Properties

Label 28.8.a.b.1.2
Level 2828
Weight 88
Character 28.1
Self dual yes
Analytic conductor 8.7478.747
Analytic rank 11
Dimension 22
CM no
Inner twists 11

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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] = [28,8,Mod(1,28)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(28, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) 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("28.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: N N == 28=227 28 = 2^{2} \cdot 7
Weight: k k == 8 8
Character orbit: [χ][\chi] == 28.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 8.746780713568.74678071356
Analytic rank: 11
Dimension: 22
Coefficient field: Q(1009)\Q(\sqrt{1009})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2x252 x^{2} - x - 252 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: yes
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.2
Root 15.3824-15.3824 of defining polynomial
Character χ\chi == 28.1

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) == q+38.7648q3496.412q5343.000q7684.293q9+4035.19q1114469.7q1319243.3q1526750.1q17+36610.0q1913296.3q2134920.1q23+168300.q25111305.q27+35874.3q29+198598.q31+156423.q33+170269.q35114413.q37560915.q39+245990.q41+30224.0q43+339692.q45305233.q47+117649.q491.03696e6q511.21053e6q532.00312e6q55+1.41918e6q571.53557e6q5945086.4q61+234713.q63+7.18295e6q65+3.08501e6q671.35367e6q691.69051e6q713.86932e6q73+6.52412e6q751.38407e6q77+2.00910e6q792.81816e6q818.28720e6q83+1.32791e7q85+1.39066e6q876.50624e6q89+4.96311e6q91+7.69862e6q931.81737e7q951.14882e7q972.76125e6q99+O(q100)q+38.7648 q^{3} -496.412 q^{5} -343.000 q^{7} -684.293 q^{9} +4035.19 q^{11} -14469.7 q^{13} -19243.3 q^{15} -26750.1 q^{17} +36610.0 q^{19} -13296.3 q^{21} -34920.1 q^{23} +168300. q^{25} -111305. q^{27} +35874.3 q^{29} +198598. q^{31} +156423. q^{33} +170269. q^{35} -114413. q^{37} -560915. q^{39} +245990. q^{41} +30224.0 q^{43} +339692. q^{45} -305233. q^{47} +117649. q^{49} -1.03696e6 q^{51} -1.21053e6 q^{53} -2.00312e6 q^{55} +1.41918e6 q^{57} -1.53557e6 q^{59} -45086.4 q^{61} +234713. q^{63} +7.18295e6 q^{65} +3.08501e6 q^{67} -1.35367e6 q^{69} -1.69051e6 q^{71} -3.86932e6 q^{73} +6.52412e6 q^{75} -1.38407e6 q^{77} +2.00910e6 q^{79} -2.81816e6 q^{81} -8.28720e6 q^{83} +1.32791e7 q^{85} +1.39066e6 q^{87} -6.50624e6 q^{89} +4.96311e6 q^{91} +7.69862e6 q^{93} -1.81737e7 q^{95} -1.14882e7 q^{97} -2.76125e6 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+14q3294q5686q72258q93492q1116170q1324256q1529232q173206q194802q219360q23+131146q2518172q27+184704q29+165060q31++9084332q99+O(q100) 2 q + 14 q^{3} - 294 q^{5} - 686 q^{7} - 2258 q^{9} - 3492 q^{11} - 16170 q^{13} - 24256 q^{15} - 29232 q^{17} - 3206 q^{19} - 4802 q^{21} - 9360 q^{23} + 131146 q^{25} - 18172 q^{27} + 184704 q^{29} + 165060 q^{31}+ \cdots + 9084332 q^{99}+O(q^{100}) Copy content Toggle raw display

Coefficient data

For each nn we display the coefficients of the qq-expansion ana_n, the Satake parameters αp\alpha_p, and the Satake angles θp=Arg(αp)\theta_p = \textrm{Arg}(\alpha_p).



Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)
Significant digits:
nn ana_n an/n(k1)/2a_n / n^{(k-1)/2} αn \alpha_n θn \theta_n
pp apa_p ap/p(k1)/2a_p / p^{(k-1)/2} αp \alpha_p θp \theta_p
22 0 0
33 38.7648 0.828920 0.414460 0.910067i 0.363970π-0.363970\pi
0.414460 + 0.910067i 0.363970π0.363970\pi
44 0 0
55 −496.412 −1.77602 −0.888009 0.459825i 0.847912π-0.847912\pi
−0.888009 + 0.459825i 0.847912π0.847912\pi
66 0 0
77 −343.000 −0.377964
88 0 0
99 −684.293 −0.312891
1010 0 0
1111 4035.19 0.914091 0.457045 0.889443i 0.348908π-0.348908\pi
0.457045 + 0.889443i 0.348908π0.348908\pi
1212 0 0
1313 −14469.7 −1.82666 −0.913331 0.407217i 0.866499π-0.866499\pi
−0.913331 + 0.407217i 0.866499π0.866499\pi
1414 0 0
1515 −19243.3 −1.47218
1616 0 0
1717 −26750.1 −1.32055 −0.660275 0.751024i 0.729560π-0.729560\pi
−0.660275 + 0.751024i 0.729560π0.729560\pi
1818 0 0
1919 36610.0 1.22451 0.612255 0.790661i 0.290263π-0.290263\pi
0.612255 + 0.790661i 0.290263π0.290263\pi
2020 0 0
2121 −13296.3 −0.313302
2222 0 0
2323 −34920.1 −0.598449 −0.299225 0.954183i 0.596728π-0.596728\pi
−0.299225 + 0.954183i 0.596728π0.596728\pi
2424 0 0
2525 168300. 2.15424
2626 0 0
2727 −111305. −1.08828
2828 0 0
2929 35874.3 0.273143 0.136571 0.990630i 0.456392π-0.456392\pi
0.136571 + 0.990630i 0.456392π0.456392\pi
3030 0 0
3131 198598. 1.19732 0.598660 0.801004i 0.295700π-0.295700\pi
0.598660 + 0.801004i 0.295700π0.295700\pi
3232 0 0
3333 156423. 0.757708
3434 0 0
3535 170269. 0.671272
