Properties

Label 3200.2.c.e.2049.2
Level 32003200
Weight 22
Character 3200.2049
Analytic conductor 25.55225.552
Analytic rank 00
Dimension 22
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3200,2,Mod(2049,3200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3200.2049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 3200=2752 3200 = 2^{7} \cdot 5^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 3200.c (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: 25.552128646825.5521286468
Analytic rank: 00
Dimension: 22
Coefficient field: Q(i)\Q(i)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2+1 x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 2049.2
Root 1.00000i1.00000i of defining polynomial
Character χ\chi == 3200.2049
Dual form 3200.2.c.e.2049.1

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+2.00000iq3+4.00000iq71.00000q92.00000q11+2.00000iq13+2.00000iq172.00000q198.00000q21+4.00000iq23+4.00000iq27+6.00000q294.00000iq3310.0000iq374.00000q396.00000q41+6.00000iq43+8.00000iq479.00000q494.00000q516.00000iq534.00000iq5714.0000q59+2.00000q614.00000iq6310.0000iq678.00000q69+12.0000q71+14.0000iq738.00000iq77+8.00000q7911.0000q816.00000iq83+12.0000iq87+2.00000q898.00000q91+2.00000iq97+2.00000q99+O(q100)q+2.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +2.00000i q^{13} +2.00000i q^{17} -2.00000 q^{19} -8.00000 q^{21} +4.00000i q^{23} +4.00000i q^{27} +6.00000 q^{29} -4.00000i q^{33} -10.0000i q^{37} -4.00000 q^{39} -6.00000 q^{41} +6.00000i q^{43} +8.00000i q^{47} -9.00000 q^{49} -4.00000 q^{51} -6.00000i q^{53} -4.00000i q^{57} -14.0000 q^{59} +2.00000 q^{61} -4.00000i q^{63} -10.0000i q^{67} -8.00000 q^{69} +12.0000 q^{71} +14.0000i q^{73} -8.00000i q^{77} +8.00000 q^{79} -11.0000 q^{81} -6.00000i q^{83} +12.0000i q^{87} +2.00000 q^{89} -8.00000 q^{91} +2.00000i q^{97} +2.00000 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q94q114q1916q21+12q298q3912q4118q498q5128q59+4q6116q69+24q71+16q7922q81+4q8916q91+4q99+O(q100) 2 q - 2 q^{9} - 4 q^{11} - 4 q^{19} - 16 q^{21} + 12 q^{29} - 8 q^{39} - 12 q^{41} - 18 q^{49} - 8 q^{51} - 28 q^{59} + 4 q^{61} - 16 q^{69} + 24 q^{71} + 16 q^{79} - 22 q^{81} + 4 q^{89} - 16 q^{91} + 4 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/3200Z)×\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times.

nn 901901 11511151 21772177
χ(n)\chi(n) 11 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 0 0
33 2.00000i 1.15470i 0.816497 + 0.577350i 0.195913π0.195913\pi
−0.816497 + 0.577350i 0.804087π0.804087\pi
44 0 0
55 0 0
66 0 0
77 4.00000i 1.51186i 0.654654 + 0.755929i 0.272814π0.272814\pi
−0.654654 + 0.755929i 0.727186π0.727186\pi
88 0 0
99 −1.00000 −0.333333
1010 0 0
1111 −2.00000 −0.603023 −0.301511 0.953463i 0.597491π-0.597491\pi
−0.301511 + 0.953463i 0.597491π0.597491\pi
1212 0 0
1313 2.00000i 0.554700i 0.960769 + 0.277350i 0.0894562π0.0894562\pi
−0.960769 + 0.277350i 0.910544π0.910544\pi
1414 0 0
1515 0 0
1616 0 0
1717 2.00000i 0.485071i 0.970143 + 0.242536i 0.0779791π0.0779791\pi
−0.970143 + 0.242536i 0.922021π0.922021\pi
1818 0 0
1919 −2.00000 −0.458831 −0.229416 0.973329i 0.573682π-0.573682\pi
−0.229416 + 0.973329i 0.573682π0.573682\pi
2020 0 0
2121 −8.00000 −1.74574
2222 0 0
2323 4.00000i 0.834058i 0.908893 + 0.417029i 0.136929π0.136929\pi
−0.908893 + 0.417029i 0.863071π0.863071\pi
2424 0 0
2525 0 0
2626 0 0
2727 4.00000i 0.769800i
2828 0 0
2929 6.00000 1.11417 0.557086 0.830455i 0.311919π-0.311919\pi
0.557086 + 0.830455i 0.311919π0.311919\pi
3030 0 0
3131 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
3232 0 0
3333 − 4.00000i − 0.696311i
3434 0 0
3535 0 0
3636 0 0
3737 − 10.0000i − 1.64399i −0.569495 0.821995i 0.692861π-0.692861\pi
0.569495 0.821995i 0.307139π-0.307139\pi
3838 0 0
3939 −4.00000 −0.640513
4040 0 0
4141 −6.00000 −0.937043 −0.468521 0.883452i 0.655213π-0.655213\pi
−0.468521 + 0.883452i 0.655213π0.655213\pi
4242 0 0
4343 6.00000i 0.914991i 0.889212 + 0.457496i 0.151253π0.151253\pi
−0.889212 + 0.457496i 0.848747π0.848747\pi
4444 0 0
4545 0 0
4646 0 0
4747 8.00000i 1.16692i 0.812142 + 0.583460i 0.198301π0.198301\pi
−0.812142 + 0.583460i 0.801699π0.801699\pi
4848 0 0
4949 −9.00000 −1.28571
5050 0 0
5151 −4.00000 −0.560112
5252 0 0
5353 − 6.00000i − 0.824163i −0.911147 0.412082i 0.864802π-0.864802\pi
0.911147 0.412082i 0.135198π-0.135198\pi
5454 0 0
5555 0 0
5656 0 0
5757 − 4.00000i − 0.529813i
5858 0 0
5959 −14.0000 −1.82264 −0.911322 0.411693i 0.864937π-0.864937\pi
