# Properties

 Label 8470.2.a.cf.1.1 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ 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 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.381966 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.85410 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.381966 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.85410 q^{9} -1.00000 q^{10} +0.381966 q^{12} +2.00000 q^{13} +1.00000 q^{14} -0.381966 q^{15} +1.00000 q^{16} +0.618034 q^{17} -2.85410 q^{18} +0.145898 q^{19} -1.00000 q^{20} +0.381966 q^{21} +6.00000 q^{23} +0.381966 q^{24} +1.00000 q^{25} +2.00000 q^{26} -2.23607 q^{27} +1.00000 q^{28} -1.23607 q^{29} -0.381966 q^{30} +3.23607 q^{31} +1.00000 q^{32} +0.618034 q^{34} -1.00000 q^{35} -2.85410 q^{36} +2.47214 q^{37} +0.145898 q^{38} +0.763932 q^{39} -1.00000 q^{40} +5.32624 q^{41} +0.381966 q^{42} -4.85410 q^{43} +2.85410 q^{45} +6.00000 q^{46} +4.76393 q^{47} +0.381966 q^{48} +1.00000 q^{49} +1.00000 q^{50} +0.236068 q^{51} +2.00000 q^{52} -3.23607 q^{53} -2.23607 q^{54} +1.00000 q^{56} +0.0557281 q^{57} -1.23607 q^{58} -5.38197 q^{59} -0.381966 q^{60} -12.4721 q^{61} +3.23607 q^{62} -2.85410 q^{63} +1.00000 q^{64} -2.00000 q^{65} -5.09017 q^{67} +0.618034 q^{68} +2.29180 q^{69} -1.00000 q^{70} +3.70820 q^{71} -2.85410 q^{72} +7.14590 q^{73} +2.47214 q^{74} +0.381966 q^{75} +0.145898 q^{76} +0.763932 q^{78} +0.472136 q^{79} -1.00000 q^{80} +7.70820 q^{81} +5.32624 q^{82} -5.32624 q^{83} +0.381966 q^{84} -0.618034 q^{85} -4.85410 q^{86} -0.472136 q^{87} +1.90983 q^{89} +2.85410 q^{90} +2.00000 q^{91} +6.00000 q^{92} +1.23607 q^{93} +4.76393 q^{94} -0.145898 q^{95} +0.381966 q^{96} +10.5623 q^{97} +1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 2 q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + q^{9} + O(q^{10})$$ $$2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 2 q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + q^{9} - 2 q^{10} + 3 q^{12} + 4 q^{13} + 2 q^{14} - 3 q^{15} + 2 q^{16} - q^{17} + q^{18} + 7 q^{19} - 2 q^{20} + 3 q^{21} + 12 q^{23} + 3 q^{24} + 2 q^{25} + 4 q^{26} + 2 q^{28} + 2 q^{29} - 3 q^{30} + 2 q^{31} + 2 q^{32} - q^{34} - 2 q^{35} + q^{36} - 4 q^{37} + 7 q^{38} + 6 q^{39} - 2 q^{40} - 5 q^{41} + 3 q^{42} - 3 q^{43} - q^{45} + 12 q^{46} + 14 q^{47} + 3 q^{48} + 2 q^{49} + 2 q^{50} - 4 q^{51} + 4 q^{52} - 2 q^{53} + 2 q^{56} + 18 q^{57} + 2 q^{58} - 13 q^{59} - 3 q^{60} - 16 q^{61} + 2 q^{62} + q^{63} + 2 q^{64} - 4 q^{65} + q^{67} - q^{68} + 18 q^{69} - 2 q^{70} - 6 q^{71} + q^{72} + 21 q^{73} - 4 q^{74} + 3 q^{75} + 7 q^{76} + 6 q^{78} - 8 q^{79} - 2 q^{80} + 2 q^{81} - 5 q^{82} + 5 q^{83} + 3 q^{84} + q^{85} - 3 q^{86} + 8 q^{87} + 15 q^{89} - q^{90} + 4 q^{91} + 12 q^{92} - 2 q^{93} + 14 q^{94} - 7 q^{95} + 3 q^{96} + q^{97} + 2 q^{98} + O(q^{100})$$

## 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
$$3$$ 0.381966 0.220528 0.110264 0.993902i $$-0.464830\pi$$
0.110264 + 0.993902i $$0.464830\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 0.381966 0.155937
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ −2.85410 −0.951367
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ 0.381966 0.110264
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 1.00000 0.267261
$$15$$ −0.381966 −0.0986232
$$16$$ 1.00000 0.250000
$$17$$ 0.618034 0.149895 0.0749476 0.997187i $$-0.476121\pi$$
0.0749476 + 0.997187i $$0.476121\pi$$
$$18$$ −2.85410 −0.672718
$$19$$ 0.145898 0.0334713 0.0167357 0.999860i $$-0.494673\pi$$
0.0167357 + 0.999860i $$0.494673\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0.381966 0.0833518
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0.381966 0.0779685
$$25$$ 1.00000 0.200000
$$26$$ 2.00000 0.392232
$$27$$ −2.23607 −0.430331
$$28$$ 1.00000 0.188982
$$29$$ −1.23607 −0.229532 −0.114766 0.993393i $$-0.536612\pi$$
−0.114766 + 0.993393i $$0.536612\pi$$
$$30$$ −0.381966 −0.0697371
$$31$$ 3.23607 0.581215 0.290607 0.956842i $$-0.406143\pi$$
0.290607 + 0.956842i $$0.406143\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 0.618034 0.105992
$$35$$ −1.00000 −0.169031
$$36$$ −2.85410 −0.475684
$$37$$ 2.47214 0.406417 0.203208 0.979136i $$-0.434863\pi$$
0.203208 + 0.979136i $$0.434863\pi$$
$$38$$ 0.145898 0.0236678
$$39$$ 0.763932 0.122327
$$40$$ −1.00000 −0.158114
$$41$$ 5.32624 0.831819 0.415909 0.909406i $$-0.363463\pi$$
0.415909 + 0.909406i $$0.363463\pi$$
$$42$$ 0.381966 0.0589386
$$43$$ −4.85410 −0.740244 −0.370122 0.928983i $$-0.620684\pi$$
−0.370122 + 0.928983i $$0.620684\pi$$
$$44$$ 0 0
$$45$$ 2.85410 0.425464
$$46$$ 6.00000 0.884652
$$47$$ 4.76393 0.694891 0.347445 0.937700i $$-0.387049\pi$$
0.347445 + 0.937700i $$0.387049\pi$$
$$48$$ 0.381966 0.0551320
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ 0.236068 0.0330561
$$52$$ 2.00000 0.277350
$$53$$ −3.23607 −0.444508 −0.222254 0.974989i $$-0.571341\pi$$
−0.222254 + 0.974989i $$0.571341\pi$$
$$54$$ −2.23607 −0.304290
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0.0557281 0.00738137
