Properties

Label 256.8.b.c.129.1
Level 256256
Weight 88
Character 256.129
Analytic conductor 79.97179.971
Analytic rank 00
Dimension 22
Inner twists 22

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [256,8,Mod(129,256)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("256.129"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: N N == 256=28 256 = 2^{8}
Weight: k k == 8 8
Character orbit: [χ][\chi] == 256.b (of order 22, degree 11, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-912] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 79.970566523979.9705665239
Analytic rank: 00
Dimension: 22
Coefficient field: Q(i)\Q(i)
Copy content comment:defining polynomial
 
Copy content 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,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 129.1
Root 1.00000i-1.00000i of defining polynomial
Character χ\chi == 256.129
Dual form 256.8.b.c.129.2

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
f(q)f(q) == q84.0000iq382.0000iq5456.000q74869.00q9+2524.00iq11+10778.0iq136888.00q1511150.0q17+4124.00iq19+38304.0iq21+81704.0q23+71401.0q25+225288.iq2799798.0iq29+40480.0q31+212016.q33+37392.0iq35419442.iq37+905352.q39141402.q41+690428.iq43+399258.iq45+682032.q47615607.q49+936600.iq51+1.81312e6iq53+206968.q55+346416.q57+966028.iq591.88767e6iq61+2.22026e6q63+883796.q65+2.96587e6iq676.86314e6iq692.54823e6q71+1.68033e6q735.99768e6iq751.15094e6iq774.03806e6q79+8.27569e6q815.38576e6iq83+914300.iq858.38303e6q87+6.47305e6q894.91477e6iq913.40032e6iq93+338168.q956.06576e6q971.22894e7iq99+O(q100)q-84.0000i q^{3} -82.0000i q^{5} -456.000 q^{7} -4869.00 q^{9} +2524.00i q^{11} +10778.0i q^{13} -6888.00 q^{15} -11150.0 q^{17} +4124.00i q^{19} +38304.0i q^{21} +81704.0 q^{23} +71401.0 q^{25} +225288. i q^{27} -99798.0i q^{29} +40480.0 q^{31} +212016. q^{33} +37392.0i q^{35} -419442. i q^{37} +905352. q^{39} -141402. q^{41} +690428. i q^{43} +399258. i q^{45} +682032. q^{47} -615607. q^{49} +936600. i q^{51} +1.81312e6i q^{53} +206968. q^{55} +346416. q^{57} +966028. i q^{59} -1.88767e6i q^{61} +2.22026e6 q^{63} +883796. q^{65} +2.96587e6i q^{67} -6.86314e6i q^{69} -2.54823e6 q^{71} +1.68033e6 q^{73} -5.99768e6i q^{75} -1.15094e6i q^{77} -4.03806e6 q^{79} +8.27569e6 q^{81} -5.38576e6i q^{83} +914300. i q^{85} -8.38303e6 q^{87} +6.47305e6 q^{89} -4.91477e6i q^{91} -3.40032e6i q^{93} +338168. q^{95} -6.06576e6 q^{97} -1.22894e7i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q912q79738q913776q1522300q17+163408q23+142802q25+80960q31+424032q33+1810704q39282804q41+1364064q471231214q49+413936q55+12131516q97+O(q100) 2 q - 912 q^{7} - 9738 q^{9} - 13776 q^{15} - 22300 q^{17} + 163408 q^{23} + 142802 q^{25} + 80960 q^{31} + 424032 q^{33} + 1810704 q^{39} - 282804 q^{41} + 1364064 q^{47} - 1231214 q^{49} + 413936 q^{55}+ \cdots - 12131516 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/256Z)×\left(\mathbb{Z}/256\mathbb{Z}\right)^\times.

nn 55 255255
χ(n)\chi(n) 1-1 11

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 − 84.0000i − 1.79620i −0.439790 0.898100i 0.644947π-0.644947\pi
0.439790 0.898100i 0.355053π-0.355053\pi
44 0 0
55 − 82.0000i − 0.293372i −0.989183 0.146686i 0.953139π-0.953139\pi
0.989183 0.146686i 0.0468607π-0.0468607\pi
66 0 0
77 −456.000 −0.502483 −0.251242 0.967924i 0.580839π-0.580839\pi
−0.251242 + 0.967924i 0.580839π0.580839\pi
88 0 0
99 −4869.00 −2.22634
1010 0 0
1111 2524.00i 0.571762i 0.958265 + 0.285881i 0.0922861π0.0922861\pi
−0.958265 + 0.285881i 0.907714π0.907714\pi
1212 0 0
1313 10778.0i 1.36062i 0.732925 + 0.680309i 0.238155π0.238155\pi
−0.732925 + 0.680309i 0.761845π0.761845\pi
1414 0 0
1515 −6888.00 −0.526955
1616 0 0
1717 −11150.0 −0.550432 −0.275216 0.961382i 0.588749π-0.588749\pi
−0.275216 + 0.961382i 0.588749π0.588749\pi
1818 0 0
1919 4124.00i 0.137937i 0.997619 + 0.0689685i 0.0219708π0.0219708\pi
−0.997619 + 0.0689685i 0.978029π0.978029\pi
2020 0 0
2121 38304.0i 0.902561i
2222 0 0
2323 81704.0 1.40022 0.700109 0.714036i 0.253135π-0.253135\pi
0.700109 + 0.714036i 0.253135π0.253135\pi
2424 0 0
2525 71401.0 0.913933
2626 0 0
2727 225288.i 2.20275i
2828 0 0
2929 − 99798.0i − 0.759852i −0.925017 0.379926i 0.875949π-0.875949\pi
0.925017 0.379926i 0.124051π-0.124051\pi
3030 0 0
3131 40480.0 0.244048 0.122024 0.992527i 0.461062π-0.461062\pi
0.122024 + 0.992527i 0.461062π0.461062\pi
3232 0 0
3333 212016. 1.02700
3434 0 0
3535 37392.0i 0.147415i
3636 0 0
3737 − 419442.i − 1.36134i −0.732591 0.680669i 0.761689π-0.761689\pi
0.732591 0.680669i 0.238311π-0.238311\pi
3838 0 0
3939 905352. 2.44394
4040 0 0
4141 −141402. −0.320414 −0.160207 0.987083i 0.551216π-0.551216\pi
−0.160207 + 0.987083i 0.551216π0.551216\pi
4242 0 0
4343 690428.i 1.32428i 0.749382 + 0.662138i 0.230351π0.230351\pi
