Properties

Label 144.9.q.b.65.5
Level 144144
Weight 99
Character 144.65
Analytic conductor 58.66358.663
Analytic rank 00
Dimension 1616
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,9,Mod(65,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.65");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: N N == 144=2432 144 = 2^{4} \cdot 3^{2}
Weight: k k == 9 9
Character orbit: [χ][\chi] == 144.q (of order 66, degree 22, not 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: 58.662519848858.6625198488
Analytic rank: 00
Dimension: 1616
Relative dimension: 88 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q[x]/(x16)\mathbb{Q}[x]/(x^{16} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x164x15150208x141927740x13+8702363206x12+239206241152x11++81 ⁣ ⁣61 x^{16} - 4 x^{15} - 150208 x^{14} - 1927740 x^{13} + 8702363206 x^{12} + 239206241152 x^{11} + \cdots + 81\!\cdots\!61 Copy content Toggle raw display
Coefficient ring: Z[a1,,a19]\Z[a_1, \ldots, a_{19}]
Coefficient ring index: 232325 2^{32}\cdot 3^{25}
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

Embedding invariants

Embedding label 65.5
Root 220.3330.866025i220.333 - 0.866025i of defining polynomial
Character χ\chi == 144.65
Dual form 144.9.q.b.113.5

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+(7.3817680.6629i)q3+(935.250539.967i)q5+(2056.093561.25i)q7+(6452.02+1190.87i)q9+(8419.534861.02i)q11+(4116.027129.16i)q13+(50459.171454.1i)q1587809.9iq17123643.q19+(302439.139562.i)q21+(92029.4+53133.2i)q23+(387816.671717.i)q25+(143686.+511648.i)q27+(430885.+248772.i)q29+(577977.+1.00109e6i)q31+(329953.+715027.i)q334.44088e6iq35+1.20145e6q37+(544675.+384636.i)q39+(1.53952e6888843.i)q41+(528726.915780.i)q43+(5.39122e6+4.59764e6i)q45+(3.80689e6+2.19791e6i)q47+(5.57261e69.65205e6i)q49+(7.08301e6+648192.i)q51+1.04810e6iq531.04991e7q55+(912700.+9.97337e6i)q57+(8.70048e65.02323e6i)q59+(4.15770e6+7.20135e6i)q61+(9.02495e6+2.54258e7i)q63+(7.69902e64.44503e6i)q65+(6.19183e61.07246e7i)q67+(4.96522e6+7.03114e6i)q69+1.69579e7iq712.42736e7q73+(5.70454e72.63239e7i)q75+(3.46226e7+1.99894e7i)q77+(6.39004e6+1.10679e7i)q79+(4.02104e71.53670e7i)q81+(2.79188e7+1.61189e7i)q83+(4.74144e78.21242e7i)q85+(1.68860e73.65928e7i)q87+2.10088e7iq893.38516e7q91+(7.64841e75.40111e7i)q93+(1.15637e8+6.67629e7i)q95+(4.46651e7+7.73622e7i)q97+(6.01118e7+2.13368e7i)q99+O(q100)q+(-7.38176 - 80.6629i) q^{3} +(935.250 - 539.967i) q^{5} +(2056.09 - 3561.25i) q^{7} +(-6452.02 + 1190.87i) q^{9} +(-8419.53 - 4861.02i) q^{11} +(-4116.02 - 7129.16i) q^{13} +(-50459.1 - 71454.1i) q^{15} -87809.9i q^{17} -123643. q^{19} +(-302439. - 139562. i) q^{21} +(-92029.4 + 53133.2i) q^{23} +(387816. - 671717. i) q^{25} +(143686. + 511648. i) q^{27} +(430885. + 248772. i) q^{29} +(577977. + 1.00109e6i) q^{31} +(-329953. + 715027. i) q^{33} -4.44088e6i q^{35} +1.20145e6 q^{37} +(-544675. + 384636. i) q^{39} +(1.53952e6 - 888843. i) q^{41} +(528726. - 915780. i) q^{43} +(-5.39122e6 + 4.59764e6i) q^{45} +(3.80689e6 + 2.19791e6i) q^{47} +(-5.57261e6 - 9.65205e6i) q^{49} +(-7.08301e6 + 648192. i) q^{51} +1.04810e6i q^{53} -1.04991e7 q^{55} +(912700. + 9.97337e6i) q^{57} +(8.70048e6 - 5.02323e6i) q^{59} +(-4.15770e6 + 7.20135e6i) q^{61} +(-9.02495e6 + 2.54258e7i) q^{63} +(-7.69902e6 - 4.44503e6i) q^{65} +(-6.19183e6 - 1.07246e7i) q^{67} +(4.96522e6 + 7.03114e6i) q^{69} +1.69579e7i q^{71} -2.42736e7 q^{73} +(-5.70454e7 - 2.63239e7i) q^{75} +(-3.46226e7 + 1.99894e7i) q^{77} +(-6.39004e6 + 1.10679e7i) q^{79} +(4.02104e7 - 1.53670e7i) q^{81} +(2.79188e7 + 1.61189e7i) q^{83} +(-4.74144e7 - 8.21242e7i) q^{85} +(1.68860e7 - 3.65928e7i) q^{87} +2.10088e7i q^{89} -3.38516e7 q^{91} +(7.64841e7 - 5.40111e7i) q^{93} +(-1.15637e8 + 6.67629e7i) q^{95} +(-4.46651e7 + 7.73622e7i) q^{97} +(6.01118e7 + 2.13368e7i) q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 16q126q3882q5+1846q728662q945756q113370q13128754q15362180q19299166q211311138q23+963394q25+208656q272851290q29542438q31+366888330q99+O(q100) 16 q - 126 q^{3} - 882 q^{5} + 1846 q^{7} - 28662 q^{9} - 45756 q^{11} - 3370 q^{13} - 128754 q^{15} - 362180 q^{19} - 299166 q^{21} - 1311138 q^{23} + 963394 q^{25} + 208656 q^{27} - 2851290 q^{29} - 542438 q^{31}+ \cdots - 366888330 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/144Z)×\left(\mathbb{Z}/144\mathbb{Z}\right)^\times.

nn 3737 6565 127127
χ(n)\chi(n) 11 e(16)e\left(\frac{1}{6}\right) 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 −7.38176 80.6629i −0.0911329 0.995839i
44 0 0
55 935.250 539.967i 1.49640 0.863947i 0.496408 0.868089i 0.334652π-0.334652\pi
0.999991 + 0.00414227i 0.00131853π0.00131853\pi
66 0 0
77 2056.09 3561.25i 0.856347 1.48324i −0.0190419 0.999819i 0.506062π-0.506062\pi
0.875389 0.483419i 0.160605π-0.160605\pi
88 0 0
99 −6452.02 + 1190.87i −0.983390 + 0.181507i
1010 0 0
1111 −8419.53 4861.02i −0.575065 0.332014i 0.184105 0.982907i 0.441061π-0.441061\pi
−0.759170 + 0.650893i 0.774395π0.774395\pi
1212 0 0
1313 −4116.02 7129.16i −0.144113 0.249612i 0.784928 0.619586i 0.212700π-0.212700\pi
−0.929042 + 0.369975i 0.879366π0.879366\pi
1414 0 0
1515 −50459.1 71454.1i −0.996723 1.41144i
1616 0 0
1717 87809.9i 1.05135i −0.850685 0.525676i 0.823813π-0.823813\pi
0.850685 0.525676i 0.176187π-0.176187\pi
1818 0 0
1919 −123643. −0.948754 −0.474377 0.880322i 0.657327π-0.657327\pi
−0.474377 + 0.880322i 0.657327π0.657327\pi
2020 0 0
2121 −302439. 139562.i −1.55511 0.717612i
2222 0 0
2323 −92029.4 + 53133.2i −0.328863 + 0.189869i −0.655336 0.755337i 0.727473π-0.727473\pi
0.326473 + 0.945206i 0.394140π0.394140\pi
2424 0 0
2525 387816. 671717.i 0.992808 1.71959i
2626 0 0
2727 143686. + 511648.i 0.270371 + 0.962756i
2828 0 0
2929 430885. + 248772.i 0.609213 + 0.351729i 0.772657 0.634823i 0.218927π-0.218927\pi
−0.163444 + 0.986553i 0.552260π0.552260\pi
3030 0 0
3131 577977. + 1.00109e6i 0.625841 + 1.08399i 0.988378 + 0.152019i 0.0485775π0.0485775\pi
−0.362536 + 0.931970i 0.618089π0.618089\pi
