# Properties

 Label 4998.2.a.m.1.1 Level $4998$ Weight $2$ Character 4998.1 Self dual yes Analytic conductor $39.909$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4998.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.9092309302$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 714) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4998.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} +3.00000 q^{29} +1.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} +4.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} -1.00000 q^{41} -4.00000 q^{43} -2.00000 q^{44} -1.00000 q^{45} -8.00000 q^{46} +9.00000 q^{47} +1.00000 q^{48} +4.00000 q^{50} -1.00000 q^{51} -1.00000 q^{52} -1.00000 q^{54} +2.00000 q^{55} -4.00000 q^{57} -3.00000 q^{58} -11.0000 q^{59} -1.00000 q^{60} -3.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +2.00000 q^{66} -12.0000 q^{67} -1.00000 q^{68} +8.00000 q^{69} +4.00000 q^{71} -1.00000 q^{72} -6.00000 q^{73} -4.00000 q^{74} -4.00000 q^{75} -4.00000 q^{76} +1.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} +1.00000 q^{82} -9.00000 q^{83} +1.00000 q^{85} +4.00000 q^{86} +3.00000 q^{87} +2.00000 q^{88} +6.00000 q^{89} +1.00000 q^{90} +8.00000 q^{92} +3.00000 q^{93} -9.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} -8.00000 q^{97} -2.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536
$$18$$ −1.00000 −0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 2.00000 0.426401
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −4.00000 −0.800000
$$26$$ 1.00000 0.196116
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −2.00000 −0.348155
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 4.00000 0.648886
$$39$$ −1.00000 −0.160128
$$40$$ 1.00000 0.158114
$$41$$ −1.00000 −0.156174 −0.0780869 0.996947i $$-0.524881\pi$$
−0.0780869 + 0.996947i $$0.524881\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ −1.00000 −0.149071
$$46$$ −8.00000 −1.17954
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 0 0
$$50$$ 4.00000 0.565685
$$51$$ −1.00000 −0.140028
$$52$$ −1.00000 −0.138675
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 2.00000 0.269680
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ −3.00000 −0.393919
$$59$$ −11.0000 −1.43208 −0.716039 0.698060i $$-0.754047\pi$$
−0.716039 + 0.698060i $$0.754047\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ −3.00000 −0.381000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 1.00000 0.124035
$$66$$ 2.00000 0.246183
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 4.00000 0.474713 0.237356 0.971423i $$-0.423719\pi$$
0.237356 + 0.971423i $$0.423719\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ −4.00000 −0.461880
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ 1.00000 0.113228
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 1.00000 0.110432
$$83$$ −9.00000 −0.987878 −0.493939 0.869496i $$-0.664443\pi$$
−0.493939 + 0.869496i $$0.664443\pi$$
$$84$$ 0 0
$$85$$ 1.00000 0.108465
$$86$$ 4.00000 0.431331
$$87$$ 3.00000 0.321634
$$88$$ 2.00000 0.213201
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 8.00000 0.834058
$$93$$ 3.00000 0.311086
$$94$$ −9.00000 −0.928279
$$95$$ 4.00000 0.410391
$$96$$ −1.00000 −0.102062
$$97$$ −8.00000 −0.812277 −0.406138 0.913812i $$-0.633125\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ −4.00000 −0.400000
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 1.00000 0.0990148
$$103$$ −14.0000 −1.37946 −0.689730 0.724066i $$-0.742271\pi$$
−0.689730 + 0.724066i $$0.742271\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ 15.0000 1.41108 0.705541 0.708669i $$-0.250704\pi$$
0.705541 + 0.708669i $$0.250704\pi$$
$$114$$ 4.00000 0.374634
$$115$$ −8.00000 −0.746004
$$116$$ 3.00000 0.278543
$$117$$ −1.00000 −0.0924500
$$118$$ 11.0000 1.01263
$$119$$ 0 0
$$120$$ 1.00000 0.0912871
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ −1.00000 −0.0901670
$$124$$ 3.00000 0.269408
$$125$$ 9.00000 0.804984
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ −1.00000 −0.0877058
