# Properties

 Label 8470.2.a.f.1.1 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} -6.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} +2.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +1.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} +2.00000 q^{42} -7.00000 q^{43} +1.00000 q^{45} +6.00000 q^{46} -2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} +9.00000 q^{53} -4.00000 q^{54} -1.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} +3.00000 q^{59} -2.00000 q^{60} -7.00000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +11.0000 q^{67} +3.00000 q^{68} +12.0000 q^{69} -1.00000 q^{70} -3.00000 q^{71} -1.00000 q^{72} +11.0000 q^{73} -2.00000 q^{74} -2.00000 q^{75} -1.00000 q^{76} +4.00000 q^{78} +11.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -12.0000 q^{83} -2.00000 q^{84} +3.00000 q^{85} +7.00000 q^{86} -12.0000 q^{87} -1.00000 q^{90} +2.00000 q^{91} -6.00000 q^{92} +8.00000 q^{93} -1.00000 q^{95} +2.00000 q^{96} -1.00000 q^{97} -1.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 2.00000 0.816497
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ −2.00000 −0.577350
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ −2.00000 −0.516398
$$16$$ 1.00000 0.250000
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −2.00000 −0.436436
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 2.00000 0.408248
$$25$$ 1.00000 0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 4.00000 0.769800
$$28$$ 1.00000 0.188982
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 2.00000 0.365148
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −3.00000 −0.514496
$$35$$ 1.00000 0.169031
$$36$$ 1.00000 0.166667
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 1.00000 0.162221
$$39$$ −4.00000 −0.640513
$$40$$ −1.00000 −0.158114
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 2.00000 0.308607
$$43$$ −7.00000 −1.06749 −0.533745 0.845645i $$-0.679216\pi$$
−0.533745 + 0.845645i $$0.679216\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 6.00000 0.884652
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −2.00000 −0.288675
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ −6.00000 −0.840168
$$52$$ 2.00000 0.277350
$$53$$ 9.00000 1.23625 0.618123 0.786082i $$-0.287894\pi$$
0.618123 + 0.786082i $$0.287894\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ 2.00000 0.264906
$$58$$ −6.00000 −0.787839
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ −2.00000 −0.258199
$$61$$ −7.00000 −0.896258 −0.448129 0.893969i $$-0.647910\pi$$
−0.448129 + 0.893969i $$0.647910\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ 11.0000 1.34386 0.671932 0.740613i $$-0.265465\pi$$
0.671932 + 0.740613i $$0.265465\pi$$
$$68$$ 3.00000 0.363803
$$69$$ 12.0000 1.44463
$$70$$ −1.00000 −0.119523
$$71$$ −3.00000 −0.356034 −0.178017 0.984027i $$-0.556968\pi$$
−0.178017 + 0.984027i $$0.556968\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ −2.00000 −0.230940
$$76$$ −1.00000 −0.114708
$$77$$ 0 0
$$78$$ 4.00000 0.452911
$$79$$ 11.0000 1.23760 0.618798 0.785550i $$-0.287620\pi$$
0.618798 + 0.785550i $$0.287620\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 3.00000 0.325396
$$86$$ 7.00000 0.754829
$$87$$ −12.0000 −1.28654
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 2.00000 0.209657
$$92$$ −6.00000 −0.625543
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 2.00000 0.204124
$$97$$ −1.00000 −0.101535 −0.0507673 0.998711i $$-0.516167\pi$$
−0.0507673 + 0.998711i $$0.516167\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 6.00000 0.594089
$$103$$ −1.00000 −0.0985329 −0.0492665 0.998786i $$-0.515688\pi$$
−0.0492665 + 0.998786i $$0.515688\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ −2.00000 −0.195180
$$106$$ −9.00000 −0.874157
$$107$$ −9.00000 −0.870063 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$108$$ 4.00000 0.384900
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 1.00000 0.0944911
$$113$$ 12.0000 1.12887 0.564433 0.825479i $$-0.309095\pi$$
0.564433 + 0.825479i $$0.309095\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ −6.00000 −0.559503
