Properties

Label 208.2.w.b.17.1
Level 208208
Weight 22
Character 208.17
Analytic conductor 1.6611.661
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] = [208,2,Mod(17,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([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("208.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 208=2413 208 = 2^{4} \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 208.w (of order 66, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2] 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: no
Analytic conductor: 1.660888362041.66088836204
Analytic rank: 00
Dimension: 22
Coefficient field: Q(ζ6)\Q(\zeta_{6})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2x+1 x^{2} - x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a9]\Z[a_1, \ldots, a_{9}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

Embedding invariants

Embedding label 17.1
Root 0.500000+0.866025i0.500000 + 0.866025i of defining polynomial
Character χ\chi == 208.17
Dual form 208.2.w.b.49.1

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+(1.000001.73205i)q31.73205iq5+(0.5000000.866025i)q9+(2.500002.59808i)q13+(3.000001.73205i)q15+(1.50000+2.59808i)q17+(3.000001.73205i)q19+(3.00000+5.19615i)q23+2.00000q25+4.00000q27+(1.50000+2.59808i)q293.46410iq31+(7.50000+4.33013i)q37+(7.00000+1.73205i)q39+(4.500002.59808i)q41+(4.00000+6.92820i)q43+(1.50000+0.866025i)q45+3.46410iq47+(3.50000+6.06218i)q49+6.00000q513.00000q536.92820iq57+(6.00000+3.46410i)q59+(0.5000000.866025i)q61+(4.50000+4.33013i)q65+(3.000001.73205i)q67+(6.00000+10.3923i)q69+(3.00000+1.73205i)q71+1.73205iq73+(2.000003.46410i)q754.00000q79+(5.500009.52628i)q8113.8564iq83+(4.500002.59808i)q85+(3.00000+5.19615i)q87+(6.000003.46410i)q89+(6.000003.46410i)q93+(3.000005.19615i)q95+(6.000003.46410i)q97+O(q100)q+(1.00000 - 1.73205i) q^{3} -1.73205i q^{5} +(-0.500000 - 0.866025i) q^{9} +(-2.50000 - 2.59808i) q^{13} +(-3.00000 - 1.73205i) q^{15} +(1.50000 + 2.59808i) q^{17} +(3.00000 - 1.73205i) q^{19} +(-3.00000 + 5.19615i) q^{23} +2.00000 q^{25} +4.00000 q^{27} +(-1.50000 + 2.59808i) q^{29} -3.46410i q^{31} +(7.50000 + 4.33013i) q^{37} +(-7.00000 + 1.73205i) q^{39} +(-4.50000 - 2.59808i) q^{41} +(4.00000 + 6.92820i) q^{43} +(-1.50000 + 0.866025i) q^{45} +3.46410i q^{47} +(-3.50000 + 6.06218i) q^{49} +6.00000 q^{51} -3.00000 q^{53} -6.92820i q^{57} +(-6.00000 + 3.46410i) q^{59} +(-0.500000 - 0.866025i) q^{61} +(-4.50000 + 4.33013i) q^{65} +(-3.00000 - 1.73205i) q^{67} +(6.00000 + 10.3923i) q^{69} +(-3.00000 + 1.73205i) q^{71} +1.73205i q^{73} +(2.00000 - 3.46410i) q^{75} -4.00000 q^{79} +(5.50000 - 9.52628i) q^{81} -13.8564i q^{83} +(4.50000 - 2.59808i) q^{85} +(3.00000 + 5.19615i) q^{87} +(-6.00000 - 3.46410i) q^{89} +(-6.00000 - 3.46410i) q^{93} +(-3.00000 - 5.19615i) q^{95} +(6.00000 - 3.46410i) q^{97} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+2q3q95q136q15+3q17+6q196q23+4q25+8q273q29+15q3714q399q41+8q433q457q49+12q516q53++12q97+O(q100) 2 q + 2 q^{3} - q^{9} - 5 q^{13} - 6 q^{15} + 3 q^{17} + 6 q^{19} - 6 q^{23} + 4 q^{25} + 8 q^{27} - 3 q^{29} + 15 q^{37} - 14 q^{39} - 9 q^{41} + 8 q^{43} - 3 q^{45} - 7 q^{49} + 12 q^{51} - 6 q^{53}+ \cdots + 12 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/208Z)×\left(\mathbb{Z}/208\mathbb{Z}\right)^\times.

nn 5353 7979 145145
χ(n)\chi(n) 11 11 e(16)e\left(\frac{1}{6}\right)

Coefficient data

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



Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)
Significant digits:
nn ana_n an/n(k1)/2a_n / n^{(k-1)/2} αn \alpha_n θn \theta_n
pp apa_p ap/p(k1)/2a_p / p^{(k-1)/2} αp \alpha_p θp \theta_p
22 0 0
33 1.00000 1.73205i 0.577350 1.00000i −0.418432 0.908248i 0.637420π-0.637420\pi
0.995782 0.0917517i 0.0292466π-0.0292466\pi
44 0 0
55 1.73205i 0.774597i −0.921954 0.387298i 0.873408π-0.873408\pi
0.921954 0.387298i 0.126592π-0.126592\pi
66 0 0
77 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
88 0 0
99 −0.500000 0.866025i −0.166667 0.288675i
1010 0 0
1111 0 0 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
1212 0 0
1313 −2.50000 2.59808i −0.693375 0.720577i
1414 0 0
1515 −3.00000 1.73205i −0.774597 0.447214i
1616 0 0
1717 1.50000 + 2.59808i 0.363803 + 0.630126i 0.988583 0.150675i 0.0481447π-0.0481447\pi
−0.624780 + 0.780801i 0.714811π0.714811\pi
1818 0 0
1919 3.00000 1.73205i 0.688247 0.397360i −0.114708 0.993399i 0.536593π-0.536593\pi
0.802955 + 0.596040i 0.203260π0.203260\pi
2020 0 0
2121 0 0
2222 0 0
2323 −3.00000 + 5.19615i −0.625543 + 1.08347i 0.362892 + 0.931831i 0.381789π0.381789\pi
−0.988436 + 0.151642i 0.951544π0.951544\pi
2424 0 0
2525 2.00000 0.400000
2626 0 0
2727 4.00000 0.769800
2828 0 0
2929 −1.50000 + 2.59808i −0.278543 + 0.482451i −0.971023 0.238987i 0.923185π-0.923185\pi
0.692480 + 0.721437i 0.256518π0.256518\pi
3030 0 0
3131 3.46410i 0.622171i −0.950382 0.311086i 0.899307π-0.899307\pi
0.950382 0.311086i 0.100693π-0.100693\pi
3232 0 0
3333 0 0
3434 0 0
3535 0 0
3636 0 0
3737 7.50000 + 4.33013i 1.23299 + 0.711868i 0.967653 0.252286i 0.0811825π-0.0811825\pi
0.265340 + 0.964155i 0.414516π0.414516\pi
3838 0 0
3939 −7.00000 + 1.73205i −1.12090 + 0.277350i
4040 0 0
4141 −4.50000 2.59808i −0.702782 0.405751i 0.105601 0.994409i 0.466323π-0.466323\pi
−0.808383 + 0.588657i 0.799657π0.799657\pi
4242 0 0