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ −2.00000 −0.174078
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ −1.00000 −0.0860663
$$136$$ 1.00000 0.0857493
$$137$$ 20.0000 1.70872 0.854358 0.519685i $$-0.173951\pi$$
0.854358 + 0.519685i $$0.173951\pi$$
$$138$$ −8.00000 −0.681005
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 9.00000 0.757937
$$142$$ −4.00000 −0.335673
$$143$$ 2.00000 0.167248
$$144$$ 1.00000 0.0833333
$$145$$ −3.00000 −0.249136
$$146$$ 6.00000 0.496564
$$147$$ 0 0
$$148$$ 4.00000 0.328798
$$149$$ −2.00000 −0.163846 −0.0819232 0.996639i $$-0.526106\pi$$
−0.0819232 + 0.996639i $$0.526106\pi$$
$$150$$ 4.00000 0.326599
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 4.00000 0.324443
$$153$$ −1.00000 −0.0808452
$$154$$ 0 0
$$155$$ −3.00000 −0.240966
$$156$$ −1.00000 −0.0800641
$$157$$ −1.00000 −0.0798087 −0.0399043 0.999204i $$-0.512705\pi$$
−0.0399043 + 0.999204i $$0.512705\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ −1.00000 −0.0780869
$$165$$ 2.00000 0.155700
$$166$$ 9.00000 0.698535
$$167$$ −2.00000 −0.154765 −0.0773823 0.997001i $$-0.524656\pi$$
−0.0773823 + 0.997001i $$0.524656\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ −1.00000 −0.0766965
$$171$$ −4.00000 −0.305888
$$172$$ −4.00000 −0.304997
$$173$$ −21.0000 −1.59660 −0.798300 0.602260i $$-0.794267\pi$$
−0.798300 + 0.602260i $$0.794267\pi$$
$$174$$ −3.00000 −0.227429
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ −11.0000 −0.826811
$$178$$ −6.00000 −0.449719
$$179$$ −21.0000 −1.56961 −0.784807 0.619740i $$-0.787238\pi$$
−0.784807 + 0.619740i $$0.787238\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −8.00000 −0.589768
$$185$$ −4.00000 −0.294086
$$186$$ −3.00000 −0.219971
$$187$$ 2.00000 0.146254
$$188$$ 9.00000 0.656392
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 8.00000 0.574367
$$195$$ 1.00000 0.0716115
$$196$$ 0 0
$$197$$ 1.00000 0.0712470 0.0356235 0.999365i $$-0.488658\pi$$
0.0356235 + 0.999365i $$0.488658\pi$$
$$198$$ 2.00000 0.142134
$$199$$ 5.00000 0.354441 0.177220 0.984171i $$-0.443289\pi$$
0.177220 + 0.984171i $$0.443289\pi$$
$$200$$ 4.00000 0.282843
$$201$$ −12.0000 −0.846415
$$202$$ 14.0000 0.985037
$$203$$ 0 0
$$204$$ −1.00000 −0.0700140
$$205$$ 1.00000 0.0698430
$$206$$ 14.0000 0.975426
$$207$$ 8.00000 0.556038
$$208$$ −1.00000 −0.0693375
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ 3.00000 0.206529 0.103264 0.994654i $$-0.467071\pi$$
0.103264 + 0.994654i $$0.467071\pi$$
$$212$$ 0 0
$$213$$ 4.00000 0.274075
$$214$$ 12.0000 0.820303
$$215$$ 4.00000 0.272798
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 16.0000 1.08366
$$219$$ −6.00000 −0.405442
$$220$$ 2.00000 0.134840
$$221$$ 1.00000 0.0672673
$$222$$ −4.00000 −0.268462
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ −4.00000 −0.266667
$$226$$ −15.0000 −0.997785
$$227$$ −10.0000 −0.663723 −0.331862 0.943328i $$-0.607677\pi$$
−0.331862 + 0.943328i $$0.607677\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ 9.00000 0.594737 0.297368 0.954763i $$-0.403891\pi$$
0.297368 + 0.954763i $$0.403891\pi$$
$$230$$ 8.00000 0.527504
$$231$$ 0 0
$$232$$ −3.00000 −0.196960
$$233$$ −11.0000 −0.720634 −0.360317 0.932830i $$-0.617331\pi$$
−0.360317 + 0.932830i $$0.617331\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ −9.00000 −0.587095
$$236$$ −11.0000 −0.716039
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 13.0000 0.840900 0.420450 0.907316i $$-0.361872\pi$$
0.420450 + 0.907316i $$0.361872\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 1.00000 0.0637577
$$247$$ 4.00000 0.254514
$$248$$ −3.00000 −0.190500
$$249$$ −9.00000 −0.570352
$$250$$ −9.00000 −0.569210
$$251$$ −23.0000 −1.45175 −0.725874 0.687828i $$-0.758564\pi$$
−0.725874 + 0.687828i $$0.758564\pi$$
$$252$$ 0 0
$$253$$ −16.0000 −1.00591
$$254$$ 8.00000 0.501965
$$255$$ 1.00000 0.0626224
$$256$$ 1.00000 0.0625000
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 0 0
$$260$$ 1.00000 0.0620174
$$261$$ 3.00000 0.185695
$$262$$ −6.00000 −0.370681
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6.00000 0.367194
