Properties

Label 252.2.j.a.85.1
Level 252252
Weight 22
Character 252.85
Analytic conductor 2.0122.012
Analytic rank 00
Dimension 66
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,2,Mod(85,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.85");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 252=22327 252 = 2^{2} \cdot 3^{2} \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 252.j (of order 33, 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: 2.012230130942.01223013094
Analytic rank: 00
Dimension: 66
Relative dimension: 33 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x63x5+10x415x3+19x212x+3 x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

Embedding invariants

Embedding label 85.1
Root 0.500000+1.41036i0.500000 + 1.41036i of defining polynomial
Character χ\chi == 252.85
Dual form 252.2.j.a.169.1

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+(1.710530.272169i)q3+(0.119562+0.207087i)q5+(0.500000+0.866025i)q7+(2.85185+0.931107i)q9+(2.56238+4.43818i)q11+(2.442824.23109i)q13+(0.2608770.321688i)q15+3.70370q17+3.66019q19+(0.6195621.61745i)q21+(3.71053+6.42683i)q23+(2.47141+4.28061i)q25+(4.624762.36887i)q27+(1.732293.00041i)q29+(0.3586850.621261i)q31+(3.175118.28905i)q330.239123q35+4.60301q37+(5.33009+6.57256i)q39+(2.80150+4.85235i)q41+(6.2443310.8155i)q43+(0.533792+0.479256i)q45+(2.169913.75839i)q47+(0.500000+0.866025i)q49+(6.335301.00803i)q510.942820q531.22545q55+(6.260880.996189i)q57+(3.78947+6.56355i)q59+(2.754044.77014i)q61+(0.619562+2.93533i)q63+(0.584135+1.01175i)q65+(0.3300950.571741i)q67+(8.096179.98342i)q69+13.7414q713.66019q73+(3.062387.99476i)q75+(2.56238+4.43818i)q77+(3.11273+5.39140i)q79+(7.26608+5.31075i)q81+(4.851858.40365i)q83+(0.442820+0.766987i)q85+(2.14652+5.60377i)q877.48865q89+4.88564q91+(0.782630+0.965064i)q93+(0.437618+0.757977i)q95+(8.57442+14.8513i)q97+(3.17511+15.0429i)q99+O(q100)q+(-1.71053 - 0.272169i) q^{3} +(-0.119562 + 0.207087i) q^{5} +(0.500000 + 0.866025i) q^{7} +(2.85185 + 0.931107i) q^{9} +(2.56238 + 4.43818i) q^{11} +(2.44282 - 4.23109i) q^{13} +(0.260877 - 0.321688i) q^{15} +3.70370 q^{17} +3.66019 q^{19} +(-0.619562 - 1.61745i) q^{21} +(-3.71053 + 6.42683i) q^{23} +(2.47141 + 4.28061i) q^{25} +(-4.62476 - 2.36887i) q^{27} +(-1.73229 - 3.00041i) q^{29} +(0.358685 - 0.621261i) q^{31} +(-3.17511 - 8.28905i) q^{33} -0.239123 q^{35} +4.60301 q^{37} +(-5.33009 + 6.57256i) q^{39} +(-2.80150 + 4.85235i) q^{41} +(-6.24433 - 10.8155i) q^{43} +(-0.533792 + 0.479256i) q^{45} +(-2.16991 - 3.75839i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-6.33530 - 1.00803i) q^{51} -0.942820 q^{53} -1.22545 q^{55} +(-6.26088 - 0.996189i) q^{57} +(-3.78947 + 6.56355i) q^{59} +(-2.75404 - 4.77014i) q^{61} +(0.619562 + 2.93533i) q^{63} +(0.584135 + 1.01175i) q^{65} +(0.330095 - 0.571741i) q^{67} +(8.09617 - 9.98342i) q^{69} +13.7414 q^{71} -3.66019 q^{73} +(-3.06238 - 7.99476i) q^{75} +(-2.56238 + 4.43818i) q^{77} +(3.11273 + 5.39140i) q^{79} +(7.26608 + 5.31075i) q^{81} +(-4.85185 - 8.40365i) q^{83} +(-0.442820 + 0.766987i) q^{85} +(2.14652 + 5.60377i) q^{87} -7.48865 q^{89} +4.88564 q^{91} +(-0.782630 + 0.965064i) q^{93} +(-0.437618 + 0.757977i) q^{95} +(8.57442 + 14.8513i) q^{97} +(3.17511 + 15.0429i) q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q2q3q5+3q7+8q92q113q13+q15+4q17+6q194q2114q23+6q25+7q27q29+3q31+8q332q356q3724q39+8q99+O(q100) 6 q - 2 q^{3} - q^{5} + 3 q^{7} + 8 q^{9} - 2 q^{11} - 3 q^{13} + q^{15} + 4 q^{17} + 6 q^{19} - 4 q^{21} - 14 q^{23} + 6 q^{25} + 7 q^{27} - q^{29} + 3 q^{31} + 8 q^{33} - 2 q^{35} - 6 q^{37} - 24 q^{39}+ \cdots - 8 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/252Z)×\left(\mathbb{Z}/252\mathbb{Z}\right)^\times.

nn 2929 7373 127127
χ(n)\chi(n) 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 −1.71053 0.272169i −0.987577 0.157137i
44 0 0
55 −0.119562 + 0.207087i −0.0534696 + 0.0926120i −0.891521 0.452979i 0.850361π-0.850361\pi
0.838052 + 0.545591i 0.183695π0.183695\pi
66 0 0
77 0.500000 + 0.866025i 0.188982 + 0.327327i
88 0 0
99 2.85185 + 0.931107i 0.950616 + 0.310369i
1010 0 0
1111 2.56238 + 4.43818i 0.772587 + 1.33816i 0.936141 + 0.351626i 0.114371π0.114371\pi
−0.163554 + 0.986534i 0.552296π0.552296\pi
1212 0 0
1313 2.44282 4.23109i 0.677516 1.17349i −0.298210 0.954500i 0.596390π-0.596390\pi
0.975727 0.218993i 0.0702770π-0.0702770\pi
1414 0 0
1515 0.260877 0.321688i 0.0673581 0.0830595i
1616 0 0
1717 3.70370 0.898278 0.449139 0.893462i 0.351731π-0.351731\pi
0.449139 + 0.893462i 0.351731π0.351731\pi
1818 0 0
1919 3.66019 0.839705 0.419853 0.907592i 0.362082π-0.362082\pi
0.419853 + 0.907592i 0.362082π0.362082\pi
2020 0 0
2121 −0.619562 1.61745i −0.135199 0.352956i
2222 0 0
2323 −3.71053 + 6.42683i −0.773700 + 1.34009i 0.161823 + 0.986820i 0.448263π0.448263\pi
−0.935522 + 0.353267i 0.885071π0.885071\pi
2424 0 0
2525 2.47141 + 4.28061i 0.494282 + 0.856122i
2626 0 0
2727 −4.62476 2.36887i −0.890036 0.455890i
2828 0 0
2929 −1.73229 3.00041i −0.321678 0.557162i 0.659157 0.752006i 0.270913π-0.270913\pi
−0.980834 + 0.194844i 0.937580π0.937580\pi
3030 0 0
3131 0.358685 0.621261i 0.0644217 0.111582i −0.832016 0.554752i 0.812813π-0.812813\pi
0.896437 + 0.443171i 0.146146π0.146146\pi
3232 0 0
3333 −3.17511 8.28905i −0.552715 1.44294i
3434 0 0
3535 −0.239123 −0.0404192
3636 0 0
3737 4.60301 0.756730 0.378365 0.925656i 0.376486π-0.376486\pi
0.378365 + 0.925656i 0.376486π0.376486\pi
3838 0 0
3939 −5.33009 + 6.57256i −0.853498 + 1.05245i
4040 0 0
4141 −2.80150 + 4.85235i −0.437522 + 0.757810i −0.997498 0.0706992i 0.977477π-0.977477\pi
0.559976 + 0.828509i 0.310810π0.310810\pi
4242 0 0
4343 −6.24433 10.8155i −0.952251 1.64935i −0.740538 0.672015i 0.765429π-0.765429\pi
−0.211713 0.977332i 0.567904π-0.567904\pi
4444 0 0