4343 4.00000 + 6.92820i 0.609994 + 1.05654i 0.991241 + 0.132068i 0.0421616π0.0421616\pi
−0.381246 + 0.924473i 0.624505π0.624505\pi
4444 0 0
4545 −1.50000 + 0.866025i −0.223607 + 0.129099i
4646 0 0
4747 3.46410i 0.505291i 0.967559 + 0.252646i 0.0813007π0.0813007\pi
−0.967559 + 0.252646i 0.918699π0.918699\pi
4848 0 0
4949 −3.50000 + 6.06218i −0.500000 + 0.866025i
5050 0 0
5151 6.00000 0.840168
5252 0 0
5353 −3.00000 −0.412082 −0.206041 0.978543i 0.566058π-0.566058\pi
−0.206041 + 0.978543i 0.566058π0.566058\pi
5454 0 0
5555 0 0
5656 0 0
5757 6.92820i 0.917663i
5858 0 0
5959 −6.00000 + 3.46410i −0.781133 + 0.450988i −0.836832 0.547460i 0.815595π-0.815595\pi
0.0556984 + 0.998448i 0.482261π0.482261\pi
6060 0 0
6161 −0.500000 0.866025i −0.0640184 0.110883i 0.832240 0.554416i 0.187058π-0.187058\pi
−0.896258 + 0.443533i 0.853725π0.853725\pi
6262 0 0
6363 0 0
6464 0 0
6565 −4.50000 + 4.33013i −0.558156 + 0.537086i
6666 0 0
6767 −3.00000 1.73205i −0.366508 0.211604i 0.305424 0.952217i 0.401202π-0.401202\pi
−0.671932 + 0.740613i 0.734535π0.734535\pi
6868 0 0
6969 6.00000 + 10.3923i 0.722315 + 1.25109i
7070 0 0
7171 −3.00000 + 1.73205i −0.356034 + 0.205557i −0.667340 0.744753i 0.732567π-0.732567\pi
0.311305 + 0.950310i 0.399234π0.399234\pi
7272 0 0
7373 1.73205i 0.202721i 0.994850 + 0.101361i 0.0323196π0.0323196\pi
−0.994850 + 0.101361i 0.967680π0.967680\pi
7474 0 0
7575 2.00000 3.46410i 0.230940 0.400000i
7676 0 0
7777 0 0
7878 0 0
7979 −4.00000 −0.450035 −0.225018 0.974355i 0.572244π-0.572244\pi
−0.225018 + 0.974355i 0.572244π0.572244\pi
8080 0 0
8181 5.50000 9.52628i 0.611111 1.05848i
8282 0 0
8383 13.8564i 1.52094i −0.649374 0.760469i 0.724969π-0.724969\pi
0.649374 0.760469i 0.275031π-0.275031\pi
8484 0 0
8585 4.50000 2.59808i 0.488094 0.281801i
8686 0 0
8787 3.00000 + 5.19615i 0.321634 + 0.557086i
8888 0 0
8989 −6.00000 3.46410i −0.635999 0.367194i 0.147073 0.989126i 0.453015π-0.453015\pi
−0.783072 + 0.621932i 0.786348π0.786348\pi
9090 0 0
9191 0 0
9292 0 0
9393 −6.00000 3.46410i −0.622171 0.359211i
9494 0 0
9595 −3.00000 5.19615i −0.307794 0.533114i
9696 0 0
9797 6.00000 3.46410i 0.609208 0.351726i −0.163448 0.986552i 0.552261π-0.552261\pi
0.772655 + 0.634826i 0.218928π0.218928\pi
9898 0 0
9999 0 0
100100 0 0
101101 1.50000 2.59808i 0.149256 0.258518i −0.781697 0.623658i 0.785646π-0.785646\pi
0.930953 + 0.365140i 0.118979π0.118979\pi
102102 0 0
103103 10.0000 0.985329 0.492665 0.870219i 0.336023π-0.336023\pi
0.492665 + 0.870219i 0.336023π0.336023\pi
104104 0 0
105105 0 0
106106 0 0
107107 3.00000 5.19615i 0.290021 0.502331i −0.683793 0.729676i 0.739671π-0.739671\pi
0.973814 + 0.227345i 0.0730044π0.0730044\pi
108108 0 0
109109 13.8564i 1.32720i −0.748086 0.663602i 0.769027π-0.769027\pi
0.748086 0.663602i 0.230973π-0.230973\pi
110110 0 0
111111 15.0000 8.66025i 1.42374 0.821995i
112112 0 0
113113 7.50000 + 12.9904i 0.705541 + 1.22203i 0.966496 + 0.256681i 0.0826291π0.0826291\pi
−0.260955 + 0.965351i 0.584038π0.584038\pi
114114 0 0
115115 9.00000 + 5.19615i 0.839254 + 0.484544i
116116 0 0
117117 −1.00000 + 3.46410i −0.0924500 + 0.320256i
118118 0 0
119119 0 0
120120 0 0
121121 −5.50000 9.52628i −0.500000 0.866025i
122122 0 0
123123 −9.00000 + 5.19615i −0.811503 + 0.468521i
124124 0 0
125125 12.1244i 1.08444i
126126 0 0
127127 −1.00000 + 1.73205i −0.0887357 + 0.153695i −0.906977 0.421180i 0.861616π-0.861616\pi
0.818241 + 0.574875i 0.194949π0.194949\pi
128128 0 0
129129 16.0000 1.40872
130130 0 0
131131 −18.0000 −1.57267 −0.786334 0.617802i 0.788023π-0.788023\pi
−0.786334 + 0.617802i 0.788023π0.788023\pi
132132 0 0
133133 0 0
134134 0 0
135135 6.92820i 0.596285i
136136 0 0
137137 −13.5000 + 7.79423i −1.15338 + 0.665906i −0.949709 0.313133i 0.898621π-0.898621\pi
−0.203674 + 0.979039i 0.565288π0.565288\pi
138138 0 0
139139 −2.00000 3.46410i −0.169638 0.293821i 0.768655 0.639664i 0.220926π-0.220926\pi
−0.938293 + 0.345843i 0.887593π0.887593\pi
140140 0 0
141141 6.00000 + 3.46410i 0.505291 + 0.291730i
142142 0 0
143143 0 0
144144 0 0
145145 4.50000 + 2.59808i 0.373705 + 0.215758i
146146 0 0
147147 7.00000 + 12.1244i 0.577350 + 1.00000i
148148 0 0
149149 −16.5000 + 9.52628i −1.35173 + 0.780423i −0.988492 0.151272i 0.951663π-0.951663\pi
−0.363241 + 0.931695i 0.618330π0.618330\pi
150150 0 0
151151 17.3205i 1.40952i −0.709444 0.704761i 0.751054π-0.751054\pi
0.709444 0.704761i 0.248946π-0.248946\pi
152152 0 0
153153 1.50000 2.59808i 0.121268 0.210042i
154154 0 0
155155 −6.00000 −0.481932
156156 0 0
157157 −13.0000 −1.03751 −0.518756 0.854922i 0.673605π-0.673605\pi
−0.518756 + 0.854922i 0.673605π0.673605\pi
158158 0 0
159159 −3.00000 + 5.19615i −0.237915 + 0.412082i
160160 0 0
161161 0 0
162162 0 0
163163 −18.0000 + 10.3923i −1.40987 + 0.813988i −0.995375 0.0960641i 0.969375π-0.969375\pi
−0.414494 + 0.910052i 0.636041π0.636041\pi
164164 0 0
165165 0 0
166166 0 0
167167 12.0000 + 6.92820i 0.928588 + 0.536120i 0.886365 0.462988i 0.153223π-0.153223\pi
0.0422232 + 0.999108i 0.486556π0.486556\pi
168168 0 0
169169 −0.500000 + 12.9904i −0.0384615 + 0.999260i
170170 0 0
171171 −3.00000 1.73205i −0.229416 0.132453i
172172 0 0
173173 −3.00000 5.19615i −0.228086 0.395056i 0.729155 0.684349i 0.239913π-0.239913\pi
−0.957241 + 0.289292i 0.906580π0.906580\pi
174174 0 0
175175 0 0
176176 0 0
177177 13.8564i 1.04151i
178178 0 0
179179 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
180180 0 0