$$116$$ 6.00000 0.557086
$$117$$ 2.00000 0.184900
$$118$$ −3.00000 −0.276172
$$119$$ 3.00000 0.275010
$$120$$ 2.00000 0.182574
$$121$$ 0 0
$$122$$ 7.00000 0.633750
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 1.00000 0.0894427
$$126$$ −1.00000 −0.0890871
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 14.0000 1.23263
$$130$$ −2.00000 −0.175412
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ −1.00000 −0.0867110
$$134$$ −11.0000 −0.950255
$$135$$ 4.00000 0.344265
$$136$$ −3.00000 −0.257248
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ −12.0000 −1.02151
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ 0 0
$$142$$ 3.00000 0.251754
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 6.00000 0.498273
$$146$$ −11.0000 −0.910366
$$147$$ −2.00000 −0.164957
$$148$$ 2.00000 0.164399
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 2.00000 0.163299
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 3.00000 0.242536
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ −4.00000 −0.320256
$$157$$ 8.00000 0.638470 0.319235 0.947676i $$-0.396574\pi$$
0.319235 + 0.947676i $$0.396574\pi$$
$$158$$ −11.0000 −0.875113
$$159$$ −18.0000 −1.42749
$$160$$ −1.00000 −0.0790569
$$161$$ −6.00000 −0.472866
$$162$$ 11.0000 0.864242
$$163$$ 5.00000 0.391630 0.195815 0.980641i $$-0.437265\pi$$
0.195815 + 0.980641i $$0.437265\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 9.00000 0.696441 0.348220 0.937413i $$-0.386786\pi$$
0.348220 + 0.937413i $$0.386786\pi$$
$$168$$ 2.00000 0.154303
$$169$$ −9.00000 −0.692308
$$170$$ −3.00000 −0.230089
$$171$$ −1.00000 −0.0764719
$$172$$ −7.00000 −0.533745
$$173$$ 12.0000 0.912343 0.456172 0.889892i $$-0.349220\pi$$
0.456172 + 0.889892i $$0.349220\pi$$
$$174$$ 12.0000 0.909718
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ −6.00000 −0.450988
$$178$$ 0 0
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ 14.0000 1.03491
$$184$$ 6.00000 0.442326
$$185$$ 2.00000 0.147043
$$186$$ −8.00000 −0.586588
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 4.00000 0.290957
$$190$$ 1.00000 0.0725476
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −2.00000 −0.144338
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 1.00000 0.0717958
$$195$$ −4.00000 −0.286446
$$196$$ 1.00000 0.0714286
$$197$$ −9.00000 −0.641223 −0.320612 0.947211i $$-0.603888\pi$$
−0.320612 + 0.947211i $$0.603888\pi$$
$$198$$ 0 0
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −22.0000 −1.55176
$$202$$ −6.00000 −0.422159
$$203$$ 6.00000 0.421117
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ 1.00000 0.0696733
$$207$$ −6.00000 −0.417029
$$208$$ 2.00000 0.138675
$$209$$ 0 0
$$210$$ 2.00000 0.138013
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 9.00000 0.618123
$$213$$ 6.00000 0.411113
$$214$$ 9.00000 0.615227
$$215$$ −7.00000 −0.477396
$$216$$ −4.00000 −0.272166
$$217$$ −4.00000 −0.271538
$$218$$ 16.0000 1.08366
$$219$$ −22.0000 −1.48662
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ 4.00000 0.268462
$$223$$ 11.0000 0.736614 0.368307 0.929704i $$-0.379937\pi$$
0.368307 + 0.929704i $$0.379937\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 1.00000 0.0666667
$$226$$ −12.0000 −0.798228
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 2.00000 0.132453
$$229$$ −25.0000 −1.65205 −0.826023 0.563636i $$-0.809402\pi$$
−0.826023 + 0.563636i $$0.809402\pi$$
$$230$$ 6.00000 0.395628
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 24.0000 1.57229 0.786146 0.618041i $$-0.212073\pi$$
0.786146 + 0.618041i $$0.212073\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 3.00000 0.195283
$$237$$ −22.0000 −1.42905
$$238$$ −3.00000 −0.194461
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ −2.00000 −0.129099
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ −7.00000 −0.448129
$$245$$ 1.00000 0.0638877
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 4.00000 0.254000
$$249$$ 24.0000 1.52094
$$250$$ −1.00000 −0.0632456
$$251$$ −15.0000 −0.946792 −0.473396 0.880850i $$-0.656972\pi$$
−0.473396 + 0.880850i $$0.656972\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 0 0
