# Properties

 Label 3450.2.d.d.2899.2 Level $3450$ Weight $2$ Character 3450.2899 Analytic conductor $27.548$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.5483886973$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 690) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2899.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3450.2899 Dual form 3450.2.d.d.2899.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} +6.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -1.00000i q^{18} +2.00000 q^{21} -2.00000i q^{22} -1.00000i q^{23} +1.00000 q^{24} -6.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} -2.00000 q^{29} +1.00000i q^{32} -2.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} -6.00000 q^{39} -6.00000 q^{41} +2.00000i q^{42} +4.00000i q^{43} +2.00000 q^{44} +1.00000 q^{46} +1.00000i q^{48} +3.00000 q^{49} +4.00000 q^{51} -6.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} -2.00000i q^{58} -8.00000 q^{61} +2.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -4.00000i q^{67} +4.00000i q^{68} +1.00000 q^{69} +16.0000 q^{71} +1.00000i q^{72} -6.00000i q^{73} +8.00000 q^{74} +4.00000i q^{77} -6.00000i q^{78} -14.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -14.0000i q^{83} -2.00000 q^{84} -4.00000 q^{86} -2.00000i q^{87} +2.00000i q^{88} +8.00000 q^{89} +12.0000 q^{91} +1.00000i q^{92} -1.00000 q^{96} -6.00000i q^{97} +3.00000i q^{98} +2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 4 q^{11} + 4 q^{14} + 2 q^{16} + 4 q^{21} + 2 q^{24} - 12 q^{26} - 4 q^{29} + 8 q^{34} + 2 q^{36} - 12 q^{39} - 12 q^{41} + 4 q^{44} + 2 q^{46} + 6 q^{49} + 8 q^{51} + 2 q^{54} - 4 q^{56} - 16 q^{61} - 2 q^{64} + 4 q^{66} + 2 q^{69} + 32 q^{71} + 16 q^{74} - 28 q^{79} + 2 q^{81} - 4 q^{84} - 8 q^{86} + 16 q^{89} + 24 q^{91} - 2 q^{96} + 4 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ 1.00000i 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 6.00000i 1.66410i 0.554700 + 0.832050i $$0.312833\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 4.00000i − 0.970143i −0.874475 0.485071i $$-0.838794\pi$$
0.874475 0.485071i $$-0.161206\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ − 2.00000i − 0.426401i
$$23$$ − 1.00000i − 0.208514i
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −6.00000 −1.17670
$$27$$ − 1.00000i − 0.192450i
$$28$$ 2.00000i 0.377964i
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 2.00000i − 0.348155i
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 0 0
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 2.00000i 0.308607i
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ − 6.00000i − 0.832050i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ 0 0
$$58$$ − 2.00000i − 0.262613i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 0 0
$$63$$ 2.00000i 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 4.00000i 0.485071i
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 4.00000i 0.455842i
$$78$$ − 6.00000i − 0.679366i
$$79$$ −14.0000 −1.57512 −0.787562 0.616236i $$-0.788657\pi$$
−0.787562 + 0.616236i $$0.788657\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 6.00000i − 0.662589i
$$83$$ − 14.0000i − 1.53670i −0.640030 0.768350i $$-0.721078\pi$$
0.640030 0.768350i $$-0.278922\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ − 2.00000i − 0.214423i
$$88$$ 2.00000i 0.213201i
$$89$$ 8.00000 0.847998 0.423999 0.905663i $$-0.360626\pi$$
0.423999 + 0.905663i $$0.360626\pi$$
$$90$$ 0 0
$$91$$ 12.0000 1.25794
$$92$$ 1.00000i 0.104257i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ − 6.00000i − 0.609208i −0.952479 0.304604i $$-0.901476\pi$$
0.952479 0.304604i $$-0.0985241\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 4.00000i 0.396059i
$$103$$ 2.00000i 0.197066i 0.995134 + 0.0985329i $$0.0314150\pi$$
