Properties

Label 448.4.j.b.335.37
Level 448448
Weight 44
Character 448.335
Analytic conductor 26.43326.433
Analytic rank 00
Dimension 8888
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,4,Mod(111,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.111");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 448=267 448 = 2^{6} \cdot 7
Weight: k k == 4 4
Character orbit: [χ][\chi] == 448.j (of order 44, degree 22, 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: 26.432855682626.4328556826
Analytic rank: 00
Dimension: 8888
Relative dimension: 4444 over Q(i)\Q(i)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

Embedding invariants

Embedding label 335.37
Character χ\chi == 448.335
Dual form 448.4.j.b.111.37

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+(4.797024.79702i)q3+(2.046242.04624i)q5+(13.3740+12.8115i)q719.0229iq9+(34.5515+34.5515i)q11+(7.325637.32563i)q1319.6317iq15+133.707iq17+(13.673513.6735i)q19+(2.69841+125.613i)q21129.773q23+116.626iq25+(38.2664+38.2664i)q27+(85.790985.7909i)q29239.073q31+331.489iq33+(1.15105+53.5820i)q35+(50.797550.7975i)q3770.2824q39+118.276q41+(75.475075.4750i)q43+(38.925438.9254i)q45169.043q47+(14.7299342.684i)q49+(641.398+641.398i)q51+(482.264+482.264i)q53+141.401iq55131.184iq57+(285.231285.231i)q59+(313.070313.070i)q61+(243.712+254.413i)q6329.9800q65+(58.2711+58.2711i)q67+(622.523+622.523i)q69+374.109q71+610.003q73+(559.457+559.457i)q75+(19.4359904.751i)q77460.899iq79+880.748q81+(875.405+875.405i)q83+(273.598+273.598i)q85823.082iq87633.231q89+(191.826+4.12080i)q91+(1146.84+1146.84i)q9355.9584iq95+946.138iq97+(657.269+657.269i)q99+O(q100)q+(4.79702 - 4.79702i) q^{3} +(2.04624 - 2.04624i) q^{5} +(-13.3740 + 12.8115i) q^{7} -19.0229i q^{9} +(-34.5515 + 34.5515i) q^{11} +(-7.32563 - 7.32563i) q^{13} -19.6317i q^{15} +133.707i q^{17} +(13.6735 - 13.6735i) q^{19} +(-2.69841 + 125.613i) q^{21} -129.773 q^{23} +116.626i q^{25} +(38.2664 + 38.2664i) q^{27} +(85.7909 - 85.7909i) q^{29} -239.073 q^{31} +331.489i q^{33} +(-1.15105 + 53.5820i) q^{35} +(-50.7975 - 50.7975i) q^{37} -70.2824 q^{39} +118.276 q^{41} +(75.4750 - 75.4750i) q^{43} +(-38.9254 - 38.9254i) q^{45} -169.043 q^{47} +(14.7299 - 342.684i) q^{49} +(641.398 + 641.398i) q^{51} +(482.264 + 482.264i) q^{53} +141.401i q^{55} -131.184i q^{57} +(-285.231 - 285.231i) q^{59} +(-313.070 - 313.070i) q^{61} +(243.712 + 254.413i) q^{63} -29.9800 q^{65} +(58.2711 + 58.2711i) q^{67} +(-622.523 + 622.523i) q^{69} +374.109 q^{71} +610.003 q^{73} +(559.457 + 559.457i) q^{75} +(19.4359 - 904.751i) q^{77} -460.899i q^{79} +880.748 q^{81} +(-875.405 + 875.405i) q^{83} +(273.598 + 273.598i) q^{85} -823.082i q^{87} -633.231 q^{89} +(191.826 + 4.12080i) q^{91} +(-1146.84 + 1146.84i) q^{93} -55.9584i q^{95} +946.138i q^{97} +(657.269 + 657.269i) q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 88q+4q7112q11+52q21160q23+528q29476q35896q37+8q3940q431376q491504q511560q538q65648q67+456q71+292q77+7592q99+O(q100) 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77}+ \cdots - 7592 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/448Z)×\left(\mathbb{Z}/448\mathbb{Z}\right)^\times.

nn 127127 129129 197197
χ(n)\chi(n) 1-1 1-1 e(14)e\left(\frac{1}{4}\right)

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 0
33 4.79702 4.79702i 0.923188 0.923188i −0.0740657 0.997253i 0.523597π-0.523597\pi
0.997253 + 0.0740657i 0.0235974π0.0235974\pi
44 0 0
55 2.04624 2.04624i 0.183021 0.183021i −0.609650 0.792671i 0.708690π-0.708690\pi
0.792671 + 0.609650i 0.208690π0.208690\pi
66 0 0
77 −13.3740 + 12.8115i −0.722130 + 0.691757i
88 0 0
99 19.0229i 0.704551i
1010 0 0
1111 −34.5515 + 34.5515i −0.947061 + 0.947061i −0.998668 0.0516064i 0.983566π-0.983566\pi
0.0516064 + 0.998668i 0.483566π0.483566\pi
1212 0 0
1313 −7.32563 7.32563i −0.156290 0.156290i 0.624631 0.780920i 0.285250π-0.285250\pi
−0.780920 + 0.624631i 0.785250π0.785250\pi
1414 0 0
1515 19.6317i 0.337926i
1616 0 0
1717 133.707i 1.90758i 0.300477 + 0.953789i 0.402854π0.402854\pi
−0.300477 + 0.953789i 0.597146π0.597146\pi
1818 0 0
1919 13.6735 13.6735i 0.165100 0.165100i −0.619722 0.784822i 0.712754π-0.712754\pi
0.784822 + 0.619722i 0.212754π0.212754\pi
2020 0 0
2121 −2.69841 + 125.613i −0.0280401 + 1.30528i
2222 0 0
2323 −129.773 −1.17650 −0.588250 0.808679i 0.700183π-0.700183\pi
−0.588250 + 0.808679i 0.700183π0.700183\pi
2424 0 0
2525 116.626i 0.933006i
2626 0 0
2727 38.2664 + 38.2664i 0.272755 + 0.272755i
2828 0 0
2929 85.7909 85.7909i 0.549344 0.549344i −0.376907 0.926251i 0.623012π-0.623012\pi
0.926251 + 0.376907i 0.123012π0.123012\pi
3030 0 0
3131 −239.073 −1.38512 −0.692560 0.721360i 0.743517π-0.743517\pi
−0.692560 + 0.721360i 0.743517π0.743517\pi
3232 0 0
3333 331.489i 1.74863i
3434 0 0
3535 −1.15105 + 53.5820i −0.00555893 + 0.258772i
3636 0 0
3737 −50.7975 50.7975i −0.225704 0.225704i 0.585191 0.810895i 0.301019π-0.301019\pi
−0.810895 + 0.585191i 0.801019π0.801019\pi
3838 0 0
3939 −70.2824 −0.288569
4040 0 0
4141 118.276 0.450527 0.225264 0.974298i 0.427676π-0.427676\pi
0.225264 + 0.974298i 0.427676π0.427676\pi
4242 0 0
4343 75.4750 75.4750i 0.267671 0.267671i −0.560490 0.828161i 0.689387π-0.689387\pi
0.828161 + 0.560490i 0.189387π0.189387\pi
4444 0 0
4545 −38.9254 38.9254i −0.128948 0.128948i
