# Properties

 Label 5070.2.b.h.1351.2 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ 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 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.h.1351.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000i q^{18} +2.00000i q^{19} +1.00000i q^{20} +2.00000i q^{21} +5.00000 q^{22} +1.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} +5.00000 q^{29} -1.00000 q^{30} +11.0000i q^{31} +1.00000i q^{32} +5.00000i q^{33} +2.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} +3.00000i q^{37} -2.00000 q^{38} -1.00000 q^{40} +2.00000i q^{41} -2.00000 q^{42} +11.0000 q^{43} +5.00000i q^{44} -1.00000i q^{45} +1.00000i q^{46} +9.00000i q^{47} -1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} -2.00000 q^{51} +6.00000 q^{53} -1.00000i q^{54} -5.00000 q^{55} -2.00000 q^{56} -2.00000i q^{57} +5.00000i q^{58} -15.0000i q^{59} -1.00000i q^{60} +10.0000 q^{61} -11.0000 q^{62} -2.00000i q^{63} -1.00000 q^{64} -5.00000 q^{66} -16.0000i q^{67} -2.00000 q^{68} -1.00000 q^{69} -2.00000i q^{70} -1.00000i q^{72} -6.00000i q^{73} -3.00000 q^{74} +1.00000 q^{75} -2.00000i q^{76} -10.0000 q^{77} -11.0000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -2.00000 q^{82} -6.00000i q^{83} -2.00000i q^{84} -2.00000i q^{85} +11.0000i q^{86} -5.00000 q^{87} -5.00000 q^{88} +2.00000i q^{89} +1.00000 q^{90} -1.00000 q^{92} -11.0000i q^{93} -9.00000 q^{94} +2.00000 q^{95} -1.00000i q^{96} +2.00000i q^{97} +3.00000i q^{98} -5.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{4} + 2q^{9} + 2q^{10} + 2q^{12} + 4q^{14} + 2q^{16} + 4q^{17} + 10q^{22} + 2q^{23} - 2q^{25} - 2q^{27} + 10q^{29} - 2q^{30} - 4q^{35} - 2q^{36} - 4q^{38} - 2q^{40} - 4q^{42} + 22q^{43} - 2q^{48} + 6q^{49} - 4q^{51} + 12q^{53} - 10q^{55} - 4q^{56} + 20q^{61} - 22q^{62} - 2q^{64} - 10q^{66} - 4q^{68} - 2q^{69} - 6q^{74} + 2q^{75} - 20q^{77} - 22q^{79} + 2q^{81} - 4q^{82} - 10q^{87} - 10q^{88} + 2q^{90} - 2q^{92} - 18q^{94} + 4q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ −1.00000 −0.500000
$$5$$ − 1.00000i − 0.447214i
$$6$$ − 1.00000i − 0.408248i
$$7$$ − 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
$$10$$ 1.00000 0.316228
$$11$$ − 5.00000i − 1.50756i −0.657129 0.753778i $$-0.728229\pi$$
0.657129 0.753778i $$-0.271771\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ 2.00000 0.534522
$$15$$ 1.00000i 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 2.00000i 0.458831i 0.973329 + 0.229416i $$0.0736815\pi$$
−0.973329 + 0.229416i $$0.926318\pi$$
$$20$$ 1.00000i 0.223607i
$$21$$ 2.00000i 0.436436i
$$22$$ 5.00000 1.06600
$$23$$ 1.00000 0.208514 0.104257 0.994550i $$-0.466753\pi$$
0.104257 + 0.994550i $$0.466753\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 2.00000i 0.377964i
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 11.0000i 1.97566i 0.155543 + 0.987829i $$0.450287\pi$$
−0.155543 + 0.987829i $$0.549713\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 5.00000i 0.870388i
$$34$$ 2.00000i 0.342997i
$$35$$ −2.00000 −0.338062
$$36$$ −1.00000 −0.166667
$$37$$ 3.00000i 0.493197i 0.969118 + 0.246598i $$0.0793129\pi$$
−0.969118 + 0.246598i $$0.920687\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 2.00000i 0.312348i 0.987730 + 0.156174i $$0.0499160\pi$$
−0.987730 + 0.156174i $$0.950084\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 11.0000 1.67748 0.838742 0.544529i $$-0.183292\pi$$
0.838742 + 0.544529i $$0.183292\pi$$
$$44$$ 5.00000i 0.753778i
$$45$$ − 1.00000i − 0.149071i
$$46$$ 1.00000i 0.147442i
$$47$$ 9.00000i 1.31278i 0.754420 + 0.656392i $$0.227918\pi$$
−0.754420 + 0.656392i $$0.772082\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 3.00000 0.428571
$$50$$ − 1.00000i − 0.141421i
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ −5.00000 −0.674200
$$56$$ −2.00000 −0.267261
$$57$$ − 2.00000i − 0.264906i
$$58$$ 5.00000i 0.656532i
$$59$$ − 15.0000i − 1.95283i −0.215894 0.976417i $$-0.569267\pi$$
0.215894 0.976417i $$-0.430733\pi$$
$$60$$ − 1.00000i − 0.129099i
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ −11.0000 −1.39700
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −5.00000 −0.615457
$$67$$ − 16.0000i − 1.95471i −0.211604 0.977356i $$-0.567869\pi$$
0.211604 0.977356i $$-0.432131\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ −1.00000 −0.120386
$$70$$ − 2.00000i − 0.239046i
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ −3.00000 −0.348743
$$75$$ 1.00000 0.115470
$$76$$ − 2.00000i − 0.229416i
$$77$$ −10.0000 −1.13961
$$78$$ 0 0
$$79$$ −11.0000 −1.23760 −0.618798 0.785550i $$-0.712380\pi$$
−0.618798 + 0.785550i $$0.712380\pi$$
$$80$$ − 1.00000i − 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ −2.00000 −0.220863
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ − 2.00000i − 0.218218i
$$85$$ − 2.00000i − 0.216930i
$$86$$ 11.0000i 1.18616i
$$87$$ −5.00000 −0.536056
