Properties

Label 39.2.j.a.4.1
Level 3939
Weight 22
Character 39.4
Analytic conductor 0.3110.311
Analytic rank 00
Dimension 22
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [39,2,Mod(4,39)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(39, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("39.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 39=313 39 = 3 \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 39.j (of order 66, degree 22, minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 0.3114165678830.311416567883
Analytic rank: 00
Dimension: 22
Coefficient field: Q(ζ6)\Q(\zeta_{6})
comment: defining polynomial
 
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: yes
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

Embedding invariants

Embedding label 4.1
Root 0.500000+0.866025i0.500000 + 0.866025i of defining polynomial
Character χ\chi == 39.4
Dual form 39.2.j.a.10.1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q+(0.5000000.866025i)q3+(1.000001.73205i)q4+3.46410iq5+(1.50000+0.866025i)q7+(0.5000000.866025i)q9+(3.000001.73205i)q112.00000q12+(3.500000.866025i)q13+(3.00000+1.73205i)q15+(2.00000+3.46410i)q16+(3.000001.73205i)q19+(6.000003.46410i)q20+1.73205iq21+(3.000005.19615i)q237.00000q251.00000q27+(3.00000+1.73205i)q28+(3.00000+5.19615i)q29+1.73205iq31+(3.00000+1.73205i)q33+(3.000005.19615i)q35+(1.00000+1.73205i)q36+(1.000003.46410i)q39+(6.000003.46410i)q41+(0.500000+0.866025i)q43+6.92820iq44+(3.000001.73205i)q45+3.46410iq47+(2.00000+3.46410i)q48+(2.00000+3.46410i)q49+(5.000005.19615i)q52+12.0000q53+(6.0000010.3923i)q553.46410iq57+(3.00000+1.73205i)q596.92820iq60+(0.5000000.866025i)q61+(1.50000+0.866025i)q63+8.00000q64+(3.00000+12.1244i)q65+(7.50000+4.33013i)q67+(3.000005.19615i)q69+(9.00000+5.19615i)q71+1.73205iq73+(3.50000+6.06218i)q75+(6.000003.46410i)q76+6.00000q7711.0000q79+(12.00006.92820i)q80+(0.500000+0.866025i)q8113.8564iq83+(3.000001.73205i)q84+(3.00000+5.19615i)q87+(6.00000+3.46410i)q89+(4.50000+4.33013i)q9112.0000q92+(1.50000+0.866025i)q93+(6.00000+10.3923i)q95+(4.50000+2.59808i)q97+3.46410iq99+O(q100)q+(0.500000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +3.46410i q^{5} +(-1.50000 + 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-3.00000 - 1.73205i) q^{11} -2.00000 q^{12} +(3.50000 - 0.866025i) q^{13} +(3.00000 + 1.73205i) q^{15} +(-2.00000 + 3.46410i) q^{16} +(3.00000 - 1.73205i) q^{19} +(6.00000 - 3.46410i) q^{20} +1.73205i q^{21} +(3.00000 - 5.19615i) q^{23} -7.00000 q^{25} -1.00000 q^{27} +(3.00000 + 1.73205i) q^{28} +(-3.00000 + 5.19615i) q^{29} +1.73205i q^{31} +(-3.00000 + 1.73205i) q^{33} +(-3.00000 - 5.19615i) q^{35} +(-1.00000 + 1.73205i) q^{36} +(1.00000 - 3.46410i) q^{39} +(-6.00000 - 3.46410i) q^{41} +(0.500000 + 0.866025i) q^{43} +6.92820i q^{44} +(3.00000 - 1.73205i) q^{45} +3.46410i q^{47} +(2.00000 + 3.46410i) q^{48} +(-2.00000 + 3.46410i) q^{49} +(-5.00000 - 5.19615i) q^{52} +12.0000 q^{53} +(6.00000 - 10.3923i) q^{55} -3.46410i q^{57} +(-3.00000 + 1.73205i) q^{59} -6.92820i q^{60} +(-0.500000 - 0.866025i) q^{61} +(1.50000 + 0.866025i) q^{63} +8.00000 q^{64} +(3.00000 + 12.1244i) q^{65} +(7.50000 + 4.33013i) q^{67} +(-3.00000 - 5.19615i) q^{69} +(-9.00000 + 5.19615i) q^{71} +1.73205i q^{73} +(-3.50000 + 6.06218i) q^{75} +(-6.00000 - 3.46410i) q^{76} +6.00000 q^{77} -11.0000 q^{79} +(-12.0000 - 6.92820i) q^{80} +(-0.500000 + 0.866025i) q^{81} -13.8564i q^{83} +(3.00000 - 1.73205i) q^{84} +(3.00000 + 5.19615i) q^{87} +(6.00000 + 3.46410i) q^{89} +(-4.50000 + 4.33013i) q^{91} -12.0000 q^{92} +(1.50000 + 0.866025i) q^{93} +(6.00000 + 10.3923i) q^{95} +(-4.50000 + 2.59808i) q^{97} +3.46410i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+q32q43q7q96q114q12+7q13+6q154q16+6q19+12q20+6q2314q252q27+6q286q296q336q352q36+9q97+O(q100) 2 q + q^{3} - 2 q^{4} - 3 q^{7} - q^{9} - 6 q^{11} - 4 q^{12} + 7 q^{13} + 6 q^{15} - 4 q^{16} + 6 q^{19} + 12 q^{20} + 6 q^{23} - 14 q^{25} - 2 q^{27} + 6 q^{28} - 6 q^{29} - 6 q^{33} - 6 q^{35} - 2 q^{36}+ \cdots - 9 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/39Z)×\left(\mathbb{Z}/39\mathbb{Z}\right)^\times.

nn 1414 2828
χ(n)\chi(n) 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 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
33 0.500000 0.866025i 0.288675 0.500000i
44 −1.00000 1.73205i −0.500000 0.866025i
55 3.46410i 1.54919i 0.632456 + 0.774597i 0.282047π0.282047\pi
−0.632456 + 0.774597i 0.717953π0.717953\pi
66 0 0
77 −1.50000 + 0.866025i −0.566947 + 0.327327i −0.755929 0.654654i 0.772814π-0.772814\pi
0.188982 + 0.981981i 0.439481π0.439481\pi
88 0 0
99 −0.500000 0.866025i −0.166667 0.288675i
1010 0 0
1111 −3.00000 1.73205i −0.904534 0.522233i −0.0258656 0.999665i 0.508234π-0.508234\pi
−0.878668 + 0.477432i 0.841568π0.841568\pi
1212 −2.00000 −0.577350
1313 3.50000 0.866025i 0.970725 0.240192i
1414 0 0
1515 3.00000 + 1.73205i 0.774597 + 0.447214i
1616 −2.00000 + 3.46410i −0.500000 + 0.866025i
1717 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\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 6.00000 3.46410i 1.34164 0.774597i
2121 1.73205i 0.377964i
2222 0 0
2323 3.00000 5.19615i 0.625543 1.08347i −0.362892 0.931831i 0.618211π-0.618211\pi
0.988436 0.151642i 0.0484560π-0.0484560\pi
2424 0 0
2525 −7.00000 −1.40000
2626 0 0
2727 −1.00000 −0.192450
2828 3.00000 + 1.73205i 0.566947 + 0.327327i
2929 −3.00000 + 5.19615i −0.557086 + 0.964901i 0.440652 + 0.897678i 0.354747π0.354747\pi
−0.997738 + 0.0672232i 0.978586π0.978586\pi
3030 0 0
3131 1.73205i 0.311086i 0.987829 + 0.155543i 0.0497126π0.0497126\pi
−0.987829 + 0.155543i 0.950287π0.950287\pi
3232 0 0
3333 −3.00000 + 1.73205i −0.522233 + 0.301511i
3434 0 0
3535 −3.00000 5.19615i −0.507093 0.878310i
3636 −1.00000 + 1.73205i −0.166667 + 0.288675i
3737 0 0 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
3838 0 0
3939 1.00000 3.46410i 0.160128 0.554700i