3636 0 0
3737 −114413. −0.371338 −0.185669 0.982612i 0.559445π-0.559445\pi
−0.185669 + 0.982612i 0.559445π0.559445\pi
3838 0 0
3939 −560915. −1.51416
4040 0 0
4141 245990. 0.557409 0.278704 0.960377i 0.410095π-0.410095\pi
0.278704 + 0.960377i 0.410095π0.410095\pi
4242 0 0
4343 30224.0 0.0579711 0.0289856 0.999580i 0.490772π-0.490772\pi
0.0289856 + 0.999580i 0.490772π0.490772\pi
4444 0 0
4545 339692. 0.555701
4646 0 0
4747 −305233. −0.428833 −0.214417 0.976742i 0.568785π-0.568785\pi
−0.214417 + 0.976742i 0.568785π0.568785\pi
4848 0 0
4949 117649. 0.142857
5050 0 0
5151 −1.03696e6 −1.09463
5252 0 0
5353 −1.21053e6 −1.11689 −0.558444 0.829542i 0.688601π-0.688601\pi
−0.558444 + 0.829542i 0.688601π0.688601\pi
5454 0 0
5555 −2.00312e6 −1.62344
5656 0 0
5757 1.41918e6 1.01502
5858 0 0
5959 −1.53557e6 −0.973391 −0.486696 0.873572i 0.661798π-0.661798\pi
−0.486696 + 0.873572i 0.661798π0.661798\pi
6060 0 0
6161 −45086.4 −0.0254326 −0.0127163 0.999919i 0.504048π-0.504048\pi
−0.0127163 + 0.999919i 0.504048π0.504048\pi
6262 0 0
6363 234713. 0.118262
6464 0 0
6565 7.18295e6 3.24419
6666 0 0
6767 3.08501e6 1.25312 0.626562 0.779371i 0.284461π-0.284461\pi
0.626562 + 0.779371i 0.284461π0.284461\pi
6868 0 0
6969 −1.35367e6 −0.496067
7070 0 0
7171 −1.69051e6 −0.560550 −0.280275 0.959920i 0.590426π-0.590426\pi
−0.280275 + 0.959920i 0.590426π0.590426\pi
7272 0 0
7373 −3.86932e6 −1.16414 −0.582069 0.813139i 0.697757π-0.697757\pi
−0.582069 + 0.813139i 0.697757π0.697757\pi
7474 0 0
7575 6.52412e6 1.78570
7676 0 0
7777 −1.38407e6 −0.345494
7878 0 0
7979 2.00910e6 0.458464 0.229232 0.973372i 0.426378π-0.426378\pi
0.229232 + 0.973372i 0.426378π0.426378\pi
8080 0 0
8181 −2.81816e6 −0.589208
8282 0 0
8383 −8.28720e6 −1.59087 −0.795435 0.606039i 0.792758π-0.792758\pi
−0.795435 + 0.606039i 0.792758π0.792758\pi
8484 0 0
8585 1.32791e7 2.34532
8686 0 0
8787 1.39066e6 0.226414
8888 0 0
8989 −6.50624e6 −0.978285 −0.489142 0.872204i 0.662690π-0.662690\pi
−0.489142 + 0.872204i 0.662690π0.662690\pi
9090 0 0
9191 4.96311e6 0.690414
9292 0 0
9393 7.69862e6 0.992482
9494 0 0
9595 −1.81737e7 −2.17475
9696 0 0
9797 −1.14882e7 −1.27806 −0.639028 0.769183i 0.720663π-0.720663\pi
−0.639028 + 0.769183i 0.720663π0.720663\pi
9898 0 0
9999 −2.76125e6 −0.286011
100100 0 0
101101 −6.90453e6 −0.666821 −0.333410 0.942782i 0.608199π-0.608199\pi
−0.333410 + 0.942782i 0.608199π0.608199\pi
102102 0 0
103103 4.17396e6 0.376372 0.188186 0.982133i 0.439739π-0.439739\pi
0.188186 + 0.982133i 0.439739π0.439739\pi
104104 0 0
105105 6.60045e6 0.556431
106106 0 0
107107 −6.65685e6 −0.525322 −0.262661 0.964888i 0.584600π-0.584600\pi
−0.262661 + 0.964888i 0.584600π0.584600\pi
108108 0 0
109109 −1.11394e7 −0.823893 −0.411946 0.911208i 0.635151π-0.635151\pi
−0.411946 + 0.911208i 0.635151π0.635151\pi
110110 0 0
111111 −4.43520e6 −0.307810
112112 0 0
113113 4.18354e6 0.272753 0.136377 0.990657i 0.456454π-0.456454\pi
0.136377 + 0.990657i 0.456454π0.456454\pi
114114 0 0
115115 1.73347e7 1.06286
116116 0 0
117117 9.90153e6 0.571547
118118 0 0
119119 9.17530e6 0.499121
120120 0 0
121121 −3.20444e6 −0.164439
122122 0 0
123123 9.53574e6 0.462047
124124 0 0
125125 −4.47641e7 −2.04996
126126 0 0
127127 3.65300e7 1.58247 0.791237 0.611510i 0.209438π-0.209438\pi
0.791237 + 0.611510i 0.209438π0.209438\pi
128128 0 0
129129 1.17162e6 0.0480534
130130 0 0
131131 5.08168e7 1.97496 0.987479 0.157748i 0.0504234π-0.0504234\pi
0.987479 + 0.157748i 0.0504234π0.0504234\pi
132132 0 0
133133 −1.25572e7 −0.462821
134134 0 0
135135 5.52532e7 1.93281
136136 0 0
137137 3.03952e7 1.00991 0.504955 0.863146i 0.331509π-0.331509\pi
0.504955 + 0.863146i 0.331509π0.331509\pi
138138 0 0
139139 −2.92052e7 −0.922377 −0.461189 0.887302i 0.652577π-0.652577\pi
−0.461189 + 0.887302i 0.652577π0.652577\pi
140140 0 0
141141 −1.18323e7 −0.355469
142142 0 0
143143 −5.83880e7 −1.66973
144144 0 0
145145 −1.78084e7 −0.485107
146146 0 0
147147 4.56064e6 0.118417
148148 0 0
149149 −3.70950e7 −0.918677 −0.459339 0.888261i 0.651914π-0.651914\pi
−0.459339 + 0.888261i 0.651914π0.651914\pi
150150 0 0
151151 1.82394e7 0.431114 0.215557 0.976491i 0.430843π-0.430843\pi
0.215557 + 0.976491i 0.430843π0.430843\pi
152152 0 0
153153 1.83049e7 0.413189
154154 0 0
155155 −9.85867e7 −2.12646
156156 0 0
157157 2.12064e7 0.437340 0.218670 0.975799i 0.429828π-0.429828\pi
0.218670 + 0.975799i 0.429828π0.429828\pi
158158 0 0
159159 −4.69258e7 −0.925810
160160 0 0
161161 1.19776e7 0.226193
162162 0 0
163163 4.50953e7 0.815595 0.407797 0.913072i 0.366297π-0.366297\pi
0.407797 + 0.913072i 0.366297π0.366297\pi
164164 0 0
