# Properties

 Label 3380.2.f.e.3041.1 Level $3380$ Weight $2$ Character 3380.3041 Analytic conductor $26.989$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3380 = 2^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3380.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.9894358832$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 260) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3041.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3380.3041 Dual form 3380.2.f.e.3041.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000i q^{5} -1.00000i q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000i q^{5} -1.00000i q^{7} -2.00000 q^{9} +3.00000i q^{11} -1.00000i q^{15} +3.00000 q^{17} +7.00000i q^{19} -1.00000i q^{21} +3.00000 q^{23} -1.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +4.00000i q^{31} +3.00000i q^{33} -1.00000 q^{35} -7.00000i q^{37} +9.00000i q^{41} -11.0000 q^{43} +2.00000i q^{45} +6.00000 q^{49} +3.00000 q^{51} -6.00000 q^{53} +3.00000 q^{55} +7.00000i q^{57} -3.00000i q^{59} +11.0000 q^{61} +2.00000i q^{63} +7.00000i q^{67} +3.00000 q^{69} +3.00000i q^{71} +2.00000i q^{73} -1.00000 q^{75} +3.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -3.00000i q^{85} +3.00000 q^{87} +15.0000i q^{89} +4.00000i q^{93} +7.00000 q^{95} +7.00000i q^{97} -6.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 4 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} - 4 q^{9} + 6 q^{17} + 6 q^{23} - 2 q^{25} - 10 q^{27} + 6 q^{29} - 2 q^{35} - 22 q^{43} + 12 q^{49} + 6 q^{51} - 12 q^{53} + 6 q^{55} + 22 q^{61} + 6 q^{69} - 2 q^{75} + 6 q^{77} + 16 q^{79} + 2 q^{81} + 6 q^{87} + 14 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$1691$$ $$1861$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 0 0
$$5$$ − 1.00000i − 0.447214i
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i −0.981981 0.188982i $$-0.939481\pi$$
0.981981 0.188982i $$-0.0605189\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 3.00000i 0.904534i 0.891883 + 0.452267i $$0.149385\pi$$
−0.891883 + 0.452267i $$0.850615\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ − 1.00000i − 0.258199i
$$16$$ 0 0
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ 7.00000i 1.60591i 0.596040 + 0.802955i $$0.296740\pi$$
−0.596040 + 0.802955i $$0.703260\pi$$
$$20$$ 0 0
$$21$$ − 1.00000i − 0.218218i
$$22$$ 0 0
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 4.00000i 0.718421i 0.933257 + 0.359211i $$0.116954\pi$$
−0.933257 + 0.359211i $$0.883046\pi$$
$$32$$ 0 0
$$33$$ 3.00000i 0.522233i
$$34$$ 0 0
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ − 7.00000i − 1.15079i −0.817875 0.575396i $$-0.804848\pi$$
0.817875 0.575396i $$-0.195152\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 9.00000i 1.40556i 0.711405 + 0.702782i $$0.248059\pi$$
−0.711405 + 0.702782i $$0.751941\pi$$
$$42$$ 0 0
$$43$$ −11.0000 −1.67748 −0.838742 0.544529i $$-0.816708\pi$$
−0.838742 + 0.544529i $$0.816708\pi$$
$$44$$ 0 0
$$45$$ 2.00000i 0.298142i
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ 6.00000 0.857143
$$50$$ 0 0
$$51$$ 3.00000 0.420084
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ 0 0
$$57$$ 7.00000i 0.927173i
$$58$$ 0 0
$$59$$ − 3.00000i − 0.390567i −0.980747 0.195283i $$-0.937437\pi$$
0.980747 0.195283i $$-0.0625627\pi$$
$$60$$ 0 0
$$61$$ 11.0000 1.40841 0.704203 0.709999i $$-0.251305\pi$$
0.704203 + 0.709999i $$0.251305\pi$$
$$62$$ 0 0
$$63$$ 2.00000i 0.251976i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.00000i 0.855186i 0.903971 + 0.427593i $$0.140638\pi$$
−0.903971 + 0.427593i $$0.859362\pi$$
$$68$$ 0 0
$$69$$ 3.00000 0.361158
$$70$$ 0 0
$$71$$ 3.00000i 0.356034i 0.984027 + 0.178017i $$0.0569683\pi$$
−0.984027 + 0.178017i $$0.943032\pi$$
$$72$$ 0 0
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 3.00000 0.341882
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 0 0
$$85$$ − 3.00000i − 0.325396i
$$86$$ 0 0
$$87$$ 3.00000 0.321634