$$88$$ −5.00000 −0.533002
$$89$$ 2.00000i 0.212000i 0.994366 + 0.106000i $$0.0338043\pi$$
−0.994366 + 0.106000i $$0.966196\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ −1.00000 −0.104257
$$93$$ − 11.0000i − 1.14065i
$$94$$ −9.00000 −0.928279
$$95$$ 2.00000 0.205196
$$96$$ − 1.00000i − 0.102062i
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ − 5.00000i − 0.502519i
$$100$$ 1.00000 0.100000
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ − 2.00000i − 0.198030i
$$103$$ −10.0000 −0.985329 −0.492665 0.870219i $$-0.663977\pi$$
−0.492665 + 0.870219i $$0.663977\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 6.00000i 0.582772i
$$107$$ 10.0000 0.966736 0.483368 0.875417i $$-0.339413\pi$$
0.483368 + 0.875417i $$0.339413\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 2.00000i 0.191565i 0.995402 + 0.0957826i $$0.0305354\pi$$
−0.995402 + 0.0957826i $$0.969465\pi$$
$$110$$ − 5.00000i − 0.476731i
$$111$$ − 3.00000i − 0.284747i
$$112$$ − 2.00000i − 0.188982i
$$113$$ 11.0000 1.03479 0.517396 0.855746i $$-0.326901\pi$$
0.517396 + 0.855746i $$0.326901\pi$$
$$114$$ 2.00000 0.187317
$$115$$ − 1.00000i − 0.0932505i
$$116$$ −5.00000 −0.464238
$$117$$ 0 0
$$118$$ 15.0000 1.38086
$$119$$ − 4.00000i − 0.366679i
$$120$$ 1.00000 0.0912871
$$121$$ −14.0000 −1.27273
$$122$$ 10.0000i 0.905357i
$$123$$ − 2.00000i − 0.180334i
$$124$$ − 11.0000i − 0.987829i
$$125$$ 1.00000i 0.0894427i
$$126$$ 2.00000 0.178174
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −11.0000 −0.968496
$$130$$ 0 0
$$131$$ −1.00000 −0.0873704 −0.0436852 0.999045i $$-0.513910\pi$$
−0.0436852 + 0.999045i $$0.513910\pi$$
$$132$$ − 5.00000i − 0.435194i
$$133$$ 4.00000 0.346844
$$134$$ 16.0000 1.38219
$$135$$ 1.00000i 0.0860663i
$$136$$ − 2.00000i − 0.171499i
$$137$$ − 11.0000i − 0.939793i −0.882721 0.469897i $$-0.844291\pi$$
0.882721 0.469897i $$-0.155709\pi$$
$$138$$ − 1.00000i − 0.0851257i
$$139$$ 2.00000 0.169638 0.0848189 0.996396i $$-0.472969\pi$$
0.0848189 + 0.996396i $$0.472969\pi$$
$$140$$ 2.00000 0.169031
$$141$$ − 9.00000i − 0.757937i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ − 5.00000i − 0.415227i
$$146$$ 6.00000 0.496564
$$147$$ −3.00000 −0.247436
$$148$$ − 3.00000i − 0.246598i
$$149$$ − 17.0000i − 1.39269i −0.717705 0.696347i $$-0.754807\pi$$
0.717705 0.696347i $$-0.245193\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ 8.00000i 0.651031i 0.945537 + 0.325515i $$0.105538\pi$$
−0.945537 + 0.325515i $$0.894462\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 2.00000 0.161690
$$154$$ − 10.0000i − 0.805823i
$$155$$ 11.0000 0.883541
$$156$$ 0 0
$$157$$ −7.00000 −0.558661 −0.279330 0.960195i $$-0.590112\pi$$
−0.279330 + 0.960195i $$0.590112\pi$$
$$158$$ − 11.0000i − 0.875113i
$$159$$ −6.00000 −0.475831
$$160$$ 1.00000 0.0790569
$$161$$ − 2.00000i − 0.157622i
$$162$$ 1.00000i 0.0785674i
$$163$$ − 15.0000i − 1.17489i −0.809264 0.587445i $$-0.800134\pi$$
0.809264 0.587445i $$-0.199866\pi$$
$$164$$ − 2.00000i − 0.156174i
$$165$$ 5.00000 0.389249
$$166$$ 6.00000 0.465690
$$167$$ 3.00000i 0.232147i 0.993241 + 0.116073i $$0.0370308\pi$$
−0.993241 + 0.116073i $$0.962969\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 0 0
$$170$$ 2.00000 0.153393
$$171$$ 2.00000i 0.152944i
$$172$$ −11.0000 −0.838742
$$173$$ −20.0000 −1.52057 −0.760286 0.649589i $$-0.774941\pi$$
−0.760286 + 0.649589i $$0.774941\pi$$
$$174$$ − 5.00000i − 0.379049i
$$175$$ 2.00000i 0.151186i
$$176$$ − 5.00000i − 0.376889i
$$177$$ 15.0000i 1.12747i
$$178$$ −2.00000 −0.149906
$$179$$ 13.0000 0.971666 0.485833 0.874052i $$-0.338516\pi$$
0.485833 + 0.874052i $$0.338516\pi$$
$$180$$ 1.00000i 0.0745356i
$$181$$ 16.0000 1.18927 0.594635 0.803996i $$-0.297296\pi$$
0.594635 + 0.803996i $$0.297296\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ − 1.00000i − 0.0737210i
$$185$$ 3.00000 0.220564
$$186$$ 11.0000 0.806559
$$187$$ − 10.0000i − 0.731272i
$$188$$ − 9.00000i − 0.656392i
$$189$$ 2.00000i 0.145479i
$$190$$ 2.00000i 0.145095i
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ − 24.0000i − 1.72756i −0.503871 0.863779i $$-0.668091\pi$$
0.503871 0.863779i $$-0.331909\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 8.00000i 0.569976i 0.958531 + 0.284988i $$0.0919897\pi$$
−0.958531 + 0.284988i $$0.908010\pi$$
$$198$$ 5.00000 0.355335
$$199$$ −24.0000 −1.70131 −0.850657 0.525720i $$-0.823796\pi$$
−0.850657 + 0.525720i $$0.823796\pi$$
$$200$$ 1.00000i 0.0707107i
$$201$$ 16.0000i 1.12855i
$$202$$ − 2.00000i − 0.140720i
$$203$$ − 10.0000i − 0.701862i
$$204$$ 2.00000 0.140028
$$205$$ 2.00000 0.139686
$$206$$ − 10.0000i − 0.696733i
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ 10.0000 0.691714
$$210$$ 2.00000i 0.138013i
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ 10.0000i 0.683586i
$$215$$ − 11.0000i − 0.750194i
$$216$$ 1.00000i 0.0680414i
$$217$$ 22.0000 1.49346
$$218$$ −2.00000 −0.135457
