# Properties

 Label 8470.2.a.co.1.3 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.4400.1 Defining polynomial: $$x^{4} - 7 x^{2} + 11$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 1.3 Root $$1.54336$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +0.543362 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.543362 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.70476 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +0.543362 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.543362 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.70476 q^{9} -1.00000 q^{10} +0.543362 q^{12} -1.65861 q^{13} +1.00000 q^{14} +0.543362 q^{15} +1.00000 q^{16} +2.04615 q^{17} +2.70476 q^{18} +3.70476 q^{19} +1.00000 q^{20} -0.543362 q^{21} -4.99442 q^{23} -0.543362 q^{24} +1.00000 q^{25} +1.65861 q^{26} -3.09975 q^{27} -1.00000 q^{28} +7.13632 q^{29} -0.543362 q^{30} -10.1809 q^{31} -1.00000 q^{32} -2.04615 q^{34} -1.00000 q^{35} -2.70476 q^{36} +9.13632 q^{37} -3.70476 q^{38} -0.901224 q^{39} -1.00000 q^{40} +5.04615 q^{41} +0.543362 q^{42} +5.78688 q^{43} -2.70476 q^{45} +4.99442 q^{46} -4.56444 q^{47} +0.543362 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.11180 q^{51} -1.65861 q^{52} +8.88262 q^{53} +3.09975 q^{54} +1.00000 q^{56} +2.01302 q^{57} -7.13632 q^{58} -14.0846 q^{59} +0.543362 q^{60} -3.30516 q^{61} +10.1809 q^{62} +2.70476 q^{63} +1.00000 q^{64} -1.65861 q^{65} -3.18992 q^{67} +2.04615 q^{68} -2.71378 q^{69} +1.00000 q^{70} -7.80795 q^{71} +2.70476 q^{72} -10.9046 q^{73} -9.13632 q^{74} +0.543362 q^{75} +3.70476 q^{76} +0.901224 q^{78} +11.2054 q^{79} +1.00000 q^{80} +6.42999 q^{81} -5.04615 q^{82} +4.07812 q^{83} -0.543362 q^{84} +2.04615 q^{85} -5.78688 q^{86} +3.87760 q^{87} -3.32679 q^{89} +2.70476 q^{90} +1.65861 q^{91} -4.99442 q^{92} -5.53191 q^{93} +4.56444 q^{94} +3.70476 q^{95} -0.543362 q^{96} +9.19639 q^{97} -1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} + O(q^{10})$$ $$4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} - 4 q^{10} - 4 q^{12} + 14 q^{13} + 4 q^{14} - 4 q^{15} + 4 q^{16} + 12 q^{17} - 6 q^{18} - 2 q^{19} + 4 q^{20} + 4 q^{21} + 4 q^{24} + 4 q^{25} - 14 q^{26} - 22 q^{27} - 4 q^{28} + 10 q^{29} + 4 q^{30} - 18 q^{31} - 4 q^{32} - 12 q^{34} - 4 q^{35} + 6 q^{36} + 18 q^{37} + 2 q^{38} - 30 q^{39} - 4 q^{40} + 24 q^{41} - 4 q^{42} + 10 q^{43} + 6 q^{45} - 8 q^{47} - 4 q^{48} + 4 q^{49} - 4 q^{50} + 14 q^{52} + 20 q^{53} + 22 q^{54} + 4 q^{56} + 30 q^{57} - 10 q^{58} - 14 q^{59} - 4 q^{60} + 14 q^{61} + 18 q^{62} - 6 q^{63} + 4 q^{64} + 14 q^{65} + 12 q^{68} - 4 q^{69} + 4 q^{70} - 14 q^{71} - 6 q^{72} + 30 q^{73} - 18 q^{74} - 4 q^{75} - 2 q^{76} + 30 q^{78} + 8 q^{79} + 4 q^{80} + 16 q^{81} - 24 q^{82} + 8 q^{83} + 4 q^{84} + 12 q^{85} - 10 q^{86} + 2 q^{87} - 4 q^{89} - 6 q^{90} - 14 q^{91} - 2 q^{93} + 8 q^{94} - 2 q^{95} + 4 q^{96} - 10 q^{97} - 4 q^{98} + O(q^{100})$$

## 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
$$3$$ 0.543362 0.313710 0.156855 0.987622i $$-0.449864\pi$$
0.156855 + 0.987622i $$0.449864\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −0.543362 −0.221827
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ −2.70476 −0.901586
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ 0.543362 0.156855
$$13$$ −1.65861 −0.460015 −0.230008 0.973189i $$-0.573875\pi$$
−0.230008 + 0.973189i $$0.573875\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0.543362 0.140295
$$16$$ 1.00000 0.250000
$$17$$ 2.04615 0.496264 0.248132 0.968726i $$-0.420183\pi$$
0.248132 + 0.968726i $$0.420183\pi$$
$$18$$ 2.70476 0.637518
$$19$$ 3.70476 0.849930 0.424965 0.905210i $$-0.360286\pi$$
0.424965 + 0.905210i $$0.360286\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −0.543362 −0.118571
$$22$$ 0 0
$$23$$ −4.99442 −1.04141 −0.520705 0.853737i $$-0.674331\pi$$
−0.520705 + 0.853737i $$0.674331\pi$$
$$24$$ −0.543362 −0.110913
$$25$$ 1.00000 0.200000
$$26$$ 1.65861 0.325280
$$27$$ −3.09975 −0.596547
$$28$$ −1.00000 −0.188982
$$29$$ 7.13632 1.32518 0.662591 0.748982i $$-0.269457\pi$$
0.662591 + 0.748982i $$0.269457\pi$$
$$30$$ −0.543362 −0.0992039
$$31$$ −10.1809 −1.82854 −0.914271 0.405102i $$-0.867236\pi$$
−0.914271 + 0.405102i $$0.867236\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −2.04615 −0.350912
$$35$$ −1.00000 −0.169031
$$36$$ −2.70476 −0.450793
$$37$$ 9.13632 1.50200 0.751001 0.660301i $$-0.229571\pi$$
0.751001 + 0.660301i $$0.229571\pi$$
$$38$$ −3.70476 −0.600991
$$39$$ −0.901224 −0.144311
$$40$$ −1.00000 −0.158114
$$41$$ 5.04615 0.788076 0.394038 0.919094i $$-0.371078\pi$$
0.394038 + 0.919094i $$0.371078\pi$$
$$42$$ 0.543362 0.0838426
$$43$$ 5.78688 0.882491 0.441245 0.897387i $$-0.354537\pi$$
0.441245 + 0.897387i $$0.354537\pi$$
$$44$$ 0 0
$$45$$ −2.70476 −0.403201
$$46$$ 4.99442 0.736388
$$47$$ −4.56444 −0.665791 −0.332896 0.942964i $$-0.608026\pi$$
−0.332896 + 0.942964i $$0.608026\pi$$
$$48$$ 0.543362 0.0784275
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 1.11180 0.155683
$$52$$ −1.65861 −0.230008