$$88$$ 0 0
$$89$$ 15.0000i 1.59000i 0.606612 + 0.794998i $$0.292528\pi$$
−0.606612 + 0.794998i $$0.707472\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 4.00000i 0.414781i
$$94$$ 0 0
$$95$$ 7.00000 0.718185
$$96$$ 0 0
$$97$$ 7.00000i 0.710742i 0.934725 + 0.355371i $$0.115646\pi$$
−0.934725 + 0.355371i $$0.884354\pi$$
$$98$$ 0 0
$$99$$ − 6.00000i − 0.603023i
$$100$$ 0 0
$$101$$ 9.00000 0.895533 0.447767 0.894150i $$-0.352219\pi$$
0.447767 + 0.894150i $$0.352219\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ −1.00000 −0.0975900
$$106$$ 0 0
$$107$$ 9.00000 0.870063 0.435031 0.900415i $$-0.356737\pi$$
0.435031 + 0.900415i $$0.356737\pi$$
$$108$$ 0 0
$$109$$ − 2.00000i − 0.191565i −0.995402 0.0957826i $$-0.969465\pi$$
0.995402 0.0957826i $$-0.0305354\pi$$
$$110$$ 0 0
$$111$$ − 7.00000i − 0.664411i
$$112$$ 0 0
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ 0 0
$$115$$ − 3.00000i − 0.279751i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 3.00000i − 0.275010i
$$120$$ 0 0
$$121$$ 2.00000 0.181818
$$122$$ 0 0
$$123$$ 9.00000i 0.811503i
$$124$$ 0 0
$$125$$ 1.00000i 0.0894427i
$$126$$ 0 0
$$127$$ 19.0000 1.68598 0.842989 0.537931i $$-0.180794\pi$$
0.842989 + 0.537931i $$0.180794\pi$$
$$128$$ 0 0
$$129$$ −11.0000 −0.968496
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 7.00000 0.606977
$$134$$ 0 0
$$135$$ 5.00000i 0.430331i
$$136$$ 0 0
$$137$$ − 15.0000i − 1.28154i −0.767734 0.640768i $$-0.778616\pi$$
0.767734 0.640768i $$-0.221384\pi$$
$$138$$ 0 0
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ − 3.00000i − 0.249136i
$$146$$ 0 0
$$147$$ 6.00000 0.494872
$$148$$ 0 0
$$149$$ 21.0000i 1.72039i 0.509968 + 0.860194i $$0.329657\pi$$
−0.509968 + 0.860194i $$0.670343\pi$$
$$150$$ 0 0
$$151$$ 8.00000i 0.651031i 0.945537 + 0.325515i $$0.105538\pi$$
−0.945537 + 0.325515i $$0.894462\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ − 3.00000i − 0.236433i
$$162$$ 0 0
$$163$$ − 1.00000i − 0.0783260i −0.999233 0.0391630i $$-0.987531\pi$$
0.999233 0.0391630i $$-0.0124692\pi$$
$$164$$ 0 0
$$165$$ 3.00000 0.233550
$$166$$ 0 0
$$167$$ − 3.00000i − 0.232147i −0.993241 0.116073i $$-0.962969\pi$$
0.993241 0.116073i $$-0.0370308\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ − 14.0000i − 1.07061i
$$172$$ 0 0
$$173$$ 3.00000 0.228086 0.114043 0.993476i $$-0.463620\pi$$
0.114043 + 0.993476i $$0.463620\pi$$
$$174$$ 0 0
$$175$$ 1.00000i 0.0755929i
$$176$$ 0 0
$$177$$ − 3.00000i − 0.225494i
$$178$$ 0 0
$$179$$ 21.0000 1.56961 0.784807 0.619740i $$-0.212762\pi$$
0.784807 + 0.619740i $$0.212762\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 11.0000 0.813143
$$184$$ 0 0
$$185$$ −7.00000 −0.514650
$$186$$ 0 0
$$187$$ 9.00000i 0.658145i
$$188$$ 0 0
$$189$$ 5.00000i 0.363696i
$$190$$ 0 0
$$191$$ −3.00000 −0.217072 −0.108536 0.994092i $$-0.534616\pi$$
−0.108536 + 0.994092i $$0.534616\pi$$
$$192$$ 0 0
$$193$$ 5.00000i 0.359908i 0.983675 + 0.179954i $$0.0575949\pi$$
−0.983675 + 0.179954i $$0.942405\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 21.0000i − 1.49619i −0.663593 0.748094i $$-0.730969\pi$$
0.663593 0.748094i $$-0.269031\pi$$
$$198$$ 0 0
$$199$$ −17.0000 −1.20510 −0.602549 0.798082i $$-0.705848\pi$$
−0.602549 + 0.798082i $$0.705848\pi$$
$$200$$ 0 0
$$201$$ 7.00000i 0.493742i
$$202$$ 0 0
$$203$$ − 3.00000i − 0.210559i
$$204$$ 0 0
$$205$$ 9.00000 0.628587
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ 0 0
$$209$$ −21.0000 −1.45260
$$210$$ 0 0
$$211$$ 11.0000 0.757271 0.378636 0.925546i $$-0.376393\pi$$
0.378636 + 0.925546i $$0.376393\pi$$
$$212$$ 0 0
$$213$$ 3.00000i 0.205557i
$$214$$ 0 0
$$215$$ 11.0000i 0.750194i
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ 0 0
$$219$$ 2.00000i 0.135147i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 19.0000i 1.27233i 0.771551 + 0.636167i $$0.219481\pi$$
−0.771551 + 0.636167i $$0.780519\pi$$
$$224$$ 0 0
$$225$$ 2.00000 0.133333
$$226$$ 0 0
