Properties

Label 4205.2.a.g.1.3
Level 42054205
Weight 22
Character 4205.1
Self dual yes
Analytic conductor 33.57733.577
Analytic rank 11
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4205,2,Mod(1,4205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4205.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 4205=5292 4205 = 5 \cdot 29^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 4205.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: 33.577094049933.5770940499
Analytic rank: 11
Dimension: 44
Coefficient field: Q(ζ24)+\Q(\zeta_{24})^+
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x44x2+1 x^{4} - 4x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 145)
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.3
Root 0.5176380.517638 of defining polynomial
Character χ\chi == 4205.1

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+0.517638q21.41421q31.73205q4+1.00000q50.732051q6+0.732051q71.93185q81.00000q9+0.517638q105.27792q11+2.44949q12+1.46410q13+0.378937q141.41421q15+2.46410q16+6.31319q170.517638q18+4.24264q191.73205q201.03528q212.73205q228.19615q23+2.73205q24+1.00000q25+0.757875q26+5.65685q271.26795q280.732051q30+4.24264q31+5.13922q32+7.46410q33+3.26795q34+0.732051q35+1.73205q36+4.24264q37+2.19615q382.07055q391.93185q408.76268q410.535898q42+4.24264q43+9.14162q441.00000q454.24264q46+8.38375q473.48477q486.46410q49+0.517638q508.92820q512.53590q52+2.92820q545.27792q551.41421q566.00000q57+6.00000q59+2.44949q603.10583q61+2.19615q620.732051q632.26795q64+1.46410q65+3.86370q6611.1244q6710.9348q68+11.5911q69+0.378937q706.00000q71+1.93185q721.13681q73+2.19615q741.41421q757.34847q763.86370q771.07180q7815.8338q79+2.46410q805.00000q814.53590q82+2.19615q83+1.79315q84+6.31319q85+2.19615q86+10.1962q88+2.07055q890.517638q90+1.07180q91+14.1962q926.00000q93+4.33975q94+4.24264q957.26795q967.34847q973.34607q98+5.27792q99+O(q100)q+0.517638 q^{2} -1.41421 q^{3} -1.73205 q^{4} +1.00000 q^{5} -0.732051 q^{6} +0.732051 q^{7} -1.93185 q^{8} -1.00000 q^{9} +0.517638 q^{10} -5.27792 q^{11} +2.44949 q^{12} +1.46410 q^{13} +0.378937 q^{14} -1.41421 q^{15} +2.46410 q^{16} +6.31319 q^{17} -0.517638 q^{18} +4.24264 q^{19} -1.73205 q^{20} -1.03528 q^{21} -2.73205 q^{22} -8.19615 q^{23} +2.73205 q^{24} +1.00000 q^{25} +0.757875 q^{26} +5.65685 q^{27} -1.26795 q^{28} -0.732051 q^{30} +4.24264 q^{31} +5.13922 q^{32} +7.46410 q^{33} +3.26795 q^{34} +0.732051 q^{35} +1.73205 q^{36} +4.24264 q^{37} +2.19615 q^{38} -2.07055 q^{39} -1.93185 q^{40} -8.76268 q^{41} -0.535898 q^{42} +4.24264 q^{43} +9.14162 q^{44} -1.00000 q^{45} -4.24264 q^{46} +8.38375 q^{47} -3.48477 q^{48} -6.46410 q^{49} +0.517638 q^{50} -8.92820 q^{51} -2.53590 q^{52} +2.92820 q^{54} -5.27792 q^{55} -1.41421 q^{56} -6.00000 q^{57} +6.00000 q^{59} +2.44949 q^{60} -3.10583 q^{61} +2.19615 q^{62} -0.732051 q^{63} -2.26795 q^{64} +1.46410 q^{65} +3.86370 q^{66} -11.1244 q^{67} -10.9348 q^{68} +11.5911 q^{69} +0.378937 q^{70} -6.00000 q^{71} +1.93185 q^{72} -1.13681 q^{73} +2.19615 q^{74} -1.41421 q^{75} -7.34847 q^{76} -3.86370 q^{77} -1.07180 q^{78} -15.8338 q^{79} +2.46410 q^{80} -5.00000 q^{81} -4.53590 q^{82} +2.19615 q^{83} +1.79315 q^{84} +6.31319 q^{85} +2.19615 q^{86} +10.1962 q^{88} +2.07055 q^{89} -0.517638 q^{90} +1.07180 q^{91} +14.1962 q^{92} -6.00000 q^{93} +4.33975 q^{94} +4.24264 q^{95} -7.26795 q^{96} -7.34847 q^{97} -3.34607 q^{98} +5.27792 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+4q5+4q64q74q98q134q164q2212q23+4q24+4q2512q28+4q30+16q33+20q344q3512q3816q424q45+36q96+O(q100) 4 q + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{9} - 8 q^{13} - 4 q^{16} - 4 q^{22} - 12 q^{23} + 4 q^{24} + 4 q^{25} - 12 q^{28} + 4 q^{30} + 16 q^{33} + 20 q^{34} - 4 q^{35} - 12 q^{38} - 16 q^{42} - 4 q^{45}+ \cdots - 36 q^{96}+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 0.517638 0.366025 0.183013 0.983111i 0.441415π-0.441415\pi
0.183013 + 0.983111i 0.441415π0.441415\pi
33 −1.41421 −0.816497 −0.408248 0.912871i 0.633860π-0.633860\pi
−0.408248 + 0.912871i 0.633860π0.633860\pi
44 −1.73205 −0.866025
55 1.00000 0.447214
66 −0.732051 −0.298858
77 0.732051 0.276689 0.138345 0.990384i 0.455822π-0.455822\pi
0.138345 + 0.990384i 0.455822π0.455822\pi
88 −1.93185 −0.683013
99 −1.00000 −0.333333
1010 0.517638 0.163692
1111 −5.27792 −1.59135 −0.795676 0.605723i 0.792884π-0.792884\pi
−0.795676 + 0.605723i 0.792884π0.792884\pi
1212 2.44949 0.707107
1313 1.46410 0.406069 0.203034 0.979172i 0.434920π-0.434920\pi
0.203034 + 0.979172i 0.434920π0.434920\pi
1414 0.378937 0.101275
1515 −1.41421 −0.365148
1616 2.46410 0.616025
1717 6.31319 1.53117 0.765587 0.643332i 0.222449π-0.222449\pi
0.765587 + 0.643332i 0.222449π0.222449\pi
1818 −0.517638 −0.122008
1919 4.24264 0.973329 0.486664 0.873589i 0.338214π-0.338214\pi
0.486664 + 0.873589i 0.338214π0.338214\pi
2020 −1.73205 −0.387298
2121 −1.03528 −0.225916
2222 −2.73205 −0.582475
2323 −8.19615 −1.70902 −0.854508 0.519438i 0.826141π-0.826141\pi
−0.854508 + 0.519438i 0.826141π0.826141\pi
2424 2.73205 0.557678
2525 1.00000 0.200000
2626 0.757875 0.148631
2727 5.65685 1.08866
2828 −1.26795 −0.239620
2929 0 0
3030 −0.732051 −0.133654
3131 4.24264 0.762001 0.381000 0.924575i 0.375580π-0.375580\pi
0.381000 + 0.924575i 0.375580π0.375580\pi
3232 5.13922 0.908494
3333 7.46410 1.29933
3434 3.26795 0.560449
3535 0.732051 0.123739
3636 1.73205 0.288675
