Properties

Label 3042.2.b.p.1351.4
Level 30423042
Weight 22
Character 3042.1351
Analytic conductor 24.29024.290
Analytic rank 00
Dimension 66
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3042,2,Mod(1351,3042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3042.1351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 3042=232132 3042 = 2 \cdot 3^{2} \cdot 13^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 3042.b (of order 22, degree 11, not minimal)

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: no
Analytic conductor: 24.290492294924.2904922949
Analytic rank: 00
Dimension: 66
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x6+5x4+6x2+1 x^{6} + 5x^{4} + 6x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 1351.4
Root 1.80194i-1.80194i of defining polynomial
Character χ\chi == 3042.1351
Dual form 3042.2.b.p.1351.3

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+1.00000iq21.00000q41.80194iq5+4.85086iq71.00000iq8+1.80194q10+6.04892iq114.85086q14+1.00000q164.89008q17+4.49396iq19+1.80194iq206.04892q226.71379q23+1.75302q254.85086iq28+3.55496q293.82908iq31+1.00000iq324.89008iq34+8.74094q353.78017iq374.49396q381.80194q405.70171iq41+3.78017q436.04892iq446.71379iq463.82371iq4716.5308q49+1.75302iq502.78986q53+10.8998q55+4.85086q56+3.55496iq588.41119iq59+0.219833q61+3.82908q621.00000q64+11.4819iq67+4.89008q68+8.74094iq70+13.5254iq710.417895iq73+3.78017q744.49396iq7629.3424q7710.5526q791.80194iq80+5.70171q823.42327iq83+8.81163iq85+3.78017iq86+6.04892q8817.3599iq89+6.71379q92+3.82371q94+8.09783q95+1.00969iq9716.5308iq98+O(q100)q+1.00000i q^{2} -1.00000 q^{4} -1.80194i q^{5} +4.85086i q^{7} -1.00000i q^{8} +1.80194 q^{10} +6.04892i q^{11} -4.85086 q^{14} +1.00000 q^{16} -4.89008 q^{17} +4.49396i q^{19} +1.80194i q^{20} -6.04892 q^{22} -6.71379 q^{23} +1.75302 q^{25} -4.85086i q^{28} +3.55496 q^{29} -3.82908i q^{31} +1.00000i q^{32} -4.89008i q^{34} +8.74094 q^{35} -3.78017i q^{37} -4.49396 q^{38} -1.80194 q^{40} -5.70171i q^{41} +3.78017 q^{43} -6.04892i q^{44} -6.71379i q^{46} -3.82371i q^{47} -16.5308 q^{49} +1.75302i q^{50} -2.78986 q^{53} +10.8998 q^{55} +4.85086 q^{56} +3.55496i q^{58} -8.41119i q^{59} +0.219833 q^{61} +3.82908 q^{62} -1.00000 q^{64} +11.4819i q^{67} +4.89008 q^{68} +8.74094i q^{70} +13.5254i q^{71} -0.417895i q^{73} +3.78017 q^{74} -4.49396i q^{76} -29.3424 q^{77} -10.5526 q^{79} -1.80194i q^{80} +5.70171 q^{82} -3.42327i q^{83} +8.81163i q^{85} +3.78017i q^{86} +6.04892 q^{88} -17.3599i q^{89} +6.71379 q^{92} +3.82371 q^{94} +8.09783 q^{95} +1.00969i q^{97} -16.5308i q^{98} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q6q4+2q102q14+6q1628q1718q2224q23+20q25+22q29+24q358q382q40+20q4324q49+30q53+20q55+2q56+4q61++12q95+O(q100) 6 q - 6 q^{4} + 2 q^{10} - 2 q^{14} + 6 q^{16} - 28 q^{17} - 18 q^{22} - 24 q^{23} + 20 q^{25} + 22 q^{29} + 24 q^{35} - 8 q^{38} - 2 q^{40} + 20 q^{43} - 24 q^{49} + 30 q^{53} + 20 q^{55} + 2 q^{56} + 4 q^{61}+ \cdots + 12 q^{95}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/3042Z)×\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times.

nn 677677 847847
χ(n)\chi(n) 11 1-1

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 1.00000i 0.707107i
33 0 0
44 −1.00000 −0.500000
55 − 1.80194i − 0.805851i −0.915233 0.402926i 0.867993π-0.867993\pi
0.915233 0.402926i 0.132007π-0.132007\pi
66 0 0
77 4.85086i 1.83345i 0.399518 + 0.916725i 0.369178π0.369178\pi
−0.399518 + 0.916725i 0.630822π0.630822\pi
88 − 1.00000i − 0.353553i
99 0 0
1010 1.80194 0.569823
1111 6.04892i 1.82382i 0.410393 + 0.911909i 0.365391π0.365391\pi
−0.410393 + 0.911909i 0.634609π0.634609\pi
1212 0 0
1313 0 0
1414 −4.85086 −1.29645
1515 0 0
1616 1.00000 0.250000
1717 −4.89008 −1.18602 −0.593010 0.805195i 0.702060π-0.702060\pi
−0.593010 + 0.805195i 0.702060π0.702060\pi
1818 0 0
1919 4.49396i 1.03098i 0.856894 + 0.515492i 0.172391π0.172391\pi
−0.856894 + 0.515492i 0.827609π0.827609\pi
2020 1.80194i 0.402926i
2121 0 0
2222 −6.04892 −1.28963
2323 −6.71379 −1.39992 −0.699961 0.714181i 0.746799π-0.746799\pi
−0.699961 + 0.714181i 0.746799π0.746799\pi
2424 0 0
2525 1.75302 0.350604
2626 0 0
2727 0 0
2828 − 4.85086i − 0.916725i
2929 3.55496 0.660139 0.330070 0.943957i 0.392928π-0.392928\pi
0.330070 + 0.943957i 0.392928π0.392928\pi
3030 0 0
3131 − 3.82908i − 0.687724i −0.939020 0.343862i 0.888265π-0.888265\pi
0.939020 0.343862i 0.111735π-0.111735\pi
3232 1.00000i 0.176777i
3333 0 0
3434 − 4.89008i − 0.838642i
3535 8.74094 1.47749
3636 0 0
3737 − 3.78017i − 0.621456i −0.950499 0.310728i 0.899427π-0.899427\pi
0.950499 0.310728i 0.100573π-0.100573\pi
3838 −4.49396 −0.729016
3939 0 0
4040 −1.80194 −0.284911
4141 − 5.70171i − 0.890458i −0.895417 0.445229i 0.853122π-0.853122\pi
0.895417 0.445229i 0.146878π-0.146878\pi
4242 0 0
