Properties

Label 65.6.c.b.51.7
Level 6565
Weight 66
Character 65.51
Analytic conductor 10.42510.425
Analytic rank 00
Dimension 1414
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [65,6,Mod(51,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("65.51");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 65=513 65 = 5 \cdot 13
Weight: k k == 6 6
Character orbit: [χ][\chi] == 65.c (of order 22, degree 11, minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 10.424948287810.4249482878
Analytic rank: 00
Dimension: 1414
Coefficient field: Q[x]/(x14+)\mathbb{Q}[x]/(x^{14} + \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x14+227x12+16583x10+514217x8+6872896x6+35265600x4+63141120x2+26873856 x^{14} + 227x^{12} + 16583x^{10} + 514217x^{8} + 6872896x^{6} + 35265600x^{4} + 63141120x^{2} + 26873856 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 243254 2^{4}\cdot 3^{2}\cdot 5^{4}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 51.7
Root 0.780788i0.780788i of defining polynomial
Character χ\chi == 65.51
Dual form 65.6.c.b.51.8

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q1.21921iq211.3769q3+30.5135q4+25.0000iq5+13.8708iq66.14028iq776.2172iq8113.566q9+30.4803q10+391.252iq11347.149q12+(472.460+384.804i)q137.48630q14284.422iq15+883.508q16+1547.69q17+138.461iq18+3012.26iq19+762.838iq20+69.8572iq21+477.019q22+921.321q23+867.115iq24625.000q25+(469.158576.029i)q26+4056.62q27187.361iq285248.50q29346.771q30+1749.89iq313516.13iq324451.23iq331886.96iq34+153.507q353465.31q36+2782.37iq37+3672.59q38+(5375.134377.88i)q39+1905.43q40+7301.71iq41+85.1708q422464.65q43+11938.5iq442839.16iq451123.29iq4622645.6iq4710051.6q48+16769.3q49+762.007iq5017607.9q51+(14416.4+11741.7i)q5227383.4q534945.87iq549781.30q55467.995q5634270.2iq57+6399.03iq5847642.5iq598678.73iq6025808.7q61+2133.49q62+697.329iq63+23985.3q64+(9620.11+11811.5i)q655426.99q66+36207.0iq67+47225.4q6810481.8q69187.157iq70+25265.2iq71+8655.71iq7260584.3iq73+3392.30q74+7110.56q75+91914.8iq76+2402.40q77+(5337.56+6553.42i)q78+87635.7q79+22087.7iq8018555.1q81+8902.33q82+82465.4iq83+2131.59iq84+38692.2iq85+3004.93iq86+59711.6q87+29820.1q88+26062.4iq893461.54q90+(2362.802901.04i)q91+28112.7q9219908.3iq9327609.7q9475306.6q95+40002.7iq96135691.iq9720445.3iq9844433.1iq99+O(q100)q-1.21921i q^{2} -11.3769 q^{3} +30.5135 q^{4} +25.0000i q^{5} +13.8708i q^{6} -6.14028i q^{7} -76.2172i q^{8} -113.566 q^{9} +30.4803 q^{10} +391.252i q^{11} -347.149 q^{12} +(472.460 + 384.804i) q^{13} -7.48630 q^{14} -284.422i q^{15} +883.508 q^{16} +1547.69 q^{17} +138.461i q^{18} +3012.26i q^{19} +762.838i q^{20} +69.8572i q^{21} +477.019 q^{22} +921.321 q^{23} +867.115i q^{24} -625.000 q^{25} +(469.158 - 576.029i) q^{26} +4056.62 q^{27} -187.361i q^{28} -5248.50 q^{29} -346.771 q^{30} +1749.89i q^{31} -3516.13i q^{32} -4451.23i q^{33} -1886.96i q^{34} +153.507 q^{35} -3465.31 q^{36} +2782.37i q^{37} +3672.59 q^{38} +(-5375.13 - 4377.88i) q^{39} +1905.43 q^{40} +7301.71i q^{41} +85.1708 q^{42} -2464.65 q^{43} +11938.5i q^{44} -2839.16i q^{45} -1123.29i q^{46} -22645.6i q^{47} -10051.6 q^{48} +16769.3 q^{49} +762.007i q^{50} -17607.9 q^{51} +(14416.4 + 11741.7i) q^{52} -27383.4 q^{53} -4945.87i q^{54} -9781.30 q^{55} -467.995 q^{56} -34270.2i q^{57} +6399.03i q^{58} -47642.5i q^{59} -8678.73i q^{60} -25808.7 q^{61} +2133.49 q^{62} +697.329i q^{63} +23985.3 q^{64} +(-9620.11 + 11811.5i) q^{65} -5426.99 q^{66} +36207.0i q^{67} +47225.4 q^{68} -10481.8 q^{69} -187.157i q^{70} +25265.2i q^{71} +8655.71i q^{72} -60584.3i q^{73} +3392.30 q^{74} +7110.56 q^{75} +91914.8i q^{76} +2402.40 q^{77} +(-5337.56 + 6553.42i) q^{78} +87635.7 q^{79} +22087.7i q^{80} -18555.1 q^{81} +8902.33 q^{82} +82465.4i q^{83} +2131.59i q^{84} +38692.2i q^{85} +3004.93i q^{86} +59711.6 q^{87} +29820.1 q^{88} +26062.4i q^{89} -3461.54 q^{90} +(2362.80 - 2901.04i) q^{91} +28112.7 q^{92} -19908.3i q^{93} -27609.7 q^{94} -75306.6 q^{95} +40002.7i q^{96} -135691. i q^{97} -20445.3i q^{98} -44433.1i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 14q84q338q4+1118q9+550q10+168q121132q13+6440q143434q16+228q17+2328q22+8104q238750q25142q26+13680q2716700q29++318800q95+O(q100) 14 q - 84 q^{3} - 38 q^{4} + 1118 q^{9} + 550 q^{10} + 168 q^{12} - 1132 q^{13} + 6440 q^{14} - 3434 q^{16} + 228 q^{17} + 2328 q^{22} + 8104 q^{23} - 8750 q^{25} - 142 q^{26} + 13680 q^{27} - 16700 q^{29}+ \cdots + 318800 q^{95}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/65Z)×\left(\mathbb{Z}/65\mathbb{Z}\right)^\times.