165165 −7.76503e7 −1.34570
166166 0 0
167167 6.32823e7 1.05142 0.525708 0.850665i 0.323801π-0.323801\pi
0.525708 + 0.850665i 0.323801π0.323801\pi
168168 0 0
169169 1.46624e8 2.33670
170170 0 0
171171 −2.50520e7 −0.383138
172172 0 0
173173 −3.34193e7 −0.490722 −0.245361 0.969432i 0.578907π-0.578907\pi
−0.245361 + 0.969432i 0.578907π0.578907\pi
174174 0 0
175175 −5.77270e7 −0.814227
176176 0 0
177177 −5.95260e7 −0.806863
178178 0 0
179179 −1.29965e8 −1.69372 −0.846861 0.531814i 0.821511π-0.821511\pi
−0.846861 + 0.531814i 0.821511π0.821511\pi
180180 0 0
181181 1.91607e7 0.240180 0.120090 0.992763i 0.461682π-0.461682\pi
0.120090 + 0.992763i 0.461682π0.461682\pi
182182 0 0
183183 −1.74776e6 −0.0210816
184184 0 0
185185 5.67961e7 0.659504
186186 0 0
187187 −1.07942e8 −1.20710
188188 0 0
189189 3.81776e7 0.411332
190190 0 0
191191 3.54364e7 0.367987 0.183994 0.982927i 0.441097π-0.441097\pi
0.183994 + 0.982927i 0.441097π0.441097\pi
192192 0 0
193193 1.80339e8 1.80567 0.902837 0.429983i 0.141481π-0.141481\pi
0.902837 + 0.429983i 0.141481π0.141481\pi
194194 0 0
195195 2.78445e8 2.68917
196196 0 0
197197 −1.39463e8 −1.29965 −0.649826 0.760083i 0.725158π-0.725158\pi
−0.649826 + 0.760083i 0.725158π0.725158\pi
198198 0 0
199199 1.46312e8 1.31612 0.658060 0.752965i 0.271377π-0.271377\pi
0.658060 + 0.752965i 0.271377π0.271377\pi
200200 0 0
201201 1.19590e8 1.03874
202202 0 0
203203 −1.23049e7 −0.103238
204204 0 0
205205 −1.22112e8 −0.989969
206206 0 0
207207 2.38956e7 0.187250
208208 0 0
209209 1.47728e8 1.11931
210210 0 0
211211 −1.02935e8 −0.754351 −0.377176 0.926142i 0.623105π-0.623105\pi
−0.377176 + 0.926142i 0.623105π0.623105\pi
212212 0 0
213213 −6.55324e7 −0.464651
214214 0 0
215215 −1.50035e7 −0.102958
216216 0 0
217217 −6.81193e7 −0.452544
218218 0 0
219219 −1.49993e8 −0.964978
220220 0 0
221221 3.87067e8 2.41220
222222 0 0
223223 −3.26446e8 −1.97126 −0.985632 0.168906i 0.945977π-0.945977\pi
−0.985632 + 0.168906i 0.945977π0.945977\pi
224224 0 0
225225 −1.15167e8 −0.674044
226226 0 0
227227 −8.99538e7 −0.510422 −0.255211 0.966885i 0.582145π-0.582145\pi
−0.255211 + 0.966885i 0.582145π0.582145\pi
228228 0 0
229229 −1.01929e8 −0.560886 −0.280443 0.959871i 0.590481π-0.590481\pi
−0.280443 + 0.959871i 0.590481π0.590481\pi
230230 0 0
231231 −5.36531e7 −0.286387
232232 0 0
233233 −4.26915e7 −0.221103 −0.110552 0.993870i 0.535262π-0.535262\pi
−0.110552 + 0.993870i 0.535262π0.535262\pi
234234 0 0
235235 1.51521e8 0.761616
236236 0 0
237237 7.78821e7 0.380030
238238 0 0
239239 −6.02163e7 −0.285313 −0.142656 0.989772i 0.545564π-0.545564\pi
−0.142656 + 0.989772i 0.545564π0.545564\pi
240240 0 0
241241 −4.71536e7 −0.216998 −0.108499 0.994097i 0.534604π-0.534604\pi
−0.108499 + 0.994097i 0.534604π0.534604\pi
242242 0 0
243243 1.34179e8 0.599876
244244 0 0
245245 −5.84024e7 −0.253717
246246 0 0
247247 −5.29736e8 −2.23677
248248 0 0
249249 −3.21251e8 −1.31870
250250 0 0
251251 1.94014e8 0.774416 0.387208 0.921992i 0.373440π-0.373440\pi
0.387208 + 0.921992i 0.373440π0.373440\pi
252252 0 0
253253 −1.40909e8 −0.547037
254254 0 0
255255 5.14761e8 1.94408
256256 0 0
257257 −6.78743e7 −0.249425 −0.124712 0.992193i 0.539801π-0.539801\pi
−0.124712 + 0.992193i 0.539801π0.539801\pi
258258 0 0
259259 3.92437e7 0.140353
260260 0 0
261261 −2.45485e7 −0.0854640
262262 0 0
263263 1.61200e8 0.546410 0.273205 0.961956i 0.411916π-0.411916\pi
0.273205 + 0.961956i 0.411916π0.411916\pi
264264 0 0
265265 6.00921e8 1.98361
266266 0 0
267267 −2.52213e8 −0.810920
268268 0 0
269269 −2.64104e8 −0.827261 −0.413631 0.910445i 0.635739π-0.635739\pi
−0.413631 + 0.910445i 0.635739π0.635739\pi
270270 0 0
271271 −4.92433e7 −0.150299 −0.0751493 0.997172i 0.523943π-0.523943\pi
−0.0751493 + 0.997172i 0.523943π0.523943\pi
272272 0 0
273273 1.92394e8 0.572298
274274 0 0
275275 6.79123e8 1.96917
276276 0 0
277277 −2.85803e8 −0.807957 −0.403978 0.914769i 0.632373π-0.632373\pi
−0.403978 + 0.914769i 0.632373π0.632373\pi
278278 0 0
279279 −1.35900e8 −0.374631
280280 0 0
281281 5.50944e8 1.48127 0.740637 0.671905i 0.234524π-0.234524\pi
0.740637 + 0.671905i 0.234524π0.234524\pi
282282 0 0
283283 2.84145e8 0.745226 0.372613 0.927987i 0.378462π-0.378462\pi
0.372613 + 0.927987i 0.378462π0.378462\pi
284284 0 0
285285 −7.04498e8 −1.80270
286286 0 0
287287 −8.43745e7 −0.210681
288288 0 0
289289 3.05231e8 0.743852
290290 0 0
291291 −4.45337e8 −1.05941
292292 0 0
293293 4.33632e7 0.100713 0.0503564 0.998731i 0.483964π-0.483964\pi
0.0503564 + 0.998731i 0.483964π0.483964\pi
294294 0 0
295295 7.62276e8 1.72876
296296 0 0
297297 −4.49136e8 −0.994788
298298 0 0
299299 5.05283e8 1.09317
