Properties

Label 169.10.a.f.1.5
Level 169169
Weight 1010
Character 169.1
Self dual yes
Analytic conductor 87.04187.041
Analytic rank 00
Dimension 2020
Inner twists 22

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [169,10,Mod(1,169)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("169.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")
 
Level: N N == 169=132 169 = 13^{2}
Weight: k k == 10 10
Character orbit: [χ][\chi] == 169.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 87.041056311787.0410563117
Analytic rank: 00
Dimension: 2020
Coefficient field: Q[x]/(x20)\mathbb{Q}[x]/(x^{20} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x207679x18+24599364x1642662336000x14+43527566862400x12++25 ⁣ ⁣36 x^{20} - 7679 x^{18} + 24599364 x^{16} - 42662336000 x^{14} + 43527566862400 x^{12} + \cdots + 25\!\cdots\!36 Copy content Toggle raw display
Coefficient ring: Z[a1,,a19]\Z[a_1, \ldots, a_{19}]
Coefficient ring index: 2253101312 2^{25}\cdot 3^{10}\cdot 13^{12}
Twist minimal: no (minimal twist has level 13)
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.5
Root 27.6771-27.6771 of defining polynomial
Character χ\chi == 169.1

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
f(q)f(q) == q27.6771q244.2389q3+254.024q4+2467.22q5+1224.41q63580.38q7+7140.04q817725.9q968285.5q10+79291.0q1111237.7q12+99094.7q14109147.q15327676.q16+8421.95q17+490603.q18+434724.q19+626732.q20+158392.q212.19455e6q22+1.39043e6q23315867.q24+4.13403e6q25+1.65493e6q27909503.q28+2.73548e6q29+3.02087e6q30+1.77534e6q31+5.41344e6q323.50775e6q33233095.q348.83357e6q354.50281e6q36+1.57016e7q371.20319e7q38+1.76160e7q40112436.q414.38384e6q422.68150e7q43+2.01418e7q444.37337e7q453.84831e7q463.59357e7q47+1.44960e7q482.75345e7q491.14418e8q50372578.q513.83324e7q534.58037e7q54+1.95628e8q552.55641e7q561.92317e7q577.57103e7q58+646636.q592.77259e7q60+4.42458e7q614.91363e7q62+6.34656e7q63+1.79417e7q64+9.70844e7q661.65124e8q67+2.13938e6q686.15110e7q69+2.44488e8q701.69848e8q711.26564e8q72+8.20084e7q734.34574e8q741.82885e8q75+1.10430e8q762.83892e8q77+5.10327e8q798.08448e8q80+2.75687e8q81+3.11191e6q82+4.20094e8q83+4.02354e7q84+2.07788e7q85+7.42162e8q861.21015e8q87+5.66141e8q883.97908e8q89+1.21042e9q90+3.53202e8q927.85390e7q93+9.94598e8q94+1.07256e9q952.39484e8q96+2.57864e8q97+7.62076e8q981.40551e9q99+O(q100)q-27.6771 q^{2} -44.2389 q^{3} +254.024 q^{4} +2467.22 q^{5} +1224.41 q^{6} -3580.38 q^{7} +7140.04 q^{8} -17725.9 q^{9} -68285.5 q^{10} +79291.0 q^{11} -11237.7 q^{12} +99094.7 q^{14} -109147. q^{15} -327676. q^{16} +8421.95 q^{17} +490603. q^{18} +434724. q^{19} +626732. q^{20} +158392. q^{21} -2.19455e6 q^{22} +1.39043e6 q^{23} -315867. q^{24} +4.13403e6 q^{25} +1.65493e6 q^{27} -909503. q^{28} +2.73548e6 q^{29} +3.02087e6 q^{30} +1.77534e6 q^{31} +5.41344e6 q^{32} -3.50775e6 q^{33} -233095. q^{34} -8.83357e6 q^{35} -4.50281e6 q^{36} +1.57016e7 q^{37} -1.20319e7 q^{38} +1.76160e7 q^{40} -112436. q^{41} -4.38384e6 q^{42} -2.68150e7 q^{43} +2.01418e7 q^{44} -4.37337e7 q^{45} -3.84831e7 q^{46} -3.59357e7 q^{47} +1.44960e7 q^{48} -2.75345e7 q^{49} -1.14418e8 q^{50} -372578. q^{51} -3.83324e7 q^{53} -4.58037e7 q^{54} +1.95628e8 q^{55} -2.55641e7 q^{56} -1.92317e7 q^{57} -7.57103e7 q^{58} +646636. q^{59} -2.77259e7 q^{60} +4.42458e7 q^{61} -4.91363e7 q^{62} +6.34656e7 q^{63} +1.79417e7 q^{64} +9.70844e7 q^{66} -1.65124e8 q^{67} +2.13938e6 q^{68} -6.15110e7 q^{69} +2.44488e8 q^{70} -1.69848e8 q^{71} -1.26564e8 q^{72} +8.20084e7 q^{73} -4.34574e8 q^{74} -1.82885e8 q^{75} +1.10430e8 q^{76} -2.83892e8 q^{77} +5.10327e8 q^{79} -8.08448e8 q^{80} +2.75687e8 q^{81} +3.11191e6 q^{82} +4.20094e8 q^{83} +4.02354e7 q^{84} +2.07788e7 q^{85} +7.42162e8 q^{86} -1.21015e8 q^{87} +5.66141e8 q^{88} -3.97908e8 q^{89} +1.21042e9 q^{90} +3.53202e8 q^{92} -7.85390e7 q^{93} +9.94598e8 q^{94} +1.07256e9 q^{95} -2.39484e8 q^{96} +2.57864e8 q^{97} +7.62076e8 q^{98} -1.40551e9 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 20q+326q3+5118q4+129526q9+88390q10+427652q12+473556q14+1189618q1699312q175073532q22+6252378q23+1529274q25+18052718q27+5424828q29++9251202540q95+O(q100) 20 q + 326 q^{3} + 5118 q^{4} + 129526 q^{9} + 88390 q^{10} + 427652 q^{12} + 473556 q^{14} + 1189618 q^{16} - 99312 q^{17} - 5073532 q^{22} + 6252378 q^{23} + 1529274 q^{25} + 18052718 q^{27} + 5424828 q^{29}+ \cdots + 9251202540 q^{95}+O(q^{100}) Copy content Toggle raw display

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 −27.6771 −1.22317 −0.611584 0.791179i 0.709467π-0.709467\pi
−0.611584 + 0.791179i 0.709467π0.709467\pi
33 −44.2389 −0.315325 −0.157663 0.987493i 0.550396π-0.550396\pi
−0.157663 + 0.987493i 0.550396π0.550396\pi
44 254.024 0.496141
55 2467.22 1.76540 0.882698 0.469941i 0.155725π-0.155725\pi
0.882698 + 0.469941i 0.155725π0.155725\pi
66 1224.41 0.385696
77 −3580.38 −0.563622 −0.281811 0.959470i 0.590935π-0.590935\pi
−0.281811 + 0.959470i 0.590935π0.590935\pi
88 7140.04 0.616305
99 −17725.9 −0.900570
1010 −68285.5 −2.15938
1111 79291.0 1.63289 0.816445 0.577424i 0.195942π-0.195942\pi
0.816445 + 0.577424i 0.195942π0.195942\pi
1212 −11237.7 −0.156446
1313 0 0
1414 99094.7 0.689405
1515 −109147. −0.556674
1616 −327676. −1.24999
1717 8421.95 0.0244564 0.0122282 0.999925i 0.496108π-0.496108\pi
0.0122282 + 0.999925i 0.496108π0.496108\pi
1818 490603. 1.10155
1919 434724. 0.765284 0.382642 0.923897i 0.375014π-0.375014\pi
0.382642 + 0.923897i 0.375014π0.375014\pi
2020 626732. 0.875885
2121 158392. 0.177724
2222 −2.19455e6 −1.99730
2323 1.39043e6 1.03603 0.518016 0.855371i 0.326671π-0.326671\pi
0.518016 + 0.855371i 0.326671π0.326671\pi
2424 −315867. −0.194336
2525 4.13403e6 2.11662
2626 0 0