4646 0 0
4747 −169.043 −0.524625 −0.262313 0.964983i 0.584485π-0.584485\pi
−0.262313 + 0.964983i 0.584485π0.584485\pi
4848 0 0
4949 14.7299 342.684i 0.0429442 0.999077i
5050 0 0
5151 641.398 + 641.398i 1.76105 + 1.76105i
5252 0 0
5353 482.264 + 482.264i 1.24989 + 1.24989i 0.955769 + 0.294118i 0.0950257π0.0950257\pi
0.294118 + 0.955769i 0.404974π0.404974\pi
5454 0 0
5555 141.401i 0.346665i
5656 0 0
5757 131.184i 0.304837i
5858 0 0
5959 −285.231 285.231i −0.629389 0.629389i 0.318525 0.947914i 0.396812π-0.396812\pi
−0.947914 + 0.318525i 0.896812π0.896812\pi
6060 0 0
6161 −313.070 313.070i −0.657124 0.657124i 0.297575 0.954699i 0.403822π-0.403822\pi
−0.954699 + 0.297575i 0.903822π0.903822\pi
6262 0 0
6363 243.712 + 254.413i 0.487378 + 0.508778i
6464 0 0
6565 −29.9800 −0.0572086
6666 0 0
6767 58.2711 + 58.2711i 0.106253 + 0.106253i 0.758235 0.651982i 0.226062π-0.226062\pi
−0.651982 + 0.758235i 0.726062π0.726062\pi
6868 0 0
6969 −622.523 + 622.523i −1.08613 + 1.08613i
7070 0 0
7171 374.109 0.625331 0.312666 0.949863i 0.398778π-0.398778\pi
0.312666 + 0.949863i 0.398778π0.398778\pi
7272 0 0
7373 610.003 0.978020 0.489010 0.872278i 0.337358π-0.337358\pi
0.489010 + 0.872278i 0.337358π0.337358\pi
7474 0 0
7575 559.457 + 559.457i 0.861340 + 0.861340i
7676 0 0
7777 19.4359 904.751i 0.0287652 1.33904i
7878 0 0
7979 460.899i 0.656395i −0.944609 0.328198i 0.893559π-0.893559\pi
0.944609 0.328198i 0.106441π-0.106441\pi
8080 0 0
8181 880.748 1.20816
8282 0 0
8383 −875.405 + 875.405i −1.15769 + 1.15769i −0.172717 + 0.984971i 0.555255π0.555255\pi
−0.984971 + 0.172717i 0.944745π0.944745\pi
8484 0 0
8585 273.598 + 273.598i 0.349128 + 0.349128i
8686 0 0
8787 823.082i 1.01430i
8888 0 0
8989 −633.231 −0.754184 −0.377092 0.926176i 0.623076π-0.623076\pi
−0.377092 + 0.926176i 0.623076π0.623076\pi
9090 0 0
9191 191.826 + 4.12080i 0.220976 + 0.00474700i
9292 0 0
9393 −1146.84 + 1146.84i −1.27873 + 1.27873i
9494 0 0
9595 55.9584i 0.0604338i
9696 0 0
9797 946.138i 0.990368i 0.868788 + 0.495184i 0.164899π0.164899\pi
−0.868788 + 0.495184i 0.835101π0.835101\pi
9898 0 0
9999 657.269 + 657.269i 0.667253 + 0.667253i
100100 0 0
101101 141.100 141.100i 0.139010 0.139010i −0.634178 0.773187i 0.718661π-0.718661\pi
0.773187 + 0.634178i 0.218661π0.218661\pi
102102 0 0
103103 891.445i 0.852784i 0.904539 + 0.426392i 0.140216π0.140216\pi
−0.904539 + 0.426392i 0.859784π0.859784\pi
104104 0 0
105105 251.512 + 262.556i 0.233763 + 0.244027i
106106 0 0
107107 727.560 727.560i 0.657345 0.657345i −0.297406 0.954751i 0.596122π-0.596122\pi
0.954751 + 0.297406i 0.0961215π0.0961215\pi
108108 0 0
109109 −1136.70 + 1136.70i −0.998867 + 0.998867i −0.999999 0.00113245i 0.999640π-0.999640\pi
0.00113245 + 0.999999i 0.499640π0.499640\pi
110110 0 0
111111 −487.353 −0.416735
112112 0 0
113113 −556.606 −0.463373 −0.231686 0.972791i 0.574424π-0.574424\pi
−0.231686 + 0.972791i 0.574424π0.574424\pi
114114 0 0
115115 −265.546 + 265.546i −0.215325 + 0.215325i
116116 0 0
117117 −139.355 + 139.355i −0.110114 + 0.110114i
118118 0 0
119119 −1713.00 1788.21i −1.31958 1.37752i
120120 0 0
121121 1056.61i 0.793849i
122122 0 0
123123 567.373 567.373i 0.415921 0.415921i
124124 0 0
125125 494.425 + 494.425i 0.353781 + 0.353781i
126126 0 0
127127 1588.68i 1.11002i 0.831843 + 0.555011i 0.187286π0.187286\pi
−0.831843 + 0.555011i 0.812714π0.812714\pi
128128 0 0
129129 724.111i 0.494220i
130130 0 0
131131 −1500.21 + 1500.21i −1.00057 + 1.00057i −0.000565274 1.00000i 0.500180π0.500180\pi
−1.00000 0.000565274i 0.999820π0.999820\pi
132132 0 0
133133 −7.69157 + 358.047i −0.00501461 + 0.233433i
134134 0 0
135135 156.605 0.0998399
136136 0 0
137137 2259.10i 1.40882i −0.709795 0.704408i 0.751212π-0.751212\pi
0.709795 0.704408i 0.248788π-0.248788\pi
138138 0 0
139139 2145.82 + 2145.82i 1.30940 + 1.30940i 0.921850 + 0.387547i 0.126677π0.126677\pi
0.387547 + 0.921850i 0.373323π0.373323\pi
140140 0 0
141141 −810.901 + 810.901i −0.484328 + 0.484328i
142142 0 0
143143 506.223 0.296031
144144 0 0
145145 351.098i 0.201083i
146146 0 0
147147 −1573.20 1714.52i −0.882690 0.961982i
148148 0 0
149149 −1085.99 1085.99i −0.597100 0.597100i 0.342440 0.939540i 0.388747π-0.388747\pi
−0.939540 + 0.342440i 0.888747π0.888747\pi
150150 0 0
151151 −257.765 −0.138918 −0.0694590 0.997585i 0.522127π-0.522127\pi
−0.0694590 + 0.997585i 0.522127π0.522127\pi
152152 0 0
153153 2543.50 1.34399
154154 0 0
155155 −489.200 + 489.200i −0.253507 + 0.253507i
156156 0 0
157157 −1532.56 1532.56i −0.779053 0.779053i 0.200617 0.979670i 0.435705π-0.435705\pi
−0.979670 + 0.200617i 0.935705π0.935705\pi
158158 0 0
159159 4626.86 2.30776
160160 0 0
161161 1735.59 1662.59i 0.849586 0.813852i
162162 0 0
163163 177.457 + 177.457i 0.0852729 + 0.0852729i 0.748457 0.663184i 0.230795π-0.230795\pi
−0.663184 + 0.748457i 0.730795π0.730795\pi
164164 0 0
165165 678.306 + 678.306i 0.320037 + 0.320037i
166166 0 0
167167 1722.29i 0.798054i −0.916939 0.399027i 0.869348π-0.869348\pi
0.916939 0.399027i 0.130652π-0.130652\pi
168168 0 0
169169 2089.67i 0.951147i
170170 0 0
171171 −260.108 260.108i −0.116322 0.116322i
172172 0 0
173173 1795.20 + 1795.20i 0.788939 + 0.788939i 0.981320 0.192381i 0.0616210π-0.0616210\pi
−0.192381 + 0.981320i 0.561621π0.561621\pi
174174 0 0
