# Properties

 Label 8470.2.a.bt.1.2 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$67.6332905120$$ Analytic rank: $$1$$ Dimension: $$2$$ 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 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +2.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.61803 q^{6} -1.00000 q^{7} -1.00000 q^{8} +3.85410 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +2.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.61803 q^{6} -1.00000 q^{7} -1.00000 q^{8} +3.85410 q^{9} +1.00000 q^{10} +2.61803 q^{12} -2.00000 q^{13} +1.00000 q^{14} -2.61803 q^{15} +1.00000 q^{16} +1.61803 q^{17} -3.85410 q^{18} -6.85410 q^{19} -1.00000 q^{20} -2.61803 q^{21} +6.00000 q^{23} -2.61803 q^{24} +1.00000 q^{25} +2.00000 q^{26} +2.23607 q^{27} -1.00000 q^{28} -3.23607 q^{29} +2.61803 q^{30} -1.23607 q^{31} -1.00000 q^{32} -1.61803 q^{34} +1.00000 q^{35} +3.85410 q^{36} -6.47214 q^{37} +6.85410 q^{38} -5.23607 q^{39} +1.00000 q^{40} +10.3262 q^{41} +2.61803 q^{42} -1.85410 q^{43} -3.85410 q^{45} -6.00000 q^{46} +9.23607 q^{47} +2.61803 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.23607 q^{51} -2.00000 q^{52} +1.23607 q^{53} -2.23607 q^{54} +1.00000 q^{56} -17.9443 q^{57} +3.23607 q^{58} -7.61803 q^{59} -2.61803 q^{60} +3.52786 q^{61} +1.23607 q^{62} -3.85410 q^{63} +1.00000 q^{64} +2.00000 q^{65} +6.09017 q^{67} +1.61803 q^{68} +15.7082 q^{69} -1.00000 q^{70} -9.70820 q^{71} -3.85410 q^{72} -13.8541 q^{73} +6.47214 q^{74} +2.61803 q^{75} -6.85410 q^{76} +5.23607 q^{78} +8.47214 q^{79} -1.00000 q^{80} -5.70820 q^{81} -10.3262 q^{82} -10.3262 q^{83} -2.61803 q^{84} -1.61803 q^{85} +1.85410 q^{86} -8.47214 q^{87} +13.0902 q^{89} +3.85410 q^{90} +2.00000 q^{91} +6.00000 q^{92} -3.23607 q^{93} -9.23607 q^{94} +6.85410 q^{95} -2.61803 q^{96} -9.56231 q^{97} -1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 3q^{3} + 2q^{4} - 2q^{5} - 3q^{6} - 2q^{7} - 2q^{8} + q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 3q^{3} + 2q^{4} - 2q^{5} - 3q^{6} - 2q^{7} - 2q^{8} + q^{9} + 2q^{10} + 3q^{12} - 4q^{13} + 2q^{14} - 3q^{15} + 2q^{16} + q^{17} - q^{18} - 7q^{19} - 2q^{20} - 3q^{21} + 12q^{23} - 3q^{24} + 2q^{25} + 4q^{26} - 2q^{28} - 2q^{29} + 3q^{30} + 2q^{31} - 2q^{32} - q^{34} + 2q^{35} + q^{36} - 4q^{37} + 7q^{38} - 6q^{39} + 2q^{40} + 5q^{41} + 3q^{42} + 3q^{43} - q^{45} - 12q^{46} + 14q^{47} + 3q^{48} + 2q^{49} - 2q^{50} + 4q^{51} - 4q^{52} - 2q^{53} + 2q^{56} - 18q^{57} + 2q^{58} - 13q^{59} - 3q^{60} + 16q^{61} - 2q^{62} - q^{63} + 2q^{64} + 4q^{65} + q^{67} + q^{68} + 18q^{69} - 2q^{70} - 6q^{71} - q^{72} - 21q^{73} + 4q^{74} + 3q^{75} - 7q^{76} + 6q^{78} + 8q^{79} - 2q^{80} + 2q^{81} - 5q^{82} - 5q^{83} - 3q^{84} - q^{85} - 3q^{86} - 8q^{87} + 15q^{89} + q^{90} + 4q^{91} + 12q^{92} - 2q^{93} - 14q^{94} + 7q^{95} - 3q^{96} + q^{97} - 2q^{98} + O(q^{100})$$

## 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
$$3$$ 2.61803 1.51152 0.755761 0.654847i $$-0.227267\pi$$
0.755761 + 0.654847i $$0.227267\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −2.61803 −1.06881
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 3.85410 1.28470
$$10$$ 1.00000 0.316228
$$11$$ 0 0
$$12$$ 2.61803 0.755761
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 1.00000 0.267261
$$15$$ −2.61803 −0.675973
$$16$$ 1.00000 0.250000
$$17$$ 1.61803 0.392431 0.196215 0.980561i $$-0.437135\pi$$
0.196215 + 0.980561i $$0.437135\pi$$
$$18$$ −3.85410 −0.908421
$$19$$ −6.85410 −1.57244 −0.786219 0.617947i $$-0.787964\pi$$
−0.786219 + 0.617947i $$0.787964\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −2.61803 −0.571302
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ −2.61803 −0.534404
$$25$$ 1.00000 0.200000
$$26$$ 2.00000 0.392232
$$27$$ 2.23607 0.430331
$$28$$ −1.00000 −0.188982
$$29$$ −3.23607 −0.600923 −0.300461 0.953794i $$-0.597141\pi$$
−0.300461 + 0.953794i $$0.597141\pi$$
$$30$$ 2.61803 0.477985
$$31$$ −1.23607 −0.222004 −0.111002 0.993820i $$-0.535406\pi$$
−0.111002 + 0.993820i $$0.535406\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −1.61803 −0.277491
$$35$$ 1.00000 0.169031
$$36$$ 3.85410 0.642350
$$37$$ −6.47214 −1.06401 −0.532006 0.846740i $$-0.678562\pi$$
−0.532006 + 0.846740i $$0.678562\pi$$
$$38$$ 6.85410 1.11188
$$39$$ −5.23607 −0.838442
$$40$$ 1.00000 0.158114
$$41$$ 10.3262 1.61269 0.806344 0.591447i $$-0.201443\pi$$
0.806344 + 0.591447i $$0.201443\pi$$
$$42$$ 2.61803 0.403971
$$43$$ −1.85410 −0.282748 −0.141374 0.989956i $$-0.545152\pi$$
−0.141374 + 0.989956i $$0.545152\pi$$
$$44$$ 0 0
$$45$$ −3.85410 −0.574536
$$46$$ −6.00000 −0.884652
$$47$$ 9.23607 1.34722 0.673609 0.739087i $$-0.264743\pi$$
0.673609 + 0.739087i $$0.264743\pi$$
$$48$$ 2.61803 0.377881
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 4.23607 0.593168
$$52$$ −2.00000 −0.277350
$$53$$ 1.23607 0.169787 0.0848935 0.996390i $$-0.472945\pi$$
0.0848935 + 0.996390i $$0.472945\pi$$
$$54$$ −2.23607 −0.304290
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ −17.9443 −2.37678