2727 1.65493e6 0.599297
2828 −909503. −0.279636
2929 2.73548e6 0.718195 0.359098 0.933300i 0.383084π-0.383084\pi
0.359098 + 0.933300i 0.383084π0.383084\pi
3030 3.02087e6 0.680905
3131 1.77534e6 0.345266 0.172633 0.984986i 0.444773π-0.444773\pi
0.172633 + 0.984986i 0.444773π0.444773\pi
3232 5.41344e6 0.912637
3333 −3.50775e6 −0.514891
3434 −233095. −0.0299143
3535 −8.83357e6 −0.995016
3636 −4.50281e6 −0.446809
3737 1.57016e7 1.37732 0.688661 0.725084i 0.258199π-0.258199\pi
0.688661 + 0.725084i 0.258199π0.258199\pi
3838 −1.20319e7 −0.936071
3939 0 0
4040 1.76160e7 1.08802
4141 −112436. −0.00621411 −0.00310706 0.999995i 0.500989π-0.500989\pi
−0.00310706 + 0.999995i 0.500989π0.500989\pi
4242 −4.38384e6 −0.217387
4343 −2.68150e7 −1.19611 −0.598053 0.801457i 0.704059π-0.704059\pi
−0.598053 + 0.801457i 0.704059π0.704059\pi
4444 2.01418e7 0.810143
4545 −4.37337e7 −1.58986
4646 −3.84831e7 −1.26724
4747 −3.59357e7 −1.07420 −0.537101 0.843518i 0.680481π-0.680481\pi
−0.537101 + 0.843518i 0.680481π0.680481\pi
4848 1.44960e7 0.394152
4949 −2.75345e7 −0.682330
5050 −1.14418e8 −2.58898
5151 −372578. −0.00771172
5252 0 0
5353 −3.83324e7 −0.667305 −0.333652 0.942696i 0.608281π-0.608281\pi
−0.333652 + 0.942696i 0.608281π0.608281\pi
5454 −4.58037e7 −0.733042
5555 1.95628e8 2.88270
5656 −2.55641e7 −0.347363
5757 −1.92317e7 −0.241313
5858 −7.57103e7 −0.878474
5959 646636. 0.00694745 0.00347373 0.999994i 0.498894π-0.498894\pi
0.00347373 + 0.999994i 0.498894π0.498894\pi
6060 −2.77259e7 −0.276188
6161 4.42458e7 0.409155 0.204577 0.978850i 0.434418π-0.434418\pi
0.204577 + 0.978850i 0.434418π0.434418\pi
6262 −4.91363e7 −0.422318
6363 6.34656e7 0.507581
6464 1.79417e7 0.133676
6565 0 0
6666 9.70844e7 0.629798
6767 −1.65124e8 −1.00109 −0.500545 0.865711i 0.666867π-0.666867\pi
−0.500545 + 0.865711i 0.666867π0.666867\pi
6868 2.13938e6 0.0121338
6969 −6.15110e7 −0.326687
7070 2.44488e8 1.21707
7171 −1.69848e8 −0.793228 −0.396614 0.917985i 0.629815π-0.629815\pi
−0.396614 + 0.917985i 0.629815π0.629815\pi
7272 −1.26564e8 −0.555026
7373 8.20084e7 0.337991 0.168996 0.985617i 0.445948π-0.445948\pi
0.168996 + 0.985617i 0.445948π0.445948\pi
7474 −4.34574e8 −1.68470
7575 −1.82885e8 −0.667424
7676 1.10430e8 0.379688
7777 −2.83892e8 −0.920333
7878 0 0
7979 5.10327e8 1.47410 0.737050 0.675839i 0.236218π-0.236218\pi
0.737050 + 0.675839i 0.236218π0.236218\pi
8080 −8.08448e8 −2.20672
8181 2.75687e8 0.711597
8282 3.11191e6 0.00760090
8383 4.20094e8 0.971618 0.485809 0.874065i 0.338525π-0.338525\pi
0.485809 + 0.874065i 0.338525π0.338525\pi
8484 4.02354e7 0.0881762
8585 2.07788e7 0.0431752
8686 7.42162e8 1.46304
8787 −1.21015e8 −0.226465
8888 5.66141e8 1.00636
8989 −3.97908e8 −0.672245 −0.336123 0.941818i 0.609116π-0.609116\pi
−0.336123 + 0.941818i 0.609116π0.609116\pi
9090 1.21042e9 1.94467
9191 0 0
9292 3.53202e8 0.514018
9393 −7.85390e7 −0.108871
9494 9.94598e8 1.31393
9595 1.07256e9 1.35103
9696 −2.39484e8 −0.287777
9797 2.57864e8 0.295746 0.147873 0.989006i 0.452757π-0.452757\pi
0.147873 + 0.989006i 0.452757π0.452757\pi
9898 7.62076e8 0.834604
9999 −1.40551e9 −1.47053
100100 1.05014e9 1.05014
101101 −1.65112e9 −1.57882 −0.789409 0.613868i 0.789613π-0.789613\pi
−0.789409 + 0.613868i 0.789613π0.789613\pi
102102 1.03119e7 0.00943273
103103 −9.79825e8 −0.857790 −0.428895 0.903354i 0.641097π-0.641097\pi
−0.428895 + 0.903354i 0.641097π0.641097\pi
104104 0 0
105105 3.90787e8 0.313754
106106 1.06093e9 0.816226
107107 −6.09542e8 −0.449549 −0.224774 0.974411i 0.572165π-0.572165\pi
−0.224774 + 0.974411i 0.572165π0.572165\pi
108108 4.20392e8 0.297336
109109 5.84277e8 0.396460 0.198230 0.980156i 0.436481π-0.436481\pi
0.198230 + 0.980156i 0.436481π0.436481\pi
110110 −5.41442e9 −3.52602
111111 −6.94620e8 −0.434304
112112 1.17321e9 0.704519
113113 3.38612e9 1.95366 0.976831 0.214013i 0.0686533π-0.0686533\pi
0.976831 + 0.214013i 0.0686533π0.0686533\pi
114114 5.32279e8 0.295167
115115 3.43049e9 1.82901
116116 6.94878e8 0.356326
117117 0 0
118118 −1.78970e7 −0.00849790
119119 −3.01538e7 −0.0137842
120120 −7.79313e8 −0.343081
121121 3.92911e9 1.66633
122122 −1.22460e9 −0.500465
123123 4.97405e6 0.00195946
124124 4.50979e8 0.171300
125125 5.38075e9 1.97128
126126 −1.75654e9 −0.620857
127127 −1.23693e9 −0.421920 −0.210960 0.977495i 0.567659π-0.567659\pi
−0.210960 + 0.977495i 0.567659π0.567659\pi
128128 −3.26825e9 −1.07615
129129 1.18627e9 0.377162
130130 0 0
131131 −3.86097e9 −1.14545 −0.572724 0.819748i 0.694114π-0.694114\pi
−0.572724 + 0.819748i 0.694114π0.694114\pi
132132 −8.91052e8 −0.255458
133133 −1.55648e9 −0.431331
134134 4.57015e9 1.22450
135135 4.08307e9 1.05800
136136 6.01330e7 0.0150726
137137 5.58699e9 1.35499 0.677494 0.735529i 0.263066π-0.263066\pi
0.677494 + 0.735529i 0.263066π0.263066\pi
138138 1.70245e9 0.399593
139139 3.62704e9 0.824111 0.412055 0.911159i 0.364811π-0.364811\pi
0.412055 + 0.911159i 0.364811π0.364811\pi
140140 −2.24394e9 −0.493668
141141 1.58976e9 0.338723
142142 4.70091e9 0.970252
143143 0 0
144144 5.80836e9 1.12570
145145 6.74902e9 1.26790
146146 −2.26976e9 −0.413420
147147 1.21809e9 0.215156
148148 3.98858e9 0.683345
149149 −4.54812e8 −0.0755951 −0.0377975 0.999285i 0.512034π-0.512034\pi
−0.0377975 + 0.999285i 0.512034π0.512034\pi
150150 5.06173e9 0.816372
151151 8.49413e9 1.32960 0.664802 0.747019i 0.268516π-0.268516\pi
0.664802 + 0.747019i 0.268516π0.268516\pi
152152 3.10395e9 0.471648
153153 −1.49287e8 −0.0220247
154154 7.85732e9 1.12572
155155 4.38014e9 0.609531
156156 0 0
157157 8.94981e9 1.17562 0.587808 0.809000i 0.299991π-0.299991\pi
0.587808 + 0.809000i 0.299991π0.299991\pi
158158 −1.41244e10 −1.80307
159159 1.69578e9 0.210418