181181 11.0000 0.817624 0.408812 0.912619i 0.365943π-0.365943\pi
0.408812 + 0.912619i 0.365943π0.365943\pi
182182 0 0
183183 −2.00000 −0.147844
184184 0 0
185185 7.50000 12.9904i 0.551411 0.955072i
186186 0 0
187187 0 0
188188 0 0
189189 0 0
190190 0 0
191191 9.00000 + 15.5885i 0.651217 + 1.12794i 0.982828 + 0.184525i 0.0590746π0.0590746\pi
−0.331611 + 0.943416i 0.607592π0.607592\pi
192192 0 0
193193 −4.50000 2.59808i −0.323917 0.187014i 0.329220 0.944253i 0.393214π-0.393214\pi
−0.653137 + 0.757240i 0.726548π0.726548\pi
194194 0 0
195195 3.00000 + 12.1244i 0.214834 + 0.868243i
196196 0 0
197197 12.0000 + 6.92820i 0.854965 + 0.493614i 0.862323 0.506359i 0.169009π-0.169009\pi
−0.00735824 + 0.999973i 0.502342π0.502342\pi
198198 0 0
199199 −1.00000 1.73205i −0.0708881 0.122782i 0.828403 0.560133i 0.189250π-0.189250\pi
−0.899291 + 0.437351i 0.855917π0.855917\pi
200200 0 0
201201 −6.00000 + 3.46410i −0.423207 + 0.244339i
202202 0 0
203203 0 0
204204 0 0
205205 −4.50000 + 7.79423i −0.314294 + 0.544373i
206206 0 0
207207 6.00000 0.417029
208208 0 0
209209 0 0
210210 0 0
211211 5.00000 8.66025i 0.344214 0.596196i −0.640996 0.767544i 0.721479π-0.721479\pi
0.985211 + 0.171347i 0.0548120π0.0548120\pi
212212 0 0
213213 6.92820i 0.474713i
214214 0 0
215215 12.0000 6.92820i 0.818393 0.472500i
216216 0 0
217217 0 0
218218 0 0
219219 3.00000 + 1.73205i 0.202721 + 0.117041i
220220 0 0
221221 3.00000 10.3923i 0.201802 0.699062i
222222 0 0
223223 9.00000 + 5.19615i 0.602685 + 0.347960i 0.770097 0.637927i 0.220208π-0.220208\pi
−0.167412 + 0.985887i 0.553541π0.553541\pi
224224 0 0
225225 −1.00000 1.73205i −0.0666667 0.115470i
226226 0 0
227227 −21.0000 + 12.1244i −1.39382 + 0.804722i −0.993736 0.111757i 0.964352π-0.964352\pi
−0.400083 + 0.916479i 0.631019π0.631019\pi
228228 0 0
229229 0 0 1.00000 00
−1.00000 π\pi
230230 0 0
231231 0 0
232232 0 0
233233 6.00000 0.393073 0.196537 0.980497i 0.437031π-0.437031\pi
0.196537 + 0.980497i 0.437031π0.437031\pi
234234 0 0
235235 6.00000 0.391397
236236 0 0
237237 −4.00000 + 6.92820i −0.259828 + 0.450035i
238238 0 0
239239 20.7846i 1.34444i 0.740349 + 0.672222i 0.234660π0.234660\pi
−0.740349 + 0.672222i 0.765340π0.765340\pi
240240 0 0
241241 −1.50000 + 0.866025i −0.0966235 + 0.0557856i −0.547533 0.836784i 0.684433π-0.684433\pi
0.450910 + 0.892570i 0.351100π0.351100\pi
242242 0 0
243243 −5.00000 8.66025i −0.320750 0.555556i
244244 0 0
245245 10.5000 + 6.06218i 0.670820 + 0.387298i
246246 0 0
247247 −12.0000 3.46410i −0.763542 0.220416i
248248 0 0
249249 −24.0000 13.8564i −1.52094 0.878114i
250250 0 0
251251 −9.00000 15.5885i −0.568075 0.983935i −0.996756 0.0804789i 0.974355π-0.974355\pi
0.428681 0.903456i 0.358978π-0.358978\pi
252252 0 0
253253 0 0
254254 0 0
255255 10.3923i 0.650791i
256256 0 0
257257 −1.50000 + 2.59808i −0.0935674 + 0.162064i −0.909010 0.416775i 0.863160π-0.863160\pi
0.815442 + 0.578838i 0.196494π0.196494\pi
258258 0 0
259259 0 0
260260 0 0
261261 3.00000 0.185695
262262 0 0
263263 6.00000 10.3923i 0.369976 0.640817i −0.619586 0.784929i 0.712699π-0.712699\pi
0.989561 + 0.144112i 0.0460326π0.0460326\pi
264264 0 0
265265 5.19615i 0.319197i
266266 0 0
267267 −12.0000 + 6.92820i −0.734388 + 0.423999i
268268 0 0
269269 3.00000 + 5.19615i 0.182913 + 0.316815i 0.942871 0.333157i 0.108114π-0.108114\pi
−0.759958 + 0.649972i 0.774781π0.774781\pi
270270 0 0
271271 −18.0000 10.3923i −1.09342 0.631288i −0.158937 0.987289i 0.550807π-0.550807\pi
−0.934485 + 0.356001i 0.884140π0.884140\pi
272272 0 0
273273 0 0
274274 0 0
275275 0 0
276276 0 0
277277 3.50000 + 6.06218i 0.210295 + 0.364241i 0.951807 0.306699i 0.0992243π-0.0992243\pi
−0.741512 + 0.670940i 0.765891π0.765891\pi
278278 0 0
279279 −3.00000 + 1.73205i −0.179605 + 0.103695i
280280 0 0
281281 22.5167i 1.34323i −0.740900 0.671616i 0.765601π-0.765601\pi
0.740900 0.671616i 0.234399π-0.234399\pi
282282 0 0
283283 2.00000 3.46410i 0.118888 0.205919i −0.800439 0.599414i 0.795400π-0.795400\pi
0.919327 + 0.393494i 0.128734π0.128734\pi
284284 0 0
285285 −12.0000 −0.710819
286286 0 0
287287 0 0
288288 0 0
289289 4.00000 6.92820i 0.235294 0.407541i
290290 0 0
291291 13.8564i 0.812277i
292292 0 0
293293 4.50000 2.59808i 0.262893 0.151781i −0.362761 0.931882i 0.618166π-0.618166\pi
0.625653 + 0.780101i 0.284832π0.284832\pi
294294 0 0
295295 6.00000 + 10.3923i 0.349334 + 0.605063i
296296 0 0
297297 0 0
298298 0 0
299299 21.0000 5.19615i 1.21446 0.300501i
300300 0 0
301301 0 0
302302 0 0
303303 −3.00000 5.19615i −0.172345 0.298511i
304304 0 0
305305 −1.50000 + 0.866025i −0.0858898 + 0.0495885i
306306 0 0
307307 17.3205i 0.988534i 0.869310 + 0.494267i 0.164563π0.164563\pi
−0.869310 + 0.494267i 0.835437π0.835437\pi
308308 0 0
309309 10.0000 17.3205i 0.568880 0.985329i
310310 0 0
311311 30.0000 1.70114 0.850572 0.525859i 0.176256π-0.176256\pi
0.850572 + 0.525859i 0.176256π0.176256\pi
312312 0 0
313313 10.0000 0.565233 0.282617 0.959233i 0.408798π-0.408798\pi
0.282617 + 0.959233i 0.408798π0.408798\pi
314314 0 0
315315 0 0
316316 0 0
317317 5.19615i 0.291845i −0.989296 0.145922i 0.953385π-0.953385\pi
0.989296 0.145922i 0.0466150π-0.0466150\pi
318318 0 0
319319 0 0
320320 0 0
321321 −6.00000 10.3923i −0.334887 0.580042i
322322 0 0
323323 9.00000 + 5.19615i 0.500773 + 0.289122i
324324 0 0
325325 −5.00000 5.19615i −0.277350 0.288231i
326326 0 0
327327 −24.0000 13.8564i −1.32720 0.766261i
