Properties

Label 338.8.b.d.337.2
Level 338338
Weight 88
Character 338.337
Analytic conductor 105.586105.586
Analytic rank 00
Dimension 22
Inner twists 22

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [338,8,Mod(337,338)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(338, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("338.337"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: N N == 338=2132 338 = 2 \cdot 13^{2}
Weight: k k == 8 8
Character orbit: [χ][\chi] == 338.b (of order 22, degree 11, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 105.586138614105.586138614
Analytic rank: 00
Dimension: 22
Coefficient field: Q(i)\Q(i)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2+1 x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 2)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 337.2
Root 1.00000i-1.00000i of defining polynomial
Character χ\chi == 338.337
Dual form 338.8.b.d.337.1

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
f(q)f(q) == q+8.00000iq2+12.0000q364.0000q4+210.000iq5+96.0000iq6+1016.00iq7512.000iq82043.00q91680.00q10+1092.00iq11768.000q128128.00q14+2520.00iq15+4096.00q1614706.0q1716344.0iq18+39940.0iq1913440.0iq20+12192.0iq218736.00q2268712.0q236144.00iq24+34025.0q2550760.0q2765024.0iq28102570.q2920160.0q30227552.iq31+32768.0iq32+13104.0iq33117648.iq34213360.q35+130752.q36+160526.iq37319520.q38+107520.q4010842.0iq4197536.0q42+630748.q4369888.0iq44429030.iq45549696.iq46+472656.iq47+49152.0q48208713.q49+272200.iq50176472.q511.49402e6q53406080.iq54229320.q55+520192.q56+479280.iq57820560.iq58+2.64066e6iq59161280.iq60+827702.q61+1.82042e6q622.07569e6iq63262144.q64104832.q66+126004.iq67+941184.q68824544.q691.70688e6iq70+1.41473e6iq71+1.04602e6iq72+980282.iq731.28421e6q74+408300.q752.55616e6iq761.10947e6q773.56680e6q79+860160.iq80+3.85892e6q81+86736.0q825.67289e6iq83780288.iq843.08826e6iq85+5.04598e6iq861.23084e6q87+559104.q881.19512e7iq89+3.43224e6q90+4.39757e6q922.73062e6iq933.78125e6q948.38740e6q95+393216.iq968.68215e6iq971.66970e6iq982.23096e6iq99+O(q100)q+8.00000i q^{2} +12.0000 q^{3} -64.0000 q^{4} +210.000i q^{5} +96.0000i q^{6} +1016.00i q^{7} -512.000i q^{8} -2043.00 q^{9} -1680.00 q^{10} +1092.00i q^{11} -768.000 q^{12} -8128.00 q^{14} +2520.00i q^{15} +4096.00 q^{16} -14706.0 q^{17} -16344.0i q^{18} +39940.0i q^{19} -13440.0i q^{20} +12192.0i q^{21} -8736.00 q^{22} -68712.0 q^{23} -6144.00i q^{24} +34025.0 q^{25} -50760.0 q^{27} -65024.0i q^{28} -102570. q^{29} -20160.0 q^{30} -227552. i q^{31} +32768.0i q^{32} +13104.0i q^{33} -117648. i q^{34} -213360. q^{35} +130752. q^{36} +160526. i q^{37} -319520. q^{38} +107520. q^{40} -10842.0i q^{41} -97536.0 q^{42} +630748. q^{43} -69888.0i q^{44} -429030. i q^{45} -549696. i q^{46} +472656. i q^{47} +49152.0 q^{48} -208713. q^{49} +272200. i q^{50} -176472. q^{51} -1.49402e6 q^{53} -406080. i q^{54} -229320. q^{55} +520192. q^{56} +479280. i q^{57} -820560. i q^{58} +2.64066e6i q^{59} -161280. i q^{60} +827702. q^{61} +1.82042e6 q^{62} -2.07569e6i q^{63} -262144. q^{64} -104832. q^{66} +126004. i q^{67} +941184. q^{68} -824544. q^{69} -1.70688e6i q^{70} +1.41473e6i q^{71} +1.04602e6i q^{72} +980282. i q^{73} -1.28421e6 q^{74} +408300. q^{75} -2.55616e6i q^{76} -1.10947e6 q^{77} -3.56680e6 q^{79} +860160. i q^{80} +3.85892e6 q^{81} +86736.0 q^{82} -5.67289e6i q^{83} -780288. i q^{84} -3.08826e6i q^{85} +5.04598e6i q^{86} -1.23084e6 q^{87} +559104. q^{88} -1.19512e7i q^{89} +3.43224e6 q^{90} +4.39757e6 q^{92} -2.73062e6i q^{93} -3.78125e6 q^{94} -8.38740e6 q^{95} +393216. i q^{96} -8.68215e6i q^{97} -1.66970e6i q^{98} -2.23096e6i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+24q3128q44086q93360q101536q1216256q14+8192q1629412q1717472q22137424q23+68050q25101520q27205140q2940320q30+16774800q95+O(q100) 2 q + 24 q^{3} - 128 q^{4} - 4086 q^{9} - 3360 q^{10} - 1536 q^{12} - 16256 q^{14} + 8192 q^{16} - 29412 q^{17} - 17472 q^{22} - 137424 q^{23} + 68050 q^{25} - 101520 q^{27} - 205140 q^{29} - 40320 q^{30}+ \cdots - 16774800 q^{95}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/338Z)×\left(\mathbb{Z}/338\mathbb{Z}\right)^\times.

nn 171171
χ(n)\chi(n) 1-1

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 8.00000i 0.707107i
33 12.0000 0.256600 0.128300 0.991735i 0.459048π-0.459048\pi
0.128300 + 0.991735i 0.459048π0.459048\pi
44 −64.0000 −0.500000
55 210.000i 0.751319i 0.926758 + 0.375659i 0.122584π0.122584\pi
−0.926758 + 0.375659i 0.877416π0.877416\pi
66 96.0000i 0.181444i
77 1016.00i 1.11957i 0.828638 + 0.559784i 0.189116π0.189116\pi
−0.828638 + 0.559784i 0.810884π0.810884\pi
88 − 512.000i − 0.353553i
99 −2043.00 −0.934156
1010 −1680.00 −0.531263
1111 1092.00i 0.247371i 0.992321 + 0.123685i 0.0394713π0.0394713\pi
−0.992321 + 0.123685i 0.960529π0.960529\pi
1212 −768.000 −0.128300
1313 0 0
1414 −8128.00 −0.791654
1515 2520.00i 0.192789i
1616 4096.00 0.250000
1717 −14706.0 −0.725978 −0.362989 0.931793i 0.618244π-0.618244\pi
−0.362989 + 0.931793i 0.618244π0.618244\pi
1818 − 16344.0i − 0.660548i
1919 39940.0i 1.33589i 0.744211 + 0.667945i 0.232826π0.232826\pi
−0.744211 + 0.667945i 0.767174π0.767174\pi
2020 − 13440.0i − 0.375659i
2121 12192.0i 0.287281i
2222 −8736.00 −0.174917
2323 −68712.0 −1.17757 −0.588783 0.808291i 0.700393π-0.700393\pi
−0.588783 + 0.808291i 0.700393π0.700393\pi
2424 − 6144.00i − 0.0907218i
2525 34025.0 0.435520
2626 0 0
2727 −50760.0 −0.496305
2828 − 65024.0i − 0.559784i
2929 −102570. −0.780957 −0.390479 0.920612i 0.627690π-0.627690\pi
−0.390479 + 0.920612i 0.627690π0.627690\pi
3030 −20160.0 −0.136322
3131 − 227552.i − 1.37188i −0.727660 0.685938i 0.759392π-0.759392\pi
0.727660 0.685938i 0.240608π-0.240608\pi
3232 32768.0i 0.176777i
3333 13104.0i 0.0634753i
3434 − 117648.i − 0.513344i
3535 −213360. −0.841153
3636 130752. 0.467078
3737 160526.i 0.521002i 0.965474 + 0.260501i 0.0838877π0.0838877\pi
−0.965474 + 0.260501i 0.916112π0.916112\pi
3838 −319520. −0.944616
3939 0 0
4040 107520. 0.265631