−0.995134 + 0.0985329i $$0.968585\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ − 18.0000i − 1.74013i −0.492941 0.870063i $$-0.664078\pi$$
0.492941 0.870063i $$-0.335922\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ 8.00000 0.759326
$$112$$ − 2.00000i − 0.188982i
$$113$$ − 12.0000i − 1.12887i −0.825479 0.564433i $$-0.809095\pi$$
0.825479 0.564433i $$-0.190905\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ − 6.00000i − 0.554700i
$$118$$ 0 0
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ − 8.00000i − 0.724286i
$$123$$ − 6.00000i − 0.541002i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 2.00000i 0.174078i
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ − 8.00000i − 0.683486i −0.939793 0.341743i $$-0.888983\pi$$
0.939793 0.341743i $$-0.111017\pi$$
$$138$$ 1.00000i 0.0851257i
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 16.0000i 1.34269i
$$143$$ − 12.0000i − 1.00349i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ 3.00000i 0.247436i
$$148$$ 8.00000i 0.657596i
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ 4.00000i 0.323381i
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ 6.00000 0.480384
$$157$$ − 16.0000i − 1.27694i −0.769647 0.638470i $$-0.779568\pi$$
0.769647 0.638470i $$-0.220432\pi$$
$$158$$ − 14.0000i − 1.11378i
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ −2.00000 −0.157622
$$162$$ 1.00000i 0.0785674i
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 14.0000 1.08661
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ − 2.00000i − 0.154303i
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 4.00000i − 0.304997i
$$173$$ − 18.0000i − 1.36851i −0.729241 0.684257i $$-0.760127\pi$$
0.729241 0.684257i $$-0.239873\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ 8.00000i 0.599625i
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ −12.0000 −0.891953 −0.445976 0.895045i $$-0.647144\pi$$
−0.445976 + 0.895045i $$0.647144\pi$$
$$182$$ 12.0000i 0.889499i
$$183$$ − 8.00000i − 0.591377i
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 8.00000i 0.585018i
$$188$$ 0 0
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 2.00000i 0.143963i 0.997406 + 0.0719816i $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 2.00000i 0.142134i
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 6.00000i 0.422159i
$$203$$ 4.00000i 0.280745i
$$204$$ −4.00000 −0.280056
$$205$$ 0 0
$$206$$ −2.00000 −0.139347
$$207$$ 1.00000i 0.0695048i
$$208$$ 6.00000i 0.416025i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 16.0000i 1.09630i
$$214$$ 18.0000 1.23045
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 4.00000i 0.270914i
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 24.0000 1.61441
$$222$$ 8.00000i 0.536925i
$$223$$ 16.0000i 1.07144i 0.844396 + 0.535720i $$0.179960\pi$$
−0.844396 + 0.535720i $$0.820040\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ 12.0000 0.798228
$$227$$ − 18.0000i − 1.19470i −0.801980 0.597351i $$-0.796220\pi$$
0.801980 0.597351i $$-0.203780\pi$$
$$228$$ 0 0
$$229$$ 20.0000 1.32164 0.660819 0.750546i $$-0.270209\pi$$
0.660819 + 0.750546i $$0.270209\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 2.00000i 0.131306i
$$233$$ 26.0000i 1.70332i 0.524097 + 0.851658i $$0.324403\pi$$
−0.524097 + 0.851658i $$0.675597\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 14.0000i − 0.909398i
$$238$$ − 8.00000i − 0.518563i
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ − 7.00000i − 0.449977i
$$243$$ 1.00000i 0.0641500i
$$244$$ 8.00000 0.512148
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 14.0000 0.887214
$$250$$ 0 0
$$251$$ 6.00000 0.378717 0.189358 0.981908i $$-0.439359\pi$$
0.189358 + 0.981908i $$0.439359\pi$$
$$252$$ − 2.00000i − 0.125988i
$$253$$ 2.00000i 0.125739i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 14.0000i 0.873296i 0.899632 + 0.436648i $$0.143834\pi$$