175175 −1494.15 1559.76i −0.645414 0.673752i
176176 0 0
177177 −2736.52 −1.16209
178178 0 0
179179 953.910 + 953.910i 0.398316 + 0.398316i 0.877639 0.479323i 0.159118π-0.159118\pi
−0.479323 + 0.877639i 0.659118π0.659118\pi
180180 0 0
181181 −1780.34 + 1780.34i −0.731114 + 0.731114i −0.970840 0.239727i 0.922942π-0.922942\pi
0.239727 + 0.970840i 0.422942π0.422942\pi
182182 0 0
183183 −3003.61 −1.21330
184184 0 0
185185 −207.888 −0.0826173
186186 0 0
187187 −4619.79 4619.79i −1.80659 1.80659i
188188 0 0
189189 −1002.03 21.5256i −0.385645 0.00828442i
190190 0 0
191191 2064.00i 0.781916i −0.920409 0.390958i 0.872144π-0.872144\pi
0.920409 0.390958i 0.127856π-0.127856\pi
192192 0 0
193193 1142.67 0.426172 0.213086 0.977033i 0.431649π-0.431649\pi
0.213086 + 0.977033i 0.431649π0.431649\pi
194194 0 0
195195 −143.815 + 143.815i −0.0528143 + 0.0528143i
196196 0 0
197197 837.511 + 837.511i 0.302894 + 0.302894i 0.842145 0.539251i 0.181292π-0.181292\pi
−0.539251 + 0.842145i 0.681292π0.681292\pi
198198 0 0
199199 2165.33i 0.771338i −0.922637 0.385669i 0.873971π-0.873971\pi
0.922637 0.385669i 0.126029π-0.126029\pi
200200 0 0
201201 559.056 0.196183
202202 0 0
203203 −48.2590 + 2246.48i −0.0166853 + 0.776710i
204204 0 0
205205 242.021 242.021i 0.0824561 0.0824561i
206206 0 0
207207 2468.65i 0.828904i
208208 0 0
209209 944.877i 0.312720i
210210 0 0
211211 −130.423 130.423i −0.0425531 0.0425531i 0.685510 0.728063i 0.259579π-0.259579\pi
−0.728063 + 0.685510i 0.759579π0.759579\pi
212212 0 0
213213 1794.61 1794.61i 0.577298 0.577298i
214214 0 0
215215 308.880i 0.0979789i
216216 0 0
217217 3197.37 3062.89i 1.00024 0.958167i
218218 0 0
219219 2926.20 2926.20i 0.902896 0.902896i
220220 0 0
221221 979.491 979.491i 0.298134 0.298134i
222222 0 0
223223 −1221.92 −0.366933 −0.183466 0.983026i 0.558732π-0.558732\pi
−0.183466 + 0.983026i 0.558732π0.558732\pi
224224 0 0
225225 2218.56 0.657351
226226 0 0
227227 3515.16 3515.16i 1.02780 1.02780i 0.0281934 0.999602i 0.491025π-0.491025\pi
0.999602 0.0281934i 0.00897542π-0.00897542\pi
228228 0 0
229229 3345.39 3345.39i 0.965369 0.965369i −0.0340515 0.999420i 0.510841π-0.510841\pi
0.999420 + 0.0340515i 0.0108410π0.0108410\pi
230230 0 0
231231 −4246.88 4433.34i −1.20963 1.26274i
232232 0 0
233233 537.847i 0.151225i −0.997137 0.0756127i 0.975909π-0.975909\pi
0.997137 0.0756127i 0.0240913π-0.0240913\pi
234234 0 0
235235 −345.902 + 345.902i −0.0960177 + 0.0960177i
236236 0 0
237237 −2210.95 2210.95i −0.605976 0.605976i
238238 0 0
239239 6403.30i 1.73303i −0.499149 0.866516i 0.666354π-0.666354\pi
0.499149 0.866516i 0.333646π-0.333646\pi
240240 0 0
241241 1648.67i 0.440664i −0.975425 0.220332i 0.929286π-0.929286\pi
0.975425 0.220332i 0.0707140π-0.0707140\pi
242242 0 0
243243 3191.77 3191.77i 0.842603 0.842603i
244244 0 0
245245 −671.072 731.354i −0.174993 0.190712i
246246 0 0
247247 −200.333 −0.0516069
248248 0 0
249249 8398.68i 2.13753i
250250 0 0
251251 3775.90 + 3775.90i 0.949534 + 0.949534i 0.998786 0.0492528i 0.0156840π-0.0156840\pi
−0.0492528 + 0.998786i 0.515684π0.515684\pi
252252 0 0
253253 4483.84 4483.84i 1.11422 1.11422i
254254 0 0
255255 2624.91 0.644620
256256 0 0
257257 6959.86i 1.68928i 0.535337 + 0.844638i 0.320185π0.320185\pi
−0.535337 + 0.844638i 0.679815π0.679815\pi
258258 0 0
259259 1330.16 + 28.5745i 0.319120 + 0.00685534i
260260 0 0
261261 −1631.99 1631.99i −0.387041 0.387041i
262262 0 0
263263 4461.40 1.04602 0.523008 0.852328i 0.324810π-0.324810\pi
0.523008 + 0.852328i 0.324810π0.324810\pi
264264 0 0
265265 1973.66 0.457512
266266 0 0
267267 −3037.62 + 3037.62i −0.696253 + 0.696253i
268268 0 0
269269 −2252.15 2252.15i −0.510469 0.510469i 0.404201 0.914670i 0.367550π-0.367550\pi
−0.914670 + 0.404201i 0.867550π0.867550\pi
270270 0 0
271271 −6670.05 −1.49512 −0.747559 0.664196i 0.768774π-0.768774\pi
−0.747559 + 0.664196i 0.768774π0.768774\pi
272272 0 0
273273 939.960 900.425i 0.208384 0.199620i
274274 0 0
275275 −4029.60 4029.60i −0.883614 0.883614i
276276 0 0
277277 −1517.50 1517.50i −0.329161 0.329161i 0.523106 0.852267i 0.324773π-0.324773\pi
−0.852267 + 0.523106i 0.824773π0.824773\pi
278278 0 0
279279 4547.85i 0.975888i
280280 0 0
281281 559.278i 0.118732i −0.998236 0.0593661i 0.981092π-0.981092\pi
0.998236 0.0593661i 0.0189079π-0.0189079\pi
282282 0 0
283283 −25.8379 25.8379i −0.00542721 0.00542721i 0.704388 0.709815i 0.251222π-0.251222\pi
−0.709815 + 0.704388i 0.751222π0.751222\pi
284284 0 0
285285 −268.434 268.434i −0.0557917 0.0557917i
286286 0 0
287287 −1581.83 + 1515.30i −0.325339 + 0.311655i
288288 0 0
289289 −12964.7 −2.63885
290290 0 0
291291 4538.65 + 4538.65i 0.914296 + 0.914296i
292292 0 0
293293 −3862.53 + 3862.53i −0.770142 + 0.770142i −0.978131 0.207989i 0.933308π-0.933308\pi
0.207989 + 0.978131i 0.433308π0.433308\pi
294294 0 0
295295 −1167.30 −0.230383
296296 0 0
297297 −2644.33 −0.516631
298298 0 0
299299 950.667 + 950.667i 0.183875 + 0.183875i
300300 0 0
301301 −42.4561 + 1976.36i −0.00812999 + 0.378456i
302302 0 0
303303 1353.72i 0.256664i
304304 0 0
305305 −1281.23 −0.240535
306306 0 0
307307 −2044.42 + 2044.42i −0.380069 + 0.380069i −0.871127 0.491058i 0.836610π-0.836610\pi
0.491058 + 0.871127i 0.336610π0.336610\pi
308308 0 0
309309 4276.28 + 4276.28i 0.787279 + 0.787279i
310310 0 0