−0.911322 + 0.411693i 0.864937π0.864937\pi
6060 0 0
6161 2.00000 0.256074 0.128037 0.991769i 0.459132π-0.459132\pi
0.128037 + 0.991769i 0.459132π0.459132\pi
6262 0 0
6363 − 4.00000i − 0.503953i
6464 0 0
6565 0 0
6666 0 0
6767 − 10.0000i − 1.22169i −0.791748 0.610847i 0.790829π-0.790829\pi
0.791748 0.610847i 0.209171π-0.209171\pi
6868 0 0
6969 −8.00000 −0.963087
7070 0 0
7171 12.0000 1.42414 0.712069 0.702109i 0.247758π-0.247758\pi
0.712069 + 0.702109i 0.247758π0.247758\pi
7272 0 0
7373 14.0000i 1.63858i 0.573382 + 0.819288i 0.305631π0.305631\pi
−0.573382 + 0.819288i 0.694369π0.694369\pi
7474 0 0
7575 0 0
7676 0 0
7777 − 8.00000i − 0.911685i
7878 0 0
7979 8.00000 0.900070 0.450035 0.893011i 0.351411π-0.351411\pi
0.450035 + 0.893011i 0.351411π0.351411\pi
8080 0 0
8181 −11.0000 −1.22222
8282 0 0
8383 − 6.00000i − 0.658586i −0.944228 0.329293i 0.893190π-0.893190\pi
0.944228 0.329293i 0.106810π-0.106810\pi
8484 0 0
8585 0 0
8686 0 0
8787 12.0000i 1.28654i
8888 0 0
8989 2.00000 0.212000 0.106000 0.994366i 0.466196π-0.466196\pi
0.106000 + 0.994366i 0.466196π0.466196\pi
9090 0 0
9191 −8.00000 −0.838628
9292 0 0
9393 0 0
9494 0 0
9595 0 0
9696 0 0
9797 2.00000i 0.203069i 0.994832 + 0.101535i 0.0323753π0.0323753\pi
−0.994832 + 0.101535i 0.967625π0.967625\pi
9898 0 0
9999 2.00000 0.201008
100100 0 0
101101 −6.00000 −0.597022 −0.298511 0.954406i 0.596490π-0.596490\pi
−0.298511 + 0.954406i 0.596490π0.596490\pi
102102 0 0
103103 − 4.00000i − 0.394132i −0.980390 0.197066i 0.936859π-0.936859\pi
0.980390 0.197066i 0.0631413π-0.0631413\pi
104104 0 0
105105 0 0
106106 0 0
107107 2.00000i 0.193347i 0.995316 + 0.0966736i 0.0308203π0.0308203\pi
−0.995316 + 0.0966736i 0.969180π0.969180\pi
108108 0 0
109109 6.00000 0.574696 0.287348 0.957826i 0.407226π-0.407226\pi
0.287348 + 0.957826i 0.407226π0.407226\pi
110110 0 0
111111 20.0000 1.89832
112112 0 0
113113 2.00000i 0.188144i 0.995565 + 0.0940721i 0.0299884π0.0299884\pi
−0.995565 + 0.0940721i 0.970012π0.970012\pi
114114 0 0
115115 0 0
116116 0 0
117117 − 2.00000i − 0.184900i
118118 0 0
119119 −8.00000 −0.733359
120120 0 0
121121 −7.00000 −0.636364
122122 0 0
123123 − 12.0000i − 1.08200i
124124 0 0
125125 0 0
126126 0 0
127127 − 16.0000i − 1.41977i −0.704317 0.709885i 0.748747π-0.748747\pi
0.704317 0.709885i 0.251253π-0.251253\pi
128128 0 0
129129 −12.0000 −1.05654
130130 0 0
131131 −6.00000 −0.524222 −0.262111 0.965038i 0.584419π-0.584419\pi
−0.262111 + 0.965038i 0.584419π0.584419\pi
132132 0 0
133133 − 8.00000i − 0.693688i
134134 0 0
135135 0 0
136136 0 0
137137 − 10.0000i − 0.854358i −0.904167 0.427179i 0.859507π-0.859507\pi
0.904167 0.427179i 0.140493π-0.140493\pi
138138 0 0
139139 10.0000 0.848189 0.424094 0.905618i 0.360592π-0.360592\pi
0.424094 + 0.905618i 0.360592π0.360592\pi
140140 0 0
141141 −16.0000 −1.34744
142142 0 0
143143 − 4.00000i − 0.334497i
144144 0 0
145145 0 0
146146 0 0
147147 − 18.0000i − 1.48461i
148148 0 0
149149 −18.0000 −1.47462 −0.737309 0.675556i 0.763904π-0.763904\pi
−0.737309 + 0.675556i 0.763904π0.763904\pi
150150 0 0
151151 4.00000 0.325515 0.162758 0.986666i 0.447961π-0.447961\pi
0.162758 + 0.986666i 0.447961π0.447961\pi
152152 0 0
153153 − 2.00000i − 0.161690i
154154 0 0
155155 0 0
156156 0 0
157157 − 18.0000i − 1.43656i −0.695756 0.718278i 0.744931π-0.744931\pi
0.695756 0.718278i 0.255069π-0.255069\pi
158158 0 0
159159 12.0000 0.951662
160160 0 0
161161 −16.0000 −1.26098
162162 0 0
163163 2.00000i 0.156652i 0.996928 + 0.0783260i 0.0249575π0.0249575\pi
−0.996928 + 0.0783260i 0.975042π0.975042\pi
164164 0 0
165165 0 0
166166 0 0
167167 20.0000i 1.54765i 0.633402 + 0.773823i 0.281658π0.281658\pi
−0.633402 + 0.773823i 0.718342π0.718342\pi
168168 0 0
169169 9.00000 0.692308
170170 0 0
171171 2.00000 0.152944
172172 0 0
173173 18.0000i 1.36851i 0.729241 + 0.684257i 0.239873π0.239873\pi
−0.729241 + 0.684257i 0.760127π0.760127\pi
174174 0 0
175175 0 0
176176 0 0
177177 − 28.0000i − 2.10461i
178178 0 0
179179 6.00000 0.448461 0.224231 0.974536i 0.428013π-0.428013\pi
0.224231 + 0.974536i 0.428013π0.428013\pi
180180 0 0
181181 2.00000 0.148659 0.0743294 0.997234i 0.476318π-0.476318\pi
0.0743294 + 0.997234i 0.476318π0.476318\pi
182182 0 0
183183 4.00000i 0.295689i
184184 0 0
185185 0 0
186186 0 0
187187 − 4.00000i − 0.292509i
188188 0 0
189189 −16.0000 −1.16383
190190 0 0
191191 −16.0000 −1.15772 −0.578860 0.815427i 0.696502π-0.696502\pi
−0.578860 + 0.815427i 0.696502π0.696502\pi