$$53$$ 8.88262 1.22012 0.610061 0.792354i $$-0.291145\pi$$
0.610061 + 0.792354i $$0.291145\pi$$
$$54$$ 3.09975 0.421822
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 2.01302 0.266632
$$58$$ −7.13632 −0.937045
$$59$$ −14.0846 −1.83366 −0.916829 0.399280i $$-0.869260\pi$$
−0.916829 + 0.399280i $$0.869260\pi$$
$$60$$ 0.543362 0.0701477
$$61$$ −3.30516 −0.423183 −0.211591 0.977358i $$-0.567865\pi$$
−0.211591 + 0.977358i $$0.567865\pi$$
$$62$$ 10.1809 1.29298
$$63$$ 2.70476 0.340767
$$64$$ 1.00000 0.125000
$$65$$ −1.65861 −0.205725
$$66$$ 0 0
$$67$$ −3.18992 −0.389711 −0.194855 0.980832i $$-0.562424\pi$$
−0.194855 + 0.980832i $$0.562424\pi$$
$$68$$ 2.04615 0.248132
$$69$$ −2.71378 −0.326701
$$70$$ 1.00000 0.119523
$$71$$ −7.80795 −0.926633 −0.463317 0.886193i $$-0.653341\pi$$
−0.463317 + 0.886193i $$0.653341\pi$$
$$72$$ 2.70476 0.318759
$$73$$ −10.9046 −1.27629 −0.638143 0.769918i $$-0.720297\pi$$
−0.638143 + 0.769918i $$0.720297\pi$$
$$74$$ −9.13632 −1.06208
$$75$$ 0.543362 0.0627420
$$76$$ 3.70476 0.424965
$$77$$ 0 0
$$78$$ 0.901224 0.102044
$$79$$ 11.2054 1.26071 0.630354 0.776308i $$-0.282910\pi$$
0.630354 + 0.776308i $$0.282910\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 6.42999 0.714443
$$82$$ −5.04615 −0.557254
$$83$$ 4.07812 0.447632 0.223816 0.974631i $$-0.428149\pi$$
0.223816 + 0.974631i $$0.428149\pi$$
$$84$$ −0.543362 −0.0592856
$$85$$ 2.04615 0.221936
$$86$$ −5.78688 −0.624015
$$87$$ 3.87760 0.415723
$$88$$ 0 0
$$89$$ −3.32679 −0.352639 −0.176320 0.984333i $$-0.556419\pi$$
−0.176320 + 0.984333i $$0.556419\pi$$
$$90$$ 2.70476 0.285107
$$91$$ 1.65861 0.173869
$$92$$ −4.99442 −0.520705
$$93$$ −5.53191 −0.573632
$$94$$ 4.56444 0.470786
$$95$$ 3.70476 0.380100
$$96$$ −0.543362 −0.0554566
$$97$$ 9.19639 0.933752 0.466876 0.884323i $$-0.345379\pi$$
0.466876 + 0.884323i $$0.345379\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 2.73328 0.271972 0.135986 0.990711i $$-0.456580\pi$$
0.135986 + 0.990711i $$0.456580\pi$$
$$102$$ −1.11180 −0.110085
$$103$$ −1.54736 −0.152466 −0.0762332 0.997090i $$-0.524289\pi$$
−0.0762332 + 0.997090i $$0.524289\pi$$
$$104$$ 1.65861 0.162640
$$105$$ −0.543362 −0.0530267
$$106$$ −8.88262 −0.862757
$$107$$ 12.2089 1.18028 0.590138 0.807303i $$-0.299073\pi$$
0.590138 + 0.807303i $$0.299073\pi$$
$$108$$ −3.09975 −0.298273
$$109$$ −1.32376 −0.126794 −0.0633968 0.997988i $$-0.520193\pi$$
−0.0633968 + 0.997988i $$0.520193\pi$$
$$110$$ 0 0
$$111$$ 4.96433 0.471193
$$112$$ −1.00000 −0.0944911
$$113$$ 3.45851 0.325349 0.162675 0.986680i $$-0.447988\pi$$
0.162675 + 0.986680i $$0.447988\pi$$
$$114$$ −2.01302 −0.188537
$$115$$ −4.99442 −0.465732
$$116$$ 7.13632 0.662591
$$117$$ 4.48613 0.414743
$$118$$ 14.0846 1.29659
$$119$$ −2.04615 −0.187570
$$120$$ −0.543362 −0.0496019
$$121$$ 0 0
$$122$$ 3.30516 0.299236
$$123$$ 2.74189 0.247228
$$124$$ −10.1809 −0.914271
$$125$$ 1.00000 0.0894427
$$126$$ −2.70476 −0.240959
$$127$$ −18.2239 −1.61711 −0.808557 0.588418i $$-0.799751\pi$$
−0.808557 + 0.588418i $$0.799751\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 3.14437 0.276846
$$130$$ 1.65861 0.145470
$$131$$ −9.47001 −0.827398 −0.413699 0.910414i $$-0.635763\pi$$
−0.413699 + 0.910414i $$0.635763\pi$$
$$132$$ 0 0
$$133$$ −3.70476 −0.321243
$$134$$ 3.18992 0.275567
$$135$$ −3.09975 −0.266784
$$136$$ −2.04615 −0.175456
$$137$$ 11.4807 0.980866 0.490433 0.871479i $$-0.336839\pi$$
0.490433 + 0.871479i $$0.336839\pi$$
$$138$$ 2.71378 0.231012
$$139$$ −4.86312 −0.412485 −0.206242 0.978501i $$-0.566124\pi$$
−0.206242 + 0.978501i $$0.566124\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ −2.48014 −0.208866
$$142$$ 7.80795 0.655229
$$143$$ 0 0
$$144$$ −2.70476 −0.225396
$$145$$ 7.13632 0.592639
$$146$$ 10.9046 0.902471
$$147$$ 0.543362 0.0448157
$$148$$ 9.13632 0.751001
$$149$$ 7.60901 0.623355 0.311677 0.950188i $$-0.399109\pi$$
0.311677 + 0.950188i $$0.399109\pi$$
$$150$$ −0.543362 −0.0443653
$$151$$ 10.8640 0.884102 0.442051 0.896990i $$-0.354251\pi$$
0.442051 + 0.896990i $$0.354251\pi$$
$$152$$ −3.70476 −0.300496
$$153$$ −5.53434 −0.447425
$$154$$ 0 0
$$155$$ −10.1809 −0.817749
$$156$$ −0.901224 −0.0721557
$$157$$ 9.67811 0.772397 0.386199 0.922416i $$-0.373788\pi$$
0.386199 + 0.922416i $$0.373788\pi$$
$$158$$ −11.2054 −0.891455
$$159$$ 4.82648 0.382765
$$160$$ −1.00000 −0.0790569
$$161$$ 4.99442 0.393616
$$162$$ −6.42999 −0.505188
$$163$$ 16.3076 1.27731 0.638656 0.769492i $$-0.279491\pi$$
0.638656 + 0.769492i $$0.279491\pi$$
$$164$$ 5.04615 0.394038
$$165$$ 0 0
$$166$$ −4.07812 −0.316523
$$167$$ −9.49351 −0.734630 −0.367315 0.930097i $$-0.619723\pi$$
−0.367315 + 0.930097i $$0.619723\pi$$
$$168$$ 0.543362 0.0419213
$$169$$ −10.2490 −0.788386
$$170$$ −2.04615 −0.156933
$$171$$ −10.0205 −0.766285
$$172$$ 5.78688 0.441245
$$173$$ 1.44017 0.109494 0.0547470 0.998500i $$-0.482565\pi$$
0.0547470 + 0.998500i $$0.482565\pi$$
$$174$$ −3.87760 −0.293960
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ −7.65303 −0.575237
$$178$$ 3.32679 0.249354
$$179$$ −9.52847 −0.712191 −0.356095 0.934450i $$-0.615892\pi$$
