# Properties

 Label 3450.2.a.bt.1.1 Level $3450$ Weight $2$ Character 3450.1 Self dual yes Analytic conductor $27.548$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.5483886973$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 690) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 3450.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.96239 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.96239 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.35026 q^{11} -1.00000 q^{12} +4.96239 q^{13} -2.96239 q^{14} +1.00000 q^{16} -1.35026 q^{17} +1.00000 q^{18} -4.96239 q^{19} +2.96239 q^{21} -3.35026 q^{22} +1.00000 q^{23} -1.00000 q^{24} +4.96239 q^{26} -1.00000 q^{27} -2.96239 q^{28} +7.73813 q^{29} -4.00000 q^{31} +1.00000 q^{32} +3.35026 q^{33} -1.35026 q^{34} +1.00000 q^{36} +7.61213 q^{37} -4.96239 q^{38} -4.96239 q^{39} +4.70052 q^{41} +2.96239 q^{42} +10.3127 q^{43} -3.35026 q^{44} +1.00000 q^{46} -3.22425 q^{47} -1.00000 q^{48} +1.77575 q^{49} +1.35026 q^{51} +4.96239 q^{52} +6.96239 q^{53} -1.00000 q^{54} -2.96239 q^{56} +4.96239 q^{57} +7.73813 q^{58} +1.22425 q^{59} -11.0884 q^{61} -4.00000 q^{62} -2.96239 q^{63} +1.00000 q^{64} +3.35026 q^{66} +7.61213 q^{67} -1.35026 q^{68} -1.00000 q^{69} -2.18664 q^{71} +1.00000 q^{72} +9.92478 q^{73} +7.61213 q^{74} -4.96239 q^{76} +9.92478 q^{77} -4.96239 q^{78} -4.12601 q^{79} +1.00000 q^{81} +4.70052 q^{82} +6.38787 q^{83} +2.96239 q^{84} +10.3127 q^{86} -7.73813 q^{87} -3.35026 q^{88} +9.92478 q^{89} -14.7005 q^{91} +1.00000 q^{92} +4.00000 q^{93} -3.22425 q^{94} -1.00000 q^{96} -12.8872 q^{97} +1.77575 q^{98} -3.35026 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 2q^{7} + 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 2q^{7} + 3q^{8} + 3q^{9} - 3q^{12} + 4q^{13} + 2q^{14} + 3q^{16} + 6q^{17} + 3q^{18} - 4q^{19} - 2q^{21} + 3q^{23} - 3q^{24} + 4q^{26} - 3q^{27} + 2q^{28} + 14q^{29} - 12q^{31} + 3q^{32} + 6q^{34} + 3q^{36} + 22q^{37} - 4q^{38} - 4q^{39} - 6q^{41} - 2q^{42} + 10q^{43} + 3q^{46} - 8q^{47} - 3q^{48} + 7q^{49} - 6q^{51} + 4q^{52} + 10q^{53} - 3q^{54} + 2q^{56} + 4q^{57} + 14q^{58} + 2q^{59} - 14q^{61} - 12q^{62} + 2q^{63} + 3q^{64} + 22q^{67} + 6q^{68} - 3q^{69} + 6q^{71} + 3q^{72} + 8q^{73} + 22q^{74} - 4q^{76} + 8q^{77} - 4q^{78} - 4q^{79} + 3q^{81} - 6q^{82} + 20q^{83} - 2q^{84} + 10q^{86} - 14q^{87} + 8q^{89} - 24q^{91} + 3q^{92} + 12q^{93} - 8q^{94} - 3q^{96} - 6q^{97} + 7q^{98} + O(q^{100})$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ −2.96239 −1.11968 −0.559839 0.828602i $$-0.689137\pi$$
−0.559839 + 0.828602i $$0.689137\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.35026 −1.01014 −0.505071 0.863078i $$-0.668534\pi$$
−0.505071 + 0.863078i $$0.668534\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 4.96239 1.37632 0.688159 0.725559i $$-0.258419\pi$$
0.688159 + 0.725559i $$0.258419\pi$$
$$14$$ −2.96239 −0.791732
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −1.35026 −0.327487 −0.163743 0.986503i $$-0.552357\pi$$
−0.163743 + 0.986503i $$0.552357\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −4.96239 −1.13845 −0.569225 0.822182i $$-0.692757\pi$$
−0.569225 + 0.822182i $$0.692757\pi$$
$$20$$ 0 0
$$21$$ 2.96239 0.646446
$$22$$ −3.35026 −0.714278
$$23$$ 1.00000 0.208514
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 4.96239 0.973204
$$27$$ −1.00000 −0.192450
$$28$$ −2.96239 −0.559839
$$29$$ 7.73813 1.43694 0.718468 0.695560i $$-0.244844\pi$$
0.718468 + 0.695560i $$0.244844\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 3.35026 0.583206
$$34$$ −1.35026 −0.231568
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 7.61213 1.25143 0.625713 0.780053i $$-0.284808\pi$$
0.625713 + 0.780053i $$0.284808\pi$$
$$38$$ −4.96239 −0.805006
$$39$$ −4.96239 −0.794618
$$40$$ 0 0
$$41$$ 4.70052 0.734098 0.367049 0.930202i $$-0.380368\pi$$
0.367049 + 0.930202i $$0.380368\pi$$
$$42$$ 2.96239 0.457106
$$43$$ 10.3127 1.57266 0.786332 0.617804i $$-0.211977\pi$$
0.786332 + 0.617804i $$0.211977\pi$$
$$44$$ −3.35026 −0.505071
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ −3.22425 −0.470306 −0.235153 0.971958i $$-0.575559\pi$$
−0.235153 + 0.971958i $$0.575559\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.77575 0.253678
$$50$$ 0 0
$$51$$ 1.35026 0.189074
$$52$$ 4.96239 0.688159
$$53$$ 6.96239 0.956358 0.478179 0.878263i $$-0.341297\pi$$
0.478179 + 0.878263i $$0.341297\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −2.96239 −0.395866
$$57$$ 4.96239 0.657284
$$58$$ 7.73813 1.01607
$$59$$ 1.22425 0.159384 0.0796921 0.996820i $$-0.474606\pi$$
0.0796921 + 0.996820i $$0.474606\pi$$
$$60$$ 0 0
$$61$$ −11.0884 −1.41972 −0.709862 0.704341i $$-0.751243\pi$$
−0.709862 + 0.704341i $$0.751243\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ −2.96239 −0.373226