$$268$$ −12.0000 −0.733017
$$269$$ 5.00000 0.304855 0.152428 0.988315i $$-0.451291\pi$$
0.152428 + 0.988315i $$0.451291\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ −18.0000 −1.09342 −0.546711 0.837321i $$-0.684120\pi$$
−0.546711 + 0.837321i $$0.684120\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ 0 0
$$274$$ −20.0000 −1.20824
$$275$$ 8.00000 0.482418
$$276$$ 8.00000 0.481543
$$277$$ 28.0000 1.68236 0.841178 0.540758i $$-0.181862\pi$$
0.841178 + 0.540758i $$0.181862\pi$$
$$278$$ 0 0
$$279$$ 3.00000 0.179605
$$280$$ 0 0
$$281$$ 20.0000 1.19310 0.596550 0.802576i $$-0.296538\pi$$
0.596550 + 0.802576i $$0.296538\pi$$
$$282$$ −9.00000 −0.535942
$$283$$ 13.0000 0.772770 0.386385 0.922338i $$-0.373724\pi$$
0.386385 + 0.922338i $$0.373724\pi$$
$$284$$ 4.00000 0.237356
$$285$$ 4.00000 0.236940
$$286$$ −2.00000 −0.118262
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ 1.00000 0.0588235
$$290$$ 3.00000 0.176166
$$291$$ −8.00000 −0.468968
$$292$$ −6.00000 −0.351123
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 11.0000 0.640445
$$296$$ −4.00000 −0.232495
$$297$$ −2.00000 −0.116052
$$298$$ 2.00000 0.115857
$$299$$ −8.00000 −0.462652
$$300$$ −4.00000 −0.230940
$$301$$ 0 0
$$302$$ 4.00000 0.230174
$$303$$ −14.0000 −0.804279
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 0 0
$$309$$ −14.0000 −0.796432
$$310$$ 3.00000 0.170389
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ −8.00000 −0.452187 −0.226093 0.974106i $$-0.572595\pi$$
−0.226093 + 0.974106i $$0.572595\pi$$
$$314$$ 1.00000 0.0564333
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −11.0000 −0.617822 −0.308911 0.951091i $$-0.599964\pi$$
−0.308911 + 0.951091i $$0.599964\pi$$
$$318$$ 0 0
$$319$$ −6.00000 −0.335936
$$320$$ −1.00000 −0.0559017
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 4.00000 0.222566
$$324$$ 1.00000 0.0555556
$$325$$ 4.00000 0.221880
$$326$$ −16.0000 −0.886158
$$327$$ −16.0000 −0.884802
$$328$$ 1.00000 0.0552158
$$329$$ 0 0
$$330$$ −2.00000 −0.110096
$$331$$ −22.0000 −1.20923 −0.604615 0.796518i $$-0.706673\pi$$
−0.604615 + 0.796518i $$0.706673\pi$$
$$332$$ −9.00000 −0.493939
$$333$$ 4.00000 0.219199
$$334$$ 2.00000 0.109435
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ 34.0000 1.85210 0.926049 0.377403i $$-0.123183\pi$$
0.926049 + 0.377403i $$0.123183\pi$$
$$338$$ 12.0000 0.652714
$$339$$ 15.0000 0.814688
$$340$$ 1.00000 0.0542326
$$341$$ −6.00000 −0.324918
$$342$$ 4.00000 0.216295
$$343$$ 0 0
$$344$$ 4.00000 0.215666
$$345$$ −8.00000 −0.430706
$$346$$ 21.0000 1.12897
$$347$$ −4.00000 −0.214731 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$348$$ 3.00000 0.160817
$$349$$ −27.0000 −1.44528 −0.722638 0.691226i $$-0.757071\pi$$
−0.722638 + 0.691226i $$0.757071\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 2.00000 0.106600
$$353$$ −28.0000 −1.49029 −0.745145 0.666903i $$-0.767620\pi$$
−0.745145 + 0.666903i $$0.767620\pi$$
$$354$$ 11.0000 0.584643
$$355$$ −4.00000 −0.212298
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 21.0000 1.10988
$$359$$ 15.0000 0.791670 0.395835 0.918322i $$-0.370455\pi$$
0.395835 + 0.918322i $$0.370455\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −3.00000 −0.157895
$$362$$ 16.0000 0.840941
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ 6.00000 0.314054
$$366$$ 0 0
$$367$$ 32.0000 1.67039 0.835193 0.549957i $$-0.185356\pi$$
0.835193 + 0.549957i $$0.185356\pi$$
$$368$$ 8.00000 0.417029
$$369$$ −1.00000 −0.0520579
$$370$$ 4.00000 0.207950
$$371$$ 0 0
$$372$$ 3.00000 0.155543
$$373$$ 19.0000 0.983783 0.491891 0.870657i $$-0.336306\pi$$
0.491891 + 0.870657i $$0.336306\pi$$
$$374$$ −2.00000 −0.103418
$$375$$ 9.00000 0.464758
$$376$$ −9.00000 −0.464140
$$377$$ −3.00000 −0.154508
$$378$$ 0 0
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 4.00000 0.205196
$$381$$ −8.00000 −0.409852
$$382$$ −3.00000 −0.153493
$$383$$ −24.0000 −1.22634 −0.613171 0.789950i $$-0.710106\pi$$
−0.613171 + 0.789950i $$0.710106\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ −4.00000 −0.203331
$$388$$ −8.00000 −0.406138