3232 0 0
3333 −329953. + 715027.i −0.278225 + 0.602929i
3434 0 0
3535 4.44088e6i 2.95935i
3636 0 0
3737 1.20145e6 0.641061 0.320531 0.947238i 0.396139π-0.396139\pi
0.320531 + 0.947238i 0.396139π0.396139\pi
3838 0 0
3939 −544675. + 384636.i −0.235439 + 0.166261i
4040 0 0
4141 1.53952e6 888843.i 0.544816 0.314550i −0.202212 0.979342i 0.564813π-0.564813\pi
0.747029 + 0.664792i 0.231480π0.231480\pi
4242 0 0
4343 528726. 915780.i 0.154652 0.267866i −0.778280 0.627917i 0.783908π-0.783908\pi
0.932932 + 0.360052i 0.117241π0.117241\pi
4444 0 0
4545 −5.39122e6 + 4.59764e6i −1.31473 + 1.12120i
4646 0 0
4747 3.80689e6 + 2.19791e6i 0.780151 + 0.450420i 0.836484 0.547992i 0.184607π-0.184607\pi
−0.0563327 + 0.998412i 0.517941π0.517941\pi
4848 0 0
4949 −5.57261e6 9.65205e6i −0.966662 1.67431i
5050 0 0
5151 −7.08301e6 + 648192.i −1.04698 + 0.0958127i
5252 0 0
5353 1.04810e6i 0.132831i 0.997792 + 0.0664157i 0.0211563π0.0211563\pi
−0.997792 + 0.0664157i 0.978844π0.978844\pi
5454 0 0
5555 −1.04991e7 −1.14737
5656 0 0
5757 912700. + 9.97337e6i 0.0864627 + 0.944806i
5858 0 0
5959 8.70048e6 5.02323e6i 0.718018 0.414548i −0.0960047 0.995381i 0.530606π-0.530606\pi
0.814023 + 0.580833i 0.197273π0.197273\pi
6060 0 0
6161 −4.15770e6 + 7.20135e6i −0.300285 + 0.520109i −0.976200 0.216870i 0.930415π-0.930415\pi
0.675915 + 0.736979i 0.263749π0.263749\pi
6262 0 0
6363 −9.02495e6 + 2.54258e7i −0.572905 + 1.61403i
6464 0 0
6565 −7.69902e6 4.44503e6i −0.431302 0.249013i
6666 0 0
6767 −6.19183e6 1.07246e7i −0.307270 0.532207i 0.670494 0.741915i 0.266082π-0.266082\pi
−0.977764 + 0.209708i 0.932749π0.932749\pi
6868 0 0
6969 4.96522e6 + 7.03114e6i 0.219049 + 0.310191i
7070 0 0
7171 1.69579e7i 0.667328i 0.942692 + 0.333664i 0.108285π0.108285\pi
−0.942692 + 0.333664i 0.891715π0.891715\pi
7272 0 0
7373 −2.42736e7 −0.854757 −0.427378 0.904073i 0.640563π-0.640563\pi
−0.427378 + 0.904073i 0.640563π0.640563\pi
7474 0 0
7575 −5.70454e7 2.63239e7i −1.80292 0.831965i
7676 0 0
7777 −3.46226e7 + 1.99894e7i −0.984911 + 0.568639i
7878 0 0
7979 −6.39004e6 + 1.10679e7i −0.164057 + 0.284155i −0.936320 0.351148i 0.885791π-0.885791\pi
0.772263 + 0.635303i 0.219125π0.219125\pi
8080 0 0
8181 4.02104e7 1.53670e7i 0.934110 0.356985i
8282 0 0
8383 2.79188e7 + 1.61189e7i 0.588281 + 0.339644i 0.764418 0.644722i 0.223027π-0.223027\pi
−0.176136 + 0.984366i 0.556360π0.556360\pi
8484 0 0
8585 −4.74144e7 8.21242e7i −0.908312 1.57324i
8686 0 0
8787 1.68860e7 3.65928e7i 0.294746 0.638732i
8888 0 0
8989 2.10088e7i 0.334843i 0.985885 + 0.167422i 0.0535441π0.0535441\pi
−0.985885 + 0.167422i 0.946456π0.946456\pi
9090 0 0
9191 −3.38516e7 −0.493644
9292 0 0
9393 7.64841e7 5.40111e7i 1.02244 0.722024i
9494 0 0
9595 −1.15637e8 + 6.67629e7i −1.41972 + 0.819673i
9696 0 0
9797 −4.46651e7 + 7.73622e7i −0.504523 + 0.873860i 0.495463 + 0.868629i 0.334998π0.334998\pi
−0.999986 + 0.00523116i 0.998335π0.998335\pi
9898 0 0
9999 6.01118e7 + 2.13368e7i 0.625776 + 0.222121i
100100 0 0
101101 −3.08430e7 1.78072e7i −0.296395 0.171124i 0.344427 0.938813i 0.388073π-0.388073\pi
−0.640822 + 0.767689i 0.721406π0.721406\pi
102102 0 0
103103 4.35300e6 + 7.53962e6i 0.0386759 + 0.0669886i 0.884715 0.466132i 0.154353π-0.154353\pi
−0.846040 + 0.533120i 0.821019π0.821019\pi
104104 0 0
105105 −3.58215e8 + 3.27815e7i −2.94704 + 0.269694i
106106 0 0
107107 2.34341e8i 1.78778i −0.448290 0.893888i 0.647967π-0.647967\pi
0.448290 0.893888i 0.352033π-0.352033\pi
108108 0 0
109109 2.66492e8 1.88789 0.943947 0.330098i 0.107082π-0.107082\pi
0.943947 + 0.330098i 0.107082π0.107082\pi
110110 0 0
111111 −8.86883e6 9.69126e7i −0.0584217 0.638393i
112112 0 0
113113 −2.47560e8 + 1.42929e8i −1.51833 + 0.876610i −0.518565 + 0.855038i 0.673534π0.673534\pi
−0.999767 + 0.0215714i 0.993133π0.993133\pi
114114 0 0
115115 −5.73803e7 + 9.93856e7i −0.328074 + 0.568241i
116116 0 0
117117 3.50465e7 + 4.10958e7i 0.187026 + 0.219308i
118118 0 0
119119 −3.12713e8 1.80545e8i −1.55940 0.900322i
120120 0 0
121121 −5.99205e7 1.03785e8i −0.279534 0.484166i
122122 0 0
123123 −8.30610e7 1.17621e8i −0.362892 0.513883i
124124 0 0
125125 4.15781e8i 1.70304i
126126 0 0
127127 8.18752e7 0.314730 0.157365 0.987541i 0.449700π-0.449700\pi
0.157365 + 0.987541i 0.449700π0.449700\pi
128128 0 0
129129 −7.77724e7 3.58885e7i −0.280845 0.129597i
130130 0 0
131131 −3.27800e8 + 1.89256e8i −1.11308 + 0.642634i −0.939624 0.342208i 0.888825π-0.888825\pi
−0.173451 + 0.984843i 0.555492π0.555492\pi
132132 0 0
133133 −2.54220e8 + 4.40322e8i −0.812463 + 1.40723i
134134 0 0
135135 4.10656e8 + 4.00933e8i 1.23635 + 1.20708i
136136 0 0
137137 8.36881e7 + 4.83174e7i 0.237565 + 0.137158i 0.614057 0.789262i 0.289537π-0.289537\pi
−0.376492 + 0.926420i 0.622870π0.622870\pi
138138 0 0
139139 2.13230e8 + 3.69325e8i 0.571200 + 0.989348i 0.996443 + 0.0842686i 0.0268554π0.0268554\pi
−0.425243 + 0.905079i 0.639811π0.639811\pi
140140 0 0
141141 1.49188e8 3.23299e8i 0.377449 0.817953i
142142 0 0
143143 8.00322e7i 0.191391i
144144 0 0
145145 5.37313e8 1.21550
146146 0 0
147147 −7.37427e8 + 5.20752e8i −1.57925 + 1.11522i
148148 0 0
149149 3.73436e8 2.15603e8i 0.757653 0.437431i −0.0707992 0.997491i 0.522555π-0.522555\pi
0.828453 + 0.560059i 0.189222π0.189222\pi
150150 0 0
151151 4.20280e8 7.27947e8i 0.808409 1.40021i −0.105556 0.994413i 0.533662π-0.533662\pi
0.913965 0.405793i 0.133004π-0.133004\pi
152152 0 0
153153 1.04570e8 + 5.66551e8i 0.190828 + 1.03389i
154154 0 0
155155 1.08111e9 + 6.24177e8i 1.87302 + 1.08139i
156156 0 0
157157 2.10919e8 + 3.65323e8i 0.347150 + 0.601282i 0.985742 0.168264i 0.0538160π-0.0538160\pi
−0.638592 + 0.769546i 0.720483π0.720483\pi
158158 0 0
159159 8.45431e7 7.73685e6i 0.132279 0.0121053i
160160 0 0
161161 4.36987e8i 0.650376i
162162 0 0
163163 5.50818e8 0.780293 0.390146 0.920753i 0.372424π-0.372424\pi
0.390146 + 0.920753i 0.372424π0.372424\pi
164164 0 0
165165 7.75022e7 + 8.46892e8i 0.104563 + 1.14259i
166166 0 0