300300 0 0
301301 −1.03668e7 −0.0219110
302302 0 0
303303 −2.67652e8 −0.552741
304304 0 0
305305 2.23814e7 0.0451688
306306 0 0
307307 −4.07582e8 −0.803954 −0.401977 0.915650i 0.631677π-0.631677\pi
−0.401977 + 0.915650i 0.631677π0.631677\pi
308308 0 0
309309 1.61802e8 0.311983
310310 0 0
311311 7.68264e8 1.44827 0.724134 0.689659i 0.242240π-0.242240\pi
0.724134 + 0.689659i 0.242240π0.242240\pi
312312 0 0
313313 4.31310e8 0.795031 0.397515 0.917596i 0.369873π-0.369873\pi
0.397515 + 0.917596i 0.369873π0.369873\pi
314314 0 0
315315 −1.16514e8 −0.210035
316316 0 0
317317 1.57984e8 0.278551 0.139276 0.990254i 0.455523π-0.455523\pi
0.139276 + 0.990254i 0.455523π0.455523\pi
318318 0 0
319319 1.44759e8 0.249677
320320 0 0
321321 −2.58051e8 −0.435450
322322 0 0
323323 −9.79323e8 −1.61703
324324 0 0
325325 −2.43526e9 −3.93507
326326 0 0
327327 −4.31818e8 −0.682941
328328 0 0
329329 1.04695e8 0.162084
330330 0 0
331331 9.75366e8 1.47832 0.739162 0.673527i 0.235222π-0.235222\pi
0.739162 + 0.673527i 0.235222π0.235222\pi
332332 0 0
333333 7.82922e7 0.116189
334334 0 0
335335 −1.53144e9 −2.22557
336336 0 0
337337 −9.07880e8 −1.29218 −0.646091 0.763260i 0.723597π-0.723597\pi
−0.646091 + 0.763260i 0.723597π0.723597\pi
338338 0 0
339339 1.62174e8 0.226090
340340 0 0
341341 8.01382e8 1.09446
342342 0 0
343343 −4.03536e7 −0.0539949
344344 0 0
345345 6.71977e8 0.881024
346346 0 0
347347 −5.79720e7 −0.0744843 −0.0372422 0.999306i 0.511857π-0.511857\pi
−0.0372422 + 0.999306i 0.511857π0.511857\pi
348348 0 0
349349 −1.33879e9 −1.68587 −0.842935 0.538016i 0.819174π-0.819174\pi
−0.842935 + 0.538016i 0.819174π0.819174\pi
350350 0 0
351351 1.61055e9 1.98792
352352 0 0
353353 5.82234e8 0.704509 0.352254 0.935904i 0.385415π-0.385415\pi
0.352254 + 0.935904i 0.385415π0.385415\pi
354354 0 0
355355 8.39192e8 0.995548
356356 0 0
357357 3.55678e8 0.413731
358358 0 0
359359 −3.20435e8 −0.365518 −0.182759 0.983158i 0.558503π-0.558503\pi
−0.182759 + 0.983158i 0.558503π0.558503\pi
360360 0 0
361361 4.46421e8 0.499424
362362 0 0
363363 −1.24219e8 −0.136306
364364 0 0
365365 1.92078e9 2.06753
366366 0 0
367367 6.55277e8 0.691981 0.345990 0.938238i 0.387543π-0.387543\pi
0.345990 + 0.938238i 0.387543π0.387543\pi
368368 0 0
369369 −1.68329e8 −0.174408
370370 0 0
371371 4.15211e8 0.422144
372372 0 0
373373 −5.42928e8 −0.541703 −0.270851 0.962621i 0.587305π-0.587305\pi
−0.270851 + 0.962621i 0.587305π0.587305\pi
374374 0 0
375375 −1.73527e9 −1.69925
376376 0 0
377377 −5.19090e8 −0.498940
378378 0 0
379379 −1.57480e9 −1.48590 −0.742949 0.669348i 0.766573π-0.766573\pi
−0.742949 + 0.669348i 0.766573π0.766573\pi
380380 0 0
381381 1.41608e9 1.31174
382382 0 0
383383 −8.88822e8 −0.808386 −0.404193 0.914674i 0.632448π-0.632448\pi
−0.404193 + 0.914674i 0.632448π0.632448\pi
384384 0 0
385385 6.87069e8 0.613603
386386 0 0
387387 −2.06820e7 −0.0181387
388388 0 0
389389 1.34673e9 1.15999 0.579997 0.814619i 0.303054π-0.303054\pi
0.579997 + 0.814619i 0.303054π0.303054\pi
390390 0 0
391391 9.34116e8 0.790282
392392 0 0
393393 1.96990e9 1.63708
394394 0 0
395395 −9.97340e8 −0.814242
396396 0 0
397397 2.15927e9 1.73197 0.865985 0.500070i 0.166693π-0.166693\pi
0.865985 + 0.500070i 0.166693π0.166693\pi
398398 0 0
399399 −4.86778e8 −0.383642
400400 0 0
401401 −1.87617e9 −1.45300 −0.726500 0.687166i 0.758854π-0.758854\pi
−0.726500 + 0.687166i 0.758854π0.758854\pi
402402 0 0
403403 −2.87366e9 −2.18710
404404 0 0
405405 1.39897e9 1.04644
406406 0 0
407407 −4.61678e8 −0.339437
408408 0 0
409409 2.27449e6 0.00164381 0.000821906 1.00000i 0.499738π-0.499738\pi
0.000821906 1.00000i 0.499738π0.499738\pi
410410 0 0
411411 1.17826e9 0.837135
412412 0 0
413413 5.26700e8 0.367907
414414 0 0
415415 4.11387e9 2.82542
416416 0 0
417417 −1.13213e9 −0.764577
418418 0 0
419419 −2.40944e9 −1.60017 −0.800087 0.599884i 0.795213π-0.795213\pi
−0.800087 + 0.599884i 0.795213π0.795213\pi
420420 0 0
421421 −1.08307e9 −0.707404 −0.353702 0.935358i 0.615077π-0.615077\pi
−0.353702 + 0.935358i 0.615077π0.615077\pi
422422 0 0
423423 2.08869e8 0.134178
424424 0 0
425425 −4.50205e9 −2.84479
426426 0 0
427427 1.54646e7 0.00961262
428428 0 0
429429 −2.26340e9 −1.38408
430430 0 0
431431 5.05987e8 0.304417 0.152208 0.988348i 0.451361π-0.451361\pi
0.152208 + 0.988348i 0.451361π0.451361\pi
432432 0 0
433433 3.73323e8 0.220992 0.110496 0.993877i 0.464756π-0.464756\pi
0.110496 + 0.993877i 0.464756π0.464756\pi
434434 0 0
435435 −6.90339e8 −0.402115
436436 0 0
437437 −1.27842e9 −0.732807
438438 0 0
439439 −8.88930e6 −0.00501466 −0.00250733 0.999997i 0.500798π-0.500798\pi
−0.00250733 + 0.999997i 0.500798π0.500798\pi