3737 4.24264 0.697486 0.348743 0.937218i 0.386609π-0.386609\pi
0.348743 + 0.937218i 0.386609π0.386609\pi
3838 2.19615 0.356263
3939 −2.07055 −0.331554
4040 −1.93185 −0.305453
4141 −8.76268 −1.36850 −0.684251 0.729247i 0.739871π-0.739871\pi
−0.684251 + 0.729247i 0.739871π0.739871\pi
4242 −0.535898 −0.0826909
4343 4.24264 0.646997 0.323498 0.946229i 0.395141π-0.395141\pi
0.323498 + 0.946229i 0.395141π0.395141\pi
4444 9.14162 1.37815
4545 −1.00000 −0.149071
4646 −4.24264 −0.625543
4747 8.38375 1.22289 0.611447 0.791285i 0.290588π-0.290588\pi
0.611447 + 0.791285i 0.290588π0.290588\pi
4848 −3.48477 −0.502983
4949 −6.46410 −0.923443
5050 0.517638 0.0732051
5151 −8.92820 −1.25020
5252 −2.53590 −0.351666
5353 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
5454 2.92820 0.398478
5555 −5.27792 −0.711674
5656 −1.41421 −0.188982
5757 −6.00000 −0.794719
5858 0 0
5959 6.00000 0.781133 0.390567 0.920575i 0.372279π-0.372279\pi
0.390567 + 0.920575i 0.372279π0.372279\pi
6060 2.44949 0.316228
6161 −3.10583 −0.397661 −0.198830 0.980034i 0.563714π-0.563714\pi
−0.198830 + 0.980034i 0.563714π0.563714\pi
6262 2.19615 0.278912
6363 −0.732051 −0.0922297
6464 −2.26795 −0.283494
6565 1.46410 0.181599
6666 3.86370 0.475589
6767 −11.1244 −1.35906 −0.679528 0.733649i 0.737815π-0.737815\pi
−0.679528 + 0.733649i 0.737815π0.737815\pi
6868 −10.9348 −1.32604
6969 11.5911 1.39541
7070 0.378937 0.0452917
7171 −6.00000 −0.712069 −0.356034 0.934473i 0.615871π-0.615871\pi
−0.356034 + 0.934473i 0.615871π0.615871\pi
7272 1.93185 0.227671
7373 −1.13681 −0.133054 −0.0665269 0.997785i 0.521192π-0.521192\pi
−0.0665269 + 0.997785i 0.521192π0.521192\pi
7474 2.19615 0.255298
7575 −1.41421 −0.163299
7676 −7.34847 −0.842927
7777 −3.86370 −0.440310
7878 −1.07180 −0.121357
7979 −15.8338 −1.78144 −0.890718 0.454556i 0.849798π-0.849798\pi
−0.890718 + 0.454556i 0.849798π0.849798\pi
8080 2.46410 0.275495
8181 −5.00000 −0.555556
8282 −4.53590 −0.500906
8383 2.19615 0.241059 0.120530 0.992710i 0.461541π-0.461541\pi
0.120530 + 0.992710i 0.461541π0.461541\pi
8484 1.79315 0.195649
8585 6.31319 0.684762
8686 2.19615 0.236817
8787 0 0
8888 10.1962 1.08691
8989 2.07055 0.219478 0.109739 0.993960i 0.464998π-0.464998\pi
0.109739 + 0.993960i 0.464998π0.464998\pi
9090 −0.517638 −0.0545638
9191 1.07180 0.112355
9292 14.1962 1.48005
9393 −6.00000 −0.622171
9494 4.33975 0.447611
9595 4.24264 0.435286
9696 −7.26795 −0.741782
9797 −7.34847 −0.746124 −0.373062 0.927806i 0.621692π-0.621692\pi
−0.373062 + 0.927806i 0.621692π0.621692\pi
9898 −3.34607 −0.338004
9999 5.27792 0.530451
100100 −1.73205 −0.173205
101101 −4.14110 −0.412055 −0.206028 0.978546i 0.566054π-0.566054\pi
−0.206028 + 0.978546i 0.566054π0.566054\pi
102102 −4.62158 −0.457604
103103 −0.196152 −0.0193275 −0.00966374 0.999953i 0.503076π-0.503076\pi
−0.00966374 + 0.999953i 0.503076π0.503076\pi
104104 −2.82843 −0.277350
105105 −1.03528 −0.101033
106106 0 0
107107 2.19615 0.212310 0.106155 0.994350i 0.466146π-0.466146\pi
0.106155 + 0.994350i 0.466146π0.466146\pi
108108 −9.79796 −0.942809
109109 1.46410 0.140236 0.0701178 0.997539i 0.477662π-0.477662\pi
0.0701178 + 0.997539i 0.477662π0.477662\pi
110110 −2.73205 −0.260491
111111 −6.00000 −0.569495
112112 1.80385 0.170448
113113 13.7632 1.29473 0.647366 0.762179i 0.275870π-0.275870\pi
0.647366 + 0.762179i 0.275870π0.275870\pi
114114 −3.10583 −0.290887
115115 −8.19615 −0.764295
116116 0 0
117117 −1.46410 −0.135356
118118 3.10583 0.285915
119119 4.62158 0.423659
120120 2.73205 0.249401
121121 16.8564 1.53240
122122 −1.60770 −0.145554
123123 12.3923 1.11738
124124 −7.34847 −0.659912
125125 1.00000 0.0894427
126126 −0.378937 −0.0337584
127127 −12.7279 −1.12942 −0.564710 0.825289i 0.691012π-0.691012\pi
−0.564710 + 0.825289i 0.691012π0.691012\pi
128128 −11.4524 −1.01226
129129 −6.00000 −0.528271
130130 0.757875 0.0664700
131131 2.17209 0.189776 0.0948881 0.995488i 0.469751π-0.469751\pi
0.0948881 + 0.995488i 0.469751π0.469751\pi
132132 −12.9282 −1.12526
133133 3.10583 0.269309
134134 −5.75839 −0.497449
135135 5.65685 0.486864
136136 −12.1962 −1.04581
137137 9.89949 0.845771 0.422885 0.906183i 0.361017π-0.361017\pi
0.422885 + 0.906183i 0.361017π0.361017\pi
138138 6.00000 0.510754
139139 −13.4641 −1.14201 −0.571005 0.820947i 0.693446π-0.693446\pi
−0.571005 + 0.820947i 0.693446π0.693446\pi
140140 −1.26795 −0.107161
141141 −11.8564 −0.998490
142142 −3.10583 −0.260635
143143 −7.72741 −0.646198
144144 −2.46410 −0.205342
145145 0 0
146146 −0.588457 −0.0487011
147147 9.14162 0.753988
148148 −7.34847 −0.604040
149149 18.0000 1.47462 0.737309 0.675556i 0.236096π-0.236096\pi
0.737309 + 0.675556i 0.236096π0.236096\pi
150150 −0.732051 −0.0597717
151151 −18.3923 −1.49674 −0.748372 0.663279i 0.769164π-0.769164\pi
−0.748372 + 0.663279i 0.769164π0.769164\pi
152152 −8.19615 −0.664796
153153 −6.31319 −0.510391
154154 −2.00000 −0.161165
155155 4.24264 0.340777
156156 3.58630 0.287134
157157 7.34847 0.586472 0.293236 0.956040i 0.405268π-0.405268\pi
0.293236 + 0.956040i 0.405268π0.405268\pi
158158 −8.19615 −0.652051
159159 0 0
160160 5.13922 0.406291
161161 −6.00000 −0.472866
162162 −2.58819 −0.203347
163163 4.24264 0.332309 0.166155 0.986100i 0.446865π-0.446865\pi
0.166155 + 0.986100i 0.446865π0.446865\pi
164164 15.1774 1.18516
165165 7.46410 0.581080
166166 1.13681 0.0882337
167167 −3.80385 −0.294351 −0.147175 0.989110i 0.547018π-0.547018\pi