$$58$$ 3.23607 0.424917
$$59$$ −7.61803 −0.991784 −0.495892 0.868384i $$-0.665159\pi$$
−0.495892 + 0.868384i $$0.665159\pi$$
$$60$$ −2.61803 −0.337987
$$61$$ 3.52786 0.451697 0.225848 0.974162i $$-0.427485\pi$$
0.225848 + 0.974162i $$0.427485\pi$$
$$62$$ 1.23607 0.156981
$$63$$ −3.85410 −0.485571
$$64$$ 1.00000 0.125000
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ 6.09017 0.744033 0.372016 0.928226i $$-0.378667\pi$$
0.372016 + 0.928226i $$0.378667\pi$$
$$68$$ 1.61803 0.196215
$$69$$ 15.7082 1.89105
$$70$$ −1.00000 −0.119523
$$71$$ −9.70820 −1.15215 −0.576076 0.817396i $$-0.695417\pi$$
−0.576076 + 0.817396i $$0.695417\pi$$
$$72$$ −3.85410 −0.454210
$$73$$ −13.8541 −1.62150 −0.810750 0.585393i $$-0.800940\pi$$
−0.810750 + 0.585393i $$0.800940\pi$$
$$74$$ 6.47214 0.752371
$$75$$ 2.61803 0.302305
$$76$$ −6.85410 −0.786219
$$77$$ 0 0
$$78$$ 5.23607 0.592868
$$79$$ 8.47214 0.953190 0.476595 0.879123i $$-0.341871\pi$$
0.476595 + 0.879123i $$0.341871\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ −5.70820 −0.634245
$$82$$ −10.3262 −1.14034
$$83$$ −10.3262 −1.13345 −0.566726 0.823906i $$-0.691790\pi$$
−0.566726 + 0.823906i $$0.691790\pi$$
$$84$$ −2.61803 −0.285651
$$85$$ −1.61803 −0.175500
$$86$$ 1.85410 0.199933
$$87$$ −8.47214 −0.908308
$$88$$ 0 0
$$89$$ 13.0902 1.38756 0.693778 0.720189i $$-0.255945\pi$$
0.693778 + 0.720189i $$0.255945\pi$$
$$90$$ 3.85410 0.406258
$$91$$ 2.00000 0.209657
$$92$$ 6.00000 0.625543
$$93$$ −3.23607 −0.335565
$$94$$ −9.23607 −0.952628
$$95$$ 6.85410 0.703216
$$96$$ −2.61803 −0.267202
$$97$$ −9.56231 −0.970905 −0.485453 0.874263i $$-0.661345\pi$$
−0.485453 + 0.874263i $$0.661345\pi$$
$$98$$ −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 $$0.646794\pi$$
$$102$$ −4.23607 −0.419433
$$103$$ −16.9443 −1.66957 −0.834784 0.550577i $$-0.814408\pi$$
−0.834784 + 0.550577i $$0.814408\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 2.61803 0.255494
$$106$$ −1.23607 −0.120058
$$107$$ −15.3820 −1.48703 −0.743515 0.668719i $$-0.766843\pi$$
−0.743515 + 0.668719i $$0.766843\pi$$
$$108$$ 2.23607 0.215166
$$109$$ 0.472136 0.0452224 0.0226112 0.999744i $$-0.492802\pi$$
0.0226112 + 0.999744i $$0.492802\pi$$
$$110$$ 0 0
$$111$$ −16.9443 −1.60828
$$112$$ −1.00000 −0.0944911
$$113$$ −9.32624 −0.877339 −0.438669 0.898649i $$-0.644550\pi$$
−0.438669 + 0.898649i $$0.644550\pi$$
$$114$$ 17.9443 1.68064
$$115$$ −6.00000 −0.559503
$$116$$ −3.23607 −0.300461
$$117$$ −7.70820 −0.712624
$$118$$ 7.61803 0.701297
$$119$$ −1.61803 −0.148325
$$120$$ 2.61803 0.238993
$$121$$ 0 0
$$122$$ −3.52786 −0.319398
$$123$$ 27.0344 2.43761
$$124$$ −1.23607 −0.111002
$$125$$ −1.00000 −0.0894427
$$126$$ 3.85410 0.343351
$$127$$ 11.4164 1.01304 0.506521 0.862228i $$-0.330931\pi$$
0.506521 + 0.862228i $$0.330931\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −4.85410 −0.427380
$$130$$ −2.00000 −0.175412
$$131$$ 7.79837 0.681347 0.340674 0.940182i $$-0.389345\pi$$
0.340674 + 0.940182i $$0.389345\pi$$
$$132$$ 0 0
$$133$$ 6.85410 0.594326
$$134$$ −6.09017 −0.526111
$$135$$ −2.23607 −0.192450
$$136$$ −1.61803 −0.138745
$$137$$ −1.09017 −0.0931395 −0.0465698 0.998915i $$-0.514829\pi$$
−0.0465698 + 0.998915i $$0.514829\pi$$
$$138$$ −15.7082 −1.33717
$$139$$ −18.4721 −1.56679 −0.783393 0.621527i $$-0.786513\pi$$
−0.783393 + 0.621527i $$0.786513\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ 24.1803 2.03635
$$142$$ 9.70820 0.814694
$$143$$ 0 0
$$144$$ 3.85410 0.321175
$$145$$ 3.23607 0.268741
$$146$$ 13.8541 1.14657
$$147$$ 2.61803 0.215932
$$148$$ −6.47214 −0.532006
$$149$$ −24.1803 −1.98093 −0.990465 0.137762i $$-0.956009\pi$$
−0.990465 + 0.137762i $$0.956009\pi$$
$$150$$ −2.61803 −0.213762
$$151$$ −2.29180 −0.186504 −0.0932519 0.995643i $$-0.529726\pi$$
−0.0932519 + 0.995643i $$0.529726\pi$$
$$152$$ 6.85410 0.555941
$$153$$ 6.23607 0.504156
$$154$$ 0 0
$$155$$ 1.23607 0.0992834
$$156$$ −5.23607 −0.419221
$$157$$ 0.291796 0.0232879 0.0116439 0.999932i $$-0.496294\pi$$
0.0116439 + 0.999932i $$0.496294\pi$$
$$158$$ −8.47214 −0.674007
$$159$$ 3.23607 0.256637
$$160$$ 1.00000 0.0790569
$$161$$ −6.00000 −0.472866
$$162$$ 5.70820 0.448479
$$163$$ 2.85410 0.223551 0.111775 0.993734i $$-0.464346\pi$$
0.111775 + 0.993734i $$0.464346\pi$$
$$164$$ 10.3262 0.806344
$$165$$ 0 0
$$166$$ 10.3262 0.801471
$$167$$ −6.47214 −0.500829 −0.250414 0.968139i $$-0.580567\pi$$
−0.250414 + 0.968139i $$0.580567\pi$$
$$168$$ 2.61803 0.201986
$$169$$ −9.00000 −0.692308
$$170$$ 1.61803 0.124098
$$171$$ −26.4164 −2.02011
$$172$$ −1.85410 −0.141374
$$173$$ −12.0000 −0.912343 −0.456172 0.889892i $$-0.650780\pi$$
−0.456172 + 0.889892i $$0.650780\pi$$
$$174$$ 8.47214 0.642271
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ −19.9443 −1.49910
$$178$$ −13.0902 −0.981150
$$179$$ −6.38197 −0.477011 −0.238505 0.971141i $$-0.576657\pi$$
−0.238505 + 0.971141i $$0.576657\pi$$
$$180$$ −3.85410 −0.287268
$$181$$ 3.52786 0.262224 0.131112 0.991368i $$-0.458145\pi$$