4545 −0.533792 + 0.479256i −0.0795730 + 0.0714432i
4646 0 0
4747 −2.16991 3.75839i −0.316513 0.548217i 0.663245 0.748403i 0.269179π-0.269179\pi
−0.979758 + 0.200186i 0.935845π0.935845\pi
4848 0 0
4949 −0.500000 + 0.866025i −0.0714286 + 0.123718i
5050 0 0
5151 −6.33530 1.00803i −0.887119 0.141152i
5252 0 0
5353 −0.942820 −0.129506 −0.0647531 0.997901i 0.520626π-0.520626\pi
−0.0647531 + 0.997901i 0.520626π0.520626\pi
5454 0 0
5555 −1.22545 −0.165240
5656 0 0
5757 −6.26088 0.996189i −0.829273 0.131948i
5858 0 0
5959 −3.78947 + 6.56355i −0.493347 + 0.854501i −0.999971 0.00766579i 0.997560π-0.997560\pi
0.506624 + 0.862167i 0.330893π0.330893\pi
6060 0 0
6161 −2.75404 4.77014i −0.352619 0.610754i 0.634089 0.773260i 0.281375π-0.281375\pi
−0.986707 + 0.162507i 0.948042π0.948042\pi
6262 0 0
6363 0.619562 + 2.93533i 0.0780574 + 0.369816i
6464 0 0
6565 0.584135 + 1.01175i 0.0724530 + 0.125492i
6666 0 0
6767 0.330095 0.571741i 0.0403275 0.0698493i −0.845157 0.534518i 0.820493π-0.820493\pi
0.885485 + 0.464669i 0.153827π0.153827\pi
6868 0 0
6969 8.09617 9.98342i 0.974665 1.20186i
7070 0 0
7171 13.7414 1.63081 0.815405 0.578891i 0.196514π-0.196514\pi
0.815405 + 0.578891i 0.196514π0.196514\pi
7272 0 0
7373 −3.66019 −0.428393 −0.214196 0.976791i 0.568713π-0.568713\pi
−0.214196 + 0.976791i 0.568713π0.568713\pi
7474 0 0
7575 −3.06238 7.99476i −0.353613 0.923156i
7676 0 0
7777 −2.56238 + 4.43818i −0.292010 + 0.505777i
7878 0 0
7979 3.11273 + 5.39140i 0.350209 + 0.606580i 0.986286 0.165046i 0.0527772π-0.0527772\pi
−0.636077 + 0.771626i 0.719444π0.719444\pi
8080 0 0
8181 7.26608 + 5.31075i 0.807342 + 0.590084i
8282 0 0
8383 −4.85185 8.40365i −0.532560 0.922420i −0.999277 0.0380138i 0.987897π-0.987897\pi
0.466718 0.884406i 0.345436π-0.345436\pi
8484 0 0
8585 −0.442820 + 0.766987i −0.0480306 + 0.0831914i
8686 0 0
8787 2.14652 + 5.60377i 0.230131 + 0.600788i
8888 0 0
8989 −7.48865 −0.793795 −0.396898 0.917863i 0.629913π-0.629913\pi
−0.396898 + 0.917863i 0.629913π0.629913\pi
9090 0 0
9191 4.88564 0.512154
9292 0 0
9393 −0.782630 + 0.965064i −0.0811550 + 0.100072i
9494 0 0
9595 −0.437618 + 0.757977i −0.0448987 + 0.0777668i
9696 0 0
9797 8.57442 + 14.8513i 0.870600 + 1.50792i 0.861377 + 0.507967i 0.169603π0.169603\pi
0.00922376 + 0.999957i 0.497064π0.497064\pi
9898 0 0
9999 3.17511 + 15.0429i 0.319110 + 1.51186i
100100 0 0
101101 −3.59097 6.21975i −0.357315 0.618888i 0.630196 0.776436i 0.282974π-0.282974\pi
−0.987511 + 0.157548i 0.949641π0.949641\pi
102102 0 0
103103 6.41423 11.1098i 0.632013 1.09468i −0.355127 0.934818i 0.615562π-0.615562\pi
0.987140 0.159860i 0.0511044π-0.0511044\pi
104104 0 0
105105 0.409028 + 0.0650819i 0.0399171 + 0.00635134i
106106 0 0
107107 7.56526 0.731361 0.365681 0.930740i 0.380836π-0.380836\pi
0.365681 + 0.930740i 0.380836π0.380836\pi
108108 0 0
109109 −6.98057 −0.668617 −0.334309 0.942464i 0.608503π-0.608503\pi
−0.334309 + 0.942464i 0.608503π0.608503\pi
110110 0 0
111111 −7.87360 1.25280i −0.747329 0.118910i
112112 0 0
113113 9.78495 16.9480i 0.920491 1.59434i 0.121834 0.992550i 0.461122π-0.461122\pi
0.798657 0.601787i 0.205544π-0.205544\pi
114114 0 0
115115 −0.887275 1.53681i −0.0827388 0.143308i
116116 0 0
117117 10.9061 9.79190i 1.00827 0.905261i
118118 0 0
119119 1.85185 + 3.20750i 0.169759 + 0.294031i
120120 0 0
121121 −7.63160 + 13.2183i −0.693782 + 1.20167i
122122 0 0
123123 6.11273 7.53762i 0.551166 0.679645i
124124 0 0
125125 −2.37756 −0.212655
126126 0 0
127127 −16.8090 −1.49156 −0.745780 0.666192i 0.767923π-0.767923\pi
−0.745780 + 0.666192i 0.767923π0.767923\pi
128128 0 0
129129 7.73749 + 20.1998i 0.681248 + 1.77849i
130130 0 0
131131 −2.44966 + 4.24293i −0.214027 + 0.370706i −0.952971 0.303061i 0.901992π-0.901992\pi
0.738944 + 0.673767i 0.235325π0.235325\pi
132132 0 0
133133 1.83009 + 3.16982i 0.158689 + 0.274858i
134134 0 0
135135 1.04351 0.674501i 0.0898108 0.0580518i
136136 0 0
137137 −2.72257 4.71563i −0.232605 0.402884i 0.725969 0.687727i 0.241392π-0.241392\pi
−0.958574 + 0.284844i 0.908058π0.908058\pi
138138 0 0
139139 −2.83009 + 4.90187i −0.240046 + 0.415771i −0.960727 0.277495i 0.910496π-0.910496\pi
0.720681 + 0.693266i 0.243829π0.243829\pi
140140 0 0
141141 2.68878 + 7.01942i 0.226436 + 0.591142i
142142 0 0
143143 25.0377 2.09376
144144 0 0
145145 0.828460 0.0687999
146146 0 0
147147 1.09097 1.34528i 0.0899818 0.110957i
148148 0 0
149149 −1.14132 + 1.97682i −0.0935002 + 0.161947i −0.908982 0.416836i 0.863139π-0.863139\pi
0.815481 + 0.578783i 0.196472π0.196472\pi
150150 0 0
151151 −5.63160 9.75422i −0.458293 0.793787i 0.540578 0.841294i 0.318206π-0.318206\pi
−0.998871 + 0.0475071i 0.984872π0.984872\pi
152152 0 0
153153 10.5624 + 3.44854i 0.853918 + 0.278798i
154154 0 0
155155 0.0857699 + 0.148558i 0.00688921 + 0.0119325i
156156 0 0
157157 −2.77292 + 4.80283i −0.221303 + 0.383308i −0.955204 0.295949i 0.904364π-0.904364\pi
0.733901 + 0.679256i 0.237698π0.237698\pi
158158 0 0
159159 1.61273 + 0.256606i 0.127897 + 0.0203502i
160160 0 0
161161 −7.42107 −0.584862
162162 0 0
163163 6.66019 0.521666 0.260833 0.965384i 0.416003π-0.416003\pi
0.260833 + 0.965384i 0.416003π0.416003\pi
164164 0 0
165165 2.09617 + 0.333529i 0.163187 + 0.0259652i
166166 0 0
167167 −2.20370 + 3.81691i −0.170527 + 0.295362i −0.938604 0.344996i 0.887880π-0.887880\pi
0.768077 + 0.640357i 0.221214π0.221214\pi
168168 0 0
169169 −5.43474 9.41325i −0.418057 0.724096i
170170 0 0
171171 10.4383 + 3.40803i 0.798237 + 0.260619i
172172 0 0
173173 −12.6654 21.9371i −0.962932 1.66785i −0.715072 0.699051i 0.753606π-0.753606\pi
−0.247860 0.968796i 0.579727π-0.579727\pi
174174 0 0
175175 −2.47141 + 4.28061i −0.186821 + 0.323584i
176176 0 0
177177 8.26840 10.1958i 0.621491 0.766363i
178178 0 0
179179 9.54583 0.713489 0.356744 0.934202i 0.383887π-0.383887\pi