−0.147175 + 0.989110i 0.547018π0.547018\pi
168168 2.00000 0.154303
169169 −10.8564 −0.835108
170170 3.26795 0.250640
171171 −4.24264 −0.324443
172172 −7.34847 −0.560316
173173 −20.7846 −1.58022 −0.790112 0.612962i 0.789978π-0.789978\pi
−0.790112 + 0.612962i 0.789978π0.789978\pi
174174 0 0
175175 0.732051 0.0553378
176176 −13.0053 −0.980313
177177 −8.48528 −0.637793
178178 1.07180 0.0803346
179179 −16.3923 −1.22522 −0.612609 0.790386i 0.709880π-0.709880\pi
−0.612609 + 0.790386i 0.709880π0.709880\pi
180180 1.73205 0.129099
181181 −5.85641 −0.435303 −0.217652 0.976027i 0.569840π-0.569840\pi
−0.217652 + 0.976027i 0.569840π0.569840\pi
182182 0.554803 0.0411247
183183 4.39230 0.324689
184184 15.8338 1.16728
185185 4.24264 0.311925
186186 −3.10583 −0.227730
187187 −33.3205 −2.43664
188188 −14.5211 −1.05906
189189 4.14110 0.301221
190190 2.19615 0.159326
191191 19.2170 1.39049 0.695246 0.718772i 0.255295π-0.255295\pi
0.695246 + 0.718772i 0.255295π0.255295\pi
192192 3.20736 0.231472
193193 −12.7279 −0.916176 −0.458088 0.888907i 0.651466π-0.651466\pi
−0.458088 + 0.888907i 0.651466π0.651466\pi
194194 −3.80385 −0.273100
195195 −2.07055 −0.148275
196196 11.1962 0.799725
197197 −1.60770 −0.114544 −0.0572718 0.998359i 0.518240π-0.518240\pi
−0.0572718 + 0.998359i 0.518240π0.518240\pi
198198 2.73205 0.194158
199199 −19.4641 −1.37977 −0.689887 0.723917i 0.742340π-0.742340\pi
−0.689887 + 0.723917i 0.742340π0.742340\pi
200200 −1.93185 −0.136603
201201 15.7322 1.10966
202202 −2.14359 −0.150823
203203 0 0
204204 15.4641 1.08270
205205 −8.76268 −0.612012
206206 −0.101536 −0.00707435
207207 8.19615 0.569672
208208 3.60770 0.250149
209209 −22.3923 −1.54891
210210 −0.535898 −0.0369805
211211 −22.0454 −1.51767 −0.758834 0.651284i 0.774231π-0.774231\pi
−0.758834 + 0.651284i 0.774231π0.774231\pi
212212 0 0
213213 8.48528 0.581402
214214 1.13681 0.0777109
215215 4.24264 0.289346
216216 −10.9282 −0.743570
217217 3.10583 0.210837
218218 0.757875 0.0513298
219219 1.60770 0.108638
220220 9.14162 0.616328
221221 9.24316 0.621762
222222 −3.10583 −0.208450
223223 −9.66025 −0.646898 −0.323449 0.946246i 0.604842π-0.604842\pi
−0.323449 + 0.946246i 0.604842π0.604842\pi
224224 3.76217 0.251370
225225 −1.00000 −0.0666667
226226 7.12436 0.473905
227227 −15.8038 −1.04894 −0.524469 0.851429i 0.675736π-0.675736\pi
−0.524469 + 0.851429i 0.675736π0.675736\pi
228228 10.3923 0.688247
229229 −25.4558 −1.68217 −0.841085 0.540903i 0.818082π-0.818082\pi
−0.841085 + 0.540903i 0.818082π0.818082\pi
230230 −4.24264 −0.279751
231231 5.46410 0.359511
232232 0 0
233233 22.3923 1.46697 0.733484 0.679706i 0.237893π-0.237893\pi
0.733484 + 0.679706i 0.237893π0.237893\pi
234234 −0.757875 −0.0495438
235235 8.38375 0.546895
236236 −10.3923 −0.676481
237237 22.3923 1.45454
238238 2.39230 0.155070
239239 6.00000 0.388108 0.194054 0.980991i 0.437836π-0.437836\pi
0.194054 + 0.980991i 0.437836π0.437836\pi
240240 −3.48477 −0.224941
241241 −8.92820 −0.575116 −0.287558 0.957763i 0.592843π-0.592843\pi
−0.287558 + 0.957763i 0.592843π0.592843\pi
242242 8.72552 0.560898
243243 −9.89949 −0.635053
244244 5.37945 0.344384
245245 −6.46410 −0.412976
246246 6.41473 0.408988
247247 6.21166 0.395238
248248 −8.19615 −0.520456
249249 −3.10583 −0.196824
250250 0.517638 0.0327383
251251 −13.7632 −0.868725 −0.434363 0.900738i 0.643026π-0.643026\pi
−0.434363 + 0.900738i 0.643026π0.643026\pi
252252 1.26795 0.0798733
253253 43.2586 2.71965
254254 −6.58846 −0.413397
255255 −8.92820 −0.559106
256256 −1.39230 −0.0870191
257257 12.0000 0.748539 0.374270 0.927320i 0.377893π-0.377893\pi
0.374270 + 0.927320i 0.377893π0.377893\pi
258258 −3.10583 −0.193360
259259 3.10583 0.192987
260260 −2.53590 −0.157270
261261 0 0
262262 1.12436 0.0694629
263263 −4.79744 −0.295823 −0.147912 0.989001i 0.547255π-0.547255\pi
−0.147912 + 0.989001i 0.547255π0.547255\pi
264264 −14.4195 −0.887461
265265 0 0
266266 1.60770 0.0985741
267267 −2.92820 −0.179203
268268 19.2679 1.17698
269269 32.7028 1.99392 0.996962 0.0778925i 0.0248191π-0.0248191\pi
0.996962 + 0.0778925i 0.0248191π0.0248191\pi
270270 2.92820 0.178205
271271 −9.62209 −0.584501 −0.292250 0.956342i 0.594404π-0.594404\pi
−0.292250 + 0.956342i 0.594404π0.594404\pi
272272 15.5563 0.943242
273273 −1.51575 −0.0917373
274274 5.12436 0.309574
275275 −5.27792 −0.318270
276276 −20.0764 −1.20846
277277 2.92820 0.175939 0.0879693 0.996123i 0.471962π-0.471962\pi
0.0879693 + 0.996123i 0.471962π0.471962\pi
278278 −6.96953 −0.418005
279279 −4.24264 −0.254000
280280 −1.41421 −0.0845154
281281 −4.39230 −0.262023 −0.131011 0.991381i 0.541822π-0.541822\pi
−0.131011 + 0.991381i 0.541822π0.541822\pi
282282 −6.13733 −0.365473
283283 20.9808 1.24718 0.623588 0.781753i 0.285674π-0.285674\pi
0.623588 + 0.781753i 0.285674π0.285674\pi
284284 10.3923 0.616670
285285 −6.00000 −0.355409
286286 −4.00000 −0.236525
287287 −6.41473 −0.378649
288288 −5.13922 −0.302831
289289 22.8564 1.34449
290290 0 0
291291 10.3923 0.609208
292292 1.96902 0.115228
293293 11.6926 0.683092 0.341546 0.939865i 0.389050π-0.389050\pi
0.341546 + 0.939865i 0.389050π0.389050\pi
294294 4.73205 0.275979
295295 6.00000 0.349334
296296 −8.19615 −0.476392
297297 −29.8564 −1.73244
298298 9.31749 0.539747
299299 −12.0000 −0.693978
300300 2.44949 0.141421
301301 3.10583 0.179017
302302 −9.52056 −0.547847
303303 5.85641 0.336442
304304 10.4543 0.599595
305305 −3.10583 −0.177839
306306 −3.26795 −0.186816