−0.356095 + 0.934450i $$0.615892\pi$$
$$180$$ −2.70476 −0.201601
$$181$$ −11.5431 −0.857992 −0.428996 0.903306i $$-0.641133\pi$$
−0.428996 + 0.903306i $$0.641133\pi$$
$$182$$ −1.65861 −0.122944
$$183$$ −1.79590 −0.132757
$$184$$ 4.99442 0.368194
$$185$$ 9.13632 0.671716
$$186$$ 5.53191 0.405619
$$187$$ 0 0
$$188$$ −4.56444 −0.332896
$$189$$ 3.09975 0.225474
$$190$$ −3.70476 −0.268771
$$191$$ 25.8516 1.87055 0.935276 0.353918i $$-0.115151\pi$$
0.935276 + 0.353918i $$0.115151\pi$$
$$192$$ 0.543362 0.0392138
$$193$$ 25.6289 1.84481 0.922403 0.386230i $$-0.126223\pi$$
0.922403 + 0.386230i $$0.126223\pi$$
$$194$$ −9.19639 −0.660263
$$195$$ −0.901224 −0.0645380
$$196$$ 1.00000 0.0714286
$$197$$ 10.5223 0.749682 0.374841 0.927089i $$-0.377697\pi$$
0.374841 + 0.927089i $$0.377697\pi$$
$$198$$ 0 0
$$199$$ 18.3423 1.30025 0.650125 0.759827i $$-0.274716\pi$$
0.650125 + 0.759827i $$0.274716\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −1.73328 −0.122256
$$202$$ −2.73328 −0.192313
$$203$$ −7.13632 −0.500871
$$204$$ 1.11180 0.0778416
$$205$$ 5.04615 0.352438
$$206$$ 1.54736 0.107810
$$207$$ 13.5087 0.938920
$$208$$ −1.65861 −0.115004
$$209$$ 0 0
$$210$$ 0.543362 0.0374955
$$211$$ −23.6304 −1.62679 −0.813393 0.581715i $$-0.802382\pi$$
−0.813393 + 0.581715i $$0.802382\pi$$
$$212$$ 8.88262 0.610061
$$213$$ −4.24254 −0.290694
$$214$$ −12.2089 −0.834581
$$215$$ 5.78688 0.394662
$$216$$ 3.09975 0.210911
$$217$$ 10.1809 0.691124
$$218$$ 1.32376 0.0896566
$$219$$ −5.92514 −0.400384
$$220$$ 0 0
$$221$$ −3.39376 −0.228289
$$222$$ −4.96433 −0.333184
$$223$$ 2.76854 0.185395 0.0926974 0.995694i $$-0.470451\pi$$
0.0926974 + 0.995694i $$0.470451\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −2.70476 −0.180317
$$226$$ −3.45851 −0.230057
$$227$$ 1.59296 0.105728 0.0528642 0.998602i $$-0.483165\pi$$
0.0528642 + 0.998602i $$0.483165\pi$$
$$228$$ 2.01302 0.133316
$$229$$ 23.5747 1.55786 0.778929 0.627113i $$-0.215763\pi$$
0.778929 + 0.627113i $$0.215763\pi$$
$$230$$ 4.99442 0.329323
$$231$$ 0 0
$$232$$ −7.13632 −0.468522
$$233$$ 3.72239 0.243862 0.121931 0.992539i $$-0.461091\pi$$
0.121931 + 0.992539i $$0.461091\pi$$
$$234$$ −4.48613 −0.293268
$$235$$ −4.56444 −0.297751
$$236$$ −14.0846 −0.916829
$$237$$ 6.08860 0.395497
$$238$$ 2.04615 0.132632
$$239$$ 15.0088 0.970836 0.485418 0.874282i $$-0.338667\pi$$
0.485418 + 0.874282i $$0.338667\pi$$
$$240$$ 0.543362 0.0350739
$$241$$ 23.4709 1.51189 0.755947 0.654633i $$-0.227177\pi$$
0.755947 + 0.654633i $$0.227177\pi$$
$$242$$ 0 0
$$243$$ 12.7931 0.820675
$$244$$ −3.30516 −0.211591
$$245$$ 1.00000 0.0638877
$$246$$ −2.74189 −0.174816
$$247$$ −6.14474 −0.390980
$$248$$ 10.1809 0.646488
$$249$$ 2.21589 0.140427
$$250$$ −1.00000 −0.0632456
$$251$$ 23.8705 1.50669 0.753346 0.657624i $$-0.228438\pi$$
0.753346 + 0.657624i $$0.228438\pi$$
$$252$$ 2.70476 0.170384
$$253$$ 0 0
$$254$$ 18.2239 1.14347
$$255$$ 1.11180 0.0696236
$$256$$ 1.00000 0.0625000
$$257$$ 3.13935 0.195827 0.0979136 0.995195i $$-0.468783\pi$$
0.0979136 + 0.995195i $$0.468783\pi$$
$$258$$ −3.14437 −0.195760
$$259$$ −9.13632 −0.567703
$$260$$ −1.65861 −0.102862
$$261$$ −19.3020 −1.19476
$$262$$ 9.47001 0.585059
$$263$$ 18.0997 1.11608 0.558039 0.829815i $$-0.311554\pi$$
0.558039 + 0.829815i $$0.311554\pi$$
$$264$$ 0 0
$$265$$ 8.88262 0.545655
$$266$$ 3.70476 0.227153
$$267$$ −1.80765 −0.110627
$$268$$ −3.18992 −0.194855
$$269$$ −5.65771 −0.344957 −0.172478 0.985013i $$-0.555177\pi$$
−0.172478 + 0.985013i $$0.555177\pi$$
$$270$$ 3.09975 0.188645
$$271$$ 31.7788 1.93042 0.965211 0.261472i $$-0.0842080\pi$$
0.965211 + 0.261472i $$0.0842080\pi$$
$$272$$ 2.04615 0.124066
$$273$$ 0.901224 0.0545446
$$274$$ −11.4807 −0.693577
$$275$$ 0 0
$$276$$ −2.71378 −0.163350
$$277$$ −2.40678 −0.144610 −0.0723048 0.997383i $$-0.523035\pi$$
−0.0723048 + 0.997383i $$0.523035\pi$$
$$278$$ 4.86312 0.291671
$$279$$ 27.5369 1.64859
$$280$$ 1.00000 0.0597614
$$281$$ 11.2228 0.669495 0.334748 0.942308i $$-0.391349\pi$$
0.334748 + 0.942308i $$0.391349\pi$$
$$282$$ 2.48014 0.147690
$$283$$ 31.2559 1.85797 0.928986 0.370116i $$-0.120682\pi$$
0.928986 + 0.370116i $$0.120682\pi$$
$$284$$ −7.80795 −0.463317
$$285$$ 2.01302 0.119241
$$286$$ 0 0
$$287$$ −5.04615 −0.297865
$$288$$ 2.70476 0.159379
$$289$$ −12.8133 −0.753722
$$290$$ −7.13632 −0.419059
$$291$$ 4.99697 0.292928
$$292$$ −10.9046 −0.638143
$$293$$ 2.27230 0.132749 0.0663745 0.997795i $$-0.478857\pi$$
0.0663745 + 0.997795i $$0.478857\pi$$
$$294$$ −0.543362 −0.0316895
$$295$$ −14.0846 −0.820037
$$296$$ −9.13632 −0.531038
$$297$$ 0 0
$$298$$ −7.60901 −0.440778
$$299$$ 8.28379 0.479064
$$300$$ 0.543362 0.0313710
$$301$$ −5.78688 −0.333550
$$302$$ −10.8640 −0.625154
$$303$$ 1.48516 0.0853202
$$304$$ 3.70476 0.212482
$$305$$ −3.30516 −0.189253
$$306$$ 5.53434 0.316377
$$307$$ 5.55583 0.317088 0.158544 0.987352i $$-0.449320\pi$$
0.158544 + 0.987352i $$0.449320\pi$$
$$308$$ 0 0
$$309$$ −0.840779 −0.0478302
$$310$$ 10.1809 0.578236
$$311$$ −20.3446 −1.15364 −0.576818 0.816873i $$-0.695706\pi$$
−0.576818 + 0.816873i $$0.695706\pi$$
