# Properties

 Label 5550.2.a.k.1.1 Level $5550$ Weight $2$ Character 5550.1 Self dual yes Analytic conductor $44.317$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.3169731218$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1110) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5550.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -5.00000 q^{14} +1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} -3.00000 q^{19} -5.00000 q^{21} +5.00000 q^{22} -3.00000 q^{23} +1.00000 q^{24} -1.00000 q^{26} -1.00000 q^{27} +5.00000 q^{28} +6.00000 q^{29} -6.00000 q^{31} -1.00000 q^{32} +5.00000 q^{33} -5.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} +3.00000 q^{38} -1.00000 q^{39} +5.00000 q^{42} -4.00000 q^{43} -5.00000 q^{44} +3.00000 q^{46} -1.00000 q^{48} +18.0000 q^{49} -5.00000 q^{51} +1.00000 q^{52} +3.00000 q^{53} +1.00000 q^{54} -5.00000 q^{56} +3.00000 q^{57} -6.00000 q^{58} -10.0000 q^{59} +10.0000 q^{61} +6.00000 q^{62} +5.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} +14.0000 q^{67} +5.00000 q^{68} +3.00000 q^{69} +6.00000 q^{71} -1.00000 q^{72} +9.00000 q^{73} -1.00000 q^{74} -3.00000 q^{76} -25.0000 q^{77} +1.00000 q^{78} +1.00000 q^{81} -5.00000 q^{83} -5.00000 q^{84} +4.00000 q^{86} -6.00000 q^{87} +5.00000 q^{88} +13.0000 q^{89} +5.00000 q^{91} -3.00000 q^{92} +6.00000 q^{93} +1.00000 q^{96} +4.00000 q^{97} -18.0000 q^{98} -5.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 5.00000 1.88982 0.944911 0.327327i $$-0.106148\pi$$
0.944911 + 0.327327i $$0.106148\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ −5.00000 −1.33631
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 5.00000 1.21268 0.606339 0.795206i $$-0.292637\pi$$
0.606339 + 0.795206i $$0.292637\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −3.00000 −0.688247 −0.344124 0.938924i $$-0.611824\pi$$
−0.344124 + 0.938924i $$0.611824\pi$$
$$20$$ 0 0
$$21$$ −5.00000 −1.09109
$$22$$ 5.00000 1.06600
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ −1.00000 −0.192450
$$28$$ 5.00000 0.944911
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 5.00000 0.870388
$$34$$ −5.00000 −0.857493
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 1.00000 0.164399
$$38$$ 3.00000 0.486664
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 5.00000 0.771517
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −5.00000 −0.753778
$$45$$ 0 0
$$46$$ 3.00000 0.442326
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 18.0000 2.57143
$$50$$ 0 0
$$51$$ −5.00000 −0.700140
$$52$$ 1.00000 0.138675
$$53$$ 3.00000 0.412082 0.206041 0.978543i $$-0.433942\pi$$
0.206041 + 0.978543i $$0.433942\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −5.00000 −0.668153
$$57$$ 3.00000 0.397360
$$58$$ −6.00000 −0.787839
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 5.00000 0.629941
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −5.00000 −0.615457
$$67$$ 14.0000 1.71037 0.855186 0.518321i $$-0.173443\pi$$
0.855186 + 0.518321i $$0.173443\pi$$
$$68$$ 5.00000 0.606339
$$69$$ 3.00000 0.361158
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 9.00000 1.05337 0.526685 0.850060i $$-0.323435\pi$$
0.526685 + 0.850060i $$0.323435\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ −3.00000 −0.344124
$$77$$ −25.0000 −2.84901
$$78$$ 1.00000 0.113228
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −5.00000 −0.548821 −0.274411 0.961613i $$-0.588483\pi$$
−0.274411 + 0.961613i $$0.588483\pi$$
$$84$$ −5.00000 −0.545545
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ −6.00000 −0.643268
$$88$$ 5.00000 0.533002
$$89$$ 13.0000 1.37800 0.688999 0.724763i $$-0.258051\pi$$
0.688999 + 0.724763i $$0.258051\pi$$
$$90$$ 0 0
$$91$$ 5.00000 0.524142
$$92$$ −3.00000 −0.312772
$$93$$ 6.00000 0.622171
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 4.00000 0.406138 0.203069 0.979164i $$-0.434908\pi$$
0.203069 + 0.979164i $$0.434908\pi$$
$$98$$ −18.0000 −1.81827
$$99$$ −5.00000 −0.502519
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 5.00000 0.495074
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −3.00000 −0.291386
$$107$$ −15.0000 −1.45010 −0.725052 0.688694i $$-0.758184\pi$$