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 3.35026 0.412389
$$67$$ 7.61213 0.929969 0.464985 0.885319i $$-0.346060\pi$$
0.464985 + 0.885319i $$0.346060\pi$$
$$68$$ −1.35026 −0.163743
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −2.18664 −0.259507 −0.129753 0.991546i $$-0.541419\pi$$
−0.129753 + 0.991546i $$0.541419\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 9.92478 1.16161 0.580804 0.814044i $$-0.302738\pi$$
0.580804 + 0.814044i $$0.302738\pi$$
$$74$$ 7.61213 0.884892
$$75$$ 0 0
$$76$$ −4.96239 −0.569225
$$77$$ 9.92478 1.13103
$$78$$ −4.96239 −0.561880
$$79$$ −4.12601 −0.464212 −0.232106 0.972690i $$-0.574562\pi$$
−0.232106 + 0.972690i $$0.574562\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 4.70052 0.519086
$$83$$ 6.38787 0.701160 0.350580 0.936533i $$-0.385984\pi$$
0.350580 + 0.936533i $$0.385984\pi$$
$$84$$ 2.96239 0.323223
$$85$$ 0 0
$$86$$ 10.3127 1.11204
$$87$$ −7.73813 −0.829615
$$88$$ −3.35026 −0.357139
$$89$$ 9.92478 1.05202 0.526012 0.850477i $$-0.323687\pi$$
0.526012 + 0.850477i $$0.323687\pi$$
$$90$$ 0 0
$$91$$ −14.7005 −1.54103
$$92$$ 1.00000 0.104257
$$93$$ 4.00000 0.414781
$$94$$ −3.22425 −0.332556
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ −12.8872 −1.30849 −0.654247 0.756281i $$-0.727014\pi$$
−0.654247 + 0.756281i $$0.727014\pi$$
$$98$$ 1.77575 0.179377
$$99$$ −3.35026 −0.336714
$$100$$ 0 0
$$101$$ 4.26187 0.424071 0.212036 0.977262i $$-0.431991\pi$$
0.212036 + 0.977262i $$0.431991\pi$$
$$102$$ 1.35026 0.133696
$$103$$ 0.261865 0.0258023 0.0129012 0.999917i $$-0.495893\pi$$
0.0129012 + 0.999917i $$0.495893\pi$$
$$104$$ 4.96239 0.486602
$$105$$ 0 0
$$106$$ 6.96239 0.676247
$$107$$ 9.08840 0.878608 0.439304 0.898338i $$-0.355225\pi$$
0.439304 + 0.898338i $$0.355225\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 18.9380 1.81393 0.906963 0.421210i $$-0.138394\pi$$
0.906963 + 0.421210i $$0.138394\pi$$
$$110$$ 0 0
$$111$$ −7.61213 −0.722511
$$112$$ −2.96239 −0.279919
$$113$$ 6.64974 0.625555 0.312777 0.949826i $$-0.398741\pi$$
0.312777 + 0.949826i $$0.398741\pi$$
$$114$$ 4.96239 0.464770
$$115$$ 0 0
$$116$$ 7.73813 0.718468
$$117$$ 4.96239 0.458773
$$118$$ 1.22425 0.112702
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 0.224254 0.0203867
$$122$$ −11.0884 −1.00390
$$123$$ −4.70052 −0.423832
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ −2.96239 −0.263911
$$127$$ −0.186642 −0.0165618 −0.00828091 0.999966i $$-0.502636\pi$$
−0.00828091 + 0.999966i $$0.502636\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −10.3127 −0.907978
$$130$$ 0 0
$$131$$ 7.92478 0.692391 0.346196 0.938162i $$-0.387473\pi$$
0.346196 + 0.938162i $$0.387473\pi$$
$$132$$ 3.35026 0.291603
$$133$$ 14.7005 1.27470
$$134$$ 7.61213 0.657588
$$135$$ 0 0
$$136$$ −1.35026 −0.115784
$$137$$ −9.19982 −0.785993 −0.392997 0.919540i $$-0.628562\pi$$
−0.392997 + 0.919540i $$0.628562\pi$$
$$138$$ −1.00000 −0.0851257
$$139$$ 8.62530 0.731588 0.365794 0.930696i $$-0.380797\pi$$
0.365794 + 0.930696i $$0.380797\pi$$
$$140$$ 0 0
$$141$$ 3.22425 0.271531
$$142$$ −2.18664 −0.183499
$$143$$ −16.6253 −1.39028
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 9.92478 0.821380
$$147$$ −1.77575 −0.146461
$$148$$ 7.61213 0.625713
$$149$$ 4.64974 0.380921 0.190461 0.981695i $$-0.439002\pi$$
0.190461 + 0.981695i $$0.439002\pi$$
$$150$$ 0 0
$$151$$ 10.7005 0.870796 0.435398 0.900238i $$-0.356608\pi$$
0.435398 + 0.900238i $$0.356608\pi$$
$$152$$ −4.96239 −0.402503
$$153$$ −1.35026 −0.109162
$$154$$ 9.92478 0.799761
$$155$$ 0 0
$$156$$ −4.96239 −0.397309
$$157$$ 17.0132 1.35780 0.678900 0.734231i $$-0.262457\pi$$
0.678900 + 0.734231i $$0.262457\pi$$
$$158$$ −4.12601 −0.328248
$$159$$ −6.96239 −0.552153
$$160$$ 0 0
$$161$$ −2.96239 −0.233469
$$162$$ 1.00000 0.0785674
$$163$$ −12.6253 −0.988890 −0.494445 0.869209i $$-0.664629\pi$$
−0.494445 + 0.869209i $$0.664629\pi$$
$$164$$ 4.70052 0.367049
$$165$$ 0 0
$$166$$ 6.38787 0.495795
$$167$$ 24.6253 1.90556 0.952781 0.303657i $$-0.0982076\pi$$
0.952781 + 0.303657i $$0.0982076\pi$$
$$168$$ 2.96239 0.228553
$$169$$ 11.6253 0.894254
$$170$$ 0 0
$$171$$ −4.96239 −0.379483
$$172$$ 10.3127 0.786332
$$173$$ 4.44851 0.338214 0.169107 0.985598i $$-0.445912\pi$$
0.169107 + 0.985598i $$0.445912\pi$$
$$174$$ −7.73813 −0.586626
$$175$$ 0 0
$$176$$ −3.35026 −0.252535
$$177$$ −1.22425 −0.0920205
$$178$$ 9.92478 0.743894
$$179$$ −13.8496 −1.03516 −0.517582 0.855634i $$-0.673168\pi$$
−0.517582 + 0.855634i $$0.673168\pi$$
$$180$$ 0 0
$$181$$ −22.6859 −1.68623 −0.843116 0.537732i $$-0.819281\pi$$
−0.843116 + 0.537732i $$0.819281\pi$$
$$182$$ −14.7005 −1.08968
$$183$$ 11.0884 0.819678
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ 4.52373 0.330808
$$188$$ −3.22425 −0.235153
$$189$$ 2.96239 0.215482
$$190$$ 0 0
$$191$$ −19.3258 −1.39837 −0.699184 0.714942i $$-0.746453\pi$$
