Properties

Label 686.2.a.c.1.2
Level 686686
Weight 22
Character 686.1
Self dual yes
Analytic conductor 5.4785.478
Analytic rank 11
Dimension 33
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] = [686,2,Mod(1,686)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(686, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("686.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 686=273 686 = 2 \cdot 7^{3}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 686.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: 5.477737578665.47773757866
Analytic rank: 11
Dimension: 33
Coefficient field: Q(ζ14)+\Q(\zeta_{14})^+
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x3x22x+1 x^{3} - x^{2} - 2x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.2
Root 1.24698-1.24698 of defining polynomial
Character χ\chi == 686.1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q+1.00000q21.55496q3+1.00000q41.44504q51.55496q6+1.00000q80.582105q91.44504q100.307979q111.55496q12+0.603875q13+2.24698q15+1.00000q165.54288q170.582105q183.89008q191.44504q200.307979q222.91185q231.55496q242.91185q25+0.603875q26+5.57002q270.594187q29+2.24698q308.07069q31+1.00000q32+0.478894q335.54288q340.582105q367.04892q373.89008q380.939001q391.44504q40+8.87800q4110.9487q430.307979q44+0.841166q452.91185q46+7.36658q471.55496q482.91185q50+8.61894q51+0.603875q52+10.9487q53+5.57002q54+0.445042q55+6.04892q570.594187q5811.6407q59+2.24698q601.52781q618.07069q62+1.00000q640.872625q65+0.478894q66+1.65279q675.54288q68+4.52781q699.44265q710.582105q72+13.1685q737.04892q74+4.52781q753.89008q760.939001q78+10.0368q791.44504q806.91484q81+8.87800q82+2.44504q83+8.00969q8510.9487q86+0.923936q870.307979q88+9.35690q89+0.841166q902.91185q92+12.5496q93+7.36658q94+5.62133q951.55496q9613.1347q97+0.179276q99+O(q100)q+1.00000 q^{2} -1.55496 q^{3} +1.00000 q^{4} -1.44504 q^{5} -1.55496 q^{6} +1.00000 q^{8} -0.582105 q^{9} -1.44504 q^{10} -0.307979 q^{11} -1.55496 q^{12} +0.603875 q^{13} +2.24698 q^{15} +1.00000 q^{16} -5.54288 q^{17} -0.582105 q^{18} -3.89008 q^{19} -1.44504 q^{20} -0.307979 q^{22} -2.91185 q^{23} -1.55496 q^{24} -2.91185 q^{25} +0.603875 q^{26} +5.57002 q^{27} -0.594187 q^{29} +2.24698 q^{30} -8.07069 q^{31} +1.00000 q^{32} +0.478894 q^{33} -5.54288 q^{34} -0.582105 q^{36} -7.04892 q^{37} -3.89008 q^{38} -0.939001 q^{39} -1.44504 q^{40} +8.87800 q^{41} -10.9487 q^{43} -0.307979 q^{44} +0.841166 q^{45} -2.91185 q^{46} +7.36658 q^{47} -1.55496 q^{48} -2.91185 q^{50} +8.61894 q^{51} +0.603875 q^{52} +10.9487 q^{53} +5.57002 q^{54} +0.445042 q^{55} +6.04892 q^{57} -0.594187 q^{58} -11.6407 q^{59} +2.24698 q^{60} -1.52781 q^{61} -8.07069 q^{62} +1.00000 q^{64} -0.872625 q^{65} +0.478894 q^{66} +1.65279 q^{67} -5.54288 q^{68} +4.52781 q^{69} -9.44265 q^{71} -0.582105 q^{72} +13.1685 q^{73} -7.04892 q^{74} +4.52781 q^{75} -3.89008 q^{76} -0.939001 q^{78} +10.0368 q^{79} -1.44504 q^{80} -6.91484 q^{81} +8.87800 q^{82} +2.44504 q^{83} +8.00969 q^{85} -10.9487 q^{86} +0.923936 q^{87} -0.307979 q^{88} +9.35690 q^{89} +0.841166 q^{90} -2.91185 q^{92} +12.5496 q^{93} +7.36658 q^{94} +5.62133 q^{95} -1.55496 q^{96} -13.1347 q^{97} +0.179276 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 3q+3q25q3+3q44q55q6+3q8+4q94q106q115q127q13+2q15+3q16+2q17+4q1811q194q206q225q23+36q99+O(q100) 3 q + 3 q^{2} - 5 q^{3} + 3 q^{4} - 4 q^{5} - 5 q^{6} + 3 q^{8} + 4 q^{9} - 4 q^{10} - 6 q^{11} - 5 q^{12} - 7 q^{13} + 2 q^{15} + 3 q^{16} + 2 q^{17} + 4 q^{18} - 11 q^{19} - 4 q^{20} - 6 q^{22} - 5 q^{23}+ \cdots - 36 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 1.00000 0.707107
33 −1.55496 −0.897755 −0.448878 0.893593i 0.648176π-0.648176\pi
−0.448878 + 0.893593i 0.648176π0.648176\pi
44 1.00000 0.500000
55 −1.44504 −0.646242 −0.323121 0.946358i 0.604732π-0.604732\pi
−0.323121 + 0.946358i 0.604732π0.604732\pi
66 −1.55496 −0.634809
77 0 0
88 1.00000 0.353553
99 −0.582105 −0.194035
1010 −1.44504 −0.456962
1111 −0.307979 −0.0928590 −0.0464295 0.998922i 0.514784π-0.514784\pi
−0.0464295 + 0.998922i 0.514784π0.514784\pi
1212 −1.55496 −0.448878
1313 0.603875 0.167485 0.0837425 0.996487i 0.473313π-0.473313\pi
0.0837425 + 0.996487i 0.473313π0.473313\pi
1414 0 0
1515 2.24698 0.580168
1616 1.00000 0.250000
1717 −5.54288 −1.34435 −0.672173 0.740395i 0.734639π-0.734639\pi
−0.672173 + 0.740395i 0.734639π0.734639\pi
1818 −0.582105 −0.137204
1919 −3.89008 −0.892446 −0.446223 0.894922i 0.647231π-0.647231\pi
−0.446223 + 0.894922i 0.647231π0.647231\pi
2020 −1.44504 −0.323121
2121 0 0
2222 −0.307979 −0.0656612
2323 −2.91185 −0.607164 −0.303582 0.952805i 0.598183π-0.598183\pi
−0.303582 + 0.952805i 0.598183π0.598183\pi
2424 −1.55496 −0.317404
2525 −2.91185 −0.582371
2626 0.603875 0.118430
2727 5.57002 1.07195
2828 0 0
2929 −0.594187 −0.110338 −0.0551689 0.998477i 0.517570π-0.517570\pi
−0.0551689 + 0.998477i 0.517570π0.517570\pi
3030 2.24698 0.410240
3131 −8.07069 −1.44954 −0.724769 0.688992i 0.758054π-0.758054\pi
−0.724769 + 0.688992i 0.758054π0.758054\pi
3232 1.00000 0.176777
3333 0.478894 0.0833647
3434 −5.54288 −0.950595
3535 0 0
3636 −0.582105 −0.0970175
3737 −7.04892 −1.15883 −0.579417 0.815031i 0.696720π-0.696720\pi