$$389$$ −2.00000 −0.101404 −0.0507020 0.998714i $$-0.516146\pi$$
−0.0507020 + 0.998714i $$0.516146\pi$$
$$390$$ −1.00000 −0.0506370
$$391$$ −8.00000 −0.404577
$$392$$ 0 0
$$393$$ 6.00000 0.302660
$$394$$ −1.00000 −0.0503793
$$395$$ 0 0
$$396$$ −2.00000 −0.100504
$$397$$ 8.00000 0.401508 0.200754 0.979642i $$-0.435661\pi$$
0.200754 + 0.979642i $$0.435661\pi$$
$$398$$ −5.00000 −0.250627
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 12.0000 0.598506
$$403$$ −3.00000 −0.149441
$$404$$ −14.0000 −0.696526
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −8.00000 −0.396545
$$408$$ 1.00000 0.0495074
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ −1.00000 −0.0493865
$$411$$ 20.0000 0.986527
$$412$$ −14.0000 −0.689730
$$413$$ 0 0
$$414$$ −8.00000 −0.393179
$$415$$ 9.00000 0.441793
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ −8.00000 −0.391293
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ 17.0000 0.828529 0.414265 0.910156i $$-0.364039\pi$$
0.414265 + 0.910156i $$0.364039\pi$$
$$422$$ −3.00000 −0.146038
$$423$$ 9.00000 0.437595
$$424$$ 0 0
$$425$$ 4.00000 0.194029
$$426$$ −4.00000 −0.193801
$$427$$ 0 0
$$428$$ −12.0000 −0.580042
$$429$$ 2.00000 0.0965609
$$430$$ −4.00000 −0.192897
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −19.0000 −0.913082 −0.456541 0.889702i $$-0.650912\pi$$
−0.456541 + 0.889702i $$0.650912\pi$$
$$434$$ 0 0
$$435$$ −3.00000 −0.143839
$$436$$ −16.0000 −0.766261
$$437$$ −32.0000 −1.53077
$$438$$ 6.00000 0.286691
$$439$$ 1.00000 0.0477274 0.0238637 0.999715i $$-0.492403\pi$$
0.0238637 + 0.999715i $$0.492403\pi$$
$$440$$ −2.00000 −0.0953463
$$441$$ 0 0
$$442$$ −1.00000 −0.0475651
$$443$$ 33.0000 1.56788 0.783939 0.620838i $$-0.213208\pi$$
0.783939 + 0.620838i $$0.213208\pi$$
$$444$$ 4.00000 0.189832
$$445$$ −6.00000 −0.284427
$$446$$ −16.0000 −0.757622
$$447$$ −2.00000 −0.0945968
$$448$$ 0 0
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 4.00000 0.188562
$$451$$ 2.00000 0.0941763
$$452$$ 15.0000 0.705541
$$453$$ −4.00000 −0.187936
$$454$$ 10.0000 0.469323
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ 1.00000 0.0467780 0.0233890 0.999726i $$-0.492554\pi$$
0.0233890 + 0.999726i $$0.492554\pi$$
$$458$$ −9.00000 −0.420542
$$459$$ −1.00000 −0.0466760
$$460$$ −8.00000 −0.373002
$$461$$ 20.0000 0.931493 0.465746 0.884918i $$-0.345786\pi$$
0.465746 + 0.884918i $$0.345786\pi$$
$$462$$ 0 0
$$463$$ −18.0000 −0.836531 −0.418265 0.908325i $$-0.637362\pi$$
−0.418265 + 0.908325i $$0.637362\pi$$
$$464$$ 3.00000 0.139272
$$465$$ −3.00000 −0.139122
$$466$$ 11.0000 0.509565
$$467$$ 3.00000 0.138823 0.0694117 0.997588i $$-0.477888\pi$$
0.0694117 + 0.997588i $$0.477888\pi$$
$$468$$ −1.00000 −0.0462250
$$469$$ 0 0
$$470$$ 9.00000 0.415139
$$471$$ −1.00000 −0.0460776
$$472$$ 11.0000 0.506316
$$473$$ 8.00000 0.367840
$$474$$ 0 0
$$475$$ 16.0000 0.734130
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −13.0000 −0.594606
$$479$$ −26.0000 −1.18797 −0.593985 0.804476i $$-0.702446\pi$$
−0.593985 + 0.804476i $$0.702446\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ −4.00000 −0.182384
$$482$$ 18.0000 0.819878
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ 8.00000 0.363261
$$486$$ −1.00000 −0.0453609
$$487$$ −27.0000 −1.22349 −0.611743 0.791056i $$-0.709531\pi$$
−0.611743 + 0.791056i $$0.709531\pi$$
$$488$$ 0 0
$$489$$ 16.0000 0.723545
$$490$$ 0 0
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ −1.00000 −0.0450835
$$493$$ −3.00000 −0.135113
$$494$$ −4.00000 −0.179969
$$495$$ 2.00000 0.0898933
$$496$$ 3.00000 0.134704
$$497$$ 0 0
$$498$$ 9.00000 0.403300
$$499$$ 13.0000 0.581960 0.290980 0.956729i $$-0.406019\pi$$
0.290980 + 0.956729i $$0.406019\pi$$
$$500$$ 9.00000 0.402492
$$501$$ −2.00000 −0.0893534
$$502$$ 23.0000 1.02654
$$503$$ −6.00000 −0.267527 −0.133763 0.991013i $$-0.542706\pi$$
−0.133763 + 0.991013i $$0.542706\pi$$
$$504$$ 0 0
$$505$$ 14.0000 0.622992
$$506$$ 16.0000 0.711287
$$507$$ −12.0000 −0.532939
$$508$$ −8.00000 −0.354943
$$509$$ −28.0000 −1.24108 −0.620539 0.784176i $$-0.713086\pi$$
−0.620539 + 0.784176i $$0.713086\pi$$
