Properties

Label 3840.2.f.m.769.7
Level 38403840
Weight 22
Character 3840.769
Analytic conductor 30.66330.663
Analytic rank 00
Dimension 1212
Inner twists 44

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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] = [3840,2,Mod(769,3840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3840.769"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 3840=2835 3840 = 2^{8} \cdot 3 \cdot 5
Weight: k k == 2 2
Character orbit: [χ][\chi] == 3840.f (of order 22, degree 11, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-4,0,0,0, 0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 30.662554376230.6625543762
Analytic rank: 00
Dimension: 1212
Coefficient field: 12.0.180227832610816.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x12+x108x6+16x2+64 x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 212 2^{12}
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 769.7
Root 0.4501291.34067i-0.450129 - 1.34067i of defining polynomial
Character χ\chi == 3840.769
Dual form 3840.2.f.m.769.1

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+1.00000iq3+(2.221580.254102i)q5+2.64265iq71.00000q91.51363q113.87086iq13+(0.2541022.22158i)q15+3.31415iq17+7.08582q192.64265q214.82778iq23+(4.87086+1.12902i)q251.00000iq27+2.18513q29+7.36266q311.51363iq33+(0.6715025.87086i)q357.87086iq37+3.87086q398.72532q411.01641iq43+(2.22158+0.254102i)q45+7.08582iq47+0.0164068q493.31415q51+4.50820iq53+(3.36266+0.384617i)q55+7.08582iq57+6.79893q593.60104q612.64265iq63+(0.983593+8.59945i)q651.01641iq67+4.82778q696.72532q71+15.5146iq73+(1.12902+4.87086i)q754.00000iq777.36266q79+1.00000q81+7.74173iq83+(0.8421317.36266i)q85+2.18513iq87+14.7581q89+10.2293q91+7.36266iq93+(15.74171.80052i)q95+11.1444iq97+1.51363q99+O(q100)q+1.00000i q^{3} +(-2.22158 - 0.254102i) q^{5} +2.64265i q^{7} -1.00000 q^{9} -1.51363 q^{11} -3.87086i q^{13} +(0.254102 - 2.22158i) q^{15} +3.31415i q^{17} +7.08582 q^{19} -2.64265 q^{21} -4.82778i q^{23} +(4.87086 + 1.12902i) q^{25} -1.00000i q^{27} +2.18513 q^{29} +7.36266 q^{31} -1.51363i q^{33} +(0.671502 - 5.87086i) q^{35} -7.87086i q^{37} +3.87086 q^{39} -8.72532 q^{41} -1.01641i q^{43} +(2.22158 + 0.254102i) q^{45} +7.08582i q^{47} +0.0164068 q^{49} -3.31415 q^{51} +4.50820i q^{53} +(3.36266 + 0.384617i) q^{55} +7.08582i q^{57} +6.79893 q^{59} -3.60104 q^{61} -2.64265i q^{63} +(-0.983593 + 8.59945i) q^{65} -1.01641i q^{67} +4.82778 q^{69} -6.72532 q^{71} +15.5146i q^{73} +(-1.12902 + 4.87086i) q^{75} -4.00000i q^{77} -7.36266 q^{79} +1.00000 q^{81} +7.74173i q^{83} +(0.842131 - 7.36266i) q^{85} +2.18513i q^{87} +14.7581 q^{89} +10.2293 q^{91} +7.36266i q^{93} +(-15.7417 - 1.80052i) q^{95} +11.1444i q^{97} +1.51363 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 12q12q94q25+32q3116q39+8q4112q4916q5524q65+32q7132q79+12q81+40q8964q95+O(q100) 12 q - 12 q^{9} - 4 q^{25} + 32 q^{31} - 16 q^{39} + 8 q^{41} - 12 q^{49} - 16 q^{55} - 24 q^{65} + 32 q^{71} - 32 q^{79} + 12 q^{81} + 40 q^{89} - 64 q^{95}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/3840Z)×\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times.

nn 511511 15371537 25612561 28212821
χ(n)\chi(n) 11 1-1 11 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 1.00000i 0.577350i
44 0 0
55 −2.22158 0.254102i −0.993522 0.113638i
66 0 0
77 2.64265i 0.998827i 0.866364 + 0.499414i 0.166451π0.166451\pi
−0.866364 + 0.499414i 0.833549π0.833549\pi
88 0 0
99 −1.00000 −0.333333
1010 0 0
1111 −1.51363 −0.456377 −0.228189 0.973617i 0.573280π-0.573280\pi
−0.228189 + 0.973617i 0.573280π0.573280\pi
1212 0 0
1313 3.87086i 1.07358i −0.843714 0.536792i 0.819636π-0.819636\pi
0.843714 0.536792i 0.180364π-0.180364\pi
1414 0 0
1515 0.254102 2.22158i 0.0656088 0.573610i
1616 0 0
1717 3.31415i 0.803800i 0.915684 + 0.401900i 0.131650π0.131650\pi
−0.915684 + 0.401900i 0.868350π0.868350\pi
1818 0 0
1919 7.08582 1.62560 0.812799 0.582545i 0.197943π-0.197943\pi
0.812799 + 0.582545i 0.197943π0.197943\pi
2020 0 0
2121 −2.64265 −0.576673
2222 0 0
2323 4.82778i 1.00666i −0.864094 0.503331i 0.832108π-0.832108\pi
0.864094 0.503331i 0.167892π-0.167892\pi
2424 0 0
2525 4.87086 + 1.12902i 0.974173 + 0.225803i
2626 0 0
2727 1.00000i 0.192450i
2828 0 0
2929 2.18513 0.405769 0.202885 0.979203i 0.434968π-0.434968\pi
0.202885 + 0.979203i 0.434968π0.434968\pi
3030 0 0
3131 7.36266 1.32237 0.661187 0.750222i 0.270053π-0.270053\pi
0.661187 + 0.750222i 0.270053π0.270053\pi
3232 0 0
3333 1.51363i 0.263490i
3434 0 0
3535 0.671502 5.87086i 0.113504 0.992357i
3636 0 0
3737 7.87086i 1.29396i −0.762506 0.646981i 0.776031π-0.776031\pi
0.762506 0.646981i 0.223969π-0.223969\pi
3838 0 0
3939 3.87086 0.619834
4040 0 0
4141 −8.72532 −1.36267 −0.681333 0.731973i 0.738600π-0.738600\pi
−0.681333 + 0.731973i 0.738600π0.738600\pi
4242 0 0
4343 1.01641i 0.155001i −0.996992 0.0775003i 0.975306π-0.975306\pi
0.996992 0.0775003i 0.0246939π-0.0246939\pi
4444 0 0
4545 2.22158 + 0.254102i 0.331174 + 0.0378792i
4646 0 0
4747 7.08582i 1.03357i 0.856114 + 0.516786i 0.172872π0.172872\pi