0.356744 + 0.934202i 0.383887π0.383887\pi
180180 0 0
181181 12.3743 0.919774 0.459887 0.887978i 0.347890π-0.347890\pi
0.459887 + 0.887978i 0.347890π0.347890\pi
182182 0 0
183183 3.41260 + 8.90904i 0.252266 + 0.658575i
184184 0 0
185185 −0.550343 + 0.953223i −0.0404621 + 0.0700823i
186186 0 0
187187 9.49028 + 16.4377i 0.693998 + 1.20204i
188188 0 0
189189 −0.260877 5.18960i −0.0189760 0.377488i
190190 0 0
191191 −6.58414 11.4041i −0.476411 0.825169i 0.523223 0.852196i 0.324729π-0.324729\pi
−0.999635 + 0.0270270i 0.991396π0.991396\pi
192192 0 0
193193 −5.57442 + 9.65518i −0.401256 + 0.694995i −0.993878 0.110486i 0.964759π-0.964759\pi
0.592622 + 0.805481i 0.298093π0.298093\pi
194194 0 0
195195 −0.723815 1.88962i −0.0518335 0.135318i
196196 0 0
197197 −0.144194 −0.0102734 −0.00513669 0.999987i 0.501635π-0.501635\pi
−0.00513669 + 0.999987i 0.501635π0.501635\pi
198198 0 0
199199 −19.4692 −1.38014 −0.690068 0.723744i 0.742419π-0.742419\pi
−0.690068 + 0.723744i 0.742419π0.742419\pi
200200 0 0
201201 −0.720248 + 0.888141i −0.0508024 + 0.0626446i
202202 0 0
203203 1.73229 3.00041i 0.121583 0.210587i
204204 0 0
205205 −0.669905 1.16031i −0.0467882 0.0810395i
206206 0 0
207207 −16.5659 + 14.8734i −1.15141 + 1.03378i
208208 0 0
209209 9.37880 + 16.2446i 0.648745 + 1.12366i
210210 0 0
211211 1.61436 2.79615i 0.111137 0.192495i −0.805092 0.593150i 0.797884π-0.797884\pi
0.916229 + 0.400655i 0.131217π0.131217\pi
212212 0 0
213213 −23.5052 3.73999i −1.61055 0.256260i
214214 0 0
215215 2.98633 0.203666
216216 0 0
217217 0.717370 0.0486982
218218 0 0
219219 6.26088 + 0.996189i 0.423071 + 0.0673162i
220220 0 0
221221 9.04746 15.6707i 0.608598 1.05412i
222222 0 0
223223 −10.3856 17.9885i −0.695474 1.20460i −0.970021 0.243022i 0.921861π-0.921861\pi
0.274547 0.961574i 0.411472π-0.411472\pi
224224 0 0
225225 3.06238 + 14.5088i 0.204159 + 0.967253i
226226 0 0
227227 −10.9714 19.0030i −0.728198 1.26128i −0.957644 0.287955i 0.907025π-0.907025\pi
0.229446 0.973321i 0.426309π-0.426309\pi
228228 0 0
229229 −11.3856 + 19.7205i −0.752384 + 1.30317i 0.194280 + 0.980946i 0.437763π0.437763\pi
−0.946664 + 0.322222i 0.895570π0.895570\pi
230230 0 0
231231 5.59097 6.89425i 0.367859 0.453608i
232232 0 0
233233 −25.7817 −1.68901 −0.844507 0.535544i 0.820106π-0.820106\pi
−0.844507 + 0.535544i 0.820106π0.820106\pi
234234 0 0
235235 1.03775 0.0676953
236236 0 0
237237 −3.85705 10.0694i −0.250542 0.654075i
238238 0 0
239239 13.6488 23.6405i 0.882870 1.52918i 0.0347345 0.999397i 0.488941π-0.488941\pi
0.848136 0.529779i 0.177725π-0.177725\pi
240240 0 0
241241 −5.01724 8.69011i −0.323189 0.559779i 0.657955 0.753057i 0.271422π-0.271422\pi
−0.981144 + 0.193277i 0.938088π0.938088\pi
242242 0 0
243243 −10.9834 11.0618i −0.704589 0.709616i
244244 0 0
245245 −0.119562 0.207087i −0.00763851 0.0132303i
246246 0 0
247247 8.94119 15.4866i 0.568914 0.985388i
248248 0 0
249249 6.01204 + 15.6952i 0.380997 + 0.994645i
250250 0 0
251251 −28.3171 −1.78736 −0.893680 0.448705i 0.851885π-0.851885\pi
−0.893680 + 0.448705i 0.851885π0.851885\pi
252252 0 0
253253 −38.0312 −2.39100
254254 0 0
255255 0.966208 1.19143i 0.0605063 0.0746105i
256256 0 0
257257 −14.4315 + 24.9960i −0.900210 + 1.55921i −0.0729899 + 0.997333i 0.523254π0.523254\pi
−0.827221 + 0.561877i 0.810079π0.810079\pi
258258 0 0
259259 2.30150 + 3.98632i 0.143009 + 0.247698i
260260 0 0
261261 −2.14652 10.1697i −0.132866 0.629486i
262262 0 0
263263 0.604645 + 1.04728i 0.0372840 + 0.0645778i 0.884065 0.467363i 0.154796π-0.154796\pi
−0.846781 + 0.531941i 0.821463π0.821463\pi
264264 0 0
265265 0.112725 0.195246i 0.00692465 0.0119938i
266266 0 0
267267 12.8096 + 2.03818i 0.783934 + 0.124734i
268268 0 0
269269 9.01367 0.549573 0.274787 0.961505i 0.411393π-0.411393\pi
0.274787 + 0.961505i 0.411393π0.411393\pi
270270 0 0
271271 17.6030 1.06931 0.534653 0.845071i 0.320442π-0.320442\pi
0.534653 + 0.845071i 0.320442π0.320442\pi
272272 0 0
273273 −8.35705 1.32972i −0.505792 0.0804782i
274274 0 0
275275 −12.6654 + 21.9371i −0.763752 + 1.32286i
276276 0 0
277277 −0.727085 1.25935i −0.0436863 0.0756669i 0.843355 0.537356i 0.180577π-0.180577\pi
−0.887042 + 0.461689i 0.847244π0.847244\pi
278278 0 0
279279 1.60138 1.43777i 0.0958718 0.0860768i
280280 0 0
281281 10.1482 + 17.5771i 0.605388 + 1.04856i 0.991990 + 0.126316i 0.0403154π0.0403154\pi
−0.386602 + 0.922247i 0.626351π0.626351\pi
282282 0 0
283283 2.30150 3.98632i 0.136810 0.236962i −0.789477 0.613780i 0.789648π-0.789648\pi
0.926288 + 0.376817i 0.122982π0.122982\pi
284284 0 0
285285 0.954858 1.17744i 0.0565609 0.0697455i
286286 0 0
287287 −5.60301 −0.330735
288288 0 0
289289 −3.28263 −0.193096
290290 0 0
291291 −10.6248 27.7374i −0.622835 1.62599i
292292 0 0
293293 3.53667 6.12569i 0.206614 0.357867i −0.744031 0.668145i 0.767089π-0.767089\pi
0.950646 + 0.310278i 0.100422π0.100422\pi
294294 0 0
295295 −0.906150 1.56950i −0.0527581 0.0913797i
296296 0 0
297297 −1.33693 26.5955i −0.0775766 1.54323i
298298 0 0
299299 18.1283 + 31.3992i 1.04839 + 1.81586i
300300 0 0
301301 6.24433 10.8155i 0.359917 0.623394i
302302 0 0
303303 4.44966 + 11.6164i 0.255626 + 0.667347i
304304 0 0
305305 1.31711 0.0754175
306306 0 0
307307 15.7518 0.899006 0.449503 0.893279i 0.351601π-0.351601\pi
0.449503 + 0.893279i 0.351601π0.351601\pi
308308 0 0
309309 −13.9955 + 17.2579i −0.796175 + 0.981767i
310310 0 0
311311 9.81191 16.9947i 0.556382 0.963682i −0.441412 0.897304i 0.645522π-0.645522\pi
0.997795 0.0663780i 0.0211443π-0.0211443\pi
312312 0 0
313313 12.7427 + 22.0710i 0.720259 + 1.24753i 0.960896 + 0.276911i 0.0893106π0.0893106\pi
−0.240636 + 0.970615i 0.577356π0.577356\pi
314314 0 0
315315 −0.681943 0.222649i −0.0384232 0.0125449i
316316 0 0
317317 −4.14132 7.17297i −0.232599 0.402874i 0.725973 0.687723i 0.241390π-0.241390\pi