−0.579417 + 0.815031i 0.696720π0.696720\pi
3838 −3.89008 −0.631055
3939 −0.939001 −0.150361
4040 −1.44504 −0.228481
4141 8.87800 1.38651 0.693255 0.720692i 0.256176π-0.256176\pi
0.693255 + 0.720692i 0.256176π0.256176\pi
4242 0 0
4343 −10.9487 −1.66966 −0.834830 0.550508i 0.814434π-0.814434\pi
−0.834830 + 0.550508i 0.814434π0.814434\pi
4444 −0.307979 −0.0464295
4545 0.841166 0.125394
4646 −2.91185 −0.429329
4747 7.36658 1.07453 0.537263 0.843415i 0.319458π-0.319458\pi
0.537263 + 0.843415i 0.319458π0.319458\pi
4848 −1.55496 −0.224439
4949 0 0
5050 −2.91185 −0.411798
5151 8.61894 1.20689
5252 0.603875 0.0837425
5353 10.9487 1.50392 0.751959 0.659210i 0.229109π-0.229109\pi
0.751959 + 0.659210i 0.229109π0.229109\pi
5454 5.57002 0.757984
5555 0.445042 0.0600094
5656 0 0
5757 6.04892 0.801199
5858 −0.594187 −0.0780205
5959 −11.6407 −1.51549 −0.757746 0.652550i 0.773699π-0.773699\pi
−0.757746 + 0.652550i 0.773699π0.773699\pi
6060 2.24698 0.290084
6161 −1.52781 −0.195616 −0.0978081 0.995205i 0.531183π-0.531183\pi
−0.0978081 + 0.995205i 0.531183π0.531183\pi
6262 −8.07069 −1.02498
6363 0 0
6464 1.00000 0.125000
6565 −0.872625 −0.108236
6666 0.478894 0.0589477
6767 1.65279 0.201921 0.100960 0.994890i 0.467808π-0.467808\pi
0.100960 + 0.994890i 0.467808π0.467808\pi
6868 −5.54288 −0.672173
6969 4.52781 0.545084
7070 0 0
7171 −9.44265 −1.12064 −0.560318 0.828277i 0.689321π-0.689321\pi
−0.560318 + 0.828277i 0.689321π0.689321\pi
7272 −0.582105 −0.0686018
7373 13.1685 1.54126 0.770629 0.637283i 0.219942π-0.219942\pi
0.770629 + 0.637283i 0.219942π0.219942\pi
7474 −7.04892 −0.819420
7575 4.52781 0.522827
7676 −3.89008 −0.446223
7777 0 0
7878 −0.939001 −0.106321
7979 10.0368 1.12923 0.564616 0.825354i 0.309024π-0.309024\pi
0.564616 + 0.825354i 0.309024π0.309024\pi
8080 −1.44504 −0.161561
8181 −6.91484 −0.768315
8282 8.87800 0.980411
8383 2.44504 0.268378 0.134189 0.990956i 0.457157π-0.457157\pi
0.134189 + 0.990956i 0.457157π0.457157\pi
8484 0 0
8585 8.00969 0.868773
8686 −10.9487 −1.18063
8787 0.923936 0.0990563
8888 −0.307979 −0.0328306
8989 9.35690 0.991829 0.495914 0.868371i 0.334833π-0.334833\pi
0.495914 + 0.868371i 0.334833π0.334833\pi
9090 0.841166 0.0886667
9191 0 0
9292 −2.91185 −0.303582
9393 12.5496 1.30133
9494 7.36658 0.759805
9595 5.62133 0.576737
9696 −1.55496 −0.158702
9797 −13.1347 −1.33362 −0.666812 0.745226i 0.732342π-0.732342\pi
−0.666812 + 0.745226i 0.732342π0.732342\pi
9898 0 0
9999 0.179276 0.0180179
100100 −2.91185 −0.291185
101101 8.74632 0.870291 0.435145 0.900360i 0.356697π-0.356697\pi
0.435145 + 0.900360i 0.356697π0.356697\pi
102102 8.61894 0.853402
103103 −4.76271 −0.469284 −0.234642 0.972082i 0.575392π-0.575392\pi
−0.234642 + 0.972082i 0.575392π0.575392\pi
104104 0.603875 0.0592149
105105 0 0
106106 10.9487 1.06343
107107 −7.71917 −0.746240 −0.373120 0.927783i 0.621712π-0.621712\pi
−0.373120 + 0.927783i 0.621712π0.621712\pi
108108 5.57002 0.535976
109109 −8.58211 −0.822017 −0.411008 0.911632i 0.634823π-0.634823\pi
−0.411008 + 0.911632i 0.634823π0.634823\pi
110110 0.445042 0.0424331
111111 10.9608 1.04035
112112 0 0
113113 18.5646 1.74642 0.873208 0.487349i 0.162036π-0.162036\pi
0.873208 + 0.487349i 0.162036π0.162036\pi
114114 6.04892 0.566533
115115 4.20775 0.392375
116116 −0.594187 −0.0551689
117117 −0.351519 −0.0324979
118118 −11.6407 −1.07161
119119 0 0
120120 2.24698 0.205120
121121 −10.9051 −0.991377
122122 −1.52781 −0.138322
123123 −13.8049 −1.24475
124124 −8.07069 −0.724769
125125 11.4330 1.02260
126126 0 0
127127 1.59850 0.141844 0.0709219 0.997482i 0.477406π-0.477406\pi
0.0709219 + 0.997482i 0.477406π0.477406\pi
128128 1.00000 0.0883883
129129 17.0248 1.49895
130130 −0.872625 −0.0765343
131131 −1.78017 −0.155534 −0.0777670 0.996972i 0.524779π-0.524779\pi
−0.0777670 + 0.996972i 0.524779π0.524779\pi
132132 0.478894 0.0416823
133133 0 0
134134 1.65279 0.142780
135135 −8.04892 −0.692741
136136 −5.54288 −0.475298
137137 0.362273 0.0309510 0.0154755 0.999880i 0.495074π-0.495074\pi
0.0154755 + 0.999880i 0.495074π0.495074\pi
138138 4.52781 0.385433
139139 −9.59419 −0.813768 −0.406884 0.913480i 0.633385π-0.633385\pi
−0.406884 + 0.913480i 0.633385π0.633385\pi
140140 0 0
141141 −11.4547 −0.964662
142142 −9.44265 −0.792409
143143 −0.185981 −0.0155525
144144 −0.582105 −0.0485088
145145 0.858625 0.0713049
146146 13.1685 1.08983
147147 0 0
148148 −7.04892 −0.579417
149149 6.80194 0.557236 0.278618 0.960402i 0.410124π-0.410124\pi
0.278618 + 0.960402i 0.410124π0.410124\pi
150150 4.52781 0.369694
151151 23.4131 1.90533 0.952666 0.304019i 0.0983287π-0.0983287\pi
0.952666 + 0.304019i 0.0983287π0.0983287\pi
152152 −3.89008 −0.315527
153153 3.22654 0.260850
154154 0 0
155155 11.6625 0.936753
156156 −0.939001 −0.0751803
157157 19.2664 1.53762 0.768811 0.639476i 0.220849π-0.220849\pi
0.768811 + 0.639476i 0.220849π0.220849\pi
158158 10.0368 0.798488
159159 −17.0248 −1.35015
160160 −1.44504 −0.114241
161161 0 0
162162 −6.91484 −0.543281
163163 0.0174584 0.00136745 0.000683724 1.00000i 0.499782π-0.499782\pi
0.000683724 1.00000i 0.499782π0.499782\pi
164164 8.87800 0.693255
165165 −0.692021 −0.0538738
166166 2.44504 0.189772
167167 1.87800 0.145324 0.0726621 0.997357i 0.476851π-0.476851\pi