nn 2727 4141
χ(n)\chi(n) 11 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 1.21921i 0.215528i −0.994176 0.107764i 0.965631π-0.965631\pi
0.994176 0.107764i 0.0343691π-0.0343691\pi
33 −11.3769 −0.729828 −0.364914 0.931041i 0.618902π-0.618902\pi
−0.364914 + 0.931041i 0.618902π0.618902\pi
44 30.5135 0.953548
55 25.0000i 0.447214i
66 13.8708i 0.157298i
77 6.14028i 0.0473634i −0.999720 0.0236817i 0.992461π-0.992461\pi
0.999720 0.0236817i 0.00753882π-0.00753882\pi
88 76.2172i 0.421045i
99 −113.566 −0.467351
1010 30.4803 0.0963871
1111 391.252i 0.974933i 0.873142 + 0.487467i 0.162079π0.162079\pi
−0.873142 + 0.487467i 0.837921π0.837921\pi
1212 −347.149 −0.695926
1313 472.460 + 384.804i 0.775366 + 0.631512i
1414 −7.48630 −0.0102081
1515 284.422i 0.326389i
1616 883.508 0.862801
1717 1547.69 1.29886 0.649428 0.760423i 0.275009π-0.275009\pi
0.649428 + 0.760423i 0.275009π0.275009\pi
1818 138.461i 0.100727i
1919 3012.26i 1.91430i 0.289601 + 0.957148i 0.406477π0.406477\pi
−0.289601 + 0.957148i 0.593523π0.593523\pi
2020 762.838i 0.426439i
2121 69.8572i 0.0345671i
2222 477.019 0.210126
2323 921.321 0.363154 0.181577 0.983377i 0.441880π-0.441880\pi
0.181577 + 0.983377i 0.441880π0.441880\pi
2424 867.115i 0.307290i
2525 −625.000 −0.200000
2626 469.158 576.029i 0.136109 0.167113i
2727 4056.62 1.07091
2828 187.361i 0.0451632i
2929 −5248.50 −1.15888 −0.579442 0.815013i 0.696730π-0.696730\pi
−0.579442 + 0.815013i 0.696730π0.696730\pi
3030 −346.771 −0.0703460
3131 1749.89i 0.327045i 0.986540 + 0.163522i 0.0522856π0.0522856\pi
−0.986540 + 0.163522i 0.947714π0.947714\pi
3232 3516.13i 0.607002i
3333 4451.23i 0.711534i
3434 1886.96i 0.279940i
3535 153.507 0.0211816
3636 −3465.31 −0.445642
3737 2782.37i 0.334127i 0.985946 + 0.167063i 0.0534284π0.0534284\pi
−0.985946 + 0.167063i 0.946572π0.946572\pi
3838 3672.59 0.412585
3939 −5375.13 4377.88i −0.565884 0.460895i
4040 1905.43 0.188297
4141 7301.71i 0.678368i 0.940720 + 0.339184i 0.110151π0.110151\pi
−0.940720 + 0.339184i 0.889849π0.889849\pi
4242 85.1708 0.00745019
4343 −2464.65 −0.203275 −0.101638 0.994821i 0.532408π-0.532408\pi
−0.101638 + 0.994821i 0.532408π0.532408\pi
4444 11938.5i 0.929645i
4545 2839.16i 0.209006i
4646 1123.29i 0.0782700i
4747 22645.6i 1.49533i −0.664073 0.747667i 0.731174π-0.731174\pi
0.664073 0.747667i 0.268826π-0.268826\pi
4848 −10051.6 −0.629696
4949 16769.3 0.997757
5050 762.007i 0.0431056i
5151 −17607.9 −0.947941
5252 14416.4 + 11741.7i 0.739349 + 0.602177i
5353 −27383.4 −1.33905 −0.669527 0.742788i 0.733503π-0.733503\pi
−0.669527 + 0.742788i 0.733503π0.733503\pi
5454 4945.87i 0.230812i
5555 −9781.30 −0.436004
5656 −467.995 −0.0199421
5757 34270.2i 1.39711i
5858 6399.03i 0.249772i
5959 47642.5i 1.78182i −0.454176 0.890912i 0.650066π-0.650066\pi
0.454176 0.890912i 0.349934π-0.349934\pi
6060 8678.73i 0.311227i
6161 −25808.7 −0.888059 −0.444030 0.896012i 0.646452π-0.646452\pi
−0.444030 + 0.896012i 0.646452π0.646452\pi
6262 2133.49 0.0704873
6363 697.329i 0.0221353i
6464 23985.3 0.731974
6565 −9620.11 + 11811.5i −0.282421 + 0.346754i
6666 −5426.99 −0.153356
6767 36207.0i 0.985384i 0.870204 + 0.492692i 0.163987π0.163987\pi
−0.870204 + 0.492692i 0.836013π0.836013\pi
6868 47225.4 1.23852
6969 −10481.8 −0.265040
7070 187.157i 0.00456522i
7171 25265.2i 0.594809i 0.954752 + 0.297404i 0.0961209π0.0961209\pi
−0.954752 + 0.297404i 0.903879π0.903879\pi
7272 8655.71i 0.196776i
7373 60584.3i 1.33062i −0.746568 0.665309i 0.768300π-0.768300\pi
0.746568 0.665309i 0.231700π-0.231700\pi
7474 3392.30 0.0720137
7575 7110.56 0.145966
7676 91914.8i 1.82537i
7777 2402.40 0.0461762
7878 −5337.56 + 6553.42i −0.0993359 + 0.121964i
7979 87635.7 1.57984 0.789920 0.613209i 0.210122π-0.210122\pi
0.789920 + 0.613209i 0.210122π0.210122\pi
8080 22087.7i 0.385856i
8181 −18555.1 −0.314232
8282 8902.33 0.146207
8383 82465.4i 1.31394i 0.753915 + 0.656972i 0.228163π0.228163\pi
−0.753915 + 0.656972i 0.771837π0.771837\pi
8484 2131.59i 0.0329614i
8585 38692.2i 0.580866i
8686 3004.93i 0.0438116i
8787 59711.6 0.845787
8888 29820.1 0.410490
8989 26062.4i 0.348771i 0.984677 + 0.174385i 0.0557938π0.0557938\pi
−0.984677 + 0.174385i 0.944206π0.944206\pi
9090 −3461.54 −0.0450467
9191 2362.80 2901.04i 0.0299105 0.0367240i
9292 28112.7 0.346285
9393 19908.3i 0.238686i
9494 −27609.7 −0.322287
9595 −75306.6 −0.856099
9696 40002.7i 0.443007i
9797 135691.i 1.46427i −0.681159 0.732135i 0.738524π-0.738524\pi
0.681159 0.732135i 0.261476π-0.261476\pi
9898 20445.3i 0.215045i
9999 44433.1i 0.455636i
100100 −19071.0 −0.190710
101101 −144713. −1.41157 −0.705787 0.708424i 0.749407π-0.749407\pi
−0.705787 + 0.708424i 0.749407π0.749407\pi
102102 21467.7i 0.204308i
103103 −6472.89 −0.0601181 −0.0300590 0.999548i 0.509570π-0.509570\pi
−0.0300590 + 0.999548i 0.509570π0.509570\pi
104104 29328.7 36009.6i 0.265895 0.326464i
105105 −1746.43 −0.0154589
106106 33386.2i 0.288604i
107107 89977.4 0.759755 0.379878 0.925037i 0.375966π-0.375966\pi
0.379878 + 0.925037i 0.375966π0.375966\pi
108108 123782. 1.02117
109109 36938.6i 0.297792i 0.988853 + 0.148896i 0.0475720π0.0475720\pi
−0.988853 + 0.148896i 0.952428π0.952428\pi
110110 11925.5i 0.0939711i
111111 31654.8i 0.243855i
112112 5424.98i 0.0408652i
113113 27703.8 0.204100 0.102050 0.994779i 0.467460π-0.467460\pi
0.102050 + 0.994779i 0.467460π0.467460\pi
114114 −41782.6 −0.301116
115115 23033.0i 0.162408i
116116 −160150. −1.10505
117117 −53655.6 43700.8i −0.362368 0.295138i
118118 −58086.3 −0.384033
119119 9503.22i 0.0615182i
120120 −21677.9 −0.137424
121121 7972.79 0.0495047
122122 31466.3i 0.191402i
123123 83070.8i 0.495092i
124124 53395.4i 0.311853i
125125 15625.0i 0.0894427i
126126 850.191 0.00477079
127127 198569. 1.09245 0.546226 0.837638i 0.316064π-0.316064\pi
0.546226 + 0.837638i 0.316064π0.316064\pi
128128 141759.i 0.764764i
129129 28040.1 0.148356
130130 14400.7 + 11728.9i 0.0747353 + 0.0608696i
131131 −102104. −0.519834 −0.259917 0.965631i 0.583695π-0.583695\pi
−0.259917 + 0.965631i 0.583695π0.583695\pi
132132 135823.i 0.678481i
133133 18496.1 0.0906675
134134 44144.0 0.212378
135135 101415.i 0.478927i
136136 117960.i 0.546876i
