# Properties

 Label 8470.2.a.cw.1.5 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8470.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.6332905120$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.19898000.1 Defining polynomial: $$x^{6} - x^{5} - 10 x^{4} + 7 x^{3} + 24 x^{2} - 15 x - 5$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$-2.05906$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +2.05906 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.05906 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.23973 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +2.05906 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.05906 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.23973 q^{9} -1.00000 q^{10} +2.05906 q^{12} -2.27257 q^{13} -1.00000 q^{14} +2.05906 q^{15} +1.00000 q^{16} -7.44890 q^{17} -1.23973 q^{18} -0.851836 q^{19} +1.00000 q^{20} +2.05906 q^{21} +0.651398 q^{23} -2.05906 q^{24} +1.00000 q^{25} +2.27257 q^{26} -3.62450 q^{27} +1.00000 q^{28} +2.36641 q^{29} -2.05906 q^{30} -0.145026 q^{31} -1.00000 q^{32} +7.44890 q^{34} +1.00000 q^{35} +1.23973 q^{36} +2.32711 q^{37} +0.851836 q^{38} -4.67936 q^{39} -1.00000 q^{40} +4.64211 q^{41} -2.05906 q^{42} -11.7588 q^{43} +1.23973 q^{45} -0.651398 q^{46} -5.27536 q^{47} +2.05906 q^{48} +1.00000 q^{49} -1.00000 q^{50} -15.3377 q^{51} -2.27257 q^{52} -0.962643 q^{53} +3.62450 q^{54} -1.00000 q^{56} -1.75398 q^{57} -2.36641 q^{58} +1.04296 q^{59} +2.05906 q^{60} -3.58885 q^{61} +0.145026 q^{62} +1.23973 q^{63} +1.00000 q^{64} -2.27257 q^{65} -0.430333 q^{67} -7.44890 q^{68} +1.34127 q^{69} -1.00000 q^{70} +10.6384 q^{71} -1.23973 q^{72} -1.16253 q^{73} -2.32711 q^{74} +2.05906 q^{75} -0.851836 q^{76} +4.67936 q^{78} +10.7068 q^{79} +1.00000 q^{80} -11.1823 q^{81} -4.64211 q^{82} +1.08367 q^{83} +2.05906 q^{84} -7.44890 q^{85} +11.7588 q^{86} +4.87258 q^{87} +3.73563 q^{89} -1.23973 q^{90} -2.27257 q^{91} +0.651398 q^{92} -0.298618 q^{93} +5.27536 q^{94} -0.851836 q^{95} -2.05906 q^{96} -1.47983 q^{97} -1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} + 3 q^{9} + O(q^{10})$$ $$6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} + 3 q^{9} - 6 q^{10} - q^{12} - 9 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} - 9 q^{17} - 3 q^{18} - 12 q^{19} + 6 q^{20} - q^{21} + 4 q^{23} + q^{24} + 6 q^{25} + 9 q^{26} - 4 q^{27} + 6 q^{28} - 15 q^{29} + q^{30} + 8 q^{31} - 6 q^{32} + 9 q^{34} + 6 q^{35} + 3 q^{36} - 4 q^{37} + 12 q^{38} + 19 q^{39} - 6 q^{40} - 4 q^{41} + q^{42} - 30 q^{43} + 3 q^{45} - 4 q^{46} - 7 q^{47} - q^{48} + 6 q^{49} - 6 q^{50} - 16 q^{51} - 9 q^{52} - 6 q^{53} + 4 q^{54} - 6 q^{56} + 14 q^{57} + 15 q^{58} + 4 q^{59} - q^{60} + 14 q^{61} - 8 q^{62} + 3 q^{63} + 6 q^{64} - 9 q^{65} + 18 q^{67} - 9 q^{68} - 10 q^{69} - 6 q^{70} + 23 q^{71} - 3 q^{72} - 23 q^{73} + 4 q^{74} - q^{75} - 12 q^{76} - 19 q^{78} - 21 q^{79} + 6 q^{80} - 18 q^{81} + 4 q^{82} - 25 q^{83} - q^{84} - 9 q^{85} + 30 q^{86} - 14 q^{87} - 18 q^{89} - 3 q^{90} - 9 q^{91} + 4 q^{92} - 24 q^{93} + 7 q^{94} - 12 q^{95} + q^{96} + 7 q^{97} - 6 q^{98} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 2.05906 1.18880 0.594400 0.804170i $$-0.297390\pi$$
0.594400 + 0.804170i $$0.297390\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −2.05906 −0.840608
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.23973 0.413245
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ 2.05906 0.594400
$$13$$ −2.27257 −0.630298 −0.315149 0.949042i $$-0.602054\pi$$
−0.315149 + 0.949042i $$0.602054\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 2.05906 0.531647
$$16$$ 1.00000 0.250000
$$17$$ −7.44890 −1.80662 −0.903312 0.428985i $$-0.858871\pi$$
−0.903312 + 0.428985i $$0.858871\pi$$
$$18$$ −1.23973 −0.292208
$$19$$ −0.851836 −0.195425 −0.0977123 0.995215i $$-0.531152\pi$$
−0.0977123 + 0.995215i $$0.531152\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 2.05906 0.449324
$$22$$ 0 0
$$23$$ 0.651398 0.135826 0.0679130 0.997691i $$-0.478366\pi$$
0.0679130 + 0.997691i $$0.478366\pi$$
$$24$$ −2.05906 −0.420304
$$25$$ 1.00000 0.200000
$$26$$ 2.27257 0.445688
$$27$$ −3.62450 −0.697534
$$28$$ 1.00000 0.188982
$$29$$ 2.36641 0.439431 0.219715 0.975564i $$-0.429487\pi$$
0.219715 + 0.975564i $$0.429487\pi$$
$$30$$ −2.05906 −0.375931
$$31$$ −0.145026 −0.0260475 −0.0130237 0.999915i $$-0.504146\pi$$
−0.0130237 + 0.999915i $$0.504146\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 7.44890 1.27748
$$35$$ 1.00000 0.169031
$$36$$ 1.23973 0.206622
$$37$$ 2.32711 0.382574 0.191287 0.981534i $$-0.438734\pi$$
0.191287 + 0.981534i $$0.438734\pi$$
$$38$$ 0.851836 0.138186
$$39$$ −4.67936 −0.749298
$$40$$ −1.00000 −0.158114
$$41$$ 4.64211 0.724976 0.362488 0.931988i $$-0.381927\pi$$
0.362488 + 0.931988i $$0.381927\pi$$
$$42$$ −2.05906 −0.317720
$$43$$ −11.7588 −1.79320 −0.896602 0.442837i $$-0.853972\pi$$
−0.896602 + 0.442837i $$0.853972\pi$$
$$44$$ 0 0
$$45$$ 1.23973 0.184809
$$46$$ −0.651398 −0.0960434
$$47$$ −5.27536 −0.769491 −0.384746 0.923023i $$-0.625711\pi$$
−0.384746 + 0.923023i $$0.625711\pi$$
$$48$$ 2.05906 0.297200