−0.725052 + 0.688694i $$0.758184\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −5.00000 −0.478913 −0.239457 0.970907i $$-0.576969\pi$$
−0.239457 + 0.970907i $$0.576969\pi$$
$$110$$ 0 0
$$111$$ −1.00000 −0.0949158
$$112$$ 5.00000 0.472456
$$113$$ 10.0000 0.940721 0.470360 0.882474i $$-0.344124\pi$$
0.470360 + 0.882474i $$0.344124\pi$$
$$114$$ −3.00000 −0.280976
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 1.00000 0.0924500
$$118$$ 10.0000 0.920575
$$119$$ 25.0000 2.29175
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ −10.0000 −0.905357
$$123$$ 0 0
$$124$$ −6.00000 −0.538816
$$125$$ 0 0
$$126$$ −5.00000 −0.445435
$$127$$ 3.00000 0.266207 0.133103 0.991102i $$-0.457506\pi$$
0.133103 + 0.991102i $$0.457506\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ 5.00000 0.435194
$$133$$ −15.0000 −1.30066
$$134$$ −14.0000 −1.20942
$$135$$ 0 0
$$136$$ −5.00000 −0.428746
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ −3.00000 −0.255377
$$139$$ 2.00000 0.169638 0.0848189 0.996396i $$-0.472969\pi$$
0.0848189 + 0.996396i $$0.472969\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −6.00000 −0.503509
$$143$$ −5.00000 −0.418121
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −9.00000 −0.744845
$$147$$ −18.0000 −1.48461
$$148$$ 1.00000 0.0821995
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 0 0
$$151$$ 15.0000 1.22068 0.610341 0.792139i $$-0.291032\pi$$
0.610341 + 0.792139i $$0.291032\pi$$
$$152$$ 3.00000 0.243332
$$153$$ 5.00000 0.404226
$$154$$ 25.0000 2.01456
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ 20.0000 1.59617 0.798087 0.602542i $$-0.205846\pi$$
0.798087 + 0.602542i $$0.205846\pi$$
$$158$$ 0 0
$$159$$ −3.00000 −0.237915
$$160$$ 0 0
$$161$$ −15.0000 −1.18217
$$162$$ −1.00000 −0.0785674
$$163$$ 23.0000 1.80150 0.900750 0.434339i $$-0.143018\pi$$
0.900750 + 0.434339i $$0.143018\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 5.00000 0.388075
$$167$$ −5.00000 −0.386912 −0.193456 0.981109i $$-0.561970\pi$$
−0.193456 + 0.981109i $$0.561970\pi$$
$$168$$ 5.00000 0.385758
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −3.00000 −0.229416
$$172$$ −4.00000 −0.304997
$$173$$ −11.0000 −0.836315 −0.418157 0.908375i $$-0.637324\pi$$
−0.418157 + 0.908375i $$0.637324\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ −5.00000 −0.376889
$$177$$ 10.0000 0.751646
$$178$$ −13.0000 −0.974391
$$179$$ 8.00000 0.597948 0.298974 0.954261i $$-0.403356\pi$$
0.298974 + 0.954261i $$0.403356\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ −5.00000 −0.370625
$$183$$ −10.0000 −0.739221
$$184$$ 3.00000 0.221163
$$185$$ 0 0
$$186$$ −6.00000 −0.439941
$$187$$ −25.0000 −1.82818
$$188$$ 0 0
$$189$$ −5.00000 −0.363696
$$190$$ 0 0
$$191$$ −7.00000 −0.506502 −0.253251 0.967401i $$-0.581500\pi$$
−0.253251 + 0.967401i $$0.581500\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ −4.00000 −0.287183
$$195$$ 0 0
$$196$$ 18.0000 1.28571
$$197$$ −23.0000 −1.63868 −0.819341 0.573306i $$-0.805660\pi$$
−0.819341 + 0.573306i $$0.805660\pi$$
$$198$$ 5.00000 0.355335
$$199$$ −24.0000 −1.70131 −0.850657 0.525720i $$-0.823796\pi$$
−0.850657 + 0.525720i $$0.823796\pi$$
$$200$$ 0 0
$$201$$ −14.0000 −0.987484
$$202$$ 6.00000 0.422159
$$203$$ 30.0000 2.10559
$$204$$ −5.00000 −0.350070
$$205$$ 0 0
$$206$$ −16.0000 −1.11477
$$207$$ −3.00000 −0.208514
$$208$$ 1.00000 0.0693375
$$209$$ 15.0000 1.03757
$$210$$ 0 0
$$211$$ −2.00000 −0.137686 −0.0688428 0.997628i $$-0.521931\pi$$
−0.0688428 + 0.997628i $$0.521931\pi$$
$$212$$ 3.00000 0.206041
$$213$$ −6.00000 −0.411113
$$214$$ 15.0000 1.02538
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ −30.0000 −2.03653
$$218$$ 5.00000 0.338643
$$219$$ −9.00000 −0.608164
$$220$$ 0 0
$$221$$ 5.00000 0.336336
$$222$$ 1.00000 0.0671156
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ −5.00000 −0.334077
$$225$$ 0 0
$$226$$ −10.0000 −0.665190
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ 3.00000 0.198680
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ 25.0000 1.64488
$$232$$ −6.00000 −0.393919
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 0 0
$$236$$ −10.0000 −0.650945
$$237$$ 0 0
$$238$$ −25.0000 −1.62051
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ −14.0000 −0.899954