0.0726621 + 0.997357i 0.476851π0.476851\pi
168168 0 0
169169 −12.6353 −0.971949
170170 8.00969 0.614315
171171 2.26444 0.173166
172172 −10.9487 −0.834830
173173 −12.0683 −0.917535 −0.458768 0.888556i 0.651709π-0.651709\pi
−0.458768 + 0.888556i 0.651709π0.651709\pi
174174 0.923936 0.0700434
175175 0 0
176176 −0.307979 −0.0232148
177177 18.1008 1.36054
178178 9.35690 0.701329
179179 −24.4306 −1.82603 −0.913013 0.407930i 0.866251π-0.866251\pi
−0.913013 + 0.407930i 0.866251π0.866251\pi
180180 0.841166 0.0626968
181181 −7.85623 −0.583949 −0.291975 0.956426i 0.594312π-0.594312\pi
−0.291975 + 0.956426i 0.594312π0.594312\pi
182182 0 0
183183 2.37568 0.175615
184184 −2.91185 −0.214665
185185 10.1860 0.748888
186186 12.5496 0.920180
187187 1.70709 0.124835
188188 7.36658 0.537263
189189 0 0
190190 5.62133 0.407814
191191 5.75063 0.416101 0.208050 0.978118i 0.433288π-0.433288\pi
0.208050 + 0.978118i 0.433288π0.433288\pi
192192 −1.55496 −0.112219
193193 21.3991 1.54034 0.770171 0.637838i 0.220171π-0.220171\pi
0.770171 + 0.637838i 0.220171π0.220171\pi
194194 −13.1347 −0.943014
195195 1.35690 0.0971693
196196 0 0
197197 −13.9312 −0.992559 −0.496280 0.868163i 0.665301π-0.665301\pi
−0.496280 + 0.868163i 0.665301π0.665301\pi
198198 0.179276 0.0127406
199199 −5.05131 −0.358078 −0.179039 0.983842i 0.557299π-0.557299\pi
−0.179039 + 0.983842i 0.557299π0.557299\pi
200200 −2.91185 −0.205899
201201 −2.57002 −0.181275
202202 8.74632 0.615389
203203 0 0
204204 8.61894 0.603447
205205 −12.8291 −0.896022
206206 −4.76271 −0.331834
207207 1.69501 0.117811
208208 0.603875 0.0418712
209209 1.19806 0.0828717
210210 0 0
211211 −1.61596 −0.111247 −0.0556235 0.998452i 0.517715π-0.517715\pi
−0.0556235 + 0.998452i 0.517715π0.517715\pi
212212 10.9487 0.751959
213213 14.6829 1.00606
214214 −7.71917 −0.527671
215215 15.8213 1.07900
216216 5.57002 0.378992
217217 0 0
218218 −8.58211 −0.581254
219219 −20.4765 −1.38367
220220 0.445042 0.0300047
221221 −3.34721 −0.225158
222222 10.9608 0.735639
223223 −27.8418 −1.86442 −0.932211 0.361915i 0.882123π-0.882123\pi
−0.932211 + 0.361915i 0.882123π0.882123\pi
224224 0 0
225225 1.69501 0.113000
226226 18.5646 1.23490
227227 −18.2892 −1.21390 −0.606948 0.794741i 0.707606π-0.707606\pi
−0.606948 + 0.794741i 0.707606π0.707606\pi
228228 6.04892 0.400599
229229 −20.5375 −1.35716 −0.678578 0.734528i 0.737403π-0.737403\pi
−0.678578 + 0.734528i 0.737403π0.737403\pi
230230 4.20775 0.277451
231231 0 0
232232 −0.594187 −0.0390103
233233 −6.44504 −0.422229 −0.211114 0.977461i 0.567709π-0.567709\pi
−0.211114 + 0.977461i 0.567709π0.567709\pi
234234 −0.351519 −0.0229795
235235 −10.6450 −0.694405
236236 −11.6407 −0.757746
237237 −15.6069 −1.01377
238238 0 0
239239 −14.4155 −0.932461 −0.466231 0.884663i 0.654388π-0.654388\pi
−0.466231 + 0.884663i 0.654388π0.654388\pi
240240 2.24698 0.145042
241241 0.339437 0.0218651 0.0109325 0.999940i 0.496520π-0.496520\pi
0.0109325 + 0.999940i 0.496520π0.496520\pi
242242 −10.9051 −0.701010
243243 −5.95779 −0.382192
244244 −1.52781 −0.0978081
245245 0 0
246246 −13.8049 −0.880170
247247 −2.34913 −0.149471
248248 −8.07069 −0.512489
249249 −3.80194 −0.240938
250250 11.4330 0.723084
251251 3.78554 0.238941 0.119471 0.992838i 0.461880π-0.461880\pi
0.119471 + 0.992838i 0.461880π0.461880\pi
252252 0 0
253253 0.896789 0.0563806
254254 1.59850 0.100299
255255 −12.4547 −0.779945
256256 1.00000 0.0625000
257257 31.4523 1.96194 0.980971 0.194152i 0.0621955π-0.0621955\pi
0.980971 + 0.194152i 0.0621955π0.0621955\pi
258258 17.0248 1.05991
259259 0 0
260260 −0.872625 −0.0541179
261261 0.345879 0.0214094
262262 −1.78017 −0.109979
263263 14.8334 0.914666 0.457333 0.889295i 0.348805π-0.348805\pi
0.457333 + 0.889295i 0.348805π0.348805\pi
264264 0.478894 0.0294739
265265 −15.8213 −0.971896
266266 0 0
267267 −14.5496 −0.890420
268268 1.65279 0.100960
269269 −12.4969 −0.761952 −0.380976 0.924585i 0.624412π-0.624412\pi
−0.380976 + 0.924585i 0.624412π0.624412\pi
270270 −8.04892 −0.489842
271271 4.35450 0.264517 0.132259 0.991215i 0.457777π-0.457777\pi
0.132259 + 0.991215i 0.457777π0.457777\pi
272272 −5.54288 −0.336086
273273 0 0
274274 0.362273 0.0218857
275275 0.896789 0.0540784
276276 4.52781 0.272542
277277 −13.2403 −0.795531 −0.397766 0.917487i 0.630214π-0.630214\pi
−0.397766 + 0.917487i 0.630214π0.630214\pi
278278 −9.59419 −0.575421
279279 4.69799 0.281261
280280 0 0
281281 12.2513 0.730851 0.365425 0.930841i 0.380924π-0.380924\pi
0.365425 + 0.930841i 0.380924π0.380924\pi
282282 −11.4547 −0.682119
283283 −5.84117 −0.347221 −0.173611 0.984814i 0.555543π-0.555543\pi
−0.173611 + 0.984814i 0.555543π0.555543\pi
284284 −9.44265 −0.560318
285285 −8.74094 −0.517769
286286 −0.185981 −0.0109973
287287 0 0
288288 −0.582105 −0.0343009
289289 13.7235 0.807264
290290 0.858625 0.0504202
291291 20.4239 1.19727
292292 13.1685 0.770629
293293 4.10752 0.239964 0.119982 0.992776i 0.461716π-0.461716\pi
0.119982 + 0.992776i 0.461716π0.461716\pi
294294 0 0
295295 16.8213 0.979375
296296 −7.04892 −0.409710
297297 −1.71545 −0.0995404
298298 6.80194 0.394026
299299 −1.75840 −0.101691
300300 4.52781 0.261413
301301 0 0
302302 23.4131 1.34727
303303 −13.6002 −0.781308
304304 −3.89008 −0.223112