$$254$$ −2.00000 −0.125491
$$255$$ −6.00000 −0.375735
$$256$$ 1.00000 0.0625000
$$257$$ 21.0000 1.30994 0.654972 0.755653i $$-0.272680\pi$$
0.654972 + 0.755653i $$0.272680\pi$$
$$258$$ −14.0000 −0.871602
$$259$$ 2.00000 0.124274
$$260$$ 2.00000 0.124035
$$261$$ 6.00000 0.371391
$$262$$ 12.0000 0.741362
$$263$$ −18.0000 −1.10993 −0.554964 0.831875i $$-0.687268\pi$$
−0.554964 + 0.831875i $$0.687268\pi$$
$$264$$ 0 0
$$265$$ 9.00000 0.552866
$$266$$ 1.00000 0.0613139
$$267$$ 0 0
$$268$$ 11.0000 0.671932
$$269$$ −15.0000 −0.914566 −0.457283 0.889321i $$-0.651177\pi$$
−0.457283 + 0.889321i $$0.651177\pi$$
$$270$$ −4.00000 −0.243432
$$271$$ 2.00000 0.121491 0.0607457 0.998153i $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ 3.00000 0.181902
$$273$$ −4.00000 −0.242091
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ 12.0000 0.722315
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ −5.00000 −0.299880
$$279$$ −4.00000 −0.239474
$$280$$ −1.00000 −0.0597614
$$281$$ −21.0000 −1.25275 −0.626377 0.779520i $$-0.715463\pi$$
−0.626377 + 0.779520i $$0.715463\pi$$
$$282$$ 0 0
$$283$$ −16.0000 −0.951101 −0.475551 0.879688i $$-0.657751\pi$$
−0.475551 + 0.879688i $$0.657751\pi$$
$$284$$ −3.00000 −0.178017
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ −8.00000 −0.470588
$$290$$ −6.00000 −0.352332
$$291$$ 2.00000 0.117242
$$292$$ 11.0000 0.643726
$$293$$ 12.0000 0.701047 0.350524 0.936554i $$-0.386004\pi$$
0.350524 + 0.936554i $$0.386004\pi$$
$$294$$ 2.00000 0.116642
$$295$$ 3.00000 0.174667
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −12.0000 −0.693978
$$300$$ −2.00000 −0.115470
$$301$$ −7.00000 −0.403473
$$302$$ −8.00000 −0.460348
$$303$$ −12.0000 −0.689382
$$304$$ −1.00000 −0.0573539
$$305$$ −7.00000 −0.400819
$$306$$ −3.00000 −0.171499
$$307$$ −34.0000 −1.94048 −0.970241 0.242140i $$-0.922151\pi$$
−0.970241 + 0.242140i $$0.922151\pi$$
$$308$$ 0 0
$$309$$ 2.00000 0.113776
$$310$$ 4.00000 0.227185
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 4.00000 0.226455
$$313$$ 2.00000 0.113047 0.0565233 0.998401i $$-0.481998\pi$$
0.0565233 + 0.998401i $$0.481998\pi$$
$$314$$ −8.00000 −0.451466
$$315$$ 1.00000 0.0563436
$$316$$ 11.0000 0.618798
$$317$$ 33.0000 1.85346 0.926732 0.375722i $$-0.122605\pi$$
0.926732 + 0.375722i $$0.122605\pi$$
$$318$$ 18.0000 1.00939
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 18.0000 1.00466
$$322$$ 6.00000 0.334367
$$323$$ −3.00000 −0.166924
$$324$$ −11.0000 −0.611111
$$325$$ 2.00000 0.110940
$$326$$ −5.00000 −0.276924
$$327$$ 32.0000 1.76960
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 26.0000 1.42909 0.714545 0.699590i $$-0.246634\pi$$
0.714545 + 0.699590i $$0.246634\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 2.00000 0.109599
$$334$$ −9.00000 −0.492458
$$335$$ 11.0000 0.600994
$$336$$ −2.00000 −0.109109
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 9.00000 0.489535
$$339$$ −24.0000 −1.30350
$$340$$ 3.00000 0.162698
$$341$$ 0 0
$$342$$ 1.00000 0.0540738
$$343$$ 1.00000 0.0539949
$$344$$ 7.00000 0.377415
$$345$$ 12.0000 0.646058
$$346$$ −12.0000 −0.645124
$$347$$ 3.00000 0.161048 0.0805242 0.996753i $$-0.474341\pi$$
0.0805242 + 0.996753i $$0.474341\pi$$
$$348$$ −12.0000 −0.643268
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ 8.00000 0.427008
$$352$$ 0 0
$$353$$ 21.0000 1.11772 0.558859 0.829263i $$-0.311239\pi$$
0.558859 + 0.829263i $$0.311239\pi$$
$$354$$ 6.00000 0.318896
$$355$$ −3.00000 −0.159223
$$356$$ 0 0
$$357$$ −6.00000 −0.317554
$$358$$ −24.0000 −1.26844
$$359$$ −3.00000 −0.158334 −0.0791670 0.996861i $$-0.525226\pi$$
−0.0791670 + 0.996861i $$0.525226\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −18.0000 −0.947368
$$362$$ 7.00000 0.367912
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ 11.0000 0.575766
$$366$$ −14.0000 −0.731792
$$367$$ 29.0000 1.51379 0.756894 0.653538i $$-0.226716\pi$$
0.756894 + 0.653538i $$0.226716\pi$$
$$368$$ −6.00000 −0.312772
$$369$$ 0 0
$$370$$ −2.00000 −0.103975
$$371$$ 9.00000 0.467257
$$372$$ 8.00000 0.414781
$$373$$ 23.0000 1.19089 0.595447 0.803394i $$-0.296975\pi$$