4141 − 10842.0i − 0.0245678i −0.999925 0.0122839i 0.996090π-0.996090\pi
0.999925 0.0122839i 0.00391018π-0.00391018\pi
4242 −97536.0 −0.203139
4343 630748. 1.20981 0.604904 0.796299i 0.293212π-0.293212\pi
0.604904 + 0.796299i 0.293212π0.293212\pi
4444 − 69888.0i − 0.123685i
4545 − 429030.i − 0.701849i
4646 − 549696.i − 0.832665i
4747 472656.i 0.664053i 0.943270 + 0.332026i 0.107732π0.107732\pi
−0.943270 + 0.332026i 0.892268π0.892268\pi
4848 49152.0 0.0641500
4949 −208713. −0.253433
5050 272200.i 0.307959i
5151 −176472. −0.186286
5252 0 0
5353 −1.49402e6 −1.37845 −0.689224 0.724548i 0.742048π-0.742048\pi
−0.689224 + 0.724548i 0.742048π0.742048\pi
5454 − 406080.i − 0.350940i
5555 −229320. −0.185854
5656 520192. 0.395827
5757 479280.i 0.342789i
5858 − 820560.i − 0.552220i
5959 2.64066e6i 1.67390i 0.547277 + 0.836952i 0.315665π0.315665\pi
−0.547277 + 0.836952i 0.684335π0.684335\pi
6060 − 161280.i − 0.0963943i
6161 827702. 0.466895 0.233448 0.972369i 0.424999π-0.424999\pi
0.233448 + 0.972369i 0.424999π0.424999\pi
6262 1.82042e6 0.970063
6363 − 2.07569e6i − 1.04585i
6464 −262144. −0.125000
6565 0 0
6666 −104832. −0.0448838
6767 126004.i 0.0511826i 0.999672 + 0.0255913i 0.00814686π0.00814686\pi
−0.999672 + 0.0255913i 0.991853π0.991853\pi
6868 941184. 0.362989
6969 −824544. −0.302164
7070 − 1.70688e6i − 0.594785i
7171 1.41473e6i 0.469104i 0.972104 + 0.234552i 0.0753622π0.0753622\pi
−0.972104 + 0.234552i 0.924638π0.924638\pi
7272 1.04602e6i 0.330274i
7373 980282.i 0.294931i 0.989067 + 0.147466i 0.0471116π0.0471116\pi
−0.989067 + 0.147466i 0.952888π0.952888\pi
7474 −1.28421e6 −0.368404
7575 408300. 0.111754
7676 − 2.55616e6i − 0.667945i
7777 −1.10947e6 −0.276948
7878 0 0
7979 −3.56680e6 −0.813924 −0.406962 0.913445i 0.633412π-0.633412\pi
−0.406962 + 0.913445i 0.633412π0.633412\pi
8080 860160.i 0.187830i
8181 3.85892e6 0.806805
8282 86736.0 0.0173720
8383 − 5.67289e6i − 1.08901i −0.838758 0.544504i 0.816718π-0.816718\pi
0.838758 0.544504i 0.183282π-0.183282\pi
8484 − 780288.i − 0.143641i
8585 − 3.08826e6i − 0.545441i
8686 5.04598e6i 0.855463i
8787 −1.23084e6 −0.200394
8888 559104. 0.0874587
8989 − 1.19512e7i − 1.79699i −0.438982 0.898496i 0.644661π-0.644661\pi
0.438982 0.898496i 0.355339π-0.355339\pi
9090 3.43224e6 0.496282
9191 0 0
9292 4.39757e6 0.588783
9393 − 2.73062e6i − 0.352023i
9494 −3.78125e6 −0.469556
9595 −8.38740e6 −1.00368
9696 393216.i 0.0453609i
9797 − 8.68215e6i − 0.965886i −0.875652 0.482943i 0.839568π-0.839568\pi
0.875652 0.482943i 0.160432π-0.160432\pi
9898 − 1.66970e6i − 0.179204i
9999 − 2.23096e6i − 0.231083i
100100 −2.17760e6 −0.217760
101101 1.00795e7 0.973455 0.486727 0.873554i 0.338190π-0.338190\pi
0.486727 + 0.873554i 0.338190π0.338190\pi
102102 − 1.41178e6i − 0.131724i
103103 −3.74799e6 −0.337962 −0.168981 0.985619i 0.554048π-0.554048\pi
−0.168981 + 0.985619i 0.554048π0.554048\pi
104104 0 0
105105 −2.56032e6 −0.215840
106106 − 1.19521e7i − 0.974710i
107107 −1.79856e7 −1.41932 −0.709661 0.704543i 0.751152π-0.751152\pi
−0.709661 + 0.704543i 0.751152π0.751152\pi
108108 3.24864e6 0.248152
109109 − 1.22570e7i − 0.906552i −0.891370 0.453276i 0.850255π-0.850255\pi
0.891370 0.453276i 0.149745π-0.149745\pi
110110 − 1.83456e6i − 0.131419i
111111 1.92631e6i 0.133689i
112112 4.16154e6i 0.279892i
113113 1.65950e7 1.08194 0.540968 0.841043i 0.318058π-0.318058\pi
0.540968 + 0.841043i 0.318058π0.318058\pi
114114 −3.83424e6 −0.242389
115115 − 1.44295e7i − 0.884727i
116116 6.56448e6 0.390479
117117 0 0
118118 −2.11253e7 −1.18363
119119 − 1.49413e7i − 0.812782i
120120 1.29024e6 0.0681610
121121 1.82947e7 0.938808
122122 6.62162e6i 0.330145i
123123 − 130104.i − 0.00630410i
124124 1.45633e7i 0.685938i
125125 2.35515e7i 1.07853i
126126 1.66055e7 0.739529
127127 −1.16826e6 −0.0506087 −0.0253043 0.999680i 0.508055π-0.508055\pi
−0.0253043 + 0.999680i 0.508055π0.508055\pi
128128 − 2.09715e6i − 0.0883883i
129129 7.56898e6 0.310437
130130 0 0
131131 −7.92383e6 −0.307954 −0.153977 0.988074i 0.549208π-0.549208\pi
−0.153977 + 0.988074i 0.549208π0.549208\pi
132132 − 838656.i − 0.0317377i
133133 −4.05790e7 −1.49562
134134 −1.00803e6 −0.0361916
135135 − 1.06596e7i − 0.372883i
136136 7.52947e6i 0.256672i
137137 − 315654.i − 0.0104879i −0.999986 0.00524396i 0.998331π-0.998331\pi
0.999986 0.00524396i 0.00166921π-0.00166921\pi
138138 − 6.59635e6i − 0.213662i
139139 3.92038e7 1.23816 0.619079 0.785329i 0.287506π-0.287506\pi
0.619079 + 0.785329i 0.287506π0.287506\pi
140140 1.36550e7 0.420576
141141 5.67187e6i 0.170396i
142142 −1.13178e7 −0.331706
143143 0 0
144144 −8.36813e6 −0.233539
145145 − 2.15397e7i − 0.586748i
146146 −7.84226e6 −0.208548
147147 −2.50456e6 −0.0650309
148148 − 1.02737e7i − 0.260501i
149149 2.18860e7i 0.542020i 0.962577 + 0.271010i 0.0873577π0.0873577\pi
−0.962577 + 0.271010i 0.912642π0.912642\pi
150150 3.26640e6i 0.0790224i
151151 − 2.94154e7i − 0.695274i −0.937629 0.347637i 0.886984π-0.886984\pi
0.937629 0.347637i 0.113016π-0.113016\pi
152152 2.04493e7 0.472308
153153 3.00444e7 0.678177
154154 − 8.87578e6i − 0.195832i
155155 4.77859e7 1.03072
156156 0 0
157157 6.05550e7 1.24882 0.624412 0.781095i 0.285339π-0.285339\pi
0.624412 + 0.781095i 0.285339π0.285339\pi
158158 − 2.85344e7i − 0.575531i
159159 −1.79282e7 −0.353710
160160 −6.88128e6 −0.132816
161161 − 6.98114e7i − 1.31837i
162162 3.08714e7i 0.570497i
163163 5.70853e7i 1.03245i 0.856454 + 0.516223i 0.172663π0.172663\pi
−0.856454 + 0.516223i 0.827337π0.827337\pi
164164 693888.i 0.0122839i
165165 −2.75184e6 −0.0476902
166166 4.53831e7 0.770045
167167 − 8.77265e7i − 1.45755i −0.684754 0.728775i 0.740090π-0.740090\pi
0.684754 0.728775i 0.259910π-0.259910\pi
168168 6.24230e6 0.101569
169169 0 0
170170 2.47061e7 0.385685
171171 − 8.15974e7i − 1.24793i
172172 −4.03679e7 −0.604904
173173 −8.56954e6 −0.125833 −0.0629167 0.998019i 0.520040π-0.520040\pi
−0.0629167 + 0.998019i 0.520040π0.520040\pi
174174 − 9.84672e6i − 0.141700i
175175 3.45694e7i 0.487594i
176176 4.47283e6i 0.0618427i