307307 −25.1512 −1.43546 −0.717728 0.696323i 0.754818π-0.754818\pi
−0.717728 + 0.696323i 0.754818π0.754818\pi
308308 6.69213 0.381320
309309 0.277401 0.0157808
310310 2.19615 0.124733
311311 0.101536 0.00575758 0.00287879 0.999996i 0.499084π-0.499084\pi
0.00287879 + 0.999996i 0.499084π0.499084\pi
312312 4.00000 0.226455
313313 −25.4641 −1.43932 −0.719658 0.694329i 0.755701π-0.755701\pi
−0.719658 + 0.694329i 0.755701π0.755701\pi
314314 3.80385 0.214664
315315 −0.732051 −0.0412464
316316 27.4249 1.54277
317317 −21.0101 −1.18005 −0.590023 0.807386i 0.700881π-0.700881\pi
−0.590023 + 0.807386i 0.700881π0.700881\pi
318318 0 0
319319 0 0
320320 −2.26795 −0.126782
321321 −3.10583 −0.173350
322322 −3.10583 −0.173081
323323 26.7846 1.49034
324324 8.66025 0.481125
325325 1.46410 0.0812137
326326 2.19615 0.121634
327327 −2.07055 −0.114502
328328 16.9282 0.934704
329329 6.13733 0.338362
330330 3.86370 0.212690
331331 18.9396 1.04101 0.520507 0.853858i 0.325743π-0.325743\pi
0.520507 + 0.853858i 0.325743π0.325743\pi
332332 −3.80385 −0.208763
333333 −4.24264 −0.232495
334334 −1.96902 −0.107740
335335 −11.1244 −0.607788
336336 −2.55103 −0.139170
337337 −18.1074 −0.986372 −0.493186 0.869924i 0.664168π-0.664168\pi
−0.493186 + 0.869924i 0.664168π0.664168\pi
338338 −5.61969 −0.305671
339339 −19.4641 −1.05714
340340 −10.9348 −0.593021
341341 −22.3923 −1.21261
342342 −2.19615 −0.118754
343343 −9.85641 −0.532196
344344 −8.19615 −0.441907
345345 11.5911 0.624044
346346 −10.7589 −0.578402
347347 −26.1962 −1.40628 −0.703142 0.711050i 0.748220π-0.748220\pi
−0.703142 + 0.711050i 0.748220π0.748220\pi
348348 0 0
349349 18.3923 0.984518 0.492259 0.870449i 0.336171π-0.336171\pi
0.492259 + 0.870449i 0.336171π0.336171\pi
350350 0.378937 0.0202551
351351 8.28221 0.442072
352352 −27.1244 −1.44573
353353 −32.7846 −1.74495 −0.872474 0.488660i 0.837486π-0.837486\pi
−0.872474 + 0.488660i 0.837486π0.837486\pi
354354 −4.39230 −0.233448
355355 −6.00000 −0.318447
356356 −3.58630 −0.190074
357357 −6.53590 −0.345916
358358 −8.48528 −0.448461
359359 −7.07107 −0.373197 −0.186598 0.982436i 0.559746π-0.559746\pi
−0.186598 + 0.982436i 0.559746π0.559746\pi
360360 1.93185 0.101818
361361 −1.00000 −0.0526316
362362 −3.03150 −0.159332
363363 −23.8386 −1.25120
364364 −1.85641 −0.0973021
365365 −1.13681 −0.0595035
366366 2.27362 0.118844
367367 1.96902 0.102782 0.0513909 0.998679i 0.483635π-0.483635\pi
0.0513909 + 0.998679i 0.483635π0.483635\pi
368368 −20.1962 −1.05280
369369 8.76268 0.456167
370370 2.19615 0.114173
371371 0 0
372372 10.3923 0.538816
373373 −22.0000 −1.13912 −0.569558 0.821951i 0.692886π-0.692886\pi
−0.569558 + 0.821951i 0.692886π0.692886\pi
374374 −17.2480 −0.891871
375375 −1.41421 −0.0730297
376376 −16.1962 −0.835253
377377 0 0
378378 2.14359 0.110255
379379 −15.8338 −0.813325 −0.406663 0.913578i 0.633308π-0.633308\pi
−0.406663 + 0.913578i 0.633308π0.633308\pi
380380 −7.34847 −0.376969
381381 18.0000 0.922168
382382 9.94744 0.508955
383383 2.19615 0.112218 0.0561091 0.998425i 0.482131π-0.482131\pi
0.0561091 + 0.998425i 0.482131π0.482131\pi
384384 16.1962 0.826506
385385 −3.86370 −0.196913
386386 −6.58846 −0.335344
387387 −4.24264 −0.215666
388388 12.7279 0.646162
389389 32.7028 1.65810 0.829048 0.559177i 0.188883π-0.188883\pi
0.829048 + 0.559177i 0.188883π0.188883\pi
390390 −1.07180 −0.0542725
391391 −51.7439 −2.61680
392392 12.4877 0.630723
393393 −3.07180 −0.154952
394394 −0.832204 −0.0419258
395395 −15.8338 −0.796682
396396 −9.14162 −0.459384
397397 30.3923 1.52535 0.762673 0.646784i 0.223887π-0.223887\pi
0.762673 + 0.646784i 0.223887π0.223887\pi
398398 −10.0754 −0.505032
399399 −4.39230 −0.219890
400400 2.46410 0.123205
401401 8.78461 0.438682 0.219341 0.975648i 0.429609π-0.429609\pi
0.219341 + 0.975648i 0.429609π0.429609\pi
402402 8.14359 0.406166
403403 6.21166 0.309425
404404 7.17260 0.356850
405405 −5.00000 −0.248452
406406 0 0
407407 −22.3923 −1.10995
408408 17.2480 0.853901
409409 17.8028 0.880290 0.440145 0.897927i 0.354927π-0.354927\pi
0.440145 + 0.897927i 0.354927π0.354927\pi
410410 −4.53590 −0.224012
411411 −14.0000 −0.690569
412412 0.339746 0.0167381
413413 4.39230 0.216131
414414 4.24264 0.208514
415415 2.19615 0.107805
416416 7.52433 0.368911
417417 19.0411 0.932447
418418 −11.5911 −0.566940
419419 18.0000 0.879358 0.439679 0.898155i 0.355092π-0.355092\pi
0.439679 + 0.898155i 0.355092π0.355092\pi
420420 1.79315 0.0874968
421421 −17.8028 −0.867654 −0.433827 0.900996i 0.642837π-0.642837\pi
−0.433827 + 0.900996i 0.642837π0.642837\pi
422422 −11.4115 −0.555505
423423 −8.38375 −0.407632
424424 0 0
425425 6.31319 0.306235
426426 4.39230 0.212808
427427 −2.27362 −0.110028
428428 −3.80385 −0.183866
429429 10.9282 0.527619
430430 2.19615 0.105908
431431 4.39230 0.211570 0.105785 0.994389i 0.466264π-0.466264\pi
0.105785 + 0.994389i 0.466264π0.466264\pi
432432 13.9391 0.670644
433433 12.7279 0.611665 0.305832 0.952085i 0.401065π-0.401065\pi
0.305832 + 0.952085i 0.401065π0.401065\pi
434434 1.60770 0.0771718
435435 0 0
436436 −2.53590 −0.121448
437437 −34.7733 −1.66343
438438 0.832204 0.0397643
439439 −8.92820 −0.426120 −0.213060 0.977039i 0.568343π-0.568343\pi
−0.213060 + 0.977039i 0.568343π0.568343\pi
440440 10.1962 0.486082
441441 6.46410 0.307814
442442 4.78461 0.227581
443443 −14.7985 −0.703097 −0.351548 0.936170i 0.614345π-0.614345\pi
−0.351548 + 0.936170i 0.614345π0.614345\pi