305305 2.20775 0.126415
306306 3.22654 0.184449
307307 −29.4010 −1.67801 −0.839003 0.544127i 0.816861π-0.816861\pi
−0.839003 + 0.544127i 0.816861π0.816861\pi
308308 0 0
309309 7.40581 0.421302
310310 11.6625 0.662384
311311 −23.0465 −1.30685 −0.653424 0.756992i 0.726668π-0.726668\pi
−0.653424 + 0.756992i 0.726668π0.726668\pi
312312 −0.939001 −0.0531605
313313 15.8931 0.898329 0.449165 0.893449i 0.351722π-0.351722\pi
0.449165 + 0.893449i 0.351722π0.351722\pi
314314 19.2664 1.08726
315315 0 0
316316 10.0368 0.564616
317317 −20.8998 −1.17385 −0.586924 0.809642i 0.699661π-0.699661\pi
−0.586924 + 0.809642i 0.699661π0.699661\pi
318318 −17.0248 −0.954701
319319 0.182997 0.0102459
320320 −1.44504 −0.0807803
321321 12.0030 0.669941
322322 0 0
323323 21.5623 1.19976
324324 −6.91484 −0.384158
325325 −1.75840 −0.0975383
326326 0.0174584 0.000966931 0
327327 13.3448 0.737970
328328 8.87800 0.490206
329329 0 0
330330 −0.692021 −0.0380945
331331 12.3123 0.676745 0.338372 0.941012i 0.390124π-0.390124\pi
0.338372 + 0.941012i 0.390124π0.390124\pi
332332 2.44504 0.134189
333333 4.10321 0.224855
334334 1.87800 0.102760
335335 −2.38835 −0.130490
336336 0 0
337337 −2.92692 −0.159439 −0.0797197 0.996817i 0.525403π-0.525403\pi
−0.0797197 + 0.996817i 0.525403π0.525403\pi
338338 −12.6353 −0.687272
339339 −28.8672 −1.56785
340340 8.00969 0.434386
341341 2.48560 0.134603
342342 2.26444 0.122447
343343 0 0
344344 −10.9487 −0.590314
345345 −6.54288 −0.352257
346346 −12.0683 −0.648796
347347 −0.262045 −0.0140673 −0.00703366 0.999975i 0.502239π-0.502239\pi
−0.00703366 + 0.999975i 0.502239π0.502239\pi
348348 0.923936 0.0495281
349349 −30.6310 −1.63964 −0.819821 0.572621i 0.805927π-0.805927\pi
−0.819821 + 0.572621i 0.805927π0.805927\pi
350350 0 0
351351 3.36360 0.179536
352352 −0.307979 −0.0164153
353353 18.5429 0.986938 0.493469 0.869763i 0.335729π-0.335729\pi
0.493469 + 0.869763i 0.335729π0.335729\pi
354354 18.1008 0.962048
355355 13.6450 0.724203
356356 9.35690 0.495914
357357 0 0
358358 −24.4306 −1.29120
359359 9.18060 0.484534 0.242267 0.970210i 0.422109π-0.422109\pi
0.242267 + 0.970210i 0.422109π0.422109\pi
360360 0.841166 0.0443334
361361 −3.86725 −0.203539
362362 −7.85623 −0.412914
363363 16.9571 0.890014
364364 0 0
365365 −19.0291 −0.996027
366366 2.37568 0.124179
367367 −24.1642 −1.26136 −0.630681 0.776042i 0.717224π-0.717224\pi
−0.630681 + 0.776042i 0.717224π0.717224\pi
368368 −2.91185 −0.151791
369369 −5.16793 −0.269032
370370 10.1860 0.529544
371371 0 0
372372 12.5496 0.650665
373373 25.9148 1.34182 0.670910 0.741539i 0.265904π-0.265904\pi
0.670910 + 0.741539i 0.265904π0.265904\pi
374374 1.70709 0.0882714
375375 −17.7778 −0.918040
376376 7.36658 0.379903
377377 −0.358815 −0.0184799
378378 0 0
379379 32.0844 1.64807 0.824033 0.566542i 0.191719π-0.191719\pi
0.824033 + 0.566542i 0.191719π0.191719\pi
380380 5.62133 0.288368
381381 −2.48560 −0.127341
382382 5.75063 0.294228
383383 −12.6280 −0.645263 −0.322631 0.946525i 0.604567π-0.604567\pi
−0.322631 + 0.946525i 0.604567π0.604567\pi
384384 −1.55496 −0.0793511
385385 0 0
386386 21.3991 1.08919
387387 6.37329 0.323973
388388 −13.1347 −0.666812
389389 0.341830 0.0173315 0.00866574 0.999962i 0.497242π-0.497242\pi
0.00866574 + 0.999962i 0.497242π0.497242\pi
390390 1.35690 0.0687091
391391 16.1400 0.816237
392392 0 0
393393 2.76809 0.139631
394394 −13.9312 −0.701845
395395 −14.5036 −0.729758
396396 0.179276 0.00900895
397397 8.78986 0.441150 0.220575 0.975370i 0.429207π-0.429207\pi
0.220575 + 0.975370i 0.429207π0.429207\pi
398398 −5.05131 −0.253199
399399 0 0
400400 −2.91185 −0.145593
401401 9.49934 0.474374 0.237187 0.971464i 0.423775π-0.423775\pi
0.237187 + 0.971464i 0.423775π0.423775\pi
402402 −2.57002 −0.128181
403403 −4.87369 −0.242776
404404 8.74632 0.435145
405405 9.99223 0.496518
406406 0 0
407407 2.17092 0.107608
408408 8.61894 0.426701
409409 −17.5526 −0.867918 −0.433959 0.900933i 0.642884π-0.642884\pi
−0.433959 + 0.900933i 0.642884π0.642884\pi
410410 −12.8291 −0.633583
411411 −0.563319 −0.0277865
412412 −4.76271 −0.234642
413413 0 0
414414 1.69501 0.0833050
415415 −3.53319 −0.173437
416416 0.603875 0.0296074
417417 14.9186 0.730565
418418 1.19806 0.0585991
419419 −19.0965 −0.932925 −0.466463 0.884541i 0.654472π-0.654472\pi
−0.466463 + 0.884541i 0.654472π0.654472\pi
420420 0 0
421421 −19.9095 −0.970328 −0.485164 0.874423i 0.661240π-0.661240\pi
−0.485164 + 0.874423i 0.661240π0.661240\pi
422422 −1.61596 −0.0786636
423423 −4.28813 −0.208496
424424 10.9487 0.531715
425425 16.1400 0.782907
426426 14.6829 0.711390
427427 0 0
428428 −7.71917 −0.373120
429429 0.289192 0.0139623
430430 15.8213 0.762972
431431 −6.21073 −0.299161 −0.149580 0.988750i 0.547792π-0.547792\pi
−0.149580 + 0.988750i 0.547792π0.547792\pi
432432 5.57002 0.267988
433433 −8.81163 −0.423460 −0.211730 0.977328i 0.567910π-0.567910\pi
−0.211730 + 0.977328i 0.567910π0.567910\pi
434434 0 0
435435 −1.33513 −0.0640144
436436 −8.58211 −0.411008
437437 11.3274 0.541861
438438 −20.4765 −0.978405
439439 −36.1672 −1.72617 −0.863083 0.505062i 0.831470π-0.831470\pi
−0.863083 + 0.505062i 0.831470π0.831470\pi
440440 0.445042 0.0212165
441441 0 0
442442 −3.34721 −0.159210