$$219$$ 6.00000i 0.405442i
$$220$$ 5.00000 0.337100
$$221$$ 0 0
$$222$$ 3.00000 0.201347
$$223$$ − 26.0000i − 1.74109i −0.492090 0.870544i $$-0.663767\pi$$
0.492090 0.870544i $$-0.336233\pi$$
$$224$$ 2.00000 0.133631
$$225$$ −1.00000 −0.0666667
$$226$$ 11.0000i 0.731709i
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 2.00000i 0.132453i
$$229$$ 10.0000i 0.660819i 0.943838 + 0.330409i $$0.107187\pi$$
−0.943838 + 0.330409i $$0.892813\pi$$
$$230$$ 1.00000 0.0659380
$$231$$ 10.0000 0.657952
$$232$$ − 5.00000i − 0.328266i
$$233$$ −1.00000 −0.0655122 −0.0327561 0.999463i $$-0.510428\pi$$
−0.0327561 + 0.999463i $$0.510428\pi$$
$$234$$ 0 0
$$235$$ 9.00000 0.587095
$$236$$ 15.0000i 0.976417i
$$237$$ 11.0000 0.714527
$$238$$ 4.00000 0.259281
$$239$$ 20.0000i 1.29369i 0.762620 + 0.646846i $$0.223912\pi$$
−0.762620 + 0.646846i $$0.776088\pi$$
$$240$$ 1.00000i 0.0645497i
$$241$$ − 7.00000i − 0.450910i −0.974254 0.225455i $$-0.927613\pi$$
0.974254 0.225455i $$-0.0723868\pi$$
$$242$$ − 14.0000i − 0.899954i
$$243$$ −1.00000 −0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ − 3.00000i − 0.191663i
$$246$$ 2.00000 0.127515
$$247$$ 0 0
$$248$$ 11.0000 0.698501
$$249$$ 6.00000i 0.380235i
$$250$$ −1.00000 −0.0632456
$$251$$ −25.0000 −1.57799 −0.788993 0.614402i $$-0.789397\pi$$
−0.788993 + 0.614402i $$0.789397\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ − 5.00000i − 0.314347i
$$254$$ − 2.00000i − 0.125491i
$$255$$ 2.00000i 0.125245i
$$256$$ 1.00000 0.0625000
$$257$$ 17.0000 1.06043 0.530215 0.847863i $$-0.322111\pi$$
0.530215 + 0.847863i $$0.322111\pi$$
$$258$$ − 11.0000i − 0.684830i
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ 5.00000 0.309492
$$262$$ − 1.00000i − 0.0617802i
$$263$$ 21.0000 1.29492 0.647458 0.762101i $$-0.275832\pi$$
0.647458 + 0.762101i $$0.275832\pi$$
$$264$$ 5.00000 0.307729
$$265$$ − 6.00000i − 0.368577i
$$266$$ 4.00000i 0.245256i
$$267$$ − 2.00000i − 0.122398i
$$268$$ 16.0000i 0.977356i
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ − 13.0000i − 0.789694i −0.918747 0.394847i $$-0.870798\pi$$
0.918747 0.394847i $$-0.129202\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ 11.0000 0.664534
$$275$$ 5.00000i 0.301511i
$$276$$ 1.00000 0.0601929
$$277$$ 11.0000 0.660926 0.330463 0.943819i $$-0.392795\pi$$
0.330463 + 0.943819i $$0.392795\pi$$
$$278$$ 2.00000i 0.119952i
$$279$$ 11.0000i 0.658553i
$$280$$ 2.00000i 0.119523i
$$281$$ − 10.0000i − 0.596550i −0.954480 0.298275i $$-0.903589\pi$$
0.954480 0.298275i $$-0.0964112\pi$$
$$282$$ 9.00000 0.535942
$$283$$ −19.0000 −1.12943 −0.564716 0.825285i $$-0.691014\pi$$
−0.564716 + 0.825285i $$0.691014\pi$$
$$284$$ 0 0
$$285$$ −2.00000 −0.118470
$$286$$ 0 0
$$287$$ 4.00000 0.236113
$$288$$ 1.00000i 0.0589256i
$$289$$ −13.0000 −0.764706
$$290$$ 5.00000 0.293610
$$291$$ − 2.00000i − 0.117242i
$$292$$ 6.00000i 0.351123i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ − 3.00000i − 0.174964i
$$295$$ −15.0000 −0.873334
$$296$$ 3.00000 0.174371
$$297$$ 5.00000i 0.290129i
$$298$$ 17.0000 0.984784
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ − 22.0000i − 1.26806i
$$302$$ −8.00000 −0.460348
$$303$$ 2.00000 0.114897
$$304$$ 2.00000i 0.114708i
$$305$$ − 10.0000i − 0.572598i
$$306$$ 2.00000i 0.114332i
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 10.0000 0.569803
$$309$$ 10.0000 0.568880
$$310$$ 11.0000i 0.624758i
$$311$$ −20.0000 −1.13410 −0.567048 0.823685i $$-0.691915\pi$$
−0.567048 + 0.823685i $$0.691915\pi$$
$$312$$ 0 0
$$313$$ 20.0000 1.13047 0.565233 0.824931i $$-0.308786\pi$$
0.565233 + 0.824931i $$0.308786\pi$$
$$314$$ − 7.00000i − 0.395033i
$$315$$ −2.00000 −0.112687
$$316$$ 11.0000 0.618798
$$317$$ − 16.0000i − 0.898650i −0.893368 0.449325i $$-0.851665\pi$$
0.893368 0.449325i $$-0.148335\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ − 25.0000i − 1.39973i
$$320$$ 1.00000i 0.0559017i
$$321$$ −10.0000 −0.558146
$$322$$ 2.00000 0.111456
$$323$$ 4.00000i 0.222566i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 15.0000 0.830773
$$327$$ − 2.00000i − 0.110600i
$$328$$ 2.00000 0.110432
$$329$$ 18.0000 0.992372
$$330$$ 5.00000i 0.275241i
$$331$$ 28.0000i 1.53902i 0.638635 + 0.769510i $$0.279499\pi$$
−0.638635 + 0.769510i $$0.720501\pi$$
$$332$$ 6.00000i 0.329293i
$$333$$ 3.00000i 0.164399i
$$334$$ −3.00000 −0.164153
$$335$$ −16.0000 −0.874173
$$336$$ 2.00000i 0.109109i
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 0 0
$$339$$ −11.0000 −0.597438
$$340$$ 2.00000i 0.108465i
$$341$$ 55.0000 2.97842
$$342$$ −2.00000 −0.108148
$$343$$ − 20.0000i − 1.07990i
$$344$$ − 11.0000i − 0.593080i
$$345$$ 1.00000i 0.0538382i
$$346$$ − 20.0000i − 1.07521i
$$347$$ 34.0000 1.82522 0.912608 0.408836i $$-0.134065\pi$$
0.912608 + 0.408836i $$0.134065\pi$$
$$348$$ 5.00000 0.268028
$$349$$ 20.0000i 1.07058i 0.844670 + 0.535288i $$0.179797\pi$$