160160 1.33561e10 1.61117
161161 −4.97826e9 −0.583931
162162 −7.63023e9 −0.870402
163163 3.72319e9 0.413115 0.206557 0.978435i 0.433774π-0.433774\pi
0.206557 + 0.978435i 0.433774π0.433774\pi
164164 −2.85615e7 −0.00308307
165165 −8.65436e9 −0.908986
166166 −1.16270e10 −1.18845
167167 3.39705e9 0.337970 0.168985 0.985619i 0.445951π-0.445951\pi
0.168985 + 0.985619i 0.445951π0.445951\pi
168168 1.13093e9 0.109532
169169 0 0
170170 −5.75097e8 −0.0528106
171171 −7.70588e9 −0.689192
172172 −6.81165e9 −0.593437
173173 −9.36068e9 −0.794511 −0.397255 0.917708i 0.630037π-0.630037\pi
−0.397255 + 0.917708i 0.630037π0.630037\pi
174174 3.34934e9 0.277005
175175 −1.48014e10 −1.19297
176176 −2.59818e10 −2.04109
177177 −2.86064e7 −0.00219071
178178 1.10130e10 0.822269
179179 1.31097e10 0.954450 0.477225 0.878781i 0.341643π-0.341643\pi
0.477225 + 0.878781i 0.341643π0.341643\pi
180180 −1.11094e10 −0.788795
181181 −1.03171e10 −0.714506 −0.357253 0.934008i 0.616287π-0.616287\pi
−0.357253 + 0.934008i 0.616287π0.616287\pi
182182 0 0
183183 −1.95738e9 −0.129017
184184 9.92771e9 0.638512
185185 3.87391e10 2.43152
186186 2.17374e9 0.133168
187187 6.67785e8 0.0399346
188188 −9.12854e9 −0.532955
189189 −5.92528e9 −0.337777
190190 −2.96853e10 −1.65254
191191 2.12665e9 0.115624 0.0578118 0.998327i 0.481588π-0.481588\pi
0.0578118 + 0.998327i 0.481588π0.481588\pi
192192 −7.93720e8 −0.0421514
193193 −2.86528e10 −1.48648 −0.743241 0.669024i 0.766712π-0.766712\pi
−0.743241 + 0.669024i 0.766712π0.766712\pi
194194 −7.13695e9 −0.361747
195195 0 0
196196 −6.99442e9 −0.338532
197197 −1.59594e10 −0.754951 −0.377476 0.926019i 0.623208π-0.623208\pi
−0.377476 + 0.926019i 0.623208π0.623208\pi
198198 3.89004e10 1.79871
199199 −1.04597e10 −0.472802 −0.236401 0.971656i 0.575968π-0.575968\pi
−0.236401 + 0.971656i 0.575968π0.575968\pi
200200 2.95171e10 1.30448
201201 7.30489e9 0.315669
202202 4.56982e10 1.93116
203203 −9.79406e9 −0.404791
204204 −9.46437e7 −0.00382610
205205 −2.77404e8 −0.0109704
206206 2.71187e10 1.04922
207207 −2.46466e10 −0.933020
208208 0 0
209209 3.44697e10 1.24962
210210 −1.08159e10 −0.383773
211211 −2.31000e10 −0.802307 −0.401154 0.916011i 0.631391π-0.631391\pi
−0.401154 + 0.916011i 0.631391π0.631391\pi
212212 −9.73734e9 −0.331077
213213 7.51389e9 0.250125
214214 1.68704e10 0.549874
215215 −6.61584e10 −2.11160
216216 1.18163e10 0.369350
217217 −6.35639e9 −0.194600
218218 −1.61711e10 −0.484937
219219 −3.62796e9 −0.106577
220220 4.96942e10 1.43022
221221 0 0
222222 1.92251e10 0.531227
223223 −1.27352e9 −0.0344853 −0.0172426 0.999851i 0.505489π-0.505489\pi
−0.0172426 + 0.999851i 0.505489π0.505489\pi
224224 −1.93822e10 −0.514383
225225 −7.32794e10 −1.90617
226226 −9.37181e10 −2.38966
227227 1.78201e10 0.445445 0.222722 0.974882i 0.428506π-0.428506\pi
0.222722 + 0.974882i 0.428506π0.428506\pi
228228 −4.88531e9 −0.119725
229229 6.06264e10 1.45681 0.728403 0.685149i 0.240263π-0.240263\pi
0.728403 + 0.685149i 0.240263π0.240263\pi
230230 −9.49461e10 −2.23718
231231 1.25591e10 0.290204
232232 1.95314e10 0.442627
233233 −1.65760e10 −0.368449 −0.184224 0.982884i 0.558977π-0.558977\pi
−0.184224 + 0.982884i 0.558977π0.558977\pi
234234 0 0
235235 −8.86612e10 −1.89639
236236 1.64261e8 0.00344691
237237 −2.25763e10 −0.464820
238238 8.34571e8 0.0168604
239239 5.26394e10 1.04357 0.521784 0.853078i 0.325267π-0.325267\pi
0.521784 + 0.853078i 0.325267π0.325267\pi
240240 3.57648e10 0.695834
241241 2.09645e10 0.400321 0.200161 0.979763i 0.435854π-0.435854\pi
0.200161 + 0.979763i 0.435854π0.435854\pi
242242 −1.08747e11 −2.03820
243243 −4.47701e10 −0.823682
244244 1.12395e10 0.202998
245245 −6.79335e10 −1.20458
246246 −1.37668e8 −0.00239676
247247 0 0
248248 1.26760e10 0.212789
249249 −1.85845e10 −0.306376
250250 −1.48924e11 −2.41121
251251 3.94060e10 0.626658 0.313329 0.949645i 0.398556π-0.398556\pi
0.313329 + 0.949645i 0.398556π0.398556\pi
252252 1.61218e10 0.251832
253253 1.10248e11 1.69173
254254 3.42348e10 0.516079
255255 −9.19229e8 −0.0136142
256256 8.12698e10 1.18263
257257 1.03941e11 1.48624 0.743120 0.669159i 0.233345π-0.233345\pi
0.743120 + 0.669159i 0.233345π0.233345\pi
258258 −3.28324e10 −0.461333
259259 −5.62176e10 −0.776289
260260 0 0
261261 −4.84889e10 −0.646785
262262 1.06861e11 1.40108
263263 1.03626e11 1.33557 0.667787 0.744352i 0.267242π-0.267242\pi
0.667787 + 0.744352i 0.267242π0.267242\pi
264264 −2.50454e10 −0.317330
265265 −9.45742e10 −1.17806
266266 4.30788e10 0.527590
267267 1.76030e10 0.211976
268268 −4.19454e10 −0.496681
269269 1.33651e11 1.55627 0.778136 0.628096i 0.216165π-0.216165\pi
0.778136 + 0.628096i 0.216165π0.216165\pi
270270 −1.13008e11 −1.29411
271271 1.18011e11 1.32911 0.664557 0.747238i 0.268620π-0.268620\pi
0.664557 + 0.747238i 0.268620π0.268620\pi
272272 −2.75967e9 −0.0305701
273273 0 0
274274 −1.54632e11 −1.65738
275275 3.27791e11 3.45621
276276 −1.56253e10 −0.162083
277277 −2.22522e10 −0.227099 −0.113549 0.993532i 0.536222π-0.536222\pi
−0.113549 + 0.993532i 0.536222π0.536222\pi
278278 −1.00386e11 −1.00803
279279 −3.14695e10 −0.310936
280280 −6.30720e10 −0.613233
281281 1.75160e10 0.167593 0.0837966 0.996483i 0.473295π-0.473295\pi
0.0837966 + 0.996483i 0.473295π0.473295\pi
282282 −4.39999e10 −0.414315
283283 9.41554e10 0.872582 0.436291 0.899806i 0.356292π-0.356292\pi
0.436291 + 0.899806i 0.356292π0.356292\pi
284284 −4.31455e10 −0.393553
285285 −4.74488e10 −0.426013
286286 0 0
287287 4.02565e8 0.00350241
288288 −9.59582e10 −0.821894
289289 −1.18517e11 −0.999402
290290 −1.86794e11 −1.55085
291291 −1.14076e10 −0.0932561
292292 2.08321e10 0.167691
293293 −4.60347e10 −0.364906 −0.182453 0.983215i 0.558404π-0.558404\pi
−0.182453 + 0.983215i 0.558404π0.558404\pi