4040 0 0
4141 −6.00000 3.46410i −0.937043 0.541002i −0.0480106 0.998847i 0.515288π-0.515288\pi
−0.889032 + 0.457845i 0.848621π0.848621\pi
4242 0 0
4343 0.500000 + 0.866025i 0.0762493 + 0.132068i 0.901629 0.432511i 0.142372π-0.142372\pi
−0.825380 + 0.564578i 0.809039π0.809039\pi
4444 6.92820i 1.04447i
4545 3.00000 1.73205i 0.447214 0.258199i
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 2.00000 + 3.46410i 0.288675 + 0.500000i
4949 −2.00000 + 3.46410i −0.285714 + 0.494872i
5050 0 0
5151 0 0
5252 −5.00000 5.19615i −0.693375 0.720577i
5353 12.0000 1.64833 0.824163 0.566352i 0.191646π-0.191646\pi
0.824163 + 0.566352i 0.191646π0.191646\pi
5454 0 0
5555 6.00000 10.3923i 0.809040 1.40130i
5656 0 0
5757 3.46410i 0.458831i
5858 0 0
5959 −3.00000 + 1.73205i −0.390567 + 0.225494i −0.682406 0.730974i 0.739066π-0.739066\pi
0.291839 + 0.956467i 0.405733π0.405733\pi
6060 6.92820i 0.894427i
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 1.50000 + 0.866025i 0.188982 + 0.109109i
6464 8.00000 1.00000
6565 3.00000 + 12.1244i 0.372104 + 1.50384i
6666 0 0
6767 7.50000 + 4.33013i 0.916271 + 0.529009i 0.882443 0.470418i 0.155897π-0.155897\pi
0.0338274 + 0.999428i 0.489230π0.489230\pi
6868 0 0
6969 −3.00000 5.19615i −0.361158 0.625543i
7070 0 0
7171 −9.00000 + 5.19615i −1.06810 + 0.616670i −0.927663 0.373419i 0.878185π-0.878185\pi
−0.140441 + 0.990089i 0.544852π0.544852\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 −3.50000 + 6.06218i −0.404145 + 0.700000i
7676 −6.00000 3.46410i −0.688247 0.397360i
7777 6.00000 0.683763
7878 0 0
7979 −11.0000 −1.23760 −0.618798 0.785550i 0.712380π-0.712380\pi
−0.618798 + 0.785550i 0.712380π0.712380\pi
8080 −12.0000 6.92820i −1.34164 0.774597i
8181 −0.500000 + 0.866025i −0.0555556 + 0.0962250i
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 3.00000 1.73205i 0.327327 0.188982i
8585 0 0
8686 0 0
8787 3.00000 + 5.19615i 0.321634 + 0.557086i
8888 0 0
8989 6.00000 + 3.46410i 0.635999 + 0.367194i 0.783072 0.621932i 0.213652π-0.213652\pi
−0.147073 + 0.989126i 0.546985π0.546985\pi
9090 0 0
9191 −4.50000 + 4.33013i −0.471728 + 0.453921i
9292 −12.0000 −1.25109
9393 1.50000 + 0.866025i 0.155543 + 0.0898027i
9494 0 0
9595 6.00000 + 10.3923i 0.615587 + 1.06623i
9696 0 0
9797 −4.50000 + 2.59808i −0.456906 + 0.263795i −0.710742 0.703452i 0.751641π-0.751641\pi
0.253837 + 0.967247i 0.418307π0.418307\pi
9898 0 0
9999 3.46410i 0.348155i
100100 7.00000 + 12.1244i 0.700000 + 1.21244i
101101 9.00000 15.5885i 0.895533 1.55111i 0.0623905 0.998052i 0.480128π-0.480128\pi
0.833143 0.553058i 0.186539π-0.186539\pi
102102 0 0
103103 −1.00000 −0.0985329 −0.0492665 0.998786i 0.515688π-0.515688\pi
−0.0492665 + 0.998786i 0.515688π0.515688\pi
104104 0 0
105105 −6.00000 −0.585540
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 1.00000 + 1.73205i 0.0962250 + 0.166667i
109109 15.5885i 1.49310i −0.665327 0.746552i 0.731708π-0.731708\pi
0.665327 0.746552i 0.268292π-0.268292\pi
110110 0 0
111111 0 0
112112 6.92820i 0.654654i
113113 −3.00000 5.19615i −0.282216 0.488813i 0.689714 0.724082i 0.257736π-0.257736\pi
−0.971930 + 0.235269i 0.924403π0.924403\pi
114114 0 0
115115 18.0000 + 10.3923i 1.67851 + 0.969087i
116116 12.0000 1.11417
117117 −2.50000 2.59808i −0.231125 0.240192i
118118 0 0
119119 0 0
120120 0 0
121121 0.500000 + 0.866025i 0.0454545 + 0.0787296i
122122 0 0
123123 −6.00000 + 3.46410i −0.541002 + 0.312348i
124124 3.00000 1.73205i 0.269408 0.155543i
125125 6.92820i 0.619677i
126126 0 0
127127 −6.50000 + 11.2583i −0.576782 + 0.999015i 0.419064 + 0.907957i 0.362358π0.362358\pi
−0.995846 + 0.0910585i 0.970975π0.970975\pi
128128 0 0
129129 1.00000 0.0880451
130130 0 0
131131 6.00000 0.524222 0.262111 0.965038i 0.415581π-0.415581\pi
0.262111 + 0.965038i 0.415581π0.415581\pi
132132 6.00000 + 3.46410i 0.522233 + 0.301511i
133133 −3.00000 + 5.19615i −0.260133 + 0.450564i
134134 0 0
135135 3.46410i 0.298142i
136136 0 0
137137 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
138138 0 0
139139 −2.50000 4.33013i −0.212047 0.367277i 0.740308 0.672268i 0.234680π-0.234680\pi
−0.952355 + 0.304991i 0.901346π0.901346\pi
140140 −6.00000 + 10.3923i −0.507093 + 0.878310i
141141 3.00000 + 1.73205i 0.252646 + 0.145865i
142142 0 0
143143 −12.0000 3.46410i −1.00349 0.289683i
144144 4.00000 0.333333
145145 −18.0000 10.3923i −1.49482 0.863034i
146146 0 0
147147 2.00000 + 3.46410i 0.164957 + 0.285714i
148148 0 0
149149 −6.00000 + 3.46410i −0.491539 + 0.283790i −0.725213 0.688525i 0.758259π-0.758259\pi
0.233674 + 0.972315i 0.424925π0.424925\pi
150150 0 0
151151 3.46410i 0.281905i −0.990016 0.140952i 0.954984π-0.954984\pi
0.990016 0.140952i 0.0450164π-0.0450164\pi
152152 0 0
153153 0 0
154154 0 0
155155 −6.00000 −0.481932
156156 −7.00000 + 1.73205i −0.560449 + 0.138675i
157157 11.0000 0.877896 0.438948 0.898513i 0.355351π-0.355351\pi
0.438948 + 0.898513i 0.355351π0.355351\pi
158158 0 0
159159 6.00000 10.3923i 0.475831 0.824163i
160160 0 0
161161 10.3923i 0.819028i
162162 0 0
163163 −16.5000 + 9.52628i −1.29238 + 0.746156i −0.979076 0.203497i 0.934769π-0.934769\pi
−0.313304 + 0.949653i 0.601436π0.601436\pi
164164 13.8564i 1.08200i
165165 −6.00000 10.3923i −0.467099 0.809040i
166166 0 0
167167 −6.00000 3.46410i −0.464294 0.268060i 0.249554 0.968361i 0.419716π-0.419716\pi
−0.713848 + 0.700301i 0.753049π0.753049\pi
168168 0 0
169169 11.5000 6.06218i 0.884615 0.466321i
170170 0 0
171171 −3.00000 1.73205i −0.229416 0.132453i
172172 1.00000 1.73205i 0.0762493 0.132068i
173173 3.00000 + 5.19615i 0.228086 + 0.395056i 0.957241 0.289292i 0.0934200π-0.0934200\pi
−0.729155 + 0.684349i 0.760087π0.760087\pi
174174 0 0
175175 10.5000 6.06218i 0.793725 0.458258i
176176 12.0000 6.92820i 0.904534 0.522233i
177177 3.46410i 0.260378i
178178 0 0
179179 −6.00000 + 10.3923i −0.448461 + 0.776757i −0.998286 0.0585225i 0.981361π-0.981361\pi