0.131112 + 0.991368i $$0.458145\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ 9.23607 0.682750
$$184$$ −6.00000 −0.442326
$$185$$ 6.47214 0.475841
$$186$$ 3.23607 0.237280
$$187$$ 0 0
$$188$$ 9.23607 0.673609
$$189$$ −2.23607 −0.162650
$$190$$ −6.85410 −0.497249
$$191$$ −1.52786 −0.110552 −0.0552762 0.998471i $$-0.517604\pi$$
−0.0552762 + 0.998471i $$0.517604\pi$$
$$192$$ 2.61803 0.188940
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 9.56231 0.686534
$$195$$ 5.23607 0.374963
$$196$$ 1.00000 0.0714286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −19.4164 −1.37639 −0.688196 0.725525i $$-0.741597\pi$$
−0.688196 + 0.725525i $$0.741597\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 15.9443 1.12462
$$202$$ 8.94427 0.629317
$$203$$ 3.23607 0.227127
$$204$$ 4.23607 0.296584
$$205$$ −10.3262 −0.721216
$$206$$ 16.9443 1.18056
$$207$$ 23.1246 1.60727
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ −2.61803 −0.180662
$$211$$ −11.2705 −0.775894 −0.387947 0.921682i $$-0.626816\pi$$
−0.387947 + 0.921682i $$0.626816\pi$$
$$212$$ 1.23607 0.0848935
$$213$$ −25.4164 −1.74150
$$214$$ 15.3820 1.05149
$$215$$ 1.85410 0.126449
$$216$$ −2.23607 −0.152145
$$217$$ 1.23607 0.0839098
$$218$$ −0.472136 −0.0319771
$$219$$ −36.2705 −2.45093
$$220$$ 0 0
$$221$$ −3.23607 −0.217681
$$222$$ 16.9443 1.13723
$$223$$ 6.29180 0.421330 0.210665 0.977558i $$-0.432437\pi$$
0.210665 + 0.977558i $$0.432437\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 3.85410 0.256940
$$226$$ 9.32624 0.620372
$$227$$ 24.2705 1.61089 0.805445 0.592670i $$-0.201926\pi$$
0.805445 + 0.592670i $$0.201926\pi$$
$$228$$ −17.9443 −1.18839
$$229$$ 11.1246 0.735135 0.367568 0.929997i $$-0.380191\pi$$
0.367568 + 0.929997i $$0.380191\pi$$
$$230$$ 6.00000 0.395628
$$231$$ 0 0
$$232$$ 3.23607 0.212458
$$233$$ −10.9098 −0.714727 −0.357363 0.933965i $$-0.616324\pi$$
−0.357363 + 0.933965i $$0.616324\pi$$
$$234$$ 7.70820 0.503901
$$235$$ −9.23607 −0.602495
$$236$$ −7.61803 −0.495892
$$237$$ 22.1803 1.44077
$$238$$ 1.61803 0.104882
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ −2.61803 −0.168993
$$241$$ 6.27051 0.403919 0.201960 0.979394i $$-0.435269\pi$$
0.201960 + 0.979394i $$0.435269\pi$$
$$242$$ 0 0
$$243$$ −21.6525 −1.38901
$$244$$ 3.52786 0.225848
$$245$$ −1.00000 −0.0638877
$$246$$ −27.0344 −1.72365
$$247$$ 13.7082 0.872232
$$248$$ 1.23607 0.0784904
$$249$$ −27.0344 −1.71324
$$250$$ 1.00000 0.0632456
$$251$$ 26.8328 1.69367 0.846836 0.531854i $$-0.178504\pi$$
0.846836 + 0.531854i $$0.178504\pi$$
$$252$$ −3.85410 −0.242786
$$253$$ 0 0
$$254$$ −11.4164 −0.716329
$$255$$ −4.23607 −0.265273
$$256$$ 1.00000 0.0625000
$$257$$ 29.5623 1.84405 0.922023 0.387135i $$-0.126535\pi$$
0.922023 + 0.387135i $$0.126535\pi$$
$$258$$ 4.85410 0.302203
$$259$$ 6.47214 0.402159
$$260$$ 2.00000 0.124035
$$261$$ −12.4721 −0.772006
$$262$$ −7.79837 −0.481785
$$263$$ −20.1803 −1.24437 −0.622187 0.782869i $$-0.713755\pi$$
−0.622187 + 0.782869i $$0.713755\pi$$
$$264$$ 0 0
$$265$$ −1.23607 −0.0759311
$$266$$ −6.85410 −0.420252
$$267$$ 34.2705 2.09732
$$268$$ 6.09017 0.372016
$$269$$ −12.1803 −0.742648 −0.371324 0.928503i $$-0.621096\pi$$
−0.371324 + 0.928503i $$0.621096\pi$$
$$270$$ 2.23607 0.136083
$$271$$ 9.41641 0.572006 0.286003 0.958229i $$-0.407673\pi$$
0.286003 + 0.958229i $$0.407673\pi$$
$$272$$ 1.61803 0.0981077
$$273$$ 5.23607 0.316901
$$274$$ 1.09017 0.0658596
$$275$$ 0 0
$$276$$ 15.7082 0.945523
$$277$$ −6.29180 −0.378037 −0.189019 0.981973i $$-0.560531\pi$$
−0.189019 + 0.981973i $$0.560531\pi$$
$$278$$ 18.4721 1.10789
$$279$$ −4.76393 −0.285209
$$280$$ −1.00000 −0.0597614
$$281$$ 24.0344 1.43377 0.716887 0.697189i $$-0.245566\pi$$
0.716887 + 0.697189i $$0.245566\pi$$
$$282$$ −24.1803 −1.43992
$$283$$ 12.0000 0.713326 0.356663 0.934233i $$-0.383914\pi$$
0.356663 + 0.934233i $$0.383914\pi$$
$$284$$ −9.70820 −0.576076
$$285$$ 17.9443 1.06293
$$286$$ 0 0
$$287$$ −10.3262 −0.609539
$$288$$ −3.85410 −0.227105
$$289$$ −14.3820 −0.845998
$$290$$ −3.23607 −0.190028
$$291$$ −25.0344 −1.46754
$$292$$ −13.8541 −0.810750
$$293$$ −16.4721 −0.962312 −0.481156 0.876635i $$-0.659783\pi$$
−0.481156 + 0.876635i $$0.659783\pi$$
$$294$$ −2.61803 −0.152687
$$295$$ 7.61803 0.443539
$$296$$ 6.47214 0.376185
$$297$$ 0 0
$$298$$ 24.1803 1.40073
$$299$$ −12.0000 −0.693978
$$300$$ 2.61803 0.151152
$$301$$ 1.85410 0.106869
$$302$$ 2.29180 0.131878
$$303$$ −23.4164 −1.34524
$$304$$ −6.85410 −0.393110
$$305$$ −3.52786 −0.202005
$$306$$ −6.23607 −0.356492
$$307$$ 27.5623 1.57306 0.786532 0.617550i $$-0.211875\pi$$
0.786532 + 0.617550i $$0.211875\pi$$
$$308$$ 0 0
$$309$$ −44.3607 −2.52359
$$310$$ −1.23607 −0.0702039
$$311$$ −29.1246 −1.65151 −0.825753 0.564032i $$-0.809249\pi$$
−0.825753 + 0.564032i $$0.809249\pi$$
$$312$$ 5.23607 0.296434
$$313$$ −0.854102 −0.0482767 −0.0241383 0.999709i $$-0.507684\pi$$
−0.0241383 + 0.999709i $$0.507684\pi$$
$$314$$ −0.291796 −0.0164670
$$315$$ 3.85410 0.217154
$$316$$ 8.47214 0.476595
