Properties

Label 624.2.q.c.289.1
Level 624624
Weight 22
Character 624.289
Analytic conductor 4.9834.983
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] = [624,2,Mod(289,624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) 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("624.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 624=24313 624 = 2^{4} \cdot 3 \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 624.q (of order 33, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1,0,-2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 4.982665086134.98266508613
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,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

Embedding invariants

Embedding label 289.1
Root 0.5000000.866025i0.500000 - 0.866025i of defining polynomial
Character χ\chi == 624.289
Dual form 624.2.q.c.529.1

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+(0.5000000.866025i)q31.00000q5+(1.000001.73205i)q7+(0.500000+0.866025i)q9+(1.000001.73205i)q11+(3.50000+0.866025i)q13+(0.500000+0.866025i)q15+(3.500006.06218i)q17+(3.00000+5.19615i)q192.00000q21+(3.000005.19615i)q234.00000q25+1.00000q27+(0.500000+0.866025i)q294.00000q31+(1.00000+1.73205i)q33+(1.00000+1.73205i)q35+(0.5000000.866025i)q37+(2.50000+2.59808i)q39+(4.500007.79423i)q41+(3.000005.19615i)q43+(0.5000000.866025i)q456.00000q47+(1.50000+2.59808i)q497.00000q519.00000q53+(1.00000+1.73205i)q55+6.00000q57+(0.500000+0.866025i)q61+(1.00000+1.73205i)q63+(3.500000.866025i)q65+(1.000001.73205i)q67+(3.00000+5.19615i)q69+(3.000005.19615i)q71+11.0000q73+(2.00000+3.46410i)q754.00000q77+4.00000q79+(0.5000000.866025i)q81+14.0000q83+(3.50000+6.06218i)q85+(0.5000000.866025i)q87+(7.00000+12.1244i)q89+(2.00000+6.92820i)q91+(2.00000+3.46410i)q93+(3.000005.19615i)q95+(1.000001.73205i)q97+2.00000q99+O(q100)q+(-0.500000 - 0.866025i) q^{3} -1.00000 q^{5} +(1.00000 - 1.73205i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(-1.00000 - 1.73205i) q^{11} +(-3.50000 + 0.866025i) q^{13} +(0.500000 + 0.866025i) q^{15} +(3.50000 - 6.06218i) q^{17} +(-3.00000 + 5.19615i) q^{19} -2.00000 q^{21} +(-3.00000 - 5.19615i) q^{23} -4.00000 q^{25} +1.00000 q^{27} +(0.500000 + 0.866025i) q^{29} -4.00000 q^{31} +(-1.00000 + 1.73205i) q^{33} +(-1.00000 + 1.73205i) q^{35} +(-0.500000 - 0.866025i) q^{37} +(2.50000 + 2.59808i) q^{39} +(-4.50000 - 7.79423i) q^{41} +(3.00000 - 5.19615i) q^{43} +(0.500000 - 0.866025i) q^{45} -6.00000 q^{47} +(1.50000 + 2.59808i) q^{49} -7.00000 q^{51} -9.00000 q^{53} +(1.00000 + 1.73205i) q^{55} +6.00000 q^{57} +(-0.500000 + 0.866025i) q^{61} +(1.00000 + 1.73205i) q^{63} +(3.50000 - 0.866025i) q^{65} +(-1.00000 - 1.73205i) q^{67} +(-3.00000 + 5.19615i) q^{69} +(3.00000 - 5.19615i) q^{71} +11.0000 q^{73} +(2.00000 + 3.46410i) q^{75} -4.00000 q^{77} +4.00000 q^{79} +(-0.500000 - 0.866025i) q^{81} +14.0000 q^{83} +(-3.50000 + 6.06218i) q^{85} +(0.500000 - 0.866025i) q^{87} +(7.00000 + 12.1244i) q^{89} +(-2.00000 + 6.92820i) q^{91} +(2.00000 + 3.46410i) q^{93} +(3.00000 - 5.19615i) q^{95} +(1.00000 - 1.73205i) q^{97} +2.00000 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2qq32q5+2q7q92q117q13+q15+7q176q194q216q238q25+2q27+q298q312q332q35q37+5q39++4q99+O(q100) 2 q - q^{3} - 2 q^{5} + 2 q^{7} - q^{9} - 2 q^{11} - 7 q^{13} + q^{15} + 7 q^{17} - 6 q^{19} - 4 q^{21} - 6 q^{23} - 8 q^{25} + 2 q^{27} + q^{29} - 8 q^{31} - 2 q^{33} - 2 q^{35} - q^{37} + 5 q^{39}+ \cdots + 4 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/624Z)×\left(\mathbb{Z}/624\mathbb{Z}\right)^\times.

nn 7979 145145 209209 469469
χ(n)\chi(n) 11 e(13)e\left(\frac{1}{3}\right) 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 −0.500000 0.866025i −0.288675 0.500000i
44 0 0
55 −1.00000 −0.447214 −0.223607 0.974679i 0.571783π-0.571783\pi
−0.223607 + 0.974679i 0.571783π0.571783\pi
66 0 0
77 1.00000 1.73205i 0.377964 0.654654i −0.612801 0.790237i 0.709957π-0.709957\pi
0.990766 + 0.135583i 0.0432908π0.0432908\pi
88 0 0
99 −0.500000 + 0.866025i −0.166667 + 0.288675i
1010 0 0
1111 −1.00000 1.73205i −0.301511 0.522233i 0.674967 0.737848i 0.264158π-0.264158\pi
−0.976478 + 0.215615i 0.930824π0.930824\pi
1212 0 0
1313 −3.50000 + 0.866025i −0.970725 + 0.240192i
1414 0 0
1515 0.500000 + 0.866025i 0.129099 + 0.223607i
1616 0 0
1717 3.50000 6.06218i 0.848875 1.47029i −0.0333386 0.999444i 0.510614π-0.510614\pi
0.882213 0.470850i 0.156053π-0.156053\pi
1818 0 0
1919 −3.00000 + 5.19615i −0.688247 + 1.19208i 0.284157 + 0.958778i 0.408286π0.408286\pi
−0.972404 + 0.233301i 0.925047π0.925047\pi
2020 0 0
2121 −2.00000 −0.436436
2222 0 0
2323 −3.00000 5.19615i −0.625543 1.08347i −0.988436 0.151642i 0.951544π-0.951544\pi
0.362892 0.931831i 0.381789π-0.381789\pi
2424 0 0
2525 −4.00000 −0.800000
2626 0 0
2727 1.00000 0.192450
2828 0 0
2929 0.500000 + 0.866025i 0.0928477 + 0.160817i 0.908708 0.417432i 0.137070π-0.137070\pi
−0.815861 + 0.578249i 0.803736π0.803736\pi
3030 0 0
3131 −4.00000 −0.718421 −0.359211 0.933257i 0.616954π-0.616954\pi
−0.359211 + 0.933257i 0.616954π0.616954\pi
3232 0 0
3333 −1.00000 + 1.73205i −0.174078 + 0.301511i
3434 0 0
3535 −1.00000 + 1.73205i −0.169031 + 0.292770i
3636 0 0
3737 −0.500000 0.866025i −0.0821995 0.142374i 0.821995 0.569495i 0.192861π-0.192861\pi
−0.904194 + 0.427121i 0.859528π0.859528\pi
3838 0 0
3939 2.50000 + 2.59808i 0.400320 + 0.416025i
4040 0 0
4141 −4.50000 7.79423i −0.702782 1.21725i −0.967486 0.252924i 0.918608π-0.918608\pi
0.264704 0.964330i 0.414726π-0.414726\pi
4242 0 0
4343 3.00000 5.19615i 0.457496 0.792406i −0.541332 0.840809i 0.682080π-0.682080\pi