440440 0 0
441441 −8.05064e7 −0.0446988
442442 0 0
443443 1.02405e9 0.559638 0.279819 0.960053i 0.409726π-0.409726\pi
0.279819 + 0.960053i 0.409726π0.409726\pi
444444 0 0
445445 3.22978e9 1.73745
446446 0 0
447447 −1.43798e9 −0.761510
448448 0 0
449449 8.32711e8 0.434142 0.217071 0.976156i 0.430350π-0.430350\pi
0.217071 + 0.976156i 0.430350π0.430350\pi
450450 0 0
451451 9.92615e8 0.509522
452452 0 0
453453 7.07047e8 0.357359
454454 0 0
455455 −2.46375e9 −1.22619
456456 0 0
457457 −2.28781e9 −1.12128 −0.560639 0.828060i 0.689444π-0.689444\pi
−0.560639 + 0.828060i 0.689444π0.689444\pi
458458 0 0
459459 2.97742e9 1.43713
460460 0 0
461461 7.84459e8 0.372921 0.186461 0.982462i 0.440298π-0.440298\pi
0.186461 + 0.982462i 0.440298π0.440298\pi
462462 0 0
463463 −2.31590e9 −1.08439 −0.542196 0.840252i 0.682407π-0.682407\pi
−0.542196 + 0.840252i 0.682407π0.682407\pi
464464 0 0
465465 −3.82169e9 −1.76267
466466 0 0
467467 −6.12659e8 −0.278362 −0.139181 0.990267i 0.544447π-0.544447\pi
−0.139181 + 0.990267i 0.544447π0.544447\pi
468468 0 0
469469 −1.05816e9 −0.473637
470470 0 0
471471 8.22063e8 0.362520
472472 0 0
473473 1.21959e8 0.0529908
474474 0 0
475475 6.16147e9 2.63789
476476 0 0
477477 8.28356e8 0.349464
478478 0 0
479479 7.28950e8 0.303056 0.151528 0.988453i 0.451581π-0.451581\pi
0.151528 + 0.988453i 0.451581π0.451581\pi
480480 0 0
481481 1.65553e9 0.678310
482482 0 0
483483 4.64308e8 0.187496
484484 0 0
485485 5.70287e9 2.26985
486486 0 0
487487 −8.68384e8 −0.340691 −0.170346 0.985384i 0.554488π-0.554488\pi
−0.170346 + 0.985384i 0.554488π0.554488\pi
488488 0 0
489489 1.74811e9 0.676063
490490 0 0
491491 −3.47503e9 −1.32487 −0.662435 0.749119i 0.730477π-0.730477\pi
−0.662435 + 0.749119i 0.730477π0.730477\pi
492492 0 0
493493 −9.59641e8 −0.360699
494494 0 0
495495 1.37072e9 0.507961
496496 0 0
497497 5.79846e8 0.211868
498498 0 0
499499 −3.68171e9 −1.32647 −0.663236 0.748410i 0.730817π-0.730817\pi
−0.663236 + 0.748410i 0.730817π0.730817\pi
500500 0 0
501501 2.45312e9 0.871540
502502 0 0
503503 3.79099e9 1.32820 0.664102 0.747642i 0.268814π-0.268814\pi
0.664102 + 0.747642i 0.268814π0.268814\pi
504504 0 0
505505 3.42749e9 1.18429
506506 0 0
507507 5.68385e9 1.93693
508508 0 0
509509 −4.09313e9 −1.37576 −0.687880 0.725824i 0.741459π-0.741459\pi
−0.687880 + 0.725824i 0.741459π0.741459\pi
510510 0 0
511511 1.32718e9 0.440003
512512 0 0
513513 −4.07488e9 −1.33261
514514 0 0
515515 −2.07200e9 −0.668445
516516 0 0
517517 −1.23167e9 −0.391993
518518 0 0
519519 −1.29549e9 −0.406770
520520 0 0
521521 1.69546e9 0.525237 0.262618 0.964900i 0.415414π-0.415414\pi
0.262618 + 0.964900i 0.415414π0.415414\pi
522522 0 0
523523 −1.53714e9 −0.469849 −0.234924 0.972014i 0.575484π-0.575484\pi
−0.234924 + 0.972014i 0.575484π0.575484\pi
524524 0 0
525525 −2.23777e9 −0.674929
526526 0 0
527527 −5.31254e9 −1.58112
528528 0 0
529529 −2.18542e9 −0.641858
530530 0 0
531531 1.05078e9 0.304566
532532 0 0
533533 −3.55940e9 −1.01820
534534 0 0
535535 3.30454e9 0.932982
536536 0 0
537537 −5.03808e9 −1.40396
538538 0 0
539539 4.74736e8 0.130584
540540 0 0
541541 −4.34509e9 −1.17980 −0.589901 0.807476i 0.700833π-0.700833\pi
−0.589901 + 0.807476i 0.700833π0.700833\pi
542542 0 0
543543 7.42760e8 0.199090
544544 0 0
545545 5.52976e9 1.46325
546546 0 0
547547 2.30195e9 0.601367 0.300683 0.953724i 0.402785π-0.402785\pi
0.300683 + 0.953724i 0.402785π0.402785\pi
548548 0 0
549549 3.08523e7 0.00795764
550550 0 0
551551 1.31336e9 0.334466
552552 0 0
553553 −6.89120e8 −0.173283
554554 0 0
555555 2.20169e9 0.546676
556556 0 0
557557 1.71107e9 0.419541 0.209770 0.977751i 0.432728π-0.432728\pi
0.209770 + 0.977751i 0.432728π0.432728\pi
558558 0 0
559559 −4.37332e8 −0.105894
560560 0 0
561561 −4.18434e9 −1.00059
562562 0 0
563563 2.16633e9 0.511617 0.255809 0.966727i 0.417658π-0.417658\pi
0.255809 + 0.966727i 0.417658π0.417658\pi
564564 0 0
565565 −2.07676e9 −0.484415
566566 0 0
567567 9.66630e8 0.222700
568568 0 0
569569 −2.76063e9 −0.628225 −0.314112 0.949386i 0.601707π-0.601707\pi
−0.314112 + 0.949386i 0.601707π0.601707\pi
570570 0 0
571571 7.25907e8 0.163175 0.0815877 0.996666i 0.474001π-0.474001\pi
0.0815877 + 0.996666i 0.474001π0.474001\pi
572572 0 0
573573 1.37368e9 0.305032
574574 0 0
575575 −5.87705e9 −1.28921
576576 0 0
577577 −5.37035e9 −1.16382 −0.581911 0.813252i 0.697695π-0.697695\pi
−0.581911 + 0.813252i 0.697695π0.697695\pi
578578 0 0
579579 6.99080e9 1.49676
580580 0 0
581581 2.84251e9 0.601292
582582 0 0
583583 −4.88471e9 −1.02094
584584 0 0
585585 −4.91524e9 −1.01508
586586 0 0
587587 −7.86145e9 −1.60424 −0.802120 0.597163i 0.796295π-0.796295\pi