4343 3.78017 0.576470 0.288235 0.957560i 0.406932π-0.406932\pi
0.288235 + 0.957560i 0.406932π0.406932\pi
4444 − 6.04892i − 0.911909i
4545 0 0
4646 − 6.71379i − 0.989895i
4747 − 3.82371i − 0.557745i −0.960328 0.278873i 0.910039π-0.910039\pi
0.960328 0.278873i 0.0899607π-0.0899607\pi
4848 0 0
4949 −16.5308 −2.36154
5050 1.75302i 0.247915i
5151 0 0
5252 0 0
5353 −2.78986 −0.383216 −0.191608 0.981472i 0.561370π-0.561370\pi
−0.191608 + 0.981472i 0.561370π0.561370\pi
5454 0 0
5555 10.8998 1.46973
5656 4.85086 0.648223
5757 0 0
5858 3.55496i 0.466789i
5959 − 8.41119i − 1.09504i −0.836791 0.547522i 0.815571π-0.815571\pi
0.836791 0.547522i 0.184429π-0.184429\pi
6060 0 0
6161 0.219833 0.0281467 0.0140733 0.999901i 0.495520π-0.495520\pi
0.0140733 + 0.999901i 0.495520π0.495520\pi
6262 3.82908 0.486294
6363 0 0
6464 −1.00000 −0.125000
6565 0 0
6666 0 0
6767 11.4819i 1.40273i 0.712800 + 0.701367i 0.247427π0.247427\pi
−0.712800 + 0.701367i 0.752573π0.752573\pi
6868 4.89008 0.593010
6969 0 0
7070 8.74094i 1.04474i
7171 13.5254i 1.60517i 0.596537 + 0.802586i 0.296543π0.296543\pi
−0.596537 + 0.802586i 0.703457π0.703457\pi
7272 0 0
7373 − 0.417895i − 0.0489109i −0.999701 0.0244554i 0.992215π-0.992215\pi
0.999701 0.0244554i 0.00778519π-0.00778519\pi
7474 3.78017 0.439436
7575 0 0
7676 − 4.49396i − 0.515492i
7777 −29.3424 −3.34388
7878 0 0
7979 −10.5526 −1.18726 −0.593628 0.804739i 0.702305π-0.702305\pi
−0.593628 + 0.804739i 0.702305π0.702305\pi
8080 − 1.80194i − 0.201463i
8181 0 0
8282 5.70171 0.629649
8383 − 3.42327i − 0.375753i −0.982193 0.187876i 0.939840π-0.939840\pi
0.982193 0.187876i 0.0601605π-0.0601605\pi
8484 0 0
8585 8.81163i 0.955755i
8686 3.78017i 0.407626i
8787 0 0
8888 6.04892 0.644817
8989 − 17.3599i − 1.84014i −0.391750 0.920072i 0.628130π-0.628130\pi
0.391750 0.920072i 0.371870π-0.371870\pi
9090 0 0
9191 0 0
9292 6.71379 0.699961
9393 0 0
9494 3.82371 0.394385
9595 8.09783 0.830820
9696 0 0
9797 1.00969i 0.102518i 0.998685 + 0.0512592i 0.0163235π0.0163235\pi
−0.998685 + 0.0512592i 0.983677π0.983677\pi
9898 − 16.5308i − 1.66986i
9999 0 0
100100 −1.75302 −0.175302
101101 −5.68664 −0.565842 −0.282921 0.959143i 0.591303π-0.591303\pi
−0.282921 + 0.959143i 0.591303π0.591303\pi
102102 0 0
103103 6.73556 0.663675 0.331837 0.943337i 0.392331π-0.392331\pi
0.331837 + 0.943337i 0.392331π0.392331\pi
104104 0 0
105105 0 0
106106 − 2.78986i − 0.270975i
107107 −0.0586060 −0.00566566 −0.00283283 0.999996i 0.500902π-0.500902\pi
−0.00283283 + 0.999996i 0.500902π0.500902\pi
108108 0 0
109109 15.5254i 1.48707i 0.668700 + 0.743533i 0.266851π0.266851\pi
−0.668700 + 0.743533i 0.733149π0.733149\pi
110110 10.8998i 1.03925i
111111 0 0
112112 4.85086i 0.458363i
113113 1.95646 0.184048 0.0920241 0.995757i 0.470666π-0.470666\pi
0.0920241 + 0.995757i 0.470666π0.470666\pi
114114 0 0
115115 12.0978i 1.12813i
116116 −3.55496 −0.330070
117117 0 0
118118 8.41119 0.774313
119119 − 23.7211i − 2.17451i
120120 0 0
121121 −25.5894 −2.32631
122122 0.219833i 0.0199027i
123123 0 0
124124 3.82908i 0.343862i
125125 − 12.1685i − 1.08839i
126126 0 0
127127 8.11960 0.720498 0.360249 0.932856i 0.382692π-0.382692\pi
0.360249 + 0.932856i 0.382692π0.382692\pi
128128 − 1.00000i − 0.0883883i
129129 0 0
130130 0 0
131131 −12.6679 −1.10680 −0.553398 0.832917i 0.686669π-0.686669\pi
−0.553398 + 0.832917i 0.686669π0.686669\pi
132132 0 0
133133 −21.7995 −1.89026
134134 −11.4819 −0.991883
135135 0 0
136136 4.89008i 0.419321i
137137 − 15.3599i − 1.31228i −0.754638 0.656142i 0.772187π-0.772187\pi
0.754638 0.656142i 0.227813π-0.227813\pi
138138 0 0
139139 −11.7995 −1.00082 −0.500412 0.865787i 0.666818π-0.666818\pi
−0.500412 + 0.865787i 0.666818π0.666818\pi
140140 −8.74094 −0.738744
141141 0 0
142142 −13.5254 −1.13503
143143 0 0
144144 0 0
145145 − 6.40581i − 0.531974i
146146 0.417895 0.0345852
147147 0 0
148148 3.78017i 0.310728i
149149 − 3.21313i − 0.263230i −0.991301 0.131615i 0.957984π-0.957984\pi
0.991301 0.131615i 0.0420162π-0.0420162\pi
150150 0 0
151151 9.15883i 0.745335i 0.927965 + 0.372668i 0.121557π0.121557\pi
−0.927965 + 0.372668i 0.878443π0.878443\pi
152152 4.49396 0.364508
153153 0 0
154154 − 29.3424i − 2.36448i
155155 −6.89977 −0.554203
156156 0 0
157157 −17.9323 −1.43115 −0.715577 0.698534i 0.753836π-0.753836\pi
−0.715577 + 0.698534i 0.753836π0.753836\pi
158158 − 10.5526i − 0.839517i
159159 0 0
160160 1.80194 0.142456
161161 − 32.5676i − 2.56669i
162162 0 0
163163 − 0.591794i − 0.0463529i −0.999731 0.0231764i 0.992622π-0.992622\pi
0.999731 0.0231764i 0.00737795π-0.00737795\pi
164164 5.70171i 0.445229i
165165 0 0
166166 3.42327 0.265697
167167 14.0000i 1.08335i 0.840587 + 0.541676i 0.182210π0.182210\pi
−0.840587 + 0.541676i 0.817790π0.817790\pi
168168 0 0
169169 0 0
170170 −8.81163 −0.675821
171171 0 0