$$312$$ 0.901224 0.0510218
$$313$$ 28.0901 1.58775 0.793873 0.608083i $$-0.208061\pi$$
0.793873 + 0.608083i $$0.208061\pi$$
$$314$$ −9.67811 −0.546167
$$315$$ 2.70476 0.152396
$$316$$ 11.2054 0.630354
$$317$$ 7.61646 0.427783 0.213892 0.976857i $$-0.431386\pi$$
0.213892 + 0.976857i $$0.431386\pi$$
$$318$$ −4.82648 −0.270656
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 6.63383 0.370264
$$322$$ −4.99442 −0.278328
$$323$$ 7.58049 0.421790
$$324$$ 6.42999 0.357222
$$325$$ −1.65861 −0.0920030
$$326$$ −16.3076 −0.903197
$$327$$ −0.719283 −0.0397764
$$328$$ −5.04615 −0.278627
$$329$$ 4.56444 0.251645
$$330$$ 0 0
$$331$$ 9.86518 0.542239 0.271120 0.962546i $$-0.412606\pi$$
0.271120 + 0.962546i $$0.412606\pi$$
$$332$$ 4.07812 0.223816
$$333$$ −24.7115 −1.35418
$$334$$ 9.49351 0.519462
$$335$$ −3.18992 −0.174284
$$336$$ −0.543362 −0.0296428
$$337$$ −22.8921 −1.24701 −0.623507 0.781818i $$-0.714293\pi$$
−0.623507 + 0.781818i $$0.714293\pi$$
$$338$$ 10.2490 0.557473
$$339$$ 1.87922 0.102065
$$340$$ 2.04615 0.110968
$$341$$ 0 0
$$342$$ 10.0205 0.541845
$$343$$ −1.00000 −0.0539949
$$344$$ −5.78688 −0.312008
$$345$$ −2.71378 −0.146105
$$346$$ −1.44017 −0.0774239
$$347$$ −8.07715 −0.433604 −0.216802 0.976216i $$-0.569563\pi$$
−0.216802 + 0.976216i $$0.569563\pi$$
$$348$$ 3.87760 0.207861
$$349$$ 16.3618 0.875827 0.437913 0.899017i $$-0.355718\pi$$
0.437913 + 0.899017i $$0.355718\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ 5.14127 0.274421
$$352$$ 0 0
$$353$$ −18.5642 −0.988072 −0.494036 0.869442i $$-0.664479\pi$$
−0.494036 + 0.869442i $$0.664479\pi$$
$$354$$ 7.65303 0.406754
$$355$$ −7.80795 −0.414403
$$356$$ −3.32679 −0.176320
$$357$$ −1.11180 −0.0588427
$$358$$ 9.52847 0.503595
$$359$$ 4.59048 0.242276 0.121138 0.992636i $$-0.461346\pi$$
0.121138 + 0.992636i $$0.461346\pi$$
$$360$$ 2.70476 0.142553
$$361$$ −5.27477 −0.277619
$$362$$ 11.5431 0.606692
$$363$$ 0 0
$$364$$ 1.65861 0.0869347
$$365$$ −10.9046 −0.570773
$$366$$ 1.79590 0.0938732
$$367$$ 12.7694 0.666559 0.333280 0.942828i $$-0.391845\pi$$
0.333280 + 0.942828i $$0.391845\pi$$
$$368$$ −4.99442 −0.260352
$$369$$ −13.6486 −0.710518
$$370$$ −9.13632 −0.474975
$$371$$ −8.88262 −0.461163
$$372$$ −5.53191 −0.286816
$$373$$ −22.4340 −1.16159 −0.580795 0.814050i $$-0.697258\pi$$
−0.580795 + 0.814050i $$0.697258\pi$$
$$374$$ 0 0
$$375$$ 0.543362 0.0280591
$$376$$ 4.56444 0.235393
$$377$$ −11.8364 −0.609603
$$378$$ −3.09975 −0.159434
$$379$$ −30.1347 −1.54791 −0.773957 0.633238i $$-0.781725\pi$$
−0.773957 + 0.633238i $$0.781725\pi$$
$$380$$ 3.70476 0.190050
$$381$$ −9.90220 −0.507305
$$382$$ −25.8516 −1.32268
$$383$$ 27.9554 1.42845 0.714226 0.699915i $$-0.246779\pi$$
0.714226 + 0.699915i $$0.246779\pi$$
$$384$$ −0.543362 −0.0277283
$$385$$ 0 0
$$386$$ −25.6289 −1.30447
$$387$$ −15.6521 −0.795641
$$388$$ 9.19639 0.466876
$$389$$ 4.57649 0.232037 0.116019 0.993247i $$-0.462987\pi$$
0.116019 + 0.993247i $$0.462987\pi$$
$$390$$ 0.901224 0.0456353
$$391$$ −10.2193 −0.516814
$$392$$ −1.00000 −0.0505076
$$393$$ −5.14564 −0.259563
$$394$$ −10.5223 −0.530105
$$395$$ 11.2054 0.563806
$$396$$ 0 0
$$397$$ −37.9390 −1.90411 −0.952053 0.305933i $$-0.901032\pi$$
−0.952053 + 0.305933i $$0.901032\pi$$
$$398$$ −18.3423 −0.919416
$$399$$ −2.01302 −0.100777
$$400$$ 1.00000 0.0500000
$$401$$ 15.9557 0.796791 0.398395 0.917214i $$-0.369567\pi$$
0.398395 + 0.917214i $$0.369567\pi$$
$$402$$ 1.73328 0.0864482
$$403$$ 16.8861 0.841157
$$404$$ 2.73328 0.135986
$$405$$ 6.42999 0.319509
$$406$$ 7.13632 0.354170
$$407$$ 0 0
$$408$$ −1.11180 −0.0550423
$$409$$ −24.2684 −1.19999 −0.599997 0.800002i $$-0.704832\pi$$
−0.599997 + 0.800002i $$0.704832\pi$$
$$410$$ −5.04615 −0.249212
$$411$$ 6.23820 0.307708
$$412$$ −1.54736 −0.0762332
$$413$$ 14.0846 0.693058
$$414$$ −13.5087 −0.663917
$$415$$ 4.07812 0.200187
$$416$$ 1.65861 0.0813199
$$417$$ −2.64244 −0.129401
$$418$$ 0 0
$$419$$ −25.6838 −1.25473 −0.627367 0.778724i $$-0.715867\pi$$
−0.627367 + 0.778724i $$0.715867\pi$$
$$420$$ −0.543362 −0.0265133
$$421$$ −9.36362 −0.456355 −0.228178 0.973620i $$-0.573277\pi$$
−0.228178 + 0.973620i $$0.573277\pi$$
$$422$$ 23.6304 1.15031
$$423$$ 12.3457 0.600268
$$424$$ −8.88262 −0.431378
$$425$$ 2.04615 0.0992528
$$426$$ 4.24254 0.205552
$$427$$ 3.30516 0.159948
$$428$$ 12.2089 0.590138
$$429$$ 0 0
$$430$$ −5.78688 −0.279068
$$431$$ 16.9902 0.818387 0.409194 0.912448i $$-0.365810\pi$$
0.409194 + 0.912448i $$0.365810\pi$$
$$432$$ −3.09975 −0.149137
$$433$$ 6.70663 0.322300 0.161150 0.986930i $$-0.448480\pi$$
0.161150 + 0.986930i $$0.448480\pi$$
$$434$$ −10.1809 −0.488699
$$435$$ 3.87760 0.185917
$$436$$ −1.32376 −0.0633968
$$437$$ −18.5031 −0.885125
$$438$$ 5.92514 0.283114
$$439$$ 12.8474 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$440$$ 0 0
$$441$$ −2.70476 −0.128798
$$442$$ 3.39376 0.161425
$$443$$ 0.678365 0.0322301 0.0161151 0.999870i $$-0.494870\pi$$
0.0161151 + 0.999870i $$0.494870\pi$$
$$444$$ 4.96433 0.235597
$$445$$ −3.32679 −0.157705
$$446$$ −2.76854 −0.131094
$$447$$ 4.13445 0.195553
$$448$$ −1.00000 −0.0472456