$$317$$ −20.7639 −1.16622 −0.583109 0.812394i $$-0.698164\pi$$
−0.583109 + 0.812394i $$0.698164\pi$$
$$318$$ −3.23607 −0.181470
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ −40.2705 −2.24768
$$322$$ 6.00000 0.334367
$$323$$ −11.0902 −0.617074
$$324$$ −5.70820 −0.317122
$$325$$ −2.00000 −0.110940
$$326$$ −2.85410 −0.158074
$$327$$ 1.23607 0.0683547
$$328$$ −10.3262 −0.570171
$$329$$ −9.23607 −0.509201
$$330$$ 0 0
$$331$$ 6.03444 0.331683 0.165841 0.986152i $$-0.446966\pi$$
0.165841 + 0.986152i $$0.446966\pi$$
$$332$$ −10.3262 −0.566726
$$333$$ −24.9443 −1.36694
$$334$$ 6.47214 0.354140
$$335$$ −6.09017 −0.332742
$$336$$ −2.61803 −0.142825
$$337$$ 15.6180 0.850769 0.425384 0.905013i $$-0.360139\pi$$
0.425384 + 0.905013i $$0.360139\pi$$
$$338$$ 9.00000 0.489535
$$339$$ −24.4164 −1.32612
$$340$$ −1.61803 −0.0877502
$$341$$ 0 0
$$342$$ 26.4164 1.42844
$$343$$ −1.00000 −0.0539949
$$344$$ 1.85410 0.0999665
$$345$$ −15.7082 −0.845701
$$346$$ 12.0000 0.645124
$$347$$ −5.32624 −0.285927 −0.142964 0.989728i $$-0.545663\pi$$
−0.142964 + 0.989728i $$0.545663\pi$$
$$348$$ −8.47214 −0.454154
$$349$$ −5.52786 −0.295900 −0.147950 0.988995i $$-0.547267\pi$$
−0.147950 + 0.988995i $$0.547267\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ −4.47214 −0.238705
$$352$$ 0 0
$$353$$ −28.4508 −1.51429 −0.757143 0.653249i $$-0.773405\pi$$
−0.757143 + 0.653249i $$0.773405\pi$$
$$354$$ 19.9443 1.06003
$$355$$ 9.70820 0.515258
$$356$$ 13.0902 0.693778
$$357$$ −4.23607 −0.224196
$$358$$ 6.38197 0.337297
$$359$$ −18.6525 −0.984440 −0.492220 0.870471i $$-0.663814\pi$$
−0.492220 + 0.870471i $$0.663814\pi$$
$$360$$ 3.85410 0.203129
$$361$$ 27.9787 1.47256
$$362$$ −3.52786 −0.185420
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ 13.8541 0.725157
$$366$$ −9.23607 −0.482777
$$367$$ 28.1803 1.47100 0.735501 0.677524i $$-0.236947\pi$$
0.735501 + 0.677524i $$0.236947\pi$$
$$368$$ 6.00000 0.312772
$$369$$ 39.7984 2.07182
$$370$$ −6.47214 −0.336470
$$371$$ −1.23607 −0.0641735
$$372$$ −3.23607 −0.167782
$$373$$ 24.4721 1.26712 0.633560 0.773694i $$-0.281593\pi$$
0.633560 + 0.773694i $$0.281593\pi$$
$$374$$ 0 0
$$375$$ −2.61803 −0.135195
$$376$$ −9.23607 −0.476314
$$377$$ 6.47214 0.333332
$$378$$ 2.23607 0.115011
$$379$$ −27.8541 −1.43077 −0.715385 0.698731i $$-0.753748\pi$$
−0.715385 + 0.698731i $$0.753748\pi$$
$$380$$ 6.85410 0.351608
$$381$$ 29.8885 1.53124
$$382$$ 1.52786 0.0781723
$$383$$ −24.0000 −1.22634 −0.613171 0.789950i $$-0.710106\pi$$
−0.613171 + 0.789950i $$0.710106\pi$$
$$384$$ −2.61803 −0.133601
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ −7.14590 −0.363246
$$388$$ −9.56231 −0.485453
$$389$$ −32.8328 −1.66469 −0.832345 0.554258i $$-0.813002\pi$$
−0.832345 + 0.554258i $$0.813002\pi$$
$$390$$ −5.23607 −0.265139
$$391$$ 9.70820 0.490965
$$392$$ −1.00000 −0.0505076
$$393$$ 20.4164 1.02987
$$394$$ 6.00000 0.302276
$$395$$ −8.47214 −0.426279
$$396$$ 0 0
$$397$$ −16.9443 −0.850409 −0.425204 0.905097i $$-0.639798\pi$$
−0.425204 + 0.905097i $$0.639798\pi$$
$$398$$ 19.4164 0.973257
$$399$$ 17.9443 0.898337
$$400$$ 1.00000 0.0500000
$$401$$ −26.5066 −1.32368 −0.661838 0.749647i $$-0.730223\pi$$
−0.661838 + 0.749647i $$0.730223\pi$$
$$402$$ −15.9443 −0.795228
$$403$$ 2.47214 0.123146
$$404$$ −8.94427 −0.444994
$$405$$ 5.70820 0.283643
$$406$$ −3.23607 −0.160603
$$407$$ 0 0
$$408$$ −4.23607 −0.209717
$$409$$ −27.8885 −1.37900 −0.689500 0.724286i $$-0.742170\pi$$
−0.689500 + 0.724286i $$0.742170\pi$$
$$410$$ 10.3262 0.509977
$$411$$ −2.85410 −0.140782
$$412$$ −16.9443 −0.834784
$$413$$ 7.61803 0.374859
$$414$$ −23.1246 −1.13651
$$415$$ 10.3262 0.506895
$$416$$ 2.00000 0.0980581
$$417$$ −48.3607 −2.36823
$$418$$ 0 0
$$419$$ 12.3262 0.602176 0.301088 0.953596i $$-0.402650\pi$$
0.301088 + 0.953596i $$0.402650\pi$$
$$420$$ 2.61803 0.127747
$$421$$ −6.36068 −0.310001 −0.155000 0.987914i $$-0.549538\pi$$
−0.155000 + 0.987914i $$0.549538\pi$$
$$422$$ 11.2705 0.548640
$$423$$ 35.5967 1.73077
$$424$$ −1.23607 −0.0600288
$$425$$ 1.61803 0.0784862
$$426$$ 25.4164 1.23143
$$427$$ −3.52786 −0.170725
$$428$$ −15.3820 −0.743515
$$429$$ 0 0
$$430$$ −1.85410 −0.0894127
$$431$$ −31.2361 −1.50459 −0.752294 0.658827i $$-0.771053\pi$$
−0.752294 + 0.658827i $$0.771053\pi$$
$$432$$ 2.23607 0.107583
$$433$$ 16.4508 0.790577 0.395289 0.918557i $$-0.370645\pi$$
0.395289 + 0.918557i $$0.370645\pi$$
$$434$$ −1.23607 −0.0593332
$$435$$ 8.47214 0.406208
$$436$$ 0.472136 0.0226112
$$437$$ −41.1246 −1.96726
$$438$$ 36.2705 1.73307
$$439$$ 28.6525 1.36751 0.683754 0.729713i $$-0.260346\pi$$
0.683754 + 0.729713i $$0.260346\pi$$
$$440$$ 0 0
$$441$$ 3.85410 0.183529
$$442$$ 3.23607 0.153924
$$443$$ 28.0344 1.33196 0.665978 0.745971i $$-0.268014\pi$$
0.665978 + 0.745971i $$0.268014\pi$$
$$444$$ −16.9443 −0.804140
$$445$$ −13.0902 −0.620534
$$446$$ −6.29180 −0.297925
$$447$$ −63.3050 −2.99422
$$448$$ −1.00000 −0.0472456
$$449$$ 24.4508 1.15391 0.576953 0.816777i $$-0.304241\pi$$
0.576953 + 0.816777i $$0.304241\pi$$
$$450$$ −3.85410 −0.181684