167167 −5.25550e8 + 3.03427e8i −0.675691 + 0.390111i −0.798230 0.602353i 0.794230π-0.794230\pi
0.122538 + 0.992464i 0.460897π0.460897\pi
168168 0 0
169169 3.73982e8 6.47756e8i 0.458463 0.794081i
170170 0 0
171171 7.97744e8 1.47242e8i 0.932995 0.172206i
172172 0 0
173173 −8.26617e8 4.77248e8i −0.922826 0.532794i −0.0382907 0.999267i 0.512191π-0.512191\pi
−0.884536 + 0.466473i 0.845525π0.845525\pi
174174 0 0
175175 −1.59477e9 2.76222e9i −1.70038 2.94514i
176176 0 0
177177 −4.69413e8 6.64726e8i −0.478258 0.677251i
178178 0 0
179179 9.44019e8i 0.919536i −0.888039 0.459768i 0.847933π-0.847933\pi
0.888039 0.459768i 0.152067π-0.152067\pi
180180 0 0
181181 −8.68230e8 −0.808948 −0.404474 0.914550i 0.632545π-0.632545\pi
−0.404474 + 0.914550i 0.632545π0.632545\pi
182182 0 0
183183 6.11573e8 + 2.82214e8i 0.545311 + 0.251637i
184184 0 0
185185 1.12366e9 6.48744e8i 0.959284 0.553843i
186186 0 0
187187 −4.26845e8 + 7.39318e8i −0.349063 + 0.604595i
188188 0 0
189189 2.11754e9 + 5.40291e8i 1.65953 + 0.423429i
190190 0 0
191191 6.09956e7 + 3.52159e7i 0.0458316 + 0.0264609i 0.522741 0.852492i 0.324910π-0.324910\pi
−0.476909 + 0.878953i 0.658243π0.658243\pi
192192 0 0
193193 −1.36871e9 2.37068e9i −0.986467 1.70861i −0.635225 0.772327i 0.719093π-0.719093\pi
−0.351242 0.936285i 0.614241π-0.614241\pi
194194 0 0
195195 −3.01717e8 + 6.53838e8i −0.208671 + 0.452201i
196196 0 0
197197 5.47276e8i 0.363364i 0.983357 + 0.181682i 0.0581541π0.0581541\pi
−0.983357 + 0.181682i 0.941846π0.941846\pi
198198 0 0
199199 2.66467e9 1.69915 0.849574 0.527469i 0.176859π-0.176859\pi
0.849574 + 0.527469i 0.176859π0.176859\pi
200200 0 0
201201 −8.19368e8 + 5.78617e8i −0.501990 + 0.354493i
202202 0 0
203203 1.77188e9 1.02299e9i 1.04340 0.602405i
204204 0 0
205205 9.59891e8 1.66258e9i 0.543509 0.941385i
206206 0 0
207207 5.30501e8 4.52411e8i 0.288938 0.246406i
208208 0 0
209209 1.04101e9 + 6.01029e8i 0.545595 + 0.315000i
210210 0 0
211211 9.39137e8 + 1.62663e9i 0.473804 + 0.820653i 0.999550 0.0299884i 0.00954702π-0.00954702\pi
−0.525746 + 0.850642i 0.676214π0.676214\pi
212212 0 0
213213 1.36788e9 1.25179e8i 0.664551 0.0608155i
214214 0 0
215215 1.14198e9i 0.534446i
216216 0 0
217217 4.75349e9 2.14375
218218 0 0
219219 1.79182e8 + 1.95798e9i 0.0778964 + 0.851200i
220220 0 0
221221 −6.26011e8 + 3.61427e8i −0.262430 + 0.151514i
222222 0 0
223223 −1.45052e9 + 2.51237e9i −0.586548 + 1.01593i 0.408132 + 0.912923i 0.366180π0.366180\pi
−0.994680 + 0.103009i 0.967153π0.967153\pi
224224 0 0
225225 −1.70227e9 + 4.79577e9i −0.664198 + 1.87123i
226226 0 0
227227 −3.60966e9 2.08404e9i −1.35945 0.784877i −0.369898 0.929072i 0.620607π-0.620607\pi
−0.989549 + 0.144195i 0.953941π0.953941\pi
228228 0 0
229229 1.80018e9 + 3.11801e9i 0.654599 + 1.13380i 0.981994 + 0.188911i 0.0604957π0.0604957\pi
−0.327396 + 0.944887i 0.606171π0.606171\pi
230230 0 0
231231 1.86798e9 + 2.64520e9i 0.656030 + 0.928991i
232232 0 0
233233 2.86365e9i 0.971618i 0.874065 + 0.485809i 0.161475π0.161475\pi
−0.874065 + 0.485809i 0.838525π0.838525\pi
234234 0 0
235235 4.74719e9 1.55656
236236 0 0
237237 9.39937e8 + 4.33739e8i 0.297924 + 0.137479i
238238 0 0
239239 −1.21500e9 + 7.01479e8i −0.372378 + 0.214992i −0.674497 0.738278i 0.735639π-0.735639\pi
0.302119 + 0.953270i 0.402306π0.402306\pi
240240 0 0
241241 2.21808e9 3.84182e9i 0.657519 1.13886i −0.323737 0.946147i 0.604939π-0.604939\pi
0.981256 0.192709i 0.0617273π-0.0617273\pi
242242 0 0
243243 −1.53637e9 3.13005e9i −0.440627 0.897690i
244244 0 0
245245 −1.04236e10 6.01805e9i −2.89303 1.67029i
246246 0 0
247247 5.08916e8 + 8.81468e8i 0.136728 + 0.236820i
248248 0 0
249249 1.09411e9 2.37100e9i 0.284619 0.616786i
250250 0 0
251251 2.42084e9i 0.609916i 0.952366 + 0.304958i 0.0986425π0.0986425\pi
−0.952366 + 0.304958i 0.901357π0.901357\pi
252252 0 0
253253 1.03313e9 0.252157
254254 0 0
255255 −6.27438e9 + 4.43081e9i −1.48392 + 1.04791i
256256 0 0
257257 −2.08915e9 + 1.20617e9i −0.478891 + 0.276488i −0.719954 0.694022i 0.755837π-0.755837\pi
0.241063 + 0.970509i 0.422504π0.422504\pi
258258 0 0
259259 2.47029e9 4.27867e9i 0.548971 0.950846i
260260 0 0
261261 −3.07633e9 1.09195e9i −0.662935 0.235310i
262262 0 0
263263 −5.65030e9 3.26220e9i −1.18100 0.681849i −0.224752 0.974416i 0.572157π-0.572157\pi
−0.956245 + 0.292567i 0.905490π0.905490\pi
264264 0 0
265265 5.65941e8 + 9.80238e8i 0.114759 + 0.198769i
266266 0 0
267267 1.69463e9 1.55082e8i 0.333450 0.0305152i
268268 0 0
269269 6.64127e9i 1.26836i −0.773186 0.634180i 0.781338π-0.781338\pi
0.773186 0.634180i 0.218662π-0.218662\pi
270270 0 0
271271 2.72229e9 0.504728 0.252364 0.967632i 0.418792π-0.418792\pi
0.252364 + 0.967632i 0.418792π0.418792\pi
272272 0 0
273273 2.49885e8 + 2.73057e9i 0.0449872 + 0.491590i
274274 0 0
275275 −6.53045e9 + 3.77036e9i −1.14186 + 0.659252i
276276 0 0
277277 −5.00946e9 + 8.67663e9i −0.850886 + 1.47378i 0.0295233 + 0.999564i 0.490601π0.490601\pi
−0.880410 + 0.474214i 0.842732π0.842732\pi
278278 0 0
279279 −4.92128e9 5.77073e9i −0.812198 0.952388i
280280 0 0
281281 −6.83019e9 3.94341e9i −1.09549 0.632480i −0.160456 0.987043i 0.551296π-0.551296\pi
−0.935032 + 0.354563i 0.884630π0.884630\pi
282282 0 0
283283 −4.86855e8 8.43258e8i −0.0759021 0.131466i 0.825576 0.564291i 0.190850π-0.190850\pi
−0.901478 + 0.432824i 0.857517π0.857517\pi
284284 0 0
285285 6.23889e9 + 8.83477e9i 0.945645 + 1.33911i
286286 0 0
287287 7.31016e9i 1.07746i
288288 0 0
289289 −7.34822e8 −0.105339
290290 0 0
291291 6.56997e9 + 3.03175e9i 0.916203 + 0.422787i
292292 0 0
293293 −4.59420e9 + 2.65246e9i −0.623361 + 0.359898i −0.778176 0.628046i 0.783855π-0.783855\pi
0.154815 + 0.987943i 0.450522π0.450522\pi
294294 0 0
295295 5.42475e9 9.39595e9i 0.716295 1.24066i
296296 0 0
297297 1.27736e9 5.00630e9i 0.164167 0.643414i
298298 0 0
299299 7.57590e8 + 4.37395e8i 0.0947871 + 0.0547254i
300300 0 0
301301 −2.17422e9 3.76585e9i −0.264872 0.458772i
302302 0 0
303303 −1.20871e9 + 2.61933e9i −0.143400 + 0.310756i
304304 0 0