−0.899632 + 0.436648i $$0.856166\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ −16.0000 −0.994192
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 12.0000i 0.741362i
$$263$$ − 4.00000i − 0.246651i −0.992366 0.123325i $$-0.960644\pi$$
0.992366 0.123325i $$-0.0393559\pi$$
$$264$$ −2.00000 −0.123091
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 8.00000i 0.489592i
$$268$$ 4.00000i 0.244339i
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ − 4.00000i − 0.242536i
$$273$$ 12.0000i 0.726273i
$$274$$ 8.00000 0.483298
$$275$$ 0 0
$$276$$ −1.00000 −0.0601929
$$277$$ − 10.0000i − 0.600842i −0.953807 0.300421i $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971271\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 8.00000 0.477240 0.238620 0.971113i $$-0.423305\pi$$
0.238620 + 0.971113i $$0.423305\pi$$
$$282$$ 0 0
$$283$$ 16.0000i 0.951101i 0.879688 + 0.475551i $$0.157751\pi$$
−0.879688 + 0.475551i $$0.842249\pi$$
$$284$$ −16.0000 −0.949425
$$285$$ 0 0
$$286$$ 12.0000 0.709575
$$287$$ 12.0000i 0.708338i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 6.00000 0.351726
$$292$$ 6.00000i 0.351123i
$$293$$ 10.0000i 0.584206i 0.956387 + 0.292103i $$0.0943550\pi$$
−0.956387 + 0.292103i $$0.905645\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ −8.00000 −0.464991
$$297$$ 2.00000i 0.116052i
$$298$$ − 10.0000i − 0.579284i
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ 12.0000i 0.690522i
$$303$$ 6.00000i 0.344691i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −4.00000 −0.228665
$$307$$ 28.0000i 1.59804i 0.601302 + 0.799022i $$0.294649\pi$$
−0.601302 + 0.799022i $$0.705351\pi$$
$$308$$ − 4.00000i − 0.227921i
$$309$$ −2.00000 −0.113776
$$310$$ 0 0
$$311$$ −32.0000 −1.81455 −0.907277 0.420534i $$-0.861843\pi$$
−0.907277 + 0.420534i $$0.861843\pi$$
$$312$$ 6.00000i 0.339683i
$$313$$ 34.0000i 1.92179i 0.276907 + 0.960897i $$0.410691\pi$$
−0.276907 + 0.960897i $$0.589309\pi$$
$$314$$ 16.0000 0.902932
$$315$$ 0 0
$$316$$ 14.0000 0.787562
$$317$$ − 22.0000i − 1.23564i −0.786318 0.617822i $$-0.788015\pi$$
0.786318 0.617822i $$-0.211985\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ 4.00000 0.223957
$$320$$ 0 0
$$321$$ 18.0000 1.00466
$$322$$ − 2.00000i − 0.111456i
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 4.00000i 0.221201i
$$328$$ 6.00000i 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 14.0000i 0.768350i
$$333$$ 8.00000i 0.438397i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ − 34.0000i − 1.85210i −0.377403 0.926049i $$-0.623183\pi$$
0.377403 0.926049i $$-0.376817\pi$$
$$338$$ − 23.0000i − 1.25104i
$$339$$ 12.0000 0.651751
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ − 20.0000i − 1.07990i
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ − 8.00000i − 0.429463i −0.976673 0.214731i $$-0.931112\pi$$
0.976673 0.214731i $$-0.0688876\pi$$
$$348$$ 2.00000i 0.107211i
$$349$$ 6.00000 0.321173 0.160586 0.987022i $$-0.448662\pi$$
0.160586 + 0.987022i $$0.448662\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ − 2.00000i − 0.106600i
$$353$$ − 26.0000i − 1.38384i −0.721974 0.691920i $$-0.756765\pi$$
0.721974 0.691920i $$-0.243235\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −8.00000 −0.423999
$$357$$ − 8.00000i − 0.423405i
$$358$$ 20.0000i 1.05703i
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ − 12.0000i − 0.630706i
$$363$$ − 7.00000i − 0.367405i
$$364$$ −12.0000 −0.628971
$$365$$ 0 0
$$366$$ 8.00000 0.418167
$$367$$ 10.0000i 0.521996i 0.965339 + 0.260998i $$0.0840516\pi$$
−0.965339 + 0.260998i $$0.915948\pi$$
$$368$$ − 1.00000i − 0.0521286i
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ − 32.0000i − 1.65690i −0.560065 0.828449i $$-0.689224\pi$$
0.560065 0.828449i $$-0.310776\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 12.0000i − 0.618031i
$$378$$ − 2.00000i − 0.102869i