−0.749382 + 0.662138i 0.769649π0.769649\pi
4444 0 0
4545 399258.i 0.653145i
4646 0 0
4747 682032. 0.958213 0.479107 0.877757i 0.340961π-0.340961\pi
0.479107 + 0.877757i 0.340961π0.340961\pi
4848 0 0
4949 −615607. −0.747510
5050 0 0
5151 936600.i 0.988686i
5252 0 0
5353 1.81312e6i 1.67286i 0.548071 + 0.836432i 0.315362π0.315362\pi
−0.548071 + 0.836432i 0.684638π0.684638\pi
5454 0 0
5555 206968. 0.167739
5656 0 0
5757 346416. 0.247763
5858 0 0
5959 966028.i 0.612361i 0.951973 + 0.306181i 0.0990511π0.0990511\pi
−0.951973 + 0.306181i 0.900949π0.900949\pi
6060 0 0
6161 − 1.88767e6i − 1.06481i −0.846490 0.532404i 0.821289π-0.821289\pi
0.846490 0.532404i 0.178711π-0.178711\pi
6262 0 0
6363 2.22026e6 1.11870
6464 0 0
6565 883796. 0.399168
6666 0 0
6767 2.96587e6i 1.20473i 0.798220 + 0.602365i 0.205775π0.205775\pi
−0.798220 + 0.602365i 0.794225π0.794225\pi
6868 0 0
6969 − 6.86314e6i − 2.51507i
7070 0 0
7171 −2.54823e6 −0.844957 −0.422479 0.906373i 0.638840π-0.638840\pi
−0.422479 + 0.906373i 0.638840π0.638840\pi
7272 0 0
7373 1.68033e6 0.505549 0.252775 0.967525i 0.418657π-0.418657\pi
0.252775 + 0.967525i 0.418657π0.418657\pi
7474 0 0
7575 − 5.99768e6i − 1.64161i
7676 0 0
7777 − 1.15094e6i − 0.287301i
7878 0 0
7979 −4.03806e6 −0.921464 −0.460732 0.887539i 0.652413π-0.652413\pi
−0.460732 + 0.887539i 0.652413π0.652413\pi
8080 0 0
8181 8.27569e6 1.73024
8282 0 0
8383 − 5.38576e6i − 1.03389i −0.856019 0.516945i 0.827069π-0.827069\pi
0.856019 0.516945i 0.172931π-0.172931\pi
8484 0 0
8585 914300.i 0.161481i
8686 0 0
8787 −8.38303e6 −1.36485
8888 0 0
8989 6.47305e6 0.973293 0.486647 0.873599i 0.338220π-0.338220\pi
0.486647 + 0.873599i 0.338220π0.338220\pi
9090 0 0
9191 − 4.91477e6i − 0.683688i
9292 0 0
9393 − 3.40032e6i − 0.438359i
9494 0 0
9595 338168. 0.0404669
9696 0 0
9797 −6.06576e6 −0.674814 −0.337407 0.941359i 0.609550π-0.609550\pi
−0.337407 + 0.941359i 0.609550π0.609550\pi
9898 0 0
9999 − 1.22894e7i − 1.27293i
100100 0 0
101101 9.70069e6i 0.936866i 0.883499 + 0.468433i 0.155181π0.155181\pi
−0.883499 + 0.468433i 0.844819π0.844819\pi
102102 0 0
103103 4.10159e6 0.369847 0.184924 0.982753i 0.440796π-0.440796\pi
0.184924 + 0.982753i 0.440796π0.440796\pi
104104 0 0
105105 3.14093e6 0.264786
106106 0 0
107107 − 72900.0i − 0.00575287i −0.999996 0.00287643i 0.999084π-0.999084\pi
0.999996 0.00287643i 0.000915598π-0.000915598\pi
108108 0 0
109109 − 9.55841e6i − 0.706957i −0.935443 0.353478i 0.884999π-0.884999\pi
0.935443 0.353478i 0.115001π-0.115001\pi
110110 0 0
111111 −3.52331e7 −2.44524
112112 0 0
113113 9.33890e6 0.608865 0.304433 0.952534i 0.401533π-0.401533\pi
0.304433 + 0.952534i 0.401533π0.401533\pi
114114 0 0
115115 − 6.69973e6i − 0.410785i
116116 0 0
117117 − 5.24781e7i − 3.02920i
118118 0 0
119119 5.08440e6 0.276583
120120 0 0
121121 1.31166e7 0.673089
122122 0 0
123123 1.18778e7i 0.575529i
124124 0 0
125125 − 1.22611e7i − 0.561495i
126126 0 0
127127 3.59794e7 1.55862 0.779311 0.626637i 0.215569π-0.215569\pi
0.779311 + 0.626637i 0.215569π0.215569\pi
128128 0 0
129129 5.79960e7 2.37867
130130 0 0
131131 − 676052.i − 0.0262743i −0.999914 0.0131371i 0.995818π-0.995818\pi
0.999914 0.0131371i 0.00418180π-0.00418180\pi
132132 0 0
133133 − 1.88054e6i − 0.0693111i
134134 0 0
135135 1.84736e7 0.646225
136136 0 0
137137 2.95841e7 0.982962 0.491481 0.870888i 0.336456π-0.336456\pi
0.491481 + 0.870888i 0.336456π0.336456\pi
138138 0 0
139139 3.19084e7i 1.00775i 0.863776 + 0.503876i 0.168093π0.168093\pi
−0.863776 + 0.503876i 0.831907π0.831907\pi
140140 0 0
141141 − 5.72907e7i − 1.72114i
142142 0 0
143143 −2.72037e7 −0.777949
144144 0 0
145145 −8.18344e6 −0.222919
146146 0 0
147147 5.17110e7i 1.34268i
148148 0 0
149149 − 1.16603e7i − 0.288773i −0.989521 0.144386i 0.953879π-0.953879\pi
0.989521 0.144386i 0.0461208π-0.0461208\pi
150150 0 0
151151 −1.76295e7 −0.416698 −0.208349 0.978055i 0.566809π-0.566809\pi
−0.208349 + 0.978055i 0.566809π0.566809\pi
152152 0 0
153153 5.42894e7 1.22545
154154 0 0
155155 − 3.31936e6i − 0.0715968i
156156 0 0
157157 − 6.34658e6i − 0.130885i −0.997856 0.0654427i 0.979154π-0.979154\pi
0.997856 0.0654427i 0.0208460π-0.0208460\pi
158158 0 0
159159 1.52302e8 3.00480
160160 0 0
161161 −3.72570e7 −0.703587
162162 0 0
163163 8.04234e7i 1.45454i 0.686351 + 0.727271i 0.259211π0.259211\pi
−0.686351 + 0.727271i 0.740789π0.740789\pi
164164 0 0
165165 − 1.73853e7i − 0.301293i
166166 0 0
167167 1.14767e8 1.90682 0.953411 0.301673i 0.0975451π-0.0975451\pi
0.953411 + 0.301673i 0.0975451π0.0975451\pi
168168 0 0
169169 −5.34168e7 −0.851283
170170 0 0
171171 − 2.00798e7i − 0.307095i
172172 0 0
173173 6.33755e7i 0.930594i 0.885155 + 0.465297i 0.154053π0.154053\pi
−0.885155 + 0.465297i 0.845947π0.845947\pi
174174 0 0
175175 −3.25589e7 −0.459236