−0.699184 + 0.714942i $$0.746453\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 5.92478 0.426475 0.213237 0.977000i $$-0.431599\pi$$
0.213237 + 0.977000i $$0.431599\pi$$
$$194$$ −12.8872 −0.925245
$$195$$ 0 0
$$196$$ 1.77575 0.126839
$$197$$ −8.07522 −0.575336 −0.287668 0.957730i $$-0.592880\pi$$
−0.287668 + 0.957730i $$0.592880\pi$$
$$198$$ −3.35026 −0.238093
$$199$$ 15.9756 1.13248 0.566239 0.824241i $$-0.308398\pi$$
0.566239 + 0.824241i $$0.308398\pi$$
$$200$$ 0 0
$$201$$ −7.61213 −0.536918
$$202$$ 4.26187 0.299864
$$203$$ −22.9234 −1.60890
$$204$$ 1.35026 0.0945372
$$205$$ 0 0
$$206$$ 0.261865 0.0182450
$$207$$ 1.00000 0.0695048
$$208$$ 4.96239 0.344080
$$209$$ 16.6253 1.15000
$$210$$ 0 0
$$211$$ 21.7743 1.49901 0.749503 0.662000i $$-0.230292\pi$$
0.749503 + 0.662000i $$0.230292\pi$$
$$212$$ 6.96239 0.478179
$$213$$ 2.18664 0.149826
$$214$$ 9.08840 0.621270
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 11.8496 0.804400
$$218$$ 18.9380 1.28264
$$219$$ −9.92478 −0.670654
$$220$$ 0 0
$$221$$ −6.70052 −0.450726
$$222$$ −7.61213 −0.510893
$$223$$ 3.81336 0.255361 0.127681 0.991815i $$-0.459247\pi$$
0.127681 + 0.991815i $$0.459247\pi$$
$$224$$ −2.96239 −0.197933
$$225$$ 0 0
$$226$$ 6.64974 0.442334
$$227$$ 16.9380 1.12421 0.562106 0.827065i $$-0.309991\pi$$
0.562106 + 0.827065i $$0.309991\pi$$
$$228$$ 4.96239 0.328642
$$229$$ −26.9380 −1.78011 −0.890055 0.455853i $$-0.849334\pi$$
−0.890055 + 0.455853i $$0.849334\pi$$
$$230$$ 0 0
$$231$$ −9.92478 −0.653002
$$232$$ 7.73813 0.508033
$$233$$ 27.4010 1.79510 0.897551 0.440910i $$-0.145344\pi$$
0.897551 + 0.440910i $$0.145344\pi$$
$$234$$ 4.96239 0.324401
$$235$$ 0 0
$$236$$ 1.22425 0.0796921
$$237$$ 4.12601 0.268013
$$238$$ 4.00000 0.259281
$$239$$ 7.48612 0.484237 0.242118 0.970247i $$-0.422158\pi$$
0.242118 + 0.970247i $$0.422158\pi$$
$$240$$ 0 0
$$241$$ −13.0738 −0.842158 −0.421079 0.907024i $$-0.638348\pi$$
−0.421079 + 0.907024i $$0.638348\pi$$
$$242$$ 0.224254 0.0144156
$$243$$ −1.00000 −0.0641500
$$244$$ −11.0884 −0.709862
$$245$$ 0 0
$$246$$ −4.70052 −0.299694
$$247$$ −24.6253 −1.56687
$$248$$ −4.00000 −0.254000
$$249$$ −6.38787 −0.404815
$$250$$ 0 0
$$251$$ −5.02302 −0.317050 −0.158525 0.987355i $$-0.550674\pi$$
−0.158525 + 0.987355i $$0.550674\pi$$
$$252$$ −2.96239 −0.186613
$$253$$ −3.35026 −0.210629
$$254$$ −0.186642 −0.0117110
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −19.4010 −1.21020 −0.605102 0.796148i $$-0.706868\pi$$
−0.605102 + 0.796148i $$0.706868\pi$$
$$258$$ −10.3127 −0.642038
$$259$$ −22.5501 −1.40119
$$260$$ 0 0
$$261$$ 7.73813 0.478979
$$262$$ 7.92478 0.489594
$$263$$ −15.4763 −0.954308 −0.477154 0.878820i $$-0.658332\pi$$
−0.477154 + 0.878820i $$0.658332\pi$$
$$264$$ 3.35026 0.206194
$$265$$ 0 0
$$266$$ 14.7005 0.901347
$$267$$ −9.92478 −0.607387
$$268$$ 7.61213 0.464985
$$269$$ −20.2130 −1.23241 −0.616204 0.787587i $$-0.711330\pi$$
−0.616204 + 0.787587i $$0.711330\pi$$
$$270$$ 0 0
$$271$$ 30.3996 1.84665 0.923323 0.384024i $$-0.125462\pi$$
0.923323 + 0.384024i $$0.125462\pi$$
$$272$$ −1.35026 −0.0818716
$$273$$ 14.7005 0.889716
$$274$$ −9.19982 −0.555781
$$275$$ 0 0
$$276$$ −1.00000 −0.0601929
$$277$$ −20.8119 −1.25047 −0.625234 0.780437i $$-0.714997\pi$$
−0.625234 + 0.780437i $$0.714997\pi$$
$$278$$ 8.62530 0.517311
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 19.8496 1.18413 0.592063 0.805892i $$-0.298314\pi$$
0.592063 + 0.805892i $$0.298314\pi$$
$$282$$ 3.22425 0.192002
$$283$$ −12.3879 −0.736383 −0.368191 0.929750i $$-0.620023\pi$$
−0.368191 + 0.929750i $$0.620023\pi$$
$$284$$ −2.18664 −0.129753
$$285$$ 0 0
$$286$$ −16.6253 −0.983075
$$287$$ −13.9248 −0.821954
$$288$$ 1.00000 0.0589256
$$289$$ −15.1768 −0.892753
$$290$$ 0 0
$$291$$ 12.8872 0.755459
$$292$$ 9.92478 0.580804
$$293$$ −9.03761 −0.527983 −0.263991 0.964525i $$-0.585039\pi$$
−0.263991 + 0.964525i $$0.585039\pi$$
$$294$$ −1.77575 −0.103564
$$295$$ 0 0
$$296$$ 7.61213 0.442446
$$297$$ 3.35026 0.194402
$$298$$ 4.64974 0.269352
$$299$$ 4.96239 0.286982
$$300$$ 0 0
$$301$$ −30.5501 −1.76088
$$302$$ 10.7005 0.615746
$$303$$ −4.26187 −0.244838
$$304$$ −4.96239 −0.284613
$$305$$ 0 0
$$306$$ −1.35026 −0.0771893
$$307$$ 30.5501 1.74359 0.871793 0.489875i $$-0.162958\pi$$
0.871793 + 0.489875i $$0.162958\pi$$
$$308$$ 9.92478 0.565517
$$309$$ −0.261865 −0.0148970
$$310$$ 0 0
$$311$$ 14.4387 0.818741 0.409371 0.912368i $$-0.365748\pi$$
0.409371 + 0.912368i $$0.365748\pi$$
$$312$$ −4.96239 −0.280940
$$313$$ −27.9610 −1.58045 −0.790224 0.612818i $$-0.790036\pi$$
−0.790224 + 0.612818i $$0.790036\pi$$
$$314$$ 17.0132 0.960109
$$315$$ 0 0
$$316$$ −4.12601 −0.232106
$$317$$ −1.47627 −0.0829156 −0.0414578 0.999140i $$-0.513200\pi$$
−0.0414578 + 0.999140i $$0.513200\pi$$
$$318$$ −6.96239 −0.390431
$$319$$ −25.9248 −1.45151
$$320$$ 0 0