−0.802120 + 0.597163i 0.796295π0.796295\pi
588588 0 0
589589 7.27069e9 1.46613
590590 0 0
591591 −5.40625e9 −1.07731
592592 0 0
593593 5.44412e9 1.07210 0.536051 0.844185i 0.319915π-0.319915\pi
0.536051 + 0.844185i 0.319915π0.319915\pi
594594 0 0
595595 −4.55473e9 −0.886448
596596 0 0
597597 5.67177e9 1.09096
598598 0 0
599599 7.97408e9 1.51596 0.757978 0.652280i 0.226188π-0.226188\pi
0.757978 + 0.652280i 0.226188π0.226188\pi
600600 0 0
601601 −6.27378e8 −0.117888 −0.0589439 0.998261i 0.518773π-0.518773\pi
−0.0589439 + 0.998261i 0.518773π0.518773\pi
602602 0 0
603603 −2.11105e9 −0.392092
604604 0 0
605605 1.59072e9 0.292046
606606 0 0
607607 3.79758e9 0.689202 0.344601 0.938749i 0.388014π-0.388014\pi
0.344601 + 0.938749i 0.388014π0.388014\pi
608608 0 0
609609 −4.76995e8 −0.0855763
610610 0 0
611611 4.41663e9 0.783334
612612 0 0
613613 1.06120e10 1.86074 0.930369 0.366625i 0.119487π-0.119487\pi
0.930369 + 0.366625i 0.119487π0.119487\pi
614614 0 0
615615 −4.73366e9 −0.820605
616616 0 0
617617 2.26828e8 0.0388776 0.0194388 0.999811i 0.493812π-0.493812\pi
0.0194388 + 0.999811i 0.493812π0.493812\pi
618618 0 0
619619 4.91801e9 0.833435 0.416718 0.909036i 0.363180π-0.363180\pi
0.416718 + 0.909036i 0.363180π0.363180\pi
620620 0 0
621621 3.88678e9 0.651282
622622 0 0
623623 2.23164e9 0.369757
624624 0 0
625625 9.07300e9 1.48652
626626 0 0
627627 5.72665e9 0.927821
628628 0 0
629629 3.06057e9 0.490371
630630 0 0
631631 6.22228e9 0.985931 0.492966 0.870049i 0.335913π-0.335913\pi
0.492966 + 0.870049i 0.335913π0.335913\pi
632632 0 0
633633 −3.99024e9 −0.625297
634634 0 0
635635 −1.81339e10 −2.81050
636636 0 0
637637 −1.70235e9 −0.260952
638638 0 0
639639 1.15681e9 0.175391
640640 0 0
641641 −2.71355e9 −0.406944 −0.203472 0.979081i 0.565223π-0.565223\pi
−0.203472 + 0.979081i 0.565223π0.565223\pi
642642 0 0
643643 −5.96086e9 −0.884240 −0.442120 0.896956i 0.645774π-0.645774\pi
−0.442120 + 0.896956i 0.645774π0.645774\pi
644644 0 0
645645 −5.81609e8 −0.0853438
646646 0 0
647647 1.87137e9 0.271640 0.135820 0.990734i 0.456633π-0.456633\pi
0.135820 + 0.990734i 0.456633π0.456633\pi
648648 0 0
649649 −6.19631e9 −0.889768
650650 0 0
651651 −2.64063e9 −0.375123
652652 0 0
653653 8.68243e9 1.22024 0.610120 0.792309i 0.291121π-0.291121\pi
0.610120 + 0.792309i 0.291121π0.291121\pi
654654 0 0
655655 −2.52261e10 −3.50756
656656 0 0
657657 2.64775e9 0.364249
658658 0 0
659659 1.17028e10 1.59291 0.796457 0.604696i 0.206705π-0.206705\pi
0.796457 + 0.604696i 0.206705π0.206705\pi
660660 0 0
661661 −4.23524e9 −0.570391 −0.285195 0.958469i 0.592059π-0.592059\pi
−0.285195 + 0.958469i 0.592059π0.592059\pi
662662 0 0
663663 1.50046e10 1.99952
664664 0 0
665665 6.23357e9 0.821979
666666 0 0
667667 −1.25273e9 −0.163462
668668 0 0
669669 −1.26546e10 −1.63402
670670 0 0
671671 −1.81932e8 −0.0232477
672672 0 0
673673 −1.64366e9 −0.207854 −0.103927 0.994585i 0.533141π-0.533141\pi
−0.103927 + 0.994585i 0.533141π0.533141\pi
674674 0 0
675675 −1.87327e10 −2.34442
676676 0 0
677677 1.08838e10 1.34810 0.674048 0.738688i 0.264554π-0.264554\pi
0.674048 + 0.738688i 0.264554π0.264554\pi
678678 0 0
679679 3.94045e9 0.483060
680680 0 0
681681 −3.48704e9 −0.423099
682682 0 0
683683 −3.85277e9 −0.462701 −0.231351 0.972870i 0.574314π-0.574314\pi
−0.231351 + 0.972870i 0.574314π0.574314\pi
684684 0 0
685685 −1.50885e10 −1.79362
686686 0 0
687687 −3.95126e9 −0.464930
688688 0 0
689689 1.75160e10 2.04018
690690 0 0
691691 −1.55439e10 −1.79220 −0.896100 0.443852i 0.853612π-0.853612\pi
−0.896100 + 0.443852i 0.853612π0.853612\pi
692692 0 0
693693 9.47109e8 0.108102
694694 0 0
695695 1.44978e10 1.63816
696696 0 0
697697 −6.58026e9 −0.736086
698698 0 0
699699 −1.65493e9 −0.183277
700700 0 0
701701 −7.75099e9 −0.849854 −0.424927 0.905228i 0.639700π-0.639700\pi
−0.424927 + 0.905228i 0.639700π0.639700\pi
702702 0 0
703703 −4.18867e9 −0.454707
704704 0 0
705705 5.87369e9 0.631319
706706 0 0
707707 2.36825e9 0.252035
708708 0 0
709709 −1.14664e10 −1.20827 −0.604137 0.796881i 0.706482π-0.706482\pi
−0.604137 + 0.796881i 0.706482π0.706482\pi
710710 0 0
711711 −1.37481e9 −0.143450
712712 0 0
713713 −6.93507e9 −0.716535
714714 0 0
715715 2.89845e10 2.96548
716716 0 0
717717 −2.33427e9 −0.236502
718718 0 0
719719 −1.63852e10 −1.64399 −0.821997 0.569492i 0.807140π-0.807140\pi
−0.821997 + 0.569492i 0.807140π0.807140\pi
720720 0 0
721721 −1.43167e9 −0.142255
722722 0 0
723723 −1.82790e9 −0.179874
724724 0 0
725725 6.03765e9 0.588416
726726 0 0
727727 1.66277e10 1.60495 0.802474 0.596687i 0.203517π-0.203517\pi
0.802474 + 0.596687i 0.203517π0.203517\pi