$$227$$ − 27.0000i − 1.79205i −0.444001 0.896026i $$-0.646441\pi$$
0.444001 0.896026i $$-0.353559\pi$$
$$228$$ 0 0
$$229$$ − 22.0000i − 1.45380i −0.686743 0.726900i $$-0.740960\pi$$
0.686743 0.726900i $$-0.259040\pi$$
$$230$$ 0 0
$$231$$ 3.00000 0.197386
$$232$$ 0 0
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ − 1.00000i − 0.0644157i −0.999481 0.0322078i $$-0.989746\pi$$
0.999481 0.0322078i $$-0.0102538\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 0 0
$$245$$ − 6.00000i − 0.383326i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 12.0000i 0.760469i
$$250$$ 0 0
$$251$$ −21.0000 −1.32551 −0.662754 0.748837i $$-0.730613\pi$$
−0.662754 + 0.748837i $$0.730613\pi$$
$$252$$ 0 0
$$253$$ 9.00000i 0.565825i
$$254$$ 0 0
$$255$$ − 3.00000i − 0.187867i
$$256$$ 0 0
$$257$$ −9.00000 −0.561405 −0.280702 0.959795i $$-0.590567\pi$$
−0.280702 + 0.959795i $$0.590567\pi$$
$$258$$ 0 0
$$259$$ −7.00000 −0.434959
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ −3.00000 −0.184988 −0.0924940 0.995713i $$-0.529484\pi$$
−0.0924940 + 0.995713i $$0.529484\pi$$
$$264$$ 0 0
$$265$$ 6.00000i 0.368577i
$$266$$ 0 0
$$267$$ 15.0000i 0.917985i
$$268$$ 0 0
$$269$$ 27.0000 1.64622 0.823110 0.567883i $$-0.192237\pi$$
0.823110 + 0.567883i $$0.192237\pi$$
$$270$$ 0 0
$$271$$ 23.0000i 1.39715i 0.715537 + 0.698575i $$0.246182\pi$$
−0.715537 + 0.698575i $$0.753818\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 3.00000i − 0.180907i
$$276$$ 0 0
$$277$$ 19.0000 1.14160 0.570800 0.821089i $$-0.306633\pi$$
0.570800 + 0.821089i $$0.306633\pi$$
$$278$$ 0 0
$$279$$ − 8.00000i − 0.478947i
$$280$$ 0 0
$$281$$ 6.00000i 0.357930i 0.983855 + 0.178965i $$0.0572749\pi$$
−0.983855 + 0.178965i $$0.942725\pi$$
$$282$$ 0 0
$$283$$ −5.00000 −0.297219 −0.148610 0.988896i $$-0.547480\pi$$
−0.148610 + 0.988896i $$0.547480\pi$$
$$284$$ 0 0
$$285$$ 7.00000 0.414644
$$286$$ 0 0
$$287$$ 9.00000 0.531253
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 7.00000i 0.410347i
$$292$$ 0 0
$$293$$ − 27.0000i − 1.57736i −0.614806 0.788678i $$-0.710766\pi$$
0.614806 0.788678i $$-0.289234\pi$$
$$294$$ 0 0
$$295$$ −3.00000 −0.174667
$$296$$ 0 0
$$297$$ − 15.0000i − 0.870388i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 11.0000i 0.634029i
$$302$$ 0 0
$$303$$ 9.00000 0.517036
$$304$$ 0 0
$$305$$ − 11.0000i − 0.629858i
$$306$$ 0 0
$$307$$ 20.0000i 1.14146i 0.821138 + 0.570730i $$0.193340\pi$$
−0.821138 + 0.570730i $$0.806660\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ 0 0
$$315$$ 2.00000 0.112687
$$316$$ 0 0
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 0 0
$$319$$ 9.00000i 0.503903i
$$320$$ 0 0
$$321$$ 9.00000 0.502331
$$322$$ 0 0
$$323$$ 21.0000i 1.16847i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 2.00000i − 0.110600i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 19.0000i 1.04433i 0.852843 + 0.522167i $$0.174876\pi$$
−0.852843 + 0.522167i $$0.825124\pi$$
$$332$$ 0 0
$$333$$ 14.0000i 0.767195i
$$334$$ 0 0
$$335$$ 7.00000 0.382451
$$336$$ 0 0
$$337$$ 34.0000 1.85210 0.926049 0.377403i $$-0.123183\pi$$
0.926049 + 0.377403i $$0.123183\pi$$
$$338$$ 0 0
$$339$$ 9.00000 0.488813
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ − 13.0000i − 0.701934i
$$344$$ 0 0
$$345$$ − 3.00000i − 0.161515i
$$346$$ 0 0
$$347$$ −33.0000 −1.77153 −0.885766 0.464131i $$-0.846367\pi$$
−0.885766 + 0.464131i $$0.846367\pi$$
$$348$$ 0 0
$$349$$ − 1.00000i − 0.0535288i −0.999642 0.0267644i $$-0.991480\pi$$
0.999642 0.0267644i $$-0.00852039\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 9.00000i − 0.479022i −0.970894 0.239511i $$-0.923013\pi$$
0.970894 0.239511i $$-0.0769871\pi$$
$$354$$ 0 0
$$355$$ 3.00000 0.159223
$$356$$ 0 0
$$357$$ − 3.00000i − 0.158777i
$$358$$ 0 0
$$359$$ − 24.0000i − 1.26667i −0.773877 0.633336i $$-0.781685\pi$$
0.773877 0.633336i $$-0.218315\pi$$
$$360$$ 0 0
$$361$$ −30.0000 −1.57895
$$362$$ 0 0
$$363$$ 2.00000 0.104973
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 0 0