$$321$$ −9.08840 −0.507265
$$322$$ −2.96239 −0.165087
$$323$$ 6.70052 0.372827
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −12.6253 −0.699251
$$327$$ −18.9380 −1.04727
$$328$$ 4.70052 0.259543
$$329$$ 9.55149 0.526591
$$330$$ 0 0
$$331$$ 23.1754 1.27383 0.636917 0.770932i $$-0.280209\pi$$
0.636917 + 0.770932i $$0.280209\pi$$
$$332$$ 6.38787 0.350580
$$333$$ 7.61213 0.417142
$$334$$ 24.6253 1.34744
$$335$$ 0 0
$$336$$ 2.96239 0.161612
$$337$$ 21.0376 1.14599 0.572996 0.819558i $$-0.305781\pi$$
0.572996 + 0.819558i $$0.305781\pi$$
$$338$$ 11.6253 0.632333
$$339$$ −6.64974 −0.361164
$$340$$ 0 0
$$341$$ 13.4010 0.725707
$$342$$ −4.96239 −0.268335
$$343$$ 15.4763 0.835640
$$344$$ 10.3127 0.556021
$$345$$ 0 0
$$346$$ 4.44851 0.239153
$$347$$ −3.37470 −0.181163 −0.0905817 0.995889i $$-0.528873\pi$$
−0.0905817 + 0.995889i $$0.528873\pi$$
$$348$$ −7.73813 −0.414808
$$349$$ −4.44851 −0.238123 −0.119062 0.992887i $$-0.537989\pi$$
−0.119062 + 0.992887i $$0.537989\pi$$
$$350$$ 0 0
$$351$$ −4.96239 −0.264873
$$352$$ −3.35026 −0.178570
$$353$$ 35.4010 1.88421 0.942104 0.335321i $$-0.108845\pi$$
0.942104 + 0.335321i $$0.108845\pi$$
$$354$$ −1.22425 −0.0650684
$$355$$ 0 0
$$356$$ 9.92478 0.526012
$$357$$ −4.00000 −0.211702
$$358$$ −13.8496 −0.731972
$$359$$ −34.5501 −1.82348 −0.911742 0.410764i $$-0.865262\pi$$
−0.911742 + 0.410764i $$0.865262\pi$$
$$360$$ 0 0
$$361$$ 5.62530 0.296068
$$362$$ −22.6859 −1.19235
$$363$$ −0.224254 −0.0117703
$$364$$ −14.7005 −0.770517
$$365$$ 0 0
$$366$$ 11.0884 0.579600
$$367$$ −22.8119 −1.19077 −0.595387 0.803439i $$-0.703001\pi$$
−0.595387 + 0.803439i $$0.703001\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 4.70052 0.244699
$$370$$ 0 0
$$371$$ −20.6253 −1.07081
$$372$$ 4.00000 0.207390
$$373$$ −10.8364 −0.561087 −0.280543 0.959841i $$-0.590515\pi$$
−0.280543 + 0.959841i $$0.590515\pi$$
$$374$$ 4.52373 0.233917
$$375$$ 0 0
$$376$$ −3.22425 −0.166278
$$377$$ 38.3996 1.97768
$$378$$ 2.96239 0.152369
$$379$$ −32.4387 −1.66626 −0.833131 0.553076i $$-0.813454\pi$$
−0.833131 + 0.553076i $$0.813454\pi$$
$$380$$ 0 0
$$381$$ 0.186642 0.00956198
$$382$$ −19.3258 −0.988795
$$383$$ 32.9986 1.68615 0.843074 0.537797i $$-0.180743\pi$$
0.843074 + 0.537797i $$0.180743\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 5.92478 0.301563
$$387$$ 10.3127 0.524221
$$388$$ −12.8872 −0.654247
$$389$$ 29.1246 1.47668 0.738338 0.674431i $$-0.235611\pi$$
0.738338 + 0.674431i $$0.235611\pi$$
$$390$$ 0 0
$$391$$ −1.35026 −0.0682857
$$392$$ 1.77575 0.0896887
$$393$$ −7.92478 −0.399752
$$394$$ −8.07522 −0.406824
$$395$$ 0 0
$$396$$ −3.35026 −0.168357
$$397$$ −13.7381 −0.689497 −0.344749 0.938695i $$-0.612036\pi$$
−0.344749 + 0.938695i $$0.612036\pi$$
$$398$$ 15.9756 0.800783
$$399$$ −14.7005 −0.735947
$$400$$ 0 0
$$401$$ −1.44992 −0.0724057 −0.0362028 0.999344i $$-0.511526\pi$$
−0.0362028 + 0.999344i $$0.511526\pi$$
$$402$$ −7.61213 −0.379658
$$403$$ −19.8496 −0.988777
$$404$$ 4.26187 0.212036
$$405$$ 0 0
$$406$$ −22.9234 −1.13767
$$407$$ −25.5026 −1.26412
$$408$$ 1.35026 0.0668479
$$409$$ −17.8496 −0.882604 −0.441302 0.897359i $$-0.645483\pi$$
−0.441302 + 0.897359i $$0.645483\pi$$
$$410$$ 0 0
$$411$$ 9.19982 0.453793
$$412$$ 0.261865 0.0129012
$$413$$ −3.62672 −0.178459
$$414$$ 1.00000 0.0491473
$$415$$ 0 0
$$416$$ 4.96239 0.243301
$$417$$ −8.62530 −0.422383
$$418$$ 16.6253 0.813170
$$419$$ 17.9003 0.874489 0.437244 0.899343i $$-0.355954\pi$$
0.437244 + 0.899343i $$0.355954\pi$$
$$420$$ 0 0
$$421$$ 7.61213 0.370992 0.185496 0.982645i $$-0.440611\pi$$
0.185496 + 0.982645i $$0.440611\pi$$
$$422$$ 21.7743 1.05996
$$423$$ −3.22425 −0.156769
$$424$$ 6.96239 0.338123
$$425$$ 0 0
$$426$$ 2.18664 0.105943
$$427$$ 32.8481 1.58963
$$428$$ 9.08840 0.439304
$$429$$ 16.6253 0.802677
$$430$$ 0 0
$$431$$ −22.9525 −1.10558 −0.552792 0.833319i $$-0.686438\pi$$
−0.552792 + 0.833319i $$0.686438\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −33.7645 −1.62262 −0.811309 0.584618i $$-0.801244\pi$$
−0.811309 + 0.584618i $$0.801244\pi$$
$$434$$ 11.8496 0.568797
$$435$$ 0 0
$$436$$ 18.9380 0.906963
$$437$$ −4.96239 −0.237383
$$438$$ −9.92478 −0.474224
$$439$$ −23.4763 −1.12046 −0.560231 0.828337i $$-0.689287\pi$$
−0.560231 + 0.828337i $$0.689287\pi$$
$$440$$ 0 0
$$441$$ 1.77575 0.0845593
$$442$$ −6.70052 −0.318711
$$443$$ 30.5501 1.45148 0.725739 0.687970i $$-0.241498\pi$$
0.725739 + 0.687970i $$0.241498\pi$$
$$444$$ −7.61213 −0.361256
$$445$$ 0 0
$$446$$ 3.81336 0.180568
$$447$$ −4.64974 −0.219925
$$448$$ −2.96239 −0.139960
$$449$$ 32.7005 1.54323 0.771617 0.636088i $$-0.219448\pi$$
0.771617 + 0.636088i $$0.219448\pi$$
$$450$$ 0 0
$$451$$ −15.7480 −0.741544
$$452$$ 6.64974 0.312777
$$453$$ −10.7005 −0.502754
$$454$$ 16.9380 0.794937
$$455$$ 0 0
$$456$$ 4.96239 0.232385
$$457$$ −5.81336 −0.271937 −0.135969 0.990713i $$-0.543415\pi$$