137137 388748.i 1.76957i −0.466001 0.884784i 0.654306π-0.654306\pi
0.466001 0.884784i 0.345694π-0.345694\pi
138138 12779.5i 0.0571236i
139139 −245256. −1.07667 −0.538334 0.842732i 0.680946π-0.680946\pi
−0.538334 + 0.842732i 0.680946π0.680946\pi
140140 4684.04 0.0201976
141141 257636.i 1.09134i
142142 30803.7 0.128198
143143 −150555. + 184851.i −0.615682 + 0.755930i
144144 −100337. −0.403231
145145 131213.i 0.518269i
146146 −73865.1 −0.286786
147147 −190782. −0.728191
148148 84900.0i 0.318606i
149149 455367.i 1.68034i −0.542327 0.840168i 0.682457π-0.682457\pi
0.542327 0.840168i 0.317543π-0.317543\pi
150150 8669.27i 0.0314597i
151151 265698.i 0.948301i 0.880444 + 0.474150i 0.157245π0.157245\pi
−0.880444 + 0.474150i 0.842755π0.842755\pi
152152 229586. 0.806004
153153 −175765. −0.607022
154154 2929.03i 0.00995226i
155155 −43747.3 −0.146259
156156 −164014. 133584.i −0.539597 0.439485i
157157 160333. 0.519127 0.259564 0.965726i 0.416421π-0.416421\pi
0.259564 + 0.965726i 0.416421π0.416421\pi
158158 106846.i 0.340500i
159159 311538. 0.977279
160160 87903.3 0.271460
161161 5657.16i 0.0172002i
162162 22622.5i 0.0677258i
163163 417031.i 1.22942i 0.788755 + 0.614708i 0.210726π0.210726\pi
−0.788755 + 0.614708i 0.789274π0.789274\pi
164164 222801.i 0.646856i
165165 111281. 0.318208
166166 100543. 0.283192
167167 84930.5i 0.235653i 0.993034 + 0.117826i 0.0375926π0.0375926\pi
−0.993034 + 0.117826i 0.962407π0.962407\pi
168168 5324.32 0.0145543
169169 75144.3 + 363609.i 0.202386 + 0.979306i
170170 47173.9 0.125193
171171 342092.i 0.894648i
172172 −75205.2 −0.193833
173173 −122126. −0.310236 −0.155118 0.987896i 0.549576π-0.549576\pi
−0.155118 + 0.987896i 0.549576π0.549576\pi
174174 72801.1i 0.182291i
175175 3837.67i 0.00947268i
176176 345674.i 0.841173i
177177 542024.i 1.30043i
178178 31775.6 0.0751699
179179 459412. 1.07169 0.535846 0.844316i 0.319993π-0.319993\pi
0.535846 + 0.844316i 0.319993π0.319993\pi
180180 86632.7i 0.199297i
181181 334446. 0.758804 0.379402 0.925232i 0.376130π-0.376130\pi
0.379402 + 0.925232i 0.376130π0.376130\pi
182182 −3536.98 2880.76i −0.00791505 0.00644657i
183183 293623. 0.648130
184184 70220.5i 0.152904i
185185 −69559.3 −0.149426
186186 −24272.5 −0.0514436
187187 605536.i 1.26630i
188188 690996.i 1.42587i
189189 24908.7i 0.0507221i
190190 91814.7i 0.184513i
191191 −51647.5 −0.102439 −0.0512195 0.998687i 0.516311π-0.516311\pi
−0.0512195 + 0.998687i 0.516311π0.516311\pi
192192 −272879. −0.534215
193193 661873.i 1.27903i 0.768778 + 0.639516i 0.220865π0.220865\pi
−0.768778 + 0.639516i 0.779135π0.779135\pi
194194 −165436. −0.315592
195195 109447. 134378.i 0.206118 0.253071i
196196 511690. 0.951409
197197 418033.i 0.767442i −0.923449 0.383721i 0.874643π-0.874643\pi
0.923449 0.383721i 0.125357π-0.125357\pi
198198 −54173.3 −0.0982025
199199 846082. 1.51454 0.757268 0.653104i 0.226534π-0.226534\pi
0.757268 + 0.653104i 0.226534π0.226534\pi
200200 47635.8i 0.0842089i
201201 411923.i 0.719161i
202202 176436.i 0.304234i
203203 32227.2i 0.0548887i
204204 −537278. −0.903907
205205 −182543. −0.303375
206206 7891.82i 0.0129571i
207207 −104631. −0.169721
208208 417422. + 339978.i 0.668986 + 0.544869i
209209 −1.17855e6 −1.86631
210210 2129.27i 0.00333183i
211211 469546. 0.726059 0.363029 0.931778i 0.381742π-0.381742\pi
0.363029 + 0.931778i 0.381742π0.381742\pi
212212 −835565. −1.27685
213213 287440.i 0.434108i
214214 109701.i 0.163749i
215215 61616.3i 0.0909075i
216216 309184.i 0.450903i
217217 10744.8 0.0154899
218218 45035.9 0.0641827
219219 689261.i 0.971122i
220220 −298462. −0.415750
221221 731220. + 595556.i 1.00709 + 0.820242i
222222 −38593.8 −0.0525576
223223 739991.i 0.996470i −0.867042 0.498235i 0.833982π-0.833982\pi
0.867042 0.498235i 0.166018π-0.166018\pi
224224 −21590.0 −0.0287497
225225 70979.0 0.0934703
226226 33776.8i 0.0439894i
227227 102602.i 0.132157i −0.997814 0.0660786i 0.978951π-0.978951\pi
0.997814 0.0660786i 0.0210488π-0.0210488\pi
228228 1.04570e6i 1.33221i
229229 82633.6i 0.104128i 0.998644 + 0.0520640i 0.0165800π0.0165800\pi
−0.998644 + 0.0520640i 0.983420π0.983420\pi
230230 28082.1 0.0350034
231231 −27331.8 −0.0337006
232232 400026.i 0.487942i
233233 −145160. −0.175169 −0.0875846 0.996157i 0.527915π-0.527915\pi
−0.0875846 + 0.996157i 0.527915π0.527915\pi
234234 −53280.5 + 65417.5i −0.0636105 + 0.0781006i
235235 566139. 0.668734
236236 1.45374e6i 1.69905i
237237 −997022. −1.15301
238238 −11586.4 −0.0132589
239239 305532.i 0.345989i 0.984923 + 0.172994i 0.0553442π0.0553442\pi
−0.984923 + 0.172994i 0.944656π0.944656\pi
240240 251289.i 0.281609i
241241 1.35283e6i 1.50037i −0.661226 0.750187i 0.729963π-0.729963\pi
0.661226 0.750187i 0.270037π-0.270037\pi
242242 9720.52i 0.0106697i
243243 −774659. −0.841579
244244 −787515. −0.846807
245245 419232.i 0.446210i
246246 −101281. −0.106706
247247 −1.15913e6 + 1.42317e6i −1.20890 + 1.48428i
248248 133372. 0.137700
249249 938200.i 0.958952i
250250 −19050.2 −0.0192774
251251 275636. 0.276155 0.138077 0.990421i 0.455908π-0.455908\pi
0.138077 + 0.990421i 0.455908π0.455908\pi
252252 21278.0i 0.0211071i
253253 360469.i 0.354051i
254254 242098.i 0.235454i
255255 440197.i 0.423932i
256256 594696. 0.567146
257257 611818. 0.577816 0.288908 0.957357i 0.406708π-0.406708\pi
0.288908 + 0.957357i 0.406708π0.406708\pi
258258 34186.8i 0.0319749i
259259 17084.5 0.0158254
260260 −293543. + 360411.i −0.269302 + 0.330647i
261261 596053. 0.541606
262262 124487.i 0.112039i
263263 −505146. −0.450327 −0.225163 0.974321i 0.572292π-0.572292\pi
−0.225163 + 0.974321i 0.572292π0.572292\pi
264264 −339261. −0.299587
265265 684586.i 0.598843i
266266 22550.7i 0.0195414i
267267 296509.i 0.254543i
268268 1.10480e6i 0.939610i
269269 325423. 0.274200 0.137100 0.990557i 0.456222π-0.456222\pi
0.137100 + 0.990557i 0.456222π0.456222\pi
270270 123647. 0.103222
271271 314685.i 0.260288i 0.991495 + 0.130144i 0.0415439π0.0415439\pi
−0.991495 + 0.130144i 0.958456π0.958456\pi
272272 1.36739e6 1.12065
273273 −26881.4 + 33004.8i −0.0218295 + 0.0268022i
274274 −473967. −0.381392
275275 244533.i 0.194987i
276276 −319836. −0.252728