$$367$$ 5.00000 0.260998 0.130499 0.991448i $$-0.458342\pi$$
0.130499 + 0.991448i $$0.458342\pi$$
$$368$$ 0 0
$$369$$ − 18.0000i − 0.937043i
$$370$$ 0 0
$$371$$ 6.00000i 0.311504i
$$372$$ 0 0
$$373$$ −31.0000 −1.60512 −0.802560 0.596572i $$-0.796529\pi$$
−0.802560 + 0.596572i $$0.796529\pi$$
$$374$$ 0 0
$$375$$ 1.00000i 0.0516398i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 1.00000i 0.0513665i 0.999670 + 0.0256833i $$0.00817614\pi$$
−0.999670 + 0.0256833i $$0.991824\pi$$
$$380$$ 0 0
$$381$$ 19.0000 0.973399
$$382$$ 0 0
$$383$$ 9.00000i 0.459879i 0.973205 + 0.229939i $$0.0738528\pi$$
−0.973205 + 0.229939i $$0.926147\pi$$
$$384$$ 0 0
$$385$$ − 3.00000i − 0.152894i
$$386$$ 0 0
$$387$$ 22.0000 1.11832
$$388$$ 0 0
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ 9.00000 0.455150
$$392$$ 0 0
$$393$$ −12.0000 −0.605320
$$394$$ 0 0
$$395$$ − 8.00000i − 0.402524i
$$396$$ 0 0
$$397$$ 5.00000i 0.250943i 0.992097 + 0.125471i $$0.0400443\pi$$
−0.992097 + 0.125471i $$0.959956\pi$$
$$398$$ 0 0
$$399$$ 7.00000 0.350438
$$400$$ 0 0
$$401$$ − 33.0000i − 1.64794i −0.566632 0.823971i $$-0.691754\pi$$
0.566632 0.823971i $$-0.308246\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 1.00000i − 0.0496904i
$$406$$ 0 0
$$407$$ 21.0000 1.04093
$$408$$ 0 0
$$409$$ 25.0000i 1.23617i 0.786111 + 0.618085i $$0.212091\pi$$
−0.786111 + 0.618085i $$0.787909\pi$$
$$410$$ 0 0
$$411$$ − 15.0000i − 0.739895i
$$412$$ 0 0
$$413$$ −3.00000 −0.147620
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ 5.00000 0.244851
$$418$$ 0 0
$$419$$ −9.00000 −0.439679 −0.219839 0.975536i $$-0.570553\pi$$
−0.219839 + 0.975536i $$0.570553\pi$$
$$420$$ 0 0
$$421$$ − 2.00000i − 0.0974740i −0.998812 0.0487370i $$-0.984480\pi$$
0.998812 0.0487370i $$-0.0155196\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −3.00000 −0.145521
$$426$$ 0 0
$$427$$ − 11.0000i − 0.532327i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 9.00000i 0.433515i 0.976226 + 0.216757i $$0.0695480\pi$$
−0.976226 + 0.216757i $$0.930452\pi$$
$$432$$ 0 0
$$433$$ −29.0000 −1.39365 −0.696826 0.717241i $$-0.745405\pi$$
−0.696826 + 0.717241i $$0.745405\pi$$
$$434$$ 0 0
$$435$$ − 3.00000i − 0.143839i
$$436$$ 0 0
$$437$$ 21.0000i 1.00457i
$$438$$ 0 0
$$439$$ −11.0000 −0.525001 −0.262501 0.964932i $$-0.584547\pi$$
−0.262501 + 0.964932i $$0.584547\pi$$
$$440$$ 0 0
$$441$$ −12.0000 −0.571429
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 15.0000 0.711068
$$446$$ 0 0
$$447$$ 21.0000i 0.993266i
$$448$$ 0 0
$$449$$ 3.00000i 0.141579i 0.997491 + 0.0707894i $$0.0225518\pi$$
−0.997491 + 0.0707894i $$0.977448\pi$$
$$450$$ 0 0
$$451$$ −27.0000 −1.27138
$$452$$ 0 0
$$453$$ 8.00000i 0.375873i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 5.00000i − 0.233890i −0.993138 0.116945i $$-0.962690\pi$$
0.993138 0.116945i $$-0.0373101\pi$$
$$458$$ 0 0
$$459$$ −15.0000 −0.700140
$$460$$ 0 0
$$461$$ − 27.0000i − 1.25752i −0.777601 0.628758i $$-0.783564\pi$$
0.777601 0.628758i $$-0.216436\pi$$
$$462$$ 0 0
$$463$$ − 4.00000i − 0.185896i −0.995671 0.0929479i $$-0.970371\pi$$
0.995671 0.0929479i $$-0.0296290\pi$$
$$464$$ 0 0
$$465$$ 4.00000 0.185496
$$466$$ 0 0
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 0 0
$$469$$ 7.00000 0.323230
$$470$$ 0 0
$$471$$ −10.0000 −0.460776
$$472$$ 0 0
$$473$$ − 33.0000i − 1.51734i
$$474$$ 0 0
$$475$$ − 7.00000i − 0.321182i
$$476$$ 0 0
$$477$$ 12.0000 0.549442
$$478$$ 0 0
$$479$$ 39.0000i 1.78196i 0.454047 + 0.890978i $$0.349980\pi$$
−0.454047 + 0.890978i $$0.650020\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ − 3.00000i − 0.136505i
$$484$$ 0 0
$$485$$ 7.00000 0.317854
$$486$$ 0 0
$$487$$ − 11.0000i − 0.498458i −0.968445 0.249229i $$-0.919823\pi$$
0.968445 0.249229i $$-0.0801771\pi$$
$$488$$ 0 0
$$489$$ − 1.00000i − 0.0452216i
$$490$$ 0 0
$$491$$ 9.00000 0.406164 0.203082 0.979162i $$-0.434904\pi$$
0.203082 + 0.979162i $$0.434904\pi$$
$$492$$ 0 0
$$493$$ 9.00000 0.405340
$$494$$ 0 0