−0.135969 + 0.990713i $$0.543415\pi$$
$$458$$ −26.9380 −1.25873
$$459$$ 1.35026 0.0630248
$$460$$ 0 0
$$461$$ −23.9902 −1.11733 −0.558666 0.829392i $$-0.688687\pi$$
−0.558666 + 0.829392i $$0.688687\pi$$
$$462$$ −9.92478 −0.461742
$$463$$ 35.3620 1.64341 0.821706 0.569911i $$-0.193022\pi$$
0.821706 + 0.569911i $$0.193022\pi$$
$$464$$ 7.73813 0.359234
$$465$$ 0 0
$$466$$ 27.4010 1.26933
$$467$$ 1.98541 0.0918739 0.0459369 0.998944i $$-0.485373\pi$$
0.0459369 + 0.998944i $$0.485373\pi$$
$$468$$ 4.96239 0.229386
$$469$$ −22.5501 −1.04127
$$470$$ 0 0
$$471$$ −17.0132 −0.783926
$$472$$ 1.22425 0.0563508
$$473$$ −34.5501 −1.58861
$$474$$ 4.12601 0.189514
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ 6.96239 0.318786
$$478$$ 7.48612 0.342407
$$479$$ 11.0738 0.505975 0.252988 0.967470i $$-0.418587\pi$$
0.252988 + 0.967470i $$0.418587\pi$$
$$480$$ 0 0
$$481$$ 37.7743 1.72236
$$482$$ −13.0738 −0.595496
$$483$$ 2.96239 0.134793
$$484$$ 0.224254 0.0101934
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −17.4372 −0.790157 −0.395078 0.918647i $$-0.629283\pi$$
−0.395078 + 0.918647i $$0.629283\pi$$
$$488$$ −11.0884 −0.501948
$$489$$ 12.6253 0.570936
$$490$$ 0 0
$$491$$ 14.8773 0.671404 0.335702 0.941968i $$-0.391027\pi$$
0.335702 + 0.941968i $$0.391027\pi$$
$$492$$ −4.70052 −0.211916
$$493$$ −10.4485 −0.470577
$$494$$ −24.6253 −1.10794
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 6.47768 0.290564
$$498$$ −6.38787 −0.286247
$$499$$ −16.1016 −0.720805 −0.360403 0.932797i $$-0.617361\pi$$
−0.360403 + 0.932797i $$0.617361\pi$$
$$500$$ 0 0
$$501$$ −24.6253 −1.10018
$$502$$ −5.02302 −0.224188
$$503$$ −38.8021 −1.73010 −0.865050 0.501686i $$-0.832713\pi$$
−0.865050 + 0.501686i $$0.832713\pi$$
$$504$$ −2.96239 −0.131955
$$505$$ 0 0
$$506$$ −3.35026 −0.148937
$$507$$ −11.6253 −0.516298
$$508$$ −0.186642 −0.00828091
$$509$$ 23.4372 1.03884 0.519419 0.854520i $$-0.326148\pi$$
0.519419 + 0.854520i $$0.326148\pi$$
$$510$$ 0 0
$$511$$ −29.4010 −1.30063
$$512$$ 1.00000 0.0441942
$$513$$ 4.96239 0.219095
$$514$$ −19.4010 −0.855743
$$515$$ 0 0
$$516$$ −10.3127 −0.453989
$$517$$ 10.8021 0.475076
$$518$$ −22.5501 −0.990794
$$519$$ −4.44851 −0.195268
$$520$$ 0 0
$$521$$ 21.1490 0.926556 0.463278 0.886213i $$-0.346673\pi$$
0.463278 + 0.886213i $$0.346673\pi$$
$$522$$ 7.73813 0.338689
$$523$$ −11.7626 −0.514341 −0.257171 0.966366i $$-0.582790\pi$$
−0.257171 + 0.966366i $$0.582790\pi$$
$$524$$ 7.92478 0.346196
$$525$$ 0 0
$$526$$ −15.4763 −0.674797
$$527$$ 5.40105 0.235273
$$528$$ 3.35026 0.145801
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 1.22425 0.0531281
$$532$$ 14.7005 0.637349
$$533$$ 23.3258 1.01035
$$534$$ −9.92478 −0.429487
$$535$$ 0 0
$$536$$ 7.61213 0.328794
$$537$$ 13.8496 0.597652
$$538$$ −20.2130 −0.871444
$$539$$ −5.94921 −0.256251
$$540$$ 0 0
$$541$$ −21.1754 −0.910401 −0.455200 0.890389i $$-0.650432\pi$$
−0.455200 + 0.890389i $$0.650432\pi$$
$$542$$ 30.3996 1.30578
$$543$$ 22.6859 0.973547
$$544$$ −1.35026 −0.0578920
$$545$$ 0 0
$$546$$ 14.7005 0.629124
$$547$$ −5.29948 −0.226589 −0.113295 0.993561i $$-0.536140\pi$$
−0.113295 + 0.993561i $$0.536140\pi$$
$$548$$ −9.19982 −0.392997
$$549$$ −11.0884 −0.473241
$$550$$ 0 0
$$551$$ −38.3996 −1.63588
$$552$$ −1.00000 −0.0425628
$$553$$ 12.2228 0.519768
$$554$$ −20.8119 −0.884215
$$555$$ 0 0
$$556$$ 8.62530 0.365794
$$557$$ 7.99015 0.338554 0.169277 0.985569i $$-0.445857\pi$$
0.169277 + 0.985569i $$0.445857\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ 51.1754 2.16449
$$560$$ 0 0
$$561$$ −4.52373 −0.190992
$$562$$ 19.8496 0.837303
$$563$$ −15.1636 −0.639070 −0.319535 0.947574i $$-0.603527\pi$$
−0.319535 + 0.947574i $$0.603527\pi$$
$$564$$ 3.22425 0.135766
$$565$$ 0 0
$$566$$ −12.3879 −0.520701
$$567$$ −2.96239 −0.124409
$$568$$ −2.18664 −0.0917495
$$569$$ −25.5223 −1.06995 −0.534976 0.844868i $$-0.679679\pi$$
−0.534976 + 0.844868i $$0.679679\pi$$
$$570$$ 0 0
$$571$$ 6.76590 0.283144 0.141572 0.989928i $$-0.454784\pi$$
0.141572 + 0.989928i $$0.454784\pi$$
$$572$$ −16.6253 −0.695139
$$573$$ 19.3258 0.807348
$$574$$ −13.9248 −0.581209
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −4.00000 −0.166522 −0.0832611 0.996528i $$-0.526534\pi$$
−0.0832611 + 0.996528i $$0.526534\pi$$
$$578$$ −15.1768 −0.631271
$$579$$ −5.92478 −0.246225
$$580$$ 0 0
$$581$$ −18.9234 −0.785073
$$582$$ 12.8872 0.534190
$$583$$ −23.3258 −0.966057
$$584$$ 9.92478 0.410690
$$585$$ 0 0
$$586$$ −9.03761 −0.373340
$$587$$ 12.8773 0.531504 0.265752 0.964041i $$-0.414380\pi$$
0.265752 + 0.964041i $$0.414380\pi$$
$$588$$ −1.77575 −0.0732305
$$589$$ 19.8496 0.817887
$$590$$ 0 0
$$591$$ 8.07522 0.332170
$$592$$ 7.61213 0.312856
$$593$$ −16.2981 −0.669281 −0.334641 0.942346i $$-0.608615\pi$$
−0.334641 + 0.942346i $$0.608615\pi$$
$$594$$ 3.35026 0.137463