−0.856114 + 0.516786i 0.827128π0.827128\pi
4848 0 0
4949 0.0164068 0.00234382
5050 0 0
5151 −3.31415 −0.464074
5252 0 0
5353 4.50820i 0.619249i 0.950859 + 0.309625i 0.100203π0.100203\pi
−0.950859 + 0.309625i 0.899797π0.899797\pi
5454 0 0
5555 3.36266 + 0.384617i 0.453421 + 0.0518617i
5656 0 0
5757 7.08582i 0.938539i
5858 0 0
5959 6.79893 0.885145 0.442573 0.896733i 0.354066π-0.354066\pi
0.442573 + 0.896733i 0.354066π0.354066\pi
6060 0 0
6161 −3.60104 −0.461065 −0.230533 0.973065i 0.574047π-0.574047\pi
−0.230533 + 0.973065i 0.574047π0.574047\pi
6262 0 0
6363 2.64265i 0.332942i
6464 0 0
6565 −0.983593 + 8.59945i −0.122000 + 1.06663i
6666 0 0
6767 1.01641i 0.124174i −0.998071 0.0620869i 0.980224π-0.980224\pi
0.998071 0.0620869i 0.0197756π-0.0197756\pi
6868 0 0
6969 4.82778 0.581197
7070 0 0
7171 −6.72532 −0.798149 −0.399074 0.916919i 0.630669π-0.630669\pi
−0.399074 + 0.916919i 0.630669π0.630669\pi
7272 0 0
7373 15.5146i 1.81585i 0.419132 + 0.907925i 0.362334π0.362334\pi
−0.419132 + 0.907925i 0.637666π0.637666\pi
7474 0 0
7575 −1.12902 + 4.87086i −0.130368 + 0.562439i
7676 0 0
7777 4.00000i 0.455842i
7878 0 0
7979 −7.36266 −0.828364 −0.414182 0.910194i 0.635932π-0.635932\pi
−0.414182 + 0.910194i 0.635932π0.635932\pi
8080 0 0
8181 1.00000 0.111111
8282 0 0
8383 7.74173i 0.849765i 0.905248 + 0.424883i 0.139685π0.139685\pi
−0.905248 + 0.424883i 0.860315π0.860315\pi
8484 0 0
8585 0.842131 7.36266i 0.0913420 0.798593i
8686 0 0
8787 2.18513i 0.234271i
8888 0 0
8989 14.7581 1.56436 0.782180 0.623053i 0.214108π-0.214108\pi
0.782180 + 0.623053i 0.214108π0.214108\pi
9090 0 0
9191 10.2293 1.07233
9292 0 0
9393 7.36266i 0.763472i
9494 0 0
9595 −15.7417 1.80052i −1.61507 0.184729i
9696 0 0
9797 11.1444i 1.13154i 0.824563 + 0.565769i 0.191421π0.191421\pi
−0.824563 + 0.565769i 0.808579π0.808579\pi
9898 0 0
9999 1.51363 0.152126
100100 0 0
101101 13.3295 1.32633 0.663167 0.748471i 0.269212π-0.269212\pi
0.663167 + 0.748471i 0.269212π0.269212\pi
102102 0 0
103103 0.958386i 0.0944326i 0.998885 + 0.0472163i 0.0150350π0.0150350\pi
−0.998885 + 0.0472163i 0.984965π0.984965\pi
104104 0 0
105105 5.87086 + 0.671502i 0.572938 + 0.0655318i
106106 0 0
107107 4.00000i 0.386695i 0.981130 + 0.193347i 0.0619344π0.0619344\pi
−0.981130 + 0.193347i 0.938066π0.938066\pi
108108 0 0
109109 −0.769233 −0.0736792 −0.0368396 0.999321i 0.511729π-0.511729\pi
−0.0368396 + 0.999321i 0.511729π0.511729\pi
110110 0 0
111111 7.87086 0.747069
112112 0 0
113113 14.4585i 1.36014i 0.733146 + 0.680071i 0.238051π0.238051\pi
−0.733146 + 0.680071i 0.761949π0.761949\pi
114114 0 0
115115 −1.22675 + 10.7253i −0.114395 + 1.00014i
116116 0 0
117117 3.87086i 0.357862i
118118 0 0
119119 −8.75814 −0.802857
120120 0 0
121121 −8.70892 −0.791720
122122 0 0
123123 8.72532i 0.786736i
124124 0 0
125125 −10.5341 3.74590i −0.942203 0.335043i
126126 0 0
127127 11.5290i 1.02303i −0.859274 0.511516i 0.829084π-0.829084\pi
0.859274 0.511516i 0.170916π-0.170916\pi
128128 0 0
129129 1.01641 0.0894896
130130 0 0
131131 −7.37270 −0.644156 −0.322078 0.946713i 0.604381π-0.604381\pi
−0.322078 + 0.946713i 0.604381π0.604381\pi
132132 0 0
133133 18.7253i 1.62369i
134134 0 0
135135 −0.254102 + 2.22158i −0.0218696 + 0.191203i
136136 0 0
137137 3.88792i 0.332167i 0.986112 + 0.166084i 0.0531122π0.0531122\pi
−0.986112 + 0.166084i 0.946888π0.946888\pi
138138 0 0
139139 14.6291 1.24083 0.620414 0.784275i 0.286965π-0.286965\pi
0.620414 + 0.784275i 0.286965π0.286965\pi
140140 0 0
141141 −7.08582 −0.596733
142142 0 0
143143 5.85907i 0.489960i
144144 0 0
145145 −4.85446 0.555246i −0.403141 0.0461107i
146146 0 0
147147 0.0164068i 0.00135321i
148148 0 0
149149 −11.0715 −0.907010 −0.453505 0.891254i 0.649827π-0.649827\pi
−0.453505 + 0.891254i 0.649827π0.649827\pi
150150 0 0
151151 0.637339 0.0518659 0.0259329 0.999664i 0.491744π-0.491744\pi
0.0259329 + 0.999664i 0.491744π0.491744\pi
152152 0 0
153153 3.31415i 0.267933i
154154 0 0
155155 −16.3568 1.87086i −1.31381 0.150271i
156156 0 0
157157 0.129135i 0.0103061i −0.999987 0.00515306i 0.998360π-0.998360\pi
0.999987 0.00515306i 0.00164028π-0.00164028\pi
158158 0 0
159159 −4.50820 −0.357524
160160 0 0
161161 12.7581 1.00548
162162 0 0
163163 19.4835i 1.52606i −0.646362 0.763031i 0.723710π-0.723710\pi
0.646362 0.763031i 0.276290π-0.276290\pi
164164 0 0
165165 −0.384617 + 3.36266i −0.0299424 + 0.261783i
166166 0 0
167167 1.80052i 0.139328i 0.997571 + 0.0696641i 0.0221928π0.0221928\pi
−0.997571 + 0.0696641i 0.977807π0.977807\pi
168168 0 0
169169 −1.98359 −0.152584
170170 0 0
171171 −7.08582 −0.541866
172172 0 0
173173 23.2335i 1.76641i 0.468985 + 0.883206i 0.344620π0.344620\pi
−0.468985 + 0.883206i 0.655380π0.655380\pi
174174 0 0
175175 −2.98359 + 12.8720i −0.225538 + 0.973031i
176176 0 0
177177 6.79893i 0.511039i
178178 0 0
179179 −2.85664 −0.213515 −0.106757 0.994285i 0.534047π-0.534047\pi
−0.106757 + 0.994285i 0.534047π0.534047\pi
180180 0 0
181181 −5.28530 −0.392853 −0.196427 0.980519i 0.562934π-0.562934\pi
−0.196427 + 0.980519i 0.562934π0.562934\pi