$$451$$ 0 0
$$452$$ −9.32624 −0.438669
$$453$$ −6.00000 −0.281905
$$454$$ −24.2705 −1.13907
$$455$$ −2.00000 −0.0937614
$$456$$ 17.9443 0.840318
$$457$$ 31.2705 1.46277 0.731386 0.681963i $$-0.238874\pi$$
0.731386 + 0.681963i $$0.238874\pi$$
$$458$$ −11.1246 −0.519819
$$459$$ 3.61803 0.168875
$$460$$ −6.00000 −0.279751
$$461$$ −6.94427 −0.323427 −0.161713 0.986838i $$-0.551702\pi$$
−0.161713 + 0.986838i $$0.551702\pi$$
$$462$$ 0 0
$$463$$ −7.41641 −0.344670 −0.172335 0.985038i $$-0.555131\pi$$
−0.172335 + 0.985038i $$0.555131\pi$$
$$464$$ −3.23607 −0.150231
$$465$$ 3.23607 0.150069
$$466$$ 10.9098 0.505388
$$467$$ −17.8885 −0.827783 −0.413892 0.910326i $$-0.635831\pi$$
−0.413892 + 0.910326i $$0.635831\pi$$
$$468$$ −7.70820 −0.356312
$$469$$ −6.09017 −0.281218
$$470$$ 9.23607 0.426028
$$471$$ 0.763932 0.0352001
$$472$$ 7.61803 0.350648
$$473$$ 0 0
$$474$$ −22.1803 −1.01878
$$475$$ −6.85410 −0.314488
$$476$$ −1.61803 −0.0741625
$$477$$ 4.76393 0.218125
$$478$$ 0 0
$$479$$ 37.8885 1.73117 0.865586 0.500761i $$-0.166946\pi$$
0.865586 + 0.500761i $$0.166946\pi$$
$$480$$ 2.61803 0.119496
$$481$$ 12.9443 0.590208
$$482$$ −6.27051 −0.285614
$$483$$ −15.7082 −0.714748
$$484$$ 0 0
$$485$$ 9.56231 0.434202
$$486$$ 21.6525 0.982176
$$487$$ −26.8328 −1.21591 −0.607955 0.793971i $$-0.708010\pi$$
−0.607955 + 0.793971i $$0.708010\pi$$
$$488$$ −3.52786 −0.159699
$$489$$ 7.47214 0.337902
$$490$$ 1.00000 0.0451754
$$491$$ 32.5066 1.46700 0.733501 0.679689i $$-0.237885\pi$$
0.733501 + 0.679689i $$0.237885\pi$$
$$492$$ 27.0344 1.21881
$$493$$ −5.23607 −0.235821
$$494$$ −13.7082 −0.616761
$$495$$ 0 0
$$496$$ −1.23607 −0.0555011
$$497$$ 9.70820 0.435472
$$498$$ 27.0344 1.21144
$$499$$ −15.2016 −0.680518 −0.340259 0.940332i $$-0.610515\pi$$
−0.340259 + 0.940332i $$0.610515\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −16.9443 −0.757014
$$502$$ −26.8328 −1.19761
$$503$$ 17.1246 0.763549 0.381774 0.924256i $$-0.375313\pi$$
0.381774 + 0.924256i $$0.375313\pi$$
$$504$$ 3.85410 0.171675
$$505$$ 8.94427 0.398015
$$506$$ 0 0
$$507$$ −23.5623 −1.04644
$$508$$ 11.4164 0.506521
$$509$$ −25.5279 −1.13150 −0.565751 0.824576i $$-0.691414\pi$$
−0.565751 + 0.824576i $$0.691414\pi$$
$$510$$ 4.23607 0.187576
$$511$$ 13.8541 0.612869
$$512$$ −1.00000 −0.0441942
$$513$$ −15.3262 −0.676670
$$514$$ −29.5623 −1.30394
$$515$$ 16.9443 0.746654
$$516$$ −4.85410 −0.213690
$$517$$ 0 0
$$518$$ −6.47214 −0.284369
$$519$$ −31.4164 −1.37903
$$520$$ −2.00000 −0.0877058
$$521$$ 35.7984 1.56836 0.784178 0.620536i $$-0.213085\pi$$
0.784178 + 0.620536i $$0.213085\pi$$
$$522$$ 12.4721 0.545891
$$523$$ −5.97871 −0.261431 −0.130715 0.991420i $$-0.541727\pi$$
−0.130715 + 0.991420i $$0.541727\pi$$
$$524$$ 7.79837 0.340674
$$525$$ −2.61803 −0.114260
$$526$$ 20.1803 0.879905
$$527$$ −2.00000 −0.0871214
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 1.23607 0.0536914
$$531$$ −29.3607 −1.27414
$$532$$ 6.85410 0.297163
$$533$$ −20.6525 −0.894558
$$534$$ −34.2705 −1.48303
$$535$$ 15.3820 0.665020
$$536$$ −6.09017 −0.263055
$$537$$ −16.7082 −0.721012
$$538$$ 12.1803 0.525132
$$539$$ 0 0
$$540$$ −2.23607 −0.0962250
$$541$$ −14.8328 −0.637713 −0.318856 0.947803i $$-0.603299\pi$$
−0.318856 + 0.947803i $$0.603299\pi$$
$$542$$ −9.41641 −0.404469
$$543$$ 9.23607 0.396358
$$544$$ −1.61803 −0.0693726
$$545$$ −0.472136 −0.0202241
$$546$$ −5.23607 −0.224083
$$547$$ 17.9098 0.765769 0.382885 0.923796i $$-0.374931\pi$$
0.382885 + 0.923796i $$0.374931\pi$$
$$548$$ −1.09017 −0.0465698
$$549$$ 13.5967 0.580295
$$550$$ 0 0
$$551$$ 22.1803 0.944914
$$552$$ −15.7082 −0.668586
$$553$$ −8.47214 −0.360272
$$554$$ 6.29180 0.267313
$$555$$ 16.9443 0.719244
$$556$$ −18.4721 −0.783393
$$557$$ −38.1803 −1.61775 −0.808876 0.587979i $$-0.799924\pi$$
−0.808876 + 0.587979i $$0.799924\pi$$
$$558$$ 4.76393 0.201673
$$559$$ 3.70820 0.156840
$$560$$ 1.00000 0.0422577
$$561$$ 0 0
$$562$$ −24.0344 −1.01383
$$563$$ −3.49342 −0.147230 −0.0736151 0.997287i $$-0.523454\pi$$
−0.0736151 + 0.997287i $$0.523454\pi$$
$$564$$ 24.1803 1.01818
$$565$$ 9.32624 0.392358
$$566$$ −12.0000 −0.504398
$$567$$ 5.70820 0.239722
$$568$$ 9.70820 0.407347
$$569$$ 8.09017 0.339158 0.169579 0.985517i $$-0.445759\pi$$
0.169579 + 0.985517i $$0.445759\pi$$
$$570$$ −17.9443 −0.751603
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ 0 0
$$573$$ −4.00000 −0.167102
$$574$$ 10.3262 0.431009
$$575$$ 6.00000 0.250217
$$576$$ 3.85410 0.160588
$$577$$ 25.2148 1.04971 0.524853 0.851193i $$-0.324120\pi$$
0.524853 + 0.851193i $$0.324120\pi$$
$$578$$ 14.3820 0.598211
$$579$$ 5.23607 0.217604
$$580$$ 3.23607 0.134370
$$581$$ 10.3262 0.428405
$$582$$ 25.0344 1.03771
$$583$$ 0 0
$$584$$ 13.8541 0.573287
$$585$$ 7.70820 0.318695
$$586$$ 16.4721 0.680458
$$587$$ 2.74265 0.113201 0.0566006 0.998397i $$-0.481974\pi$$
0.0566006 + 0.998397i $$0.481974\pi$$
$$588$$ 2.61803 0.107966
$$589$$ 8.47214 0.349088
$$590$$ −7.61803 −0.313629
$$591$$ −15.7082 −0.646149
$$592$$ −6.47214 −0.266003