328328 0 0
329329 0 0
330330 0 0
331331 24.0000 13.8564i 1.31916 0.761617i 0.335566 0.942017i 0.391072π-0.391072\pi
0.983593 + 0.180400i 0.0577391π0.0577391\pi
332332 0 0
333333 8.66025i 0.474579i
334334 0 0
335335 −3.00000 + 5.19615i −0.163908 + 0.283896i
336336 0 0
337337 −23.0000 −1.25289 −0.626445 0.779466i 0.715491π-0.715491\pi
−0.626445 + 0.779466i 0.715491π0.715491\pi
338338 0 0
339339 30.0000 1.62938
340340 0 0
341341 0 0
342342 0 0
343343 0 0
344344 0 0
345345 18.0000 10.3923i 0.969087 0.559503i
346346 0 0
347347 −15.0000 25.9808i −0.805242 1.39472i −0.916127 0.400887i 0.868702π-0.868702\pi
0.110885 0.993833i 0.464631π-0.464631\pi
348348 0 0
349349 12.0000 + 6.92820i 0.642345 + 0.370858i 0.785517 0.618840i 0.212397π-0.212397\pi
−0.143172 + 0.989698i 0.545730π0.545730\pi
350350 0 0
351351 −10.0000 10.3923i −0.533761 0.554700i
352352 0 0
353353 28.5000 + 16.4545i 1.51690 + 0.875784i 0.999803 + 0.0198582i 0.00632149π0.00632149\pi
0.517099 + 0.855926i 0.327012π0.327012\pi
354354 0 0
355355 3.00000 + 5.19615i 0.159223 + 0.275783i
356356 0 0
357357 0 0
358358 0 0
359359 6.92820i 0.365657i 0.983145 + 0.182828i 0.0585252π0.0585252\pi
−0.983145 + 0.182828i 0.941475π0.941475\pi
360360 0 0
361361 −3.50000 + 6.06218i −0.184211 + 0.319062i
362362 0 0
363363 −22.0000 −1.15470
364364 0 0
365365 3.00000 0.157027
366366 0 0
367367 −11.0000 + 19.0526i −0.574195 + 0.994535i 0.421933 + 0.906627i 0.361352π0.361352\pi
−0.996129 + 0.0879086i 0.971982π0.971982\pi
368368 0 0
369369 5.19615i 0.270501i
370370 0 0
371371 0 0
372372 0 0
373373 −9.50000 16.4545i −0.491891 0.851981i 0.508065 0.861319i 0.330361π-0.330361\pi
−0.999956 + 0.00933789i 0.997028π0.997028\pi
374374 0 0
375375 −21.0000 12.1244i −1.08444 0.626099i
376376 0 0
377377 10.5000 2.59808i 0.540778 0.133808i
378378 0 0
379379 21.0000 + 12.1244i 1.07870 + 0.622786i 0.930545 0.366178i 0.119334π-0.119334\pi
0.148153 + 0.988964i 0.452667π0.452667\pi
380380 0 0
381381 2.00000 + 3.46410i 0.102463 + 0.177471i
382382 0 0
383383 18.0000 10.3923i 0.919757 0.531022i 0.0361995 0.999345i 0.488475π-0.488475\pi
0.883558 + 0.468323i 0.155141π0.155141\pi
384384 0 0
385385 0 0
386386 0 0
387387 4.00000 6.92820i 0.203331 0.352180i
388388 0 0
389389 −9.00000 −0.456318 −0.228159 0.973624i 0.573271π-0.573271\pi
−0.228159 + 0.973624i 0.573271π0.573271\pi
390390 0 0
391391 −18.0000 −0.910299
392392 0 0
393393 −18.0000 + 31.1769i −0.907980 + 1.57267i
394394 0 0
395395 6.92820i 0.348596i
396396 0 0
397397 −12.0000 + 6.92820i −0.602263 + 0.347717i −0.769931 0.638127i 0.779710π-0.779710\pi
0.167668 + 0.985843i 0.446376π0.446376\pi
398398 0 0
399399 0 0
400400 0 0
401401 −1.50000 0.866025i −0.0749064 0.0432472i 0.462079 0.886839i 0.347104π-0.347104\pi
−0.536985 + 0.843592i 0.680437π0.680437\pi
402402 0 0
403403 −9.00000 + 8.66025i −0.448322 + 0.431398i
404404 0 0
405405 −16.5000 9.52628i −0.819892 0.473365i
406406 0 0
407407 0 0
408408 0 0
409409 13.5000 7.79423i 0.667532 0.385400i −0.127609 0.991825i 0.540730π-0.540730\pi
0.795141 + 0.606425i 0.207397π0.207397\pi
410410 0 0
411411 31.1769i 1.53784i
412412 0 0
413413 0 0
414414 0 0
415415 −24.0000 −1.17811
416416 0 0
417417 −8.00000 −0.391762
418418 0 0
419419 9.00000 15.5885i 0.439679 0.761546i −0.557986 0.829851i 0.688426π-0.688426\pi
0.997665 + 0.0683046i 0.0217590π0.0217590\pi
420420 0 0
421421 15.5885i 0.759735i 0.925041 + 0.379867i 0.124030π0.124030\pi
−0.925041 + 0.379867i 0.875970π0.875970\pi
422422 0 0
423423 3.00000 1.73205i 0.145865 0.0842152i
424424 0 0
425425 3.00000 + 5.19615i 0.145521 + 0.252050i
426426 0 0
427427 0 0
428428 0 0
429429 0 0
430430 0 0
431431 6.00000 + 3.46410i 0.289010 + 0.166860i 0.637495 0.770454i 0.279971π-0.279971\pi
−0.348485 + 0.937314i 0.613304π0.613304\pi
432432 0 0
433433 8.50000 + 14.7224i 0.408484 + 0.707515i 0.994720 0.102625i 0.0327243π-0.0327243\pi
−0.586236 + 0.810140i 0.699391π0.699391\pi
434434 0 0
435435 9.00000 5.19615i 0.431517 0.249136i
436436 0 0
437437 20.7846i 0.994263i
438438 0 0
439439 −14.0000 + 24.2487i −0.668184 + 1.15733i 0.310228 + 0.950662i 0.399595π0.399595\pi
−0.978412 + 0.206666i 0.933739π0.933739\pi
440440 0 0
441441 7.00000 0.333333
442442 0 0
443443 12.0000 0.570137 0.285069 0.958507i 0.407984π-0.407984\pi
0.285069 + 0.958507i 0.407984π0.407984\pi
444444 0 0
445445 −6.00000 + 10.3923i −0.284427 + 0.492642i
446446 0 0
447447 38.1051i 1.80231i
448448 0 0
449449 −6.00000 + 3.46410i −0.283158 + 0.163481i −0.634852 0.772634i 0.718939π-0.718939\pi
0.351694 + 0.936115i 0.385606π0.385606\pi
450450 0 0
451451 0 0
452452 0 0
453453 −30.0000 17.3205i −1.40952 0.813788i
454454 0 0
455455 0 0
456456 0 0
457457 1.50000 + 0.866025i 0.0701670 + 0.0405110i 0.534673 0.845059i 0.320435π-0.320435\pi
−0.464506 + 0.885570i 0.653768π0.653768\pi
458458 0 0
459459 6.00000 + 10.3923i 0.280056 + 0.485071i
460460 0 0
461461 19.5000 11.2583i 0.908206 0.524353i 0.0283522 0.999598i 0.490974π-0.490974\pi
0.879853 + 0.475245i 0.157641π0.157641\pi
462462 0 0
463463 13.8564i 0.643962i −0.946746 0.321981i 0.895651π-0.895651\pi
0.946746 0.321981i 0.104349π-0.104349\pi
464464 0 0
465465 −6.00000 + 10.3923i −0.278243 + 0.481932i
466466 0 0
467467 −12.0000 −0.555294 −0.277647 0.960683i 0.589555π-0.589555\pi
−0.277647 + 0.960683i 0.589555π0.589555\pi
468468 0 0
469469 0 0
470470 0 0
471471 −13.0000 + 22.5167i −0.599008 + 1.03751i