−0.844670 + 0.535288i $$0.820203\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 0 0
$$352$$ 5.00000 0.266501
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ −15.0000 −0.797241
$$355$$ 0 0
$$356$$ − 2.00000i − 0.106000i
$$357$$ 4.00000i 0.211702i
$$358$$ 13.0000i 0.687071i
$$359$$ 12.0000i 0.633336i 0.948536 + 0.316668i $$0.102564\pi$$
−0.948536 + 0.316668i $$0.897436\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 15.0000 0.789474
$$362$$ 16.0000i 0.840941i
$$363$$ 14.0000 0.734809
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ − 10.0000i − 0.522708i
$$367$$ −16.0000 −0.835193 −0.417597 0.908633i $$-0.637127\pi$$
−0.417597 + 0.908633i $$0.637127\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 2.00000i 0.104116i
$$370$$ 3.00000i 0.155963i
$$371$$ − 12.0000i − 0.623009i
$$372$$ 11.0000i 0.570323i
$$373$$ −19.0000 −0.983783 −0.491891 0.870657i $$-0.663694\pi$$
−0.491891 + 0.870657i $$0.663694\pi$$
$$374$$ 10.0000 0.517088
$$375$$ − 1.00000i − 0.0516398i
$$376$$ 9.00000 0.464140
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ 2.00000i 0.102733i 0.998680 + 0.0513665i $$0.0163577\pi$$
−0.998680 + 0.0513665i $$0.983642\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ 2.00000 0.102463
$$382$$ 4.00000i 0.204658i
$$383$$ − 31.0000i − 1.58403i −0.610504 0.792013i $$-0.709033\pi$$
0.610504 0.792013i $$-0.290967\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 10.0000i 0.509647i
$$386$$ 24.0000 1.22157
$$387$$ 11.0000 0.559161
$$388$$ − 2.00000i − 0.101535i
$$389$$ −5.00000 −0.253510 −0.126755 0.991934i $$-0.540456\pi$$
−0.126755 + 0.991934i $$0.540456\pi$$
$$390$$ 0 0
$$391$$ 2.00000 0.101144
$$392$$ − 3.00000i − 0.151523i
$$393$$ 1.00000 0.0504433
$$394$$ −8.00000 −0.403034
$$395$$ 11.0000i 0.553470i
$$396$$ 5.00000i 0.251259i
$$397$$ 3.00000i 0.150566i 0.997162 + 0.0752828i $$0.0239860\pi$$
−0.997162 + 0.0752828i $$0.976014\pi$$
$$398$$ − 24.0000i − 1.20301i
$$399$$ −4.00000 −0.200250
$$400$$ −1.00000 −0.0500000
$$401$$ 36.0000i 1.79775i 0.438201 + 0.898877i $$0.355616\pi$$
−0.438201 + 0.898877i $$0.644384\pi$$
$$402$$ −16.0000 −0.798007
$$403$$ 0 0
$$404$$ 2.00000 0.0995037
$$405$$ − 1.00000i − 0.0496904i
$$406$$ 10.0000 0.496292
$$407$$ 15.0000 0.743522
$$408$$ 2.00000i 0.0990148i
$$409$$ − 30.0000i − 1.48340i −0.670729 0.741702i $$-0.734019\pi$$
0.670729 0.741702i $$-0.265981\pi$$
$$410$$ 2.00000i 0.0987730i
$$411$$ 11.0000i 0.542590i
$$412$$ 10.0000 0.492665
$$413$$ −30.0000 −1.47620
$$414$$ 1.00000i 0.0491473i
$$415$$ −6.00000 −0.294528
$$416$$ 0 0
$$417$$ −2.00000 −0.0979404
$$418$$ 10.0000i 0.489116i
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ 16.0000i 0.779792i 0.920859 + 0.389896i $$0.127489\pi$$
−0.920859 + 0.389896i $$0.872511\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ 9.00000i 0.437595i
$$424$$ − 6.00000i − 0.291386i
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ − 20.0000i − 0.967868i
$$428$$ −10.0000 −0.483368
$$429$$ 0 0
$$430$$ 11.0000 0.530467
$$431$$ − 40.0000i − 1.92673i −0.268190 0.963366i $$-0.586425\pi$$
0.268190 0.963366i $$-0.413575\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −28.0000 −1.34559 −0.672797 0.739827i $$-0.734907\pi$$
−0.672797 + 0.739827i $$0.734907\pi$$
$$434$$ 22.0000i 1.05603i
$$435$$ 5.00000i 0.239732i
$$436$$ − 2.00000i − 0.0957826i
$$437$$ 2.00000i 0.0956730i
$$438$$ −6.00000 −0.286691
$$439$$ 4.00000 0.190910 0.0954548 0.995434i $$-0.469569\pi$$
0.0954548 + 0.995434i $$0.469569\pi$$
$$440$$ 5.00000i 0.238366i
$$441$$ 3.00000 0.142857
$$442$$ 0 0
$$443$$ 20.0000 0.950229 0.475114 0.879924i $$-0.342407\pi$$
0.475114 + 0.879924i $$0.342407\pi$$
$$444$$ 3.00000i 0.142374i
$$445$$ 2.00000 0.0948091
$$446$$ 26.0000 1.23114
$$447$$ 17.0000i 0.804072i
$$448$$ 2.00000i 0.0944911i
$$449$$ 12.0000i 0.566315i 0.959073 + 0.283158i $$0.0913819\pi$$
−0.959073 + 0.283158i $$0.908618\pi$$
$$450$$ − 1.00000i − 0.0471405i
$$451$$ 10.0000 0.470882
$$452$$ −11.0000 −0.517396
$$453$$ − 8.00000i − 0.375873i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ − 38.0000i − 1.77757i −0.458329 0.888783i $$-0.651552\pi$$
0.458329 0.888783i $$-0.348448\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ −2.00000 −0.0933520
$$460$$ 1.00000i 0.0466252i
$$461$$ 39.0000i 1.81641i 0.418524 + 0.908206i $$0.362547\pi$$
−0.418524 + 0.908206i $$0.637453\pi$$
$$462$$ 10.0000i 0.465242i
$$463$$ 14.0000i 0.650635i 0.945605 + 0.325318i $$0.105471\pi$$
−0.945605 + 0.325318i $$0.894529\pi$$
$$464$$ 5.00000 0.232119
$$465$$ −11.0000 −0.510113
$$466$$ − 1.00000i − 0.0463241i
$$467$$ −6.00000 −0.277647 −0.138823 0.990317i $$-0.544332\pi$$
−0.138823 + 0.990317i $$0.544332\pi$$
$$468$$ 0 0
$$469$$ −32.0000 −1.47762
$$470$$ 9.00000i 0.415139i
$$471$$ 7.00000 0.322543
$$472$$ −15.0000 −0.690431
$$473$$ − 55.0000i − 2.52890i
$$474$$ 11.0000i 0.505247i
$$475$$ − 2.00000i − 0.0917663i
$$476$$ 4.00000i 0.183340i