$$449$$ 7.59525 0.358442 0.179221 0.983809i $$-0.442642\pi$$
0.179221 + 0.983809i $$0.442642\pi$$
$$450$$ 2.70476 0.127504
$$451$$ 0 0
$$452$$ 3.45851 0.162675
$$453$$ 5.90310 0.277352
$$454$$ −1.59296 −0.0747612
$$455$$ 1.65861 0.0777567
$$456$$ −2.01302 −0.0942685
$$457$$ −33.0502 −1.54602 −0.773012 0.634391i $$-0.781251\pi$$
−0.773012 + 0.634391i $$0.781251\pi$$
$$458$$ −23.5747 −1.10157
$$459$$ −6.34255 −0.296045
$$460$$ −4.99442 −0.232866
$$461$$ 20.2500 0.943136 0.471568 0.881830i $$-0.343688\pi$$
0.471568 + 0.881830i $$0.343688\pi$$
$$462$$ 0 0
$$463$$ 39.3213 1.82742 0.913708 0.406371i $$-0.133206\pi$$
0.913708 + 0.406371i $$0.133206\pi$$
$$464$$ 7.13632 0.331295
$$465$$ −5.53191 −0.256536
$$466$$ −3.72239 −0.172436
$$467$$ −18.8167 −0.870734 −0.435367 0.900253i $$-0.643381\pi$$
−0.435367 + 0.900253i $$0.643381\pi$$
$$468$$ 4.48613 0.207372
$$469$$ 3.18992 0.147297
$$470$$ 4.56444 0.210542
$$471$$ 5.25872 0.242309
$$472$$ 14.0846 0.648296
$$473$$ 0 0
$$474$$ −6.08860 −0.279658
$$475$$ 3.70476 0.169986
$$476$$ −2.04615 −0.0937851
$$477$$ −24.0253 −1.10005
$$478$$ −15.0088 −0.686485
$$479$$ −7.56631 −0.345713 −0.172857 0.984947i $$-0.555300\pi$$
−0.172857 + 0.984947i $$0.555300\pi$$
$$480$$ −0.543362 −0.0248010
$$481$$ −15.1536 −0.690943
$$482$$ −23.4709 −1.06907
$$483$$ 2.71378 0.123481
$$484$$ 0 0
$$485$$ 9.19639 0.417587
$$486$$ −12.7931 −0.580305
$$487$$ 7.92350 0.359048 0.179524 0.983754i $$-0.442544\pi$$
0.179524 + 0.983754i $$0.442544\pi$$
$$488$$ 3.30516 0.149618
$$489$$ 8.86095 0.400706
$$490$$ −1.00000 −0.0451754
$$491$$ −15.1101 −0.681909 −0.340954 0.940080i $$-0.610750\pi$$
−0.340954 + 0.940080i $$0.610750\pi$$
$$492$$ 2.74189 0.123614
$$493$$ 14.6020 0.657640
$$494$$ 6.14474 0.276465
$$495$$ 0 0
$$496$$ −10.1809 −0.457136
$$497$$ 7.80795 0.350235
$$498$$ −2.21589 −0.0992966
$$499$$ −5.96549 −0.267052 −0.133526 0.991045i $$-0.542630\pi$$
−0.133526 + 0.991045i $$0.542630\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −5.15841 −0.230461
$$502$$ −23.8705 −1.06539
$$503$$ −25.5217 −1.13795 −0.568977 0.822353i $$-0.692661\pi$$
−0.568977 + 0.822353i $$0.692661\pi$$
$$504$$ −2.70476 −0.120479
$$505$$ 2.73328 0.121629
$$506$$ 0 0
$$507$$ −5.56893 −0.247325
$$508$$ −18.2239 −0.808557
$$509$$ −27.7639 −1.23061 −0.615306 0.788289i $$-0.710967\pi$$
−0.615306 + 0.788289i $$0.710967\pi$$
$$510$$ −1.11180 −0.0492313
$$511$$ 10.9046 0.482391
$$512$$ −1.00000 −0.0441942
$$513$$ −11.4838 −0.507023
$$514$$ −3.13935 −0.138471
$$515$$ −1.54736 −0.0681850
$$516$$ 3.14437 0.138423
$$517$$ 0 0
$$518$$ 9.13632 0.401427
$$519$$ 0.782532 0.0343494
$$520$$ 1.65861 0.0727348
$$521$$ 23.8699 1.04576 0.522880 0.852407i $$-0.324858\pi$$
0.522880 + 0.852407i $$0.324858\pi$$
$$522$$ 19.3020 0.844826
$$523$$ 2.38789 0.104415 0.0522075 0.998636i $$-0.483374\pi$$
0.0522075 + 0.998636i $$0.483374\pi$$
$$524$$ −9.47001 −0.413699
$$525$$ −0.543362 −0.0237143
$$526$$ −18.0997 −0.789187
$$527$$ −20.8316 −0.907440
$$528$$ 0 0
$$529$$ 1.94427 0.0845336
$$530$$ −8.88262 −0.385837
$$531$$ 38.0954 1.65320
$$532$$ −3.70476 −0.160622
$$533$$ −8.36958 −0.362527
$$534$$ 1.80765 0.0782248
$$535$$ 12.2089 0.527835
$$536$$ 3.18992 0.137783
$$537$$ −5.17741 −0.223422
$$538$$ 5.65771 0.243921
$$539$$ 0 0
$$540$$ −3.09975 −0.133392
$$541$$ 24.8483 1.06831 0.534155 0.845386i $$-0.320630\pi$$
0.534155 + 0.845386i $$0.320630\pi$$
$$542$$ −31.7788 −1.36501
$$543$$ −6.27208 −0.269161
$$544$$ −2.04615 −0.0877280
$$545$$ −1.32376 −0.0567038
$$546$$ −0.901224 −0.0385688
$$547$$ 33.1504 1.41741 0.708704 0.705506i $$-0.249280\pi$$
0.708704 + 0.705506i $$0.249280\pi$$
$$548$$ 11.4807 0.490433
$$549$$ 8.93967 0.381536
$$550$$ 0 0
$$551$$ 26.4383 1.12631
$$552$$ 2.71378 0.115506
$$553$$ −11.2054 −0.476503
$$554$$ 2.40678 0.102254
$$555$$ 4.96433 0.210724
$$556$$ −4.86312 −0.206242
$$557$$ 28.0256 1.18748 0.593741 0.804657i $$-0.297651\pi$$
0.593741 + 0.804657i $$0.297651\pi$$
$$558$$ −27.5369 −1.16573
$$559$$ −9.59816 −0.405959
$$560$$ −1.00000 −0.0422577
$$561$$ 0 0
$$562$$ −11.2228 −0.473405
$$563$$ 3.17711 0.133899 0.0669495 0.997756i $$-0.478673\pi$$
0.0669495 + 0.997756i $$0.478673\pi$$
$$564$$ −2.48014 −0.104433
$$565$$ 3.45851 0.145501
$$566$$ −31.2559 −1.31378
$$567$$ −6.42999 −0.270034
$$568$$ 7.80795 0.327614
$$569$$ 18.7562 0.786302 0.393151 0.919474i $$-0.371385\pi$$
0.393151 + 0.919474i $$0.371385\pi$$
$$570$$ −2.01302 −0.0843163
$$571$$ −20.5329 −0.859276 −0.429638 0.903001i $$-0.641359\pi$$
−0.429638 + 0.903001i $$0.641359\pi$$
$$572$$ 0 0
$$573$$ 14.0468 0.586811
$$574$$ 5.04615 0.210622
$$575$$ −4.99442 −0.208282
$$576$$ −2.70476 −0.112698
$$577$$ 16.7729 0.698264 0.349132 0.937074i $$-0.386476\pi$$
0.349132 + 0.937074i $$0.386476\pi$$
$$578$$ 12.8133 0.532962
$$579$$ 13.9257 0.578734
$$580$$ 7.13632 0.296320
$$581$$ −4.07812 −0.169189
$$582$$ −4.99697 −0.207131
$$583$$ 0 0
$$584$$ 10.9046 0.451235
$$585$$ 4.48613 0.185479
$$586$$ −2.27230 −0.0938677
$$587$$ 25.0595 1.03432 0.517158 0.855890i $$-0.326990\pi$$
0.517158 + 0.855890i $$0.326990\pi$$