172172 −3.78017 −0.288235
173173 0.982542 0.0747013 0.0373506 0.999302i 0.488108π-0.488108\pi
0.0373506 + 0.999302i 0.488108π0.488108\pi
174174 0 0
175175 8.50365i 0.642815i
176176 6.04892i 0.455954i
177177 0 0
178178 17.3599 1.30118
179179 −9.94331 −0.743198 −0.371599 0.928393i 0.621190π-0.621190\pi
−0.371599 + 0.928393i 0.621190π0.621190\pi
180180 0 0
181181 5.87800 0.436908 0.218454 0.975847i 0.429899π-0.429899\pi
0.218454 + 0.975847i 0.429899π0.429899\pi
182182 0 0
183183 0 0
184184 6.71379i 0.494947i
185185 −6.81163 −0.500801
186186 0 0
187187 − 29.5797i − 2.16308i
188188 3.82371i 0.278873i
189189 0 0
190190 8.09783i 0.587479i
191191 11.0422 0.798986 0.399493 0.916736i 0.369186π-0.369186\pi
0.399493 + 0.916736i 0.369186π0.369186\pi
192192 0 0
193193 20.9638i 1.50900i 0.656298 + 0.754502i 0.272122π0.272122\pi
−0.656298 + 0.754502i 0.727878π0.727878\pi
194194 −1.00969 −0.0724914
195195 0 0
196196 16.5308 1.18077
197197 15.7409i 1.12150i 0.827987 + 0.560748i 0.189486π0.189486\pi
−0.827987 + 0.560748i 0.810514π0.810514\pi
198198 0 0
199199 1.51142 0.107142 0.0535708 0.998564i 0.482940π-0.482940\pi
0.0535708 + 0.998564i 0.482940π0.482940\pi
200200 − 1.75302i − 0.123957i
201201 0 0
202202 − 5.68664i − 0.400111i
203203 17.2446i 1.21033i
204204 0 0
205205 −10.2741 −0.717576
206206 6.73556i 0.469289i
207207 0 0
208208 0 0
209209 −27.1836 −1.88033
210210 0 0
211211 5.65817 0.389524 0.194762 0.980850i 0.437606π-0.437606\pi
0.194762 + 0.980850i 0.437606π0.437606\pi
212212 2.78986 0.191608
213213 0 0
214214 − 0.0586060i − 0.00400622i
215215 − 6.81163i − 0.464549i
216216 0 0
217217 18.5743 1.26091
218218 −15.5254 −1.05151
219219 0 0
220220 −10.8998 −0.734863
221221 0 0
222222 0 0
223223 1.97584i 0.132312i 0.997809 + 0.0661559i 0.0210735π0.0210735\pi
−0.997809 + 0.0661559i 0.978927π0.978927\pi
224224 −4.85086 −0.324111
225225 0 0
226226 1.95646i 0.130142i
227227 25.0640i 1.66355i 0.555109 + 0.831777i 0.312676π0.312676\pi
−0.555109 + 0.831777i 0.687324π0.687324\pi
228228 0 0
229229 − 24.5133i − 1.61989i −0.586508 0.809943i 0.699498π-0.699498\pi
0.586508 0.809943i 0.300502π-0.300502\pi
230230 −12.0978 −0.797708
231231 0 0
232232 − 3.55496i − 0.233394i
233233 8.37196 0.548465 0.274233 0.961663i 0.411576π-0.411576\pi
0.274233 + 0.961663i 0.411576π0.411576\pi
234234 0 0
235235 −6.89008 −0.449460
236236 8.41119i 0.547522i
237237 0 0
238238 23.7211 1.53761
239239 − 14.0194i − 0.906838i −0.891297 0.453419i 0.850204π-0.850204\pi
0.891297 0.453419i 0.149796π-0.149796\pi
240240 0 0
241241 − 21.1183i − 1.36035i −0.733051 0.680174i 0.761904π-0.761904\pi
0.733051 0.680174i 0.238096π-0.238096\pi
242242 − 25.5894i − 1.64495i
243243 0 0
244244 −0.219833 −0.0140733
245245 29.7875i 1.90305i
246246 0 0
247247 0 0
248248 −3.82908 −0.243147
249249 0 0
250250 12.1685 0.769605
251251 −26.1129 −1.64823 −0.824116 0.566421i 0.808328π-0.808328\pi
−0.824116 + 0.566421i 0.808328π0.808328\pi
252252 0 0
253253 − 40.6112i − 2.55320i
254254 8.11960i 0.509469i
255255 0 0
256256 1.00000 0.0625000
257257 −16.5918 −1.03497 −0.517484 0.855693i 0.673131π-0.673131\pi
−0.517484 + 0.855693i 0.673131π0.673131\pi
258258 0 0
259259 18.3370 1.13941
260260 0 0
261261 0 0
262262 − 12.6679i − 0.782623i
263263 9.64742 0.594885 0.297443 0.954740i 0.403866π-0.403866\pi
0.297443 + 0.954740i 0.403866π0.403866\pi
264264 0 0
265265 5.02715i 0.308815i
266266 − 21.7995i − 1.33662i
267267 0 0
268268 − 11.4819i − 0.701367i
269269 10.1981 0.621787 0.310893 0.950445i 0.399372π-0.399372\pi
0.310893 + 0.950445i 0.399372π0.399372\pi
270270 0 0
271271 27.6799i 1.68144i 0.541473 + 0.840718i 0.317867π0.317867\pi
−0.541473 + 0.840718i 0.682133π0.682133\pi
272272 −4.89008 −0.296505
273273 0 0
274274 15.3599 0.927924
275275 10.6039i 0.639438i
276276 0 0
277277 −6.32842 −0.380238 −0.190119 0.981761i 0.560887π-0.560887\pi
−0.190119 + 0.981761i 0.560887π0.560887\pi
278278 − 11.7995i − 0.707690i
279279 0 0
280280 − 8.74094i − 0.522371i
281281 9.26205i 0.552527i 0.961082 + 0.276264i 0.0890963π0.0890963\pi
−0.961082 + 0.276264i 0.910904π0.910904\pi
282282 0 0
283283 −0.561663 −0.0333874 −0.0166937 0.999861i 0.505314π-0.505314\pi
−0.0166937 + 0.999861i 0.505314π0.505314\pi
284284 − 13.5254i − 0.802586i
285285 0 0
286286 0 0
287287 27.6582 1.63261
288288 0 0
289289 6.91292 0.406642
290290 6.40581 0.376162
291291 0 0
292292 0.417895i 0.0244554i
293293 11.7506i 0.686479i 0.939248 + 0.343239i 0.111524π0.111524\pi
−0.939248 + 0.343239i 0.888476π0.888476\pi
294294 0 0
295295 −15.1564 −0.882442
296296 −3.78017 −0.219718
297297 0 0
298298 3.21313 0.186131
299299 0 0
300300 0 0
301301 18.3370i 1.05693i
302302 −9.15883 −0.527032
303303 0 0
304304 4.49396i 0.257746i
305305 − 0.396125i − 0.0226820i
306306 0 0
307307 1.09054i 0.0622403i 0.999516 + 0.0311202i 0.00990746π0.00990746\pi