182182 0 0
183183 3.60104i 0.266196i
184184 0 0
185185 −2.00000 + 17.4858i −0.147043 + 1.28558i
186186 0 0
187187 5.01641i 0.366836i
188188 0 0
189189 2.64265 0.192224
190190 0 0
191191 5.96719 0.431770 0.215885 0.976419i 0.430736π-0.430736\pi
0.215885 + 0.976419i 0.430736π0.430736\pi
192192 0 0
193193 14.9409i 1.07547i 0.843115 + 0.537733i 0.180719π0.180719\pi
−0.843115 + 0.537733i 0.819281π0.819281\pi
194194 0 0
195195 −8.59945 0.983593i −0.615819 0.0704366i
196196 0 0
197197 3.23353i 0.230379i 0.993344 + 0.115190i 0.0367476π0.0367476\pi
−0.993344 + 0.115190i 0.963252π0.963252\pi
198198 0 0
199199 8.12080 0.575668 0.287834 0.957680i 0.407065π-0.407065\pi
0.287834 + 0.957680i 0.407065π0.407065\pi
200200 0 0
201201 1.01641 0.0716918
202202 0 0
203203 5.77454i 0.405293i
204204 0 0
205205 19.3840 + 2.21712i 1.35384 + 0.154850i
206206 0 0
207207 4.82778i 0.335554i
208208 0 0
209209 −10.7253 −0.741886
210210 0 0
211211 13.7141 0.944119 0.472059 0.881567i 0.343511π-0.343511\pi
0.472059 + 0.881567i 0.343511π0.343511\pi
212212 0 0
213213 6.72532i 0.460812i
214214 0 0
215215 −0.258271 + 2.25803i −0.0176139 + 0.153997i
216216 0 0
217217 19.4569i 1.32082i
218218 0 0
219219 −15.5146 −1.04838
220220 0 0
221221 12.8286 0.862947
222222 0 0
223223 9.84472i 0.659251i −0.944112 0.329626i 0.893077π-0.893077\pi
0.944112 0.329626i 0.106923π-0.106923\pi
224224 0 0
225225 −4.87086 1.12902i −0.324724 0.0752677i
226226 0 0
227227 5.70892i 0.378914i −0.981889 0.189457i 0.939327π-0.939327\pi
0.981889 0.189457i 0.0606728π-0.0606728\pi
228228 0 0
229229 −0.769233 −0.0508324 −0.0254162 0.999677i 0.508091π-0.508091\pi
−0.0254162 + 0.999677i 0.508091π0.508091\pi
230230 0 0
231231 4.00000 0.263181
232232 0 0
233233 18.4008i 1.20548i 0.797939 + 0.602739i 0.205924π0.205924\pi
−0.797939 + 0.602739i 0.794076π0.794076\pi
234234 0 0
235235 1.80052 15.7417i 0.117453 1.02688i
236236 0 0
237237 7.36266i 0.478256i
238238 0 0
239239 10.0328 0.648969 0.324484 0.945891i 0.394809π-0.394809\pi
0.324484 + 0.945891i 0.394809π0.394809\pi
240240 0 0
241241 10.7581 0.692992 0.346496 0.938051i 0.387371π-0.387371\pi
0.346496 + 0.938051i 0.387371π0.387371\pi
242242 0 0
243243 1.00000i 0.0641500i
244244 0 0
245245 −0.0364490 0.00416898i −0.00232864 0.000266347i
246246 0 0
247247 27.4282i 1.74522i
248248 0 0
249249 −7.74173 −0.490612
250250 0 0
251251 −12.6580 −0.798966 −0.399483 0.916741i 0.630810π-0.630810\pi
−0.399483 + 0.916741i 0.630810π0.630810\pi
252252 0 0
253253 7.30749i 0.459418i
254254 0 0
255255 7.36266 + 0.842131i 0.461068 + 0.0527363i
256256 0 0
257257 13.3110i 0.830316i −0.909749 0.415158i 0.863726π-0.863726\pi
0.909749 0.415158i 0.136274π-0.136274\pi
258258 0 0
259259 20.7999 1.29244
260260 0 0
261261 −2.18513 −0.135256
262262 0 0
263263 18.4256i 1.13617i 0.822969 + 0.568087i 0.192316π0.192316\pi
−0.822969 + 0.568087i 0.807684π0.807684\pi
264264 0 0
265265 1.14554 10.0153i 0.0703701 0.615238i
266266 0 0
267267 14.7581i 0.903183i
268268 0 0
269269 3.86940 0.235921 0.117961 0.993018i 0.462364π-0.462364\pi
0.117961 + 0.993018i 0.462364π0.462364\pi
270270 0 0
271271 17.3955 1.05670 0.528350 0.849027i 0.322811π-0.322811\pi
0.528350 + 0.849027i 0.322811π0.322811\pi
272272 0 0
273273 10.2293i 0.619108i
274274 0 0
275275 −7.37270 1.70892i −0.444591 0.103052i
276276 0 0
277277 0.887271i 0.0533110i 0.999645 + 0.0266555i 0.00848571π0.00848571\pi
−0.999645 + 0.0266555i 0.991514π0.991514\pi
278278 0 0
279279 −7.36266 −0.440791
280280 0 0
281281 13.4835 0.804356 0.402178 0.915562i 0.368253π-0.368253\pi
0.402178 + 0.915562i 0.368253π0.368253\pi
282282 0 0
283283 28.4342i 1.69024i 0.534577 + 0.845120i 0.320471π0.320471\pi
−0.534577 + 0.845120i 0.679529π0.679529\pi
284284 0 0
285285 1.80052 15.7417i 0.106653 0.932460i
286286 0 0
287287 23.0580i 1.36107i
288288 0 0
289289 6.01641 0.353906
290290 0 0
291291 −11.1444 −0.653294
292292 0 0
293293 7.99166i 0.466878i −0.972371 0.233439i 0.925002π-0.925002\pi
0.972371 0.233439i 0.0749979π-0.0749979\pi
294294 0 0
295295 −15.1044 1.72762i −0.879412 0.100586i
296296 0 0
297297 1.51363i 0.0878299i
298298 0 0
299299 −18.6877 −1.08074
300300 0 0
301301 2.68601 0.154819
302302 0 0
303303 13.3295i 0.765760i
304304 0 0
305305 8.00000 + 0.915029i 0.458079 + 0.0523944i
306306 0 0
307307 17.4506i 0.995961i 0.867188 + 0.497980i 0.165925π0.165925\pi
−0.867188 + 0.497980i 0.834075π0.834075\pi
308308 0 0
309309 −0.958386 −0.0545207
310310 0 0
311311 21.4506 1.21635 0.608177 0.793801i 0.291901π-0.291901\pi
0.608177 + 0.793801i 0.291901π0.291901\pi
312312 0 0
313313 7.73879i 0.437422i −0.975790 0.218711i 0.929815π-0.929815\pi
0.975790 0.218711i 0.0701853π-0.0701853\pi
314314 0 0
315315 −0.671502 + 5.87086i −0.0378348 + 0.330786i
316316 0 0
317317 11.2335i 0.630938i −0.948936 0.315469i 0.897838π-0.897838\pi
0.948936 0.315469i 0.102162π-0.102162\pi
318318 0 0
319319 −3.30749 −0.185184
320320 0 0
321321 −4.00000 −0.223258
322322 0 0
323323 23.4835i 1.30665i
324324 0 0
325325 4.37027 18.8545i 0.242419 1.04586i
326326 0 0
327327 0.769233i 0.0425387i
