 0.973580i 0.426668π-0.426668\pi
0.228348 + 0.973580i 0.426668π0.426668\pi
168168 −1.81069e7 −0.294619
169169 −3.53148e7 −0.562799
170170 0 0
171171 2.38975e7 0.365482
172172 4.64537e7 0.696099
173173 −6.19491e7 −0.909649 −0.454825 0.890581i 0.650298π-0.650298\pi
−0.454825 + 0.890581i 0.650298π0.650298\pi
174174 1.58565e7 0.228184
175175 0 0
176176 −2.23533e7 −0.309063
177177 −6.28913e7 −0.852480
178178 8.21709e7 1.09206
179179 1.04816e7 0.136598 0.0682988 0.997665i 0.478243π-0.478243\pi
0.0682988 + 0.997665i 0.478243π0.478243\pi
180180 0 0
181181 −9.64546e7 −1.20906 −0.604530 0.796582i 0.706639π-0.706639\pi
−0.604530 + 0.796582i 0.706639π0.706639\pi
182182 5.48835e7 0.674826
183183 −7.11593e7 −0.858328
184184 −4.27448e7 −0.505849
185185 0 0
186186 −5.28594e7 −0.602319
187187 −2.65823e7 −0.297267
188188 5.12946e7 0.563014
189189 −2.57811e7 −0.277769
190190 0 0
191191 9.71449e7 1.00880 0.504398 0.863471i 0.331714π-0.331714\pi
0.504398 + 0.863471i 0.331714π0.331714\pi
192192 −7.07789e6 −0.0721688
193193 −5.17514e7 −0.518169 −0.259085 0.965855i 0.583421π-0.583421\pi
−0.259085 + 0.965855i 0.583421π0.583421\pi
194194 −4.15746e7 −0.408811
195195 0 0
196196 5.70923e7 0.541604
197197 −1.55737e8 −1.45131 −0.725657 0.688057i 0.758464π-0.758464\pi
−0.725657 + 0.688057i 0.758464π0.758464\pi
198198 −3.18272e7 −0.291387
199199 1.49558e8 1.34532 0.672658 0.739954i 0.265153π-0.265153\pi
0.672658 + 0.739954i 0.265153π0.265153\pi
200200 0 0
201201 5.66695e6 0.0492224
202202 2.27892e7 0.194536
203203 −9.61530e7 −0.806728
204204 −8.41695e6 −0.0694143
205205 0 0
206206 −1.19492e8 −0.952364
207207 −6.08613e7 −0.476919
208208 2.14537e7 0.165303
209209 −1.78898e8 −1.35548
210210 0 0
211211 −4.83513e6 −0.0354340 −0.0177170 0.999843i 0.505640π-0.505640\pi
−0.0177170 + 0.999843i 0.505640π0.505640\pi
212212 −7.83329e7 −0.564636
213213 −1.01676e8 −0.720924
214214 −8.34536e6 −0.0582099
215215 0 0
216216 −1.00777e7 −0.0680414
217217 3.20537e8 2.12946
218218 4.18467e7 0.273567
219219 −7.61246e7 −0.489746
220220 0 0
221221 2.55125e7 0.158994
222222 −2.32280e7 −0.142488
223223 2.61843e8 1.58116 0.790578 0.612361i 0.209780π-0.209780\pi
0.790578 + 0.612361i 0.209780π0.209780\pi
224224 4.29200e7 0.255147
225225 0 0
226226 1.26546e8 0.729237
227227 −1.28619e8 −0.729820 −0.364910 0.931043i 0.618900π-0.618900\pi
−0.364910 + 0.931043i 0.618900π0.618900\pi
228228 −5.66459e7 −0.316517
229229 1.01465e8 0.558330 0.279165 0.960243i 0.409942π-0.409942\pi
0.279165 + 0.960243i 0.409942π0.409942\pi
230230 0 0
231231 1.92999e8 1.03018
232232 −3.75858e7 −0.197613
233233 −2.74718e8 −1.42279 −0.711397 0.702791i 0.751937π-0.751937\pi
−0.711397 + 0.702791i 0.751937π0.751937\pi
234234 3.05464e7 0.155849
235235 0 0
236236 1.49076e8 0.738270
237237 2.34518e8 1.14434
238238 5.10399e7 0.245409
239239 −3.77861e7 −0.179035 −0.0895177 0.995985i 0.528533π-0.528533\pi
−0.0895177 + 0.995985i 0.528533π0.528533\pi
240240 0 0
241241 1.70494e8 0.784600 0.392300 0.919837i 0.371680π-0.371680\pi
0.392300 + 0.919837i 0.371680π0.371680\pi
242242 8.23637e7 0.373578
243243 −1.43489e7 −0.0641500
244244 1.68674e8 0.743334
245245 0 0
246246 −8.57031e7 −0.367049
247247 1.71699e8 0.724983
248248 1.25296e8 0.521624
249249 1.55876e8 0.639854
250250 0 0
251251 −3.41080e8 −1.36144 −0.680719 0.732544i 0.738333π-0.738333\pi
−0.680719 + 0.732544i 0.738333π0.738333\pi
252252 6.11106e7 0.240555
253253 4.55612e8 1.76878
254254 3.39976e7 0.130176
255255 0 0
256256 1.67772e7 0.0625000
257257 −1.94713e8 −0.715530 −0.357765 0.933812i 0.616461π-0.616461\pi
−0.357765 + 0.933812i 0.616461π0.616461\pi
258258 −1.56781e8 −0.568362
259259 1.40854e8 0.503755
260260 0 0
261261 −5.35157e7 −0.186311
262262 4.47324e7 0.153663
263263 4.85601e8 1.64602 0.823009 0.568029i 0.192294π-0.192294\pi
0.823009 + 0.568029i 0.192294π0.192294\pi
264264 7.54424e7 0.252349
265265 0 0
266266 3.43498e8 1.11902
267267 −2.77327e8 −0.891666
268268 −1.34328e7 −0.0426279