0.998828 + 0.0484030i 0.0154132π0.0154132\pi
4444 0 0
4545 0.500000 0.866025i 0.0745356 0.129099i
4646 0 0
4747 −6.00000 −0.875190 −0.437595 0.899172i 0.644170π-0.644170\pi
−0.437595 + 0.899172i 0.644170π0.644170\pi
4848 0 0
4949 1.50000 + 2.59808i 0.214286 + 0.371154i
5050 0 0
5151 −7.00000 −0.980196
5252 0 0
5353 −9.00000 −1.23625 −0.618123 0.786082i 0.712106π-0.712106\pi
−0.618123 + 0.786082i 0.712106π0.712106\pi
5454 0 0
5555 1.00000 + 1.73205i 0.134840 + 0.233550i
5656 0 0
5757 6.00000 0.794719
5858 0 0
5959 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
6060 0 0
6161 −0.500000 + 0.866025i −0.0640184 + 0.110883i −0.896258 0.443533i 0.853725π-0.853725\pi
0.832240 + 0.554416i 0.187058π0.187058\pi
6262 0 0
6363 1.00000 + 1.73205i 0.125988 + 0.218218i
6464 0 0
6565 3.50000 0.866025i 0.434122 0.107417i
6666 0 0
6767 −1.00000 1.73205i −0.122169 0.211604i 0.798454 0.602056i 0.205652π-0.205652\pi
−0.920623 + 0.390453i 0.872318π0.872318\pi
6868 0 0
6969 −3.00000 + 5.19615i −0.361158 + 0.625543i
7070 0 0
7171 3.00000 5.19615i 0.356034 0.616670i −0.631260 0.775571i 0.717462π-0.717462\pi
0.987294 + 0.158901i 0.0507952π0.0507952\pi
7272 0 0
7373 11.0000 1.28745 0.643726 0.765256i 0.277388π-0.277388\pi
0.643726 + 0.765256i 0.277388π0.277388\pi
7474 0 0
7575 2.00000 + 3.46410i 0.230940 + 0.400000i
7676 0 0
7777 −4.00000 −0.455842
7878 0 0
7979 4.00000 0.450035 0.225018 0.974355i 0.427756π-0.427756\pi
0.225018 + 0.974355i 0.427756π0.427756\pi
8080 0 0
8181 −0.500000 0.866025i −0.0555556 0.0962250i
8282 0 0
8383 14.0000 1.53670 0.768350 0.640030i 0.221078π-0.221078\pi
0.768350 + 0.640030i 0.221078π0.221078\pi
8484 0 0
8585 −3.50000 + 6.06218i −0.379628 + 0.657536i
8686 0 0
8787 0.500000 0.866025i 0.0536056 0.0928477i
8888 0 0
8989 7.00000 + 12.1244i 0.741999 + 1.28518i 0.951584 + 0.307389i 0.0994552π0.0994552\pi
−0.209585 + 0.977790i 0.567211π0.567211\pi
9090 0 0
9191 −2.00000 + 6.92820i −0.209657 + 0.726273i
9292 0 0
9393 2.00000 + 3.46410i 0.207390 + 0.359211i
9494 0 0
9595 3.00000 5.19615i 0.307794 0.533114i
9696 0 0
9797 1.00000 1.73205i 0.101535 0.175863i −0.810782 0.585348i 0.800958π-0.800958\pi
0.912317 + 0.409484i 0.134291π0.134291\pi
9898 0 0
9999 2.00000 0.201008
100100 0 0
101101 −1.50000 2.59808i −0.149256 0.258518i 0.781697 0.623658i 0.214354π-0.214354\pi
−0.930953 + 0.365140i 0.881021π0.881021\pi
102102 0 0
103103 −6.00000 −0.591198 −0.295599 0.955312i 0.595519π-0.595519\pi
−0.295599 + 0.955312i 0.595519π0.595519\pi
104104 0 0
105105 2.00000 0.195180
106106 0 0
107107 −3.00000 5.19615i −0.290021 0.502331i 0.683793 0.729676i 0.260329π-0.260329\pi
−0.973814 + 0.227345i 0.926996π0.926996\pi
108108 0 0
109109 −2.00000 −0.191565 −0.0957826 0.995402i 0.530535π-0.530535\pi
−0.0957826 + 0.995402i 0.530535π0.530535\pi
110110 0 0
111111 −0.500000 + 0.866025i −0.0474579 + 0.0821995i
112112 0 0
113113 7.50000 12.9904i 0.705541 1.22203i −0.260955 0.965351i 0.584038π-0.584038\pi
0.966496 0.256681i 0.0826291π-0.0826291\pi
114114 0 0
115115 3.00000 + 5.19615i 0.279751 + 0.484544i
116116 0 0
117117 1.00000 3.46410i 0.0924500 0.320256i
118118 0 0
119119 −7.00000 12.1244i −0.641689 1.11144i
120120 0 0
121121 3.50000 6.06218i 0.318182 0.551107i
122122 0 0
123123 −4.50000 + 7.79423i −0.405751 + 0.702782i
124124 0 0
125125 9.00000 0.804984
126126 0 0
127127 10.0000 + 17.3205i 0.887357 + 1.53695i 0.842989 + 0.537931i 0.180794π0.180794\pi
0.0443678 + 0.999015i 0.485873π0.485873\pi
128128 0 0
129129 −6.00000 −0.528271
130130 0 0
131131 8.00000 0.698963 0.349482 0.936943i 0.386358π-0.386358\pi
0.349482 + 0.936943i 0.386358π0.386358\pi
132132 0 0
133133 6.00000 + 10.3923i 0.520266 + 0.901127i
134134 0 0
135135 −1.00000 −0.0860663
136136 0 0
137137 1.50000 2.59808i 0.128154 0.221969i −0.794808 0.606861i 0.792428π-0.792428\pi
0.922961 + 0.384893i 0.125762π0.125762\pi
138138 0 0
139139 6.00000 10.3923i 0.508913 0.881464i −0.491033 0.871141i 0.663381π-0.663381\pi
0.999947 0.0103230i 0.00328598π-0.00328598\pi
140140 0 0
141141 3.00000 + 5.19615i 0.252646 + 0.437595i
142142 0 0
143143 5.00000 + 5.19615i 0.418121 + 0.434524i
144144 0 0
145145 −0.500000 0.866025i −0.0415227 0.0719195i
146146 0 0
147147 1.50000 2.59808i 0.123718 0.214286i
148148 0 0
149149 −1.50000 + 2.59808i −0.122885 + 0.212843i −0.920904 0.389789i 0.872548π-0.872548\pi
0.798019 + 0.602632i 0.205881π0.205881\pi
150150 0 0
151151 2.00000 0.162758 0.0813788 0.996683i 0.474068π-0.474068\pi
0.0813788 + 0.996683i 0.474068π0.474068\pi
152152 0 0
153153 3.50000 + 6.06218i 0.282958 + 0.490098i
154154 0 0
155155 4.00000 0.321288
156156 0 0
157157 −3.00000 −0.239426 −0.119713 0.992809i 0.538197π-0.538197\pi
−0.119713 + 0.992809i 0.538197π0.538197\pi
158158 0 0
159159 4.50000 + 7.79423i 0.356873 + 0.618123i
160160 0 0
161161 −12.0000 −0.945732
162162 0 0
163163 −2.00000 + 3.46410i −0.156652 + 0.271329i −0.933659 0.358162i 0.883403π-0.883403\pi
0.777007 + 0.629492i 0.216737π0.216737\pi
164164 0 0
165165 1.00000 1.73205i 0.0778499 0.134840i
166166 0 0
167167 8.00000 + 13.8564i 0.619059 + 1.07224i 0.989658 + 0.143448i 0.0458190π0.0458190\pi
−0.370599 + 0.928793i 0.620848π0.620848\pi
168168 0 0
169169 11.5000 6.06218i 0.884615 0.466321i
170170 0 0
171171 −3.00000 5.19615i −0.229416 0.397360i
172172 0 0
173173 3.00000 5.19615i 0.228086 0.395056i −0.729155 0.684349i 0.760087π-0.760087\pi
0.957241 + 0.289292i 0.0934200π0.0934200\pi
174174 0 0
175175 −4.00000 + 6.92820i −0.302372 + 0.523723i
176176 0 0
177177 0 0
178178 0 0
179179 −1.00000 1.73205i −0.0747435 0.129460i 0.826231 0.563331i 0.190480π-0.190480\pi