472472 0 0
473473 0 0
474474 0 0
475475 6.00000 3.46410i 0.275299 0.158944i
476476 0 0
477477 1.50000 + 2.59808i 0.0686803 + 0.118958i
478478 0 0
479479 −21.0000 12.1244i −0.959514 0.553976i −0.0634909 0.997982i 0.520223π-0.520223\pi
−0.896024 + 0.444006i 0.853557π0.853557\pi
480480 0 0
481481 −7.50000 30.3109i −0.341971 1.38206i
482482 0 0
483483 0 0
484484 0 0
485485 −6.00000 10.3923i −0.272446 0.471890i
486486 0 0
487487 6.00000 3.46410i 0.271886 0.156973i −0.357858 0.933776i 0.616493π-0.616493\pi
0.629744 + 0.776802i 0.283160π0.283160\pi
488488 0 0
489489 41.5692i 1.87983i
490490 0 0
491491 6.00000 10.3923i 0.270776 0.468998i −0.698285 0.715820i 0.746053π-0.746053\pi
0.969061 + 0.246822i 0.0793863π0.0793863\pi
492492 0 0
493493 −9.00000 −0.405340
494494 0 0
495495 0 0
496496 0 0
497497 0 0
498498 0 0
499499 31.1769i 1.39567i −0.716258 0.697835i 0.754147π-0.754147\pi
0.716258 0.697835i 0.245853π-0.245853\pi
500500 0 0
501501 24.0000 13.8564i 1.07224 0.619059i
502502 0 0
503503 18.0000 + 31.1769i 0.802580 + 1.39011i 0.917912 + 0.396783i 0.129873π0.129873\pi
−0.115332 + 0.993327i 0.536793π0.536793\pi
504504 0 0
505505 −4.50000 2.59808i −0.200247 0.115613i
506506 0 0
507507 22.0000 + 13.8564i 0.977054 + 0.615385i
508508 0 0
509509 16.5000 + 9.52628i 0.731350 + 0.422245i 0.818916 0.573914i 0.194576π-0.194576\pi
−0.0875661 + 0.996159i 0.527909π0.527909\pi
510510 0 0
511511 0 0
512512 0 0
513513 12.0000 6.92820i 0.529813 0.305888i
514514 0 0
515515 17.3205i 0.763233i
516516 0 0
517517 0 0
518518 0 0
519519 −12.0000 −0.526742
520520 0 0
521521 9.00000 0.394297 0.197149 0.980374i 0.436832π-0.436832\pi
0.197149 + 0.980374i 0.436832π0.436832\pi
522522 0 0
523523 −8.00000 + 13.8564i −0.349816 + 0.605898i −0.986216 0.165460i 0.947089π-0.947089\pi
0.636401 + 0.771358i 0.280422π0.280422\pi
524524 0 0
525525 0 0
526526 0 0
527527 9.00000 5.19615i 0.392046 0.226348i
528528 0 0
529529 −6.50000 11.2583i −0.282609 0.489493i
530530 0 0
531531 6.00000 + 3.46410i 0.260378 + 0.150329i
532532 0 0
533533 4.50000 + 18.1865i 0.194917 + 0.787746i
534534 0 0
535535 −9.00000 5.19615i −0.389104 0.224649i
536536 0 0
537537 0 0
538538 0 0
539539 0 0
540540 0 0
541541 29.4449i 1.26593i 0.774179 + 0.632967i 0.218163π0.218163\pi
−0.774179 + 0.632967i 0.781837π0.781837\pi
542542 0 0
543543 11.0000 19.0526i 0.472055 0.817624i
544544 0 0
545545 −24.0000 −1.02805
546546 0 0
547547 22.0000 0.940652 0.470326 0.882493i 0.344136π-0.344136\pi
0.470326 + 0.882493i 0.344136π0.344136\pi
548548 0 0
549549 −0.500000 + 0.866025i −0.0213395 + 0.0369611i
550550 0 0
551551 10.3923i 0.442727i
552552 0 0
553553 0 0
554554 0 0
555555 −15.0000 25.9808i −0.636715 1.10282i
556556 0 0
557557 −13.5000 7.79423i −0.572013 0.330252i 0.185940 0.982561i 0.440467π-0.440467\pi
−0.757953 + 0.652309i 0.773800π0.773800\pi
558558 0 0
559559 8.00000 27.7128i 0.338364 1.17213i
560560 0 0
561561 0 0
562562 0 0
563563 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
564564 0 0
565565 22.5000 12.9904i 0.946582 0.546509i
566566 0 0
567567 0 0
568568 0 0
569569 21.0000 36.3731i 0.880366 1.52484i 0.0294311 0.999567i 0.490630π-0.490630\pi
0.850935 0.525271i 0.176036π-0.176036\pi
570570 0 0
571571 −40.0000 −1.67395 −0.836974 0.547243i 0.815677π-0.815677\pi
−0.836974 + 0.547243i 0.815677π0.815677\pi
572572 0 0
573573 36.0000 1.50392
574574 0 0
575575 −6.00000 + 10.3923i −0.250217 + 0.433389i
576576 0 0
577577 19.0526i 0.793168i −0.917998 0.396584i 0.870195π-0.870195\pi
0.917998 0.396584i 0.129805π-0.129805\pi
578578 0 0
579579 −9.00000 + 5.19615i −0.374027 + 0.215945i
580580 0 0
581581 0 0
582582 0 0
583583 0 0
584584 0 0
585585 6.00000 + 1.73205i 0.248069 + 0.0716115i
586586 0 0
587587 18.0000 + 10.3923i 0.742940 + 0.428936i 0.823137 0.567843i 0.192222π-0.192222\pi
−0.0801976 + 0.996779i 0.525555π0.525555\pi
588588 0 0
589589 −6.00000 10.3923i −0.247226 0.428207i
590590 0 0
591591 24.0000 13.8564i 0.987228 0.569976i
592592 0 0
593593 25.9808i 1.06690i −0.845831 0.533451i 0.820895π-0.820895\pi
0.845831 0.533451i 0.179105π-0.179105\pi
594594 0 0
595595 0 0
596596 0 0
597597 −4.00000 −0.163709
598598 0 0
599599 −30.0000 −1.22577 −0.612883 0.790173i 0.709990π-0.709990\pi
−0.612883 + 0.790173i 0.709990π0.709990\pi
600600 0 0
601601 −12.5000 + 21.6506i −0.509886 + 0.883148i 0.490049 + 0.871695i 0.336979π0.336979\pi
−0.999934 + 0.0114528i 0.996354π0.996354\pi
602602 0 0
603603 3.46410i 0.141069i
604604 0 0
605605 −16.5000 + 9.52628i −0.670820 + 0.387298i
606606 0 0
607607 −17.0000 29.4449i −0.690009 1.19513i −0.971834 0.235665i 0.924273π-0.924273\pi
0.281826 0.959466i 0.409060π-0.409060\pi
608608 0 0
609609 0 0
610610 0 0
611611 9.00000 8.66025i 0.364101 0.350356i
612612 0 0
613613 −10.5000 6.06218i −0.424091 0.244849i 0.272735 0.962089i 0.412072π-0.412072\pi
−0.696826 + 0.717240i 0.745405π0.745405\pi
614614 0 0
615615 9.00000 + 15.5885i 0.362915 + 0.628587i
616616 0 0
617617 −19.5000 + 11.2583i −0.785040 + 0.453243i −0.838214 0.545342i 0.816400π-0.816400\pi
0.0531732 + 0.998585i 0.483066π0.483066\pi
618618 0 0
619619 20.7846i 0.835404i 0.908584 + 0.417702i 0.137164π0.137164\pi
−0.908584 + 0.417702i 0.862836π0.862836\pi
620620 0 0
621621 −12.0000 + 20.7846i −0.481543 + 0.834058i
622622 0 0
623623 0 0
624624 0 0
625625 −11.0000 −0.440000
626626 0 0
627627 0 0
628628 0 0
629629 25.9808i 1.03592i
630630 0 0