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ −15.3377 −2.14771
$$52$$ −2.27257 −0.315149
$$53$$ −0.962643 −0.132229 −0.0661146 0.997812i $$-0.521060\pi$$
−0.0661146 + 0.997812i $$0.521060\pi$$
$$54$$ 3.62450 0.493231
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ −1.75398 −0.232321
$$58$$ −2.36641 −0.310724
$$59$$ 1.04296 0.135782 0.0678911 0.997693i $$-0.478373\pi$$
0.0678911 + 0.997693i $$0.478373\pi$$
$$60$$ 2.05906 0.265824
$$61$$ −3.58885 −0.459505 −0.229752 0.973249i $$-0.573792\pi$$
−0.229752 + 0.973249i $$0.573792\pi$$
$$62$$ 0.145026 0.0184183
$$63$$ 1.23973 0.156192
$$64$$ 1.00000 0.125000
$$65$$ −2.27257 −0.281878
$$66$$ 0 0
$$67$$ −0.430333 −0.0525735 −0.0262868 0.999654i $$-0.508368\pi$$
−0.0262868 + 0.999654i $$0.508368\pi$$
$$68$$ −7.44890 −0.903312
$$69$$ 1.34127 0.161470
$$70$$ −1.00000 −0.119523
$$71$$ 10.6384 1.26255 0.631276 0.775558i $$-0.282532\pi$$
0.631276 + 0.775558i $$0.282532\pi$$
$$72$$ −1.23973 −0.146104
$$73$$ −1.16253 −0.136063 −0.0680316 0.997683i $$-0.521672\pi$$
−0.0680316 + 0.997683i $$0.521672\pi$$
$$74$$ −2.32711 −0.270521
$$75$$ 2.05906 0.237760
$$76$$ −0.851836 −0.0977123
$$77$$ 0 0
$$78$$ 4.67936 0.529833
$$79$$ 10.7068 1.20461 0.602303 0.798268i $$-0.294250\pi$$
0.602303 + 0.798268i $$0.294250\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −11.1823 −1.24247
$$82$$ −4.64211 −0.512636
$$83$$ 1.08367 0.118948 0.0594741 0.998230i $$-0.481058\pi$$
0.0594741 + 0.998230i $$0.481058\pi$$
$$84$$ 2.05906 0.224662
$$85$$ −7.44890 −0.807947
$$86$$ 11.7588 1.26799
$$87$$ 4.87258 0.522395
$$88$$ 0 0
$$89$$ 3.73563 0.395976 0.197988 0.980204i $$-0.436559\pi$$
0.197988 + 0.980204i $$0.436559\pi$$
$$90$$ −1.23973 −0.130679
$$91$$ −2.27257 −0.238230
$$92$$ 0.651398 0.0679130
$$93$$ −0.298618 −0.0309652
$$94$$ 5.27536 0.544112
$$95$$ −0.851836 −0.0873965
$$96$$ −2.05906 −0.210152
$$97$$ −1.47983 −0.150254 −0.0751270 0.997174i $$-0.523936\pi$$
−0.0751270 + 0.997174i $$0.523936\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −13.9523 −1.38830 −0.694151 0.719830i $$-0.744220\pi$$
−0.694151 + 0.719830i $$0.744220\pi$$
$$102$$ 15.3377 1.51866
$$103$$ −1.40558 −0.138496 −0.0692481 0.997599i $$-0.522060\pi$$
−0.0692481 + 0.997599i $$0.522060\pi$$
$$104$$ 2.27257 0.222844
$$105$$ 2.05906 0.200944
$$106$$ 0.962643 0.0935001
$$107$$ −18.9282 −1.82985 −0.914927 0.403618i $$-0.867752\pi$$
−0.914927 + 0.403618i $$0.867752\pi$$
$$108$$ −3.62450 −0.348767
$$109$$ −10.8423 −1.03850 −0.519252 0.854621i $$-0.673789\pi$$
−0.519252 + 0.854621i $$0.673789\pi$$
$$110$$ 0 0
$$111$$ 4.79166 0.454804
$$112$$ 1.00000 0.0944911
$$113$$ −17.1609 −1.61436 −0.807179 0.590307i $$-0.799007\pi$$
−0.807179 + 0.590307i $$0.799007\pi$$
$$114$$ 1.75398 0.164276
$$115$$ 0.651398 0.0607432
$$116$$ 2.36641 0.219715
$$117$$ −2.81738 −0.260467
$$118$$ −1.04296 −0.0960125
$$119$$ −7.44890 −0.682839
$$120$$ −2.05906 −0.187966
$$121$$ 0 0
$$122$$ 3.58885 0.324919
$$123$$ 9.55840 0.861852
$$124$$ −0.145026 −0.0130237
$$125$$ 1.00000 0.0894427
$$126$$ −1.23973 −0.110444
$$127$$ 11.1188 0.986631 0.493316 0.869850i $$-0.335785\pi$$
0.493316 + 0.869850i $$0.335785\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −24.2121 −2.13176
$$130$$ 2.27257 0.199318
$$131$$ −10.9162 −0.953749 −0.476874 0.878971i $$-0.658230\pi$$
−0.476874 + 0.878971i $$0.658230\pi$$
$$132$$ 0 0
$$133$$ −0.851836 −0.0738635
$$134$$ 0.430333 0.0371751
$$135$$ −3.62450 −0.311947
$$136$$ 7.44890 0.638738
$$137$$ −4.81357 −0.411251 −0.205626 0.978631i $$-0.565923\pi$$
−0.205626 + 0.978631i $$0.565923\pi$$
$$138$$ −1.34127 −0.114176
$$139$$ −10.1432 −0.860336 −0.430168 0.902749i $$-0.641546\pi$$
−0.430168 + 0.902749i $$0.641546\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ −10.8623 −0.914771
$$142$$ −10.6384 −0.892759
$$143$$ 0 0
$$144$$ 1.23973 0.103311
$$145$$ 2.36641 0.196519
$$146$$ 1.16253 0.0962113
$$147$$ 2.05906 0.169829
$$148$$ 2.32711 0.191287
$$149$$ 1.88432 0.154370 0.0771848 0.997017i $$-0.475407\pi$$
0.0771848 + 0.997017i $$0.475407\pi$$
$$150$$ −2.05906 −0.168122
$$151$$ 14.3826 1.17044 0.585218 0.810876i $$-0.301009\pi$$
0.585218 + 0.810876i $$0.301009\pi$$
$$152$$ 0.851836 0.0690930
$$153$$ −9.23466 −0.746578
$$154$$ 0 0
$$155$$ −0.145026 −0.0116488
$$156$$ −4.67936 −0.374649
$$157$$ −12.6590 −1.01030 −0.505150 0.863031i $$-0.668563\pi$$
−0.505150 + 0.863031i $$0.668563\pi$$
$$158$$ −10.7068 −0.851784
$$159$$ −1.98214 −0.157194
$$160$$ −1.00000 −0.0790569
$$161$$ 0.651398 0.0513374
$$162$$ 11.1823 0.878561
$$163$$ −8.95282 −0.701239 −0.350620 0.936518i $$-0.614029\pi$$
−0.350620 + 0.936518i $$0.614029\pi$$
$$164$$ 4.64211 0.362488
$$165$$ 0 0
$$166$$ −1.08367 −0.0841091
$$167$$ 6.36465 0.492511 0.246256 0.969205i $$-0.420800\pi$$
0.246256 + 0.969205i $$0.420800\pi$$
$$168$$ −2.05906 −0.158860
$$169$$ −7.83543 −0.602725
$$170$$ 7.44890 0.571305
$$171$$ −1.05605 −0.0807582
$$172$$ −11.7588 −0.896602
$$173$$ −24.4230 −1.85684 −0.928421 0.371529i $$-0.878834\pi$$
−0.928421 + 0.371529i $$0.878834\pi$$
$$174$$ −4.87258 −0.369389
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 2.14753 0.161418