−0.900975 + 0.433872i 0.857147π0.857147\pi
180180 0 0
181181 −7.00000 −0.520306 −0.260153 0.965567i 0.583773π-0.583773\pi
−0.260153 + 0.965567i 0.583773π0.583773\pi
182182 0 0
183183 1.00000 0.0739221
184184 0 0
185185 0.500000 + 0.866025i 0.0367607 + 0.0636715i
186186 0 0
187187 −14.0000 −1.02378
188188 0 0
189189 1.00000 1.73205i 0.0727393 0.125988i
190190 0 0
191191 −2.00000 + 3.46410i −0.144715 + 0.250654i −0.929267 0.369410i 0.879560π-0.879560\pi
0.784552 + 0.620063i 0.212893π0.212893\pi
192192 0 0
193193 4.50000 + 7.79423i 0.323917 + 0.561041i 0.981293 0.192522i 0.0616668π-0.0616668\pi
−0.657376 + 0.753563i 0.728333π0.728333\pi
194194 0 0
195195 −2.50000 2.59808i −0.179029 0.186052i
196196 0 0
197197 −3.00000 5.19615i −0.213741 0.370211i 0.739141 0.673550i 0.235232π-0.235232\pi
−0.952882 + 0.303340i 0.901898π0.901898\pi
198198 0 0
199199 7.00000 12.1244i 0.496217 0.859473i −0.503774 0.863836i 0.668055π-0.668055\pi
0.999990 + 0.00436292i 0.00138876π0.00138876\pi
200200 0 0
201201 −1.00000 + 1.73205i −0.0705346 + 0.122169i
202202 0 0
203203 2.00000 0.140372
204204 0 0
205205 4.50000 + 7.79423i 0.314294 + 0.544373i
206206 0 0
207207 6.00000 0.417029
208208 0 0
209209 12.0000 0.830057
210210 0 0
211211 −4.00000 6.92820i −0.275371 0.476957i 0.694857 0.719148i 0.255467π-0.255467\pi
−0.970229 + 0.242190i 0.922134π0.922134\pi
212212 0 0
213213 −6.00000 −0.411113
214214 0 0
215215 −3.00000 + 5.19615i −0.204598 + 0.354375i
216216 0 0
217217 −4.00000 + 6.92820i −0.271538 + 0.470317i
218218 0 0
219219 −5.50000 9.52628i −0.371656 0.643726i
220220 0 0
221221 −7.00000 + 24.2487i −0.470871 + 1.63114i
222222 0 0
223223 8.00000 + 13.8564i 0.535720 + 0.927894i 0.999128 + 0.0417488i 0.0132929π0.0132929\pi
−0.463409 + 0.886145i 0.653374π0.653374\pi
224224 0 0
225225 2.00000 3.46410i 0.133333 0.230940i
226226 0 0
227227 −7.00000 + 12.1244i −0.464606 + 0.804722i −0.999184 0.0403978i 0.987137π-0.987137\pi
0.534577 + 0.845120i 0.320471π0.320471\pi
228228 0 0
229229 −22.0000 −1.45380 −0.726900 0.686743i 0.759040π-0.759040\pi
−0.726900 + 0.686743i 0.759040π0.759040\pi
230230 0 0
231231 2.00000 + 3.46410i 0.131590 + 0.227921i
232232 0 0
233233 10.0000 0.655122 0.327561 0.944830i 0.393773π-0.393773\pi
0.327561 + 0.944830i 0.393773π0.393773\pi
234234 0 0
235235 6.00000 0.391397
236236 0 0
237237 −2.00000 3.46410i −0.129914 0.225018i
238238 0 0
239239 −30.0000 −1.94054 −0.970269 0.242028i 0.922188π-0.922188\pi
−0.970269 + 0.242028i 0.922188π0.922188\pi
240240 0 0
241241 −3.50000 + 6.06218i −0.225455 + 0.390499i −0.956456 0.291877i 0.905720π-0.905720\pi
0.731001 + 0.682376i 0.239053π0.239053\pi
242242 0 0
243243 −0.500000 + 0.866025i −0.0320750 + 0.0555556i
244244 0 0
245245 −1.50000 2.59808i −0.0958315 0.165985i
246246 0 0
247247 6.00000 20.7846i 0.381771 1.32249i
248248 0 0
249249 −7.00000 12.1244i −0.443607 0.768350i
250250 0 0
251251 6.00000 10.3923i 0.378717 0.655956i −0.612159 0.790735i 0.709699π-0.709699\pi
0.990876 + 0.134778i 0.0430322π0.0430322\pi
252252 0 0
253253 −6.00000 + 10.3923i −0.377217 + 0.653359i
254254 0 0
255255 7.00000 0.438357
256256 0 0
257257 3.50000 + 6.06218i 0.218324 + 0.378148i 0.954296 0.298864i 0.0966077π-0.0966077\pi
−0.735972 + 0.677012i 0.763274π0.763274\pi
258258 0 0
259259 −2.00000 −0.124274
260260 0 0
261261 −1.00000 −0.0618984
262262 0 0
263263 −15.0000 25.9808i −0.924940 1.60204i −0.791658 0.610964i 0.790782π-0.790782\pi
−0.133281 0.991078i 0.542551π-0.542551\pi
264264 0 0
265265 9.00000 0.552866
266266 0 0
267267 7.00000 12.1244i 0.428393 0.741999i
268268 0 0
269269 7.00000 12.1244i 0.426798 0.739235i −0.569789 0.821791i 0.692975π-0.692975\pi
0.996586 + 0.0825561i 0.0263084π0.0263084\pi
270270 0 0
271271 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
272272 0 0
273273 7.00000 1.73205i 0.423659 0.104828i
274274 0 0
275275 4.00000 + 6.92820i 0.241209 + 0.417786i
276276 0 0
277277 15.5000 26.8468i 0.931305 1.61307i 0.150210 0.988654i 0.452005π-0.452005\pi
0.781094 0.624413i 0.214662π-0.214662\pi
278278 0 0
279279 2.00000 3.46410i 0.119737 0.207390i
280280 0 0
281281 −19.0000 −1.13344 −0.566722 0.823909i 0.691789π-0.691789\pi
−0.566722 + 0.823909i 0.691789π0.691789\pi
282282 0 0
283283 −9.00000 15.5885i −0.534994 0.926638i −0.999164 0.0408910i 0.986980π-0.986980\pi
0.464169 0.885747i 0.346353π-0.346353\pi
284284 0 0
285285 −6.00000 −0.355409
286286 0 0
287287 −18.0000 −1.06251
288288 0 0
289289 −16.0000 27.7128i −0.941176 1.63017i
290290 0 0
291291 −2.00000 −0.117242
292292 0 0
293293 4.50000 7.79423i 0.262893 0.455344i −0.704117 0.710084i 0.748657π-0.748657\pi
0.967009 + 0.254741i 0.0819901π0.0819901\pi
294294 0 0
295295 0 0
296296 0 0
297297 −1.00000 1.73205i −0.0580259 0.100504i
298298 0 0
299299 15.0000 + 15.5885i 0.867472 + 0.901504i
300300 0 0
301301 −6.00000 10.3923i −0.345834 0.599002i
302302 0 0
303303 −1.50000 + 2.59808i −0.0861727 + 0.149256i
304304 0 0
305305 0.500000 0.866025i 0.0286299 0.0495885i
306306 0 0
307307 −14.0000 −0.799022 −0.399511 0.916728i 0.630820π-0.630820\pi
−0.399511 + 0.916728i 0.630820π0.630820\pi
308308 0 0
309309 3.00000 + 5.19615i 0.170664 + 0.295599i
310310 0 0
311311 −18.0000 −1.02069 −0.510343 0.859971i 0.670482π-0.670482\pi
−0.510343 + 0.859971i 0.670482π0.670482\pi
312312 0 0
313313 6.00000 0.339140 0.169570 0.985518i 0.445762π-0.445762\pi
0.169570 + 0.985518i 0.445762π0.445762\pi
314314 0 0
315315 −1.00000 1.73205i −0.0563436 0.0975900i
316316 0 0
317317 −25.0000 −1.40414 −0.702070 0.712108i 0.747741π-0.747741\pi
−0.702070 + 0.712108i 0.747741π0.747741\pi
