Properties

Label 78.6.b.a.25.5
Level 7878
Weight 66
Character 78.25
Analytic conductor 12.51012.510
Analytic rank 00
Dimension 66
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [78,6,Mod(25,78)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78, 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("78.25");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 78=2313 78 = 2 \cdot 3 \cdot 13
Weight: k k == 6 6
Character orbit: [χ][\chi] == 78.b (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: 12.509937945412.5099379454
Analytic rank: 00
Dimension: 66
Coefficient field: Q[x]/(x6)\mathbb{Q}[x]/(x^{6} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x62x5+2x42946x3+131769x21332936x+6741792 x^{6} - 2x^{5} + 2x^{4} - 2946x^{3} + 131769x^{2} - 1332936x + 6741792 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 21132 2^{11}\cdot 3^{2}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 25.5
Root 15.0768+15.0768i-15.0768 + 15.0768i of defining polynomial
Character χ\chi == 78.25
Dual form 78.6.b.a.25.2

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q+4.00000iq29.00000q316.0000q4+9.73803iq536.0000iq6105.184iq764.0000iq8+81.0000q938.9521q10269.350iq11+144.000q12+(227.464+565.290i)q13+420.736q1487.6423iq15+256.000q16+1663.75q17+324.000iq18+2.82742iq19155.809iq20+946.657iq21+1077.40q22+2151.14q23+576.000iq24+3030.17q25+(2261.16+909.856i)q26729.000q27+1682.95iq28+220.829q29+350.569q30+788.106iq31+1024.00iq32+2424.15iq33+6654.98iq34+1024.29q351296.00q36+980.445iq3711.3097q38+(2047.185087.61i)q39+623.234q4014809.7iq413786.63q42+14142.2q43+4309.61iq44+788.781iq45+8604.55iq4625181.2iq472304.00q48+5743.31q49+12120.7iq5014973.7q51+(3639.429044.65i)q5230900.9q532916.00iq54+2622.94q556731.78q5625.4468iq57+883.317iq5825094.7iq59+1402.28iq60+12060.8q613152.42q628519.91iq634096.00q64+(5504.81+2215.05i)q659696.61q6614450.5iq6726619.9q6819360.2q69+4097.14iq70+33547.5iq715184.00iq72+25805.7iq733921.78q7427271.5q7545.2388iq7628331.4q77+(20350.58188.70i)q78+15620.3q79+2492.94iq80+6561.00q81+59238.7q822819.78iq8315146.5iq84+16201.6iq85+56568.8iq861987.46q8717238.4q88+16227.2iq893155.12q90+(59459.623925.6i)q9134418.2q927092.95iq93+100725.q9427.5335q959216.00iq96+126289.iq97+22973.2iq9821817.4iq99+O(q100)q+4.00000i q^{2} -9.00000 q^{3} -16.0000 q^{4} +9.73803i q^{5} -36.0000i q^{6} -105.184i q^{7} -64.0000i q^{8} +81.0000 q^{9} -38.9521 q^{10} -269.350i q^{11} +144.000 q^{12} +(227.464 + 565.290i) q^{13} +420.736 q^{14} -87.6423i q^{15} +256.000 q^{16} +1663.75 q^{17} +324.000i q^{18} +2.82742i q^{19} -155.809i q^{20} +946.657i q^{21} +1077.40 q^{22} +2151.14 q^{23} +576.000i q^{24} +3030.17 q^{25} +(-2261.16 + 909.856i) q^{26} -729.000 q^{27} +1682.95i q^{28} +220.829 q^{29} +350.569 q^{30} +788.106i q^{31} +1024.00i q^{32} +2424.15i q^{33} +6654.98i q^{34} +1024.29 q^{35} -1296.00 q^{36} +980.445i q^{37} -11.3097 q^{38} +(-2047.18 - 5087.61i) q^{39} +623.234 q^{40} -14809.7i q^{41} -3786.63 q^{42} +14142.2 q^{43} +4309.61i q^{44} +788.781i q^{45} +8604.55i q^{46} -25181.2i q^{47} -2304.00 q^{48} +5743.31 q^{49} +12120.7i q^{50} -14973.7 q^{51} +(-3639.42 - 9044.65i) q^{52} -30900.9 q^{53} -2916.00i q^{54} +2622.94 q^{55} -6731.78 q^{56} -25.4468i q^{57} +883.317i q^{58} -25094.7i q^{59} +1402.28i q^{60} +12060.8 q^{61} -3152.42 q^{62} -8519.91i q^{63} -4096.00 q^{64} +(-5504.81 + 2215.05i) q^{65} -9696.61 q^{66} -14450.5i q^{67} -26619.9 q^{68} -19360.2 q^{69} +4097.14i q^{70} +33547.5i q^{71} -5184.00i q^{72} +25805.7i q^{73} -3921.78 q^{74} -27271.5 q^{75} -45.2388i q^{76} -28331.4 q^{77} +(20350.5 - 8188.70i) q^{78} +15620.3 q^{79} +2492.94i q^{80} +6561.00 q^{81} +59238.7 q^{82} -2819.78i q^{83} -15146.5i q^{84} +16201.6i q^{85} +56568.8i q^{86} -1987.46 q^{87} -17238.4 q^{88} +16227.2i q^{89} -3155.12 q^{90} +(59459.6 - 23925.6i) q^{91} -34418.2 q^{92} -7092.95i q^{93} +100725. q^{94} -27.5335 q^{95} -9216.00i q^{96} +126289. i q^{97} +22973.2i q^{98} -21817.4i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q54q396q4+486q9+320q10+864q12+530q131360q14+1536q16836q171296q22416q23+718q251360q264374q27+18788q292880q30++567168q95+O(q100) 6 q - 54 q^{3} - 96 q^{4} + 486 q^{9} + 320 q^{10} + 864 q^{12} + 530 q^{13} - 1360 q^{14} + 1536 q^{16} - 836 q^{17} - 1296 q^{22} - 416 q^{23} + 718 q^{25} - 1360 q^{26} - 4374 q^{27} + 18788 q^{29} - 2880 q^{30}+ \cdots + 567168 q^{95}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/78Z)×\left(\mathbb{Z}/78\mathbb{Z}\right)^\times.