$$495$$ −6.00000 −0.269680
$$496$$ 0 0
$$497$$ 3.00000 0.134568
$$498$$ 0 0
$$499$$ − 32.0000i − 1.43252i −0.697835 0.716258i $$-0.745853\pi$$
0.697835 0.716258i $$-0.254147\pi$$
$$500$$ 0 0
$$501$$ − 3.00000i − 0.134030i
$$502$$ 0 0
$$503$$ 15.0000 0.668817 0.334408 0.942428i $$-0.391463\pi$$
0.334408 + 0.942428i $$0.391463\pi$$
$$504$$ 0 0
$$505$$ − 9.00000i − 0.400495i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 9.00000i 0.398918i 0.979906 + 0.199459i $$0.0639185\pi$$
−0.979906 + 0.199459i $$0.936082\pi$$
$$510$$ 0 0
$$511$$ 2.00000 0.0884748
$$512$$ 0 0
$$513$$ − 35.0000i − 1.54529i
$$514$$ 0 0
$$515$$ 8.00000i 0.352522i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 3.00000 0.131685
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 0 0
$$523$$ 29.0000 1.26808 0.634041 0.773300i $$-0.281395\pi$$
0.634041 + 0.773300i $$0.281395\pi$$
$$524$$ 0 0
$$525$$ 1.00000i 0.0436436i
$$526$$ 0 0
$$527$$ 12.0000i 0.522728i
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 6.00000i 0.260378i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ − 9.00000i − 0.389104i
$$536$$ 0 0
$$537$$ 21.0000 0.906217
$$538$$ 0 0
$$539$$ 18.0000i 0.775315i
$$540$$ 0 0
$$541$$ − 22.0000i − 0.945854i −0.881102 0.472927i $$-0.843197\pi$$
0.881102 0.472927i $$-0.156803\pi$$
$$542$$ 0 0
$$543$$ −2.00000 −0.0858282
$$544$$ 0 0
$$545$$ −2.00000 −0.0856706
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 0 0
$$549$$ −22.0000 −0.938937
$$550$$ 0 0
$$551$$ 21.0000i 0.894630i
$$552$$ 0 0
$$553$$ − 8.00000i − 0.340195i
$$554$$ 0 0
$$555$$ −7.00000 −0.297133
$$556$$ 0 0
$$557$$ − 39.0000i − 1.65248i −0.563316 0.826242i $$-0.690475\pi$$
0.563316 0.826242i $$-0.309525\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 9.00000i 0.379980i
$$562$$ 0 0
$$563$$ −39.0000 −1.64365 −0.821827 0.569737i $$-0.807045\pi$$
−0.821827 + 0.569737i $$0.807045\pi$$
$$564$$ 0 0
$$565$$ − 9.00000i − 0.378633i
$$566$$ 0 0
$$567$$ − 1.00000i − 0.0419961i
$$568$$ 0 0
$$569$$ −27.0000 −1.13190 −0.565949 0.824440i $$-0.691490\pi$$
−0.565949 + 0.824440i $$0.691490\pi$$
$$570$$ 0 0
$$571$$ 40.0000 1.67395 0.836974 0.547243i $$-0.184323\pi$$
0.836974 + 0.547243i $$0.184323\pi$$
$$572$$ 0 0
$$573$$ −3.00000 −0.125327
$$574$$ 0 0
$$575$$ −3.00000 −0.125109
$$576$$ 0 0
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ 0 0
$$579$$ 5.00000i 0.207793i
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ 0 0
$$583$$ − 18.0000i − 0.745484i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 33.0000i − 1.36206i −0.732257 0.681028i $$-0.761533\pi$$
0.732257 0.681028i $$-0.238467\pi$$
$$588$$ 0 0
$$589$$ −28.0000 −1.15372
$$590$$ 0 0
$$591$$ − 21.0000i − 0.863825i
$$592$$ 0 0
$$593$$ − 6.00000i − 0.246390i −0.992382 0.123195i $$-0.960686\pi$$
0.992382 0.123195i $$-0.0393141\pi$$
$$594$$ 0 0
$$595$$ −3.00000 −0.122988
$$596$$ 0 0
$$597$$ −17.0000 −0.695764
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 35.0000 1.42768 0.713840 0.700309i $$-0.246954\pi$$
0.713840 + 0.700309i $$0.246954\pi$$
$$602$$ 0 0
$$603$$ − 14.0000i − 0.570124i
$$604$$ 0 0
$$605$$ − 2.00000i − 0.0813116i
$$606$$ 0 0
$$607$$ −13.0000 −0.527654 −0.263827 0.964570i $$-0.584985\pi$$
−0.263827 + 0.964570i $$0.584985\pi$$
$$608$$ 0 0
$$609$$ − 3.00000i − 0.121566i
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 43.0000i 1.73675i 0.495905 + 0.868377i $$0.334836\pi$$
−0.495905 + 0.868377i $$0.665164\pi$$
$$614$$ 0 0
$$615$$ 9.00000 0.362915
$$616$$ 0 0
$$617$$ − 33.0000i − 1.32853i −0.747497 0.664265i $$-0.768745\pi$$
0.747497 0.664265i $$-0.231255\pi$$
$$618$$ 0 0
$$619$$ 44.0000i 1.76851i 0.467005 + 0.884255i $$0.345333\pi$$
−0.467005 + 0.884255i $$0.654667\pi$$
$$620$$ 0 0
$$621$$ −15.0000 −0.601929
$$622$$ 0 0
$$623$$ 15.0000 0.600962
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −21.0000 −0.838659
$$628$$ 0 0
$$629$$ − 21.0000i − 0.837325i
$$630$$ 0 0
$$631$$ 17.0000i 0.676759i 0.941010 + 0.338380i $$0.109879\pi$$