$$477$$ 6.00000 0.274721
$$478$$ −20.0000 −0.914779
$$479$$ − 24.0000i − 1.09659i −0.836286 0.548294i $$-0.815277\pi$$
0.836286 0.548294i $$-0.184723\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ 7.00000 0.318841
$$483$$ 2.00000i 0.0910032i
$$484$$ 14.0000 0.636364
$$485$$ 2.00000 0.0908153
$$486$$ − 1.00000i − 0.0453609i
$$487$$ − 2.00000i − 0.0906287i −0.998973 0.0453143i $$-0.985571\pi$$
0.998973 0.0453143i $$-0.0144289\pi$$
$$488$$ − 10.0000i − 0.452679i
$$489$$ 15.0000i 0.678323i
$$490$$ 3.00000 0.135526
$$491$$ 16.0000 0.722070 0.361035 0.932552i $$-0.382424\pi$$
0.361035 + 0.932552i $$0.382424\pi$$
$$492$$ 2.00000i 0.0901670i
$$493$$ 10.0000 0.450377
$$494$$ 0 0
$$495$$ −5.00000 −0.224733
$$496$$ 11.0000i 0.493915i
$$497$$ 0 0
$$498$$ −6.00000 −0.268866
$$499$$ − 28.0000i − 1.25345i −0.779240 0.626726i $$-0.784395\pi$$
0.779240 0.626726i $$-0.215605\pi$$
$$500$$ − 1.00000i − 0.0447214i
$$501$$ − 3.00000i − 0.134030i
$$502$$ − 25.0000i − 1.11580i
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 2.00000i 0.0889988i
$$506$$ 5.00000 0.222277
$$507$$ 0 0
$$508$$ 2.00000 0.0887357
$$509$$ − 29.0000i − 1.28540i −0.766117 0.642701i $$-0.777814\pi$$
0.766117 0.642701i $$-0.222186\pi$$
$$510$$ −2.00000 −0.0885615
$$511$$ −12.0000 −0.530849
$$512$$ 1.00000i 0.0441942i
$$513$$ − 2.00000i − 0.0883022i
$$514$$ 17.0000i 0.749838i
$$515$$ 10.0000i 0.440653i
$$516$$ 11.0000 0.484248
$$517$$ 45.0000 1.97910
$$518$$ 6.00000i 0.263625i
$$519$$ 20.0000 0.877903
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 5.00000i 0.218844i
$$523$$ −31.0000 −1.35554 −0.677768 0.735276i $$-0.737052\pi$$
−0.677768 + 0.735276i $$0.737052\pi$$
$$524$$ 1.00000 0.0436852
$$525$$ − 2.00000i − 0.0872872i
$$526$$ 21.0000i 0.915644i
$$527$$ 22.0000i 0.958335i
$$528$$ 5.00000i 0.217597i
$$529$$ −22.0000 −0.956522
$$530$$ 6.00000 0.260623
$$531$$ − 15.0000i − 0.650945i
$$532$$ −4.00000 −0.173422
$$533$$ 0 0
$$534$$ 2.00000 0.0865485
$$535$$ − 10.0000i − 0.432338i
$$536$$ −16.0000 −0.691095
$$537$$ −13.0000 −0.560991
$$538$$ 14.0000i 0.603583i
$$539$$ − 15.0000i − 0.646096i
$$540$$ − 1.00000i − 0.0430331i
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 13.0000 0.558398
$$543$$ −16.0000 −0.686626
$$544$$ 2.00000i 0.0857493i
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ 11.0000i 0.469897i
$$549$$ 10.0000 0.426790
$$550$$ −5.00000 −0.213201
$$551$$ 10.0000i 0.426014i
$$552$$ 1.00000i 0.0425628i
$$553$$ 22.0000i 0.935535i
$$554$$ 11.0000i 0.467345i
$$555$$ −3.00000 −0.127343
$$556$$ −2.00000 −0.0848189
$$557$$ 26.0000i 1.10166i 0.834619 + 0.550828i $$0.185688\pi$$
−0.834619 + 0.550828i $$0.814312\pi$$
$$558$$ −11.0000 −0.465667
$$559$$ 0 0
$$560$$ −2.00000 −0.0845154
$$561$$ 10.0000i 0.422200i
$$562$$ 10.0000 0.421825
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 9.00000i 0.378968i
$$565$$ − 11.0000i − 0.462773i
$$566$$ − 19.0000i − 0.798630i
$$567$$ − 2.00000i − 0.0839921i
$$568$$ 0 0
$$569$$ 22.0000 0.922288 0.461144 0.887325i $$-0.347439\pi$$
0.461144 + 0.887325i $$0.347439\pi$$
$$570$$ − 2.00000i − 0.0837708i
$$571$$ 16.0000 0.669579 0.334790 0.942293i $$-0.391335\pi$$
0.334790 + 0.942293i $$0.391335\pi$$
$$572$$ 0 0
$$573$$ −4.00000 −0.167102
$$574$$ 4.00000i 0.166957i
$$575$$ −1.00000 −0.0417029
$$576$$ −1.00000 −0.0416667
$$577$$ 22.0000i 0.915872i 0.888985 + 0.457936i $$0.151411\pi$$
−0.888985 + 0.457936i $$0.848589\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ 24.0000i 0.997406i
$$580$$ 5.00000i 0.207614i
$$581$$ −12.0000 −0.497844
$$582$$ 2.00000 0.0829027
$$583$$ − 30.0000i − 1.24247i
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 18.0000i 0.742940i 0.928445 + 0.371470i $$0.121146\pi$$
−0.928445 + 0.371470i $$0.878854\pi$$
$$588$$ 3.00000 0.123718
$$589$$ −22.0000 −0.906494
$$590$$ − 15.0000i − 0.617540i
$$591$$ − 8.00000i − 0.329076i
$$592$$ 3.00000i 0.123299i
$$593$$ 31.0000i 1.27302i 0.771270 + 0.636509i $$0.219622\pi$$
−0.771270 + 0.636509i $$0.780378\pi$$
$$594$$ −5.00000 −0.205152
$$595$$ −4.00000 −0.163984
$$596$$ 17.0000i 0.696347i
$$597$$ 24.0000 0.982255
$$598$$ 0 0
$$599$$ 42.0000 1.71607 0.858037 0.513588i $$-0.171684\pi$$
0.858037 + 0.513588i $$0.171684\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ −3.00000 −0.122373 −0.0611863 0.998126i $$-0.519488\pi$$
−0.0611863 + 0.998126i $$0.519488\pi$$
$$602$$ 22.0000 0.896653
$$603$$ − 16.0000i − 0.651570i
$$604$$ − 8.00000i − 0.325515i
$$605$$ 14.0000i 0.569181i
$$606$$ 2.00000i 0.0812444i
$$607$$ 18.0000 0.730597 0.365299 0.930890i $$-0.380967\pi$$
0.365299 + 0.930890i $$0.380967\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 10.0000i 0.405220i
$$610$$ 10.0000 0.404888
$$611$$ 0 0
$$612$$ −2.00000 −0.0808452
$$613$$ 29.0000i 1.17130i 0.810564 + 0.585649i $$0.199160\pi$$
−0.810564 + 0.585649i $$0.800840\pi$$
$$614$$ 0 0
$$615$$ −2.00000 −0.0806478
$$616$$ 10.0000i 0.402911i