nn 5353 6767
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 4.00000i 0.707107i
33 −9.00000 −0.577350
44 −16.0000 −0.500000
55 9.73803i 0.174199i 0.996200 + 0.0870996i 0.0277598π0.0277598\pi
−0.996200 + 0.0870996i 0.972240π0.972240\pi
66 36.0000i 0.408248i
77 105.184i 0.811344i −0.914019 0.405672i 0.867038π-0.867038\pi
0.914019 0.405672i 0.132962π-0.132962\pi
88 64.0000i 0.353553i
99 81.0000 0.333333
1010 −38.9521 −0.123177
1111 269.350i 0.671175i −0.942009 0.335588i 0.891065π-0.891065\pi
0.942009 0.335588i 0.108935π-0.108935\pi
1212 144.000 0.288675
1313 227.464 + 565.290i 0.373297 + 0.927712i
1414 420.736 0.573707
1515 87.6423i 0.100574i
1616 256.000 0.250000
1717 1663.75 1.39625 0.698127 0.715974i 0.254017π-0.254017\pi
0.698127 + 0.715974i 0.254017π0.254017\pi
1818 324.000i 0.235702i
1919 2.82742i 0.00179683i 1.00000 0.000898415i 0.000285974π0.000285974\pi
−1.00000 0.000898415i 0.999714π0.999714\pi
2020 155.809i 0.0870996i
2121 946.657i 0.468430i
2222 1077.40 0.474592
2323 2151.14 0.847908 0.423954 0.905684i 0.360642π-0.360642\pi
0.423954 + 0.905684i 0.360642π0.360642\pi
2424 576.000i 0.204124i
2525 3030.17 0.969655
2626 −2261.16 + 909.856i −0.655991 + 0.263961i
2727 −729.000 −0.192450
2828 1682.95i 0.405672i
2929 220.829 0.0487598 0.0243799 0.999703i 0.492239π-0.492239\pi
0.0243799 + 0.999703i 0.492239π0.492239\pi
3030 350.569 0.0711165
3131 788.106i 0.147292i 0.997284 + 0.0736462i 0.0234636π0.0234636\pi
−0.997284 + 0.0736462i 0.976536π0.976536\pi
3232 1024.00i 0.176777i
3333 2424.15i 0.387503i
3434 6654.98i 0.987301i
3535 1024.29 0.141335
3636 −1296.00 −0.166667
3737 980.445i 0.117739i 0.998266 + 0.0588693i 0.0187495π0.0187495\pi
−0.998266 + 0.0588693i 0.981250π0.981250\pi
3838 −11.3097 −0.00127055
3939 −2047.18 5087.61i −0.215523 0.535615i
4040 623.234 0.0615887
4141 14809.7i 1.37590i −0.725759 0.687949i 0.758511π-0.758511\pi
0.725759 0.687949i 0.241489π-0.241489\pi
4242 −3786.63 −0.331230
4343 14142.2 1.16640 0.583198 0.812330i 0.301801π-0.301801\pi
0.583198 + 0.812330i 0.301801π0.301801\pi
4444 4309.61i 0.335588i
4545 788.781i 0.0580664i
4646 8604.55i 0.599561i
4747 25181.2i 1.66277i −0.555699 0.831383i 0.687549π-0.687549\pi
0.555699 0.831383i 0.312451π-0.312451\pi
4848 −2304.00 −0.144338
4949 5743.31 0.341721
5050 12120.7i 0.685649i
5151 −14973.7 −0.806128
5252 −3639.42 9044.65i −0.186648 0.463856i
5353 −30900.9 −1.51106 −0.755530 0.655114i 0.772621π-0.772621\pi
−0.755530 + 0.655114i 0.772621π0.772621\pi
5454 2916.00i 0.136083i
5555 2622.94 0.116918
5656 −6731.78 −0.286853
5757 25.4468i 0.00103740i
5858 883.317i 0.0344784i
5959 25094.7i 0.938538i −0.883055 0.469269i 0.844517π-0.844517\pi
0.883055 0.469269i 0.155483π-0.155483\pi
6060 1402.28i 0.0502870i
6161 12060.8 0.415003 0.207501 0.978235i 0.433467π-0.433467\pi
0.207501 + 0.978235i 0.433467π0.433467\pi
6262 −3152.42 −0.104151
6363 8519.91i 0.270448i
6464 −4096.00 −0.125000
6565 −5504.81 + 2215.05i −0.161607 + 0.0650280i
6666 −9696.61 −0.274006
6767 14450.5i 0.393276i −0.980476 0.196638i 0.936998π-0.936998\pi
0.980476 0.196638i 0.0630023π-0.0630023\pi
6868 −26619.9 −0.698127
6969 −19360.2 −0.489540
7070 4097.14i 0.0999393i
7171 33547.5i 0.789795i 0.918725 + 0.394897i 0.129220π0.129220\pi
−0.918725 + 0.394897i 0.870780π0.870780\pi
7272 5184.00i 0.117851i
7373 25805.7i 0.566772i 0.959006 + 0.283386i 0.0914578π0.0914578\pi
−0.959006 + 0.283386i 0.908542π0.908542\pi
7474 −3921.78 −0.0832538
7575 −27271.5 −0.559830
7676 45.2388i 0.000898415i
7777 −28331.4 −0.544554
7878 20350.5 8188.70i 0.378737 0.152398i
7979 15620.3 0.281593 0.140797 0.990039i 0.455034π-0.455034\pi
0.140797 + 0.990039i 0.455034π0.455034\pi
8080 2492.94i 0.0435498i
8181 6561.00 0.111111
8282 59238.7 0.972906
8383 2819.78i 0.0449284i −0.999748 0.0224642i 0.992849π-0.992849\pi
0.999748 0.0224642i 0.00715117π-0.00715117\pi
8484 15146.5i 0.234215i
8585 16201.6i 0.243226i
8686 56568.8i 0.824766i
8787 −1987.46 −0.0281515
8888 −17238.4 −0.237296
8989 16227.2i 0.217154i 0.994088 + 0.108577i 0.0346294π0.0346294\pi
−0.994088 + 0.108577i 0.965371π0.965371\pi
9090 −3155.12 −0.0410591
9191 59459.6 23925.6i 0.752693 0.302872i
9292 −34418.2 −0.423954
9393 7092.95i 0.0850393i
9494 100725. 1.17575
9595 −27.5335 −0.000313006
9696 9216.00i 0.102062i
9797 126289.i 1.36281i 0.731906 + 0.681406i 0.238631π0.238631\pi
−0.731906 + 0.681406i 0.761369π0.761369\pi
9898 22973.2i 0.241633i
9999 21817.4i 0.223725i
100100 −48482.7 −0.484827
101101 92398.8 0.901287 0.450643 0.892704i 0.351195π-0.351195\pi
0.450643 + 0.892704i 0.351195π0.351195\pi
102102 59894.8i 0.570018i
103103 148091. 1.37542 0.687711 0.725985i 0.258616π-0.258616\pi
0.687711 + 0.725985i 0.258616π0.258616\pi
104104 36178.6 14557.7i 0.327996 0.131980i
105105 −9218.57 −0.0816001
106106 123604.i 1.06848i
107107 88180.1 0.744580 0.372290 0.928117i 0.378573π-0.378573\pi
0.372290 + 0.928117i 0.378573π0.378573\pi
108108 11664.0 0.0962250
109109 61467.2i 0.495538i 0.968819 + 0.247769i 0.0796974π0.0796974\pi
−0.968819 + 0.247769i 0.920303π0.920303\pi
110110 10491.8i 0.0826736i
111111 8824.01i 0.0679764i
112112 26927.1i 0.202836i
113113 −82256.1 −0.605999 −0.303000 0.952991i 0.597988π-0.597988\pi
−0.303000 + 0.952991i 0.597988π0.597988\pi
114114 101.787 0.000733553
115115 20947.8i 0.147705i
116116 −3533.27 −0.0243799
117117 18424.6 + 45788.5i 0.124432 + 0.309237i
118118 100379. 0.663647
119119 175000.i 1.13284i
120120 −5609.11 −0.0355583
121121 88501.4 0.549524
122122 48243.1i 0.293451i
123123 133287.i 0.794375i
124124 12609.7i 0.0736462i
125125 59939.2i 0.343112i
126126 34079.6 0.191236
127127 −143150. −0.787556 −0.393778 0.919206i 0.628832π-0.628832\pi
−0.393778 + 0.919206i 0.628832π0.628832\pi
128128 16384.0i 0.0883883i
129129 −127280. −0.673419
130130 −8860.20 22019.3i −0.0459817 0.114273i
131131 −141329. −0.719539 −0.359769 0.933041i 0.617145π-0.617145\pi
−0.359769 + 0.933041i 0.617145π0.617145\pi
132132 38786.4i 0.193752i
133133 297.400 0.00145785
134134 57802.2 0.278088
135135 7099.02i 0.0335247i
136136 106480.i 0.493650i
137137 3958.33i 0.0180182i −0.999959 0.00900909i 0.997132π-0.997132\pi
0.999959 0.00900909i 0.00286772π-0.00286772\pi