−0.999516 + 0.0311202i 0.990093π0.990093\pi
308308 29.3424 1.67194
309309 0 0
310310 − 6.89977i − 0.391881i
311311 −8.09783 −0.459186 −0.229593 0.973287i 0.573740π-0.573740\pi
−0.229593 + 0.973287i 0.573740π0.573740\pi
312312 0 0
313313 19.4252 1.09798 0.548988 0.835830i 0.315013π-0.315013\pi
0.548988 + 0.835830i 0.315013π0.315013\pi
314314 − 17.9323i − 1.01198i
315315 0 0
316316 10.5526 0.593628
317317 24.5719i 1.38010i 0.723763 + 0.690049i 0.242411π0.242411\pi
−0.723763 + 0.690049i 0.757589π0.757589\pi
318318 0 0
319319 21.5036i 1.20397i
320320 1.80194i 0.100731i
321321 0 0
322322 32.5676 1.81492
323323 − 21.9758i − 1.22277i
324324 0 0
325325 0 0
326326 0.591794 0.0327764
327327 0 0
328328 −5.70171 −0.314824
329329 18.5483 1.02260
330330 0 0
331331 − 25.9758i − 1.42776i −0.700267 0.713881i 0.746936π-0.746936\pi
0.700267 0.713881i 0.253064π-0.253064\pi
332332 3.42327i 0.187876i
333333 0 0
334334 −14.0000 −0.766046
335335 20.6896 1.13040
336336 0 0
337337 −1.87263 −0.102008 −0.0510042 0.998698i 0.516242π-0.516242\pi
−0.0510042 + 0.998698i 0.516242π0.516242\pi
338338 0 0
339339 0 0
340340 − 8.81163i − 0.477878i
341341 23.1618 1.25428
342342 0 0
343343 − 46.2325i − 2.49632i
344344 − 3.78017i − 0.203813i
345345 0 0
346346 0.982542i 0.0528218i
347347 −16.6853 −0.895715 −0.447857 0.894105i 0.647813π-0.647813\pi
−0.447857 + 0.894105i 0.647813π0.647813\pi
348348 0 0
349349 4.01938i 0.215152i 0.994197 + 0.107576i 0.0343090π0.0343090\pi
−0.994197 + 0.107576i 0.965691π0.965691\pi
350350 −8.50365 −0.454539
351351 0 0
352352 −6.04892 −0.322408
353353 − 21.6039i − 1.14986i −0.818203 0.574929i 0.805030π-0.805030\pi
0.818203 0.574929i 0.194970π-0.194970\pi
354354 0 0
355355 24.3720 1.29353
356356 17.3599i 0.920072i
357357 0 0
358358 − 9.94331i − 0.525520i
359359 0.835790i 0.0441113i 0.999757 + 0.0220556i 0.00702110π0.00702110\pi
−0.999757 + 0.0220556i 0.992979π0.992979\pi
360360 0 0
361361 −1.19567 −0.0629300
362362 5.87800i 0.308941i
363363 0 0
364364 0 0
365365 −0.753020 −0.0394149
366366 0 0
367367 −21.1250 −1.10272 −0.551358 0.834269i 0.685890π-0.685890\pi
−0.551358 + 0.834269i 0.685890π0.685890\pi
368368 −6.71379 −0.349981
369369 0 0
370370 − 6.81163i − 0.354120i
371371 − 13.5332i − 0.702608i
372372 0 0
373373 21.3840 1.10722 0.553612 0.832775i 0.313249π-0.313249\pi
0.553612 + 0.832775i 0.313249π0.313249\pi
374374 29.5797 1.52953
375375 0 0
376376 −3.82371 −0.197193
377377 0 0
378378 0 0
379379 − 13.5496i − 0.695995i −0.937495 0.347998i 0.886862π-0.886862\pi
0.937495 0.347998i 0.113138π-0.113138\pi
380380 −8.09783 −0.415410
381381 0 0
382382 11.0422i 0.564969i
383383 − 9.10992i − 0.465495i −0.972537 0.232747i 0.925228π-0.925228\pi
0.972537 0.232747i 0.0747716π-0.0747716\pi
384384 0 0
385385 52.8732i 2.69467i
386386 −20.9638 −1.06703
387387 0 0
388388 − 1.00969i − 0.0512592i
389389 14.8498 0.752914 0.376457 0.926434i 0.377142π-0.377142\pi
0.376457 + 0.926434i 0.377142π0.377142\pi
390390 0 0
391391 32.8310 1.66034
392392 16.5308i 0.834931i
393393 0 0
394394 −15.7409 −0.793017
395395 19.0151i 0.956752i
396396 0 0
397397 33.7453i 1.69363i 0.531891 + 0.846813i 0.321482π0.321482\pi
−0.531891 + 0.846813i 0.678518π0.678518\pi
398398 1.51142i 0.0757605i
399399 0 0
400400 1.75302 0.0876510
401401 − 19.7017i − 0.983856i −0.870636 0.491928i 0.836292π-0.836292\pi
0.870636 0.491928i 0.163708π-0.163708\pi
402402 0 0
403403 0 0
404404 5.68664 0.282921
405405 0 0
406406 −17.2446 −0.855834
407407 22.8659 1.13342
408408 0 0
409409 35.9366i 1.77695i 0.458924 + 0.888475i 0.348235π0.348235\pi
−0.458924 + 0.888475i 0.651765π0.651765\pi
410410 − 10.2741i − 0.507403i
411411 0 0
412412 −6.73556 −0.331837
413413 40.8015 2.00771
414414 0 0
415415 −6.16852 −0.302801
416416 0 0
417417 0 0
418418 − 27.1836i − 1.32959i
419419 18.7928 0.918090 0.459045 0.888413i 0.348192π-0.348192\pi
0.459045 + 0.888413i 0.348192π0.348192\pi
420420 0 0
421421 − 3.90217i − 0.190180i −0.995469 0.0950900i 0.969686π-0.969686\pi
0.995469 0.0950900i 0.0303139π-0.0303139\pi
422422 5.65817i 0.275435i
423423 0 0
424424 2.78986i 0.135487i
425425 −8.57242 −0.415823
426426 0 0
427427 1.06638i 0.0516055i
428428 0.0586060 0.00283283
429429 0 0
430430 6.81163 0.328486
431431 18.8310i 0.907058i 0.891242 + 0.453529i 0.149835π0.149835\pi
−0.891242 + 0.453529i 0.850165π0.850165\pi
432432 0 0
433433 −24.8364 −1.19356 −0.596780 0.802405i 0.703554π-0.703554\pi
−0.596780 + 0.802405i 0.703554π0.703554\pi
434434 18.5743i 0.891597i
435435 0 0
436436 − 15.5254i − 0.743533i
437437 − 30.1715i − 1.44330i
438438 0 0
439439 −22.2784 −1.06329 −0.531646 0.846967i 0.678426π-0.678426\pi
−0.531646 + 0.846967i 0.678426π0.678426\pi
440440 − 10.8998i − 0.519626i
441441 0 0
442442 0 0
443443 26.6359 1.26551 0.632756 0.774352i 0.281924π-0.281924\pi