$$617$$ − 41.0000i − 1.65060i −0.564696 0.825299i $$-0.691007\pi$$
0.564696 0.825299i $$-0.308993\pi$$
$$618$$ 10.0000i 0.402259i
$$619$$ − 10.0000i − 0.401934i −0.979598 0.200967i $$-0.935592\pi$$
0.979598 0.200967i $$-0.0644084\pi$$
$$620$$ −11.0000 −0.441771
$$621$$ −1.00000 −0.0401286
$$622$$ − 20.0000i − 0.801927i
$$623$$ 4.00000 0.160257
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 20.0000i 0.799361i
$$627$$ −10.0000 −0.399362
$$628$$ 7.00000 0.279330
$$629$$ 6.00000i 0.239236i
$$630$$ − 2.00000i − 0.0796819i
$$631$$ 40.0000i 1.59237i 0.605050 + 0.796187i $$0.293153\pi$$
−0.605050 + 0.796187i $$0.706847\pi$$
$$632$$ 11.0000i 0.437557i
$$633$$ 4.00000 0.158986
$$634$$ 16.0000 0.635441
$$635$$ 2.00000i 0.0793676i
$$636$$ 6.00000 0.237915
$$637$$ 0 0
$$638$$ 25.0000 0.989759
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ − 10.0000i − 0.394669i
$$643$$ − 8.00000i − 0.315489i −0.987480 0.157745i $$-0.949578\pi$$
0.987480 0.157745i $$-0.0504223\pi$$
$$644$$ 2.00000i 0.0788110i
$$645$$ 11.0000i 0.433125i
$$646$$ −4.00000 −0.157378
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −75.0000 −2.94401
$$650$$ 0 0
$$651$$ −22.0000 −0.862248
$$652$$ 15.0000i 0.587445i
$$653$$ 34.0000 1.33052 0.665261 0.746611i $$-0.268320\pi$$
0.665261 + 0.746611i $$0.268320\pi$$
$$654$$ 2.00000 0.0782062
$$655$$ 1.00000i 0.0390732i
$$656$$ 2.00000i 0.0780869i
$$657$$ − 6.00000i − 0.234082i
$$658$$ 18.0000i 0.701713i
$$659$$ −39.0000 −1.51922 −0.759612 0.650376i $$-0.774611\pi$$
−0.759612 + 0.650376i $$0.774611\pi$$
$$660$$ −5.00000 −0.194625
$$661$$ 32.0000i 1.24466i 0.782757 + 0.622328i $$0.213813\pi$$
−0.782757 + 0.622328i $$0.786187\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ − 4.00000i − 0.155113i
$$666$$ −3.00000 −0.116248
$$667$$ 5.00000 0.193601
$$668$$ − 3.00000i − 0.116073i
$$669$$ 26.0000i 1.00522i
$$670$$ − 16.0000i − 0.618134i
$$671$$ − 50.0000i − 1.93023i
$$672$$ −2.00000 −0.0771517
$$673$$ −8.00000 −0.308377 −0.154189 0.988041i $$-0.549276\pi$$
−0.154189 + 0.988041i $$0.549276\pi$$
$$674$$ 2.00000i 0.0770371i
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −4.00000 −0.153732 −0.0768662 0.997041i $$-0.524491\pi$$
−0.0768662 + 0.997041i $$0.524491\pi$$
$$678$$ − 11.0000i − 0.422452i
$$679$$ 4.00000 0.153506
$$680$$ −2.00000 −0.0766965
$$681$$ 0 0
$$682$$ 55.0000i 2.10606i
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ − 2.00000i − 0.0764719i
$$685$$ −11.0000 −0.420288
$$686$$ 20.0000 0.763604
$$687$$ − 10.0000i − 0.381524i
$$688$$ 11.0000 0.419371
$$689$$ 0 0
$$690$$ −1.00000 −0.0380693
$$691$$ 30.0000i 1.14125i 0.821209 + 0.570627i $$0.193300\pi$$
−0.821209 + 0.570627i $$0.806700\pi$$
$$692$$ 20.0000 0.760286
$$693$$ −10.0000 −0.379869
$$694$$ 34.0000i 1.29062i
$$695$$ − 2.00000i − 0.0758643i
$$696$$ 5.00000i 0.189525i
$$697$$ 4.00000i 0.151511i
$$698$$ −20.0000 −0.757011
$$699$$ 1.00000 0.0378235
$$700$$ − 2.00000i − 0.0755929i
$$701$$ −13.0000 −0.491003 −0.245502 0.969396i $$-0.578953\pi$$
−0.245502 + 0.969396i $$0.578953\pi$$
$$702$$ 0 0
$$703$$ −6.00000 −0.226294
$$704$$ 5.00000i 0.188445i
$$705$$ −9.00000 −0.338960
$$706$$ −18.0000 −0.677439
$$707$$ 4.00000i 0.150435i
$$708$$ − 15.0000i − 0.563735i
$$709$$ − 8.00000i − 0.300446i −0.988652 0.150223i $$-0.952001\pi$$
0.988652 0.150223i $$-0.0479992\pi$$
$$710$$ 0 0
$$711$$ −11.0000 −0.412532
$$712$$ 2.00000 0.0749532
$$713$$ 11.0000i 0.411953i
$$714$$ −4.00000 −0.149696
$$715$$ 0 0
$$716$$ −13.0000 −0.485833
$$717$$ − 20.0000i − 0.746914i
$$718$$ −12.0000 −0.447836
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ − 1.00000i − 0.0372678i
$$721$$ 20.0000i 0.744839i
$$722$$ 15.0000i 0.558242i
$$723$$ 7.00000i 0.260333i
$$724$$ −16.0000 −0.594635
$$725$$ −5.00000 −0.185695
$$726$$ 14.0000i 0.519589i
$$727$$ 40.0000 1.48352 0.741759 0.670667i $$-0.233992\pi$$
0.741759 + 0.670667i $$0.233992\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ − 6.00000i − 0.222070i
$$731$$ 22.0000 0.813699
$$732$$ 10.0000 0.369611
$$733$$ − 6.00000i − 0.221615i −0.993842 0.110808i $$-0.964656\pi$$
0.993842 0.110808i $$-0.0353437\pi$$
$$734$$ − 16.0000i − 0.590571i
$$735$$ 3.00000i 0.110657i
$$736$$ 1.00000i 0.0368605i
$$737$$ −80.0000 −2.94684
$$738$$ −2.00000 −0.0736210
$$739$$ 44.0000i 1.61857i 0.587419 + 0.809283i $$0.300144\pi$$
−0.587419 + 0.809283i $$0.699856\pi$$
$$740$$ −3.00000 −0.110282
$$741$$ 0 0
$$742$$ 12.0000 0.440534
$$743$$ − 51.0000i − 1.87101i −0.353315 0.935504i $$-0.614946\pi$$
0.353315 0.935504i $$-0.385054\pi$$
$$744$$ −11.0000 −0.403280
$$745$$ −17.0000 −0.622832
$$746$$ − 19.0000i − 0.695639i
$$747$$ − 6.00000i − 0.219529i
$$748$$ 10.0000i 0.365636i
$$749$$ − 20.0000i − 0.730784i
$$750$$ 1.00000 0.0365148
$$751$$ 23.0000 0.839282 0.419641 0.907690i $$-0.362156\pi$$
0.419641 + 0.907690i $$0.362156\pi$$
$$752$$ 9.00000i 0.328196i
$$753$$ 25.0000 0.911051
$$754$$ 0 0
$$755$$ 8.00000 0.291150