192192 0 0
193193 − 2.00000i − 0.143963i −0.997406 0.0719816i 0.977068π-0.977068\pi
0.997406 0.0719816i 0.0229323π-0.0229323\pi
194194 0 0
195195 0 0
196196 0 0
197197 14.0000i 0.997459i 0.866758 + 0.498729i 0.166200π0.166200\pi
−0.866758 + 0.498729i 0.833800π0.833800\pi
198198 0 0
199199 4.00000 0.283552 0.141776 0.989899i 0.454719π-0.454719\pi
0.141776 + 0.989899i 0.454719π0.454719\pi
200200 0 0
201201 20.0000 1.41069
202202 0 0
203203 24.0000i 1.68447i
204204 0 0
205205 0 0
206206 0 0
207207 − 4.00000i − 0.278019i
208208 0 0
209209 4.00000 0.276686
210210 0 0
211211 −22.0000 −1.51454 −0.757271 0.653101i 0.773468π-0.773468\pi
−0.757271 + 0.653101i 0.773468π0.773468\pi
212212 0 0
213213 24.0000i 1.64445i
214214 0 0
215215 0 0
216216 0 0
217217 0 0
218218 0 0
219219 −28.0000 −1.89206
220220 0 0
221221 −4.00000 −0.269069
222222 0 0
223223 0 0 1.00000 00
−1.00000 π\pi
224224 0 0
225225 0 0
226226 0 0
227227 − 18.0000i − 1.19470i −0.801980 0.597351i 0.796220π-0.796220\pi
0.801980 0.597351i 0.203780π-0.203780\pi
228228 0 0
229229 14.0000 0.925146 0.462573 0.886581i 0.346926π-0.346926\pi
0.462573 + 0.886581i 0.346926π0.346926\pi
230230 0 0
231231 16.0000 1.05272
232232 0 0
233233 − 18.0000i − 1.17922i −0.807688 0.589610i 0.799282π-0.799282\pi
0.807688 0.589610i 0.200718π-0.200718\pi
234234 0 0
235235 0 0
236236 0 0
237237 16.0000i 1.03931i
238238 0 0
239239 24.0000 1.55243 0.776215 0.630468i 0.217137π-0.217137\pi
0.776215 + 0.630468i 0.217137π0.217137\pi
240240 0 0
241241 −2.00000 −0.128831 −0.0644157 0.997923i 0.520518π-0.520518\pi
−0.0644157 + 0.997923i 0.520518π0.520518\pi
242242 0 0
243243 − 10.0000i − 0.641500i
244244 0 0
245245 0 0
246246 0 0
247247 − 4.00000i − 0.254514i
248248 0 0
249249 12.0000 0.760469
250250 0 0
251251 −18.0000 −1.13615 −0.568075 0.822977i 0.692312π-0.692312\pi
−0.568075 + 0.822977i 0.692312π0.692312\pi
252252 0 0
253253 − 8.00000i − 0.502956i
254254 0 0
255255 0 0
256256 0 0
257257 − 18.0000i − 1.12281i −0.827541 0.561405i 0.810261π-0.810261\pi
0.827541 0.561405i 0.189739π-0.189739\pi
258258 0 0
259259 40.0000 2.48548
260260 0 0
261261 −6.00000 −0.371391
262262 0 0
263263 12.0000i 0.739952i 0.929041 + 0.369976i 0.120634π0.120634\pi
−0.929041 + 0.369976i 0.879366π0.879366\pi
264264 0 0
265265 0 0
266266 0 0
267267 4.00000i 0.244796i
268268 0 0
269269 −10.0000 −0.609711 −0.304855 0.952399i 0.598608π-0.598608\pi
−0.304855 + 0.952399i 0.598608π0.598608\pi
270270 0 0
271271 8.00000 0.485965 0.242983 0.970031i 0.421874π-0.421874\pi
0.242983 + 0.970031i 0.421874π0.421874\pi
272272 0 0
273273 − 16.0000i − 0.968364i
274274 0 0
275275 0 0
276276 0 0
277277 6.00000i 0.360505i 0.983620 + 0.180253i 0.0576915π0.0576915\pi
−0.983620 + 0.180253i 0.942309π0.942309\pi
278278 0 0
279279 0 0
280280 0 0
281281 −18.0000 −1.07379 −0.536895 0.843649i 0.680403π-0.680403\pi
−0.536895 + 0.843649i 0.680403π0.680403\pi
282282 0 0
283283 6.00000i 0.356663i 0.983970 + 0.178331i 0.0570699π0.0570699\pi
−0.983970 + 0.178331i 0.942930π0.942930\pi
284284 0 0
285285 0 0
286286 0 0
287287 − 24.0000i − 1.41668i
288288 0 0
289289 13.0000 0.764706
290290 0 0
291291 −4.00000 −0.234484
292292 0 0
293293 − 14.0000i − 0.817889i −0.912559 0.408944i 0.865897π-0.865897\pi
0.912559 0.408944i 0.134103π-0.134103\pi
294294 0 0
295295 0 0
296296 0 0
297297 − 8.00000i − 0.464207i
298298 0 0
299299 −8.00000 −0.462652
300300 0 0
301301 −24.0000 −1.38334
302302 0 0
303303 − 12.0000i − 0.689382i
304304 0 0
305305 0 0
306306 0 0
307307 − 18.0000i − 1.02731i −0.857996 0.513657i 0.828290π-0.828290\pi
0.857996 0.513657i 0.171710π-0.171710\pi
308308 0 0
309309 8.00000 0.455104
310310 0 0
311311 −28.0000 −1.58773 −0.793867 0.608091i 0.791935π-0.791935\pi
−0.793867 + 0.608091i 0.791935π0.791935\pi
312312 0 0
313313 10.0000i 0.565233i 0.959233 + 0.282617i 0.0912024π0.0912024\pi
−0.959233 + 0.282617i 0.908798π0.908798\pi
314314 0 0
315315 0 0
316316 0 0
317317 6.00000i 0.336994i 0.985702 + 0.168497i 0.0538913π0.0538913\pi
−0.985702 + 0.168497i 0.946109π0.946109\pi
318318 0 0
319319 −12.0000 −0.671871
320320 0 0
321321 −4.00000 −0.223258
322322 0 0
323323 − 4.00000i − 0.222566i
324324 0 0
325325 0 0
326326 0 0
327327 12.0000i 0.663602i
328328 0 0
329329 −32.0000 −1.76422
330330 0 0
331331 14.0000 0.769510 0.384755 0.923019i 0.374286π-0.374286\pi
0.384755 + 0.923019i 0.374286π0.374286\pi
332332 0 0
333333 10.0000i 0.547997i
334334 0 0
335335 0 0
336336 0 0
337337 14.0000i 0.762629i 0.924445 + 0.381314i 0.124528π0.124528\pi