0.595447 + 0.803394i $$0.296975\pi$$
$$374$$ 0 0
$$375$$ −2.00000 −0.103280
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ −4.00000 −0.205738
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ −1.00000 −0.0512989
$$381$$ −4.00000 −0.204926
$$382$$ 0 0
$$383$$ 21.0000 1.07305 0.536525 0.843884i $$-0.319737\pi$$
0.536525 + 0.843884i $$0.319737\pi$$
$$384$$ 2.00000 0.102062
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ −7.00000 −0.355830
$$388$$ −1.00000 −0.0507673
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ 4.00000 0.202548
$$391$$ −18.0000 −0.910299
$$392$$ −1.00000 −0.0505076
$$393$$ 24.0000 1.21064
$$394$$ 9.00000 0.453413
$$395$$ 11.0000 0.553470
$$396$$ 0 0
$$397$$ −16.0000 −0.803017 −0.401508 0.915855i $$-0.631514\pi$$
−0.401508 + 0.915855i $$0.631514\pi$$
$$398$$ −2.00000 −0.100251
$$399$$ 2.00000 0.100125
$$400$$ 1.00000 0.0500000
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 22.0000 1.09726
$$403$$ −8.00000 −0.398508
$$404$$ 6.00000 0.298511
$$405$$ −11.0000 −0.546594
$$406$$ −6.00000 −0.297775
$$407$$ 0 0
$$408$$ 6.00000 0.297044
$$409$$ −4.00000 −0.197787 −0.0988936 0.995098i $$-0.531530\pi$$
−0.0988936 + 0.995098i $$0.531530\pi$$
$$410$$ 0 0
$$411$$ −24.0000 −1.18383
$$412$$ −1.00000 −0.0492665
$$413$$ 3.00000 0.147620
$$414$$ 6.00000 0.294884
$$415$$ −12.0000 −0.589057
$$416$$ −2.00000 −0.0980581
$$417$$ −10.0000 −0.489702
$$418$$ 0 0
$$419$$ 9.00000 0.439679 0.219839 0.975536i $$-0.429447\pi$$
0.219839 + 0.975536i $$0.429447\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ 8.00000 0.389896 0.194948 0.980814i $$-0.437546\pi$$
0.194948 + 0.980814i $$0.437546\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 0 0
$$424$$ −9.00000 −0.437079
$$425$$ 3.00000 0.145521
$$426$$ −6.00000 −0.290701
$$427$$ −7.00000 −0.338754
$$428$$ −9.00000 −0.435031
$$429$$ 0 0
$$430$$ 7.00000 0.337570
$$431$$ 3.00000 0.144505 0.0722525 0.997386i $$-0.476981\pi$$
0.0722525 + 0.997386i $$0.476981\pi$$
$$432$$ 4.00000 0.192450
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 4.00000 0.192006
$$435$$ −12.0000 −0.575356
$$436$$ −16.0000 −0.766261
$$437$$ 6.00000 0.287019
$$438$$ 22.0000 1.05120
$$439$$ −10.0000 −0.477274 −0.238637 0.971109i $$-0.576701\pi$$
−0.238637 + 0.971109i $$0.576701\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ −6.00000 −0.285391
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ −11.0000 −0.520865
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ 21.0000 0.991051 0.495526 0.868593i $$-0.334975\pi$$
0.495526 + 0.868593i $$0.334975\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 0 0
$$452$$ 12.0000 0.564433
$$453$$ −16.0000 −0.751746
$$454$$ 0 0
$$455$$ 2.00000 0.0937614
$$456$$ −2.00000 −0.0936586
$$457$$ 8.00000 0.374224 0.187112 0.982339i $$-0.440087\pi$$
0.187112 + 0.982339i $$0.440087\pi$$
$$458$$ 25.0000 1.16817
$$459$$ 12.0000 0.560112
$$460$$ −6.00000 −0.279751
$$461$$ 9.00000 0.419172 0.209586 0.977790i $$-0.432788\pi$$
0.209586 + 0.977790i $$0.432788\pi$$
$$462$$ 0 0
$$463$$ −22.0000 −1.02243 −0.511213 0.859454i $$-0.670804\pi$$
−0.511213 + 0.859454i $$0.670804\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 8.00000 0.370991
$$466$$ −24.0000 −1.11178
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 11.0000 0.507933
$$470$$ 0 0
$$471$$ −16.0000 −0.737241
$$472$$ −3.00000 −0.138086
$$473$$ 0 0
$$474$$ 22.0000 1.01049
$$475$$ −1.00000 −0.0458831
$$476$$ 3.00000 0.137505
$$477$$ 9.00000 0.412082
$$478$$ 24.0000 1.09773
$$479$$ 36.0000 1.64488 0.822441 0.568850i $$-0.192612\pi$$
0.822441 + 0.568850i $$0.192612\pi$$
$$480$$ 2.00000 0.0912871
$$481$$ 4.00000 0.182384
$$482$$ −8.00000 −0.364390
$$483$$ 12.0000 0.546019
$$484$$ 0 0
$$485$$ −1.00000 −0.0454077
$$486$$ −10.0000 −0.453609
$$487$$ −4.00000 −0.181257 −0.0906287 0.995885i $$-0.528888\pi$$
−0.0906287 + 0.995885i $$0.528888\pi$$
$$488$$ 7.00000 0.316875
$$489$$ −10.0000 −0.452216
$$490$$ −1.00000 −0.0451754
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ 18.0000 0.810679
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ −3.00000 −0.134568
$$498$$ −24.0000 −1.07547