$$58$$ −1.23607 −0.162304
$$59$$ −5.38197 −0.700672 −0.350336 0.936624i $$-0.613933\pi$$
−0.350336 + 0.936624i $$0.613933\pi$$
$$60$$ −0.381966 −0.0493116
$$61$$ −12.4721 −1.59689 −0.798447 0.602066i $$-0.794345\pi$$
−0.798447 + 0.602066i $$0.794345\pi$$
$$62$$ 3.23607 0.410981
$$63$$ −2.85410 −0.359583
$$64$$ 1.00000 0.125000
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ −5.09017 −0.621863 −0.310932 0.950432i $$-0.600641\pi$$
−0.310932 + 0.950432i $$0.600641\pi$$
$$68$$ 0.618034 0.0749476
$$69$$ 2.29180 0.275900
$$70$$ −1.00000 −0.119523
$$71$$ 3.70820 0.440083 0.220041 0.975491i $$-0.429381\pi$$
0.220041 + 0.975491i $$0.429381\pi$$
$$72$$ −2.85410 −0.336359
$$73$$ 7.14590 0.836364 0.418182 0.908363i $$-0.362667\pi$$
0.418182 + 0.908363i $$0.362667\pi$$
$$74$$ 2.47214 0.287380
$$75$$ 0.381966 0.0441056
$$76$$ 0.145898 0.0167357
$$77$$ 0 0
$$78$$ 0.763932 0.0864983
$$79$$ 0.472136 0.0531194 0.0265597 0.999647i $$-0.491545\pi$$
0.0265597 + 0.999647i $$0.491545\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 7.70820 0.856467
$$82$$ 5.32624 0.588185
$$83$$ −5.32624 −0.584631 −0.292315 0.956322i $$-0.594426\pi$$
−0.292315 + 0.956322i $$0.594426\pi$$
$$84$$ 0.381966 0.0416759
$$85$$ −0.618034 −0.0670352
$$86$$ −4.85410 −0.523431
$$87$$ −0.472136 −0.0506183
$$88$$ 0 0
$$89$$ 1.90983 0.202442 0.101221 0.994864i $$-0.467725\pi$$
0.101221 + 0.994864i $$0.467725\pi$$
$$90$$ 2.85410 0.300849
$$91$$ 2.00000 0.209657
$$92$$ 6.00000 0.625543
$$93$$ 1.23607 0.128174
$$94$$ 4.76393 0.491362
$$95$$ −0.145898 −0.0149688
$$96$$ 0.381966 0.0389842
$$97$$ 10.5623 1.07244 0.536220 0.844078i $$-0.319852\pi$$
0.536220 + 0.844078i $$0.319852\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −8.94427 −0.889988 −0.444994 0.895533i $$-0.646794\pi$$
−0.444994 + 0.895533i $$0.646794\pi$$
$$102$$ 0.236068 0.0233742
$$103$$ 0.944272 0.0930419 0.0465209 0.998917i $$-0.485187\pi$$
0.0465209 + 0.998917i $$0.485187\pi$$
$$104$$ 2.00000 0.196116
$$105$$ −0.381966 −0.0372761
$$106$$ −3.23607 −0.314315
$$107$$ 17.6180 1.70320 0.851600 0.524192i $$-0.175633\pi$$
0.851600 + 0.524192i $$0.175633\pi$$
$$108$$ −2.23607 −0.215166
$$109$$ 8.47214 0.811483 0.405742 0.913988i $$-0.367013\pi$$
0.405742 + 0.913988i $$0.367013\pi$$
$$110$$ 0 0
$$111$$ 0.944272 0.0896263
$$112$$ 1.00000 0.0944911
$$113$$ 6.32624 0.595122 0.297561 0.954703i $$-0.403827\pi$$
0.297561 + 0.954703i $$0.403827\pi$$
$$114$$ 0.0557281 0.00521941
$$115$$ −6.00000 −0.559503
$$116$$ −1.23607 −0.114766
$$117$$ −5.70820 −0.527724
$$118$$ −5.38197 −0.495450
$$119$$ 0.618034 0.0566551
$$120$$ −0.381966 −0.0348686
$$121$$ 0 0
$$122$$ −12.4721 −1.12917
$$123$$ 2.03444 0.183439
$$124$$ 3.23607 0.290607
$$125$$ −1.00000 −0.0894427
$$126$$ −2.85410 −0.254264
$$127$$ 15.4164 1.36798 0.683992 0.729489i $$-0.260242\pi$$
0.683992 + 0.729489i $$0.260242\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −1.85410 −0.163245
$$130$$ −2.00000 −0.175412
$$131$$ 16.7984 1.46768 0.733840 0.679322i $$-0.237726\pi$$
0.733840 + 0.679322i $$0.237726\pi$$
$$132$$ 0 0
$$133$$ 0.145898 0.0126510
$$134$$ −5.09017 −0.439724
$$135$$ 2.23607 0.192450
$$136$$ 0.618034 0.0529960
$$137$$ 10.0902 0.862061 0.431031 0.902337i $$-0.358150\pi$$
0.431031 + 0.902337i $$0.358150\pi$$
$$138$$ 2.29180 0.195091
$$139$$ 9.52786 0.808143 0.404071 0.914727i $$-0.367595\pi$$
0.404071 + 0.914727i $$0.367595\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ 1.81966 0.153243
$$142$$ 3.70820 0.311186
$$143$$ 0 0
$$144$$ −2.85410 −0.237842
$$145$$ 1.23607 0.102650
$$146$$ 7.14590 0.591399
$$147$$ 0.381966 0.0315040
$$148$$ 2.47214 0.203208
$$149$$ 1.81966 0.149072 0.0745362 0.997218i $$-0.476252\pi$$
0.0745362 + 0.997218i $$0.476252\pi$$
$$150$$ 0.381966 0.0311874
$$151$$ 15.7082 1.27832 0.639158 0.769076i $$-0.279283\pi$$
0.639158 + 0.769076i $$0.279283\pi$$
$$152$$ 0.145898 0.0118339
$$153$$ −1.76393 −0.142605
$$154$$ 0 0
$$155$$ −3.23607 −0.259927
$$156$$ 0.763932 0.0611635
$$157$$ 13.7082 1.09403 0.547017 0.837122i $$-0.315763\pi$$
0.547017 + 0.837122i $$0.315763\pi$$
$$158$$ 0.472136 0.0375611
$$159$$ −1.23607 −0.0980266
$$160$$ −1.00000 −0.0790569
$$161$$ 6.00000 0.472866
$$162$$ 7.70820 0.605614
$$163$$ −3.85410 −0.301877 −0.150938 0.988543i $$-0.548229\pi$$
−0.150938 + 0.988543i $$0.548229\pi$$
$$164$$ 5.32624 0.415909
$$165$$ 0 0
$$166$$ −5.32624 −0.413396
$$167$$ −2.47214 −0.191300 −0.0956498 0.995415i $$-0.530493\pi$$
−0.0956498 + 0.995415i $$0.530493\pi$$
$$168$$ 0.381966 0.0294693
$$169$$ −9.00000 −0.692308
$$170$$ −0.618034 −0.0474010
$$171$$ −0.416408 −0.0318435
$$172$$ −4.85410 −0.370122
$$173$$ 12.0000 0.912343 0.456172 0.889892i $$-0.349220\pi$$
0.456172 + 0.889892i $$0.349220\pi$$
$$174$$ −0.472136 −0.0357925
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ −2.05573 −0.154518
$$178$$ 1.90983 0.143148
$$179$$ −8.61803 −0.644142 −0.322071 0.946715i $$-0.604379\pi$$
−0.322071 + 0.946715i $$0.604379\pi$$
$$180$$ 2.85410 0.212732
$$181$$ 12.4721 0.927047 0.463523 0.886085i $$-0.346585\pi$$