$$178$$ −3.73563 −0.279997
$$179$$ 5.80767 0.434086 0.217043 0.976162i $$-0.430359\pi$$
0.217043 + 0.976162i $$0.430359\pi$$
$$180$$ 1.23973 0.0924044
$$181$$ −12.8812 −0.957450 −0.478725 0.877965i $$-0.658901\pi$$
−0.478725 + 0.877965i $$0.658901\pi$$
$$182$$ 2.27257 0.168454
$$183$$ −7.38966 −0.546259
$$184$$ −0.651398 −0.0480217
$$185$$ 2.32711 0.171093
$$186$$ 0.298618 0.0218957
$$187$$ 0 0
$$188$$ −5.27536 −0.384746
$$189$$ −3.62450 −0.263643
$$190$$ 0.851836 0.0617987
$$191$$ −5.65871 −0.409450 −0.204725 0.978820i $$-0.565630\pi$$
−0.204725 + 0.978820i $$0.565630\pi$$
$$192$$ 2.05906 0.148600
$$193$$ 13.7878 0.992470 0.496235 0.868188i $$-0.334715\pi$$
0.496235 + 0.868188i $$0.334715\pi$$
$$194$$ 1.47983 0.106246
$$195$$ −4.67936 −0.335096
$$196$$ 1.00000 0.0714286
$$197$$ 2.82825 0.201504 0.100752 0.994912i $$-0.467875\pi$$
0.100752 + 0.994912i $$0.467875\pi$$
$$198$$ 0 0
$$199$$ −0.513194 −0.0363793 −0.0181897 0.999835i $$-0.505790\pi$$
−0.0181897 + 0.999835i $$0.505790\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −0.886082 −0.0624994
$$202$$ 13.9523 0.981677
$$203$$ 2.36641 0.166089
$$204$$ −15.3377 −1.07386
$$205$$ 4.64211 0.324219
$$206$$ 1.40558 0.0979316
$$207$$ 0.807561 0.0561294
$$208$$ −2.27257 −0.157574
$$209$$ 0 0
$$210$$ −2.05906 −0.142089
$$211$$ −20.4146 −1.40540 −0.702698 0.711488i $$-0.748021\pi$$
−0.702698 + 0.711488i $$0.748021\pi$$
$$212$$ −0.962643 −0.0661146
$$213$$ 21.9052 1.50092
$$214$$ 18.9282 1.29390
$$215$$ −11.7588 −0.801945
$$216$$ 3.62450 0.246616
$$217$$ −0.145026 −0.00984502
$$218$$ 10.8423 0.734333
$$219$$ −2.39371 −0.161752
$$220$$ 0 0
$$221$$ 16.9281 1.13871
$$222$$ −4.79166 −0.321595
$$223$$ 15.3296 1.02654 0.513272 0.858226i $$-0.328433\pi$$
0.513272 + 0.858226i $$0.328433\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 1.23973 0.0826490
$$226$$ 17.1609 1.14152
$$227$$ −28.8027 −1.91170 −0.955850 0.293856i $$-0.905061\pi$$
−0.955850 + 0.293856i $$0.905061\pi$$
$$228$$ −1.75398 −0.116160
$$229$$ 7.06918 0.467145 0.233572 0.972339i $$-0.424958\pi$$
0.233572 + 0.972339i $$0.424958\pi$$
$$230$$ −0.651398 −0.0429519
$$231$$ 0 0
$$232$$ −2.36641 −0.155362
$$233$$ 25.1088 1.64493 0.822466 0.568814i $$-0.192598\pi$$
0.822466 + 0.568814i $$0.192598\pi$$
$$234$$ 2.81738 0.184178
$$235$$ −5.27536 −0.344127
$$236$$ 1.04296 0.0678911
$$237$$ 22.0459 1.43203
$$238$$ 7.44890 0.482840
$$239$$ 9.09178 0.588098 0.294049 0.955790i $$-0.404997\pi$$
0.294049 + 0.955790i $$0.404997\pi$$
$$240$$ 2.05906 0.132912
$$241$$ 11.8409 0.762739 0.381370 0.924423i $$-0.375452\pi$$
0.381370 + 0.924423i $$0.375452\pi$$
$$242$$ 0 0
$$243$$ −12.1515 −0.779518
$$244$$ −3.58885 −0.229752
$$245$$ 1.00000 0.0638877
$$246$$ −9.55840 −0.609421
$$247$$ 1.93586 0.123176
$$248$$ 0.145026 0.00920917
$$249$$ 2.23134 0.141406
$$250$$ −1.00000 −0.0632456
$$251$$ −12.2824 −0.775260 −0.387630 0.921815i $$-0.626706\pi$$
−0.387630 + 0.921815i $$0.626706\pi$$
$$252$$ 1.23973 0.0780959
$$253$$ 0 0
$$254$$ −11.1188 −0.697654
$$255$$ −15.3377 −0.960487
$$256$$ 1.00000 0.0625000
$$257$$ −23.8811 −1.48966 −0.744830 0.667254i $$-0.767469\pi$$
−0.744830 + 0.667254i $$0.767469\pi$$
$$258$$ 24.2121 1.50738
$$259$$ 2.32711 0.144600
$$260$$ −2.27257 −0.140939
$$261$$ 2.93372 0.181592
$$262$$ 10.9162 0.674402
$$263$$ −10.5902 −0.653019 −0.326509 0.945194i $$-0.605872\pi$$
−0.326509 + 0.945194i $$0.605872\pi$$
$$264$$ 0 0
$$265$$ −0.962643 −0.0591347
$$266$$ 0.851836 0.0522294
$$267$$ 7.69189 0.470736
$$268$$ −0.430333 −0.0262868
$$269$$ 16.5719 1.01041 0.505203 0.863001i $$-0.331418\pi$$
0.505203 + 0.863001i $$0.331418\pi$$
$$270$$ 3.62450 0.220580
$$271$$ −29.4244 −1.78740 −0.893702 0.448661i $$-0.851901\pi$$
−0.893702 + 0.448661i $$0.851901\pi$$
$$272$$ −7.44890 −0.451656
$$273$$ −4.67936 −0.283208
$$274$$ 4.81357 0.290799
$$275$$ 0 0
$$276$$ 1.34127 0.0807349
$$277$$ 29.6933 1.78410 0.892049 0.451939i $$-0.149267\pi$$
0.892049 + 0.451939i $$0.149267\pi$$
$$278$$ 10.1432 0.608349
$$279$$ −0.179794 −0.0107640
$$280$$ −1.00000 −0.0597614
$$281$$ −23.8076 −1.42024 −0.710121 0.704080i $$-0.751360\pi$$
−0.710121 + 0.704080i $$0.751360\pi$$
$$282$$ 10.8623 0.646841
$$283$$ −17.5036 −1.04048 −0.520241 0.854019i $$-0.674158\pi$$
−0.520241 + 0.854019i $$0.674158\pi$$
$$284$$ 10.6384 0.631276
$$285$$ −1.75398 −0.103897
$$286$$ 0 0
$$287$$ 4.64211 0.274015
$$288$$ −1.23973 −0.0730521
$$289$$ 38.4861 2.26389
$$290$$ −2.36641 −0.138960
$$291$$ −3.04706 −0.178622
$$292$$ −1.16253 −0.0680316
$$293$$ 23.1174 1.35053 0.675267 0.737573i $$-0.264028\pi$$
0.675267 + 0.737573i $$0.264028\pi$$
$$294$$ −2.05906 −0.120087
$$295$$ 1.04296 0.0607237
$$296$$ −2.32711 −0.135261
$$297$$ 0 0
$$298$$ −1.88432 −0.109156
$$299$$ −1.48035 −0.0856107
$$300$$ 2.05906 0.118880
$$301$$ −11.7588 −0.677768
$$302$$ −14.3826 −0.827623
$$303$$ −28.7285 −1.65041
$$304$$ −0.851836 −0.0488561
$$305$$ −3.58885 −0.205497
$$306$$ 9.23466 0.527910
$$307$$ −17.1933 −0.981271 −0.490635 0.871365i $$-0.663235\pi$$
−0.490635 + 0.871365i $$0.663235\pi$$
$$308$$ 0 0
$$309$$ −2.89418 −0.164644
$$310$$ 0.145026 0.00823693