0.632756 + 0.774352i 0.281924π0.281924\pi
444444 0 0
445445 −31.2814 −1.48288
446446 −1.97584 −0.0935586
447447 0 0
448448 − 4.85086i − 0.229181i
449449 21.8538i 1.03135i 0.856785 + 0.515673i 0.172458π0.172458\pi
−0.856785 + 0.515673i 0.827542π0.827542\pi
450450 0 0
451451 34.4892 1.62403
452452 −1.95646 −0.0920241
453453 0 0
454454 −25.0640 −1.17631
455455 0 0
456456 0 0
457457 20.7278i 0.969605i 0.874624 + 0.484803i 0.161109π0.161109\pi
−0.874624 + 0.484803i 0.838891π0.838891\pi
458458 24.5133 1.14543
459459 0 0
460460 − 12.0978i − 0.564064i
461461 10.7832i 0.502221i 0.967958 + 0.251111i 0.0807958π0.0807958\pi
−0.967958 + 0.251111i 0.919204π0.919204\pi
462462 0 0
463463 − 23.1594i − 1.07631i −0.842846 0.538155i 0.819122π-0.819122\pi
0.842846 0.538155i 0.180878π-0.180878\pi
464464 3.55496 0.165035
465465 0 0
466466 8.37196i 0.387824i
467467 −13.3515 −0.617835 −0.308917 0.951089i 0.599967π-0.599967\pi
−0.308917 + 0.951089i 0.599967π0.599967\pi
468468 0 0
469469 −55.6969 −2.57185
470470 − 6.89008i − 0.317816i
471471 0 0
472472 −8.41119 −0.387156
473473 22.8659i 1.05138i
474474 0 0
475475 7.87800i 0.361468i
476476 23.7211i 1.08725i
477477 0 0
478478 14.0194 0.641231
479479 − 4.38537i − 0.200373i −0.994969 0.100186i 0.968056π-0.968056\pi
0.994969 0.100186i 0.0319439π-0.0319439\pi
480480 0 0
481481 0 0
482482 21.1183 0.961911
483483 0 0
484484 25.5894 1.16315
485485 1.81940 0.0826145
486486 0 0
487487 − 7.33214i − 0.332251i −0.986105 0.166126i 0.946874π-0.946874\pi
0.986105 0.166126i 0.0531257π-0.0531257\pi
488488 − 0.219833i − 0.00995135i
489489 0 0
490490 −29.7875 −1.34566
491491 16.8062 0.758455 0.379228 0.925303i 0.376190π-0.376190\pi
0.379228 + 0.925303i 0.376190π0.376190\pi
492492 0 0
493493 −17.3840 −0.782938
494494 0 0
495495 0 0
496496 − 3.82908i − 0.171931i
497497 −65.6098 −2.94300
498498 0 0
499499 6.59658i 0.295303i 0.989039 + 0.147652i 0.0471715π0.0471715\pi
−0.989039 + 0.147652i 0.952829π0.952829\pi
500500 12.1685i 0.544193i
501501 0 0
502502 − 26.1129i − 1.16548i
503503 −27.8297 −1.24086 −0.620432 0.784260i 0.713043π-0.713043\pi
−0.620432 + 0.784260i 0.713043π0.713043\pi
504504 0 0
505505 10.2470i 0.455985i
506506 40.6112 1.80539
507507 0 0
508508 −8.11960 −0.360249
509509 − 4.13275i − 0.183181i −0.995797 0.0915905i 0.970805π-0.970805\pi
0.995797 0.0915905i 0.0291951π-0.0291951\pi
510510 0 0
511511 2.02715 0.0896757
512512 1.00000i 0.0441942i
513513 0 0
514514 − 16.5918i − 0.731833i
515515 − 12.1371i − 0.534823i
516516 0 0
517517 23.1293 1.01723
518518 18.3370i 0.805683i
519519 0 0
520520 0 0
521521 11.7995 0.516947 0.258474 0.966018i 0.416780π-0.416780\pi
0.258474 + 0.966018i 0.416780π0.416780\pi
522522 0 0
523523 38.2887 1.67425 0.837124 0.547013i 0.184235π-0.184235\pi
0.837124 + 0.547013i 0.184235π0.184235\pi
524524 12.6679 0.553398
525525 0 0
526526 9.64742i 0.420647i
527527 18.7245i 0.815654i
528528 0 0
529529 22.0750 0.959783
530530 −5.02715 −0.218365
531531 0 0
532532 21.7995 0.945130
533533 0 0
534534 0 0
535535 0.105604i 0.00456568i
536536 11.4819 0.495942
537537 0 0
538538 10.1981i 0.439670i
539539 − 99.9934i − 4.30702i
540540 0 0
541541 17.4142i 0.748694i 0.927289 + 0.374347i 0.122133π0.122133\pi
−0.927289 + 0.374347i 0.877867π0.877867\pi
542542 −27.6799 −1.18896
543543 0 0
544544 − 4.89008i − 0.209661i
545545 27.9758 1.19835
546546 0 0
547547 16.9444 0.724489 0.362245 0.932083i 0.382010π-0.382010\pi
0.362245 + 0.932083i 0.382010π0.382010\pi
548548 15.3599i 0.656142i
549549 0 0
550550 −10.6039 −0.452151
551551 15.9758i 0.680594i
552552 0 0
553553 − 51.1890i − 2.17678i
554554 − 6.32842i − 0.268869i
555555 0 0
556556 11.7995 0.500412
557557 11.9758i 0.507432i 0.967279 + 0.253716i 0.0816529π0.0816529\pi
−0.967279 + 0.253716i 0.918347π0.918347\pi
558558 0 0
559559 0 0
560560 8.74094 0.369372
561561 0 0
562562 −9.26205 −0.390696
563563 −42.2911 −1.78236 −0.891179 0.453652i 0.850121π-0.850121\pi
−0.891179 + 0.453652i 0.850121π0.850121\pi
564564 0 0
565565 − 3.52542i − 0.148315i
566566 − 0.561663i − 0.0236085i
567567 0 0
568568 13.5254 0.567514
569569 −8.26934 −0.346669 −0.173334 0.984863i 0.555454π-0.555454\pi
−0.173334 + 0.984863i 0.555454π0.555454\pi
570570 0 0
571571 16.8552 0.705367 0.352683 0.935743i 0.385269π-0.385269\pi
0.352683 + 0.935743i 0.385269π0.385269\pi
572572 0 0
573573 0 0
574574 27.6582i 1.15443i
575575 −11.7694 −0.490818
576576 0 0
577577 37.9995i 1.58194i 0.611854 + 0.790970i 0.290424π0.290424\pi
−0.611854 + 0.790970i 0.709576π0.709576\pi
578578 6.91292i 0.287540i
579579 0 0
580580 6.40581i 0.265987i
581581 16.6058 0.688924
582582 0 0
583583 − 16.8756i − 0.698916i
584584 −0.417895 −0.0172926
585585 0 0
586586 −11.7506 −0.485414
587587 31.4282i 1.29718i 0.761138 + 0.648590i 0.224641π0.224641\pi
−0.761138 + 0.648590i 0.775359π0.775359\pi