177177 3.16879e7i 0.429524i
178178 9.56095e7 1.27067
179179 −1.88041e7 −0.245056 −0.122528 0.992465i 0.539100π-0.539100\pi
−0.122528 + 0.992465i 0.539100π0.539100\pi
180180 2.74579e7i 0.350925i
181181 5.99625e7 0.751631 0.375816 0.926694i 0.377363π-0.377363\pi
0.375816 + 0.926694i 0.377363π0.377363\pi
182182 0 0
183183 9.93242e6 0.119805
184184 3.51805e7i 0.416332i
185185 −3.37105e7 −0.391439
186186 2.18450e7 0.248918
187187 − 1.60590e7i − 0.179586i
188188 − 3.02500e7i − 0.332026i
189189 − 5.15722e7i − 0.555647i
190190 − 6.70992e7i − 0.709708i
191191 9.39861e7 0.975993 0.487997 0.872845i 0.337728π-0.337728\pi
0.487997 + 0.872845i 0.337728π0.337728\pi
192192 −3.14573e6 −0.0320750
193193 − 3.51946e7i − 0.352391i −0.984355 0.176196i 0.943621π-0.943621\pi
0.984355 0.176196i 0.0563791π-0.0563791\pi
194194 6.94572e7 0.682985
195195 0 0
196196 1.33576e7 0.126717
197197 − 1.02985e8i − 0.959718i −0.877346 0.479859i 0.840688π-0.840688\pi
0.877346 0.479859i 0.159312π-0.159312\pi
198198 1.78476e7 0.163400
199199 −8.36376e7 −0.752342 −0.376171 0.926550i 0.622760π-0.622760\pi
−0.376171 + 0.926550i 0.622760π0.622760\pi
200200 − 1.74208e7i − 0.153980i
201201 1.51205e6i 0.0131335i
202202 8.06363e7i 0.688337i
203203 − 1.04211e8i − 0.874335i
204204 1.12942e7 0.0931430
205205 2.27682e6 0.0184582
206206 − 2.99839e7i − 0.238975i
207207 1.40379e8 1.10003
208208 0 0
209209 −4.36145e7 −0.330460
210210 − 2.04826e7i − 0.152622i
211211 −9.74010e7 −0.713797 −0.356899 0.934143i 0.616166π-0.616166\pi
−0.356899 + 0.934143i 0.616166π0.616166\pi
212212 9.56172e7 0.689224
213213 1.69767e7i 0.120372i
214214 − 1.43885e8i − 1.00361i
215215 1.32457e8i 0.908951i
216216 2.59891e7i 0.175470i
217217 2.31193e8 1.53591
218218 9.80562e7 0.641029
219219 1.17634e7i 0.0756794i
220220 1.46765e7 0.0929271
221221 0 0
222222 −1.54105e7 −0.0945325
223223 1.46457e7i 0.0884390i 0.999022 + 0.0442195i 0.0140801π0.0140801\pi
−0.999022 + 0.0442195i 0.985920π0.985920\pi
224224 −3.32923e7 −0.197914
225225 −6.95131e7 −0.406844
226226 1.32760e8i 0.765045i
227227 1.84541e8i 1.04713i 0.851985 + 0.523567i 0.175399π0.175399\pi
−0.851985 + 0.523567i 0.824601π0.824601\pi
228228 − 3.06739e7i − 0.171395i
229229 − 8.75461e6i − 0.0481740i −0.999710 0.0240870i 0.992332π-0.992332\pi
0.999710 0.0240870i 0.00766787π-0.00766787\pi
230230 1.15436e8 0.625597
231231 −1.33137e7 −0.0710650
232232 5.25158e7i 0.276110i
233233 1.19556e8 0.619193 0.309597 0.950868i 0.399806π-0.399806\pi
0.309597 + 0.950868i 0.399806π0.399806\pi
234234 0 0
235235 −9.92578e7 −0.498915
236236 − 1.69002e8i − 0.836952i
237237 −4.28016e7 −0.208853
238238 1.19530e8 0.574723
239239 − 3.96209e8i − 1.87729i −0.344883 0.938646i 0.612081π-0.612081\pi
0.344883 0.938646i 0.387919π-0.387919\pi
240240 1.03219e7i 0.0481971i
241241 − 2.56606e8i − 1.18089i −0.807080 0.590443i 0.798953π-0.798953\pi
0.807080 0.590443i 0.201047π-0.201047\pi
242242 1.46358e8i 0.663837i
243243 1.57319e8 0.703331
244244 −5.29729e7 −0.233448
245245 − 4.38297e7i − 0.190409i
246246 1.04083e6 0.00445767
247247 0 0
248248 −1.16507e8 −0.485031
249249 − 6.80747e7i − 0.279440i
250250 −1.88412e8 −0.762638
251251 7.34775e7 0.293290 0.146645 0.989189i 0.453153π-0.453153\pi
0.146645 + 0.989189i 0.453153π0.453153\pi
252252 1.32844e8i 0.522926i
253253 − 7.50335e7i − 0.291295i
254254 − 9.34605e6i − 0.0357857i
255255 − 3.70591e7i − 0.139960i
256256 1.67772e7 0.0625000
257257 2.02701e8 0.744886 0.372443 0.928055i 0.378520π-0.378520\pi
0.372443 + 0.928055i 0.378520π0.378520\pi
258258 6.05518e7i 0.219512i
259259 −1.63094e8 −0.583297
260260 0 0
261261 2.09551e8 0.729536
262262 − 6.33906e7i − 0.217756i
263263 1.54254e8 0.522867 0.261434 0.965221i 0.415805π-0.415805\pi
0.261434 + 0.965221i 0.415805π0.415805\pi
264264 6.70925e6 0.0224419
265265 − 3.13744e8i − 1.03565i
266266 − 3.24632e8i − 1.05756i
267267 − 1.43414e8i − 0.461108i
268268 − 8.06426e6i − 0.0255913i
269269 −6.24018e8 −1.95463 −0.977315 0.211793i 0.932070π-0.932070\pi
−0.977315 + 0.211793i 0.932070π0.932070\pi
270270 8.52768e7 0.263668
271271 − 3.87983e8i − 1.18419i −0.805869 0.592094i 0.798302π-0.798302\pi
0.805869 0.592094i 0.201698π-0.201698\pi
272272 −6.02358e7 −0.181494
273273 0 0
274274 2.52523e6 0.00741608
275275 3.71553e7i 0.107735i
276276 5.27708e7 0.151082
277277 −4.53952e8 −1.28331 −0.641654 0.766994i 0.721752π-0.721752\pi
−0.641654 + 0.766994i 0.721752π0.721752\pi
278278 3.13630e8i 0.875510i
279279 4.64889e8i 1.28155i
280280 1.09240e8i 0.297392i
281281 3.33770e8i 0.897377i 0.893688 + 0.448689i 0.148109π0.148109\pi
−0.893688 + 0.448689i 0.851891π0.851891\pi
282282 −4.53750e7 −0.120488
283283 −5.37695e8 −1.41021 −0.705104 0.709104i 0.749100π-0.749100\pi
−0.705104 + 0.709104i 0.749100π0.749100\pi
284284 − 9.05426e7i − 0.234552i
285285 −1.00649e8 −0.257544
286286 0 0
287287 1.10155e7 0.0275053
288288 − 6.69450e7i − 0.165137i
289289 −1.94072e8 −0.472956
290290 1.72318e8 0.414894
291291 − 1.04186e8i − 0.247847i
292292 − 6.27380e7i − 0.147466i
293293 3.35600e8i 0.779445i 0.920932 + 0.389722i 0.127429π0.127429\pi
−0.920932 + 0.389722i 0.872571π0.872571\pi
294294 − 2.00364e7i − 0.0459838i
295295 −5.54539e8 −1.25764
296296 8.21893e7 0.184202
297297 − 5.54299e7i − 0.122771i
298298 −1.75088e8 −0.383266
299299 0 0
300300 −2.61312e7 −0.0558772
301301 6.40840e8i 1.35446i
302302 2.35324e8 0.491633
303303 1.20954e8 0.249789
304304 1.63594e8i 0.333972i
305305 1.73817e8i 0.350787i
306306 2.40355e8i 0.479543i
307307 2.15029e8i 0.424143i 0.977254 + 0.212072i 0.0680210π0.0680210\pi
−0.977254 + 0.212072i 0.931979π0.931979\pi
308308 7.10062e7 0.138474
309309 −4.49759e7 −0.0867212
310310 3.82287e8i 0.728826i
311311 −7.92062e8 −1.49313 −0.746565 0.665313i 0.768298π-0.768298\pi
−0.746565 + 0.665313i 0.768298π0.768298\pi
312312 0 0
313313 −1.18457e8 −0.218352 −0.109176 0.994022i 0.534821π-0.534821\pi
−0.109176 + 0.994022i 0.534821π0.534821\pi
314314 4.84440e8i 0.883051i
315315 4.35894e8 0.785768
316316 2.28275e8 0.406962