$$311$$ −13.9706 −0.792200 −0.396100 0.918207i $$-0.629637\pi$$
−0.396100 + 0.918207i $$0.629637\pi$$
$$312$$ 4.67936 0.264917
$$313$$ 10.4793 0.592327 0.296163 0.955137i $$-0.404293\pi$$
0.296163 + 0.955137i $$0.404293\pi$$
$$314$$ 12.6590 0.714390
$$315$$ 1.23973 0.0698511
$$316$$ 10.7068 0.602303
$$317$$ −32.2756 −1.81278 −0.906390 0.422442i $$-0.861173\pi$$
−0.906390 + 0.422442i $$0.861173\pi$$
$$318$$ 1.98214 0.111153
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ −38.9743 −2.17533
$$322$$ −0.651398 −0.0363010
$$323$$ 6.34524 0.353059
$$324$$ −11.1823 −0.621237
$$325$$ −2.27257 −0.126060
$$326$$ 8.95282 0.495851
$$327$$ −22.3250 −1.23457
$$328$$ −4.64211 −0.256318
$$329$$ −5.27536 −0.290840
$$330$$ 0 0
$$331$$ 27.1275 1.49106 0.745532 0.666470i $$-0.232196\pi$$
0.745532 + 0.666470i $$0.232196\pi$$
$$332$$ 1.08367 0.0594741
$$333$$ 2.88500 0.158097
$$334$$ −6.36465 −0.348258
$$335$$ −0.430333 −0.0235116
$$336$$ 2.05906 0.112331
$$337$$ 25.8297 1.40703 0.703517 0.710678i $$-0.251612\pi$$
0.703517 + 0.710678i $$0.251612\pi$$
$$338$$ 7.83543 0.426191
$$339$$ −35.3353 −1.91915
$$340$$ −7.44890 −0.403973
$$341$$ 0 0
$$342$$ 1.05605 0.0571047
$$343$$ 1.00000 0.0539949
$$344$$ 11.7588 0.633993
$$345$$ 1.34127 0.0722115
$$346$$ 24.4230 1.31299
$$347$$ 6.20850 0.333290 0.166645 0.986017i $$-0.446707\pi$$
0.166645 + 0.986017i $$0.446707\pi$$
$$348$$ 4.87258 0.261198
$$349$$ −10.3551 −0.554295 −0.277148 0.960827i $$-0.589389\pi$$
−0.277148 + 0.960827i $$0.589389\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ 8.23692 0.439654
$$352$$ 0 0
$$353$$ 27.4798 1.46260 0.731300 0.682055i $$-0.238914\pi$$
0.731300 + 0.682055i $$0.238914\pi$$
$$354$$ −2.14753 −0.114140
$$355$$ 10.6384 0.564630
$$356$$ 3.73563 0.197988
$$357$$ −15.3377 −0.811759
$$358$$ −5.80767 −0.306945
$$359$$ 0.0307419 0.00162250 0.000811248 1.00000i $$-0.499742\pi$$
0.000811248 1.00000i $$0.499742\pi$$
$$360$$ −1.23973 −0.0653397
$$361$$ −18.2744 −0.961809
$$362$$ 12.8812 0.677019
$$363$$ 0 0
$$364$$ −2.27257 −0.119115
$$365$$ −1.16253 −0.0608493
$$366$$ 7.38966 0.386264
$$367$$ −10.2929 −0.537285 −0.268643 0.963240i $$-0.586575\pi$$
−0.268643 + 0.963240i $$0.586575\pi$$
$$368$$ 0.651398 0.0339565
$$369$$ 5.75499 0.299593
$$370$$ −2.32711 −0.120981
$$371$$ −0.962643 −0.0499779
$$372$$ −0.298618 −0.0154826
$$373$$ −12.2144 −0.632438 −0.316219 0.948686i $$-0.602413\pi$$
−0.316219 + 0.948686i $$0.602413\pi$$
$$374$$ 0 0
$$375$$ 2.05906 0.106329
$$376$$ 5.27536 0.272056
$$377$$ −5.37782 −0.276972
$$378$$ 3.62450 0.186424
$$379$$ 0.699608 0.0359364 0.0179682 0.999839i $$-0.494280\pi$$
0.0179682 + 0.999839i $$0.494280\pi$$
$$380$$ −0.851836 −0.0436983
$$381$$ 22.8942 1.17291
$$382$$ 5.65871 0.289525
$$383$$ −37.4527 −1.91374 −0.956871 0.290513i $$-0.906174\pi$$
−0.956871 + 0.290513i $$0.906174\pi$$
$$384$$ −2.05906 −0.105076
$$385$$ 0 0
$$386$$ −13.7878 −0.701783
$$387$$ −14.5778 −0.741032
$$388$$ −1.47983 −0.0751270
$$389$$ −8.64880 −0.438512 −0.219256 0.975667i $$-0.570363\pi$$
−0.219256 + 0.975667i $$0.570363\pi$$
$$390$$ 4.67936 0.236949
$$391$$ −4.85220 −0.245386
$$392$$ −1.00000 −0.0505076
$$393$$ −22.4770 −1.13382
$$394$$ −2.82825 −0.142485
$$395$$ 10.7068 0.538716
$$396$$ 0 0
$$397$$ 9.10554 0.456994 0.228497 0.973545i $$-0.426619\pi$$
0.228497 + 0.973545i $$0.426619\pi$$
$$398$$ 0.513194 0.0257241
$$399$$ −1.75398 −0.0878090
$$400$$ 1.00000 0.0500000
$$401$$ 19.2528 0.961437 0.480718 0.876875i $$-0.340376\pi$$
0.480718 + 0.876875i $$0.340376\pi$$
$$402$$ 0.886082 0.0441938
$$403$$ 0.329582 0.0164177
$$404$$ −13.9523 −0.694151
$$405$$ −11.1823 −0.555651
$$406$$ −2.36641 −0.117443
$$407$$ 0 0
$$408$$ 15.3377 0.759331
$$409$$ 26.7614 1.32327 0.661634 0.749827i $$-0.269863\pi$$
0.661634 + 0.749827i $$0.269863\pi$$
$$410$$ −4.64211 −0.229258
$$411$$ −9.91144 −0.488895
$$412$$ −1.40558 −0.0692481
$$413$$ 1.04296 0.0513209
$$414$$ −0.807561 −0.0396894
$$415$$ 1.08367 0.0531953
$$416$$ 2.27257 0.111422
$$417$$ −20.8855 −1.02277
$$418$$ 0 0
$$419$$ −24.4437 −1.19415 −0.597075 0.802185i $$-0.703671\pi$$
−0.597075 + 0.802185i $$0.703671\pi$$
$$420$$ 2.05906 0.100472
$$421$$ −7.85402 −0.382781 −0.191391 0.981514i $$-0.561300\pi$$
−0.191391 + 0.981514i $$0.561300\pi$$
$$422$$ 20.4146 0.993765
$$423$$ −6.54005 −0.317988
$$424$$ 0.962643 0.0467501
$$425$$ −7.44890 −0.361325
$$426$$ −21.9052 −1.06131
$$427$$ −3.58885 −0.173677
$$428$$ −18.9282 −0.914927
$$429$$ 0 0
$$430$$ 11.7588 0.567061
$$431$$ −23.8170 −1.14722 −0.573612 0.819127i $$-0.694458\pi$$
−0.573612 + 0.819127i $$0.694458\pi$$
$$432$$ −3.62450 −0.174384
$$433$$ 15.4390 0.741952 0.370976 0.928642i $$-0.379023\pi$$
0.370976 + 0.928642i $$0.379023\pi$$
$$434$$ 0.145026 0.00696148
$$435$$ 4.87258 0.233622
$$436$$ −10.8423 −0.519252
$$437$$ −0.554884 −0.0265437
$$438$$ 2.39371 0.114376
$$439$$ 29.0621 1.38706 0.693530 0.720428i $$-0.256054\pi$$
0.693530 + 0.720428i $$0.256054\pi$$
$$440$$ 0 0
$$441$$ 1.23973 0.0590350
$$442$$ −16.9281 −0.805190
$$443$$ 34.7126 1.64924 0.824622 0.565685i $$-0.191388\pi$$
0.824622 + 0.565685i $$0.191388\pi$$
$$444$$ 4.79166 0.227402
$$445$$ 3.73563 0.177086