−0.958572 + 0.284849i 0.908056π0.908056\pi
318318 0 0
319319 8.87756 15.3764i 0.497048 0.860912i
320320 0 0
321321 −12.9406 2.05903i −0.722276 0.114924i
322322 0 0
323323 13.5562 0.754289
324324 0 0
325325 24.1488 1.33954
326326 0 0
327327 11.9405 + 1.89989i 0.660311 + 0.105064i
328328 0 0
329329 2.16991 3.75839i 0.119631 0.207207i
330330 0 0
331331 5.99028 + 10.3755i 0.329256 + 0.570288i 0.982364 0.186976i 0.0598688π-0.0598688\pi
−0.653109 + 0.757264i 0.726535π0.726535\pi
332332 0 0
333333 13.1271 + 4.28590i 0.719360 + 0.234866i
334334 0 0
335335 0.0789334 + 0.136717i 0.00431259 + 0.00746963i
336336 0 0
337337 6.46006 11.1892i 0.351902 0.609512i −0.634681 0.772774i 0.718868π-0.718868\pi
0.986583 + 0.163262i 0.0522017π0.0522017\pi
338338 0 0
339339 −21.3502 + 26.3270i −1.15958 + 1.42989i
340340 0 0
341341 3.67635 0.199086
342342 0 0
343343 −1.00000 −0.0539949
344344 0 0
345345 1.09944 + 2.87024i 0.0591920 + 0.154529i
346346 0 0
347347 −8.09329 + 14.0180i −0.434471 + 0.752526i −0.997252 0.0740802i 0.976398π-0.976398\pi
0.562781 + 0.826606i 0.309731π0.309731\pi
348348 0 0
349349 9.05718 + 15.6875i 0.484820 + 0.839732i 0.999848 0.0174409i 0.00555188π-0.00555188\pi
−0.515028 + 0.857173i 0.672219π0.672219\pi
350350 0 0
351351 −21.3204 + 13.7811i −1.13800 + 0.735578i
352352 0 0
353353 5.84897 + 10.1307i 0.311309 + 0.539203i 0.978646 0.205553i 0.0658992π-0.0658992\pi
−0.667337 + 0.744756i 0.732566π0.732566\pi
354354 0 0
355355 −1.64295 + 2.84567i −0.0871987 + 0.151033i
356356 0 0
357357 −2.29467 5.99054i −0.121447 0.317053i
358358 0 0
359359 35.7245 1.88547 0.942734 0.333547i 0.108245π-0.108245\pi
0.942734 + 0.333547i 0.108245π0.108245\pi
360360 0 0
361361 −5.60301 −0.294895
362362 0 0
363363 16.6517 20.5333i 0.873989 1.07772i
364364 0 0
365365 0.437618 0.757977i 0.0229060 0.0396743i
366366 0 0
367367 −8.52696 14.7691i −0.445103 0.770942i 0.552956 0.833210i 0.313500π-0.313500\pi
−0.998059 + 0.0622687i 0.980166π0.980166\pi
368368 0 0
369369 −12.5075 + 11.2297i −0.651116 + 0.584593i
370370 0 0
371371 −0.471410 0.816506i −0.0244744 0.0423909i
372372 0 0
373373 12.9617 22.4503i 0.671131 1.16243i −0.306453 0.951886i 0.599142π-0.599142\pi
0.977584 0.210547i 0.0675246π-0.0675246\pi
374374 0 0
375375 4.06690 + 0.647097i 0.210014 + 0.0334160i
376376 0 0
377377 −16.9267 −0.871767
378378 0 0
379379 26.8446 1.37892 0.689458 0.724326i 0.257849π-0.257849\pi
0.689458 + 0.724326i 0.257849π0.257849\pi
380380 0 0
381381 28.7524 + 4.57489i 1.47303 + 0.234379i
382382 0 0
383383 −12.4263 + 21.5229i −0.634953 + 1.09977i 0.351573 + 0.936161i 0.385647π0.385647\pi
−0.986525 + 0.163610i 0.947686π0.947686\pi
384384 0 0
385385 −0.612725 1.06127i −0.0312274 0.0540874i
386386 0 0
387387 −7.73749 36.6583i −0.393319 1.86344i
388388 0 0
389389 9.12640 + 15.8074i 0.462727 + 0.801466i 0.999096 0.0425174i 0.0135378π-0.0135378\pi
−0.536369 + 0.843984i 0.680204π0.680204\pi
390390 0 0
391391 −13.7427 + 23.8030i −0.694998 + 1.20377i
392392 0 0
393393 5.34501 6.59095i 0.269620 0.332470i
394394 0 0
395395 −1.48865 −0.0749021
396396 0 0
397397 12.3743 0.621048 0.310524 0.950566i 0.399496π-0.399496\pi
0.310524 + 0.950566i 0.399496π0.399496\pi
398398 0 0
399399 −2.26771 5.92017i −0.113528 0.296379i
400400 0 0
401401 5.48113 9.49359i 0.273714 0.474087i −0.696096 0.717949i 0.745081π-0.745081\pi
0.969810 + 0.243862i 0.0784144π0.0784144\pi
402402 0 0
403403 −1.75241 3.03526i −0.0872935 0.151197i
404404 0 0
405405 −1.96853 + 0.869747i −0.0978171 + 0.0432181i
406406 0 0
407407 11.7947 + 20.4290i 0.584640 + 1.01263i
408408 0 0
409409 6.66019 11.5358i 0.329325 0.570408i −0.653053 0.757312i 0.726512π-0.726512\pi
0.982378 + 0.186904i 0.0598454π0.0598454\pi
410410 0 0
411411 3.37360 + 8.80724i 0.166408 + 0.434429i
412412 0 0
413413 −7.57893 −0.372935
414414 0 0
415415 2.32038 0.113903
416416 0 0
417417 6.17511 7.61455i 0.302396 0.372886i
418418 0 0
419419 −2.35705 + 4.08253i −0.115149 + 0.199445i −0.917839 0.396952i 0.870068π-0.870068\pi
0.802690 + 0.596396i 0.203401π0.203401\pi
420420 0 0
421421 −9.65856 16.7291i −0.470729 0.815327i 0.528710 0.848802i 0.322676π-0.322676\pi
−0.999440 + 0.0334755i 0.989342π0.989342\pi
422422 0 0
423423 −2.68878 12.7388i −0.130733 0.619380i
424424 0 0
425425 9.15335 + 15.8541i 0.444003 + 0.769036i
426426 0 0
427427 2.75404 4.77014i 0.133277 0.230843i
428428 0 0
429429 −42.8279 6.81449i −2.06775 0.329007i
430430 0 0
431431 −30.2794 −1.45851 −0.729253 0.684244i 0.760132π-0.760132\pi
−0.729253 + 0.684244i 0.760132π0.760132\pi
432432 0 0
433433 34.2060 1.64384 0.821918 0.569606i 0.192904π-0.192904\pi
0.821918 + 0.569606i 0.192904π0.192904\pi
434434 0 0
435435 −1.41711 0.225481i −0.0679452 0.0108110i
436436 0 0
437437 −13.5813 + 23.5234i −0.649680 + 1.12528i
438438 0 0
439439 0.311220 + 0.539049i 0.0148537 + 0.0257274i 0.873357 0.487081i 0.161938π-0.161938\pi
−0.858503 + 0.512809i 0.828605π0.828605\pi
440440 0 0
441441 −2.23229 + 2.00422i −0.106299 + 0.0954390i
442442 0 0
443443 2.58934 + 4.48486i 0.123023 + 0.213082i 0.920958 0.389661i 0.127408π-0.127408\pi
−0.797935 + 0.602743i 0.794074π0.794074\pi
444444 0 0
445445 0.895355 1.55080i 0.0424439 0.0735150i
446446 0 0
447447 2.49028 3.07078i 0.117786 0.145243i
448448 0 0
449449 −10.4977 −0.495416 −0.247708 0.968835i 0.579677π-0.579677\pi
−0.247708 + 0.968835i 0.579677π0.579677\pi
450450 0 0
451451 −28.7141 −1.35209
452452 0 0
453453 6.97825 + 18.2177i 0.327867 + 0.855940i
454454 0 0
455455 −0.584135 + 1.01175i −0.0273847 + 0.0474317i
456456 0 0
457457 1.60464 + 2.77933i 0.0750621 + 0.130011i 0.901113 0.433584i 0.142751π-0.142751\pi
−0.826051 + 0.563595i 0.809418π0.809418\pi
458458 0 0
459459 −17.1287 8.77359i −0.799500 0.409516i
460460 0 0
461461 18.1150 + 31.3762i 0.843702 + 1.46133i 0.886744 + 0.462261i 0.152962π0.152962\pi