−0.924445 + 0.381314i 0.875472π0.875472\pi
338338 0 0
339339 −4.00000 −0.217250
340340 0 0
341341 0 0
342342 0 0
343343 − 8.00000i − 0.431959i
344344 0 0
345345 0 0
346346 0 0
347347 18.0000i 0.966291i 0.875540 + 0.483145i 0.160506π0.160506\pi
−0.875540 + 0.483145i 0.839494π0.839494\pi
348348 0 0
349349 −10.0000 −0.535288 −0.267644 0.963518i 0.586245π-0.586245\pi
−0.267644 + 0.963518i 0.586245π0.586245\pi
350350 0 0
351351 −8.00000 −0.427008
352352 0 0
353353 18.0000i 0.958043i 0.877803 + 0.479022i 0.159008π0.159008\pi
−0.877803 + 0.479022i 0.840992π0.840992\pi
354354 0 0
355355 0 0
356356 0 0
357357 − 16.0000i − 0.846810i
358358 0 0
359359 4.00000 0.211112 0.105556 0.994413i 0.466338π-0.466338\pi
0.105556 + 0.994413i 0.466338π0.466338\pi
360360 0 0
361361 −15.0000 −0.789474
362362 0 0
363363 − 14.0000i − 0.734809i
364364 0 0
365365 0 0
366366 0 0
367367 8.00000i 0.417597i 0.977959 + 0.208798i 0.0669552π0.0669552\pi
−0.977959 + 0.208798i 0.933045π0.933045\pi
368368 0 0
369369 6.00000 0.312348
370370 0 0
371371 24.0000 1.24602
372372 0 0
373373 10.0000i 0.517780i 0.965907 + 0.258890i 0.0833568π0.0833568\pi
−0.965907 + 0.258890i 0.916643π0.916643\pi
374374 0 0
375375 0 0
376376 0 0
377377 12.0000i 0.618031i
378378 0 0
379379 2.00000 0.102733 0.0513665 0.998680i 0.483642π-0.483642\pi
0.0513665 + 0.998680i 0.483642π0.483642\pi
380380 0 0
381381 32.0000 1.63941
382382 0 0
383383 0 0 1.00000 00
−1.00000 π\pi
384384 0 0
385385 0 0
386386 0 0
387387 − 6.00000i − 0.304997i
388388 0 0
389389 −10.0000 −0.507020 −0.253510 0.967333i 0.581585π-0.581585\pi
−0.253510 + 0.967333i 0.581585π0.581585\pi
390390 0 0
391391 −8.00000 −0.404577
392392 0 0
393393 − 12.0000i − 0.605320i
394394 0 0
395395 0 0
396396 0 0
397397 6.00000i 0.301131i 0.988600 + 0.150566i 0.0481095π0.0481095\pi
−0.988600 + 0.150566i 0.951890π0.951890\pi
398398 0 0
399399 16.0000 0.801002
400400 0 0
401401 30.0000 1.49813 0.749064 0.662497i 0.230503π-0.230503\pi
0.749064 + 0.662497i 0.230503π0.230503\pi
402402 0 0
403403 0 0
404404 0 0
405405 0 0
406406 0 0
407407 20.0000i 0.991363i
408408 0 0
409409 −10.0000 −0.494468 −0.247234 0.968956i 0.579522π-0.579522\pi
−0.247234 + 0.968956i 0.579522π0.579522\pi
410410 0 0
411411 20.0000 0.986527
412412 0 0
413413 − 56.0000i − 2.75558i
414414 0 0
415415 0 0
416416 0 0
417417 20.0000i 0.979404i
418418 0 0
419419 −26.0000 −1.27018 −0.635092 0.772437i 0.719038π-0.719038\pi
−0.635092 + 0.772437i 0.719038π0.719038\pi
420420 0 0
421421 34.0000 1.65706 0.828529 0.559946i 0.189178π-0.189178\pi
0.828529 + 0.559946i 0.189178π0.189178\pi
422422 0 0
423423 − 8.00000i − 0.388973i
424424 0 0
425425 0 0
426426 0 0
427427 8.00000i 0.387147i
428428 0 0
429429 8.00000 0.386244
430430 0 0
431431 40.0000 1.92673 0.963366 0.268190i 0.0864254π-0.0864254\pi
0.963366 + 0.268190i 0.0864254π0.0864254\pi
432432 0 0
433433 30.0000i 1.44171i 0.693087 + 0.720854i 0.256250π0.256250\pi
−0.693087 + 0.720854i 0.743750π0.743750\pi
434434 0 0
435435 0 0
436436 0 0
437437 − 8.00000i − 0.382692i
438438 0 0
439439 −36.0000 −1.71819 −0.859093 0.511819i 0.828972π-0.828972\pi
−0.859093 + 0.511819i 0.828972π0.828972\pi
440440 0 0
441441 9.00000 0.428571
442442 0 0
443443 6.00000i 0.285069i 0.989790 + 0.142534i 0.0455251π0.0455251\pi
−0.989790 + 0.142534i 0.954475π0.954475\pi
444444 0 0
445445 0 0
446446 0 0
447447 − 36.0000i − 1.70274i
448448 0 0
449449 34.0000 1.60456 0.802280 0.596948i 0.203620π-0.203620\pi
0.802280 + 0.596948i 0.203620π0.203620\pi
450450 0 0
451451 12.0000 0.565058
452452 0 0
453453 8.00000i 0.375873i
454454 0 0
455455 0 0
456456 0 0
457457 6.00000i 0.280668i 0.990104 + 0.140334i 0.0448177π0.0448177\pi
−0.990104 + 0.140334i 0.955182π0.955182\pi
458458 0 0
459459 −8.00000 −0.373408
460460 0 0
461461 10.0000 0.465746 0.232873 0.972507i 0.425187π-0.425187\pi
0.232873 + 0.972507i 0.425187π0.425187\pi
462462 0 0
463463 − 8.00000i − 0.371792i −0.982569 0.185896i 0.940481π-0.940481\pi
0.982569 0.185896i 0.0595187π-0.0595187\pi
464464 0 0
465465 0 0
466466 0 0
467467 14.0000i 0.647843i 0.946084 + 0.323921i 0.105001π0.105001\pi
−0.946084 + 0.323921i 0.894999π0.894999\pi
468468 0 0
469469 40.0000 1.84703
470470 0 0
471471 36.0000 1.65879
472472 0 0
473473 − 12.0000i − 0.551761i
474474 0 0
475475 0 0
476476 0 0
477477 6.00000i 0.274721i
478478 0 0
479479 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
480480 0 0
481481 20.0000 0.911922
482482 0 0