−0.941010 + 0.338380i $$0.890121\pi$$
$$632$$ 0 0
$$633$$ 11.0000 0.437211
$$634$$ 0 0
$$635$$ − 19.0000i − 0.753992i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ − 6.00000i − 0.237356i
$$640$$ 0 0
$$641$$ −27.0000 −1.06644 −0.533218 0.845978i $$-0.679017\pi$$
−0.533218 + 0.845978i $$0.679017\pi$$
$$642$$ 0 0
$$643$$ − 11.0000i − 0.433798i −0.976194 0.216899i $$-0.930406\pi$$
0.976194 0.216899i $$-0.0695942\pi$$
$$644$$ 0 0
$$645$$ 11.0000i 0.433125i
$$646$$ 0 0
$$647$$ −45.0000 −1.76913 −0.884566 0.466415i $$-0.845546\pi$$
−0.884566 + 0.466415i $$0.845546\pi$$
$$648$$ 0 0
$$649$$ 9.00000 0.353281
$$650$$ 0 0
$$651$$ 4.00000 0.156772
$$652$$ 0 0
$$653$$ −39.0000 −1.52619 −0.763094 0.646288i $$-0.776321\pi$$
−0.763094 + 0.646288i $$0.776321\pi$$
$$654$$ 0 0
$$655$$ 12.0000i 0.468879i
$$656$$ 0 0
$$657$$ − 4.00000i − 0.156055i
$$658$$ 0 0
$$659$$ −39.0000 −1.51922 −0.759612 0.650376i $$-0.774611\pi$$
−0.759612 + 0.650376i $$0.774611\pi$$
$$660$$ 0 0
$$661$$ − 1.00000i − 0.0388955i −0.999811 0.0194477i $$-0.993809\pi$$
0.999811 0.0194477i $$-0.00619080\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 7.00000i − 0.271448i
$$666$$ 0 0
$$667$$ 9.00000 0.348481
$$668$$ 0 0
$$669$$ 19.0000i 0.734582i
$$670$$ 0 0
$$671$$ 33.0000i 1.27395i
$$672$$ 0 0
$$673$$ −17.0000 −0.655302 −0.327651 0.944799i $$-0.606257\pi$$
−0.327651 + 0.944799i $$0.606257\pi$$
$$674$$ 0 0
$$675$$ 5.00000 0.192450
$$676$$ 0 0
$$677$$ −42.0000 −1.61419 −0.807096 0.590421i $$-0.798962\pi$$
−0.807096 + 0.590421i $$0.798962\pi$$
$$678$$ 0 0
$$679$$ 7.00000 0.268635
$$680$$ 0 0
$$681$$ − 27.0000i − 1.03464i
$$682$$ 0 0
$$683$$ 51.0000i 1.95146i 0.218975 + 0.975730i $$0.429729\pi$$
−0.218975 + 0.975730i $$0.570271\pi$$
$$684$$ 0 0
$$685$$ −15.0000 −0.573121
$$686$$ 0 0
$$687$$ − 22.0000i − 0.839352i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 23.0000i − 0.874961i −0.899228 0.437481i $$-0.855871\pi$$
0.899228 0.437481i $$-0.144129\pi$$
$$692$$ 0 0
$$693$$ −6.00000 −0.227921
$$694$$ 0 0
$$695$$ − 5.00000i − 0.189661i
$$696$$ 0 0
$$697$$ 27.0000i 1.02270i
$$698$$ 0 0
$$699$$ −18.0000 −0.680823
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 0 0
$$703$$ 49.0000 1.84807
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 9.00000i − 0.338480i
$$708$$ 0 0
$$709$$ − 37.0000i − 1.38956i −0.719220 0.694782i $$-0.755501\pi$$
0.719220 0.694782i $$-0.244499\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ 0 0
$$713$$ 12.0000i 0.449404i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −9.00000 −0.335643 −0.167822 0.985817i $$-0.553673\pi$$
−0.167822 + 0.985817i $$0.553673\pi$$
$$720$$ 0 0
$$721$$ 8.00000i 0.297936i
$$722$$ 0 0
$$723$$ − 1.00000i − 0.0371904i
$$724$$ 0 0
$$725$$ −3.00000 −0.111417
$$726$$ 0 0
$$727$$ 52.0000 1.92857 0.964287 0.264861i $$-0.0853260\pi$$
0.964287 + 0.264861i $$0.0853260\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −33.0000 −1.22055
$$732$$ 0 0
$$733$$ 34.0000i 1.25582i 0.778287 + 0.627909i $$0.216089\pi$$
−0.778287 + 0.627909i $$0.783911\pi$$
$$734$$ 0 0
$$735$$ − 6.00000i − 0.221313i
$$736$$ 0 0
$$737$$ −21.0000 −0.773545
$$738$$ 0 0
$$739$$ 47.0000i 1.72892i 0.502699 + 0.864461i $$0.332340\pi$$
−0.502699 + 0.864461i $$0.667660\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 21.0000i − 0.770415i −0.922830 0.385208i $$-0.874130\pi$$
0.922830 0.385208i $$-0.125870\pi$$
$$744$$ 0 0
$$745$$ 21.0000 0.769380
$$746$$ 0 0
$$747$$ − 24.0000i − 0.878114i
$$748$$ 0 0
$$749$$ − 9.00000i − 0.328853i
$$750$$ 0 0
$$751$$ 13.0000 0.474377 0.237188 0.971464i $$-0.423774\pi$$
0.237188 + 0.971464i $$0.423774\pi$$
$$752$$ 0 0
$$753$$ −21.0000 −0.765283
$$754$$ 0 0
$$755$$ 8.00000 0.291150
$$756$$ 0 0
$$757$$ 29.0000 1.05402 0.527011 0.849858i $$-0.323312\pi$$
0.527011 + 0.849858i $$0.323312\pi$$
$$758$$ 0 0
$$759$$ 9.00000i 0.326679i
$$760$$ 0 0
$$761$$ 3.00000i 0.108750i 0.998521 + 0.0543750i $$0.0173166\pi$$
−0.998521 + 0.0543750i $$0.982683\pi$$