−0.0430418 + 0.999073i 0.513705π0.513705\pi
462462 0 0
463463 −14.5253 + 25.1586i −0.675049 + 1.16922i 0.301406 + 0.953496i 0.402544π0.402544\pi
−0.976455 + 0.215723i 0.930789π0.930789\pi
464464 0 0
465465 −0.106279 0.277457i −0.00492859 0.0128668i
466466 0 0
467467 −37.4933 −1.73498 −0.867491 0.497452i 0.834269π-0.834269\pi
−0.867491 + 0.497452i 0.834269π0.834269\pi
468468 0 0
469469 0.660190 0.0304847
470470 0 0
471471 6.05034 7.46070i 0.278785 0.343771i
472472 0 0
473473 32.0007 55.4268i 1.47139 2.54853i
474474 0 0
475475 9.04583 + 15.6678i 0.415051 + 0.718890i
476476 0 0
477477 −2.68878 0.877867i −0.123111 0.0401948i
478478 0 0
479479 −14.9549 25.9026i −0.683305 1.18352i −0.973966 0.226693i 0.927209π-0.927209\pi
0.290661 0.956826i 0.406125π-0.406125\pi
480480 0 0
481481 11.2443 19.4757i 0.512697 0.888017i
482482 0 0
483483 12.6940 + 2.01978i 0.577596 + 0.0919033i
484484 0 0
485485 −4.10069 −0.186203
486486 0 0
487487 21.2632 0.963528 0.481764 0.876301i 0.339996π-0.339996\pi
0.481764 + 0.876301i 0.339996π0.339996\pi
488488 0 0
489489 −11.3925 1.81270i −0.515186 0.0819729i
490490 0 0
491491 10.6985 18.5303i 0.482816 0.836262i −0.516989 0.855992i 0.672947π-0.672947\pi
0.999805 + 0.0197296i 0.00628054π0.00628054\pi
492492 0 0
493493 −6.41586 11.1126i −0.288956 0.500487i
494494 0 0
495495 −3.49480 1.14103i −0.157079 0.0512853i
496496 0 0
497497 6.87072 + 11.9004i 0.308194 + 0.533808i
498498 0 0
499499 −7.28263 + 12.6139i −0.326015 + 0.564675i −0.981717 0.190345i 0.939039π-0.939039\pi
0.655702 + 0.755020i 0.272373π0.272373\pi
500500 0 0
501501 4.80834 5.92918i 0.214821 0.264896i
502502 0 0
503503 2.92339 0.130348 0.0651738 0.997874i 0.479240π-0.479240\pi
0.0651738 + 0.997874i 0.479240π0.479240\pi
504504 0 0
505505 1.71737 0.0764220
506506 0 0
507507 6.73431 + 17.5808i 0.299081 + 0.780792i
508508 0 0
509509 9.62025 16.6628i 0.426410 0.738564i −0.570141 0.821547i 0.693111π-0.693111\pi
0.996551 + 0.0829830i 0.0264447π0.0264447\pi
510510 0 0
511511 −1.83009 3.16982i −0.0809586 0.140224i
512512 0 0
513513 −16.9275 8.67053i −0.747368 0.382813i
514514 0 0
515515 1.53379 + 2.65661i 0.0675869 + 0.117064i
516516 0 0
517517 11.1202 19.2608i 0.489068 0.847091i
518518 0 0
519519 15.6940 + 40.9713i 0.688889 + 1.79844i
520520 0 0
521521 27.7486 1.21569 0.607844 0.794057i 0.292035π-0.292035\pi
0.607844 + 0.794057i 0.292035π0.292035\pi
522522 0 0
523523 −2.73680 −0.119672 −0.0598360 0.998208i 0.519058π-0.519058\pi
−0.0598360 + 0.998208i 0.519058π0.519058\pi
524524 0 0
525525 5.39248 6.64948i 0.235347 0.290207i
526526 0 0
527527 1.32846 2.30096i 0.0578686 0.100231i
528528 0 0
529529 −16.0361 27.7754i −0.697222 1.20762i
530530 0 0
531531 −16.9184 + 15.1898i −0.734194 + 0.659183i
532532 0 0
533533 13.6871 + 23.7068i 0.592856 + 1.02686i
534534 0 0
535535 −0.904515 + 1.56667i −0.0391056 + 0.0677329i
536536 0 0
537537 −16.3285 2.59808i −0.704625 0.112115i
538538 0 0
539539 −5.12476 −0.220739
540540 0 0
541541 −11.9773 −0.514944 −0.257472 0.966286i 0.582890π-0.582890\pi
−0.257472 + 0.966286i 0.582890π0.582890\pi
542542 0 0
543543 −21.1666 3.36789i −0.908347 0.144530i
544544 0 0
545545 0.834608 1.44558i 0.0357507 0.0619220i
546546 0 0
547547 10.7346 + 18.5929i 0.458979 + 0.794975i 0.998907 0.0467363i 0.0148821π-0.0148821\pi
−0.539928 + 0.841711i 0.681549π0.681549\pi
548548 0 0
549549 −3.41260 16.1680i −0.145646 0.690034i
550550 0 0
551551 −6.34050 10.9821i −0.270114 0.467852i
552552 0 0
553553 −3.11273 + 5.39140i −0.132367 + 0.229266i
554554 0 0
555555 1.20082 1.48073i 0.0509719 0.0628536i
556556 0 0
557557 −31.8493 −1.34950 −0.674748 0.738048i 0.735748π-0.735748\pi
−0.674748 + 0.738048i 0.735748π0.735748\pi
558558 0 0
559559 −61.0150 −2.58066
560560 0 0
561561 −11.7596 30.7001i −0.496492 1.29616i
562562 0 0
563563 −17.7742 + 30.7857i −0.749091 + 1.29746i 0.199167 + 0.979966i 0.436176π0.436176\pi
−0.948259 + 0.317499i 0.897157π0.897157\pi
564564 0 0
565565 2.33981 + 4.05267i 0.0984366 + 0.170497i
566566 0 0
567567 −0.966208 + 8.94799i −0.0405769 + 0.375780i
568568 0 0
569569 −10.8743 18.8348i −0.455874 0.789597i 0.542864 0.839821i 0.317340π-0.317340\pi
−0.998738 + 0.0502237i 0.984007π0.984007\pi
570570 0 0
571571 4.79987 8.31362i 0.200868 0.347914i −0.747940 0.663766i 0.768957π-0.768957\pi
0.948808 + 0.315852i 0.102290π0.102290\pi
572572 0 0
573573 8.15856 + 21.2990i 0.340829 + 0.889779i
574574 0 0
575575 −36.6810 −1.52970
576576 0 0
577577 13.0183 0.541960 0.270980 0.962585i 0.412652π-0.412652\pi
0.270980 + 0.962585i 0.412652π0.412652\pi
578578 0 0
579579 12.1631 14.9983i 0.505480 0.623309i
580580 0 0
581581 4.85185 8.40365i 0.201289 0.348642i
582582 0 0
583583 −2.41586 4.18440i −0.100055 0.173300i
584584 0 0
585585 0.723815 + 3.42926i 0.0299261 + 0.141782i
586586 0 0
587587 −8.64364 14.9712i −0.356761 0.617928i 0.630657 0.776062i 0.282786π-0.282786\pi
−0.987418 + 0.158134i 0.949452π0.949452\pi
588588 0 0
589589 1.31285 2.27393i 0.0540952 0.0936957i
590590 0 0
591591 0.246648 + 0.0392450i 0.0101457 + 0.00161432i
592592 0 0
593593 12.4153 0.509836 0.254918 0.966963i 0.417952π-0.417952\pi
0.254918 + 0.966963i 0.417952π0.417952\pi
594594 0 0
595595 −0.885640 −0.0363077
596596 0 0
597597 33.3027 + 5.29891i 1.36299 + 0.216870i
598598 0 0
599599 3.94282 6.82916i 0.161099 0.279032i −0.774164 0.632985i 0.781829π-0.781829\pi
0.935263 + 0.353953i 0.115163π0.115163\pi
600600 0 0
601601 −11.1413 19.2973i −0.454464 0.787154i 0.544193 0.838960i 0.316836π-0.316836\pi
−0.998657 + 0.0518055i 0.983502π0.983502\pi
602602 0 0
603603 1.47373 1.32317i 0.0600151 0.0538835i
604604 0 0
605605 −1.82489 3.16081i −0.0741925 0.128505i
606606 0 0
607607 −11.0458 + 19.1319i −0.448336 + 0.776541i −0.998278 0.0586617i 0.981317π-0.981317\pi
0.549942 + 0.835203i 0.314650π0.314650\pi
608608 0 0