317317 5.07310e7i 0.0894470i 0.998999 + 0.0447235i 0.0142407π0.0142407\pi
−0.998999 + 0.0447235i 0.985759π0.985759\pi
318318 − 1.43426e8i − 0.250111i
319319 − 1.12006e8i − 0.193186i
320320 − 5.50502e7i − 0.0939149i
321321 −2.15827e8 −0.364198
322322 5.58491e8 0.932225
323323 − 5.87358e8i − 0.969826i
324324 −2.46971e8 −0.403402
325325 0 0
326326 −4.56682e8 −0.730050
327327 − 1.47084e8i − 0.232621i
328328 −5.55110e6 −0.00868602
329329 −4.80218e8 −0.743453
330330 − 2.20147e7i − 0.0337221i
331331 − 2.73757e8i − 0.414923i −0.978243 0.207461i 0.933480π-0.933480\pi
0.978243 0.207461i 0.0665201π-0.0665201\pi
332332 3.63065e8i 0.544504i
333333 − 3.27955e8i − 0.486697i
334334 7.01812e8 1.03064
335335 −2.64608e7 −0.0384545
336336 4.99384e7i 0.0718203i
337337 9.18512e7 0.130732 0.0653658 0.997861i 0.479179π-0.479179\pi
0.0653658 + 0.997861i 0.479179π0.479179\pi
338338 0 0
339339 1.99140e8 0.277625
340340 1.97649e8i 0.272720i
341341 2.48487e8 0.339362
342342 6.52779e8 0.882419
343343 6.24667e8i 0.835833i
344344 − 3.22943e8i − 0.427732i
345345 − 1.73154e8i − 0.227021i
346346 − 6.85563e7i − 0.0889777i
347347 −1.36700e9 −1.75637 −0.878187 0.478318i 0.841247π-0.841247\pi
−0.878187 + 0.478318i 0.841247π0.841247\pi
348348 7.87738e7 0.100197
349349 1.13143e9i 1.42475i 0.701797 + 0.712377i 0.252381π0.252381\pi
−0.701797 + 0.712377i 0.747619π0.747619\pi
350350 −2.76555e8 −0.344781
351351 0 0
352352 −3.57827e7 −0.0437294
353353 4.48395e7i 0.0542562i 0.999632 + 0.0271281i 0.00863620π0.00863620\pi
−0.999632 + 0.0271281i 0.991364π0.991364\pi
354354 −2.53503e8 −0.303719
355355 −2.97093e8 −0.352446
356356 7.64876e8i 0.898496i
357357 − 1.79296e8i − 0.208560i
358358 − 1.50432e8i − 0.173281i
359359 3.98281e8i 0.454317i 0.973858 + 0.227158i 0.0729435π0.0729435\pi
−0.973858 + 0.227158i 0.927057π0.927057\pi
360360 −2.19663e8 −0.248141
361361 −7.01332e8 −0.784600
362362 4.79700e8i 0.531483i
363363 2.19536e8 0.240898
364364 0 0
365365 −2.05859e8 −0.221588
366366 7.94594e7i 0.0847152i
367367 1.63472e9 1.72628 0.863140 0.504964i 0.168494π-0.168494\pi
0.863140 + 0.504964i 0.168494π0.168494\pi
368368 −2.81444e8 −0.294391
369369 2.21502e7i 0.0229501i
370370 − 2.69684e8i − 0.276789i
371371 − 1.51792e9i − 1.54327i
372372 1.74760e8i 0.176012i
373373 −1.54633e9 −1.54284 −0.771421 0.636325i 0.780454π-0.780454\pi
−0.771421 + 0.636325i 0.780454π0.780454\pi
374374 1.28472e8 0.126986
375375 2.82618e8i 0.276752i
376376 2.42000e8 0.234778
377377 0 0
378378 4.12577e8 0.392902
379379 1.05688e9i 0.997216i 0.866828 + 0.498608i 0.166155π0.166155\pi
−0.866828 + 0.498608i 0.833845π0.833845\pi
380380 5.36794e8 0.501839
381381 −1.40191e7 −0.0129862
382382 7.51889e8i 0.690132i
383383 − 2.24910e8i − 0.204556i −0.994756 0.102278i 0.967387π-0.967387\pi
0.994756 0.102278i 0.0326132π-0.0326132\pi
384384 − 2.51658e7i − 0.0226805i
385385 − 2.32989e8i − 0.208077i
386386 2.81556e8 0.249178
387387 −1.28862e9 −1.13015
388388 5.55657e8i 0.482943i
389389 −1.01788e9 −0.876746 −0.438373 0.898793i 0.644445π-0.644445\pi
−0.438373 + 0.898793i 0.644445π0.644445\pi
390390 0 0
391391 1.01048e9 0.854887
392392 1.06861e8i 0.0896021i
393393 −9.50859e7 −0.0790210
394394 8.23883e8 0.678623
395395 − 7.49028e8i − 0.611517i
396396 1.42781e8i 0.115541i
397397 − 1.47565e9i − 1.18363i −0.806072 0.591817i 0.798411π-0.798411\pi
0.806072 0.591817i 0.201589π-0.201589\pi
398398 − 6.69100e8i − 0.531986i
399399 −4.86948e8 −0.383776
400400 1.39366e8 0.108880
401401 2.74912e8i 0.212906i 0.994318 + 0.106453i 0.0339494π0.0339494\pi
−0.994318 + 0.106453i 0.966051π0.966051\pi
402402 −1.20964e7 −0.00928676
403403 0 0
404404 −6.45090e8 −0.486727
405405 8.10373e8i 0.606167i
406406 8.33689e8 0.618248
407407 −1.75294e8 −0.128881
408408 9.03537e7i 0.0658620i
409409 1.63427e9i 1.18112i 0.806995 + 0.590558i 0.201092π0.201092\pi
−0.806995 + 0.590558i 0.798908π0.798908\pi
410410 1.82146e7i 0.0130519i
411411 − 3.78785e6i − 0.00269120i
412412 2.39871e8 0.168981
413413 −2.68291e9 −1.87405
414414 1.12303e9i 0.777839i
415415 1.19131e9 0.818192
416416 0 0
417417 4.70445e8 0.317712
418418 − 3.48916e8i − 0.233670i
419419 −1.11280e9 −0.739039 −0.369519 0.929223i 0.620478π-0.620478\pi
−0.369519 + 0.929223i 0.620478π0.620478\pi
420420 1.63860e8 0.107920
421421 − 9.22528e8i − 0.602549i −0.953537 0.301274i 0.902588π-0.902588\pi
0.953537 0.301274i 0.0974120π-0.0974120\pi
422422 − 7.79208e8i − 0.504731i
423423 − 9.65636e8i − 0.620329i
424424 7.64937e8i 0.487355i
425425 −5.00372e8 −0.316178
426426 −1.35814e8 −0.0851159
427427 8.40945e8i 0.522721i
428428 1.15108e9 0.709661
429429 0 0
430430 −1.05966e9 −0.642726
431431 9.81508e8i 0.590505i 0.955419 + 0.295252i 0.0954037π0.0954037\pi
−0.955419 + 0.295252i 0.904596π0.904596\pi
432432 −2.07913e8 −0.124076
433433 −2.84998e9 −1.68707 −0.843537 0.537071i 0.819531π-0.819531\pi
−0.843537 + 0.537071i 0.819531π0.819531\pi
434434 1.84954e9i 1.08605i
435435 − 2.58476e8i − 0.150560i
436436 7.84450e8i 0.453276i
437437 − 2.74436e9i − 1.57310i
438438 −9.41071e7 −0.0535134
439439 1.05622e9 0.595838 0.297919 0.954591i 0.403708π-0.403708\pi
0.297919 + 0.954591i 0.403708π0.403708\pi
440440 1.17412e8i 0.0657094i
441441 4.26401e8 0.236746
442442 0 0
443443 1.82325e9 0.996401 0.498201 0.867062i 0.333994π-0.333994\pi
0.498201 + 0.867062i 0.333994π0.333994\pi
444444 − 1.23284e8i − 0.0668446i
445445 2.50975e9 1.35011
446446 −1.17166e8 −0.0625358
447447 2.62633e8i 0.139082i
448448 − 2.66338e8i − 0.139946i
449449 1.84846e9i 0.963713i 0.876250 + 0.481856i 0.160037π0.160037\pi
−0.876250 + 0.481856i 0.839963π0.839963\pi
450450 − 5.56105e8i − 0.287682i
451451 1.18395e7 0.00607735
452452 −1.06208e9 −0.540968
453453 − 3.52985e8i − 0.178407i
454454 −1.47633e9 −0.740435
455455 0 0
456456 2.45391e8 0.121194
457457 2.98066e9i 1.46085i 0.682993 + 0.730425i 0.260678π0.260678\pi
−0.682993 + 0.730425i 0.739322π0.739322\pi
458458 7.00369e7 0.0340642
459459 7.46477e8 0.360306
460460 9.23489e8i 0.442364i
461461 2.52781e9i 1.20169i 0.799367 + 0.600843i 0.205168π0.205168\pi