443443 20.2881 0.963918 0.481959 0.876194i 0.339925π-0.339925\pi
0.481959 + 0.876194i 0.339925π0.339925\pi
444444 10.9608 0.520175
445445 −13.5211 −0.640962
446446 −27.8418 −1.31835
447447 −10.5767 −0.500262
448448 0 0
449449 −31.3448 −1.47925 −0.739627 0.673017i 0.764998π-0.764998\pi
−0.739627 + 0.673017i 0.764998π0.764998\pi
450450 1.69501 0.0799033
451451 −2.73423 −0.128750
452452 18.5646 0.873208
453453 −36.4064 −1.71052
454454 −18.2892 −0.858354
455455 0 0
456456 6.04892 0.283267
457457 8.78315 0.410858 0.205429 0.978672i 0.434141π-0.434141\pi
0.205429 + 0.978672i 0.434141π0.434141\pi
458458 −20.5375 −0.959654
459459 −30.8740 −1.44107
460460 4.20775 0.196187
461461 16.0968 0.749701 0.374851 0.927085i 0.377694π-0.377694\pi
0.374851 + 0.927085i 0.377694π0.377694\pi
462462 0 0
463463 −1.68963 −0.0785237 −0.0392618 0.999229i 0.512501π-0.512501\pi
−0.0392618 + 0.999229i 0.512501π0.512501\pi
464464 −0.594187 −0.0275844
465465 −18.1347 −0.840975
466466 −6.44504 −0.298561
467467 33.5623 1.55308 0.776538 0.630070i 0.216974π-0.216974\pi
0.776538 + 0.630070i 0.216974π0.216974\pi
468468 −0.351519 −0.0162490
469469 0 0
470470 −10.6450 −0.491018
471471 −29.9584 −1.38041
472472 −11.6407 −0.535807
473473 3.37196 0.155043
474474 −15.6069 −0.716847
475475 11.3274 0.519735
476476 0 0
477477 −6.37329 −0.291813
478478 −14.4155 −0.659350
479479 7.94331 0.362939 0.181470 0.983397i 0.441915π-0.441915\pi
0.181470 + 0.983397i 0.441915π0.441915\pi
480480 2.24698 0.102560
481481 −4.25667 −0.194087
482482 0.339437 0.0154610
483483 0 0
484484 −10.9051 −0.495689
485485 18.9801 0.861844
486486 −5.95779 −0.270251
487487 35.8998 1.62677 0.813387 0.581723i 0.197621π-0.197621\pi
0.813387 + 0.581723i 0.197621π0.197621\pi
488488 −1.52781 −0.0691608
489489 −0.0271471 −0.00122763
490490 0 0
491491 −6.17092 −0.278490 −0.139245 0.990258i 0.544467π-0.544467\pi
−0.139245 + 0.990258i 0.544467π0.544467\pi
492492 −13.8049 −0.622374
493493 3.29350 0.148332
494494 −2.34913 −0.105692
495495 −0.259061 −0.0116439
496496 −8.07069 −0.362385
497497 0 0
498498 −3.80194 −0.170369
499499 5.19673 0.232638 0.116319 0.993212i 0.462891π-0.462891\pi
0.116319 + 0.993212i 0.462891π0.462891\pi
500500 11.4330 0.511298
501501 −2.92021 −0.130466
502502 3.78554 0.168957
503503 −11.4058 −0.508560 −0.254280 0.967131i 0.581838π-0.581838\pi
−0.254280 + 0.967131i 0.581838π0.581838\pi
504504 0 0
505505 −12.6388 −0.562419
506506 0.896789 0.0398671
507507 19.6474 0.872572
508508 1.59850 0.0709219
509509 −42.1323 −1.86748 −0.933740 0.357951i 0.883475π-0.883475\pi
−0.933740 + 0.357951i 0.883475π0.883475\pi
510510 −12.4547 −0.551505
511511 0 0
512512 1.00000 0.0441942
513513 −21.6679 −0.956659
514514 31.4523 1.38730
515515 6.88231 0.303271
516516 17.0248 0.749473
517517 −2.26875 −0.0997795
518518 0 0
519519 18.7657 0.823722
520520 −0.872625 −0.0382672
521521 21.4722 0.940714 0.470357 0.882476i 0.344125π-0.344125\pi
0.470357 + 0.882476i 0.344125π0.344125\pi
522522 0.345879 0.0151387
523523 −12.0479 −0.526816 −0.263408 0.964685i 0.584847π-0.584847\pi
−0.263408 + 0.964685i 0.584847π0.584847\pi
524524 −1.78017 −0.0777670
525525 0 0
526526 14.8334 0.646767
527527 44.7348 1.94868
528528 0.478894 0.0208412
529529 −14.5211 −0.631352
530530 −15.8213 −0.687234
531531 6.77612 0.294059
532532 0 0
533533 5.36121 0.232220
534534 −14.5496 −0.629622
535535 11.1545 0.482252
536536 1.65279 0.0713898
537537 37.9885 1.63933
538538 −12.4969 −0.538781
539539 0 0
540540 −8.04892 −0.346370
541541 −25.5375 −1.09794 −0.548971 0.835841i 0.684980π-0.684980\pi
−0.548971 + 0.835841i 0.684980π0.684980\pi
542542 4.35450 0.187042
543543 12.2161 0.524244
544544 −5.54288 −0.237649
545545 12.4015 0.531222
546546 0 0
547547 −24.1497 −1.03257 −0.516284 0.856417i 0.672685π-0.672685\pi
−0.516284 + 0.856417i 0.672685π0.672685\pi
548548 0.362273 0.0154755
549549 0.889347 0.0379564
550550 0.896789 0.0382392
551551 2.31144 0.0984705
552552 4.52781 0.192716
553553 0 0
554554 −13.2403 −0.562525
555555 −15.8388 −0.672318
556556 −9.59419 −0.406884
557557 8.36765 0.354549 0.177274 0.984161i 0.443272π-0.443272\pi
0.177274 + 0.984161i 0.443272π0.443272\pi
558558 4.69799 0.198882
559559 −6.61165 −0.279643
560560 0 0
561561 −2.65445 −0.112071
562562 12.2513 0.516790
563563 36.8568 1.55333 0.776665 0.629914i 0.216910π-0.216910\pi
0.776665 + 0.629914i 0.216910π0.216910\pi
564564 −11.4547 −0.482331
565565 −26.8267 −1.12861
566566 −5.84117 −0.245523
567567 0 0
568568 −9.44265 −0.396205
569569 −39.8135 −1.66907 −0.834535 0.550954i 0.814264π-0.814264\pi
−0.834535 + 0.550954i 0.814264π0.814264\pi
570570 −8.74094 −0.366118
571571 −12.1062 −0.506629 −0.253314 0.967384i 0.581521π-0.581521\pi
−0.253314 + 0.967384i 0.581521π0.581521\pi
572572 −0.185981 −0.00777624
573573 −8.94198 −0.373557
574574 0 0
575575 8.47889 0.353594
576576 −0.582105 −0.0242544
577577 8.17928 0.340508 0.170254 0.985400i 0.445541π-0.445541\pi
0.170254 + 0.985400i 0.445541π0.445541\pi
578578 13.7235 0.570822
579579 −33.2747 −1.38285
580580 0.858625 0.0356525
581581 0 0
582582 20.4239 0.846596
583583 −3.37196 −0.139652
584584 13.1685 0.544917
585585 0.507960 0.0210016
586586 4.10752 0.169680
587587 11.5386 0.476248 0.238124 0.971235i 0.423468π-0.423468\pi