138138 77440.9i 0.346157i
139139 −445579. −1.95608 −0.978042 0.208408i 0.933172π-0.933172\pi
−0.978042 + 0.208408i 0.933172π0.933172\pi
140140 −16388.6 −0.0706677
141141 226631.i 0.959999i
142142 −134190. −0.558469
143143 152261. 61267.5i 0.622657 0.250547i
144144 20736.0 0.0833333
145145 2150.44i 0.00849391i
146146 −103223. −0.400769
147147 −51689.7 −0.197293
148148 15687.1i 0.0588693i
149149 49619.3i 0.183098i 0.995801 + 0.0915492i 0.0291819π0.0291819\pi
−0.995801 + 0.0915492i 0.970818π0.970818\pi
150150 109086.i 0.395860i
151151 402323.i 1.43593i 0.696081 + 0.717964i 0.254926π0.254926\pi
−0.696081 + 0.717964i 0.745074π0.745074\pi
152152 180.955 0.000635275
153153 134763. 0.465418
154154 113325.i 0.385058i
155155 −7674.60 −0.0256582
156156 32754.8 + 81401.8i 0.107761 + 0.267807i
157157 −562429. −1.82104 −0.910519 0.413468i 0.864317π-0.864317\pi
−0.910519 + 0.413468i 0.864317π0.864317\pi
158158 62481.3i 0.199116i
159159 278108. 0.872411
160160 −9971.74 −0.0307944
161161 226265.i 0.687945i
162162 26244.0i 0.0785674i
163163 313878.i 0.925319i −0.886536 0.462659i 0.846895π-0.846895\pi
0.886536 0.462659i 0.153105π-0.153105\pi
164164 236955.i 0.687949i
165165 −23606.5 −0.0675027
166166 11279.1 0.0317691
167167 602228.i 1.67097i −0.549510 0.835487i 0.685186π-0.685186\pi
0.549510 0.835487i 0.314814π-0.314814\pi
168168 60586.0 0.165615
169169 −267813. + 257166.i −0.721299 + 0.692624i
170170 −64806.4 −0.171987
171171 229.021i 0.000598943i
172172 −226275. −0.583198
173173 304021. 0.772304 0.386152 0.922435i 0.373804π-0.373804\pi
0.386152 + 0.922435i 0.373804π0.373804\pi
174174 7949.85i 0.0199061i
175175 318726.i 0.786723i
176176 68953.7i 0.167794i
177177 225852.i 0.541865i
178178 −64908.7 −0.153551
179179 406190. 0.947539 0.473769 0.880649i 0.342893π-0.342893\pi
0.473769 + 0.880649i 0.342893π0.342893\pi
180180 12620.5i 0.0290332i
181181 −240967. −0.546714 −0.273357 0.961913i 0.588134π-0.588134\pi
−0.273357 + 0.961913i 0.588134π0.588134\pi
182182 95702.4 + 237838.i 0.214163 + 0.532235i
183183 −108547. −0.239602
184184 137673.i 0.299781i
185185 −9547.60 −0.0205100
186186 28371.8 0.0601319
187187 448130.i 0.937131i
188188 402899.i 0.831383i
189189 76679.2i 0.156143i
190190 110.134i 0.000221329i
191191 226140. 0.448532 0.224266 0.974528i 0.428001π-0.428001\pi
0.224266 + 0.974528i 0.428001π0.428001\pi
192192 36864.0 0.0721688
193193 406075.i 0.784717i −0.919812 0.392359i 0.871659π-0.871659\pi
0.919812 0.392359i 0.128341π-0.128341\pi
194194 −505156. −0.963653
195195 49543.3 19935.5i 0.0933037 0.0375439i
196196 −91892.9 −0.170861
197197 407390.i 0.747902i −0.927449 0.373951i 0.878003π-0.878003\pi
0.927449 0.373951i 0.121997π-0.121997\pi
198198 87269.5 0.158197
199199 202865. 0.363140 0.181570 0.983378i 0.441882π-0.441882\pi
0.181570 + 0.983378i 0.441882π0.441882\pi
200200 193931.i 0.342825i
201201 130055.i 0.227058i
202202 369595.i 0.637306i
203203 23227.7i 0.0395609i
204204 239579. 0.403064
205205 144217. 0.239680
206206 592364.i 0.972570i
207207 174242. 0.282636
208208 58230.8 + 144714.i 0.0933242 + 0.231928i
209209 761.568 0.00120599
210210 36874.3i 0.0577000i
211211 657970. 1.01742 0.508710 0.860938i 0.330123π-0.330123\pi
0.508710 + 0.860938i 0.330123π0.330123\pi
212212 494415. 0.755530
213213 301927.i 0.455988i
214214 352720.i 0.526497i
215215 137717.i 0.203185i
216216 46656.0i 0.0680414i
217217 82896.2 0.119505
218218 −245869. −0.350398
219219 232251.i 0.327226i
220220 −41967.1 −0.0584591
221221 378442. + 940499.i 0.521217 + 1.29532i
222222 35296.0 0.0480666
223223 978060.i 1.31705i −0.752558 0.658527i 0.771180π-0.771180\pi
0.752558 0.658527i 0.228820π-0.228820\pi
224224 107709. 0.143427
225225 245444. 0.323218
226226 329024.i 0.428506i
227227 812137.i 1.04608i 0.852308 + 0.523040i 0.175202π0.175202\pi
−0.852308 + 0.523040i 0.824798π0.824798\pi
228228 407.149i 0.000518700i
229229 1.36329e6i 1.71790i 0.512058 + 0.858951i 0.328883π0.328883\pi
−0.512058 + 0.858951i 0.671117π0.671117\pi
230230 −83791.4 −0.104443
231231 254982. 0.314398
232232 14133.1i 0.0172392i
233233 931926. 1.12458 0.562292 0.826939i 0.309920π-0.309920\pi
0.562292 + 0.826939i 0.309920π0.309920\pi
234234 −183154. + 73698.3i −0.218664 + 0.0879869i
235235 245215. 0.289653
236236 401515.i 0.469269i
237237 −140583. −0.162578
238238 699998. 0.801041
239239 885874.i 1.00318i −0.865107 0.501588i 0.832749π-0.832749\pi
0.865107 0.501588i 0.167251π-0.167251\pi
240240 22436.4i 0.0251435i
241241 1.32690e6i 1.47161i 0.677191 + 0.735807i 0.263197π0.263197\pi
−0.677191 + 0.735807i 0.736803π0.736803\pi
242242 354006.i 0.388572i
243243 −59049.0 −0.0641500
244244 −192973. −0.207501
245245 55928.5i 0.0595275i
246246 −533148. −0.561708
247247 −1598.32 + 643.137i −0.00166694 + 0.000670751i
248248 50438.8 0.0520757
249249 25378.1i 0.0259394i
250250 −239757. −0.242617
251251 −547596. −0.548625 −0.274313 0.961641i 0.588450π-0.588450\pi
−0.274313 + 0.961641i 0.588450π0.588450\pi
252252 136319.i 0.135224i
253253 579409.i 0.569094i
254254 572599.i 0.556886i
255255 145814.i 0.140427i
256256 65536.0 0.0625000
257257 −1.65509e6 −1.56311 −0.781554 0.623837i 0.785573π-0.785573\pi
−0.781554 + 0.623837i 0.785573π0.785573\pi
258258 509119.i 0.476179i
259259 103127. 0.0955265
260260 88077.0 35440.8i 0.0808033 0.0325140i
261261 17887.2 0.0162533
262262 565317.i 0.508791i
263263 −668188. −0.595675 −0.297838 0.954617i 0.596265π-0.596265\pi
−0.297838 + 0.954617i 0.596265π0.596265\pi
264264 155146. 0.137003
265265 300914.i 0.263226i
266266 1189.60i 0.00103085i
267267 146045.i 0.125374i
268268 231209.i 0.196638i
269269 −384942. −0.324350 −0.162175 0.986762i 0.551851π-0.551851\pi
−0.162175 + 0.986762i 0.551851π0.551851\pi
270270 28396.1 0.0237055
271271 1.05344e6i 0.871334i −0.900108 0.435667i 0.856513π-0.856513\pi
0.900108 0.435667i 0.143487π-0.143487\pi
272272 425919. 0.349064
273273 −535136. + 215330.i −0.434568 + 0.174863i
274274 15833.3 0.0127408
275275 816177.i 0.650808i
276276 309764. 0.244770
277277 −1.62051e6 −1.26897 −0.634485 0.772936i 0.718788π-0.718788\pi
−0.634485 + 0.772936i 0.718788π0.718788\pi
278278 1.78231e6i 1.38316i
279279 63836.6i 0.0490975i