328328 0 0
329329 −18.7253 −1.03236
330330 0 0
331331 8.00084 0.439766 0.219883 0.975526i 0.429432π-0.429432\pi
0.219883 + 0.975526i 0.429432π0.429432\pi
332332 0 0
333333 7.87086i 0.431321i
334334 0 0
335335 −0.258271 + 2.25803i −0.0141108 + 0.123369i
336336 0 0
337337 21.5692i 1.17495i 0.809243 + 0.587474i 0.199877π0.199877\pi
−0.809243 + 0.587474i 0.800123π0.800123\pi
338338 0 0
339339 −14.4585 −0.785279
340340 0 0
341341 −11.1444 −0.603501
342342 0 0
343343 18.5419i 1.00117i
344344 0 0
345345 −10.7253 1.22675i −0.577432 0.0660459i
346346 0 0
347347 21.7089i 1.16540i −0.812689 0.582698i 0.801997π-0.801997\pi
0.812689 0.582698i 0.198003π-0.198003\pi
348348 0 0
349349 −24.7422 −1.32442 −0.662211 0.749318i 0.730382π-0.730382\pi
−0.662211 + 0.749318i 0.730382π0.730382\pi
350350 0 0
351351 −3.87086 −0.206611
352352 0 0
353353 3.31415i 0.176394i 0.996103 + 0.0881972i 0.0281106π0.0281106\pi
−0.996103 + 0.0881972i 0.971889π0.971889\pi
354354 0 0
355355 14.9409 + 1.70892i 0.792979 + 0.0906998i
356356 0 0
357357 8.75814i 0.463530i
358358 0 0
359359 16.7581 0.884461 0.442230 0.896902i 0.354187π-0.354187\pi
0.442230 + 0.896902i 0.354187π0.354187\pi
360360 0 0
361361 31.2088 1.64257
362362 0 0
363363 8.70892i 0.457100i
364364 0 0
365365 3.94229 34.4671i 0.206349 1.80409i
366366 0 0
367367 28.5324i 1.48938i 0.667411 + 0.744690i 0.267403π0.267403\pi
−0.667411 + 0.744690i 0.732597π0.732597\pi
368368 0 0
369369 8.72532 0.454222
370370 0 0
371371 −11.9136 −0.618523
372372 0 0
373373 37.5798i 1.94581i 0.231211 + 0.972904i 0.425731π0.425731\pi
−0.231211 + 0.972904i 0.574269π0.574269\pi
374374 0 0
375375 3.74590 10.5341i 0.193437 0.543981i
376376 0 0
377377 8.45836i 0.435628i
378378 0 0
379379 6.74456 0.346445 0.173222 0.984883i 0.444582π-0.444582\pi
0.173222 + 0.984883i 0.444582π0.444582\pi
380380 0 0
381381 11.5290 0.590648
382382 0 0
383383 21.8312i 1.11552i −0.830001 0.557762i 0.811660π-0.811660\pi
0.830001 0.557762i 0.188340π-0.188340\pi
384384 0 0
385385 −1.01641 + 8.88633i −0.0518009 + 0.452889i
386386 0 0
387387 1.01641i 0.0516669i
388388 0 0
389389 8.81344 0.446859 0.223429 0.974720i 0.428275π-0.428275\pi
0.223429 + 0.974720i 0.428275π0.428275\pi
390390 0 0
391391 16.0000 0.809155
392392 0 0
393393 7.37270i 0.371904i
394394 0 0
395395 16.3568 + 1.87086i 0.822998 + 0.0941334i
396396 0 0
397397 0.821644i 0.0412372i 0.999787 + 0.0206186i 0.00656356π0.00656356\pi
−0.999787 + 0.0206186i 0.993436π0.993436\pi
398398 0 0
399399 −18.7253 −0.937439
400400 0 0
401401 −12.7253 −0.635472 −0.317736 0.948179i 0.602923π-0.602923\pi
−0.317736 + 0.948179i 0.602923π0.602923\pi
402402 0 0
403403 28.4999i 1.41968i
404404 0 0
405405 −2.22158 0.254102i −0.110391 0.0126264i
406406 0 0
407407 11.9136i 0.590535i
408408 0 0
409409 2.25827 0.111664 0.0558321 0.998440i 0.482219π-0.482219\pi
0.0558321 + 0.998440i 0.482219π0.482219\pi
410410 0 0
411411 −3.88792 −0.191777
412412 0 0
413413 17.9672i 0.884107i
414414 0 0
415415 1.96719 17.1989i 0.0965654 0.844261i
416416 0 0
417417 14.6291i 0.716392i
418418 0 0
419419 −33.4579 −1.63453 −0.817263 0.576264i 0.804510π-0.804510\pi
−0.817263 + 0.576264i 0.804510π0.804510\pi
420420 0 0
421421 11.3398 0.552669 0.276335 0.961061i 0.410880π-0.410880\pi
0.276335 + 0.961061i 0.410880π0.410880\pi
422422 0 0
423423 7.08582i 0.344524i
424424 0 0
425425 −3.74173 + 16.1428i −0.181501 + 0.783040i
426426 0 0
427427 9.51627i 0.460525i
428428 0 0
429429 −5.85907 −0.282878
430430 0 0
431431 −10.6597 −0.513459 −0.256730 0.966483i 0.582645π-0.582645\pi
−0.256730 + 0.966483i 0.582645π0.582645\pi
432432 0 0
433433 26.5132i 1.27414i −0.770805 0.637072i 0.780146π-0.780146\pi
0.770805 0.637072i 0.219854π-0.219854\pi
434434 0 0
435435 0.555246 4.85446i 0.0266220 0.232753i
436436 0 0
437437 34.2088i 1.63643i
438438 0 0
439439 −32.8789 −1.56923 −0.784613 0.619986i 0.787138π-0.787138\pi
−0.784613 + 0.619986i 0.787138π0.787138\pi
440440 0 0
441441 −0.0164068 −0.000781274
442442 0 0
443443 5.70892i 0.271239i −0.990761 0.135619i 0.956698π-0.956698\pi
0.990761 0.135619i 0.0433024π-0.0433024\pi
444444 0 0
445445 −32.7864 3.75007i −1.55423 0.177770i
446446 0 0
447447 11.0715i 0.523662i
448448 0 0
449449 2.00000 0.0943858 0.0471929 0.998886i 0.484972π-0.484972\pi
0.0471929 + 0.998886i 0.484972π0.484972\pi
450450 0 0
451451 13.2069 0.621890
452452 0 0
453453 0.637339i 0.0299448i
454454 0 0
455455 −22.7253 2.59929i −1.06538 0.121857i
456456 0 0
457457 3.94229i 0.184413i 0.995740 + 0.0922064i 0.0293920π0.0293920\pi
−0.995740 + 0.0922064i 0.970608π0.970608\pi
458458 0 0
459459 3.31415 0.154691
460460 0 0
461461 −33.8969 −1.57874 −0.789369 0.613920i 0.789592π-0.789592\pi
−0.789369 + 0.613920i 0.789592π0.789592\pi
462462 0 0
463463 22.8688i 1.06280i 0.847120 + 0.531402i 0.178335π0.178335\pi
−0.847120 + 0.531402i 0.821665π0.821665\pi
464464 0 0
465465 1.87086 16.3568i 0.0867593 0.758527i
466466 0 0
467467 15.7417i 0.728440i −0.931313 0.364220i 0.881336π-0.881336\pi
0.931313 0.364220i 0.118664π-0.118664\pi
468468 0 0
469469 2.68601 0.124028