$$446$$ −15.3296 −0.725877
$$447$$ 3.87993 0.183515
$$448$$ 1.00000 0.0472456
$$449$$ −30.7259 −1.45005 −0.725024 0.688724i $$-0.758171\pi$$
−0.725024 + 0.688724i $$0.758171\pi$$
$$450$$ −1.23973 −0.0584416
$$451$$ 0 0
$$452$$ −17.1609 −0.807179
$$453$$ 29.6146 1.39141
$$454$$ 28.8027 1.35178
$$455$$ −2.27257 −0.106540
$$456$$ 1.75398 0.0821378
$$457$$ 20.4932 0.958633 0.479317 0.877642i $$-0.340885\pi$$
0.479317 + 0.877642i $$0.340885\pi$$
$$458$$ −7.06918 −0.330321
$$459$$ 26.9985 1.26018
$$460$$ 0.651398 0.0303716
$$461$$ 17.5629 0.817988 0.408994 0.912537i $$-0.365880\pi$$
0.408994 + 0.912537i $$0.365880\pi$$
$$462$$ 0 0
$$463$$ 13.6361 0.633721 0.316861 0.948472i $$-0.397371\pi$$
0.316861 + 0.948472i $$0.397371\pi$$
$$464$$ 2.36641 0.109858
$$465$$ −0.298618 −0.0138481
$$466$$ −25.1088 −1.16314
$$467$$ −30.7210 −1.42160 −0.710799 0.703395i $$-0.751666\pi$$
−0.710799 + 0.703395i $$0.751666\pi$$
$$468$$ −2.81738 −0.130234
$$469$$ −0.430333 −0.0198709
$$470$$ 5.27536 0.243334
$$471$$ −26.0657 −1.20105
$$472$$ −1.04296 −0.0480063
$$473$$ 0 0
$$474$$ −22.0459 −1.01260
$$475$$ −0.851836 −0.0390849
$$476$$ −7.44890 −0.341420
$$477$$ −1.19342 −0.0546430
$$478$$ −9.09178 −0.415848
$$479$$ 34.6829 1.58470 0.792352 0.610064i $$-0.208856\pi$$
0.792352 + 0.610064i $$0.208856\pi$$
$$480$$ −2.05906 −0.0939829
$$481$$ −5.28852 −0.241136
$$482$$ −11.8409 −0.539338
$$483$$ 1.34127 0.0610298
$$484$$ 0 0
$$485$$ −1.47983 −0.0671956
$$486$$ 12.1515 0.551202
$$487$$ 35.5251 1.60980 0.804899 0.593412i $$-0.202220\pi$$
0.804899 + 0.593412i $$0.202220\pi$$
$$488$$ 3.58885 0.162460
$$489$$ −18.4344 −0.833633
$$490$$ −1.00000 −0.0451754
$$491$$ 26.0609 1.17611 0.588055 0.808821i $$-0.299894\pi$$
0.588055 + 0.808821i $$0.299894\pi$$
$$492$$ 9.55840 0.430926
$$493$$ −17.6271 −0.793886
$$494$$ −1.93586 −0.0870983
$$495$$ 0 0
$$496$$ −0.145026 −0.00651187
$$497$$ 10.6384 0.477200
$$498$$ −2.23134 −0.0999889
$$499$$ 13.6585 0.611439 0.305719 0.952122i $$-0.401103\pi$$
0.305719 + 0.952122i $$0.401103\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 13.1052 0.585497
$$502$$ 12.2824 0.548191
$$503$$ 33.6925 1.50227 0.751136 0.660147i $$-0.229506\pi$$
0.751136 + 0.660147i $$0.229506\pi$$
$$504$$ −1.23973 −0.0552222
$$505$$ −13.9523 −0.620867
$$506$$ 0 0
$$507$$ −16.1336 −0.716519
$$508$$ 11.1188 0.493316
$$509$$ 25.0801 1.11166 0.555829 0.831297i $$-0.312401\pi$$
0.555829 + 0.831297i $$0.312401\pi$$
$$510$$ 15.3377 0.679167
$$511$$ −1.16253 −0.0514271
$$512$$ −1.00000 −0.0441942
$$513$$ 3.08748 0.136315
$$514$$ 23.8811 1.05335
$$515$$ −1.40558 −0.0619374
$$516$$ −24.2121 −1.06588
$$517$$ 0 0
$$518$$ −2.32711 −0.102247
$$519$$ −50.2884 −2.20741
$$520$$ 2.27257 0.0996588
$$521$$ −23.2513 −1.01866 −0.509330 0.860572i $$-0.670107\pi$$
−0.509330 + 0.860572i $$0.670107\pi$$
$$522$$ −2.93372 −0.128405
$$523$$ 0.198677 0.00868753 0.00434376 0.999991i $$-0.498617\pi$$
0.00434376 + 0.999991i $$0.498617\pi$$
$$524$$ −10.9162 −0.476874
$$525$$ 2.05906 0.0898648
$$526$$ 10.5902 0.461754
$$527$$ 1.08029 0.0470580
$$528$$ 0 0
$$529$$ −22.5757 −0.981551
$$530$$ 0.962643 0.0418145
$$531$$ 1.29300 0.0561113
$$532$$ −0.851836 −0.0369318
$$533$$ −10.5495 −0.456951
$$534$$ −7.69189 −0.332861
$$535$$ −18.9282 −0.818336
$$536$$ 0.430333 0.0185876
$$537$$ 11.9584 0.516041
$$538$$ −16.5719 −0.714465
$$539$$ 0 0
$$540$$ −3.62450 −0.155973
$$541$$ 32.5444 1.39919 0.699597 0.714538i $$-0.253363\pi$$
0.699597 + 0.714538i $$0.253363\pi$$
$$542$$ 29.4244 1.26389
$$543$$ −26.5231 −1.13822
$$544$$ 7.44890 0.319369
$$545$$ −10.8423 −0.464433
$$546$$ 4.67936 0.200258
$$547$$ −24.7703 −1.05910 −0.529550 0.848279i $$-0.677639\pi$$
−0.529550 + 0.848279i $$0.677639\pi$$
$$548$$ −4.81357 −0.205626
$$549$$ −4.44922 −0.189888
$$550$$ 0 0
$$551$$ −2.01579 −0.0858755
$$552$$ −1.34127 −0.0570882
$$553$$ 10.7068 0.455298
$$554$$ −29.6933 −1.26155
$$555$$ 4.79166 0.203395
$$556$$ −10.1432 −0.430168
$$557$$ 37.3936 1.58442 0.792208 0.610251i $$-0.208931\pi$$
0.792208 + 0.610251i $$0.208931\pi$$
$$558$$ 0.179794 0.00761128
$$559$$ 26.7228 1.13025
$$560$$ 1.00000 0.0422577
$$561$$ 0 0
$$562$$ 23.8076 1.00426
$$563$$ 17.2949 0.728894 0.364447 0.931224i $$-0.381258\pi$$
0.364447 + 0.931224i $$0.381258\pi$$
$$564$$ −10.8623 −0.457385
$$565$$ −17.1609 −0.721963
$$566$$ 17.5036 0.735732
$$567$$ −11.1823 −0.469611
$$568$$ −10.6384 −0.446379
$$569$$ 38.4117 1.61030 0.805152 0.593068i $$-0.202084\pi$$
0.805152 + 0.593068i $$0.202084\pi$$
$$570$$ 1.75398 0.0734662
$$571$$ −5.08249 −0.212696 −0.106348 0.994329i $$-0.533916\pi$$
−0.106348 + 0.994329i $$0.533916\pi$$
$$572$$ 0 0
$$573$$ −11.6516 −0.486754
$$574$$ −4.64211 −0.193758
$$575$$ 0.651398 0.0271652
$$576$$ 1.23973 0.0516556
$$577$$ 41.3365 1.72086 0.860430 0.509568i $$-0.170195\pi$$
0.860430 + 0.509568i $$0.170195\pi$$
$$578$$ −38.4861 −1.60081
$$579$$ 28.3900 1.17985
$$580$$ 2.36641 0.0982597
$$581$$ 1.08367 0.0449582
$$582$$ 3.04706 0.126305
$$583$$ 0 0
$$584$$ 1.16253 0.0481056
$$585$$ −2.81738 −0.116484
$$586$$ −23.1174 −0.954972
$$587$$ −6.92122 −0.285669 −0.142835 0.989747i $$-0.545622\pi$$