280280 65554.3i 0.0499696i
281281 563009.i 0.425353i 0.977123 + 0.212677i 0.0682181π0.0682181\pi
−0.977123 + 0.212677i 0.931782π0.931782\pi
282282 −906522. −0.678822
283283 361892. 0.268604 0.134302 0.990940i 0.457121π-0.457121\pi
0.134302 + 0.990940i 0.457121π0.457121\pi
284284 536760.i 0.394897i
285285 247.802 0.000180714
286286 245070. + 609045.i 0.177164 + 0.440285i
287287 −1.55774e6 −1.11633
288288 82944.0i 0.0589256i
289289 1.34819e6 0.949526
290290 −8601.77 −0.00600610
291291 1.13660e6i 0.786820i
292292 412891.i 0.283386i
293293 757318.i 0.515358i 0.966231 + 0.257679i 0.0829577π0.0829577\pi
−0.966231 + 0.257679i 0.917042π0.917042\pi
294294 206759.i 0.139507i
295295 244373. 0.163493
296296 62748.5 0.0416269
297297 196356.i 0.129168i
298298 −198477. −0.129470
299299 489306. + 1.21602e6i 0.316521 + 0.786614i
300300 436345. 0.279915
301301 1.48753e6i 0.946348i
302302 −1.60929e6 −1.01535
303303 −831589. −0.520358
304304 723.821i 0.000449207i
305305 117448.i 0.0722932i
306306 539053.i 0.329100i
307307 1.96280e6i 1.18858i 0.804249 + 0.594292i 0.202568π0.202568\pi
−0.804249 + 0.594292i 0.797432π0.797432\pi
308308 453302. 0.272277
309309 −1.33282e6 −0.794100
310310 30698.4i 0.0181431i
311311 834419. 0.489197 0.244598 0.969624i 0.421344π-0.421344\pi
0.244598 + 0.969624i 0.421344π0.421344\pi
312312 −325607. + 131019.i −0.189368 + 0.0761989i
313313 1.61834e6 0.933703 0.466851 0.884336i 0.345388π-0.345388\pi
0.466851 + 0.884336i 0.345388π0.345388\pi
314314 2.24972e6i 1.28767i
315315 82967.2 0.0471118
316316 −249925. −0.140797
317317 660304.i 0.369059i −0.982827 0.184529i 0.940924π-0.940924\pi
0.982827 0.184529i 0.0590761π-0.0590761\pi
318318 1.11243e6i 0.616888i
319319 59480.4i 0.0327263i
320320 39887.0i 0.0217749i
321321 −793621. −0.429883
322322 905062. 0.486450
323323 4704.11i 0.00250883i
324324 −104976. −0.0555556
325325 689255. + 1.71293e6i 0.361969 + 0.899560i
326326 1.25551e6 0.654299
327327 553205.i 0.286099i
328328 −947819. −0.486453
329329 −2.64866e6 −1.34908
330330 94425.9i 0.0477316i
331331 2.73497e6i 1.37209i 0.727558 + 0.686046i 0.240655π0.240655\pi
−0.727558 + 0.686046i 0.759345π0.759345\pi
332332 45116.5i 0.0224642i
333333 79416.1i 0.0392462i
334334 2.40891e6 1.18156
335335 140720. 0.0685083
336336 242344.i 0.117107i
337337 −880707. −0.422432 −0.211216 0.977439i 0.567742π-0.567742\pi
−0.211216 + 0.977439i 0.567742π0.567742\pi
338338 −1.02867e6 1.07125e6i −0.489759 0.510035i
339339 740305. 0.349874
340340 259226.i 0.121613i
341341 212277. 0.0988590
342342 −916.085 −0.000423517
343343 2.37193e6i 1.08860i
344344 905101.i 0.412383i
345345 188531.i 0.0852774i
346346 1.21608e6i 0.546102i
347347 −33079.1 −0.0147479 −0.00737395 0.999973i 0.502347π-0.502347\pi
−0.00737395 + 0.999973i 0.502347π0.502347\pi
348348 31799.4 0.0140757
349349 4.06448e6i 1.78624i 0.449814 + 0.893122i 0.351490π0.351490\pi
−0.449814 + 0.893122i 0.648510π0.648510\pi
350350 1.27490e6 0.556297
351351 −165821. 412097.i −0.0718410 0.178538i
352352 275815. 0.118648
353353 3.57770e6i 1.52815i −0.645125 0.764077i 0.723195π-0.723195\pi
0.645125 0.764077i 0.276805π-0.276805\pi
354354 −903409. −0.383157
355355 −326686. −0.137582
356356 259635.i 0.108577i
357357 1.57500e6i 0.654047i
358358 1.62476e6i 0.670011i
359359 3.31455e6i 1.35734i 0.734444 + 0.678669i 0.237443π0.237443\pi
−0.734444 + 0.678669i 0.762557π0.762557\pi
360360 50482.0 0.0205296
361361 2.47609e6 0.999997
362362 963867.i 0.386585i
363363 −796513. −0.317268
364364 −951353. + 382809.i −0.376347 + 0.151436i
365365 −251297. −0.0987313
366366 434188.i 0.169424i
367367 192406. 0.0745681 0.0372840 0.999305i 0.488129π-0.488129\pi
0.0372840 + 0.999305i 0.488129π0.488129\pi
368368 550691. 0.211977
369369 1.19958e6i 0.458632i
370370 38190.4i 0.0145027i
371371 3.25029e6i 1.22599i
372372 113487.i 0.0425197i
373373 1.60161e6 0.596053 0.298027 0.954558i 0.403672π-0.403672\pi
0.298027 + 0.954558i 0.403672π0.403672\pi
374374 1.79252e6 0.662652
375375 539453.i 0.198096i
376376 −1.61160e6 −0.587877
377377 50230.7 + 124833.i 0.0182019 + 0.0452350i
378378 −306717. −0.110410
379379 3.76006e6i 1.34461i −0.740274 0.672305i 0.765304π-0.765304\pi
0.740274 0.672305i 0.234696π-0.234696\pi
380380 440.537 0.000156503
381381 1.28835e6 0.454695
382382 904560.i 0.317160i
383383 4.69693e6i 1.63613i 0.575128 + 0.818063i 0.304952π0.304952\pi
−0.575128 + 0.818063i 0.695048π0.695048\pi
384384 147456.i 0.0510310i
385385 275892.i 0.0948608i
386386 1.62430e6 0.554879
387387 1.14552e6 0.388799
388388 2.02062e6i 0.681406i
389389 4.85881e6 1.62801 0.814003 0.580860i 0.197284π-0.197284\pi
0.814003 + 0.580860i 0.197284π0.197284\pi
390390 79741.8 + 198173.i 0.0265476 + 0.0659757i
391391 3.57894e6 1.18389
392392 367572.i 0.120817i
393393 1.27196e6 0.415426
394394 1.62956e6 0.528846
395395 152111.i 0.0490533i
396396 349078.i 0.111863i
397397 3.91926e6i 1.24804i −0.781409 0.624019i 0.785499π-0.785499\pi
0.781409 0.624019i 0.214501π-0.214501\pi
398398 811460.i 0.256779i
399399 −2676.60 −0.000841688
400400 775724. 0.242414
401401 5.01898e6i 1.55867i −0.626606 0.779336i 0.715557π-0.715557\pi
0.626606 0.779336i 0.284443π-0.284443\pi
402402 −520219. −0.160554
403403 −445509. + 179266.i −0.136645 + 0.0549838i
404404 −1.47838e6 −0.450643
405405 63891.2i 0.0193555i
406406 92910.9 0.0279738
407407 264083. 0.0790232
408408 958317.i 0.285009i
409409 2.97886e6i 0.880526i −0.897869 0.440263i 0.854885π-0.854885\pi
0.897869 0.440263i 0.145115π-0.145115\pi
410410 576868.i 0.169480i
411411 35625.0i 0.0104028i
412412 −2.36946e6 −0.687711
413413 −2.63956e6 −0.761477
414414 696968.i 0.199854i
415415 27459.1 0.00782648
416416 −578857. + 232923.i −0.163998 + 0.0659902i
417417 4.01021e6 1.12935
418418 3046.27i 0.000852762i
419419 −4.76887e6 −1.32703 −0.663515 0.748163i 0.730936π-0.730936\pi
−0.663515 + 0.748163i 0.730936π0.730936\pi
420420 147497. 0.0408000
421421 651294.i 0.179090i 0.995983 + 0.0895451i 0.0285413π0.0285413\pi
−0.995983 + 0.0895451i 0.971459π0.971459\pi
422422 2.63188e6i 0.719424i
423423 2.03968e6i 0.554256i
424424 1.97766e6i 0.534241i
425425 5.04143e6 1.35388
426426 1.20771e6 0.322432
427427 1.26860e6i 0.336710i