$$510$$ −1.00000 −0.0442807
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ 14.0000 0.617514
$$515$$ 14.0000 0.616914
$$516$$ −4.00000 −0.176090
$$517$$ −18.0000 −0.791639
$$518$$ 0 0
$$519$$ −21.0000 −0.921798
$$520$$ −1.00000 −0.0438529
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ −3.00000 −0.131306
$$523$$ 40.0000 1.74908 0.874539 0.484955i $$-0.161164\pi$$
0.874539 + 0.484955i $$0.161164\pi$$
$$524$$ 6.00000 0.262111
$$525$$ 0 0
$$526$$ 8.00000 0.348817
$$527$$ −3.00000 −0.130682
$$528$$ −2.00000 −0.0870388
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ −11.0000 −0.477359
$$532$$ 0 0
$$533$$ 1.00000 0.0433148
$$534$$ −6.00000 −0.259645
$$535$$ 12.0000 0.518805
$$536$$ 12.0000 0.518321
$$537$$ −21.0000 −0.906217
$$538$$ −5.00000 −0.215565
$$539$$ 0 0
$$540$$ −1.00000 −0.0430331
$$541$$ −32.0000 −1.37579 −0.687894 0.725811i $$-0.741464\pi$$
−0.687894 + 0.725811i $$0.741464\pi$$
$$542$$ 18.0000 0.773166
$$543$$ −16.0000 −0.686626
$$544$$ 1.00000 0.0428746
$$545$$ 16.0000 0.685365
$$546$$ 0 0
$$547$$ −41.0000 −1.75303 −0.876517 0.481371i $$-0.840139\pi$$
−0.876517 + 0.481371i $$0.840139\pi$$
$$548$$ 20.0000 0.854358
$$549$$ 0 0
$$550$$ −8.00000 −0.341121
$$551$$ −12.0000 −0.511217
$$552$$ −8.00000 −0.340503
$$553$$ 0 0
$$554$$ −28.0000 −1.18961
$$555$$ −4.00000 −0.169791
$$556$$ 0 0
$$557$$ 16.0000 0.677942 0.338971 0.940797i $$-0.389921\pi$$
0.338971 + 0.940797i $$0.389921\pi$$
$$558$$ −3.00000 −0.127000
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ 2.00000 0.0844401
$$562$$ −20.0000 −0.843649
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 9.00000 0.378968
$$565$$ −15.0000 −0.631055
$$566$$ −13.0000 −0.546431
$$567$$ 0 0
$$568$$ −4.00000 −0.167836
$$569$$ −2.00000 −0.0838444 −0.0419222 0.999121i $$-0.513348\pi$$
−0.0419222 + 0.999121i $$0.513348\pi$$
$$570$$ −4.00000 −0.167542
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ 3.00000 0.125327
$$574$$ 0 0
$$575$$ −32.0000 −1.33449
$$576$$ 1.00000 0.0416667
$$577$$ 15.0000 0.624458 0.312229 0.950007i $$-0.398924\pi$$
0.312229 + 0.950007i $$0.398924\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ −14.0000 −0.581820
$$580$$ −3.00000 −0.124568
$$581$$ 0 0
$$582$$ 8.00000 0.331611
$$583$$ 0 0
$$584$$ 6.00000 0.248282
$$585$$ 1.00000 0.0413449
$$586$$ 0 0
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ −12.0000 −0.494451
$$590$$ −11.0000 −0.452863
$$591$$ 1.00000 0.0411345
$$592$$ 4.00000 0.164399
$$593$$ −22.0000 −0.903432 −0.451716 0.892162i $$-0.649188\pi$$
−0.451716 + 0.892162i $$0.649188\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ −2.00000 −0.0819232
$$597$$ 5.00000 0.204636
$$598$$ 8.00000 0.327144
$$599$$ 11.0000 0.449448 0.224724 0.974422i $$-0.427852\pi$$
0.224724 + 0.974422i $$0.427852\pi$$
$$600$$ 4.00000 0.163299
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 0 0
$$603$$ −12.0000 −0.488678
$$604$$ −4.00000 −0.162758
$$605$$ 7.00000 0.284590
$$606$$ 14.0000 0.568711
$$607$$ −1.00000 −0.0405887 −0.0202944 0.999794i $$-0.506460\pi$$
−0.0202944 + 0.999794i $$0.506460\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −9.00000 −0.364101
$$612$$ −1.00000 −0.0404226
$$613$$ −26.0000 −1.05013 −0.525065 0.851062i $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 1.00000 0.0403239
$$616$$ 0 0
$$617$$ 17.0000 0.684394 0.342197 0.939628i $$-0.388829\pi$$
0.342197 + 0.939628i $$0.388829\pi$$
$$618$$ 14.0000 0.563163
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ −3.00000 −0.120483
$$621$$ 8.00000 0.321029
$$622$$ −18.0000 −0.721734
$$623$$ 0 0
$$624$$ −1.00000 −0.0400320
$$625$$ 11.0000 0.440000
$$626$$ 8.00000 0.319744
$$627$$ 8.00000 0.319489
$$628$$ −1.00000 −0.0399043
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ −38.0000 −1.51276 −0.756378 0.654135i $$-0.773033\pi$$
−0.756378 + 0.654135i $$0.773033\pi$$
$$632$$ 0 0
$$633$$ 3.00000 0.119239
$$634$$ 11.0000 0.436866
$$635$$ 8.00000 0.317470
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 6.00000 0.237542
$$639$$ 4.00000 0.158238
$$640$$ 1.00000 0.0395285