$$588$$ 0.543362 0.0224079
$$589$$ −37.7178 −1.55413
$$590$$ 14.0846 0.579854
$$591$$ 5.71741 0.235183
$$592$$ 9.13632 0.375500
$$593$$ 44.2586 1.81748 0.908741 0.417360i $$-0.137044\pi$$
0.908741 + 0.417360i $$0.137044\pi$$
$$594$$ 0 0
$$595$$ −2.04615 −0.0838840
$$596$$ 7.60901 0.311677
$$597$$ 9.96650 0.407902
$$598$$ −8.28379 −0.338749
$$599$$ 2.39133 0.0977072 0.0488536 0.998806i $$-0.484443\pi$$
0.0488536 + 0.998806i $$0.484443\pi$$
$$600$$ −0.543362 −0.0221827
$$601$$ −34.6639 −1.41397 −0.706984 0.707229i $$-0.749945\pi$$
−0.706984 + 0.707229i $$0.749945\pi$$
$$602$$ 5.78688 0.235856
$$603$$ 8.62796 0.351358
$$604$$ 10.8640 0.442051
$$605$$ 0 0
$$606$$ −1.48516 −0.0603305
$$607$$ 27.2676 1.10676 0.553379 0.832929i $$-0.313338\pi$$
0.553379 + 0.832929i $$0.313338\pi$$
$$608$$ −3.70476 −0.150248
$$609$$ −3.87760 −0.157128
$$610$$ 3.30516 0.133822
$$611$$ 7.57061 0.306274
$$612$$ −5.53434 −0.223712
$$613$$ 0.453748 0.0183267 0.00916336 0.999958i $$-0.497083\pi$$
0.00916336 + 0.999958i $$0.497083\pi$$
$$614$$ −5.55583 −0.224215
$$615$$ 2.74189 0.110564
$$616$$ 0 0
$$617$$ 35.9399 1.44688 0.723442 0.690385i $$-0.242559\pi$$
0.723442 + 0.690385i $$0.242559\pi$$
$$618$$ 0.840779 0.0338211
$$619$$ −44.4805 −1.78782 −0.893912 0.448243i $$-0.852050\pi$$
−0.893912 + 0.448243i $$0.852050\pi$$
$$620$$ −10.1809 −0.408875
$$621$$ 15.4815 0.621249
$$622$$ 20.3446 0.815743
$$623$$ 3.32679 0.133285
$$624$$ −0.901224 −0.0360778
$$625$$ 1.00000 0.0400000
$$626$$ −28.0901 −1.12271
$$627$$ 0 0
$$628$$ 9.67811 0.386199
$$629$$ 18.6943 0.745390
$$630$$ −2.70476 −0.107760
$$631$$ 19.3612 0.770756 0.385378 0.922759i $$-0.374071\pi$$
0.385378 + 0.922759i $$0.374071\pi$$
$$632$$ −11.2054 −0.445728
$$633$$ −12.8399 −0.510339
$$634$$ −7.61646 −0.302488
$$635$$ −18.2239 −0.723195
$$636$$ 4.82648 0.191382
$$637$$ −1.65861 −0.0657164
$$638$$ 0 0
$$639$$ 21.1186 0.835440
$$640$$ −1.00000 −0.0395285
$$641$$ 24.0572 0.950203 0.475102 0.879931i $$-0.342411\pi$$
0.475102 + 0.879931i $$0.342411\pi$$
$$642$$ −6.63383 −0.261816
$$643$$ −14.4357 −0.569286 −0.284643 0.958634i $$-0.591875\pi$$
−0.284643 + 0.958634i $$0.591875\pi$$
$$644$$ 4.99442 0.196808
$$645$$ 3.14437 0.123809
$$646$$ −7.58049 −0.298250
$$647$$ −14.6590 −0.576306 −0.288153 0.957584i $$-0.593041\pi$$
−0.288153 + 0.957584i $$0.593041\pi$$
$$648$$ −6.42999 −0.252594
$$649$$ 0 0
$$650$$ 1.65861 0.0650560
$$651$$ 5.53191 0.216813
$$652$$ 16.3076 0.638656
$$653$$ 24.1399 0.944668 0.472334 0.881420i $$-0.343412\pi$$
0.472334 + 0.881420i $$0.343412\pi$$
$$654$$ 0.719283 0.0281262
$$655$$ −9.47001 −0.370024
$$656$$ 5.04615 0.197019
$$657$$ 29.4943 1.15068
$$658$$ −4.56444 −0.177940
$$659$$ −25.1109 −0.978182 −0.489091 0.872233i $$-0.662671\pi$$
−0.489091 + 0.872233i $$0.662671\pi$$
$$660$$ 0 0
$$661$$ −29.7048 −1.15538 −0.577692 0.816255i $$-0.696047\pi$$
−0.577692 + 0.816255i $$0.696047\pi$$
$$662$$ −9.86518 −0.383421
$$663$$ −1.84404 −0.0716166
$$664$$ −4.07812 −0.158262
$$665$$ −3.70476 −0.143664
$$666$$ 24.7115 0.957552
$$667$$ −35.6418 −1.38006
$$668$$ −9.49351 −0.367315
$$669$$ 1.50432 0.0581602
$$670$$ 3.18992 0.123237
$$671$$ 0 0
$$672$$ 0.543362 0.0209606
$$673$$ −49.0571 −1.89101 −0.945505 0.325607i $$-0.894431\pi$$
−0.945505 + 0.325607i $$0.894431\pi$$
$$674$$ 22.8921 0.881772
$$675$$ −3.09975 −0.119309
$$676$$ −10.2490 −0.394193
$$677$$ 27.6246 1.06170 0.530849 0.847466i $$-0.321873\pi$$
0.530849 + 0.847466i $$0.321873\pi$$
$$678$$ −1.87922 −0.0721711
$$679$$ −9.19639 −0.352925
$$680$$ −2.04615 −0.0784663
$$681$$ 0.865553 0.0331681
$$682$$ 0 0
$$683$$ 6.50934 0.249073 0.124536 0.992215i $$-0.460256\pi$$
0.124536 + 0.992215i $$0.460256\pi$$
$$684$$ −10.0205 −0.383142
$$685$$ 11.4807 0.438657
$$686$$ 1.00000 0.0381802
$$687$$ 12.8096 0.488716
$$688$$ 5.78688 0.220623
$$689$$ −14.7328 −0.561275
$$690$$ 2.71378 0.103312
$$691$$ 19.8453 0.754949 0.377475 0.926020i $$-0.376793\pi$$
0.377475 + 0.926020i $$0.376793\pi$$
$$692$$ 1.44017 0.0547470
$$693$$ 0 0
$$694$$ 8.07715 0.306604
$$695$$ −4.86312 −0.184469
$$696$$ −3.87760 −0.146980
$$697$$ 10.3252 0.391094
$$698$$ −16.3618 −0.619303
$$699$$ 2.02260 0.0765019
$$700$$ −1.00000 −0.0377964
$$701$$ −9.40172 −0.355098 −0.177549 0.984112i $$-0.556817\pi$$
−0.177549 + 0.984112i $$0.556817\pi$$
$$702$$ −5.14127 −0.194045
$$703$$ 33.8479 1.27660
$$704$$ 0 0
$$705$$ −2.48014 −0.0934075
$$706$$ 18.5642 0.698672
$$707$$ −2.73328 −0.102796
$$708$$ −7.65303 −0.287619
$$709$$ −26.9787 −1.01321 −0.506603 0.862180i $$-0.669099\pi$$
−0.506603 + 0.862180i $$0.669099\pi$$
$$710$$ 7.80795 0.293027
$$711$$ −30.3079 −1.13664
$$712$$ 3.32679 0.124677
$$713$$ 50.8477 1.90426
$$714$$ 1.11180 0.0416081
$$715$$ 0 0
$$716$$ −9.52847 −0.356095
$$717$$ 8.15519 0.304561
$$718$$ −4.59048 −0.171315
$$719$$ −23.7054 −0.884061 −0.442030 0.897000i $$-0.645742\pi$$
−0.442030 + 0.897000i $$0.645742\pi$$
$$720$$ −2.70476 −0.100800
$$721$$ 1.54736 0.0576269
$$722$$ 5.27477 0.196307
$$723$$ 12.7532 0.474296
$$724$$ −11.5431 −0.428996
$$725$$ 7.13632 0.265036
$$726$$ 0 0