311311 9371.35i 1.70868i 0.519711 + 0.854342i 0.326040π0.326040\pi
−0.519711 + 0.854342i 0.673960π0.673960\pi
312312 0 0
313313 −8978.95 −1.62147 −0.810735 0.585413i 0.800932π-0.800932\pi
−0.810735 + 0.585413i 0.800932π0.800932\pi
314314 0 0
315315 1019.28 + 21.8962i 0.182318 + 0.00391655i
316316 0 0
317317 −1767.75 + 1767.75i −0.313207 + 0.313207i −0.846151 0.532944i 0.821086π-0.821086\pi
0.532944 + 0.846151i 0.321086π0.321086\pi
318318 0 0
319319 5928.41i 1.04052i
320320 0 0
321321 6980.25i 1.21371i
322322 0 0
323323 1828.24 + 1828.24i 0.314942 + 0.314942i
324324 0 0
325325 854.357 854.357i 0.145819 0.145819i
326326 0 0
327327 10905.6i 1.84428i
328328 0 0
329329 2260.78 2165.69i 0.378848 0.362913i
330330 0 0
331331 −4730.32 + 4730.32i −0.785504 + 0.785504i −0.980754 0.195250i 0.937448π-0.937448\pi
0.195250 + 0.980754i 0.437448π0.437448\pi
332332 0 0
333333 −966.314 + 966.314i −0.159020 + 0.159020i
334334 0 0
335335 238.473 0.0388931
336336 0 0
337337 1717.91 0.277688 0.138844 0.990314i 0.455661π-0.455661\pi
0.138844 + 0.990314i 0.455661π0.455661\pi
338338 0 0
339339 −2670.05 + 2670.05i −0.427780 + 0.427780i
340340 0 0
341341 8260.32 8260.32i 1.31179 1.31179i
342342 0 0
343343 4193.30 + 4771.78i 0.660108 + 0.751171i
344344 0 0
345345 2547.66i 0.397570i
346346 0 0
347347 7793.80 7793.80i 1.20574 1.20574i 0.233352 0.972392i 0.425031π-0.425031\pi
0.972392 0.233352i 0.0749695π-0.0749695\pi
348348 0 0
349349 4211.57 + 4211.57i 0.645960 + 0.645960i 0.952014 0.306054i 0.0990088π-0.0990088\pi
−0.306054 + 0.952014i 0.599009π0.599009\pi
350350 0 0
351351 560.652i 0.0852574i
352352 0 0
353353 7172.87i 1.08151i 0.841180 + 0.540755i 0.181862π0.181862\pi
−0.841180 + 0.540755i 0.818138π0.818138\pi
354354 0 0
355355 765.516 765.516i 0.114449 0.114449i
356356 0 0
357357 −16795.4 360.798i −2.48993 0.0534887i
358358 0 0
359359 12810.1 1.88326 0.941630 0.336650i 0.109294π-0.109294\pi
0.941630 + 0.336650i 0.109294π0.109294\pi
360360 0 0
361361 6485.07i 0.945484i
362362 0 0
363363 −5068.60 5068.60i −0.732872 0.732872i
364364 0 0
365365 1248.21 1248.21i 0.178998 0.178998i
366366 0 0
367367 −5502.08 −0.782578 −0.391289 0.920268i 0.627971π-0.627971\pi
−0.391289 + 0.920268i 0.627971π0.627971\pi
368368 0 0
369369 2249.95i 0.317419i
370370 0 0
371371 −12628.3 271.282i −1.76720 0.0379630i
372372 0 0
373373 −5026.12 5026.12i −0.697702 0.697702i 0.266213 0.963914i 0.414228π-0.414228\pi
−0.963914 + 0.266213i 0.914228π0.914228\pi
374374 0 0
375375 4743.53 0.653213
376376 0 0
377377 −1256.94 −0.171713
378378 0 0
379379 −6001.11 + 6001.11i −0.813342 + 0.813342i −0.985133 0.171792i 0.945044π-0.945044\pi
0.171792 + 0.985133i 0.445044π0.445044\pi
380380 0 0
381381 7620.96 + 7620.96i 1.02476 + 1.02476i
382382 0 0
383383 12452.4 1.66133 0.830665 0.556773i 0.187961π-0.187961\pi
0.830665 + 0.556773i 0.187961π0.187961\pi
384384 0 0
385385 −1811.57 1891.11i −0.239808 0.250337i
386386 0 0
387387 −1435.75 1435.75i −0.188588 0.188588i
388388 0 0
389389 7568.58 + 7568.58i 0.986483 + 0.986483i 0.999910 0.0134265i 0.00427393π-0.00427393\pi
−0.0134265 + 0.999910i 0.504274π0.504274\pi
390390 0 0
391391 17351.6i 2.24426i
392392 0 0
393393 14393.1i 1.84742i
394394 0 0
395395 −943.111 943.111i −0.120134 0.120134i
396396 0 0
397397 −746.476 746.476i −0.0943692 0.0943692i 0.658346 0.752715i 0.271256π-0.271256\pi
−0.752715 + 0.658346i 0.771256π0.771256\pi
398398 0 0
399399 1680.66 + 1754.46i 0.210873 + 0.220132i
400400 0 0
401401 8402.79 1.04642 0.523211 0.852203i 0.324734π-0.324734\pi
0.523211 + 0.852203i 0.324734π0.324734\pi
402402 0 0
403403 1751.36 + 1751.36i 0.216480 + 0.216480i
404404 0 0
405405 1802.22 1802.22i 0.221119 0.221119i
406406 0 0
407407 3510.26 0.427511
408408 0 0
409409 −8430.15 −1.01918 −0.509589 0.860418i 0.670203π-0.670203\pi
−0.509589 + 0.860418i 0.670203π0.670203\pi
410410 0 0
411411 −10837.0 10837.0i −1.30060 1.30060i
412412 0 0
413413 7468.95 + 160.448i 0.889886 + 0.0191165i
414414 0 0
415415 3582.58i 0.423764i
416416 0 0
417417 20587.1 2.41764
418418 0 0
419419 8672.14 8672.14i 1.01113 1.01113i 0.0111887 0.999937i 0.496438π-0.496438\pi
0.999937 0.0111887i 0.00356153π-0.00356153\pi
420420 0 0
421421 10398.8 + 10398.8i 1.20382 + 1.20382i 0.972997 + 0.230818i 0.0741403π0.0741403\pi
0.230818 + 0.972997i 0.425860π0.425860\pi
422422 0 0
423423 3215.68i 0.369625i
424424 0 0
425425 −15593.7 −1.77978
426426 0 0
427427 8197.92 + 176.108i 0.929099 + 0.0199589i
428428 0 0
429429 2428.36 2428.36i 0.273293 0.273293i
430430 0 0
431431 12243.6i 1.36834i 0.729322 + 0.684170i 0.239835π0.239835\pi
−0.729322 + 0.684170i 0.760165π0.760165\pi
432432 0 0
433433 8787.77i 0.975320i −0.873034 0.487660i 0.837851π-0.837851\pi
0.873034 0.487660i 0.162149π-0.162149\pi
434434 0 0
435435 −1684.22 1684.22i −0.185638 0.185638i
436436 0 0
437437 −1774.44 + 1774.44i −0.194240 + 0.194240i
438438 0 0
439439 6163.76i 0.670115i 0.942198 + 0.335057i 0.108756π0.108756\pi
−0.942198 + 0.335057i 0.891244π0.891244\pi
440440 0 0
441441 −6518.83 280.204i −0.703901 0.0302564i
442442 0 0
443443 −705.371 + 705.371i −0.0756505 + 0.0756505i −0.743920 0.668269i 0.767035π-0.767035\pi
0.668269 + 0.743920i 0.267035π0.267035\pi
444444 0 0
445445 −1295.74 + 1295.74i −0.138032 + 0.138032i
446446 0 0
447447 −10419.1 −1.10247