$$756$$ − 2.00000i − 0.0727393i
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ −2.00000 −0.0726433
$$759$$ 5.00000i 0.181489i
$$760$$ − 2.00000i − 0.0725476i
$$761$$ 20.0000i 0.724999i 0.931984 + 0.362500i $$0.118077\pi$$
−0.931984 + 0.362500i $$0.881923\pi$$
$$762$$ 2.00000i 0.0724524i
$$763$$ 4.00000 0.144810
$$764$$ −4.00000 −0.144715
$$765$$ − 2.00000i − 0.0723102i
$$766$$ 31.0000 1.12008
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ − 43.0000i − 1.55062i −0.631581 0.775310i $$-0.717594\pi$$
0.631581 0.775310i $$-0.282406\pi$$
$$770$$ −10.0000 −0.360375
$$771$$ −17.0000 −0.612240
$$772$$ 24.0000i 0.863779i
$$773$$ − 32.0000i − 1.15096i −0.817816 0.575480i $$-0.804815\pi$$
0.817816 0.575480i $$-0.195185\pi$$
$$774$$ 11.0000i 0.395387i
$$775$$ − 11.0000i − 0.395132i
$$776$$ 2.00000 0.0717958
$$777$$ −6.00000 −0.215249
$$778$$ − 5.00000i − 0.179259i
$$779$$ −4.00000 −0.143315
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 2.00000i 0.0715199i
$$783$$ −5.00000 −0.178685
$$784$$ 3.00000 0.107143
$$785$$ 7.00000i 0.249841i
$$786$$ 1.00000i 0.0356688i
$$787$$ − 31.0000i − 1.10503i −0.833503 0.552515i $$-0.813668\pi$$
0.833503 0.552515i $$-0.186332\pi$$
$$788$$ − 8.00000i − 0.284988i
$$789$$ −21.0000 −0.747620
$$790$$ −11.0000 −0.391362
$$791$$ − 22.0000i − 0.782230i
$$792$$ −5.00000 −0.177667
$$793$$ 0 0
$$794$$ −3.00000 −0.106466
$$795$$ 6.00000i 0.212798i
$$796$$ 24.0000 0.850657
$$797$$ −12.0000 −0.425062 −0.212531 0.977154i $$-0.568171\pi$$
−0.212531 + 0.977154i $$0.568171\pi$$
$$798$$ − 4.00000i − 0.141598i
$$799$$ 18.0000i 0.636794i
$$800$$ − 1.00000i − 0.0353553i
$$801$$ 2.00000i 0.0706665i
$$802$$ −36.0000 −1.27120
$$803$$ −30.0000 −1.05868
$$804$$ − 16.0000i − 0.564276i
$$805$$ −2.00000 −0.0704907
$$806$$ 0 0
$$807$$ −14.0000 −0.492823
$$808$$ 2.00000i 0.0703598i
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ − 20.0000i − 0.702295i −0.936320 0.351147i $$-0.885792\pi$$
0.936320 0.351147i $$-0.114208\pi$$
$$812$$ 10.0000i 0.350931i
$$813$$ 13.0000i 0.455930i
$$814$$ 15.0000i 0.525750i
$$815$$ −15.0000 −0.525427
$$816$$ −2.00000 −0.0700140
$$817$$ 22.0000i 0.769683i
$$818$$ 30.0000 1.04893
$$819$$ 0 0
$$820$$ −2.00000 −0.0698430
$$821$$ 51.0000i 1.77991i 0.456046 + 0.889956i $$0.349265\pi$$
−0.456046 + 0.889956i $$0.650735\pi$$
$$822$$ −11.0000 −0.383669
$$823$$ 2.00000 0.0697156 0.0348578 0.999392i $$-0.488902\pi$$
0.0348578 + 0.999392i $$0.488902\pi$$
$$824$$ 10.0000i 0.348367i
$$825$$ − 5.00000i − 0.174078i
$$826$$ − 30.0000i − 1.04383i
$$827$$ − 50.0000i − 1.73867i −0.494223 0.869335i $$-0.664547\pi$$
0.494223 0.869335i $$-0.335453\pi$$
$$828$$ −1.00000 −0.0347524
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ − 6.00000i − 0.208263i
$$831$$ −11.0000 −0.381586
$$832$$ 0 0
$$833$$ 6.00000 0.207888
$$834$$ − 2.00000i − 0.0692543i
$$835$$ 3.00000 0.103819
$$836$$ −10.0000 −0.345857
$$837$$ − 11.0000i − 0.380216i
$$838$$ − 4.00000i − 0.138178i
$$839$$ − 54.0000i − 1.86429i −0.362089 0.932144i $$-0.617936\pi$$
0.362089 0.932144i $$-0.382064\pi$$
$$840$$ − 2.00000i − 0.0690066i
$$841$$ −4.00000 −0.137931
$$842$$ −16.0000 −0.551396
$$843$$ 10.0000i 0.344418i
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ −9.00000 −0.309426
$$847$$ 28.0000i 0.962091i
$$848$$ 6.00000 0.206041
$$849$$ 19.0000 0.652078
$$850$$ − 2.00000i − 0.0685994i
$$851$$ 3.00000i 0.102839i
$$852$$ 0 0
$$853$$ − 49.0000i − 1.67773i −0.544341 0.838864i $$-0.683220\pi$$
0.544341 0.838864i $$-0.316780\pi$$
$$854$$ 20.0000 0.684386
$$855$$ 2.00000 0.0683986
$$856$$ − 10.0000i − 0.341793i
$$857$$ 25.0000 0.853984 0.426992 0.904255i $$-0.359573\pi$$
0.426992 + 0.904255i $$0.359573\pi$$
$$858$$ 0 0
$$859$$ −50.0000 −1.70598 −0.852989 0.521929i $$-0.825213\pi$$
−0.852989 + 0.521929i $$0.825213\pi$$
$$860$$ 11.0000i 0.375097i
$$861$$ −4.00000 −0.136320
$$862$$ 40.0000 1.36241
$$863$$ − 39.0000i − 1.32758i −0.747921 0.663788i $$-0.768948\pi$$
0.747921 0.663788i $$-0.231052\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 20.0000i 0.680020i
$$866$$ − 28.0000i − 0.951479i
$$867$$ 13.0000 0.441503
$$868$$ −22.0000 −0.746729
$$869$$ 55.0000i 1.86575i
$$870$$ −5.00000 −0.169516
$$871$$ 0 0
$$872$$ 2.00000 0.0677285
$$873$$ 2.00000i 0.0676897i
$$874$$ −2.00000 −0.0676510
$$875$$ 2.00000 0.0676123
$$876$$ − 6.00000i − 0.202721i
$$877$$ 19.0000i 0.641584i 0.947150 + 0.320792i $$0.103949\pi$$
−0.947150 + 0.320792i $$0.896051\pi$$
$$878$$ 4.00000i 0.134993i
$$879$$ − 6.00000i − 0.202375i
$$880$$ −5.00000 −0.168550
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ 1.00000 0.0336527 0.0168263 0.999858i $$-0.494644\pi$$
0.0168263 + 0.999858i $$0.494644\pi$$
$$884$$ 0 0
$$885$$ 15.0000 0.504219
$$886$$ 20.0000i 0.671913i
$$887$$ 3.00000 0.100730 0.0503651 0.998731i $$-0.483962\pi$$
0.0503651 + 0.998731i $$0.483962\pi$$
$$888$$ −3.00000 −0.100673
$$889$$ 4.00000i 0.134156i