318318 0 0
319319 1.00000 1.73205i 0.0559893 0.0969762i
320320 0 0
321321 −3.00000 + 5.19615i −0.167444 + 0.290021i
322322 0 0
323323 21.0000 + 36.3731i 1.16847 + 2.02385i
324324 0 0
325325 14.0000 3.46410i 0.776580 0.192154i
326326 0 0
327327 1.00000 + 1.73205i 0.0553001 + 0.0957826i
328328 0 0
329329 −6.00000 + 10.3923i −0.330791 + 0.572946i
330330 0 0
331331 −2.00000 + 3.46410i −0.109930 + 0.190404i −0.915742 0.401768i 0.868396π-0.868396\pi
0.805812 + 0.592172i 0.201729π0.201729\pi
332332 0 0
333333 1.00000 0.0547997
334334 0 0
335335 1.00000 + 1.73205i 0.0546358 + 0.0946320i
336336 0 0
337337 −33.0000 −1.79762 −0.898812 0.438334i 0.855569π-0.855569\pi
−0.898812 + 0.438334i 0.855569π0.855569\pi
338338 0 0
339339 −15.0000 −0.814688
340340 0 0
341341 4.00000 + 6.92820i 0.216612 + 0.375183i
342342 0 0
343343 20.0000 1.07990
344344 0 0
345345 3.00000 5.19615i 0.161515 0.279751i
346346 0 0
347347 9.00000 15.5885i 0.483145 0.836832i −0.516667 0.856186i 0.672828π-0.672828\pi
0.999813 + 0.0193540i 0.00616095π0.00616095\pi
348348 0 0
349349 13.0000 + 22.5167i 0.695874 + 1.20529i 0.969885 + 0.243563i 0.0783162π0.0783162\pi
−0.274011 + 0.961727i 0.588351π0.588351\pi
350350 0 0
351351 −3.50000 + 0.866025i −0.186816 + 0.0462250i
352352 0 0
353353 5.50000 + 9.52628i 0.292735 + 0.507033i 0.974456 0.224580i 0.0721011π-0.0721011\pi
−0.681720 + 0.731613i 0.738768π0.738768\pi
354354 0 0
355355 −3.00000 + 5.19615i −0.159223 + 0.275783i
356356 0 0
357357 −7.00000 + 12.1244i −0.370479 + 0.641689i
358358 0 0
359359 18.0000 0.950004 0.475002 0.879985i 0.342447π-0.342447\pi
0.475002 + 0.879985i 0.342447π0.342447\pi
360360 0 0
361361 −8.50000 14.7224i −0.447368 0.774865i
362362 0 0
363363 −7.00000 −0.367405
364364 0 0
365365 −11.0000 −0.575766
366366 0 0
367367 5.00000 + 8.66025i 0.260998 + 0.452062i 0.966507 0.256639i 0.0826151π-0.0826151\pi
−0.705509 + 0.708700i 0.749282π0.749282\pi
368368 0 0
369369 9.00000 0.468521
370370 0 0
371371 −9.00000 + 15.5885i −0.467257 + 0.809312i
372372 0 0
373373 5.50000 9.52628i 0.284779 0.493252i −0.687776 0.725923i 0.741413π-0.741413\pi
0.972556 + 0.232671i 0.0747464π0.0747464\pi
374374 0 0
375375 −4.50000 7.79423i −0.232379 0.402492i
376376 0 0
377377 −2.50000 2.59808i −0.128757 0.133808i
378378 0 0
379379 −18.0000 31.1769i −0.924598 1.60145i −0.792207 0.610253i 0.791068π-0.791068\pi
−0.132391 0.991198i 0.542266π-0.542266\pi
380380 0 0
381381 10.0000 17.3205i 0.512316 0.887357i
382382 0 0
383383 4.00000 6.92820i 0.204390 0.354015i −0.745548 0.666452i 0.767812π-0.767812\pi
0.949938 + 0.312437i 0.101145π0.101145\pi
384384 0 0
385385 4.00000 0.203859
386386 0 0
387387 3.00000 + 5.19615i 0.152499 + 0.264135i
388388 0 0
389389 19.0000 0.963338 0.481669 0.876353i 0.340031π-0.340031\pi
0.481669 + 0.876353i 0.340031π0.340031\pi
390390 0 0
391391 −42.0000 −2.12403
392392 0 0
393393 −4.00000 6.92820i −0.201773 0.349482i
394394 0 0
395395 −4.00000 −0.201262
396396 0 0
397397 17.0000 29.4449i 0.853206 1.47780i −0.0250943 0.999685i 0.507989π-0.507989\pi
0.878300 0.478110i 0.158678π-0.158678\pi
398398 0 0
399399 6.00000 10.3923i 0.300376 0.520266i
400400 0 0
401401 −0.500000 0.866025i −0.0249688 0.0432472i 0.853271 0.521468i 0.174615π-0.174615\pi
−0.878240 + 0.478220i 0.841282π0.841282\pi
402402 0 0
403403 14.0000 3.46410i 0.697390 0.172559i
404404 0 0
405405 0.500000 + 0.866025i 0.0248452 + 0.0430331i
406406 0 0
407407 −1.00000 + 1.73205i −0.0495682 + 0.0858546i
408408 0 0
409409 −3.50000 + 6.06218i −0.173064 + 0.299755i −0.939490 0.342578i 0.888700π-0.888700\pi
0.766426 + 0.642333i 0.222033π0.222033\pi
410410 0 0
411411 −3.00000 −0.147979
412412 0 0
413413 0 0
414414 0 0
415415 −14.0000 −0.687233
416416 0 0
417417 −12.0000 −0.587643
418418 0 0
419419 8.00000 + 13.8564i 0.390826 + 0.676930i 0.992559 0.121768i 0.0388562π-0.0388562\pi
−0.601733 + 0.798697i 0.705523π0.705523\pi
420420 0 0
421421 −19.0000 −0.926003 −0.463002 0.886357i 0.653228π-0.653228\pi
−0.463002 + 0.886357i 0.653228π0.653228\pi
422422 0 0
423423 3.00000 5.19615i 0.145865 0.252646i
424424 0 0
425425 −14.0000 + 24.2487i −0.679100 + 1.17624i
426426 0 0
427427 1.00000 + 1.73205i 0.0483934 + 0.0838198i
428428 0 0
429429 2.00000 6.92820i 0.0965609 0.334497i
430430 0 0
431431 −15.0000 25.9808i −0.722525 1.25145i −0.959985 0.280052i 0.909648π-0.909648\pi
0.237460 0.971397i 0.423685π-0.423685\pi
432432 0 0
433433 −9.50000 + 16.4545i −0.456541 + 0.790752i −0.998775 0.0494752i 0.984245π-0.984245\pi
0.542234 + 0.840227i 0.317578π0.317578\pi
434434 0 0
435435 −0.500000 + 0.866025i −0.0239732 + 0.0415227i
436436 0 0
437437 36.0000 1.72211
438438 0 0
439439 7.00000 + 12.1244i 0.334092 + 0.578664i 0.983310 0.181938i 0.0582371π-0.0582371\pi
−0.649218 + 0.760602i 0.724904π0.724904\pi
440440 0 0
441441 −3.00000 −0.142857
442442 0 0
443443 4.00000 0.190046 0.0950229 0.995475i 0.469708π-0.469708\pi
0.0950229 + 0.995475i 0.469708π0.469708\pi
444444 0 0
445445 −7.00000 12.1244i −0.331832 0.574750i
446446 0 0
447447 3.00000 0.141895
448448 0 0
449449 −17.0000 + 29.4449i −0.802280 + 1.38959i 0.115833 + 0.993269i 0.463046π0.463046\pi
−0.918112 + 0.396320i 0.870287π0.870287\pi
450450 0 0
451451 −9.00000 + 15.5885i −0.423793 + 0.734032i
452452 0 0
453453 −1.00000 1.73205i −0.0469841 0.0813788i
454454 0 0
455455 2.00000 6.92820i 0.0937614 0.324799i
456456 0 0
457457 6.50000 + 11.2583i 0.304057 + 0.526642i 0.977051 0.213006i 0.0683253π-0.0683253\pi
−0.672994 + 0.739648i 0.734992π0.734992\pi
458458 0 0
459459 3.50000 6.06218i 0.163366 0.282958i
460460 0 0