470470 0 0
471471 0.129135 0.00595024
472472 0 0
473473 1.53847i 0.0707388i
474474 0 0
475475 34.5140 + 8.00000i 1.58361 + 0.367065i
476476 0 0
477477 4.50820i 0.206416i
478478 0 0
479479 20.6925 0.945465 0.472732 0.881206i 0.343268π-0.343268\pi
0.472732 + 0.881206i 0.343268π0.343268\pi
480480 0 0
481481 −30.4671 −1.38918
482482 0 0
483483 12.7581i 0.580515i
484484 0 0
485485 2.83180 24.7581i 0.128586 1.12421i
486486 0 0
487487 30.8401i 1.39750i 0.715366 + 0.698750i 0.246260π0.246260\pi
−0.715366 + 0.698750i 0.753740π0.753740\pi
488488 0 0
489489 19.4835 0.881072
490490 0 0
491491 −10.9737 −0.495238 −0.247619 0.968858i 0.579648π-0.579648\pi
−0.247619 + 0.968858i 0.579648π0.579648\pi
492492 0 0
493493 7.24186i 0.326157i
494494 0 0
495495 −3.36266 0.384617i −0.151140 0.0172872i
496496 0 0
497497 17.7727i 0.797213i
498498 0 0
499499 3.71729 0.166409 0.0832044 0.996533i 0.473485π-0.473485\pi
0.0832044 + 0.996533i 0.473485π0.473485\pi
500500 0 0
501501 −1.80052 −0.0804412
502502 0 0
503503 39.9451i 1.78107i −0.454919 0.890533i 0.650332π-0.650332\pi
0.454919 0.890533i 0.349668π-0.349668\pi
504504 0 0
505505 −29.6126 3.38705i −1.31774 0.150722i
506506 0 0
507507 1.98359i 0.0880945i
508508 0 0
509509 0.0728979 0.00323114 0.00161557 0.999999i 0.499486π-0.499486\pi
0.00161557 + 0.999999i 0.499486π0.499486\pi
510510 0 0
511511 −40.9997 −1.81372
512512 0 0
513513 7.08582i 0.312846i
514514 0 0
515515 0.243528 2.12914i 0.0107311 0.0938209i
516516 0 0
517517 10.7253i 0.471699i
518518 0 0
519519 −23.2335 −1.01984
520520 0 0
521521 11.9672 0.524292 0.262146 0.965028i 0.415570π-0.415570\pi
0.262146 + 0.965028i 0.415570π0.415570\pi
522522 0 0
523523 16.0656i 0.702501i −0.936282 0.351250i 0.885757π-0.885757\pi
0.936282 0.351250i 0.114243π-0.114243\pi
524524 0 0
525525 −12.8720 2.98359i −0.561779 0.130215i
526526 0 0
527527 24.4010i 1.06292i
528528 0 0
529529 −0.307491 −0.0133692
530530 0 0
531531 −6.79893 −0.295048
532532 0 0
533533 33.7745i 1.46294i
534534 0 0
535535 1.01641 8.88633i 0.0439431 0.384190i
536536 0 0
537537 2.85664i 0.123273i
538538 0 0
539539 −0.0248338 −0.00106967
540540 0 0
541541 15.8559 0.681698 0.340849 0.940118i 0.389285π-0.389285\pi
0.340849 + 0.940118i 0.389285π0.389285\pi
542542 0 0
543543 5.28530i 0.226814i
544544 0 0
545545 1.70892 + 0.195463i 0.0732019 + 0.00837274i
546546 0 0
547547 4.95078i 0.211680i 0.994383 + 0.105840i 0.0337531π0.0337531\pi
−0.994383 + 0.105840i 0.966247π0.966247\pi
548548 0 0
549549 3.60104 0.153688
550550 0 0
551551 15.4835 0.659618
552552 0 0
553553 19.4569i 0.827393i
554554 0 0
555555 −17.4858 2.00000i −0.742230 0.0848953i
556556 0 0
557557 1.26634i 0.0536565i 0.999640 + 0.0268283i 0.00854073π0.00854073\pi
−0.999640 + 0.0268283i 0.991459π0.991459\pi
558558 0 0
559559 −3.93437 −0.166406
560560 0 0
561561 5.01641 0.211793
562562 0 0
563563 5.70892i 0.240602i −0.992737 0.120301i 0.961614π-0.961614\pi
0.992737 0.120301i 0.0383860π-0.0383860\pi
564564 0 0
565565 3.67393 32.1208i 0.154564 1.35133i
566566 0 0
567567 2.64265i 0.110981i
568568 0 0
569569 −2.75814 −0.115627 −0.0578135 0.998327i 0.518413π-0.518413\pi
−0.0578135 + 0.998327i 0.518413π0.518413\pi
570570 0 0
571571 25.7735 1.07859 0.539294 0.842118i 0.318691π-0.318691\pi
0.539294 + 0.842118i 0.318691π0.318691\pi
572572 0 0
573573 5.96719i 0.249283i
574574 0 0
575575 5.45065 23.5155i 0.227308 0.980663i
576576 0 0
577577 32.7135i 1.36188i −0.732338 0.680941i 0.761571π-0.761571\pi
0.732338 0.680941i 0.238429π-0.238429\pi
578578 0 0
579579 −14.9409 −0.620921
580580 0 0
581581 −20.4587 −0.848769
582582 0 0
583583 6.82376i 0.282611i
584584 0 0
585585 0.983593 8.59945i 0.0406666 0.355543i
586586 0 0
587587 43.4835i 1.79475i 0.441264 + 0.897377i 0.354530π0.354530\pi
−0.441264 + 0.897377i 0.645470π0.645470\pi
588588 0 0
589589 52.1705 2.14965
590590 0 0
591591 −3.23353 −0.133009
592592 0 0
593593 7.83021i 0.321548i −0.986991 0.160774i 0.948601π-0.948601\pi
0.986991 0.160774i 0.0513991π-0.0513991\pi
594594 0 0
595595 19.4569 + 2.22546i 0.797656 + 0.0912348i
596596 0 0
597597 8.12080i 0.332362i
598598 0 0
599599 32.7581 1.33846 0.669231 0.743055i 0.266624π-0.266624\pi
0.669231 + 0.743055i 0.266624π0.266624\pi
600600 0 0
601601 −17.8074 −0.726377 −0.363189 0.931716i 0.618312π-0.618312\pi
−0.363189 + 0.931716i 0.618312π0.618312\pi
602602 0 0
603603 1.01641i 0.0413913i
604604 0 0
605605 19.3476 + 2.21295i 0.786591 + 0.0899692i
606606 0 0
607607 3.41188i 0.138484i 0.997600 + 0.0692420i 0.0220581π0.0220581\pi
−0.997600 + 0.0692420i 0.977942π0.977942\pi
608608 0 0
609609 −5.77454 −0.233996
610610 0 0
611611 27.4282 1.10963
612612 0 0
613613 36.6290i 1.47943i 0.672920 + 0.739716i 0.265040π0.265040\pi
−0.672920 + 0.739716i 0.734960π0.734960\pi
614614 0 0
615615 −2.21712 + 19.3840i −0.0894029 + 0.781640i
616616 0 0
617617 40.3979i 1.62636i −0.582012 0.813180i 0.697734π-0.697734\pi
0.582012 0.813180i 0.302266π-0.302266\pi
618618 0 0
619619 24.5172 0.985430 0.492715 0.870191i 0.336004π-0.336004\pi
0.492715 + 0.870191i 0.336004π0.336004\pi
620620 0 0