728728 0 0
729729 1.13647e10 1.08646
730730 0 0
731731 −8.08495e8 −0.0765537
732732 0 0
733733 1.45773e10 1.36714 0.683568 0.729886i 0.260427π-0.260427\pi
0.683568 + 0.729886i 0.260427π0.260427\pi
734734 0 0
735735 −2.26396e9 −0.210311
736736 0 0
737737 1.24486e10 1.14547
738738 0 0
739739 4.30003e9 0.391937 0.195968 0.980610i 0.437215π-0.437215\pi
0.195968 + 0.980610i 0.437215π0.437215\pi
740740 0 0
741741 −2.05351e10 −1.85410
742742 0 0
743743 −9.14731e9 −0.818150 −0.409075 0.912501i 0.634148π-0.634148\pi
−0.409075 + 0.912501i 0.634148π0.634148\pi
744744 0 0
745745 1.84144e10 1.63159
746746 0 0
747747 5.67088e9 0.497769
748748 0 0
749749 2.28330e9 0.198553
750750 0 0
751751 1.71874e9 0.148071 0.0740354 0.997256i 0.476412π-0.476412\pi
0.0740354 + 0.997256i 0.476412π0.476412\pi
752752 0 0
753753 7.52089e9 0.641929
754754 0 0
755755 −9.05428e9 −0.765666
756756 0 0
757757 −1.58386e10 −1.32703 −0.663517 0.748161i 0.730937π-0.730937\pi
−0.663517 + 0.748161i 0.730937π0.730937\pi
758758 0 0
759759 −5.46230e9 −0.453450
760760 0 0
761761 3.29329e9 0.270884 0.135442 0.990785i 0.456755π-0.456755\pi
0.135442 + 0.990785i 0.456755π0.456755\pi
762762 0 0
763763 3.82083e9 0.311402
764764 0 0
765765 −9.08680e9 −0.733831
766766 0 0
767767 2.22193e10 1.77806
768768 0 0
769769 −1.80581e10 −1.43196 −0.715978 0.698123i 0.754019π-0.754019\pi
−0.715978 + 0.698123i 0.754019π0.754019\pi
770770 0 0
771771 −2.63113e9 −0.206753
772772 0 0
773773 −1.57529e10 −1.22668 −0.613342 0.789817i 0.710175π-0.710175\pi
−0.613342 + 0.789817i 0.710175π0.710175\pi
774774 0 0
775775 3.34242e10 2.57932
776776 0 0
777777 1.52127e9 0.116341
778778 0 0
779779 9.00569e9 0.682552
780780 0 0
781781 −6.82154e9 −0.512394
782782 0 0
783783 −3.99298e9 −0.297257
784784 0 0
785785 −1.05271e10 −0.776724
786786 0 0
787787 1.02224e10 0.747551 0.373776 0.927519i 0.378063π-0.378063\pi
0.373776 + 0.927519i 0.378063π0.378063\pi
788788 0 0
789789 6.24886e9 0.452930
790790 0 0
791791 −1.43495e9 −0.103091
792792 0 0
793793 6.52387e8 0.0464568
794794 0 0
795795 2.32946e10 1.64426
796796 0 0
797797 1.44391e10 1.01027 0.505134 0.863041i 0.331443π-0.331443\pi
0.505134 + 0.863041i 0.331443π0.331443\pi
798798 0 0
799799 8.16502e9 0.566296
800800 0 0
801801 4.45218e9 0.306097
802802 0 0
803803 −1.56134e10 −1.06413
804804 0 0
805805 −5.94582e9 −0.401722
806806 0 0
807807 −1.02379e10 −0.685733
808808 0 0
809809 −1.00885e10 −0.669897 −0.334949 0.942236i 0.608719π-0.608719\pi
−0.334949 + 0.942236i 0.608719π0.608719\pi
810810 0 0
811811 1.35981e9 0.0895172 0.0447586 0.998998i 0.485748π-0.485748\pi
0.0447586 + 0.998998i 0.485748π0.485748\pi
812812 0 0
813813 −1.90890e9 −0.124585
814814 0 0
815815 −2.23859e10 −1.44851
816816 0 0
817817 1.10650e9 0.0709862
818818 0 0
819819 −3.39623e9 −0.216024
820820 0 0
821821 −1.49802e10 −0.944751 −0.472375 0.881397i 0.656603π-0.656603\pi
−0.472375 + 0.881397i 0.656603π0.656603\pi
822822 0 0
823823 −1.96016e10 −1.22572 −0.612861 0.790191i 0.709981π-0.709981\pi
−0.612861 + 0.790191i 0.709981π0.709981\pi
824824 0 0
825825 2.63260e10 1.63229
826826 0 0
827827 1.68768e9 0.103758 0.0518789 0.998653i 0.483479π-0.483479\pi
0.0518789 + 0.998653i 0.483479π0.483479\pi
828828 0 0
829829 −1.64541e9 −0.100307 −0.0501536 0.998742i 0.515971π-0.515971\pi
−0.0501536 + 0.998742i 0.515971π0.515971\pi
830830 0 0
831831 −1.10791e10 −0.669732
832832 0 0
833833 −3.14713e9 −0.188650
834834 0 0
835835 −3.14141e10 −1.86733
836836 0 0
837837 −2.21050e10 −1.30302
838838 0 0
839839 −3.49725e9 −0.204437 −0.102219 0.994762i 0.532594π-0.532594\pi
−0.102219 + 0.994762i 0.532594π0.532594\pi
840840 0 0
841841 −1.59629e10 −0.925393
842842 0 0
843843 2.13572e10 1.22786
844844 0 0
845845 −7.27861e10 −4.15002
846846 0 0
847847 1.09912e9 0.0621519
848848 0 0
849849 1.10148e10 0.617733
850850 0 0
851851 3.99531e9 0.222227
852852 0 0
853853 2.86798e10 1.58218 0.791088 0.611702i 0.209515π-0.209515\pi
0.791088 + 0.611702i 0.209515π0.209515\pi
854854 0 0
855855 1.24361e10 0.680461
856856 0 0
857857 3.67333e10 1.99355 0.996776 0.0802366i 0.0255676π-0.0255676\pi
0.996776 + 0.0802366i 0.0255676π0.0255676\pi
858858 0 0
859859 5.54734e9 0.298613 0.149306 0.988791i 0.452296π-0.452296\pi
0.149306 + 0.988791i 0.452296π0.452296\pi
860860 0 0
861861 −3.27076e9 −0.174638
862862 0 0
863863 5.26057e9 0.278609 0.139305 0.990250i 0.455513π-0.455513\pi
0.139305 + 0.990250i 0.455513π0.455513\pi
864864 0 0
865865 1.65897e10 0.871532
866866 0 0
867867 1.18322e10 0.616594
868868 0 0
869869 8.10707e9 0.419078
870870 0 0
871871 −4.46392e10 −2.28904
872872 0 0
873873 7.86129e9 0.399893
874874 0 0