609609 −3.77975 + 4.66082i −0.153163 + 0.188866i
610610 0 0
611611 −21.2028 −0.857771
612612 0 0
613613 −29.5264 −1.19256 −0.596280 0.802777i 0.703355π-0.703355\pi
−0.596280 + 0.802777i 0.703355π0.703355\pi
614614 0 0
615615 0.830095 + 2.16708i 0.0334727 + 0.0873849i
616616 0 0
617617 5.01655 8.68892i 0.201959 0.349803i −0.747201 0.664598i 0.768603π-0.768603\pi
0.949159 + 0.314796i 0.101936π0.101936\pi
618618 0 0
619619 19.1283 + 33.1312i 0.768833 + 1.33166i 0.938196 + 0.346103i 0.112495π0.112495\pi
−0.169364 + 0.985554i 0.554171π0.554171\pi
620620 0 0
621621 32.3847 20.9328i 1.29955 0.840004i
622622 0 0
623623 −3.74433 6.48536i −0.150013 0.259831i
624624 0 0
625625 −12.0728 + 20.9107i −0.482911 + 0.836427i
626626 0 0
627627 −11.6215 30.3395i −0.464118 1.21164i
628628 0 0
629629 17.0482 0.679754
630630 0 0
631631 −23.0377 −0.917118 −0.458559 0.888664i 0.651634π-0.651634\pi
−0.458559 + 0.888664i 0.651634π0.651634\pi
632632 0 0
633633 −3.52244 + 4.34354i −0.140004 + 0.172640i
634634 0 0
635635 2.00972 3.48093i 0.0797531 0.138136i
636636 0 0
637637 2.44282 + 4.23109i 0.0967881 + 0.167642i
638638 0 0
639639 39.1885 + 12.7948i 1.55027 + 0.506153i
640640 0 0
641641 8.68646 + 15.0454i 0.343094 + 0.594257i 0.985006 0.172522i 0.0551916π-0.0551916\pi
−0.641911 + 0.766779i 0.721858π0.721858\pi
642642 0 0
643643 9.47949 16.4190i 0.373835 0.647501i −0.616317 0.787498i 0.711376π-0.711376\pi
0.990152 + 0.139997i 0.0447094π0.0447094\pi
644644 0 0
645645 −5.10821 0.812785i −0.201136 0.0320034i
646646 0 0
647647 19.0194 0.747731 0.373865 0.927483i 0.378032π-0.378032\pi
0.373865 + 0.927483i 0.378032π0.378032\pi
648648 0 0
649649 −38.8402 −1.52461
650650 0 0
651651 −1.22708 0.195246i −0.0480933 0.00765228i
652652 0 0
653653 3.59329 6.22377i 0.140616 0.243555i −0.787112 0.616810i 0.788425π-0.788425\pi
0.927729 + 0.373255i 0.121758π0.121758\pi
654654 0 0
655655 −0.585770 1.01458i −0.0228879 0.0396430i
656656 0 0
657657 −10.4383 3.40803i −0.407237 0.132960i
658658 0 0
659659 −12.7261 22.0423i −0.495740 0.858647i 0.504248 0.863559i 0.331770π-0.331770\pi
−0.999988 + 0.00491209i 0.998436π0.998436\pi
660660 0 0
661661 −4.14295 + 7.17580i −0.161142 + 0.279106i −0.935279 0.353912i 0.884851π-0.884851\pi
0.774136 + 0.633019i 0.218184π0.218184\pi
662662 0 0
663663 −19.7411 + 24.3428i −0.766679 + 0.945395i
664664 0 0
665665 −0.875237 −0.0339402
666666 0 0
667667 25.7108 0.995527
668668 0 0
669669 12.8691 + 33.5965i 0.497548 + 1.29892i
670670 0 0
671671 14.1138 24.4458i 0.544857 0.943721i
672672 0 0
673673 5.91586 + 10.2466i 0.228040 + 0.394977i 0.957227 0.289338i 0.0934350π-0.0934350\pi
−0.729187 + 0.684314i 0.760102π0.760102\pi
674674 0 0
675675 −1.28947 25.6513i −0.0496316 0.987317i
676676 0 0
677677 −6.80314 11.7834i −0.261466 0.452872i 0.705166 0.709042i 0.250873π-0.250873\pi
−0.966632 + 0.256170i 0.917539π0.917539\pi
678678 0 0
679679 −8.57442 + 14.8513i −0.329056 + 0.569942i
680680 0 0
681681 13.5949 + 35.4914i 0.520959 + 1.36003i
682682 0 0
683683 3.58142 0.137039 0.0685196 0.997650i 0.478172π-0.478172\pi
0.0685196 + 0.997650i 0.478172π0.478172\pi
684684 0 0
685685 1.30206 0.0497492
686686 0 0
687687 24.8428 30.6338i 0.947813 1.16875i
688688 0 0
689689 −2.30314 + 3.98916i −0.0877426 + 0.151975i
690690 0 0
691691 −5.85868 10.1475i −0.222875 0.386031i 0.732805 0.680439i 0.238211π-0.238211\pi
−0.955680 + 0.294408i 0.904877π0.904877\pi
692692 0 0
693693 −11.4399 + 10.2712i −0.434567 + 0.390169i
694694 0 0
695695 −0.676742 1.17215i −0.0256703 0.0444622i
696696 0 0
697697 −10.3759 + 17.9716i −0.393016 + 0.680724i
698698 0 0
699699 44.1004 + 7.01697i 1.66803 + 0.265406i
700700 0 0
701701 −10.5926 −0.400077 −0.200039 0.979788i 0.564107π-0.564107\pi
−0.200039 + 0.979788i 0.564107π0.564107\pi
702702 0 0
703703 16.8479 0.635430
704704 0 0
705705 −1.77511 0.282443i −0.0668543 0.0106374i
706706 0 0
707707 3.59097 6.21975i 0.135052 0.233918i
708708 0 0
709709 −19.1488 33.1668i −0.719150 1.24560i −0.961337 0.275374i 0.911198π-0.911198\pi
0.242187 0.970230i 0.422135π-0.422135\pi
710710 0 0
711711 3.85705 + 18.2737i 0.144651 + 0.685318i
712712 0 0
713713 2.66182 + 4.61042i 0.0996861 + 0.172661i
714714 0 0
715715 −2.99355 + 5.18499i −0.111953 + 0.193908i
716716 0 0
717717 −29.7810 + 36.7230i −1.11219 + 1.37145i
718718 0 0
719719 41.6752 1.55422 0.777112 0.629362i 0.216684π-0.216684\pi
0.777112 + 0.629362i 0.216684π0.216684\pi
720720 0 0
721721 12.8285 0.477757
722722 0 0
723723 6.21698 + 16.2303i 0.231212 + 0.603610i
724724 0 0
725725 8.56238 14.8305i 0.317999 0.550790i
726726 0 0
727727 16.4126 + 28.4274i 0.608709 + 1.05432i 0.991453 + 0.130461i 0.0416458π0.0416458\pi
−0.382744 + 0.923854i 0.625021π0.625021\pi
728728 0 0
729729 15.7769 + 21.9110i 0.584329 + 0.811517i
730730 0 0
731731 −23.1271 40.0573i −0.855386 1.48157i
732732 0 0
733733 −4.64884 + 8.05203i −0.171709 + 0.297408i −0.939017 0.343870i 0.888262π-0.888262\pi
0.767309 + 0.641278i 0.221595π0.221595\pi
734734 0 0
735735 0.148152 + 0.386770i 0.00546465 + 0.0142662i
736736 0 0
737737 3.38332 0.124626
738738 0 0
739739 11.3776 0.418530 0.209265 0.977859i 0.432893π-0.432893\pi
0.209265 + 0.977859i 0.432893π0.432893\pi
740740 0 0
741741 −19.5092 + 24.0568i −0.716687 + 0.883749i
742742 0 0
743743 1.16182 2.01234i 0.0426232 0.0738256i −0.843927 0.536458i 0.819762π-0.819762\pi
0.886550 + 0.462633i 0.153095π0.153095\pi
744744 0 0
745745 −0.272915 0.472703i −0.00999883 0.0173185i
746746 0 0
747747 −6.01204 28.4835i −0.219969 1.04216i
748748 0 0
749749 3.78263 + 6.55171i 0.138214 + 0.239394i
750750 0 0
751751 5.56690 9.64215i 0.203139 0.351847i −0.746399 0.665498i 0.768219π-0.768219\pi
0.949538 + 0.313652i 0.101552π0.101552\pi
752752 0 0
753753 48.4374 + 7.70703i 1.76516 + 0.280860i
754754 0 0
755755 2.69329 0.0980190
756756 0 0