294294 −3.37134e10 −0.263172
295295 1.59539e9 0.0122650
296296 1.12110e11 0.848850
297297 1.31221e11 0.978586
298298 1.25879e10 0.0924655
299299 0 0
300300 −4.64571e10 −0.331136
301301 9.60079e10 0.674152
302302 −2.35093e11 −1.62633
303303 7.30436e10 0.497841
304304 −1.42449e11 −0.956593
305305 1.09164e11 0.722320
306306 4.13183e9 0.0269399
307307 8.99981e9 0.0578243 0.0289122 0.999582i 0.490796π-0.490796\pi
0.0289122 + 0.999582i 0.490796π0.490796\pi
308308 −7.21154e10 −0.456615
309309 4.33463e10 0.270483
310310 −1.21230e11 −0.745559
311311 7.38621e9 0.0447713 0.0223857 0.999749i 0.492874π-0.492874\pi
0.0223857 + 0.999749i 0.492874π0.492874\pi
312312 0 0
313313 2.33361e11 1.37429 0.687145 0.726521i 0.258864π-0.258864\pi
0.687145 + 0.726521i 0.258864π0.258864\pi
314314 −2.47705e11 −1.43798
315315 1.56583e11 0.896082
316316 1.29635e11 0.731361
317317 1.01774e10 0.0566070 0.0283035 0.999599i 0.490990π-0.490990\pi
0.0283035 + 0.999599i 0.490990π0.490990\pi
318318 −4.69344e10 −0.257376
319319 2.16899e11 1.17273
320320 4.42660e10 0.235991
321321 2.69655e10 0.141754
322322 1.37784e11 0.714246
323323 3.66122e9 0.0187161
324324 7.00311e10 0.353052
325325 0 0
326326 −1.03047e11 −0.505309
327327 −2.58478e10 −0.125014
328328 −8.02799e8 −0.00382979
329329 1.28664e11 0.605444
330330 2.39528e11 1.11184
331331 3.78785e11 1.73447 0.867236 0.497897i 0.165894π-0.165894\pi
0.867236 + 0.497897i 0.165894π0.165894\pi
332332 1.06714e11 0.482059
333333 −2.78325e11 −1.24037
334334 −9.40206e10 −0.413394
335335 −4.07396e11 −1.76732
336336 −5.19013e10 −0.222153
337337 −1.27459e11 −0.538314 −0.269157 0.963096i 0.586745π-0.586745\pi
−0.269157 + 0.963096i 0.586745π0.586745\pi
338338 0 0
339339 −1.49798e11 −0.616039
340340 5.27831e9 0.0214210
341341 1.40768e11 0.563781
342342 2.13277e11 0.842997
343343 2.43065e11 0.948199
344344 −1.91460e11 −0.737166
345345 −1.51761e11 −0.576732
346346 2.59077e11 0.971820
347347 −4.97009e10 −0.184027 −0.0920136 0.995758i 0.529330π-0.529330\pi
−0.0920136 + 0.995758i 0.529330π0.529330\pi
348348 −3.07406e10 −0.112359
349349 −6.73686e10 −0.243077 −0.121538 0.992587i 0.538783π-0.538783\pi
−0.121538 + 0.992587i 0.538783π0.538783\pi
350350 4.09660e11 1.45921
351351 0 0
352352 4.29237e11 1.49024
353353 −5.64977e11 −1.93662 −0.968311 0.249749i 0.919652π-0.919652\pi
−0.968311 + 0.249749i 0.919652π0.919652\pi
354354 7.91744e8 0.00267960
355355 −4.19052e11 −1.40036
356356 −1.01078e11 −0.333528
357357 1.33397e9 0.00434649
358358 −3.62838e11 −1.16745
359359 5.75596e11 1.82891 0.914456 0.404686i 0.132619π-0.132619\pi
0.914456 + 0.404686i 0.132619π0.132619\pi
360360 −3.12260e11 −0.979840
361361 −1.33703e11 −0.414341
362362 2.85549e11 0.873961
363363 −1.73820e11 −0.525435
364364 0 0
365365 2.02332e11 0.596688
366366 5.41748e10 0.157809
367367 6.01271e11 1.73011 0.865053 0.501680i 0.167285π-0.167285\pi
0.865053 + 0.501680i 0.167285π0.167285\pi
368368 −4.55610e11 −1.29503
369369 1.99304e9 0.00559624
370370 −1.07219e12 −2.97415
371371 1.37244e11 0.376108
372372 −1.99508e10 −0.0540153
373373 1.21541e11 0.325112 0.162556 0.986699i 0.448026π-0.448026\pi
0.162556 + 0.986699i 0.448026π0.448026\pi
374374 −1.84824e10 −0.0488467
375375 −2.38039e11 −0.621594
376376 −2.56582e11 −0.662036
377377 0 0
378378 1.63995e11 0.413159
379379 −1.30959e11 −0.326030 −0.163015 0.986624i 0.552122π-0.552122\pi
−0.163015 + 0.986624i 0.552122π0.552122\pi
380380 2.72455e11 0.670300
381381 5.47206e10 0.133042
382382 −5.88597e10 −0.141427
383383 −6.95780e11 −1.65226 −0.826128 0.563483i 0.809461π-0.809461\pi
−0.826128 + 0.563483i 0.809461π0.809461\pi
384384 1.44584e11 0.339336
385385 −7.00423e11 −1.62475
386386 7.93029e11 1.81822
387387 4.75320e11 1.07718
388388 6.55038e10 0.146732
389389 −1.49432e10 −0.0330880 −0.0165440 0.999863i 0.505266π-0.505266\pi
−0.0165440 + 0.999863i 0.505266π0.505266\pi
390390 0 0
391391 1.17101e10 0.0253376
392392 −1.96597e11 −0.420523
393393 1.70805e11 0.361189
394394 4.41711e11 0.923433
395395 1.25909e12 2.60237
396396 −3.57032e11 −0.729590
397397 −6.44574e11 −1.30231 −0.651157 0.758943i 0.725716π-0.725716\pi
−0.651157 + 0.758943i 0.725716π0.725716\pi
398398 2.89494e11 0.578316
399399 6.88568e10 0.136009
400400 −1.35462e12 −2.64575
401401 −8.71275e11 −1.68270 −0.841348 0.540494i 0.818237π-0.818237\pi
−0.841348 + 0.540494i 0.818237π0.818237\pi
402402 −2.02178e11 −0.386116
403403 0 0
404404 −4.19424e11 −0.783316
405405 6.80179e11 1.25625
406406 2.71072e11 0.495127
407407 1.24499e12 2.24901
408408 −2.66022e9 −0.00475277
409409 1.40219e11 0.247772 0.123886 0.992296i 0.460464π-0.460464\pi
0.123886 + 0.992296i 0.460464π0.460464\pi
410410 7.67776e9 0.0134186
411411 −2.47162e11 −0.427261
412412 −2.48899e11 −0.425584
413413 −2.31520e9 −0.00391574
414414 6.82148e11 1.14124
415415 1.03646e12 1.71529
416416 0 0
417417 −1.60456e11 −0.259863
418418 −9.54023e11 −1.52850
419419 5.39944e11 0.855826 0.427913 0.903820i 0.359249π-0.359249\pi
0.427913 + 0.903820i 0.359249π0.359249\pi
420420 9.92694e10 0.155666
421421 −8.59651e11 −1.33368 −0.666842 0.745199i 0.732354π-0.732354\pi
−0.666842 + 0.745199i 0.732354π0.732354\pi
422422 6.39342e11 0.981357
423423 6.36994e11 0.967394
424424 −2.73694e11 −0.411263
425425 3.48166e10 0.0517649
426426 −2.07963e11 −0.305945
427427 −1.58417e11 −0.230609
428428 −1.54838e11 −0.223039
429429 0 0
430430 1.83107e12 2.58284
431431 2.53427e11 0.353758 0.176879 0.984233i 0.443400π-0.443400\pi
0.176879 + 0.984233i 0.443400π0.443400\pi
432432 −5.42281e11 −0.749113
433433 1.55645e11 0.212785 0.106392 0.994324i 0.466070π-0.466070\pi
0.106392 + 0.994324i 0.466070π0.466070\pi
434434 1.75927e11 0.238028
435435 −2.98569e11 −0.399800