269269 −2.89744e7 −0.0907572 −0.0453786 0.998970i 0.514449π-0.514449\pi
−0.0453786 + 0.998970i 0.514449π0.514449\pi
270270 0 0
271271 3.02460e8 0.923156 0.461578 0.887100i 0.347283π-0.347283\pi
0.461578 + 0.887100i 0.347283π0.347283\pi
272272 1.99513e7 0.0601146
273273 −1.85232e8 −0.550993
274274 −1.76939e8 −0.519634
275275 0 0
276276 1.44264e8 0.413024
277277 −1.78252e8 −0.503912 −0.251956 0.967739i 0.581074π-0.581074\pi
−0.251956 + 0.967739i 0.581074π0.581074\pi
278278 −2.04112e8 −0.569786
279279 1.78400e8 0.491792
280280 0 0
281281 2.21487e7 0.0595492 0.0297746 0.999557i 0.490521π-0.490521\pi
0.0297746 + 0.999557i 0.490521π0.490521\pi
282282 −1.73119e8 −0.459699
283283 1.08099e8 0.283511 0.141756 0.989902i 0.454725π-0.454725\pi
0.141756 + 0.989902i 0.454725π0.454725\pi
284284 2.41010e8 0.624339
285285 0 0
286286 −2.28672e8 −0.578006
287287 5.19699e8 1.29767
288288 2.38879e7 0.0589256
289289 −3.86613e8 −0.942180
290290 0 0
291291 1.40314e8 0.333793
292292 1.80444e8 0.424132
293293 −5.08345e8 −1.18065 −0.590326 0.807165i 0.701001π-0.701001\pi
−0.590326 + 0.807165i 0.701001π0.701001\pi
294294 −1.92687e8 −0.442217
295295 0 0
296296 5.50591e7 0.123398
297297 1.07417e8 0.237917
298298 −1.31201e7 −0.0287197
299299 −4.37276e8 −0.946034
300300 0 0
301301 9.50714e8 2.00940
302302 −3.72636e8 −0.778503
303303 −7.69137e7 −0.158838
304304 1.34272e8 0.274111
305305 0 0
306306 2.84072e7 0.0566765
307307 −4.60852e8 −0.909027 −0.454514 0.890740i 0.650187π-0.650187\pi
−0.454514 + 0.890740i 0.650187π0.650187\pi
308308 −4.57479e8 −0.892161
309309 4.03285e8 0.777602
310310 0 0
311311 7.63906e8 1.44005 0.720027 0.693946i 0.244129π-0.244129\pi
0.720027 + 0.693946i 0.244129π0.244129\pi
312312 −7.24062e7 −0.134969
313313 4.96971e8 0.916064 0.458032 0.888936i 0.348554π-0.348554\pi
0.458032 + 0.888936i 0.348554π0.348554\pi
314314 −4.97488e8 −0.906836
315315 0 0
316316 −5.55893e8 −0.991030
317317 4.18865e8 0.738528 0.369264 0.929325i 0.379610π-0.379610\pi
0.369264 + 0.929325i 0.379610π0.379610\pi
318318 2.64373e8 0.461023
319319 4.00622e8 0.690984
320320 0 0
321321 2.81656e7 0.0475282
322322 −8.74808e8 −1.46022
323323 1.59675e8 0.263649
324324 3.40122e7 0.0555556
325325 0 0
326326 4.69491e8 0.750527
327327 −1.41233e8 −0.223366
328328 2.03148e8 0.317873
329329 1.04979e9 1.62523
330330 0 0
331331 −8.00262e8 −1.21293 −0.606463 0.795112i 0.707412π-0.707412\pi
−0.606463 + 0.795112i 0.707412π0.707412\pi
332332 −3.69483e8 −0.554129
333333 7.83946e7 0.116341
334334 2.19900e8 0.322933
335335 0 0
336336 −1.44855e8 −0.208327
337337 1.38535e9 1.97177 0.985883 0.167437i 0.0535491π-0.0535491\pi
0.985883 + 0.167437i 0.0535491π0.0535491\pi
338338 −2.82518e8 −0.397959
339339 −4.27093e8 −0.595420
340340 0 0
341341 −1.33552e9 −1.82394
342342 1.91180e8 0.258435
343343 8.97545e7 0.120096
344344 3.71630e8 0.492216
345345 0 0
346346 −4.95593e8 −0.643219
347347 9.62881e8 1.23714 0.618571 0.785729i 0.287712π-0.287712\pi
0.618571 + 0.785729i 0.287712π0.287712\pi
348348 1.26852e8 0.161350
349349 −5.40052e8 −0.680058 −0.340029 0.940415i 0.610437π-0.610437\pi
−0.340029 + 0.940415i 0.610437π0.610437\pi
350350 0 0
351351 −1.03094e8 −0.127250
352352 −1.78826e8 −0.218541
353353 −1.02599e9 −1.24145 −0.620727 0.784027i 0.713163π-0.713163\pi
−0.620727 + 0.784027i 0.713163π0.713163\pi
354354 −5.03131e8 −0.602795
355355 0 0
356356 6.57367e8 0.772206
357357 −1.72260e8 −0.200376
358358 8.38530e7 0.0965891
359359 −7.47383e8 −0.852536 −0.426268 0.904597i 0.640172π-0.640172\pi
−0.426268 + 0.904597i 0.640172π0.640172\pi
360360 0 0
361361 1.80735e8 0.202194
362362 −7.71637e8 −0.854935
363363 −2.77977e8 −0.305026
364364 4.39068e8 0.477174
365365 0 0
366366 −5.69275e8 −0.606929
367367 4.77968e8 0.504740 0.252370 0.967631i 0.418790π-0.418790\pi
0.252370 + 0.967631i 0.418790π0.418790\pi
368368 −3.41959e8 −0.357689
369369 2.89248e8 0.299694
370370 0 0
371371 −1.60315e9 −1.62991
372372 −4.22875e8 −0.425904
373373 −1.41621e9 −1.41302 −0.706509 0.707704i 0.749731π-0.749731\pi