0.549825 + 0.835280i 0.314694π0.314694\pi
180180 −6.00000 3.46410i −0.447214 0.258199i
181181 14.0000 1.04061 0.520306 0.853980i 0.325818π-0.325818\pi
0.520306 + 0.853980i 0.325818π0.325818\pi
182182 0 0
183183 −1.00000 −0.0739221
184184 0 0
185185 0 0
186186 0 0
187187 0 0
188188 6.00000 3.46410i 0.437595 0.252646i
189189 1.50000 0.866025i 0.109109 0.0629941i
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 4.00000 6.92820i 0.288675 0.500000i
193193 −13.5000 7.79423i −0.971751 0.561041i −0.0719816 0.997406i 0.522932π-0.522932\pi
−0.899770 + 0.436365i 0.856266π0.856266\pi
194194 0 0
195195 12.0000 + 3.46410i 0.859338 + 0.248069i
196196 8.00000 0.571429
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 −3.50000 6.06218i −0.248108 0.429736i 0.714893 0.699234i 0.246476π-0.246476\pi
−0.963001 + 0.269498i 0.913142π0.913142\pi
200200 0 0
201201 7.50000 4.33013i 0.529009 0.305424i
202202 0 0
203203 10.3923i 0.729397i
204204 0 0
205205 12.0000 20.7846i 0.838116 1.45166i
206206 0 0
207207 −6.00000 −0.417029
208208 −4.00000 + 13.8564i −0.277350 + 0.960769i
209209 −12.0000 −0.830057
210210 0 0
211211 −6.50000 + 11.2583i −0.447478 + 0.775055i −0.998221 0.0596196i 0.981011π-0.981011\pi
0.550743 + 0.834675i 0.314345π0.314345\pi
212212 −12.0000 20.7846i −0.824163 1.42749i
213213 10.3923i 0.712069i
214214 0 0
215215 −3.00000 + 1.73205i −0.204598 + 0.118125i
216216 0 0
217217 −1.50000 2.59808i −0.101827 0.176369i
218218 0 0
219219 1.50000 + 0.866025i 0.101361 + 0.0585206i
220220 −24.0000 −1.61808
221221 0 0
222222 0 0
223223 15.0000 + 8.66025i 1.00447 + 0.579934i 0.909569 0.415553i 0.136412π-0.136412\pi
0.0949052 + 0.995486i 0.469745π0.469745\pi
224224 0 0
225225 3.50000 + 6.06218i 0.233333 + 0.404145i
226226 0 0
227227 18.0000 10.3923i 1.19470 0.689761i 0.235333 0.971915i 0.424382π-0.424382\pi
0.959369 + 0.282153i 0.0910487π0.0910487\pi
228228 −6.00000 + 3.46410i −0.397360 + 0.229416i
229229 27.7128i 1.83131i 0.401960 + 0.915657i 0.368329π0.368329\pi
−0.401960 + 0.915657i 0.631671π0.631671\pi
230230 0 0
231231 3.00000 5.19615i 0.197386 0.341882i
232232 0 0
233233 −18.0000 −1.17922 −0.589610 0.807688i 0.700718π-0.700718\pi
−0.589610 + 0.807688i 0.700718π0.700718\pi
234234 0 0
235235 −12.0000 −0.782794
236236 6.00000 + 3.46410i 0.390567 + 0.225494i
237237 −5.50000 + 9.52628i −0.357263 + 0.618798i
238238 0 0
239239 0 0 1.00000 00
−1.00000 π\pi
240240 −12.0000 + 6.92820i −0.774597 + 0.447214i
241241 18.0000 10.3923i 1.15948 0.669427i 0.208302 0.978065i 0.433206π-0.433206\pi
0.951180 + 0.308637i 0.0998729π0.0998729\pi
242242 0 0
243243 0.500000 + 0.866025i 0.0320750 + 0.0555556i
244244 −1.00000 + 1.73205i −0.0640184 + 0.110883i
245245 −12.0000 6.92820i −0.766652 0.442627i
246246 0 0
247247 9.00000 8.66025i 0.572656 0.551039i
248248 0 0
249249 −12.0000 6.92820i −0.760469 0.439057i
250250 0 0
251251 −6.00000 10.3923i −0.378717 0.655956i 0.612159 0.790735i 0.290301π-0.290301\pi
−0.990876 + 0.134778i 0.956968π0.956968\pi
252252 3.46410i 0.218218i
253253 −18.0000 + 10.3923i −1.13165 + 0.653359i
254254 0 0
255255 0 0
256256 −8.00000 13.8564i −0.500000 0.866025i
257257 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
258258 0 0
259259 0 0
260260 18.0000 17.3205i 1.11631 1.07417i
261261 6.00000 0.371391
262262 0 0
263263 −6.00000 + 10.3923i −0.369976 + 0.640817i −0.989561 0.144112i 0.953967π-0.953967\pi
0.619586 + 0.784929i 0.287301π0.287301\pi
264264 0 0
265265 41.5692i 2.55358i
266266 0 0
267267 6.00000 3.46410i 0.367194 0.212000i
268268 17.3205i 1.05802i
269269 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
270270 0 0
271271 −4.50000 2.59808i −0.273356 0.157822i 0.357056 0.934083i 0.383781π-0.383781\pi
−0.630412 + 0.776261i 0.717114π0.717114\pi
272272 0 0
273273 1.50000 + 6.06218i 0.0907841 + 0.366900i
274274 0 0
275275 21.0000 + 12.1244i 1.26635 + 0.731126i
276276 −6.00000 + 10.3923i −0.361158 + 0.625543i
277277 5.00000 + 8.66025i 0.300421 + 0.520344i 0.976231 0.216731i 0.0695395π-0.0695395\pi
−0.675810 + 0.737075i 0.736206π0.736206\pi
278278 0 0
279279 1.50000 0.866025i 0.0898027 0.0518476i
280280 0 0
281281 24.2487i 1.44656i −0.690557 0.723278i 0.742634π-0.742634\pi
0.690557 0.723278i 0.257366π-0.257366\pi
282282 0 0
283283 5.50000 9.52628i 0.326941 0.566279i −0.654962 0.755662i 0.727315π-0.727315\pi
0.981903 + 0.189383i 0.0606488π0.0606488\pi
284284 18.0000 + 10.3923i 1.06810 + 0.616670i
285285 12.0000 0.710819
286286 0 0
287287 12.0000 0.708338
288288 0 0
289289 8.50000 14.7224i 0.500000 0.866025i
290290 0 0
291291 5.19615i 0.304604i
292292 3.00000 1.73205i 0.175562 0.101361i
293293 −15.0000 + 8.66025i −0.876309 + 0.505937i −0.869440 0.494039i 0.835520π-0.835520\pi
−0.00686959 + 0.999976i 0.502187π0.502187\pi
294294 0 0
295295 −6.00000 10.3923i −0.349334 0.605063i
296296 0 0
297297 3.00000 + 1.73205i 0.174078 + 0.100504i
298298 0 0
299299 6.00000 20.7846i 0.346989 1.20201i
300300 14.0000 0.808290
301301 −1.50000 0.866025i −0.0864586 0.0499169i
302302 0 0
303303 −9.00000 15.5885i −0.517036 0.895533i
304304 13.8564i 0.794719i
305305 3.00000 1.73205i 0.171780 0.0991769i
306306 0 0
307307 1.73205i 0.0988534i −0.998778 0.0494267i 0.984261π-0.984261\pi
0.998778 0.0494267i 0.0157394π-0.0157394\pi
308308 −6.00000 10.3923i −0.341882 0.592157i
309309 −0.500000 + 0.866025i −0.0284440 + 0.0492665i
310310 0 0
311311 −24.0000 −1.36092 −0.680458 0.732787i 0.738219π-0.738219\pi
−0.680458 + 0.732787i 0.738219π0.738219\pi
312312 0 0
313313 13.0000 0.734803 0.367402 0.930062i 0.380247π-0.380247\pi
0.367402 + 0.930062i 0.380247π0.380247\pi
314314 0 0
315315 −3.00000 + 5.19615i −0.169031 + 0.292770i
316316 11.0000 + 19.0526i 0.618798 + 1.07179i
317317 6.92820i 0.389127i 0.980890 + 0.194563i 0.0623290π0.0623290\pi
−0.980890 + 0.194563i 0.937671π0.937671\pi
318318 0 0
319319 18.0000 10.3923i 1.00781 0.581857i
320320 27.7128i 1.54919i
321321 −3.00000 5.19615i −0.167444 0.290021i