631631 42.0000 24.2487i 1.67199 0.965326i 0.705473 0.708737i 0.250735π-0.250735\pi
0.966521 0.256589i 0.0825987π-0.0825987\pi
632632 0 0
633633 −10.0000 17.3205i −0.397464 0.688428i
634634 0 0
635635 3.00000 + 1.73205i 0.119051 + 0.0687343i
636636 0 0
637637 24.5000 6.06218i 0.970725 0.240192i
638638 0 0
639639 3.00000 + 1.73205i 0.118678 + 0.0685189i
640640 0 0
641641 −16.5000 28.5788i −0.651711 1.12880i −0.982708 0.185164i 0.940718π-0.940718\pi
0.330997 0.943632i 0.392615π-0.392615\pi
642642 0 0
643643 −12.0000 + 6.92820i −0.473234 + 0.273222i −0.717592 0.696463i 0.754756π-0.754756\pi
0.244359 + 0.969685i 0.421423π0.421423\pi
644644 0 0
645645 27.7128i 1.09119i
646646 0 0
647647 9.00000 15.5885i 0.353827 0.612845i −0.633090 0.774078i 0.718214π-0.718214\pi
0.986916 + 0.161233i 0.0515470π0.0515470\pi
648648 0 0
649649 0 0
650650 0 0
651651 0 0
652652 0 0
653653 15.0000 25.9808i 0.586995 1.01671i −0.407628 0.913148i 0.633644π-0.633644\pi
0.994623 0.103558i 0.0330227π-0.0330227\pi
654654 0 0
655655 31.1769i 1.21818i
656656 0 0
657657 1.50000 0.866025i 0.0585206 0.0337869i
658658 0 0
659659 −6.00000 10.3923i −0.233727 0.404827i 0.725175 0.688565i 0.241759π-0.241759\pi
−0.958902 + 0.283738i 0.908425π0.908425\pi
660660 0 0
661661 −40.5000 23.3827i −1.57527 0.909481i −0.995506 0.0946945i 0.969813π-0.969813\pi
−0.579761 0.814787i 0.696854π-0.696854\pi
662662 0 0
663663 −15.0000 15.5885i −0.582552 0.605406i
664664 0 0
665665 0 0
666666 0 0
667667 −9.00000 15.5885i −0.348481 0.603587i
668668 0 0
669669 18.0000 10.3923i 0.695920 0.401790i
670670 0 0
671671 0 0
672672 0 0
673673 9.50000 16.4545i 0.366198 0.634274i −0.622770 0.782405i 0.713993π-0.713993\pi
0.988968 + 0.148132i 0.0473259π0.0473259\pi
674674 0 0
675675 8.00000 0.307920
676676 0 0
677677 −6.00000 −0.230599 −0.115299 0.993331i 0.536783π-0.536783\pi
−0.115299 + 0.993331i 0.536783π0.536783\pi
678678 0 0
679679 0 0
680680 0 0
681681 48.4974i 1.85843i
682682 0 0
683683 −21.0000 + 12.1244i −0.803543 + 0.463926i −0.844708 0.535227i 0.820226π-0.820226\pi
0.0411658 + 0.999152i 0.486893π0.486893\pi
684684 0 0
685685 13.5000 + 23.3827i 0.515808 + 0.893407i
686686 0 0
687687 0 0
688688 0 0
689689 7.50000 + 7.79423i 0.285727 + 0.296936i
690690 0 0
691691 −12.0000 6.92820i −0.456502 0.263561i 0.254071 0.967186i 0.418230π-0.418230\pi
−0.710572 + 0.703624i 0.751564π0.751564\pi
692692 0 0
693693 0 0
694694 0 0
695695 −6.00000 + 3.46410i −0.227593 + 0.131401i
696696 0 0
697697 15.5885i 0.590455i
698698 0 0
699699 6.00000 10.3923i 0.226941 0.393073i
700700 0 0
701701 18.0000 0.679851 0.339925 0.940452i 0.389598π-0.389598\pi
0.339925 + 0.940452i 0.389598π0.389598\pi
702702 0 0
703703 30.0000 1.13147
704704 0 0
705705 6.00000 10.3923i 0.225973 0.391397i
706706 0 0
707707 0 0
708708 0 0
709709 −4.50000 + 2.59808i −0.169001 + 0.0975728i −0.582115 0.813107i 0.697775π-0.697775\pi
0.413114 + 0.910679i 0.364441π0.364441\pi
710710 0 0
711711 2.00000 + 3.46410i 0.0750059 + 0.129914i
712712 0 0
713713 18.0000 + 10.3923i 0.674105 + 0.389195i
714714 0 0
715715 0 0
716716 0 0
717717 36.0000 + 20.7846i 1.34444 + 0.776215i
718718 0 0
719719 −24.0000 41.5692i −0.895049 1.55027i −0.833744 0.552151i 0.813807π-0.813807\pi
−0.0613050 0.998119i 0.519526π-0.519526\pi
720720 0 0
721721 0 0
722722 0 0
723723 3.46410i 0.128831i
724724 0 0
725725 −3.00000 + 5.19615i −0.111417 + 0.192980i
726726 0 0
727727 32.0000 1.18681 0.593407 0.804902i 0.297782π-0.297782\pi
0.593407 + 0.804902i 0.297782π0.297782\pi
728728 0 0
729729 13.0000 0.481481
730730 0 0
731731 −12.0000 + 20.7846i −0.443836 + 0.768747i
732732 0 0
733733 12.1244i 0.447823i 0.974609 + 0.223912i 0.0718827π0.0718827\pi
−0.974609 + 0.223912i 0.928117π0.928117\pi
734734 0 0
735735 21.0000 12.1244i 0.774597 0.447214i
736736 0 0
737737 0 0
738738 0 0
739739 18.0000 + 10.3923i 0.662141 + 0.382287i 0.793092 0.609102i 0.208470π-0.208470\pi
−0.130951 + 0.991389i 0.541803π0.541803\pi
740740 0 0
741741 −18.0000 + 17.3205i −0.661247 + 0.636285i
742742 0 0
743743 30.0000 + 17.3205i 1.10059 + 0.635428i 0.936377 0.350997i 0.114157π-0.114157\pi
0.164216 + 0.986424i 0.447490π0.447490\pi
744744 0 0
745745 16.5000 + 28.5788i 0.604513 + 1.04705i
746746 0 0
747747 −12.0000 + 6.92820i −0.439057 + 0.253490i
748748 0 0
749749 0 0
750750 0 0
751751 −8.00000 + 13.8564i −0.291924 + 0.505627i −0.974265 0.225407i 0.927629π-0.927629\pi
0.682341 + 0.731034i 0.260962π0.260962\pi
752752 0 0
753753 −36.0000 −1.31191
754754 0 0
755755 −30.0000 −1.09181
756756 0 0
757757 13.0000 22.5167i 0.472493 0.818382i −0.527011 0.849858i 0.676688π-0.676688\pi
0.999505 + 0.0314762i 0.0100208π0.0100208\pi
758758 0 0
759759 0 0
760760 0 0
761761 30.0000 17.3205i 1.08750 0.627868i 0.154590 0.987979i 0.450594π-0.450594\pi
0.932910 + 0.360111i 0.117261π0.117261\pi
762762 0 0
763763 0 0
764764 0 0
765765 −4.50000 2.59808i −0.162698 0.0939336i
766766 0 0
767767 24.0000 + 6.92820i 0.866590 + 0.250163i
768768 0 0
769769 6.00000 + 3.46410i 0.216366 + 0.124919i 0.604266 0.796782i 0.293466π-0.293466\pi
−0.387901 + 0.921701i 0.626800π0.626800\pi
770770 0 0
771771 3.00000 + 5.19615i 0.108042 + 0.187135i
772772 0 0
773773 −30.0000 + 17.3205i −1.07903 + 0.622975i −0.930633 0.365953i 0.880743π-0.880743\pi
−0.148392 + 0.988929i 0.547410π0.547410\pi
774774 0 0
775775 6.92820i 0.248868i
776776 0 0
777777 0 0
778778 0 0