−0.799367 + 0.600843i 0.794832π0.794832\pi
462462 − 1.06509e8i − 0.0502505i
463463 − 8.90291e8i − 0.416868i −0.978036 0.208434i 0.933163π-0.933163\pi
0.978036 0.208434i 0.0668366π-0.0668366\pi
464464 −4.20127e8 −0.195239
465465 5.73431e8 0.264482
466466 9.56450e8i 0.437836i
467467 −2.65667e9 −1.20706 −0.603529 0.797341i 0.706239π-0.706239\pi
−0.603529 + 0.797341i 0.706239π0.706239\pi
468468 0 0
469469 −1.28020e8 −0.0573024
470470 − 7.94062e8i − 0.352786i
471471 7.26660e8 0.320448
472472 1.35202e9 0.591814
473473 6.88777e8i 0.299271i
474474 − 3.42413e8i − 0.147681i
475475 1.35896e9i 0.581806i
476476 9.56243e8i 0.406391i
477477 3.05228e9 1.28769
478478 3.16967e9 1.32745
479479 1.30093e9i 0.540855i 0.962740 + 0.270428i 0.0871651π0.0871651\pi
−0.962740 + 0.270428i 0.912835π0.912835\pi
480480 −8.25754e7 −0.0340805
481481 0 0
482482 2.05285e9 0.835012
483483 − 8.37737e8i − 0.338293i
484484 −1.17086e9 −0.469404
485485 1.82325e9 0.725689
486486 1.25855e9i 0.497330i
487487 1.07447e9i 0.421542i 0.977535 + 0.210771i 0.0675975π0.0675975\pi
−0.977535 + 0.210771i 0.932402π0.932402\pi
488488 − 4.23783e8i − 0.165072i
489489 6.85024e8i 0.264926i
490490 3.50638e8 0.134640
491491 7.83344e8 0.298653 0.149327 0.988788i 0.452289π-0.452289\pi
0.149327 + 0.988788i 0.452289π0.452289\pi
492492 8.32666e6i 0.00315205i
493493 1.50839e9 0.566958
494494 0 0
495495 4.68501e8 0.173617
496496 − 9.32053e8i − 0.342969i
497497 −1.43736e9 −0.525193
498498 5.44598e8 0.197594
499499 6.23188e8i 0.224526i 0.993679 + 0.112263i 0.0358100π0.0358100\pi
−0.993679 + 0.112263i 0.964190π0.964190\pi
500500 − 1.50730e9i − 0.539267i
501501 − 1.05272e9i − 0.374007i
502502 5.87820e8i 0.207387i
503503 −2.70927e9 −0.949215 −0.474607 0.880198i 0.657410π-0.657410\pi
−0.474607 + 0.880198i 0.657410π0.657410\pi
504504 −1.06275e9 −0.369764
505505 2.11670e9i 0.731375i
506506 6.00268e8 0.205977
507507 0 0
508508 7.47684e7 0.0253043
509509 − 3.49943e9i − 1.17621i −0.808784 0.588106i 0.799874π-0.799874\pi
0.808784 0.588106i 0.200126π-0.200126\pi
510510 2.96473e8 0.0989668
511511 −9.95967e8 −0.330196
512512 1.34218e8i 0.0441942i
513513 − 2.02735e9i − 0.663008i
514514 1.62161e9i 0.526714i
515515 − 7.87078e8i − 0.253918i
516516 −4.84414e8 −0.155218
517517 −5.16140e8 −0.164267
518518 − 1.30476e9i − 0.412453i
519519 −1.02835e8 −0.0322889
520520 0 0
521521 −1.37683e9 −0.426530 −0.213265 0.976994i 0.568410π-0.568410\pi
−0.213265 + 0.976994i 0.568410π0.568410\pi
522522 1.67640e9i 0.515860i
523523 −2.86154e9 −0.874669 −0.437334 0.899299i 0.644077π-0.644077\pi
−0.437334 + 0.899299i 0.644077π0.644077\pi
524524 5.07125e8 0.153977
525525 4.14833e8i 0.125117i
526526 1.23403e9i 0.369723i
527527 3.34638e9i 0.995951i
528528 5.36740e7i 0.0158688i
529529 1.31651e9 0.386661
530530 2.50995e9 0.732318
531531 − 5.39487e9i − 1.56369i
532532 2.59706e9 0.747810
533533 0 0
534534 1.14731e9 0.326053
535535 − 3.77697e9i − 1.06636i
536536 6.45140e7 0.0180958
537537 −2.25649e8 −0.0628815
538538 − 4.99215e9i − 1.38213i
539539 − 2.27915e8i − 0.0626919i
540540 6.82214e8i 0.186442i
541541 5.34467e9i 1.45121i 0.688111 + 0.725605i 0.258440π0.258440\pi
−0.688111 + 0.725605i 0.741560π0.741560\pi
542542 3.10387e9 0.837347
543543 7.19550e8 0.192869
544544 − 4.81886e8i − 0.128336i
545545 2.57398e9 0.681109
546546 0 0
547547 −3.37135e9 −0.880740 −0.440370 0.897816i 0.645153π-0.645153\pi
−0.440370 + 0.897816i 0.645153π0.645153\pi
548548 2.02019e7i 0.00524396i
549549 −1.69100e9 −0.436153
550550 −2.97242e8 −0.0761801
551551 − 4.09665e9i − 1.04327i
552552 4.22167e8i 0.106831i
553553 − 3.62387e9i − 0.911244i
554554 − 3.63162e9i − 0.907436i
555555 −4.04526e8 −0.100443
556556 −2.50904e9 −0.619079
557557 − 5.61106e9i − 1.37579i −0.725811 0.687894i 0.758535π-0.758535\pi
0.725811 0.687894i 0.241465π-0.241465\pi
558558 −3.71911e9 −0.906190
559559 0 0
560560 −8.73923e8 −0.210288
561561 − 1.92707e8i − 0.0460817i
562562 −2.67016e9 −0.634542
563563 −6.69690e9 −1.58159 −0.790795 0.612081i 0.790333π-0.790333\pi
−0.790795 + 0.612081i 0.790333π0.790333\pi
564564 − 3.63000e8i − 0.0851980i
565565 3.48494e9i 0.812879i
566566 − 4.30156e9i − 0.997168i
567567 3.92066e9i 0.903273i
568568 7.24341e8 0.165853
569569 −1.96850e9 −0.447964 −0.223982 0.974593i 0.571906π-0.571906\pi
−0.223982 + 0.974593i 0.571906π0.571906\pi
570570 − 8.05190e8i − 0.182111i
571571 −1.02926e9 −0.231365 −0.115682 0.993286i 0.536906π-0.536906\pi
−0.115682 + 0.993286i 0.536906π0.536906\pi
572572 0 0
573573 1.12783e9 0.250440
574574 8.81238e7i 0.0194492i
575575 −2.33793e9 −0.512853
576576 5.35560e8 0.116770
577577 − 3.31179e9i − 0.717708i −0.933394 0.358854i 0.883168π-0.883168\pi
0.933394 0.358854i 0.116832π-0.116832\pi
578578 − 1.55258e9i − 0.334431i
579579 − 4.22335e8i − 0.0904236i
580580 1.37854e9i 0.293374i
581581 5.76366e9 1.21922
582582 8.33486e8 0.175254
583583 − 1.63147e9i − 0.340988i
584584 5.01904e8 0.104274
585585 0 0
586586 −2.68480e9 −0.551151
587587 5.59411e8i 0.114156i 0.998370 + 0.0570778i 0.0181783π0.0181783\pi
−0.998370 + 0.0570778i 0.981822π0.981822\pi
588588 1.60292e8 0.0325155
589589 9.08843e9 1.83267
590590 − 4.43631e9i − 0.889282i
591591 − 1.23582e9i − 0.246264i
592592 6.57514e8i 0.130250i
593593 − 3.02459e9i − 0.595628i −0.954624 0.297814i 0.903742π-0.903742\pi
0.954624 0.297814i 0.0962575π-0.0962575\pi
594594 4.43439e8 0.0868124
595595 3.13767e9 0.610658
596596 − 1.40071e9i − 0.271010i
597597 −1.00365e9 −0.193051
598598 0 0
599599 −5.63246e9 −1.07079 −0.535395 0.844602i 0.679837π-0.679837\pi
−0.535395 + 0.844602i 0.679837π0.679837\pi
600600 − 2.09050e8i − 0.0395112i
601601 3.40792e8 0.0640366 0.0320183 0.999487i 0.489807π-0.489807\pi
0.0320183 + 0.999487i 0.489807π0.489807\pi
602602 −5.12672e9 −0.957749
603603 − 2.57426e8i − 0.0478126i
604604 1.88259e9i 0.347637i
605605 3.84189e9i 0.705344i
606606 9.67636e8i 0.176627i
607607 3.85420e9 0.699477 0.349739 0.936847i 0.386270π-0.386270\pi
0.349739 + 0.936847i 0.386270π0.386270\pi