461461 −9.50000 + 16.4545i −0.442459 + 0.766362i −0.997871 0.0652135i 0.979227π-0.979227\pi
0.555412 + 0.831575i 0.312560π0.312560\pi
462462 0 0
463463 26.0000 1.20832 0.604161 0.796862i 0.293508π-0.293508\pi
0.604161 + 0.796862i 0.293508π0.293508\pi
464464 0 0
465465 −2.00000 3.46410i −0.0927478 0.160644i
466466 0 0
467467 −6.00000 −0.277647 −0.138823 0.990317i 0.544332π-0.544332\pi
−0.138823 + 0.990317i 0.544332π0.544332\pi
468468 0 0
469469 −4.00000 −0.184703
470470 0 0
471471 1.50000 + 2.59808i 0.0691164 + 0.119713i
472472 0 0
473473 −12.0000 −0.551761
474474 0 0
475475 12.0000 20.7846i 0.550598 0.953663i
476476 0 0
477477 4.50000 7.79423i 0.206041 0.356873i
478478 0 0
479479 12.0000 + 20.7846i 0.548294 + 0.949673i 0.998392 + 0.0566937i 0.0180558π0.0180558\pi
−0.450098 + 0.892979i 0.648611π0.648611\pi
480480 0 0
481481 2.50000 + 2.59808i 0.113990 + 0.118462i
482482 0 0
483483 6.00000 + 10.3923i 0.273009 + 0.472866i
484484 0 0
485485 −1.00000 + 1.73205i −0.0454077 + 0.0786484i
486486 0 0
487487 9.00000 15.5885i 0.407829 0.706380i −0.586817 0.809719i 0.699619π-0.699619\pi
0.994646 + 0.103339i 0.0329526π0.0329526\pi
488488 0 0
489489 4.00000 0.180886
490490 0 0
491491 3.00000 + 5.19615i 0.135388 + 0.234499i 0.925746 0.378147i 0.123439π-0.123439\pi
−0.790358 + 0.612646i 0.790105π0.790105\pi
492492 0 0
493493 7.00000 0.315264
494494 0 0
495495 −2.00000 −0.0898933
496496 0 0
497497 −6.00000 10.3923i −0.269137 0.466159i
498498 0 0
499499 −24.0000 −1.07439 −0.537194 0.843459i 0.680516π-0.680516\pi
−0.537194 + 0.843459i 0.680516π0.680516\pi
500500 0 0
501501 8.00000 13.8564i 0.357414 0.619059i
502502 0 0
503503 −1.00000 + 1.73205i −0.0445878 + 0.0772283i −0.887458 0.460889i 0.847531π-0.847531\pi
0.842870 + 0.538117i 0.180864π0.180864\pi
504504 0 0
505505 1.50000 + 2.59808i 0.0667491 + 0.115613i
506506 0 0
507507 −11.0000 6.92820i −0.488527 0.307692i
508508 0 0
509509 −3.50000 6.06218i −0.155135 0.268701i 0.777973 0.628297i 0.216248π-0.216248\pi
−0.933108 + 0.359596i 0.882915π0.882915\pi
510510 0 0
511511 11.0000 19.0526i 0.486611 0.842836i
512512 0 0
513513 −3.00000 + 5.19615i −0.132453 + 0.229416i
514514 0 0
515515 6.00000 0.264392
516516 0 0
517517 6.00000 + 10.3923i 0.263880 + 0.457053i
518518 0 0
519519 −6.00000 −0.263371
520520 0 0
521521 −3.00000 −0.131432 −0.0657162 0.997838i 0.520933π-0.520933\pi
−0.0657162 + 0.997838i 0.520933π0.520933\pi
522522 0 0
523523 7.00000 + 12.1244i 0.306089 + 0.530161i 0.977503 0.210921i 0.0676463π-0.0676463\pi
−0.671414 + 0.741082i 0.734313π0.734313\pi
524524 0 0
525525 8.00000 0.349149
526526 0 0
527527 −14.0000 + 24.2487i −0.609850 + 1.05629i
528528 0 0
529529 −6.50000 + 11.2583i −0.282609 + 0.489493i
530530 0 0
531531 0 0
532532 0 0
533533 22.5000 + 23.3827i 0.974583 + 1.01282i
534534 0 0
535535 3.00000 + 5.19615i 0.129701 + 0.224649i
536536 0 0
537537 −1.00000 + 1.73205i −0.0431532 + 0.0747435i
538538 0 0
539539 3.00000 5.19615i 0.129219 0.223814i
540540 0 0
541541 45.0000 1.93470 0.967351 0.253442i 0.0815627π-0.0815627\pi
0.967351 + 0.253442i 0.0815627π0.0815627\pi
542542 0 0
543543 3.50000 + 6.06218i 0.150199 + 0.260153i
544544 0 0
545545 2.00000 0.0856706
546546 0 0
547547 26.0000 1.11168 0.555840 0.831289i 0.312397π-0.312397\pi
0.555840 + 0.831289i 0.312397π0.312397\pi
548548 0 0
549549 −0.500000 0.866025i −0.0213395 0.0369611i
550550 0 0
551551 −6.00000 −0.255609
552552 0 0
553553 4.00000 6.92820i 0.170097 0.294617i
554554 0 0
555555 0.500000 0.866025i 0.0212238 0.0367607i
556556 0 0
557557 4.50000 + 7.79423i 0.190671 + 0.330252i 0.945473 0.325701i 0.105600π-0.105600\pi
−0.754802 + 0.655953i 0.772267π0.772267\pi
558558 0 0
559559 −6.00000 + 20.7846i −0.253773 + 0.879095i
560560 0 0
561561 7.00000 + 12.1244i 0.295540 + 0.511891i
562562 0 0
563563 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
564564 0 0
565565 −7.50000 + 12.9904i −0.315527 + 0.546509i
566566 0 0
567567 −2.00000 −0.0839921
568568 0 0
569569 −11.0000 19.0526i −0.461144 0.798725i 0.537874 0.843025i 0.319228π-0.319228\pi
−0.999018 + 0.0443003i 0.985894π0.985894\pi
570570 0 0
571571 −26.0000 −1.08807 −0.544033 0.839064i 0.683103π-0.683103\pi
−0.544033 + 0.839064i 0.683103π0.683103\pi
572572 0 0
573573 4.00000 0.167102
574574 0 0
575575 12.0000 + 20.7846i 0.500435 + 0.866778i
576576 0 0
577577 11.0000 0.457936 0.228968 0.973434i 0.426465π-0.426465\pi
0.228968 + 0.973434i 0.426465π0.426465\pi
578578 0 0
579579 4.50000 7.79423i 0.187014 0.323917i
580580 0 0
581581 14.0000 24.2487i 0.580818 1.00601i
582582 0 0
583583 9.00000 + 15.5885i 0.372742 + 0.645608i
584584 0 0
585585 −1.00000 + 3.46410i −0.0413449 + 0.143223i
586586 0 0
587587 8.00000 + 13.8564i 0.330195 + 0.571915i 0.982550 0.185999i 0.0595520π-0.0595520\pi
−0.652355 + 0.757914i 0.726219π0.726219\pi
588588 0 0
589589 12.0000 20.7846i 0.494451 0.856415i
590590 0 0
591591 −3.00000 + 5.19615i −0.123404 + 0.213741i
592592 0 0
593593 13.0000 0.533846 0.266923 0.963718i 0.413993π-0.413993\pi
0.266923 + 0.963718i 0.413993π0.413993\pi
594594 0 0
595595 7.00000 + 12.1244i 0.286972 + 0.497050i
596596 0 0
597597 −14.0000 −0.572982
598598 0 0
599599 16.0000 0.653742 0.326871 0.945069i 0.394006π-0.394006\pi
0.326871 + 0.945069i 0.394006π0.394006\pi
600600 0 0
601601 2.50000 + 4.33013i 0.101977 + 0.176630i 0.912499 0.409079i 0.134150π-0.134150\pi
−0.810522 + 0.585708i 0.800816π0.800816\pi
602602 0 0
603603 2.00000 0.0814463
604604 0 0
605605 −3.50000 + 6.06218i −0.142295 + 0.246463i
606606 0 0
607607 4.00000 6.92820i 0.162355 0.281207i −0.773358 0.633970i 0.781424π-0.781424\pi