779779 −18.0000 −0.644917
780780 0 0
781781 0 0
782782 0 0
783783 −6.00000 + 10.3923i −0.214423 + 0.371391i
784784 0 0
785785 22.5167i 0.803654i
786786 0 0
787787 −33.0000 + 19.0526i −1.17632 + 0.679150i −0.955161 0.296087i 0.904318π-0.904318\pi
−0.221162 + 0.975237i 0.570985π0.570985\pi
788788 0 0
789789 −12.0000 20.7846i −0.427211 0.739952i
790790 0 0
791791 0 0
792792 0 0
793793 −1.00000 + 3.46410i −0.0355110 + 0.123014i
794794 0 0
795795 9.00000 + 5.19615i 0.319197 + 0.184289i
796796 0 0
797797 −21.0000 36.3731i −0.743858 1.28840i −0.950726 0.310031i 0.899660π-0.899660\pi
0.206868 0.978369i 0.433673π-0.433673\pi
798798 0 0
799799 −9.00000 + 5.19615i −0.318397 + 0.183827i
800800 0 0
801801 6.92820i 0.244796i
802802 0 0
803803 0 0
804804 0 0
805805 0 0
806806 0 0
807807 12.0000 0.422420
808808 0 0
809809 −16.5000 + 28.5788i −0.580109 + 1.00478i 0.415357 + 0.909659i 0.363657π0.363657\pi
−0.995466 + 0.0951198i 0.969677π0.969677\pi
810810 0 0
811811 38.1051i 1.33805i 0.743239 + 0.669026i 0.233288π0.233288\pi
−0.743239 + 0.669026i 0.766712π0.766712\pi
812812 0 0
813813 −36.0000 + 20.7846i −1.26258 + 0.728948i
814814 0 0
815815 18.0000 + 31.1769i 0.630512 + 1.09208i
816816 0 0
817817 24.0000 + 13.8564i 0.839654 + 0.484774i
818818 0 0
819819 0 0
820820 0 0
821821 −36.0000 20.7846i −1.25641 0.725388i −0.284034 0.958814i 0.591673π-0.591673\pi
−0.972375 + 0.233426i 0.925006π0.925006\pi
822822 0 0
823823 −2.00000 3.46410i −0.0697156 0.120751i 0.829060 0.559159i 0.188876π-0.188876\pi
−0.898776 + 0.438408i 0.855543π0.855543\pi
824824 0 0
825825 0 0
826826 0 0
827827 20.7846i 0.722752i −0.932420 0.361376i 0.882307π-0.882307\pi
0.932420 0.361376i 0.117693π-0.117693\pi
828828 0 0
829829 −12.5000 + 21.6506i −0.434143 + 0.751958i −0.997225 0.0744432i 0.976282π-0.976282\pi
0.563082 + 0.826401i 0.309615π0.309615\pi
830830 0 0
831831 14.0000 0.485655
832832 0 0
833833 −21.0000 −0.727607
834834 0 0
835835 12.0000 20.7846i 0.415277 0.719281i
836836 0 0
837837 13.8564i 0.478947i
838838 0 0
839839 −39.0000 + 22.5167i −1.34643 + 0.777361i −0.987742 0.156096i 0.950109π-0.950109\pi
−0.358688 + 0.933458i 0.616776π0.616776\pi
840840 0 0
841841 10.0000 + 17.3205i 0.344828 + 0.597259i
842842 0 0
843843 −39.0000 22.5167i −1.34323 0.775515i
844844 0 0
845845 22.5000 + 0.866025i 0.774024 + 0.0297922i
846846 0 0
847847 0 0
848848 0 0
849849 −4.00000 6.92820i −0.137280 0.237775i
850850 0 0
851851 −45.0000 + 25.9808i −1.54258 + 0.890609i
852852 0 0
853853 25.9808i 0.889564i 0.895639 + 0.444782i 0.146719π0.146719\pi
−0.895639 + 0.444782i 0.853281π0.853281\pi
854854 0 0
855855 −3.00000 + 5.19615i −0.102598 + 0.177705i
856856 0 0
857857 3.00000 0.102478 0.0512390 0.998686i 0.483683π-0.483683\pi
0.0512390 + 0.998686i 0.483683π0.483683\pi
858858 0 0
859859 14.0000 0.477674 0.238837 0.971060i 0.423234π-0.423234\pi
0.238837 + 0.971060i 0.423234π0.423234\pi
860860 0 0
861861 0 0
862862 0 0
863863 27.7128i 0.943355i 0.881771 + 0.471678i 0.156351π0.156351\pi
−0.881771 + 0.471678i 0.843649π0.843649\pi
864864 0 0
865865 −9.00000 + 5.19615i −0.306009 + 0.176674i
866866 0 0
867867 −8.00000 13.8564i −0.271694 0.470588i
868868 0 0
869869 0 0
870870 0 0
871871 3.00000 + 12.1244i 0.101651 + 0.410818i
872872 0 0
873873 −6.00000 3.46410i −0.203069 0.117242i
874874 0 0
875875 0 0
876876 0 0
877877 10.5000 6.06218i 0.354560 0.204705i −0.312132 0.950039i 0.601043π-0.601043\pi
0.666692 + 0.745334i 0.267710π0.267710\pi
878878 0 0
879879 10.3923i 0.350524i
880880 0 0
881881 −13.5000 + 23.3827i −0.454827 + 0.787783i −0.998678 0.0513987i 0.983632π-0.983632\pi
0.543852 + 0.839181i 0.316965π0.316965\pi
882882 0 0
883883 −10.0000 −0.336527 −0.168263 0.985742i 0.553816π-0.553816\pi
−0.168263 + 0.985742i 0.553816π0.553816\pi
884884 0 0
885885 24.0000 0.806751
886886 0 0
887887 18.0000 31.1769i 0.604381 1.04682i −0.387768 0.921757i 0.626754π-0.626754\pi
0.992149 0.125061i 0.0399128π-0.0399128\pi
888888 0 0
889889 0 0
890890 0 0
891891 0 0
892892 0 0
893893 6.00000 + 10.3923i 0.200782 + 0.347765i
894894 0 0
895895 0 0
896896 0 0
897897 12.0000 41.5692i 0.400668 1.38796i
898898 0 0
899899 9.00000 + 5.19615i 0.300167 + 0.173301i
900900 0 0
901901 −4.50000 7.79423i −0.149917 0.259663i
902902 0 0
903903 0 0
904904 0 0
905905 19.0526i 0.633328i
906906 0 0
907907 14.0000 24.2487i 0.464862 0.805165i −0.534333 0.845274i 0.679437π-0.679437\pi
0.999195 + 0.0401089i 0.0127705π0.0127705\pi
908908 0 0
909909 −3.00000 −0.0995037
910910 0 0
911911 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
912912 0 0
913913 0 0
914914 0 0
915915 3.46410i 0.114520i
916916 0 0
917917 0 0
918918 0 0
919919 11.0000 + 19.0526i 0.362857 + 0.628486i 0.988430 0.151680i 0.0484682π-0.0484682\pi
−0.625573 + 0.780165i 0.715135π0.715135\pi
920920 0 0
921921 30.0000 + 17.3205i 0.988534 + 0.570730i
922922 0 0
923923 12.0000 + 3.46410i 0.394985 + 0.114022i
924924 0 0
925925 15.0000 + 8.66025i 0.493197 + 0.284747i
926926 0 0
927927 −5.00000 8.66025i −0.164222 0.284440i
928928 0 0
929929 −40.5000 + 23.3827i −1.32876 + 0.767161i −0.985108 0.171935i 0.944998π-0.944998\pi
−0.343654 + 0.939096i 0.611665π0.611665\pi
930930 0 0
931931 24.2487i 0.794719i
932932 0 0
933933 30.0000 51.9615i 0.982156 1.70114i
934934 0 0
935935 0 0
936936 0 0
937937 7.00000 0.228680 0.114340 0.993442i 0.463525π-0.463525\pi
0.114340 + 0.993442i 0.463525π0.463525\pi
938938 0 0