436436 1.48420e11 0.196700
437437 6.04453e11 0.792859
438438 1.00412e11 0.130362
439439 8.15703e11 1.04819 0.524097 0.851658i 0.324403π-0.324403\pi
0.524097 + 0.851658i 0.324403π0.324403\pi
440440 1.39679e12 1.77662
441441 4.88074e11 0.614486
442442 0 0
443443 −4.25413e11 −0.524800 −0.262400 0.964959i 0.584514π-0.584514\pi
−0.262400 + 0.964959i 0.584514π0.584514\pi
444444 −1.76450e11 −0.215476
445445 −9.81725e11 −1.18678
446446 3.52474e10 0.0421813
447447 2.01204e10 0.0238370
448448 −6.42381e10 −0.0753427
449449 5.63347e11 0.654135 0.327068 0.945001i 0.393939π-0.393939\pi
0.327068 + 0.945001i 0.393939π0.393939\pi
450450 2.02817e12 2.33156
451451 −8.91518e9 −0.0101470
452452 8.60156e11 0.969291
453453 −3.75771e11 −0.419258
454454 −4.93209e11 −0.544854
455455 0 0
456456 −1.37315e11 −0.148722
457457 −8.80745e11 −0.944556 −0.472278 0.881450i 0.656568π-0.656568\pi
−0.472278 + 0.881450i 0.656568π0.656568\pi
458458 −1.67796e12 −1.78192
459459 1.39377e10 0.0146567
460460 8.71426e11 0.907445
461461 −5.49345e11 −0.566488 −0.283244 0.959048i 0.591411π-0.591411\pi
−0.283244 + 0.959048i 0.591411π0.591411\pi
462462 −3.47599e11 −0.354968
463463 4.06010e11 0.410603 0.205302 0.978699i 0.434182π-0.434182\pi
0.205302 + 0.978699i 0.434182π0.434182\pi
464464 −8.96352e11 −0.897734
465465 −1.93773e11 −0.192200
466466 4.58775e11 0.450675
467467 1.33161e12 1.29554 0.647769 0.761837i 0.275702π-0.275702\pi
0.647769 + 0.761837i 0.275702π0.275702\pi
468468 0 0
469469 5.91206e11 0.564236
470470 2.45389e12 2.31961
471471 −3.95930e11 −0.370701
472472 4.61700e9 0.00428175
473473 −2.12619e12 −1.95311
474474 6.24847e11 0.568554
475475 1.79716e12 1.61982
476476 −7.65979e9 −0.00683889
477477 6.79476e11 0.600955
478478 −1.45691e12 −1.27646
479479 9.28564e11 0.805939 0.402969 0.915214i 0.367978π-0.367978\pi
0.402969 + 0.915214i 0.367978π0.367978\pi
480480 −5.90860e11 −0.508041
481481 0 0
482482 −5.80238e11 −0.489660
483483 2.20233e11 0.184128
484484 9.98090e11 0.826733
485485 6.36207e11 0.522108
486486 1.23911e12 1.00750
487487 1.47674e12 1.18966 0.594829 0.803852i 0.297220π-0.297220\pi
0.594829 + 0.803852i 0.297220π0.297220\pi
488488 3.15916e11 0.252164
489489 −1.64710e11 −0.130265
490490 1.88020e12 1.47341
491491 1.91948e12 1.49045 0.745225 0.666813i 0.232342π-0.232342\pi
0.745225 + 0.666813i 0.232342π0.232342\pi
492492 1.26353e9 0.000972170 0
493493 2.30381e10 0.0175645
494494 0 0
495495 −3.46769e12 −2.59607
496496 −5.81736e11 −0.431577
497497 6.08121e11 0.447081
498498 5.14366e11 0.374749
499499 −9.40035e11 −0.678721 −0.339361 0.940656i 0.610211π-0.610211\pi
−0.339361 + 0.940656i 0.610211π0.610211\pi
500500 1.36684e12 0.978032
501501 −1.50282e11 −0.106570
502502 −1.09065e12 −0.766509
503503 4.07654e11 0.283946 0.141973 0.989871i 0.454655π-0.454655\pi
0.141973 + 0.989871i 0.454655π0.454655\pi
504504 4.53146e11 0.312825
505505 −4.07366e12 −2.78724
506506 −3.05136e12 −2.06927
507507 0 0
508508 −3.14211e11 −0.209332
509509 8.18527e11 0.540509 0.270255 0.962789i 0.412892π-0.412892\pi
0.270255 + 0.962789i 0.412892π0.412892\pi
510510 2.54416e10 0.0166525
511511 −2.93621e11 −0.190499
512512 −5.75969e11 −0.370411
513513 7.19437e11 0.458632
514514 −2.87680e12 −1.81792
515515 −2.41744e12 −1.51434
516516 3.01340e11 0.187125
517517 −2.84938e12 −1.75405
518518 1.55594e12 0.949532
519519 4.14106e11 0.250529
520520 0 0
521521 −1.15230e11 −0.0685165 −0.0342582 0.999413i 0.510907π-0.510907\pi
−0.0342582 + 0.999413i 0.510907π0.510907\pi
522522 1.34203e12 0.791127
523523 −7.61800e11 −0.445229 −0.222615 0.974907i 0.571459π-0.571459\pi
−0.222615 + 0.974907i 0.571459π0.571459\pi
524524 −9.80779e11 −0.568304
525525 6.54797e11 0.376175
526526 −2.86807e12 −1.63363
527527 1.49518e10 0.00844396
528528 1.14940e12 0.643606
529529 1.32139e11 0.0733638
530530 2.61754e12 1.44096
531531 −1.14622e10 −0.00625667
532532 −3.95383e11 −0.214001
533533 0 0
534534 −4.87201e11 −0.259282
535535 −1.50387e12 −0.793632
536536 −1.17899e12 −0.616976
537537 −5.79958e11 −0.300962
538538 −3.69907e12 −1.90358
539539 −2.18324e12 −1.11417
540540 1.03720e12 0.524915
541541 −3.17517e12 −1.59360 −0.796800 0.604243i 0.793476π-0.793476\pi
−0.796800 + 0.604243i 0.793476π0.793476\pi
542542 −3.26622e12 −1.62573
543543 4.56419e11 0.225302
544544 4.55917e10 0.0223198
545545 1.44154e12 0.699909
546546 0 0
547547 1.41950e12 0.677942 0.338971 0.940797i 0.389921π-0.389921\pi
0.338971 + 0.940797i 0.389921π0.389921\pi
548548 1.41923e12 0.672264
549549 −7.84297e11 −0.368472
550550 −9.07232e12 −4.22753
551551 1.18918e12 0.549623
552552 −4.39191e11 −0.201339
553553 −1.82717e12 −0.830835
554554 6.15878e11 0.277780
555555 −1.71378e12 −0.766718
556556 9.21355e11 0.408875
557557 −1.79642e11 −0.0790786 −0.0395393 0.999218i 0.512589π-0.512589\pi
−0.0395393 + 0.999218i 0.512589π0.512589\pi
558558 8.70986e11 0.380327
559559 0 0
560560 2.89455e12 1.24376
561561 −2.95421e10 −0.0125924
562562 −4.84793e11 −0.204995
563563 2.86519e12 1.20189 0.600946 0.799290i 0.294791π-0.294791\pi
0.600946 + 0.799290i 0.294791π0.294791\pi
564564 4.03836e11 0.168054
565565 8.35429e12 3.44899
566566 −2.60595e12 −1.06732
567567 −9.87065e11 −0.401072
568568 −1.21272e12 −0.488870
569569 −1.82502e12 −0.729897 −0.364948 0.931028i 0.618913π-0.618913\pi
−0.364948 + 0.931028i 0.618913π0.618913\pi
570570 1.31325e12 0.521086
571571 −2.25174e12 −0.886454 −0.443227 0.896409i 0.646166π-0.646166\pi
−0.443227 + 0.896409i 0.646166π0.646166\pi
572572 0 0
573573 −9.40808e10 −0.0364590
574574 −1.11418e10 −0.00428404
575575 5.74807e12 2.19289
576576 −3.18033e11 −0.120385
577577 2.42445e12 0.910587 0.455293 0.890342i 0.349534π-0.349534\pi
0.455293 + 0.890342i 0.349534π0.349534\pi