$$595$$ 0 0
$$596$$ 4.64974 0.190461
$$597$$ −15.9756 −0.653836
$$598$$ 4.96239 0.202927
$$599$$ 9.91493 0.405113 0.202556 0.979271i $$-0.435075\pi$$
0.202556 + 0.979271i $$0.435075\pi$$
$$600$$ 0 0
$$601$$ −7.40105 −0.301895 −0.150948 0.988542i $$-0.548232\pi$$
−0.150948 + 0.988542i $$0.548232\pi$$
$$602$$ −30.5501 −1.24513
$$603$$ 7.61213 0.309990
$$604$$ 10.7005 0.435398
$$605$$ 0 0
$$606$$ −4.26187 −0.173126
$$607$$ 13.9610 0.566658 0.283329 0.959023i $$-0.408561\pi$$
0.283329 + 0.959023i $$0.408561\pi$$
$$608$$ −4.96239 −0.201251
$$609$$ 22.9234 0.928902
$$610$$ 0 0
$$611$$ −16.0000 −0.647291
$$612$$ −1.35026 −0.0545811
$$613$$ 41.3865 1.67158 0.835792 0.549047i $$-0.185009\pi$$
0.835792 + 0.549047i $$0.185009\pi$$
$$614$$ 30.5501 1.23290
$$615$$ 0 0
$$616$$ 9.92478 0.399881
$$617$$ −29.3014 −1.17963 −0.589815 0.807539i $$-0.700799\pi$$
−0.589815 + 0.807539i $$0.700799\pi$$
$$618$$ −0.261865 −0.0105338
$$619$$ −19.0376 −0.765186 −0.382593 0.923917i $$-0.624969\pi$$
−0.382593 + 0.923917i $$0.624969\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ 14.4387 0.578937
$$623$$ −29.4010 −1.17793
$$624$$ −4.96239 −0.198655
$$625$$ 0 0
$$626$$ −27.9610 −1.11755
$$627$$ −16.6253 −0.663951
$$628$$ 17.0132 0.678900
$$629$$ −10.2784 −0.409825
$$630$$ 0 0
$$631$$ 12.4993 0.497589 0.248794 0.968556i $$-0.419966\pi$$
0.248794 + 0.968556i $$0.419966\pi$$
$$632$$ −4.12601 −0.164124
$$633$$ −21.7743 −0.865452
$$634$$ −1.47627 −0.0586302
$$635$$ 0 0
$$636$$ −6.96239 −0.276077
$$637$$ 8.81194 0.349142
$$638$$ −25.9248 −1.02637
$$639$$ −2.18664 −0.0865022
$$640$$ 0 0
$$641$$ −34.0263 −1.34396 −0.671980 0.740569i $$-0.734556\pi$$
−0.671980 + 0.740569i $$0.734556\pi$$
$$642$$ −9.08840 −0.358690
$$643$$ 7.34041 0.289478 0.144739 0.989470i $$-0.453766\pi$$
0.144739 + 0.989470i $$0.453766\pi$$
$$644$$ −2.96239 −0.116734
$$645$$ 0 0
$$646$$ 6.70052 0.263629
$$647$$ 14.9525 0.587845 0.293922 0.955829i $$-0.405039\pi$$
0.293922 + 0.955829i $$0.405039\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −4.10157 −0.161001
$$650$$ 0 0
$$651$$ −11.8496 −0.464421
$$652$$ −12.6253 −0.494445
$$653$$ −20.4485 −0.800212 −0.400106 0.916469i $$-0.631027\pi$$
−0.400106 + 0.916469i $$0.631027\pi$$
$$654$$ −18.9380 −0.740532
$$655$$ 0 0
$$656$$ 4.70052 0.183525
$$657$$ 9.92478 0.387202
$$658$$ 9.55149 0.372356
$$659$$ 38.4241 1.49679 0.748395 0.663254i $$-0.230825\pi$$
0.748395 + 0.663254i $$0.230825\pi$$
$$660$$ 0 0
$$661$$ −39.3112 −1.52903 −0.764515 0.644606i $$-0.777021\pi$$
−0.764515 + 0.644606i $$0.777021\pi$$
$$662$$ 23.1754 0.900737
$$663$$ 6.70052 0.260227
$$664$$ 6.38787 0.247898
$$665$$ 0 0
$$666$$ 7.61213 0.294964
$$667$$ 7.73813 0.299622
$$668$$ 24.6253 0.952781
$$669$$ −3.81336 −0.147433
$$670$$ 0 0
$$671$$ 37.1490 1.43412
$$672$$ 2.96239 0.114277
$$673$$ −25.5515 −0.984938 −0.492469 0.870330i $$-0.663905\pi$$
−0.492469 + 0.870330i $$0.663905\pi$$
$$674$$ 21.0376 0.810339
$$675$$ 0 0
$$676$$ 11.6253 0.447127
$$677$$ −36.2130 −1.39178 −0.695889 0.718149i $$-0.744990\pi$$
−0.695889 + 0.718149i $$0.744990\pi$$
$$678$$ −6.64974 −0.255382
$$679$$ 38.1768 1.46509
$$680$$ 0 0
$$681$$ −16.9380 −0.649064
$$682$$ 13.4010 0.513153
$$683$$ 15.4763 0.592183 0.296092 0.955160i $$-0.404317\pi$$
0.296092 + 0.955160i $$0.404317\pi$$
$$684$$ −4.96239 −0.189742
$$685$$ 0 0
$$686$$ 15.4763 0.590887
$$687$$ 26.9380 1.02775
$$688$$ 10.3127 0.393166
$$689$$ 34.5501 1.31625
$$690$$ 0 0
$$691$$ −37.6531 −1.43239 −0.716195 0.697900i $$-0.754118\pi$$
−0.716195 + 0.697900i $$0.754118\pi$$
$$692$$ 4.44851 0.169107
$$693$$ 9.92478 0.377011
$$694$$ −3.37470 −0.128102
$$695$$ 0 0
$$696$$ −7.73813 −0.293313
$$697$$ −6.34694 −0.240407
$$698$$ −4.44851 −0.168378
$$699$$ −27.4010 −1.03640
$$700$$ 0 0
$$701$$ 28.3488 1.07072 0.535361 0.844624i $$-0.320176\pi$$
0.535361 + 0.844624i $$0.320176\pi$$
$$702$$ −4.96239 −0.187293
$$703$$ −37.7743 −1.42469
$$704$$ −3.35026 −0.126268
$$705$$ 0 0
$$706$$ 35.4010 1.33234
$$707$$ −12.6253 −0.474823
$$708$$ −1.22425 −0.0460103
$$709$$ −43.5633 −1.63605 −0.818026 0.575181i $$-0.804932\pi$$
−0.818026 + 0.575181i $$0.804932\pi$$
$$710$$ 0 0
$$711$$ −4.12601 −0.154737
$$712$$ 9.92478 0.371947
$$713$$ −4.00000 −0.149801
$$714$$ −4.00000 −0.149696
$$715$$ 0 0
$$716$$ −13.8496 −0.517582
$$717$$ −7.48612 −0.279574
$$718$$ −34.5501 −1.28940
$$719$$ 32.3634 1.20695 0.603476 0.797381i $$-0.293782\pi$$
0.603476 + 0.797381i $$0.293782\pi$$
$$720$$ 0 0
$$721$$ −0.775746 −0.0288903
$$722$$ 5.62530 0.209352
$$723$$ 13.0738 0.486220
$$724$$ −22.6859 −0.843116
$$725$$ 0 0
$$726$$ −0.224254 −0.00832284
$$727$$ 6.96239 0.258221 0.129110 0.991630i $$-0.458788\pi$$
0.129110 + 0.991630i $$0.458788\pi$$
$$728$$ −14.7005 −0.544838
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −13.9248 −0.515026
$$732$$ 11.0884 0.409839