448448 0 0
449449 1808.14 0.190048 0.0950240 0.995475i 0.469707π-0.469707\pi
0.0950240 + 0.995475i 0.469707π0.469707\pi
450450 0 0
451451 −4086.62 + 4086.62i −0.426677 + 0.426677i
452452 0 0
453453 −1236.51 + 1236.51i −0.128247 + 0.128247i
454454 0 0
455455 400.954 384.089i 0.0413121 0.0395745i
456456 0 0
457457 13173.0i 1.34838i −0.738560 0.674188i 0.764494π-0.764494\pi
0.738560 0.674188i 0.235506π-0.235506\pi
458458 0 0
459459 −5116.51 + 5116.51i −0.520301 + 0.520301i
460460 0 0
461461 −2780.05 2780.05i −0.280867 0.280867i 0.552588 0.833455i 0.313640π-0.313640\pi
−0.833455 + 0.552588i 0.813640π0.813640\pi
462462 0 0
463463 3630.01i 0.364365i 0.983265 + 0.182182i 0.0583161π0.0583161\pi
−0.983265 + 0.182182i 0.941684π0.941684\pi
464464 0 0
465465 4693.41i 0.468068i
466466 0 0
467467 39.0209 39.0209i 0.00386654 0.00386654i −0.705171 0.709037i 0.749130π-0.749130\pi
0.709037 + 0.705171i 0.249130π0.249130\pi
468468 0 0
469469 −1525.86 32.7786i −0.150230 0.00322723i
470470 0 0
471471 −14703.4 −1.43842
472472 0 0
473473 5215.55i 0.507001i
474474 0 0
475475 1594.68 + 1594.68i 0.154040 + 0.154040i
476476 0 0
477477 9174.04 9174.04i 0.880609 0.880609i
478478 0 0
479479 1088.16 0.103798 0.0518990 0.998652i 0.483473π-0.483473\pi
0.0518990 + 0.998652i 0.483473π0.483473\pi
480480 0 0
481481 744.247i 0.0705504i
482482 0 0
483483 350.181 16301.1i 0.0329892 1.53567i
484484 0 0
485485 1936.03 + 1936.03i 0.181259 + 0.181259i
486486 0 0
487487 17740.1 1.65068 0.825339 0.564638i 0.190984π-0.190984\pi
0.825339 + 0.564638i 0.190984π0.190984\pi
488488 0 0
489489 1702.53 0.157446
490490 0 0
491491 −5821.10 + 5821.10i −0.535036 + 0.535036i −0.922067 0.387031i 0.873501π-0.873501\pi
0.387031 + 0.922067i 0.373501π0.373501\pi
492492 0 0
493493 11470.9 + 11470.9i 1.04792 + 1.04792i
494494 0 0
495495 2689.86 0.244243
496496 0 0
497497 −5003.34 + 4792.90i −0.451571 + 0.432577i
498498 0 0
499499 −4283.85 4283.85i −0.384312 0.384312i 0.488341 0.872653i 0.337602π-0.337602\pi
−0.872653 + 0.488341i 0.837602π0.837602\pi
500500 0 0
501501 −8261.88 8261.88i −0.736753 0.736753i
502502 0 0
503503 743.585i 0.0659142i 0.999457 + 0.0329571i 0.0104925π0.0104925\pi
−0.999457 + 0.0329571i 0.989508π0.989508\pi
504504 0 0
505505 577.449i 0.0508834i
506506 0 0
507507 −10024.2 10024.2i −0.878087 0.878087i
508508 0 0
509509 6189.71 + 6189.71i 0.539006 + 0.539006i 0.923237 0.384231i 0.125533π-0.125533\pi
−0.384231 + 0.923237i 0.625533π0.625533\pi
510510 0 0
511511 −8158.20 + 7815.06i −0.706258 + 0.676552i
512512 0 0
513513 1046.47 0.0900638
514514 0 0
515515 1824.11 + 1824.11i 0.156078 + 0.156078i
516516 0 0
517517 5840.68 5840.68i 0.496852 0.496852i
518518 0 0
519519 17223.2 1.45668
520520 0 0
521521 11779.3 0.990521 0.495260 0.868745i 0.335073π-0.335073\pi
0.495260 + 0.868745i 0.335073π0.335073\pi
522522 0 0
523523 1261.69 + 1261.69i 0.105488 + 0.105488i 0.757881 0.652393i 0.226235π-0.226235\pi
−0.652393 + 0.757881i 0.726235π0.726235\pi
524524 0 0
525525 −14649.7 314.705i −1.21784 0.0261616i
526526 0 0
527527 31965.8i 2.64222i
528528 0 0
529529 4673.97 0.384151
530530 0 0
531531 −5425.92 + 5425.92i −0.443437 + 0.443437i
532532 0 0
533533 −866.447 866.447i −0.0704127 0.0704127i
534534 0 0
535535 2977.53i 0.240616i
536536 0 0
537537 9151.86 0.735441
538538 0 0
539539 11331.3 + 12349.2i 0.905517 + 0.986858i
540540 0 0
541541 3180.80 3180.80i 0.252778 0.252778i −0.569330 0.822109i 0.692797π-0.692797\pi
0.822109 + 0.569330i 0.192797π0.192797\pi
542542 0 0
543543 17080.7i 1.34991i
544544 0 0
545545 4651.94i 0.365628i
546546 0 0
547547 2706.48 + 2706.48i 0.211555 + 0.211555i 0.804928 0.593373i 0.202204π-0.202204\pi
−0.593373 + 0.804928i 0.702204π0.702204\pi
548548 0 0
549549 −5955.50 + 5955.50i −0.462977 + 0.462977i
550550 0 0
551551 2346.12i 0.181394i
552552 0 0
553553 5904.82 + 6164.09i 0.454066 + 0.474003i
554554 0 0
555555 −997.242 + 997.242i −0.0762713 + 0.0762713i
556556 0 0
557557 −1976.55 + 1976.55i −0.150358 + 0.150358i −0.778278 0.627920i 0.783906π-0.783906\pi
0.627920 + 0.778278i 0.283906π0.283906\pi
558558 0 0
559559 −1105.80 −0.0836682
560560 0 0
561561 −44322.5 −3.33565
562562 0 0
563563 1969.01 1969.01i 0.147396 0.147396i −0.629558 0.776954i 0.716764π-0.716764\pi
0.776954 + 0.629558i 0.216764π0.216764\pi
564564 0 0
565565 −1138.95 + 1138.95i −0.0848071 + 0.0848071i
566566 0 0
567567 −11779.2 + 11283.7i −0.872448 + 0.835752i
568568 0 0
569569 12025.5i 0.886004i 0.896521 + 0.443002i 0.146087π0.146087\pi
−0.896521 + 0.443002i 0.853913π0.853913\pi
570570 0 0
571571 3067.62 3067.62i 0.224826 0.224826i −0.585701 0.810527i 0.699181π-0.699181\pi
0.810527 + 0.585701i 0.199181π0.199181\pi
572572 0 0
573573 −9901.06 9901.06i −0.721855 0.721855i
574574 0 0
575575 15134.8i 1.09768i
576576 0 0
577577 4574.91i 0.330080i −0.986287 0.165040i 0.947225π-0.947225\pi
0.986287 0.165040i 0.0527753π-0.0527753\pi
578578 0 0
579579 5481.42 5481.42i 0.393437 0.393437i
580580 0 0
581581 492.431 22923.0i 0.0351626 1.63684i
582582 0 0
583583 −33325.9 −2.36744
584584 0 0
585585 570.306i 0.0403064i
586586 0 0
587587 −1826.09 1826.09i −0.128400 0.128400i 0.639986 0.768386i 0.278940π-0.278940\pi
−0.768386 + 0.639986i 0.778940π0.778940\pi
588588 0 0
589589 −3268.95 + 3268.95i −0.228684 + 0.228684i