428428 −1.41088e6 −0.372290
429429 −1.37035e6 + 551407.i −0.359491 + 0.144654i
430430 −550869. −0.143674
431431 5.22621e6i 1.35517i 0.735444 + 0.677585i 0.236973π0.236973\pi
−0.735444 + 0.677585i 0.763027π0.763027\pi
432432 −186624. −0.0481125
433433 −6.33506e6 −1.62379 −0.811897 0.583801i 0.801565π-0.801565\pi
−0.811897 + 0.583801i 0.801565π0.801565\pi
434434 331585.i 0.0845026i
435435 19354.0i 0.00490396i
436436 983475.i 0.247769i
437437 6082.18i 0.00152355i
438438 929005. 0.231384
439439 −7.44321e6 −1.84331 −0.921657 0.388006i 0.873164π-0.873164\pi
−0.921657 + 0.388006i 0.873164π0.873164\pi
440440 167868.i 0.0413368i
441441 465208. 0.113907
442442 −3.76200e6 + 1.51377e6i −0.915931 + 0.368556i
443443 327031. 0.0791734 0.0395867 0.999216i 0.487396π-0.487396\pi
0.0395867 + 0.999216i 0.487396π0.487396\pi
444444 141184.i 0.0339882i
445445 −158021. −0.0378281
446446 3.91224e6 0.931297
447447 446574.i 0.105712i
448448 430834.i 0.101418i
449449 5.82118e6i 1.36268i −0.731965 0.681342i 0.761396π-0.761396\pi
0.731965 0.681342i 0.238604π-0.238604\pi
450450 981775.i 0.228550i
451451 −3.98899e6 −0.923468
452452 1.31610e6 0.303000
453453 3.62091e6i 0.829033i
454454 −3.24855e6 −0.739690
455455 232988. + 579019.i 0.0527601 + 0.131119i
456456 −1628.60 −0.000366776
457457 8.15788e6i 1.82720i −0.406612 0.913601i 0.633290π-0.633290\pi
0.406612 0.913601i 0.366710π-0.366710\pi
458458 −5.45315e6 −1.21474
459459 −1.21287e6 −0.268709
460460 335165.i 0.0738524i
461461 3.15011e6i 0.690357i 0.938537 + 0.345179i 0.112182π0.112182\pi
−0.938537 + 0.345179i 0.887818π0.887818\pi
462462 1.01993e6i 0.222313i
463463 202622.i 0.0439273i 0.999759 + 0.0219637i 0.00699182π0.00699182\pi
−0.999759 + 0.0219637i 0.993008π0.993008\pi
464464 56532.3 0.0121899
465465 69071.4 0.0148138
466466 3.72771e6i 0.795201i
467467 91749.8 0.0194676 0.00973382 0.999953i 0.496902π-0.496902\pi
0.00973382 + 0.999953i 0.496902π0.496902\pi
468468 −294793. 732616.i −0.0622161 0.154619i
469469 −1.51997e6 −0.319082
470470 980860.i 0.204815i
471471 5.06186e6 1.05138
472472 −1.60606e6 −0.331823
473473 3.80921e6i 0.782856i
474474 562331.i 0.114960i
475475 8567.58i 0.00174230i
476476 2.79999e6i 0.566421i
477477 −2.50298e6 −0.503687
478478 3.54350e6 0.709353
479479 2.82910e6i 0.563390i 0.959504 + 0.281695i 0.0908967π0.0908967\pi
−0.959504 + 0.281695i 0.909103π0.909103\pi
480480 89745.7 0.0177791
481481 −554236. + 223016.i −0.109228 + 0.0439514i
482482 −5.30758e6 −1.04059
483483 2.03639e6i 0.397185i
484484 −1.41602e6 −0.274762
485485 −1.22981e6 −0.237401
486486 236196.i 0.0453609i
487487 1.30176e6i 0.248719i −0.992237 0.124359i 0.960312π-0.960312\pi
0.992237 0.124359i 0.0396876π-0.0396876\pi
488488 771890.i 0.146726i
489489 2.82490e6i 0.534233i
490490 −223714. −0.0420923
491491 −6.07696e6 −1.13758 −0.568791 0.822482i 0.692588π-0.692588\pi
−0.568791 + 0.822482i 0.692588π0.692588\pi
492492 2.13259e6i 0.397187i
493493 367404. 0.0680810
494494 −2572.55 6393.26i −0.000474292 0.00117870i
495495 212458. 0.0389727
496496 201755.i 0.0368231i
497497 3.52866e6 0.640795
498498 −101512. −0.0183419
499499 3.78708e6i 0.680853i 0.940271 + 0.340426i 0.110571π0.110571\pi
−0.940271 + 0.340426i 0.889429π0.889429\pi
500500 959028.i 0.171556i
501501 5.42005e6i 0.964737i
502502 2.19038e6i 0.387937i
503503 2.28832e6 0.403271 0.201636 0.979461i 0.435374π-0.435374\pi
0.201636 + 0.979461i 0.435374π0.435374\pi
504504 −545274. −0.0956178
505505 899783.i 0.157003i
506506 2.31764e6 0.402410
507507 2.41032e6 2.31450e6i 0.416442 0.399886i
508508 2.29040e6 0.393778
509509 3.70029e6i 0.633055i −0.948583 0.316528i 0.897483π-0.897483\pi
0.948583 0.316528i 0.102517π-0.102517\pi
510510 583258. 0.0992968
511511 2.71435e6 0.459847
512512 262144.i 0.0441942i
513513 2061.19i 0.000345800i
514514 6.62037e6i 1.10528i
515515 1.44212e6i 0.239597i
516516 2.03648e6 0.336709
517517 −6.78256e6 −1.11601
518518 412509.i 0.0675474i
519519 −2.73619e6 −0.445890
520520 141763. + 352308.i 0.0229909 + 0.0571366i
521521 6.13756e6 0.990608 0.495304 0.868720i 0.335057π-0.335057\pi
0.495304 + 0.868720i 0.335057π0.335057\pi
522522 71548.7i 0.0114928i
523523 4.10572e6 0.656349 0.328175 0.944617i 0.393567π-0.393567\pi
0.328175 + 0.944617i 0.393567π0.393567\pi
524524 2.26127e6 0.359769
525525 2.86853e6i 0.454215i
526526 2.67275e6i 0.421206i
527527 1.31121e6i 0.205658i
528528 620583.i 0.0968758i
529529 −1.80895e6 −0.281053
530530 1.20366e6 0.186129
531531 2.03267e6i 0.312846i
532532 −4758.40 −0.000728923
533533 8.37177e6 3.36867e6i 1.27644 0.513618i
534534 584178. 0.0886528
535535 858701.i 0.129705i
536536 −924834. −0.139044
537537 −3.65571e6 −0.547062
538538 1.53977e6i 0.229350i
539539 1.54696e6i 0.229355i
540540 113584.i 0.0167623i
541541 2.51266e6i 0.369098i 0.982823 + 0.184549i 0.0590824π0.0590824\pi
−0.982823 + 0.184549i 0.940918π0.940918\pi
542542 4.21374e6 0.616126
543543 2.16870e6 0.315646
544544 1.70368e6i 0.246825i
545545 −598569. −0.0863223
546546 −861321. 2.14054e6i −0.123647 0.307286i
547547 −2.29454e6 −0.327889 −0.163945 0.986470i 0.552422π-0.552422\pi
−0.163945 + 0.986470i 0.552422π0.552422\pi
548548 63333.4i 0.00900909i
549549 976924. 0.138334
550550 3.26471e6 0.460191
551551 624.378i 8.76130e-5i
552552 1.23905e6i 0.173078i
553553 1.64301e6i 0.228469i
554554 6.48202e6i 0.897297i
555555 85928.4 0.0118414
556556 7.12926e6 0.978042
557557 5.96324e6i 0.814412i −0.913336 0.407206i 0.866503π-0.866503\pi
0.913336 0.407206i 0.133497π-0.133497\pi
558558 −255346. −0.0347171
559559 3.21684e6 + 7.99445e6i 0.435412 + 1.08208i
560560 262217. 0.0353339
561561 4.03317e6i 0.541053i
562562 −2.25204e6 −0.300770
563563 −7.37754e6 −0.980935 −0.490468 0.871459i 0.663174π-0.663174\pi
−0.490468 + 0.871459i 0.663174π0.663174\pi
564564 3.62609e6i 0.479999i
565565 801012.i 0.105565i
566566 1.44757e6i 0.189932i
567567 690113.i 0.0901493i
568568 2.14704e6 0.279235
569569 2.60845e6 0.337756 0.168878 0.985637i 0.445986π-0.445986\pi
0.168878 + 0.985637i 0.445986π0.445986\pi
570570 991.208i 0.000127784i
571571 1.09867e7 1.41019 0.705095 0.709113i 0.250905π-0.250905\pi
0.705095 + 0.709113i 0.250905π0.250905\pi
572572 −2.43618e6 + 980280.i −0.311329 + 0.125274i