277277 −1.70118e6 −1.33214 −0.666072 0.745887i 0.732026π-0.732026\pi
−0.666072 + 0.745887i 0.732026π0.732026\pi
278278 299018.i 0.232052i
279279 198729.i 0.152845i
280280 11699.9i 0.00891838i
281281 973161.i 0.735222i −0.929980 0.367611i 0.880176π-0.880176\pi
0.929980 0.367611i 0.119824π-0.119824\pi
282282 314113. 0.235214
283283 855504. 0.634974 0.317487 0.948263i 0.397161π-0.397161\pi
0.317487 + 0.948263i 0.397161π0.397161\pi
284284 770931.i 0.567178i
285285 856755. 0.624805
286286 225373. + 183559.i 0.162924 + 0.132697i
287287 44834.5 0.0321298
288288 399315.i 0.283683i
289289 975477. 0.687025
290290 −159976. −0.111702
291291 1.54374e6i 1.06867i
292292 1.84864e6i 1.26881i
293293 434109.i 0.295413i −0.989031 0.147707i 0.952811π-0.952811\pi
0.989031 0.147707i 0.0471891π-0.0471891\pi
294294 232604.i 0.156946i
295295 1.19106e6 0.796856
296296 212065. 0.140682
297297 1.58716e6i 1.04407i
298298 −555189. −0.362160
299299 435287. + 354528.i 0.281578 + 0.229336i
300300 216968. 0.139185
301301 15133.6i 0.00962781i
302302 323942. 0.204386
303303 1.64638e6 1.03021
304304 2.66136e6i 1.65165i
305305 645218.i 0.397152i
306306 214295.i 0.130830i
307307 1.33224e6i 0.806748i −0.915035 0.403374i 0.867837π-0.867837\pi
0.915035 0.403374i 0.132163π-0.132163\pi
308308 73305.6 0.0440312
309309 73641.3 0.0438758
310310 53337.2i 0.0315229i
311311 −1.22090e6 −0.715781 −0.357891 0.933764i 0.616504π-0.616504\pi
−0.357891 + 0.933764i 0.616504π0.616504\pi
312312 −333670. + 409677.i −0.194057 + 0.238262i
313313 −3.36877e6 −1.94362 −0.971808 0.235772i 0.924238π-0.924238\pi
−0.971808 + 0.235772i 0.924238π0.924238\pi
314314 195480.i 0.111887i
315315 −17433.2 −0.00989922
316316 2.67407e6 1.50645
317317 181109.i 0.101226i −0.998718 0.0506131i 0.983882π-0.983882\pi
0.998718 0.0506131i 0.0161175π-0.0161175\pi
318318 379831.i 0.210631i
319319 2.05349e6i 1.12984i
320320 599633.i 0.327349i
321321 −1.02366e6 −0.554491
322322 −6897.28 −0.00370713
323323 4.66204e6i 2.48639i
324324 −566180. −0.299635
325325 −295288. 240503.i −0.155073 0.126302i
326326 508449. 0.264974
327327 420246.i 0.217337i
328328 556516. 0.285623
329329 −139050. −0.0708241
330330 135675.i 0.0685827i
331331 1.72873e6i 0.867277i −0.901087 0.433639i 0.857229π-0.857229\pi
0.901087 0.433639i 0.142771π-0.142771\pi
332332 2.51631e6i 1.25291i
333333 315984.i 0.156154i
334334 103548. 0.0507898
335335 −905175. −0.440677
336336 61719.4i 0.0298245i
337337 2.31320e6 1.10953 0.554764 0.832008i 0.312809π-0.312809\pi
0.554764 + 0.832008i 0.312809π0.312809\pi
338338 443317. 91616.8i 0.211068 0.0436198i
339339 −315183. −0.148958
340340 1.18063e6i 0.553883i
341341 −684649. −0.318847
342342 −417082. −0.192822
343343 206168.i 0.0946205i
344344 187849.i 0.0855880i
345345 262044.i 0.118530i
346346 148897.i 0.0668646i
347347 1.45965e6 0.650766 0.325383 0.945582i 0.394507π-0.394507\pi
0.325383 + 0.945582i 0.394507π0.394507\pi
348348 1.82201e6 0.806498
349349 441041.i 0.193828i 0.995293 + 0.0969138i 0.0308971π0.0308971\pi
−0.995293 + 0.0969138i 0.969103π0.969103\pi
350350 4678.93 0.00204163
351351 1.91659e6 + 1.56100e6i 0.830350 + 0.676295i
352352 1.37569e6 0.591787
353353 1.75552e6i 0.749843i −0.927057 0.374921i 0.877670π-0.877670\pi
0.927057 0.374921i 0.122330π-0.122330\pi
354354 660842. 0.280278
355355 −631631. −0.266006
356356 795257.i 0.332569i
357357 108117.i 0.0448977i
358358 560121.i 0.230980i
359359 2.60457e6i 1.06659i −0.845928 0.533297i 0.820953π-0.820953\pi
0.845928 0.533297i 0.179047π-0.179047\pi
360360 −216393. −0.0880008
361361 −6.59763e6 −2.66453
362362 407761.i 0.163544i
363363 −90705.5 −0.0361299
364364 72097.5 88520.8i 0.0285211 0.0350181i
365365 1.51461e6 0.595070
366366 357989.i 0.139690i
367367 −1.40036e6 −0.542717 −0.271359 0.962478i 0.587473π-0.587473\pi
−0.271359 + 0.962478i 0.587473π0.587473\pi
368368 813994. 0.313330
369369 829229.i 0.317036i
370370 84807.5i 0.0322055i
371371 168142.i 0.0634221i
372372 607473.i 0.227599i
373373 −698420. −0.259923 −0.129961 0.991519i 0.541485π-0.541485\pi
−0.129961 + 0.991519i 0.541485π0.541485\pi
374374 738276. 0.272923
375375 177764.i 0.0652778i
376376 −1.72598e6 −0.629603
377377 −2.47971e6 2.01965e6i −0.898560 0.731850i
378378 −30369.0 −0.0109320
379379 860183.i 0.307605i 0.988102 + 0.153802i 0.0491519π0.0491519\pi
−0.988102 + 0.153802i 0.950848π0.950848\pi
380380 −2.29787e6 −0.816331
381381 −2.25910e6 −0.797302
382382 62969.2i 0.0220785i
383383 3.53928e6i 1.23287i −0.787406 0.616435i 0.788576π-0.788576\pi
0.787406 0.616435i 0.211424π-0.211424\pi
384384 1.61278e6i 0.558146i
385385 60059.9i 0.0206506i
386386 806964. 0.275668
387387 279902. 0.0950010
388388 4.14041e6i 1.39625i
389389 854347. 0.286260 0.143130 0.989704i 0.454283π-0.454283\pi
0.143130 + 0.989704i 0.454283π0.454283\pi
390390 −163835. 133439.i −0.0545439 0.0444244i
391391 1.42592e6 0.471685
392392 1.27811e6i 0.420100i
393393 1.16163e6 0.379390
394394 −509671. −0.165405
395395 2.19089e6i 0.706526i
396396 1.35581e6i 0.434471i
397397 4.61900e6i 1.47086i −0.677601 0.735430i 0.736980π-0.736980\pi
0.677601 0.735430i 0.263020π-0.263020\pi
398398 1.03155e6i 0.326425i
399399 −210428. −0.0661717
400400 −552192. −0.172560
401401 147425.i 0.0457835i 0.999738 + 0.0228917i 0.00728730π0.00728730\pi
−0.999738 + 0.0228917i 0.992713π0.992713\pi
402402 −502221. −0.154999
403403 −673366. + 826754.i −0.206533 + 0.253579i
404404 −4.41570e6 −1.34600
405405 463876.i 0.140529i
406406 39291.8 0.0118301
407407 −1.08861e6 −0.325751
408408 1.34202e6i 0.399125i
409409 5.88390e6i 1.73923i −0.493731 0.869615i 0.664367π-0.664367\pi
0.493731 0.869615i 0.335633π-0.335633\pi
410410 222558.i 0.0653859i
411411 4.42275e6i 1.29148i
412412 −197511. −0.0573254
413413 −292538. −0.0843932
414414 127567.i 0.0365796i
415415 −2.06164e6 −0.587613
416416 1.35302e6 1.66123e6i 0.383329 0.470649i
417417 2.79025e6 0.785782
418418 1.43691e6i 0.402243i
419419 875261. 0.243558 0.121779 0.992557i 0.461140π-0.461140\pi
0.121779 + 0.992557i 0.461140π0.461140\pi
420420 −53289.8 −0.0147408
421421 268396.i 0.0738025i −0.999319 0.0369012i 0.988251π-0.988251\pi
0.999319 0.0369012i 0.0117487π-0.0117487\pi
422422 572475.i 0.156486i