−0.706509 + 0.707704i 0.749731π0.749731\pi
374374 −2.12658e8 −0.210200
375375 0 0
376376 4.10357e8 0.398111
377377 −3.84500e8 −0.369574
378378 −2.06248e8 −0.196413
379379 4.23809e8 0.399883 0.199941 0.979808i 0.435925π-0.435925\pi
0.199941 + 0.979808i 0.435925π0.435925\pi
380380 0 0
381381 −1.14742e8 −0.106288
382382 7.77159e8 0.713327
383383 8.56505e8 0.778994 0.389497 0.921028i 0.372649π-0.372649\pi
0.389497 + 0.921028i 0.372649π0.372649\pi
384384 −5.66231e7 −0.0510310
385385 0 0
386386 −4.14011e8 −0.366401
387387 5.29137e8 0.464066
388388 −3.32597e8 −0.289073
389389 −1.08932e8 −0.0938282 −0.0469141 0.998899i 0.514939π-0.514939\pi
−0.0469141 + 0.998899i 0.514939π0.514939\pi
390390 0 0
391391 −4.06653e8 −0.344037
392392 4.56739e8 0.382972
393393 −1.50972e8 −0.125465
394394 −1.24590e9 −1.02623
395395 0 0
396396 −2.54618e8 −0.206042
397397 −2.13997e9 −1.71649 −0.858245 0.513239i 0.828445π-0.828445\pi
−0.858245 + 0.513239i 0.828445π0.828445\pi
398398 1.19646e9 0.951281
399399 −1.15931e9 −0.913677
400400 0 0
401401 −9.00462e8 −0.697365 −0.348682 0.937241i 0.613371π-0.613371\pi
−0.348682 + 0.937241i 0.613371π0.613371\pi
402402 4.53356e7 0.0348055
403403 1.28177e9 0.975535
404404 1.82314e8 0.137558
405405 0 0
406406 −7.69224e8 −0.570443
407407 −5.86868e8 −0.431479
408408 −6.73356e7 −0.0490833
409409 −2.15143e9 −1.55487 −0.777436 0.628962i 0.783480π-0.783480\pi
−0.777436 + 0.628962i 0.783480π0.783480\pi
410410 0 0
411411 5.97171e8 0.424279
412412 −9.55935e8 −0.673423
413413 3.05096e9 2.13114
414414 −4.86890e8 −0.337233
415415 0 0
416416 1.71630e8 0.116887
417417 6.88878e8 0.465228
418418 −1.43119e9 −0.958472
419419 7.96072e7 0.0528693 0.0264346 0.999651i 0.491585π-0.491585\pi
0.0264346 + 0.999651i 0.491585π0.491585\pi
420420 0 0
421421 3.29700e8 0.215344 0.107672 0.994186i 0.465660π-0.465660\pi
0.107672 + 0.994186i 0.465660π0.465660\pi
422422 −3.86810e7 −0.0250556
423423 5.84278e8 0.375343
424424 −6.26663e8 −0.399258
425425 0 0
426426 −8.13408e8 −0.509771
427427 3.45205e9 2.14575
428428 −6.67629e7 −0.0411606
429429 7.71769e8 0.471940
430430 0 0
431431 5.57393e8 0.335345 0.167672 0.985843i 0.446375π-0.446375\pi
0.167672 + 0.985843i 0.446375π0.446375\pi
432432 −8.06216e7 −0.0481125
433433 −1.00562e9 −0.595287 −0.297644 0.954677i 0.596201π-0.596201\pi
−0.297644 + 0.954677i 0.596201π0.596201\pi
434434 2.56429e9 1.50575
435435 0 0
436436 3.34774e8 0.193441
437437 −2.73677e9 −1.56875
438438 −6.08997e8 −0.346303
439439 2.43906e9 1.37593 0.687967 0.725742i 0.258504π-0.258504\pi
0.687967 + 0.725742i 0.258504π0.258504\pi
440440 0 0
441441 6.50317e8 0.361069
442442 2.04100e8 0.112426
443443 1.15960e9 0.633715 0.316858 0.948473i 0.397372π-0.397372\pi
0.316858 + 0.948473i 0.397372π0.397372\pi
444444 −1.85824e8 −0.100754
445445 0 0
446446 2.09475e9 1.11805
447447 4.42804e7 0.0234496
448448 3.43360e8 0.180416
449449 6.32037e8 0.329519 0.164759 0.986334i 0.447315π-0.447315\pi
0.164759 + 0.986334i 0.447315π0.447315\pi
450450 0 0
451451 −2.16533e9 −1.11149
452452 1.01237e9 0.515649
453453 1.25765e9 0.635645
454454 −1.02895e9 −0.516061
455455 0 0
456456 −4.53167e8 −0.223811
457457 −1.78534e9 −0.875015 −0.437507 0.899215i 0.644139π-0.644139\pi
−0.437507 + 0.899215i 0.644139π0.644139\pi
458458 8.11718e8 0.394799
459459 −9.58743e7 −0.0462762
460460 0 0
461461 −1.23239e9 −0.585862 −0.292931 0.956134i 0.594631π-0.594631\pi
−0.292931 + 0.956134i 0.594631π0.594631\pi
462462 1.54399e9 0.728447
463463 −3.27184e9 −1.53200 −0.766000 0.642841i 0.777756π-0.777756\pi
−0.766000 + 0.642841i 0.777756π0.777756\pi
464464 −3.00686e8 −0.139734
465465 0 0
466466 −2.19775e9 −1.00607
467467 4.20893e8 0.191233 0.0956165 0.995418i 0.469518π-0.469518\pi
0.0956165 + 0.995418i 0.469518π0.469518\pi
468468 2.44371e8 0.110202
469469 −2.74913e8 −0.123052
470470 0 0
471471 1.67902e9 0.740428
472472 1.19261e9 0.522036
473473 −3.96116e9 −1.72111
474474 1.87614e9 0.809172
475475 0 0
476476 4.08320e8 0.173531