0.463523 + 0.886085i $$0.346585\pi$$
$$182$$ 2.00000 0.148250
$$183$$ −4.76393 −0.352160
$$184$$ 6.00000 0.442326
$$185$$ −2.47214 −0.181755
$$186$$ 1.23607 0.0906329
$$187$$ 0 0
$$188$$ 4.76393 0.347445
$$189$$ −2.23607 −0.162650
$$190$$ −0.145898 −0.0105846
$$191$$ −10.4721 −0.757737 −0.378869 0.925450i $$-0.623687\pi$$
−0.378869 + 0.925450i $$0.623687\pi$$
$$192$$ 0.381966 0.0275660
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 10.5623 0.758329
$$195$$ −0.763932 −0.0547063
$$196$$ 1.00000 0.0714286
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ 7.41641 0.525735 0.262868 0.964832i $$-0.415332\pi$$
0.262868 + 0.964832i $$0.415332\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −1.94427 −0.137138
$$202$$ −8.94427 −0.629317
$$203$$ −1.23607 −0.0867550
$$204$$ 0.236068 0.0165281
$$205$$ −5.32624 −0.372001
$$206$$ 0.944272 0.0657905
$$207$$ −17.1246 −1.19024
$$208$$ 2.00000 0.138675
$$209$$ 0 0
$$210$$ −0.381966 −0.0263582
$$211$$ −22.2705 −1.53317 −0.766583 0.642146i $$-0.778044\pi$$
−0.766583 + 0.642146i $$0.778044\pi$$
$$212$$ −3.23607 −0.222254
$$213$$ 1.41641 0.0970507
$$214$$ 17.6180 1.20434
$$215$$ 4.85410 0.331047
$$216$$ −2.23607 −0.152145
$$217$$ 3.23607 0.219679
$$218$$ 8.47214 0.573805
$$219$$ 2.72949 0.184442
$$220$$ 0 0
$$221$$ 1.23607 0.0831469
$$222$$ 0.944272 0.0633754
$$223$$ 19.7082 1.31976 0.659879 0.751371i $$-0.270607\pi$$
0.659879 + 0.751371i $$0.270607\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −2.85410 −0.190273
$$226$$ 6.32624 0.420815
$$227$$ 9.27051 0.615305 0.307653 0.951499i $$-0.400457\pi$$
0.307653 + 0.951499i $$0.400457\pi$$
$$228$$ 0.0557281 0.00369068
$$229$$ −29.1246 −1.92461 −0.962304 0.271975i $$-0.912323\pi$$
−0.962304 + 0.271975i $$0.912323\pi$$
$$230$$ −6.00000 −0.395628
$$231$$ 0 0
$$232$$ −1.23607 −0.0811518
$$233$$ 22.0902 1.44718 0.723588 0.690233i $$-0.242492\pi$$
0.723588 + 0.690233i $$0.242492\pi$$
$$234$$ −5.70820 −0.373157
$$235$$ −4.76393 −0.310765
$$236$$ −5.38197 −0.350336
$$237$$ 0.180340 0.0117143
$$238$$ 0.618034 0.0400612
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ −0.381966 −0.0246558
$$241$$ 27.2705 1.75665 0.878324 0.478066i $$-0.158662\pi$$
0.878324 + 0.478066i $$0.158662\pi$$
$$242$$ 0 0
$$243$$ 9.65248 0.619207
$$244$$ −12.4721 −0.798447
$$245$$ −1.00000 −0.0638877
$$246$$ 2.03444 0.129711
$$247$$ 0.291796 0.0185665
$$248$$ 3.23607 0.205491
$$249$$ −2.03444 −0.128928
$$250$$ −1.00000 −0.0632456
$$251$$ −26.8328 −1.69367 −0.846836 0.531854i $$-0.821496\pi$$
−0.846836 + 0.531854i $$0.821496\pi$$
$$252$$ −2.85410 −0.179792
$$253$$ 0 0
$$254$$ 15.4164 0.967311
$$255$$ −0.236068 −0.0147832
$$256$$ 1.00000 0.0625000
$$257$$ 9.43769 0.588707 0.294354 0.955697i $$-0.404896\pi$$
0.294354 + 0.955697i $$0.404896\pi$$
$$258$$ −1.85410 −0.115431
$$259$$ 2.47214 0.153611
$$260$$ −2.00000 −0.124035
$$261$$ 3.52786 0.218369
$$262$$ 16.7984 1.03781
$$263$$ −2.18034 −0.134446 −0.0672228 0.997738i $$-0.521414\pi$$
−0.0672228 + 0.997738i $$0.521414\pi$$
$$264$$ 0 0
$$265$$ 3.23607 0.198790
$$266$$ 0.145898 0.00894558
$$267$$ 0.729490 0.0446441
$$268$$ −5.09017 −0.310932
$$269$$ 10.1803 0.620706 0.310353 0.950621i $$-0.399553\pi$$
0.310353 + 0.950621i $$0.399553\pi$$
$$270$$ 2.23607 0.136083
$$271$$ 17.4164 1.05797 0.528986 0.848631i $$-0.322573\pi$$
0.528986 + 0.848631i $$0.322573\pi$$
$$272$$ 0.618034 0.0374738
$$273$$ 0.763932 0.0462353
$$274$$ 10.0902 0.609569
$$275$$ 0 0
$$276$$ 2.29180 0.137950
$$277$$ 19.7082 1.18415 0.592076 0.805882i $$-0.298309\pi$$
0.592076 + 0.805882i $$0.298309\pi$$
$$278$$ 9.52786 0.571443
$$279$$ −9.23607 −0.552949
$$280$$ −1.00000 −0.0597614
$$281$$ 5.03444 0.300330 0.150165 0.988661i $$-0.452020\pi$$
0.150165 + 0.988661i $$0.452020\pi$$
$$282$$ 1.81966 0.108359
$$283$$ −12.0000 −0.713326 −0.356663 0.934233i $$-0.616086\pi$$
−0.356663 + 0.934233i $$0.616086\pi$$
$$284$$ 3.70820 0.220041
$$285$$ −0.0557281 −0.00330105
$$286$$ 0 0
$$287$$ 5.32624 0.314398
$$288$$ −2.85410 −0.168180
$$289$$ −16.6180 −0.977531
$$290$$ 1.23607 0.0725844
$$291$$ 4.03444 0.236503
$$292$$ 7.14590 0.418182
$$293$$ 7.52786 0.439783 0.219891 0.975524i $$-0.429430\pi$$
0.219891 + 0.975524i $$0.429430\pi$$
$$294$$ 0.381966 0.0222767
$$295$$ 5.38197 0.313350
$$296$$ 2.47214 0.143690
$$297$$ 0 0
$$298$$ 1.81966 0.105410
$$299$$ 12.0000 0.693978
$$300$$ 0.381966 0.0220528
$$301$$ −4.85410 −0.279786
$$302$$ 15.7082 0.903906
$$303$$ −3.41641 −0.196268
$$304$$ 0.145898 0.00836783
$$305$$ 12.4721 0.714152
$$306$$ −1.76393 −0.100837
$$307$$ −7.43769 −0.424492 −0.212246 0.977216i $$-0.568078\pi$$
−0.212246 + 0.977216i $$0.568078\pi$$
$$308$$ 0 0
$$309$$ 0.360680 0.0205184
$$310$$ −3.23607 −0.183796
$$311$$ 11.1246 0.630819 0.315409 0.948956i $$-0.397858\pi$$
0.315409 + 0.948956i $$0.397858\pi$$
$$312$$ 0.763932 0.0432491
$$313$$ 5.85410 0.330893 0.165447 0.986219i $$-0.447093\pi$$
0.165447 + 0.986219i $$0.447093\pi$$
$$314$$ 13.7082 0.773599
$$315$$ 2.85410 0.160810
$$316$$ 0.472136 0.0265597
$$317$$ −25.2361 −1.41740 −0.708699 0.705511i $$-0.750718\pi$$