0.238124 + 0.971235i 0.423468π0.423468\pi
588588 0 0
589589 31.3957 1.29364
590590 16.8213 0.692523
591591 21.6625 0.891075
592592 −7.04892 −0.289709
593593 38.6679 1.58790 0.793949 0.607984i 0.208022π-0.208022\pi
0.793949 + 0.607984i 0.208022π0.208022\pi
594594 −1.71545 −0.0703857
595595 0 0
596596 6.80194 0.278618
597597 7.85458 0.321466
598598 −1.75840 −0.0719062
599599 25.5894 1.04555 0.522777 0.852469i 0.324896π-0.324896\pi
0.522777 + 0.852469i 0.324896π0.324896\pi
600600 4.52781 0.184847
601601 −0.881723 −0.0359662 −0.0179831 0.999838i 0.505725π-0.505725\pi
−0.0179831 + 0.999838i 0.505725π0.505725\pi
602602 0 0
603603 −0.962099 −0.0391797
604604 23.4131 0.952666
605605 15.7584 0.640670
606606 −13.6002 −0.552469
607607 7.62804 0.309613 0.154806 0.987945i 0.450525π-0.450525\pi
0.154806 + 0.987945i 0.450525π0.450525\pi
608608 −3.89008 −0.157764
609609 0 0
610610 2.20775 0.0893892
611611 4.44850 0.179967
612612 3.22654 0.130425
613613 30.1575 1.21805 0.609025 0.793151i 0.291561π-0.291561\pi
0.609025 + 0.793151i 0.291561π0.291561\pi
614614 −29.4010 −1.18653
615615 19.9487 0.804409
616616 0 0
617617 −1.54958 −0.0623838 −0.0311919 0.999513i 0.509930π-0.509930\pi
−0.0311919 + 0.999513i 0.509930π0.509930\pi
618618 7.40581 0.297905
619619 −6.86294 −0.275845 −0.137922 0.990443i 0.544042π-0.544042\pi
−0.137922 + 0.990443i 0.544042π0.544042\pi
620620 11.6625 0.468377
621621 −16.2191 −0.650850
622622 −23.0465 −0.924081
623623 0 0
624624 −0.939001 −0.0375901
625625 −1.96184 −0.0784735
626626 15.8931 0.635215
627627 −1.86294 −0.0743985
628628 19.2664 0.768811
629629 39.0713 1.55787
630630 0 0
631631 −9.08708 −0.361751 −0.180875 0.983506i 0.557893π-0.557893\pi
−0.180875 + 0.983506i 0.557893π0.557893\pi
632632 10.0368 0.399244
633633 2.51275 0.0998727
634634 −20.8998 −0.830036
635635 −2.30990 −0.0916655
636636 −17.0248 −0.675075
637637 0 0
638638 0.182997 0.00724491
639639 5.49662 0.217443
640640 −1.44504 −0.0571203
641641 −24.0954 −0.951713 −0.475856 0.879523i 0.657862π-0.657862\pi
−0.475856 + 0.879523i 0.657862π0.657862\pi
642642 12.0030 0.473720
643643 44.6915 1.76246 0.881231 0.472685i 0.156715π-0.156715\pi
0.881231 + 0.472685i 0.156715π0.156715\pi
644644 0 0
645645 −24.6015 −0.968682
646646 21.5623 0.848356
647647 12.1558 0.477896 0.238948 0.971032i 0.423197π-0.423197\pi
0.238948 + 0.971032i 0.423197π0.423197\pi
648648 −6.91484 −0.271640
649649 3.58509 0.140727
650650 −1.75840 −0.0689700
651651 0 0
652652 0.0174584 0.000683724 0
653653 −27.2868 −1.06781 −0.533907 0.845543i 0.679277π-0.679277\pi
−0.533907 + 0.845543i 0.679277π0.679277\pi
654654 13.3448 0.521824
655655 2.57242 0.100513
656656 8.87800 0.346628
657657 −7.66547 −0.299058
658658 0 0
659659 −0.126310 −0.00492033 −0.00246016 0.999997i 0.500783π-0.500783\pi
−0.00246016 + 0.999997i 0.500783π0.500783\pi
660660 −0.692021 −0.0269369
661661 15.5603 0.605227 0.302613 0.953113i 0.402141π-0.402141\pi
0.302613 + 0.953113i 0.402141π0.402141\pi
662662 12.3123 0.478531
663663 5.20477 0.202136
664664 2.44504 0.0948860
665665 0 0
666666 4.10321 0.158996
667667 1.73019 0.0669930
668668 1.87800 0.0726621
669669 43.2928 1.67380
670670 −2.38835 −0.0922702
671671 0.470533 0.0181647
672672 0 0
673673 20.5295 0.791353 0.395676 0.918390i 0.370510π-0.370510\pi
0.395676 + 0.918390i 0.370510π0.370510\pi
674674 −2.92692 −0.112741
675675 −16.2191 −0.624273
676676 −12.6353 −0.485974
677677 38.8998 1.49504 0.747520 0.664239i 0.231244π-0.231244\pi
0.747520 + 0.664239i 0.231244π0.231244\pi
678678 −28.8672 −1.10864
679679 0 0
680680 8.00969 0.307158
681681 28.4389 1.08978
682682 2.48560 0.0951785
683683 −32.0344 −1.22576 −0.612882 0.790174i 0.709990π-0.709990\pi
−0.612882 + 0.790174i 0.709990π0.709990\pi
684684 2.26444 0.0865830
685685 −0.523499 −0.0200019
686686 0 0
687687 31.9350 1.21839
688688 −10.9487 −0.417415
689689 6.61165 0.251884
690690 −6.54288 −0.249083
691691 −3.86964 −0.147208 −0.0736040 0.997288i 0.523450π-0.523450\pi
−0.0736040 + 0.997288i 0.523450π0.523450\pi
692692 −12.0683 −0.458768
693693 0 0
694694 −0.262045 −0.00994710
695695 13.8640 0.525892
696696 0.923936 0.0350217
697697 −49.2097 −1.86395
698698 −30.6310 −1.15940
699699 10.0218 0.379058
700700 0 0
701701 38.5411 1.45568 0.727838 0.685749i 0.240525π-0.240525\pi
0.727838 + 0.685749i 0.240525π0.240525\pi
702702 3.36360 0.126951
703703 27.4209 1.03420
704704 −0.307979 −0.0116074
705705 16.5526 0.623406
706706 18.5429 0.697870
707707 0 0
708708 18.1008 0.680270
709709 −2.56033 −0.0961554 −0.0480777 0.998844i 0.515310π-0.515310\pi
−0.0480777 + 0.998844i 0.515310π0.515310\pi
710710 13.6450 0.512089
711711 −5.84249 −0.219111
712712 9.35690 0.350664
713713 23.5007 0.880107
714714 0 0
715715 0.268750 0.0100507
716716 −24.4306 −0.913013
717717 22.4155 0.837122
718718 9.18060 0.342617
719719 38.6752 1.44234 0.721170 0.692758i 0.243605π-0.243605\pi
0.721170 + 0.692758i 0.243605π0.243605\pi
720720 0.841166 0.0313484
721721 0 0
722722 −3.86725 −0.143924
723723 −0.527811 −0.0196295
724724 −7.85623 −0.291975
725725 1.73019 0.0642575
726726 16.9571 0.629335
727727 19.0597 0.706884 0.353442 0.935456i 0.385011π-0.385011\pi
0.353442 + 0.935456i 0.385011π0.385011\pi