621621 −4.82778 −0.193732
622622 0 0
623623 39.0006i 1.56252i
624624 0 0
625625 22.4506 + 10.9986i 0.898026 + 0.439943i
626626 0 0
627627 10.7253i 0.428328i
628628 0 0
629629 26.0852 1.04009
630630 0 0
631631 18.7805 0.747640 0.373820 0.927501i 0.378048π-0.378048\pi
0.373820 + 0.927501i 0.378048π0.378048\pi
632632 0 0
633633 13.7141i 0.545087i
634634 0 0
635635 −2.92953 + 25.6126i −0.116255 + 1.01640i
636636 0 0
637637 0.0635083i 0.00251629i
638638 0 0
639639 6.72532 0.266050
640640 0 0
641641 −15.5163 −0.612856 −0.306428 0.951894i 0.599134π-0.599134\pi
−0.306428 + 0.951894i 0.599134π0.599134\pi
642642 0 0
643643 17.4506i 0.688186i 0.938936 + 0.344093i 0.111814π0.111814\pi
−0.938936 + 0.344093i 0.888186π0.888186\pi
644644 0 0
645645 −2.25803 0.258271i −0.0889099 0.0101694i
646646 0 0
647647 13.1403i 0.516600i −0.966065 0.258300i 0.916838π-0.916838\pi
0.966065 0.258300i 0.0831624π-0.0831624\pi
648648 0 0
649649 −10.2911 −0.403960
650650 0 0
651651 −19.4569 −0.762577
652652 0 0
653653 14.7993i 0.579141i 0.957157 + 0.289570i 0.0935124π0.0935124\pi
−0.957157 + 0.289570i 0.906488π0.906488\pi
654654 0 0
655655 16.3791 + 1.87342i 0.639983 + 0.0732004i
656656 0 0
657657 15.5146i 0.605284i
658658 0 0
659659 7.99614 0.311485 0.155743 0.987798i 0.450223π-0.450223\pi
0.155743 + 0.987798i 0.450223π0.450223\pi
660660 0 0
661661 0.915029 0.0355905 0.0177953 0.999842i 0.494335π-0.494335\pi
0.0177953 + 0.999842i 0.494335π0.494335\pi
662662 0 0
663663 12.8286i 0.498223i
664664 0 0
665665 4.75814 41.5999i 0.184513 1.61317i
666666 0 0
667667 10.5494i 0.408473i
668668 0 0
669669 9.84472 0.380619
670670 0 0
671671 5.45065 0.210420
672672 0 0
673673 34.3978i 1.32594i −0.748647 0.662969i 0.769296π-0.769296\pi
0.748647 0.662969i 0.230704π-0.230704\pi
674674 0 0
675675 1.12902 4.87086i 0.0434559 0.187480i
676676 0 0
677677 40.1676i 1.54377i 0.635764 + 0.771884i 0.280685π0.280685\pi
−0.635764 + 0.771884i 0.719315π0.719315\pi
678678 0 0
679679 −29.4506 −1.13021
680680 0 0
681681 5.70892 0.218766
682682 0 0
683683 33.2580i 1.27258i −0.771449 0.636291i 0.780468π-0.780468\pi
0.771449 0.636291i 0.219532π-0.219532\pi
684684 0 0
685685 0.987927 8.63734i 0.0377468 0.330016i
686686 0 0
687687 0.769233i 0.0293481i
688688 0 0
689689 17.4506 0.664817
690690 0 0
691691 50.2241 1.91062 0.955308 0.295611i 0.0955233π-0.0955233\pi
0.955308 + 0.295611i 0.0955233π0.0955233\pi
692692 0 0
693693 4.00000i 0.151947i
694694 0 0
695695 −32.4999 3.71729i −1.23279 0.141005i
696696 0 0
697697 28.9170i 1.09531i
698698 0 0
699699 −18.4008 −0.695983
700700 0 0
701701 23.7543 0.897188 0.448594 0.893736i 0.351925π-0.351925\pi
0.448594 + 0.893736i 0.351925π0.351925\pi
702702 0 0
703703 55.7715i 2.10346i
704704 0 0
705705 15.7417 + 1.80052i 0.592868 + 0.0678114i
706706 0 0
707707 35.2252i 1.32478i
708708 0 0
709709 −36.3146 −1.36382 −0.681911 0.731435i 0.738851π-0.738851\pi
−0.681911 + 0.731435i 0.738851π0.738851\pi
710710 0 0
711711 7.36266 0.276121
712712 0 0
713713 35.5453i 1.33118i
714714 0 0
715715 1.48880 13.0164i 0.0556779 0.486786i
716716 0 0
717717 10.0328i 0.374682i
718718 0 0
719719 30.7253 1.14586 0.572931 0.819604i 0.305806π-0.305806\pi
0.572931 + 0.819604i 0.305806π0.305806\pi
720720 0 0
721721 −2.53268 −0.0943219
722722 0 0
723723 10.7581i 0.400099i
724724 0 0
725725 10.6435 + 2.46705i 0.395289 + 0.0916240i
726726 0 0
727727 5.47445i 0.203036i 0.994834 + 0.101518i 0.0323700π0.0323700\pi
−0.994834 + 0.101518i 0.967630π0.967630\pi
728728 0 0
729729 −1.00000 −0.0370370
730730 0 0
731731 3.36852 0.124589
732732 0 0
733733 17.1455i 0.633285i 0.948545 + 0.316643i 0.102556π0.102556\pi
−0.948545 + 0.316643i 0.897444π0.897444\pi
734734 0 0
735735 0.00416898 0.0364490i 0.000153775 0.00134444i
736736 0 0
737737 1.53847i 0.0566701i
738738 0 0
739739 11.6019 0.426782 0.213391 0.976967i 0.431549π-0.431549\pi
0.213391 + 0.976967i 0.431549π0.431549\pi
740740 0 0
741741 27.4282 1.00760
742742 0 0
743743 23.6613i 0.868048i 0.900901 + 0.434024i 0.142907π0.142907\pi
−0.900901 + 0.434024i 0.857093π0.857093\pi
744744 0 0
745745 24.5962 + 2.81328i 0.901135 + 0.103071i
746746 0 0
747747 7.74173i 0.283255i
748748 0 0
749749 −10.5706 −0.386241
750750 0 0
751751 11.4283 0.417024 0.208512 0.978020i 0.433138π-0.433138\pi
0.208512 + 0.978020i 0.433138π0.433138\pi
752752 0 0
753753 12.6580i 0.461283i
754754 0 0
755755 −1.41590 0.161949i −0.0515299 0.00589392i
756756 0 0
757757 19.1784i 0.697049i −0.937300 0.348525i 0.886683π-0.886683\pi
0.937300 0.348525i 0.113317π-0.113317\pi
758758 0 0
759759 −7.30749 −0.265245
760760 0 0
761761 −4.03281 −0.146189 −0.0730947 0.997325i 0.523288π-0.523288\pi
−0.0730947 + 0.997325i 0.523288π0.523288\pi
762762 0 0
763763 2.03281i 0.0735928i
764764 0 0
765765 −0.842131 + 7.36266i −0.0304473 + 0.266198i
766766 0 0
767767 26.3177i 0.950279i
768768 0 0
769769 2.95078 0.106408 0.0532039 0.998584i 0.483057π-0.483057\pi
0.0532039 + 0.998584i 0.483057π0.483057\pi
770770 0 0
771771 13.3110 0.479383
772772 0 0
773773 45.2663i 1.62812i 0.580783 + 0.814059i 0.302747π0.302747\pi