322322 0 0
323323 0 0
324324 2.00000 0.111111
325325 −24.5000 + 6.06218i −1.35902 + 0.336269i
326326 0 0
327327 −13.5000 7.79423i −0.746552 0.431022i
328328 0 0
329329 −3.00000 5.19615i −0.165395 0.286473i
330330 0 0
331331 4.50000 2.59808i 0.247342 0.142803i −0.371204 0.928551i 0.621055π-0.621055\pi
0.618547 + 0.785748i 0.287722π0.287722\pi
332332 −24.0000 + 13.8564i −1.31717 + 0.760469i
333333 0 0
334334 0 0
335335 −15.0000 + 25.9808i −0.819538 + 1.41948i
336336 −6.00000 3.46410i −0.327327 0.188982i
337337 −5.00000 −0.272367 −0.136184 0.990684i 0.543484π-0.543484\pi
−0.136184 + 0.990684i 0.543484π0.543484\pi
338338 0 0
339339 −6.00000 −0.325875
340340 0 0
341341 3.00000 5.19615i 0.162459 0.281387i
342342 0 0
343343 19.0526i 1.02874i
344344 0 0
345345 18.0000 10.3923i 0.969087 0.559503i
346346 0 0
347347 12.0000 + 20.7846i 0.644194 + 1.11578i 0.984487 + 0.175457i 0.0561403π0.0561403\pi
−0.340293 + 0.940319i 0.610526π0.610526\pi
348348 6.00000 10.3923i 0.321634 0.557086i
349349 −16.5000 9.52628i −0.883225 0.509930i −0.0115044 0.999934i 0.503662π-0.503662\pi
−0.871720 + 0.490004i 0.836995π0.836995\pi
350350 0 0
351351 −3.50000 + 0.866025i −0.186816 + 0.0462250i
352352 0 0
353353 9.00000 + 5.19615i 0.479022 + 0.276563i 0.720009 0.693965i 0.244138π-0.244138\pi
−0.240987 + 0.970528i 0.577471π0.577471\pi
354354 0 0
355355 −18.0000 31.1769i −0.955341 1.65470i
356356 13.8564i 0.734388i
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 1.00000 0.0524864
364364 12.0000 + 3.46410i 0.628971 + 0.181568i
365365 −6.00000 −0.314054
366366 0 0
367367 −8.50000 + 14.7224i −0.443696 + 0.768505i −0.997960 0.0638362i 0.979666π-0.979666\pi
0.554264 + 0.832341i 0.313000π0.313000\pi
368368 12.0000 + 20.7846i 0.625543 + 1.08347i
369369 6.92820i 0.360668i
370370 0 0
371371 −18.0000 + 10.3923i −0.934513 + 0.539542i
372372 3.46410i 0.179605i
373373 5.50000 + 9.52628i 0.284779 + 0.493252i 0.972556 0.232671i 0.0747464π-0.0747464\pi
−0.687776 + 0.725923i 0.741413π0.741413\pi
374374 0 0
375375 −6.00000 3.46410i −0.309839 0.178885i
376376 0 0
377377 −6.00000 + 20.7846i −0.309016 + 1.07046i
378378 0 0
379379 −19.5000 11.2583i −1.00165 0.578302i −0.0929123 0.995674i 0.529618π-0.529618\pi
−0.908735 + 0.417373i 0.862951π0.862951\pi
380380 12.0000 20.7846i 0.615587 1.06623i
381381 6.50000 + 11.2583i 0.333005 + 0.576782i
382382 0 0
383383 24.0000 13.8564i 1.22634 0.708029i 0.260080 0.965587i 0.416251π-0.416251\pi
0.966263 + 0.257558i 0.0829178π0.0829178\pi
384384 0 0
385385 20.7846i 1.05928i
386386 0 0
387387 0.500000 0.866025i 0.0254164 0.0440225i
388388 9.00000 + 5.19615i 0.456906 + 0.263795i
389389 12.0000 0.608424 0.304212 0.952604i 0.401607π-0.401607\pi
0.304212 + 0.952604i 0.401607π0.401607\pi
390390 0 0
391391 0 0
392392 0 0
393393 3.00000 5.19615i 0.151330 0.262111i
394394 0 0
395395 38.1051i 1.91728i
396396 6.00000 3.46410i 0.301511 0.174078i
397397 13.5000 7.79423i 0.677546 0.391181i −0.121384 0.992606i 0.538733π-0.538733\pi
0.798930 + 0.601424i 0.205400π0.205400\pi
398398 0 0
399399 3.00000 + 5.19615i 0.150188 + 0.260133i
400400 14.0000 24.2487i 0.700000 1.21244i
401401 0 0 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
402402 0 0
403403 1.50000 + 6.06218i 0.0747203 + 0.301979i
404404 −36.0000 −1.79107
405405 −3.00000 1.73205i −0.149071 0.0860663i
406406 0 0
407407 0 0
408408 0 0
409409 7.50000 4.33013i 0.370851 0.214111i −0.302979 0.952997i 0.597981π-0.597981\pi
0.673830 + 0.738886i 0.264648π0.264648\pi
410410 0 0
411411 0 0
412412 1.00000 + 1.73205i 0.0492665 + 0.0853320i
413413 3.00000 5.19615i 0.147620 0.255686i
414414 0 0
415415 48.0000 2.35623
416416 0 0
417417 −5.00000 −0.244851
418418 0 0
419419 6.00000 10.3923i 0.293119 0.507697i −0.681426 0.731887i 0.738640π-0.738640\pi
0.974546 + 0.224189i 0.0719734π0.0719734\pi
420420 6.00000 + 10.3923i 0.292770 + 0.507093i
421421 12.1244i 0.590905i −0.955357 0.295452i 0.904530π-0.904530\pi
0.955357 0.295452i 0.0954704π-0.0954704\pi
422422 0 0
423423 3.00000 1.73205i 0.145865 0.0842152i
424424 0 0
425425 0 0
426426 0 0
427427 1.50000 + 0.866025i 0.0725901 + 0.0419099i
428428 −12.0000 −0.580042
429429 −9.00000 + 8.66025i −0.434524 + 0.418121i
430430 0 0
431431 18.0000 + 10.3923i 0.867029 + 0.500580i 0.866360 0.499420i 0.166454π-0.166454\pi
0.000669521 1.00000i 0.499787π0.499787\pi
432432 2.00000 3.46410i 0.0962250 0.166667i
433433 11.5000 + 19.9186i 0.552655 + 0.957226i 0.998082 + 0.0619079i 0.0197185π0.0197185\pi
−0.445427 + 0.895318i 0.646948π0.646948\pi
434434 0 0
435435 −18.0000 + 10.3923i −0.863034 + 0.498273i
436436 −27.0000 + 15.5885i −1.29307 + 0.746552i
437437 20.7846i 0.994263i
438438 0 0
439439 −17.5000 + 30.3109i −0.835229 + 1.44666i 0.0586141 + 0.998281i 0.481332π0.481332\pi
−0.893843 + 0.448379i 0.852001π0.852001\pi
440440 0 0
441441 4.00000 0.190476
442442 0 0
443443 24.0000 1.14027 0.570137 0.821549i 0.306890π-0.306890\pi
0.570137 + 0.821549i 0.306890π0.306890\pi
444444 0 0
445445 −12.0000 + 20.7846i −0.568855 + 0.985285i
446446 0 0
447447 6.92820i 0.327693i
448448 −12.0000 + 6.92820i −0.566947 + 0.327327i
449449 33.0000 19.0526i 1.55737 0.899146i 0.559859 0.828588i 0.310855π-0.310855\pi
0.997508 0.0705577i 0.0224779π-0.0224779\pi
450450 0 0
451451 12.0000 + 20.7846i 0.565058 + 0.978709i
452452 −6.00000 + 10.3923i −0.282216 + 0.488813i
453453 −3.00000 1.73205i −0.140952 0.0813788i
454454 0 0
455455 −15.0000 15.5885i −0.703211 0.730798i
456456 0 0
457457 −31.5000 18.1865i −1.47351 0.850730i −0.473953 0.880550i 0.657173π-0.657173\pi
−0.999555 + 0.0298202i 0.990507π0.990507\pi
458458 0 0
459459 0 0
460460 41.5692i 1.93817i
461461 −36.0000 + 20.7846i −1.67669 + 0.968036i −0.712938 + 0.701228i 0.752636π0.752636\pi
−0.963750 + 0.266808i 0.914031π0.914031\pi
462462 0 0
463463 36.3731i 1.69040i −0.534450 0.845200i 0.679481π-0.679481\pi
0.534450 0.845200i 0.320519π-0.320519\pi