444444 10.3923 0.493197
445445 2.07055 0.0981536
446446 −5.00052 −0.236781
447447 −25.4558 −1.20402
448448 −1.66025 −0.0784396
449449 30.6322 1.44562 0.722812 0.691045i 0.242849π-0.242849\pi
0.722812 + 0.691045i 0.242849π0.242849\pi
450450 −0.517638 −0.0244017
451451 46.2487 2.17777
452452 −23.8386 −1.12127
453453 26.0106 1.22209
454454 −8.18067 −0.383938
455455 1.07180 0.0502466
456456 11.5911 0.542803
457457 −26.9282 −1.25965 −0.629824 0.776738i 0.716873π-0.716873\pi
−0.629824 + 0.776738i 0.716873π0.716873\pi
458458 −13.1769 −0.615717
459459 35.7128 1.66693
460460 14.1962 0.661899
461461 −22.6274 −1.05386 −0.526932 0.849907i 0.676658π-0.676658\pi
−0.526932 + 0.849907i 0.676658π0.676658\pi
462462 2.82843 0.131590
463463 21.6603 1.00664 0.503319 0.864101i 0.332112π-0.332112\pi
0.503319 + 0.864101i 0.332112π0.332112\pi
464464 0 0
465465 −6.00000 −0.278243
466466 11.5911 0.536948
467467 −21.7680 −1.00730 −0.503652 0.863907i 0.668010π-0.668010\pi
−0.503652 + 0.863907i 0.668010π0.668010\pi
468468 2.53590 0.117222
469469 −8.14359 −0.376036
470470 4.33975 0.200178
471471 −10.3923 −0.478852
472472 −11.5911 −0.533524
473473 −22.3923 −1.02960
474474 11.5911 0.532397
475475 4.24264 0.194666
476476 −8.00481 −0.366900
477477 0 0
478478 3.10583 0.142057
479479 −4.52004 −0.206526 −0.103263 0.994654i 0.532928π-0.532928\pi
−0.103263 + 0.994654i 0.532928π0.532928\pi
480480 −7.26795 −0.331735
481481 6.21166 0.283227
482482 −4.62158 −0.210507
483483 8.48528 0.386094
484484 −29.1962 −1.32710
485485 −7.34847 −0.333677
486486 −5.12436 −0.232445
487487 −22.5885 −1.02358 −0.511790 0.859110i 0.671018π-0.671018\pi
−0.511790 + 0.859110i 0.671018π0.671018\pi
488488 6.00000 0.271607
489489 −6.00000 −0.271329
490490 −3.34607 −0.151160
491491 −1.41421 −0.0638226 −0.0319113 0.999491i 0.510159π-0.510159\pi
−0.0319113 + 0.999491i 0.510159π0.510159\pi
492492 −21.4641 −0.967676
493493 0 0
494494 3.21539 0.144667
495495 5.27792 0.237225
496496 10.4543 0.469412
497497 −4.39230 −0.197022
498498 −1.60770 −0.0720425
499499 5.85641 0.262169 0.131084 0.991371i 0.458154π-0.458154\pi
0.131084 + 0.991371i 0.458154π0.458154\pi
500500 −1.73205 −0.0774597
501501 5.37945 0.240336
502502 −7.12436 −0.317976
503503 −14.7985 −0.659831 −0.329916 0.944010i 0.607020π-0.607020\pi
−0.329916 + 0.944010i 0.607020π0.607020\pi
504504 1.41421 0.0629941
505505 −4.14110 −0.184277
506506 22.3923 0.995459
507507 15.3533 0.681863
508508 22.0454 0.978107
509509 20.7846 0.921262 0.460631 0.887592i 0.347623π-0.347623\pi
0.460631 + 0.887592i 0.347623π0.347623\pi
510510 −4.62158 −0.204647
511511 −0.832204 −0.0368145
512512 22.1841 0.980408
513513 24.0000 1.05963
514514 6.21166 0.273984
515515 −0.196152 −0.00864351
516516 10.3923 0.457496
517517 −44.2487 −1.94606
518518 1.60770 0.0706381
519519 29.3939 1.29025
520520 −2.82843 −0.124035
521521 32.7846 1.43632 0.718160 0.695878i 0.244985π-0.244985\pi
0.718160 + 0.695878i 0.244985π0.244985\pi
522522 0 0
523523 −21.6603 −0.947137 −0.473568 0.880757i 0.657034π-0.657034\pi
−0.473568 + 0.880757i 0.657034π0.657034\pi
524524 −3.76217 −0.164351
525525 −1.03528 −0.0451832
526526 −2.48334 −0.108279
527527 26.7846 1.16676
528528 18.3923 0.800422
529529 44.1769 1.92074
530530 0 0
531531 −6.00000 −0.260378
532532 −5.37945 −0.233229
533533 −12.8295 −0.555706
534534 −1.51575 −0.0655929
535535 2.19615 0.0949479
536536 21.4906 0.928253
537537 23.1822 1.00039
538538 16.9282 0.729827
539539 34.1170 1.46952
540540 −9.79796 −0.421637
541541 25.4558 1.09443 0.547216 0.836991i 0.315688π-0.315688\pi
0.547216 + 0.836991i 0.315688π0.315688\pi
542542 −4.98076 −0.213942
543543 8.28221 0.355424
544544 32.4449 1.39106
545545 1.46410 0.0627152
546546 −0.784610 −0.0335782
547547 21.5167 0.919986 0.459993 0.887923i 0.347852π-0.347852\pi
0.459993 + 0.887923i 0.347852π0.347852\pi
548548 −17.1464 −0.732459
549549 3.10583 0.132554
550550 −2.73205 −0.116495
551551 0 0
552552 −22.3923 −0.953080
553553 −11.5911 −0.492904
554554 1.51575 0.0643980
555555 −6.00000 −0.254686
556556 23.3205 0.989010
557557 22.3923 0.948792 0.474396 0.880311i 0.342666π-0.342666\pi
0.474396 + 0.880311i 0.342666π0.342666\pi
558558 −2.19615 −0.0929705
559559 6.21166 0.262725
560560 1.80385 0.0762265
561561 47.1223 1.98951
562562 −2.27362 −0.0959071
563563 16.8690 0.710945 0.355472 0.934687i 0.384320π-0.384320\pi
0.355472 + 0.934687i 0.384320π0.384320\pi
564564 20.5359 0.864717
565565 13.7632 0.579022
566566 10.8604 0.456498
567567 −3.66025 −0.153716
568568 11.5911 0.486352
569569 −15.9353 −0.668042 −0.334021 0.942566i 0.608406π-0.608406\pi
−0.334021 + 0.942566i 0.608406π0.608406\pi
570570 −3.10583 −0.130089
571571 −11.6077 −0.485767 −0.242883 0.970055i 0.578093π-0.578093\pi
−0.242883 + 0.970055i 0.578093π0.578093\pi
572572 13.3843 0.559624
573573 −27.1769 −1.13533
574574 −3.32051 −0.138595
575575 −8.19615 −0.341803
576576 2.26795 0.0944979
577577 44.3954 1.84821 0.924103 0.382144i 0.124814π-0.124814\pi
0.924103 + 0.382144i 0.124814π0.124814\pi
578578 11.8313 0.492119
579579 18.0000 0.748054
580580 0 0
581581 1.60770 0.0666984
582582 5.37945 0.222985
583583 0 0
584584 2.19615 0.0908774
585585 −1.46410 −0.0605332
586586 6.05256 0.250029
587587 −44.1962 −1.82417 −0.912085 0.410001i 0.865528π-0.865528\pi
−0.912085 + 0.410001i 0.865528π0.865528\pi
588588 −15.8338 −0.652973
589589 18.0000 0.741677
590590 3.10583 0.127865
591591 2.27362 0.0935244