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 12.0000i 0.613171i 0.951843 + 0.306586i $$0.0991866\pi$$
−0.951843 + 0.306586i $$0.900813\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ − 4.00000i − 0.203331i
$$388$$ 6.00000i 0.304604i
$$389$$ −10.0000 −0.507020 −0.253510 0.967333i $$-0.581585\pi$$
−0.253510 + 0.967333i $$0.581585\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ − 3.00000i − 0.151523i
$$393$$ 12.0000i 0.605320i
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ −2.00000 −0.100504
$$397$$ − 22.0000i − 1.10415i −0.833795 0.552074i $$-0.813837\pi$$
0.833795 0.552074i $$-0.186163\pi$$
$$398$$ 2.00000i 0.100251i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −24.0000 −1.19850 −0.599251 0.800561i $$-0.704535\pi$$
−0.599251 + 0.800561i $$0.704535\pi$$
$$402$$ 4.00000i 0.199502i
$$403$$ 0 0
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ −4.00000 −0.198517
$$407$$ 16.0000i 0.793091i
$$408$$ − 4.00000i − 0.198030i
$$409$$ −38.0000 −1.87898 −0.939490 0.342578i $$-0.888700\pi$$
−0.939490 + 0.342578i $$0.888700\pi$$
$$410$$ 0 0
$$411$$ 8.00000 0.394611
$$412$$ − 2.00000i − 0.0985329i
$$413$$ 0 0
$$414$$ −1.00000 −0.0491473
$$415$$ 0 0
$$416$$ −6.00000 −0.294174
$$417$$ 4.00000i 0.195881i
$$418$$ 0 0
$$419$$ 14.0000 0.683945 0.341972 0.939710i $$-0.388905\pi$$
0.341972 + 0.939710i $$0.388905\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$422$$ − 20.0000i − 0.973585i
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ −16.0000 −0.775203
$$427$$ 16.0000i 0.774294i
$$428$$ 18.0000i 0.870063i
$$429$$ 12.0000 0.579365
$$430$$ 0 0
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ − 30.0000i − 1.44171i −0.693087 0.720854i $$-0.743750\pi$$
0.693087 0.720854i $$-0.256250\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ 0 0
$$438$$ 6.00000i 0.286691i
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 24.0000i 1.14156i
$$443$$ − 4.00000i − 0.190046i −0.995475 0.0950229i $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ −8.00000 −0.379663
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ − 10.0000i − 0.472984i
$$448$$ 2.00000i 0.0944911i
$$449$$ 22.0000 1.03824 0.519122 0.854700i $$-0.326259\pi$$
0.519122 + 0.854700i $$0.326259\pi$$
$$450$$ 0 0
$$451$$ 12.0000 0.565058
$$452$$ 12.0000i 0.564433i
$$453$$ 12.0000i 0.563809i
$$454$$ 18.0000 0.844782
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 18.0000i 0.842004i 0.907060 + 0.421002i $$0.138322\pi$$
−0.907060 + 0.421002i $$0.861678\pi$$
$$458$$ 20.0000i 0.934539i
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ −10.0000 −0.465746 −0.232873 0.972507i $$-0.574813\pi$$
−0.232873 + 0.972507i $$0.574813\pi$$
$$462$$ − 4.00000i − 0.186097i
$$463$$ 36.0000i 1.67306i 0.547920 + 0.836531i $$0.315420\pi$$
−0.547920 + 0.836531i $$0.684580\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −26.0000 −1.20443
$$467$$ − 30.0000i − 1.38823i −0.719862 0.694117i $$-0.755795\pi$$
0.719862 0.694117i $$-0.244205\pi$$
$$468$$ 6.00000i 0.277350i
$$469$$ −8.00000 −0.369406
$$470$$ 0 0
$$471$$ 16.0000 0.737241
$$472$$ 0 0
$$473$$ − 8.00000i − 0.367840i
$$474$$ 14.0000 0.643041
$$475$$ 0 0
$$476$$ 8.00000 0.366679
$$477$$ 6.00000i 0.274721i
$$478$$ − 16.0000i − 0.731823i
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ 48.0000 2.18861
$$482$$ − 14.0000i − 0.637683i
$$483$$ − 2.00000i − 0.0910032i
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 12.0000i 0.543772i 0.962329 + 0.271886i $$0.0876473\pi$$
−0.962329 + 0.271886i $$0.912353\pi$$
$$488$$ 8.00000i 0.362143i
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 6.00000i 0.270501i
$$493$$ 8.00000i 0.360302i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 32.0000i − 1.43540i
$$498$$ 14.0000i 0.627355i
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 6.00000i 0.267793i
$$503$$ − 4.00000i − 0.178351i −0.996016 0.0891756i $$-0.971577\pi$$