578578 3.28021e12 1.22244
579579 1.26757e12 0.468725
580580 1.71441e12 0.629056
581581 −1.50410e12 −0.547625
582582 3.15731e11 0.114068
583583 −3.03941e12 −1.08963
584584 5.85543e11 0.208306
585585 0 0
586586 1.27411e12 0.446341
587587 −3.06448e12 −1.06533 −0.532666 0.846325i 0.678810π-0.678810\pi
−0.532666 + 0.846325i 0.678810π0.678810\pi
588588 3.09425e11 0.106748
589589 7.71782e11 0.264226
590590 −4.41558e10 −0.0150022
591591 7.06027e11 0.238055
592592 −5.14503e12 −1.72163
593593 −4.24429e12 −1.40948 −0.704740 0.709466i 0.748936π-0.748936\pi
−0.704740 + 0.709466i 0.748936π0.748936\pi
594594 −3.63182e12 −1.19698
595595 −7.43959e10 −0.0243345
596596 −1.15533e11 −0.0375058
597597 4.62724e11 0.149086
598598 0 0
599599 −5.54247e12 −1.75907 −0.879535 0.475835i 0.842146π-0.842146\pi
−0.879535 + 0.475835i 0.842146π0.842146\pi
600600 −1.30580e12 −0.411337
601601 1.14070e12 0.356646 0.178323 0.983972i 0.442933π-0.442933\pi
0.178323 + 0.983972i 0.442933π0.442933\pi
602602 −2.65722e12 −0.824601
603603 2.92697e12 0.901551
604604 2.15771e12 0.659671
605605 9.69397e12 2.94173
606606 −2.02164e12 −0.608943
607607 −1.62034e11 −0.0484459 −0.0242229 0.999707i 0.507711π-0.507711\pi
−0.0242229 + 0.999707i 0.507711π0.507711\pi
608608 2.35335e12 0.698426
609609 4.33278e11 0.127641
610610 −3.02134e12 −0.883519
611611 0 0
612612 −3.79224e10 −0.0109273
613613 4.29927e12 1.22977 0.614883 0.788619i 0.289203π-0.289203\pi
0.614883 + 0.788619i 0.289203π0.289203\pi
614614 −2.49089e11 −0.0707289
615615 1.22721e10 0.00345923
616616 −2.02700e12 −0.567205
617617 −2.36564e12 −0.657152 −0.328576 0.944477i 0.606569π-0.606569\pi
−0.328576 + 0.944477i 0.606569π0.606569\pi
618618 −1.19970e12 −0.330846
619619 5.26560e11 0.144158 0.0720792 0.997399i 0.477037π-0.477037\pi
0.0720792 + 0.997399i 0.477037π0.477037\pi
620620 1.11266e12 0.302413
621621 2.30106e12 0.620892
622622 −2.04429e11 −0.0547629
623623 1.42466e12 0.378892
624624 0 0
625625 5.20121e12 1.36347
626626 −6.45876e12 −1.68099
627627 −1.52490e12 −0.394038
628628 2.27347e12 0.583271
629629 1.32238e11 0.0336843
630630 −4.33377e12 −1.09606
631631 −6.16163e12 −1.54726 −0.773630 0.633638i 0.781561π-0.781561\pi
−0.773630 + 0.633638i 0.781561π0.781561\pi
632632 3.64375e12 0.908494
633633 1.02192e12 0.252988
634634 −2.81681e11 −0.0692399
635635 −3.05178e12 −0.744856
636636 4.30769e11 0.104397
637637 0 0
638638 −6.00314e12 −1.43445
639639 3.01071e12 0.714358
640640 −8.06349e12 −1.89982
641641 −4.86391e12 −1.13795 −0.568977 0.822354i 0.692661π-0.692661\pi
−0.568977 + 0.822354i 0.692661π0.692661\pi
642642 −7.46327e11 −0.173389
643643 −5.33732e12 −1.23133 −0.615664 0.788009i 0.711112π-0.711112\pi
−0.615664 + 0.788009i 0.711112π0.711112\pi
644644 −1.26460e12 −0.289712
645645 2.92677e12 0.665840
646646 −1.01332e11 −0.0228929
647647 −2.91982e12 −0.655069 −0.327535 0.944839i 0.606218π-0.606218\pi
−0.327535 + 0.944839i 0.606218π0.606218\pi
648648 1.96842e12 0.438560
649649 5.12724e10 0.0113444
650650 0 0
651651 2.81200e11 0.0613621
652652 9.45779e11 0.204963
653653 1.57692e12 0.339391 0.169696 0.985497i 0.445722π-0.445722\pi
0.169696 + 0.985497i 0.445722π0.445722\pi
654654 7.15392e11 0.152913
655655 −9.52585e12 −2.02217
656656 3.68427e10 0.00776755
657657 −1.45367e12 −0.304385
658658 −3.56104e12 −0.740560
659659 8.06220e12 1.66521 0.832606 0.553866i 0.186848π-0.186848\pi
0.832606 + 0.553866i 0.186848π0.186848\pi
660660 −2.19842e12 −0.450985
661661 2.48474e12 0.506262 0.253131 0.967432i 0.418540π-0.418540\pi
0.253131 + 0.967432i 0.418540π0.418540\pi
662662 −1.04837e13 −2.12155
663663 0 0
664664 2.99949e12 0.598813
665665 −3.84017e12 −0.761470
666666 7.70323e12 1.51719
667667 3.80349e12 0.744074
668668 8.62932e11 0.167680
669669 5.63391e10 0.0108741
670670 1.12756e13 2.16173
671671 3.50829e12 0.668104
672672 8.57446e11 0.162198
673673 3.04990e12 0.573083 0.286542 0.958068i 0.407494π-0.407494\pi
0.286542 + 0.958068i 0.407494π0.407494\pi
674674 3.52770e12 0.658449
675675 6.84152e12 1.26849
676676 0 0
677677 −9.40636e12 −1.72097 −0.860483 0.509479i 0.829838π-0.829838\pi
−0.860483 + 0.509479i 0.829838π0.829838\pi
678678 4.14599e12 0.753519
679679 −9.23253e11 −0.166689
680680 1.48361e11 0.0266091
681681 −7.88341e11 −0.140460
682682 −3.89607e12 −0.689599
683683 1.50847e12 0.265243 0.132621 0.991167i 0.457661π-0.457661\pi
0.132621 + 0.991167i 0.457661π0.457661\pi
684684 −1.95748e12 −0.341936
685685 1.37843e13 2.39209
686686 −6.72735e12 −1.15981
687687 −2.68204e12 −0.459368
688688 8.78663e12 1.49511
689689 0 0
690690 4.20031e12 0.705440
691691 9.80911e12 1.63674 0.818368 0.574695i 0.194879π-0.194879\pi
0.818368 + 0.574695i 0.194879π0.194879\pi
692692 −2.37784e12 −0.394189
693693 5.03225e12 0.828824
694694 1.37558e12 0.225096
695695 8.94868e12 1.45488
696696 −8.64049e11 −0.139571
697697 −9.46932e8 −0.000151975 0
698698 1.86457e12 0.297324
699699 7.33302e11 0.116181
700700 −3.75991e12 −0.591883
701701 −2.97437e11 −0.0465226 −0.0232613 0.999729i 0.507405π-0.507405\pi
−0.0232613 + 0.999729i 0.507405π0.507405\pi
702702 0 0
703703 6.82585e12 1.05404
704704 1.42261e12 0.218278
705705 3.92227e12 0.597980
706706 1.56370e13 2.36881
707707 5.91163e12 0.889857
708708 −7.26672e9 −0.00108690
709709 1.52039e12 0.225969 0.112984 0.993597i 0.463959π-0.463959\pi
0.112984 + 0.993597i 0.463959π0.463959\pi
710710 1.15982e13 1.71288
711711 −9.04602e12 −1.32753
712712 −2.84108e12 −0.414308
713713 2.46848e12 0.357707
714714 −3.69205e10 −0.00531649
715715 0 0
716716 3.33017e12 0.473542
717717 −2.32871e12 −0.329063
718718 −1.59309e13 −2.23707
719719 4.71840e12 0.658438 0.329219 0.944254i 0.393215π-0.393215\pi
0.329219 + 0.944254i 0.393215π0.393215\pi