0.935713 + 0.352763i 0.114758π0.114758\pi
608608 0 0
609609 −1.00000 1.73205i −0.0405220 0.0701862i
610610 0 0
611611 21.0000 5.19615i 0.849569 0.210214i
612612 0 0
613613 11.5000 + 19.9186i 0.464481 + 0.804504i 0.999178 0.0405396i 0.0129077π-0.0129077\pi
−0.534697 + 0.845044i 0.679574π0.679574\pi
614614 0 0
615615 4.50000 7.79423i 0.181458 0.314294i
616616 0 0
617617 −6.50000 + 11.2583i −0.261680 + 0.453243i −0.966689 0.255956i 0.917610π-0.917610\pi
0.705008 + 0.709199i 0.250943π0.250943\pi
618618 0 0
619619 24.0000 0.964641 0.482321 0.875995i 0.339794π-0.339794\pi
0.482321 + 0.875995i 0.339794π0.339794\pi
620620 0 0
621621 −3.00000 5.19615i −0.120386 0.208514i
622622 0 0
623623 28.0000 1.12180
624624 0 0
625625 11.0000 0.440000
626626 0 0
627627 −6.00000 10.3923i −0.239617 0.415029i
628628 0 0
629629 −7.00000 −0.279108
630630 0 0
631631 10.0000 17.3205i 0.398094 0.689519i −0.595397 0.803432i 0.703005π-0.703005\pi
0.993491 + 0.113913i 0.0363385π0.0363385\pi
632632 0 0
633633 −4.00000 + 6.92820i −0.158986 + 0.275371i
634634 0 0
635635 −10.0000 17.3205i −0.396838 0.687343i
636636 0 0
637637 −7.50000 7.79423i −0.297161 0.308819i
638638 0 0
639639 3.00000 + 5.19615i 0.118678 + 0.205557i
640640 0 0
641641 15.5000 26.8468i 0.612213 1.06038i −0.378653 0.925539i 0.623613π-0.623613\pi
0.990867 0.134846i 0.0430539π-0.0430539\pi
642642 0 0
643643 −8.00000 + 13.8564i −0.315489 + 0.546443i −0.979541 0.201243i 0.935502π-0.935502\pi
0.664052 + 0.747686i 0.268835π0.268835\pi
644644 0 0
645645 6.00000 0.236250
646646 0 0
647647 −16.0000 27.7128i −0.629025 1.08950i −0.987748 0.156059i 0.950121π-0.950121\pi
0.358723 0.933444i 0.383212π-0.383212\pi
648648 0 0
649649 0 0
650650 0 0
651651 8.00000 0.313545
652652 0 0
653653 3.00000 + 5.19615i 0.117399 + 0.203341i 0.918736 0.394872i 0.129211π-0.129211\pi
−0.801337 + 0.598213i 0.795878π0.795878\pi
654654 0 0
655655 −8.00000 −0.312586
656656 0 0
657657 −5.50000 + 9.52628i −0.214575 + 0.371656i
658658 0 0
659659 −4.00000 + 6.92820i −0.155818 + 0.269884i −0.933357 0.358951i 0.883135π-0.883135\pi
0.777539 + 0.628835i 0.216468π0.216468\pi
660660 0 0
661661 −22.5000 38.9711i −0.875149 1.51580i −0.856604 0.515974i 0.827430π-0.827430\pi
−0.0185442 0.999828i 0.505903π-0.505903\pi
662662 0 0
663663 24.5000 6.06218i 0.951501 0.235435i
664664 0 0
665665 −6.00000 10.3923i −0.232670 0.402996i
666666 0 0
667667 3.00000 5.19615i 0.116160 0.201196i
668668 0 0
669669 8.00000 13.8564i 0.309298 0.535720i
670670 0 0
671671 2.00000 0.0772091
672672 0 0
673673 14.5000 + 25.1147i 0.558934 + 0.968102i 0.997586 + 0.0694449i 0.0221228π0.0221228\pi
−0.438652 + 0.898657i 0.644544π0.644544\pi
674674 0 0
675675 −4.00000 −0.153960
676676 0 0
677677 34.0000 1.30673 0.653363 0.757045i 0.273358π-0.273358\pi
0.653363 + 0.757045i 0.273358π0.273358\pi
678678 0 0
679679 −2.00000 3.46410i −0.0767530 0.132940i
680680 0 0
681681 14.0000 0.536481
682682 0 0
683683 −12.0000 + 20.7846i −0.459167 + 0.795301i −0.998917 0.0465244i 0.985185π-0.985185\pi
0.539750 + 0.841825i 0.318519π0.318519\pi
684684 0 0
685685 −1.50000 + 2.59808i −0.0573121 + 0.0992674i
686686 0 0
687687 11.0000 + 19.0526i 0.419676 + 0.726900i
688688 0 0
689689 31.5000 7.79423i 1.20005 0.296936i
690690 0 0
691691 21.0000 + 36.3731i 0.798878 + 1.38370i 0.920348 + 0.391102i 0.127906π0.127906\pi
−0.121470 + 0.992595i 0.538761π0.538761\pi
692692 0 0
693693 2.00000 3.46410i 0.0759737 0.131590i
694694 0 0
695695 −6.00000 + 10.3923i −0.227593 + 0.394203i
696696 0 0
697697 −63.0000 −2.38630
698698 0 0
699699 −5.00000 8.66025i −0.189117 0.327561i
700700 0 0
701701 2.00000 0.0755390 0.0377695 0.999286i 0.487975π-0.487975\pi
0.0377695 + 0.999286i 0.487975π0.487975\pi
702702 0 0
703703 6.00000 0.226294
704704 0 0
705705 −3.00000 5.19615i −0.112987 0.195698i
706706 0 0
707707 −6.00000 −0.225653
708708 0 0
709709 5.50000 9.52628i 0.206557 0.357767i −0.744071 0.668101i 0.767108π-0.767108\pi
0.950628 + 0.310334i 0.100441π0.100441\pi
710710 0 0
711711 −2.00000 + 3.46410i −0.0750059 + 0.129914i
712712 0 0
713713 12.0000 + 20.7846i 0.449404 + 0.778390i
714714 0 0
715715 −5.00000 5.19615i −0.186989 0.194325i
716716 0 0
717717 15.0000 + 25.9808i 0.560185 + 0.970269i
718718 0 0
719719 −24.0000 + 41.5692i −0.895049 + 1.55027i −0.0613050 + 0.998119i 0.519526π0.519526\pi
−0.833744 + 0.552151i 0.813807π0.813807\pi
720720 0 0
721721 −6.00000 + 10.3923i −0.223452 + 0.387030i
722722 0 0
723723 7.00000 0.260333
724724 0 0
725725 −2.00000 3.46410i −0.0742781 0.128654i
726726 0 0
727727 −14.0000 −0.519231 −0.259616 0.965712i 0.583596π-0.583596\pi
−0.259616 + 0.965712i 0.583596π0.583596\pi
728728 0 0
729729 1.00000 0.0370370
730730 0 0
731731 −21.0000 36.3731i −0.776713 1.34531i
732732 0 0
733733 −15.0000 −0.554038 −0.277019 0.960864i 0.589346π-0.589346\pi
−0.277019 + 0.960864i 0.589346π0.589346\pi
734734 0 0
735735 −1.50000 + 2.59808i −0.0553283 + 0.0958315i
736736 0 0
737737 −2.00000 + 3.46410i −0.0736709 + 0.127602i
738738 0 0
739739 −8.00000 13.8564i −0.294285 0.509716i 0.680534 0.732717i 0.261748π-0.261748\pi
−0.974818 + 0.223001i 0.928415π0.928415\pi
740740 0 0
741741 −21.0000 + 5.19615i −0.771454 + 0.190885i
742742 0 0
743743 −18.0000 31.1769i −0.660356 1.14377i −0.980522 0.196409i 0.937072π-0.937072\pi
0.320166 0.947361i 0.396261π-0.396261\pi
744744 0 0
745745 1.50000 2.59808i 0.0549557 0.0951861i
746746 0 0
747747 −7.00000 + 12.1244i −0.256117 + 0.443607i
748748 0 0
749749 −12.0000 −0.438470
750750 0 0
751751 −17.0000 29.4449i −0.620339 1.07446i −0.989423 0.145062i 0.953662π-0.953662\pi
0.369084 0.929396i 0.379672π-0.379672\pi