483483 − 32.0000i − 1.45605i
484484 0 0
485485 0 0
486486 0 0
487487 20.0000i 0.906287i 0.891438 + 0.453143i 0.149697π0.149697\pi
−0.891438 + 0.453143i 0.850303π0.850303\pi
488488 0 0
489489 −4.00000 −0.180886
490490 0 0
491491 −10.0000 −0.451294 −0.225647 0.974209i 0.572450π-0.572450\pi
−0.225647 + 0.974209i 0.572450π0.572450\pi
492492 0 0
493493 12.0000i 0.540453i
494494 0 0
495495 0 0
496496 0 0
497497 48.0000i 2.15309i
498498 0 0
499499 22.0000 0.984855 0.492428 0.870353i 0.336110π-0.336110\pi
0.492428 + 0.870353i 0.336110π0.336110\pi
500500 0 0
501501 −40.0000 −1.78707
502502 0 0
503503 20.0000i 0.891756i 0.895094 + 0.445878i 0.147108π0.147108\pi
−0.895094 + 0.445878i 0.852892π0.852892\pi
504504 0 0
505505 0 0
506506 0 0
507507 18.0000i 0.799408i
508508 0 0
509509 14.0000 0.620539 0.310270 0.950649i 0.399581π-0.399581\pi
0.310270 + 0.950649i 0.399581π0.399581\pi
510510 0 0
511511 −56.0000 −2.47729
512512 0 0
513513 − 8.00000i − 0.353209i
514514 0 0
515515 0 0
516516 0 0
517517 − 16.0000i − 0.703679i
518518 0 0
519519 −36.0000 −1.58022
520520 0 0
521521 −22.0000 −0.963837 −0.481919 0.876216i 0.660060π-0.660060\pi
−0.481919 + 0.876216i 0.660060π0.660060\pi
522522 0 0
523523 14.0000i 0.612177i 0.952003 + 0.306089i 0.0990204π0.0990204\pi
−0.952003 + 0.306089i 0.900980π0.900980\pi
524524 0 0
525525 0 0
526526 0 0
527527 0 0
528528 0 0
529529 7.00000 0.304348
530530 0 0
531531 14.0000 0.607548
532532 0 0
533533 − 12.0000i − 0.519778i
534534 0 0
535535 0 0
536536 0 0
537537 12.0000i 0.517838i
538538 0 0
539539 18.0000 0.775315
540540 0 0
541541 34.0000 1.46177 0.730887 0.682498i 0.239107π-0.239107\pi
0.730887 + 0.682498i 0.239107π0.239107\pi
542542 0 0
543543 4.00000i 0.171656i
544544 0 0
545545 0 0
546546 0 0
547547 38.0000i 1.62476i 0.583127 + 0.812381i 0.301829π0.301829\pi
−0.583127 + 0.812381i 0.698171π0.698171\pi
548548 0 0
549549 −2.00000 −0.0853579
550550 0 0
551551 −12.0000 −0.511217
552552 0 0
553553 32.0000i 1.36078i
554554 0 0
555555 0 0
556556 0 0
557557 − 2.00000i − 0.0847427i −0.999102 0.0423714i 0.986509π-0.986509\pi
0.999102 0.0423714i 0.0134913π-0.0134913\pi
558558 0 0
559559 −12.0000 −0.507546
560560 0 0
561561 8.00000 0.337760
562562 0 0
563563 18.0000i 0.758610i 0.925272 + 0.379305i 0.123837π0.123837\pi
−0.925272 + 0.379305i 0.876163π0.876163\pi
564564 0 0
565565 0 0
566566 0 0
567567 − 44.0000i − 1.84783i
568568 0 0
569569 −26.0000 −1.08998 −0.544988 0.838444i 0.683466π-0.683466\pi
−0.544988 + 0.838444i 0.683466π0.683466\pi
570570 0 0
571571 38.0000 1.59025 0.795125 0.606445i 0.207405π-0.207405\pi
0.795125 + 0.606445i 0.207405π0.207405\pi
572572 0 0
573573 − 32.0000i − 1.33682i
574574 0 0
575575 0 0
576576 0 0
577577 − 2.00000i − 0.0832611i −0.999133 0.0416305i 0.986745π-0.986745\pi
0.999133 0.0416305i 0.0132552π-0.0132552\pi
578578 0 0
579579 4.00000 0.166234
580580 0 0
581581 24.0000 0.995688
582582 0 0
583583 12.0000i 0.496989i
584584 0 0
585585 0 0
586586 0 0
587587 34.0000i 1.40333i 0.712507 + 0.701665i 0.247560π0.247560\pi
−0.712507 + 0.701665i 0.752440π0.752440\pi
588588 0 0
589589 0 0
590590 0 0
591591 −28.0000 −1.15177
592592 0 0
593593 18.0000i 0.739171i 0.929197 + 0.369586i 0.120500π0.120500\pi
−0.929197 + 0.369586i 0.879500π0.879500\pi
594594 0 0
595595 0 0
596596 0 0
597597 8.00000i 0.327418i
598598 0 0
599599 12.0000 0.490307 0.245153 0.969484i 0.421162π-0.421162\pi
0.245153 + 0.969484i 0.421162π0.421162\pi
600600 0 0
601601 30.0000 1.22373 0.611863 0.790964i 0.290420π-0.290420\pi
0.611863 + 0.790964i 0.290420π0.290420\pi
602602 0 0
603603 10.0000i 0.407231i
604604 0 0
605605 0 0
606606 0 0
607607 16.0000i 0.649420i 0.945814 + 0.324710i 0.105267π0.105267\pi
−0.945814 + 0.324710i 0.894733π0.894733\pi
608608 0 0
609609 −48.0000 −1.94506
610610 0 0
611611 −16.0000 −0.647291
612612 0 0
613613 34.0000i 1.37325i 0.727013 + 0.686624i 0.240908π0.240908\pi
−0.727013 + 0.686624i 0.759092π0.759092\pi
614614 0 0
615615 0 0
616616 0 0
617617 2.00000i 0.0805170i 0.999189 + 0.0402585i 0.0128181π0.0128181\pi
−0.999189 + 0.0402585i 0.987182π0.987182\pi
618618 0 0
619619 −46.0000 −1.84890 −0.924448 0.381308i 0.875474π-0.875474\pi
−0.924448 + 0.381308i 0.875474π0.875474\pi
620620 0 0
621621 −16.0000 −0.642058
622622 0 0
623623 8.00000i 0.320513i
624624 0 0
625625 0 0
626626 0 0
627627 8.00000i 0.319489i
628628 0 0
629629 20.0000 0.797452
630630 0 0
631631 −44.0000 −1.75161 −0.875806 0.482663i 0.839670π-0.839670\pi
−0.875806 + 0.482663i 0.839670π0.839670\pi