608608 −1.30875e9 −0.236154
609609 − 1.25053e9i − 0.224355i
610610 −1.39054e9 −0.248044
611611 0 0
612612 −1.92284e9 −0.339088
613613 − 9.22245e9i − 1.61709i −0.588434 0.808545i 0.700255π-0.700255\pi
0.588434 0.808545i 0.299745π-0.299745\pi
614614 −1.72023e9 −0.299915
615615 2.73218e7 0.00473639
616616 5.68050e8i 0.0979160i
617617 − 6.53611e9i − 1.12027i −0.828402 0.560133i 0.810750π-0.810750\pi
0.828402 0.560133i 0.189250π-0.189250\pi
618618 − 3.59807e8i − 0.0613211i
619619 1.36559e9i 0.231420i 0.993283 + 0.115710i 0.0369144π0.0369144\pi
−0.993283 + 0.115710i 0.963086π0.963086\pi
620620 −3.05830e9 −0.515358
621621 3.48782e9 0.584431
622622 − 6.33649e9i − 1.05580i
623623 1.21424e10 2.01186
624624 0 0
625625 −2.28761e9 −0.374802
626626 − 9.47659e8i − 0.154398i
627627 −5.23374e8 −0.0847960
628628 −3.87552e9 −0.624412
629629 − 2.36070e9i − 0.378236i
630630 3.48716e9i 0.555622i
631631 1.54079e9i 0.244141i 0.992521 + 0.122070i 0.0389533π0.0389533\pi
−0.992521 + 0.122070i 0.961047π0.961047\pi
632632 1.82620e9i 0.287766i
633633 −1.16881e9 −0.183160
634634 −4.05848e8 −0.0632486
635635 − 2.45334e8i − 0.0380233i
636636 1.14741e9 0.176855
637637 0 0
638638 8.96052e8 0.136603
639639 − 2.89029e9i − 0.438216i
640640 4.40402e8 0.0664078
641641 4.54018e9 0.680879 0.340440 0.940266i 0.389424π-0.389424\pi
0.340440 + 0.940266i 0.389424π0.389424\pi
642642 − 1.72661e9i − 0.257527i
643643 − 1.14054e10i − 1.69189i −0.533272 0.845944i 0.679038π-0.679038\pi
0.533272 0.845944i 0.320962π-0.320962\pi
644644 4.46793e9i 0.659183i
645645 1.58948e9i 0.233237i
646646 4.69886e9 0.685770
647647 1.26393e10 1.83468 0.917338 0.398109i 0.130334π-0.130334\pi
0.917338 + 0.398109i 0.130334π0.130334\pi
648648 − 1.97577e9i − 0.285248i
649649 −2.88360e9 −0.414075
650650 0 0
651651 2.77431e9 0.394114
652652 − 3.65346e9i − 0.516223i
653653 −1.05004e10 −1.47575 −0.737873 0.674940i 0.764170π-0.764170\pi
−0.737873 + 0.674940i 0.764170π0.764170\pi
654654 1.17667e9 0.164488
655655 − 1.66400e9i − 0.231371i
656656 − 4.44088e7i − 0.00614194i
657657 − 2.00272e9i − 0.275512i
658658 − 3.84175e9i − 0.525700i
659659 9.64818e9 1.31325 0.656624 0.754219i 0.271984π-0.271984\pi
0.656624 + 0.754219i 0.271984π0.271984\pi
660660 1.76118e8 0.0238451
661661 − 6.58299e9i − 0.886580i −0.896378 0.443290i 0.853811π-0.853811\pi
0.896378 0.443290i 0.146189π-0.146189\pi
662662 2.19006e9 0.293395
663663 0 0
664664 −2.90452e9 −0.385023
665665 − 8.52160e9i − 1.12369i
666666 2.62364e9 0.344147
667667 7.04779e9 0.919629
668668 5.61450e9i 0.728775i
669669 1.75749e8i 0.0226935i
670670 − 2.11687e8i − 0.0271914i
671671 9.03851e8i 0.115496i
672672 −3.99507e8 −0.0507846
673673 8.54649e9 1.08077 0.540387 0.841416i 0.318278π-0.318278\pi
0.540387 + 0.841416i 0.318278π0.318278\pi
674674 7.34810e8i 0.0924411i
675675 −1.72711e9 −0.216151
676676 0 0
677677 8.71305e9 1.07922 0.539610 0.841915i 0.318572π-0.318572\pi
0.539610 + 0.841915i 0.318572π0.318572\pi
678678 1.59312e9i 0.196311i
679679 8.82106e9 1.08138
680680 −1.58119e9 −0.192842
681681 2.21449e9i 0.268695i
682682 1.98789e9i 0.239965i
683683 1.46109e10i 1.75470i 0.479849 + 0.877351i 0.340692π0.340692\pi
−0.479849 + 0.877351i 0.659308π0.659308\pi
684684 5.22223e9i 0.623965i
685685 6.62873e7 0.00787977
686686 −4.99734e9 −0.591023
687687 − 1.05055e8i − 0.0123615i
688688 2.58354e9 0.302452
689689 0 0
690690 1.38523e9 0.160528
691691 1.47348e10i 1.69891i 0.527662 + 0.849454i 0.323069π0.323069\pi
−0.527662 + 0.849454i 0.676931π0.676931\pi
692692 5.48451e8 0.0629167
693693 2.26665e9 0.258713
694694 − 1.09360e10i − 1.24194i
695695 8.23279e9i 0.930252i
696696 6.30190e8i 0.0708499i
697697 1.59442e8i 0.0178357i
698698 −9.05146e9 −1.00745
699699 1.43467e9 0.158885
700700 − 2.21244e9i − 0.243797i
701701 −1.31502e9 −0.144185 −0.0720923 0.997398i 0.522968π-0.522968\pi
−0.0720923 + 0.997398i 0.522968π0.522968\pi
702702 0 0
703703 −6.41141e9 −0.696001
704704 − 2.86261e8i − 0.0309213i
705705 −1.19109e9 −0.128022
706706 −3.58716e8 −0.0383649
707707 1.02408e10i 1.08985i
708708 − 2.02803e9i − 0.214762i
709709 6.64028e8i 0.0699721i 0.999388 + 0.0349860i 0.0111387π0.0111387\pi
−0.999388 + 0.0349860i 0.988861π0.988861\pi
710710 − 2.37674e9i − 0.249217i
711711 7.28697e9 0.760332
712712 −6.11901e9 −0.635333
713713 1.56356e10i 1.61547i
714714 1.43436e9 0.147474
715715 0 0
716716 1.20346e9 0.122528
717717 − 4.75451e9i − 0.481713i
718718 −3.18624e9 −0.321250
719719 −4.95034e9 −0.496689 −0.248344 0.968672i 0.579886π-0.579886\pi
−0.248344 + 0.968672i 0.579886π0.579886\pi
720720 − 1.75731e9i − 0.175462i
721721 − 3.80796e9i − 0.378372i
722722 − 5.61065e9i − 0.554796i
723723 − 3.07928e9i − 0.303015i
724724 −3.83760e9 −0.375816
725725 −3.48994e9 −0.340123
726726 1.75629e9i 0.170341i
727727 −8.81101e9 −0.850463 −0.425231 0.905085i 0.639807π-0.639807\pi
−0.425231 + 0.905085i 0.639807π0.639807\pi
728728 0 0
729729 −6.55163e9 −0.626330
730730 − 1.64687e9i − 0.156686i
731731 −9.27578e9 −0.878293
732732 −6.35675e8 −0.0599027
733733 1.49414e8i 0.0140129i 0.999975 + 0.00700643i 0.00223023π0.00223023\pi
−0.999975 + 0.00700643i 0.997770π0.997770\pi
734734 1.30777e10i 1.22066i
735735 − 5.25957e8i − 0.0488590i
736736 − 2.25155e9i − 0.208166i
737737 −1.37596e8 −0.0126611
738738 −1.77202e8 −0.0162282
739739 − 4.70806e9i − 0.429127i −0.976710 0.214564i 0.931167π-0.931167\pi
0.976710 0.214564i 0.0688329π-0.0688329\pi
740740 2.15747e9 0.195719
741741 0 0
742742 1.21434e10 1.09125
743743 − 1.69676e9i − 0.151761i −0.997117 0.0758805i 0.975823π-0.975823\pi
0.997117 0.0758805i 0.0241768π-0.0241768\pi
744744 −1.39808e9 −0.124459
745745 −4.59607e9 −0.407230
746746 − 1.23707e10i − 1.09095i
747747 1.15897e10i 1.01730i
748748 1.02777e9i 0.0897928i
749749 − 1.82733e10i − 1.58903i
750750 −2.26094e9 −0.195693
751751 −1.06650e10 −0.918800 −0.459400 0.888229i 0.651936π-0.651936\pi
−0.459400 + 0.888229i 0.651936π0.651936\pi
752752 1.93600e9i 0.166013i
753753 8.81731e8 0.0752581
754754 0 0
755755 6.17724e9 0.522373