0.996016 0.0891756i $$-0.0284232\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ −2.00000 −0.0889108
$$507$$ − 23.0000i − 1.02147i
$$508$$ 0 0
$$509$$ 14.0000 0.620539 0.310270 0.950649i $$-0.399581\pi$$
0.310270 + 0.950649i $$0.399581\pi$$
$$510$$ 0 0
$$511$$ −12.0000 −0.530849
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ −14.0000 −0.617514
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ − 16.0000i − 0.703000i
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ −4.00000 −0.175243 −0.0876216 0.996154i $$-0.527927\pi$$
−0.0876216 + 0.996154i $$0.527927\pi$$
$$522$$ 2.00000i 0.0875376i
$$523$$ − 4.00000i − 0.174908i −0.996169 0.0874539i $$-0.972127\pi$$
0.996169 0.0874539i $$-0.0278730\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 4.00000 0.174408
$$527$$ 0 0
$$528$$ − 2.00000i − 0.0870388i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 36.0000i − 1.55933i
$$534$$ −8.00000 −0.346194
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ 20.0000i 0.863064i
$$538$$ 18.0000i 0.776035i
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ 26.0000 1.11783 0.558914 0.829226i $$-0.311218\pi$$
0.558914 + 0.829226i $$0.311218\pi$$
$$542$$ 20.0000i 0.859074i
$$543$$ − 12.0000i − 0.514969i
$$544$$ 4.00000 0.171499
$$545$$ 0 0
$$546$$ −12.0000 −0.513553
$$547$$ 12.0000i 0.513083i 0.966533 + 0.256541i $$0.0825830\pi$$
−0.966533 + 0.256541i $$0.917417\pi$$
$$548$$ 8.00000i 0.341743i
$$549$$ 8.00000 0.341432
$$550$$ 0 0
$$551$$ 0 0
$$552$$ − 1.00000i − 0.0425628i
$$553$$ 28.0000i 1.19068i
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ − 22.0000i − 0.932170i −0.884740 0.466085i $$-0.845664\pi$$
0.884740 0.466085i $$-0.154336\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 8.00000i 0.337460i
$$563$$ 10.0000i 0.421450i 0.977545 + 0.210725i $$0.0675824\pi$$
−0.977545 + 0.210725i $$0.932418\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −16.0000 −0.672530
$$567$$ − 2.00000i − 0.0839921i
$$568$$ − 16.0000i − 0.671345i
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 12.0000i 0.501745i
$$573$$ 0 0
$$574$$ −12.0000 −0.500870
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ −28.0000 −1.16164
$$582$$ 6.00000i 0.248708i
$$583$$ 12.0000i 0.496989i
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ −10.0000 −0.413096
$$587$$ 8.00000i 0.330195i 0.986277 + 0.165098i $$0.0527939\pi$$
−0.986277 + 0.165098i $$0.947206\pi$$
$$588$$ − 3.00000i − 0.123718i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ − 8.00000i − 0.328798i
$$593$$ − 30.0000i − 1.23195i −0.787765 0.615976i $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ −2.00000 −0.0820610
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 2.00000i 0.0818546i
$$598$$ 6.00000i 0.245358i
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 8.00000i 0.326056i
$$603$$ 4.00000i 0.162893i
$$604$$ −12.0000 −0.488273
$$605$$ 0 0
$$606$$ −6.00000 −0.243733
$$607$$ 40.0000i 1.62355i 0.583970 + 0.811775i $$0.301498\pi$$
−0.583970 + 0.811775i $$0.698502\pi$$
$$608$$ 0 0
$$609$$ −4.00000 −0.162088
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 4.00000i − 0.161690i
$$613$$ 4.00000i 0.161558i 0.996732 + 0.0807792i $$0.0257409\pi$$
−0.996732 + 0.0807792i $$0.974259\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 0 0
$$616$$ 4.00000 0.161165
$$617$$ 28.0000i 1.12724i 0.826035 + 0.563619i $$0.190591\pi$$
−0.826035 + 0.563619i $$0.809409\pi$$
$$618$$ − 2.00000i − 0.0804518i
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ − 32.0000i − 1.28308i
$$623$$ − 16.0000i − 0.641026i
$$624$$ −6.00000 −0.240192
$$625$$ 0 0
$$626$$ −34.0000 −1.35891
$$627$$ 0 0
$$628$$ 16.0000i 0.638470i
$$629$$ −32.0000 −1.27592
$$630$$ 0 0
$$631$$ −22.0000 −0.875806 −0.437903 0.899022i $$-0.644279\pi$$
−0.437903 + 0.899022i $$0.644279\pi$$
$$632$$ 14.0000i 0.556890i
$$633$$ − 20.0000i − 0.794929i