720720 1.43305e13 1.98730
721721 3.50815e12 0.483469
722722 3.70051e12 0.506809
723723 −9.27448e11 −0.126231
724724 −2.62080e12 −0.354496
725725 1.13086e13 1.52015
726726 4.81083e12 0.642696
727727 −4.23948e12 −0.562870 −0.281435 0.959580i 0.590810π-0.590810\pi
−0.281435 + 0.959580i 0.590810π0.590810\pi
728728 0 0
729729 −3.44577e12 −0.451869
730730 −5.59998e12 −0.729850
731731 −2.25834e11 −0.0292524
732732 −4.97222e11 −0.0640105
733733 −2.32506e12 −0.297485 −0.148743 0.988876i 0.547523π-0.547523\pi
−0.148743 + 0.988876i 0.547523π0.547523\pi
734734 −1.66415e13 −2.11621
735735 3.00530e12 0.379835
736736 7.52700e12 0.945522
737737 −1.30928e13 −1.63467
738738 −5.51615e10 −0.00684515
739739 3.94677e12 0.486790 0.243395 0.969927i 0.421739π-0.421739\pi
0.243395 + 0.969927i 0.421739π0.421739\pi
740740 9.84067e12 1.20637
741741 0 0
742742 −3.79853e12 −0.460043
743743 −6.17403e12 −0.743222 −0.371611 0.928388i 0.621195π-0.621195\pi
−0.371611 + 0.928388i 0.621195π0.621195\pi
744744 −5.60771e11 −0.0670977
745745 −1.12212e12 −0.133455
746746 −3.36391e12 −0.397667
747747 −7.44656e12 −0.875010
748748 1.69633e11 0.0198132
749749 2.18239e12 0.253376
750750 6.58823e12 0.760314
751751 −5.93809e12 −0.681188 −0.340594 0.940211i 0.610628π-0.610628\pi
−0.340594 + 0.940211i 0.610628π0.610628\pi
752752 1.17753e13 1.34274
753753 −1.74328e12 −0.197601
754754 0 0
755755 2.09568e13 2.34728
756756 −1.50516e12 −0.167585
757757 −5.64069e12 −0.624311 −0.312155 0.950031i 0.601051π-0.601051\pi
−0.312155 + 0.950031i 0.601051π0.601051\pi
758758 3.62456e12 0.398790
759759 −4.87727e12 −0.533444
760760 7.65810e12 0.832645
761761 −3.67608e12 −0.397332 −0.198666 0.980067i 0.563661π-0.563661\pi
−0.198666 + 0.980067i 0.563661π0.563661\pi
762762 −1.51451e12 −0.162733
763763 −2.09193e12 −0.223454
764764 5.40221e11 0.0573656
765765 −3.68323e11 −0.0388823
766766 1.92572e13 2.02099
767767 0 0
768768 −3.59529e12 −0.372913
769769 8.01554e11 0.0826541 0.0413270 0.999146i 0.486841π-0.486841\pi
0.0413270 + 0.999146i 0.486841π0.486841\pi
770770 1.93857e13 1.98734
771771 −4.59824e12 −0.468649
772772 −7.27851e12 −0.737504
773773 −9.66089e12 −0.973216 −0.486608 0.873620i 0.661766π-0.661766\pi
−0.486608 + 0.873620i 0.661766π0.661766\pi
774774 −1.31555e13 −1.31757
775775 7.33930e12 0.730797
776776 1.84116e12 0.182270
777777 2.48700e12 0.244783
778778 4.13585e11 0.0404722
779779 −4.88787e10 −0.00475556
780780 0 0
781781 −1.34674e13 −1.29525
782782 −3.24103e11 −0.0309922
783783 4.52703e12 0.430413
784784 9.02239e12 0.852902
785785 2.20811e13 2.07543
786786 −4.72740e12 −0.441795
787787 −1.24505e12 −0.115691 −0.0578456 0.998326i 0.518423π-0.518423\pi
−0.0578456 + 0.998326i 0.518423π0.518423\pi
788788 −4.05408e12 −0.374562
789789 −4.58430e12 −0.421140
790790 −3.48479e13 −3.18313
791791 −1.21236e13 −1.10113
792792 −1.00354e13 −0.906296
793793 0 0
794794 1.78400e13 1.59295
795795 4.18386e12 0.371471
796796 −2.65701e12 −0.234576
797797 9.96620e12 0.874918 0.437459 0.899238i 0.355879π-0.355879\pi
0.437459 + 0.899238i 0.355879π0.355879\pi
798798 −1.90576e12 −0.166362
799799 −3.02649e11 −0.0262711
800800 2.23793e13 1.93171
801801 7.05328e12 0.605404
802802 2.41144e13 2.05822
803803 6.50253e12 0.551902
804804 1.85562e12 0.156616
805805 −1.22825e13 −1.03087
806806 0 0
807807 −5.91255e12 −0.490732
808808 −1.17890e13 −0.973033
809809 1.65385e13 1.35746 0.678732 0.734386i 0.262530π-0.262530\pi
0.678732 + 0.734386i 0.262530π0.262530\pi
810810 −1.88254e13 −1.53660
811811 7.93499e12 0.644099 0.322049 0.946723i 0.395628π-0.395628\pi
0.322049 + 0.946723i 0.395628π0.395628\pi
812812 −2.48793e12 −0.200833
813813 −5.22069e12 −0.419103
814814 −3.44578e13 −2.75092
815815 9.18590e12 0.729311
816816 1.22085e11 0.00963953
817817 −1.16571e13 −0.915360
818818 −3.88087e12 −0.303067
819819 0 0
820820 −7.04674e10 −0.00544284
821821 −4.39935e12 −0.337944 −0.168972 0.985621i 0.554045π-0.554045\pi
−0.168972 + 0.985621i 0.554045π0.554045\pi
822822 6.84074e12 0.522613
823823 −1.96888e13 −1.49596 −0.747979 0.663723i 0.768976π-0.768976\pi
−0.747979 + 0.663723i 0.768976π0.768976\pi
824824 −6.99598e12 −0.528660
825825 −1.45011e13 −1.08983
826826 6.40782e10 0.00478961
827827 −3.94021e12 −0.292917 −0.146459 0.989217i 0.546787π-0.546787\pi
−0.146459 + 0.989217i 0.546787π0.546787\pi
828828 −6.26084e12 −0.462909
829829 8.09989e12 0.595640 0.297820 0.954622i 0.403740π-0.403740\pi
0.297820 + 0.954622i 0.403740π0.403740\pi
830830 −2.86863e13 −2.09809
831831 9.84413e11 0.0716099
832832 0 0
833833 −2.31894e11 −0.0166873
834834 4.44097e12 0.317856
835835 8.38125e12 0.596650
836836 8.75613e12 0.619989
837837 2.93806e12 0.206917
838838 −1.49441e13 −1.04682
839839 −1.62696e13 −1.13357 −0.566784 0.823866i 0.691813π-0.691813\pi
−0.566784 + 0.823866i 0.691813π0.691813\pi
840840 2.79024e12 0.193368
841841 −7.02429e12 −0.484195
842842 2.37927e13 1.63132
843843 −7.74888e11 −0.0528463
844844 −5.86795e12 −0.398057
845845 0 0
846846 −1.76302e13 −1.18329
847847 −1.40677e13 −0.939180
848848 1.25606e13 0.834121
849849 −4.16533e12 −0.275147
850850 −9.63623e11 −0.0633172
851851 2.18319e13 1.42695
852852 1.90871e12 0.124097
853853 2.11246e13 1.36621 0.683106 0.730320i 0.260629π-0.260629\pi
0.683106 + 0.730320i 0.260629π0.260629\pi
854854 4.38452e12 0.282073
855855 −1.90121e13 −1.21670
856856 −4.35215e12 −0.277059
857857 −1.66234e13 −1.05270 −0.526350 0.850268i 0.676440π-0.676440\pi
−0.526350 + 0.850268i 0.676440π0.676440\pi
858858 0 0
859859 2.12853e13 1.33386 0.666930 0.745121i 0.267608π-0.267608\pi
0.666930 + 0.745121i 0.267608π0.267608\pi
860860 −1.68058e13 −1.04765
861861 −1.78090e10 −0.00110440
862862 −7.01415e12 −0.432705