423423 2.57177e6i 0.698847i
424424 2.08709e6i 0.563802i
425425 −967304. −0.259771
426426 −350450. −0.0935625
427427 158473.i 0.0420615i
428428 2.74553e6 0.724463
429429 1.71285e6 2.10303e6i 0.449342 0.551699i
430430 −75123.3 −0.0195931
431431 5.07495e6i 1.31595i 0.753041 + 0.657974i 0.228586π0.228586\pi
−0.753041 + 0.657974i 0.771414π0.771414\pi
432432 3.58405e6 0.923985
433433 192569. 0.0493590 0.0246795 0.999695i 0.492143π-0.492143\pi
0.0246795 + 0.999695i 0.492143π0.492143\pi
434434 13100.2i 0.00333852i
435435 1.49279e6i 0.378247i
436436 1.12713e6i 0.283959i
437437 2.77526e6i 0.695185i
438438 840356. 0.209304
439439 7.75585e6 1.92074 0.960368 0.278734i 0.0899148π-0.0899148\pi
0.960368 + 0.278734i 0.0899148π0.0899148\pi
440440 745504.i 0.183577i
441441 −1.90443e6 −0.466303
442442 726109. 891512.i 0.176785 0.217056i
443443 −1.65318e6 −0.400230 −0.200115 0.979772i 0.564132π-0.564132\pi
−0.200115 + 0.979772i 0.564132π0.564132\pi
444444 965898.i 0.232527i
445445 −651561. −0.155975
446446 −902205. −0.214767
447447 5.18066e6i 1.22636i
448448 147277.i 0.0346688i
449449 4.12655e6i 0.965988i 0.875624 + 0.482994i 0.160451π0.160451\pi
−0.875624 + 0.482994i 0.839549π0.839549\pi
450450 86538.4i 0.0201455i
451451 −2.85681e6 −0.661363
452452 845341. 0.194619
453453 3.02282e6i 0.692096i
454454 −125094. −0.0284836
455455 72525.9 + 59070.1i 0.0164235 + 0.0133764i
456456 −2.61198e6 −0.588244
457457 2.76471e6i 0.619239i −0.950861 0.309620i 0.899798π-0.899798\pi
0.950861 0.309620i 0.100202π-0.100202\pi
458458 100748. 0.0224425
459459 6.27837e6 1.39096
460460 702819.i 0.154863i
461461 4.25293e6i 0.932043i 0.884773 + 0.466022i 0.154313π0.154313\pi
−0.884773 + 0.466022i 0.845687π0.845687\pi
462462 33323.2i 0.00726344i
463463 2.81006e6i 0.609204i 0.952480 + 0.304602i 0.0985235π0.0985235\pi
−0.952480 + 0.304602i 0.901477π0.901477\pi
464464 −4.63709e6 −0.999887
465465 497708. 0.106744
466466 176981.i 0.0377539i
467467 8.91425e6 1.89144 0.945720 0.324982i 0.105358π-0.105358\pi
0.945720 + 0.324982i 0.105358π0.105358\pi
468468 −1.63722e6 1.33347e6i −0.345535 0.281428i
469469 222321. 0.0466711
470470 690243.i 0.144131i
471471 −1.82409e6 −0.378873
472472 −3.63118e6 −0.750228
473473 964301.i 0.198180i
474474 1.21558e6i 0.248507i
475475 1.88266e6i 0.382859i
476476 289977.i 0.0586605i
477477 3.10984e6 0.625809
478478 372508. 0.0745703
479479 1.04773e6i 0.208647i 0.994543 + 0.104323i 0.0332677π0.0332677\pi
−0.994543 + 0.104323i 0.966732π0.966732\pi
480480 −1.00007e6 −0.198119
481481 −1.07067e6 + 1.31456e6i −0.211005 + 0.259070i
482482 −1.64938e6 −0.323373
483483 64360.9i 0.0125532i
484484 243278. 0.0472051
485485 3.39227e6 0.654842
486486 944473.i 0.181384i
487487 2.64473e6i 0.505311i 0.967556 + 0.252656i 0.0813039π0.0813039\pi
−0.967556 + 0.252656i 0.918696π0.918696\pi
488488 1.96707e6i 0.373913i
489489 4.74451e6i 0.897262i
490490 511133. 0.0961709
491491 −2.58589e6 −0.484069 −0.242034 0.970268i 0.577815π-0.577815\pi
−0.242034 + 0.970268i 0.577815π0.577815\pi
492492 2.53478e6i 0.472093i
493493 −8.12304e6 −1.50522
494494 1.73515e6 + 1.41323e6i 0.319904 + 0.260552i
495495 1.11083e6 0.203767
496496 1.54604e6i 0.282174i
497497 155135. 0.0281722
498498 −1.14386e6 −0.206681
499499 8.62141e6i 1.54998i −0.631972 0.774992i 0.717754π-0.717754\pi
0.631972 0.774992i 0.282246π-0.282246\pi
500500 476774.i 0.0852879i
501501 966245.i 0.171986i
502502 336059.i 0.0595191i
503503 4.82702e6 0.850665 0.425333 0.905037i 0.360157π-0.360157\pi
0.425333 + 0.905037i 0.360157π0.360157\pi
504504 53148.5 0.00931997
505505 3.61782e6i 0.631275i
506506 439488. 0.0763081
507507 −854909. 4.13674e6i −0.147707 0.714725i
508508 6.05905e6 1.04171
509509 7.34279e6i 1.25622i 0.778124 + 0.628111i 0.216172π0.216172\pi
−0.778124 + 0.628111i 0.783828π0.783828\pi
510510 −536693. −0.0913693
511511 −372005. −0.0630225
512512 5.26136e6i 0.887000i
513513 1.22196e7i 2.05005i
514514 745936.i 0.124536i
515515 161822.i 0.0268856i
516516 855602. 0.141464
517517 8.86012e6 1.45785
518518 20829.7i 0.00341081i
519519 1.38941e6 0.226419
520520 900240. + 733218.i 0.145999 + 0.118912i
521521 1.06181e7 1.71377 0.856885 0.515508i 0.172397π-0.172397\pi
0.856885 + 0.515508i 0.172397π0.172397\pi
522522 726715.i 0.116731i
523523 −7.02911e6 −1.12369 −0.561844 0.827243i 0.689908π-0.689908\pi
−0.561844 + 0.827243i 0.689908π0.689908\pi
524524 −3.11556e6 −0.495687
525525 43660.8i 0.00691342i
526526 615880.i 0.0970581i
527527 2.70828e6i 0.424784i
528528 3.93270e6i 0.613912i
529529 −5.58751e6 −0.868119
530530 −834655. −0.129068
531531 5.41059e6i 0.832738i
532532 564382. 0.0864558
533533 −2.80973e6 + 3.44977e6i −0.428397 + 0.525983i
534534 −361508. −0.0548611
535535 2.24943e6i 0.339773i
536536 2.75960e6 0.414891
537537 −5.22668e6 −0.782150
538538 396760.i 0.0590979i
539539 6.56102e6i 0.972746i
540540 3.09454e6i 0.456680i
541541 1.98960e6i 0.292263i 0.989265 + 0.146131i 0.0466822π0.0466822\pi
−0.989265 + 0.146131i 0.953318π0.953318\pi
542542 383668. 0.0560993
543543 −3.80496e6 −0.553796
544544 5.44187e6i 0.788408i
545545 −923464. −0.133177
546546 40239.8 + 32774.1i 0.00577663 + 0.00470488i
547547 −7.00138e6 −1.00050 −0.500248 0.865882i 0.666758π-0.666758\pi
−0.500248 + 0.865882i 0.666758π0.666758\pi
548548 1.18621e7i 1.68737i
549549 2.93100e6 0.415036
550550 −298137. −0.0420251
551551 1.58099e7i 2.21845i
552552 798891.i 0.111594i
553553 538107.i 0.0748266i
554554 2.07410e6i 0.287115i
555555 791369. 0.109055
556556 −7.48361e6 −1.02665
557557 2.38525e6i 0.325759i −0.986646 0.162879i 0.947922π-0.947922\pi
0.986646 0.162879i 0.0520781π-0.0520781\pi
558558 −242292. −0.0329423
559559 −1.16445e6 948409.i −0.157613 0.128371i
560560 135625. 0.0182755
561561 6.88911e6i 0.924179i
562562 −1.18649e6 −0.158461
563563 1.10441e7 1.46845 0.734226 0.678905i 0.237545π-0.237545\pi
0.734226 + 0.678905i 0.237545π0.237545\pi
564564 7.86138e6i 1.04064i
565565 692595.i 0.0912764i
566566 1.04304e6i 0.136855i
567567 113933.i 0.0148831i
568568 1.92564e6 0.250441
569569 −1.05466e7 −1.36563 −0.682815 0.730591i 0.739245π-0.739245\pi
−0.682815 + 0.730591i 0.739245π0.739245\pi
570570 1.04457e6i 0.134663i