632632 0 0
633633 − 44.0000i − 1.74884i
634634 0 0
635635 0 0
636636 0 0
637637 − 18.0000i − 0.713186i
638638 0 0
639639 −12.0000 −0.474713
640640 0 0
641641 30.0000 1.18493 0.592464 0.805597i 0.298155π-0.298155\pi
0.592464 + 0.805597i 0.298155π0.298155\pi
642642 0 0
643643 42.0000i 1.65632i 0.560493 + 0.828159i 0.310612π0.310612\pi
−0.560493 + 0.828159i 0.689388π0.689388\pi
644644 0 0
645645 0 0
646646 0 0
647647 − 12.0000i − 0.471769i −0.971781 0.235884i 0.924201π-0.924201\pi
0.971781 0.235884i 0.0757987π-0.0757987\pi
648648 0 0
649649 28.0000 1.09910
650650 0 0
651651 0 0
652652 0 0
653653 42.0000i 1.64359i 0.569785 + 0.821794i 0.307026π0.307026\pi
−0.569785 + 0.821794i 0.692974π0.692974\pi
654654 0 0
655655 0 0
656656 0 0
657657 − 14.0000i − 0.546192i
658658 0 0
659659 6.00000 0.233727 0.116863 0.993148i 0.462716π-0.462716\pi
0.116863 + 0.993148i 0.462716π0.462716\pi
660660 0 0
661661 34.0000 1.32245 0.661223 0.750189i 0.270038π-0.270038\pi
0.661223 + 0.750189i 0.270038π0.270038\pi
662662 0 0
663663 − 8.00000i − 0.310694i
664664 0 0
665665 0 0
666666 0 0
667667 24.0000i 0.929284i
668668 0 0
669669 0 0
670670 0 0
671671 −4.00000 −0.154418
672672 0 0
673673 − 2.00000i − 0.0770943i −0.999257 0.0385472i 0.987727π-0.987727\pi
0.999257 0.0385472i 0.0122730π-0.0122730\pi
674674 0 0
675675 0 0
676676 0 0
677677 22.0000i 0.845529i 0.906240 + 0.422764i 0.138940π0.138940\pi
−0.906240 + 0.422764i 0.861060π0.861060\pi
678678 0 0
679679 −8.00000 −0.307012
680680 0 0
681681 36.0000 1.37952
682682 0 0
683683 − 42.0000i − 1.60709i −0.595247 0.803543i 0.702946π-0.702946\pi
0.595247 0.803543i 0.297054π-0.297054\pi
684684 0 0
685685 0 0
686686 0 0
687687 28.0000i 1.06827i
688688 0 0
689689 12.0000 0.457164
690690 0 0
691691 −6.00000 −0.228251 −0.114125 0.993466i 0.536407π-0.536407\pi
−0.114125 + 0.993466i 0.536407π0.536407\pi
692692 0 0
693693 8.00000i 0.303895i
694694 0 0
695695 0 0
696696 0 0
697697 − 12.0000i − 0.454532i
698698 0 0
699699 36.0000 1.36165
700700 0 0
701701 2.00000 0.0755390 0.0377695 0.999286i 0.487975π-0.487975\pi
0.0377695 + 0.999286i 0.487975π0.487975\pi
702702 0 0
703703 20.0000i 0.754314i
704704 0 0
705705 0 0
706706 0 0
707707 − 24.0000i − 0.902613i
708708 0 0
709709 22.0000 0.826227 0.413114 0.910679i 0.364441π-0.364441\pi
0.413114 + 0.910679i 0.364441π0.364441\pi
710710 0 0
711711 −8.00000 −0.300023
712712 0 0
713713 0 0
714714 0 0
715715 0 0
716716 0 0
717717 48.0000i 1.79259i
718718 0 0
719719 −24.0000 −0.895049 −0.447524 0.894272i 0.647694π-0.647694\pi
−0.447524 + 0.894272i 0.647694π0.647694\pi
720720 0 0
721721 16.0000 0.595871
722722 0 0
723723 − 4.00000i − 0.148762i
724724 0 0
725725 0 0
726726 0 0
727727 12.0000i 0.445055i 0.974926 + 0.222528i 0.0714308π0.0714308\pi
−0.974926 + 0.222528i 0.928569π0.928569\pi
728728 0 0
729729 −13.0000 −0.481481
730730 0 0
731731 −12.0000 −0.443836
732732 0 0
733733 − 6.00000i − 0.221615i −0.993842 0.110808i 0.964656π-0.964656\pi
0.993842 0.110808i 0.0353437π-0.0353437\pi
734734 0 0
735735 0 0
736736 0 0
737737 20.0000i 0.736709i
738738 0 0
739739 −18.0000 −0.662141 −0.331070 0.943606i 0.607410π-0.607410\pi
−0.331070 + 0.943606i 0.607410π0.607410\pi
740740 0 0
741741 8.00000 0.293887
742742 0 0
743743 44.0000i 1.61420i 0.590412 + 0.807102i 0.298965π0.298965\pi
−0.590412 + 0.807102i 0.701035π0.701035\pi
744744 0 0
745745 0 0
746746 0 0
747747 6.00000i 0.219529i
748748 0 0
749749 −8.00000 −0.292314
750750 0 0
751751 8.00000 0.291924 0.145962 0.989290i 0.453372π-0.453372\pi
0.145962 + 0.989290i 0.453372π0.453372\pi
752752 0 0
753753 − 36.0000i − 1.31191i
754754 0 0
755755 0 0
756756 0 0
757757 46.0000i 1.67190i 0.548807 + 0.835949i 0.315082π0.315082\pi
−0.548807 + 0.835949i 0.684918π0.684918\pi
758758 0 0
759759 16.0000 0.580763
760760 0 0
761761 10.0000 0.362500 0.181250 0.983437i 0.441986π-0.441986\pi
0.181250 + 0.983437i 0.441986π0.441986\pi
762762 0 0
763763 24.0000i 0.868858i
764764 0 0
765765 0 0
766766 0 0
767767 − 28.0000i − 1.01102i
768768 0 0
769769 34.0000 1.22607 0.613036 0.790055i 0.289948π-0.289948\pi
0.613036 + 0.790055i 0.289948π0.289948\pi
770770 0 0
771771 36.0000 1.29651
772772 0 0
773773 − 54.0000i − 1.94225i −0.238581 0.971123i 0.576682π-0.576682\pi
0.238581 0.971123i 0.423318π-0.423318\pi
774774 0 0
775775 0 0
776776 0 0
777777 80.0000i 2.86998i
778778 0 0
779779 12.0000 0.429945
780780 0 0
781781 −24.0000 −0.858788
782782 0 0
783783 24.0000i 0.857690i
784784 0 0