573573 −2.03526e6 −0.258960
574574 6.23097e6i 0.789362i
575575 6.51831e6 0.822177
576576 −331776. −0.0416667
577577 1.10048e7i 1.37607i 0.725677 + 0.688036i 0.241527π0.241527\pi
−0.725677 + 0.688036i 0.758473π0.758473\pi
578578 5.39276e6i 0.671416i
579579 3.65468e6i 0.453057i
580580 34407.1i 0.00424696i
581581 −296596. −0.0364523
582582 4.54640e6 0.556365
583583 8.32318e6i 1.01419i
584584 1.65157e6 0.200384
585585 −445890. + 179419.i −0.0538689 + 0.0216760i
586586 −3.02927e6 −0.364413
587587 1.09851e7i 1.31586i 0.753080 + 0.657929i 0.228567π0.228567\pi
−0.753080 + 0.657929i 0.771433π0.771433\pi
588588 827036. 0.0986464
589589 −2228.31 −0.000264659
590590 977492.i 0.115607i
591591 3.66651e6i 0.431801i
592592 250994.i 0.0294347i
593593 3.90620e6i 0.456161i −0.973642 0.228081i 0.926755π-0.926755\pi
0.973642 0.228081i 0.0732449π-0.0732449\pi
594594 −785426. −0.0913354
595595 1.70415e6 0.197340
596596 793908.i 0.0915492i
597597 −1.82578e6 −0.209659
598598 −4.86407e6 + 1.95722e6i −0.556220 + 0.223814i
599599 −1.69973e7 −1.93559 −0.967797 0.251733i 0.918999π-0.918999\pi
−0.967797 + 0.251733i 0.918999π0.918999\pi
600600 1.74538e6i 0.197930i
601601 1.07099e7 1.20948 0.604739 0.796424i 0.293278π-0.293278\pi
0.604739 + 0.796424i 0.293278π0.293278\pi
602602 5.95014e6 0.669169
603603 1.17049e6i 0.131092i
604604 6.43717e6i 0.717964i
605605 861829.i 0.0957267i
606606 3.32636e6i 0.367949i
607607 −1.43102e7 −1.57642 −0.788212 0.615404i 0.788993π-0.788993\pi
−0.788212 + 0.615404i 0.788993π0.788993\pi
608608 −2895.28 −0.000317638
609609 209050.i 0.0228405i
610610 −469793. −0.0511190
611611 1.42347e7 5.72781e6i 1.54257 0.620705i
612612 −2.15621e6 −0.232709
613613 1.67594e7i 1.80139i 0.434450 + 0.900696i 0.356943π0.356943\pi
−0.434450 + 0.900696i 0.643057π0.643057\pi
614614 −7.85120e6 −0.840456
615615 −1.29795e6 −0.138379
616616 1.81321e6i 0.192529i
617617 1.26793e7i 1.34085i −0.741976 0.670427i 0.766111π-0.766111\pi
0.741976 0.670427i 0.233889π-0.233889\pi
618618 5.33128e6i 0.561513i
619619 5.43928e6i 0.570577i −0.958442 0.285289i 0.907911π-0.907911\pi
0.958442 0.285289i 0.0920895π-0.0920895\pi
620620 122794. 0.0128291
621621 −1.56818e6 −0.163180
622622 3.33768e6i 0.345914i
623623 1.70684e6 0.176187
624624 −524077. 1.30243e6i −0.0538807 0.133904i
625625 8.88559e6 0.909885
626626 6.47335e6i 0.660227i
627627 −6854.11 −0.000696277
628628 8.99887e6 0.910519
629629 1.63121e6i 0.164393i
630630 331869.i 0.0333131i
631631 5.28439e6i 0.528350i −0.964475 0.264175i 0.914900π-0.914900\pi
0.964475 0.264175i 0.0850996π-0.0850996\pi
632632 999700.i 0.0995582i
633633 −5.92173e6 −0.587407
634634 2.64121e6 0.260964
635635 1.39400e6i 0.137192i
636636 −4.44973e6 −0.436206
637637 1.30639e6 + 3.24664e6i 0.127563 + 0.317019i
638638 237922. 0.0231410
639639 2.71735e6i 0.263265i
640640 159548. 0.0153972
641641 8.01855e6 0.770816 0.385408 0.922746i 0.374061π-0.374061\pi
0.385408 + 0.922746i 0.374061π0.374061\pi
642642 3.17448e6i 0.303973i
643643 1.39227e7i 1.32799i −0.747735 0.663997i 0.768859π-0.768859\pi
0.747735 0.663997i 0.231141π-0.231141\pi
644644 3.62025e6i 0.343972i
645645 1.23945e6i 0.117309i
646646 −18816.5 −0.00177401
647647 −9.55517e6 −0.897383 −0.448692 0.893687i 0.648110π-0.648110\pi
−0.448692 + 0.893687i 0.648110π0.648110\pi
648648 419904.i 0.0392837i
649649 −6.75926e6 −0.629923
650650 −6.85170e6 + 2.75702e6i −0.636085 + 0.255951i
651651 −746066. −0.0689961
652652 5.02204e6i 0.462659i
653653 −1.87340e7 −1.71928 −0.859640 0.510901i 0.829312π-0.829312\pi
−0.859640 + 0.510901i 0.829312π0.829312\pi
654654 2.21282e6 0.202303
655655 1.37627e6i 0.125343i
656656 3.79128e6i 0.343974i
657657 2.09026e6i 0.188924i
658658 1.05946e7i 0.953941i
659659 −3.39787e6 −0.304785 −0.152392 0.988320i 0.548698π-0.548698\pi
−0.152392 + 0.988320i 0.548698π0.548698\pi
660660 377704. 0.0337514
661661 1.81823e7i 1.61862i 0.587382 + 0.809310i 0.300159π0.300159\pi
−0.587382 + 0.809310i 0.699841π0.699841\pi
662662 −1.09399e7 −0.970215
663663 −3.40598e6 8.46449e6i −0.300925 0.747854i
664664 −180466. −0.0158846
665665 2896.09i 0.000253956i
666666 −317664. −0.0277513
667667 475034. 0.0413438
668668 9.63564e6i 0.835487i
669669 8.80254e6i 0.760401i
670670 562879.i 0.0484427i
671671 3.24858e6i 0.278540i
672672 −969377. −0.0828074
673673 686844. 0.0584548 0.0292274 0.999573i 0.490695π-0.490695\pi
0.0292274 + 0.999573i 0.490695π0.490695\pi
674674 3.52283e6i 0.298704i
675675 −2.20899e6 −0.186610
676676 4.28501e6 4.11466e6i 0.360650 0.346312i
677677 −6.64858e6 −0.557516 −0.278758 0.960361i 0.589923π-0.589923\pi
−0.278758 + 0.960361i 0.589923π0.589923\pi
678678 2.96122e6i 0.247398i
679679 1.32836e7 1.10571
680680 1.03690e6 0.0859935
681681 7.30923e6i 0.603954i
682682 849106.i 0.0699038i
683683 1.66853e7i 1.36862i 0.729193 + 0.684308i 0.239896π0.239896\pi
−0.729193 + 0.684308i 0.760104π0.760104\pi
684684 3664.34i 0.000299472i
685685 38546.4 0.00313875
686686 9.48773e6 0.769754
687687 1.22696e7i 0.991831i
688688 3.62040e6 0.291599
689689 −7.02885e6 1.74680e7i −0.564074 1.40183i
690690 754122. 0.0603002
691691 1.75290e7i 1.39657i −0.715822 0.698283i 0.753948π-0.753948\pi
0.715822 0.698283i 0.246052π-0.246052\pi
692692 −4.86434e6 −0.386152
693693 −2.29484e6 −0.181518
694694 132317.i 0.0104283i
695695 4.33906e6i 0.340748i
696696 127198.i 0.00995305i
697697 2.46395e7i 1.92110i
698698 −1.62579e7 −1.26307
699699 −8.38734e6 −0.649279
700700 5.09961e6i 0.393362i
701701 7.06379e6 0.542929 0.271464 0.962449i 0.412492π-0.412492\pi
0.271464 + 0.962449i 0.412492π0.412492\pi
702702 1.64839e6 663285.i 0.126246 0.0507993i
703703 −2772.13 −0.000211556
704704 1.10326e6i 0.0838969i
705705 −2.20694e6 −0.167231
706706 1.43108e7 1.08057
707707 9.71889e6i 0.731253i
708708 3.61364e6i 0.270933i
709709 9.07731e6i 0.678175i 0.940755 + 0.339087i 0.110118π0.110118\pi
−0.940755 + 0.339087i 0.889882π0.889882\pi
710710 1.30675e6i 0.0972849i
711711 1.26525e6 0.0938644
712712 1.03854e6 0.0767755
713713 1.69532e6i 0.124890i
714714 −6.29998e6 −0.462481
715715 596625. + 1.48272e6i 0.0436452 + 0.108466i
716716 −6.49904e6 −0.473769
717717 7.97287e6i 0.579184i
718718 −1.32582e7 −0.959783