757757 52.1639 1.89593 0.947964 0.318376i 0.103138π-0.103138\pi
0.947964 + 0.318376i 0.103138π0.103138\pi
758758 0 0
759759 65.0537 + 10.3509i 2.36130 + 0.375714i
760760 0 0
761761 23.5127 40.7252i 0.852336 1.47629i −0.0267592 0.999642i 0.508519π-0.508519\pi
0.879095 0.476647i 0.158148π-0.158148\pi
762762 0 0
763763 −3.49028 6.04535i −0.126357 0.218856i
764764 0 0
765765 −1.97700 + 1.77502i −0.0714787 + 0.0641759i
766766 0 0
767767 18.5140 + 32.0671i 0.668501 + 1.15788i
768768 0 0
769769 3.30314 5.72121i 0.119114 0.206312i −0.800303 0.599596i 0.795328π-0.795328\pi
0.919417 + 0.393284i 0.128661π0.128661\pi
770770 0 0
771771 31.4887 38.8288i 1.13404 1.39838i
772772 0 0
773773 19.0870 0.686512 0.343256 0.939242i 0.388470π-0.388470\pi
0.343256 + 0.939242i 0.388470π0.388470\pi
774774 0 0
775775 3.54583 0.127370
776776 0 0
777777 −2.85185 7.44514i −0.102309 0.267093i
778778 0 0
779779 −10.2540 + 17.7605i −0.367389 + 0.636337i
780780 0 0
781781 35.2108 + 60.9869i 1.25994 + 2.18228i
782782 0 0
783783 0.903827 + 17.9797i 0.0323001 + 0.642544i
784784 0 0
785785 −0.663069 1.14847i −0.0236659 0.0409906i
786786 0 0
787787 −25.4503 + 44.0813i −0.907207 + 1.57133i −0.0892796 + 0.996007i 0.528456π0.528456\pi
−0.817927 + 0.575322i 0.804877π0.804877\pi
788788 0 0
789789 −0.749229 1.95596i −0.0266733 0.0696342i
790790 0 0
791791 19.5699 0.695826
792792 0 0
793793 −26.9105 −0.955620
794794 0 0
795795 −0.245960 + 0.303294i −0.00872330 + 0.0107567i
796796 0 0
797797 −4.38727 + 7.59898i −0.155405 + 0.269170i −0.933207 0.359341i 0.883002π-0.883002\pi
0.777801 + 0.628510i 0.216335π0.216335\pi
798798 0 0
799799 −8.03667 13.9199i −0.284317 0.492451i
800800 0 0
801801 −21.3565 6.97274i −0.754595 0.246370i
802802 0 0
803803 −9.37880 16.2446i −0.330971 0.573258i
804804 0 0
805805 0.887275 1.53681i 0.0312723 0.0541653i
806806 0 0
807807 −15.4182 2.45324i −0.542746 0.0863581i
808808 0 0
809809 −9.51384 −0.334489 −0.167244 0.985915i 0.553487π-0.553487\pi
−0.167244 + 0.985915i 0.553487π0.553487\pi
810810 0 0
811811 −25.0118 −0.878282 −0.439141 0.898418i 0.644717π-0.644717\pi
−0.439141 + 0.898418i 0.644717π0.644717\pi
812812 0 0
813813 −30.1105 4.79099i −1.05602 0.168027i
814814 0 0
815815 −0.796303 + 1.37924i −0.0278933 + 0.0483126i
816816 0 0
817817 −22.8554 39.5867i −0.799610 1.38496i
818818 0 0
819819 13.9331 + 4.54906i 0.486862 + 0.158957i
820820 0 0
821821 17.7970 + 30.8253i 0.621119 + 1.07581i 0.989278 + 0.146047i 0.0466551π0.0466551\pi
−0.368158 + 0.929763i 0.620012π0.620012\pi
822822 0 0
823823 −8.00000 + 13.8564i −0.278862 + 0.483004i −0.971102 0.238664i 0.923291π-0.923291\pi
0.692240 + 0.721668i 0.256624π0.256624\pi
824824 0 0
825825 27.6352 34.0770i 0.962133 1.18641i
826826 0 0
827827 −25.4531 −0.885090 −0.442545 0.896746i 0.645924π-0.645924\pi
−0.442545 + 0.896746i 0.645924π0.645924\pi
828828 0 0
829829 −17.5458 −0.609392 −0.304696 0.952450i 0.598555π-0.598555\pi
−0.304696 + 0.952450i 0.598555π0.598555\pi
830830 0 0
831831 0.900948 + 2.35205i 0.0312535 + 0.0815916i
832832 0 0
833833 −1.85185 + 3.20750i −0.0641627 + 0.111133i
834834 0 0
835835 −0.526955 0.912713i −0.0182360 0.0315857i
836836 0 0
837837 −3.13052 + 2.02350i −0.108207 + 0.0699425i
838838 0 0
839839 −12.0562 20.8820i −0.416227 0.720927i 0.579329 0.815094i 0.303315π-0.303315\pi
−0.995556 + 0.0941668i 0.969981π0.969981\pi
840840 0 0
841841 8.49837 14.7196i 0.293047 0.507572i
842842 0 0
843843 −12.5748 32.8282i −0.433099 1.13066i
844844 0 0
845845 2.59915 0.0894133
846846 0 0
847847 −15.2632 −0.524450
848848 0 0
849849 −5.02175 + 6.19234i −0.172346 + 0.212521i
850850 0 0
851851 −17.0796 + 29.5828i −0.585482 + 1.01408i
852852 0 0
853853 −16.2616 28.1659i −0.556785 0.964381i −0.997762 0.0668621i 0.978701π-0.978701\pi
0.440977 0.897518i 0.354632π-0.354632\pi
854854 0 0
855855 −1.95378 + 1.75417i −0.0668178 + 0.0599912i
856856 0 0
857857 0.299870 + 0.519390i 0.0102434 + 0.0177420i 0.871102 0.491103i 0.163406π-0.163406\pi
−0.860858 + 0.508845i 0.830073π0.830073\pi
858858 0 0
859859 13.2174 22.8932i 0.450971 0.781104i −0.547476 0.836822i 0.684411π-0.684411\pi
0.998447 + 0.0557171i 0.0177445π0.0177445\pi
860860 0 0
861861 9.58414 + 1.52496i 0.326626 + 0.0519706i
862862 0 0
863863 19.8454 0.675545 0.337773 0.941228i 0.390326π-0.390326\pi
0.337773 + 0.941228i 0.390326π0.390326\pi
864864 0 0
865865 6.05718 0.205950
866866 0 0
867867 5.61505 + 0.893429i 0.190697 + 0.0303424i
868868 0 0
869869 −15.9520 + 27.6296i −0.541134 + 0.937271i
870870 0 0
871871 −1.61273 2.79332i −0.0546451 0.0946481i
872872 0 0
873873 10.6248 + 50.3374i 0.359594 + 1.70366i
874874 0 0
875875 −1.18878 2.05903i −0.0401881 0.0696078i
876876 0 0
877877 −10.2352 + 17.7278i −0.345617 + 0.598626i −0.985466 0.169875i 0.945664π-0.945664\pi
0.639849 + 0.768501i 0.278997π0.278997\pi
878878 0 0
879879 −7.71681 + 9.51563i −0.260282 + 0.320954i
880880 0 0
881881 −31.1683 −1.05009 −0.525043 0.851076i 0.675951π-0.675951\pi
−0.525043 + 0.851076i 0.675951π0.675951\pi
882882 0 0
883883 −2.64187 −0.0889060 −0.0444530 0.999011i 0.514154π-0.514154\pi
−0.0444530 + 0.999011i 0.514154π0.514154\pi
884884 0 0
885885 1.12283 + 2.93130i 0.0377436 + 0.0985347i
886886 0 0
887887 11.5825 20.0615i 0.388902 0.673599i −0.603400 0.797439i 0.706188π-0.706188\pi
0.992302 + 0.123840i 0.0395210π0.0395210\pi
888888 0 0
889889 −8.40451 14.5570i −0.281878 0.488228i
890890 0 0
891891 −4.95159 + 45.8563i −0.165884 + 1.53624i
892892 0 0
893893 −7.94226 13.7564i −0.265778 0.460341i
894894 0 0
895895 −1.14132 + 1.97682i −0.0381500 + 0.0660777i
896896 0 0
897897 −22.4632 58.6433i −0.750026 1.95804i
898898 0 0
899899 −2.48538 −0.0828921
900900 0 0
901901 −3.49192 −0.116333
902902 0 0
903903 −13.6248 + 16.8007i −0.453404 + 0.559094i
904904 0 0
905905 −1.47949 + 2.56255i −0.0491799 + 0.0851821i