$$243$$ −1.00000 −0.0641500
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3.00000 −0.190885
$$248$$ 6.00000 0.381000
$$249$$ 5.00000 0.316862
$$250$$ 0 0
$$251$$ 30.0000 1.89358 0.946792 0.321847i $$-0.104304\pi$$
0.946792 + 0.321847i $$0.104304\pi$$
$$252$$ 5.00000 0.314970
$$253$$ 15.0000 0.943042
$$254$$ −3.00000 −0.188237
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −15.0000 −0.935674 −0.467837 0.883815i $$-0.654967\pi$$
−0.467837 + 0.883815i $$0.654967\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ 5.00000 0.310685
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 20.0000 1.23560
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ −5.00000 −0.307729
$$265$$ 0 0
$$266$$ 15.0000 0.919709
$$267$$ −13.0000 −0.795587
$$268$$ 14.0000 0.855186
$$269$$ 9.00000 0.548740 0.274370 0.961624i $$-0.411531\pi$$
0.274370 + 0.961624i $$0.411531\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 5.00000 0.303170
$$273$$ −5.00000 −0.302614
$$274$$ −2.00000 −0.120824
$$275$$ 0 0
$$276$$ 3.00000 0.180579
$$277$$ −25.0000 −1.50210 −0.751052 0.660243i $$-0.770453\pi$$
−0.751052 + 0.660243i $$0.770453\pi$$
$$278$$ −2.00000 −0.119952
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ 1.00000 0.0596550 0.0298275 0.999555i $$-0.490504\pi$$
0.0298275 + 0.999555i $$0.490504\pi$$
$$282$$ 0 0
$$283$$ −5.00000 −0.297219 −0.148610 0.988896i $$-0.547480\pi$$
−0.148610 + 0.988896i $$0.547480\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 5.00000 0.295656
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ −4.00000 −0.234484
$$292$$ 9.00000 0.526685
$$293$$ 11.0000 0.642627 0.321313 0.946973i $$-0.395876\pi$$
0.321313 + 0.946973i $$0.395876\pi$$
$$294$$ 18.0000 1.04978
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ 5.00000 0.290129
$$298$$ −2.00000 −0.115857
$$299$$ −3.00000 −0.173494
$$300$$ 0 0
$$301$$ −20.0000 −1.15278
$$302$$ −15.0000 −0.863153
$$303$$ 6.00000 0.344691
$$304$$ −3.00000 −0.172062
$$305$$ 0 0
$$306$$ −5.00000 −0.285831
$$307$$ 30.0000 1.71219 0.856095 0.516818i $$-0.172884\pi$$
0.856095 + 0.516818i $$0.172884\pi$$
$$308$$ −25.0000 −1.42451
$$309$$ −16.0000 −0.910208
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ −24.0000 −1.35656 −0.678280 0.734803i $$-0.737274\pi$$
−0.678280 + 0.734803i $$0.737274\pi$$
$$314$$ −20.0000 −1.12867
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 10.0000 0.561656 0.280828 0.959758i $$-0.409391\pi$$
0.280828 + 0.959758i $$0.409391\pi$$
$$318$$ 3.00000 0.168232
$$319$$ −30.0000 −1.67968
$$320$$ 0 0
$$321$$ 15.0000 0.837218
$$322$$ 15.0000 0.835917
$$323$$ −15.0000 −0.834622
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −23.0000 −1.27385
$$327$$ 5.00000 0.276501
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ −5.00000 −0.274411
$$333$$ 1.00000 0.0547997
$$334$$ 5.00000 0.273588
$$335$$ 0 0
$$336$$ −5.00000 −0.272772
$$337$$ 9.00000 0.490261 0.245131 0.969490i $$-0.421169\pi$$
0.245131 + 0.969490i $$0.421169\pi$$
$$338$$ 12.0000 0.652714
$$339$$ −10.0000 −0.543125
$$340$$ 0 0
$$341$$ 30.0000 1.62459
$$342$$ 3.00000 0.162221
$$343$$ 55.0000 2.96972
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 11.0000 0.591364
$$347$$ 36.0000 1.93258 0.966291 0.257454i $$-0.0828835\pi$$
0.966291 + 0.257454i $$0.0828835\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ 24.0000 1.28469 0.642345 0.766415i $$-0.277962\pi$$
0.642345 + 0.766415i $$0.277962\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 5.00000 0.266501
$$353$$ 26.0000 1.38384 0.691920 0.721974i $$-0.256765\pi$$
0.691920 + 0.721974i $$0.256765\pi$$
$$354$$ −10.0000 −0.531494
$$355$$ 0 0
$$356$$ 13.0000 0.688999
$$357$$ −25.0000 −1.32314
$$358$$ −8.00000 −0.422813
$$359$$ 36.0000 1.90001 0.950004 0.312239i $$-0.101079\pi$$
0.950004 + 0.312239i $$0.101079\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ 10.0000 0.525588
$$363$$ −14.0000 −0.734809
$$364$$ 5.00000 0.262071
$$365$$ 0 0
$$366$$ 10.0000 0.522708
$$367$$ 5.00000 0.260998 0.130499 0.991448i $$-0.458342\pi$$
0.130499 + 0.991448i $$0.458342\pi$$
$$368$$ −3.00000 −0.156386
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 15.0000 0.778761
$$372$$ 6.00000 0.311086