305305 8.98008e9i 1.03772i
306306 0 0
307307 −1.34241e10 −1.51124 −0.755618 0.655013i 0.772663π-0.772663\pi
−0.755618 + 0.655013i 0.772663π0.772663\pi
308308 0 0
309309 5.76035e8 4.06782e8i 0.0631852 0.0446198i
310310 0 0
311311 −1.98291e9 + 1.14483e9i −0.211964 + 0.122377i −0.602224 0.798327i 0.705718π-0.705718\pi
0.390260 + 0.920705i 0.372385π0.372385\pi
312312 0 0
313313 8.94161e9 1.54873e10i 0.931619 1.61361i 0.151065 0.988524i 0.451730π-0.451730\pi
0.780555 0.625088i 0.214937π-0.214937\pi
314314 0 0
315315 5.28851e9 + 2.86527e10i 0.537144 + 2.91020i
316316 0 0
317317 5.77404e9 + 3.33364e9i 0.571798 + 0.330128i 0.757867 0.652409i 0.226242π-0.226242\pi
−0.186069 + 0.982537i 0.559575π0.559575\pi
318318 0 0
319319 −2.41856e9 4.18908e9i −0.233558 0.404535i
320320 0 0
321321 −1.89026e10 + 1.72985e9i −1.78034 + 0.162925i
322322 0 0
323323 1.08570e10i 0.997474i
324324 0 0
325325 −6.38503e9 −0.572308
326326 0 0
327327 −1.96718e9 2.14960e10i −0.172049 1.88004i
328328 0 0
329329 1.56546e10 9.03820e9i 1.33616 0.771433i
330330 0 0
331331 7.84292e9 1.35843e10i 0.653380 1.13169i −0.328918 0.944359i 0.606684π-0.606684\pi
0.982297 0.187328i 0.0599828π-0.0599828\pi
332332 0 0
333333 −7.75179e9 + 1.43077e9i −0.630413 + 0.116357i
334334 0 0
335335 −1.15818e10 6.68676e9i −0.919597 0.530929i
336336 0 0
337337 −1.29188e9 2.23761e9i −0.100162 0.173486i 0.811589 0.584229i 0.198603π-0.198603\pi
−0.911751 + 0.410743i 0.865270π0.865270\pi
338338 0 0
339339 1.33565e10 + 1.89139e10i 1.01133 + 1.43213i
340340 0 0
341341 1.12382e10i 0.831152i
342342 0 0
343343 −2.21253e10 −1.59850
344344 0 0
345345 8.44030e9 + 3.89482e9i 0.595774 + 0.274923i
346346 0 0
347347 −1.45626e10 + 8.40775e9i −1.00444 + 0.579912i −0.909558 0.415577i 0.863580π-0.863580\pi
−0.0948784 + 0.995489i 0.530246π0.530246\pi
348348 0 0
349349 1.22366e10 2.11945e10i 0.824822 1.42863i −0.0772336 0.997013i 0.524609π-0.524609\pi
0.902055 0.431620i 0.142058π-0.142058\pi
350350 0 0
351351 3.05620e9 3.13032e9i 0.201351 0.206234i
352352 0 0
353353 −9.05088e9 5.22553e9i −0.582897 0.336536i 0.179387 0.983779i 0.442589π-0.442589\pi
−0.762284 + 0.647243i 0.775922π0.775922\pi
354354 0 0
355355 9.15672e9 + 1.58599e10i 0.576536 + 0.998590i
356356 0 0
357357 −1.22549e10 + 2.65571e10i −0.754463 + 1.63496i
358358 0 0
359359 2.37497e10i 1.42982i −0.699219 0.714908i 0.746469π-0.746469\pi
0.699219 0.714908i 0.253531π-0.253531\pi
360360 0 0
361361 −1.69607e9 −0.0998656
362362 0 0
363363 −7.92931e9 + 5.59948e9i −0.456677 + 0.322494i
364364 0 0
365365 −2.27019e10 + 1.31069e10i −1.27906 + 0.738464i
366366 0 0
367367 5.17433e9 8.96220e9i 0.285226 0.494027i −0.687438 0.726243i 0.741265π-0.741265\pi
0.972664 + 0.232217i 0.0745979π0.0745979\pi
368368 0 0
369369 −8.87452e9 + 7.56820e9i −0.478674 + 0.408213i
370370 0 0
371371 3.73256e9 + 2.15499e9i 0.197020 + 0.113750i
372372 0 0
373373 4.02672e9 + 6.97448e9i 0.208025 + 0.360310i 0.951092 0.308907i 0.0999631π-0.0999631\pi
−0.743067 + 0.669217i 0.766630π0.766630\pi
374374 0 0
375375 −3.35382e10 + 3.06920e9i −1.69595 + 0.155203i
376376 0 0
377377 4.09580e9i 0.202756i
378378 0 0
379379 2.63218e10 1.27573 0.637864 0.770149i 0.279818π-0.279818\pi
0.637864 + 0.770149i 0.279818π0.279818\pi
380380 0 0
381381 −6.04383e8 6.60429e9i −0.0286822 0.313420i
382382 0 0
383383 2.64941e10 1.52964e10i 1.23127 0.710876i 0.263978 0.964529i 0.414965π-0.414965\pi
0.967295 + 0.253653i 0.0816321π0.0816321\pi
384384 0 0
385385 −2.15872e10 + 3.73901e10i −0.982547 + 1.70182i
386386 0 0
387387 −2.32077e9 + 6.53827e9i −0.103464 + 0.291487i
388388 0 0
389389 3.76791e10 + 2.17540e10i 1.64552 + 0.950039i 0.978824 + 0.204704i 0.0656233π0.0656233\pi
0.666691 + 0.745334i 0.267710π0.267710\pi
390390 0 0
391391 4.66562e9 + 8.08109e9i 0.199619 + 0.345751i
392392 0 0
393393 1.76857e10 + 2.50443e10i 0.741398 + 1.04988i
394394 0 0
395395 1.38016e10i 0.566947i
396396 0 0
397397 4.31419e10 1.73675 0.868375 0.495908i 0.165165π-0.165165\pi
0.868375 + 0.495908i 0.165165π0.165165\pi
398398 0 0
399399 3.73943e10 + 1.72558e10i 1.47541 + 0.680838i
400400 0 0
401401 −1.57455e10 + 9.09069e9i −0.608948 + 0.351576i −0.772554 0.634950i 0.781021π-0.781021\pi
0.163606 + 0.986526i 0.447688π0.447688\pi
402402 0 0
403403 4.75794e9 8.24099e9i 0.180384 0.312434i
404404 0 0
405405 2.93091e10 3.60843e10i 1.08939 1.34121i
406406 0 0
407407 −1.01157e10 5.84028e9i −0.368652 0.212841i
408408 0 0
409409 3.23724e9 + 5.60707e9i 0.115686 + 0.200374i 0.918054 0.396456i 0.129760π-0.129760\pi
−0.802368 + 0.596830i 0.796427π0.796427\pi
410410 0 0
411411 3.27965e9 7.10720e9i 0.114937 0.249076i
412412 0 0
413413 4.13128e10i 1.41999i
414414 0 0
415415 3.48148e10 1.17374
416416 0 0
417417 2.82168e10 1.99260e10i 0.933176 0.658985i
418418 0 0
419419 3.28096e10 1.89427e10i 1.06450 0.614589i 0.137826 0.990456i 0.455989π-0.455989\pi
0.926673 + 0.375868i 0.122655π0.122655\pi
420420 0 0
421421 2.38781e10 4.13581e10i 0.760102 1.31654i −0.182696 0.983170i 0.558482π-0.558482\pi
0.942798 0.333366i 0.108184π-0.108184\pi
422422 0 0
423423 −2.71795e10 9.64744e9i −0.848947 0.301336i
424424 0 0
425425 −5.89834e10 3.40541e10i −1.80790 1.04379i
426426 0 0
427427 1.70972e10 + 2.96133e10i 0.514297 + 0.890789i
428428 0 0
429429 6.45563e9 5.90779e8i 0.190594 0.0174420i
430430 0 0
431431 2.73395e10i 0.792285i 0.918189 + 0.396143i 0.129651π0.129651\pi
−0.918189 + 0.396143i 0.870349π0.870349\pi
432432 0 0
433433 −6.96958e9 −0.198269 −0.0991345 0.995074i 0.531607π-0.531607\pi
−0.0991345 + 0.995074i 0.531607π0.531607\pi
434434 0 0
435435 −3.96632e9 4.33413e10i −0.110772 1.21044i
436436 0 0
437437 1.13788e10 6.56953e9i 0.312010 0.180139i
438438 0 0
439439 −4.26214e9 + 7.38223e9i −0.114754 + 0.198760i −0.917682 0.397317i 0.869941π-0.869941\pi
0.802927 + 0.596077i 0.203275π0.203275\pi
440440 0 0
441441 4.74489e10 + 5.56389e10i 1.25450 + 1.47104i
442442 0 0
443443 1.77214e10 + 1.02315e10i 0.460134 + 0.265658i 0.712101 0.702077i 0.247744π-0.247744\pi