$$499$$ 32.0000 1.43252 0.716258 0.697835i $$-0.245853\pi$$
0.716258 + 0.697835i $$0.245853\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −18.0000 −0.804181
$$502$$ 15.0000 0.669483
$$503$$ 21.0000 0.936344 0.468172 0.883637i $$-0.344913\pi$$
0.468172 + 0.883637i $$0.344913\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 6.00000 0.266996
$$506$$ 0 0
$$507$$ 18.0000 0.799408
$$508$$ 2.00000 0.0887357
$$509$$ −21.0000 −0.930809 −0.465404 0.885098i $$-0.654091\pi$$
−0.465404 + 0.885098i $$0.654091\pi$$
$$510$$ 6.00000 0.265684
$$511$$ 11.0000 0.486611
$$512$$ −1.00000 −0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ −21.0000 −0.926270
$$515$$ −1.00000 −0.0440653
$$516$$ 14.0000 0.616316
$$517$$ 0 0
$$518$$ −2.00000 −0.0878750
$$519$$ −24.0000 −1.05348
$$520$$ −2.00000 −0.0877058
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ −2.00000 −0.0872872
$$526$$ 18.0000 0.784837
$$527$$ −12.0000 −0.522728
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ −9.00000 −0.390935
$$531$$ 3.00000 0.130189
$$532$$ −1.00000 −0.0433555
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −9.00000 −0.389104
$$536$$ −11.0000 −0.475128
$$537$$ −48.0000 −2.07135
$$538$$ 15.0000 0.646696
$$539$$ 0 0
$$540$$ 4.00000 0.172133
$$541$$ 26.0000 1.11783 0.558914 0.829226i $$-0.311218\pi$$
0.558914 + 0.829226i $$0.311218\pi$$
$$542$$ −2.00000 −0.0859074
$$543$$ 14.0000 0.600798
$$544$$ −3.00000 −0.128624
$$545$$ −16.0000 −0.685365
$$546$$ 4.00000 0.171184
$$547$$ 35.0000 1.49649 0.748246 0.663421i $$-0.230896\pi$$
0.748246 + 0.663421i $$0.230896\pi$$
$$548$$ 12.0000 0.512615
$$549$$ −7.00000 −0.298753
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ −12.0000 −0.510754
$$553$$ 11.0000 0.467768
$$554$$ 22.0000 0.934690
$$555$$ −4.00000 −0.169791
$$556$$ 5.00000 0.212047
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 4.00000 0.169334
$$559$$ −14.0000 −0.592137
$$560$$ 1.00000 0.0422577
$$561$$ 0 0
$$562$$ 21.0000 0.885832
$$563$$ 30.0000 1.26435 0.632175 0.774826i $$-0.282163\pi$$
0.632175 + 0.774826i $$0.282163\pi$$
$$564$$ 0 0
$$565$$ 12.0000 0.504844
$$566$$ 16.0000 0.672530
$$567$$ −11.0000 −0.461957
$$568$$ 3.00000 0.125877
$$569$$ −9.00000 −0.377300 −0.188650 0.982044i $$-0.560411\pi$$
−0.188650 + 0.982044i $$0.560411\pi$$
$$570$$ −2.00000 −0.0837708
$$571$$ −22.0000 −0.920671 −0.460336 0.887745i $$-0.652271\pi$$
−0.460336 + 0.887745i $$0.652271\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −6.00000 −0.250217
$$576$$ 1.00000 0.0416667
$$577$$ 11.0000 0.457936 0.228968 0.973434i $$-0.426465\pi$$
0.228968 + 0.973434i $$0.426465\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 8.00000 0.332469
$$580$$ 6.00000 0.249136
$$581$$ −12.0000 −0.497844
$$582$$ −2.00000 −0.0829027
$$583$$ 0 0
$$584$$ −11.0000 −0.455183
$$585$$ 2.00000 0.0826898
$$586$$ −12.0000 −0.495715
$$587$$ 30.0000 1.23823 0.619116 0.785299i $$-0.287491\pi$$
0.619116 + 0.785299i $$0.287491\pi$$
$$588$$ −2.00000 −0.0824786
$$589$$ 4.00000 0.164817
$$590$$ −3.00000 −0.123508
$$591$$ 18.0000 0.740421
$$592$$ 2.00000 0.0821995
$$593$$ 39.0000 1.60154 0.800769 0.598973i $$-0.204424\pi$$
0.800769 + 0.598973i $$0.204424\pi$$
$$594$$ 0 0
$$595$$ 3.00000 0.122988
$$596$$ 0 0
$$597$$ −4.00000 −0.163709
$$598$$ 12.0000 0.490716
$$599$$ 27.0000 1.10319 0.551595 0.834112i $$-0.314019\pi$$
0.551595 + 0.834112i $$0.314019\pi$$
$$600$$ 2.00000 0.0816497
$$601$$ −40.0000 −1.63163 −0.815817 0.578310i $$-0.803712\pi$$
−0.815817 + 0.578310i $$0.803712\pi$$
$$602$$ 7.00000 0.285299
$$603$$ 11.0000 0.447955
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 12.0000 0.487467
$$607$$ 5.00000 0.202944 0.101472 0.994838i $$-0.467645\pi$$
0.101472 + 0.994838i $$0.467645\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ −12.0000 −0.486265
$$610$$ 7.00000 0.283422
$$611$$ 0 0
$$612$$ 3.00000 0.121268
$$613$$ −1.00000 −0.0403896 −0.0201948 0.999796i $$-0.506429\pi$$
−0.0201948 + 0.999796i $$0.506429\pi$$
$$614$$ 34.0000 1.37213
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ −2.00000 −0.0804518
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ −24.0000 −0.963087
$$622$$ 0 0