590590 0 0
591591 8035.12 0.559257
592592 0 0
593593 5705.55i 0.395107i −0.980292 0.197554i 0.936700π-0.936700\pi
0.980292 0.197554i 0.0632997π-0.0632997\pi
594594 0 0
595595 −7164.31 153.904i −0.493627 0.0106041i
596596 0 0
597597 −10387.1 10387.1i −0.712090 0.712090i
598598 0 0
599599 −10497.7 −0.716070 −0.358035 0.933708i 0.616553π-0.616553\pi
−0.358035 + 0.933708i 0.616553π0.616553\pi
600600 0 0
601601 −9375.74 −0.636347 −0.318173 0.948033i 0.603069π-0.603069\pi
−0.318173 + 0.948033i 0.603069π0.603069\pi
602602 0 0
603603 1108.48 1108.48i 0.0748606 0.0748606i
604604 0 0
605605 −2162.09 2162.09i −0.145291 0.145291i
606606 0 0
607607 −5702.60 −0.381320 −0.190660 0.981656i 0.561063π-0.561063\pi
−0.190660 + 0.981656i 0.561063π0.561063\pi
608608 0 0
609609 10544.9 + 11007.9i 0.701646 + 0.732453i
610610 0 0
611611 1238.34 + 1238.34i 0.0819934 + 0.0819934i
612612 0 0
613613 7424.25 + 7424.25i 0.489172 + 0.489172i 0.908045 0.418873i 0.137575π-0.137575\pi
−0.418873 + 0.908045i 0.637575π0.637575\pi
614614 0 0
615615 2321.96i 0.152245i
616616 0 0
617617 22808.8i 1.48825i −0.668043 0.744123i 0.732868π-0.732868\pi
0.668043 0.744123i 0.267132π-0.267132\pi
618618 0 0
619619 15749.6 + 15749.6i 1.02267 + 1.02267i 0.999737 + 0.0229291i 0.00729921π0.00729921\pi
0.0229291 + 0.999737i 0.492701π0.492701\pi
620620 0 0
621621 −4965.94 4965.94i −0.320896 0.320896i
622622 0 0
623623 8468.86 8112.65i 0.544619 0.521712i
624624 0 0
625625 −12554.8 −0.803507
626626 0 0
627627 4532.60 + 4532.60i 0.288699 + 0.288699i
628628 0 0
629629 6792.00 6792.00i 0.430548 0.430548i
630630 0 0
631631 7632.20 0.481510 0.240755 0.970586i 0.422605π-0.422605\pi
0.240755 + 0.970586i 0.422605π0.422605\pi
632632 0 0
633633 −1251.29 −0.0785689
634634 0 0
635635 3250.83 + 3250.83i 0.203158 + 0.203158i
636636 0 0
637637 −2618.28 + 2402.47i −0.162857 + 0.149434i
638638 0 0
639639 7116.62i 0.440578i
640640 0 0
641641 18811.7 1.15915 0.579576 0.814918i 0.303218π-0.303218\pi
0.579576 + 0.814918i 0.303218π0.303218\pi
642642 0 0
643643 14360.3 14360.3i 0.880740 0.880740i −0.112869 0.993610i 0.536004π-0.536004\pi
0.993610 + 0.112869i 0.0360042π0.0360042\pi
644644 0 0
645645 −1481.71 1481.71i −0.0904529 0.0904529i
646646 0 0
647647 13880.5i 0.843430i 0.906728 + 0.421715i 0.138572π0.138572\pi
−0.906728 + 0.421715i 0.861428π0.861428\pi
648648 0 0
649649 19710.4 1.19214
650650 0 0
651651 645.117 30030.6i 0.0388389 1.80797i
652652 0 0
653653 6926.86 6926.86i 0.415113 0.415113i −0.468402 0.883515i 0.655170π-0.655170\pi
0.883515 + 0.468402i 0.155170π0.155170\pi
654654 0 0
655655 6139.59i 0.366250i
656656 0 0
657657 11604.0i 0.689065i
658658 0 0
659659 21306.6 + 21306.6i 1.25947 + 1.25947i 0.951348 + 0.308118i 0.0996992π0.0996992\pi
0.308118 + 0.951348i 0.400301π0.400301\pi
660660 0 0
661661 9192.89 9192.89i 0.540941 0.540941i −0.382864 0.923805i 0.625062π-0.625062\pi
0.923805 + 0.382864i 0.125062π0.125062\pi
662662 0 0
663663 9397.29i 0.550468i
664664 0 0
665665 716.912 + 748.389i 0.0418055 + 0.0436410i
666666 0 0
667667 −11133.3 + 11133.3i −0.646303 + 0.646303i
668668 0 0
669669 −5861.59 + 5861.59i −0.338748 + 0.338748i
670670 0 0
671671 21634.1 1.24467
672672 0 0
673673 −24859.6 −1.42388 −0.711938 0.702242i 0.752182π-0.752182\pi
−0.711938 + 0.702242i 0.752182π0.752182\pi
674674 0 0
675675 −4462.85 + 4462.85i −0.254482 + 0.254482i
676676 0 0
677677 −11098.7 + 11098.7i −0.630070 + 0.630070i −0.948086 0.318016i 0.896984π-0.896984\pi
0.318016 + 0.948086i 0.396984π0.396984\pi
678678 0 0
679679 −12121.5 12653.7i −0.685094 0.715175i
680680 0 0
681681 33724.7i 1.89770i
682682 0 0
683683 5129.88 5129.88i 0.287393 0.287393i −0.548656 0.836048i 0.684860π-0.684860\pi
0.836048 + 0.548656i 0.184860π0.184860\pi
684684 0 0
685685 −4622.66 4622.66i −0.257844 0.257844i
686686 0 0
687687 32095.8i 1.78243i
688688 0 0
689689 7065.77i 0.390688i
690690 0 0
691691 1243.53 1243.53i 0.0684606 0.0684606i −0.672047 0.740508i 0.734585π-0.734585\pi
0.740508 + 0.672047i 0.234585π0.234585\pi
692692 0 0
693693 −17211.0 369.726i −0.943420 0.0202666i
694694 0 0
695695 8781.73 0.479295
696696 0 0
697697 15814.4i 0.859416i
698698 0 0
699699 −2580.07 2580.07i −0.139609 0.139609i
700700 0 0
701701 8513.11 8513.11i 0.458681 0.458681i −0.439541 0.898223i 0.644859π-0.644859\pi
0.898223 + 0.439541i 0.144859π0.144859\pi
702702 0 0
703703 −1389.15 −0.0745276
704704 0 0
705705 3318.60i 0.177285i
706706 0 0
707707 −79.3713 + 3694.78i −0.00422216 + 0.196544i
708708 0 0
709709 20499.4 + 20499.4i 1.08585 + 1.08585i 0.995950 + 0.0899043i 0.0286561π0.0286561\pi
0.0899043 + 0.995950i 0.471344π0.471344\pi
710710 0 0
711711 −8767.63 −0.462464
712712 0 0
713713 31025.1 1.62959
714714 0 0
715715 1035.85 1035.85i 0.0541801 0.0541801i
716716 0 0
717717 −30716.8 30716.8i −1.59991 1.59991i
718718 0 0
719719 −14074.7 −0.730037 −0.365018 0.931000i 0.618937π-0.618937\pi
−0.365018 + 0.931000i 0.618937π0.618937\pi
720720 0 0
721721 −11420.8 11922.2i −0.589919 0.615821i
722722 0 0
723723 −7908.69 7908.69i −0.406815 0.406815i
724724 0 0
725725 10005.4 + 10005.4i 0.512541 + 0.512541i
726726 0 0
727727 95.2264i 0.00485798i 0.999997 + 0.00242899i 0.000773172π0.000773172\pi
−0.999997 + 0.00242899i 0.999227π0.999227\pi
728728 0 0
729729 6841.84i 0.347602i
730730 0 0