$$733$$ −9.68735 −0.357810 −0.178905 0.983866i $$-0.557256\pi$$
−0.178905 + 0.983866i $$0.557256\pi$$
$$734$$ −22.8119 −0.842004
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ −25.5026 −0.939401
$$738$$ 4.70052 0.173029
$$739$$ −42.9234 −1.57896 −0.789481 0.613775i $$-0.789650\pi$$
−0.789481 + 0.613775i $$0.789650\pi$$
$$740$$ 0 0
$$741$$ 24.6253 0.904633
$$742$$ −20.6253 −0.757179
$$743$$ 16.9986 0.623618 0.311809 0.950145i $$-0.399065\pi$$
0.311809 + 0.950145i $$0.399065\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ −10.8364 −0.396748
$$747$$ 6.38787 0.233720
$$748$$ 4.52373 0.165404
$$749$$ −26.9234 −0.983758
$$750$$ 0 0
$$751$$ −17.9003 −0.653193 −0.326596 0.945164i $$-0.605902\pi$$
−0.326596 + 0.945164i $$0.605902\pi$$
$$752$$ −3.22425 −0.117576
$$753$$ 5.02302 0.183049
$$754$$ 38.3996 1.39843
$$755$$ 0 0
$$756$$ 2.96239 0.107741
$$757$$ −12.3879 −0.450245 −0.225122 0.974330i $$-0.572278\pi$$
−0.225122 + 0.974330i $$0.572278\pi$$
$$758$$ −32.4387 −1.17823
$$759$$ 3.35026 0.121607
$$760$$ 0 0
$$761$$ 38.5764 1.39839 0.699197 0.714929i $$-0.253541\pi$$
0.699197 + 0.714929i $$0.253541\pi$$
$$762$$ 0.186642 0.00676134
$$763$$ −56.1016 −2.03101
$$764$$ −19.3258 −0.699184
$$765$$ 0 0
$$766$$ 32.9986 1.19229
$$767$$ 6.07522 0.219364
$$768$$ −1.00000 −0.0360844
$$769$$ 34.2228 1.23411 0.617054 0.786921i $$-0.288326\pi$$
0.617054 + 0.786921i $$0.288326\pi$$
$$770$$ 0 0
$$771$$ 19.4010 0.698712
$$772$$ 5.92478 0.213237
$$773$$ 13.5613 0.487768 0.243884 0.969804i $$-0.421578\pi$$
0.243884 + 0.969804i $$0.421578\pi$$
$$774$$ 10.3127 0.370681
$$775$$ 0 0
$$776$$ −12.8872 −0.462622
$$777$$ 22.5501 0.808980
$$778$$ 29.1246 1.04417
$$779$$ −23.3258 −0.835734
$$780$$ 0 0
$$781$$ 7.32582 0.262139
$$782$$ −1.35026 −0.0482853
$$783$$ −7.73813 −0.276538
$$784$$ 1.77575 0.0634195
$$785$$ 0 0
$$786$$ −7.92478 −0.282667
$$787$$ −37.0132 −1.31938 −0.659689 0.751539i $$-0.729312\pi$$
−0.659689 + 0.751539i $$0.729312\pi$$
$$788$$ −8.07522 −0.287668
$$789$$ 15.4763 0.550970
$$790$$ 0 0
$$791$$ −19.6991 −0.700420
$$792$$ −3.35026 −0.119046
$$793$$ −55.0249 −1.95399
$$794$$ −13.7381 −0.487548
$$795$$ 0 0
$$796$$ 15.9756 0.566239
$$797$$ −47.6893 −1.68924 −0.844620 0.535366i $$-0.820174\pi$$
−0.844620 + 0.535366i $$0.820174\pi$$
$$798$$ −14.7005 −0.520393
$$799$$ 4.35359 0.154019
$$800$$ 0 0
$$801$$ 9.92478 0.350675
$$802$$ −1.44992 −0.0511985
$$803$$ −33.2506 −1.17339
$$804$$ −7.61213 −0.268459
$$805$$ 0 0
$$806$$ −19.8496 −0.699171
$$807$$ 20.2130 0.711531
$$808$$ 4.26187 0.149932
$$809$$ −9.17538 −0.322589 −0.161295 0.986906i $$-0.551567\pi$$
−0.161295 + 0.986906i $$0.551567\pi$$
$$810$$ 0 0
$$811$$ −38.5501 −1.35368 −0.676838 0.736132i $$-0.736650\pi$$
−0.676838 + 0.736132i $$0.736650\pi$$
$$812$$ −22.9234 −0.804452
$$813$$ −30.3996 −1.06616
$$814$$ −25.5026 −0.893866
$$815$$ 0 0
$$816$$ 1.35026 0.0472686
$$817$$ −51.1754 −1.79040
$$818$$ −17.8496 −0.624095
$$819$$ −14.7005 −0.513678
$$820$$ 0 0
$$821$$ 43.4372 1.51597 0.757985 0.652272i $$-0.226184\pi$$
0.757985 + 0.652272i $$0.226184\pi$$
$$822$$ 9.19982 0.320880
$$823$$ 3.51247 0.122437 0.0612184 0.998124i $$-0.480501\pi$$
0.0612184 + 0.998124i $$0.480501\pi$$
$$824$$ 0.261865 0.00912250
$$825$$ 0 0
$$826$$ −3.62672 −0.126190
$$827$$ 9.56325 0.332547 0.166273 0.986080i $$-0.446827\pi$$
0.166273 + 0.986080i $$0.446827\pi$$
$$828$$ 1.00000 0.0347524
$$829$$ 30.5764 1.06196 0.530982 0.847383i $$-0.321823\pi$$
0.530982 + 0.847383i $$0.321823\pi$$
$$830$$ 0 0
$$831$$ 20.8119 0.721958
$$832$$ 4.96239 0.172040
$$833$$ −2.39772 −0.0830762
$$834$$ −8.62530 −0.298670
$$835$$ 0 0
$$836$$ 16.6253 0.574998
$$837$$ 4.00000 0.138260
$$838$$ 17.9003 0.618357
$$839$$ 22.1768 0.765628 0.382814 0.923825i $$-0.374955\pi$$
0.382814 + 0.923825i $$0.374955\pi$$
$$840$$ 0 0
$$841$$ 30.8787 1.06478
$$842$$ 7.61213 0.262331
$$843$$ −19.8496 −0.683655
$$844$$ 21.7743 0.749503
$$845$$ 0 0
$$846$$ −3.22425 −0.110852
$$847$$ −0.664327 −0.0228265
$$848$$ 6.96239 0.239089
$$849$$ 12.3879 0.425151
$$850$$ 0 0
$$851$$ 7.61213 0.260940
$$852$$ 2.18664 0.0749131
$$853$$ −29.7381 −1.01821 −0.509107 0.860703i $$-0.670024\pi$$
−0.509107 + 0.860703i $$0.670024\pi$$
$$854$$ 32.8481 1.12404
$$855$$ 0 0
$$856$$ 9.08840 0.310635
$$857$$ 22.2520 0.760114 0.380057 0.924963i $$-0.375904\pi$$
0.380057 + 0.924963i $$0.375904\pi$$
$$858$$ 16.6253 0.567578
$$859$$ 11.0738 0.377833 0.188917 0.981993i $$-0.439502\pi$$
0.188917 + 0.981993i $$0.439502\pi$$
$$860$$ 0 0
$$861$$ 13.9248 0.474555
$$862$$ −22.9525 −0.781767
$$863$$ −22.8218 −0.776863 −0.388431 0.921478i $$-0.626983\pi$$
−0.388431 + 0.921478i $$0.626983\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −33.7645 −1.14736
$$867$$ 15.1768 0.515431
$$868$$ 11.8496 0.402200
$$869$$ 13.8232 0.468920
$$870$$ 0 0
$$871$$ 37.7743 1.27993
$$872$$ 18.9380 0.641320