−0.142835 + 0.989747i $$0.545622\pi$$
$$588$$ 2.05906 0.0849143
$$589$$ 0.123538 0.00509032
$$590$$ −1.04296 −0.0429381
$$591$$ 5.82354 0.239548
$$592$$ 2.32711 0.0956436
$$593$$ 22.7439 0.933979 0.466989 0.884263i $$-0.345339\pi$$
0.466989 + 0.884263i $$0.345339\pi$$
$$594$$ 0 0
$$595$$ −7.44890 −0.305375
$$596$$ 1.88432 0.0771848
$$597$$ −1.05670 −0.0432477
$$598$$ 1.48035 0.0605359
$$599$$ 4.13028 0.168759 0.0843794 0.996434i $$-0.473109\pi$$
0.0843794 + 0.996434i $$0.473109\pi$$
$$600$$ −2.05906 −0.0840608
$$601$$ 3.87084 0.157895 0.0789474 0.996879i $$-0.474844\pi$$
0.0789474 + 0.996879i $$0.474844\pi$$
$$602$$ 11.7588 0.479254
$$603$$ −0.533499 −0.0217257
$$604$$ 14.3826 0.585218
$$605$$ 0 0
$$606$$ 28.7285 1.16702
$$607$$ 29.7726 1.20843 0.604216 0.796820i $$-0.293486\pi$$
0.604216 + 0.796820i $$0.293486\pi$$
$$608$$ 0.851836 0.0345465
$$609$$ 4.87258 0.197447
$$610$$ 3.58885 0.145308
$$611$$ 11.9886 0.485008
$$612$$ −9.23466 −0.373289
$$613$$ −26.1253 −1.05519 −0.527595 0.849496i $$-0.676906\pi$$
−0.527595 + 0.849496i $$0.676906\pi$$
$$614$$ 17.1933 0.693863
$$615$$ 9.55840 0.385432
$$616$$ 0 0
$$617$$ 13.3021 0.535524 0.267762 0.963485i $$-0.413716\pi$$
0.267762 + 0.963485i $$0.413716\pi$$
$$618$$ 2.89418 0.116421
$$619$$ −9.29198 −0.373476 −0.186738 0.982410i $$-0.559792\pi$$
−0.186738 + 0.982410i $$0.559792\pi$$
$$620$$ −0.145026 −0.00582439
$$621$$ −2.36099 −0.0947432
$$622$$ 13.9706 0.560170
$$623$$ 3.73563 0.149665
$$624$$ −4.67936 −0.187324
$$625$$ 1.00000 0.0400000
$$626$$ −10.4793 −0.418838
$$627$$ 0 0
$$628$$ −12.6590 −0.505150
$$629$$ −17.3344 −0.691168
$$630$$ −1.23973 −0.0493922
$$631$$ −29.7104 −1.18275 −0.591376 0.806396i $$-0.701415\pi$$
−0.591376 + 0.806396i $$0.701415\pi$$
$$632$$ −10.7068 −0.425892
$$633$$ −42.0348 −1.67073
$$634$$ 32.2756 1.28183
$$635$$ 11.1188 0.441235
$$636$$ −1.98214 −0.0785970
$$637$$ −2.27257 −0.0900425
$$638$$ 0 0
$$639$$ 13.1889 0.521743
$$640$$ −1.00000 −0.0395285
$$641$$ −17.1408 −0.677020 −0.338510 0.940963i $$-0.609923\pi$$
−0.338510 + 0.940963i $$0.609923\pi$$
$$642$$ 38.9743 1.53819
$$643$$ −25.7316 −1.01476 −0.507378 0.861724i $$-0.669385\pi$$
−0.507378 + 0.861724i $$0.669385\pi$$
$$644$$ 0.651398 0.0256687
$$645$$ −24.2121 −0.953352
$$646$$ −6.34524 −0.249650
$$647$$ −21.6382 −0.850686 −0.425343 0.905032i $$-0.639847\pi$$
−0.425343 + 0.905032i $$0.639847\pi$$
$$648$$ 11.1823 0.439281
$$649$$ 0 0
$$650$$ 2.27257 0.0891375
$$651$$ −0.298618 −0.0117038
$$652$$ −8.95282 −0.350620
$$653$$ 38.5661 1.50921 0.754604 0.656181i $$-0.227829\pi$$
0.754604 + 0.656181i $$0.227829\pi$$
$$654$$ 22.3250 0.872975
$$655$$ −10.9162 −0.426529
$$656$$ 4.64211 0.181244
$$657$$ −1.44122 −0.0562274
$$658$$ 5.27536 0.205655
$$659$$ −41.3104 −1.60923 −0.804613 0.593799i $$-0.797627\pi$$
−0.804613 + 0.593799i $$0.797627\pi$$
$$660$$ 0 0
$$661$$ −17.6230 −0.685456 −0.342728 0.939435i $$-0.611351\pi$$
−0.342728 + 0.939435i $$0.611351\pi$$
$$662$$ −27.1275 −1.05434
$$663$$ 34.8561 1.35370
$$664$$ −1.08367 −0.0420546
$$665$$ −0.851836 −0.0330328
$$666$$ −2.88500 −0.111791
$$667$$ 1.54147 0.0596861
$$668$$ 6.36465 0.246256
$$669$$ 31.5645 1.22036
$$670$$ 0.430333 0.0166252
$$671$$ 0 0
$$672$$ −2.05906 −0.0794300
$$673$$ −23.9060 −0.921508 −0.460754 0.887528i $$-0.652421\pi$$
−0.460754 + 0.887528i $$0.652421\pi$$
$$674$$ −25.8297 −0.994923
$$675$$ −3.62450 −0.139507
$$676$$ −7.83543 −0.301363
$$677$$ −4.64307 −0.178448 −0.0892238 0.996012i $$-0.528439\pi$$
−0.0892238 + 0.996012i $$0.528439\pi$$
$$678$$ 35.3353 1.35704
$$679$$ −1.47983 −0.0567906
$$680$$ 7.44890 0.285652
$$681$$ −59.3064 −2.27263
$$682$$ 0 0
$$683$$ −25.5818 −0.978859 −0.489430 0.872043i $$-0.662795\pi$$
−0.489430 + 0.872043i $$0.662795\pi$$
$$684$$ −1.05605 −0.0403791
$$685$$ −4.81357 −0.183917
$$686$$ −1.00000 −0.0381802
$$687$$ 14.5559 0.555341
$$688$$ −11.7588 −0.448301
$$689$$ 2.18767 0.0833437
$$690$$ −1.34127 −0.0510612
$$691$$ 22.5390 0.857424 0.428712 0.903441i $$-0.358967\pi$$
0.428712 + 0.903441i $$0.358967\pi$$
$$692$$ −24.4230 −0.928421
$$693$$ 0 0
$$694$$ −6.20850 −0.235671
$$695$$ −10.1432 −0.384754
$$696$$ −4.87258 −0.184695
$$697$$ −34.5786 −1.30976
$$698$$ 10.3551 0.391946
$$699$$ 51.7006 1.95550
$$700$$ 1.00000 0.0377964
$$701$$ 34.9336 1.31942 0.659711 0.751519i $$-0.270679\pi$$
0.659711 + 0.751519i $$0.270679\pi$$
$$702$$ −8.23692 −0.310882
$$703$$ −1.98232 −0.0747645
$$704$$ 0 0
$$705$$ −10.8623 −0.409098
$$706$$ −27.4798 −1.03422
$$707$$ −13.9523 −0.524729
$$708$$ 2.14753 0.0807089
$$709$$ −2.18225 −0.0819563 −0.0409781 0.999160i $$-0.513047\pi$$
−0.0409781 + 0.999160i $$0.513047\pi$$
$$710$$ −10.6384 −0.399254
$$711$$ 13.2735 0.497797
$$712$$ −3.73563 −0.139999
$$713$$ −0.0944698 −0.00353792
$$714$$ 15.3377 0.574001
$$715$$ 0 0
$$716$$ 5.80767 0.217043
$$717$$ 18.7205 0.699131
$$718$$ −0.0307419 −0.00114728
$$719$$ 23.1198 0.862224 0.431112 0.902299i $$-0.358121\pi$$
0.431112 + 0.902299i $$0.358121\pi$$
$$720$$ 1.23973 0.0462022
$$721$$ −1.40558 −0.0523466
$$722$$ 18.2744 0.680102
$$723$$ 24.3811 0.906745
$$724$$ −12.8812 −0.478725
$$725$$ 2.36641 0.0878861