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 12.0000 0.473602
$$643$$ −45.0000 −1.77463 −0.887313 0.461167i $$-0.847431\pi$$
−0.887313 + 0.461167i $$0.847431\pi$$
$$644$$ 0 0
$$645$$ 4.00000 0.157500
$$646$$ −4.00000 −0.157378
$$647$$ −33.0000 −1.29736 −0.648682 0.761060i $$-0.724679\pi$$
−0.648682 + 0.761060i $$0.724679\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 22.0000 0.863576
$$650$$ −4.00000 −0.156893
$$651$$ 0 0
$$652$$ 16.0000 0.626608
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 16.0000 0.625650
$$655$$ −6.00000 −0.234439
$$656$$ −1.00000 −0.0390434
$$657$$ −6.00000 −0.234082
$$658$$ 0 0
$$659$$ 39.0000 1.51922 0.759612 0.650376i $$-0.225389\pi$$
0.759612 + 0.650376i $$0.225389\pi$$
$$660$$ 2.00000 0.0778499
$$661$$ −47.0000 −1.82809 −0.914044 0.405615i $$-0.867057\pi$$
−0.914044 + 0.405615i $$0.867057\pi$$
$$662$$ 22.0000 0.855054
$$663$$ 1.00000 0.0388368
$$664$$ 9.00000 0.349268
$$665$$ 0 0
$$666$$ −4.00000 −0.154997
$$667$$ 24.0000 0.929284
$$668$$ −2.00000 −0.0773823
$$669$$ 16.0000 0.618596
$$670$$ −12.0000 −0.463600
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ −34.0000 −1.30963
$$675$$ −4.00000 −0.153960
$$676$$ −12.0000 −0.461538
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ −15.0000 −0.576072
$$679$$ 0 0
$$680$$ −1.00000 −0.0383482
$$681$$ −10.0000 −0.383201
$$682$$ 6.00000 0.229752
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ −20.0000 −0.764161
$$686$$ 0 0
$$687$$ 9.00000 0.343371
$$688$$ −4.00000 −0.152499
$$689$$ 0 0
$$690$$ 8.00000 0.304555
$$691$$ 47.0000 1.78796 0.893982 0.448103i $$-0.147900\pi$$
0.893982 + 0.448103i $$0.147900\pi$$
$$692$$ −21.0000 −0.798300
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ −3.00000 −0.113715
$$697$$ 1.00000 0.0378777
$$698$$ 27.0000 1.02197
$$699$$ −11.0000 −0.416058
$$700$$ 0 0
$$701$$ 20.0000 0.755390 0.377695 0.925930i $$-0.376717\pi$$
0.377695 + 0.925930i $$0.376717\pi$$
$$702$$ 1.00000 0.0377426
$$703$$ −16.0000 −0.603451
$$704$$ −2.00000 −0.0753778
$$705$$ −9.00000 −0.338960
$$706$$ 28.0000 1.05379
$$707$$ 0 0
$$708$$ −11.0000 −0.413405
$$709$$ −42.0000 −1.57734 −0.788672 0.614815i $$-0.789231\pi$$
−0.788672 + 0.614815i $$0.789231\pi$$
$$710$$ 4.00000 0.150117
$$711$$ 0 0
$$712$$ −6.00000 −0.224860
$$713$$ 24.0000 0.898807
$$714$$ 0 0
$$715$$ −2.00000 −0.0747958
$$716$$ −21.0000 −0.784807
$$717$$ 13.0000 0.485494
$$718$$ −15.0000 −0.559795
$$719$$ 50.0000 1.86469 0.932343 0.361576i $$-0.117761\pi$$
0.932343 + 0.361576i $$0.117761\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 0 0
$$722$$ 3.00000 0.111648
$$723$$ −18.0000 −0.669427
$$724$$ −16.0000 −0.594635
$$725$$ −12.0000 −0.445669
$$726$$ 7.00000 0.259794
$$727$$ −26.0000 −0.964287 −0.482143 0.876092i $$-0.660142\pi$$
−0.482143 + 0.876092i $$0.660142\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −6.00000 −0.222070
$$731$$ 4.00000 0.147945
$$732$$ 0 0
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ −8.00000 −0.294884
$$737$$ 24.0000 0.884051
$$738$$ 1.00000 0.0368105
$$739$$ 42.0000 1.54499 0.772497 0.635018i $$-0.219007\pi$$
0.772497 + 0.635018i $$0.219007\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ 4.00000 0.146944
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ −3.00000 −0.109985
$$745$$ 2.00000 0.0732743
$$746$$ −19.0000 −0.695639
$$747$$ −9.00000 −0.329293
$$748$$ 2.00000 0.0731272
$$749$$ 0 0
$$750$$ −9.00000 −0.328634
$$751$$ −27.0000 −0.985244 −0.492622 0.870243i $$-0.663961\pi$$
−0.492622 + 0.870243i $$0.663961\pi$$
$$752$$ 9.00000 0.328196
$$753$$ −23.0000 −0.838167
$$754$$ 3.00000 0.109254
$$755$$ 4.00000 0.145575
$$756$$ 0 0
$$757$$ 27.0000 0.981332 0.490666 0.871348i $$-0.336754\pi$$
0.490666 + 0.871348i $$0.336754\pi$$
$$758$$ −8.00000 −0.290573
$$759$$ −16.0000 −0.580763
$$760$$ −4.00000 −0.145095
$$761$$ 36.0000 1.30500 0.652499 0.757789i $$-0.273720\pi$$
0.652499 + 0.757789i $$0.273720\pi$$
$$762$$ 8.00000 0.289809
$$763$$ 0 0
$$764$$ 3.00000 0.108536
$$765$$ 1.00000 0.0361551
$$766$$ 24.0000 0.867155
$$767$$ 11.0000 0.397187
$$768$$ 1.00000 0.0360844