$$634$$ 22.0000 0.873732
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 18.0000i 0.713186i
$$638$$ 4.00000i 0.158362i
$$639$$ −16.0000 −0.632950
$$640$$ 0 0
$$641$$ 16.0000 0.631962 0.315981 0.948766i $$-0.397666\pi$$
0.315981 + 0.948766i $$0.397666\pi$$
$$642$$ 18.0000i 0.710403i
$$643$$ 28.0000i 1.10421i 0.833774 + 0.552106i $$0.186176\pi$$
−0.833774 + 0.552106i $$0.813824\pi$$
$$644$$ 2.00000 0.0788110
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 4.00000i − 0.156652i
$$653$$ − 22.0000i − 0.860927i −0.902608 0.430463i $$-0.858350\pi$$
0.902608 0.430463i $$-0.141650\pi$$
$$654$$ −4.00000 −0.156412
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 6.00000i 0.234082i
$$658$$ 0 0
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ 20.0000i 0.777322i
$$663$$ 24.0000i 0.932083i
$$664$$ −14.0000 −0.543305
$$665$$ 0 0
$$666$$ −8.00000 −0.309994
$$667$$ 2.00000i 0.0774403i
$$668$$ 0 0
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ 16.0000 0.617673
$$672$$ 2.00000i 0.0771517i
$$673$$ − 42.0000i − 1.61898i −0.587133 0.809491i $$-0.699743\pi$$
0.587133 0.809491i $$-0.300257\pi$$
$$674$$ 34.0000 1.30963
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 2.00000i 0.0768662i 0.999261 + 0.0384331i $$0.0122367\pi$$
−0.999261 + 0.0384331i $$0.987763\pi$$
$$678$$ 12.0000i 0.460857i
$$679$$ −12.0000 −0.460518
$$680$$ 0 0
$$681$$ 18.0000 0.689761
$$682$$ 0 0
$$683$$ − 48.0000i − 1.83667i −0.395805 0.918334i $$-0.629534\pi$$
0.395805 0.918334i $$-0.370466\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 20.0000i 0.763048i
$$688$$ 4.00000i 0.152499i
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 18.0000i 0.684257i
$$693$$ − 4.00000i − 0.151947i
$$694$$ 8.00000 0.303676
$$695$$ 0 0
$$696$$ −2.00000 −0.0758098
$$697$$ 24.0000i 0.909065i
$$698$$ 6.00000i 0.227103i
$$699$$ −26.0000 −0.983410
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 6.00000i 0.226455i
$$703$$ 0 0
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ 26.0000 0.978523
$$707$$ − 12.0000i − 0.451306i
$$708$$ 0 0
$$709$$ −16.0000 −0.600893 −0.300446 0.953799i $$-0.597136\pi$$
−0.300446 + 0.953799i $$0.597136\pi$$
$$710$$ 0 0
$$711$$ 14.0000 0.525041
$$712$$ − 8.00000i − 0.299813i
$$713$$ 0 0
$$714$$ 8.00000 0.299392
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ − 16.0000i − 0.597531i
$$718$$ − 24.0000i − 0.895672i
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 4.00000 0.148968
$$722$$ − 19.0000i − 0.707107i
$$723$$ − 14.0000i − 0.520666i
$$724$$ 12.0000 0.445976
$$725$$ 0 0
$$726$$ 7.00000 0.259794
$$727$$ 38.0000i 1.40934i 0.709534 + 0.704671i $$0.248905\pi$$
−0.709534 + 0.704671i $$0.751095\pi$$
$$728$$ − 12.0000i − 0.444750i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 16.0000 0.591781
$$732$$ 8.00000i 0.295689i
$$733$$ − 36.0000i − 1.32969i −0.746981 0.664845i $$-0.768498\pi$$
0.746981 0.664845i $$-0.231502\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ 8.00000i 0.294684i
$$738$$ 6.00000i 0.220863i
$$739$$ −12.0000 −0.441427 −0.220714 0.975339i $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 12.0000i − 0.440534i
$$743$$ − 4.00000i − 0.146746i −0.997305 0.0733729i $$-0.976624\pi$$
0.997305 0.0733729i $$-0.0233763\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 32.0000 1.17160
$$747$$ 14.0000i 0.512233i
$$748$$ − 8.00000i − 0.292509i
$$749$$ −36.0000 −1.31541
$$750$$ 0 0
$$751$$ −50.0000 −1.82453 −0.912263 0.409605i $$-0.865667\pi$$
−0.912263 + 0.409605i $$0.865667\pi$$
$$752$$ 0 0
$$753$$ 6.00000i 0.218652i
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ 40.0000i 1.45382i 0.686730 + 0.726912i $$0.259045\pi$$
−0.686730 + 0.726912i $$0.740955\pi$$
$$758$$ − 16.0000i − 0.581146i
$$759$$ −2.00000 −0.0725954
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ − 8.00000i − 0.289619i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −12.0000 −0.433578