588588 0 0
589589 17.2078 0.709033
590590 − 15.1564i − 0.623981i
591591 0 0
592592 − 3.78017i − 0.155364i
593593 41.8866i 1.72008i 0.510229 + 0.860039i 0.329561π0.329561\pi
−0.510229 + 0.860039i 0.670439π0.670439\pi
594594 0 0
595595 −42.7439 −1.75233
596596 3.21313i 0.131615i
597597 0 0
598598 0 0
599599 −36.0194 −1.47171 −0.735856 0.677138i 0.763220π-0.763220\pi
−0.735856 + 0.677138i 0.763220π0.763220\pi
600600 0 0
601601 5.13946 0.209643 0.104821 0.994491i 0.466573π-0.466573\pi
0.104821 + 0.994491i 0.466573π0.466573\pi
602602 −18.3370 −0.747362
603603 0 0
604604 − 9.15883i − 0.372668i
605605 46.1105i 1.87466i
606606 0 0
607607 −17.6517 −0.716462 −0.358231 0.933633i 0.616620π-0.616620\pi
−0.358231 + 0.933633i 0.616620π0.616620\pi
608608 −4.49396 −0.182254
609609 0 0
610610 0.396125 0.0160386
611611 0 0
612612 0 0
613613 45.8974i 1.85378i 0.375336 + 0.926889i 0.377527π0.377527\pi
−0.375336 + 0.926889i 0.622473π0.622473\pi
614614 −1.09054 −0.0440106
615615 0 0
616616 29.3424i 1.18224i
617617 − 5.12929i − 0.206498i −0.994656 0.103249i 0.967076π-0.967076\pi
0.994656 0.103249i 0.0329238π-0.0329238\pi
618618 0 0
619619 − 43.5448i − 1.75021i −0.483930 0.875107i 0.660791π-0.660791\pi
0.483930 0.875107i 0.339209π-0.339209\pi
620620 6.89977 0.277102
621621 0 0
622622 − 8.09783i − 0.324694i
623623 84.2103 3.37381
624624 0 0
625625 −13.1618 −0.526473
626626 19.4252i 0.776387i
627627 0 0
628628 17.9323 0.715577
629629 18.4853i 0.737059i
630630 0 0
631631 30.4655i 1.21281i 0.795155 + 0.606406i 0.207389π0.207389\pi
−0.795155 + 0.606406i 0.792611π0.792611\pi
632632 10.5526i 0.419759i
633633 0 0
634634 −24.5719 −0.975877
635635 − 14.6310i − 0.580614i
636636 0 0
637637 0 0
638638 −21.5036 −0.851338
639639 0 0
640640 −1.80194 −0.0712278
641641 32.3370 1.27724 0.638618 0.769524i 0.279506π-0.279506\pi
0.638618 + 0.769524i 0.279506π0.279506\pi
642642 0 0
643643 − 1.79092i − 0.0706270i −0.999376 0.0353135i 0.988757π-0.988757\pi
0.999376 0.0353135i 0.0112430π-0.0112430\pi
644644 32.5676i 1.28334i
645645 0 0
646646 21.9758 0.864628
647647 −41.4685 −1.63029 −0.815147 0.579254i 0.803344π-0.803344\pi
−0.815147 + 0.579254i 0.803344π0.803344\pi
648648 0 0
649649 50.8786 1.99716
650650 0 0
651651 0 0
652652 0.591794i 0.0231764i
653653 −2.39181 −0.0935989 −0.0467994 0.998904i 0.514902π-0.514902\pi
−0.0467994 + 0.998904i 0.514902π0.514902\pi
654654 0 0
655655 22.8267i 0.891913i
656656 − 5.70171i − 0.222614i
657657 0 0
658658 18.5483i 0.723086i
659659 1.76032 0.0685722 0.0342861 0.999412i 0.489084π-0.489084\pi
0.0342861 + 0.999412i 0.489084π0.489084\pi
660660 0 0
661661 − 44.7138i − 1.73916i −0.493788 0.869582i 0.664388π-0.664388\pi
0.493788 0.869582i 0.335612π-0.335612\pi
662662 25.9758 1.00958
663663 0 0
664664 −3.42327 −0.132849
665665 39.2814i 1.52327i
666666 0 0
667667 −23.8672 −0.924144
668668 − 14.0000i − 0.541676i
669669 0 0
670670 20.6896i 0.799310i
671671 1.32975i 0.0513344i
672672 0 0
673673 5.84356 0.225253 0.112626 0.993637i 0.464074π-0.464074\pi
0.112626 + 0.993637i 0.464074π0.464074\pi
674674 − 1.87263i − 0.0721308i
675675 0 0
676676 0 0
677677 27.8883 1.07183 0.535917 0.844271i 0.319966π-0.319966\pi
0.535917 + 0.844271i 0.319966π0.319966\pi
678678 0 0
679679 −4.89785 −0.187962
680680 8.81163 0.337910
681681 0 0
682682 23.1618i 0.886912i
683683 28.3803i 1.08594i 0.839751 + 0.542971i 0.182701π0.182701\pi
−0.839751 + 0.542971i 0.817299π0.817299\pi
684684 0 0
685685 −27.6775 −1.05750
686686 46.2325 1.76517
687687 0 0
688688 3.78017 0.144118
689689 0 0
690690 0 0
691691 − 31.3142i − 1.19125i −0.803263 0.595624i 0.796905π-0.796905\pi
0.803263 0.595624i 0.203095π-0.203095\pi
692692 −0.982542 −0.0373506
693693 0 0
694694 − 16.6853i − 0.633366i
695695 21.2620i 0.806515i
696696 0 0
697697 27.8818i 1.05610i
698698 −4.01938 −0.152136
699699 0 0
700700 − 8.50365i − 0.321408i
701701 22.6568 0.855737 0.427869 0.903841i 0.359265π-0.359265\pi
0.427869 + 0.903841i 0.359265π0.359265\pi
702702 0 0
703703 16.9879 0.640711
704704 − 6.04892i − 0.227977i
705705 0 0
706706 21.6039 0.813073
707707 − 27.5851i − 1.03744i
708708 0 0
709709 − 24.4155i − 0.916943i −0.888709 0.458472i 0.848397π-0.848397\pi
0.888709 0.458472i 0.151603π-0.151603\pi
710710 24.3720i 0.914663i
711711 0 0
712712 −17.3599 −0.650589
713713 25.7077i 0.962760i
714714 0 0
715715 0 0
716716 9.94331 0.371599
717717 0 0
718718 −0.835790 −0.0311914
719719 −13.2707 −0.494912 −0.247456 0.968899i 0.579595π-0.579595\pi
−0.247456 + 0.968899i 0.579595π0.579595\pi
720720 0 0
721721 32.6732i 1.21681i
722722 − 1.19567i − 0.0444982i
723723 0 0
724724 −5.87800 −0.218454
725725 6.23191 0.231447
726726 0 0
727727 −25.7904 −0.956515 −0.478257 0.878220i 0.658731π-0.658731\pi
−0.478257 + 0.878220i 0.658731π0.658731\pi
728728 0 0
729729 0 0