477477 −8.92260e8 −0.376424
478478 −3.02288e8 −0.126597
479479 −2.91500e7 −0.0121189 −0.00605946 0.999982i 0.501929π-0.501929\pi
−0.00605946 + 0.999982i 0.501929π0.501929\pi
480480 0 0
481481 5.63250e8 0.230777
482482 1.36395e9 0.554796
483483 2.95248e9 1.19226
484484 6.58909e8 0.264160
485485 0 0
486486 −1.14791e8 −0.0453609
487487 −2.58798e9 −1.01534 −0.507668 0.861553i 0.669492π-0.669492\pi
−0.507668 + 0.861553i 0.669492π0.669492\pi
488488 1.34939e9 0.525616
489489 −1.58453e9 −0.612802
490490 0 0
491491 −2.01157e9 −0.766918 −0.383459 0.923558i 0.625267π-0.625267\pi
−0.383459 + 0.923558i 0.625267π0.625267\pi
492492 −6.85625e8 −0.259543
493493 −3.57573e8 −0.134400
494494 1.37359e9 0.512640
495495 0 0
496496 1.00237e9 0.368844
497497 4.93246e9 1.80226
498498 1.24701e9 0.452445
499499 −6.31261e8 −0.227435 −0.113717 0.993513i 0.536276π-0.536276\pi
−0.113717 + 0.993513i 0.536276π0.536276\pi
500500 0 0
501501 −7.42162e8 −0.263674
502502 −2.72864e9 −0.962682
503503 −1.57252e9 −0.550944 −0.275472 0.961309i 0.588834π-0.588834\pi
−0.275472 + 0.961309i 0.588834π0.588834\pi
504504 4.88885e8 0.170098
505505 0 0
506506 3.64490e9 1.25071
507507 9.53500e8 0.324932
508508 2.71981e8 0.0920481
509509 −2.18665e9 −0.734965 −0.367482 0.930031i 0.619780π-0.619780\pi
−0.367482 + 0.930031i 0.619780π0.619780\pi
510510 0 0
511511 3.69293e9 1.22433
512512 1.34218e8 0.0441942
513513 −6.45232e8 −0.211011
514514 −1.55770e9 −0.505956
515515 0 0
516516 −1.25425e9 −0.401893
517517 −4.37394e9 −1.39205
518518 1.12683e9 0.356208
519519 1.67263e9 0.525186
520520 0 0
521521 5.81389e9 1.80109 0.900543 0.434766i 0.143169π-0.143169\pi
0.900543 + 0.434766i 0.143169π0.143169\pi
522522 −4.28126e8 −0.131742
523523 −2.36475e9 −0.722819 −0.361409 0.932407i 0.617704π-0.617704\pi
−0.361409 + 0.932407i 0.617704π0.617704\pi
524524 3.57859e8 0.108656
525525 0 0
526526 3.88481e9 1.16391
527527 1.19201e9 0.354766
528528 6.03539e8 0.178438
529529 3.56509e9 1.04707
530530 0 0
531531 1.69807e9 0.492180
532532 2.74798e9 0.791268
533533 2.07819e9 0.594483
534534 −2.21861e9 −0.630503
535535 0 0
536536 −1.07462e8 −0.0301425
537537 −2.83004e8 −0.0788647
538538 −2.31795e8 −0.0641751
539539 −4.86832e9 −1.33912
540540 0 0
541541 5.64040e9 1.53151 0.765754 0.643134i 0.222366π-0.222366\pi
0.765754 + 0.643134i 0.222366π0.222366\pi
542542 2.41968e9 0.652770
543543 2.60428e9 0.698051
544544 1.59610e8 0.0425074
545545 0 0
546546 −1.48185e9 −0.389611
547547 −1.78063e9 −0.465177 −0.232589 0.972575i 0.574720π-0.574720\pi
−0.232589 + 0.972575i 0.574720π0.574720\pi
548548 −1.41552e9 −0.367437
549549 1.92130e9 0.495556
550550 0 0
551551 −2.40646e9 −0.612841
552552 1.15411e9 0.292052
553553 −1.13768e10 −2.86077
554554 −1.42602e9 −0.356320
555555 0 0
556556 −1.63290e9 −0.402900
557557 −1.56532e9 −0.383805 −0.191902 0.981414i 0.561466π-0.561466\pi
−0.191902 + 0.981414i 0.561466π0.561466\pi
558558 1.42720e9 0.347749
559559 3.80174e9 0.920537
560560 0 0
561561 7.17722e8 0.171627
562562 1.77190e8 0.0421077
563563 2.73932e9 0.646939 0.323470 0.946239i 0.395151π-0.395151\pi
0.323470 + 0.946239i 0.395151π0.395151\pi
564564 −1.38495e9 −0.325056
565565 0 0
566566 8.64794e8 0.200473
567567 6.96088e8 0.160370
568568 1.92808e9 0.441474
569569 7.38630e8 0.168087 0.0840435 0.996462i 0.473217π-0.473217\pi
0.0840435 + 0.996462i 0.473217π0.473217\pi
570570 0 0
571571 3.23235e9 0.726594 0.363297 0.931673i 0.381651π-0.381651\pi
0.363297 + 0.931673i 0.381651π0.381651\pi
572572 −1.82938e9 −0.408712
573573 −2.62291e9 −0.582429
574574 4.15759e9 0.917594
575575 0 0
576576 1.91103e8 0.0416667
577577 −2.65831e9 −0.576090 −0.288045 0.957617i 0.593005π-0.593005\pi
−0.288045 + 0.957617i 0.593005π0.593005\pi
578578 −3.09290e9 −0.666222
579579 1.39729e9 0.299165
580580 0 0
581581 −7.56178e9 −1.59959
582582 1.12252e9 0.236027
583583 6.67952e9 1.39606
584584 1.44355e9 0.299907
585585 0 0
586586 −4.06676e9 −0.834848
587587 −9.42375e9 −1.92305 −0.961524 0.274720i 0.911415π-0.911415\pi