−0.708699 + 0.705511i $$0.750718\pi$$
$$318$$ −1.23607 −0.0693153
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ 6.72949 0.375604
$$322$$ 6.00000 0.334367
$$323$$ 0.0901699 0.00501719
$$324$$ 7.70820 0.428234
$$325$$ 2.00000 0.110940
$$326$$ −3.85410 −0.213459
$$327$$ 3.23607 0.178955
$$328$$ 5.32624 0.294092
$$329$$ 4.76393 0.262644
$$330$$ 0 0
$$331$$ −23.0344 −1.26609 −0.633044 0.774116i $$-0.718195\pi$$
−0.633044 + 0.774116i $$0.718195\pi$$
$$332$$ −5.32624 −0.292315
$$333$$ −7.05573 −0.386652
$$334$$ −2.47214 −0.135269
$$335$$ 5.09017 0.278106
$$336$$ 0.381966 0.0208380
$$337$$ −13.3820 −0.728962 −0.364481 0.931211i $$-0.618754\pi$$
−0.364481 + 0.931211i $$0.618754\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 2.41641 0.131241
$$340$$ −0.618034 −0.0335176
$$341$$ 0 0
$$342$$ −0.416408 −0.0225168
$$343$$ 1.00000 0.0539949
$$344$$ −4.85410 −0.261716
$$345$$ −2.29180 −0.123386
$$346$$ 12.0000 0.645124
$$347$$ −10.3262 −0.554341 −0.277171 0.960821i $$-0.589397\pi$$
−0.277171 + 0.960821i $$0.589397\pi$$
$$348$$ −0.472136 −0.0253091
$$349$$ 14.4721 0.774676 0.387338 0.921938i $$-0.373395\pi$$
0.387338 + 0.921938i $$0.373395\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ −4.47214 −0.238705
$$352$$ 0 0
$$353$$ 27.4508 1.46106 0.730531 0.682880i $$-0.239273\pi$$
0.730531 + 0.682880i $$0.239273\pi$$
$$354$$ −2.05573 −0.109261
$$355$$ −3.70820 −0.196811
$$356$$ 1.90983 0.101221
$$357$$ 0.236068 0.0124940
$$358$$ −8.61803 −0.455477
$$359$$ −12.6525 −0.667772 −0.333886 0.942613i $$-0.608360\pi$$
−0.333886 + 0.942613i $$0.608360\pi$$
$$360$$ 2.85410 0.150424
$$361$$ −18.9787 −0.998880
$$362$$ 12.4721 0.655521
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ −7.14590 −0.374033
$$366$$ −4.76393 −0.249015
$$367$$ 5.81966 0.303784 0.151892 0.988397i $$-0.451463\pi$$
0.151892 + 0.988397i $$0.451463\pi$$
$$368$$ 6.00000 0.312772
$$369$$ −15.2016 −0.791365
$$370$$ −2.47214 −0.128520
$$371$$ −3.23607 −0.168008
$$372$$ 1.23607 0.0640871
$$373$$ −15.5279 −0.804002 −0.402001 0.915639i $$-0.631685\pi$$
−0.402001 + 0.915639i $$0.631685\pi$$
$$374$$ 0 0
$$375$$ −0.381966 −0.0197246
$$376$$ 4.76393 0.245681
$$377$$ −2.47214 −0.127321
$$378$$ −2.23607 −0.115011
$$379$$ −21.1459 −1.08619 −0.543096 0.839671i $$-0.682748\pi$$
−0.543096 + 0.839671i $$0.682748\pi$$
$$380$$ −0.145898 −0.00748441
$$381$$ 5.88854 0.301679
$$382$$ −10.4721 −0.535801
$$383$$ −24.0000 −1.22634 −0.613171 0.789950i $$-0.710106\pi$$
−0.613171 + 0.789950i $$0.710106\pi$$
$$384$$ 0.381966 0.0194921
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ 13.8541 0.704244
$$388$$ 10.5623 0.536220
$$389$$ 20.8328 1.05627 0.528133 0.849162i $$-0.322892\pi$$
0.528133 + 0.849162i $$0.322892\pi$$
$$390$$ −0.763932 −0.0386832
$$391$$ 3.70820 0.187532
$$392$$ 1.00000 0.0505076
$$393$$ 6.41641 0.323665
$$394$$ 6.00000 0.302276
$$395$$ −0.472136 −0.0237557
$$396$$ 0 0
$$397$$ 0.944272 0.0473916 0.0236958 0.999719i $$-0.492457\pi$$
0.0236958 + 0.999719i $$0.492457\pi$$
$$398$$ 7.41641 0.371751
$$399$$ 0.0557281 0.00278989
$$400$$ 1.00000 0.0500000
$$401$$ 11.5066 0.574611 0.287306 0.957839i $$-0.407240\pi$$
0.287306 + 0.957839i $$0.407240\pi$$
$$402$$ −1.94427 −0.0969715
$$403$$ 6.47214 0.322400
$$404$$ −8.94427 −0.444994
$$405$$ −7.70820 −0.383024
$$406$$ −1.23607 −0.0613450
$$407$$ 0 0
$$408$$ 0.236068 0.0116871
$$409$$ −7.88854 −0.390063 −0.195032 0.980797i $$-0.562481\pi$$
−0.195032 + 0.980797i $$0.562481\pi$$
$$410$$ −5.32624 −0.263044
$$411$$ 3.85410 0.190109
$$412$$ 0.944272 0.0465209
$$413$$ −5.38197 −0.264829
$$414$$ −17.1246 −0.841629
$$415$$ 5.32624 0.261455
$$416$$ 2.00000 0.0980581
$$417$$ 3.63932 0.178218
$$418$$ 0 0
$$419$$ −3.32624 −0.162497 −0.0812487 0.996694i $$-0.525891\pi$$
−0.0812487 + 0.996694i $$0.525891\pi$$
$$420$$ −0.381966 −0.0186380
$$421$$ 38.3607 1.86959 0.934793 0.355194i $$-0.115585\pi$$
0.934793 + 0.355194i $$0.115585\pi$$
$$422$$ −22.2705 −1.08411
$$423$$ −13.5967 −0.661096
$$424$$ −3.23607 −0.157157
$$425$$ 0.618034 0.0299791
$$426$$ 1.41641 0.0686252
$$427$$ −12.4721 −0.603569
$$428$$ 17.6180 0.851600
$$429$$ 0 0
$$430$$ 4.85410 0.234086
$$431$$ 26.7639 1.28917 0.644587 0.764531i $$-0.277030\pi$$
0.644587 + 0.764531i $$0.277030\pi$$
$$432$$ −2.23607 −0.107583
$$433$$ −39.4508 −1.89589 −0.947943 0.318439i $$-0.896841\pi$$
−0.947943 + 0.318439i $$0.896841\pi$$
$$434$$ 3.23607 0.155336
$$435$$ 0.472136 0.0226372
$$436$$ 8.47214 0.405742
$$437$$ 0.875388 0.0418755
$$438$$ 2.72949 0.130420
$$439$$ 2.65248 0.126596 0.0632979 0.997995i $$-0.479838\pi$$
0.0632979 + 0.997995i $$0.479838\pi$$
$$440$$ 0 0
$$441$$ −2.85410 −0.135910
$$442$$ 1.23607 0.0587938
$$443$$ −1.03444 −0.0491478 −0.0245739 0.999698i $$-0.507823\pi$$
−0.0245739 + 0.999698i $$0.507823\pi$$
$$444$$ 0.944272 0.0448132
$$445$$ −1.90983 −0.0905346
$$446$$ 19.7082 0.933211
$$447$$ 0.695048 0.0328747
$$448$$ 1.00000 0.0472456
$$449$$ −31.4508 −1.48426 −0.742129 0.670257i $$-0.766184\pi$$
−0.742129 + 0.670257i $$0.766184\pi$$
$$450$$ −2.85410 −0.134544
$$451$$ 0 0