752752 0 0
753753 −12.0000 −0.437304
754754 0 0
755755 −2.00000 −0.0727875
756756 0 0
757757 25.0000 + 43.3013i 0.908640 + 1.57381i 0.815955 + 0.578116i 0.196212π0.196212\pi
0.0926859 + 0.995695i 0.470455π0.470455\pi
758758 0 0
759759 12.0000 0.435572
760760 0 0
761761 −25.0000 + 43.3013i −0.906249 + 1.56967i −0.0870179 + 0.996207i 0.527734π0.527734\pi
−0.819231 + 0.573463i 0.805600π0.805600\pi
762762 0 0
763763 −2.00000 + 3.46410i −0.0724049 + 0.125409i
764764 0 0
765765 −3.50000 6.06218i −0.126543 0.219179i
766766 0 0
767767 0 0
768768 0 0
769769 −15.0000 25.9808i −0.540914 0.936890i −0.998852 0.0479061i 0.984745π-0.984745\pi
0.457938 0.888984i 0.348588π-0.348588\pi
770770 0 0
771771 3.50000 6.06218i 0.126049 0.218324i
772772 0 0
773773 7.00000 12.1244i 0.251773 0.436083i −0.712241 0.701935i 0.752320π-0.752320\pi
0.964014 + 0.265852i 0.0856532π0.0856532\pi
774774 0 0
775775 16.0000 0.574737
776776 0 0
777777 1.00000 + 1.73205i 0.0358748 + 0.0621370i
778778 0 0
779779 54.0000 1.93475
780780 0 0
781781 −12.0000 −0.429394
782782 0 0
783783 0.500000 + 0.866025i 0.0178685 + 0.0309492i
784784 0 0
785785 3.00000 0.107075
786786 0 0
787787 14.0000 24.2487i 0.499046 0.864373i −0.500953 0.865474i 0.667017π-0.667017\pi
0.999999 + 0.00110111i 0.000350496π0.000350496\pi
788788 0 0
789789 −15.0000 + 25.9808i −0.534014 + 0.924940i
790790 0 0
791791 −15.0000 25.9808i −0.533339 0.923770i
792792 0 0
793793 1.00000 3.46410i 0.0355110 0.123014i
794794 0 0
795795 −4.50000 7.79423i −0.159599 0.276433i
796796 0 0
797797 1.00000 1.73205i 0.0354218 0.0613524i −0.847771 0.530362i 0.822056π-0.822056\pi
0.883193 + 0.469010i 0.155389π0.155389\pi
798798 0 0
799799 −21.0000 + 36.3731i −0.742927 + 1.28679i
800800 0 0
801801 −14.0000 −0.494666
802802 0 0
803803 −11.0000 19.0526i −0.388182 0.672350i
804804 0 0
805805 12.0000 0.422944
806806 0 0
807807 −14.0000 −0.492823
808808 0 0
809809 −16.5000 28.5788i −0.580109 1.00478i −0.995466 0.0951198i 0.969677π-0.969677\pi
0.415357 0.909659i 0.363657π-0.363657\pi
810810 0 0
811811 28.0000 0.983213 0.491606 0.870817i 0.336410π-0.336410\pi
0.491606 + 0.870817i 0.336410π0.336410\pi
812812 0 0
813813 0 0
814814 0 0
815815 2.00000 3.46410i 0.0700569 0.121342i
816816 0 0
817817 18.0000 + 31.1769i 0.629740 + 1.09074i
818818 0 0
819819 −5.00000 5.19615i −0.174714 0.181568i
820820 0 0
821821 −25.0000 43.3013i −0.872506 1.51122i −0.859396 0.511311i 0.829160π-0.829160\pi
−0.0131101 0.999914i 0.504173π-0.504173\pi
822822 0 0
823823 −12.0000 + 20.7846i −0.418294 + 0.724506i −0.995768 0.0919029i 0.970705π-0.970705\pi
0.577474 + 0.816409i 0.304038π0.304038\pi
824824 0 0
825825 4.00000 6.92820i 0.139262 0.241209i
826826 0 0
827827 −16.0000 −0.556375 −0.278187 0.960527i 0.589734π-0.589734\pi
−0.278187 + 0.960527i 0.589734π0.589734\pi
828828 0 0
829829 −8.50000 14.7224i −0.295217 0.511331i 0.679818 0.733381i 0.262059π-0.262059\pi
−0.975035 + 0.222049i 0.928725π0.928725\pi
830830 0 0
831831 −31.0000 −1.07538
832832 0 0
833833 21.0000 0.727607
834834 0 0
835835 −8.00000 13.8564i −0.276851 0.479521i
836836 0 0
837837 −4.00000 −0.138260
838838 0 0
839839 6.00000 10.3923i 0.207143 0.358782i −0.743670 0.668546i 0.766917π-0.766917\pi
0.950813 + 0.309764i 0.100250π0.100250\pi
840840 0 0
841841 14.0000 24.2487i 0.482759 0.836162i
842842 0 0
843843 9.50000 + 16.4545i 0.327197 + 0.566722i
844844 0 0
845845 −11.5000 + 6.06218i −0.395612 + 0.208545i
846846 0 0
847847 −7.00000 12.1244i −0.240523 0.416598i
848848 0 0
849849 −9.00000 + 15.5885i −0.308879 + 0.534994i
850850 0 0
851851 −3.00000 + 5.19615i −0.102839 + 0.178122i
852852 0 0
853853 21.0000 0.719026 0.359513 0.933140i 0.382943π-0.382943\pi
0.359513 + 0.933140i 0.382943π0.382943\pi
854854 0 0
855855 3.00000 + 5.19615i 0.102598 + 0.177705i
856856 0 0
857857 −31.0000 −1.05894 −0.529470 0.848329i 0.677609π-0.677609\pi
−0.529470 + 0.848329i 0.677609π0.677609\pi
858858 0 0
859859 −34.0000 −1.16007 −0.580033 0.814593i 0.696960π-0.696960\pi
−0.580033 + 0.814593i 0.696960π0.696960\pi
860860 0 0
861861 9.00000 + 15.5885i 0.306719 + 0.531253i
862862 0 0
863863 −10.0000 −0.340404 −0.170202 0.985409i 0.554442π-0.554442\pi
−0.170202 + 0.985409i 0.554442π0.554442\pi
864864 0 0
865865 −3.00000 + 5.19615i −0.102003 + 0.176674i
866866 0 0
867867 −16.0000 + 27.7128i −0.543388 + 0.941176i
868868 0 0
869869 −4.00000 6.92820i −0.135691 0.235023i
870870 0 0
871871 5.00000 + 5.19615i 0.169419 + 0.176065i
872872 0 0
873873 1.00000 + 1.73205i 0.0338449 + 0.0586210i
874874 0 0
875875 9.00000 15.5885i 0.304256 0.526986i
876876 0 0
877877 −8.50000 + 14.7224i −0.287025 + 0.497141i −0.973098 0.230391i 0.925999π-0.925999\pi
0.686074 + 0.727532i 0.259333π0.259333\pi
878878 0 0
879879 −9.00000 −0.303562
880880 0 0
881881 −18.5000 32.0429i −0.623281 1.07955i −0.988871 0.148778i 0.952466π-0.952466\pi
0.365590 0.930776i 0.380867π-0.380867\pi
882882 0 0
883883 −16.0000 −0.538443 −0.269221 0.963078i 0.586766π-0.586766\pi
−0.269221 + 0.963078i 0.586766π0.586766\pi
884884 0 0
885885 0 0
886886 0 0
887887 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
888888 0 0
889889 40.0000 1.34156
890890 0 0
891891 −1.00000 + 1.73205i −0.0335013 + 0.0580259i
892892 0 0
893893 18.0000 31.1769i 0.602347 1.04330i
894894 0 0
895895 1.00000 + 1.73205i 0.0334263 + 0.0578961i
896896 0 0
897897 6.00000 20.7846i 0.200334 0.693978i
898898 0 0
899899 −2.00000 3.46410i −0.0667037 0.115534i
900900 0 0
901901 −31.5000 + 54.5596i −1.04942 + 1.81764i
902902 0 0
903903 −6.00000 + 10.3923i −0.199667 + 0.345834i
904904 0 0
905905 7.00000 0.232688