719719 −1.03755e7 −0.748492 −0.374246 0.927330i 0.622098π-0.622098\pi
−0.374246 + 0.927330i 0.622098π0.622098\pi
720720 201928.i 0.0145166i
721721 1.55768e7i 1.11594i
722722 9.90436e6i 0.707104i
723723 1.19421e7i 0.849637i
724724 3.85547e6 0.273357
725725 669150. 0.0472801
726726 3.18605e6i 0.224342i
727727 −2.94915e6 −0.206948 −0.103474 0.994632i 0.532996π-0.532996\pi
−0.103474 + 0.994632i 0.532996π0.532996\pi
728728 −1.53124e6 3.80541e6i −0.107081 0.266117i
729729 531441. 0.0370370
730730 1.00519e6i 0.0698136i
731731 2.35290e7 1.62858
732732 1.73675e6 0.119801
733733 1.71605e7i 1.17970i 0.807513 + 0.589849i 0.200813π0.200813\pi
−0.807513 + 0.589849i 0.799187π0.799187\pi
734734 769623.i 0.0527276i
735735 503356.i 0.0343682i
736736 2.20276e6i 0.149890i
737737 −3.89226e6 −0.263957
738738 4.79834e6 0.324302
739739 2.20849e7i 1.48760i 0.668405 + 0.743798i 0.266977π0.266977\pi
−0.668405 + 0.743798i 0.733023π0.733023\pi
740740 152762. 0.0102550
741741 14384.8 5788.23i 0.000962409 0.000387258i
742742 −1.30011e7 −0.866906
743743 2.21459e7i 1.47171i 0.677140 + 0.735854i 0.263219π0.263219\pi
−0.677140 + 0.735854i 0.736781π0.736781\pi
744744 −453949. −0.0300659
745745 −483194. −0.0318956
746746 6.40644e6i 0.421473i
747747 228402.i 0.0149761i
748748 7.17008e6i 0.468565i
749749 9.27514e6i 0.604110i
750750 2.15781e6 0.140075
751751 6.38710e6 0.413242 0.206621 0.978421i 0.433753π-0.433753\pi
0.206621 + 0.978421i 0.433753π0.433753\pi
752752 6.44638e6i 0.415692i
753753 4.92836e6 0.316749
754754 −499331. + 200923.i −0.0319860 + 0.0128707i
755755 −3.91783e6 −0.250137
756756 1.22687e6i 0.0780716i
757757 −2.47266e6 −0.156828 −0.0784141 0.996921i 0.524986π-0.524986\pi
−0.0784141 + 0.996921i 0.524986π0.524986\pi
758758 1.50402e7 0.950783
759759 5.21469e6i 0.328567i
760760 1762.15i 0.000110664i
761761 1.59009e7i 0.995315i −0.867374 0.497657i 0.834194π-0.834194\pi
0.867374 0.497657i 0.165806π-0.165806\pi
762762 5.15339e6i 0.321518i
763763 6.46537e6 0.402052
764764 −3.61824e6 −0.224266
765765 1.31233e6i 0.0810755i
766766 −1.87877e7 −1.15692
767767 1.41858e7 5.70814e6i 0.870693 0.350353i
768768 −589824. −0.0360844
769769 4.80537e6i 0.293029i 0.989209 + 0.146515i 0.0468055π0.0468055\pi
−0.989209 + 0.146515i 0.953194π0.953194\pi
770770 1.10357e6 0.0670767
771771 1.48958e7 0.902461
772772 6.49720e6i 0.392359i
773773 2.22008e7i 1.33635i 0.744004 + 0.668175i 0.232924π0.232924\pi
−0.744004 + 0.668175i 0.767076π0.767076\pi
774774 4.58207e6i 0.274922i
775775 2.38810e6i 0.142823i
776776 8.08249e6 0.481827
777777 −928145. −0.0551523
778778 1.94353e7i 1.15117i
779779 41873.2 0.00247225
780780 −792693. + 318967.i −0.0466518 + 0.0187720i
781781 9.03603e6 0.530090
782782 1.43158e7i 0.837140i
783783 −160985. −0.00938382
784784 1.47029e6 0.0854303
785785 5.47695e6i 0.317223i
786786 5.08786e6i 0.293750i
787787 2.85397e7i 1.64253i −0.570548 0.821264i 0.693269π-0.693269\pi
0.570548 0.821264i 0.306731π-0.306731\pi
788788 6.51824e6i 0.373951i
789789 6.01370e6 0.343913
790790 −608444. −0.0346859
791791 8.65203e6i 0.491674i
792792 −1.39631e6 −0.0790987
793793 2.74339e6 + 6.81785e6i 0.154919 + 0.385003i
794794 1.56770e7 0.882496
795795 2.70823e6i 0.151973i
796796 −3.24584e6 −0.181570
797797 −2.10013e7 −1.17112 −0.585558 0.810630i 0.699125π-0.699125\pi
−0.585558 + 0.810630i 0.699125π0.699125\pi
798798 10706.4i 0.000595163i
799799 4.18951e7i 2.32165i
800800 3.10289e6i 0.171412i
801801 1.31440e6i 0.0723847i
802802 2.00759e7 1.10215
803803 6.95078e6 0.380403
804804 2.08088e6i 0.113529i
805805 2.20338e6 0.119839
806806 −717063. 1.78203e6i −0.0388794 0.0966225i
807807 3.46448e6 0.187264
808808 5.91352e6i 0.318653i
809809 3.59907e7 1.93339 0.966694 0.255935i 0.0823834π-0.0823834\pi
0.966694 + 0.255935i 0.0823834π0.0823834\pi
810810 −255565. −0.0136864
811811 3.20415e7i 1.71065i 0.518093 + 0.855324i 0.326642π0.326642\pi
−0.518093 + 0.855324i 0.673358π0.673358\pi
812812 371644.i 0.0197805i
813813 9.48092e6i 0.503065i
814814 1.05633e6i 0.0558779i
815815 3.05655e6 0.161190
816816 −3.83327e6 −0.201532
817817 39986.0i 0.00209581i
818818 1.19154e7 0.622626
819819 4.81622e6 1.93797e6i 0.250898 0.100957i
820820 −2.30747e6 −0.119840
821821 2.92849e7i 1.51630i 0.652079 + 0.758151i 0.273897π0.273897\pi
−0.652079 + 0.758151i 0.726103π0.726103\pi
822822 −142500. −0.00735589
823823 −1.27788e6 −0.0657646 −0.0328823 0.999459i 0.510469π-0.510469\pi
−0.0328823 + 0.999459i 0.510469π0.510469\pi
824824 9.47783e6i 0.486285i
825825 7.34560e6i 0.375744i
826826 1.05583e7i 0.538446i
827827 3.34508e7i 1.70076i −0.526169 0.850380i 0.676372π-0.676372\pi
0.526169 0.850380i 0.323628π-0.323628\pi
828828 −2.78787e6 −0.141318
829829 −1.65727e7 −0.837541 −0.418770 0.908092i 0.637539π-0.637539\pi
−0.418770 + 0.908092i 0.637539π0.637539\pi
830830 109837.i 0.00553416i
831831 1.45845e7 0.732640
832832 −931692. 2.31543e6i −0.0466621 0.115964i
833833 9.55540e6 0.477129
834834 1.60408e7i 0.798568i
835835 5.86451e6 0.291082
836836 −12185.1 −0.000602994
837837 574529.i 0.0283464i
838838 1.90755e7i 0.938352i
839839 7.23411e6i 0.354797i −0.984139 0.177399i 0.943232π-0.943232\pi
0.984139 0.177399i 0.0567682π-0.0567682\pi
840840 589989.i 0.0288500i
841841 −2.04624e7 −0.997622
842842 −2.60518e6 −0.126636
843843 5.06708e6i 0.245578i
844844 −1.05275e7 −0.508710
845845 −2.50429e6 2.60797e6i −0.120654 0.125650i
846846 8.15870e6 0.391918
847847 9.30894e6i 0.445853i
848848 −7.91064e6 −0.377765
849849 −3.25703e6 −0.155079
850850 2.01657e7i 0.957341i
851851 2.10907e6i 0.0998315i
852852 4.83084e6i 0.227994i
853853 3.08382e6i 0.145116i 0.997364 + 0.0725582i 0.0231163π0.0231163\pi
−0.997364 + 0.0725582i 0.976884π0.976884\pi
854854 5.07441e6 0.238090
855855 −2230.22 −0.000104335
856856 5.64353e6i 0.263249i
857857 9.15615e6 0.425854 0.212927 0.977068i 0.431700π-0.431700\pi
0.212927 + 0.977068i 0.431700π0.431700\pi
858858 −2.20563e6 5.48140e6i −0.102286 0.254199i
859859 −3.40397e7 −1.57399 −0.786996 0.616958i 0.788365π-0.788365\pi
−0.786996 + 0.616958i 0.788365π0.788365\pi
860860 2.20348e6i 0.101593i
861861 1.40197e7 0.644511
862862 −2.09048e7 −0.958250
863863 1.92683e7i 0.880676i 0.897832 + 0.440338i 0.145141π0.145141\pi