$$373$$ −14.0000 −0.724893 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ 25.0000 1.29272
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.00000 0.309016
$$378$$ 5.00000 0.257172
$$379$$ −26.0000 −1.33553 −0.667765 0.744372i $$-0.732749\pi$$
−0.667765 + 0.744372i $$0.732749\pi$$
$$380$$ 0 0
$$381$$ −3.00000 −0.153695
$$382$$ 7.00000 0.358151
$$383$$ −9.00000 −0.459879 −0.229939 0.973205i $$-0.573853\pi$$
−0.229939 + 0.973205i $$0.573853\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ −4.00000 −0.203331
$$388$$ 4.00000 0.203069
$$389$$ 34.0000 1.72387 0.861934 0.507020i $$-0.169253\pi$$
0.861934 + 0.507020i $$0.169253\pi$$
$$390$$ 0 0
$$391$$ −15.0000 −0.758583
$$392$$ −18.0000 −0.909137
$$393$$ 20.0000 1.00887
$$394$$ 23.0000 1.15872
$$395$$ 0 0
$$396$$ −5.00000 −0.251259
$$397$$ 36.0000 1.80679 0.903394 0.428811i $$-0.141067\pi$$
0.903394 + 0.428811i $$0.141067\pi$$
$$398$$ 24.0000 1.20301
$$399$$ 15.0000 0.750939
$$400$$ 0 0
$$401$$ −27.0000 −1.34832 −0.674158 0.738587i $$-0.735493\pi$$
−0.674158 + 0.738587i $$0.735493\pi$$
$$402$$ 14.0000 0.698257
$$403$$ −6.00000 −0.298881
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ −30.0000 −1.48888
$$407$$ −5.00000 −0.247841
$$408$$ 5.00000 0.247537
$$409$$ 24.0000 1.18672 0.593362 0.804936i $$-0.297800\pi$$
0.593362 + 0.804936i $$0.297800\pi$$
$$410$$ 0 0
$$411$$ −2.00000 −0.0986527
$$412$$ 16.0000 0.788263
$$413$$ −50.0000 −2.46034
$$414$$ 3.00000 0.147442
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ −2.00000 −0.0979404
$$418$$ −15.0000 −0.733674
$$419$$ −19.0000 −0.928211 −0.464105 0.885780i $$-0.653624\pi$$
−0.464105 + 0.885780i $$0.653624\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 2.00000 0.0973585
$$423$$ 0 0
$$424$$ −3.00000 −0.145693
$$425$$ 0 0
$$426$$ 6.00000 0.290701
$$427$$ 50.0000 2.41967
$$428$$ −15.0000 −0.725052
$$429$$ 5.00000 0.241402
$$430$$ 0 0
$$431$$ −33.0000 −1.58955 −0.794777 0.606902i $$-0.792412\pi$$
−0.794777 + 0.606902i $$0.792412\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −21.0000 −1.00920 −0.504598 0.863355i $$-0.668359\pi$$
−0.504598 + 0.863355i $$0.668359\pi$$
$$434$$ 30.0000 1.44005
$$435$$ 0 0
$$436$$ −5.00000 −0.239457
$$437$$ 9.00000 0.430528
$$438$$ 9.00000 0.430037
$$439$$ 36.0000 1.71819 0.859093 0.511819i $$-0.171028\pi$$
0.859093 + 0.511819i $$0.171028\pi$$
$$440$$ 0 0
$$441$$ 18.0000 0.857143
$$442$$ −5.00000 −0.237826
$$443$$ 28.0000 1.33032 0.665160 0.746701i $$-0.268363\pi$$
0.665160 + 0.746701i $$0.268363\pi$$
$$444$$ −1.00000 −0.0474579
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ −2.00000 −0.0945968
$$448$$ 5.00000 0.236228
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 10.0000 0.470360
$$453$$ −15.0000 −0.704761
$$454$$ 8.00000 0.375459
$$455$$ 0 0
$$456$$ −3.00000 −0.140488
$$457$$ −4.00000 −0.187112 −0.0935561 0.995614i $$-0.529823\pi$$
−0.0935561 + 0.995614i $$0.529823\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ −5.00000 −0.233380
$$460$$ 0 0
$$461$$ 26.0000 1.21094 0.605470 0.795868i $$-0.292985\pi$$
0.605470 + 0.795868i $$0.292985\pi$$
$$462$$ −25.0000 −1.16311
$$463$$ 28.0000 1.30127 0.650635 0.759390i $$-0.274503\pi$$
0.650635 + 0.759390i $$0.274503\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −26.0000 −1.20443
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ 70.0000 3.23230
$$470$$ 0 0
$$471$$ −20.0000 −0.921551
$$472$$ 10.0000 0.460287
$$473$$ 20.0000 0.919601
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 25.0000 1.14587
$$477$$ 3.00000 0.137361
$$478$$ −12.0000 −0.548867
$$479$$ −21.0000 −0.959514 −0.479757 0.877401i $$-0.659275\pi$$
−0.479757 + 0.877401i $$0.659275\pi$$
$$480$$ 0 0
$$481$$ 1.00000 0.0455961
$$482$$ 18.0000 0.819878
$$483$$ 15.0000 0.682524
$$484$$ 14.0000 0.636364
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −26.0000 −1.17817 −0.589086 0.808070i $$-0.700512\pi$$
−0.589086 + 0.808070i $$0.700512\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ −23.0000 −1.04010
$$490$$ 0 0
$$491$$ −9.00000 −0.406164 −0.203082 0.979162i $$-0.565096\pi$$
−0.203082 + 0.979162i $$0.565096\pi$$
$$492$$ 0 0
$$493$$ 30.0000 1.35113
$$494$$ 3.00000 0.134976
$$495$$ 0 0
$$496$$ −6.00000 −0.269408
$$497$$ 30.0000 1.34568
$$498$$ −5.00000 −0.224055