464464 −12.0000 20.7846i −0.557086 0.964901i
465465 −3.00000 + 5.19615i −0.139122 + 0.240966i
466466 0 0
467467 6.00000 0.277647 0.138823 0.990317i 0.455668π-0.455668\pi
0.138823 + 0.990317i 0.455668π0.455668\pi
468468 −2.00000 + 6.92820i −0.0924500 + 0.320256i
469469 −15.0000 −0.692636
470470 0 0
471471 5.50000 9.52628i 0.253427 0.438948i
472472 0 0
473473 3.46410i 0.159280i
474474 0 0
475475 −21.0000 + 12.1244i −0.963546 + 0.556304i
476476 0 0
477477 −6.00000 10.3923i −0.274721 0.475831i
478478 0 0
479479 30.0000 + 17.3205i 1.37073 + 0.791394i 0.991021 0.133710i 0.0426889π-0.0426889\pi
0.379714 + 0.925104i 0.376022π0.376022\pi
480480 0 0
481481 0 0
482482 0 0
483483 9.00000 + 5.19615i 0.409514 + 0.236433i
484484 1.00000 1.73205i 0.0454545 0.0787296i
485485 −9.00000 15.5885i −0.408669 0.707835i
486486 0 0
487487 −21.0000 + 12.1244i −0.951601 + 0.549407i −0.893578 0.448908i 0.851813π-0.851813\pi
−0.0580230 + 0.998315i 0.518480π0.518480\pi
488488 0 0
489489 19.0526i 0.861586i
490490 0 0
491491 −3.00000 + 5.19615i −0.135388 + 0.234499i −0.925746 0.378147i 0.876561π-0.876561\pi
0.790358 + 0.612646i 0.209895π0.209895\pi
492492 12.0000 + 6.92820i 0.541002 + 0.312348i
493493 0 0
494494 0 0
495495 −12.0000 −0.539360
496496 −6.00000 3.46410i −0.269408 0.155543i
497497 9.00000 15.5885i 0.403705 0.699238i
498498 0 0
499499 31.1769i 1.39567i 0.716258 + 0.697835i 0.245853π0.245853\pi
−0.716258 + 0.697835i 0.754147π0.754147\pi
500500 −12.0000 + 6.92820i −0.536656 + 0.309839i
501501 −6.00000 + 3.46410i −0.268060 + 0.154765i
502502 0 0
503503 15.0000 + 25.9808i 0.668817 + 1.15842i 0.978235 + 0.207499i 0.0665323π0.0665323\pi
−0.309418 + 0.950926i 0.600134π0.600134\pi
504504 0 0
505505 54.0000 + 31.1769i 2.40297 + 1.38735i
506506 0 0
507507 0.500000 12.9904i 0.0222058 0.576923i
508508 26.0000 1.15356
509509 −15.0000 8.66025i −0.664863 0.383859i 0.129264 0.991610i 0.458738π-0.458738\pi
−0.794128 + 0.607751i 0.792072π0.792072\pi
510510 0 0
511511 −1.50000 2.59808i −0.0663561 0.114932i
512512 0 0
513513 −3.00000 + 1.73205i −0.132453 + 0.0764719i
514514 0 0
515515 3.46410i 0.152647i
516516 −1.00000 1.73205i −0.0440225 0.0762493i
517517 6.00000 10.3923i 0.263880 0.457053i
518518 0 0
519519 6.00000 0.263371
520520 0 0
521521 −24.0000 −1.05146 −0.525730 0.850652i 0.676208π-0.676208\pi
−0.525730 + 0.850652i 0.676208π0.676208\pi
522522 0 0
523523 14.0000 24.2487i 0.612177 1.06032i −0.378695 0.925521i 0.623627π-0.623627\pi
0.990873 0.134801i 0.0430394π-0.0430394\pi
524524 −6.00000 10.3923i −0.262111 0.453990i
525525 12.1244i 0.529150i
526526 0 0
527527 0 0
528528 13.8564i 0.603023i
529529 −6.50000 11.2583i −0.282609 0.489493i
530530 0 0
531531 3.00000 + 1.73205i 0.130189 + 0.0751646i
532532 12.0000 0.520266
533533 −24.0000 6.92820i −1.03956 0.300094i
534534 0 0
535535 18.0000 + 10.3923i 0.778208 + 0.449299i
536536 0 0
537537 6.00000 + 10.3923i 0.258919 + 0.448461i
538538 0 0
539539 12.0000 6.92820i 0.516877 0.298419i
540540 −6.00000 + 3.46410i −0.258199 + 0.149071i
541541 29.4449i 1.26593i −0.774179 0.632967i 0.781837π-0.781837\pi
0.774179 0.632967i 0.218163π-0.218163\pi
542542 0 0
543543 7.00000 12.1244i 0.300399 0.520306i
544544 0 0
545545 54.0000 2.31311
546546 0 0
547547 −19.0000 −0.812381 −0.406191 0.913788i 0.633143π-0.633143\pi
−0.406191 + 0.913788i 0.633143π0.633143\pi
548548 0 0
549549 −0.500000 + 0.866025i −0.0213395 + 0.0369611i
550550 0 0
551551 20.7846i 0.885454i
552552 0 0
553553 16.5000 9.52628i 0.701651 0.405099i
554554 0 0
555555 0 0
556556 −5.00000 + 8.66025i −0.212047 + 0.367277i
557557 −24.0000 13.8564i −1.01691 0.587115i −0.103704 0.994608i 0.533070π-0.533070\pi
−0.913208 + 0.407493i 0.866403π0.866403\pi
558558 0 0
559559 2.50000 + 2.59808i 0.105739 + 0.109887i
560560 24.0000 1.01419
561561 0 0
562562 0 0
563563 21.0000 + 36.3731i 0.885044 + 1.53294i 0.845663 + 0.533718i 0.179206π0.179206\pi
0.0393818 + 0.999224i 0.487461π0.487461\pi
564564 6.92820i 0.291730i
565565 18.0000 10.3923i 0.757266 0.437208i
566566 0 0
567567 1.73205i 0.0727393i
568568 0 0
569569 3.00000 5.19615i 0.125767 0.217834i −0.796266 0.604947i 0.793194π-0.793194\pi
0.922032 + 0.387113i 0.126528π0.126528\pi
570570 0 0
571571 −44.0000 −1.84134 −0.920671 0.390339i 0.872358π-0.872358\pi
−0.920671 + 0.390339i 0.872358π0.872358\pi
572572 6.00000 + 24.2487i 0.250873 + 1.01389i
573573 18.0000 0.751961
574574 0 0
575575 −21.0000 + 36.3731i −0.875761 + 1.51686i
576576 −4.00000 6.92820i −0.166667 0.288675i
577577 34.6410i 1.44212i −0.692870 0.721062i 0.743654π-0.743654\pi
0.692870 0.721062i 0.256346π-0.256346\pi
578578 0 0
579579 −13.5000 + 7.79423i −0.561041 + 0.323917i
580580 41.5692i 1.72607i
581581 12.0000 + 20.7846i 0.497844 + 0.862291i
582582 0 0
583583 −36.0000 20.7846i −1.49097 0.860811i
584584 0 0
585585 9.00000 8.66025i 0.372104 0.358057i
586586 0 0
587587 −27.0000 15.5885i −1.11441 0.643404i −0.174441 0.984668i 0.555812π-0.555812\pi
−0.939968 + 0.341263i 0.889145π0.889145\pi
588588 4.00000 6.92820i 0.164957 0.285714i
589589 3.00000 + 5.19615i 0.123613 + 0.214104i
590590 0 0
591591 12.0000 6.92820i 0.493614 0.284988i
592592 0 0
593593 3.46410i 0.142254i 0.997467 + 0.0711268i 0.0226595π0.0226595\pi
−0.997467 + 0.0711268i 0.977341π0.977341\pi
594594 0 0
595595 0 0
596596 12.0000 + 6.92820i 0.491539 + 0.283790i
597597 −7.00000 −0.286491
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 13.0000 22.5167i 0.530281 0.918474i −0.469095 0.883148i 0.655420π-0.655420\pi
0.999376 0.0353259i 0.0112469π-0.0112469\pi
602602 0 0
603603 8.66025i 0.352673i
604604 −6.00000 + 3.46410i −0.244137 + 0.140952i
605605 −3.00000 + 1.73205i −0.121967 + 0.0704179i
606606 0 0
607607 −4.00000 6.92820i −0.162355 0.281207i 0.773358 0.633970i 0.218576π-0.218576\pi
−0.935713 + 0.352763i 0.885242π0.885242\pi
608608 0 0
609609 −9.00000 5.19615i −0.364698 0.210559i
610610 0 0
611611 3.00000 + 12.1244i 0.121367 + 0.490499i