−0.251967 + 0.967736i 0.581077π0.581077\pi
444444 0 0
445445 1.13441e10 + 1.96485e10i 0.289287 + 0.501060i
446446 0 0
447447 −2.01478e10 2.85309e10i −0.504658 0.714636i
448448 0 0
449449 1.52992e10i 0.376430i 0.982128 + 0.188215i 0.0602702π0.0602702\pi
−0.982128 + 0.188215i 0.939730π0.939730\pi
450450 0 0
451451 −1.72827e10 −0.417740
452452 0 0
453453 −6.18207e10 2.85275e10i −1.46805 0.677440i
454454 0 0
455455 −3.16597e10 + 1.82788e10i −0.738689 + 0.426483i
456456 0 0
457457 −1.20106e10 + 2.08029e10i −0.275359 + 0.476935i −0.970226 0.242203i 0.922130π-0.922130\pi
0.694867 + 0.719138i 0.255463π0.255463\pi
458458 0 0
459459 4.49278e10 1.26171e10i 1.01219 0.284255i
460460 0 0
461461 4.96796e10 + 2.86825e10i 1.09995 + 0.635058i 0.936209 0.351445i 0.114310π-0.114310\pi
0.163744 + 0.986503i 0.447643π0.447643\pi
462462 0 0
463463 −3.03202e10 5.25162e10i −0.659795 1.14280i −0.980669 0.195676i 0.937310π-0.937310\pi
0.320874 0.947122i 0.396023π-0.396023\pi
464464 0 0
465465 4.23675e10 9.18128e10i 0.906194 1.96377i
466466 0 0
467467 1.01091e10i 0.212543i −0.994337 0.106271i 0.966109π-0.966109\pi
0.994337 0.106271i 0.0338912π-0.0338912\pi
468468 0 0
469469 −5.09238e10 −1.05252
470470 0 0
471471 2.79111e10 1.97101e10i 0.567143 0.400502i
472472 0 0
473473 −8.90324e9 + 5.14029e9i −0.177870 + 0.102693i
474474 0 0
475475 −4.79505e10 + 8.30528e10i −0.941931 + 1.63147i
476476 0 0
477477 −1.24815e9 6.76238e9i −0.0241099 0.130625i
478478 0 0
479479 −1.87971e10 1.08525e10i −0.357067 0.206153i 0.310727 0.950499i 0.399428π-0.399428\pi
−0.667793 + 0.744347i 0.732761π0.732761\pi
480480 0 0
481481 −4.94520e9 8.56534e9i −0.0923855 0.160016i
482482 0 0
483483 3.52486e10 3.22573e9i 0.647670 0.0592706i
484484 0 0
485485 9.64707e10i 1.74353i
486486 0 0
487487 5.82792e9 0.103609 0.0518045 0.998657i 0.483503π-0.483503\pi
0.0518045 + 0.998657i 0.483503π0.483503\pi
488488 0 0
489489 −4.06601e9 4.44306e10i −0.0711103 0.777046i
490490 0 0
491491 2.15718e10 1.24545e10i 0.371159 0.214288i −0.302806 0.953052i 0.597923π-0.597923\pi
0.673964 + 0.738764i 0.264590π0.264590\pi
492492 0 0
493493 2.18446e10 3.78360e10i 0.369791 0.640497i
494494 0 0
495495 6.77407e10 1.25031e10i 1.12831 0.208256i
496496 0 0
497497 6.03915e10 + 3.48670e10i 0.989806 + 0.571465i
498498 0 0
499499 8.19755e9 + 1.41986e10i 0.132215 + 0.229004i 0.924530 0.381109i 0.124458π-0.124458\pi
−0.792315 + 0.610112i 0.791124π0.791124\pi
500500 0 0
501501 2.83548e10 + 4.01526e10i 0.450065 + 0.637328i
502502 0 0
503503 4.61285e10i 0.720606i 0.932835 + 0.360303i 0.117327π0.117327\pi
−0.932835 + 0.360303i 0.882673π0.882673\pi
504504 0 0
505505 −3.84612e10 −0.591367
506506 0 0
507507 −5.50105e10 2.53849e10i −0.832557 0.384188i
508508 0 0
509509 −6.69443e9 + 3.86503e9i −0.0997339 + 0.0575814i −0.549037 0.835798i 0.685006π-0.685006\pi
0.449303 + 0.893379i 0.351672π0.351672\pi
510510 0 0
511511 −4.99087e10 + 8.64444e10i −0.731969 + 1.26781i
512512 0 0
513513 −1.77657e10 6.32615e10i −0.256516 0.913419i
514514 0 0
515515 8.14229e9 + 4.70095e9i 0.115749 + 0.0668278i
516516 0 0
517517 −2.13681e10 3.70107e10i −0.299092 0.518042i
518518 0 0
519519 −3.23943e10 + 7.02003e10i −0.446477 + 0.967541i
520520 0 0
521521 3.97902e10i 0.540039i 0.962855 + 0.270019i 0.0870301π0.0870301\pi
−0.962855 + 0.270019i 0.912970π0.912970\pi
522522 0 0
523523 3.49978e10 0.467772 0.233886 0.972264i 0.424856π-0.424856\pi
0.233886 + 0.972264i 0.424856π0.424856\pi
524524 0 0
525525 −2.11037e11 + 1.49029e11i −2.77792 + 1.96170i
526526 0 0
527527 8.79053e10 5.07521e10i 1.13965 0.657979i
528528 0 0
529529 −3.35092e10 + 5.80397e10i −0.427899 + 0.741143i
530530 0 0
531531 −5.01537e10 + 4.27711e10i −0.630848 + 0.537988i
532532 0 0
533533 −1.26734e10 7.31699e9i −0.157031 0.0906617i
534534 0 0
535535 −1.26536e11 2.19167e11i −1.54454 2.67523i
536536 0 0
537537 −7.61474e10 + 6.96853e9i −0.915709 + 0.0838000i
538538 0 0
539539 1.08354e11i 1.28378i
540540 0 0
541541 2.34001e10 0.273168 0.136584 0.990629i 0.456388π-0.456388\pi
0.136584 + 0.990629i 0.456388π0.456388\pi
542542 0 0
543543 6.40907e9 + 7.00340e10i 0.0737218 + 0.805582i
544544 0 0
545545 2.49236e11 1.43897e11i 2.82504 1.63104i
546546 0 0
547547 1.16091e10 2.01075e10i 0.129673 0.224600i −0.793877 0.608078i 0.791941π-0.791941\pi
0.923550 + 0.383478i 0.125274π0.125274\pi
548548 0 0
549549 1.82497e10 5.14145e10i 0.200894 0.565974i
550550 0 0
551551 −5.32757e10 3.07588e10i −0.577994 0.333705i
552552 0 0
553553 2.62770e10 + 4.55131e10i 0.280980 + 0.486671i
554554 0 0
555555 −6.06242e10 8.58486e10i −0.638960 0.904819i
556556 0 0
557557 3.65617e10i 0.379844i −0.981799 0.189922i 0.939176π-0.939176\pi
0.981799 0.189922i 0.0608236π-0.0608236\pi
558558 0 0
559559 −8.70499e9 −0.0891499
560560 0 0
561561 6.27864e10 + 2.89731e10i 0.633891 + 0.292512i
562562 0 0
563563 1.15433e11 6.66450e10i 1.14893 0.663337i 0.200306 0.979733i 0.435806π-0.435806\pi
0.948627 + 0.316397i 0.102473π0.102473\pi
564564 0 0
565565 −1.54354e11 + 2.67348e11i −1.51469 + 2.62352i
566566 0 0
567567 2.79503e10 1.74795e11i 0.270430 1.69121i
568568 0 0
569569 −1.42643e11 8.23549e10i −1.36082 0.785670i −0.371087 0.928598i 0.621015π-0.621015\pi
−0.989733 + 0.142928i 0.954348π0.954348\pi
570570 0 0
571571 6.84597e10 + 1.18576e11i 0.644007 + 1.11545i 0.984530 + 0.175216i 0.0560625π0.0560625\pi
−0.340523 + 0.940236i 0.610604π0.610604\pi
572572 0 0
573573 2.39036e9 5.18004e9i 0.0221740 0.0480524i
574574 0 0
575575 8.24235e10i 0.754015i
576576 0 0
577577 −1.01555e11 −0.916218 −0.458109 0.888896i 0.651473π-0.651473\pi
−0.458109 + 0.888896i 0.651473π0.651473\pi
578578 0 0
579579 −1.81122e11 + 1.27904e11i −1.61160 + 1.13807i
580580 0 0
581581 1.14807e11 6.62840e10i 1.00755 0.581707i
582582 0 0
583583 5.09485e9 8.82453e9i 0.0441018 0.0763866i
584584 0 0
585585 5.49677e10 + 1.95109e10i 0.469336 + 0.166592i
586586 0 0
587587 3.25669e10 + 1.88025e10i 0.274299 + 0.158367i 0.630840 0.775913i 0.282711π-0.282711\pi