$$769$$ 46.0000 1.65880 0.829401 0.558653i $$-0.188682\pi$$
0.829401 + 0.558653i $$0.188682\pi$$
$$770$$ 0 0
$$771$$ −14.0000 −0.504198
$$772$$ −14.0000 −0.503871
$$773$$ −46.0000 −1.65451 −0.827253 0.561830i $$-0.810097\pi$$
−0.827253 + 0.561830i $$0.810097\pi$$
$$774$$ 4.00000 0.143777
$$775$$ −12.0000 −0.431053
$$776$$ 8.00000 0.287183
$$777$$ 0 0
$$778$$ 2.00000 0.0717035
$$779$$ 4.00000 0.143315
$$780$$ 1.00000 0.0358057
$$781$$ −8.00000 −0.286263
$$782$$ 8.00000 0.286079
$$783$$ 3.00000 0.107211
$$784$$ 0 0
$$785$$ 1.00000 0.0356915
$$786$$ −6.00000 −0.214013
$$787$$ −19.0000 −0.677277 −0.338638 0.940917i $$-0.609966\pi$$
−0.338638 + 0.940917i $$0.609966\pi$$
$$788$$ 1.00000 0.0356235
$$789$$ −8.00000 −0.284808
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 2.00000 0.0710669
$$793$$ 0 0
$$794$$ −8.00000 −0.283909
$$795$$ 0 0
$$796$$ 5.00000 0.177220
$$797$$ −40.0000 −1.41687 −0.708436 0.705775i $$-0.750599\pi$$
−0.708436 + 0.705775i $$0.750599\pi$$
$$798$$ 0 0
$$799$$ −9.00000 −0.318397
$$800$$ 4.00000 0.141421
$$801$$ 6.00000 0.212000
$$802$$ −30.0000 −1.05934
$$803$$ 12.0000 0.423471
$$804$$ −12.0000 −0.423207
$$805$$ 0 0
$$806$$ 3.00000 0.105670
$$807$$ 5.00000 0.176008
$$808$$ 14.0000 0.492518
$$809$$ −38.0000 −1.33601 −0.668004 0.744157i $$-0.732851\pi$$
−0.668004 + 0.744157i $$0.732851\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −11.0000 −0.386262 −0.193131 0.981173i $$-0.561864\pi$$
−0.193131 + 0.981173i $$0.561864\pi$$
$$812$$ 0 0
$$813$$ −18.0000 −0.631288
$$814$$ 8.00000 0.280400
$$815$$ −16.0000 −0.560456
$$816$$ −1.00000 −0.0350070
$$817$$ 16.0000 0.559769
$$818$$ 26.0000 0.909069
$$819$$ 0 0
$$820$$ 1.00000 0.0349215
$$821$$ −2.00000 −0.0698005 −0.0349002 0.999391i $$-0.511111\pi$$
−0.0349002 + 0.999391i $$0.511111\pi$$
$$822$$ −20.0000 −0.697580
$$823$$ 35.0000 1.22002 0.610012 0.792392i $$-0.291165\pi$$
0.610012 + 0.792392i $$0.291165\pi$$
$$824$$ 14.0000 0.487713
$$825$$ 8.00000 0.278524
$$826$$ 0 0
$$827$$ −46.0000 −1.59958 −0.799788 0.600282i $$-0.795055\pi$$
−0.799788 + 0.600282i $$0.795055\pi$$
$$828$$ 8.00000 0.278019
$$829$$ 54.0000 1.87550 0.937749 0.347314i $$-0.112906\pi$$
0.937749 + 0.347314i $$0.112906\pi$$
$$830$$ −9.00000 −0.312395
$$831$$ 28.0000 0.971309
$$832$$ −1.00000 −0.0346688
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 2.00000 0.0692129
$$836$$ 8.00000 0.276686
$$837$$ 3.00000 0.103695
$$838$$ 4.00000 0.138178
$$839$$ 26.0000 0.897620 0.448810 0.893627i $$-0.351848\pi$$
0.448810 + 0.893627i $$0.351848\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ −17.0000 −0.585859
$$843$$ 20.0000 0.688837
$$844$$ 3.00000 0.103264
$$845$$ 12.0000 0.412813
$$846$$ −9.00000 −0.309426
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 13.0000 0.446159
$$850$$ −4.00000 −0.137199
$$851$$ 32.0000 1.09695
$$852$$ 4.00000 0.137038
$$853$$ −16.0000 −0.547830 −0.273915 0.961754i $$-0.588319\pi$$
−0.273915 + 0.961754i $$0.588319\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ 12.0000 0.410152
$$857$$ 21.0000 0.717346 0.358673 0.933463i $$-0.383229\pi$$
0.358673 + 0.933463i $$0.383229\pi$$
$$858$$ −2.00000 −0.0682789
$$859$$ 8.00000 0.272956 0.136478 0.990643i $$-0.456422\pi$$
0.136478 + 0.990643i $$0.456422\pi$$
$$860$$ 4.00000 0.136399
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 21.0000 0.714021
$$866$$ 19.0000 0.645646
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 3.00000 0.101710
$$871$$ 12.0000 0.406604
$$872$$ 16.0000 0.541828
$$873$$ −8.00000 −0.270759
$$874$$ 32.0000 1.08242
$$875$$ 0 0
$$876$$ −6.00000 −0.202721
$$877$$ 30.0000 1.01303 0.506514 0.862232i $$-0.330934\pi$$
0.506514 + 0.862232i $$0.330934\pi$$
$$878$$ −1.00000 −0.0337484
$$879$$ 0 0
$$880$$ 2.00000 0.0674200
$$881$$ 35.0000 1.17918 0.589590 0.807703i $$-0.299289\pi$$
0.589590 + 0.807703i $$0.299289\pi$$
$$882$$ 0 0
$$883$$ 4.00000 0.134611 0.0673054 0.997732i $$-0.478560\pi$$
0.0673054 + 0.997732i $$0.478560\pi$$
$$884$$ 1.00000 0.0336336
$$885$$ 11.0000 0.369761
$$886$$ −33.0000 −1.10866
$$887$$ 26.0000 0.872995 0.436497 0.899706i $$-0.356219\pi$$