$$452$$ 6.32624 0.297561
$$453$$ 6.00000 0.281905
$$454$$ 9.27051 0.435087
$$455$$ −2.00000 −0.0937614
$$456$$ 0.0557281 0.00260971
$$457$$ 2.27051 0.106210 0.0531050 0.998589i $$-0.483088\pi$$
0.0531050 + 0.998589i $$0.483088\pi$$
$$458$$ −29.1246 −1.36090
$$459$$ −1.38197 −0.0645046
$$460$$ −6.00000 −0.279751
$$461$$ −10.9443 −0.509726 −0.254863 0.966977i $$-0.582030\pi$$
−0.254863 + 0.966977i $$0.582030\pi$$
$$462$$ 0 0
$$463$$ 19.4164 0.902357 0.451178 0.892434i $$-0.351004\pi$$
0.451178 + 0.892434i $$0.351004\pi$$
$$464$$ −1.23607 −0.0573830
$$465$$ −1.23607 −0.0573213
$$466$$ 22.0902 1.02331
$$467$$ 17.8885 0.827783 0.413892 0.910326i $$-0.364169\pi$$
0.413892 + 0.910326i $$0.364169\pi$$
$$468$$ −5.70820 −0.263862
$$469$$ −5.09017 −0.235042
$$470$$ −4.76393 −0.219744
$$471$$ 5.23607 0.241265
$$472$$ −5.38197 −0.247725
$$473$$ 0 0
$$474$$ 0.180340 0.00828329
$$475$$ 0.145898 0.00669426
$$476$$ 0.618034 0.0283275
$$477$$ 9.23607 0.422891
$$478$$ 0 0
$$479$$ −2.11146 −0.0964749 −0.0482374 0.998836i $$-0.515360\pi$$
−0.0482374 + 0.998836i $$0.515360\pi$$
$$480$$ −0.381966 −0.0174343
$$481$$ 4.94427 0.225439
$$482$$ 27.2705 1.24214
$$483$$ 2.29180 0.104280
$$484$$ 0 0
$$485$$ −10.5623 −0.479610
$$486$$ 9.65248 0.437845
$$487$$ 26.8328 1.21591 0.607955 0.793971i $$-0.291990\pi$$
0.607955 + 0.793971i $$0.291990\pi$$
$$488$$ −12.4721 −0.564587
$$489$$ −1.47214 −0.0665723
$$490$$ −1.00000 −0.0451754
$$491$$ 5.50658 0.248508 0.124254 0.992250i $$-0.460346\pi$$
0.124254 + 0.992250i $$0.460346\pi$$
$$492$$ 2.03444 0.0917197
$$493$$ −0.763932 −0.0344058
$$494$$ 0.291796 0.0131285
$$495$$ 0 0
$$496$$ 3.23607 0.145304
$$497$$ 3.70820 0.166336
$$498$$ −2.03444 −0.0911655
$$499$$ −39.7984 −1.78162 −0.890810 0.454376i $$-0.849862\pi$$
−0.890810 + 0.454376i $$0.849862\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −0.944272 −0.0421870
$$502$$ −26.8328 −1.19761
$$503$$ 23.1246 1.03108 0.515538 0.856867i $$-0.327592\pi$$
0.515538 + 0.856867i $$0.327592\pi$$
$$504$$ −2.85410 −0.127132
$$505$$ 8.94427 0.398015
$$506$$ 0 0
$$507$$ −3.43769 −0.152673
$$508$$ 15.4164 0.683992
$$509$$ −34.4721 −1.52795 −0.763975 0.645246i $$-0.776755\pi$$
−0.763975 + 0.645246i $$0.776755\pi$$
$$510$$ −0.236068 −0.0104533
$$511$$ 7.14590 0.316116
$$512$$ 1.00000 0.0441942
$$513$$ −0.326238 −0.0144038
$$514$$ 9.43769 0.416279
$$515$$ −0.944272 −0.0416096
$$516$$ −1.85410 −0.0816223
$$517$$ 0 0
$$518$$ 2.47214 0.108619
$$519$$ 4.58359 0.201197
$$520$$ −2.00000 −0.0877058
$$521$$ 11.2016 0.490752 0.245376 0.969428i $$-0.421089\pi$$
0.245376 + 0.969428i $$0.421089\pi$$
$$522$$ 3.52786 0.154410
$$523$$ −40.9787 −1.79187 −0.895937 0.444181i $$-0.853495\pi$$
−0.895937 + 0.444181i $$0.853495\pi$$
$$524$$ 16.7984 0.733840
$$525$$ 0.381966 0.0166704
$$526$$ −2.18034 −0.0950673
$$527$$ 2.00000 0.0871214
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 3.23607 0.140566
$$531$$ 15.3607 0.666597
$$532$$ 0.145898 0.00632548
$$533$$ 10.6525 0.461410
$$534$$ 0.729490 0.0315681
$$535$$ −17.6180 −0.761694
$$536$$ −5.09017 −0.219862
$$537$$ −3.29180 −0.142051
$$538$$ 10.1803 0.438906
$$539$$ 0 0
$$540$$ 2.23607 0.0962250
$$541$$ −38.8328 −1.66955 −0.834777 0.550589i $$-0.814403\pi$$
−0.834777 + 0.550589i $$0.814403\pi$$
$$542$$ 17.4164 0.748099
$$543$$ 4.76393 0.204440
$$544$$ 0.618034 0.0264980
$$545$$ −8.47214 −0.362906
$$546$$ 0.763932 0.0326933
$$547$$ −29.0902 −1.24381 −0.621903 0.783094i $$-0.713640\pi$$
−0.621903 + 0.783094i $$0.713640\pi$$
$$548$$ 10.0902 0.431031
$$549$$ 35.5967 1.51923
$$550$$ 0 0
$$551$$ −0.180340 −0.00768274
$$552$$ 2.29180 0.0975453
$$553$$ 0.472136 0.0200773
$$554$$ 19.7082 0.837321
$$555$$ −0.944272 −0.0400821
$$556$$ 9.52786 0.404071
$$557$$ 15.8197 0.670301 0.335150 0.942165i $$-0.391213\pi$$
0.335150 + 0.942165i $$0.391213\pi$$
$$558$$ −9.23607 −0.390994
$$559$$ −9.70820 −0.410613
$$560$$ −1.00000 −0.0422577
$$561$$ 0 0
$$562$$ 5.03444 0.212365
$$563$$ 41.5066 1.74929 0.874647 0.484761i $$-0.161093\pi$$
0.874647 + 0.484761i $$0.161093\pi$$
$$564$$ 1.81966 0.0766215
$$565$$ −6.32624 −0.266147
$$566$$ −12.0000 −0.504398
$$567$$ 7.70820 0.323714
$$568$$ 3.70820 0.155593
$$569$$ 3.09017 0.129547 0.0647733 0.997900i $$-0.479368\pi$$
0.0647733 + 0.997900i $$0.479368\pi$$
$$570$$ −0.0557281 −0.00233419
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ 0 0
$$573$$ −4.00000 −0.167102
$$574$$ 5.32624 0.222313
$$575$$ 6.00000 0.250217
$$576$$ −2.85410 −0.118921
$$577$$ −26.2148 −1.09134 −0.545668 0.838002i $$-0.683724\pi$$
−0.545668 + 0.838002i $$0.683724\pi$$
$$578$$ −16.6180 −0.691219
$$579$$ −0.763932 −0.0317479
$$580$$ 1.23607 0.0513249
$$581$$ −5.32624 −0.220970
$$582$$ 4.03444 0.167233
$$583$$ 0 0
$$584$$ 7.14590 0.295699
$$585$$ 5.70820 0.236005
$$586$$ 7.52786 0.310973
$$587$$ −39.7426 −1.64035 −0.820177 0.572109i $$-0.806125\pi$$
−0.820177 + 0.572109i $$0.806125\pi$$
$$588$$ 0.381966 0.0157520
$$589$$ 0.472136 0.0194540
$$590$$ 5.38197 0.221572
$$591$$ 2.29180 0.0942719
$$592$$ 2.47214 0.101604