−0.961524 + 0.274720i 0.911415π0.911415\pi
588588 −1.54149e9 −0.312695
589589 8.02219e9 1.61767
590590 0 0
591591 4.20491e9 0.837916
592592 4.40472e8 0.0872555
593593 −2.34902e9 −0.462589 −0.231294 0.972884i 0.574296π-0.574296\pi
−0.231294 + 0.972884i 0.574296π0.574296\pi
594594 8.59336e8 0.168233
595595 0 0
596596 −1.04961e8 −0.0203079
597597 −4.03807e9 −0.776718
598598 −3.49821e9 −0.668947
599599 2.66914e9 0.507431 0.253715 0.967279i 0.418347π-0.418347\pi
0.253715 + 0.967279i 0.418347π0.418347\pi
600600 0 0
601601 −1.24294e9 −0.233555 −0.116777 0.993158i 0.537256π-0.537256\pi
−0.116777 + 0.993158i 0.537256π0.537256\pi
602602 7.60571e9 1.42086
603603 −1.53008e8 −0.0284186
604604 −2.98109e9 −0.550485
605605 0 0
606606 −6.15310e8 −0.112315
607607 −1.29515e9 −0.235050 −0.117525 0.993070i 0.537496π-0.537496\pi
−0.117525 + 0.993070i 0.537496π0.537496\pi
608608 1.07417e9 0.193826
609609 2.59613e9 0.465764
610610 0 0
611611 4.19792e9 0.744543
612612 2.27258e8 0.0400764
613613 −5.13495e9 −0.900377 −0.450188 0.892934i 0.648643π-0.648643\pi
−0.450188 + 0.892934i 0.648643π0.648643\pi
614614 −3.68681e9 −0.642779
615615 0 0
616616 −3.65983e9 −0.630853
617617 1.20368e9 0.206306 0.103153 0.994665i 0.467107π-0.467107\pi
0.103153 + 0.994665i 0.467107π0.467107\pi
618618 3.22628e9 0.549848
619619 −9.74534e9 −1.65150 −0.825752 0.564034i 0.809249π-0.809249\pi
−0.825752 + 0.564034i 0.809249π0.809249\pi
620620 0 0
621621 1.64325e9 0.275350
622622 6.11125e9 1.01827
623623 1.34536e10 2.22910
624624 −5.79250e8 −0.0954377
625625 0 0
626626 3.97577e9 0.647755
627627 4.83026e9 0.782589
628628 −3.97990e9 −0.641230
629629 5.23805e8 0.0839252
630630 0 0
631631 −9.82029e9 −1.55604 −0.778021 0.628238i 0.783776π-0.783776\pi
−0.778021 + 0.628238i 0.783776π0.783776\pi
632632 −4.44715e9 −0.700764
633633 1.30549e8 0.0204578
634634 3.35092e9 0.522218
635635 0 0
636636 2.11499e9 0.325993
637637 4.67240e9 0.716229
638638 3.20498e9 0.488599
639639 2.74525e9 0.416226
640640 0 0
641641 −5.05603e9 −0.758240 −0.379120 0.925347i 0.623773π-0.623773\pi
−0.379120 + 0.925347i 0.623773π0.623773\pi
642642 2.25325e8 0.0336075
643643 1.13650e10 1.68590 0.842951 0.537990i 0.180816π-0.180816\pi
0.842951 + 0.537990i 0.180816π0.180816\pi
644644 −6.99847e9 −1.03253
645645 0 0
646646 1.27740e9 0.186428
647647 −1.08126e8 −0.0156952 −0.00784758 0.999969i 0.502498π-0.502498\pi
−0.00784758 + 0.999969i 0.502498π0.502498\pi
648648 2.72098e8 0.0392837
649649 −1.27118e10 −1.82538
650650 0 0
651651 −8.65449e9 −1.22944
652652 3.75593e9 0.530702
653653 −3.37806e9 −0.474757 −0.237378 0.971417i 0.576288π-0.576288\pi
−0.237378 + 0.971417i 0.576288π0.576288\pi
654654 −1.12986e9 −0.157944
655655 0 0
656656 1.62518e9 0.224770
657657 2.05536e9 0.282755
658658 8.39829e9 1.14921
659659 −2.43494e9 −0.331428 −0.165714 0.986174i 0.552993π-0.552993\pi
−0.165714 + 0.986174i 0.552993π0.552993\pi
660660 0 0
661661 6.61695e9 0.891154 0.445577 0.895244i 0.352999π-0.352999\pi
0.445577 + 0.895244i 0.352999π0.352999\pi
662662 −6.40210e9 −0.857668
663663 −6.88838e8 −0.0917951
664664 −2.95586e9 −0.391829
665665 0 0
666666 6.27157e8 0.0822653
667667 6.12868e9 0.799700
668668 1.75920e9 0.228348
669669 −7.06977e9 −0.912881
670670 0 0
671671 −1.43830e10 −1.83790
672672 −1.15884e9 −0.147309
673673 −4.76258e9 −0.602267 −0.301134 0.953582i 0.597365π-0.597365\pi
−0.301134 + 0.953582i 0.597365π0.597365\pi
674674 1.10828e10 1.39425
675675 0 0
676676 −2.26015e9 −0.281400
677677 2.85988e9 0.354232 0.177116 0.984190i 0.443323π-0.443323\pi
0.177116 + 0.984190i 0.443323π0.443323\pi
678678 −3.41674e9 −0.421025
679679 −6.80688e9 −0.834456
680680 0 0
681681 3.47272e9 0.421362
682682 −1.06841e10 −1.28972
683683 5.11045e9 0.613744 0.306872 0.951751i 0.400718π-0.400718\pi
0.306872 + 0.951751i 0.400718π0.400718\pi
684684 1.52944e9 0.182741
685685 0 0
686686 7.18036e8 0.0849204
687687 −2.73955e9 −0.322352
688688 2.97304e9 0.348049
689689 −6.41071e9 −0.746688
690690 0 0