−0.580783 + 0.814059i 0.697253π0.697253\pi
774774 0 0
775775 35.8625 + 8.31256i 1.28822 + 0.298596i
776776 0 0
777777 20.7999i 0.746193i
778778 0 0
779779 −61.8260 −2.21515
780780 0 0
781781 10.1797 0.364257
782782 0 0
783783 2.18513i 0.0780903i
784784 0 0
785785 −0.0328135 + 0.286885i −0.00117116 + 0.0102394i
786786 0 0
787787 52.9997i 1.88924i −0.328171 0.944618i 0.606432π-0.606432\pi
0.328171 0.944618i 0.393568π-0.393568\pi
788788 0 0
789789 −18.4256 −0.655970
790790 0 0
791791 −38.2088 −1.35855
792792 0 0
793793 13.9391i 0.494993i
794794 0 0
795795 10.0153 + 1.14554i 0.355208 + 0.0406282i
796796 0 0
797797 16.5738i 0.587075i −0.955948 0.293538i 0.905167π-0.905167\pi
0.955948 0.293538i 0.0948326π-0.0948326\pi
798798 0 0
799799 −23.4835 −0.830785
800800 0 0
801801 −14.7581 −0.521453
802802 0 0
803803 23.4835i 0.828713i
804804 0 0
805805 −28.3433 3.24186i −0.998969 0.114261i
806806 0 0
807807 3.86940i 0.136209i
808808 0 0
809809 −37.5491 −1.32016 −0.660078 0.751197i 0.729477π-0.729477\pi
−0.660078 + 0.751197i 0.729477π0.729477\pi
810810 0 0
811811 32.1102 1.12754 0.563771 0.825931i 0.309350π-0.309350\pi
0.563771 + 0.825931i 0.309350π0.309350\pi
812812 0 0
813813 17.3955i 0.610086i
814814 0 0
815815 −4.95078 + 43.2841i −0.173418 + 1.51618i
816816 0 0
817817 7.20207i 0.251969i
818818 0 0
819819 −10.2293 −0.357442
820820 0 0
821821 −29.3809 −1.02540 −0.512699 0.858568i 0.671354π-0.671354\pi
−0.512699 + 0.858568i 0.671354π0.671354\pi
822822 0 0
823823 28.3866i 0.989495i −0.869037 0.494748i 0.835260π-0.835260\pi
0.869037 0.494748i 0.164740π-0.164740\pi
824824 0 0
825825 1.70892 7.37270i 0.0594968 0.256684i
826826 0 0
827827 1.45065i 0.0504439i 0.999682 + 0.0252219i 0.00802924π0.00802924\pi
−0.999682 + 0.0252219i 0.991971π0.991971\pi
828828 0 0
829829 −37.4621 −1.30111 −0.650556 0.759458i 0.725464π-0.725464\pi
−0.650556 + 0.759458i 0.725464π0.725464\pi
830830 0 0
831831 −0.887271 −0.0307791
832832 0 0
833833 0.0543744i 0.00188396i
834834 0 0
835835 0.457515 4.00000i 0.0158329 0.138426i
836836 0 0
837837 7.36266i 0.254491i
838838 0 0
839839 48.7581 1.68332 0.841659 0.540010i 0.181579π-0.181579\pi
0.841659 + 0.540010i 0.181579π0.181579\pi
840840 0 0
841841 −24.2252 −0.835351
842842 0 0
843843 13.4835i 0.464395i
844844 0 0
845845 4.40672 + 0.504034i 0.151596 + 0.0173393i
846846 0 0
847847 23.0146i 0.790791i
848848 0 0
849849 −28.4342 −0.975861
850850 0 0
851851 −37.9988 −1.30258
852852 0 0
853853 4.37073i 0.149651i −0.997197 0.0748255i 0.976160π-0.976160\pi
0.997197 0.0748255i 0.0238400π-0.0238400\pi
854854 0 0
855855 15.7417 + 1.80052i 0.538356 + 0.0615764i
856856 0 0
857857 20.5130i 0.700712i −0.936617 0.350356i 0.886061π-0.886061\pi
0.936617 0.350356i 0.113939π-0.113939\pi
858858 0 0
859859 10.1131 0.345054 0.172527 0.985005i 0.444807π-0.444807\pi
0.172527 + 0.985005i 0.444807π0.444807\pi
860860 0 0
861861 23.0580 0.785813
862862 0 0
863863 13.2861i 0.452266i 0.974096 + 0.226133i 0.0726083π0.0726083\pi
−0.974096 + 0.226133i 0.927392π0.927392\pi
864864 0 0
865865 5.90368 51.6152i 0.200731 1.75497i
866866 0 0
867867 6.01641i 0.204328i
868868 0 0
869869 11.1444 0.378047
870870 0 0
871871 −3.93437 −0.133311
872872 0 0
873873 11.1444i 0.377180i
874874 0 0
875875 9.89909 27.8381i 0.334650 0.941098i
876876 0 0
877877 33.6454i 1.13612i −0.822986 0.568062i 0.807693π-0.807693\pi
0.822986 0.568062i 0.192307π-0.192307\pi
878878 0 0
879879 7.99166 0.269552
880880 0 0
881881 −32.7909 −1.10476 −0.552378 0.833594i 0.686279π-0.686279\pi
−0.552378 + 0.833594i 0.686279π0.686279\pi
882882 0 0
883883 33.4506i 1.12570i 0.826558 + 0.562852i 0.190296π0.190296\pi
−0.826558 + 0.562852i 0.809704π0.809704\pi
884884 0 0
885885 1.72762 15.1044i 0.0580733 0.507729i
886886 0 0
887887 34.8924i 1.17157i −0.810466 0.585785i 0.800786π-0.800786\pi
0.810466 0.585785i 0.199214π-0.199214\pi
888888 0 0
889889 30.4671 1.02183
890890 0 0
891891 −1.51363 −0.0507086
892892 0 0
893893 50.2088i 1.68017i
894894 0 0
895895 6.34625 + 0.725876i 0.212132 + 0.0242634i
896896 0 0
897897 18.6877i 0.623964i
898898 0 0
899899 16.0884 0.536578
900900 0 0
901901 −14.9409 −0.497752
902902 0 0
903903 2.68601i 0.0893847i
904904 0 0
905905 11.7417 + 1.34300i 0.390308 + 0.0446429i
906906 0 0
907907 30.9836i 1.02879i 0.857552 + 0.514397i 0.171984π0.171984\pi
−0.857552 + 0.514397i 0.828016π0.828016\pi
908908 0 0
909909 −13.3295 −0.442112
910910 0 0
911911 −16.0000 −0.530104 −0.265052 0.964234i 0.585389π-0.585389\pi
−0.265052 + 0.964234i 0.585389π0.585389\pi
912912 0 0
913913 11.7181i 0.387814i
914914 0 0
915915 −0.915029 + 8.00000i −0.0302499 + 0.264472i
916916 0 0
917917 19.4835i 0.643400i
918918 0 0
919919 15.6043 0.514737 0.257368 0.966313i 0.417145π-0.417145\pi
0.257368 + 0.966313i 0.417145π0.417145\pi
920920 0 0
921921 −17.4506 −0.575018
922922 0 0
923923 26.0328i 0.856880i
924924 0 0
925925 8.88633 38.3379i 0.292181 1.26054i
926926 0 0
927927 0.958386i 0.0314775i
928928 0 0
929929 38.9341 1.27739 0.638693 0.769461i 0.279475π-0.279475\pi
0.638693 + 0.769461i 0.279475π0.279475\pi