176176 0 0
177177 8.11464e7 1.09992
178178 0 0
179179 − 1.13228e7i − 0.147559i −0.997275 0.0737796i 0.976494π-0.976494\pi
0.997275 0.0737796i 0.0235061π-0.0235061\pi
180180 0 0
181181 − 5.22650e6i − 0.0655143i −0.999463 0.0327571i 0.989571π-0.989571\pi
0.999463 0.0327571i 0.0104288π-0.0104288\pi
182182 0 0
183183 −1.58564e8 −1.91261
184184 0 0
185185 −3.43942e7 −0.399379
186186 0 0
187187 − 2.81426e7i − 0.314716i
188188 0 0
189189 − 1.02731e8i − 1.10684i
190190 0 0
191191 8.50301e7 0.882990 0.441495 0.897264i 0.354448π-0.354448\pi
0.441495 + 0.897264i 0.354448π0.354448\pi
192192 0 0
193193 1.15092e8 1.15237 0.576186 0.817319i 0.304540π-0.304540\pi
0.576186 + 0.817319i 0.304540π0.304540\pi
194194 0 0
195195 − 7.42389e7i − 0.716985i
196196 0 0
197197 − 1.38522e8i − 1.29088i −0.763810 0.645441i 0.776674π-0.776674\pi
0.763810 0.645441i 0.223326π-0.223326\pi
198198 0 0
199199 −2.19614e7 −0.197548 −0.0987742 0.995110i 0.531492π-0.531492\pi
−0.0987742 + 0.995110i 0.531492π0.531492\pi
200200 0 0
201201 2.49133e8 2.16394
202202 0 0
203203 4.55079e7i 0.381813i
204204 0 0
205205 1.15950e7i 0.0940007i
206206 0 0
207207 −3.97817e8 −3.11736
208208 0 0
209209 −1.04090e7 −0.0788671
210210 0 0
211211 − 6.10208e7i − 0.447187i −0.974682 0.223594i 0.928221π-0.928221\pi
0.974682 0.223594i 0.0717789π-0.0717789\pi
212212 0 0
213213 2.14051e8i 1.51771i
214214 0 0
215215 5.66151e7 0.388506
216216 0 0
217217 −1.84589e7 −0.122630
218218 0 0
219219 − 1.41147e8i − 0.908068i
220220 0 0
221221 − 1.20175e8i − 0.748928i
222222 0 0
223223 4.22448e7 0.255098 0.127549 0.991832i 0.459289π-0.459289\pi
0.127549 + 0.991832i 0.459289π0.459289\pi
224224 0 0
225225 −3.47651e8 −2.03472
226226 0 0
227227 − 2.39102e8i − 1.35673i −0.734726 0.678364i 0.762689π-0.762689\pi
0.734726 0.678364i 0.237311π-0.237311\pi
228228 0 0
229229 − 4.67889e7i − 0.257465i −0.991679 0.128733i 0.958909π-0.958909\pi
0.991679 0.128733i 0.0410909π-0.0410909\pi
230230 0 0
231231 −9.66793e7 −0.516050
232232 0 0
233233 −3.45225e8 −1.78795 −0.893977 0.448113i 0.852096π-0.852096\pi
−0.893977 + 0.448113i 0.852096π0.852096\pi
234234 0 0
235235 − 5.59266e7i − 0.281113i
236236 0 0
237237 3.39197e8i 1.65513i
238238 0 0
239239 −2.34413e8 −1.11068 −0.555340 0.831624i 0.687412π-0.687412\pi
−0.555340 + 0.831624i 0.687412π0.687412\pi
240240 0 0
241241 −1.09557e8 −0.504175 −0.252087 0.967705i 0.581117π-0.581117\pi
−0.252087 + 0.967705i 0.581117π0.581117\pi
242242 0 0
243243 − 2.02453e8i − 0.905112i
244244 0 0
245245 5.04798e7i 0.219299i
246246 0 0
247247 −4.44485e7 −0.187680
248248 0 0
249249 −4.52404e8 −1.85707
250250 0 0
251251 3.94031e8i 1.57280i 0.617720 + 0.786398i 0.288057π0.288057\pi
−0.617720 + 0.786398i 0.711943π0.711943\pi
252252 0 0
253253 2.06221e8i 0.800591i
254254 0 0
255255 7.68012e7 0.290053
256256 0 0
257257 3.19064e8 1.17250 0.586248 0.810131i 0.300604π-0.300604\pi
0.586248 + 0.810131i 0.300604π0.300604\pi
258258 0 0
259259 1.91266e8i 0.684050i
260260 0 0
261261 4.85916e8i 1.69169i
262262 0 0
263263 2.19359e8 0.743549 0.371774 0.928323i 0.378750π-0.378750\pi
0.371774 + 0.928323i 0.378750π0.378750\pi
264264 0 0
265265 1.48676e8 0.490772
266266 0 0
267267 − 5.43736e8i − 1.74823i
268268 0 0
269269 1.48033e8i 0.463687i 0.972753 + 0.231844i 0.0744757π0.0744757\pi
−0.972753 + 0.231844i 0.925524π0.925524\pi
270270 0 0
271271 3.69934e8 1.12910 0.564549 0.825399i 0.309050π-0.309050\pi
0.564549 + 0.825399i 0.309050π0.309050\pi
272272 0 0
273273 −4.12841e8 −1.22804
274274 0 0
275275 1.80216e8i 0.522552i
276276 0 0
277277 − 3.95860e8i − 1.11908i −0.828803 0.559541i 0.810977π-0.810977\pi
0.828803 0.559541i 0.189023π-0.189023\pi
278278 0 0
279279 −1.97097e8 −0.543332
280280 0 0
281281 5.97760e8 1.60714 0.803572 0.595208i 0.202930π-0.202930\pi
0.803572 + 0.595208i 0.202930π0.202930\pi
282282 0 0
283283 − 8.05797e7i − 0.211336i −0.994401 0.105668i 0.966302π-0.966302\pi
0.994401 0.105668i 0.0336981π-0.0336981\pi
284284 0 0
285285 − 2.84061e7i − 0.0726867i
286286 0 0
287287 6.44793e7 0.161003
288288 0 0
289289 −2.86016e8 −0.697025
290290 0 0
291291 5.09524e8i 1.21210i
292292 0 0
293293 7.54530e8i 1.75243i 0.481924 + 0.876213i 0.339938π0.339938\pi
−0.481924 + 0.876213i 0.660062π0.660062\pi
294294 0 0
295295 7.92143e7 0.179650
296296 0 0
297297 −5.68627e8 −1.25945
298298 0 0
299299 8.80606e8i 1.90516i
300300 0 0
301301 − 3.14835e8i − 0.665427i
302302 0 0
303303 8.14858e8 1.68280
304304 0 0
305305 −1.54789e8 −0.312385
306306 0 0
307307 8.20472e8i 1.61838i 0.587549 + 0.809188i 0.300093π0.300093\pi
−0.587549 + 0.809188i 0.699907π0.699907\pi
308308 0 0
309309 − 3.44534e8i − 0.664320i
310310 0 0
311311 6.53503e8 1.23193 0.615965 0.787773i 0.288766π-0.288766\pi
0.615965 + 0.787773i 0.288766π0.288766\pi
312312 0 0