612612 0 0
613613 7.50000 + 4.33013i 0.302922 + 0.174892i 0.643755 0.765232i 0.277376π-0.277376\pi
−0.340833 + 0.940124i 0.610709π0.610709\pi
614614 0 0
615615 −12.0000 20.7846i −0.483887 0.838116i
616616 0 0
617617 −9.00000 + 5.19615i −0.362326 + 0.209189i −0.670101 0.742270i 0.733749π-0.733749\pi
0.307774 + 0.951459i 0.400416π0.400416\pi
618618 0 0
619619 25.9808i 1.04425i 0.852867 + 0.522127i 0.174861π0.174861\pi
−0.852867 + 0.522127i 0.825139π0.825139\pi
620620 6.00000 + 10.3923i 0.240966 + 0.417365i
621621 −3.00000 + 5.19615i −0.120386 + 0.208514i
622622 0 0
623623 −12.0000 −0.480770
624624 10.0000 + 10.3923i 0.400320 + 0.416025i
625625 −11.0000 −0.440000
626626 0 0
627627 −6.00000 + 10.3923i −0.239617 + 0.415029i
628628 −11.0000 19.0526i −0.438948 0.760280i
629629 0 0
630630 0 0
631631 1.50000 0.866025i 0.0597141 0.0344759i −0.469846 0.882749i 0.655690π-0.655690\pi
0.529560 + 0.848273i 0.322357π0.322357\pi
632632 0 0
633633 6.50000 + 11.2583i 0.258352 + 0.447478i
634634 0 0
635635 −39.0000 22.5167i −1.54767 0.893546i
636636 −24.0000 −0.951662
637637 −4.00000 + 13.8564i −0.158486 + 0.549011i
638638 0 0
639639 9.00000 + 5.19615i 0.356034 + 0.205557i
640640 0 0
641641 −9.00000 15.5885i −0.355479 0.615707i 0.631721 0.775196i 0.282349π-0.282349\pi
−0.987200 + 0.159489i 0.949015π0.949015\pi
642642 0 0
643643 16.5000 9.52628i 0.650696 0.375680i −0.138027 0.990429i 0.544076π-0.544076\pi
0.788723 + 0.614749i 0.210743π0.210743\pi
644644 18.0000 10.3923i 0.709299 0.409514i
645645 3.46410i 0.136399i
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 12.0000 0.471041
650650 0 0
651651 −3.00000 −0.117579
652652 33.0000 + 19.0526i 1.29238 + 0.746156i
653653 −18.0000 + 31.1769i −0.704394 + 1.22005i 0.262515 + 0.964928i 0.415448π0.415448\pi
−0.966910 + 0.255119i 0.917885π0.917885\pi
654654 0 0
655655 20.7846i 0.812122i
656656 24.0000 13.8564i 0.937043 0.541002i
657657 1.50000 0.866025i 0.0585206 0.0337869i
658658 0 0
659659 −24.0000 41.5692i −0.934907 1.61931i −0.774799 0.632207i 0.782149π-0.782149\pi
−0.160108 0.987099i 0.551184π-0.551184\pi
660660 −12.0000 + 20.7846i −0.467099 + 0.809040i
661661 −22.5000 12.9904i −0.875149 0.505267i −0.00609283 0.999981i 0.501939π-0.501939\pi
−0.869056 + 0.494714i 0.835273π0.835273\pi
662662 0 0
663663 0 0
664664 0 0
665665 −18.0000 10.3923i −0.698010 0.402996i
666666 0 0
667667 18.0000 + 31.1769i 0.696963 + 1.20717i
668668 13.8564i 0.536120i
669669 15.0000 8.66025i 0.579934 0.334825i
670670 0 0
671671 3.46410i 0.133730i
672672 0 0
673673 0.500000 0.866025i 0.0192736 0.0333828i −0.856228 0.516599i 0.827198π-0.827198\pi
0.875501 + 0.483216i 0.160531π0.160531\pi
674674 0 0
675675 7.00000 0.269430
676676 −22.0000 13.8564i −0.846154 0.532939i
677677 48.0000 1.84479 0.922395 0.386248i 0.126229π-0.126229\pi
0.922395 + 0.386248i 0.126229π0.126229\pi
678678 0 0
679679 4.50000 7.79423i 0.172694 0.299115i
680680 0 0
681681 20.7846i 0.796468i
682682 0 0
683683 21.0000 12.1244i 0.803543 0.463926i −0.0411658 0.999152i 0.513107π-0.513107\pi
0.844708 + 0.535227i 0.179774π0.179774\pi
684684 6.92820i 0.264906i
685685 0 0
686686 0 0
687687 24.0000 + 13.8564i 0.915657 + 0.528655i
688688 −4.00000 −0.152499
689689 42.0000 10.3923i 1.60007 0.395915i
690690 0 0
691691 37.5000 + 21.6506i 1.42657 + 0.823629i 0.996848 0.0793336i 0.0252792π-0.0252792\pi
0.429719 + 0.902963i 0.358613π0.358613\pi
692692 6.00000 10.3923i 0.228086 0.395056i
693693 −3.00000 5.19615i −0.113961 0.197386i
694694 0 0
695695 15.0000 8.66025i 0.568982 0.328502i
696696 0 0
697697 0 0
698698 0 0
699699 −9.00000 + 15.5885i −0.340411 + 0.589610i
700700 −21.0000 12.1244i −0.793725 0.458258i
701701 −12.0000 −0.453234 −0.226617 0.973984i 0.572767π-0.572767\pi
−0.226617 + 0.973984i 0.572767π0.572767\pi
702702 0 0
703703 0 0
704704 −24.0000 13.8564i −0.904534 0.522233i
705705 −6.00000 + 10.3923i −0.225973 + 0.391397i
706706 0 0
707707 31.1769i 1.17253i
708708 6.00000 3.46410i 0.225494 0.130189i
709709 −16.5000 + 9.52628i −0.619671 + 0.357767i −0.776741 0.629821i 0.783128π-0.783128\pi
0.157070 + 0.987587i 0.449795π0.449795\pi
710710 0 0
711711 5.50000 + 9.52628i 0.206266 + 0.357263i
712712 0 0
713713 9.00000 + 5.19615i 0.337053 + 0.194597i
714714 0 0
715715 12.0000 41.5692i 0.448775 1.55460i
716716 24.0000 0.896922
717717 0 0
718718 0 0
719719 3.00000 + 5.19615i 0.111881 + 0.193784i 0.916529 0.399969i 0.130979π-0.130979\pi
−0.804648 + 0.593753i 0.797646π0.797646\pi
720720 13.8564i 0.516398i
721721 1.50000 0.866025i 0.0558629 0.0322525i
722722 0 0
723723 20.7846i 0.772988i
724724 −14.0000 24.2487i −0.520306 0.901196i
725725 21.0000 36.3731i 0.779920 1.35086i
726726 0 0
727727 −35.0000 −1.29808 −0.649039 0.760755i 0.724829π-0.724829\pi
−0.649039 + 0.760755i 0.724829π0.724829\pi
728728 0 0
729729 1.00000 0.0370370
730730 0 0
731731 0 0
732732 1.00000 + 1.73205i 0.0369611 + 0.0640184i
733733 39.8372i 1.47142i 0.677297 + 0.735710i 0.263151π0.263151\pi
−0.677297 + 0.735710i 0.736849π0.736849\pi
734734 0 0
735735 −12.0000 + 6.92820i −0.442627 + 0.255551i
736736 0 0
737737 −15.0000 25.9808i −0.552532 0.957014i
738738 0 0
739739 39.0000 + 22.5167i 1.43464 + 0.828289i 0.997470 0.0710909i 0.0226481π-0.0226481\pi
0.437168 + 0.899380i 0.355981π0.355981\pi
740740 0 0
741741 −3.00000 12.1244i −0.110208 0.445399i
742742 0 0
743743 −9.00000 5.19615i −0.330178 0.190628i 0.325742 0.945459i 0.394386π-0.394386\pi
−0.655920 + 0.754830i 0.727719π0.727719\pi
744744 0 0
745745 −12.0000 20.7846i −0.439646 0.761489i
746746 0 0
747747 −12.0000 + 6.92820i −0.439057 + 0.253490i
748748 0 0
749749 10.3923i 0.379727i
750750 0 0
751751 −4.00000 + 6.92820i −0.145962 + 0.252814i −0.929731 0.368238i 0.879961π-0.879961\pi
0.783769 + 0.621052i 0.213294π0.213294\pi
752752 −12.0000 6.92820i −0.437595 0.252646i
753753 −12.0000 −0.437304
754754 0 0
755755 12.0000 0.436725
756756 −3.00000 1.73205i −0.109109 0.0629941i