−0.356541 + 0.934280i 0.616044π0.616044\pi
588588 0 0
589589 −7.14626e10 1.23777e11i −0.593769 1.02844i
590590 0 0
591591 4.41449e10 4.03986e9i 0.361852 0.0331144i
592592 0 0
593593 1.30215e11i 1.05304i 0.850163 + 0.526519i 0.176503π0.176503\pi
−0.850163 + 0.526519i 0.823497π0.823497\pi
594594 0 0
595595 −3.89953e11 −3.11132
596596 0 0
597597 −1.96700e10 2.14940e11i −0.154848 1.69208i
598598 0 0
599599 −1.13848e11 + 6.57304e10i −0.884341 + 0.510575i −0.872087 0.489350i 0.837234π-0.837234\pi
−0.0122537 + 0.999925i 0.503901π0.503901\pi
600600 0 0
601601 2.15092e10 3.72550e10i 0.164864 0.285553i −0.771743 0.635935i 0.780615π-0.780615\pi
0.936607 + 0.350382i 0.113948π0.113948\pi
602602 0 0
603603 5.27213e10 + 6.18214e10i 0.398765 + 0.467595i
604604 0 0
605605 −1.12081e11 6.47102e10i −0.836588 0.483004i
606606 0 0
607607 1.29606e10 + 2.24485e10i 0.0954710 + 0.165361i 0.909805 0.415036i 0.136231π-0.136231\pi
−0.814334 + 0.580396i 0.802898π0.802898\pi
608608 0 0
609609 −9.55972e10 1.35373e11i −0.694986 0.984156i
610610 0 0
611611 3.61866e10i 0.259646i
612612 0 0
613613 1.28485e11 0.909933 0.454967 0.890508i 0.349651π-0.349651\pi
0.454967 + 0.890508i 0.349651π0.349651\pi
614614 0 0
615615 −1.41194e11 6.51549e10i −0.986999 0.455456i
616616 0 0
617617 −1.02289e11 + 5.90563e10i −0.705808 + 0.407498i −0.809507 0.587110i 0.800265π-0.800265\pi
0.103699 + 0.994609i 0.466932π0.466932\pi
618618 0 0
619619 6.40218e10 1.10889e11i 0.436079 0.755312i −0.561304 0.827610i 0.689700π-0.689700\pi
0.997383 + 0.0722982i 0.0230333π0.0230333\pi
620620 0 0
621621 −4.04089e10 3.94521e10i −0.271713 0.265280i
622622 0 0
623623 7.48177e10 + 4.31960e10i 0.496652 + 0.286742i
624624 0 0
625625 −7.30176e10 1.26470e11i −0.478528 0.828835i
626626 0 0
627627 4.07962e10 8.84077e10i 0.263967 0.572032i
628628 0 0
629629 1.05499e11i 0.673980i
630630 0 0
631631 2.40613e11 1.51775 0.758877 0.651233i 0.225748π-0.225748\pi
0.758877 + 0.651233i 0.225748π0.225748\pi
632632 0 0
633633 1.24277e11 8.77610e10i 0.774059 0.546621i
634634 0 0
635635 7.65738e10 4.42099e10i 0.470961 0.271910i
636636 0 0
637637 −4.58740e10 + 7.94561e10i −0.278618 + 0.482580i
638638 0 0
639639 −2.01947e10 1.09413e11i −0.121125 0.656244i
640640 0 0
641641 1.87111e11 + 1.08029e11i 1.10833 + 0.639893i 0.938396 0.345563i 0.112312π-0.112312\pi
0.169932 + 0.985456i 0.445645π0.445645\pi
642642 0 0
643643 1.85669e10 + 3.21588e10i 0.108616 + 0.188129i 0.915210 0.402977i 0.132025π-0.132025\pi
−0.806594 + 0.591106i 0.798691π0.798691\pi
644644 0 0
645645 −9.21153e10 + 8.42981e9i −0.532222 + 0.0487056i
646646 0 0
647647 5.88598e10i 0.335894i −0.985796 0.167947i 0.946286π-0.946286\pi
0.985796 0.167947i 0.0537137π-0.0537137\pi
648648 0 0
649649 −9.76719e10 −0.550543
650650 0 0
651651 −3.50892e10 3.83431e11i −0.195366 2.13483i
652652 0 0
653653 −1.48938e11 + 8.59893e10i −0.819129 + 0.472925i −0.850116 0.526595i 0.823468π-0.823468\pi
0.0309867 + 0.999520i 0.490135π0.490135\pi
654654 0 0
655655 −2.04384e11 + 3.54003e11i −1.11040 + 1.92328i
656656 0 0
657657 1.56614e11 2.89067e10i 0.840559 0.155145i
658658 0 0
659659 −2.34850e11 1.35591e11i −1.24523 0.718934i −0.275075 0.961423i 0.588703π-0.588703\pi
−0.970154 + 0.242489i 0.922036π0.922036\pi
660660 0 0
661661 1.21597e11 + 2.10613e11i 0.636968 + 1.10326i 0.986095 + 0.166185i 0.0531451π0.0531451\pi
−0.349126 + 0.937076i 0.613522π0.613522\pi
662662 0 0
663663 3.37749e10 + 4.78279e10i 0.174799 + 0.247530i
664664 0 0
665665 5.49082e11i 2.80770i
666666 0 0
667667 −5.28721e10 −0.267130
668668 0 0
669669 2.13363e11 + 9.84574e10i 1.06516 + 0.491523i
670670 0 0
671671 7.00118e10 4.04213e10i 0.345367 0.199398i
672672 0 0
673673 1.44609e10 2.50471e10i 0.0704914 0.122095i −0.828625 0.559804i 0.810877π-0.810877\pi
0.899117 + 0.437709i 0.144210π0.144210\pi
674674 0 0
675675 3.99406e11 + 1.01909e11i 1.92398 + 0.490904i
676676 0 0
677677 1.58076e11 + 9.12654e10i 0.752509 + 0.434462i 0.826600 0.562790i 0.190272π-0.190272\pi
−0.0740905 + 0.997252i 0.523605π0.523605\pi
678678 0 0
679679 1.83671e11 + 3.18127e11i 0.864095 + 1.49666i
680680 0 0
681681 −1.41459e11 + 3.06549e11i −0.657721 + 1.42532i
682682 0 0
683683 3.71514e11i 1.70723i 0.520903 + 0.853616i 0.325595π0.325595\pi
−0.520903 + 0.853616i 0.674405π0.674405\pi
684684 0 0
685685 1.04359e11 0.473989
686686 0 0
687687 2.38219e11 1.68225e11i 1.06942 0.755201i
688688 0 0
689689 7.47209e9 4.31401e9i 0.0331562 0.0191428i
690690 0 0
691691 9.88238e10 1.71168e11i 0.433460 0.750775i −0.563708 0.825974i 0.690626π-0.690626\pi
0.997169 + 0.0751987i 0.0239591π0.0239591\pi
692692 0 0
693693 1.99581e11 1.70203e11i 0.865339 0.737962i
694694 0 0
695695 3.98846e11 + 2.30274e11i 1.70949 + 0.986973i
696696 0 0
697697 −7.80492e10 1.35185e11i −0.330702 0.572793i
698698 0 0
699699 2.30990e11 2.11388e10i 0.967575 0.0885464i
700700 0 0
701701 6.70096e10i 0.277501i −0.990327 0.138751i 0.955691π-0.955691\pi
0.990327 0.138751i 0.0443087π-0.0443087\pi
702702 0 0
703703 −1.48551e11 −0.608209
704704 0 0
705705 −3.50426e10 3.82922e11i −0.141854 1.55008i
706706 0 0
707707 −1.26832e11 + 7.32264e10i −0.507634 + 0.293083i
708708 0 0
709709 5.70291e10 9.87774e10i 0.225690 0.390906i −0.730836 0.682553i 0.760870π-0.760870\pi
0.956526 + 0.291647i 0.0942031π0.0942031\pi
710710 0 0
711711 2.80483e10 7.90198e10i 0.109756 0.309213i
712712 0 0
713713 −1.06382e11 6.14196e10i −0.411632 0.237656i
714714 0 0
715715 4.32147e10 + 7.48501e10i 0.165351 + 0.286397i
716716 0 0
717717 6.55522e10 + 9.28271e10i 0.248034 + 0.351235i
718718 0 0
719719 2.31072e11i 0.864633i −0.901722 0.432317i 0.857696π-0.857696\pi
0.901722 0.432317i 0.142304π-0.142304\pi
720720 0 0
721721 3.58007e10 0.132480
722722 0 0
723723 −3.26266e11 1.50557e11i −1.19404 0.550995i
724724 0 0
725725 3.34208e11 1.92955e11i 1.20966 0.698400i
726726 0 0
727727 −1.70637e11 + 2.95551e11i −0.610850 + 1.05802i 0.380248 + 0.924885i 0.375839π0.375839\pi
−0.991097 + 0.133138i 0.957495π0.957495\pi