313313 −6.63587e8 −1.22319 −0.611594 0.791172i 0.709471π-0.709471\pi
−0.611594 + 0.791172i 0.709471π0.709471\pi
314314 0 0
315315 − 1.82062e8i − 0.328195i
316316 0 0
317317 3.54718e8i 0.625426i 0.949848 + 0.312713i 0.101238π0.101238\pi
−0.949848 + 0.312713i 0.898762π0.898762\pi
318318 0 0
319319 2.51890e8 0.434454
320320 0 0
321321 −6.12360e6 −0.0103333
322322 0 0
323323 − 4.59826e7i − 0.0759250i
324324 0 0
325325 7.69560e8i 1.24351i
326326 0 0
327327 −8.02906e8 −1.26984
328328 0 0
329329 −3.11007e8 −0.481486
330330 0 0
331331 3.05543e8i 0.463100i 0.972823 + 0.231550i 0.0743797π0.0743797\pi
−0.972823 + 0.231550i 0.925620π0.925620\pi
332332 0 0
333333 2.04226e9i 3.03080i
334334 0 0
335335 2.43201e8 0.353434
336336 0 0
337337 3.54965e7 0.0505220 0.0252610 0.999681i 0.491958π-0.491958\pi
0.0252610 + 0.999681i 0.491958π0.491958\pi
338338 0 0
339339 − 7.84467e8i − 1.09364i
340340 0 0
341341 1.02172e8i 0.139537i
342342 0 0
343343 6.56252e8 0.878095
344344 0 0
345345 −5.62777e8 −0.737853
346346 0 0
347347 1.90594e8i 0.244882i 0.992476 + 0.122441i 0.0390723π0.0390723\pi
−0.992476 + 0.122441i 0.960928π0.960928\pi
348348 0 0
349349 − 8.60864e8i − 1.08404i −0.840366 0.542020i 0.817660π-0.817660\pi
0.840366 0.542020i 0.182340π-0.182340\pi
350350 0 0
351351 −2.42815e9 −2.99710
352352 0 0
353353 −1.04544e9 −1.26500 −0.632498 0.774562i 0.717970π-0.717970\pi
−0.632498 + 0.774562i 0.717970π0.717970\pi
354354 0 0
355355 2.08955e8i 0.247887i
356356 0 0
357357 − 4.27090e8i − 0.496798i
358358 0 0
359359 7.63303e8 0.870696 0.435348 0.900262i 0.356625π-0.356625\pi
0.435348 + 0.900262i 0.356625π0.356625\pi
360360 0 0
361361 8.76864e8 0.980973
362362 0 0
363363 − 1.10179e9i − 1.20900i
364364 0 0
365365 − 1.37787e8i − 0.148314i
366366 0 0
367367 1.38692e9 1.46460 0.732302 0.680980i 0.238446π-0.238446\pi
0.732302 + 0.680980i 0.238446π0.238446\pi
368368 0 0
369369 6.88486e8 0.713351
370370 0 0
371371 − 8.26782e8i − 0.840586i
372372 0 0
373373 4.77105e8i 0.476029i 0.971262 + 0.238015i 0.0764966π0.0764966\pi
−0.971262 + 0.238015i 0.923503π0.923503\pi
374374 0 0
375375 −1.02994e9 −1.00856
376376 0 0
377377 1.07562e9 1.03387
378378 0 0
379379 3.92468e8i 0.370311i 0.982709 + 0.185156i 0.0592789π0.0592789\pi
−0.982709 + 0.185156i 0.940721π0.940721\pi
380380 0 0
381381 − 3.02227e9i − 2.79960i
382382 0 0
383383 −2.10409e9 −1.91368 −0.956839 0.290617i 0.906139π-0.906139\pi
−0.956839 + 0.290617i 0.906139π0.906139\pi
384384 0 0
385385 −9.43774e7 −0.0842860
386386 0 0
387387 − 3.36169e9i − 2.94829i
388388 0 0
389389 − 1.26019e9i − 1.08546i −0.839907 0.542730i 0.817391π-0.817391\pi
0.839907 0.542730i 0.182609π-0.182609\pi
390390 0 0
391391 −9.11000e8 −0.770725
392392 0 0
393393 −5.67884e7 −0.0471939
394394 0 0
395395 3.31121e8i 0.270332i
396396 0 0
397397 9.81298e8i 0.787107i 0.919302 + 0.393554i 0.128754π0.128754\pi
−0.919302 + 0.393554i 0.871246π0.871246\pi
398398 0 0
399399 −1.57966e8 −0.124497
400400 0 0
401401 9.09981e8 0.704737 0.352369 0.935861i 0.385376π-0.385376\pi
0.352369 + 0.935861i 0.385376π0.385376\pi
402402 0 0
403403 4.36293e8i 0.332056i
404404 0 0
405405 − 6.78606e8i − 0.507604i
406406 0 0
407407 1.05867e9 0.778361
408408 0 0
409409 3.55609e7 0.0257004 0.0128502 0.999917i 0.495910π-0.495910\pi
0.0128502 + 0.999917i 0.495910π0.495910\pi
410410 0 0
411411 − 2.48507e9i − 1.76560i
412412 0 0
413413 − 4.40509e8i − 0.307701i
414414 0 0
415415 −4.41633e8 −0.303314
416416 0 0
417417 2.68031e9 1.81013
418418 0 0
419419 2.65360e9i 1.76233i 0.472813 + 0.881163i 0.343239π0.343239\pi
−0.472813 + 0.881163i 0.656761π0.656761\pi
420420 0 0
421421 − 1.12113e9i − 0.732264i −0.930563 0.366132i 0.880682π-0.880682\pi
0.930563 0.366132i 0.119318π-0.119318\pi
422422 0 0
423423 −3.32081e9 −2.13331
424424 0 0
425425 −7.96121e8 −0.503058
426426 0 0
427427 8.60778e8i 0.535049i
428428 0 0
429429 2.28511e9i 1.39735i
430430 0 0
431431 1.06344e9 0.639799 0.319900 0.947451i 0.396351π-0.396351\pi
0.319900 + 0.947451i 0.396351π0.396351\pi
432432 0 0
433433 −7.05962e8 −0.417901 −0.208951 0.977926i 0.567005π-0.567005\pi
−0.208951 + 0.977926i 0.567005π0.567005\pi
434434 0 0
435435 6.87409e8i 0.400408i
436436 0 0
437437 3.36947e8i 0.193142i
438438 0 0
439439 1.48506e9 0.837760 0.418880 0.908042i 0.362423π-0.362423\pi
0.418880 + 0.908042i 0.362423π0.362423\pi
440440 0 0
441441 2.99739e9 1.66421
442442 0 0
443443 7.22153e8i 0.394654i 0.980338 + 0.197327i 0.0632260π0.0632260\pi
−0.980338 + 0.197327i 0.936774π0.936774\pi
444444 0 0
445445 − 5.30790e8i − 0.285537i
446446 0 0
447447 −9.79462e8 −0.518694
448448 0 0
449449 −1.22968e9 −0.641109 −0.320554 0.947230i 0.603869π-0.603869\pi
−0.320554 + 0.947230i 0.603869π0.603869\pi
450450 0 0
451451 − 3.56899e8i − 0.183201i
452452 0 0
453453 1.48088e9i 0.748473i