$$767$$ 0 0
$$768$$ 1.00000i 0.0360844i
$$769$$ −6.00000 −0.216366 −0.108183 0.994131i $$-0.534503\pi$$
−0.108183 + 0.994131i $$0.534503\pi$$
$$770$$ 0 0
$$771$$ −14.0000 −0.504198
$$772$$ − 2.00000i − 0.0719816i
$$773$$ 18.0000i 0.647415i 0.946157 + 0.323708i $$0.104929\pi$$
−0.946157 + 0.323708i $$0.895071\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ − 16.0000i − 0.573997i
$$778$$ − 10.0000i − 0.358517i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ − 4.00000i − 0.143040i
$$783$$ 2.00000i 0.0714742i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ −12.0000 −0.428026
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 4.00000 0.142404
$$790$$ 0 0
$$791$$ −24.0000 −0.853342
$$792$$ − 2.00000i − 0.0710669i
$$793$$ − 48.0000i − 1.70453i
$$794$$ 22.0000 0.780751
$$795$$ 0 0
$$796$$ −2.00000 −0.0708881
$$797$$ − 46.0000i − 1.62940i −0.579880 0.814702i $$-0.696901\pi$$
0.579880 0.814702i $$-0.303099\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −8.00000 −0.282666
$$802$$ − 24.0000i − 0.847469i
$$803$$ 12.0000i 0.423471i
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 18.0000i 0.633630i
$$808$$ − 6.00000i − 0.211079i
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 0 0
$$811$$ −44.0000 −1.54505 −0.772524 0.634985i $$-0.781006\pi$$
−0.772524 + 0.634985i $$0.781006\pi$$
$$812$$ − 4.00000i − 0.140372i
$$813$$ 20.0000i 0.701431i
$$814$$ −16.0000 −0.560800
$$815$$ 0 0
$$816$$ 4.00000 0.140028
$$817$$ 0 0
$$818$$ − 38.0000i − 1.32864i
$$819$$ −12.0000 −0.419314
$$820$$ 0 0
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ 8.00000i 0.279032i
$$823$$ 4.00000i 0.139431i 0.997567 + 0.0697156i $$0.0222092\pi$$
−0.997567 + 0.0697156i $$0.977791\pi$$
$$824$$ 2.00000 0.0696733
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 22.0000i − 0.765015i −0.923952 0.382507i $$-0.875061\pi$$
0.923952 0.382507i $$-0.124939\pi$$
$$828$$ − 1.00000i − 0.0347524i
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ − 6.00000i − 0.208013i
$$833$$ − 12.0000i − 0.415775i
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 14.0000i 0.483622i
$$839$$ 20.0000 0.690477 0.345238 0.938515i $$-0.387798\pi$$
0.345238 + 0.938515i $$0.387798\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 8.00000i 0.275535i
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 14.0000i 0.481046i
$$848$$ − 6.00000i − 0.206041i
$$849$$ −16.0000 −0.549119
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ − 16.0000i − 0.548151i
$$853$$ − 26.0000i − 0.890223i −0.895475 0.445112i $$-0.853164\pi$$
0.895475 0.445112i $$-0.146836\pi$$
$$854$$ −16.0000 −0.547509
$$855$$ 0 0
$$856$$ −18.0000 −0.615227
$$857$$ 30.0000i 1.02478i 0.858753 + 0.512390i $$0.171240\pi$$
−0.858753 + 0.512390i $$0.828760\pi$$
$$858$$ 12.0000i 0.409673i
$$859$$ 12.0000 0.409435 0.204717 0.978821i $$-0.434372\pi$$
0.204717 + 0.978821i $$0.434372\pi$$
$$860$$ 0 0
$$861$$ −12.0000 −0.408959
$$862$$ − 36.0000i − 1.22616i
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 30.0000 1.01944
$$867$$ 1.00000i 0.0339618i
$$868$$ 0 0
$$869$$ 28.0000 0.949835
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ − 4.00000i − 0.135457i
$$873$$ 6.00000i 0.203069i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −6.00000 −0.202721
$$877$$ − 2.00000i − 0.0675352i −0.999430 0.0337676i $$-0.989249\pi$$
0.999430 0.0337676i $$-0.0107506\pi$$
$$878$$ − 32.0000i − 1.07995i
$$879$$ −10.0000 −0.337292
$$880$$ 0 0
$$881$$ −56.0000 −1.88669 −0.943344 0.331816i $$-0.892339\pi$$
−0.943344 + 0.331816i $$0.892339\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ 36.0000i 1.21150i 0.795656 + 0.605748i $$0.207126\pi$$
−0.795656 + 0.605748i $$0.792874\pi$$
$$884$$ −24.0000 −0.807207
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ 48.0000i 1.61168i 0.592132 + 0.805841i $$0.298286\pi$$