571571 −2.56217e6 −0.328865 −0.164433 0.986388i 0.552579π-0.552579\pi
−0.164433 + 0.986388i 0.552579π0.552579\pi
572572 −4.59398e6 + 5.64046e6i −0.587082 + 0.720816i
573573 587588. 0.0747629
574574 54662.8i 0.00692487i
575575 −575826. −0.0726309
576576 −2.72393e6 −0.342089
577577 1.48196e7i 1.85309i −0.376183 0.926545i 0.622764π-0.622764\pi
0.376183 0.926545i 0.377236π-0.377236\pi
578578 1.18931e6i 0.148073i
579579 7.53006e6i 0.933474i
580580 4.00376e6i 0.494194i
581581 506360. 0.0622328
582582 1.88215e6 0.230328
583583 1.07138e7i 1.30549i
584584 −4.61757e6 −0.560249
585585 1.09252e6 1.34139e6i 0.131990 0.162056i
586586 −529271. −0.0636699
587587 1.20402e7i 1.44224i 0.692809 + 0.721121i 0.256373π0.256373\pi
−0.692809 + 0.721121i 0.743627π0.743627\pi
588588 −5.82145e6 −0.694364
589589 −5.27113e6 −0.626060
590590 1.45216e6i 0.171745i
591591 4.75592e6i 0.560100i
592592 2.45825e6i 0.288285i
593593 2.36654e6i 0.276361i 0.990407 + 0.138180i 0.0441254π0.0441254\pi
−0.990407 + 0.138180i 0.955875π0.955875\pi
594594 1.93508e6 0.225026
595595 237581. 0.0275118
596596 1.38949e7i 1.60228i
597597 −9.62578e6 −1.10535
598598 432245. 530708.i 0.0494284 0.0606879i
599599 −292768. −0.0333392 −0.0166696 0.999861i 0.505306π-0.505306\pi
−0.0166696 + 0.999861i 0.505306π0.505306\pi
600600 541947.i 0.0614580i
601601 1.20438e7 1.36011 0.680057 0.733159i 0.261955π-0.261955\pi
0.680057 + 0.733159i 0.261955π0.261955\pi
602602 18451.1 0.00207506
603603 4.11190e6i 0.460520i
604604 8.10738e6i 0.904250i
605605 199320.i 0.0221392i
606606 2.00729e6i 0.222039i
607607 −1.05634e7 −1.16367 −0.581836 0.813306i 0.697666π-0.697666\pi
−0.581836 + 0.813306i 0.697666π0.697666\pi
608608 1.05915e7 1.16198
609609 366646.i 0.0400593i
610610 −786657. −0.0855975
611611 8.71411e6 1.06991e7i 0.944322 1.15943i
612612 −5.36321e6 −0.578824
613613 1.72742e7i 1.85673i 0.371675 + 0.928363i 0.378783π0.378783\pi
−0.371675 + 0.928363i 0.621217π0.621217\pi
614614 −1.62429e6 −0.173877
615615 2.07677e6 0.221412
616616 183104.i 0.0194422i
617617 1.23381e7i 1.30477i 0.757886 + 0.652387i 0.226232π0.226232\pi
−0.757886 + 0.652387i 0.773768π0.773768\pi
618618 89784.4i 0.00945648i
619619 3.76541e6i 0.394990i 0.980304 + 0.197495i 0.0632806π0.0632806\pi
−0.980304 + 0.197495i 0.936719π0.936719\pi
620620 −1.33488e6 −0.139465
621621 3.73745e6 0.388907
622622 1.48854e6i 0.154271i
623623 160031. 0.0165190
624624 −4.74897e6 3.86789e6i −0.488245 0.397660i
625625 390625. 0.0400000
626626 4.10724e6i 0.418904i
627627 1.34083e7 1.36209
628628 4.89232e6 0.495012
629629 4.30624e6i 0.433982i
630630 21254.8i 0.00213356i
631631 4.25951e6i 0.425879i 0.977065 + 0.212940i 0.0683037π0.0683037\pi
−0.977065 + 0.212940i 0.931696π0.931696\pi
632632 6.67935e6i 0.665183i
633633 −5.34197e6 −0.529898
634634 −220811. −0.0218171
635635 4.96423e6i 0.488560i
636636 9.50613e6 0.931882
637637 7.92283e6 + 6.45290e6i 0.773627 + 0.630095i
638638 −2.50364e6 −0.243511
639639 2.86928e6i 0.277985i
640640 3.54399e6 0.342013
641641 2.97347e6 0.285837 0.142918 0.989734i 0.454351π-0.454351\pi
0.142918 + 0.989734i 0.454351π0.454351\pi
642642 1.24806e6i 0.119508i
643643 9.08699e6i 0.866748i 0.901214 + 0.433374i 0.142677π0.142677\pi
−0.901214 + 0.433374i 0.857323π0.857323\pi
644644 172620.i 0.0164012i
645645 701002.i 0.0663468i
646646 5.68401e6 0.535888
647647 −1.70405e7 −1.60037 −0.800185 0.599753i 0.795266π-0.795266\pi
−0.800185 + 0.599753i 0.795266π0.795266\pi
648648 1.41421e6i 0.132306i
649649 1.86402e7 1.73716
650650 −293224. + 360018.i −0.0272217 + 0.0334227i
651651 −122243. −0.0113050
652652 1.27251e7i 1.17231i
653653 33156.7 0.00304291 0.00152145 0.999999i 0.499516π-0.499516\pi
0.00152145 + 0.999999i 0.499516π0.499516\pi
654654 −512369. −0.0468423
655655 2.55260e6i 0.232477i
656656 6.45112e6i 0.585296i
657657 6.88034e6i 0.621866i
658658 169531.i 0.0152646i
659659 1.21273e7 1.08780 0.543902 0.839149i 0.316946π-0.316946\pi
0.543902 + 0.839149i 0.316946π0.316946\pi
660660 3.39557e6 0.303426
661661 7.42160e6i 0.660684i −0.943861 0.330342i 0.892836π-0.892836\pi
0.943861 0.330342i 0.107164π-0.107164\pi
662662 −2.10769e6 −0.186923
663663 −8.31901e6 6.77558e6i −0.735001 0.598636i
664664 6.28528e6 0.553229
665665 462403.i 0.0405477i
666666 −385251. −0.0336557
667667 −4.83555e6 −0.420854
668668 2.59153e6i 0.224706i
669669 8.41879e6i 0.727251i
670670 1.10360e6i 0.0949783i
671671 1.00977e7i 0.865799i
672672 245627. 0.0209823
673673 1.24325e7 1.05809 0.529043 0.848595i 0.322551π-0.322551\pi
0.529043 + 0.848595i 0.322551π0.322551\pi
674674 2.82028e6i 0.239135i
675675 −2.53539e6 −0.214183
676676 2.29292e6 + 1.10950e7i 0.192984 + 0.933815i
677677 −8.45003e6 −0.708576 −0.354288 0.935136i 0.615277π-0.615277\pi
−0.354288 + 0.935136i 0.615277π0.615277\pi
678678 384275.i 0.0321047i
679679 −833179. −0.0693528
680680 2.94901e6 0.244570
681681 1.16729e6i 0.0964520i
682682 834732.i 0.0687205i
683683 9.91289e6i 0.813108i −0.913627 0.406554i 0.866730π-0.866730\pi
0.913627 0.406554i 0.133270π-0.133270\pi
684684 1.04384e7i 0.853090i
685685 9.71871e6 0.791375
686686 −251362. −0.0203934
687687 940113.i 0.0759956i
688688 −2.17754e6 −0.175386
689689 −1.29376e7 1.05373e7i −1.03826 0.845629i
690690 −319487. −0.0255465
691691 3.01942e6i 0.240563i −0.992740 0.120281i 0.961620π-0.961620\pi
0.992740 0.120281i 0.0383796π-0.0383796\pi
692692 −3.72649e6 −0.295825
693693 −272831. −0.0215805
694694 1.77962e6i 0.140258i
695695 6.13139e6i 0.481501i
696696 4.55105e6i 0.356114i
697697 1.13008e7i 0.881101i
698698 537722. 0.0417753
699699 1.65147e6 0.127843
700700 117101.i 0.00903265i
701701 7.55729e6 0.580859 0.290430 0.956896i 0.406202π-0.406202\pi
0.290430 + 0.956896i 0.406202π0.406202\pi
702702 1.90319e6 2.33673e6i 0.145761 0.178964i
703703 −8.38124e6 −0.639617
704704 9.38431e6i 0.713626i
705705 −6.44090e6 −0.488061
706706 −2.14036e6 −0.161612
707707 888577.i 0.0668570i
708708 1.65391e7i 1.24002i
709709 8.22641e6i 0.614603i 0.951612 + 0.307302i 0.0994260π0.0994260\pi
−0.951612 + 0.307302i 0.900574π0.900574\pi
710710 770091.i 0.0573319i
711711 −9.95247e6 −0.738341
712712 1.98641e6 0.146848
713713 1.61221e6i 0.118768i