454454 0 0
455455 −4.03011e8 −0.200575
456456 0 0
457457 8.85551e7 0.0434017 0.0217009 0.999765i 0.493092π-0.493092\pi
0.0217009 + 0.999765i 0.493092π0.493092\pi
458458 0 0
459459 − 2.51196e9i − 1.21246i
460460 0 0
461461 − 2.10937e8i − 0.100277i −0.998742 0.0501384i 0.984034π-0.984034\pi
0.998742 0.0501384i 0.0159662π-0.0159662\pi
462462 0 0
463463 3.29775e9 1.54413 0.772066 0.635543i 0.219224π-0.219224\pi
0.772066 + 0.635543i 0.219224π0.219224\pi
464464 0 0
465465 −2.78826e8 −0.128602
466466 0 0
467467 8.82873e7i 0.0401134i 0.999799 + 0.0200567i 0.00638467π0.00638467\pi
−0.999799 + 0.0200567i 0.993615π0.993615\pi
468468 0 0
469469 − 1.35244e9i − 0.605357i
470470 0 0
471471 −5.33113e8 −0.235096
472472 0 0
473473 −1.74264e9 −0.757171
474474 0 0
475475 2.94458e8i 0.126065i
476476 0 0
477477 − 8.82807e9i − 3.72436i
478478 0 0
479479 4.51507e9 1.87711 0.938557 0.345125i 0.112164π-0.112164\pi
0.938557 + 0.345125i 0.112164π0.112164\pi
480480 0 0
481481 4.52075e9 1.85226
482482 0 0
483483 3.12959e9i 1.26378i
484484 0 0
485485 4.97392e8i 0.197972i
486486 0 0
487487 3.31338e9 1.29993 0.649964 0.759965i 0.274784π-0.274784\pi
0.649964 + 0.759965i 0.274784π0.274784\pi
488488 0 0
489489 6.75557e9 2.61265
490490 0 0
491491 4.01694e9i 1.53147i 0.643154 + 0.765737i 0.277626π0.277626\pi
−0.643154 + 0.765737i 0.722374π0.722374\pi
492492 0 0
493493 1.11275e9i 0.418247i
494494 0 0
495495 −1.00773e9 −0.373443
496496 0 0
497497 1.16199e9 0.424577
498498 0 0
499499 2.70976e9i 0.976290i 0.872763 + 0.488145i 0.162326π0.162326\pi
−0.872763 + 0.488145i 0.837674π0.837674\pi
500500 0 0
501501 − 9.64045e9i − 3.42504i
502502 0 0
503503 3.04579e8 0.106712 0.0533558 0.998576i 0.483008π-0.483008\pi
0.0533558 + 0.998576i 0.483008π0.483008\pi
504504 0 0
505505 7.95456e8 0.274850
506506 0 0
507507 4.48701e9i 1.52908i
508508 0 0
509509 1.88202e8i 0.0632575i 0.999500 + 0.0316287i 0.0100694π0.0100694\pi
−0.999500 + 0.0316287i 0.989931π0.989931\pi
510510 0 0
511511 −7.66229e8 −0.254030
512512 0 0
513513 −9.29088e8 −0.303841
514514 0 0
515515 − 3.36331e8i − 0.108503i
516516 0 0
517517 1.72145e9i 0.547870i
518518 0 0
519519 5.32355e9 1.67153
520520 0 0
521521 −4.14963e9 −1.28552 −0.642758 0.766069i 0.722210π-0.722210\pi
−0.642758 + 0.766069i 0.722210π0.722210\pi
522522 0 0
523523 − 2.51360e9i − 0.768318i −0.923267 0.384159i 0.874491π-0.874491\pi
0.923267 0.384159i 0.125509π-0.125509\pi
524524 0 0
525525 2.73494e9i 0.824880i
526526 0 0
527527 −4.51352e8 −0.134332
528528 0 0
529529 3.27072e9 0.960613
530530 0 0
531531 − 4.70359e9i − 1.36332i
532532 0 0
533533 − 1.52403e9i − 0.435962i
534534 0 0
535535 −5.97780e6 −0.00168773
536536 0 0
537537 −9.51112e8 −0.265046
538538 0 0
539539 − 1.55379e9i − 0.427398i
540540 0 0
541541 1.32416e9i 0.359543i 0.983708 + 0.179772i 0.0575358π0.0575358\pi
−0.983708 + 0.179772i 0.942464π0.942464\pi
542542 0 0
543543 −4.39026e8 −0.117677
544544 0 0
545545 −7.83789e8 −0.207401
546546 0 0
547547 5.58047e8i 0.145786i 0.997340 + 0.0728929i 0.0232231π0.0232231\pi
−0.997340 + 0.0728929i 0.976777π0.976777\pi
548548 0 0
549549 9.19107e9i 2.37062i
550550 0 0
551551 4.11567e8 0.104812
552552 0 0
553553 1.84136e9 0.463020
554554 0 0
555555 2.88912e9i 0.717364i
556556 0 0
557557 3.30331e9i 0.809946i 0.914329 + 0.404973i 0.132719π0.132719\pi
−0.914329 + 0.404973i 0.867281π0.867281\pi
558558 0 0
559559 −7.44143e9 −1.80184
560560 0 0
561561 −2.36398e9 −0.565293
562562 0 0
563563 − 1.22011e8i − 0.0288152i −0.999896 0.0144076i 0.995414π-0.995414\pi
0.999896 0.0144076i 0.00458623π-0.00458623\pi
564564 0 0
565565 − 7.65790e8i − 0.178624i
566566 0 0
567567 −3.77371e9 −0.869417
568568 0 0
569569 −5.00925e8 −0.113993 −0.0569967 0.998374i 0.518152π-0.518152\pi
−0.0569967 + 0.998374i 0.518152π0.518152\pi
570570 0 0
571571 6.98702e9i 1.57060i 0.619116 + 0.785300i 0.287491π0.287491\pi
−0.619116 + 0.785300i 0.712509π0.712509\pi
572572 0 0
573573 − 7.14253e9i − 1.58603i
574574 0 0
575575 5.83375e9 1.27971
576576 0 0
577577 −8.16573e9 −1.76962 −0.884809 0.465954i 0.845711π-0.845711\pi
−0.884809 + 0.465954i 0.845711π0.845711\pi
578578 0 0
579579 − 9.66769e9i − 2.06989i
580580 0 0
581581 2.45591e9i 0.519512i
582582 0 0
583583 −4.57631e9 −0.956479
584584 0 0
585585 −4.30320e9 −0.888682
586586 0 0
587587 − 8.53182e9i − 1.74104i −0.492135 0.870519i 0.663783π-0.663783\pi
0.492135 0.870519i 0.336217π-0.336217\pi
588588 0 0
589589 1.66940e8i 0.0336632i
590590 0 0
591591 −1.16358e10 −2.31868
592592 0 0
593593 −1.71175e9 −0.337092 −0.168546 0.985694i 0.553907π-0.553907\pi
−0.168546 + 0.985694i 0.553907π0.553907\pi
594594 0 0
595595 − 4.16921e8i − 0.0811417i
596596 0 0
597597 1.84475e9i 0.354836i
598598 0 0
599599 −4.77362e9 −0.907516 −0.453758 0.891125i 0.649917π-0.649917\pi