$$593$$ 0.909830 0.0373622 0.0186811 0.999825i $$-0.494053\pi$$
0.0186811 + 0.999825i $$0.494053\pi$$
$$594$$ 0 0
$$595$$ −0.618034 −0.0253369
$$596$$ 1.81966 0.0745362
$$597$$ 2.83282 0.115939
$$598$$ 12.0000 0.490716
$$599$$ −20.7639 −0.848391 −0.424196 0.905571i $$-0.639443\pi$$
−0.424196 + 0.905571i $$0.639443\pi$$
$$600$$ 0.381966 0.0155937
$$601$$ −17.9098 −0.730557 −0.365279 0.930898i $$-0.619026\pi$$
−0.365279 + 0.930898i $$0.619026\pi$$
$$602$$ −4.85410 −0.197838
$$603$$ 14.5279 0.591620
$$604$$ 15.7082 0.639158
$$605$$ 0 0
$$606$$ −3.41641 −0.138782
$$607$$ 19.5967 0.795407 0.397704 0.917514i $$-0.369807\pi$$
0.397704 + 0.917514i $$0.369807\pi$$
$$608$$ 0.145898 0.00591695
$$609$$ −0.472136 −0.0191319
$$610$$ 12.4721 0.504982
$$611$$ 9.52786 0.385456
$$612$$ −1.76393 −0.0713027
$$613$$ −10.5836 −0.427467 −0.213734 0.976892i $$-0.568562\pi$$
−0.213734 + 0.976892i $$0.568562\pi$$
$$614$$ −7.43769 −0.300161
$$615$$ −2.03444 −0.0820366
$$616$$ 0 0
$$617$$ −6.03444 −0.242937 −0.121469 0.992595i $$-0.538760\pi$$
−0.121469 + 0.992595i $$0.538760\pi$$
$$618$$ 0.360680 0.0145087
$$619$$ 34.3262 1.37969 0.689844 0.723958i $$-0.257679\pi$$
0.689844 + 0.723958i $$0.257679\pi$$
$$620$$ −3.23607 −0.129964
$$621$$ −13.4164 −0.538382
$$622$$ 11.1246 0.446056
$$623$$ 1.90983 0.0765157
$$624$$ 0.763932 0.0305818
$$625$$ 1.00000 0.0400000
$$626$$ 5.85410 0.233977
$$627$$ 0 0
$$628$$ 13.7082 0.547017
$$629$$ 1.52786 0.0609199
$$630$$ 2.85410 0.113710
$$631$$ −10.2918 −0.409710 −0.204855 0.978792i $$-0.565672\pi$$
−0.204855 + 0.978792i $$0.565672\pi$$
$$632$$ 0.472136 0.0187806
$$633$$ −8.50658 −0.338106
$$634$$ −25.2361 −1.00225
$$635$$ −15.4164 −0.611781
$$636$$ −1.23607 −0.0490133
$$637$$ 2.00000 0.0792429
$$638$$ 0 0
$$639$$ −10.5836 −0.418680
$$640$$ −1.00000 −0.0395285
$$641$$ 24.6738 0.974555 0.487278 0.873247i $$-0.337990\pi$$
0.487278 + 0.873247i $$0.337990\pi$$
$$642$$ 6.72949 0.265592
$$643$$ 31.9787 1.26112 0.630559 0.776142i $$-0.282826\pi$$
0.630559 + 0.776142i $$0.282826\pi$$
$$644$$ 6.00000 0.236433
$$645$$ 1.85410 0.0730052
$$646$$ 0.0901699 0.00354769
$$647$$ −22.1803 −0.871999 −0.436000 0.899947i $$-0.643605\pi$$
−0.436000 + 0.899947i $$0.643605\pi$$
$$648$$ 7.70820 0.302807
$$649$$ 0 0
$$650$$ 2.00000 0.0784465
$$651$$ 1.23607 0.0484453
$$652$$ −3.85410 −0.150938
$$653$$ −26.8328 −1.05005 −0.525025 0.851087i $$-0.675944\pi$$
−0.525025 + 0.851087i $$0.675944\pi$$
$$654$$ 3.23607 0.126540
$$655$$ −16.7984 −0.656367
$$656$$ 5.32624 0.207955
$$657$$ −20.3951 −0.795689
$$658$$ 4.76393 0.185717
$$659$$ −7.09017 −0.276194 −0.138097 0.990419i $$-0.544099\pi$$
−0.138097 + 0.990419i $$0.544099\pi$$
$$660$$ 0 0
$$661$$ −10.1115 −0.393290 −0.196645 0.980475i $$-0.563005\pi$$
−0.196645 + 0.980475i $$0.563005\pi$$
$$662$$ −23.0344 −0.895259
$$663$$ 0.472136 0.0183362
$$664$$ −5.32624 −0.206698
$$665$$ −0.145898 −0.00565768
$$666$$ −7.05573 −0.273404
$$667$$ −7.41641 −0.287164
$$668$$ −2.47214 −0.0956498
$$669$$ 7.52786 0.291044
$$670$$ 5.09017 0.196650
$$671$$ 0 0
$$672$$ 0.381966 0.0147347
$$673$$ −32.1459 −1.23913 −0.619567 0.784944i $$-0.712692\pi$$
−0.619567 + 0.784944i $$0.712692\pi$$
$$674$$ −13.3820 −0.515454
$$675$$ −2.23607 −0.0860663
$$676$$ −9.00000 −0.346154
$$677$$ 35.5967 1.36809 0.684047 0.729438i $$-0.260218\pi$$
0.684047 + 0.729438i $$0.260218\pi$$
$$678$$ 2.41641 0.0928016
$$679$$ 10.5623 0.405344
$$680$$ −0.618034 −0.0237005
$$681$$ 3.54102 0.135692
$$682$$ 0 0
$$683$$ −48.9443 −1.87280 −0.936400 0.350934i $$-0.885864\pi$$
−0.936400 + 0.350934i $$0.885864\pi$$
$$684$$ −0.416408 −0.0159218
$$685$$ −10.0902 −0.385526
$$686$$ 1.00000 0.0381802
$$687$$ −11.1246 −0.424430
$$688$$ −4.85410 −0.185061
$$689$$ −6.47214 −0.246569
$$690$$ −2.29180 −0.0872472
$$691$$ 27.8541 1.05962 0.529810 0.848116i $$-0.322263\pi$$
0.529810 + 0.848116i $$0.322263\pi$$
$$692$$ 12.0000 0.456172
$$693$$ 0 0
$$694$$ −10.3262 −0.391979
$$695$$ −9.52786 −0.361412
$$696$$ −0.472136 −0.0178963
$$697$$ 3.29180 0.124686
$$698$$ 14.4721 0.547778
$$699$$ 8.43769 0.319143
$$700$$ 1.00000 0.0377964
$$701$$ 10.1803 0.384506 0.192253 0.981345i $$-0.438421\pi$$
0.192253 + 0.981345i $$0.438421\pi$$
$$702$$ −4.47214 −0.168790
$$703$$ 0.360680 0.0136033
$$704$$ 0 0
$$705$$ −1.81966 −0.0685324
$$706$$ 27.4508 1.03313
$$707$$ −8.94427 −0.336384
$$708$$ −2.05573 −0.0772590
$$709$$ −7.05573 −0.264983 −0.132492 0.991184i $$-0.542298\pi$$
−0.132492 + 0.991184i $$0.542298\pi$$
$$710$$ −3.70820 −0.139166
$$711$$ −1.34752 −0.0505361
$$712$$ 1.90983 0.0715739
$$713$$ 19.4164 0.727150
$$714$$ 0.236068 0.00883462
$$715$$ 0 0
$$716$$ −8.61803 −0.322071
$$717$$ 0 0
$$718$$ −12.6525 −0.472186
$$719$$ −12.3607 −0.460976 −0.230488 0.973075i $$-0.574032\pi$$
−0.230488 + 0.973075i $$0.574032\pi$$
$$720$$ 2.85410 0.106366
$$721$$ 0.944272 0.0351665
$$722$$ −18.9787 −0.706315
$$723$$ 10.4164 0.387390
$$724$$ 12.4721 0.463523
$$725$$ −1.23607 −0.0459064
$$726$$ 0 0
$$727$$ −14.8328 −0.550119 −0.275059 0.961427i $$-0.588698\pi$$
−0.275059 + 0.961427i $$0.588698\pi$$
$$728$$ 2.00000 0.0741249