$$593$$ −12.0902 −0.496484 −0.248242 0.968698i $$-0.579853\pi$$
−0.248242 + 0.968698i $$0.579853\pi$$
$$594$$ 0 0
$$595$$ 1.61803 0.0663329
$$596$$ −24.1803 −0.990465
$$597$$ −50.8328 −2.08045
$$598$$ 12.0000 0.490716
$$599$$ −25.2361 −1.03112 −0.515559 0.856854i $$-0.672416\pi$$
−0.515559 + 0.856854i $$0.672416\pi$$
$$600$$ −2.61803 −0.106881
$$601$$ 29.0902 1.18661 0.593306 0.804977i $$-0.297822\pi$$
0.593306 + 0.804977i $$0.297822\pi$$
$$602$$ −1.85410 −0.0755676
$$603$$ 23.4721 0.955859
$$604$$ −2.29180 −0.0932519
$$605$$ 0 0
$$606$$ 23.4164 0.951227
$$607$$ 29.5967 1.20129 0.600647 0.799514i $$-0.294910\pi$$
0.600647 + 0.799514i $$0.294910\pi$$
$$608$$ 6.85410 0.277971
$$609$$ 8.47214 0.343308
$$610$$ 3.52786 0.142839
$$611$$ −18.4721 −0.747303
$$612$$ 6.23607 0.252078
$$613$$ 37.4164 1.51123 0.755617 0.655013i $$-0.227337\pi$$
0.755617 + 0.655013i $$0.227337\pi$$
$$614$$ −27.5623 −1.11232
$$615$$ −27.0344 −1.09013
$$616$$ 0 0
$$617$$ 23.0344 0.927332 0.463666 0.886010i $$-0.346534\pi$$
0.463666 + 0.886010i $$0.346534\pi$$
$$618$$ 44.3607 1.78445
$$619$$ 18.6738 0.750562 0.375281 0.926911i $$-0.377546\pi$$
0.375281 + 0.926911i $$0.377546\pi$$
$$620$$ 1.23607 0.0496417
$$621$$ 13.4164 0.538382
$$622$$ 29.1246 1.16779
$$623$$ −13.0902 −0.524447
$$624$$ −5.23607 −0.209610
$$625$$ 1.00000 0.0400000
$$626$$ 0.854102 0.0341368
$$627$$ 0 0
$$628$$ 0.291796 0.0116439
$$629$$ −10.4721 −0.417551
$$630$$ −3.85410 −0.153551
$$631$$ −23.7082 −0.943809 −0.471904 0.881650i $$-0.656433\pi$$
−0.471904 + 0.881650i $$0.656433\pi$$
$$632$$ −8.47214 −0.337003
$$633$$ −29.5066 −1.17278
$$634$$ 20.7639 0.824641
$$635$$ −11.4164 −0.453046
$$636$$ 3.23607 0.128318
$$637$$ −2.00000 −0.0792429
$$638$$ 0 0
$$639$$ −37.4164 −1.48017
$$640$$ 1.00000 0.0395285
$$641$$ 40.3262 1.59279 0.796395 0.604776i $$-0.206738\pi$$
0.796395 + 0.604776i $$0.206738\pi$$
$$642$$ 40.2705 1.58935
$$643$$ −14.9787 −0.590703 −0.295351 0.955389i $$-0.595437\pi$$
−0.295351 + 0.955389i $$0.595437\pi$$
$$644$$ −6.00000 −0.236433
$$645$$ 4.85410 0.191130
$$646$$ 11.0902 0.436337
$$647$$ 0.180340 0.00708989 0.00354495 0.999994i $$-0.498872\pi$$
0.00354495 + 0.999994i $$0.498872\pi$$
$$648$$ 5.70820 0.224239
$$649$$ 0 0
$$650$$ 2.00000 0.0784465
$$651$$ 3.23607 0.126832
$$652$$ 2.85410 0.111775
$$653$$ 26.8328 1.05005 0.525025 0.851087i $$-0.324056\pi$$
0.525025 + 0.851087i $$0.324056\pi$$
$$654$$ −1.23607 −0.0483341
$$655$$ −7.79837 −0.304708
$$656$$ 10.3262 0.403172
$$657$$ −53.3951 −2.08314
$$658$$ 9.23607 0.360059
$$659$$ −4.09017 −0.159330 −0.0796652 0.996822i $$-0.525385\pi$$
−0.0796652 + 0.996822i $$0.525385\pi$$
$$660$$ 0 0
$$661$$ −45.8885 −1.78486 −0.892429 0.451188i $$-0.851000\pi$$
−0.892429 + 0.451188i $$0.851000\pi$$
$$662$$ −6.03444 −0.234535
$$663$$ −8.47214 −0.329030
$$664$$ 10.3262 0.400736
$$665$$ −6.85410 −0.265791
$$666$$ 24.9443 0.966571
$$667$$ −19.4164 −0.751806
$$668$$ −6.47214 −0.250414
$$669$$ 16.4721 0.636850
$$670$$ 6.09017 0.235284
$$671$$ 0 0
$$672$$ 2.61803 0.100993
$$673$$ 38.8541 1.49772 0.748858 0.662731i $$-0.230603\pi$$
0.748858 + 0.662731i $$0.230603\pi$$
$$674$$ −15.6180 −0.601584
$$675$$ 2.23607 0.0860663
$$676$$ −9.00000 −0.346154
$$677$$ 13.5967 0.522565 0.261283 0.965262i $$-0.415855\pi$$
0.261283 + 0.965262i $$0.415855\pi$$
$$678$$ 24.4164 0.937706
$$679$$ 9.56231 0.366968
$$680$$ 1.61803 0.0620488
$$681$$ 63.5410 2.43490
$$682$$ 0 0
$$683$$ −31.0557 −1.18831 −0.594157 0.804349i $$-0.702514\pi$$
−0.594157 + 0.804349i $$0.702514\pi$$
$$684$$ −26.4164 −1.01006
$$685$$ 1.09017 0.0416533
$$686$$ 1.00000 0.0381802
$$687$$ 29.1246 1.11117
$$688$$ −1.85410 −0.0706870
$$689$$ −2.47214 −0.0941809
$$690$$ 15.7082 0.598001
$$691$$ 21.1459 0.804428 0.402214 0.915546i $$-0.368241\pi$$
0.402214 + 0.915546i $$0.368241\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ 0 0
$$694$$ 5.32624 0.202181
$$695$$ 18.4721 0.700688
$$696$$ 8.47214 0.321135
$$697$$ 16.7082 0.632868
$$698$$ 5.52786 0.209233
$$699$$ −28.5623 −1.08033
$$700$$ −1.00000 −0.0377964
$$701$$ 12.1803 0.460045 0.230023 0.973185i $$-0.426120\pi$$
0.230023 + 0.973185i $$0.426120\pi$$
$$702$$ 4.47214 0.168790
$$703$$ 44.3607 1.67309
$$704$$ 0 0
$$705$$ −24.1803 −0.910684
$$706$$ 28.4508 1.07076
$$707$$ 8.94427 0.336384
$$708$$ −19.9443 −0.749552
$$709$$ −24.9443 −0.936802 −0.468401 0.883516i $$-0.655170\pi$$
−0.468401 + 0.883516i $$0.655170\pi$$
$$710$$ −9.70820 −0.364342
$$711$$ 32.6525 1.22456
$$712$$ −13.0902 −0.490575
$$713$$ −7.41641 −0.277747
$$714$$ 4.23607 0.158531
$$715$$ 0 0
$$716$$ −6.38197 −0.238505
$$717$$ 0 0
$$718$$ 18.6525 0.696104
$$719$$ 32.3607 1.20685 0.603425 0.797420i $$-0.293802\pi$$
0.603425 + 0.797420i $$0.293802\pi$$
$$720$$ −3.85410 −0.143634
$$721$$ 16.9443 0.631038
$$722$$ −27.9787 −1.04126
$$723$$ 16.4164 0.610533
$$724$$ 3.52786 0.131112
$$725$$ −3.23607 −0.120185
$$726$$ 0 0
$$727$$ 38.8328 1.44023 0.720115 0.693855i $$-0.244089\pi$$
0.720115 + 0.693855i $$0.244089\pi$$
$$728$$ −2.00000 −0.0741249