$$623$$ 0 0
$$624$$ −4.00000 −0.160128
$$625$$ 1.00000 0.0400000
$$626$$ −2.00000 −0.0799361
$$627$$ 0 0
$$628$$ 8.00000 0.319235
$$629$$ 6.00000 0.239236
$$630$$ −1.00000 −0.0398410
$$631$$ 29.0000 1.15447 0.577236 0.816577i $$-0.304131\pi$$
0.577236 + 0.816577i $$0.304131\pi$$
$$632$$ −11.0000 −0.437557
$$633$$ 8.00000 0.317971
$$634$$ −33.0000 −1.31060
$$635$$ 2.00000 0.0793676
$$636$$ −18.0000 −0.713746
$$637$$ 2.00000 0.0792429
$$638$$ 0 0
$$639$$ −3.00000 −0.118678
$$640$$ −1.00000 −0.0395285
$$641$$ 21.0000 0.829450 0.414725 0.909947i $$-0.363878\pi$$
0.414725 + 0.909947i $$0.363878\pi$$
$$642$$ −18.0000 −0.710403
$$643$$ 32.0000 1.26196 0.630978 0.775800i $$-0.282654\pi$$
0.630978 + 0.775800i $$0.282654\pi$$
$$644$$ −6.00000 −0.236433
$$645$$ 14.0000 0.551249
$$646$$ 3.00000 0.118033
$$647$$ 45.0000 1.76913 0.884566 0.466415i $$-0.154454\pi$$
0.884566 + 0.466415i $$0.154454\pi$$
$$648$$ 11.0000 0.432121
$$649$$ 0 0
$$650$$ −2.00000 −0.0784465
$$651$$ 8.00000 0.313545
$$652$$ 5.00000 0.195815
$$653$$ 21.0000 0.821794 0.410897 0.911682i $$-0.365216\pi$$
0.410897 + 0.911682i $$0.365216\pi$$
$$654$$ −32.0000 −1.25130
$$655$$ −12.0000 −0.468879
$$656$$ 0 0
$$657$$ 11.0000 0.429151
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ −26.0000 −1.01052
$$663$$ −12.0000 −0.466041
$$664$$ 12.0000 0.465690
$$665$$ −1.00000 −0.0387783
$$666$$ −2.00000 −0.0774984
$$667$$ −36.0000 −1.39393
$$668$$ 9.00000 0.348220
$$669$$ −22.0000 −0.850569
$$670$$ −11.0000 −0.424967
$$671$$ 0 0
$$672$$ 2.00000 0.0771517
$$673$$ 8.00000 0.308377 0.154189 0.988041i $$-0.450724\pi$$
0.154189 + 0.988041i $$0.450724\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 4.00000 0.153960
$$676$$ −9.00000 −0.346154
$$677$$ 48.0000 1.84479 0.922395 0.386248i $$-0.126229\pi$$
0.922395 + 0.386248i $$0.126229\pi$$
$$678$$ 24.0000 0.921714
$$679$$ −1.00000 −0.0383765
$$680$$ −3.00000 −0.115045
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 15.0000 0.573959 0.286980 0.957937i $$-0.407349\pi$$
0.286980 + 0.957937i $$0.407349\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 12.0000 0.458496
$$686$$ −1.00000 −0.0381802
$$687$$ 50.0000 1.90762
$$688$$ −7.00000 −0.266872
$$689$$ 18.0000 0.685745
$$690$$ −12.0000 −0.456832
$$691$$ −43.0000 −1.63580 −0.817899 0.575362i $$-0.804861\pi$$
−0.817899 + 0.575362i $$0.804861\pi$$
$$692$$ 12.0000 0.456172
$$693$$ 0 0
$$694$$ −3.00000 −0.113878
$$695$$ 5.00000 0.189661
$$696$$ 12.0000 0.454859
$$697$$ 0 0
$$698$$ −14.0000 −0.529908
$$699$$ −48.0000 −1.81553
$$700$$ 1.00000 0.0377964
$$701$$ 48.0000 1.81293 0.906467 0.422276i $$-0.138769\pi$$
0.906467 + 0.422276i $$0.138769\pi$$
$$702$$ −8.00000 −0.301941
$$703$$ −2.00000 −0.0754314
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −21.0000 −0.790345
$$707$$ 6.00000 0.225653
$$708$$ −6.00000 −0.225494
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 3.00000 0.112588
$$711$$ 11.0000 0.412532
$$712$$ 0 0
$$713$$ 24.0000 0.898807
$$714$$ 6.00000 0.224544
$$715$$ 0 0
$$716$$ 24.0000 0.896922
$$717$$ 48.0000 1.79259
$$718$$ 3.00000 0.111959
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ −1.00000 −0.0372419
$$722$$ 18.0000 0.669891
$$723$$ −16.0000 −0.595046
$$724$$ −7.00000 −0.260153
$$725$$ 6.00000 0.222834
$$726$$ 0 0
$$727$$ −4.00000 −0.148352 −0.0741759 0.997245i $$-0.523633\pi$$
−0.0741759 + 0.997245i $$0.523633\pi$$
$$728$$ −2.00000 −0.0741249
$$729$$ 13.0000 0.481481
$$730$$ −11.0000 −0.407128
$$731$$ −21.0000 −0.776713
$$732$$ 14.0000 0.517455
$$733$$ 32.0000 1.18195 0.590973 0.806691i $$-0.298744\pi$$
0.590973 + 0.806691i $$0.298744\pi$$
$$734$$ −29.0000 −1.07041
$$735$$ −2.00000 −0.0737711
$$736$$ 6.00000 0.221163
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 2.00000 0.0735215
$$741$$ 4.00000 0.146944
$$742$$ −9.00000 −0.330400
$$743$$ −6.00000 −0.220119 −0.110059 0.993925i $$-0.535104\pi$$
−0.110059 + 0.993925i $$0.535104\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ −23.0000 −0.842090
$$747$$ −12.0000 −0.439057
$$748$$ 0 0
$$749$$ −9.00000 −0.328853
$$750$$ 2.00000 0.0730297
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 0 0