730730 − 0.753020i − 0.0278705i
731731 −18.4853 −0.683705
732732 0 0
733733 − 13.1400i − 0.485339i −0.970109 0.242669i 0.921977π-0.921977\pi
0.970109 0.242669i 0.0780230π-0.0780230\pi
734734 − 21.1250i − 0.779737i
735735 0 0
736736 − 6.71379i − 0.247474i
737737 −69.4529 −2.55833
738738 0 0
739739 11.8130i 0.434547i 0.976111 + 0.217273i 0.0697163π0.0697163\pi
−0.976111 + 0.217273i 0.930284π0.930284\pi
740740 6.81163 0.250400
741741 0 0
742742 13.5332 0.496819
743743 3.50125i 0.128449i 0.997935 + 0.0642243i 0.0204573π0.0204573\pi
−0.997935 + 0.0642243i 0.979543π0.979543\pi
744744 0 0
745745 −5.78986 −0.212124
746746 21.3840i 0.782925i
747747 0 0
748748 29.5797i 1.08154i
749749 − 0.284289i − 0.0103877i
750750 0 0
751751 11.6722 0.425924 0.212962 0.977061i 0.431689π-0.431689\pi
0.212962 + 0.977061i 0.431689π0.431689\pi
752752 − 3.82371i − 0.139436i
753753 0 0
754754 0 0
755755 16.5036 0.600629
756756 0 0
757757 0.444451 0.0161538 0.00807692 0.999967i 0.497429π-0.497429\pi
0.00807692 + 0.999967i 0.497429π0.497429\pi
758758 13.5496 0.492143
759759 0 0
760760 − 8.09783i − 0.293739i
761761 − 29.4905i − 1.06903i −0.845159 0.534515i 0.820494π-0.820494\pi
0.845159 0.534515i 0.179506π-0.179506\pi
762762 0 0
763763 −75.3116 −2.72646
764764 −11.0422 −0.399493
765765 0 0
766766 9.10992 0.329155
767767 0 0
768768 0 0
769769 13.4517i 0.485082i 0.970141 + 0.242541i 0.0779810π0.0779810\pi
−0.970141 + 0.242541i 0.922019π0.922019\pi
770770 −52.8732 −1.90542
771771 0 0
772772 − 20.9638i − 0.754502i
773773 − 24.1021i − 0.866894i −0.901179 0.433447i 0.857297π-0.857297\pi
0.901179 0.433447i 0.142703π-0.142703\pi
774774 0 0
775775 − 6.71246i − 0.241119i
776776 1.00969 0.0362457
777777 0 0
778778 14.8498i 0.532391i
779779 25.6233 0.918048
780780 0 0
781781 −81.8141 −2.92754
782782 32.8310i 1.17403i
783783 0 0
784784 −16.5308 −0.590386
785785 32.3129i 1.15330i
786786 0 0
787787 16.2258i 0.578387i 0.957271 + 0.289194i 0.0933872π0.0933872\pi
−0.957271 + 0.289194i 0.906613π0.906613\pi
788788 − 15.7409i − 0.560748i
789789 0 0
790790 −19.0151 −0.676526
791791 9.49050i 0.337443i
792792 0 0
793793 0 0
794794 −33.7453 −1.19757
795795 0 0
796796 −1.51142 −0.0535708
797797 −37.2760 −1.32039 −0.660193 0.751096i 0.729525π-0.729525\pi
−0.660193 + 0.751096i 0.729525π0.729525\pi
798798 0 0
799799 18.6983i 0.661497i
800800 1.75302i 0.0619786i
801801 0 0
802802 19.7017 0.695692
803803 2.52781 0.0892045
804804 0 0
805805 −58.6848 −2.06837
806806 0 0
807807 0 0
808808 5.68664i 0.200055i
809809 24.0844 0.846763 0.423382 0.905951i 0.360843π-0.360843\pi
0.423382 + 0.905951i 0.360843π0.360843\pi
810810 0 0
811811 − 9.87800i − 0.346864i −0.984846 0.173432i 0.944514π-0.944514\pi
0.984846 0.173432i 0.0554856π-0.0554856\pi
812812 − 17.2446i − 0.605166i
813813 0 0
814814 22.8659i 0.801450i
815815 −1.06638 −0.0373535
816816 0 0
817817 16.9879i 0.594332i
818818 −35.9366 −1.25649
819819 0 0
820820 10.2741 0.358788
821821 − 31.2731i − 1.09144i −0.837968 0.545719i 0.816257π-0.816257\pi
0.837968 0.545719i 0.183743π-0.183743\pi
822822 0 0
823823 −33.8388 −1.17955 −0.589773 0.807569i 0.700783π-0.700783\pi
−0.589773 + 0.807569i 0.700783π0.700783\pi
824824 − 6.73556i − 0.234644i
825825 0 0
826826 40.8015i 1.41966i
827827 17.4028i 0.605156i 0.953125 + 0.302578i 0.0978472π0.0978472\pi
−0.953125 + 0.302578i 0.902153π0.902153\pi
828828 0 0
829829 50.9724 1.77034 0.885172 0.465264i 0.154041π-0.154041\pi
0.885172 + 0.465264i 0.154041π0.154041\pi
830830 − 6.16852i − 0.214113i
831831 0 0
832832 0 0
833833 80.8370 2.80084
834834 0 0
835835 25.2271 0.873021
836836 27.1836 0.940164
837837 0 0
838838 18.7928i 0.649188i
839839 34.6983i 1.19792i 0.800780 + 0.598958i 0.204419π0.204419\pi
−0.800780 + 0.598958i 0.795581π0.795581\pi
840840 0 0
841841 −16.3623 −0.564216
842842 3.90217 0.134477
843843 0 0
844844 −5.65817 −0.194762
845845 0 0
846846 0 0
847847 − 124.130i − 4.26517i
848848 −2.78986 −0.0958041
849849 0 0
850850 − 8.57242i − 0.294031i
851851 25.3793i 0.869990i
852852 0 0
853853 − 8.37675i − 0.286814i −0.989664 0.143407i 0.954194π-0.954194\pi
0.989664 0.143407i 0.0458059π-0.0458059\pi
854854 −1.06638 −0.0364906
855855 0 0
856856 0.0586060i 0.00200311i
857857 17.7888 0.607654 0.303827 0.952727i 0.401736π-0.401736\pi
0.303827 + 0.952727i 0.401736π0.401736\pi
858858 0 0
859859 50.1473 1.71101 0.855503 0.517798i 0.173248π-0.173248\pi
0.855503 + 0.517798i 0.173248π0.173248\pi
860860 6.81163i 0.232275i
861861 0 0
862862 −18.8310 −0.641387
863863 50.1172i 1.70601i 0.521903 + 0.853005i 0.325222π0.325222\pi
−0.521903 + 0.853005i 0.674778π0.674778\pi
864864 0 0
865865 − 1.77048i − 0.0601981i
866866 − 24.8364i − 0.843975i
867867 0 0
868868 −18.5743 −0.630454
869869 − 63.8316i − 2.16534i
870870 0 0
871871 0 0
872872 15.5254 0.525757
873873 0 0