906906 0 0
907907 6.00000 + 10.3923i 0.199227 + 0.345071i 0.948278 0.317441i 0.102824π-0.102824\pi
−0.749051 + 0.662512i 0.769490π0.769490\pi
908908 0 0
909909 3.00000 0.0995037
910910 0 0
911911 40.0000 1.32526 0.662630 0.748947i 0.269440π-0.269440\pi
0.662630 + 0.748947i 0.269440π0.269440\pi
912912 0 0
913913 −14.0000 24.2487i −0.463332 0.802515i
914914 0 0
915915 −1.00000 −0.0330590
916916 0 0
917917 8.00000 13.8564i 0.264183 0.457579i
918918 0 0
919919 12.0000 20.7846i 0.395843 0.685621i −0.597365 0.801970i 0.703786π-0.703786\pi
0.993208 + 0.116348i 0.0371189π0.0371189\pi
920920 0 0
921921 7.00000 + 12.1244i 0.230658 + 0.399511i
922922 0 0
923923 −6.00000 + 20.7846i −0.197492 + 0.684134i
924924 0 0
925925 2.00000 + 3.46410i 0.0657596 + 0.113899i
926926 0 0
927927 3.00000 5.19615i 0.0985329 0.170664i
928928 0 0
929929 13.5000 23.3827i 0.442921 0.767161i −0.554984 0.831861i 0.687276π-0.687276\pi
0.997905 + 0.0646999i 0.0206090π0.0206090\pi
930930 0 0
931931 −18.0000 −0.589926
932932 0 0
933933 9.00000 + 15.5885i 0.294647 + 0.510343i
934934 0 0
935935 14.0000 0.457849
936936 0 0
937937 −49.0000 −1.60076 −0.800380 0.599493i 0.795369π-0.795369\pi
−0.800380 + 0.599493i 0.795369π0.795369\pi
938938 0 0
939939 −3.00000 5.19615i −0.0979013 0.169570i
940940 0 0
941941 −38.0000 −1.23876 −0.619382 0.785090i 0.712617π-0.712617\pi
−0.619382 + 0.785090i 0.712617π0.712617\pi
942942 0 0
943943 −27.0000 + 46.7654i −0.879241 + 1.52289i
944944 0 0
945945 −1.00000 + 1.73205i −0.0325300 + 0.0563436i
946946 0 0
947947 −24.0000 41.5692i −0.779895 1.35082i −0.932002 0.362454i 0.881939π-0.881939\pi
0.152106 0.988364i 0.451394π-0.451394\pi
948948 0 0
949949 −38.5000 + 9.52628i −1.24976 + 0.309236i
950950 0 0
951951 12.5000 + 21.6506i 0.405340 + 0.702070i
952952 0 0
953953 −3.00000 + 5.19615i −0.0971795 + 0.168320i −0.910516 0.413473i 0.864315π-0.864315\pi
0.813337 + 0.581793i 0.197649π0.197649\pi
954954 0 0
955955 2.00000 3.46410i 0.0647185 0.112096i
956956 0 0
957957 −2.00000 −0.0646508
958958 0 0
959959 −3.00000 5.19615i −0.0968751 0.167793i
960960 0 0
961961 −15.0000 −0.483871
962962 0 0
963963 6.00000 0.193347
964964 0 0
965965 −4.50000 7.79423i −0.144860 0.250905i
966966 0 0
967967 −2.00000 −0.0643157 −0.0321578 0.999483i 0.510238π-0.510238\pi
−0.0321578 + 0.999483i 0.510238π0.510238\pi
968968 0 0
969969 21.0000 36.3731i 0.674617 1.16847i
970970 0 0
971971 18.0000 31.1769i 0.577647 1.00051i −0.418101 0.908401i 0.637304π-0.637304\pi
0.995748 0.0921142i 0.0293625π-0.0293625\pi
972972 0 0
973973 −12.0000 20.7846i −0.384702 0.666324i
974974 0 0
975975 −10.0000 10.3923i −0.320256 0.332820i
976976 0 0
977977 −16.5000 28.5788i −0.527882 0.914318i −0.999472 0.0325001i 0.989653π-0.989653\pi
0.471590 0.881818i 0.343680π-0.343680\pi
978978 0 0
979979 14.0000 24.2487i 0.447442 0.774992i
980980 0 0
981981 1.00000 1.73205i 0.0319275 0.0553001i
982982 0 0
983983 −4.00000 −0.127580 −0.0637901 0.997963i 0.520319π-0.520319\pi
−0.0637901 + 0.997963i 0.520319π0.520319\pi
984984 0 0
985985 3.00000 + 5.19615i 0.0955879 + 0.165563i
986986 0 0
987987 12.0000 0.381964
988988 0 0
989989 −36.0000 −1.14473
990990 0 0
991991 −1.00000 1.73205i −0.0317660 0.0550204i 0.849705 0.527258i 0.176780π-0.176780\pi
−0.881471 + 0.472237i 0.843446π0.843446\pi
992992 0 0
993993 4.00000 0.126936
994994 0 0
995995 −7.00000 + 12.1244i −0.221915 + 0.384368i
996996 0 0
997997 17.5000 30.3109i 0.554231 0.959955i −0.443732 0.896159i 0.646346π-0.646346\pi
0.997963 0.0637961i 0.0203207π-0.0203207\pi
998998 0 0
999999 −0.500000 0.866025i −0.0158193 0.0273998i
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 624.2.q.c.289.1 2
3.2 odd 2 1872.2.t.j.289.1 2
4.3 odd 2 39.2.e.a.16.1 2
12.11 even 2 117.2.g.b.55.1 2
13.3 even 3 8112.2.a.w.1.1 1
13.9 even 3 inner 624.2.q.c.529.1 2
13.10 even 6 8112.2.a.bc.1.1 1
20.3 even 4 975.2.bb.d.874.1 4
20.7 even 4 975.2.bb.d.874.2 4
20.19 odd 2 975.2.i.f.601.1 2
39.35 odd 6 1872.2.t.j.1153.1 2
52.3 odd 6 507.2.a.c.1.1 1
52.7 even 12 507.2.j.d.316.1 4
52.11 even 12 507.2.b.b.337.2 2
52.15 even 12 507.2.b.b.337.1 2
52.19 even 12 507.2.j.d.316.2 4
52.23 odd 6 507.2.a.b.1.1 1
52.31 even 4 507.2.j.d.361.1 4
52.35 odd 6 39.2.e.a.22.1 yes 2
52.43 odd 6 507.2.e.c.22.1 2
52.47 even 4 507.2.j.d.361.2 4
52.51 odd 2 507.2.e.c.484.1 2
156.11 odd 12 1521.2.b.c.1351.1 2
156.23 even 6 1521.2.a.d.1.1 1
156.35 even 6 117.2.g.b.100.1 2
156.107 even 6 1521.2.a.a.1.1 1
156.119 odd 12 1521.2.b.c.1351.2 2
260.87 even 12 975.2.bb.d.724.1 4
260.139 odd 6 975.2.i.f.451.1 2
260.243 even 12 975.2.bb.d.724.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.e.a.16.1 2 4.3 odd 2
39.2.e.a.22.1 yes 2 52.35 odd 6
117.2.g.b.55.1 2 12.11 even 2
117.2.g.b.100.1 2 156.35 even 6
507.2.a.b.1.1 1 52.23 odd 6
507.2.a.c.1.1 1 52.3 odd 6
507.2.b.b.337.1 2 52.15 even 12
507.2.b.b.337.2 2 52.11 even 12
507.2.e.c.22.1 2 52.43 odd 6
507.2.e.c.484.1 2 52.51 odd 2
507.2.j.d.316.1 4 52.7 even 12
507.2.j.d.316.2 4 52.19 even 12
507.2.j.d.361.1 4 52.31 even 4
507.2.j.d.361.2 4 52.47 even 4
624.2.q.c.289.1 2 1.1 even 1 trivial
624.2.q.c.529.1 2 13.9 even 3 inner
975.2.i.f.451.1 2 260.139 odd 6
975.2.i.f.601.1 2 20.19 odd 2
975.2.bb.d.724.1 4 260.87 even 12
975.2.bb.d.724.2 4 260.243 even 12
975.2.bb.d.874.1 4 20.3 even 4
975.2.bb.d.874.2 4 20.7 even 4
1521.2.a.a.1.1 1 156.107 even 6
1521.2.a.d.1.1 1 156.23 even 6
1521.2.b.c.1351.1 2 156.11 odd 12
1521.2.b.c.1351.2 2 156.119 odd 12
1872.2.t.j.289.1 2 3.2 odd 2
1872.2.t.j.1153.1 2 39.35 odd 6
8112.2.a.w.1.1 1 13.3 even 3
8112.2.a.bc.1.1 1 13.10 even 6