731731 10091.6 + 10091.6i 0.510602 + 0.510602i
732732 0 0
733733 −13958.3 13958.3i −0.703357 0.703357i 0.261772 0.965130i 0.415693π-0.415693\pi
−0.965130 + 0.261772i 0.915693π0.915693\pi
734734 0 0
735735 −6727.47 289.173i −0.337614 0.0145120i
736736 0 0
737737 −4026.71 −0.201256
738738 0 0
739739 −22132.3 22132.3i −1.10169 1.10169i −0.994207 0.107485i 0.965720π-0.965720\pi
−0.107485 0.994207i 0.534280π-0.534280\pi
740740 0 0
741741 −961.004 + 961.004i −0.0476428 + 0.0476428i
742742 0 0
743743 5532.55 0.273176 0.136588 0.990628i 0.456386π-0.456386\pi
0.136588 + 0.990628i 0.456386π0.456386\pi
744744 0 0
745745 −4444.40 −0.218564
746746 0 0
747747 16652.7 + 16652.7i 0.815651 + 0.815651i
748748 0 0
749749 −409.266 + 19051.6i −0.0199656 + 0.929412i
750750 0 0
751751 36330.7i 1.76528i 0.470048 + 0.882641i 0.344236π0.344236\pi
−0.470048 + 0.882641i 0.655764π0.655764\pi
752752 0 0
753753 36226.2 1.75320
754754 0 0
755755 −527.449 + 527.449i −0.0254250 + 0.0254250i
756756 0 0
757757 15674.3 + 15674.3i 0.752564 + 0.752564i 0.974957 0.222393i 0.0713869π-0.0713869\pi
−0.222393 + 0.974957i 0.571387π0.571387\pi
758758 0 0
759759 43018.2i 2.05726i
760760 0 0
761761 16656.7 0.793437 0.396718 0.917940i 0.370149π-0.370149\pi
0.396718 + 0.917940i 0.370149π0.370149\pi
762762 0 0
763763 639.417 29765.2i 0.0303387 1.41229i
764764 0 0
765765 5204.62 5204.62i 0.245978 0.245978i
766766 0 0
767767 4179.00i 0.196734i
768768 0 0
769769 9190.37i 0.430966i 0.976508 + 0.215483i 0.0691327π0.0691327\pi
−0.976508 + 0.215483i 0.930867π0.930867\pi
770770 0 0
771771 33386.6 + 33386.6i 1.55952 + 1.55952i
772772 0 0
773773 −3285.14 + 3285.14i −0.152857 + 0.152857i −0.779393 0.626536i 0.784472π-0.784472\pi
0.626536 + 0.779393i 0.284472π0.284472\pi
774774 0 0
775775 27882.0i 1.29233i
776776 0 0
777777 6517.88 6243.74i 0.300937 0.288279i
778778 0 0
779779 1617.24 1617.24i 0.0743822 0.0743822i
780780 0 0
781781 −12926.0 + 12926.0i −0.592227 + 0.592227i
782782 0 0
783783 6565.83 0.299672
784784 0 0
785785 −6271.96 −0.285167
786786 0 0
787787 11875.1 11875.1i 0.537867 0.537867i −0.385035 0.922902i 0.625811π-0.625811\pi
0.922902 + 0.385035i 0.125811π0.125811\pi
788788 0 0
789789 21401.5 21401.5i 0.965668 0.965668i
790790 0 0
791791 7444.08 7130.98i 0.334616 0.320541i
792792 0 0
793793 4586.87i 0.205403i
794794 0 0
795795 9467.67 9467.67i 0.422369 0.422369i
796796 0 0
797797 27267.4 + 27267.4i 1.21187 + 1.21187i 0.970411 + 0.241460i 0.0776263π0.0776263\pi
0.241460 + 0.970411i 0.422374π0.422374\pi
798798 0 0
799799 22602.3i 1.00076i
800800 0 0
801801 12045.9i 0.531361i
802802 0 0
803803 −21076.5 + 21076.5i −0.926244 + 0.926244i
804804 0 0
805805 149.375 6953.48i 0.00654008 0.304445i
806806 0 0
807807 −21607.3 −0.942518
808808 0 0
809809 37899.6i 1.64707i −0.567265 0.823535i 0.691999π-0.691999\pi
0.567265 0.823535i 0.308001π-0.308001\pi
810810 0 0
811811 −2453.46 2453.46i −0.106230 0.106230i 0.651994 0.758224i 0.273933π-0.273933\pi
−0.758224 + 0.651994i 0.773933π0.773933\pi
812812 0 0
813813 −31996.4 + 31996.4i −1.38027 + 1.38027i
814814 0 0
815815 726.238 0.0312135
816816 0 0
817817 2064.01i 0.0883850i
818818 0 0
819819 78.3895 3649.08i 0.00334451 0.155689i
820820 0 0
821821 −16123.1 16123.1i −0.685385 0.685385i 0.275823 0.961208i 0.411050π-0.411050\pi
−0.961208 + 0.275823i 0.911050π0.911050\pi
822822 0 0
823823 −24632.9 −1.04332 −0.521658 0.853155i 0.674686π-0.674686\pi
−0.521658 + 0.853155i 0.674686π0.674686\pi
824824 0 0
825825 −38660.1 −1.63148
826826 0 0
827827 −1089.80 + 1089.80i −0.0458236 + 0.0458236i −0.729647 0.683824i 0.760316π-0.760316\pi
0.683824 + 0.729647i 0.260316π0.260316\pi
828828 0 0
829829 4055.04 + 4055.04i 0.169888 + 0.169888i 0.786930 0.617042i 0.211669π-0.211669\pi
−0.617042 + 0.786930i 0.711669π0.711669\pi
830830 0 0
831831 −14559.0 −0.607755
832832 0 0
833833 45819.4 + 1969.49i 1.90582 + 0.0819194i
834834 0 0
835835 −3524.22 3524.22i −0.146061 0.146061i
836836 0 0
837837 −9148.46 9148.46i −0.377798 0.377798i
838838 0 0
839839 35349.1i 1.45457i 0.686335 + 0.727285i 0.259218π0.259218\pi
−0.686335 + 0.727285i 0.740782π0.740782\pi
840840 0 0
841841 9668.83i 0.396442i
842842 0 0
843843 −2682.87 2682.87i −0.109612 0.109612i
844844 0 0
845845 −4275.97 4275.97i −0.174080 0.174080i
846846 0 0
847847 13536.8 + 14131.2i 0.549151 + 0.573263i
848848 0 0
849849 −247.890 −0.0100207
850850 0 0
851851 6592.13 + 6592.13i 0.265541 + 0.265541i
852852 0 0
853853 −28356.9 + 28356.9i −1.13824 + 1.13824i −0.149477 + 0.988765i 0.547759π0.547759\pi
−0.988765 + 0.149477i 0.952241π0.952241\pi
854854 0 0
855855 −1064.49 −0.0425787
856856 0 0
857857 5617.28 0.223900 0.111950 0.993714i 0.464290π-0.464290\pi
0.111950 + 0.993714i 0.464290π0.464290\pi
858858 0 0
859859 26923.7 + 26923.7i 1.06941 + 1.06941i 0.997404 + 0.0720059i 0.0229400π0.0229400\pi
0.0720059 + 0.997404i 0.477060π0.477060\pi
860860 0 0
861861 −319.158 + 14857.0i −0.0126328 + 0.588066i
862862 0 0
863863 7764.09i 0.306249i −0.988207 0.153124i 0.951067π-0.951067\pi
0.988207 0.153124i 0.0489335π-0.0489335\pi
864864 0 0
865865 7346.82 0.288785
866866 0 0
867867 −62191.9 + 62191.9i −2.43616 + 2.43616i
868868 0 0
869869 15924.8 + 15924.8i 0.621646 + 0.621646i
870870 0 0
871871 853.745i 0.0332124i
872872 0 0
873873 17998.3 0.697765