728728 0 0
729729 30.0086 1.11143
730730 −19.0291 −0.704297
731731 60.6872 2.24460
732732 2.37568 0.0878077
733733 −30.2626 −1.11778 −0.558888 0.829243i 0.688772π-0.688772\pi
−0.558888 + 0.829243i 0.688772π0.688772\pi
734734 −24.1642 −0.891917
735735 0 0
736736 −2.91185 −0.107332
737737 −0.509025 −0.0187502
738738 −5.16793 −0.190234
739739 −6.89738 −0.253724 −0.126862 0.991920i 0.540491π-0.540491\pi
−0.126862 + 0.991920i 0.540491π0.540491\pi
740740 10.1860 0.374444
741741 3.65279 0.134189
742742 0 0
743743 0.440730 0.0161688 0.00808441 0.999967i 0.497427π-0.497427\pi
0.00808441 + 0.999967i 0.497427π0.497427\pi
744744 12.5496 0.460090
745745 −9.82908 −0.360110
746746 25.9148 0.948810
747747 −1.42327 −0.0520748
748748 1.70709 0.0624173
749749 0 0
750750 −17.7778 −0.649153
751751 −18.0664 −0.659251 −0.329626 0.944112i 0.606923π-0.606923\pi
−0.329626 + 0.944112i 0.606923π0.606923\pi
752752 7.36658 0.268632
753753 −5.88636 −0.214511
754754 −0.358815 −0.0130673
755755 −33.8329 −1.23131
756756 0 0
757757 36.8491 1.33930 0.669651 0.742676i 0.266444π-0.266444\pi
0.669651 + 0.742676i 0.266444π0.266444\pi
758758 32.0844 1.16536
759759 −1.39447 −0.0506160
760760 5.62133 0.203907
761761 −3.29350 −0.119389 −0.0596947 0.998217i 0.519013π-0.519013\pi
−0.0596947 + 0.998217i 0.519013π0.519013\pi
762762 −2.48560 −0.0900437
763763 0 0
764764 5.75063 0.208050
765765 −4.66248 −0.168572
766766 −12.6280 −0.456270
767767 −7.02954 −0.253822
768768 −1.55496 −0.0561097
769769 19.1752 0.691476 0.345738 0.938331i 0.387629π-0.387629\pi
0.345738 + 0.938331i 0.387629π0.387629\pi
770770 0 0
771771 −48.9071 −1.76135
772772 21.3991 0.770171
773773 −35.8297 −1.28870 −0.644352 0.764729i 0.722873π-0.722873\pi
−0.644352 + 0.764729i 0.722873π0.722873\pi
774774 6.37329 0.229083
775775 23.5007 0.844169
776776 −13.1347 −0.471507
777777 0 0
778778 0.341830 0.0122552
779779 −34.5362 −1.23739
780780 1.35690 0.0485847
781781 2.90813 0.104061
782782 16.1400 0.577167
783783 −3.30963 −0.118277
784784 0 0
785785 −27.8407 −0.993677
786786 2.76809 0.0987344
787787 2.63209 0.0938238 0.0469119 0.998899i 0.485062π-0.485062\pi
0.0469119 + 0.998899i 0.485062π0.485062\pi
788788 −13.9312 −0.496280
789789 −23.0653 −0.821147
790790 −14.5036 −0.516017
791791 0 0
792792 0.179276 0.00637029
793793 −0.922608 −0.0327628
794794 8.78986 0.311940
795795 24.6015 0.872525
796796 −5.05131 −0.179039
797797 20.2674 0.717909 0.358954 0.933355i 0.383133π-0.383133\pi
0.358954 + 0.933355i 0.383133π0.383133\pi
798798 0 0
799799 −40.8321 −1.44453
800800 −2.91185 −0.102950
801801 −5.44670 −0.192450
802802 9.49934 0.335433
803803 −4.05562 −0.143120
804804 −2.57002 −0.0906377
805805 0 0
806806 −4.87369 −0.171668
807807 19.4322 0.684047
808808 8.74632 0.307694
809809 40.7144 1.43144 0.715721 0.698387i 0.246098π-0.246098\pi
0.715721 + 0.698387i 0.246098π0.246098\pi
810810 9.99223 0.351091
811811 3.19806 0.112299 0.0561496 0.998422i 0.482118π-0.482118\pi
0.0561496 + 0.998422i 0.482118π0.482118\pi
812812 0 0
813813 −6.77107 −0.237472
814814 2.17092 0.0760905
815815 −0.0252281 −0.000883702 0
816816 8.61894 0.301723
817817 42.5913 1.49008
818818 −17.5526 −0.613711
819819 0 0
820820 −12.8291 −0.448011
821821 11.7616 0.410484 0.205242 0.978711i 0.434202π-0.434202\pi
0.205242 + 0.978711i 0.434202π0.434202\pi
822822 −0.563319 −0.0196480
823823 −23.7694 −0.828550 −0.414275 0.910152i 0.635965π-0.635965\pi
−0.414275 + 0.910152i 0.635965π0.635965\pi
824824 −4.76271 −0.165917
825825 −1.39447 −0.0485492
826826 0 0
827827 −14.2631 −0.495977 −0.247988 0.968763i 0.579770π-0.579770\pi
−0.247988 + 0.968763i 0.579770π0.579770\pi
828828 1.69501 0.0589055
829829 −35.7060 −1.24012 −0.620061 0.784554i 0.712892π-0.712892\pi
−0.620061 + 0.784554i 0.712892π0.712892\pi
830830 −3.53319 −0.122639
831831 20.5881 0.714192
832832 0.603875 0.0209356
833833 0 0
834834 14.9186 0.516587
835835 −2.71379 −0.0939146
836836 1.19806 0.0414358
837837 −44.9539 −1.55383
838838 −19.0965 −0.659678
839839 −4.55496 −0.157255 −0.0786273 0.996904i 0.525054π-0.525054\pi
−0.0786273 + 0.996904i 0.525054π0.525054\pi
840840 0 0
841841 −28.6469 −0.987826
842842 −19.9095 −0.686125
843843 −19.0502 −0.656125
844844 −1.61596 −0.0556235
845845 18.2586 0.628114
846846 −4.28813 −0.147429
847847 0 0
848848 10.9487 0.375980
849849 9.08277 0.311720
850850 16.1400 0.553599
851851 20.5254 0.703602
852852 14.6829 0.503029
853853 −38.7560 −1.32698 −0.663490 0.748185i 0.730926π-0.730926\pi
−0.663490 + 0.748185i 0.730926π0.730926\pi
854854 0 0
855855 −3.27221 −0.111907
856856 −7.71917 −0.263836
857857 −5.07905 −0.173497 −0.0867485 0.996230i 0.527648π-0.527648\pi
−0.0867485 + 0.996230i 0.527648π0.527648\pi
858858 0.289192 0.00987286
859859 −18.9396 −0.646211 −0.323105 0.946363i 0.604727π-0.604727\pi
−0.323105 + 0.946363i 0.604727π0.604727\pi
860860 15.8213 0.539502
861861 0 0
862862 −6.21073 −0.211538
863863 −47.6920 −1.62346 −0.811728 0.584036i 0.801473π-0.801473\pi
−0.811728 + 0.584036i 0.801473π0.801473\pi
864864 5.57002 0.189496
865865 17.4392 0.592950
866866 −8.81163 −0.299431
867867 −21.3394 −0.724725
868868 0 0
869869 −3.09113 −0.104859
870870 −1.33513 −0.0452650
871871 0.998081 0.0338187
872872 −8.58211 −0.290627