$$727$$ −42.5192 −1.57695 −0.788475 0.615066i $$-0.789129\pi$$
−0.788475 + 0.615066i $$0.789129\pi$$
$$728$$ −1.65861 −0.0614721
$$729$$ −12.3387 −0.456989
$$730$$ 10.9046 0.403597
$$731$$ 11.8408 0.437949
$$732$$ −1.79590 −0.0663784
$$733$$ −4.52655 −0.167192 −0.0835959 0.996500i $$-0.526641\pi$$
−0.0835959 + 0.996500i $$0.526641\pi$$
$$734$$ −12.7694 −0.471328
$$735$$ 0.543362 0.0200422
$$736$$ 4.99442 0.184097
$$737$$ 0 0
$$738$$ 13.6486 0.502412
$$739$$ 31.4047 1.15524 0.577619 0.816306i $$-0.303982\pi$$
0.577619 + 0.816306i $$0.303982\pi$$
$$740$$ 9.13632 0.335858
$$741$$ −3.33882 −0.122655
$$742$$ 8.88262 0.326091
$$743$$ 9.01673 0.330792 0.165396 0.986227i $$-0.447110\pi$$
0.165396 + 0.986227i $$0.447110\pi$$
$$744$$ 5.53191 0.202810
$$745$$ 7.60901 0.278773
$$746$$ 22.4340 0.821368
$$747$$ −11.0303 −0.403578
$$748$$ 0 0
$$749$$ −12.2089 −0.446102
$$750$$ −0.543362 −0.0198408
$$751$$ 8.52599 0.311118 0.155559 0.987827i $$-0.450282\pi$$
0.155559 + 0.987827i $$0.450282\pi$$
$$752$$ −4.56444 −0.166448
$$753$$ 12.9703 0.472665
$$754$$ 11.8364 0.431055
$$755$$ 10.8640 0.395382
$$756$$ 3.09975 0.112737
$$757$$ 31.4517 1.14313 0.571566 0.820556i $$-0.306336\pi$$
0.571566 + 0.820556i $$0.306336\pi$$
$$758$$ 30.1347 1.09454
$$759$$ 0 0
$$760$$ −3.70476 −0.134386
$$761$$ 49.4950 1.79419 0.897096 0.441835i $$-0.145672\pi$$
0.897096 + 0.441835i $$0.145672\pi$$
$$762$$ 9.90220 0.358719
$$763$$ 1.32376 0.0479235
$$764$$ 25.8516 0.935276
$$765$$ −5.53434 −0.200094
$$766$$ −27.9554 −1.01007
$$767$$ 23.3608 0.843510
$$768$$ 0.543362 0.0196069
$$769$$ −32.2787 −1.16400 −0.582000 0.813189i $$-0.697730\pi$$
−0.582000 + 0.813189i $$0.697730\pi$$
$$770$$ 0 0
$$771$$ 1.70580 0.0614330
$$772$$ 25.6289 0.922403
$$773$$ 5.16410 0.185740 0.0928699 0.995678i $$-0.470396\pi$$
0.0928699 + 0.995678i $$0.470396\pi$$
$$774$$ 15.6521 0.562603
$$775$$ −10.1809 −0.365709
$$776$$ −9.19639 −0.330131
$$777$$ −4.96433 −0.178094
$$778$$ −4.57649 −0.164075
$$779$$ 18.6948 0.669809
$$780$$ −0.901224 −0.0322690
$$781$$ 0 0
$$782$$ 10.2193 0.365443
$$783$$ −22.1208 −0.790533
$$784$$ 1.00000 0.0357143
$$785$$ 9.67811 0.345426
$$786$$ 5.14564 0.183539
$$787$$ −14.6338 −0.521638 −0.260819 0.965388i $$-0.583993\pi$$
−0.260819 + 0.965388i $$0.583993\pi$$
$$788$$ 10.5223 0.374841
$$789$$ 9.83471 0.350125
$$790$$ −11.2054 −0.398671
$$791$$ −3.45851 −0.122970
$$792$$ 0 0
$$793$$ 5.48197 0.194671
$$794$$ 37.9390 1.34641
$$795$$ 4.82648 0.171178
$$796$$ 18.3423 0.650125
$$797$$ 29.1014 1.03082 0.515412 0.856942i $$-0.327639\pi$$
0.515412 + 0.856942i $$0.327639\pi$$
$$798$$ 2.01302 0.0712603
$$799$$ −9.33952 −0.330408
$$800$$ −1.00000 −0.0353553
$$801$$ 8.99817 0.317935
$$802$$ −15.9557 −0.563416
$$803$$ 0 0
$$804$$ −1.73328 −0.0611281
$$805$$ 4.99442 0.176030
$$806$$ −16.8861 −0.594788
$$807$$ −3.07418 −0.108216
$$808$$ −2.73328 −0.0961565
$$809$$ −29.3769 −1.03284 −0.516418 0.856337i $$-0.672735\pi$$
−0.516418 + 0.856337i $$0.672735\pi$$
$$810$$ −6.42999 −0.225927
$$811$$ 27.8818 0.979064 0.489532 0.871985i $$-0.337168\pi$$
0.489532 + 0.871985i $$0.337168\pi$$
$$812$$ −7.13632 −0.250436
$$813$$ 17.2674 0.605593
$$814$$ 0 0
$$815$$ 16.3076 0.571232
$$816$$ 1.11180 0.0389208
$$817$$ 21.4390 0.750055
$$818$$ 24.2684 0.848524
$$819$$ −4.48613 −0.156758
$$820$$ 5.04615 0.176219
$$821$$ 5.70412 0.199075 0.0995375 0.995034i $$-0.468264\pi$$
0.0995375 + 0.995034i $$0.468264\pi$$
$$822$$ −6.23820 −0.217582
$$823$$ −11.2756 −0.393044 −0.196522 0.980499i $$-0.562965\pi$$
−0.196522 + 0.980499i $$0.562965\pi$$
$$824$$ 1.54736 0.0539050
$$825$$ 0 0
$$826$$ −14.0846 −0.490066
$$827$$ 10.9311 0.380110 0.190055 0.981773i $$-0.439133\pi$$
0.190055 + 0.981773i $$0.439133\pi$$
$$828$$ 13.5087 0.469460
$$829$$ 0.763376 0.0265131 0.0132566 0.999912i $$-0.495780\pi$$
0.0132566 + 0.999912i $$0.495780\pi$$
$$830$$ −4.07812 −0.141554
$$831$$ −1.30776 −0.0453655
$$832$$ −1.65861 −0.0575019
$$833$$ 2.04615 0.0708949
$$834$$ 2.64244 0.0915001
$$835$$ −9.49351 −0.328536
$$836$$ 0 0
$$837$$ 31.5582 1.09081
$$838$$ 25.6838 0.887231
$$839$$ −27.3649 −0.944740 −0.472370 0.881400i $$-0.656601\pi$$
−0.472370 + 0.881400i $$0.656601\pi$$
$$840$$ 0.543362 0.0187478
$$841$$ 21.9271 0.756106
$$842$$ 9.36362 0.322692
$$843$$ 6.09803 0.210027
$$844$$ −23.6304 −0.813393
$$845$$ −10.2490 −0.352577
$$846$$ −12.3457 −0.424454
$$847$$ 0 0
$$848$$ 8.88262 0.305031
$$849$$ 16.9833 0.582864
$$850$$ −2.04615 −0.0701824
$$851$$ −45.6307 −1.56420
$$852$$ −4.24254 −0.145347
$$853$$ 36.3022 1.24296 0.621482 0.783428i $$-0.286531\pi$$
0.621482 + 0.783428i $$0.286531\pi$$
$$854$$ −3.30516 −0.113100
$$855$$ −10.0205 −0.342693
$$856$$ −12.2089 −0.417290
$$857$$ 30.3248 1.03587 0.517937 0.855419i $$-0.326700\pi$$
0.517937 + 0.855419i $$0.326700\pi$$
$$858$$ 0 0
$$859$$ −45.7487 −1.56093 −0.780463 0.625202i $$-0.785017\pi$$
−0.780463 + 0.625202i $$0.785017\pi$$
$$860$$ 5.78688 0.197331
$$861$$ −2.74189 −0.0934432
$$862$$ −16.9902 −0.578687
$$863$$ −4.79812 −0.163330 −0.0816649 0.996660i $$-0.526024\pi$$
−0.0816649 + 0.996660i $$0.526024\pi$$
$$864$$ 3.09975 0.105456