−0.592132 + 0.805841i $$0.701714\pi$$
$$888$$ − 8.00000i − 0.268462i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ − 16.0000i − 0.535720i
$$893$$ 0 0
$$894$$ 10.0000 0.334450
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ 6.00000i 0.200334i
$$898$$ 22.0000i 0.734150i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −24.0000 −0.799556
$$902$$ 12.0000i 0.399556i
$$903$$ 8.00000i 0.266223i
$$904$$ −12.0000 −0.399114
$$905$$ 0 0
$$906$$ −12.0000 −0.398673
$$907$$ − 24.0000i − 0.796907i −0.917189 0.398453i $$-0.869547\pi$$
0.917189 0.398453i $$-0.130453\pi$$
$$908$$ 18.0000i 0.597351i
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ 0 0
$$913$$ 28.0000i 0.926665i
$$914$$ −18.0000 −0.595387
$$915$$ 0 0
$$916$$ −20.0000 −0.660819
$$917$$ − 24.0000i − 0.792550i
$$918$$ − 4.00000i − 0.132020i
$$919$$ 2.00000 0.0659739 0.0329870 0.999456i $$-0.489498\pi$$
0.0329870 + 0.999456i $$0.489498\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ − 10.0000i − 0.329332i
$$923$$ 96.0000i 3.15988i
$$924$$ 4.00000 0.131590
$$925$$ 0 0
$$926$$ −36.0000 −1.18303
$$927$$ − 2.00000i − 0.0656886i
$$928$$ − 2.00000i − 0.0656532i
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 26.0000i − 0.851658i
$$933$$ − 32.0000i − 1.04763i
$$934$$ 30.0000 0.981630
$$935$$ 0 0
$$936$$ −6.00000 −0.196116
$$937$$ 42.0000i 1.37208i 0.727564 + 0.686040i $$0.240653\pi$$
−0.727564 + 0.686040i $$0.759347\pi$$
$$938$$ − 8.00000i − 0.261209i
$$939$$ −34.0000 −1.10955
$$940$$ 0 0
$$941$$ −22.0000 −0.717180 −0.358590 0.933495i $$-0.616742\pi$$
−0.358590 + 0.933495i $$0.616742\pi$$
$$942$$ 16.0000i 0.521308i
$$943$$ 6.00000i 0.195387i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ 8.00000i 0.259965i 0.991516 + 0.129983i $$0.0414921\pi$$
−0.991516 + 0.129983i $$0.958508\pi$$
$$948$$ 14.0000i 0.454699i
$$949$$ 36.0000 1.16861
$$950$$ 0 0
$$951$$ 22.0000 0.713399
$$952$$ 8.00000i 0.259281i
$$953$$ − 44.0000i − 1.42530i −0.701520 0.712650i $$-0.747495\pi$$
0.701520 0.712650i $$-0.252505\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ 4.00000i 0.129302i
$$958$$ − 8.00000i − 0.258468i
$$959$$ −16.0000 −0.516667
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 48.0000i 1.54758i
$$963$$ 18.0000i 0.580042i
$$964$$ 14.0000 0.450910
$$965$$ 0 0
$$966$$ 2.00000 0.0643489
$$967$$ 52.0000i 1.67221i 0.548572 + 0.836104i $$0.315172\pi$$
−0.548572 + 0.836104i $$0.684828\pi$$
$$968$$ 7.00000i 0.224989i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −46.0000 −1.47621 −0.738105 0.674686i $$-0.764279\pi$$
−0.738105 + 0.674686i $$0.764279\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ − 8.00000i − 0.256468i
$$974$$ −12.0000 −0.384505
$$975$$ 0 0
$$976$$ −8.00000 −0.256074
$$977$$ 24.0000i 0.767828i 0.923369 + 0.383914i $$0.125424\pi$$
−0.923369 + 0.383914i $$0.874576\pi$$
$$978$$ − 4.00000i − 0.127906i
$$979$$ −16.0000 −0.511362
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ − 12.0000i − 0.382935i
$$983$$ 12.0000i 0.382741i 0.981518 + 0.191370i $$0.0612931\pi$$
−0.981518 + 0.191370i $$0.938707\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ −8.00000 −0.254772
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ 20.0000i 0.634681i
$$994$$ 32.0000 1.01498
$$995$$ 0 0
$$996$$ −14.0000 −0.443607
$$997$$ − 2.00000i − 0.0633406i −0.999498 0.0316703i $$-0.989917\pi$$
0.999498 0.0316703i $$-0.0100827\pi$$
$$998$$ 28.0000i 0.886325i
$$999$$ −8.00000 −0.253109
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.d.d.2899.2 2
5.2 odd 4 3450.2.a.l.1.1 1
5.3 odd 4 690.2.a.g.1.1 1
5.4 even 2 inner 3450.2.d.d.2899.1 2
15.8 even 4 2070.2.a.e.1.1 1
20.3 even 4 5520.2.a.z.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.g.1.1 1 5.3 odd 4
2070.2.a.e.1.1 1 15.8 even 4
3450.2.a.l.1.1 1 5.2 odd 4
3450.2.d.d.2899.1 2 5.4 even 2 inner
3450.2.d.d.2899.2 2 1.1 even 1 trivial
5520.2.a.z.1.1 1 20.3 even 4