757757 −17.0000 + 29.4449i −0.617876 + 1.07019i 0.371997 + 0.928234i 0.378673π0.378673\pi
−0.989873 + 0.141958i 0.954660π0.954660\pi
758758 0 0
759759 20.7846i 0.754434i
760760 0 0
761761 −18.0000 + 10.3923i −0.652499 + 0.376721i −0.789413 0.613862i 0.789615π-0.789615\pi
0.136914 + 0.990583i 0.456282π0.456282\pi
762762 0 0
763763 13.5000 + 23.3827i 0.488733 + 0.846510i
764764 18.0000 31.1769i 0.651217 1.12794i
765765 0 0
766766 0 0
767767 −9.00000 + 8.66025i −0.324971 + 0.312704i
768768 −16.0000 −0.577350
769769 −6.00000 3.46410i −0.216366 0.124919i 0.387901 0.921701i 0.373200π-0.373200\pi
−0.604266 + 0.796782i 0.706534π0.706534\pi
770770 0 0
771771 0 0
772772 31.1769i 1.12208i
773773 −45.0000 + 25.9808i −1.61854 + 0.934463i −0.631239 + 0.775589i 0.717453π0.717453\pi
−0.987299 + 0.158874i 0.949213π0.949213\pi
774774 0 0
775775 12.1244i 0.435520i
776776 0 0
777777 0 0
778778 0 0
779779 −24.0000 −0.859889
780780 −6.00000 24.2487i −0.214834 0.868243i
781781 36.0000 1.28818
782782 0 0
783783 3.00000 5.19615i 0.107211 0.185695i
784784 −8.00000 13.8564i −0.285714 0.494872i
785785 38.1051i 1.36003i
786786 0 0
787787 28.5000 16.4545i 1.01592 0.586539i 0.102997 0.994682i 0.467157π-0.467157\pi
0.912918 + 0.408143i 0.133823π0.133823\pi
788788 27.7128i 0.987228i
789789 6.00000 + 10.3923i 0.213606 + 0.369976i
790790 0 0
791791 9.00000 + 5.19615i 0.320003 + 0.184754i
792792 0 0
793793 −2.50000 2.59808i −0.0887776 0.0922604i
794794 0 0
795795 36.0000 + 20.7846i 1.27679 + 0.737154i
796796 −7.00000 + 12.1244i −0.248108 + 0.429736i
797797 −6.00000 10.3923i −0.212531 0.368114i 0.739975 0.672634i 0.234837π-0.234837\pi
−0.952506 + 0.304520i 0.901504π0.901504\pi
798798 0 0
799799 0 0
800800 0 0
801801 6.92820i 0.244796i
802802 0 0
803803 3.00000 5.19615i 0.105868 0.183368i
804804 −15.0000 8.66025i −0.529009 0.305424i
805805 −36.0000 −1.26883
806806 0 0
807807 0 0
808808 0 0
809809 −6.00000 + 10.3923i −0.210949 + 0.365374i −0.952012 0.306062i 0.900989π-0.900989\pi
0.741063 + 0.671436i 0.234322π0.234322\pi
810810 0 0
811811 25.9808i 0.912308i −0.889901 0.456154i 0.849227π-0.849227\pi
0.889901 0.456154i 0.150773π-0.150773\pi
812812 −18.0000 + 10.3923i −0.631676 + 0.364698i
813813 −4.50000 + 2.59808i −0.157822 + 0.0911185i
814814 0 0
815815 −33.0000 57.1577i −1.15594 2.00215i
816816 0 0
817817 3.00000 + 1.73205i 0.104957 + 0.0605968i
818818 0 0
819819 6.00000 + 1.73205i 0.209657 + 0.0605228i
820820 −48.0000 −1.67623
821821 21.0000 + 12.1244i 0.732905 + 0.423143i 0.819484 0.573102i 0.194260π-0.194260\pi
−0.0865789 + 0.996245i 0.527593π0.527593\pi
822822 0 0
823823 −16.0000 27.7128i −0.557725 0.966008i −0.997686 0.0679910i 0.978341π-0.978341\pi
0.439961 0.898017i 0.354992π-0.354992\pi
824824 0 0
825825 21.0000 12.1244i 0.731126 0.422116i
826826 0 0
827827 48.4974i 1.68642i 0.537584 + 0.843210i 0.319337π0.319337\pi
−0.537584 + 0.843210i 0.680663π0.680663\pi
828828 6.00000 + 10.3923i 0.208514 + 0.361158i
829829 −15.5000 + 26.8468i −0.538337 + 0.932427i 0.460657 + 0.887578i 0.347614π0.347614\pi
−0.998994 + 0.0448490i 0.985719π0.985719\pi
830830 0 0
831831 10.0000 0.346896
832832 28.0000 6.92820i 0.970725 0.240192i
833833 0 0
834834 0 0
835835 12.0000 20.7846i 0.415277 0.719281i
836836 12.0000 + 20.7846i 0.415029 + 0.718851i
837837 1.73205i 0.0598684i
838838 0 0
839839 27.0000 15.5885i 0.932144 0.538173i 0.0446547 0.999002i 0.485781π-0.485781\pi
0.887489 + 0.460829i 0.152448π0.152448\pi
840840 0 0
841841 −3.50000 6.06218i −0.120690 0.209041i
842842 0 0
843843 −21.0000 12.1244i −0.723278 0.417585i
844844 26.0000 0.894957
845845 21.0000 + 39.8372i 0.722422 + 1.37044i
846846 0 0
847847 −1.50000 0.866025i −0.0515406 0.0297570i
848848 −24.0000 + 41.5692i −0.824163 + 1.42749i
849849 −5.50000 9.52628i −0.188760 0.326941i
850850 0 0
851851 0 0
852852 18.0000 10.3923i 0.616670 0.356034i
853853 25.9808i 0.889564i −0.895639 0.444782i 0.853281π-0.853281\pi
0.895639 0.444782i 0.146719π-0.146719\pi
854854 0 0
855855 6.00000 10.3923i 0.205196 0.355409i
856856 0 0
857857 18.0000 0.614868 0.307434 0.951569i 0.400530π-0.400530\pi
0.307434 + 0.951569i 0.400530π0.400530\pi
858858 0 0
859859 13.0000 0.443554 0.221777 0.975097i 0.428814π-0.428814\pi
0.221777 + 0.975097i 0.428814π0.428814\pi
860860 6.00000 + 3.46410i 0.204598 + 0.118125i
861861 6.00000 10.3923i 0.204479 0.354169i
862862 0 0
863863 17.3205i 0.589597i −0.955559 0.294798i 0.904747π-0.904747\pi
0.955559 0.294798i 0.0952525π-0.0952525\pi
864864 0 0
865865 −18.0000 + 10.3923i −0.612018 + 0.353349i
866866 0 0
867867 −8.50000 14.7224i −0.288675 0.500000i
868868 −3.00000 + 5.19615i −0.101827 + 0.176369i
869869 33.0000 + 19.0526i 1.11945 + 0.646314i
870870 0 0
871871 30.0000 + 8.66025i 1.01651 + 0.293442i
872872 0 0
873873 4.50000 + 2.59808i 0.152302 + 0.0879316i
874874 0 0
875875 6.00000 + 10.3923i 0.202837 + 0.351324i
876876 3.46410i 0.117041i
877877 36.0000 20.7846i 1.21563 0.701846i 0.251653 0.967818i 0.419026π-0.419026\pi
0.963981 + 0.265971i 0.0856926π0.0856926\pi
878878 0 0
879879 17.3205i 0.584206i
880880 24.0000 + 41.5692i 0.809040 + 1.40130i
881881 6.00000 10.3923i 0.202145 0.350126i −0.747074 0.664741i 0.768542π-0.768542\pi
0.949219 + 0.314615i 0.101875π0.101875\pi
882882 0 0
883883 −5.00000 −0.168263 −0.0841317 0.996455i 0.526812π-0.526812\pi
−0.0841317 + 0.996455i 0.526812π0.526812\pi
884884 0 0
885885 −12.0000 −0.403376
886886 0 0
887887 21.0000 36.3731i 0.705111 1.22129i −0.261540 0.965193i 0.584230π-0.584230\pi
0.966651 0.256096i 0.0824362π-0.0824362\pi
888888 0 0
889889 22.5167i 0.755185i
890890 0 0
891891 3.00000 1.73205i 0.100504 0.0580259i
892892 34.6410i 1.15987i
893893 6.00000 + 10.3923i 0.200782 + 0.347765i
894894 0 0
895895 −36.0000 20.7846i −1.20335 0.694753i
896896 0 0
897897 −15.0000 15.5885i −0.500835 0.520483i
898898 0 0
899899 −9.00000 5.19615i −0.300167 0.173301i
900900 7.00000 12.1244i 0.233333 0.404145i