$$726$$ 0 0
$$727$$ −48.6753 −1.80527 −0.902633 0.430410i $$-0.858369\pi$$
−0.902633 + 0.430410i $$0.858369\pi$$
$$728$$ 2.27257 0.0842270
$$729$$ 8.52614 0.315783
$$730$$ 1.16253 0.0430270
$$731$$ 87.5903 3.23964
$$732$$ −7.38966 −0.273130
$$733$$ 21.8686 0.807735 0.403867 0.914818i $$-0.367666\pi$$
0.403867 + 0.914818i $$0.367666\pi$$
$$734$$ 10.2929 0.379918
$$735$$ 2.05906 0.0759496
$$736$$ −0.651398 −0.0240109
$$737$$ 0 0
$$738$$ −5.75499 −0.211844
$$739$$ −0.178113 −0.00655200 −0.00327600 0.999995i $$-0.501043\pi$$
−0.00327600 + 0.999995i $$0.501043\pi$$
$$740$$ 2.32711 0.0855463
$$741$$ 3.98605 0.146431
$$742$$ 0.962643 0.0353397
$$743$$ −37.0632 −1.35972 −0.679859 0.733343i $$-0.737959\pi$$
−0.679859 + 0.733343i $$0.737959\pi$$
$$744$$ 0.298618 0.0109479
$$745$$ 1.88432 0.0690362
$$746$$ 12.2144 0.447201
$$747$$ 1.34346 0.0491548
$$748$$ 0 0
$$749$$ −18.9282 −0.691620
$$750$$ −2.05906 −0.0751863
$$751$$ −26.4258 −0.964291 −0.482146 0.876091i $$-0.660142\pi$$
−0.482146 + 0.876091i $$0.660142\pi$$
$$752$$ −5.27536 −0.192373
$$753$$ −25.2903 −0.921628
$$754$$ 5.37782 0.195849
$$755$$ 14.3826 0.523435
$$756$$ −3.62450 −0.131822
$$757$$ 9.40549 0.341848 0.170924 0.985284i $$-0.445325\pi$$
0.170924 + 0.985284i $$0.445325\pi$$
$$758$$ −0.699608 −0.0254109
$$759$$ 0 0
$$760$$ 0.851836 0.0308993
$$761$$ −22.7380 −0.824253 −0.412127 0.911127i $$-0.635214\pi$$
−0.412127 + 0.911127i $$0.635214\pi$$
$$762$$ −22.8942 −0.829371
$$763$$ −10.8423 −0.392517
$$764$$ −5.65871 −0.204725
$$765$$ −9.23466 −0.333880
$$766$$ 37.4527 1.35322
$$767$$ −2.37021 −0.0855832
$$768$$ 2.05906 0.0743000
$$769$$ 13.3469 0.481300 0.240650 0.970612i $$-0.422639\pi$$
0.240650 + 0.970612i $$0.422639\pi$$
$$770$$ 0 0
$$771$$ −49.1726 −1.77091
$$772$$ 13.7878 0.496235
$$773$$ −2.14317 −0.0770845 −0.0385422 0.999257i $$-0.512271\pi$$
−0.0385422 + 0.999257i $$0.512271\pi$$
$$774$$ 14.5778 0.523989
$$775$$ −0.145026 −0.00520949
$$776$$ 1.47983 0.0531228
$$777$$ 4.79166 0.171900
$$778$$ 8.64880 0.310074
$$779$$ −3.95432 −0.141678
$$780$$ −4.67936 −0.167548
$$781$$ 0 0
$$782$$ 4.85220 0.173514
$$783$$ −8.57703 −0.306518
$$784$$ 1.00000 0.0357143
$$785$$ −12.6590 −0.451820
$$786$$ 22.4770 0.801729
$$787$$ 39.7496 1.41692 0.708461 0.705750i $$-0.249390\pi$$
0.708461 + 0.705750i $$0.249390\pi$$
$$788$$ 2.82825 0.100752
$$789$$ −21.8058 −0.776308
$$790$$ −10.7068 −0.380930
$$791$$ −17.1609 −0.610170
$$792$$ 0 0
$$793$$ 8.15591 0.289625
$$794$$ −9.10554 −0.323144
$$795$$ −1.98214 −0.0702993
$$796$$ −0.513194 −0.0181897
$$797$$ −45.8902 −1.62551 −0.812757 0.582603i $$-0.802034\pi$$
−0.812757 + 0.582603i $$0.802034\pi$$
$$798$$ 1.75398 0.0620903
$$799$$ 39.2957 1.39018
$$800$$ −1.00000 −0.0353553
$$801$$ 4.63119 0.163635
$$802$$ −19.2528 −0.679838
$$803$$ 0 0
$$804$$ −0.886082 −0.0312497
$$805$$ 0.651398 0.0229588
$$806$$ −0.329582 −0.0116090
$$807$$ 34.1225 1.20117
$$808$$ 13.9523 0.490839
$$809$$ 3.54263 0.124552 0.0622761 0.998059i $$-0.480164\pi$$
0.0622761 + 0.998059i $$0.480164\pi$$
$$810$$ 11.1823 0.392905
$$811$$ 31.0933 1.09183 0.545916 0.837840i $$-0.316182\pi$$
0.545916 + 0.837840i $$0.316182\pi$$
$$812$$ 2.36641 0.0830446
$$813$$ −60.5866 −2.12487
$$814$$ 0 0
$$815$$ −8.95282 −0.313604
$$816$$ −15.3377 −0.536928
$$817$$ 10.0166 0.350436
$$818$$ −26.7614 −0.935692
$$819$$ −2.81738 −0.0984473
$$820$$ 4.64211 0.162110
$$821$$ −12.0289 −0.419812 −0.209906 0.977722i $$-0.567316\pi$$
−0.209906 + 0.977722i $$0.567316\pi$$
$$822$$ 9.91144 0.345701
$$823$$ 8.79883 0.306708 0.153354 0.988171i $$-0.450993\pi$$
0.153354 + 0.988171i $$0.450993\pi$$
$$824$$ 1.40558 0.0489658
$$825$$ 0 0
$$826$$ −1.04296 −0.0362893
$$827$$ −27.7899 −0.966349 −0.483175 0.875524i $$-0.660516\pi$$
−0.483175 + 0.875524i $$0.660516\pi$$
$$828$$ 0.807561 0.0280647
$$829$$ −13.8407 −0.480708 −0.240354 0.970685i $$-0.577263\pi$$
−0.240354 + 0.970685i $$0.577263\pi$$
$$830$$ −1.08367 −0.0376148
$$831$$ 61.1403 2.12094
$$832$$ −2.27257 −0.0787872
$$833$$ −7.44890 −0.258089
$$834$$ 20.8855 0.723205
$$835$$ 6.36465 0.220258
$$836$$ 0 0
$$837$$ 0.525647 0.0181690
$$838$$ 24.4437 0.844392
$$839$$ 34.3543 1.18604 0.593022 0.805187i $$-0.297935\pi$$
0.593022 + 0.805187i $$0.297935\pi$$
$$840$$ −2.05906 −0.0710444
$$841$$ −23.4001 −0.806901
$$842$$ 7.85402 0.270667
$$843$$ −49.0213 −1.68838
$$844$$ −20.4146 −0.702698
$$845$$ −7.83543 −0.269547
$$846$$ 6.54005 0.224852
$$847$$ 0 0
$$848$$ −0.962643 −0.0330573
$$849$$ −36.0410 −1.23693
$$850$$ 7.44890 0.255495
$$851$$ 1.51588 0.0519635
$$852$$ 21.9052 0.750460
$$853$$ −40.0952 −1.37283 −0.686416 0.727209i $$-0.740817\pi$$
−0.686416 + 0.727209i $$0.740817\pi$$
$$854$$ 3.58885 0.122808
$$855$$ −1.05605 −0.0361162
$$856$$ 18.9282 0.646951
$$857$$ 26.0287 0.889123 0.444562 0.895748i $$-0.353359\pi$$
0.444562 + 0.895748i $$0.353359\pi$$
$$858$$ 0 0
$$859$$ 29.9861 1.02311 0.511556 0.859250i $$-0.329069\pi$$
0.511556 + 0.859250i $$0.329069\pi$$
$$860$$ −11.7588 −0.400973
$$861$$ 9.55840 0.325749
$$862$$ 23.8170 0.811210
$$863$$ 28.1310 0.957590 0.478795 0.877927i $$-0.341074\pi$$
0.478795 + 0.877927i $$0.341074\pi$$