874874 0 0
875875 −12946.8 278.123i −0.500207 0.0107455i
876876 0 0
877877 15619.2 15619.2i 0.601396 0.601396i −0.339287 0.940683i 0.610186π-0.610186\pi
0.940683 + 0.339287i 0.110186π0.110186\pi
878878 0 0
879879 37057.3i 1.42197i
880880 0 0
881881 45320.9i 1.73314i 0.499053 + 0.866571i 0.333681π0.333681\pi
−0.499053 + 0.866571i 0.666319π0.666319\pi
882882 0 0
883883 5998.33 + 5998.33i 0.228607 + 0.228607i 0.812110 0.583504i 0.198319π-0.198319\pi
−0.583504 + 0.812110i 0.698319π0.698319\pi
884884 0 0
885885 −5599.59 + 5599.59i −0.212687 + 0.212687i
886886 0 0
887887 25507.8i 0.965578i −0.875737 0.482789i 0.839624π-0.839624\pi
0.875737 0.482789i 0.160376π-0.160376\pi
888888 0 0
889889 −20353.5 21247.1i −0.767866 0.801581i
890890 0 0
891891 −30431.2 + 30431.2i −1.14420 + 1.14420i
892892 0 0
893893 −2311.40 + 2311.40i −0.0866158 + 0.0866158i
894894 0 0
895895 3903.86 0.145801
896896 0 0
897897 9120.74 0.339501
898898 0 0
899899 −20510.3 + 20510.3i −0.760907 + 0.760907i
900900 0 0
901901 −64482.2 + 64482.2i −2.38426 + 2.38426i
902902 0 0
903903 9276.96 + 9684.29i 0.341880 + 0.356891i
904904 0 0
905905 7286.01i 0.267619i
906906 0 0
907907 26334.9 26334.9i 0.964096 0.964096i −0.0352818 0.999377i 0.511233π-0.511233\pi
0.999377 + 0.0352818i 0.0112329π0.0112329\pi
908908 0 0
909909 −2684.13 2684.13i −0.0979393 0.0979393i
910910 0 0
911911 16395.8i 0.596285i 0.954521 + 0.298142i 0.0963671π0.0963671\pi
−0.954521 + 0.298142i 0.903633π0.903633\pi
912912 0 0
913913 60493.1i 2.19280i
914914 0 0
915915 −6146.11 + 6146.11i −0.222059 + 0.222059i
916916 0 0
917917 843.896 39283.9i 0.0303903 1.41469i
918918 0 0
919919 26851.4 0.963815 0.481908 0.876222i 0.339944π-0.339944\pi
0.481908 + 0.876222i 0.339944π0.339944\pi
920920 0 0
921921 19614.3i 0.701750i
922922 0 0
923923 −2740.58 2740.58i −0.0977327 0.0977327i
924924 0 0
925925 5924.29 5924.29i 0.210583 0.210583i
926926 0 0
927927 16957.9 0.600830
928928 0 0
929929 19627.5i 0.693171i 0.938018 + 0.346585i 0.112659π0.112659\pi
−0.938018 + 0.346585i 0.887341π0.887341\pi
930930 0 0
931931 −4484.26 4887.08i −0.157858 0.172038i
932932 0 0
933933 44954.6 + 44954.6i 1.57744 + 1.57744i
934934 0 0
935935 −18906.4 −0.661290
936936 0 0
937937 −4230.42 −0.147494 −0.0737469 0.997277i 0.523496π-0.523496\pi
−0.0737469 + 0.997277i 0.523496π0.523496\pi
938938 0 0
939939 −43072.2 + 43072.2i −1.49692 + 1.49692i
940940 0 0
941941 3133.94 + 3133.94i 0.108569 + 0.108569i 0.759305 0.650735i 0.225539π-0.225539\pi
−0.650735 + 0.759305i 0.725539π0.725539\pi
942942 0 0
943943 −15349.0 −0.530045
944944 0 0
945945 −2094.44 + 2006.35i −0.0720974 + 0.0690650i
946946 0 0
947947 −273.354 273.354i −0.00937997 0.00937997i 0.702401 0.711781i 0.252111π-0.252111\pi
−0.711781 + 0.702401i 0.752111π0.752111\pi
948948 0 0
949949 −4468.65 4468.65i −0.152854 0.152854i
950950 0 0
951951 16959.9i 0.578298i
952952 0 0
953953 20906.0i 0.710610i −0.934750 0.355305i 0.884377π-0.884377\pi
0.934750 0.355305i 0.115623π-0.115623\pi
954954 0 0
955955 −4223.44 4223.44i −0.143107 0.143107i
956956 0 0
957957 28438.7 + 28438.7i 0.960599 + 0.960599i
958958 0 0
959959 28942.5 + 30213.3i 0.974559 + 1.01735i
960960 0 0
961961 27364.8 0.918558
962962 0 0
963963 −13840.3 13840.3i −0.463133 0.463133i
964964 0 0
965965 2338.18 2338.18i 0.0779986 0.0779986i
966966 0 0
967967 −9771.57 −0.324956 −0.162478 0.986712i 0.551949π-0.551949\pi
−0.162478 + 0.986712i 0.551949π0.551949\pi
968968 0 0
969969 17540.3 0.581501
970970 0 0
971971 −9977.45 9977.45i −0.329754 0.329754i 0.522739 0.852493i 0.324910π-0.324910\pi
−0.852493 + 0.522739i 0.824910π0.824910\pi
972972 0 0
973973 −56189.5 1207.06i −1.85134 0.0397705i
974974 0 0
975975 8196.74i 0.269237i
976976 0 0
977977 15291.4 0.500732 0.250366 0.968151i 0.419449π-0.419449\pi
0.250366 + 0.968151i 0.419449π0.419449\pi
978978 0 0
979979 21879.1 21879.1i 0.714258 0.714258i
980980 0 0
981981 21623.4 + 21623.4i 0.703753 + 0.703753i
982982 0 0
983983 14778.2i 0.479504i −0.970834 0.239752i 0.922934π-0.922934\pi
0.970834 0.239752i 0.0770661π-0.0770661\pi
984984 0 0
985985 3427.50 0.110872
986986 0 0
987987 456.147 21233.9i 0.0147106 0.684785i
988988 0 0
989989 −9794.60 + 9794.60i −0.314914 + 0.314914i
990990 0 0
991991 7672.15i 0.245927i −0.992411 0.122964i 0.960760π-0.960760\pi
0.992411 0.122964i 0.0392398π-0.0392398\pi
992992 0 0
993993 45382.9i 1.45033i
994994 0 0
995995 −4430.79 4430.79i −0.141171 0.141171i
996996 0 0
997997 −28731.6 + 28731.6i −0.912677 + 0.912677i −0.996482 0.0838054i 0.973293π-0.973293\pi
0.0838054 + 0.996482i 0.473293π0.473293\pi
998998 0 0
999999 3887.68i 0.123124i
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 448.4.j.b.335.37 88
4.3 odd 2 112.4.j.b.27.9 88
7.6 odd 2 inner 448.4.j.b.335.8 88
16.3 odd 4 inner 448.4.j.b.111.8 88
16.13 even 4 112.4.j.b.83.10 yes 88
28.27 even 2 112.4.j.b.27.10 yes 88
112.13 odd 4 112.4.j.b.83.9 yes 88
112.83 even 4 inner 448.4.j.b.111.37 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.4.j.b.27.9 88 4.3 odd 2
112.4.j.b.27.10 yes 88 28.27 even 2
112.4.j.b.83.9 yes 88 112.13 odd 4
112.4.j.b.83.10 yes 88 16.13 even 4
448.4.j.b.111.8 88 16.3 odd 4 inner
448.4.j.b.111.37 88 112.83 even 4 inner
448.4.j.b.335.8 88 7.6 odd 2 inner
448.4.j.b.335.37 88 1.1 even 1 trivial