$$873$$ −12.8872 −0.436164
$$874$$ −4.96239 −0.167855
$$875$$ 0 0
$$876$$ −9.92478 −0.335327
$$877$$ −8.81194 −0.297558 −0.148779 0.988870i $$-0.547534\pi$$
−0.148779 + 0.988870i $$0.547534\pi$$
$$878$$ −23.4763 −0.792286
$$879$$ 9.03761 0.304831
$$880$$ 0 0
$$881$$ 21.4010 0.721020 0.360510 0.932755i $$-0.382603\pi$$
0.360510 + 0.932755i $$0.382603\pi$$
$$882$$ 1.77575 0.0597925
$$883$$ 6.17679 0.207866 0.103933 0.994584i $$-0.466857\pi$$
0.103933 + 0.994584i $$0.466857\pi$$
$$884$$ −6.70052 −0.225363
$$885$$ 0 0
$$886$$ 30.5501 1.02635
$$887$$ −35.5975 −1.19525 −0.597624 0.801776i $$-0.703889\pi$$
−0.597624 + 0.801776i $$0.703889\pi$$
$$888$$ −7.61213 −0.255446
$$889$$ 0.552907 0.0185439
$$890$$ 0 0
$$891$$ −3.35026 −0.112238
$$892$$ 3.81336 0.127681
$$893$$ 16.0000 0.535420
$$894$$ −4.64974 −0.155511
$$895$$ 0 0
$$896$$ −2.96239 −0.0989665
$$897$$ −4.96239 −0.165689
$$898$$ 32.7005 1.09123
$$899$$ −30.9525 −1.03232
$$900$$ 0 0
$$901$$ −9.40105 −0.313194
$$902$$ −15.7480 −0.524351
$$903$$ 30.5501 1.01664
$$904$$ 6.64974 0.221167
$$905$$ 0 0
$$906$$ −10.7005 −0.355501
$$907$$ 5.53690 0.183850 0.0919249 0.995766i $$-0.470698\pi$$
0.0919249 + 0.995766i $$0.470698\pi$$
$$908$$ 16.9380 0.562106
$$909$$ 4.26187 0.141357
$$910$$ 0 0
$$911$$ −15.4763 −0.512752 −0.256376 0.966577i $$-0.582528\pi$$
−0.256376 + 0.966577i $$0.582528\pi$$
$$912$$ 4.96239 0.164321
$$913$$ −21.4010 −0.708271
$$914$$ −5.81336 −0.192289
$$915$$ 0 0
$$916$$ −26.9380 −0.890055
$$917$$ −23.4763 −0.775255
$$918$$ 1.35026 0.0445653
$$919$$ −5.17347 −0.170657 −0.0853285 0.996353i $$-0.527194\pi$$
−0.0853285 + 0.996353i $$0.527194\pi$$
$$920$$ 0 0
$$921$$ −30.5501 −1.00666
$$922$$ −23.9902 −0.790074
$$923$$ −10.8510 −0.357164
$$924$$ −9.92478 −0.326501
$$925$$ 0 0
$$926$$ 35.3620 1.16207
$$927$$ 0.261865 0.00860078
$$928$$ 7.73813 0.254017
$$929$$ −44.3996 −1.45670 −0.728352 0.685203i $$-0.759714\pi$$
−0.728352 + 0.685203i $$0.759714\pi$$
$$930$$ 0 0
$$931$$ −8.81194 −0.288800
$$932$$ 27.4010 0.897551
$$933$$ −14.4387 −0.472700
$$934$$ 1.98541 0.0649647
$$935$$ 0 0
$$936$$ 4.96239 0.162201
$$937$$ 3.99015 0.130353 0.0651763 0.997874i $$-0.479239\pi$$
0.0651763 + 0.997874i $$0.479239\pi$$
$$938$$ −22.5501 −0.736286
$$939$$ 27.9610 0.912472
$$940$$ 0 0
$$941$$ −2.30280 −0.0750692 −0.0375346 0.999295i $$-0.511950\pi$$
−0.0375346 + 0.999295i $$0.511950\pi$$
$$942$$ −17.0132 −0.554319
$$943$$ 4.70052 0.153070
$$944$$ 1.22425 0.0398461
$$945$$ 0 0
$$946$$ −34.5501 −1.12332
$$947$$ −24.7269 −0.803515 −0.401758 0.915746i $$-0.631601\pi$$
−0.401758 + 0.915746i $$0.631601\pi$$
$$948$$ 4.12601 0.134007
$$949$$ 49.2506 1.59874
$$950$$ 0 0
$$951$$ 1.47627 0.0478713
$$952$$ 4.00000 0.129641
$$953$$ 11.6775 0.378271 0.189136 0.981951i $$-0.439431\pi$$
0.189136 + 0.981951i $$0.439431\pi$$
$$954$$ 6.96239 0.225416
$$955$$ 0 0
$$956$$ 7.48612 0.242118
$$957$$ 25.9248 0.838029
$$958$$ 11.0738 0.357779
$$959$$ 27.2534 0.880059
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 37.7743 1.21789
$$963$$ 9.08840 0.292869
$$964$$ −13.0738 −0.421079
$$965$$ 0 0
$$966$$ 2.96239 0.0953133
$$967$$ −51.2116 −1.64685 −0.823427 0.567423i $$-0.807941\pi$$
−0.823427 + 0.567423i $$0.807941\pi$$
$$968$$ 0.224254 0.00720779
$$969$$ −6.70052 −0.215252
$$970$$ 0 0
$$971$$ 14.8265 0.475806 0.237903 0.971289i $$-0.423540\pi$$
0.237903 + 0.971289i $$0.423540\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −25.5515 −0.819143
$$974$$ −17.4372 −0.558725
$$975$$ 0 0
$$976$$ −11.0884 −0.354931
$$977$$ 16.6958 0.534145 0.267073 0.963676i $$-0.413944\pi$$
0.267073 + 0.963676i $$0.413944\pi$$
$$978$$ 12.6253 0.403713
$$979$$ −33.2506 −1.06269
$$980$$ 0 0
$$981$$ 18.9380 0.604642
$$982$$ 14.8773 0.474754
$$983$$ 30.0263 0.957692 0.478846 0.877899i $$-0.341055\pi$$
0.478846 + 0.877899i $$0.341055\pi$$
$$984$$ −4.70052 −0.149847
$$985$$ 0 0
$$986$$ −10.4485 −0.332748
$$987$$ −9.55149 −0.304027
$$988$$ −24.6253 −0.783435
$$989$$ 10.3127 0.327923
$$990$$ 0 0
$$991$$ −1.40105 −0.0445057 −0.0222529 0.999752i $$-0.507084\pi$$
−0.0222529 + 0.999752i $$0.507084\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ −23.1754 −0.735448
$$994$$ 6.47768 0.205460
$$995$$ 0 0
$$996$$ −6.38787 −0.202408
$$997$$ −13.2144 −0.418504 −0.209252 0.977862i $$-0.567103\pi$$
−0.209252 + 0.977862i $$0.567103\pi$$
$$998$$ −16.1016 −0.509686
$$999$$ −7.61213 −0.240837
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 3450.2.a.bt.1.1 3
5.2 odd 4 690.2.d.c.139.4 yes 6
5.3 odd 4 690.2.d.c.139.1 6
5.4 even 2 3450.2.a.bo.1.3 3
15.2 even 4 2070.2.d.e.829.3 6
15.8 even 4 2070.2.d.e.829.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.d.c.139.1 6 5.3 odd 4
690.2.d.c.139.4 yes 6 5.2 odd 4
2070.2.d.e.829.3 6 15.2 even 4
2070.2.d.e.829.6 6 15.8 even 4
3450.2.a.bo.1.3 3 5.4 even 2
3450.2.a.bt.1.1 3 1.1 even 1 trivial