−0.453758 + 0.891125i 0.649917π0.649917\pi
600600 0 0
601601 −7.89998e8 −0.148445 −0.0742224 0.997242i 0.523647π-0.523647\pi
−0.0742224 + 0.997242i 0.523647π0.523647\pi
602602 0 0
603603 − 1.44408e10i − 2.68214i
604604 0 0
605605 − 1.07556e9i − 0.197465i
606606 0 0
607607 1.82652e9 0.331485 0.165743 0.986169i 0.446998π-0.446998\pi
0.165743 + 0.986169i 0.446998π0.446998\pi
608608 0 0
609609 3.82266e9 0.685813
610610 0 0
611611 7.35094e9i 1.30376i
612612 0 0
613613 − 6.90339e9i − 1.21046i −0.796050 0.605231i 0.793081π-0.793081\pi
0.796050 0.605231i 0.206919π-0.206919\pi
614614 0 0
615615 9.73977e8 0.168844
616616 0 0
617617 5.69235e9 0.975649 0.487825 0.872942i 0.337791π-0.337791\pi
0.487825 + 0.872942i 0.337791π0.337791\pi
618618 0 0
619619 4.28594e9i 0.726321i 0.931727 + 0.363161i 0.118302π0.118302\pi
−0.931727 + 0.363161i 0.881698π0.881698\pi
620620 0 0
621621 1.84069e10i 3.08433i
622622 0 0
623623 −2.95171e9 −0.489064
624624 0 0
625625 4.57279e9 0.749206
626626 0 0
627627 8.74354e8i 0.141661i
628628 0 0
629629 4.67678e9i 0.749324i
630630 0 0
631631 −5.61602e8 −0.0889869 −0.0444935 0.999010i 0.514167π-0.514167\pi
−0.0444935 + 0.999010i 0.514167π0.514167\pi
632632 0 0
633633 −5.12575e9 −0.803238
634634 0 0
635635 − 2.95031e9i − 0.457256i
636636 0 0
637637 − 6.63501e9i − 1.01708i
638638 0 0
639639 1.24073e10 1.88116
640640 0 0
641641 5.17445e9 0.775998 0.387999 0.921660i 0.373166π-0.373166\pi
0.387999 + 0.921660i 0.373166π0.373166\pi
642642 0 0
643643 − 1.04374e10i − 1.54830i −0.633004 0.774148i 0.718178π-0.718178\pi
0.633004 0.774148i 0.281822π-0.281822\pi
644644 0 0
645645 − 4.75567e9i − 0.697835i
646646 0 0
647647 −9.71623e8 −0.141037 −0.0705185 0.997510i 0.522465π-0.522465\pi
−0.0705185 + 0.997510i 0.522465π0.522465\pi
648648 0 0
649649 −2.43825e9 −0.350125
650650 0 0
651651 1.55055e9i 0.220268i
652652 0 0
653653 7.25223e9i 1.01924i 0.860400 + 0.509619i 0.170214π0.170214\pi
−0.860400 + 0.509619i 0.829786π0.829786\pi
654654 0 0
655655 −5.54363e7 −0.00770814
656656 0 0
657657 −8.18151e9 −1.12552
658658 0 0
659659 3.81924e9i 0.519851i 0.965629 + 0.259925i 0.0836979π0.0836979\pi
−0.965629 + 0.259925i 0.916302π0.916302\pi
660660 0 0
661661 1.07881e10i 1.45292i 0.687210 + 0.726459i 0.258835π0.258835\pi
−0.687210 + 0.726459i 0.741165π0.741165\pi
662662 0 0
663663 −1.00947e10 −1.34523
664664 0 0
665665 −1.54205e8 −0.0203339
666666 0 0
667667 − 8.15390e9i − 1.06396i
668668 0 0
669669 − 3.54857e9i − 0.458207i
670670 0 0
671671 4.76448e9 0.608817
672672 0 0
673673 −6.34833e9 −0.802798 −0.401399 0.915903i 0.631476π-0.631476\pi
−0.401399 + 0.915903i 0.631476π0.631476\pi
674674 0 0
675675 1.60858e10i 2.01316i
676676 0 0
677677 8.82566e9i 1.09317i 0.837404 + 0.546584i 0.184072π0.184072\pi
−0.837404 + 0.546584i 0.815928π0.815928\pi
678678 0 0
679679 2.76599e9 0.339083
680680 0 0
681681 −2.00846e10 −2.43696
682682 0 0
683683 − 4.92331e9i − 0.591268i −0.955301 0.295634i 0.904469π-0.904469\pi
0.955301 0.295634i 0.0955309π-0.0955309\pi
684684 0 0
685685 − 2.42590e9i − 0.288374i
686686 0 0
687687 −3.93027e9 −0.462459
688688 0 0
689689 −1.95418e10 −2.27613
690690 0 0
691691 − 5.68449e9i − 0.655418i −0.944779 0.327709i 0.893723π-0.893723\pi
0.944779 0.327709i 0.106277π-0.106277\pi
692692 0 0
693693 5.60395e9i 0.639628i
694694 0 0
695695 2.61649e9 0.295646
696696 0 0
697697 1.57663e9 0.176366
698698 0 0
699699 2.89989e10i 3.21152i
700700 0 0
701701 1.70567e9i 0.187017i 0.995618 + 0.0935085i 0.0298082π0.0298082\pi
−0.995618 + 0.0935085i 0.970192π0.970192\pi
702702 0 0
703703 1.72978e9 0.187779
704704 0 0
705705 −4.69784e9 −0.504936
706706 0 0
707707 − 4.42351e9i − 0.470760i
708708 0 0
709709 4.52189e9i 0.476495i 0.971204 + 0.238248i 0.0765730π0.0765730\pi
−0.971204 + 0.238248i 0.923427π0.923427\pi
710710 0 0
711711 1.96613e10 2.05149
712712 0 0
713713 3.30738e9 0.341720
714714 0 0
715715 2.23070e9i 0.228229i
716716 0 0
717717 1.96907e10i 1.99500i
718718 0 0
719719 3.09206e9 0.310239 0.155120 0.987896i 0.450424π-0.450424\pi
0.155120 + 0.987896i 0.450424π0.450424\pi
720720 0 0
721721 −1.87033e9 −0.185842
722722 0 0
723723 9.20280e9i 0.905599i
724724 0 0
725725 − 7.12568e9i − 0.694453i
726726 0 0
727727 1.44622e10 1.39593 0.697965 0.716132i 0.254089π-0.254089\pi
0.697965 + 0.716132i 0.254089π0.254089\pi
728728 0 0
729729 1.09288e9 0.104478
730730 0 0
731731 − 7.69827e9i − 0.728924i
732732 0 0
733733 − 3.15415e9i − 0.295814i −0.989001 0.147907i 0.952746π-0.952746\pi
0.989001 0.147907i 0.0472536π-0.0472536\pi
734734 0 0
735735 4.24030e9 0.393905
736736 0 0
737737 −7.48585e9 −0.688819
738738 0 0
739739 1.54236e10i 1.40582i 0.711277 + 0.702912i 0.248117π0.248117\pi
−0.711277 + 0.702912i 0.751883π0.751883\pi
740740 0 0
741741 3.73367e9i 0.337111i
742742 0 0