592592 10.4543 0.429669
593593 19.6077 0.805192 0.402596 0.915378i 0.368108π-0.368108\pi
0.402596 + 0.915378i 0.368108π0.368108\pi
594594 −15.4548 −0.634119
595595 4.62158 0.189466
596596 −31.1769 −1.27706
597597 27.5264 1.12658
598598 −6.21166 −0.254014
599599 7.14540 0.291953 0.145977 0.989288i 0.453368π-0.453368\pi
0.145977 + 0.989288i 0.453368π0.453368\pi
600600 2.73205 0.111536
601601 −20.0764 −0.818933 −0.409467 0.912325i 0.634285π-0.634285\pi
−0.409467 + 0.912325i 0.634285π0.634285\pi
602602 1.60770 0.0655248
603603 11.1244 0.453019
604604 31.8564 1.29622
605605 16.8564 0.685310
606606 3.03150 0.123146
607607 −12.7279 −0.516610 −0.258305 0.966063i 0.583164π-0.583164\pi
−0.258305 + 0.966063i 0.583164π0.583164\pi
608608 21.8038 0.884263
609609 0 0
610610 −1.60770 −0.0650937
611611 12.2747 0.496579
612612 10.9348 0.442012
613613 −32.9282 −1.32996 −0.664979 0.746862i 0.731559π-0.731559\pi
−0.664979 + 0.746862i 0.731559π0.731559\pi
614614 −13.0192 −0.525414
615615 12.3923 0.499706
616616 7.46410 0.300737
617617 −6.79367 −0.273503 −0.136751 0.990605i 0.543666π-0.543666\pi
−0.136751 + 0.990605i 0.543666π0.543666\pi
618618 0.143594 0.00577618
619619 −13.5601 −0.545027 −0.272514 0.962152i 0.587855π-0.587855\pi
−0.272514 + 0.962152i 0.587855π0.587855\pi
620620 −7.34847 −0.295122
621621 −46.3644 −1.86054
622622 0.0525589 0.00210742
623623 1.51575 0.0607272
624624 −5.10205 −0.204246
625625 1.00000 0.0400000
626626 −13.1812 −0.526826
627627 31.6675 1.26468
628628 −12.7279 −0.507899
629629 26.7846 1.06797
630630 −0.378937 −0.0150972
631631 17.8564 0.710852 0.355426 0.934704i 0.384336π-0.384336\pi
0.355426 + 0.934704i 0.384336π0.384336\pi
632632 30.5885 1.21674
633633 31.1769 1.23917
634634 −10.8756 −0.431927
635635 −12.7279 −0.505092
636636 0 0
637637 −9.46410 −0.374981
638638 0 0
639639 6.00000 0.237356
640640 −11.4524 −0.452696
641641 15.9353 0.629406 0.314703 0.949190i 0.398095π-0.398095\pi
0.314703 + 0.949190i 0.398095π0.398095\pi
642642 −1.60770 −0.0634507
643643 −9.94744 −0.392289 −0.196144 0.980575i 0.562842π-0.562842\pi
−0.196144 + 0.980575i 0.562842π0.562842\pi
644644 10.3923 0.409514
645645 −6.00000 −0.236250
646646 13.8647 0.545501
647647 27.3731 1.07615 0.538073 0.842898i 0.319152π-0.319152\pi
0.538073 + 0.842898i 0.319152π0.319152\pi
648648 9.65926 0.379452
649649 −31.6675 −1.24306
650650 0.757875 0.0297263
651651 −4.39230 −0.172148
652652 −7.34847 −0.287788
653653 −26.3896 −1.03270 −0.516352 0.856376i 0.672710π-0.672710\pi
−0.516352 + 0.856376i 0.672710π0.672710\pi
654654 −1.07180 −0.0419106
655655 2.17209 0.0848705
656656 −21.5921 −0.843031
657657 1.13681 0.0443513
658658 3.17691 0.123849
659659 −10.6574 −0.415152 −0.207576 0.978219i 0.566557π-0.566557\pi
−0.207576 + 0.978219i 0.566557π0.566557\pi
660660 −12.9282 −0.503230
661661 −5.85641 −0.227788 −0.113894 0.993493i 0.536332π-0.536332\pi
−0.113894 + 0.993493i 0.536332π0.536332\pi
662662 9.80385 0.381037
663663 −13.0718 −0.507667
664664 −4.24264 −0.164646
665665 3.10583 0.120439
666666 −2.19615 −0.0850992
667667 0 0
668668 6.58846 0.254915
669669 13.6617 0.528190
670670 −5.75839 −0.222466
671671 16.3923 0.632818
672672 −5.32051 −0.205243
673673 −25.3205 −0.976034 −0.488017 0.872834i 0.662280π-0.662280\pi
−0.488017 + 0.872834i 0.662280π0.662280\pi
674674 −9.37307 −0.361037
675675 5.65685 0.217732
676676 18.8038 0.723225
677677 −38.8129 −1.49170 −0.745850 0.666113i 0.767957π-0.767957\pi
−0.745850 + 0.666113i 0.767957π0.767957\pi
678678 −10.0754 −0.386942
679679 −5.37945 −0.206444
680680 −12.1962 −0.467701
681681 22.3500 0.856454
682682 −11.5911 −0.443847
683683 32.1962 1.23195 0.615976 0.787765i 0.288762π-0.288762\pi
0.615976 + 0.787765i 0.288762π0.288762\pi
684684 7.34847 0.280976
685685 9.89949 0.378240
686686 −5.10205 −0.194797
687687 36.0000 1.37349
688688 10.4543 0.398566
689689 0 0
690690 6.00000 0.228416
691691 −3.07180 −0.116857 −0.0584283 0.998292i 0.518609π-0.518609\pi
−0.0584283 + 0.998292i 0.518609π0.518609\pi
692692 36.0000 1.36851
693693 3.86370 0.146770
694694 −13.5601 −0.514735
695695 −13.4641 −0.510722
696696 0 0
697697 −55.3205 −2.09541
698698 9.52056 0.360358
699699 −31.6675 −1.19777
700700 −1.26795 −0.0479240
701701 −32.7846 −1.23826 −0.619129 0.785289i 0.712514π-0.712514\pi
−0.619129 + 0.785289i 0.712514π0.712514\pi
702702 4.28719 0.161809
703703 18.0000 0.678883
704704 11.9700 0.451138
705705 −11.8564 −0.446538
706706 −16.9706 −0.638696
707707 −3.03150 −0.114011
708708 14.6969 0.552345
709709 7.21539 0.270980 0.135490 0.990779i 0.456739π-0.456739\pi
0.135490 + 0.990779i 0.456739π0.456739\pi
710710 −3.10583 −0.116560
711711 15.8338 0.593812
712712 −4.00000 −0.149906
713713 −34.7733 −1.30227
714714 −3.38323 −0.126614
715715 −7.72741 −0.288989
716716 28.3923 1.06107
717717 −8.48528 −0.316889
718718 −3.66025 −0.136599
719719 −10.3923 −0.387568 −0.193784 0.981044i 0.562076π-0.562076\pi
−0.193784 + 0.981044i 0.562076π0.562076\pi
720720 −2.46410 −0.0918316
721721 −0.143594 −0.00534770
722722 −0.517638 −0.0192645
723723 12.6264 0.469580
724724 10.1436 0.376984
725725 0 0
726726 −12.3397 −0.457971
727727 50.6071 1.87691 0.938456 0.345398i 0.112256π-0.112256\pi
0.938456 + 0.345398i 0.112256π0.112256\pi
728728 −2.07055 −0.0767398
729729 29.0000 1.07407
730730 −0.588457 −0.0217798
731731 26.7846 0.990665
732732 −7.60770 −0.281189
733733 15.0015 0.554095 0.277047 0.960856i 0.410644π-0.410644\pi