714714 131818. 0.00967672
715715 −4.62128e6 3.76389e6i −0.338062 0.275341i
716716 1.40183e7 1.02191
717717 3.47600e6i 0.252512i
718718 −3.17552e6 −0.229881
719719 −2.42533e7 −1.74964 −0.874821 0.484446i 0.839021π-0.839021\pi
−0.874821 + 0.484446i 0.839021π0.839021\pi
720720 2.50842e6i 0.180330i
721721 39745.3i 0.00284740i
722722 8.04391e6i 0.574280i
723723 1.53910e7i 1.09501i
724724 1.02051e7 0.723556
725725 3.28031e6 0.231777
726726 110589.i 0.00778702i
727727 −4.40741e6 −0.309277 −0.154638 0.987971i 0.549421π-0.549421\pi
−0.154638 + 0.987971i 0.549421π0.549421\pi
728728 −221109. 180086.i −0.0154624 0.0125937i
729729 1.33221e7 0.928439
730730 1.84663e6i 0.128254i
731731 −3.81451e6 −0.264025
732732 8.95947e6 0.618023
733733 1.70711e7i 1.17355i −0.809751 0.586774i 0.800398π-0.800398\pi
0.809751 0.586774i 0.199602π-0.199602\pi
734734 1.70733e6i 0.116971i
735735 4.76956e6i 0.325657i
736736 3.23949e6i 0.220436i
737737 −1.41661e7 −0.960684
738738 −1.01101e6 −0.0683302
739739 6.86306e6i 0.462281i −0.972920 0.231141i 0.925754π-0.925754\pi
0.972920 0.231141i 0.0742458π-0.0742458\pi
740740 −2.12250e6 −0.142485
741741 1.31873e7 1.61913e7i 0.882289 1.08327i
742742 205000. 0.0136693
743743 7.82433e6i 0.519966i −0.965613 0.259983i 0.916283π-0.916283\pi
0.965613 0.259983i 0.0837170π-0.0837170\pi
744744 −1.51736e6 −0.100498
745745 1.13842e7 0.751469
746746 851521.i 0.0560207i
747747 9.36530e6i 0.614073i
748748 1.84770e7i 1.20747i
749749 552486.i 0.0359846i
750750 216732. 0.0140692
751751 −1.55733e7 −1.00758 −0.503790 0.863826i 0.668062π-0.668062\pi
−0.503790 + 0.863826i 0.668062π0.668062\pi
752752 2.00075e7i 1.29018i
753753 −3.13588e6 −0.201545
754754 −2.46238e6 + 3.02329e6i −0.157734 + 0.193665i
755755 −6.64245e6 −0.424093
756756 760053.i 0.0483659i
757757 4.30381e6 0.272969 0.136485 0.990642i 0.456420π-0.456420\pi
0.136485 + 0.990642i 0.456420π0.456420\pi
758758 1.04874e6 0.0662974
759759 4.10101e6i 0.258397i
760760 5.73966e6i 0.360456i
761761 2.17549e7i 1.36174i 0.732403 + 0.680871i 0.238399π0.238399\pi
−0.732403 + 0.680871i 0.761601π0.761601\pi
762762 2.75432e6i 0.171841i
763763 226813. 0.0141045
764764 −1.57595e6 −0.0976805
765765 4.39413e6i 0.271468i
766766 −4.31513e6 −0.265718
767767 1.83331e7 2.25092e7i 1.12524 1.38157i
768768 −6.76579e6 −0.413919
769769 2.85348e7i 1.74004i 0.493017 + 0.870020i 0.335894π0.335894\pi
−0.493017 + 0.870020i 0.664106π0.664106\pi
770770 73225.7 0.00445079
771771 −6.96059e6 −0.421706
772772 2.01961e7i 1.21962i
773773 2.79269e7i 1.68102i −0.541795 0.840511i 0.682255π-0.682255\pi
0.541795 0.840511i 0.317745π-0.317745\pi
774774 341259.i 0.0204754i
775775 1.09368e6i 0.0654089i
776776 −1.03420e7 −0.616523
777777 −194369. −0.0115498
778778 1.04163e6i 0.0616971i
779779 −2.19947e7 −1.29860
780780 3.33961e6 4.10035e6i 0.196544 0.241315i
781781 −9.88507e6 −0.579899
782782 1.73849e6i 0.101661i
783783 −2.12912e7 −1.24107
784784 1.48158e7 0.860865
785785 4.00832e6i 0.232161i
786786 1.41627e6i 0.0817692i
787787 2.09367e7i 1.20496i −0.798136 0.602478i 0.794180π-0.794180\pi
0.798136 0.602478i 0.205820π-0.205820\pi
788788 1.27557e7i 0.731792i
789789 5.74700e6 0.328661
790790 2.67116e6 0.152276
791791 170109.i 0.00966688i
792792 −3.38657e6 −0.191843
793793 −1.21936e7 9.93131e6i −0.688571 0.560820i
794794 −5.63153e6 −0.317012
795795 7.78846e6i 0.437052i
796796 2.58169e7 1.44418
797797 1.45712e7 0.812549 0.406274 0.913751i 0.366828π-0.366828\pi
0.406274 + 0.913751i 0.366828π0.366828\pi
798798 256557.i 0.0142619i
799799 3.50482e7i 1.94222i
800800 2.19758e6i 0.121400i
801801 2.95982e6i 0.162998i
802802 179742. 0.00986763
803803 2.37037e7 1.29726
804804 1.25692e7i 0.685754i
805805 141429. 0.00769217
806806 1.00799e6 + 820975.i 0.0546535 + 0.0445136i
807807 −3.70231e6 −0.200119
808808 1.10296e7i 0.594336i
809809 −2.59047e7 −1.39158 −0.695789 0.718246i 0.744945π-0.744945\pi
−0.695789 + 0.718246i 0.744945π0.744945\pi
810810 −565564. −0.0302879
811811 9.38778e6i 0.501200i 0.968091 + 0.250600i 0.0806279π0.0806279\pi
−0.968091 + 0.250600i 0.919372π0.919372\pi
812812 983367.i 0.0523390i
813813 3.58014e6i 0.189965i
814814 1.32725e6i 0.0702086i
815815 −1.04258e7 −0.549812
816816 −1.55567e7 −0.817884
817817 7.42418e6i 0.389129i
818818 −7.17372e6 −0.374853
819819 −268335. + 329460.i −0.0139787 + 0.0171630i
820820 −5.57002e6 −0.289283
821821 7.37712e6i 0.381970i 0.981593 + 0.190985i 0.0611681π0.0611681\pi
−0.981593 + 0.190985i 0.938832π0.938832\pi
822822 5.39227e6 0.278350
823823 2.13960e7 1.10111 0.550557 0.834798i 0.314416π-0.314416\pi
0.550557 + 0.834798i 0.314416π0.314416\pi
824824 493345.i 0.0253124i
825825 2.78202e6i 0.142307i
826826 356666.i 0.0181891i
827827 6.14027e6i 0.312194i −0.987742 0.156097i 0.950109π-0.950109\pi
0.987742 0.156097i 0.0498912π-0.0498912\pi
828828 −3.19266e6 −0.161837
829829 −1.46790e7 −0.741838 −0.370919 0.928665i 0.620957π-0.620957\pi
−0.370919 + 0.928665i 0.620957π0.620957\pi
830830 2.51357e6i 0.126647i
831831 1.93542e7 0.972237
832832 1.13321e7 + 9.22966e6i 0.567548 + 0.462251i
833833 2.59536e7 1.29594
834834 3.40190e6i 0.169358i
835835 −2.12326e6 −0.105387
836836 −3.59618e7 −1.77962
837837 7.09864e6i 0.350237i
838838 1.06713e6i 0.0524937i
839839 2.35783e7i 1.15640i 0.815896 + 0.578198i 0.196244π0.196244\pi
−0.815896 + 0.578198i 0.803756π0.803756\pi
840840 133108.i 0.00650888i
841841 7.03561e6 0.343014
842842 −327232. −0.0159065
843843 1.10715e7i 0.536586i
844844 1.43275e7 0.692331
845845 −9.09024e6 + 1.87861e6i −0.437959 + 0.0905096i
846846 3.13554e6 0.150621
847847 48955.1i 0.00234471i
848848 −2.41935e7 −1.15534
849849 −9.73298e6 −0.463422
850850 1.17935e6i 0.0559880i
851851 2.56346e6i 0.121340i
852852 8.77080e6i 0.413943i
853853 1.60837e7i 0.756856i 0.925631 + 0.378428i 0.123535π0.123535\pi
−0.925631 + 0.378428i 0.876465π0.876465\pi
854854 193212. 0.00906544
855855 8.55229e6 0.400099
856856 6.85782e6i 0.319891i
857857 2.45455e7 1.14162 0.570809 0.821083i 0.306630π-0.306630\pi
0.570809 + 0.821083i 0.306630π0.306630\pi
858858 −2.56404e6 2.08833e6i −0.118907 0.0968459i
859859 2.36147e7 1.09194 0.545971 0.837804i 0.316161π-0.316161\pi