785785 0 0
786786 0 0
787787 22.0000i 0.784215i 0.919919 + 0.392108i 0.128254π0.128254\pi
−0.919919 + 0.392108i 0.871746π0.871746\pi
788788 0 0
789789 −24.0000 −0.854423
790790 0 0
791791 −8.00000 −0.284447
792792 0 0
793793 4.00000i 0.142044i
794794 0 0
795795 0 0
796796 0 0
797797 − 18.0000i − 0.637593i −0.947823 0.318796i 0.896721π-0.896721\pi
0.947823 0.318796i 0.103279π-0.103279\pi
798798 0 0
799799 −16.0000 −0.566039
800800 0 0
801801 −2.00000 −0.0706665
802802 0 0
803803 − 28.0000i − 0.988099i
804804 0 0
805805 0 0
806806 0 0
807807 − 20.0000i − 0.704033i
808808 0 0
809809 6.00000 0.210949 0.105474 0.994422i 0.466364π-0.466364\pi
0.105474 + 0.994422i 0.466364π0.466364\pi
810810 0 0
811811 −18.0000 −0.632065 −0.316033 0.948748i 0.602351π-0.602351\pi
−0.316033 + 0.948748i 0.602351π0.602351\pi
812812 0 0
813813 16.0000i 0.561144i
814814 0 0
815815 0 0
816816 0 0
817817 − 12.0000i − 0.419827i
818818 0 0
819819 8.00000 0.279543
820820 0 0
821821 10.0000 0.349002 0.174501 0.984657i 0.444169π-0.444169\pi
0.174501 + 0.984657i 0.444169π0.444169\pi
822822 0 0
823823 − 28.0000i − 0.976019i −0.872838 0.488009i 0.837723π-0.837723\pi
0.872838 0.488009i 0.162277π-0.162277\pi
824824 0 0
825825 0 0
826826 0 0
827827 − 22.0000i − 0.765015i −0.923952 0.382507i 0.875061π-0.875061\pi
0.923952 0.382507i 0.124939π-0.124939\pi
828828 0 0
829829 14.0000 0.486240 0.243120 0.969996i 0.421829π-0.421829\pi
0.243120 + 0.969996i 0.421829π0.421829\pi
830830 0 0
831831 −12.0000 −0.416275
832832 0 0
833833 − 18.0000i − 0.623663i
834834 0 0
835835 0 0
836836 0 0
837837 0 0
838838 0 0
839839 36.0000 1.24286 0.621429 0.783470i 0.286552π-0.286552\pi
0.621429 + 0.783470i 0.286552π0.286552\pi
840840 0 0
841841 7.00000 0.241379
842842 0 0
843843 − 36.0000i − 1.23991i
844844 0 0
845845 0 0
846846 0 0
847847 − 28.0000i − 0.962091i
848848 0 0
849849 −12.0000 −0.411839
850850 0 0
851851 40.0000 1.37118
852852 0 0
853853 26.0000i 0.890223i 0.895475 + 0.445112i 0.146836π0.146836\pi
−0.895475 + 0.445112i 0.853164π0.853164\pi
854854 0 0
855855 0 0
856856 0 0
857857 6.00000i 0.204956i 0.994735 + 0.102478i 0.0326771π0.0326771\pi
−0.994735 + 0.102478i 0.967323π0.967323\pi
858858 0 0
859859 50.0000 1.70598 0.852989 0.521929i 0.174787π-0.174787\pi
0.852989 + 0.521929i 0.174787π0.174787\pi
860860 0 0
861861 48.0000 1.63584
862862 0 0
863863 − 32.0000i − 1.08929i −0.838666 0.544646i 0.816664π-0.816664\pi
0.838666 0.544646i 0.183336π-0.183336\pi
864864 0 0
865865 0 0
866866 0 0
867867 26.0000i 0.883006i
868868 0 0
869869 −16.0000 −0.542763
870870 0 0
871871 20.0000 0.677674
872872 0 0
873873 − 2.00000i − 0.0676897i
874874 0 0
875875 0 0
876876 0 0
877877 22.0000i 0.742887i 0.928456 + 0.371444i 0.121137π0.121137\pi
−0.928456 + 0.371444i 0.878863π0.878863\pi
878878 0 0
879879 28.0000 0.944417
880880 0 0
881881 −46.0000 −1.54978 −0.774890 0.632096i 0.782195π-0.782195\pi
−0.774890 + 0.632096i 0.782195π0.782195\pi
882882 0 0
883883 34.0000i 1.14419i 0.820187 + 0.572096i 0.193869π0.193869\pi
−0.820187 + 0.572096i 0.806131π0.806131\pi
884884 0 0
885885 0 0
886886 0 0
887887 − 36.0000i − 1.20876i −0.796696 0.604381i 0.793421π-0.793421\pi
0.796696 0.604381i 0.206579π-0.206579\pi
888888 0 0
889889 64.0000 2.14649
890890 0 0
891891 22.0000 0.737028
892892 0 0
893893 − 16.0000i − 0.535420i
894894 0 0
895895 0 0
896896 0 0
897897 − 16.0000i − 0.534224i
898898 0 0
899899 0 0
900900 0 0
901901 12.0000 0.399778
902902 0 0
903903 − 48.0000i − 1.59734i
904904 0 0
905905 0 0
906906 0 0
907907 − 38.0000i − 1.26177i −0.775877 0.630885i 0.782692π-0.782692\pi
0.775877 0.630885i 0.217308π-0.217308\pi
908908 0 0
909909 6.00000 0.199007
910910 0 0
911911 −24.0000 −0.795155 −0.397578 0.917568i 0.630149π-0.630149\pi
−0.397578 + 0.917568i 0.630149π0.630149\pi
912912 0 0
913913 12.0000i 0.397142i
914914 0 0
915915 0 0
916916 0 0
917917 − 24.0000i − 0.792550i
918918 0 0
919919 −36.0000 −1.18753 −0.593765 0.804638i 0.702359π-0.702359\pi
−0.593765 + 0.804638i 0.702359π0.702359\pi
920920 0 0
921921 36.0000 1.18624
922922 0 0
923923 24.0000i 0.789970i
924924 0 0
925925 0 0
926926 0 0
927927 4.00000i 0.131377i
928928 0 0
929929 −30.0000 −0.984268 −0.492134 0.870519i 0.663783π-0.663783\pi
−0.492134 + 0.870519i 0.663783π0.663783\pi
930930 0 0
931931 18.0000 0.589926
932932 0 0
933933 − 56.0000i − 1.83336i
934934 0 0
935935 0 0
936936 0 0
937937 − 46.0000i − 1.50275i −0.659873 0.751377i 0.729390π-0.729390\pi
0.659873 0.751377i 0.270610π-0.270610\pi