691691 9.23300e9 1.06456 0.532279 0.846569i 0.321336π-0.321336\pi
0.532279 + 0.846569i 0.321336π0.321336\pi
692692 −3.96475e9 −0.454825
693693 −5.21097e9 −0.594774
694694 7.70304e9 0.874791
695695 0 0
696696 1.01482e9 0.114092
697697 1.93265e9 0.216192
698698 −4.32041e9 −0.480874
699699 7.41739e9 0.821450
700700 0 0
701701 −4.27775e9 −0.469031 −0.234516 0.972112i 0.575350π-0.575350\pi
−0.234516 + 0.972112i 0.575350π0.575350\pi
702702 −8.24752e8 −0.0899795
703703 3.52520e9 0.382684
704704 −1.43061e9 −0.154532
705705 0 0
706706 −8.20790e9 −0.877841
707707 3.73121e9 0.397083
708708 −4.02505e9 −0.426240
709709 6.00248e9 0.632512 0.316256 0.948674i 0.397574π-0.397574\pi
0.316256 + 0.948674i 0.397574π0.397574\pi
710710 0 0
711711 −6.33197e9 −0.660686
712712 5.25894e9 0.546032
713713 −2.04306e10 −2.11090
714714 −1.37808e9 −0.141687
715715 0 0
716716 6.70824e8 0.0682988
717717 1.02022e9 0.103366
718718 −5.97907e9 −0.602834
719719 1.01189e9 0.101527 0.0507634 0.998711i 0.483835π-0.483835\pi
0.0507634 + 0.998711i 0.483835π0.483835\pi
720720 0 0
721721 −1.95640e10 −1.94395
722722 1.44588e9 0.142973
723723 −4.60333e9 −0.452989
724724 −6.17310e9 −0.604530
725725 0 0
726726 −2.22382e9 −0.215686
727727 1.27391e10 1.22961 0.614807 0.788678i 0.289234π-0.289234\pi
0.614807 + 0.788678i 0.289234π0.289234\pi
728728 3.51254e9 0.337413
729729 3.87420e8 0.0370370
730730 0 0
731731 3.53550e9 0.334765
732732 −4.55420e9 −0.429164
733733 1.02714e10 0.963314 0.481657 0.876360i 0.340035π-0.340035\pi
0.481657 + 0.876360i 0.340035π0.340035\pi
734734 3.82374e9 0.356905
735735 0 0
736736 −2.73567e9 −0.252925
737737 1.14543e9 0.105398
738738 2.31398e9 0.211916
739739 7.46828e9 0.680714 0.340357 0.940296i 0.389452π-0.389452\pi
0.340357 + 0.940296i 0.389452π0.389452\pi
740740 0 0
741741 −4.63587e9 −0.418569
742742 −1.28252e10 −1.15252
743743 −1.13615e10 −1.01619 −0.508093 0.861302i 0.669649π-0.669649\pi
−0.508093 + 0.861302i 0.669649π0.669649\pi
744744 −3.38300e9 −0.301160
745745 0 0
746746 −1.13297e10 −0.999155
747747 −4.20864e9 −0.369420
748748 −1.70127e9 −0.148634
749749 −1.36636e9 −0.118817
750750 0 0
751751 9.08552e9 0.782726 0.391363 0.920236i 0.372004π-0.372004\pi
0.391363 + 0.920236i 0.372004π0.372004\pi
752752 3.28285e9 0.281507
753753 9.20916e9 0.786027
754754 −3.07600e9 −0.261328
755755 0 0
756756 −1.64999e9 −0.138885
757757 −1.00537e10 −0.842348 −0.421174 0.906980i 0.638382π-0.638382\pi
−0.421174 + 0.906980i 0.638382π0.638382\pi
758758 3.39047e9 0.282760
759759 −1.23015e10 −1.02120
760760 0 0
761761 1.04527e9 0.0859768 0.0429884 0.999076i 0.486312π-0.486312\pi
0.0429884 + 0.999076i 0.486312π0.486312\pi
762762 −9.17934e8 −0.0751570
763763 6.85142e9 0.558399
764764 6.21728e9 0.504398
765765 0 0
766766 6.85204e9 0.550832
767767 1.22003e10 0.976305
768768 −4.52985e8 −0.0360844
769769 1.24461e10 0.986939 0.493470 0.869763i 0.335728π-0.335728\pi
0.493470 + 0.869763i 0.335728π0.335728\pi
770770 0 0
771771 5.25724e9 0.413112
772772 −3.31209e9 −0.259085
773773 −2.86975e9 −0.223469 −0.111734 0.993738i 0.535641π-0.535641\pi
−0.111734 + 0.993738i 0.535641π0.535641\pi
774774 4.23310e9 0.328144
775775 0 0
776776 −2.66078e9 −0.204405
777777 −3.80305e9 −0.290843
778778 −8.71459e8 −0.0663465
779779 1.30067e10 0.985795
780780 0 0
781781 −2.05511e10 −1.54368
782782 −3.25323e9 −0.243271
783783 1.44492e9 0.107567
784784 3.65391e9 0.270802
785785 0 0
786786 −1.20778e9 −0.0887171
787787 −1.10693e10 −0.809487 −0.404743 0.914430i 0.632639π-0.632639\pi
−0.404743 + 0.914430i 0.632639π0.632639\pi
788788 −9.96719e9 −0.725657
789789 −1.31112e10 −0.950329
790790 0 0
791791 2.07189e10 1.48850
792792 −2.03694e9 −0.145694
793793 1.38042e10 0.983002
794794 −1.71198e10 −1.21374
795795 0 0
796796 9.57172e9 0.672658
797797 −9.16190e9 −0.641034 −0.320517 0.947243i 0.603857π-0.603857\pi
−0.320517 + 0.947243i 0.603857π0.603857\pi
798798 −9.27445e9 −0.646067
799799 3.90393e9 0.270763
800800 0 0
801801 7.48782e9 0.514804
802802 −7.20369e9 −0.493111