$$753$$ 30.0000 1.09326
$$754$$ −12.0000 −0.437014
$$755$$ 8.00000 0.291150
$$756$$ 4.00000 0.145479
$$757$$ 47.0000 1.70824 0.854122 0.520073i $$-0.174095\pi$$
0.854122 + 0.520073i $$0.174095\pi$$
$$758$$ −8.00000 −0.290573
$$759$$ 0 0
$$760$$ 1.00000 0.0362738
$$761$$ 36.0000 1.30500 0.652499 0.757789i $$-0.273720\pi$$
0.652499 + 0.757789i $$0.273720\pi$$
$$762$$ 4.00000 0.144905
$$763$$ −16.0000 −0.579239
$$764$$ 0 0
$$765$$ 3.00000 0.108465
$$766$$ −21.0000 −0.758761
$$767$$ 6.00000 0.216647
$$768$$ −2.00000 −0.0721688
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 0 0
$$771$$ −42.0000 −1.51259
$$772$$ −4.00000 −0.143963
$$773$$ −12.0000 −0.431610 −0.215805 0.976436i $$-0.569238\pi$$
−0.215805 + 0.976436i $$0.569238\pi$$
$$774$$ 7.00000 0.251610
$$775$$ −4.00000 −0.143684
$$776$$ 1.00000 0.0358979
$$777$$ −4.00000 −0.143499
$$778$$ −12.0000 −0.430221
$$779$$ 0 0
$$780$$ −4.00000 −0.143223
$$781$$ 0 0
$$782$$ 18.0000 0.643679
$$783$$ 24.0000 0.857690
$$784$$ 1.00000 0.0357143
$$785$$ 8.00000 0.285532
$$786$$ −24.0000 −0.856052
$$787$$ 50.0000 1.78231 0.891154 0.453701i $$-0.149897\pi$$
0.891154 + 0.453701i $$0.149897\pi$$
$$788$$ −9.00000 −0.320612
$$789$$ 36.0000 1.28163
$$790$$ −11.0000 −0.391362
$$791$$ 12.0000 0.426671
$$792$$ 0 0
$$793$$ −14.0000 −0.497155
$$794$$ 16.0000 0.567819
$$795$$ −18.0000 −0.638394
$$796$$ 2.00000 0.0708881
$$797$$ −42.0000 −1.48772 −0.743858 0.668338i $$-0.767006\pi$$
−0.743858 + 0.668338i $$0.767006\pi$$
$$798$$ −2.00000 −0.0707992
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 18.0000 0.635602
$$803$$ 0 0
$$804$$ −22.0000 −0.775880
$$805$$ −6.00000 −0.211472
$$806$$ 8.00000 0.281788
$$807$$ 30.0000 1.05605
$$808$$ −6.00000 −0.211079
$$809$$ 3.00000 0.105474 0.0527372 0.998608i $$-0.483205\pi$$
0.0527372 + 0.998608i $$0.483205\pi$$
$$810$$ 11.0000 0.386501
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 6.00000 0.210559
$$813$$ −4.00000 −0.140286
$$814$$ 0 0
$$815$$ 5.00000 0.175142
$$816$$ −6.00000 −0.210042
$$817$$ 7.00000 0.244899
$$818$$ 4.00000 0.139857
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ −24.0000 −0.837606 −0.418803 0.908077i $$-0.637550\pi$$
−0.418803 + 0.908077i $$0.637550\pi$$
$$822$$ 24.0000 0.837096
$$823$$ −22.0000 −0.766872 −0.383436 0.923567i $$-0.625259\pi$$
−0.383436 + 0.923567i $$0.625259\pi$$
$$824$$ 1.00000 0.0348367
$$825$$ 0 0
$$826$$ −3.00000 −0.104383
$$827$$ −51.0000 −1.77344 −0.886722 0.462303i $$-0.847023\pi$$
−0.886722 + 0.462303i $$0.847023\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ 50.0000 1.73657 0.868286 0.496064i $$-0.165222\pi$$
0.868286 + 0.496064i $$0.165222\pi$$
$$830$$ 12.0000 0.416526
$$831$$ 44.0000 1.52634
$$832$$ 2.00000 0.0693375
$$833$$ 3.00000 0.103944
$$834$$ 10.0000 0.346272
$$835$$ 9.00000 0.311458
$$836$$ 0 0
$$837$$ −16.0000 −0.553041
$$838$$ −9.00000 −0.310900
$$839$$ 6.00000 0.207143 0.103572 0.994622i $$-0.466973\pi$$
0.103572 + 0.994622i $$0.466973\pi$$
$$840$$ 2.00000 0.0690066
$$841$$ 7.00000 0.241379
$$842$$ −8.00000 −0.275698
$$843$$ 42.0000 1.44656
$$844$$ −4.00000 −0.137686
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 9.00000 0.309061
$$849$$ 32.0000 1.09824
$$850$$ −3.00000 −0.102899
$$851$$ −12.0000 −0.411355
$$852$$ 6.00000 0.205557
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ 7.00000 0.239535
$$855$$ −1.00000 −0.0341993
$$856$$ 9.00000 0.307614
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ 47.0000 1.60362 0.801810 0.597580i $$-0.203871\pi$$
0.801810 + 0.597580i $$0.203871\pi$$
$$860$$ −7.00000 −0.238698
$$861$$ 0 0
$$862$$ −3.00000 −0.102180
$$863$$ 30.0000 1.02121 0.510606 0.859815i $$-0.329421\pi$$
0.510606 + 0.859815i $$0.329421\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 12.0000 0.408012
$$866$$ −2.00000 −0.0679628
$$867$$ 16.0000 0.543388
$$868$$ −4.00000 −0.135769
$$869$$ 0 0
$$870$$ 12.0000 0.406838
$$871$$ 22.0000 0.745442
$$872$$ 16.0000 0.541828
$$873$$ −1.00000 −0.0338449
$$874$$ −6.00000 −0.202953
$$875$$ 1.00000 0.0338062
$$876$$ −22.0000 −0.743311
$$877$$ 17.0000 0.574049 0.287025 0.957923i $$-0.407334\pi$$