$$890$$ 2.00000i 0.0670402i
$$891$$ − 5.00000i − 0.167506i
$$892$$ 26.0000i 0.870544i
$$893$$ −18.0000 −0.602347
$$894$$ −17.0000 −0.568565
$$895$$ − 13.0000i − 0.434542i
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ −12.0000 −0.400445
$$899$$ 55.0000i 1.83435i
$$900$$ 1.00000 0.0333333
$$901$$ 12.0000 0.399778
$$902$$ 10.0000i 0.332964i
$$903$$ 22.0000i 0.732114i
$$904$$ − 11.0000i − 0.365855i
$$905$$ − 16.0000i − 0.531858i
$$906$$ 8.00000 0.265782
$$907$$ −19.0000 −0.630885 −0.315442 0.948945i $$-0.602153\pi$$
−0.315442 + 0.948945i $$0.602153\pi$$
$$908$$ 0 0
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ −18.0000 −0.596367 −0.298183 0.954509i $$-0.596381\pi$$
−0.298183 + 0.954509i $$0.596381\pi$$
$$912$$ − 2.00000i − 0.0662266i
$$913$$ −30.0000 −0.992855
$$914$$ 38.0000 1.25693
$$915$$ 10.0000i 0.330590i
$$916$$ − 10.0000i − 0.330409i
$$917$$ 2.00000i 0.0660458i
$$918$$ − 2.00000i − 0.0660098i
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ −1.00000 −0.0329690
$$921$$ 0 0
$$922$$ −39.0000 −1.28440
$$923$$ 0 0
$$924$$ −10.0000 −0.328976
$$925$$ − 3.00000i − 0.0986394i
$$926$$ −14.0000 −0.460069
$$927$$ −10.0000 −0.328443
$$928$$ 5.00000i 0.164133i
$$929$$ 16.0000i 0.524943i 0.964940 + 0.262471i $$0.0845376\pi$$
−0.964940 + 0.262471i $$0.915462\pi$$
$$930$$ − 11.0000i − 0.360704i
$$931$$ 6.00000i 0.196642i
$$932$$ 1.00000 0.0327561
$$933$$ 20.0000 0.654771
$$934$$ − 6.00000i − 0.196326i
$$935$$ −10.0000 −0.327035
$$936$$ 0 0
$$937$$ −50.0000 −1.63343 −0.816714 0.577042i $$-0.804207\pi$$
−0.816714 + 0.577042i $$0.804207\pi$$
$$938$$ − 32.0000i − 1.04484i
$$939$$ −20.0000 −0.652675
$$940$$ −9.00000 −0.293548
$$941$$ 50.0000i 1.62995i 0.579494 + 0.814977i $$0.303250\pi$$
−0.579494 + 0.814977i $$0.696750\pi$$
$$942$$ 7.00000i 0.228072i
$$943$$ 2.00000i 0.0651290i
$$944$$ − 15.0000i − 0.488208i
$$945$$ 2.00000 0.0650600
$$946$$ 55.0000 1.78820
$$947$$ 8.00000i 0.259965i 0.991516 + 0.129983i $$0.0414921\pi$$
−0.991516 + 0.129983i $$0.958508\pi$$
$$948$$ −11.0000 −0.357263
$$949$$ 0 0
$$950$$ 2.00000 0.0648886
$$951$$ 16.0000i 0.518836i
$$952$$ −4.00000 −0.129641
$$953$$ 51.0000 1.65205 0.826026 0.563632i $$-0.190596\pi$$
0.826026 + 0.563632i $$0.190596\pi$$
$$954$$ 6.00000i 0.194257i
$$955$$ − 4.00000i − 0.129437i
$$956$$ − 20.0000i − 0.646846i
$$957$$ 25.0000i 0.808135i
$$958$$ 24.0000 0.775405
$$959$$ −22.0000 −0.710417
$$960$$ − 1.00000i − 0.0322749i
$$961$$ −90.0000 −2.90323
$$962$$ 0 0
$$963$$ 10.0000 0.322245
$$964$$ 7.00000i 0.225455i
$$965$$ −24.0000 −0.772587
$$966$$ −2.00000 −0.0643489
$$967$$ − 20.0000i − 0.643157i −0.946883 0.321578i $$-0.895787\pi$$
0.946883 0.321578i $$-0.104213\pi$$
$$968$$ 14.0000i 0.449977i
$$969$$ − 4.00000i − 0.128499i
$$970$$ 2.00000i 0.0642161i
$$971$$ 48.0000 1.54039 0.770197 0.637806i $$-0.220158\pi$$
0.770197 + 0.637806i $$0.220158\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ − 4.00000i − 0.128234i
$$974$$ 2.00000 0.0640841
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ 21.0000i 0.671850i 0.941889 + 0.335925i $$0.109049\pi$$
−0.941889 + 0.335925i $$0.890951\pi$$
$$978$$ −15.0000 −0.479647
$$979$$ 10.0000 0.319601
$$980$$ 3.00000i 0.0958315i
$$981$$ 2.00000i 0.0638551i
$$982$$ 16.0000i 0.510581i
$$983$$ − 3.00000i − 0.0956851i −0.998855 0.0478426i $$-0.984765\pi$$
0.998855 0.0478426i $$-0.0152346\pi$$
$$984$$ −2.00000 −0.0637577
$$985$$ 8.00000 0.254901
$$986$$ 10.0000i 0.318465i
$$987$$ −18.0000 −0.572946
$$988$$ 0 0
$$989$$ 11.0000 0.349780
$$990$$ − 5.00000i − 0.158910i
$$991$$ −17.0000 −0.540023 −0.270011 0.962857i $$-0.587027\pi$$
−0.270011 + 0.962857i $$0.587027\pi$$
$$992$$ −11.0000 −0.349250
$$993$$ − 28.0000i − 0.888553i
$$994$$ 0 0
$$995$$ 24.0000i 0.760851i
$$996$$ − 6.00000i − 0.190117i
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ 28.0000 0.886325
$$999$$ − 3.00000i − 0.0949158i
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 5070.2.b.h.1351.2 2
13.5 odd 4 5070.2.a.p.1.1 1
13.7 odd 12 390.2.i.f.211.1 yes 2
13.8 odd 4 5070.2.a.d.1.1 1
13.11 odd 12 390.2.i.f.61.1 2
13.12 even 2 inner 5070.2.b.h.1351.1 2
39.11 even 12 1170.2.i.a.451.1 2
39.20 even 12 1170.2.i.a.991.1 2
65.7 even 12 1950.2.z.e.1849.1 4
65.24 odd 12 1950.2.i.d.451.1 2
65.33 even 12 1950.2.z.e.1849.2 4
65.37 even 12 1950.2.z.e.1699.2 4
65.59 odd 12 1950.2.i.d.601.1 2
65.63 even 12 1950.2.z.e.1699.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.f.61.1 2 13.11 odd 12
390.2.i.f.211.1 yes 2 13.7 odd 12
1170.2.i.a.451.1 2 39.11 even 12
1170.2.i.a.991.1 2 39.20 even 12
1950.2.i.d.451.1 2 65.24 odd 12
1950.2.i.d.601.1 2 65.59 odd 12
1950.2.z.e.1699.1 4 65.63 even 12
1950.2.z.e.1699.2 4 65.37 even 12
1950.2.z.e.1849.1 4 65.7 even 12
1950.2.z.e.1849.2 4 65.33 even 12
5070.2.a.d.1.1 1 13.8 odd 4
5070.2.a.p.1.1 1 13.5 odd 4
5070.2.b.h.1351.1 2 13.12 even 2 inner
5070.2.b.h.1351.2 2 1.1 even 1 trivial