$$499$$ −5.00000 −0.223831 −0.111915 0.993718i $$-0.535699\pi$$
−0.111915 + 0.993718i $$0.535699\pi$$
$$500$$ 0 0
$$501$$ 5.00000 0.223384
$$502$$ −30.0000 −1.33897
$$503$$ 20.0000 0.891756 0.445878 0.895094i $$-0.352892\pi$$
0.445878 + 0.895094i $$0.352892\pi$$
$$504$$ −5.00000 −0.222718
$$505$$ 0 0
$$506$$ −15.0000 −0.666831
$$507$$ 12.0000 0.532939
$$508$$ 3.00000 0.133103
$$509$$ 15.0000 0.664863 0.332432 0.943127i $$-0.392131\pi$$
0.332432 + 0.943127i $$0.392131\pi$$
$$510$$ 0 0
$$511$$ 45.0000 1.99068
$$512$$ −1.00000 −0.0441942
$$513$$ 3.00000 0.132453
$$514$$ 15.0000 0.661622
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ −5.00000 −0.219687
$$519$$ 11.0000 0.482846
$$520$$ 0 0
$$521$$ 12.0000 0.525730 0.262865 0.964833i $$-0.415333\pi$$
0.262865 + 0.964833i $$0.415333\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ 36.0000 1.57417 0.787085 0.616844i $$-0.211589\pi$$
0.787085 + 0.616844i $$0.211589\pi$$
$$524$$ −20.0000 −0.873704
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ −30.0000 −1.30682
$$528$$ 5.00000 0.217597
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ −10.0000 −0.433963
$$532$$ −15.0000 −0.650332
$$533$$ 0 0
$$534$$ 13.0000 0.562565
$$535$$ 0 0
$$536$$ −14.0000 −0.604708
$$537$$ −8.00000 −0.345225
$$538$$ −9.00000 −0.388018
$$539$$ −90.0000 −3.87657
$$540$$ 0 0
$$541$$ 5.00000 0.214967 0.107483 0.994207i $$-0.465721\pi$$
0.107483 + 0.994207i $$0.465721\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ 10.0000 0.429141
$$544$$ −5.00000 −0.214373
$$545$$ 0 0
$$546$$ 5.00000 0.213980
$$547$$ 43.0000 1.83855 0.919274 0.393619i $$-0.128777\pi$$
0.919274 + 0.393619i $$0.128777\pi$$
$$548$$ 2.00000 0.0854358
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ −18.0000 −0.766826
$$552$$ −3.00000 −0.127688
$$553$$ 0 0
$$554$$ 25.0000 1.06215
$$555$$ 0 0
$$556$$ 2.00000 0.0848189
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 6.00000 0.254000
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 25.0000 1.05550
$$562$$ −1.00000 −0.0421825
$$563$$ −22.0000 −0.927189 −0.463595 0.886047i $$-0.653441\pi$$
−0.463595 + 0.886047i $$0.653441\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 5.00000 0.210166
$$567$$ 5.00000 0.209980
$$568$$ −6.00000 −0.251754
$$569$$ −13.0000 −0.544988 −0.272494 0.962157i $$-0.587849\pi$$
−0.272494 + 0.962157i $$0.587849\pi$$
$$570$$ 0 0
$$571$$ −10.0000 −0.418487 −0.209243 0.977864i $$-0.567100\pi$$
−0.209243 + 0.977864i $$0.567100\pi$$
$$572$$ −5.00000 −0.209061
$$573$$ 7.00000 0.292429
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −26.0000 −1.08239 −0.541197 0.840896i $$-0.682029\pi$$
−0.541197 + 0.840896i $$0.682029\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ −25.0000 −1.03717
$$582$$ 4.00000 0.165805
$$583$$ −15.0000 −0.621237
$$584$$ −9.00000 −0.372423
$$585$$ 0 0
$$586$$ −11.0000 −0.454406
$$587$$ 24.0000 0.990586 0.495293 0.868726i $$-0.335061\pi$$
0.495293 + 0.868726i $$0.335061\pi$$
$$588$$ −18.0000 −0.742307
$$589$$ 18.0000 0.741677
$$590$$ 0 0
$$591$$ 23.0000 0.946094
$$592$$ 1.00000 0.0410997
$$593$$ 24.0000 0.985562 0.492781 0.870153i $$-0.335980\pi$$
0.492781 + 0.870153i $$0.335980\pi$$
$$594$$ −5.00000 −0.205152
$$595$$ 0 0
$$596$$ 2.00000 0.0819232
$$597$$ 24.0000 0.982255
$$598$$ 3.00000 0.122679
$$599$$ 44.0000 1.79779 0.898896 0.438163i $$-0.144371\pi$$
0.898896 + 0.438163i $$0.144371\pi$$
$$600$$ 0 0
$$601$$ −13.0000 −0.530281 −0.265141 0.964210i $$-0.585418\pi$$
−0.265141 + 0.964210i $$0.585418\pi$$
$$602$$ 20.0000 0.815139
$$603$$ 14.0000 0.570124
$$604$$ 15.0000 0.610341
$$605$$ 0 0
$$606$$ −6.00000 −0.243733
$$607$$ 18.0000 0.730597 0.365299 0.930890i $$-0.380967\pi$$
0.365299 + 0.930890i $$0.380967\pi$$
$$608$$ 3.00000 0.121666
$$609$$ −30.0000 −1.21566
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 5.00000 0.202113
$$613$$ −34.0000 −1.37325 −0.686624 0.727013i $$-0.740908\pi$$
−0.686624 + 0.727013i $$0.740908\pi$$
$$614$$ −30.0000 −1.21070
$$615$$ 0 0
$$616$$ 25.0000 1.00728
$$617$$ 34.0000 1.36879 0.684394 0.729112i $$-0.260067\pi$$
0.684394 + 0.729112i $$0.260067\pi$$
$$618$$ 16.0000 0.643614
$$619$$ −32.0000 −1.28619 −0.643094 0.765787i $$-0.722350\pi$$