728728 0 0
729729 −2.41138e11 + 1.47034e11i −0.853799 + 0.520603i
730730 0 0
731731 −8.04145e10 4.64274e10i −0.281621 0.162594i
732732 0 0
733733 2.07203e11 + 3.58887e11i 0.717763 + 1.24320i 0.961884 + 0.273457i 0.0881670π0.0881670\pi
−0.244122 + 0.969745i 0.578500π0.578500\pi
734734 0 0
735735 −4.08489e11 + 8.85220e11i −1.39969 + 3.03320i
736736 0 0
737737 1.20394e11i 0.408071i
738738 0 0
739739 −2.71838e11 −0.911449 −0.455725 0.890121i 0.650620π-0.650620\pi
−0.455725 + 0.890121i 0.650620π0.650620\pi
740740 0 0
741741 6.73451e10 4.75574e10i 0.223374 0.157741i
742742 0 0
743743 2.96676e11 1.71286e11i 0.973479 0.562039i 0.0731842 0.997318i 0.476684π-0.476684\pi
0.900295 + 0.435280i 0.143351π0.143351\pi
744744 0 0
745745 2.32837e11 4.03286e11i 0.755835 1.30914i
746746 0 0
747747 −1.99328e11 7.07521e10i −0.640157 0.227225i
748748 0 0
749749 −8.34547e11 4.81826e11i −2.65170 1.53096i
750750 0 0
751751 5.25839e10 + 9.10780e10i 0.165308 + 0.286321i 0.936765 0.349960i 0.113805π-0.113805\pi
−0.771457 + 0.636282i 0.780472π0.780472\pi
752752 0 0
753753 1.95272e11 1.78700e10i 0.607378 0.0555834i
754754 0 0
755755 9.07750e11i 2.79369i
756756 0 0
757757 −4.23531e11 −1.28974 −0.644869 0.764293i 0.723088π-0.723088\pi
−0.644869 + 0.764293i 0.723088π0.723088\pi
758758 0 0
759759 −7.62628e9 8.33349e10i −0.0229798 0.251108i
760760 0 0
761761 4.62418e11 2.66977e11i 1.37878 0.796041i 0.386770 0.922176i 0.373591π-0.373591\pi
0.992013 + 0.126135i 0.0402574π0.0402574\pi
762762 0 0
763763 5.47931e11 9.49044e11i 1.61669 2.80019i
764764 0 0
765765 4.03718e11 + 4.73403e11i 1.17878 + 1.38224i
766766 0 0
767767 −7.16228e10 4.13514e10i −0.206952 0.119484i
768768 0 0
769769 −5.66789e10 9.81707e10i −0.162075 0.280722i 0.773538 0.633750i 0.218485π-0.218485\pi
−0.935613 + 0.353028i 0.885152π0.885152\pi
770770 0 0
771771 1.12715e11 + 1.59613e11i 0.318980 + 0.451701i
772772 0 0
773773 2.83942e11i 0.795265i −0.917545 0.397632i 0.869832π-0.869832\pi
0.917545 0.397632i 0.130168π-0.130168\pi
774774 0 0
775775 8.96595e11 2.48536
776776 0 0
777777 −3.63365e11 1.67677e11i −0.996918 0.460033i
778778 0 0
779779 −1.90350e11 + 1.09899e11i −0.516897 + 0.298430i
780780 0 0
781781 8.24328e10 1.42778e11i 0.221562 0.383757i
782782 0 0
783783 −6.53712e10 + 2.56207e11i −0.173916 + 0.681621i
784784 0 0
785785 3.94524e11 + 2.27779e11i 1.03895 + 0.599839i
786786 0 0
787787 −3.25568e11 5.63900e11i −0.848678 1.46995i −0.882389 0.470521i 0.844066π-0.844066\pi
0.0337110 0.999432i 0.489267π-0.489267\pi
788788 0 0
789789 −2.21430e11 + 4.79851e11i −0.571384 + 1.23822i
790790 0 0
791791 1.17550e12i 3.00273i
792792 0 0
793793 6.84528e10 0.173100
794794 0 0
795795 7.48913e10 5.28863e10i 0.187483 0.132396i
796796 0 0
797797 4.16469e11 2.40448e11i 1.03216 0.595921i 0.114561 0.993416i 0.463454π-0.463454\pi
0.917604 + 0.397496i 0.130121π0.130121\pi
798798 0 0
799799 1.92998e11 3.34283e11i 0.473550 0.820213i
800800 0 0
801801 −2.50188e10 1.35549e11i −0.0607765 0.329281i
802802 0 0
803803 2.04372e11 + 1.17994e11i 0.491541 + 0.283791i
804804 0 0
805805 2.35958e11 + 4.08692e11i 0.561890 + 0.973223i
806806 0 0
807807 −5.35705e11 + 4.90243e10i −1.26308 + 0.115589i
808808 0 0
809809 1.97505e11i 0.461089i 0.973062 + 0.230545i 0.0740508π0.0740508\pi
−0.973062 + 0.230545i 0.925949π0.925949\pi
810810 0 0
811811 1.30697e11 0.302121 0.151061 0.988524i 0.451731π-0.451731\pi
0.151061 + 0.988524i 0.451731π0.451731\pi
812812 0 0
813813 −2.00953e10 2.19588e11i −0.0459973 0.502627i
814814 0 0
815815 5.15152e11 2.97423e11i 1.16763 0.674131i
816816 0 0
817817 −6.53730e10 + 1.13229e11i −0.146727 + 0.254139i
818818 0 0
819819 2.18411e11 4.03129e10i 0.485445 0.0896001i
820820 0 0
821821 2.25465e10 + 1.30172e10i 0.0496256 + 0.0286514i 0.524608 0.851344i 0.324212π-0.324212\pi
−0.474982 + 0.879996i 0.657545π0.657545\pi
822822 0 0
823823 −1.02310e11 1.77206e11i −0.223007 0.386260i 0.732712 0.680539i 0.238254π-0.238254\pi
−0.955720 + 0.294278i 0.904921π0.904921\pi
824824 0 0
825825 3.52334e11 + 4.98933e11i 0.760570 + 1.07703i
826826 0 0
827827 2.31574e11i 0.495072i 0.968879 + 0.247536i 0.0796208π0.0796208\pi
−0.968879 + 0.247536i 0.920379π0.920379\pi
828828 0 0
829829 7.98205e11 1.69004 0.845019 0.534736i 0.179589π-0.179589\pi
0.845019 + 0.534736i 0.179589π0.179589\pi
830830 0 0
831831 7.36861e11 + 3.40029e11i 1.54519 + 0.713036i
832832 0 0
833833 −8.47545e11 + 4.89331e11i −1.76029 + 1.01630i
834834 0 0
835835 −3.27681e11 + 5.67559e11i −0.674070 + 1.16752i
836836 0 0
837837 −4.29156e11 + 4.39563e11i −0.874407 + 0.895612i
838838 0 0
839839 −5.56790e11 3.21463e11i −1.12368 0.648758i −0.181343 0.983420i 0.558045π-0.558045\pi
−0.942338 + 0.334662i 0.891378π0.891378\pi
840840 0 0
841841 −1.26349e11 2.18842e11i −0.252573 0.437469i
842842 0 0
843843 −2.67669e11 + 5.80053e11i −0.530014 + 1.14857i
844844 0 0
845845 8.07752e11i 1.58435i
846846 0 0
847847 −4.92808e11 −0.957511
848848 0 0
849849 −6.44258e10 + 4.54959e10i −0.124002 + 0.0875672i
850850 0 0
851851 −1.10569e11 + 6.38370e10i −0.210821 + 0.121718i
852852 0 0
853853 9.75384e10 1.68942e11i 0.184238 0.319110i −0.759081 0.650996i 0.774352π-0.774352\pi
0.943320 + 0.331886i 0.107685π0.107685\pi
854854 0 0
855855 6.66584e11 5.68464e11i 1.24736 1.06375i
856856 0 0
857857 −3.84904e10 2.22224e10i −0.0713557 0.0411972i 0.463898 0.885889i 0.346451π-0.346451\pi
−0.535253 + 0.844692i 0.679784π0.679784\pi
858858 0 0
859859 1.25294e11 + 2.17015e11i 0.230122 + 0.398582i 0.957844 0.287290i 0.0927543π-0.0927543\pi
−0.727722 + 0.685872i 0.759421π0.759421\pi
860860 0 0
861861 −5.89659e11 + 5.39619e10i −1.07297 + 0.0981916i
862862 0 0
863863 3.91558e11i 0.705915i −0.935639 0.352958i 0.885176π-0.885176\pi
0.935639 0.352958i 0.114824π-0.114824\pi
864864 0 0
865865 −1.03079e12 −1.84122
866866 0 0
867867 5.42428e9 + 5.92729e10i 0.00959988 + 0.104901i
868868 0 0
869869 1.07602e11 6.21242e10i 0.188687 0.108939i
870870 0 0
871871 −5.09714e10 + 8.82851e10i −0.0885633 + 0.153396i