$$762$$ 0 0
$$763$$ −2.00000 −0.0724049
$$764$$ 0 0
$$765$$ 6.00000i 0.216930i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 13.0000i 0.468792i 0.972141 + 0.234396i $$0.0753112\pi$$
−0.972141 + 0.234396i $$0.924689\pi$$
$$770$$ 0 0
$$771$$ −9.00000 −0.324127
$$772$$ 0 0
$$773$$ 27.0000i 0.971123i 0.874203 + 0.485561i $$0.161385\pi$$
−0.874203 + 0.485561i $$0.838615\pi$$
$$774$$ 0 0
$$775$$ − 4.00000i − 0.143684i
$$776$$ 0 0
$$777$$ −7.00000 −0.251124
$$778$$ 0 0
$$779$$ −63.0000 −2.25721
$$780$$ 0 0
$$781$$ −9.00000 −0.322045
$$782$$ 0 0
$$783$$ −15.0000 −0.536056
$$784$$ 0 0
$$785$$ 10.0000i 0.356915i
$$786$$ 0 0
$$787$$ − 37.0000i − 1.31891i −0.751745 0.659454i $$-0.770788\pi$$
0.751745 0.659454i $$-0.229212\pi$$
$$788$$ 0 0
$$789$$ −3.00000 −0.106803
$$790$$ 0 0
$$791$$ − 9.00000i − 0.320003i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 6.00000i 0.212798i
$$796$$ 0 0
$$797$$ 51.0000 1.80651 0.903256 0.429101i $$-0.141170\pi$$
0.903256 + 0.429101i $$0.141170\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ − 30.0000i − 1.06000i
$$802$$ 0 0
$$803$$ −6.00000 −0.211735
$$804$$ 0 0
$$805$$ −3.00000 −0.105736
$$806$$ 0 0
$$807$$ 27.0000 0.950445
$$808$$ 0 0
$$809$$ 15.0000 0.527372 0.263686 0.964609i $$-0.415062\pi$$
0.263686 + 0.964609i $$0.415062\pi$$
$$810$$ 0 0
$$811$$ − 32.0000i − 1.12367i −0.827249 0.561836i $$-0.810095\pi$$
0.827249 0.561836i $$-0.189905\pi$$
$$812$$ 0 0
$$813$$ 23.0000i 0.806645i
$$814$$ 0 0
$$815$$ −1.00000 −0.0350285
$$816$$ 0 0
$$817$$ − 77.0000i − 2.69389i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 15.0000i − 0.523504i −0.965135 0.261752i $$-0.915700\pi$$
0.965135 0.261752i $$-0.0843002\pi$$
$$822$$ 0 0
$$823$$ 13.0000 0.453152 0.226576 0.973994i $$-0.427247\pi$$
0.226576 + 0.973994i $$0.427247\pi$$
$$824$$ 0 0
$$825$$ − 3.00000i − 0.104447i
$$826$$ 0 0
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 0 0
$$829$$ −11.0000 −0.382046 −0.191023 0.981586i $$-0.561180\pi$$
−0.191023 + 0.981586i $$0.561180\pi$$
$$830$$ 0 0
$$831$$ 19.0000 0.659103
$$832$$ 0 0
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ −3.00000 −0.103819
$$836$$ 0 0
$$837$$ − 20.0000i − 0.691301i
$$838$$ 0 0
$$839$$ 21.0000i 0.725001i 0.931984 + 0.362500i $$0.118077\pi$$
−0.931984 + 0.362500i $$0.881923\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 6.00000i 0.206651i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 2.00000i − 0.0687208i
$$848$$ 0 0
$$849$$ −5.00000 −0.171600
$$850$$ 0 0
$$851$$ − 21.0000i − 0.719871i
$$852$$ 0 0
$$853$$ − 22.0000i − 0.753266i −0.926363 0.376633i $$-0.877082\pi$$
0.926363 0.376633i $$-0.122918\pi$$
$$854$$ 0 0
$$855$$ −14.0000 −0.478790
$$856$$ 0 0
$$857$$ 30.0000 1.02478 0.512390 0.858753i $$-0.328760\pi$$
0.512390 + 0.858753i $$0.328760\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 0 0
$$861$$ 9.00000 0.306719
$$862$$ 0 0
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ 0 0
$$865$$ − 3.00000i − 0.102003i
$$866$$ 0 0
$$867$$ −8.00000 −0.271694
$$868$$ 0 0
$$869$$ 24.0000i 0.814144i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ − 14.0000i − 0.473828i
$$874$$ 0 0
$$875$$ 1.00000 0.0338062
$$876$$ 0 0
$$877$$ − 41.0000i − 1.38447i −0.721671 0.692236i $$-0.756626\pi$$
0.721671 0.692236i $$-0.243374\pi$$
$$878$$ 0 0
$$879$$ − 27.0000i − 0.910687i
$$880$$ 0 0
$$881$$ −27.0000 −0.909653 −0.454827 0.890580i $$-0.650299\pi$$
−0.454827 + 0.890580i $$0.650299\pi$$
$$882$$ 0 0
$$883$$ 4.00000 0.134611 0.0673054 0.997732i $$-0.478560\pi$$
0.0673054 + 0.997732i $$0.478560\pi$$
$$884$$ 0 0
$$885$$ −3.00000 −0.100844
$$886$$ 0 0
$$887$$ 9.00000 0.302190 0.151095 0.988519i $$-0.451720\pi$$
0.151095 + 0.988519i $$0.451720\pi$$
$$888$$ 0 0
$$889$$ − 19.0000i − 0.637240i
$$890$$ 0 0
$$891$$ 3.00000i 0.100504i
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ − 21.0000i − 0.701953i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 12.0000i 0.400222i
$$900$$ 0 0