$$729$$ −39.5623 −1.46527
$$730$$ −13.8541 −0.512763
$$731$$ −3.00000 −0.110959
$$732$$ 9.23607 0.341375
$$733$$ 13.2361 0.488885 0.244443 0.969664i $$-0.421395\pi$$
0.244443 + 0.969664i $$0.421395\pi$$
$$734$$ −28.1803 −1.04016
$$735$$ −2.61803 −0.0965676
$$736$$ −6.00000 −0.221163
$$737$$ 0 0
$$738$$ −39.7984 −1.46500
$$739$$ 24.6738 0.907639 0.453820 0.891094i $$-0.350061\pi$$
0.453820 + 0.891094i $$0.350061\pi$$
$$740$$ 6.47214 0.237920
$$741$$ 35.8885 1.31840
$$742$$ 1.23607 0.0453775
$$743$$ 12.1115 0.444326 0.222163 0.975010i $$-0.428688\pi$$
0.222163 + 0.975010i $$0.428688\pi$$
$$744$$ 3.23607 0.118640
$$745$$ 24.1803 0.885899
$$746$$ −24.4721 −0.895989
$$747$$ −39.7984 −1.45615
$$748$$ 0 0
$$749$$ 15.3820 0.562045
$$750$$ 2.61803 0.0955971
$$751$$ −32.6525 −1.19151 −0.595753 0.803168i $$-0.703146\pi$$
−0.595753 + 0.803168i $$0.703146\pi$$
$$752$$ 9.23607 0.336805
$$753$$ 70.2492 2.56002
$$754$$ −6.47214 −0.235701
$$755$$ 2.29180 0.0834070
$$756$$ −2.23607 −0.0813250
$$757$$ 19.8885 0.722861 0.361431 0.932399i $$-0.382288\pi$$
0.361431 + 0.932399i $$0.382288\pi$$
$$758$$ 27.8541 1.01171
$$759$$ 0 0
$$760$$ −6.85410 −0.248624
$$761$$ 31.7984 1.15269 0.576345 0.817206i $$-0.304478\pi$$
0.576345 + 0.817206i $$0.304478\pi$$
$$762$$ −29.8885 −1.08275
$$763$$ −0.472136 −0.0170925
$$764$$ −1.52786 −0.0552762
$$765$$ −6.23607 −0.225466
$$766$$ 24.0000 0.867155
$$767$$ 15.2361 0.550143
$$768$$ 2.61803 0.0944702
$$769$$ 7.88854 0.284468 0.142234 0.989833i $$-0.454571\pi$$
0.142234 + 0.989833i $$0.454571\pi$$
$$770$$ 0 0
$$771$$ 77.3951 2.78732
$$772$$ 2.00000 0.0719816
$$773$$ −24.6525 −0.886688 −0.443344 0.896352i $$-0.646208\pi$$
−0.443344 + 0.896352i $$0.646208\pi$$
$$774$$ 7.14590 0.256854
$$775$$ −1.23607 −0.0444009
$$776$$ 9.56231 0.343267
$$777$$ 16.9443 0.607872
$$778$$ 32.8328 1.17711
$$779$$ −70.7771 −2.53585
$$780$$ 5.23607 0.187481
$$781$$ 0 0
$$782$$ −9.70820 −0.347165
$$783$$ −7.23607 −0.258596
$$784$$ 1.00000 0.0357143
$$785$$ −0.291796 −0.0104146
$$786$$ −20.4164 −0.728229
$$787$$ −51.1033 −1.82164 −0.910818 0.412807i $$-0.864548\pi$$
−0.910818 + 0.412807i $$0.864548\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ −52.8328 −1.88090
$$790$$ 8.47214 0.301425
$$791$$ 9.32624 0.331603
$$792$$ 0 0
$$793$$ −7.05573 −0.250556
$$794$$ 16.9443 0.601330
$$795$$ −3.23607 −0.114772
$$796$$ −19.4164 −0.688196
$$797$$ −6.36068 −0.225307 −0.112653 0.993634i $$-0.535935\pi$$
−0.112653 + 0.993634i $$0.535935\pi$$
$$798$$ −17.9443 −0.635220
$$799$$ 14.9443 0.528690
$$800$$ −1.00000 −0.0353553
$$801$$ 50.4508 1.78259
$$802$$ 26.5066 0.935980
$$803$$ 0 0
$$804$$ 15.9443 0.562311
$$805$$ 6.00000 0.211472
$$806$$ −2.47214 −0.0870773
$$807$$ −31.8885 −1.12253
$$808$$ 8.94427 0.314658
$$809$$ 48.8115 1.71612 0.858061 0.513548i $$-0.171669\pi$$
0.858061 + 0.513548i $$0.171669\pi$$
$$810$$ −5.70820 −0.200566
$$811$$ 29.9230 1.05074 0.525369 0.850874i $$-0.323927\pi$$
0.525369 + 0.850874i $$0.323927\pi$$
$$812$$ 3.23607 0.113564
$$813$$ 24.6525 0.864600
$$814$$ 0 0
$$815$$ −2.85410 −0.0999748
$$816$$ 4.23607 0.148292
$$817$$ 12.7082 0.444604
$$818$$ 27.8885 0.975100
$$819$$ 7.70820 0.269346
$$820$$ −10.3262 −0.360608
$$821$$ −31.0557 −1.08385 −0.541926 0.840426i $$-0.682305\pi$$
−0.541926 + 0.840426i $$0.682305\pi$$
$$822$$ 2.85410 0.0995482
$$823$$ −2.83282 −0.0987457 −0.0493729 0.998780i $$-0.515722\pi$$
−0.0493729 + 0.998780i $$0.515722\pi$$
$$824$$ 16.9443 0.590282
$$825$$ 0 0
$$826$$ −7.61803 −0.265065
$$827$$ 40.7426 1.41676 0.708380 0.705831i $$-0.249426\pi$$
0.708380 + 0.705831i $$0.249426\pi$$
$$828$$ 23.1246 0.803636
$$829$$ 34.1803 1.18713 0.593566 0.804785i $$-0.297720\pi$$
0.593566 + 0.804785i $$0.297720\pi$$
$$830$$ −10.3262 −0.358429
$$831$$ −16.4721 −0.571412
$$832$$ −2.00000 −0.0693375
$$833$$ 1.61803 0.0560616
$$834$$ 48.3607 1.67459
$$835$$ 6.47214 0.223978
$$836$$ 0 0
$$837$$ −2.76393 −0.0955355
$$838$$ −12.3262 −0.425803
$$839$$ 4.47214 0.154395 0.0771976 0.997016i $$-0.475403\pi$$
0.0771976 + 0.997016i $$0.475403\pi$$
$$840$$ −2.61803 −0.0903308
$$841$$ −18.5279 −0.638892
$$842$$ 6.36068 0.219204
$$843$$ 62.9230 2.16718
$$844$$ −11.2705 −0.387947
$$845$$ 9.00000 0.309609
$$846$$ −35.5967 −1.22384
$$847$$ 0 0
$$848$$ 1.23607 0.0424467
$$849$$ 31.4164 1.07821
$$850$$ −1.61803 −0.0554981
$$851$$ −38.8328 −1.33117
$$852$$ −25.4164 −0.870752
$$853$$ 4.29180 0.146948 0.0734741 0.997297i $$-0.476591\pi$$
0.0734741 + 0.997297i $$0.476591\pi$$
$$854$$ 3.52786 0.120721
$$855$$ 26.4164 0.903422
$$856$$ 15.3820 0.525745
$$857$$ 9.27051 0.316675 0.158337 0.987385i $$-0.449387\pi$$
0.158337 + 0.987385i $$0.449387\pi$$
$$858$$ 0 0
$$859$$ −7.43769 −0.253771 −0.126885 0.991917i $$-0.540498\pi$$
−0.126885 + 0.991917i $$0.540498\pi$$
$$860$$ 1.85410 0.0632244
$$861$$ −27.0344 −0.921331
$$862$$ 31.2361 1.06390
$$863$$ 6.76393 0.230247 0.115123 0.993351i $$-0.463274\pi$$
0.115123 + 0.993351i $$0.463274\pi$$
$$864$$ −2.23607 −0.0760726
$$865$$ 12.0000 0.408012
$$866$$ −16.4508 −0.559023