873873 7.64576 0.258770
874874 11.3274 0.383154
875875 0 0
876876 −20.4765 −0.691837
877877 −25.3752 −0.856860 −0.428430 0.903575i 0.640933π-0.640933\pi
−0.428430 + 0.903575i 0.640933π0.640933\pi
878878 −36.1672 −1.22058
879879 −6.38703 −0.215429
880880 0.445042 0.0150024
881881 4.35258 0.146642 0.0733211 0.997308i 0.476640π-0.476640\pi
0.0733211 + 0.997308i 0.476640π0.476640\pi
882882 0 0
883883 39.6088 1.33294 0.666471 0.745531i 0.267804π-0.267804\pi
0.666471 + 0.745531i 0.267804π0.267804\pi
884884 −3.34721 −0.112579
885885 −26.1564 −0.879239
886886 20.2881 0.681593
887887 −18.8619 −0.633320 −0.316660 0.948539i 0.602561π-0.602561\pi
−0.316660 + 0.948539i 0.602561π0.602561\pi
888888 10.9608 0.367819
889889 0 0
890890 −13.5211 −0.453229
891891 2.12962 0.0713450
892892 −27.8418 −0.932211
893893 −28.6566 −0.958958
894894 −10.5767 −0.353739
895895 35.3032 1.18006
896896 0 0
897897 2.73423 0.0912934
898898 −31.3448 −1.04599
899899 4.79550 0.159939
900900 1.69501 0.0565002
901901 −60.6872 −2.02178
902902 −2.73423 −0.0910400
903903 0 0
904904 18.5646 0.617451
905905 11.3526 0.377373
906906 −36.4064 −1.20952
907907 27.5502 0.914788 0.457394 0.889264i 0.348783π-0.348783\pi
0.457394 + 0.889264i 0.348783π0.348783\pi
908908 −18.2892 −0.606948
909909 −5.09128 −0.168867
910910 0 0
911911 −20.8576 −0.691042 −0.345521 0.938411i 0.612298π-0.612298\pi
−0.345521 + 0.938411i 0.612298π0.612298\pi
912912 6.04892 0.200300
913913 −0.753020 −0.0249213
914914 8.78315 0.290521
915915 −3.43296 −0.113490
916916 −20.5375 −0.678578
917917 0 0
918918 −30.8740 −1.01899
919919 38.1237 1.25758 0.628792 0.777574i 0.283550π-0.283550\pi
0.628792 + 0.777574i 0.283550π0.283550\pi
920920 4.20775 0.138725
921921 45.7174 1.50644
922922 16.0968 0.530119
923923 −5.70218 −0.187690
924924 0 0
925925 20.5254 0.674872
926926 −1.68963 −0.0555246
927927 2.77240 0.0910575
928928 −0.594187 −0.0195051
929929 −0.287273 −0.00942513 −0.00471256 0.999989i 0.501500π-0.501500\pi
−0.00471256 + 0.999989i 0.501500π0.501500\pi
930930 −18.1347 −0.594659
931931 0 0
932932 −6.44504 −0.211114
933933 35.8364 1.17323
934934 33.5623 1.09819
935935 −2.46681 −0.0806734
936936 −0.351519 −0.0114898
937937 29.3773 0.959716 0.479858 0.877346i 0.340688π-0.340688\pi
0.479858 + 0.877346i 0.340688π0.340688\pi
938938 0 0
939939 −24.7131 −0.806480
940940 −10.6450 −0.347202
941941 −25.2392 −0.822775 −0.411387 0.911461i 0.634956π-0.634956\pi
−0.411387 + 0.911461i 0.634956π0.634956\pi
942942 −29.9584 −0.976097
943943 −25.8514 −0.841839
944944 −11.6407 −0.378873
945945 0 0
946946 3.37196 0.109632
947947 −33.3773 −1.08462 −0.542309 0.840179i 0.682450π-0.682450\pi
−0.542309 + 0.840179i 0.682450π0.682450\pi
948948 −15.6069 −0.506887
949949 7.95215 0.258138
950950 11.3274 0.367508
951951 32.4983 1.05383
952952 0 0
953953 29.0683 0.941614 0.470807 0.882236i 0.343963π-0.343963\pi
0.470807 + 0.882236i 0.343963π0.343963\pi
954954 −6.37329 −0.206343
955955 −8.30990 −0.268902
956956 −14.4155 −0.466231
957957 −0.284552 −0.00919827
958958 7.94331 0.256637
959959 0 0
960960 2.24698 0.0725210
961961 34.1360 1.10116
962962 −4.25667 −0.137240
963963 4.49337 0.144797
964964 0.339437 0.0109325
965965 −30.9226 −0.995434
966966 0 0
967967 −45.7036 −1.46973 −0.734865 0.678214i 0.762754π-0.762754\pi
−0.734865 + 0.678214i 0.762754π0.762754\pi
968968 −10.9051 −0.350505
969969 −33.5284 −1.07709
970970 18.9801 0.609416
971971 30.9885 0.994469 0.497234 0.867616i 0.334349π-0.334349\pi
0.497234 + 0.867616i 0.334349π0.334349\pi
972972 −5.95779 −0.191096
973973 0 0
974974 35.8998 1.15030
975975 2.73423 0.0875656
976976 −1.52781 −0.0489040
977977 −58.6480 −1.87632 −0.938158 0.346207i 0.887469π-0.887469\pi
−0.938158 + 0.346207i 0.887469π0.887469\pi
978978 −0.0271471 −0.000868068 0
979979 −2.88172 −0.0921003
980980 0 0
981981 4.99569 0.159500
982982 −6.17092 −0.196922
983983 −1.33214 −0.0424887 −0.0212444 0.999774i 0.506763π-0.506763\pi
−0.0212444 + 0.999774i 0.506763π0.506763\pi
984984 −13.8049 −0.440085
985985 20.1312 0.641434
986986 3.29350 0.104887
987987 0 0
988988 −2.34913 −0.0747357
989989 31.8810 1.01376
990990 −0.259061 −0.00823351
991991 −56.5126 −1.79518 −0.897591 0.440829i 0.854684π-0.854684\pi
−0.897591 + 0.440829i 0.854684π0.854684\pi
992992 −8.07069 −0.256245
993993 −19.1451 −0.607551
994994 0 0
995995 7.29935 0.231405
996996 −3.80194 −0.120469
997997 −55.3032 −1.75147 −0.875735 0.482792i 0.839623π-0.839623\pi
−0.875735 + 0.482792i 0.839623π0.839623\pi
998998 5.19673 0.164500
999999 −39.2626 −1.24221
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 686.2.a.c.1.2 3
3.2 odd 2 6174.2.a.e.1.2 3
4.3 odd 2 5488.2.a.f.1.2 3
7.2 even 3 686.2.c.b.361.2 6
7.3 odd 6 686.2.c.a.667.2 6
7.4 even 3 686.2.c.b.667.2 6
7.5 odd 6 686.2.c.a.361.2 6
7.6 odd 2 686.2.a.d.1.2 yes 3
21.20 even 2 6174.2.a.c.1.2 3
28.27 even 2 5488.2.a.a.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
686.2.a.c.1.2 3 1.1 even 1 trivial
686.2.a.d.1.2 yes 3 7.6 odd 2
686.2.c.a.361.2 6 7.5 odd 6
686.2.c.a.667.2 6 7.3 odd 6
686.2.c.b.361.2 6 7.2 even 3
686.2.c.b.667.2 6 7.4 even 3
5488.2.a.a.1.2 3 28.27 even 2
5488.2.a.f.1.2 3 4.3 odd 2
6174.2.a.c.1.2 3 21.20 even 2
6174.2.a.e.1.2 3 3.2 odd 2