938938 0 0
939939 −20.0000 −0.652675
940940 0 0
941941 −38.0000 −1.23876 −0.619382 0.785090i 0.712617π-0.712617\pi
−0.619382 + 0.785090i 0.712617π0.712617\pi
942942 0 0
943943 − 24.0000i − 0.781548i
944944 0 0
945945 0 0
946946 0 0
947947 14.0000i 0.454939i 0.973785 + 0.227469i 0.0730452π0.0730452\pi
−0.973785 + 0.227469i 0.926955π0.926955\pi
948948 0 0
949949 −28.0000 −0.908918
950950 0 0
951951 −12.0000 −0.389127
952952 0 0
953953 58.0000i 1.87880i 0.342817 + 0.939402i 0.388619π0.388619\pi
−0.342817 + 0.939402i 0.611381π0.611381\pi
954954 0 0
955955 0 0
956956 0 0
957957 − 24.0000i − 0.775810i
958958 0 0
959959 40.0000 1.29167
960960 0 0
961961 −31.0000 −1.00000
962962 0 0
963963 − 2.00000i − 0.0644491i
964964 0 0
965965 0 0
966966 0 0
967967 − 28.0000i − 0.900419i −0.892923 0.450210i 0.851349π-0.851349\pi
0.892923 0.450210i 0.148651π-0.148651\pi
968968 0 0
969969 8.00000 0.256997
970970 0 0
971971 38.0000 1.21948 0.609739 0.792602i 0.291274π-0.291274\pi
0.609739 + 0.792602i 0.291274π0.291274\pi
972972 0 0
973973 40.0000i 1.28234i
974974 0 0
975975 0 0
976976 0 0
977977 − 30.0000i − 0.959785i −0.877327 0.479893i 0.840676π-0.840676\pi
0.877327 0.479893i 0.159324π-0.159324\pi
978978 0 0
979979 −4.00000 −0.127841
980980 0 0
981981 −6.00000 −0.191565
982982 0 0
983983 20.0000i 0.637901i 0.947771 + 0.318950i 0.103330π0.103330\pi
−0.947771 + 0.318950i 0.896670π0.896670\pi
984984 0 0
985985 0 0
986986 0 0
987987 − 64.0000i − 2.03714i
988988 0 0
989989 −24.0000 −0.763156
990990 0 0
991991 16.0000 0.508257 0.254128 0.967170i 0.418211π-0.418211\pi
0.254128 + 0.967170i 0.418211π0.418211\pi
992992 0 0
993993 28.0000i 0.888553i
994994 0 0
995995 0 0
996996 0 0
997997 54.0000i 1.71020i 0.518465 + 0.855099i 0.326503π0.326503\pi
−0.518465 + 0.855099i 0.673497π0.673497\pi
998998 0 0
999999 40.0000 1.26554
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 3200.2.c.e.2049.2 2
4.3 odd 2 3200.2.c.k.2049.1 2
5.2 odd 4 128.2.a.d.1.1 yes 1
5.3 odd 4 3200.2.a.h.1.1 1
5.4 even 2 inner 3200.2.c.e.2049.1 2
8.3 odd 2 3200.2.c.f.2049.2 2
8.5 even 2 3200.2.c.l.2049.1 2
15.2 even 4 1152.2.a.c.1.1 1
20.3 even 4 3200.2.a.u.1.1 1
20.7 even 4 128.2.a.b.1.1 yes 1
20.19 odd 2 3200.2.c.k.2049.2 2
35.27 even 4 6272.2.a.a.1.1 1
40.3 even 4 3200.2.a.e.1.1 1
40.13 odd 4 3200.2.a.x.1.1 1
40.19 odd 2 3200.2.c.f.2049.1 2
40.27 even 4 128.2.a.c.1.1 yes 1
40.29 even 2 3200.2.c.l.2049.2 2
40.37 odd 4 128.2.a.a.1.1 1
60.47 odd 4 1152.2.a.h.1.1 1
80.27 even 4 256.2.b.a.129.2 2
80.37 odd 4 256.2.b.c.129.1 2
80.67 even 4 256.2.b.a.129.1 2
80.77 odd 4 256.2.b.c.129.2 2
120.77 even 4 1152.2.a.m.1.1 1
120.107 odd 4 1152.2.a.r.1.1 1
140.27 odd 4 6272.2.a.g.1.1 1
160.27 even 8 1024.2.e.m.769.2 4
160.37 odd 8 1024.2.e.i.769.1 4
160.67 even 8 1024.2.e.m.257.1 4
160.77 odd 8 1024.2.e.i.257.1 4
160.107 even 8 1024.2.e.m.769.1 4
160.117 odd 8 1024.2.e.i.769.2 4
160.147 even 8 1024.2.e.m.257.2 4
160.157 odd 8 1024.2.e.i.257.2 4
240.77 even 4 2304.2.d.r.1153.2 2
240.107 odd 4 2304.2.d.b.1153.1 2
240.197 even 4 2304.2.d.r.1153.1 2
240.227 odd 4 2304.2.d.b.1153.2 2
280.27 odd 4 6272.2.a.b.1.1 1
280.237 even 4 6272.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.a.a.1.1 1 40.37 odd 4
128.2.a.b.1.1 yes 1 20.7 even 4
128.2.a.c.1.1 yes 1 40.27 even 4
128.2.a.d.1.1 yes 1 5.2 odd 4
256.2.b.a.129.1 2 80.67 even 4
256.2.b.a.129.2 2 80.27 even 4
256.2.b.c.129.1 2 80.37 odd 4
256.2.b.c.129.2 2 80.77 odd 4
1024.2.e.i.257.1 4 160.77 odd 8
1024.2.e.i.257.2 4 160.157 odd 8
1024.2.e.i.769.1 4 160.37 odd 8
1024.2.e.i.769.2 4 160.117 odd 8
1024.2.e.m.257.1 4 160.67 even 8
1024.2.e.m.257.2 4 160.147 even 8
1024.2.e.m.769.1 4 160.107 even 8
1024.2.e.m.769.2 4 160.27 even 8
1152.2.a.c.1.1 1 15.2 even 4
1152.2.a.h.1.1 1 60.47 odd 4
1152.2.a.m.1.1 1 120.77 even 4
1152.2.a.r.1.1 1 120.107 odd 4
2304.2.d.b.1153.1 2 240.107 odd 4
2304.2.d.b.1153.2 2 240.227 odd 4
2304.2.d.r.1153.1 2 240.197 even 4
2304.2.d.r.1153.2 2 240.77 even 4
3200.2.a.e.1.1 1 40.3 even 4
3200.2.a.h.1.1 1 5.3 odd 4
3200.2.a.u.1.1 1 20.3 even 4
3200.2.a.x.1.1 1 40.13 odd 4
3200.2.c.e.2049.1 2 5.4 even 2 inner
3200.2.c.e.2049.2 2 1.1 even 1 trivial
3200.2.c.f.2049.1 2 40.19 odd 2
3200.2.c.f.2049.2 2 8.3 odd 2
3200.2.c.k.2049.1 2 4.3 odd 2
3200.2.c.k.2049.2 2 20.19 odd 2
3200.2.c.l.2049.1 2 8.5 even 2
3200.2.c.l.2049.2 2 40.29 even 2
6272.2.a.a.1.1 1 35.27 even 4
6272.2.a.b.1.1 1 280.27 odd 4
6272.2.a.g.1.1 1 140.27 odd 4
6272.2.a.h.1.1 1 280.237 even 4