863863 −1.58665e13 −0.973714 −0.486857 0.873482i 0.661857π-0.661857\pi
−0.486857 + 0.873482i 0.661857π0.661857\pi
864864 8.95886e12 0.546941
865865 −2.30948e13 −1.40263
866866 −4.30782e12 −0.260272
867867 5.24306e12 0.315136
868868 −1.61468e12 −0.0965487
869869 4.04643e13 2.40704
870870 8.26354e12 0.489023
871871 0 0
872872 4.17176e12 0.244340
873873 −4.57088e12 −0.266340
874874 −1.67295e13 −0.969800
875875 −1.92652e13 −1.11106
876876 −9.21589e11 −0.0528772
877877 6.92981e12 0.395570 0.197785 0.980245i 0.436625π-0.436625\pi
0.197785 + 0.980245i 0.436625π0.436625\pi
878878 −2.25763e13 −1.28212
879879 2.03652e12 0.115064
880880 −6.41026e13 −3.60333
881881 1.50068e13 0.839259 0.419630 0.907695i 0.362160π-0.362160\pi
0.419630 + 0.907695i 0.362160π0.362160\pi
882882 −1.35085e13 −0.751620
883883 −1.63884e13 −0.907221 −0.453611 0.891200i 0.649864π-0.649864\pi
−0.453611 + 0.891200i 0.649864π0.649864\pi
884884 0 0
885885 −7.05782e10 −0.00386746
886886 1.17742e13 0.641919
887887 −2.70000e13 −1.46456 −0.732281 0.681002i 0.761544π-0.761544\pi
−0.732281 + 0.681002i 0.761544π0.761544\pi
888888 −4.95961e12 −0.267664
889889 4.42870e12 0.237803
890890 2.71713e13 1.45163
891891 2.18595e13 1.16196
892892 −3.23504e11 −0.0171095
893893 −1.56221e13 −0.822069
894894 −5.56874e11 −0.0291567
895895 3.23444e13 1.68498
896896 1.17016e13 0.606540
897897 0 0
898898 −1.55918e13 −0.800118
899899 4.85641e12 0.247968
900900 −1.86147e13 −0.945727
901901 −3.22833e11 −0.0163199
902902 2.46747e11 0.0124114
903903 −4.24728e12 −0.212577
904904 2.41770e13 1.20405
905905 −2.54546e13 −1.26139
906906 1.04003e13 0.512823
907907 −1.12074e13 −0.549884 −0.274942 0.961461i 0.588659π-0.588659\pi
−0.274942 + 0.961461i 0.588659π0.588659\pi
908908 4.52673e12 0.221003
909909 2.92676e13 1.42184
910910 0 0
911911 1.06250e13 0.511089 0.255545 0.966797i 0.417745π-0.417745\pi
0.255545 + 0.966797i 0.417745π0.417745\pi
912912 6.30177e12 0.301638
913913 3.33097e13 1.58654
914914 2.43765e13 1.15535
915915 −4.82929e12 −0.227766
916916 1.54006e13 0.722781
917917 1.38237e13 0.645600
918918 −3.85756e11 −0.0179276
919919 4.03137e13 1.86437 0.932186 0.361980i 0.117899π-0.117899\pi
0.932186 + 0.361980i 0.117899π0.117899\pi
920920 2.44938e13 1.12723
921921 −3.98141e11 −0.0182335
922922 1.52043e13 0.692910
923923 0 0
924924 3.19030e12 0.143982
925925 6.49107e13 2.91527
926926 −1.12372e13 −0.502237
927927 1.73683e13 0.772500
928928 1.48084e13 0.655452
929929 3.30900e13 1.45756 0.728779 0.684749i 0.240088π-0.240088\pi
0.728779 + 0.684749i 0.240088π0.240088\pi
930930 5.36307e12 0.235093
931931 −1.19699e13 −0.522176
932932 −4.21069e12 −0.182802
933933 −3.26758e11 −0.0141175
934934 −3.68551e13 −1.58466
935935 1.64757e12 0.0705004
936936 0 0
937937 −9.13054e12 −0.386962 −0.193481 0.981104i 0.561978π-0.561978\pi
−0.193481 + 0.981104i 0.561978π0.561978\pi
938938 −1.63629e13 −0.690156
939939 −1.03236e13 −0.433348
940940 −2.25221e13 −0.940877
941941 −2.61735e13 −1.08820 −0.544099 0.839021i 0.683128π-0.683128\pi
−0.544099 + 0.839021i 0.683128π0.683128\pi
942942 1.09582e13 0.453430
943943 −1.56335e11 −0.00643802
944944 −2.11887e11 −0.00868421
945945 −1.46189e13 −0.596311
946946 5.88468e13 2.38898
947947 −3.70875e12 −0.149849 −0.0749243 0.997189i 0.523872π-0.523872\pi
−0.0749243 + 0.997189i 0.523872π0.523872\pi
948948 −5.73492e12 −0.230616
949949 0 0
950950 −4.97403e13 −1.98131
951951 −4.50237e11 −0.0178496
952952 −2.15299e11 −0.00849525
953953 −1.82708e13 −0.717528 −0.358764 0.933428i 0.616802π-0.616802\pi
−0.358764 + 0.933428i 0.616802π0.616802\pi
954954 −1.88060e13 −0.735069
955955 5.24691e12 0.204121
956956 1.33717e13 0.517756
957957 −9.59537e12 −0.369792
958958 −2.57000e13 −0.985798
959959 −2.00035e13 −0.763701
960960 −1.95828e12 −0.0744139
961961 −2.32878e13 −0.880791
962962 0 0
963963 1.08047e13 0.404850
964964 5.32550e12 0.198616
965965 −7.06927e13 −2.62423
966966 −6.09542e12 −0.225220
967967 6.64147e12 0.244256 0.122128 0.992514i 0.461028π-0.461028\pi
0.122128 + 0.992514i 0.461028π0.461028\pi
968968 2.80540e13 1.02697
969969 −1.61968e11 −0.00590165
970970 −1.76084e13 −0.638626
971971 2.72248e13 0.982828 0.491414 0.870926i 0.336480π-0.336480\pi
0.491414 + 0.870926i 0.336480π0.336480\pi
972972 −1.13727e13 −0.408662
973973 −1.29862e13 −0.464487
974974 −4.08718e13 −1.45515
975975 0 0
976976 −1.44983e13 −0.511437
977977 −5.44450e12 −0.191175 −0.0955877 0.995421i 0.530473π-0.530473\pi
−0.0955877 + 0.995421i 0.530473π0.530473\pi
978978 4.55869e12 0.159337
979979 −3.15505e13 −1.09770
980980 −1.72567e13 −0.597642
981981 −1.03568e13 −0.357040
982982 −5.31258e13 −1.82307
983983 2.73557e13 0.934454 0.467227 0.884138i 0.345253π-0.345253\pi
0.467227 + 0.884138i 0.345253π0.345253\pi
984984 3.55149e10 0.00120763
985985 −3.93753e13 −1.33279
986986 −6.37628e11 −0.0214843
987987 −5.69193e12 −0.190912
988988 0 0
989989 −3.72843e13 −1.23920
990990 9.59756e13 3.17543
991991 −9.85914e12 −0.324719 −0.162359 0.986732i 0.551910π-0.551910\pi
−0.162359 + 0.986732i 0.551910π0.551910\pi
992992 9.61069e12 0.315103
993993 −1.67570e13 −0.546923
994994 −1.68310e13 −0.546855
995995 −2.58063e13 −0.834683
996996 −4.72091e12 −0.152005
997997 −2.13556e13 −0.684516 −0.342258 0.939606i 0.611192π-0.611192\pi
−0.342258 + 0.939606i 0.611192π0.611192\pi
998998 2.60175e13 0.830190
999999 2.59850e13 0.825425
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 169.10.a.f.1.5 20
13.2 odd 12 13.10.e.a.4.3 20
13.7 odd 12 13.10.e.a.10.3 yes 20
13.12 even 2 inner 169.10.a.f.1.16 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.10.e.a.4.3 20 13.2 odd 12
13.10.e.a.10.3 yes 20 13.7 odd 12
169.10.a.f.1.5 20 1.1 even 1 trivial
169.10.a.f.1.16 20 13.12 even 2 inner