803803 −1.53866e10 −1.04867
804804 3.62685e8 0.0246112
805805 0 0
806806 1.02542e10 0.689807
807807 7.82308e8 0.0523987
808808 1.45851e9 0.0972681
809809 −7.65555e9 −0.508343 −0.254171 0.967159i 0.581803π-0.581803\pi
−0.254171 + 0.967159i 0.581803π0.581803\pi
810810 0 0
811811 1.37797e10 0.907121 0.453561 0.891225i 0.350154π-0.350154\pi
0.453561 + 0.891225i 0.350154π0.350154\pi
812812 −6.15380e9 −0.403364
813813 −8.16641e9 −0.532984
814814 −4.69494e9 −0.305102
815815 0 0
816816 −5.38685e8 −0.0347072
817817 2.37939e10 1.52647
818818 −1.72114e10 −1.09946
819819 5.00126e9 0.318116
820820 0 0
821821 3.07675e10 1.94040 0.970200 0.242304i 0.0779032π-0.0779032\pi
0.970200 + 0.242304i 0.0779032π0.0779032\pi
822822 4.77736e9 0.300011
823823 2.78795e10 1.74335 0.871677 0.490081i 0.163033π-0.163033\pi
0.871677 + 0.490081i 0.163033π0.163033\pi
824824 −7.64748e9 −0.476182
825825 0 0
826826 2.44077e10 1.50694
827827 1.30484e10 0.802211 0.401105 0.916032i 0.368626π-0.368626\pi
0.401105 + 0.916032i 0.368626π0.368626\pi
828828 −3.89512e9 −0.238460
829829 −2.37741e10 −1.44932 −0.724659 0.689108i 0.758003π-0.758003\pi
−0.724659 + 0.689108i 0.758003π0.758003\pi
830830 0 0
831831 4.81280e9 0.290934
832832 1.37304e9 0.0826515
833833 4.34519e9 0.260466
834834 5.51103e9 0.328966
835835 0 0
836836 −1.14495e10 −0.677742
837837 −4.81681e9 −0.283936
838838 6.36858e8 0.0373842
839839 1.02523e10 0.599315 0.299658 0.954047i 0.403128π-0.403128\pi
0.299658 + 0.954047i 0.403128π0.403128\pi
840840 0 0
841841 −1.18609e10 −0.687593
842842 2.63760e9 0.152271
843843 −5.98015e8 −0.0343808
844844 −3.09448e8 −0.0177170
845845 0 0
846846 4.67422e9 0.265407
847847 1.34851e10 0.762541
848848 −5.01330e9 −0.282318
849849 −2.91868e9 −0.163685
850850 0 0
851851 −8.97785e9 −0.499366
852852 −6.50726e9 −0.360462
853853 −1.30152e10 −0.718008 −0.359004 0.933336i 0.616884π-0.616884\pi
−0.359004 + 0.933336i 0.616884π0.616884\pi
854854 2.76164e10 1.51728
855855 0 0
856856 −5.34103e8 −0.0291050
857857 −3.47955e10 −1.88838 −0.944192 0.329396i 0.893155π-0.893155\pi
−0.944192 + 0.329396i 0.893155π0.893155\pi
858858 6.17415e9 0.333712
859859 −2.22903e9 −0.119989 −0.0599943 0.998199i 0.519108π-0.519108\pi
−0.0599943 + 0.998199i 0.519108π0.519108\pi
860860 0 0
861861 −1.40319e10 −0.749212
862862 4.45915e9 0.237124
863863 2.28044e10 1.20776 0.603880 0.797075i 0.293621π-0.293621\pi
0.603880 + 0.797075i 0.293621π0.293621\pi
864864 −6.44973e8 −0.0340207
865865 0 0
866866 −8.04497e9 −0.420932
867867 1.04385e10 0.543968
868868 2.05143e10 1.06473
869869 4.74016e10 2.45033
870870 0 0
871871 −1.09933e9 −0.0563721
872872 2.67819e9 0.136783
873873 −3.78849e9 −0.192715
874874 −2.18942e10 −1.10927
875875 0 0
876876 −4.87198e9 −0.244873
877877 4.21151e9 0.210833 0.105417 0.994428i 0.466382π-0.466382\pi
0.105417 + 0.994428i 0.466382π0.466382\pi
878878 1.95125e10 0.972932
879879 1.37253e10 0.681650
880880 0 0
881881 2.60856e10 1.28524 0.642622 0.766184i 0.277847π-0.277847\pi
0.642622 + 0.766184i 0.277847π0.277847\pi
882882 5.20254e9 0.255314
883883 1.24090e9 0.0606560 0.0303280 0.999540i 0.490345π-0.490345\pi
0.0303280 + 0.999540i 0.490345π0.490345\pi
884884 1.63280e9 0.0794969
885885 0 0
886886 9.27678e9 0.448104
887887 2.74043e10 1.31852 0.659260 0.751915i 0.270870π-0.270870\pi
0.659260 + 0.751915i 0.270870π0.270870\pi
888888 −1.48659e9 −0.0712438
889889 5.56631e9 0.265712
890890 0 0
891891 −2.90026e9 −0.137361
892892 1.67580e10 0.790578
893893 2.62734e10 1.23463
894894 3.54243e8 0.0165813
895895 0 0
896896 2.74688e9 0.127574
897897 1.18065e10 0.546193
898898 5.05629e9 0.233005
899899 −1.79648e10 −0.824637
900900 0 0
901901 −5.96176e9 −0.271543
902902 −1.73226e10 −0.785943
903903 −2.56693e10 −1.16013
904904 8.09894e9 0.364619
905905 0 0
906906 1.00612e10 0.449469
907907 1.54071e10 0.685638 0.342819 0.939402i 0.388618π-0.388618\pi
0.342819 + 0.939402i 0.388618π0.388618\pi
908908 −8.23163e9 −0.364910
909909 2.07667e9 0.0917052
910910 0 0