$$901$$ −18.0000 −0.599667
$$902$$ 0 0
$$903$$ 11.0000i 0.366057i
$$904$$ 0 0
$$905$$ 2.00000i 0.0664822i
$$906$$ 0 0
$$907$$ 19.0000 0.630885 0.315442 0.948945i $$-0.397847\pi$$
0.315442 + 0.948945i $$0.397847\pi$$
$$908$$ 0 0
$$909$$ −18.0000 −0.597022
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ −36.0000 −1.19143
$$914$$ 0 0
$$915$$ − 11.0000i − 0.363649i
$$916$$ 0 0
$$917$$ 12.0000i 0.396275i
$$918$$ 0 0
$$919$$ 29.0000 0.956622 0.478311 0.878191i $$-0.341249\pi$$
0.478311 + 0.878191i $$0.341249\pi$$
$$920$$ 0 0
$$921$$ 20.0000i 0.659022i
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 7.00000i 0.230159i
$$926$$ 0 0
$$927$$ 16.0000 0.525509
$$928$$ 0 0
$$929$$ − 51.0000i − 1.67326i −0.547772 0.836628i $$-0.684524\pi$$
0.547772 0.836628i $$-0.315476\pi$$
$$930$$ 0 0
$$931$$ 42.0000i 1.37649i
$$932$$ 0 0
$$933$$ −24.0000 −0.785725
$$934$$ 0 0
$$935$$ 9.00000 0.294331
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 0 0
$$939$$ −22.0000 −0.717943
$$940$$ 0 0
$$941$$ − 42.0000i − 1.36916i −0.728937 0.684580i $$-0.759985\pi$$
0.728937 0.684580i $$-0.240015\pi$$
$$942$$ 0 0
$$943$$ 27.0000i 0.879241i
$$944$$ 0 0
$$945$$ 5.00000 0.162650
$$946$$ 0 0
$$947$$ 21.0000i 0.682408i 0.939989 + 0.341204i $$0.110835\pi$$
−0.939989 + 0.341204i $$0.889165\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 18.0000i 0.583690i
$$952$$ 0 0
$$953$$ 51.0000 1.65205 0.826026 0.563632i $$-0.190596\pi$$
0.826026 + 0.563632i $$0.190596\pi$$
$$954$$ 0 0
$$955$$ 3.00000i 0.0970777i
$$956$$ 0 0
$$957$$ 9.00000i 0.290929i
$$958$$ 0 0
$$959$$ −15.0000 −0.484375
$$960$$ 0 0
$$961$$ 15.0000 0.483871
$$962$$ 0 0
$$963$$ −18.0000 −0.580042
$$964$$ 0 0
$$965$$ 5.00000 0.160956
$$966$$ 0 0
$$967$$ 40.0000i 1.28631i 0.765735 + 0.643157i $$0.222376\pi$$
−0.765735 + 0.643157i $$0.777624\pi$$
$$968$$ 0 0
$$969$$ 21.0000i 0.674617i
$$970$$ 0 0
$$971$$ 21.0000 0.673922 0.336961 0.941519i $$-0.390601\pi$$
0.336961 + 0.941519i $$0.390601\pi$$
$$972$$ 0 0
$$973$$ − 5.00000i − 0.160293i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 9.00000i − 0.287936i −0.989582 0.143968i $$-0.954014\pi$$
0.989582 0.143968i $$-0.0459862\pi$$
$$978$$ 0 0
$$979$$ −45.0000 −1.43821
$$980$$ 0 0
$$981$$ 4.00000i 0.127710i
$$982$$ 0 0
$$983$$ − 36.0000i − 1.14822i −0.818778 0.574111i $$-0.805348\pi$$
0.818778 0.574111i $$-0.194652\pi$$
$$984$$ 0 0
$$985$$ −21.0000 −0.669116
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −33.0000 −1.04934
$$990$$ 0 0
$$991$$ −61.0000 −1.93773 −0.968864 0.247592i $$-0.920361\pi$$
−0.968864 + 0.247592i $$0.920361\pi$$
$$992$$ 0 0
$$993$$ 19.0000i 0.602947i
$$994$$ 0 0
$$995$$ 17.0000i 0.538936i
$$996$$ 0 0
$$997$$ −31.0000 −0.981780 −0.490890 0.871222i $$-0.663328\pi$$
−0.490890 + 0.871222i $$0.663328\pi$$
$$998$$ 0 0
$$999$$ 35.0000i 1.10735i
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 3380.2.f.e.3041.1 2
13.5 odd 4 3380.2.a.g.1.1 1
13.7 odd 12 260.2.i.b.81.1 yes 2
13.8 odd 4 3380.2.a.h.1.1 1
13.11 odd 12 260.2.i.b.61.1 2
13.12 even 2 inner 3380.2.f.e.3041.2 2
39.11 even 12 2340.2.q.b.1621.1 2
39.20 even 12 2340.2.q.b.2161.1 2
52.7 even 12 1040.2.q.j.81.1 2
52.11 even 12 1040.2.q.j.321.1 2
65.7 even 12 1300.2.bb.a.549.1 4
65.24 odd 12 1300.2.i.e.1101.1 2
65.33 even 12 1300.2.bb.a.549.2 4
65.37 even 12 1300.2.bb.a.1049.2 4
65.59 odd 12 1300.2.i.e.601.1 2
65.63 even 12 1300.2.bb.a.1049.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.i.b.61.1 2 13.11 odd 12
260.2.i.b.81.1 yes 2 13.7 odd 12
1040.2.q.j.81.1 2 52.7 even 12
1040.2.q.j.321.1 2 52.11 even 12
1300.2.i.e.601.1 2 65.59 odd 12
1300.2.i.e.1101.1 2 65.24 odd 12
1300.2.bb.a.549.1 4 65.7 even 12
1300.2.bb.a.549.2 4 65.33 even 12
1300.2.bb.a.1049.1 4 65.63 even 12
1300.2.bb.a.1049.2 4 65.37 even 12
2340.2.q.b.1621.1 2 39.11 even 12
2340.2.q.b.2161.1 2 39.20 even 12
3380.2.a.g.1.1 1 13.5 odd 4
3380.2.a.h.1.1 1 13.8 odd 4
3380.2.f.e.3041.1 2 1.1 even 1 trivial
3380.2.f.e.3041.2 2 13.12 even 2 inner