0.436497 + 0.899706i $$0.356219\pi$$
$$888$$ −4.00000 −0.134231
$$889$$ 0 0
$$890$$ 6.00000 0.201120
$$891$$ −2.00000 −0.0670025
$$892$$ 16.0000 0.535720
$$893$$ −36.0000 −1.20469
$$894$$ 2.00000 0.0668900
$$895$$ 21.0000 0.701953
$$896$$ 0 0
$$897$$ −8.00000 −0.267112
$$898$$ 6.00000 0.200223
$$899$$ 9.00000 0.300167
$$900$$ −4.00000 −0.133333
$$901$$ 0 0
$$902$$ −2.00000 −0.0665927
$$903$$ 0 0
$$904$$ −15.0000 −0.498893
$$905$$ 16.0000 0.531858
$$906$$ 4.00000 0.132891
$$907$$ 47.0000 1.56061 0.780305 0.625400i $$-0.215064\pi$$
0.780305 + 0.625400i $$0.215064\pi$$
$$908$$ −10.0000 −0.331862
$$909$$ −14.0000 −0.464351
$$910$$ 0 0
$$911$$ −30.0000 −0.993944 −0.496972 0.867766i $$-0.665555\pi$$
−0.496972 + 0.867766i $$0.665555\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ 18.0000 0.595713
$$914$$ −1.00000 −0.0330771
$$915$$ 0 0
$$916$$ 9.00000 0.297368
$$917$$ 0 0
$$918$$ 1.00000 0.0330049
$$919$$ 58.0000 1.91324 0.956622 0.291333i $$-0.0940987\pi$$
0.956622 + 0.291333i $$0.0940987\pi$$
$$920$$ 8.00000 0.263752
$$921$$ −20.0000 −0.659022
$$922$$ −20.0000 −0.658665
$$923$$ −4.00000 −0.131662
$$924$$ 0 0
$$925$$ −16.0000 −0.526077
$$926$$ 18.0000 0.591517
$$927$$ −14.0000 −0.459820
$$928$$ −3.00000 −0.0984798
$$929$$ 5.00000 0.164045 0.0820223 0.996630i $$-0.473862\pi$$
0.0820223 + 0.996630i $$0.473862\pi$$
$$930$$ 3.00000 0.0983739
$$931$$ 0 0
$$932$$ −11.0000 −0.360317
$$933$$ 18.0000 0.589294
$$934$$ −3.00000 −0.0981630
$$935$$ −2.00000 −0.0654070
$$936$$ 1.00000 0.0326860
$$937$$ 14.0000 0.457360 0.228680 0.973502i $$-0.426559\pi$$
0.228680 + 0.973502i $$0.426559\pi$$
$$938$$ 0 0
$$939$$ −8.00000 −0.261070
$$940$$ −9.00000 −0.293548
$$941$$ 2.00000 0.0651981 0.0325991 0.999469i $$-0.489622\pi$$
0.0325991 + 0.999469i $$0.489622\pi$$
$$942$$ 1.00000 0.0325818
$$943$$ −8.00000 −0.260516
$$944$$ −11.0000 −0.358020
$$945$$ 0 0
$$946$$ −8.00000 −0.260102
$$947$$ 14.0000 0.454939 0.227469 0.973785i $$-0.426955\pi$$
0.227469 + 0.973785i $$0.426955\pi$$
$$948$$ 0 0
$$949$$ 6.00000 0.194768
$$950$$ −16.0000 −0.519109
$$951$$ −11.0000 −0.356699
$$952$$ 0 0
$$953$$ 20.0000 0.647864 0.323932 0.946080i $$-0.394995\pi$$
0.323932 + 0.946080i $$0.394995\pi$$
$$954$$ 0 0
$$955$$ −3.00000 −0.0970777
$$956$$ 13.0000 0.420450
$$957$$ −6.00000 −0.193952
$$958$$ 26.0000 0.840022
$$959$$ 0 0
$$960$$ −1.00000 −0.0322749
$$961$$ −22.0000 −0.709677
$$962$$ 4.00000 0.128965
$$963$$ −12.0000 −0.386695
$$964$$ −18.0000 −0.579741
$$965$$ 14.0000 0.450676
$$966$$ 0 0
$$967$$ −52.0000 −1.67221 −0.836104 0.548572i $$-0.815172\pi$$
−0.836104 + 0.548572i $$0.815172\pi$$
$$968$$ 7.00000 0.224989
$$969$$ 4.00000 0.128499
$$970$$ −8.00000 −0.256865
$$971$$ 28.0000 0.898563 0.449281 0.893390i $$-0.351680\pi$$
0.449281 + 0.893390i $$0.351680\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ 27.0000 0.865136
$$975$$ 4.00000 0.128103
$$976$$ 0 0
$$977$$ −52.0000 −1.66363 −0.831814 0.555055i $$-0.812697\pi$$
−0.831814 + 0.555055i $$0.812697\pi$$
$$978$$ −16.0000 −0.511624
$$979$$ −12.0000 −0.383522
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ −28.0000 −0.893516
$$983$$ −12.0000 −0.382741 −0.191370 0.981518i $$-0.561293\pi$$
−0.191370 + 0.981518i $$0.561293\pi$$
$$984$$ 1.00000 0.0318788
$$985$$ −1.00000 −0.0318626
$$986$$ 3.00000 0.0955395
$$987$$ 0 0
$$988$$ 4.00000 0.127257
$$989$$ −32.0000 −1.01754
$$990$$ −2.00000 −0.0635642
$$991$$ 25.0000 0.794151 0.397076 0.917786i $$-0.370025\pi$$
0.397076 + 0.917786i $$0.370025\pi$$
$$992$$ −3.00000 −0.0952501
$$993$$ −22.0000 −0.698149
$$994$$ 0 0
$$995$$ −5.00000 −0.158511
$$996$$ −9.00000 −0.285176
$$997$$ 12.0000 0.380044 0.190022 0.981780i $$-0.439144\pi$$
0.190022 + 0.981780i $$0.439144\pi$$
$$998$$ −13.0000 −0.411508
$$999$$ 4.00000 0.126554
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 4998.2.a.m.1.1 1
7.3 odd 6 714.2.i.i.205.1 2
7.5 odd 6 714.2.i.i.613.1 yes 2
7.6 odd 2 4998.2.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
714.2.i.i.205.1 2 7.3 odd 6
714.2.i.i.613.1 yes 2 7.5 odd 6
4998.2.a.e.1.1 1 7.6 odd 2
4998.2.a.m.1.1 1 1.1 even 1 trivial