−0.643094 + 0.765787i $$0.722350\pi$$
$$620$$ 0 0
$$621$$ 3.00000 0.120386
$$622$$ −8.00000 −0.320771
$$623$$ 65.0000 2.60417
$$624$$ −1.00000 −0.0400320
$$625$$ 0 0
$$626$$ 24.0000 0.959233
$$627$$ −15.0000 −0.599042
$$628$$ 20.0000 0.798087
$$629$$ 5.00000 0.199363
$$630$$ 0 0
$$631$$ −4.00000 −0.159237 −0.0796187 0.996825i $$-0.525370\pi$$
−0.0796187 + 0.996825i $$0.525370\pi$$
$$632$$ 0 0
$$633$$ 2.00000 0.0794929
$$634$$ −10.0000 −0.397151
$$635$$ 0 0
$$636$$ −3.00000 −0.118958
$$637$$ 18.0000 0.713186
$$638$$ 30.0000 1.18771
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ −15.0000 −0.592003
$$643$$ −23.0000 −0.907031 −0.453516 0.891248i $$-0.649830\pi$$
−0.453516 + 0.891248i $$0.649830\pi$$
$$644$$ −15.0000 −0.591083
$$645$$ 0 0
$$646$$ 15.0000 0.590167
$$647$$ −3.00000 −0.117942 −0.0589711 0.998260i $$-0.518782\pi$$
−0.0589711 + 0.998260i $$0.518782\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 50.0000 1.96267
$$650$$ 0 0
$$651$$ 30.0000 1.17579
$$652$$ 23.0000 0.900750
$$653$$ 48.0000 1.87839 0.939193 0.343391i $$-0.111576\pi$$
0.939193 + 0.343391i $$0.111576\pi$$
$$654$$ −5.00000 −0.195515
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 9.00000 0.351123
$$658$$ 0 0
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ −47.0000 −1.82809 −0.914044 0.405615i $$-0.867057\pi$$
−0.914044 + 0.405615i $$0.867057\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ −5.00000 −0.194184
$$664$$ 5.00000 0.194038
$$665$$ 0 0
$$666$$ −1.00000 −0.0387492
$$667$$ −18.0000 −0.696963
$$668$$ −5.00000 −0.193456
$$669$$ −8.00000 −0.309298
$$670$$ 0 0
$$671$$ −50.0000 −1.93023
$$672$$ 5.00000 0.192879
$$673$$ −19.0000 −0.732396 −0.366198 0.930537i $$-0.619341\pi$$
−0.366198 + 0.930537i $$0.619341\pi$$
$$674$$ −9.00000 −0.346667
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ 9.00000 0.345898 0.172949 0.984931i $$-0.444670\pi$$
0.172949 + 0.984931i $$0.444670\pi$$
$$678$$ 10.0000 0.384048
$$679$$ 20.0000 0.767530
$$680$$ 0 0
$$681$$ 8.00000 0.306561
$$682$$ −30.0000 −1.14876
$$683$$ −48.0000 −1.83667 −0.918334 0.395805i $$-0.870466\pi$$
−0.918334 + 0.395805i $$0.870466\pi$$
$$684$$ −3.00000 −0.114708
$$685$$ 0 0
$$686$$ −55.0000 −2.09991
$$687$$ −6.00000 −0.228914
$$688$$ −4.00000 −0.152499
$$689$$ 3.00000 0.114291
$$690$$ 0 0
$$691$$ −22.0000 −0.836919 −0.418460 0.908235i $$-0.637430\pi$$
−0.418460 + 0.908235i $$0.637430\pi$$
$$692$$ −11.0000 −0.418157
$$693$$ −25.0000 −0.949671
$$694$$ −36.0000 −1.36654
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ 0 0
$$698$$ −24.0000 −0.908413
$$699$$ −26.0000 −0.983410
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 1.00000 0.0377426
$$703$$ −3.00000 −0.113147
$$704$$ −5.00000 −0.188445
$$705$$ 0 0
$$706$$ −26.0000 −0.978523
$$707$$ −30.0000 −1.12827
$$708$$ 10.0000 0.375823
$$709$$ −35.0000 −1.31445 −0.657226 0.753693i $$-0.728270\pi$$
−0.657226 + 0.753693i $$0.728270\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −13.0000 −0.487196
$$713$$ 18.0000 0.674105
$$714$$ 25.0000 0.935601
$$715$$ 0 0
$$716$$ 8.00000 0.298974
$$717$$ −12.0000 −0.448148
$$718$$ −36.0000 −1.34351
$$719$$ −26.0000 −0.969636 −0.484818 0.874615i $$-0.661114\pi$$
−0.484818 + 0.874615i $$0.661114\pi$$
$$720$$ 0 0
$$721$$ 80.0000 2.97936
$$722$$ 10.0000 0.372161
$$723$$ 18.0000 0.669427
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ 14.0000 0.519589
$$727$$ 30.0000 1.11264 0.556319 0.830969i $$-0.312213\pi$$
0.556319 + 0.830969i $$0.312213\pi$$
$$728$$ −5.00000 −0.185312
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −20.0000 −0.739727
$$732$$ −10.0000 −0.369611
$$733$$ −46.0000 −1.69905 −0.849524 0.527549i $$-0.823111\pi$$
−0.849524 + 0.527549i $$0.823111\pi$$
$$734$$ −5.00000 −0.184553
$$735$$ 0 0
$$736$$ 3.00000 0.110581
$$737$$ −70.0000 −2.57848
$$738$$ 0 0
$$739$$ 26.0000 0.956425 0.478213 0.878244i $$-0.341285\pi$$
0.478213 + 0.878244i $$0.341285\pi$$
$$740$$ 0 0
$$741$$ 3.00000 0.110208
$$742$$ −15.0000 −0.550667
$$743$$ −10.0000 −0.366864 −0.183432 0.983032i $$-0.558721\pi$$
−0.183432 + 0.983032i $$0.558721\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ 0 0
$$746$$ 14.0000 0.512576
$$747$$ −5.00000 −0.182940
$$748$$ −25.0000 −0.914091
$$749$$ −75.0000 −2.74044