$$867$$ −37.6525 −1.27875
$$868$$ 1.23607 0.0419549
$$869$$ 0 0
$$870$$ −8.47214 −0.287232
$$871$$ −12.1803 −0.412715
$$872$$ −0.472136 −0.0159885
$$873$$ −36.8541 −1.24732
$$874$$ 41.1246 1.39106
$$875$$ 1.00000 0.0338062
$$876$$ −36.2705 −1.22547
$$877$$ −18.0000 −0.607817 −0.303908 0.952701i $$-0.598292\pi$$
−0.303908 + 0.952701i $$0.598292\pi$$
$$878$$ −28.6525 −0.966974
$$879$$ −43.1246 −1.45456
$$880$$ 0 0
$$881$$ 4.14590 0.139679 0.0698394 0.997558i $$-0.477751\pi$$
0.0698394 + 0.997558i $$0.477751\pi$$
$$882$$ −3.85410 −0.129774
$$883$$ 2.43769 0.0820349 0.0410175 0.999158i $$-0.486940\pi$$
0.0410175 + 0.999158i $$0.486940\pi$$
$$884$$ −3.23607 −0.108841
$$885$$ 19.9443 0.670419
$$886$$ −28.0344 −0.941835
$$887$$ 4.36068 0.146417 0.0732086 0.997317i $$-0.476676\pi$$
0.0732086 + 0.997317i $$0.476676\pi$$
$$888$$ 16.9443 0.568613
$$889$$ −11.4164 −0.382894
$$890$$ 13.0902 0.438783
$$891$$ 0 0
$$892$$ 6.29180 0.210665
$$893$$ −63.3050 −2.11842
$$894$$ 63.3050 2.11723
$$895$$ 6.38197 0.213326
$$896$$ 1.00000 0.0334077
$$897$$ −31.4164 −1.04896
$$898$$ −24.4508 −0.815935
$$899$$ 4.00000 0.133407
$$900$$ 3.85410 0.128470
$$901$$ 2.00000 0.0666297
$$902$$ 0 0
$$903$$ 4.85410 0.161534
$$904$$ 9.32624 0.310186
$$905$$ −3.52786 −0.117270
$$906$$ 6.00000 0.199337
$$907$$ 22.2705 0.739480 0.369740 0.929135i $$-0.379447\pi$$
0.369740 + 0.929135i $$0.379447\pi$$
$$908$$ 24.2705 0.805445
$$909$$ −34.4721 −1.14337
$$910$$ 2.00000 0.0662994
$$911$$ −51.5967 −1.70948 −0.854738 0.519059i $$-0.826282\pi$$
−0.854738 + 0.519059i $$0.826282\pi$$
$$912$$ −17.9443 −0.594194
$$913$$ 0 0
$$914$$ −31.2705 −1.03434
$$915$$ −9.23607 −0.305335
$$916$$ 11.1246 0.367568
$$917$$ −7.79837 −0.257525
$$918$$ −3.61803 −0.119413
$$919$$ 8.83282 0.291368 0.145684 0.989331i $$-0.453462\pi$$
0.145684 + 0.989331i $$0.453462\pi$$
$$920$$ 6.00000 0.197814
$$921$$ 72.1591 2.37772
$$922$$ 6.94427 0.228697
$$923$$ 19.4164 0.639099
$$924$$ 0 0
$$925$$ −6.47214 −0.212803
$$926$$ 7.41641 0.243718
$$927$$ −65.3050 −2.14490
$$928$$ 3.23607 0.106229
$$929$$ −3.25735 −0.106870 −0.0534352 0.998571i $$-0.517017\pi$$
−0.0534352 + 0.998571i $$0.517017\pi$$
$$930$$ −3.23607 −0.106115
$$931$$ −6.85410 −0.224634
$$932$$ −10.9098 −0.357363
$$933$$ −76.2492 −2.49629
$$934$$ 17.8885 0.585331
$$935$$ 0 0
$$936$$ 7.70820 0.251951
$$937$$ −33.3951 −1.09097 −0.545486 0.838120i $$-0.683655\pi$$
−0.545486 + 0.838120i $$0.683655\pi$$
$$938$$ 6.09017 0.198851
$$939$$ −2.23607 −0.0729713
$$940$$ −9.23607 −0.301247
$$941$$ −34.1803 −1.11425 −0.557124 0.830430i $$-0.688095\pi$$
−0.557124 + 0.830430i $$0.688095\pi$$
$$942$$ −0.763932 −0.0248903
$$943$$ 61.9574 2.01761
$$944$$ −7.61803 −0.247946
$$945$$ 2.23607 0.0727393
$$946$$ 0 0
$$947$$ −41.9098 −1.36189 −0.680943 0.732336i $$-0.738430\pi$$
−0.680943 + 0.732336i $$0.738430\pi$$
$$948$$ 22.1803 0.720384
$$949$$ 27.7082 0.899446
$$950$$ 6.85410 0.222376
$$951$$ −54.3607 −1.76277
$$952$$ 1.61803 0.0524408
$$953$$ −29.4508 −0.954007 −0.477003 0.878902i $$-0.658277\pi$$
−0.477003 + 0.878902i $$0.658277\pi$$
$$954$$ −4.76393 −0.154238
$$955$$ 1.52786 0.0494405
$$956$$ 0 0
$$957$$ 0 0
$$958$$ −37.8885 −1.22412
$$959$$ 1.09017 0.0352034
$$960$$ −2.61803 −0.0844967
$$961$$ −29.4721 −0.950714
$$962$$ −12.9443 −0.417340
$$963$$ −59.2837 −1.91039
$$964$$ 6.27051 0.201960
$$965$$ −2.00000 −0.0643823
$$966$$ 15.7082 0.505403
$$967$$ −27.0132 −0.868685 −0.434342 0.900748i $$-0.643019\pi$$
−0.434342 + 0.900748i $$0.643019\pi$$
$$968$$ 0 0
$$969$$ −29.0344 −0.932721
$$970$$ −9.56231 −0.307027
$$971$$ −26.8328 −0.861106 −0.430553 0.902565i $$-0.641681\pi$$
−0.430553 + 0.902565i $$0.641681\pi$$
$$972$$ −21.6525 −0.694503
$$973$$ 18.4721 0.592189
$$974$$ 26.8328 0.859779
$$975$$ −5.23607 −0.167688
$$976$$ 3.52786 0.112924
$$977$$ 43.3050 1.38545 0.692724 0.721203i $$-0.256410\pi$$
0.692724 + 0.721203i $$0.256410\pi$$
$$978$$ −7.47214 −0.238933
$$979$$ 0 0
$$980$$ −1.00000 −0.0319438
$$981$$ 1.81966 0.0580973
$$982$$ −32.5066 −1.03733
$$983$$ −14.6525 −0.467341 −0.233671 0.972316i $$-0.575074\pi$$
−0.233671 + 0.972316i $$0.575074\pi$$
$$984$$ −27.0344 −0.861827
$$985$$ 6.00000 0.191176
$$986$$ 5.23607 0.166750
$$987$$ −24.1803 −0.769669
$$988$$ 13.7082 0.436116
$$989$$ −11.1246 −0.353742
$$990$$ 0 0
$$991$$ 20.6525 0.656048 0.328024 0.944669i $$-0.393617\pi$$
0.328024 + 0.944669i $$0.393617\pi$$
$$992$$ 1.23607 0.0392452
$$993$$ 15.7984 0.501346
$$994$$ −9.70820 −0.307926
$$995$$ 19.4164 0.615542
$$996$$ −27.0344 −0.856619
$$997$$ 23.1246 0.732364 0.366182 0.930543i $$-0.380665\pi$$
0.366182 + 0.930543i $$0.380665\pi$$
$$998$$ 15.2016 0.481199
$$999$$ −14.4721 −0.457878
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.bt.1.2 2
11.3 even 5 770.2.n.b.141.1 yes 4
11.4 even 5 770.2.n.b.71.1 4
11.10 odd 2 8470.2.a.cf.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.b.71.1 4 11.4 even 5
770.2.n.b.141.1 yes 4 11.3 even 5
8470.2.a.bt.1.2 2 1.1 even 1 trivial
8470.2.a.cf.1.2 2 11.10 odd 2