0.287025 + 0.957923i $$0.407334\pi$$
$$878$$ 10.0000 0.337484
$$879$$ −24.0000 −0.809500
$$880$$ 0 0
$$881$$ −12.0000 −0.404290 −0.202145 0.979356i $$-0.564791\pi$$
−0.202145 + 0.979356i $$0.564791\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ 29.0000 0.975928 0.487964 0.872864i $$-0.337740\pi$$
0.487964 + 0.872864i $$0.337740\pi$$
$$884$$ 6.00000 0.201802
$$885$$ −6.00000 −0.201688
$$886$$ 12.0000 0.403148
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 4.00000 0.134231
$$889$$ 2.00000 0.0670778
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 11.0000 0.368307
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ −1.00000 −0.0334077
$$897$$ 24.0000 0.801337
$$898$$ −21.0000 −0.700779
$$899$$ −24.0000 −0.800445
$$900$$ 1.00000 0.0333333
$$901$$ 27.0000 0.899500
$$902$$ 0 0
$$903$$ 14.0000 0.465891
$$904$$ −12.0000 −0.399114
$$905$$ −7.00000 −0.232688
$$906$$ 16.0000 0.531564
$$907$$ −1.00000 −0.0332045 −0.0166022 0.999862i $$-0.505285\pi$$
−0.0166022 + 0.999862i $$0.505285\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ −2.00000 −0.0662994
$$911$$ −21.0000 −0.695761 −0.347881 0.937539i $$-0.613099\pi$$
−0.347881 + 0.937539i $$0.613099\pi$$
$$912$$ 2.00000 0.0662266
$$913$$ 0 0
$$914$$ −8.00000 −0.264616
$$915$$ 14.0000 0.462826
$$916$$ −25.0000 −0.826023
$$917$$ −12.0000 −0.396275
$$918$$ −12.0000 −0.396059
$$919$$ 20.0000 0.659739 0.329870 0.944027i $$-0.392995\pi$$
0.329870 + 0.944027i $$0.392995\pi$$
$$920$$ 6.00000 0.197814
$$921$$ 68.0000 2.24068
$$922$$ −9.00000 −0.296399
$$923$$ −6.00000 −0.197492
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ 22.0000 0.722965
$$927$$ −1.00000 −0.0328443
$$928$$ −6.00000 −0.196960
$$929$$ 12.0000 0.393707 0.196854 0.980433i $$-0.436928\pi$$
0.196854 + 0.980433i $$0.436928\pi$$
$$930$$ −8.00000 −0.262330
$$931$$ −1.00000 −0.0327737
$$932$$ 24.0000 0.786146
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ −11.0000 −0.359163
$$939$$ −4.00000 −0.130535
$$940$$ 0 0
$$941$$ 39.0000 1.27136 0.635682 0.771951i $$-0.280719\pi$$
0.635682 + 0.771951i $$0.280719\pi$$
$$942$$ 16.0000 0.521308
$$943$$ 0 0
$$944$$ 3.00000 0.0976417
$$945$$ 4.00000 0.130120
$$946$$ 0 0
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ −22.0000 −0.714527
$$949$$ 22.0000 0.714150
$$950$$ 1.00000 0.0324443
$$951$$ −66.0000 −2.14020
$$952$$ −3.00000 −0.0972306
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ −9.00000 −0.291386
$$955$$ 0 0
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ −36.0000 −1.16311
$$959$$ 12.0000 0.387500
$$960$$ −2.00000 −0.0645497
$$961$$ −15.0000 −0.483871
$$962$$ −4.00000 −0.128965
$$963$$ −9.00000 −0.290021
$$964$$ 8.00000 0.257663
$$965$$ −4.00000 −0.128765
$$966$$ −12.0000 −0.386094
$$967$$ −22.0000 −0.707472 −0.353736 0.935345i $$-0.615089\pi$$
−0.353736 + 0.935345i $$0.615089\pi$$
$$968$$ 0 0
$$969$$ 6.00000 0.192748
$$970$$ 1.00000 0.0321081
$$971$$ 21.0000 0.673922 0.336961 0.941519i $$-0.390601\pi$$
0.336961 + 0.941519i $$0.390601\pi$$
$$972$$ 10.0000 0.320750
$$973$$ 5.00000 0.160293
$$974$$ 4.00000 0.128168
$$975$$ −4.00000 −0.128103
$$976$$ −7.00000 −0.224065
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 10.0000 0.319765
$$979$$ 0 0
$$980$$ 1.00000 0.0319438
$$981$$ −16.0000 −0.510841
$$982$$ 12.0000 0.382935
$$983$$ −36.0000 −1.14822 −0.574111 0.818778i $$-0.694652\pi$$
−0.574111 + 0.818778i $$0.694652\pi$$
$$984$$ 0 0
$$985$$ −9.00000 −0.286764
$$986$$ −18.0000 −0.573237
$$987$$ 0 0
$$988$$ −2.00000 −0.0636285
$$989$$ 42.0000 1.33552
$$990$$ 0 0
$$991$$ 5.00000 0.158830 0.0794151 0.996842i $$-0.474695\pi$$
0.0794151 + 0.996842i $$0.474695\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −52.0000 −1.65017
$$994$$ 3.00000 0.0951542
$$995$$ 2.00000 0.0634043
$$996$$ 24.0000 0.760469
$$997$$ 20.0000 0.633406 0.316703 0.948525i $$-0.397424\pi$$
0.316703 + 0.948525i $$0.397424\pi$$
$$998$$ −32.0000 −1.01294
$$999$$ 8.00000 0.253109
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.f.1.1 1
11.10 odd 2 8470.2.a.u.1.1 yes 1

By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.f.1.1 1 1.1 even 1 trivial
8470.2.a.u.1.1 yes 1 11.10 odd 2