756756 3.30062e9i 0.277824i
757757 6.22876e9 0.521874 0.260937 0.965356i 0.415968π-0.415968\pi
0.260937 + 0.965356i 0.415968π0.415968\pi
758758 −8.45506e9 −0.705138
759759 − 9.00402e8i − 0.0747464i
760760 4.29435e9i 0.354854i
761761 − 8.38334e9i − 0.689558i −0.938684 0.344779i 0.887954π-0.887954\pi
0.938684 0.344779i 0.112046π-0.112046\pi
762762 − 1.12153e8i − 0.00918263i
763763 1.24531e10 1.01495
764764 −6.01511e9 −0.487997
765765 6.30932e9i 0.509527i
766766 1.79928e9 0.144643
767767 0 0
768768 2.01327e8 0.0160375
769769 1.18649e10i 0.940852i 0.882439 + 0.470426i 0.155900π0.155900\pi
−0.882439 + 0.470426i 0.844100π0.844100\pi
770770 1.86391e9 0.147132
771771 2.43241e9 0.191138
772772 2.25245e9i 0.176196i
773773 − 5.56680e9i − 0.433488i −0.976228 0.216744i 0.930456π-0.930456\pi
0.976228 0.216744i 0.0695438π-0.0695438\pi
774774 − 1.03089e10i − 0.799136i
775775 − 7.74246e9i − 0.597479i
776776 −4.44526e9 −0.341492
777777 −1.95713e9 −0.149674
778778 − 8.14306e9i − 0.619953i
779779 4.33029e8 0.0328198
780780 0 0
781781 −1.54488e9 −0.116042
782782 8.08383e9i 0.604496i
783783 5.20645e9 0.387593
784784 −8.54888e8 −0.0633583
785785 1.27165e10i 0.938264i
786786 − 7.60687e8i − 0.0558763i
787787 1.34611e8i 0.00984395i 0.999988 + 0.00492198i 0.00156672π0.00156672\pi
−0.999988 + 0.00492198i 0.998433π0.998433\pi
788788 6.59106e9i 0.479859i
789789 1.85105e9 0.134168
790790 5.99222e9 0.432408
791791 1.68605e10i 1.21130i
792792 −1.14225e9 −0.0817001
793793 0 0
794794 1.18052e10 0.836955
795795 − 3.76493e9i − 0.265749i
796796 5.35280e9 0.376171
797797 7.41548e9 0.518842 0.259421 0.965764i 0.416468π-0.416468\pi
0.259421 + 0.965764i 0.416468π0.416468\pi
798798 − 3.89559e9i − 0.271371i
799799 − 6.95088e9i − 0.482088i
800800 1.11493e9i 0.0769898i
801801 2.44163e10i 1.67867i
802802 −2.19930e9 −0.150548
803803 −1.07047e9 −0.0729574
804804 − 9.67711e7i − 0.00656673i
805805 1.46604e10 0.990513
806806 0 0
807807 −7.48822e9 −0.501558
808808 − 5.16072e9i − 0.344168i
809809 −1.41542e10 −0.939863 −0.469932 0.882703i 0.655721π-0.655721\pi
−0.469932 + 0.882703i 0.655721π0.655721\pi
810810 −6.48299e9 −0.428625
811811 2.63708e10i 1.73600i 0.496563 + 0.868001i 0.334595π0.334595\pi
−0.496563 + 0.868001i 0.665405π0.665405\pi
812812 6.66951e9i 0.437168i
813813 − 4.65580e9i − 0.303863i
814814 − 1.40236e9i − 0.0911324i
815815 −1.19879e10 −0.775697
816816 −7.22829e8 −0.0465715
817817 2.51921e10i 1.61617i
818818 −1.30742e10 −0.835176
819819 0 0
820820 −1.45716e8 −0.00922912
821821 − 8.06264e9i − 0.508483i −0.967141 0.254241i 0.918174π-0.918174\pi
0.967141 0.254241i 0.0818258π-0.0818258\pi
822822 3.03028e7 0.00190297
823823 2.34202e10 1.46451 0.732253 0.681033i 0.238469π-0.238469\pi
0.732253 + 0.681033i 0.238469π0.238469\pi
824824 1.91897e9i 0.119488i
825825 4.45864e8i 0.0276448i
826826 − 2.14633e10i − 1.32515i
827827 5.55722e9i 0.341655i 0.985301 + 0.170828i 0.0546442π0.0546442\pi
−0.985301 + 0.170828i 0.945356π0.945356\pi
828828 −8.98423e9 −0.550015
829829 −2.84256e10 −1.73288 −0.866440 0.499281i 0.833597π-0.833597\pi
−0.866440 + 0.499281i 0.833597π0.833597\pi
830830 9.53046e9i 0.578549i
831831 −5.44743e9 −0.329297
832832 0 0
833833 3.06933e9 0.183987
834834 3.76356e9i 0.224656i
835835 1.84226e10 1.09508
836836 2.79133e9 0.165230
837837 1.15505e10i 0.680868i
838838 − 8.90238e9i − 0.522579i
839839 1.04036e10i 0.608156i 0.952647 + 0.304078i 0.0983484π0.0983484\pi
−0.952647 + 0.304078i 0.901652π0.901652\pi
840840 1.31088e9i 0.0763109i
841841 −6.72927e9 −0.390105
842842 7.38023e9 0.426066
843843 4.00524e9i 0.230267i
844844 6.23367e9 0.356899
845845 0 0
846846 7.72509e9 0.438639
847847 1.85874e10i 1.05106i
848848 −6.11950e9 −0.344612
849849 −6.45234e9 −0.361860
850850 − 4.00297e9i − 0.223572i
851851 − 1.10301e10i − 0.613514i
852852 − 1.08651e9i − 0.0601860i
853853 − 1.80580e10i − 0.996205i −0.867118 0.498102i 0.834030π-0.834030\pi
0.867118 0.498102i 0.165970π-0.165970\pi
854854 −6.72756e9 −0.369620
855855 1.71355e10 0.937593
856856 9.20861e9i 0.501806i
857857 6.34034e9 0.344096 0.172048 0.985089i 0.444962π-0.444962\pi
0.172048 + 0.985089i 0.444962π0.444962\pi
858858 0 0
859859 1.21489e10 0.653973 0.326987 0.945029i 0.393967π-0.393967\pi
0.326987 + 0.945029i 0.393967π0.393967\pi
860860 − 8.47725e9i − 0.454476i
861861 1.32186e8 0.00705786
862862 −7.85206e9 −0.417550
863863 2.87111e10i 1.52059i 0.649578 + 0.760295i 0.274946π0.274946\pi
−0.649578 + 0.760295i 0.725054π0.725054\pi
864864 − 1.66330e9i − 0.0877351i
865865 − 1.79960e9i − 0.0945411i
866866 − 2.27998e10i − 1.19294i
867867 −2.32887e9 −0.121361
868868 −1.47963e10 −0.767954
869869 − 3.89495e9i − 0.201341i
870870 2.06781e9 0.106462
871871 0 0
872872 −6.27560e9 −0.320514
873873 1.77376e10i 0.902289i
874874 2.19549e10 1.11235
875875 −2.39283e10 −1.20749
876876 − 7.52857e8i − 0.0378397i
877877 − 2.46021e10i − 1.23161i −0.787898 0.615806i 0.788831π-0.788831\pi
0.787898 0.615806i 0.211169π-0.211169\pi
878878 8.44975e9i 0.421321i
879879 4.02720e9i 0.200006i
880880 −9.39295e8 −0.0464636
881881 1.25378e10 0.617738 0.308869 0.951105i 0.400049π-0.400049\pi
0.308869 + 0.951105i 0.400049π0.400049\pi
882882 3.41121e9i 0.167405i
883883 −1.93097e10 −0.943873 −0.471937 0.881633i 0.656445π-0.656445\pi
−0.471937 + 0.881633i 0.656445π0.656445\pi
884884 0 0
885885 −6.65446e9 −0.322709
886886 1.45860e10i 0.704562i
887887 3.20268e10 1.54092 0.770462 0.637486i 0.220026π-0.220026\pi
0.770462 + 0.637486i 0.220026π0.220026\pi
888888 9.86272e8 0.0472663
889889 − 1.18695e9i − 0.0566599i
890890 2.00780e10i 0.954675i
891891 4.21394e9i 0.199580i
892892 − 9.37327e8i − 0.0442195i
893893 −1.88779e10 −0.887101
894894 −2.10106e9 −0.0983461
895895 − 3.94885e9i − 0.184115i
896896 2.13071e9 0.0989568
897897 0 0
898898 −1.47877e10 −0.681448
899899 2.33400e10i 1.07138i
900900 4.44884e9 0.203422
901901 2.19710e10 1.00072
902902 9.47157e7i 0.00429733i
903903 7.69008e9i 0.347555i
904904 − 8.49662e9i − 0.382522i
905905 1.25921e10i 0.564715i
906906 2.82388e9 0.126153