$$865$$ 1.44017 0.0489672
$$866$$ −6.70663 −0.227901
$$867$$ −6.96224 −0.236450
$$868$$ 10.1809 0.345562
$$869$$ 0 0
$$870$$ −3.87760 −0.131463
$$871$$ 5.29082 0.179273
$$872$$ 1.32376 0.0448283
$$873$$ −24.8740 −0.841858
$$874$$ 18.5031 0.625878
$$875$$ −1.00000 −0.0338062
$$876$$ −5.92514 −0.200192
$$877$$ 32.5055 1.09763 0.548816 0.835943i $$-0.315079\pi$$
0.548816 + 0.835943i $$0.315079\pi$$
$$878$$ −12.8474 −0.433578
$$879$$ 1.23468 0.0416447
$$880$$ 0 0
$$881$$ −45.5066 −1.53316 −0.766578 0.642151i $$-0.778042\pi$$
−0.766578 + 0.642151i $$0.778042\pi$$
$$882$$ 2.70476 0.0910739
$$883$$ −16.9399 −0.570071 −0.285036 0.958517i $$-0.592005\pi$$
−0.285036 + 0.958517i $$0.592005\pi$$
$$884$$ −3.39376 −0.114145
$$885$$ −7.65303 −0.257254
$$886$$ −0.678365 −0.0227901
$$887$$ −1.79960 −0.0604248 −0.0302124 0.999544i $$-0.509618\pi$$
−0.0302124 + 0.999544i $$0.509618\pi$$
$$888$$ −4.96433 −0.166592
$$889$$ 18.2239 0.611211
$$890$$ 3.32679 0.111514
$$891$$ 0 0
$$892$$ 2.76854 0.0926974
$$893$$ −16.9101 −0.565876
$$894$$ −4.13445 −0.138277
$$895$$ −9.52847 −0.318501
$$896$$ 1.00000 0.0334077
$$897$$ 4.50110 0.150287
$$898$$ −7.59525 −0.253457
$$899$$ −72.6541 −2.42315
$$900$$ −2.70476 −0.0901586
$$901$$ 18.1752 0.605503
$$902$$ 0 0
$$903$$ −3.14437 −0.104638
$$904$$ −3.45851 −0.115028
$$905$$ −11.5431 −0.383706
$$906$$ −5.90310 −0.196117
$$907$$ 18.0878 0.600595 0.300297 0.953846i $$-0.402914\pi$$
0.300297 + 0.953846i $$0.402914\pi$$
$$908$$ 1.59296 0.0528642
$$909$$ −7.39286 −0.245206
$$910$$ −1.65861 −0.0549823
$$911$$ 20.4160 0.676412 0.338206 0.941072i $$-0.390180\pi$$
0.338206 + 0.941072i $$0.390180\pi$$
$$912$$ 2.01302 0.0666579
$$913$$ 0 0
$$914$$ 33.0502 1.09320
$$915$$ −1.79590 −0.0593706
$$916$$ 23.5747 0.778929
$$917$$ 9.47001 0.312727
$$918$$ 6.34255 0.209335
$$919$$ 41.4420 1.36704 0.683522 0.729930i $$-0.260447\pi$$
0.683522 + 0.729930i $$0.260447\pi$$
$$920$$ 4.99442 0.164661
$$921$$ 3.01883 0.0994737
$$922$$ −20.2500 −0.666898
$$923$$ 12.9503 0.426265
$$924$$ 0 0
$$925$$ 9.13632 0.300400
$$926$$ −39.3213 −1.29218
$$927$$ 4.18525 0.137461
$$928$$ −7.13632 −0.234261
$$929$$ −45.4529 −1.49126 −0.745630 0.666360i $$-0.767851\pi$$
−0.745630 + 0.666360i $$0.767851\pi$$
$$930$$ 5.53191 0.181399
$$931$$ 3.70476 0.121419
$$932$$ 3.72239 0.121931
$$933$$ −11.0545 −0.361907
$$934$$ 18.8167 0.615702
$$935$$ 0 0
$$936$$ −4.48613 −0.146634
$$937$$ −24.2812 −0.793233 −0.396616 0.917984i $$-0.629816\pi$$
−0.396616 + 0.917984i $$0.629816\pi$$
$$938$$ −3.18992 −0.104155
$$939$$ 15.2631 0.498092
$$940$$ −4.56444 −0.148875
$$941$$ −28.6929 −0.935361 −0.467680 0.883898i $$-0.654910\pi$$
−0.467680 + 0.883898i $$0.654910\pi$$
$$942$$ −5.25872 −0.171338
$$943$$ −25.2026 −0.820710
$$944$$ −14.0846 −0.458414
$$945$$ 3.09975 0.100835
$$946$$ 0 0
$$947$$ 50.0233 1.62554 0.812770 0.582585i $$-0.197959\pi$$
0.812770 + 0.582585i $$0.197959\pi$$
$$948$$ 6.08860 0.197748
$$949$$ 18.0865 0.587111
$$950$$ −3.70476 −0.120198
$$951$$ 4.13849 0.134200
$$952$$ 2.04615 0.0663161
$$953$$ 59.7306 1.93486 0.967431 0.253133i $$-0.0814612\pi$$
0.967431 + 0.253133i $$0.0814612\pi$$
$$954$$ 24.0253 0.777849
$$955$$ 25.8516 0.836537
$$956$$ 15.0088 0.485418
$$957$$ 0 0
$$958$$ 7.56631 0.244456
$$959$$ −11.4807 −0.370732
$$960$$ 0.543362 0.0175369
$$961$$ 72.6506 2.34357
$$962$$ 15.1536 0.488571
$$963$$ −33.0220 −1.06412
$$964$$ 23.4709 0.755947
$$965$$ 25.6289 0.825022
$$966$$ −2.71378 −0.0873144
$$967$$ −14.2933 −0.459640 −0.229820 0.973233i $$-0.573814\pi$$
−0.229820 + 0.973233i $$0.573814\pi$$
$$968$$ 0 0
$$969$$ 4.11895 0.132320
$$970$$ −9.19639 −0.295278
$$971$$ 17.7082 0.568283 0.284142 0.958782i $$-0.408291\pi$$
0.284142 + 0.958782i $$0.408291\pi$$
$$972$$ 12.7931 0.410337
$$973$$ 4.86312 0.155905
$$974$$ −7.92350 −0.253885
$$975$$ −0.901224 −0.0288623
$$976$$ −3.30516 −0.105796
$$977$$ −24.9354 −0.797755 −0.398878 0.917004i $$-0.630600\pi$$
−0.398878 + 0.917004i $$0.630600\pi$$
$$978$$ −8.86095 −0.283342
$$979$$ 0 0
$$980$$ 1.00000 0.0319438
$$981$$ 3.58046 0.114315
$$982$$ 15.1101 0.482182
$$983$$ −50.1552 −1.59970 −0.799851 0.600199i $$-0.795088\pi$$
−0.799851 + 0.600199i $$0.795088\pi$$
$$984$$ −2.74189 −0.0874081
$$985$$ 10.5223 0.335268
$$986$$ −14.6020 −0.465022
$$987$$ 2.48014 0.0789437
$$988$$ −6.14474 −0.195490
$$989$$ −28.9021 −0.919034
$$990$$ 0 0
$$991$$ −5.56911 −0.176909 −0.0884543 0.996080i $$-0.528193\pi$$
−0.0884543 + 0.996080i $$0.528193\pi$$
$$992$$ 10.1809 0.323244
$$993$$ 5.36036 0.170106
$$994$$ −7.80795 −0.247653
$$995$$ 18.3423 0.581490
$$996$$ 2.21589 0.0702133
$$997$$ −12.2104 −0.386708 −0.193354 0.981129i $$-0.561937\pi$$
−0.193354 + 0.981129i $$0.561937\pi$$
$$998$$ 5.96549 0.188834
$$999$$ −28.3203 −0.896014
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.co.1.3 4
11.5 even 5 770.2.n.f.421.2 8
11.9 even 5 770.2.n.f.631.2 yes 8
11.10 odd 2 8470.2.a.cs.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.f.421.2 8 11.5 even 5
770.2.n.f.631.2 yes 8 11.9 even 5
8470.2.a.co.1.3 4 1.1 even 1 trivial
8470.2.a.cs.1.3 4 11.10 odd 2