$$750$$ 0 0
$$751$$ 28.0000 1.02173 0.510867 0.859660i $$-0.329324\pi$$
0.510867 + 0.859660i $$0.329324\pi$$
$$752$$ 0 0
$$753$$ −30.0000 −1.09326
$$754$$ −6.00000 −0.218507
$$755$$ 0 0
$$756$$ −5.00000 −0.181848
$$757$$ −37.0000 −1.34479 −0.672394 0.740193i $$-0.734734\pi$$
−0.672394 + 0.740193i $$0.734734\pi$$
$$758$$ 26.0000 0.944363
$$759$$ −15.0000 −0.544466
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 3.00000 0.108679
$$763$$ −25.0000 −0.905061
$$764$$ −7.00000 −0.253251
$$765$$ 0 0
$$766$$ 9.00000 0.325183
$$767$$ −10.0000 −0.361079
$$768$$ −1.00000 −0.0360844
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 15.0000 0.540212
$$772$$ 2.00000 0.0719816
$$773$$ 21.0000 0.755318 0.377659 0.925945i $$-0.376729\pi$$
0.377659 + 0.925945i $$0.376729\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ −4.00000 −0.143592
$$777$$ −5.00000 −0.179374
$$778$$ −34.0000 −1.21896
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −30.0000 −1.07348
$$782$$ 15.0000 0.536399
$$783$$ −6.00000 −0.214423
$$784$$ 18.0000 0.642857
$$785$$ 0 0
$$786$$ −20.0000 −0.713376
$$787$$ −42.0000 −1.49714 −0.748569 0.663057i $$-0.769259\pi$$
−0.748569 + 0.663057i $$0.769259\pi$$
$$788$$ −23.0000 −0.819341
$$789$$ 12.0000 0.427211
$$790$$ 0 0
$$791$$ 50.0000 1.77780
$$792$$ 5.00000 0.177667
$$793$$ 10.0000 0.355110
$$794$$ −36.0000 −1.27759
$$795$$ 0 0
$$796$$ −24.0000 −0.850657
$$797$$ 14.0000 0.495905 0.247953 0.968772i $$-0.420242\pi$$
0.247953 + 0.968772i $$0.420242\pi$$
$$798$$ −15.0000 −0.530994
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 13.0000 0.459332
$$802$$ 27.0000 0.953403
$$803$$ −45.0000 −1.58802
$$804$$ −14.0000 −0.493742
$$805$$ 0 0
$$806$$ 6.00000 0.211341
$$807$$ −9.00000 −0.316815
$$808$$ 6.00000 0.211079
$$809$$ −15.0000 −0.527372 −0.263686 0.964609i $$-0.584938\pi$$
−0.263686 + 0.964609i $$0.584938\pi$$
$$810$$ 0 0
$$811$$ 40.0000 1.40459 0.702295 0.711886i $$-0.252159\pi$$
0.702295 + 0.711886i $$0.252159\pi$$
$$812$$ 30.0000 1.05279
$$813$$ −20.0000 −0.701431
$$814$$ 5.00000 0.175250
$$815$$ 0 0
$$816$$ −5.00000 −0.175035
$$817$$ 12.0000 0.419827
$$818$$ −24.0000 −0.839140
$$819$$ 5.00000 0.174714
$$820$$ 0 0
$$821$$ 39.0000 1.36111 0.680555 0.732697i $$-0.261739\pi$$
0.680555 + 0.732697i $$0.261739\pi$$
$$822$$ 2.00000 0.0697580
$$823$$ 35.0000 1.22002 0.610012 0.792392i $$-0.291165\pi$$
0.610012 + 0.792392i $$0.291165\pi$$
$$824$$ −16.0000 −0.557386
$$825$$ 0 0
$$826$$ 50.0000 1.73972
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ −3.00000 −0.104257
$$829$$ 7.00000 0.243120 0.121560 0.992584i $$-0.461210\pi$$
0.121560 + 0.992584i $$0.461210\pi$$
$$830$$ 0 0
$$831$$ 25.0000 0.867240
$$832$$ 1.00000 0.0346688
$$833$$ 90.0000 3.11832
$$834$$ 2.00000 0.0692543
$$835$$ 0 0
$$836$$ 15.0000 0.518786
$$837$$ 6.00000 0.207390
$$838$$ 19.0000 0.656344
$$839$$ −42.0000 −1.45000 −0.725001 0.688748i $$-0.758161\pi$$
−0.725001 + 0.688748i $$0.758161\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 6.00000 0.206774
$$843$$ −1.00000 −0.0344418
$$844$$ −2.00000 −0.0688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 70.0000 2.40523
$$848$$ 3.00000 0.103020
$$849$$ 5.00000 0.171600
$$850$$ 0 0
$$851$$ −3.00000 −0.102839
$$852$$ −6.00000 −0.205557
$$853$$ −19.0000 −0.650548 −0.325274 0.945620i $$-0.605456\pi$$
−0.325274 + 0.945620i $$0.605456\pi$$
$$854$$ −50.0000 −1.71096
$$855$$ 0 0
$$856$$ 15.0000 0.512689
$$857$$ −9.00000 −0.307434 −0.153717 0.988115i $$-0.549124\pi$$
−0.153717 + 0.988115i $$0.549124\pi$$
$$858$$ −5.00000 −0.170697
$$859$$ 7.00000 0.238837 0.119418 0.992844i $$-0.461897\pi$$
0.119418 + 0.992844i $$0.461897\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 33.0000 1.12398
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 21.0000 0.713609
$$867$$ −8.00000 −0.271694
$$868$$ −30.0000 −1.01827
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 14.0000 0.474372
$$872$$ 5.00000 0.169321
$$873$$ 4.00000 0.135379
$$874$$ −9.00000 −0.304430
$$875$$ 0 0
$$876$$ −9.00000 −0.304082
$$877$$ 52.0000 1.75592 0.877958 0.478738i $$-0.158906\pi$$
0.877958 + 0.478738i $$0.158906\pi$$
$$878$$ −36.0000 −1.21494
$$879$$ −11.0000 −0.371021
$$880$$ 0 0
$$881$$ −40.0000 −1.34763 −0.673817 0.738898i $$-0.735346\pi$$