939939 10.0000 17.3205i 0.326338 0.565233i
940940 0 0
941941 20.7846i 0.677559i 0.940866 + 0.338779i 0.110014π0.110014\pi
−0.940866 + 0.338779i 0.889986π0.889986\pi
942942 0 0
943943 27.0000 15.5885i 0.879241 0.507630i
944944 0 0
945945 0 0
946946 0 0
947947 15.0000 + 8.66025i 0.487435 + 0.281420i 0.723510 0.690314i 0.242528π-0.242528\pi
−0.236075 + 0.971735i 0.575861π0.575861\pi
948948 0 0
949949 4.50000 4.33013i 0.146076 0.140562i
950950 0 0
951951 −9.00000 5.19615i −0.291845 0.168497i
952952 0 0
953953 3.00000 + 5.19615i 0.0971795 + 0.168320i 0.910516 0.413473i 0.135685π-0.135685\pi
−0.813337 + 0.581793i 0.802351π0.802351\pi
954954 0 0
955955 27.0000 15.5885i 0.873699 0.504431i
956956 0 0
957957 0 0
958958 0 0
959959 0 0
960960 0 0
961961 19.0000 0.612903
962962 0 0
963963 −6.00000 −0.193347
964964 0 0
965965 −4.50000 + 7.79423i −0.144860 + 0.250905i
966966 0 0
967967 58.8897i 1.89377i −0.321578 0.946883i 0.604213π-0.604213\pi
0.321578 0.946883i 0.395787π-0.395787\pi
968968 0 0
969969 18.0000 10.3923i 0.578243 0.333849i
970970 0 0
971971 3.00000 + 5.19615i 0.0962746 + 0.166752i 0.910140 0.414301i 0.135974π-0.135974\pi
−0.813865 + 0.581054i 0.802641π0.802641\pi
972972 0 0
973973 0 0
974974 0 0
975975 −14.0000 + 3.46410i −0.448359 + 0.110940i
976976 0 0
977977 37.5000 + 21.6506i 1.19973 + 0.692665i 0.960495 0.278296i 0.0897697π-0.0897697\pi
0.239236 + 0.970961i 0.423103π0.423103\pi
978978 0 0
979979 0 0
980980 0 0
981981 −12.0000 + 6.92820i −0.383131 + 0.221201i
982982 0 0
983983 51.9615i 1.65732i −0.559756 0.828658i 0.689105π-0.689105\pi
0.559756 0.828658i 0.310895π-0.310895\pi
984984 0 0
985985 12.0000 20.7846i 0.382352 0.662253i
986986 0 0
987987 0 0
988988 0 0
989989 −48.0000 −1.52631
990990 0 0
991991 1.00000 1.73205i 0.0317660 0.0550204i −0.849705 0.527258i 0.823220π-0.823220\pi
0.881471 + 0.472237i 0.156554π0.156554\pi
992992 0 0
993993 55.4256i 1.75888i
994994 0 0
995995 −3.00000 + 1.73205i −0.0951064 + 0.0549097i
996996 0 0
997997 −8.50000 14.7224i −0.269198 0.466264i 0.699457 0.714675i 0.253425π-0.253425\pi
−0.968655 + 0.248410i 0.920092π0.920092\pi
998998 0 0
999999 30.0000 + 17.3205i 0.949158 + 0.547997i
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 208.2.w.b.17.1 2
3.2 odd 2 1872.2.by.d.433.1 2
4.3 odd 2 13.2.e.a.4.1 2
8.3 odd 2 832.2.w.d.641.1 2
8.5 even 2 832.2.w.a.641.1 2
12.11 even 2 117.2.q.c.82.1 2
13.4 even 6 2704.2.f.b.337.2 2
13.6 odd 12 2704.2.a.o.1.1 2
13.7 odd 12 2704.2.a.o.1.2 2
13.9 even 3 2704.2.f.b.337.1 2
13.10 even 6 inner 208.2.w.b.49.1 2
20.3 even 4 325.2.m.a.199.1 4
20.7 even 4 325.2.m.a.199.2 4
20.19 odd 2 325.2.n.a.251.1 2
28.3 even 6 637.2.u.b.30.1 2
28.11 odd 6 637.2.u.c.30.1 2
28.19 even 6 637.2.k.c.459.1 2
28.23 odd 6 637.2.k.a.459.1 2
28.27 even 2 637.2.q.a.589.1 2
39.23 odd 6 1872.2.by.d.1297.1 2
52.3 odd 6 169.2.e.a.23.1 2
52.7 even 12 169.2.a.a.1.1 2
52.11 even 12 169.2.c.a.146.2 4
52.15 even 12 169.2.c.a.146.1 4
52.19 even 12 169.2.a.a.1.2 2
52.23 odd 6 13.2.e.a.10.1 yes 2
52.31 even 4 169.2.c.a.22.1 4
52.35 odd 6 169.2.b.a.168.2 2
52.43 odd 6 169.2.b.a.168.1 2
52.47 even 4 169.2.c.a.22.2 4
52.51 odd 2 169.2.e.a.147.1 2
104.75 odd 6 832.2.w.d.257.1 2
104.101 even 6 832.2.w.a.257.1 2
156.23 even 6 117.2.q.c.10.1 2
156.35 even 6 1521.2.b.a.1351.1 2
156.59 odd 12 1521.2.a.k.1.2 2
156.71 odd 12 1521.2.a.k.1.1 2
156.95 even 6 1521.2.b.a.1351.2 2
260.19 even 12 4225.2.a.v.1.1 2
260.23 even 12 325.2.m.a.49.2 4
260.59 even 12 4225.2.a.v.1.2 2
260.127 even 12 325.2.m.a.49.1 4
260.179 odd 6 325.2.n.a.101.1 2
364.23 odd 6 637.2.u.c.361.1 2
364.75 even 6 637.2.u.b.361.1 2
364.111 odd 12 8281.2.a.q.1.1 2
364.179 odd 6 637.2.k.a.569.1 2
364.279 odd 12 8281.2.a.q.1.2 2
364.283 even 6 637.2.k.c.569.1 2
364.335 even 6 637.2.q.a.491.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 2 4.3 odd 2
13.2.e.a.10.1 yes 2 52.23 odd 6
117.2.q.c.10.1 2 156.23 even 6
117.2.q.c.82.1 2 12.11 even 2
169.2.a.a.1.1 2 52.7 even 12
169.2.a.a.1.2 2 52.19 even 12
169.2.b.a.168.1 2 52.43 odd 6
169.2.b.a.168.2 2 52.35 odd 6
169.2.c.a.22.1 4 52.31 even 4
169.2.c.a.22.2 4 52.47 even 4
169.2.c.a.146.1 4 52.15 even 12
169.2.c.a.146.2 4 52.11 even 12
169.2.e.a.23.1 2 52.3 odd 6
169.2.e.a.147.1 2 52.51 odd 2
208.2.w.b.17.1 2 1.1 even 1 trivial
208.2.w.b.49.1 2 13.10 even 6 inner
325.2.m.a.49.1 4 260.127 even 12
325.2.m.a.49.2 4 260.23 even 12
325.2.m.a.199.1 4 20.3 even 4
325.2.m.a.199.2 4 20.7 even 4
325.2.n.a.101.1 2 260.179 odd 6
325.2.n.a.251.1 2 20.19 odd 2
637.2.k.a.459.1 2 28.23 odd 6
637.2.k.a.569.1 2 364.179 odd 6
637.2.k.c.459.1 2 28.19 even 6
637.2.k.c.569.1 2 364.283 even 6
637.2.q.a.491.1 2 364.335 even 6
637.2.q.a.589.1 2 28.27 even 2
637.2.u.b.30.1 2 28.3 even 6
637.2.u.b.361.1 2 364.75 even 6
637.2.u.c.30.1 2 28.11 odd 6
637.2.u.c.361.1 2 364.23 odd 6
832.2.w.a.257.1 2 104.101 even 6
832.2.w.a.641.1 2 8.5 even 2
832.2.w.d.257.1 2 104.75 odd 6
832.2.w.d.641.1 2 8.3 odd 2
1521.2.a.k.1.1 2 156.71 odd 12
1521.2.a.k.1.2 2 156.59 odd 12
1521.2.b.a.1351.1 2 156.35 even 6
1521.2.b.a.1351.2 2 156.95 even 6
1872.2.by.d.433.1 2 3.2 odd 2
1872.2.by.d.1297.1 2 39.23 odd 6
2704.2.a.o.1.1 2 13.6 odd 12
2704.2.a.o.1.2 2 13.7 odd 12
2704.2.f.b.337.1 2 13.9 even 3
2704.2.f.b.337.2 2 13.4 even 6
4225.2.a.v.1.1 2 260.19 even 12
4225.2.a.v.1.2 2 260.59 even 12
8281.2.a.q.1.1 2 364.111 odd 12
8281.2.a.q.1.2 2 364.279 odd 12