0.277047 + 0.960856i 0.410644π0.410644\pi
734734 1.01924 0.0376208
735735 9.14162 0.337194
736736 −42.1218 −1.55263
737737 58.7134 2.16274
738738 4.53590 0.166969
739739 16.6660 0.613067 0.306534 0.951860i 0.400831π-0.400831\pi
0.306534 + 0.951860i 0.400831π0.400831\pi
740740 −7.34847 −0.270135
741741 −8.78461 −0.322711
742742 0 0
743743 −0.101536 −0.00372499 −0.00186250 0.999998i 0.500593π-0.500593\pi
−0.00186250 + 0.999998i 0.500593π0.500593\pi
744744 11.5911 0.424951
745745 18.0000 0.659469
746746 −11.3880 −0.416946
747747 −2.19615 −0.0803530
748748 57.7128 2.11019
749749 1.60770 0.0587439
750750 −0.732051 −0.0267307
751751 −18.9396 −0.691115 −0.345558 0.938398i 0.612310π-0.612310\pi
−0.345558 + 0.938398i 0.612310π0.612310\pi
752752 20.6584 0.753334
753753 19.4641 0.709311
754754 0 0
755755 −18.3923 −0.669365
756756 −7.17260 −0.260865
757757 30.5307 1.10966 0.554828 0.831965i 0.312784π-0.312784\pi
0.554828 + 0.831965i 0.312784π0.312784\pi
758758 −8.19615 −0.297698
759759 −61.1769 −2.22058
760760 −8.19615 −0.297306
761761 −10.3923 −0.376721 −0.188360 0.982100i 0.560317π-0.560317\pi
−0.188360 + 0.982100i 0.560317π0.560317\pi
762762 9.31749 0.337537
763763 1.07180 0.0388016
764764 −33.2848 −1.20420
765765 −6.31319 −0.228254
766766 1.13681 0.0410747
767767 8.78461 0.317194
768768 1.96902 0.0710508
769769 14.6969 0.529985 0.264993 0.964250i 0.414630π-0.414630\pi
0.264993 + 0.964250i 0.414630π0.414630\pi
770770 −2.00000 −0.0720750
771771 −16.9706 −0.611180
772772 22.0454 0.793432
773773 18.3848 0.661254 0.330627 0.943761i 0.392740π-0.392740\pi
0.330627 + 0.943761i 0.392740π0.392740\pi
774774 −2.19615 −0.0789391
775775 4.24264 0.152400
776776 14.1962 0.509612
777777 −4.39230 −0.157573
778778 16.9282 0.606905
779779 −37.1769 −1.33200
780780 3.58630 0.128410
781781 31.6675 1.13315
782782 −26.7846 −0.957816
783783 0 0
784784 −15.9282 −0.568864
785785 7.34847 0.262278
786786 −1.59008 −0.0567162
787787 −38.5885 −1.37553 −0.687765 0.725933i 0.741408π-0.741408\pi
−0.687765 + 0.725933i 0.741408π0.741408\pi
788788 2.78461 0.0991976
789789 6.78461 0.241539
790790 −8.19615 −0.291606
791791 10.0754 0.358239
792792 −10.1962 −0.362305
793793 −4.54725 −0.161478
794794 15.7322 0.558315
795795 0 0
796796 33.7128 1.19492
797797 6.79367 0.240644 0.120322 0.992735i 0.461607π-0.461607\pi
0.120322 + 0.992735i 0.461607π0.461607\pi
798798 −2.27362 −0.0804854
799799 52.9282 1.87247
800800 5.13922 0.181699
801801 −2.07055 −0.0731594
802802 4.54725 0.160569
803803 6.00000 0.211735
804804 −27.2490 −0.960998
805805 −6.00000 −0.211472
806806 3.21539 0.113257
807807 −46.2487 −1.62803
808808 8.00000 0.281439
809809 31.8706 1.12051 0.560255 0.828320i 0.310703π-0.310703\pi
0.560255 + 0.828320i 0.310703π0.310703\pi
810810 −2.58819 −0.0909397
811811 4.67949 0.164319 0.0821596 0.996619i 0.473818π-0.473818\pi
0.0821596 + 0.996619i 0.473818π0.473818\pi
812812 0 0
813813 13.6077 0.477243
814814 −11.5911 −0.406268
815815 4.24264 0.148613
816816 −22.0000 −0.770154
817817 18.0000 0.629740
818818 9.21539 0.322209
819819 −1.07180 −0.0374516
820820 15.1774 0.530018
821821 26.7846 0.934789 0.467395 0.884049i 0.345193π-0.345193\pi
0.467395 + 0.884049i 0.345193π0.345193\pi
822822 −7.24693 −0.252766
823823 50.6071 1.76405 0.882026 0.471201i 0.156179π-0.156179\pi
0.882026 + 0.471201i 0.156179π0.156179\pi
824824 0.378937 0.0132009
825825 7.46410 0.259867
826826 2.27362 0.0791095
827827 −33.0817 −1.15036 −0.575182 0.818025i 0.695069π-0.695069\pi
−0.575182 + 0.818025i 0.695069π0.695069\pi
828828 −14.1962 −0.493350
829829 −45.5322 −1.58140 −0.790700 0.612204i 0.790283π-0.790283\pi
−0.790700 + 0.612204i 0.790283π0.790283\pi
830830 1.13681 0.0394593
831831 −4.14110 −0.143653
832832 −3.32051 −0.115118
833833 −40.8091 −1.41395
834834 9.85641 0.341299
835835 −3.80385 −0.131638
836836 38.7846 1.34139
837837 24.0000 0.829561
838838 9.31749 0.321867
839839 13.7632 0.475158 0.237579 0.971368i 0.423646π-0.423646\pi
0.237579 + 0.971368i 0.423646π0.423646\pi
840840 2.00000 0.0690066
841841 0 0
842842 −9.21539 −0.317583
843843 6.21166 0.213941
844844 38.1838 1.31434
845845 −10.8564 −0.373472
846846 −4.33975 −0.149204
847847 12.3397 0.423999
848848 0 0
849849 −29.6713 −1.01832
850850 3.26795 0.112090
851851 −34.7733 −1.19201
852852 −14.6969 −0.503509
853853 −35.9101 −1.22954 −0.614770 0.788707i 0.710751π-0.710751\pi
−0.614770 + 0.788707i 0.710751π0.710751\pi
854854 −1.17691 −0.0402732
855855 −4.24264 −0.145095
856856 −4.24264 −0.145010
857857 −44.7846 −1.52981 −0.764907 0.644141i 0.777215π-0.777215\pi
−0.764907 + 0.644141i 0.777215π0.777215\pi
858858 5.65685 0.193122
859859 5.07484 0.173151 0.0865757 0.996245i 0.472408π-0.472408\pi
0.0865757 + 0.996245i 0.472408π0.472408\pi
860860 −7.34847 −0.250581
861861 9.07180 0.309166
862862 2.27362 0.0774400
863863 38.1962 1.30021 0.650106 0.759843i 0.274724π-0.274724\pi
0.650106 + 0.759843i 0.274724π0.274724\pi
864864 29.0718 0.989043
865865 −20.7846 −0.706698
866866 6.58846 0.223885
867867 −32.3238 −1.09778
868868 −5.37945 −0.182591
869869 83.5692 2.83489
870870 0 0
871871 −16.2872 −0.551870
872872 −2.82843 −0.0957826
873873 7.34847 0.248708
874874 −18.0000 −0.608859
875875 0.732051 0.0247478
876876 −2.78461 −0.0940832
877877 −24.7846 −0.836917 −0.418458 0.908236i 0.637429π-0.637429\pi
−0.418458 + 0.908236i 0.637429π0.637429\pi
878878 −4.62158 −0.155971
879879 −16.5359 −0.557742