906906 0 0
907907 25.0264 + 43.3470i 0.830988 + 1.43931i 0.897256 + 0.441511i 0.145558π0.145558\pi
−0.0662676 + 0.997802i 0.521109π0.521109\pi
908908 0 0
909909 −4.44966 21.0814i −0.147586 0.699224i
910910 0 0
911911 5.42231 + 9.39172i 0.179649 + 0.311161i 0.941760 0.336285i 0.109170π-0.109170\pi
−0.762111 + 0.647446i 0.775837π0.775837\pi
912912 0 0
913913 24.8646 43.0667i 0.822897 1.42530i
914914 0 0
915915 −2.25296 0.358476i −0.0744806 0.0118509i
916916 0 0
917917 −4.89931 −0.161790
918918 0 0
919919 11.1910 0.369156 0.184578 0.982818i 0.440908π-0.440908\pi
0.184578 + 0.982818i 0.440908π0.440908\pi
920920 0 0
921921 −26.9441 4.28716i −0.887837 0.141267i
922922 0 0
923923 33.5679 58.1413i 1.10490 1.91374i
924924 0 0
925925 11.3759 + 19.7037i 0.374038 + 0.647853i
926926 0 0
927927 28.6368 25.7111i 0.940556 0.844462i
928928 0 0
929929 −20.6478 35.7630i −0.677431 1.17335i −0.975752 0.218879i 0.929760π-0.929760\pi
0.298321 0.954466i 0.403573π-0.403573\pi
930930 0 0
931931 −1.83009 + 3.16982i −0.0599789 + 0.103887i
932932 0 0
933933 −21.4090 + 26.3995i −0.700900 + 0.864282i
934934 0 0
935935 −4.53870 −0.148431
936936 0 0
937937 −33.5620 −1.09642 −0.548211 0.836340i 0.684691π-0.684691\pi
−0.548211 + 0.836340i 0.684691π0.684691\pi
938938 0 0
939939 −15.7898 41.2213i −0.515279 1.34521i
940940 0 0
941941 −25.2112 + 43.6671i −0.821862 + 1.42351i 0.0824315 + 0.996597i 0.473731π0.473731\pi
−0.904294 + 0.426911i 0.859602π0.859602\pi
942942 0 0
943943 −20.7902 36.0096i −0.677021 1.17263i
944944 0 0
945945 1.10589 + 0.566453i 0.0359746 + 0.0184267i
946946 0 0
947947 −10.1212 17.5304i −0.328895 0.569662i 0.653398 0.757014i 0.273343π-0.273343\pi
−0.982293 + 0.187352i 0.940009π0.940009\pi
948948 0 0
949949 −8.94119 + 15.4866i −0.290243 + 0.502716i
950950 0 0
951951 5.13160 + 13.3967i 0.166404 + 0.434419i
952952 0 0
953953 −29.3685 −0.951340 −0.475670 0.879624i 0.657794π-0.657794\pi
−0.475670 + 0.879624i 0.657794π0.657794\pi
954954 0 0
955955 3.14884 0.101894
956956 0 0
957957 −19.3703 + 23.8856i −0.626154 + 0.772113i
958958 0 0
959959 2.72257 4.71563i 0.0879164 0.152276i
960960 0 0
961961 15.2427 + 26.4011i 0.491700 + 0.851649i
962962 0 0
963963 21.5750 + 7.04407i 0.695244 + 0.226992i
964964 0 0
965965 −1.33297 2.30878i −0.0429099 0.0743222i
966966 0 0
967967 15.2157 26.3544i 0.489305 0.847501i −0.510619 0.859807i 0.670584π-0.670584\pi
0.999924 + 0.0123057i 0.00391714π0.00391714\pi
968968 0 0
969969 −23.1884 3.68958i −0.744918 0.118526i
970970 0 0
971971 7.18659 0.230629 0.115314 0.993329i 0.463212π-0.463212\pi
0.115314 + 0.993329i 0.463212π0.463212\pi
972972 0 0
973973 −5.66019 −0.181457
974974 0 0
975975 −41.3074 6.57256i −1.32290 0.210490i
976976 0 0
977977 −14.2713 + 24.7186i −0.456579 + 0.790818i −0.998777 0.0494328i 0.984259π-0.984259\pi
0.542199 + 0.840250i 0.317592π0.317592\pi
978978 0 0
979979 −19.1888 33.2359i −0.613276 1.06223i
980980 0 0
981981 −19.9075 6.49966i −0.635598 0.207518i
982982 0 0
983983 −2.20821 3.82473i −0.0704310 0.121990i 0.828659 0.559753i 0.189104π-0.189104\pi
−0.899090 + 0.437763i 0.855771π0.855771\pi
984984 0 0
985985 0.0172400 0.0298606i 0.000549313 0.000951438i
986986 0 0
987987 −4.73461 + 5.83826i −0.150704 + 0.185834i
988988 0 0
989989 92.6791 2.94702
990990 0 0
991991 −5.81341 −0.184669 −0.0923345 0.995728i 0.529433π-0.529433\pi
−0.0923345 + 0.995728i 0.529433π0.529433\pi
992992 0 0
993993 −7.42270 19.3780i −0.235552 0.614941i
994994 0 0
995995 2.32777 4.03182i 0.0737953 0.127817i
996996 0 0
997997 26.3204 + 45.5882i 0.833575 + 1.44379i 0.895186 + 0.445694i 0.147043π0.147043\pi
−0.0616108 + 0.998100i 0.519624π0.519624\pi
998998 0 0
999999 −21.2878 10.9040i −0.673517 0.344986i
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 252.2.j.a.85.1 6
3.2 odd 2 756.2.j.b.253.2 6
4.3 odd 2 1008.2.r.j.337.3 6
7.2 even 3 1764.2.l.e.949.3 6
7.3 odd 6 1764.2.i.d.373.2 6
7.4 even 3 1764.2.i.g.373.2 6
7.5 odd 6 1764.2.l.f.949.1 6
7.6 odd 2 1764.2.j.e.589.3 6
9.2 odd 6 756.2.j.b.505.2 6
9.4 even 3 2268.2.a.i.1.2 3
9.5 odd 6 2268.2.a.h.1.2 3
9.7 even 3 inner 252.2.j.a.169.1 yes 6
12.11 even 2 3024.2.r.j.1009.2 6
21.2 odd 6 5292.2.l.e.361.2 6
21.5 even 6 5292.2.l.f.361.2 6
21.11 odd 6 5292.2.i.f.1549.2 6
21.17 even 6 5292.2.i.e.1549.2 6
21.20 even 2 5292.2.j.d.1765.2 6
36.7 odd 6 1008.2.r.j.673.3 6
36.11 even 6 3024.2.r.j.2017.2 6
36.23 even 6 9072.2.a.bv.1.2 3
36.31 odd 6 9072.2.a.by.1.2 3
63.2 odd 6 5292.2.i.f.2125.2 6
63.11 odd 6 5292.2.l.e.3313.2 6
63.16 even 3 1764.2.i.g.1537.2 6
63.20 even 6 5292.2.j.d.3529.2 6
63.25 even 3 1764.2.l.e.961.3 6
63.34 odd 6 1764.2.j.e.1177.3 6
63.38 even 6 5292.2.l.f.3313.2 6
63.47 even 6 5292.2.i.e.2125.2 6
63.52 odd 6 1764.2.l.f.961.1 6
63.61 odd 6 1764.2.i.d.1537.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.j.a.85.1 6 1.1 even 1 trivial
252.2.j.a.169.1 yes 6 9.7 even 3 inner
756.2.j.b.253.2 6 3.2 odd 2
756.2.j.b.505.2 6 9.2 odd 6
1008.2.r.j.337.3 6 4.3 odd 2
1008.2.r.j.673.3 6 36.7 odd 6
1764.2.i.d.373.2 6 7.3 odd 6
1764.2.i.d.1537.2 6 63.61 odd 6
1764.2.i.g.373.2 6 7.4 even 3
1764.2.i.g.1537.2 6 63.16 even 3
1764.2.j.e.589.3 6 7.6 odd 2
1764.2.j.e.1177.3 6 63.34 odd 6
1764.2.l.e.949.3 6 7.2 even 3
1764.2.l.e.961.3 6 63.25 even 3
1764.2.l.f.949.1 6 7.5 odd 6
1764.2.l.f.961.1 6 63.52 odd 6
2268.2.a.h.1.2 3 9.5 odd 6
2268.2.a.i.1.2 3 9.4 even 3
3024.2.r.j.1009.2 6 12.11 even 2
3024.2.r.j.2017.2 6 36.11 even 6
5292.2.i.e.1549.2 6 21.17 even 6
5292.2.i.e.2125.2 6 63.47 even 6
5292.2.i.f.1549.2 6 21.11 odd 6
5292.2.i.f.2125.2 6 63.2 odd 6
5292.2.j.d.1765.2 6 21.20 even 2
5292.2.j.d.3529.2 6 63.20 even 6
5292.2.l.e.361.2 6 21.2 odd 6
5292.2.l.e.3313.2 6 63.11 odd 6
5292.2.l.f.361.2 6 21.5 even 6
5292.2.l.f.3313.2 6 63.38 even 6
9072.2.a.bv.1.2 3 36.23 even 6
9072.2.a.by.1.2 3 36.31 odd 6