−0.673817 + 0.738898i $$0.735346\pi$$
$$882$$ −18.0000 −0.606092
$$883$$ −21.0000 −0.706706 −0.353353 0.935490i $$-0.614959\pi$$
−0.353353 + 0.935490i $$0.614959\pi$$
$$884$$ 5.00000 0.168168
$$885$$ 0 0
$$886$$ −28.0000 −0.940678
$$887$$ 38.0000 1.27592 0.637958 0.770072i $$-0.279780\pi$$
0.637958 + 0.770072i $$0.279780\pi$$
$$888$$ 1.00000 0.0335578
$$889$$ 15.0000 0.503084
$$890$$ 0 0
$$891$$ −5.00000 −0.167506
$$892$$ 8.00000 0.267860
$$893$$ 0 0
$$894$$ 2.00000 0.0668900
$$895$$ 0 0
$$896$$ −5.00000 −0.167038
$$897$$ 3.00000 0.100167
$$898$$ −30.0000 −1.00111
$$899$$ −36.0000 −1.20067
$$900$$ 0 0
$$901$$ 15.0000 0.499722
$$902$$ 0 0
$$903$$ 20.0000 0.665558
$$904$$ −10.0000 −0.332595
$$905$$ 0 0
$$906$$ 15.0000 0.498342
$$907$$ −37.0000 −1.22856 −0.614282 0.789086i $$-0.710554\pi$$
−0.614282 + 0.789086i $$0.710554\pi$$
$$908$$ −8.00000 −0.265489
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −8.00000 −0.265052 −0.132526 0.991180i $$-0.542309\pi$$
−0.132526 + 0.991180i $$0.542309\pi$$
$$912$$ 3.00000 0.0993399
$$913$$ 25.0000 0.827379
$$914$$ 4.00000 0.132308
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ −100.000 −3.30229
$$918$$ 5.00000 0.165025
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ −30.0000 −0.988534
$$922$$ −26.0000 −0.856264
$$923$$ 6.00000 0.197492
$$924$$ 25.0000 0.822440
$$925$$ 0 0
$$926$$ −28.0000 −0.920137
$$927$$ 16.0000 0.525509
$$928$$ −6.00000 −0.196960
$$929$$ 2.00000 0.0656179 0.0328089 0.999462i $$-0.489555\pi$$
0.0328089 + 0.999462i $$0.489555\pi$$
$$930$$ 0 0
$$931$$ −54.0000 −1.76978
$$932$$ 26.0000 0.851658
$$933$$ −8.00000 −0.261908
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ −1.00000 −0.0326860
$$937$$ 34.0000 1.11073 0.555366 0.831606i $$-0.312578\pi$$
0.555366 + 0.831606i $$0.312578\pi$$
$$938$$ −70.0000 −2.28558
$$939$$ 24.0000 0.783210
$$940$$ 0 0
$$941$$ 50.0000 1.62995 0.814977 0.579494i $$-0.196750\pi$$
0.814977 + 0.579494i $$0.196750\pi$$
$$942$$ 20.0000 0.651635
$$943$$ 0 0
$$944$$ −10.0000 −0.325472
$$945$$ 0 0
$$946$$ −20.0000 −0.650256
$$947$$ −4.00000 −0.129983 −0.0649913 0.997886i $$-0.520702\pi$$
−0.0649913 + 0.997886i $$0.520702\pi$$
$$948$$ 0 0
$$949$$ 9.00000 0.292152
$$950$$ 0 0
$$951$$ −10.0000 −0.324272
$$952$$ −25.0000 −0.810255
$$953$$ −14.0000 −0.453504 −0.226752 0.973952i $$-0.572811\pi$$
−0.226752 + 0.973952i $$0.572811\pi$$
$$954$$ −3.00000 −0.0971286
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 30.0000 0.969762
$$958$$ 21.0000 0.678479
$$959$$ 10.0000 0.322917
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ −1.00000 −0.0322413
$$963$$ −15.0000 −0.483368
$$964$$ −18.0000 −0.579741
$$965$$ 0 0
$$966$$ −15.0000 −0.482617
$$967$$ 18.0000 0.578841 0.289420 0.957202i $$-0.406537\pi$$
0.289420 + 0.957202i $$0.406537\pi$$
$$968$$ −14.0000 −0.449977
$$969$$ 15.0000 0.481869
$$970$$ 0 0
$$971$$ −4.00000 −0.128366 −0.0641831 0.997938i $$-0.520444\pi$$
−0.0641831 + 0.997938i $$0.520444\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 10.0000 0.320585
$$974$$ 26.0000 0.833094
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ −39.0000 −1.24772 −0.623860 0.781536i $$-0.714437\pi$$
−0.623860 + 0.781536i $$0.714437\pi$$
$$978$$ 23.0000 0.735459
$$979$$ −65.0000 −2.07741
$$980$$ 0 0
$$981$$ −5.00000 −0.159638
$$982$$ 9.00000 0.287202
$$983$$ 58.0000 1.84991 0.924956 0.380073i $$-0.124101\pi$$
0.924956 + 0.380073i $$0.124101\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −30.0000 −0.955395
$$987$$ 0 0
$$988$$ −3.00000 −0.0954427
$$989$$ 12.0000 0.381578
$$990$$ 0 0
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ 6.00000 0.190500
$$993$$ −4.00000 −0.126936
$$994$$ −30.0000 −0.951542
$$995$$ 0 0
$$996$$ 5.00000 0.158431
$$997$$ 39.0000 1.23514 0.617571 0.786515i $$-0.288117\pi$$
0.617571 + 0.786515i $$0.288117\pi$$
$$998$$ 5.00000 0.158272
$$999$$ −1.00000 −0.0316386
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 5550.2.a.k.1.1 1
5.4 even 2 1110.2.a.m.1.1 1
15.14 odd 2 3330.2.a.f.1.1 1
20.19 odd 2 8880.2.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.m.1.1 1 5.4 even 2
3330.2.a.f.1.1 1 15.14 odd 2
5550.2.a.k.1.1 1 1.1 even 1 trivial
8880.2.a.i.1.1 1 20.19 odd 2