911911 1.20985e9 0.0530171 0.0265085 0.999649i 0.491561π-0.491561\pi
0.0265085 + 0.999649i 0.491561π0.491561\pi
912912 −3.62534e9 −0.158258
913913 3.15062e10 1.37009
914914 −1.42828e10 −0.618729
915915 0 0
916916 6.49374e9 0.279165
917917 7.32389e9 0.313653
918918 −7.66994e8 −0.0327222
919919 1.95758e10 0.831986 0.415993 0.909368i 0.363434π-0.363434\pi
0.415993 + 0.909368i 0.363434π0.363434\pi
920920 0 0
921921 1.24430e10 0.524827
922922 −9.85913e9 −0.414267
923923 1.97241e10 0.825640
924924 1.23519e10 0.515090
925925 0 0
926926 −2.61747e10 −1.08329
927927 −1.08887e10 −0.448949
928928 −2.40549e9 −0.0988065
929929 2.43152e10 0.994999 0.497500 0.867464i 0.334252π-0.334252\pi
0.497500 + 0.867464i 0.334252π0.334252\pi
930930 0 0
931931 2.92430e10 1.18768
932932 −1.75820e10 −0.711397
933933 −2.06255e10 −0.831416
934934 3.36714e9 0.135222
935935 0 0
936936 1.95497e9 0.0779245
937937 2.87271e9 0.114078 0.0570391 0.998372i 0.481834π-0.481834\pi
0.0570391 + 0.998372i 0.481834π0.481834\pi
938938 −2.19930e9 −0.0870111
939939 −1.34182e10 −0.528890
940940 0 0
941941 3.33166e10 1.30346 0.651729 0.758452i 0.274044π-0.274044\pi
0.651729 + 0.758452i 0.274044π0.274044\pi
942942 1.34322e10 0.523562
943943 −3.31250e10 −1.28637
944944 9.54085e9 0.369135
945945 0 0
946946 −3.16893e10 −1.21701
947947 −5.07496e9 −0.194182 −0.0970908 0.995276i 0.530954π-0.530954\pi
−0.0970908 + 0.995276i 0.530954π0.530954\pi
948948 1.50091e10 0.572171
949949 1.47674e10 0.560882
950950 0 0
951951 −1.13094e10 −0.426389
952952 3.26656e9 0.122705
953953 −2.69161e10 −1.00737 −0.503683 0.863889i 0.668022π-0.668022\pi
−0.503683 + 0.863889i 0.668022π0.668022\pi
954954 −7.13808e9 −0.266172
955955 0 0
956956 −2.41831e9 −0.0895177
957957 −1.08168e10 −0.398940
958958 −2.33200e8 −0.00856937
959959 −2.89697e10 −1.06067
960960 0 0
961961 3.23749e10 1.17673
962962 4.50600e9 0.163184
963963 −7.60471e8 −0.0274404
964964 1.09116e10 0.392300
965965 0 0
966966 2.36198e10 0.843057
967967 −1.62053e10 −0.576320 −0.288160 0.957582i 0.593043π-0.593043\pi
−0.288160 + 0.957582i 0.593043π0.593043\pi
968968 5.27127e9 0.186789
969969 −4.31121e9 −0.152218
970970 0 0
971971 −1.58919e10 −0.557068 −0.278534 0.960426i 0.589848π-0.589848\pi
−0.278534 + 0.960426i 0.589848π0.589848\pi
972972 −9.18330e8 −0.0320750
973973 −3.34186e10 −1.16304
974974 −2.07039e10 −0.717951
975975 0 0
976976 1.07951e10 0.371667
977977 −2.98228e10 −1.02310 −0.511549 0.859254i 0.670928π-0.670928\pi
−0.511549 + 0.859254i 0.670928π0.670928\pi
978978 −1.26763e10 −0.433317
979979 −5.60544e10 −1.90928
980980 0 0
981981 3.81328e9 0.128961
982982 −1.60925e10 −0.542293
983983 −3.04567e10 −1.02270 −0.511348 0.859374i 0.670854π-0.670854\pi
−0.511348 + 0.859374i 0.670854π0.670854\pi
984984 −5.48500e9 −0.183524
985985 0 0
986986 −2.86058e9 −0.0950354
987987 −2.83442e10 −0.938328
988988 1.09887e10 0.362491
989989 −6.05974e10 −1.99190
990990 0 0
991991 −4.03364e10 −1.31656 −0.658278 0.752775i 0.728715π-0.728715\pi
−0.658278 + 0.752775i 0.728715π0.728715\pi
992992 8.01896e9 0.260812
993993 2.16071e10 0.700283
994994 3.94597e10 1.27439
995995 0 0
996996 9.97604e9 0.319927
997997 1.00858e10 0.322312 0.161156 0.986929i 0.448478π-0.448478\pi
0.161156 + 0.986929i 0.448478π0.448478\pi
998998 −5.05009e9 −0.160821
999999 −2.11666e9 −0.0671693
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 150.8.a.s.1.2 2
3.2 odd 2 450.8.a.be.1.2 2
5.2 odd 4 30.8.c.b.19.4 yes 4
5.3 odd 4 30.8.c.b.19.2 4
5.4 even 2 150.8.a.r.1.1 2
15.2 even 4 90.8.c.b.19.1 4
15.8 even 4 90.8.c.b.19.3 4
15.14 odd 2 450.8.a.bh.1.1 2
20.3 even 4 240.8.f.d.49.4 4
20.7 even 4 240.8.f.d.49.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.8.c.b.19.2 4 5.3 odd 4
30.8.c.b.19.4 yes 4 5.2 odd 4
90.8.c.b.19.1 4 15.2 even 4
90.8.c.b.19.3 4 15.8 even 4
150.8.a.r.1.1 2 5.4 even 2
150.8.a.s.1.2 2 1.1 even 1 trivial
240.8.f.d.49.2 4 20.7 even 4
240.8.f.d.49.4 4 20.3 even 4
450.8.a.be.1.2 2 3.2 odd 2
450.8.a.bh.1.1 2 15.14 odd 2