$$729$$ −19.4377 −0.719915
$$730$$ −7.14590 −0.264482
$$731$$ −3.00000 −0.110959
$$732$$ −4.76393 −0.176080
$$733$$ −8.76393 −0.323703 −0.161852 0.986815i $$-0.551747\pi$$
−0.161852 + 0.986815i $$0.551747\pi$$
$$734$$ 5.81966 0.214808
$$735$$ −0.381966 −0.0140890
$$736$$ 6.00000 0.221163
$$737$$ 0 0
$$738$$ −15.2016 −0.559580
$$739$$ −40.3262 −1.48342 −0.741712 0.670718i $$-0.765986\pi$$
−0.741712 + 0.670718i $$0.765986\pi$$
$$740$$ −2.47214 −0.0908775
$$741$$ 0.111456 0.00409445
$$742$$ −3.23607 −0.118800
$$743$$ −47.8885 −1.75686 −0.878430 0.477871i $$-0.841409\pi$$
−0.878430 + 0.477871i $$0.841409\pi$$
$$744$$ 1.23607 0.0453165
$$745$$ −1.81966 −0.0666672
$$746$$ −15.5279 −0.568515
$$747$$ 15.2016 0.556198
$$748$$ 0 0
$$749$$ 17.6180 0.643749
$$750$$ −0.381966 −0.0139474
$$751$$ −1.34752 −0.0491719 −0.0245859 0.999698i $$-0.507827\pi$$
−0.0245859 + 0.999698i $$0.507827\pi$$
$$752$$ 4.76393 0.173723
$$753$$ −10.2492 −0.373502
$$754$$ −2.47214 −0.0900299
$$755$$ −15.7082 −0.571680
$$756$$ −2.23607 −0.0813250
$$757$$ −15.8885 −0.577479 −0.288739 0.957408i $$-0.593236\pi$$
−0.288739 + 0.957408i $$0.593236\pi$$
$$758$$ −21.1459 −0.768054
$$759$$ 0 0
$$760$$ −0.145898 −0.00529228
$$761$$ −7.20163 −0.261059 −0.130529 0.991444i $$-0.541668\pi$$
−0.130529 + 0.991444i $$0.541668\pi$$
$$762$$ 5.88854 0.213319
$$763$$ 8.47214 0.306712
$$764$$ −10.4721 −0.378869
$$765$$ 1.76393 0.0637751
$$766$$ −24.0000 −0.867155
$$767$$ −10.7639 −0.388663
$$768$$ 0.381966 0.0137830
$$769$$ 27.8885 1.00569 0.502843 0.864378i $$-0.332287\pi$$
0.502843 + 0.864378i $$0.332287\pi$$
$$770$$ 0 0
$$771$$ 3.60488 0.129827
$$772$$ −2.00000 −0.0719816
$$773$$ 6.65248 0.239273 0.119636 0.992818i $$-0.461827\pi$$
0.119636 + 0.992818i $$0.461827\pi$$
$$774$$ 13.8541 0.497975
$$775$$ 3.23607 0.116243
$$776$$ 10.5623 0.379165
$$777$$ 0.944272 0.0338756
$$778$$ 20.8328 0.746893
$$779$$ 0.777088 0.0278421
$$780$$ −0.763932 −0.0273532
$$781$$ 0 0
$$782$$ 3.70820 0.132605
$$783$$ 2.76393 0.0987749
$$784$$ 1.00000 0.0357143
$$785$$ −13.7082 −0.489267
$$786$$ 6.41641 0.228866
$$787$$ −36.1033 −1.28694 −0.643472 0.765469i $$-0.722507\pi$$
−0.643472 + 0.765469i $$0.722507\pi$$
$$788$$ 6.00000 0.213741
$$789$$ −0.832816 −0.0296490
$$790$$ −0.472136 −0.0167978
$$791$$ 6.32624 0.224935
$$792$$ 0 0
$$793$$ −24.9443 −0.885797
$$794$$ 0.944272 0.0335110
$$795$$ 1.23607 0.0438388
$$796$$ 7.41641 0.262868
$$797$$ 38.3607 1.35880 0.679402 0.733766i $$-0.262239\pi$$
0.679402 + 0.733766i $$0.262239\pi$$
$$798$$ 0.0557281 0.00197275
$$799$$ 2.94427 0.104161
$$800$$ 1.00000 0.0353553
$$801$$ −5.45085 −0.192596
$$802$$ 11.5066 0.406311
$$803$$ 0 0
$$804$$ −1.94427 −0.0685692
$$805$$ −6.00000 −0.211472
$$806$$ 6.47214 0.227971
$$807$$ 3.88854 0.136883
$$808$$ −8.94427 −0.314658
$$809$$ 51.8115 1.82160 0.910798 0.412852i $$-0.135467\pi$$
0.910798 + 0.412852i $$0.135467\pi$$
$$810$$ −7.70820 −0.270839
$$811$$ 34.9230 1.22631 0.613156 0.789962i $$-0.289900\pi$$
0.613156 + 0.789962i $$0.289900\pi$$
$$812$$ −1.23607 −0.0433775
$$813$$ 6.65248 0.233313
$$814$$ 0 0
$$815$$ 3.85410 0.135003
$$816$$ 0.236068 0.00826403
$$817$$ −0.708204 −0.0247769
$$818$$ −7.88854 −0.275816
$$819$$ −5.70820 −0.199461
$$820$$ −5.32624 −0.186000
$$821$$ 48.9443 1.70817 0.854083 0.520136i $$-0.174119\pi$$
0.854083 + 0.520136i $$0.174119\pi$$
$$822$$ 3.85410 0.134427
$$823$$ 50.8328 1.77192 0.885960 0.463761i $$-0.153500\pi$$
0.885960 + 0.463761i $$0.153500\pi$$
$$824$$ 0.944272 0.0328953
$$825$$ 0 0
$$826$$ −5.38197 −0.187263
$$827$$ 1.74265 0.0605977 0.0302989 0.999541i $$-0.490354\pi$$
0.0302989 + 0.999541i $$0.490354\pi$$
$$828$$ −17.1246 −0.595121
$$829$$ 11.8197 0.410514 0.205257 0.978708i $$-0.434197\pi$$
0.205257 + 0.978708i $$0.434197\pi$$
$$830$$ 5.32624 0.184876
$$831$$ 7.52786 0.261139
$$832$$ 2.00000 0.0693375
$$833$$ 0.618034 0.0214136
$$834$$ 3.63932 0.126019
$$835$$ 2.47214 0.0855518
$$836$$ 0 0
$$837$$ −7.23607 −0.250115
$$838$$ −3.32624 −0.114903
$$839$$ −4.47214 −0.154395 −0.0771976 0.997016i $$-0.524597\pi$$
−0.0771976 + 0.997016i $$0.524597\pi$$
$$840$$ −0.381966 −0.0131791
$$841$$ −27.4721 −0.947315
$$842$$ 38.3607 1.32200
$$843$$ 1.92299 0.0662311
$$844$$ −22.2705 −0.766583
$$845$$ 9.00000 0.309609
$$846$$ −13.5967 −0.467466
$$847$$ 0 0
$$848$$ −3.23607 −0.111127
$$849$$ −4.58359 −0.157308
$$850$$ 0.618034 0.0211984
$$851$$ 14.8328 0.508462
$$852$$ 1.41641 0.0485253
$$853$$ −17.7082 −0.606317 −0.303159 0.952940i $$-0.598041\pi$$
−0.303159 + 0.952940i $$0.598041\pi$$
$$854$$ −12.4721 −0.426788
$$855$$ 0.416408 0.0142408
$$856$$ 17.6180 0.602172
$$857$$ 24.2705 0.829065 0.414532 0.910035i $$-0.363945\pi$$
0.414532 + 0.910035i $$0.363945\pi$$
$$858$$ 0 0
$$859$$ −27.5623 −0.940414 −0.470207 0.882556i $$-0.655821\pi$$
−0.470207 + 0.882556i $$0.655821\pi$$
$$860$$ 4.85410 0.165524
$$861$$ 2.03444 0.0693336
$$862$$ 26.7639 0.911583
$$863$$ 11.2361 0.382480 0.191240 0.981543i $$-0.438749\pi$$
0.191240 + 0.981543i $$0.438749\pi$$
$$864$$ −2.23607 −0.0760726
$$865$$ −12.0000 −0.408012