930930 0 0
931931 0.116255 0.00381011
932932 0 0
933933 21.4506i 0.702263i
934934 0 0
935935 −1.27468 + 11.1444i −0.0416864 + 0.364460i
936936 0 0
937937 19.6027i 0.640393i −0.947351 0.320197i 0.896251π-0.896251\pi
0.947351 0.320197i 0.103749π-0.103749\pi
938938 0 0
939939 7.73879 0.252546
940940 0 0
941941 −25.0476 −0.816530 −0.408265 0.912864i 0.633866π-0.633866\pi
−0.408265 + 0.912864i 0.633866π0.633866\pi
942942 0 0
943943 42.1240i 1.37175i
944944 0 0
945945 −5.87086 0.671502i −0.190979 0.0218439i
946946 0 0
947947 7.93437i 0.257832i −0.991655 0.128916i 0.958850π-0.958850\pi
0.991655 0.128916i 0.0411498π-0.0411498\pi
948948 0 0
949949 60.0550 1.94947
950950 0 0
951951 11.2335 0.364272
952952 0 0
953953 11.4809i 0.371903i −0.982559 0.185952i 0.940463π-0.940463\pi
0.982559 0.185952i 0.0595368π-0.0595368\pi
954954 0 0
955955 −13.2566 1.51627i −0.428974 0.0490654i
956956 0 0
957957 3.30749i 0.106916i
958958 0 0
959959 −10.2744 −0.331778
960960 0 0
961961 23.2088 0.748670
962962 0 0
963963 4.00000i 0.128898i
964964 0 0
965965 3.79650 33.1924i 0.122214 1.06850i
966966 0 0
967967 15.8993i 0.511285i 0.966771 + 0.255643i 0.0822871π0.0822871\pi
−0.966771 + 0.255643i 0.917713π0.917713\pi
968968 0 0
969969 −23.4835 −0.754397
970970 0 0
971971 40.6600 1.30484 0.652421 0.757857i 0.273754π-0.273754\pi
0.652421 + 0.757857i 0.273754π0.273754\pi
972972 0 0
973973 38.6597i 1.23937i
974974 0 0
975975 18.8545 + 4.37027i 0.603826 + 0.139961i
976976 0 0
977977 26.5676i 0.849972i −0.905200 0.424986i 0.860279π-0.860279\pi
0.905200 0.424986i 0.139721π-0.139721\pi
978978 0 0
979979 −22.3384 −0.713938
980980 0 0
981981 0.769233 0.0245597
982982 0 0
983983 9.88057i 0.315141i −0.987508 0.157571i 0.949634π-0.949634\pi
0.987508 0.157571i 0.0503662π-0.0503662\pi
984984 0 0
985985 0.821644 7.18355i 0.0261798 0.228887i
986986 0 0
987987 18.7253i 0.596034i
988988 0 0
989989 −4.90699 −0.156033
990990 0 0
991991 53.0549 1.68534 0.842672 0.538427i 0.180981π-0.180981\pi
0.842672 + 0.538427i 0.180981π0.180981\pi
992992 0 0
993993 8.00084i 0.253899i
994994 0 0
995995 −18.0410 2.06351i −0.571939 0.0654176i
996996 0 0
997997 32.3051i 1.02311i −0.859250 0.511556i 0.829069π-0.829069\pi
0.859250 0.511556i 0.170931π-0.170931\pi
998998 0 0
999999 −7.87086 −0.249023
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 3840.2.f.m.769.7 12
4.3 odd 2 3840.2.f.l.769.1 12
5.4 even 2 inner 3840.2.f.m.769.1 12
8.3 odd 2 3840.2.f.l.769.12 12
8.5 even 2 inner 3840.2.f.m.769.6 12
16.3 odd 4 120.2.d.a.109.1 6
16.5 even 4 480.2.d.b.49.3 6
16.11 odd 4 120.2.d.b.109.5 yes 6
16.13 even 4 480.2.d.a.49.4 6
20.19 odd 2 3840.2.f.l.769.7 12
40.19 odd 2 3840.2.f.l.769.6 12
40.29 even 2 inner 3840.2.f.m.769.12 12
48.5 odd 4 1440.2.d.f.1009.4 6
48.11 even 4 360.2.d.e.109.2 6
48.29 odd 4 1440.2.d.e.1009.3 6
48.35 even 4 360.2.d.f.109.6 6
80.3 even 4 600.2.k.f.301.6 12
80.13 odd 4 2400.2.k.f.1201.2 12
80.19 odd 4 120.2.d.b.109.6 yes 6
80.27 even 4 600.2.k.f.301.8 12
80.29 even 4 480.2.d.b.49.4 6
80.37 odd 4 2400.2.k.f.1201.5 12
80.43 even 4 600.2.k.f.301.5 12
80.53 odd 4 2400.2.k.f.1201.8 12
80.59 odd 4 120.2.d.a.109.2 yes 6
80.67 even 4 600.2.k.f.301.7 12
80.69 even 4 480.2.d.a.49.3 6
80.77 odd 4 2400.2.k.f.1201.11 12
240.29 odd 4 1440.2.d.f.1009.3 6
240.53 even 4 7200.2.k.u.3601.4 12
240.59 even 4 360.2.d.f.109.5 6
240.77 even 4 7200.2.k.u.3601.9 12
240.83 odd 4 1800.2.k.u.901.7 12
240.107 odd 4 1800.2.k.u.901.5 12
240.149 odd 4 1440.2.d.e.1009.4 6
240.173 even 4 7200.2.k.u.3601.3 12
240.179 even 4 360.2.d.e.109.1 6
240.197 even 4 7200.2.k.u.3601.10 12
240.203 odd 4 1800.2.k.u.901.8 12
240.227 odd 4 1800.2.k.u.901.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.d.a.109.1 6 16.3 odd 4
120.2.d.a.109.2 yes 6 80.59 odd 4
120.2.d.b.109.5 yes 6 16.11 odd 4
120.2.d.b.109.6 yes 6 80.19 odd 4
360.2.d.e.109.1 6 240.179 even 4
360.2.d.e.109.2 6 48.11 even 4
360.2.d.f.109.5 6 240.59 even 4
360.2.d.f.109.6 6 48.35 even 4
480.2.d.a.49.3 6 80.69 even 4
480.2.d.a.49.4 6 16.13 even 4
480.2.d.b.49.3 6 16.5 even 4
480.2.d.b.49.4 6 80.29 even 4
600.2.k.f.301.5 12 80.43 even 4
600.2.k.f.301.6 12 80.3 even 4
600.2.k.f.301.7 12 80.67 even 4
600.2.k.f.301.8 12 80.27 even 4
1440.2.d.e.1009.3 6 48.29 odd 4
1440.2.d.e.1009.4 6 240.149 odd 4
1440.2.d.f.1009.3 6 240.29 odd 4
1440.2.d.f.1009.4 6 48.5 odd 4
1800.2.k.u.901.5 12 240.107 odd 4
1800.2.k.u.901.6 12 240.227 odd 4
1800.2.k.u.901.7 12 240.83 odd 4
1800.2.k.u.901.8 12 240.203 odd 4
2400.2.k.f.1201.2 12 80.13 odd 4
2400.2.k.f.1201.5 12 80.37 odd 4
2400.2.k.f.1201.8 12 80.53 odd 4
2400.2.k.f.1201.11 12 80.77 odd 4
3840.2.f.l.769.1 12 4.3 odd 2
3840.2.f.l.769.6 12 40.19 odd 2
3840.2.f.l.769.7 12 20.19 odd 2
3840.2.f.l.769.12 12 8.3 odd 2
3840.2.f.m.769.1 12 5.4 even 2 inner
3840.2.f.m.769.6 12 8.5 even 2 inner
3840.2.f.m.769.7 12 1.1 even 1 trivial
3840.2.f.m.769.12 12 40.29 even 2 inner
7200.2.k.u.3601.3 12 240.173 even 4
7200.2.k.u.3601.4 12 240.53 even 4
7200.2.k.u.3601.9 12 240.77 even 4
7200.2.k.u.3601.10 12 240.197 even 4