743743 −1.59520e10 −1.42677 −0.713385 0.700772i 0.752839π-0.752839\pi
−0.713385 + 0.700772i 0.752839π0.752839\pi
744744 0 0
745745 −9.56141e8 −0.0847179
746746 0 0
747747 2.62233e10i 2.30179i
748748 0 0
749749 3.32424e7i 0.00289072i
750750 0 0
751751 −6.13964e9 −0.528936 −0.264468 0.964395i 0.585196π-0.585196\pi
−0.264468 + 0.964395i 0.585196π0.585196\pi
752752 0 0
753753 3.30986e10 2.82506
754754 0 0
755755 1.44562e9i 0.122248i
756756 0 0
757757 − 1.42818e10i − 1.19660i −0.801273 0.598299i 0.795843π-0.795843\pi
0.801273 0.598299i 0.204157π-0.204157\pi
758758 0 0
759759 1.73226e10 1.43802
760760 0 0
761761 −1.47536e10 −1.21353 −0.606767 0.794880i 0.707534π-0.707534\pi
−0.606767 + 0.794880i 0.707534π0.707534\pi
762762 0 0
763763 4.35863e9i 0.355234i
764764 0 0
765765 − 4.45173e9i − 0.359512i
766766 0 0
767767 −1.04118e10 −0.833190
768768 0 0
769769 1.97592e10 1.56685 0.783424 0.621487i 0.213471π-0.213471\pi
0.783424 + 0.621487i 0.213471π0.213471\pi
770770 0 0
771771 − 2.68014e10i − 2.10604i
772772 0 0
773773 1.01370e10i 0.789374i 0.918816 + 0.394687i 0.129147π0.129147\pi
−0.918816 + 0.394687i 0.870853π0.870853\pi
774774 0 0
775775 2.89031e9 0.223043
776776 0 0
777777 1.60663e10 1.22869
778778 0 0
779779 − 5.83142e8i − 0.0441970i
780780 0 0
781781 − 6.43174e9i − 0.483114i
782782 0 0
783783 2.24833e10 1.67376
784784 0 0
785785 −5.20420e8 −0.0383981
786786 0 0
787787 − 1.27882e10i − 0.935188i −0.883943 0.467594i 0.845121π-0.845121\pi
0.883943 0.467594i 0.154879π-0.154879\pi
788788 0 0
789789 − 1.84261e10i − 1.33556i
790790 0 0
791791 −4.25854e9 −0.305945
792792 0 0
793793 2.03453e10 1.44880
794794 0 0
795795 − 1.24888e10i − 0.881524i
796796 0 0
797797 7.38617e9i 0.516791i 0.966039 + 0.258396i 0.0831938π0.0831938\pi
−0.966039 + 0.258396i 0.916806π0.916806\pi
798798 0 0
799799 −7.60466e9 −0.527431
800800 0 0
801801 −3.15173e10 −2.16688
802802 0 0
803803 4.24114e9i 0.289054i
804804 0 0
805805 3.05508e9i 0.206413i
806806 0 0
807807 1.24348e10 0.832875
808808 0 0
809809 −1.53742e10 −1.02087 −0.510437 0.859915i 0.670516π-0.670516\pi
−0.510437 + 0.859915i 0.670516π0.670516\pi
810810 0 0
811811 − 9.77882e9i − 0.643744i −0.946783 0.321872i 0.895688π-0.895688\pi
0.946783 0.321872i 0.104312π-0.104312\pi
812812 0 0
813813 − 3.10745e10i − 2.02809i
814814 0 0
815815 6.59472e9 0.426722
816816 0 0
817817 −2.84733e9 −0.182667
818818 0 0
819819 2.39300e10i 1.52212i
820820 0 0
821821 1.83470e10i 1.15708i 0.815654 + 0.578540i 0.196377π0.196377\pi
−0.815654 + 0.578540i 0.803623π0.803623\pi
822822 0 0
823823 −3.16960e10 −1.98201 −0.991004 0.133829i 0.957273π-0.957273\pi
−0.991004 + 0.133829i 0.957273π0.957273\pi
824824 0 0
825825 1.51382e10 0.938608
826826 0 0
827827 6.12845e9i 0.376774i 0.982095 + 0.188387i 0.0603260π0.0603260\pi
−0.982095 + 0.188387i 0.939674π0.939674\pi
828828 0 0
829829 1.24652e10i 0.759904i 0.925006 + 0.379952i 0.124060π0.124060\pi
−0.925006 + 0.379952i 0.875940π0.875940\pi
830830 0 0
831831 −3.32522e10 −2.01010
832832 0 0
833833 6.86402e9 0.411454
834834 0 0
835835 − 9.41091e9i − 0.559409i
836836 0 0
837837 9.11966e9i 0.537575i
838838 0 0
839839 −1.82237e10 −1.06530 −0.532648 0.846337i 0.678803π-0.678803\pi
−0.532648 + 0.846337i 0.678803π0.678803\pi
840840 0 0
841841 7.29024e9 0.422625
842842 0 0
843843 − 5.02118e10i − 2.88675i
844844 0 0
845845 4.38017e9i 0.249743i
846846 0 0
847847 −5.98117e9 −0.338216
848848 0 0
849849 −6.76870e9 −0.379602
850850 0 0
851851 − 3.42701e10i − 1.90617i
852852 0 0
853853 − 2.48619e10i − 1.37155i −0.727812 0.685777i 0.759463π-0.759463\pi
0.727812 0.685777i 0.240537π-0.240537\pi
854854 0 0
855855 −1.64654e9 −0.0900930
856856 0 0
857857 −2.96761e10 −1.61055 −0.805275 0.592902i 0.797982π-0.797982\pi
−0.805275 + 0.592902i 0.797982π0.797982\pi
858858 0 0
859859 − 1.14772e10i − 0.617819i −0.951091 0.308910i 0.900036π-0.900036\pi
0.951091 0.308910i 0.0999640π-0.0999640\pi
860860 0 0
861861 − 5.41626e9i − 0.289194i
862862 0 0
863863 −2.13485e10 −1.13066 −0.565328 0.824866i 0.691250π-0.691250\pi
−0.565328 + 0.824866i 0.691250π0.691250\pi
864864 0 0
865865 5.19679e9 0.273010
866866 0 0
867867 2.40254e10i 1.25200i
868868 0 0
869869 − 1.01921e10i − 0.526858i
870870 0 0
871871 −3.19661e10 −1.63918
872872 0 0
873873 2.95342e10 1.50236
874874 0 0
875875 5.59108e9i 0.282142i
876876 0 0
877877 7.92753e9i 0.396862i 0.980115 + 0.198431i 0.0635846π0.0635846\pi
−0.980115 + 0.198431i 0.936415π0.936415\pi
878878 0 0
879879 6.33805e10 3.14771
880880 0 0
881881 7.32045e9 0.360680 0.180340 0.983604i 0.442280π-0.442280\pi
0.180340 + 0.983604i 0.442280π0.442280\pi
882882 0 0
883883 − 3.54988e9i − 0.173521i −0.996229 0.0867604i 0.972349π-0.972349\pi
0.996229 0.0867604i 0.0276514π-0.0276514\pi
884884 0 0
885885 − 6.65400e9i − 0.322687i
886886 0 0