874874 30.1715 1.02057
875875 59.0277 1.99550
876876 0 0
877877 38.3827i 1.29609i 0.761601 + 0.648046i 0.224414π0.224414\pi
−0.761601 + 0.648046i 0.775586π0.775586\pi
878878 − 22.2784i − 0.751861i
879879 0 0
880880 10.8998 0.367431
881881 31.8383 1.07266 0.536330 0.844009i 0.319810π-0.319810\pi
0.536330 + 0.844009i 0.319810π0.319810\pi
882882 0 0
883883 −4.74871 −0.159807 −0.0799034 0.996803i 0.525461π-0.525461\pi
−0.0799034 + 0.996803i 0.525461π0.525461\pi
884884 0 0
885885 0 0
886886 26.6359i 0.894851i
887887 −22.9288 −0.769875 −0.384938 0.922943i 0.625777π-0.625777\pi
−0.384938 + 0.922943i 0.625777π0.625777\pi
888888 0 0
889889 39.3870i 1.32100i
890890 − 31.2814i − 1.04856i
891891 0 0
892892 − 1.97584i − 0.0661559i
893893 17.1836 0.575027
894894 0 0
895895 17.9172i 0.598907i
896896 4.85086 0.162056
897897 0 0
898898 −21.8538 −0.729272
899899 − 13.6122i − 0.453993i
900900 0 0
901901 13.6426 0.454502
902902 34.4892i 1.14836i
903903 0 0
904904 − 1.95646i − 0.0650709i
905905 − 10.5918i − 0.352083i
906906 0 0
907907 −43.0267 −1.42868 −0.714339 0.699800i 0.753272π-0.753272\pi
−0.714339 + 0.699800i 0.753272π0.753272\pi
908908 − 25.0640i − 0.831777i
909909 0 0
910910 0 0
911911 −6.65087 −0.220353 −0.110177 0.993912i 0.535142π-0.535142\pi
−0.110177 + 0.993912i 0.535142π0.535142\pi
912912 0 0
913913 20.7071 0.685305
914914 −20.7278 −0.685614
915915 0 0
916916 24.5133i 0.809943i
917917 − 61.4499i − 2.02926i
918918 0 0
919919 −9.11231 −0.300587 −0.150294 0.988641i 0.548022π-0.548022\pi
−0.150294 + 0.988641i 0.548022π0.548022\pi
920920 12.0978 0.398854
921921 0 0
922922 −10.7832 −0.355124
923923 0 0
924924 0 0
925925 − 6.62671i − 0.217885i
926926 23.1594 0.761066
927927 0 0
928928 3.55496i 0.116697i
929929 11.8672i 0.389352i 0.980868 + 0.194676i 0.0623655π0.0623655\pi
−0.980868 + 0.194676i 0.937634π0.937634\pi
930930 0 0
931931 − 74.2887i − 2.43471i
932932 −8.37196 −0.274233
933933 0 0
934934 − 13.3515i − 0.436875i
935935 −53.3008 −1.74312
936936 0 0
937937 19.1153 0.624469 0.312235 0.950005i 0.398922π-0.398922\pi
0.312235 + 0.950005i 0.398922π0.398922\pi
938938 − 55.6969i − 1.81857i
939939 0 0
940940 6.89008 0.224730
941941 − 0.445042i − 0.0145080i −0.999974 0.00725398i 0.997691π-0.997691\pi
0.999974 0.00725398i 0.00230903π-0.00230903\pi
942942 0 0
943943 38.2801i 1.24657i
944944 − 8.41119i − 0.273761i
945945 0 0
946946 −22.8659 −0.743435
947947 − 8.99894i − 0.292426i −0.989253 0.146213i 0.953291π-0.953291\pi
0.989253 0.146213i 0.0467085π-0.0467085\pi
948948 0 0
949949 0 0
950950 −7.87800 −0.255596
951951 0 0
952952 −23.7211 −0.768805
953953 −37.2573 −1.20688 −0.603441 0.797408i 0.706204π-0.706204\pi
−0.603441 + 0.797408i 0.706204π0.706204\pi
954954 0 0
955955 − 19.8974i − 0.643864i
956956 14.0194i 0.453419i
957957 0 0
958958 4.38537 0.141685
959959 74.5086 2.40601
960960 0 0
961961 16.3381 0.527036
962962 0 0
963963 0 0
964964 21.1183i 0.680174i
965965 37.7754 1.21603
966966 0 0
967967 − 37.2567i − 1.19809i −0.800714 0.599047i 0.795546π-0.795546\pi
0.800714 0.599047i 0.204454π-0.204454\pi
968968 25.5894i 0.822474i
969969 0 0
970970 1.81940i 0.0584173i
971971 −54.0393 −1.73421 −0.867103 0.498130i 0.834020π-0.834020\pi
−0.867103 + 0.498130i 0.834020π0.834020\pi
972972 0 0
973973 − 57.2379i − 1.83496i
974974 7.33214 0.234937
975975 0 0
976976 0.219833 0.00703667
977977 11.4470i 0.366221i 0.983092 + 0.183110i 0.0586166π0.0586166\pi
−0.983092 + 0.183110i 0.941383π0.941383\pi
978978 0 0
979979 105.008 3.35609
980980 − 29.7875i − 0.951526i
981981 0 0
982982 16.8062i 0.536309i
983983 − 0.111244i − 0.00354814i −0.999998 0.00177407i 0.999435π-0.999435\pi
0.999998 0.00177407i 0.000564704π-0.000564704\pi
984984 0 0
985985 28.3642 0.903758
986986 − 17.3840i − 0.553621i
987987 0 0
988988 0 0
989989 −25.3793 −0.807013
990990 0 0
991991 11.3709 0.361208 0.180604 0.983556i 0.442195π-0.442195\pi
0.180604 + 0.983556i 0.442195π0.442195\pi
992992 3.82908 0.121574
993993 0 0
994994 − 65.6098i − 2.08102i
995995 − 2.72348i − 0.0863401i
996996 0 0
997997 6.45042 0.204287 0.102143 0.994770i 0.467430π-0.467430\pi
0.102143 + 0.994770i 0.467430π0.467430\pi
998998 −6.59658 −0.208811
999999 0 0
Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3042.2.b.p.1351.4 6
3.2 odd 2 3042.2.b.q.1351.3 6
13.5 odd 4 3042.2.a.bf.1.1 yes 3
13.8 odd 4 3042.2.a.bc.1.3 yes 3
13.12 even 2 inner 3042.2.b.p.1351.3 6
39.5 even 4 3042.2.a.bb.1.3 3
39.8 even 4 3042.2.a.bg.1.1 yes 3
39.38 odd 2 3042.2.b.q.1351.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3042.2.a.bb.1.3 3 39.5 even 4
3042.2.a.bc.1.3 yes 3 13.8 odd 4
3042.2.a.bf.1.1 yes 3 13.5 odd 4
3042.2.a.bg.1.1 yes 3 39.8 even 4
3042.2.b.p.1351.3 6 13.12 even 2 inner
3042.2.b.p.1351.4 6 1.1 even 1 trivial
3042.2.b.q.1351.3 6 3.2 odd 2
3042.2.b.q.1351.4 6 39.38 odd 2