−0.897832 + 0.440338i 0.854859π0.854859\pi
864864 746496.i 0.0340207i
865865 2.96057e6i 0.134535i
866866 2.53402e7i 1.14820i
867867 −1.21337e7 −0.548209
868868 −1.32634e6 −0.0597524
869869 4.20734e6i 0.188998i
870870 77415.9 0.00346763
871871 8.16875e6 3.28698e6i 0.364846 0.146808i
872872 3.93390e6 0.175199
873873 1.02294e7i 0.454270i
874874 −24328.7 −0.00107731
875875 6.30466e6 0.278382
876876 3.71602e6i 0.163613i
877877 2.63479e7i 1.15677i −0.815765 0.578384i 0.803684π-0.803684\pi
0.815765 0.578384i 0.196316π-0.196316\pi
878878 2.97729e7i 1.30342i
879879 6.81586e6i 0.297542i
880880 671473. 0.0292295
881881 4.12657e6 0.179122 0.0895610 0.995981i 0.471454π-0.471454\pi
0.0895610 + 0.995981i 0.471454π0.471454\pi
882882 1.86083e6i 0.0805444i
883883 1.36506e6 0.0589182 0.0294591 0.999566i 0.490622π-0.490622\pi
0.0294591 + 0.999566i 0.490622π0.490622\pi
884884 −6.05507e6 1.50480e7i −0.260609 0.647661i
885885 −2.19936e6 −0.0943925
886886 1.30812e6i 0.0559841i
887887 −2.77026e7 −1.18225 −0.591127 0.806578i 0.701317π-0.701317\pi
−0.591127 + 0.806578i 0.701317π0.701317\pi
888888 −564736. −0.0240333
889889 1.50571e7i 0.638978i
890890 632083.i 0.0267485i
891891 1.76721e6i 0.0745750i
892892 1.56490e7i 0.658527i
893893 71197.9 0.00298771
894894 1.78629e6 0.0747496
895895 3.95549e6i 0.165060i
896896 −1.72334e6 −0.0717133
897897 −4.40376e6 1.09442e7i −0.182744 0.454152i
898898 2.32847e7 0.963564
899899 174037.i 0.00718194i
900900 −3.92710e6 −0.161609
901901 −5.14113e7 −2.10983
902902 1.59560e7i 0.652990i
903903 1.33878e7i 0.546374i
904904 5.26439e6i 0.214253i
905905 2.34654e6i 0.0952372i
906906 1.44836e7 0.586215
907907 −2.08574e7 −0.841863 −0.420931 0.907093i 0.638297π-0.638297\pi
−0.420931 + 0.907093i 0.638297π0.638297\pi
908908 1.29942e7i 0.523040i
909909 7.48430e6 0.300429
910910 −2.31608e6 + 931953.i −0.0927149 + 0.0373070i
911911 3.95317e6 0.157815 0.0789077 0.996882i 0.474857π-0.474857\pi
0.0789077 + 0.996882i 0.474857π0.474857\pi
912912 6514.38i 0.000259350i
913913 −759510. −0.0301548
914914 3.26315e7 1.29203
915915 1.05703e6i 0.0417385i
916916 2.18126e7i 0.858951i
917917 1.48656e7i 0.583793i
918918 4.85148e6i 0.190006i
919919 1.66130e7 0.648873 0.324436 0.945907i 0.394825π-0.394825\pi
0.324436 + 0.945907i 0.394825π0.394825\pi
920920 1.34066e6 0.0522215
921921 1.76652e7i 0.686230i
922922 −1.26005e7 −0.488156
923923 −1.89641e7 + 7.63084e6i −0.732702 + 0.294828i
924924 −4.07972e6 −0.157199
925925 2.97092e6i 0.114166i
926926 −810490. −0.0310613
927927 1.19954e7 0.458474
928928 226129.i 0.00861959i
929929 4.33406e7i 1.64762i 0.566869 + 0.823808i 0.308155π0.308155\pi
−0.566869 + 0.823808i 0.691845π0.691845\pi
930930 276286.i 0.0104749i
931931 16238.8i 0.000614014i
932932 −1.49108e7 −0.562292
933933 −7.50978e6 −0.282438
934934 366999.i 0.0137657i
935935 4.36391e6 0.163247
936936 2.93047e6 1.17917e6i 0.109332 0.0439934i
937937 −5.84680e6 −0.217555 −0.108778 0.994066i 0.534694π-0.534694\pi
−0.108778 + 0.994066i 0.534694π0.534694\pi
938938 6.07987e6i 0.225625i
939939 −1.45650e7 −0.539073
940940 −3.92344e6 −0.144826
941941 3.23340e7i 1.19038i 0.803585 + 0.595190i 0.202923π0.202923\pi
−0.803585 + 0.595190i 0.797077π0.797077\pi
942942 2.02475e7i 0.743435i
943943 3.18577e7i 1.16663i
944944 6.42424e6i 0.234634i
945945 −746705. −0.0272000
946946 1.52368e7 0.553563
947947 3.74163e6i 0.135577i −0.997700 0.0677884i 0.978406π-0.978406\pi
0.997700 0.0677884i 0.0215943π-0.0215943\pi
948948 2.24933e6 0.0812889
949949 −1.45877e7 + 5.86987e6i −0.525801 + 0.211574i
950950 −34270.3 −0.00123200
951951 5.94273e6i 0.213076i
952952 −1.12000e7 −0.400520
953953 −5.30640e7 −1.89264 −0.946319 0.323234i 0.895230π-0.895230\pi
−0.946319 + 0.323234i 0.895230π0.895230\pi
954954 1.00119e7i 0.356160i
955955 2.20216e6i 0.0781340i
956956 1.41740e7i 0.501588i
957957 535324.i 0.0188946i
958958 −1.13164e7 −0.398377
959959 −416354. −0.0146189
960960 358983.i 0.0125717i
961961 2.80080e7 0.978305
962962 −892064. 2.21694e6i −0.0310784 0.0772355i
963963 7.14259e6 0.248193
964964 2.12303e7i 0.735807i
965965 3.95437e6 0.136697
966966 −8.14555e6 −0.280852
967967 888502.i 0.0305557i 0.999883 + 0.0152779i 0.00486328π0.00486328\pi
−0.999883 + 0.0152779i 0.995137π0.995137\pi
968968 5.66409e6i 0.194286i
969969 42337.0i 0.00144847i
970970 4.91922e6i 0.167868i
971971 −1.92989e7 −0.656878 −0.328439 0.944525i 0.606523π-0.606523\pi
−0.328439 + 0.944525i 0.606523π0.606523\pi
972972 944784. 0.0320750
973973 4.68678e7i 1.58706i
974974 5.20704e6 0.175871
975975 −6.20329e6 1.54163e7i −0.208983 0.519361i
976976 3.08756e6 0.103751
977977 2.73358e7i 0.916212i 0.888898 + 0.458106i 0.151472π0.151472\pi
−0.888898 + 0.458106i 0.848528π0.848528\pi
978978 −1.12996e7 −0.377760
979979 4.37079e6 0.145748
980980 894856.i 0.0297638i
981981 4.97884e6i 0.165179i
982982 2.43078e7i 0.804391i
983983 2.36204e7i 0.779656i 0.920888 + 0.389828i 0.127466π0.127466\pi
−0.920888 + 0.389828i 0.872534π0.872534\pi
984984 8.53038e6 0.280854
985985 3.96717e6 0.130284
986986 1.46961e6i 0.0481406i
987987 2.38379e7 0.778889
988988 25573.0 10290.2i 0.000833470 0.000335375i
989989 3.04218e7 0.988996
990990 849833.i 0.0275579i
991991 −3.49750e7 −1.13129 −0.565644 0.824649i 0.691372π-0.691372\pi
−0.565644 + 0.824649i 0.691372π0.691372\pi
992992 −807020. −0.0260379
993993 2.46148e7i 0.792177i
994994 1.41146e7i 0.453110i
995995 1.97550e6i 0.0632587i
996996 406049.i 0.0129697i
997997 1.96013e7 0.624521 0.312261 0.949996i 0.398914π-0.398914\pi
0.312261 + 0.949996i 0.398914π0.398914\pi
998998 −1.51483e7 −0.481436
999999 714744.i 0.0226588i
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 78.6.b.a.25.5 yes 6
3.2 odd 2 234.6.b.c.181.2 6
4.3 odd 2 624.6.c.d.337.4 6
13.5 odd 4 1014.6.a.q.1.2 3
13.8 odd 4 1014.6.a.o.1.2 3
13.12 even 2 inner 78.6.b.a.25.2 6
39.38 odd 2 234.6.b.c.181.5 6
52.51 odd 2 624.6.c.d.337.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.6.b.a.25.2 6 13.12 even 2 inner
78.6.b.a.25.5 yes 6 1.1 even 1 trivial
234.6.b.c.181.2 6 3.2 odd 2
234.6.b.c.181.5 6 39.38 odd 2
624.6.c.d.337.3 6 52.51 odd 2
624.6.c.d.337.4 6 4.3 odd 2
1014.6.a.o.1.2 3 13.8 odd 4
1014.6.a.q.1.2 3 13.5 odd 4