887887 5.80634e9 0.279364 0.139682 0.990196i 0.455392π-0.455392\pi
0.139682 + 0.990196i 0.455392π0.455392\pi
888888 0 0
889889 −1.64066e10 −0.783182
890890 0 0
891891 2.08878e10i 0.989285i
892892 0 0
893893 2.81270e9i 0.132173i
894894 0 0
895895 −9.28466e8 −0.0432898
896896 0 0
897897 7.39709e10 3.42206
898898 0 0
899899 − 4.03982e9i − 0.185440i
900900 0 0
901901 − 2.02163e10i − 0.920798i
902902 0 0
903903 −2.64462e10 −1.19524
904904 0 0
905905 −4.28573e8 −0.0192201
906906 0 0
907907 − 1.78240e10i − 0.793196i −0.917992 0.396598i 0.870191π-0.870191\pi
0.917992 0.396598i 0.129809π-0.129809\pi
908908 0 0
909909 − 4.72326e10i − 2.08578i
910910 0 0
911911 −1.87703e10 −0.822538 −0.411269 0.911514i 0.634914π-0.634914\pi
−0.411269 + 0.911514i 0.634914π0.634914\pi
912912 0 0
913913 1.35937e10 0.591138
914914 0 0
915915 1.30023e10i 0.561107i
916916 0 0
917917 3.08280e8i 0.0132024i
918918 0 0
919919 3.75844e10 1.59736 0.798681 0.601754i 0.205531π-0.205531\pi
0.798681 + 0.601754i 0.205531π0.205531\pi
920920 0 0
921921 6.89197e10 2.90693
922922 0 0
923923 − 2.74648e10i − 1.14966i
924924 0 0
925925 − 2.99486e10i − 1.24417i
926926 0 0
927927 −1.99707e10 −0.823404
928928 0 0
929929 −1.92372e10 −0.787205 −0.393602 0.919281i 0.628771π-0.628771\pi
−0.393602 + 0.919281i 0.628771π0.628771\pi
930930 0 0
931931 − 2.53876e9i − 0.103109i
932932 0 0
933933 − 5.48943e10i − 2.21279i
934934 0 0
935935 −2.30769e9 −0.0923289
936936 0 0
937937 −1.04732e9 −0.0415900 −0.0207950 0.999784i 0.506620π-0.506620\pi
−0.0207950 + 0.999784i 0.506620π0.506620\pi
938938 0 0
939939 5.57413e10i 2.19709i
940940 0 0
941941 7.97861e9i 0.312150i 0.987745 + 0.156075i 0.0498842π0.0498842\pi
−0.987745 + 0.156075i 0.950116π0.950116\pi
942942 0 0
943943 −1.15531e10 −0.448650
944944 0 0
945945 −8.42397e9 −0.324717
946946 0 0
947947 − 4.26943e9i − 0.163360i −0.996659 0.0816799i 0.973971π-0.973971\pi
0.996659 0.0816799i 0.0260285π-0.0260285\pi
948948 0 0
949949 1.81106e10i 0.687860i
950950 0 0
951951 2.97963e10 1.12339
952952 0 0
953953 1.06048e10 0.396897 0.198449 0.980111i 0.436410π-0.436410\pi
0.198449 + 0.980111i 0.436410π0.436410\pi
954954 0 0
955955 − 6.97247e9i − 0.259045i
956956 0 0
957957 − 2.11588e10i − 0.780367i
958958 0 0
959959 −1.34904e10 −0.493922
960960 0 0
961961 −2.58740e10 −0.940441
962962 0 0
963963 3.54950e8i 0.0128078i
964964 0 0
965965 − 9.43750e9i − 0.338074i
966966 0 0
967967 −1.65090e10 −0.587123 −0.293562 0.955940i 0.594841π-0.594841\pi
−0.293562 + 0.955940i 0.594841π0.594841\pi
968968 0 0
969969 −3.86254e9 −0.136377
970970 0 0
971971 2.46094e10i 0.862649i 0.902197 + 0.431324i 0.141954π0.141954\pi
−0.902197 + 0.431324i 0.858046π0.858046\pi
972972 0 0
973973 − 1.45503e10i − 0.506379i
974974 0 0
975975 6.46430e10 2.23360
976976 0 0
977977 −2.53886e10 −0.870979 −0.435489 0.900194i 0.643425π-0.643425\pi
−0.435489 + 0.900194i 0.643425π0.643425\pi
978978 0 0
979979 1.63380e10i 0.556492i
980980 0 0
981981 4.65399e10i 1.57392i
982982 0 0
983983 1.87585e10 0.629884 0.314942 0.949111i 0.398015π-0.398015\pi
0.314942 + 0.949111i 0.398015π0.398015\pi
984984 0 0
985985 −1.13588e10 −0.378709
986986 0 0
987987 2.61246e10i 0.864846i
988988 0 0
989989 5.64107e10i 1.85428i
990990 0 0
991991 3.59792e9 0.117434 0.0587170 0.998275i 0.481299π-0.481299\pi
0.0587170 + 0.998275i 0.481299π0.481299\pi
992992 0 0
993993 2.56656e10 0.831821
994994 0 0
995995 1.80083e9i 0.0579552i
996996 0 0
997997 − 1.34287e10i − 0.429143i −0.976708 0.214571i 0.931165π-0.931165\pi
0.976708 0.214571i 0.0688354π-0.0688354\pi
998998 0 0
999999 9.44952e10 2.99868
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 256.8.b.c.129.1 2
4.3 odd 2 256.8.b.e.129.2 2
8.3 odd 2 256.8.b.e.129.1 2
8.5 even 2 inner 256.8.b.c.129.2 2
16.3 odd 4 8.8.a.a.1.1 1
16.5 even 4 64.8.a.a.1.1 1
16.11 odd 4 64.8.a.g.1.1 1
16.13 even 4 16.8.a.c.1.1 1
48.5 odd 4 576.8.a.k.1.1 1
48.11 even 4 576.8.a.j.1.1 1
48.29 odd 4 144.8.a.g.1.1 1
48.35 even 4 72.8.a.d.1.1 1
80.3 even 4 200.8.c.a.49.1 2
80.13 odd 4 400.8.c.b.49.2 2
80.19 odd 4 200.8.a.i.1.1 1
80.29 even 4 400.8.a.b.1.1 1
80.67 even 4 200.8.c.a.49.2 2
80.77 odd 4 400.8.c.b.49.1 2
112.83 even 4 392.8.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.a.a.1.1 1 16.3 odd 4
16.8.a.c.1.1 1 16.13 even 4
64.8.a.a.1.1 1 16.5 even 4
64.8.a.g.1.1 1 16.11 odd 4
72.8.a.d.1.1 1 48.35 even 4
144.8.a.g.1.1 1 48.29 odd 4
200.8.a.i.1.1 1 80.19 odd 4
200.8.c.a.49.1 2 80.3 even 4
200.8.c.a.49.2 2 80.67 even 4
256.8.b.c.129.1 2 1.1 even 1 trivial
256.8.b.c.129.2 2 8.5 even 2 inner
256.8.b.e.129.1 2 8.3 odd 2
256.8.b.e.129.2 2 4.3 odd 2
392.8.a.d.1.1 1 112.83 even 4
400.8.a.b.1.1 1 80.29 even 4
400.8.c.b.49.1 2 80.77 odd 4
400.8.c.b.49.2 2 80.13 odd 4
576.8.a.j.1.1 1 48.11 even 4
576.8.a.k.1.1 1 48.5 odd 4