$$864$$ 3.62450 0.123308
$$865$$ −24.4230 −0.830405
$$866$$ −15.4390 −0.524639
$$867$$ 79.2453 2.69131
$$868$$ −0.145026 −0.00492251
$$869$$ 0 0
$$870$$ −4.87258 −0.165196
$$871$$ 0.977962 0.0331370
$$872$$ 10.8423 0.367166
$$873$$ −1.83460 −0.0620917
$$874$$ 0.554884 0.0187692
$$875$$ 1.00000 0.0338062
$$876$$ −2.39371 −0.0808760
$$877$$ −15.9142 −0.537383 −0.268691 0.963226i $$-0.586591\pi$$
−0.268691 + 0.963226i $$0.586591\pi$$
$$878$$ −29.0621 −0.980800
$$879$$ 47.6002 1.60551
$$880$$ 0 0
$$881$$ 42.3031 1.42523 0.712614 0.701556i $$-0.247511\pi$$
0.712614 + 0.701556i $$0.247511\pi$$
$$882$$ −1.23973 −0.0417440
$$883$$ −38.5685 −1.29793 −0.648967 0.760817i $$-0.724799\pi$$
−0.648967 + 0.760817i $$0.724799\pi$$
$$884$$ 16.9281 0.569355
$$885$$ 2.14753 0.0721883
$$886$$ −34.7126 −1.16619
$$887$$ 11.4973 0.386041 0.193021 0.981195i $$-0.438172\pi$$
0.193021 + 0.981195i $$0.438172\pi$$
$$888$$ −4.79166 −0.160798
$$889$$ 11.1188 0.372912
$$890$$ −3.73563 −0.125219
$$891$$ 0 0
$$892$$ 15.3296 0.513272
$$893$$ 4.49375 0.150377
$$894$$ −3.87993 −0.129764
$$895$$ 5.80767 0.194129
$$896$$ −1.00000 −0.0334077
$$897$$ −3.04813 −0.101774
$$898$$ 30.7259 1.02534
$$899$$ −0.343191 −0.0114461
$$900$$ 1.23973 0.0413245
$$901$$ 7.17063 0.238888
$$902$$ 0 0
$$903$$ −24.2121 −0.805730
$$904$$ 17.1609 0.570762
$$905$$ −12.8812 −0.428185
$$906$$ −29.6146 −0.983878
$$907$$ 35.9537 1.19382 0.596911 0.802308i $$-0.296395\pi$$
0.596911 + 0.802308i $$0.296395\pi$$
$$908$$ −28.8027 −0.955850
$$909$$ −17.2971 −0.573708
$$910$$ 2.27257 0.0753350
$$911$$ 13.3132 0.441086 0.220543 0.975377i $$-0.429217\pi$$
0.220543 + 0.975377i $$0.429217\pi$$
$$912$$ −1.75398 −0.0580802
$$913$$ 0 0
$$914$$ −20.4932 −0.677856
$$915$$ −7.38966 −0.244295
$$916$$ 7.06918 0.233572
$$917$$ −10.9162 −0.360483
$$918$$ −26.9985 −0.891083
$$919$$ 57.6449 1.90153 0.950764 0.309915i $$-0.100301\pi$$
0.950764 + 0.309915i $$0.100301\pi$$
$$920$$ −0.651398 −0.0214760
$$921$$ −35.4020 −1.16653
$$922$$ −17.5629 −0.578405
$$923$$ −24.1766 −0.795783
$$924$$ 0 0
$$925$$ 2.32711 0.0765149
$$926$$ −13.6361 −0.448109
$$927$$ −1.74255 −0.0572328
$$928$$ −2.36641 −0.0776811
$$929$$ 0.549043 0.0180135 0.00900675 0.999959i $$-0.497133\pi$$
0.00900675 + 0.999959i $$0.497133\pi$$
$$930$$ 0.298618 0.00979206
$$931$$ −0.851836 −0.0279178
$$932$$ 25.1088 0.822466
$$933$$ −28.7663 −0.941767
$$934$$ 30.7210 1.00522
$$935$$ 0 0
$$936$$ 2.81738 0.0920891
$$937$$ 56.7821 1.85499 0.927495 0.373835i $$-0.121957\pi$$
0.927495 + 0.373835i $$0.121957\pi$$
$$938$$ 0.430333 0.0140509
$$939$$ 21.5776 0.704158
$$940$$ −5.27536 −0.172063
$$941$$ −29.2904 −0.954840 −0.477420 0.878675i $$-0.658428\pi$$
−0.477420 + 0.878675i $$0.658428\pi$$
$$942$$ 26.0657 0.849267
$$943$$ 3.02386 0.0984706
$$944$$ 1.04296 0.0339456
$$945$$ −3.62450 −0.117905
$$946$$ 0 0
$$947$$ 24.8536 0.807632 0.403816 0.914840i $$-0.367684\pi$$
0.403816 + 0.914840i $$0.367684\pi$$
$$948$$ 22.0459 0.716017
$$949$$ 2.64192 0.0857603
$$950$$ 0.851836 0.0276372
$$951$$ −66.4575 −2.15503
$$952$$ 7.44890 0.241420
$$953$$ −43.8496 −1.42043 −0.710214 0.703986i $$-0.751402\pi$$
−0.710214 + 0.703986i $$0.751402\pi$$
$$954$$ 1.19342 0.0386384
$$955$$ −5.65871 −0.183111
$$956$$ 9.09178 0.294049
$$957$$ 0 0
$$958$$ −34.6829 −1.12056
$$959$$ −4.81357 −0.155438
$$960$$ 2.05906 0.0664559
$$961$$ −30.9790 −0.999322
$$962$$ 5.28852 0.170509
$$963$$ −23.4659 −0.756178
$$964$$ 11.8409 0.381370
$$965$$ 13.7878 0.443846
$$966$$ −1.34127 −0.0431546
$$967$$ 40.0914 1.28925 0.644627 0.764497i $$-0.277013\pi$$
0.644627 + 0.764497i $$0.277013\pi$$
$$968$$ 0 0
$$969$$ 13.0652 0.419716
$$970$$ 1.47983 0.0475145
$$971$$ −51.9047 −1.66570 −0.832850 0.553498i $$-0.813292\pi$$
−0.832850 + 0.553498i $$0.813292\pi$$
$$972$$ −12.1515 −0.389759
$$973$$ −10.1432 −0.325176
$$974$$ −35.5251 −1.13830
$$975$$ −4.67936 −0.149860
$$976$$ −3.58885 −0.114876
$$977$$ −16.3448 −0.522918 −0.261459 0.965215i $$-0.584204\pi$$
−0.261459 + 0.965215i $$0.584204\pi$$
$$978$$ 18.4344 0.589467
$$979$$ 0 0
$$980$$ 1.00000 0.0319438
$$981$$ −13.4416 −0.429156
$$982$$ −26.0609 −0.831636
$$983$$ −39.4682 −1.25884 −0.629420 0.777065i $$-0.716708\pi$$
−0.629420 + 0.777065i $$0.716708\pi$$
$$984$$ −9.55840 −0.304711
$$985$$ 2.82825 0.0901155
$$986$$ 17.6271 0.561362
$$987$$ −10.8623 −0.345751
$$988$$ 1.93586 0.0615878
$$989$$ −7.65968 −0.243564
$$990$$ 0 0
$$991$$ 20.8327 0.661771 0.330886 0.943671i $$-0.392653\pi$$
0.330886 + 0.943671i $$0.392653\pi$$
$$992$$ 0.145026 0.00460459
$$993$$ 55.8572 1.77258
$$994$$ −10.6384 −0.337431
$$995$$ −0.513194 −0.0162693
$$996$$ 2.23134 0.0707028
$$997$$ 1.23042 0.0389678 0.0194839 0.999810i $$-0.493798\pi$$
0.0194839 + 0.999810i $$0.493798\pi$$
$$998$$ −13.6585 −0.432353
$$999$$ −8.43460 −0.266859
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8470.2.a.cw.1.5 6
11.5 even 5 770.2.n.j.421.3 12
11.9 even 5 770.2.n.j.631.3 yes 12
11.10 odd 2 8470.2.a.dc.1.5 6

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.j.421.3 12 11.5 even 5
770.2.n.j.631.3 yes 12 11.9 even 5
8470.2.a.cw.1.5 6 1.1 even 1 trivial
8470.2.a.dc.1.5 6 11.10 odd 2