$$866$$ −39.4508 −1.34059
$$867$$ −6.34752 −0.215573
$$868$$ 3.23607 0.109839
$$869$$ 0 0
$$870$$ 0.472136 0.0160069
$$871$$ −10.1803 −0.344948
$$872$$ 8.47214 0.286903
$$873$$ −30.1459 −1.02028
$$874$$ 0.875388 0.0296104
$$875$$ −1.00000 −0.0338062
$$876$$ 2.72949 0.0922209
$$877$$ 18.0000 0.607817 0.303908 0.952701i $$-0.401708\pi$$
0.303908 + 0.952701i $$0.401708\pi$$
$$878$$ 2.65248 0.0895167
$$879$$ 2.87539 0.0969844
$$880$$ 0 0
$$881$$ 10.8541 0.365684 0.182842 0.983142i $$-0.441470\pi$$
0.182842 + 0.983142i $$0.441470\pi$$
$$882$$ −2.85410 −0.0961026
$$883$$ 22.5623 0.759282 0.379641 0.925134i $$-0.376048\pi$$
0.379641 + 0.925134i $$0.376048\pi$$
$$884$$ 1.23607 0.0415735
$$885$$ 2.05573 0.0691025
$$886$$ −1.03444 −0.0347528
$$887$$ 40.3607 1.35518 0.677589 0.735440i $$-0.263025\pi$$
0.677589 + 0.735440i $$0.263025\pi$$
$$888$$ 0.944272 0.0316877
$$889$$ 15.4164 0.517050
$$890$$ −1.90983 −0.0640176
$$891$$ 0 0
$$892$$ 19.7082 0.659879
$$893$$ 0.695048 0.0232589
$$894$$ 0.695048 0.0232459
$$895$$ 8.61803 0.288069
$$896$$ 1.00000 0.0334077
$$897$$ 4.58359 0.153042
$$898$$ −31.4508 −1.04953
$$899$$ −4.00000 −0.133407
$$900$$ −2.85410 −0.0951367
$$901$$ −2.00000 −0.0666297
$$902$$ 0 0
$$903$$ −1.85410 −0.0617007
$$904$$ 6.32624 0.210408
$$905$$ −12.4721 −0.414588
$$906$$ 6.00000 0.199337
$$907$$ −11.2705 −0.374231 −0.187116 0.982338i $$-0.559914\pi$$
−0.187116 + 0.982338i $$0.559914\pi$$
$$908$$ 9.27051 0.307653
$$909$$ 25.5279 0.846706
$$910$$ −2.00000 −0.0662994
$$911$$ −2.40325 −0.0796233 −0.0398116 0.999207i $$-0.512676\pi$$
−0.0398116 + 0.999207i $$0.512676\pi$$
$$912$$ 0.0557281 0.00184534
$$913$$ 0 0
$$914$$ 2.27051 0.0751018
$$915$$ 4.76393 0.157491
$$916$$ −29.1246 −0.962304
$$917$$ 16.7984 0.554731
$$918$$ −1.38197 −0.0456117
$$919$$ 44.8328 1.47890 0.739449 0.673213i $$-0.235086\pi$$
0.739449 + 0.673213i $$0.235086\pi$$
$$920$$ −6.00000 −0.197814
$$921$$ −2.84095 −0.0936124
$$922$$ −10.9443 −0.360430
$$923$$ 7.41641 0.244114
$$924$$ 0 0
$$925$$ 2.47214 0.0812833
$$926$$ 19.4164 0.638063
$$927$$ −2.69505 −0.0885170
$$928$$ −1.23607 −0.0405759
$$929$$ −45.7426 −1.50077 −0.750384 0.661002i $$-0.770131\pi$$
−0.750384 + 0.661002i $$0.770131\pi$$
$$930$$ −1.23607 −0.0405323
$$931$$ 0.145898 0.00478161
$$932$$ 22.0902 0.723588
$$933$$ 4.24922 0.139113
$$934$$ 17.8885 0.585331
$$935$$ 0 0
$$936$$ −5.70820 −0.186578
$$937$$ −40.3951 −1.31965 −0.659826 0.751419i $$-0.729370\pi$$
−0.659826 + 0.751419i $$0.729370\pi$$
$$938$$ −5.09017 −0.166200
$$939$$ 2.23607 0.0729713
$$940$$ −4.76393 −0.155382
$$941$$ 11.8197 0.385310 0.192655 0.981267i $$-0.438290\pi$$
0.192655 + 0.981267i $$0.438290\pi$$
$$942$$ 5.23607 0.170600
$$943$$ 31.9574 1.04068
$$944$$ −5.38197 −0.175168
$$945$$ 2.23607 0.0727393
$$946$$ 0 0
$$947$$ −53.0902 −1.72520 −0.862599 0.505888i $$-0.831165\pi$$
−0.862599 + 0.505888i $$0.831165\pi$$
$$948$$ 0.180340 0.00585717
$$949$$ 14.2918 0.463931
$$950$$ 0.145898 0.00473356
$$951$$ −9.63932 −0.312576
$$952$$ 0.618034 0.0200306
$$953$$ −26.4508 −0.856827 −0.428414 0.903583i $$-0.640927\pi$$
−0.428414 + 0.903583i $$0.640927\pi$$
$$954$$ 9.23607 0.299029
$$955$$ 10.4721 0.338870
$$956$$ 0 0
$$957$$ 0 0
$$958$$ −2.11146 −0.0682181
$$959$$ 10.0902 0.325829
$$960$$ −0.381966 −0.0123279
$$961$$ −20.5279 −0.662189
$$962$$ 4.94427 0.159410
$$963$$ −50.2837 −1.62037
$$964$$ 27.2705 0.878324
$$965$$ 2.00000 0.0643823
$$966$$ 2.29180 0.0737373
$$967$$ −49.0132 −1.57616 −0.788078 0.615575i $$-0.788924\pi$$
−0.788078 + 0.615575i $$0.788924\pi$$
$$968$$ 0 0
$$969$$ 0.0344419 0.00110643
$$970$$ −10.5623 −0.339135
$$971$$ 26.8328 0.861106 0.430553 0.902565i $$-0.358319\pi$$
0.430553 + 0.902565i $$0.358319\pi$$
$$972$$ 9.65248 0.309603
$$973$$ 9.52786 0.305449
$$974$$ 26.8328 0.859779
$$975$$ 0.763932 0.0244654
$$976$$ −12.4721 −0.399223
$$977$$ −19.3050 −0.617620 −0.308810 0.951124i $$-0.599931\pi$$
−0.308810 + 0.951124i $$0.599931\pi$$
$$978$$ −1.47214 −0.0470737
$$979$$ 0 0
$$980$$ −1.00000 −0.0319438
$$981$$ −24.1803 −0.772019
$$982$$ 5.50658 0.175722
$$983$$ 16.6525 0.531131 0.265566 0.964093i $$-0.414441\pi$$
0.265566 + 0.964093i $$0.414441\pi$$
$$984$$ 2.03444 0.0648556
$$985$$ −6.00000 −0.191176
$$986$$ −0.763932 −0.0243286
$$987$$ 1.81966 0.0579204
$$988$$ 0.291796 0.00928327
$$989$$ −29.1246 −0.926109
$$990$$ 0 0
$$991$$ −10.6525 −0.338387 −0.169194 0.985583i $$-0.554116\pi$$
−0.169194 + 0.985583i $$0.554116\pi$$
$$992$$ 3.23607 0.102745
$$993$$ −8.79837 −0.279208
$$994$$ 3.70820 0.117617
$$995$$ −7.41641 −0.235116
$$996$$ −2.03444 −0.0644638
$$997$$ 17.1246 0.542342 0.271171 0.962531i $$-0.412589\pi$$
0.271171 + 0.962531i $$0.412589\pi$$
$$998$$ −39.7984 −1.25980
$$999$$ −5.52786 −0.174894
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.cf.1.1 2
11.2 odd 10 770.2.n.b.631.1 yes 4
11.6 odd 10 770.2.n.b.421.1 4
11.10 odd 2 8470.2.a.bt.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.b.421.1 4 11.6 odd 10
770.2.n.b.631.1 yes 4 11.2 odd 10
8470.2.a.bt.1.1 2 11.10 odd 2
8470.2.a.cf.1.1 2 1.1 even 1 trivial