880880 −13.0053 −0.438409
881881 30.6322 1.03203 0.516013 0.856581i 0.327416π-0.327416\pi
0.516013 + 0.856581i 0.327416π0.327416\pi
882882 3.34607 0.112668
883883 22.5885 0.760162 0.380081 0.924953i 0.375896π-0.375896\pi
0.380081 + 0.924953i 0.375896π0.375896\pi
884884 −16.0096 −0.538462
885885 −8.48528 −0.285230
886886 −7.66025 −0.257351
887887 21.0101 0.705451 0.352726 0.935727i 0.385255π-0.385255\pi
0.352726 + 0.935727i 0.385255π0.385255\pi
888888 11.5911 0.388972
889889 −9.31749 −0.312498
890890 1.07180 0.0359267
891891 26.3896 0.884084
892892 16.7321 0.560230
893893 35.5692 1.19028
894894 −13.1769 −0.440702
895895 −16.3923 −0.547934
896896 −8.38375 −0.280081
897897 16.9706 0.566631
898898 15.8564 0.529135
899899 0 0
900900 1.73205 0.0577350
901901 0 0
902902 23.9401 0.797118
903903 −4.39230 −0.146167
904904 −26.5885 −0.884319
905905 −5.85641 −0.194674
906906 13.4641 0.447315
907907 35.9101 1.19238 0.596188 0.802845i 0.296681π-0.296681\pi
0.596188 + 0.802845i 0.296681π0.296681\pi
908908 27.3731 0.908407
909909 4.14110 0.137352
910910 0.554803 0.0183915
911911 −21.0101 −0.696097 −0.348048 0.937477i 0.613156π-0.613156\pi
−0.348048 + 0.937477i 0.613156π0.613156\pi
912912 −14.7846 −0.489567
913913 −11.5911 −0.383610
914914 −13.9391 −0.461063
915915 4.39230 0.145205
916916 44.0908 1.45680
917917 1.59008 0.0525090
918918 18.4863 0.610139
919919 −23.8564 −0.786950 −0.393475 0.919335i 0.628727π-0.628727\pi
−0.393475 + 0.919335i 0.628727π0.628727\pi
920920 15.8338 0.522023
921921 35.5692 1.17205
922922 −11.7128 −0.385741
923923 −8.78461 −0.289149
924924 −9.46410 −0.311346
925925 4.24264 0.139497
926926 11.2122 0.368455
927927 0.196152 0.00644249
928928 0 0
929929 18.0000 0.590561 0.295280 0.955411i 0.404587π-0.404587\pi
0.295280 + 0.955411i 0.404587π0.404587\pi
930930 −3.10583 −0.101844
931931 −27.4249 −0.898814
932932 −38.7846 −1.27043
933933 −0.143594 −0.00470104
934934 −11.2679 −0.368699
935935 −33.3205 −1.08970
936936 2.82843 0.0924500
937937 46.2487 1.51088 0.755440 0.655218i 0.227423π-0.227423\pi
0.755440 + 0.655218i 0.227423π0.227423\pi
938938 −4.21543 −0.137639
939939 36.0117 1.17520
940940 −14.5211 −0.473625
941941 18.0000 0.586783 0.293392 0.955992i 0.405216π-0.405216\pi
0.293392 + 0.955992i 0.405216π0.405216\pi
942942 −5.37945 −0.175272
943943 71.8203 2.33879
944944 14.7846 0.481198
945945 4.14110 0.134710
946946 −11.5911 −0.376859
947947 −48.5365 −1.57723 −0.788613 0.614890i 0.789200π-0.789200\pi
−0.788613 + 0.614890i 0.789200π0.789200\pi
948948 −38.7846 −1.25967
949949 −1.66441 −0.0540290
950950 2.19615 0.0712526
951951 29.7128 0.963504
952952 −8.92820 −0.289365
953953 −47.5692 −1.54092 −0.770459 0.637489i 0.779973π-0.779973\pi
−0.770459 + 0.637489i 0.779973π0.779973\pi
954954 0 0
955955 19.2170 0.621847
956956 −10.3923 −0.336111
957957 0 0
958958 −2.33975 −0.0755938
959959 7.24693 0.234016
960960 3.20736 0.103517
961961 −13.0000 −0.419355
962962 3.21539 0.103668
963963 −2.19615 −0.0707700
964964 15.4641 0.498065
965965 −12.7279 −0.409726
966966 4.39230 0.141320
967967 59.0924 1.90028 0.950141 0.311820i 0.100939π-0.100939\pi
0.950141 + 0.311820i 0.100939π0.100939\pi
968968 −32.5641 −1.04665
969969 −37.8792 −1.21685
970970 −3.80385 −0.122134
971971 −36.4649 −1.17022 −0.585108 0.810955i 0.698948π-0.698948\pi
−0.585108 + 0.810955i 0.698948π0.698948\pi
972972 17.1464 0.549972
973973 −9.85641 −0.315982
974974 −11.6926 −0.374657
975975 −2.07055 −0.0663107
976976 −7.65308 −0.244969
977977 48.0000 1.53566 0.767828 0.640656i 0.221338π-0.221338\pi
0.767828 + 0.640656i 0.221338π0.221338\pi
978978 −3.10583 −0.0993134
979979 −10.9282 −0.349267
980980 11.1962 0.357648
981981 −1.46410 −0.0467452
982982 −0.732051 −0.0233607
983983 0.859411 0.0274109 0.0137055 0.999906i 0.495637π-0.495637\pi
0.0137055 + 0.999906i 0.495637π0.495637\pi
984984 −23.9401 −0.763182
985985 −1.60770 −0.0512254
986986 0 0
987987 −8.67949 −0.276271
988988 −10.7589 −0.342286
989989 −34.7733 −1.10573
990990 2.73205 0.0868303
991991 −61.3205 −1.94791 −0.973955 0.226741i 0.927193π-0.927193\pi
−0.973955 + 0.226741i 0.927193π0.927193\pi
992992 21.8038 0.692273
993993 −26.7846 −0.849984
994994 −2.27362 −0.0721150
995995 −19.4641 −0.617054
996996 5.37945 0.170454
997997 −30.5307 −0.966917 −0.483458 0.875367i 0.660620π-0.660620\pi
−0.483458 + 0.875367i 0.660620π0.660620\pi
998998 3.03150 0.0959604
999999 24.0000 0.759326
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 4205.2.a.g.1.3 4
29.12 odd 4 145.2.c.a.86.3 yes 4
29.17 odd 4 145.2.c.a.86.2 4
29.28 even 2 inner 4205.2.a.g.1.2 4
87.17 even 4 1305.2.d.a.811.3 4
87.41 even 4 1305.2.d.a.811.2 4
116.75 even 4 2320.2.g.e.1681.1 4
116.99 even 4 2320.2.g.e.1681.3 4
145.12 even 4 725.2.d.b.724.4 8
145.17 even 4 725.2.d.b.724.6 8
145.99 odd 4 725.2.c.d.376.2 4
145.104 odd 4 725.2.c.d.376.3 4
145.128 even 4 725.2.d.b.724.5 8
145.133 even 4 725.2.d.b.724.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.c.a.86.2 4 29.17 odd 4
145.2.c.a.86.3 yes 4 29.12 odd 4
725.2.c.d.376.2 4 145.99 odd 4
725.2.c.d.376.3 4 145.104 odd 4
725.2.d.b.724.3 8 145.133 even 4
725.2.d.b.724.4 8 145.12 even 4
725.2.d.b.724.5 8 145.128 even 4
725.2.d.b.724.6 8 145.17 even 4
1305.2.d.a.811.2 4 87.41 even 4
1305.2.d.a.811.3 4 87.17 even 4
2320.2.g.e.1681.1 4 116.75 even 4
2320.2.g.e.1681.3 4 116.99 even 4
4205.2.a.g.1.2 4 29.28 even 2 inner
4205.2.a.g.1.3 4 1.1 even 1 trivial