875875 1.53541e10 0.774811
876876 0 0
877877 2.83160e10 1.41753 0.708767 0.705443i 0.249252π-0.249252\pi
0.708767 + 0.705443i 0.249252π0.249252\pi
878878 0 0
879879 1.68096e9 0.0834828
880880 0 0
881881 −5.58205e9 −0.275029 −0.137514 0.990500i 0.543911π-0.543911\pi
−0.137514 + 0.990500i 0.543911π0.543911\pi
882882 0 0
883883 −2.81316e10 −1.37509 −0.687547 0.726140i 0.741312π-0.741312\pi
−0.687547 + 0.726140i 0.741312π0.741312\pi
884884 0 0
885885 2.95494e10 1.43300
886886 0 0
887887 −9.92105e6 −0.000477337 0 −0.000238668 1.00000i 0.500076π-0.500076\pi
−0.000238668 1.00000i 0.500076π0.500076\pi
888888 0 0
889889 −1.25298e10 −0.598119
890890 0 0
891891 −1.13718e10 −0.538589
892892 0 0
893893 −1.11746e10 −0.525111
894894 0 0
895895 6.45164e10 3.00808
896896 0 0
897897 1.95872e10 0.906147
898898 0 0
899899 7.12457e9 0.327039
900900 0 0
901901 3.23818e10 1.47491
902902 0 0
903903 −4.01867e8 −0.0181625
904904 0 0
905905 −9.51161e9 −0.426564
906906 0 0
907907 −1.61173e9 −0.0717242 −0.0358621 0.999357i 0.511418π-0.511418\pi
−0.0358621 + 0.999357i 0.511418π0.511418\pi
908908 0 0
909909 4.72472e9 0.208642
910910 0 0
911911 −2.31768e10 −1.01564 −0.507818 0.861464i 0.669548π-0.669548\pi
−0.507818 + 0.861464i 0.669548π0.669548\pi
912912 0 0
913913 −3.34404e10 −1.45420
914914 0 0
915915 8.67611e8 0.0374413
916916 0 0
917917 −1.74302e10 −0.746464
918918 0 0
919919 −1.36641e10 −0.580733 −0.290367 0.956916i 0.593777π-0.593777\pi
−0.290367 + 0.956916i 0.593777π0.593777\pi
920920 0 0
921921 −1.57998e10 −0.666414
922922 0 0
923923 2.44613e10 1.02394
924924 0 0
925925 −1.92558e10 −0.799953
926926 0 0
927927 −2.85621e9 −0.117764
928928 0 0
929929 −3.34282e10 −1.36791 −0.683956 0.729523i 0.739742π-0.739742\pi
−0.683956 + 0.729523i 0.739742π0.739742\pi
930930 0 0
931931 4.30713e9 0.174930
932932 0 0
933933 2.97816e10 1.20050
934934 0 0
935935 5.35836e10 2.14384
936936 0 0
937937 1.04114e10 0.413450 0.206725 0.978399i 0.433720π-0.433720\pi
0.206725 + 0.978399i 0.433720π0.433720\pi
938938 0 0
939939 1.67196e10 0.659017
940940 0 0
941941 5.00017e10 1.95623 0.978117 0.208055i 0.0667134π-0.0667134\pi
0.978117 + 0.208055i 0.0667134π0.0667134\pi
942942 0 0
943943 −8.58998e9 −0.333581
944944 0 0
945945 −1.89518e10 −0.730533
946946 0 0
947947 −1.82730e10 −0.699172 −0.349586 0.936904i 0.613678π-0.613678\pi
−0.349586 + 0.936904i 0.613678π0.613678\pi
948948 0 0
949949 5.59880e10 2.12649
950950 0 0
951951 6.12420e9 0.230897
952952 0 0
953953 −1.24663e10 −0.466565 −0.233282 0.972409i 0.574947π-0.574947\pi
−0.233282 + 0.972409i 0.574947π0.574947\pi
954954 0 0
955955 −1.75911e10 −0.653552
956956 0 0
957957 5.61156e9 0.206963
958958 0 0
959959 −1.04256e10 −0.381710
960960 0 0
961961 1.19287e10 0.433573
962962 0 0
963963 4.55524e9 0.164369
964964 0 0
965965 −8.95225e10 −3.20691
966966 0 0
967967 4.12045e9 0.146539 0.0732693 0.997312i 0.476657π-0.476657\pi
0.0732693 + 0.997312i 0.476657π0.476657\pi
968968 0 0
969969 −3.79632e10 −1.34039
970970 0 0
971971 −1.24238e10 −0.435499 −0.217749 0.976005i 0.569872π-0.569872\pi
−0.217749 + 0.976005i 0.569872π0.569872\pi
972972 0 0
973973 1.00174e10 0.348626
974974 0 0
975975 −9.44021e10 −3.26186
976976 0 0
977977 4.51303e10 1.54824 0.774119 0.633041i 0.218193π-0.218193\pi
0.774119 + 0.633041i 0.218193π0.218193\pi
978978 0 0
979979 −2.62539e10 −0.894241
980980 0 0
981981 7.62265e9 0.257789
982982 0 0
983983 5.75558e9 0.193264 0.0966321 0.995320i 0.469193π-0.469193\pi
0.0966321 + 0.995320i 0.469193π0.469193\pi
984984 0 0
985985 6.92312e10 2.30821
986986 0 0
987987 4.05847e9 0.134355
988988 0 0
989989 −1.05542e9 −0.0346928
990990 0 0
991991 −3.64108e10 −1.18843 −0.594214 0.804307i 0.702537π-0.702537\pi
−0.594214 + 0.804307i 0.702537π0.702537\pi
992992 0 0
993993 3.78098e10 1.22541
994994 0 0
995995 −7.26313e10 −2.33745
996996 0 0
997997 1.13033e10 0.361220 0.180610 0.983555i 0.442193π-0.442193\pi
0.180610 + 0.983555i 0.442193π0.442193\pi
998998 0 0
999999 1.27348e10 0.404121
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 28.8.a.b.1.2 2
3.2 odd 2 252.8.a.f.1.2 2
4.3 odd 2 112.8.a.h.1.1 2
7.2 even 3 196.8.e.b.165.1 4
7.3 odd 6 196.8.e.c.177.2 4
7.4 even 3 196.8.e.b.177.1 4
7.5 odd 6 196.8.e.c.165.2 4
7.6 odd 2 196.8.a.a.1.1 2
8.3 odd 2 448.8.a.q.1.2 2
8.5 even 2 448.8.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.8.a.b.1.2 2 1.1 even 1 trivial
112.8.a.h.1.1 2 4.3 odd 2
196.8.a.a.1.1 2 7.6 odd 2
196.8.e.b.165.1 4 7.2 even 3
196.8.e.b.177.1 4 7.4 even 3
196.8.e.c.165.2 4 7.5 odd 6
196.8.e.c.177.2 4 7.3 odd 6
252.8.a.f.1.2 2 3.2 odd 2
448.8.a.o.1.1 2 8.5 even 2
448.8.a.q.1.2 2 8.3 odd 2