907907 −2.33703e9 −0.104002 −0.0520008 0.998647i 0.516560π-0.516560\pi
−0.0520008 + 0.998647i 0.516560π0.516560\pi
908908 − 1.18106e10i − 0.523567i
909909 −2.05925e10 −0.909359
910910 0 0
911911 2.20343e10 0.965573 0.482786 0.875738i 0.339625π-0.339625\pi
0.482786 + 0.875738i 0.339625π0.339625\pi
912912 1.96313e9i 0.0856973i
913913 6.19480e9 0.269389
914914 −2.38453e10 −1.03298
915915 2.08581e9i 0.0900121i
916916 5.60295e8i 0.0240870i
917917 − 8.05061e9i − 0.344775i
918918 5.97181e9i 0.254775i
919919 −1.43277e10 −0.608938 −0.304469 0.952522i 0.598479π-0.598479\pi
−0.304469 + 0.952522i 0.598479π0.598479\pi
920920 −7.38791e9 −0.312798
921921 2.58035e9i 0.108835i
922922 −2.02225e10 −0.849720
923923 0 0
924924 8.52074e8 0.0355325
925925 5.46190e9i 0.226907i
926926 7.12233e9 0.294770
927927 7.65715e9 0.315710
928928 − 3.36101e9i − 0.138055i
929929 − 1.31280e10i − 0.537208i −0.963251 0.268604i 0.913438π-0.913438\pi
0.963251 0.268604i 0.0865623π-0.0865623\pi
930930 4.58745e9i 0.187017i
931931 − 8.33600e9i − 0.338558i
932932 −7.65160e9 −0.309597
933933 −9.50474e9 −0.383137
934934 − 2.12533e10i − 0.853519i
935935 3.37238e9 0.134926
936936 0 0
937937 −3.87626e10 −1.53930 −0.769652 0.638463i 0.779571π-0.779571\pi
−0.769652 + 0.638463i 0.779571π0.779571\pi
938938 − 1.02416e9i − 0.0405189i
939939 −1.42149e9 −0.0560291
940940 6.35250e9 0.249458
941941 − 2.06279e10i − 0.807035i −0.914972 0.403517i 0.867788π-0.867788\pi
0.914972 0.403517i 0.132212π-0.132212\pi
942942 5.81328e9i 0.226591i
943943 7.44976e8i 0.0289302i
944944 1.08161e10i 0.418476i
945945 1.08302e10 0.417468
946946 −5.51021e9 −0.211617
947947 − 2.11705e10i − 0.810040i −0.914308 0.405020i 0.867264π-0.867264\pi
0.914308 0.405020i 0.132736π-0.132736\pi
948948 2.73930e9 0.104427
949949 0 0
950950 −1.08717e10 −0.411399
951951 6.08771e8i 0.0229521i
952952 −7.64994e9 −0.287362
953953 −2.14876e10 −0.804196 −0.402098 0.915597i 0.631719π-0.631719\pi
−0.402098 + 0.915597i 0.631719π0.631719\pi
954954 2.44182e10i 0.910531i
955955 1.97371e10i 0.733282i
956956 2.53574e10i 0.938646i
957957 − 1.34408e9i − 0.0495715i
958958 −1.04075e10 −0.382442
959959 3.20704e8 0.0117419
960960 − 6.60603e8i − 0.0240986i
961961 −2.42673e10 −0.882043
962962 0 0
963963 3.67445e10 1.32587
964964 1.64228e10i 0.590443i
965965 7.39086e9 0.264758
966966 6.70189e9 0.239209
967967 − 3.92625e10i − 1.39632i −0.715941 0.698161i 0.754002π-0.754002\pi
0.715941 0.698161i 0.245998π-0.245998\pi
968968 − 9.36689e9i − 0.331919i
969969 − 7.04829e9i − 0.248857i
970970 1.45860e10i 0.513139i
971971 −5.62647e10 −1.97228 −0.986140 0.165917i 0.946941π-0.946941\pi
−0.986140 + 0.165917i 0.946941π0.946941\pi
972972 −1.00684e10 −0.351665
973973 3.98310e10i 1.38620i
974974 −8.59573e9 −0.298076
975975 0 0
976976 3.39027e9 0.116724
977977 8.43437e9i 0.289349i 0.989479 + 0.144674i 0.0462135π0.0462135\pi
−0.989479 + 0.144674i 0.953787π0.953787\pi
978978 −5.48019e9 −0.187331
979979 1.30507e10 0.444523
980980 2.80510e9i 0.0952045i
981981 2.50411e10i 0.846861i
982982 6.26675e9i 0.211180i
983983 − 2.24230e10i − 0.752932i −0.926430 0.376466i 0.877139π-0.877139\pi
0.926430 0.376466i 0.122861π-0.122861\pi
984984 −6.66132e7 −0.00222883
985985 2.16269e10 0.721054
986986 1.20672e10i 0.400900i
987987 −5.76262e9 −0.190770
988988 0 0
989989 −4.33400e10 −1.42463
990990 3.74801e9i 0.122766i
991991 3.46728e10 1.13170 0.565849 0.824509i 0.308548π-0.308548\pi
0.565849 + 0.824509i 0.308548π0.308548\pi
992992 7.45642e9 0.242516
993993 − 3.28508e9i − 0.106469i
994994 − 1.14989e10i − 0.371368i
995995 − 1.75639e10i − 0.565249i
996996 4.35678e9i 0.139720i
997997 −2.96474e10 −0.947444 −0.473722 0.880674i 0.657090π-0.657090\pi
−0.473722 + 0.880674i 0.657090π0.657090\pi
998998 −4.98550e9 −0.158764
999999 − 8.14830e9i − 0.258576i
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 338.8.b.d.337.2 2
13.5 odd 4 338.8.a.d.1.1 1
13.8 odd 4 2.8.a.a.1.1 1
13.12 even 2 inner 338.8.b.d.337.1 2
39.8 even 4 18.8.a.b.1.1 1
52.47 even 4 16.8.a.b.1.1 1
65.8 even 4 50.8.b.c.49.2 2
65.34 odd 4 50.8.a.g.1.1 1
65.47 even 4 50.8.b.c.49.1 2
91.34 even 4 98.8.a.a.1.1 1
91.47 even 12 98.8.c.e.67.1 2
91.60 odd 12 98.8.c.d.79.1 2
91.73 even 12 98.8.c.e.79.1 2
91.86 odd 12 98.8.c.d.67.1 2
104.21 odd 4 64.8.a.c.1.1 1
104.99 even 4 64.8.a.e.1.1 1
117.34 odd 12 162.8.c.l.109.1 2
117.47 even 12 162.8.c.a.109.1 2
117.86 even 12 162.8.c.a.55.1 2
117.112 odd 12 162.8.c.l.55.1 2
143.21 even 4 242.8.a.e.1.1 1
156.47 odd 4 144.8.a.i.1.1 1
195.8 odd 4 450.8.c.g.199.1 2
195.47 odd 4 450.8.c.g.199.2 2
195.164 even 4 450.8.a.c.1.1 1
208.21 odd 4 256.8.b.b.129.1 2
208.99 even 4 256.8.b.f.129.1 2
208.125 odd 4 256.8.b.b.129.2 2
208.203 even 4 256.8.b.f.129.2 2
221.203 odd 4 578.8.a.b.1.1 1
260.47 odd 4 400.8.c.j.49.2 2
260.99 even 4 400.8.a.l.1.1 1
260.203 odd 4 400.8.c.j.49.1 2
312.125 even 4 576.8.a.g.1.1 1
312.203 odd 4 576.8.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 1 13.8 odd 4
16.8.a.b.1.1 1 52.47 even 4
18.8.a.b.1.1 1 39.8 even 4
50.8.a.g.1.1 1 65.34 odd 4
50.8.b.c.49.1 2 65.47 even 4
50.8.b.c.49.2 2 65.8 even 4
64.8.a.c.1.1 1 104.21 odd 4
64.8.a.e.1.1 1 104.99 even 4
98.8.a.a.1.1 1 91.34 even 4
98.8.c.d.67.1 2 91.86 odd 12
98.8.c.d.79.1 2 91.60 odd 12
98.8.c.e.67.1 2 91.47 even 12
98.8.c.e.79.1 2 91.73 even 12
144.8.a.i.1.1 1 156.47 odd 4
162.8.c.a.55.1 2 117.86 even 12
162.8.c.a.109.1 2 117.47 even 12
162.8.c.l.55.1 2 117.112 odd 12
162.8.c.l.109.1 2 117.34 odd 12
242.8.a.e.1.1 1 143.21 even 4
256.8.b.b.129.1 2 208.21 odd 4
256.8.b.b.129.2 2 208.125 odd 4
256.8.b.f.129.1 2 208.99 even 4
256.8.b.f.129.2 2 208.203 even 4
338.8.a.d.1.1 1 13.5 odd 4
338.8.b.d.337.1 2 13.12 even 2 inner
338.8.b.d.337.2 2 1.1 even 1 trivial
400.8.a.l.1.1 1 260.99 even 4
400.8.c.j.49.1 2 260.203 odd 4
400.8.c.j.49.2 2 260.47 odd 4
450.8.a.c.1.1 1 195.164 even 4
450.8.c.g.199.1 2 195.8 odd 4
450.8.c.g.199.2 2 195.47 odd 4
576.8.a.f.1.1 1 312.203 odd 4
576.8.a.g.1.1 1 312.125 even 4
578.8.a.b.1.1 1 221.203 odd 4