901901 0 0
902902 0 0
903903 −1.50000 + 0.866025i −0.0499169 + 0.0288195i
904904 0 0
905905 48.4974i 1.61211i
906906 0 0
907907 22.0000 38.1051i 0.730498 1.26526i −0.226173 0.974087i 0.572621π-0.572621\pi
0.956671 0.291172i 0.0940453π-0.0940453\pi
908908 −36.0000 20.7846i −1.19470 0.689761i
909909 −18.0000 −0.597022
910910 0 0
911911 −12.0000 −0.397578 −0.198789 0.980042i 0.563701π-0.563701\pi
−0.198789 + 0.980042i 0.563701π0.563701\pi
912912 12.0000 + 6.92820i 0.397360 + 0.229416i
913913 −24.0000 + 41.5692i −0.794284 + 1.37574i
914914 0 0
915915 3.46410i 0.114520i
916916 48.0000 27.7128i 1.58596 0.915657i
917917 −9.00000 + 5.19615i −0.297206 + 0.171592i
918918 0 0
919919 16.0000 + 27.7128i 0.527791 + 0.914161i 0.999475 + 0.0323936i 0.0103130π0.0103130\pi
−0.471684 + 0.881768i 0.656354π0.656354\pi
920920 0 0
921921 −1.50000 0.866025i −0.0494267 0.0285365i
922922 0 0
923923 −27.0000 + 25.9808i −0.888716 + 0.855167i
924924 −12.0000 −0.394771
925925 0 0
926926 0 0
927927 0.500000 + 0.866025i 0.0164222 + 0.0284440i
928928 0 0
929929 −18.0000 + 10.3923i −0.590561 + 0.340960i −0.765319 0.643651i 0.777419π-0.777419\pi
0.174758 + 0.984611i 0.444086π0.444086\pi
930930 0 0
931931 13.8564i 0.454125i
932932 18.0000 + 31.1769i 0.589610 + 1.02123i
933933 −12.0000 + 20.7846i −0.392862 + 0.680458i
934934 0 0
935935 0 0
936936 0 0
937937 −14.0000 −0.457360 −0.228680 0.973502i 0.573441π-0.573441\pi
−0.228680 + 0.973502i 0.573441π0.573441\pi
938938 0 0
939939 6.50000 11.2583i 0.212119 0.367402i
940940 12.0000 + 20.7846i 0.391397 + 0.677919i
941941 10.3923i 0.338779i −0.985549 0.169390i 0.945820π-0.945820\pi
0.985549 0.169390i 0.0541797π-0.0541797\pi
942942 0 0
943943 −36.0000 + 20.7846i −1.17232 + 0.676840i
944944 13.8564i 0.450988i
945945 3.00000 + 5.19615i 0.0975900 + 0.169031i
946946 0 0
947947 −42.0000 24.2487i −1.36482 0.787977i −0.374556 0.927204i 0.622205π-0.622205\pi
−0.990260 + 0.139227i 0.955538π0.955538\pi
948948 22.0000 0.714527
949949 1.50000 + 6.06218i 0.0486921 + 0.196787i
950950 0 0
951951 6.00000 + 3.46410i 0.194563 + 0.112331i
952952 0 0
953953 −3.00000 5.19615i −0.0971795 0.168320i 0.813337 0.581793i 0.197649π-0.197649\pi
−0.910516 + 0.413473i 0.864315π0.864315\pi
954954 0 0
955955 −54.0000 + 31.1769i −1.74740 + 1.00886i
956956 0 0
957957 20.7846i 0.671871i
958958 0 0
959959 0 0
960960 24.0000 + 13.8564i 0.774597 + 0.447214i
961961 28.0000 0.903226
962962 0 0
963963 −6.00000 −0.193347
964964 −36.0000 20.7846i −1.15948 0.669427i
965965 27.0000 46.7654i 0.869161 1.50543i
966966 0 0
967967 24.2487i 0.779786i −0.920860 0.389893i 0.872512π-0.872512\pi
0.920860 0.389893i 0.127488π-0.127488\pi
968968 0 0
969969 0 0
970970 0 0
971971 −18.0000 31.1769i −0.577647 1.00051i −0.995748 0.0921142i 0.970638π-0.970638\pi
0.418101 0.908401i 0.362696π-0.362696\pi
972972 1.00000 1.73205i 0.0320750 0.0555556i
973973 7.50000 + 4.33013i 0.240439 + 0.138817i
974974 0 0
975975 −7.00000 + 24.2487i −0.224179 + 0.776580i
976976 4.00000 0.128037
977977 −24.0000 13.8564i −0.767828 0.443306i 0.0642712 0.997932i 0.479528π-0.479528\pi
−0.832099 + 0.554627i 0.812861π0.812861\pi
978978 0 0
979979 −12.0000 20.7846i −0.383522 0.664279i
980980 27.7128i 0.885253i
981981 −13.5000 + 7.79423i −0.431022 + 0.248851i
982982 0 0
983983 10.3923i 0.331463i −0.986171 0.165732i 0.947001π-0.947001\pi
0.986171 0.165732i 0.0529985π-0.0529985\pi
984984 0 0
985985 −24.0000 + 41.5692i −0.764704 + 1.32451i
986986 0 0
987987 −6.00000 −0.190982
988988 −24.0000 6.92820i −0.763542 0.220416i
989989 6.00000 0.190789
990990 0 0
991991 8.00000 13.8564i 0.254128 0.440163i −0.710530 0.703667i 0.751545π-0.751545\pi
0.964658 + 0.263504i 0.0848781π0.0848781\pi
992992 0 0
993993 5.19615i 0.164895i
994994 0 0
995995 21.0000 12.1244i 0.665745 0.384368i
996996 27.7128i 0.878114i
997997 −17.5000 30.3109i −0.554231 0.959955i −0.997963 0.0637961i 0.979679π-0.979679\pi
0.443732 0.896159i 0.353654π-0.353654\pi
998998 0 0
999999 0 0
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 39.2.j.a.4.1 2
3.2 odd 2 117.2.q.a.82.1 2
4.3 odd 2 624.2.bv.b.433.1 2
5.2 odd 4 975.2.w.d.199.1 4
5.3 odd 4 975.2.w.d.199.2 4
5.4 even 2 975.2.bc.c.901.1 2
12.11 even 2 1872.2.by.f.433.1 2
13.2 odd 12 507.2.e.f.484.2 4
13.3 even 3 507.2.j.b.361.1 2
13.4 even 6 507.2.b.c.337.1 2
13.5 odd 4 507.2.e.f.22.2 4
13.6 odd 12 507.2.a.e.1.2 2
13.7 odd 12 507.2.a.e.1.1 2
13.8 odd 4 507.2.e.f.22.1 4
13.9 even 3 507.2.b.c.337.2 2
13.10 even 6 inner 39.2.j.a.10.1 yes 2
13.11 odd 12 507.2.e.f.484.1 4
13.12 even 2 507.2.j.b.316.1 2
39.17 odd 6 1521.2.b.f.1351.2 2
39.20 even 12 1521.2.a.h.1.2 2
39.23 odd 6 117.2.q.a.10.1 2
39.32 even 12 1521.2.a.h.1.1 2
39.35 odd 6 1521.2.b.f.1351.1 2
52.7 even 12 8112.2.a.bu.1.1 2
52.19 even 12 8112.2.a.bu.1.2 2
52.23 odd 6 624.2.bv.b.49.1 2
65.23 odd 12 975.2.w.d.49.1 4
65.49 even 6 975.2.bc.c.751.1 2
65.62 odd 12 975.2.w.d.49.2 4
156.23 even 6 1872.2.by.f.1297.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.j.a.4.1 2 1.1 even 1 trivial
39.2.j.a.10.1 yes 2 13.10 even 6 inner
117.2.q.a.10.1 2 39.23 odd 6
117.2.q.a.82.1 2 3.2 odd 2
507.2.a.e.1.1 2 13.7 odd 12
507.2.a.e.1.2 2 13.6 odd 12
507.2.b.c.337.1 2 13.4 even 6
507.2.b.c.337.2 2 13.9 even 3
507.2.e.f.22.1 4 13.8 odd 4
507.2.e.f.22.2 4 13.5 odd 4
507.2.e.f.484.1 4 13.11 odd 12
507.2.e.f.484.2 4 13.2 odd 12
507.2.j.b.316.1 2 13.12 even 2
507.2.j.b.361.1 2 13.3 even 3
624.2.bv.b.49.1 2 52.23 odd 6
624.2.bv.b.433.1 2 4.3 odd 2
975.2.w.d.49.1 4 65.23 odd 12
975.2.w.d.49.2 4 65.62 odd 12
975.2.w.d.199.1 4 5.2 odd 4
975.2.w.d.199.2 4 5.3 odd 4
975.2.bc.c.751.1 2 65.49 even 6
975.2.bc.c.901.1 2 5.4 even 2
1521.2.a.h.1.1 2 39.32 even 12
1521.2.a.h.1.2 2 39.20 even 12
1521.2.b.f.1351.1 2 39.35 odd 6
1521.2.b.f.1351.2 2 39.17 odd 6
1872.2.by.f.433.1 2 12.11 even 2
1872.2.by.f.1297.1 2 156.23 even 6
8112.2.a.bu.1.1 2 52.7 even 12
8112.2.a.bu.1.2 2 52.19 even 12