872872 0 0
873873 1.96052e11 5.52333e11i 0.337531 0.950920i
874874 0 0
875875 −1.48070e12 8.54884e11i −2.52601 1.45839i
876876 0 0
877877 3.99055e10 + 6.91184e10i 0.0674582 + 0.116841i 0.897782 0.440441i 0.145178π-0.145178\pi
−0.830324 + 0.557282i 0.811844π0.811844\pi
878878 0 0
879879 2.47869e11 + 3.51002e11i 0.415209 + 0.587968i
880880 0 0
881881 5.84453e11i 0.970166i 0.874468 + 0.485083i 0.161211π0.161211\pi
−0.874468 + 0.485083i 0.838789π0.838789\pi
882882 0 0
883883 −7.52384e10 −0.123765 −0.0618823 0.998083i 0.519710π-0.519710\pi
−0.0618823 + 0.998083i 0.519710π0.519710\pi
884884 0 0
885885 −7.97949e11 3.68218e11i −1.30077 0.600249i
886886 0 0
887887 −6.05896e11 + 3.49814e11i −0.978822 + 0.565123i −0.901914 0.431915i 0.857838π-0.857838\pi
−0.0769076 + 0.997038i 0.524505π0.524505\pi
888888 0 0
889889 1.68343e11 2.91578e11i 0.269518 0.466819i
890890 0 0
891891 −4.13252e11 6.60802e10i −0.655698 0.104848i
892892 0 0
893893 −4.70694e11 2.71755e11i −0.740172 0.427338i
894894 0 0
895895 −5.09739e11 8.82894e11i −0.794430 1.37599i
896896 0 0
897897 2.96892e10 6.43382e10i 0.0458594 0.0993800i
898898 0 0
899899 5.75137e11i 0.880507i
900900 0 0
901901 9.20338e10 0.139652
902902 0 0
903903 −2.87715e11 + 2.03177e11i −0.432725 + 0.305579i
904904 0 0
905905 −8.12012e11 + 4.68815e11i −1.21051 + 0.698888i
906906 0 0
907907 5.79947e10 1.00450e11i 0.0856958 0.148429i −0.819992 0.572375i 0.806022π-0.806022\pi
0.905687 + 0.423946i 0.139355π0.139355\pi
908908 0 0
909909 2.20205e11 + 7.81624e10i 0.322532 + 0.114483i
910910 0 0
911911 −9.81176e11 5.66482e11i −1.42454 0.822456i −0.427854 0.903848i 0.640730π-0.640730\pi
−0.996682 + 0.0813913i 0.974064π0.974064\pi
912912 0 0
913913 −1.56709e11 2.71428e11i −0.225533 0.390635i
914914 0 0
915915 7.24360e11 6.62889e10i 1.03340 0.0945706i
916916 0 0
917917 1.55651e12i 2.20127i
918918 0 0
919919 3.06256e11 0.429361 0.214680 0.976684i 0.431129π-0.431129\pi
0.214680 + 0.976684i 0.431129π0.431129\pi
920920 0 0
921921 9.90936e10 + 1.08283e12i 0.137723 + 1.50495i
922922 0 0
923923 1.20896e11 6.97992e10i 0.166573 0.0961709i
924924 0 0
925925 4.65942e11 8.07035e11i 0.636451 1.10237i
926926 0 0
927927 −3.70644e10 4.34619e10i −0.0501923 0.0588559i
928928 0 0
929929 −8.75115e11 5.05248e11i −1.17490 0.678331i −0.220074 0.975483i 0.570630π-0.570630\pi
−0.954830 + 0.297152i 0.903963π0.903963\pi
930930 0 0
931931 6.89012e11 + 1.19340e12i 0.917124 + 1.58851i
932932 0 0
933933 1.06983e11 + 1.51496e11i 0.141185 + 0.199929i
934934 0 0
935935 9.21929e11i 1.20629i
936936 0 0
937937 −3.80033e11 −0.493018 −0.246509 0.969141i 0.579283π-0.579283\pi
−0.246509 + 0.969141i 0.579283π0.579283\pi
938938 0 0
939939 −1.31526e12 6.06933e11i −1.69180 0.780689i
940940 0 0
941941 4.20722e11 2.42904e11i 0.536583 0.309796i −0.207110 0.978318i 0.566406π-0.566406\pi
0.743693 + 0.668521i 0.233072π0.233072\pi
942942 0 0
943943 −9.44541e10 + 1.63599e11i −0.119447 + 0.206888i
944944 0 0
945945 2.27217e12 6.38094e11i 2.84914 0.800124i
946946 0 0
947947 −4.64773e9 2.68337e9i −0.00577885 0.00333642i 0.497108 0.867689i 0.334395π-0.334395\pi
−0.502887 + 0.864352i 0.667729π0.667729\pi
948948 0 0
949949 9.99106e10 + 1.73050e11i 0.123182 + 0.213357i
950950 0 0
951951 2.26279e11 4.90359e11i 0.276644 0.599504i
952952 0 0
953953 2.87948e11i 0.349094i −0.984649 0.174547i 0.944154π-0.944154\pi
0.984649 0.174547i 0.0558462π-0.0558462\pi
954954 0 0
955955 7.60616e10 0.0914433
956956 0 0
957957 −3.20050e11 + 2.26011e11i −0.381566 + 0.269453i
958958 0 0
959959 3.44141e11 1.98690e11i 0.406876 0.234910i
960960 0 0
961961 −2.41670e11 + 4.18585e11i −0.283354 + 0.490784i
962962 0 0
963963 2.79069e11 + 1.51197e12i 0.324494 + 1.75808i
964964 0 0
965965 −2.56017e12 1.47812e12i −2.95230 1.70451i
966966 0 0
967967 4.51417e11 + 7.81877e11i 0.516264 + 0.894195i 0.999822 + 0.0188827i 0.00601091π0.00601091\pi
−0.483558 + 0.875312i 0.660656π0.660656\pi
968968 0 0
969969 8.75761e11 8.01441e10i 0.993323 0.0909027i
970970 0 0
971971 8.08040e11i 0.908983i 0.890751 + 0.454492i 0.150179π0.150179\pi
−0.890751 + 0.454492i 0.849821π0.849821\pi
972972 0 0
973973 1.75368e12 1.95658
974974 0 0
975975 4.71328e10 + 5.15035e11i 0.0521561 + 0.569926i
976976 0 0
977977 1.39658e12 8.06317e11i 1.53281 0.884968i 0.533579 0.845750i 0.320847π-0.320847\pi
0.999231 0.0392176i 0.0124866π-0.0124866\pi
978978 0 0
979979 1.02124e11 1.76884e11i 0.111173 0.192557i
980980 0 0
981981 −1.71941e12 + 3.17357e11i −1.85653 + 0.342666i
982982 0 0
983983 −1.24469e12 7.18625e11i −1.33306 0.769641i −0.347290 0.937758i 0.612898π-0.612898\pi
−0.985767 + 0.168117i 0.946231π0.946231\pi
984984 0 0
985985 2.95511e11 + 5.11840e11i 0.313927 + 0.543737i
986986 0 0
987987 −8.44606e11 1.19603e12i −0.889991 1.26030i
988988 0 0
989989 1.12372e11i 0.117455i
990990 0 0
991991 1.11953e12 1.16075 0.580377 0.814348i 0.302905π-0.302905\pi
0.580377 + 0.814348i 0.302905π0.302905\pi
992992 0 0
993993 −1.15365e12 5.32356e11i −1.18652 0.547527i
994994 0 0
995995 2.49213e12 1.43883e12i 2.54261 1.46797i
996996 0 0
997997 −4.45668e11 + 7.71920e11i −0.451057 + 0.781253i −0.998452 0.0556212i 0.982286π-0.982286\pi
0.547395 + 0.836874i 0.315619π0.315619\pi
998998 0 0
999999 1.72632e11 + 6.14721e11i 0.173324 + 0.617186i
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 144.9.q.b.65.5 16
3.2 odd 2 432.9.q.c.305.1 16
4.3 odd 2 18.9.d.a.11.6 yes 16
9.4 even 3 432.9.q.c.17.1 16
9.5 odd 6 inner 144.9.q.b.113.5 16
12.11 even 2 54.9.d.a.35.1 16
36.7 odd 6 162.9.b.c.161.8 16
36.11 even 6 162.9.b.c.161.9 16
36.23 even 6 18.9.d.a.5.6 16
36.31 odd 6 54.9.d.a.17.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.9.d.a.5.6 16 36.23 even 6
18.9.d.a.11.6 yes 16 4.3 odd 2
54.9.d.a.17.1 16 36.31 odd 6
54.9.d.a.35.1 16 12.11 even 2
144.9.q.b.65.5 16 1.1 even 1 trivial
144.9.q.b.113.5 16 9.5 odd 6 inner
162.9.b.c.161.8 16 36.7 odd 6
162.9.b.c.161.9 16 36.11 even 6
432.9.q.c.17.1 16 9.4 even 3
432.9.q.c.305.1 16 3.2 odd 2