0.545971 + 0.837804i 0.316161π0.316161\pi
860860 1.88013e6i 0.0866846i
861861 −510077. −0.0234492
862862 6.18744e6 0.283624
863863 2.10316e7i 0.961271i −0.876920 0.480635i 0.840406π-0.840406\pi
0.876920 0.480635i 0.159594π-0.159594\pi
864864 1.42636e7i 0.650047i
865865 3.05314e6i 0.138742i
866866 234782.i 0.0106383i
867867 −1.10979e7 −0.501410
868868 327862. 0.0147704
869869 3.42876e7i 1.54024i
870870 1.82003e6 0.0815229
871871 −1.39326e7 + 1.71064e7i −0.622282 + 0.764033i
872872 2.81535e6 0.125384
873873 1.54099e7i 0.684329i
874874 3.38363e6 0.149832
875875 −95941.8 −0.00423631
876876 2.10318e7i 0.926011i
877877 2.16451e7i 0.950298i 0.879905 + 0.475149i 0.157606π0.157606\pi
−0.879905 + 0.475149i 0.842394π0.842394\pi
878878 9.45602e6i 0.413973i
879879 4.93881e6i 0.215601i
880880 −8.64186e6 −0.376184
881881 1.25222e7 0.543552 0.271776 0.962360i 0.412389π-0.412389\pi
0.271776 + 0.962360i 0.412389π0.412389\pi
882882 2.32190e6i 0.100501i
883883 4.38601e6 0.189307 0.0946537 0.995510i 0.469826π-0.469826\pi
0.0946537 + 0.995510i 0.469826π0.469826\pi
884884 2.23121e7 + 1.81725e7i 0.960307 + 0.782140i
885885 −1.35506e7 −0.581568
886886 2.01557e6i 0.0862609i
887887 3.48074e7 1.48547 0.742733 0.669587i 0.233529π-0.233529\pi
0.742733 + 0.669587i 0.233529π0.233529\pi
888888 −2.41264e6 −0.102674
889889 1.21927e6i 0.0517423i
890890 794391.i 0.0336170i
891891 7.25971e6i 0.306355i
892892 2.25797e7i 0.950181i
893893 6.82144e7 2.86251
894894 6.31632e6 0.264314
895895 1.14853e7i 0.479275i
896896 −870442. −0.0362218
897897 −4.95222e6 4.03343e6i −0.205503 0.167376i
898898 5.03114e6 0.208198
899899 9.18431e6i 0.379007i
900900 2.16582e6 0.0891283
901901 −4.23810e7 −1.73924
902902 3.48306e6i 0.142542i
903903 172174.i 0.00702664i
904904 2.11151e6i 0.0859353i
905905 8.36115e6i 0.339347i
906906 −3.68546e6 −0.149166
907907 −8.74330e6 −0.352904 −0.176452 0.984309i 0.556462π-0.556462\pi
−0.176452 + 0.984309i 0.556462π0.556462\pi
908908 3.13075e6i 0.126018i
909909 1.64345e7 0.659701
910910 72019.0 88424.4i 0.00288299 0.00353972i
911911 −2.23703e7 −0.893048 −0.446524 0.894772i 0.647338π-0.647338\pi
−0.446524 + 0.894772i 0.647338π0.647338\pi
912912 3.02780e7i 1.20542i
913913 −3.22648e7 −1.28101
914914 −3.37076e6 −0.133463
915915 7.34058e6i 0.289853i
916916 2.52144e6i 0.0992911i
917917 626947.i 0.0246211i
918918 7.65466e6i 0.299792i
919919 −3.29243e6 −0.128596 −0.0642981 0.997931i 0.520481π-0.520481\pi
−0.0642981 + 0.997931i 0.520481π0.520481\pi
920920 1.75551e6 0.0683808
921921 1.51568e7i 0.588787i
922922 5.18522e6 0.200882
923923 −9.72217e6 + 1.19368e7i −0.375629 + 0.461194i
924924 −833989. −0.0321352
925925 1.73898e6i 0.0668253i
926926 3.42606e6 0.131301
927927 735102. 0.0280963
928928 1.84544e7i 0.703446i
929929 2.50376e7i 0.951817i 0.879495 + 0.475908i 0.157881π0.157881\pi
−0.879495 + 0.475908i 0.842119π0.842119\pi
930930 606812.i 0.0230063i
931931 5.05135e7i 1.91000i
932932 −4.42935e6 −0.167032
933933 1.38901e7 0.522397
934934 1.08684e7i 0.407659i
935935 −1.51384e7 −0.566305
936936 −3.33075e6 + 4.08948e6i −0.124266 + 0.152573i
937937 −4.64120e6 −0.172696 −0.0863478 0.996265i 0.527520π-0.527520\pi
−0.0863478 + 0.996265i 0.527520π0.527520\pi
938938 271056.i 0.0100589i
939939 3.83261e7 1.41851
940940 1.72749e7 0.637670
941941 1.62911e7i 0.599760i −0.953977 0.299880i 0.903053π-0.903053\pi
0.953977 0.299880i 0.0969467π-0.0969467\pi
942942 2.22395e6i 0.0816579i
943943 6.72722e6i 0.246352i
944944 4.20926e7i 1.53736i
945945 622719. 0.0226836
946946 −1.17569e6 −0.0427134
947947 2.98948e7i 1.08323i 0.840627 + 0.541614i 0.182187π0.182187\pi
−0.840627 + 0.541614i 0.817813π0.817813\pi
948948 −3.04226e7 −1.09945
949949 2.33131e7 2.86237e7i 0.840301 1.03172i
950950 −2.29537e6 −0.0825169
951951 2.06046e6i 0.0738777i
952952 −724309. −0.0259019
953953 4.34693e7 1.55043 0.775213 0.631700i 0.217643π-0.217643\pi
0.775213 + 0.631700i 0.217643π0.217643\pi
954954 3.79155e6i 0.134879i
955955 1.29119e6i 0.0458122i
956956 9.32286e6i 0.329917i
957957 2.33623e7i 0.824586i
958958 1.27741e6 0.0449692
959959 −2.38702e6 −0.0838127
960960 6.82196e6i 0.238908i
961961 2.55670e7 0.893042
962962 1.60273e6 + 1.30537e6i 0.0558370 + 0.0454775i
963963 −1.02184e7 −0.355073
964964 4.12795e7i 1.43068i
965965 −1.65468e7 −0.572001
966966 78469.6 0.00270557
967967 2.94293e7i 1.01208i −0.862510 0.506039i 0.831109π-0.831109\pi
0.862510 0.506039i 0.168891π-0.168891\pi
968968 607664.i 0.0208437i
969969 5.30395e7i 1.81464i
970970 4.13590e6i 0.141137i
971971 −3.14967e7 −1.07205 −0.536027 0.844201i 0.680076π-0.680076\pi
−0.536027 + 0.844201i 0.680076π0.680076\pi
972972 −2.36376e7 −0.802486
973973 1.50594e6i 0.0509947i
974974 3.22448e6 0.108909
975975 3.35946e6 + 2.73617e6i 0.113177 + 0.0921790i
976976 −2.28022e7 −0.766218
977977 5.67335e6i 0.190153i −0.995470 0.0950764i 0.969690π-0.969690\pi
0.995470 0.0950764i 0.0303096π-0.0303096\pi
978978 −5.78457e6 −0.193385
979979 −1.01970e7 −0.340028
980980 1.27923e7i 0.425483i
981981 4.19498e6i 0.139174i
982982 3.15275e6i 0.104330i
983983 1.96686e7i 0.649217i −0.945848 0.324609i 0.894767π-0.894767\pi
0.945848 0.324609i 0.105233π-0.105233\pi
984984 −6.33142e6 −0.208456
985985 1.04508e7 0.343210
986986 9.90370e6i 0.324418i
987987 1.58196e6 0.0516894
988988 −3.53692e7 + 4.34261e7i −1.15274 + 1.41533i
989989 −2.27074e6 −0.0738203
990990 1.35433e6i 0.0439175i
991991 3.75674e7 1.21514 0.607571 0.794266i 0.292144π-0.292144\pi
0.607571 + 0.794266i 0.292144π0.292144\pi
992992 6.15285e6 0.198517
993993 1.96676e7i 0.632963i
994994 189143.i 0.00607189i
995995 2.11520e7i 0.677321i
996996 2.86278e7i 0.914407i
997997 −6.54052e6 −0.208389 −0.104194 0.994557i 0.533226π-0.533226\pi
−0.104194 + 0.994557i 0.533226π0.533226\pi
998998 −1.05113e7 −0.334065
999999 1.12870e7i 0.357821i
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 65.6.c.b.51.7 14
13.5 odd 4 845.6.a.i.1.5 7
13.8 odd 4 845.6.a.j.1.3 7
13.12 even 2 inner 65.6.c.b.51.8 yes 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.6.c.b.51.7 14 1.1 even 1 trivial
65.6.c.b.51.8 yes 14 13.12 even 2 inner
845.6.a.i.1.5 7 13.5 odd 4
845.6.a.j.1.3 7 13.8 odd 4