Properties

 Label 5070.2.b.l.1351.1 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 1351.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.l.1351.2

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +5.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +5.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000i q^{11} -1.00000 q^{12} +5.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +8.00000 q^{17} -1.00000i q^{18} -5.00000i q^{19} +1.00000i q^{20} +5.00000i q^{21} +3.00000 q^{22} +4.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -5.00000i q^{28} -4.00000 q^{29} -1.00000 q^{30} -2.00000i q^{31} -1.00000i q^{32} +3.00000i q^{33} -8.00000i q^{34} +5.00000 q^{35} -1.00000 q^{36} +7.00000i q^{37} -5.00000 q^{38} +1.00000 q^{40} +6.00000i q^{41} +5.00000 q^{42} -6.00000 q^{43} -3.00000i q^{44} -1.00000i q^{45} -4.00000i q^{46} +3.00000i q^{47} +1.00000 q^{48} -18.0000 q^{49} +1.00000i q^{50} +8.00000 q^{51} +1.00000 q^{53} -1.00000i q^{54} +3.00000 q^{55} -5.00000 q^{56} -5.00000i q^{57} +4.00000i q^{58} -12.0000i q^{59} +1.00000i q^{60} +2.00000 q^{61} -2.00000 q^{62} +5.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +8.00000i q^{67} -8.00000 q^{68} +4.00000 q^{69} -5.00000i q^{70} +2.00000i q^{71} +1.00000i q^{72} +7.00000 q^{74} -1.00000 q^{75} +5.00000i q^{76} -15.0000 q^{77} -2.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} +8.00000i q^{83} -5.00000i q^{84} -8.00000i q^{85} +6.00000i q^{86} -4.00000 q^{87} -3.00000 q^{88} +11.0000i q^{89} -1.00000 q^{90} -4.00000 q^{92} -2.00000i q^{93} +3.00000 q^{94} -5.00000 q^{95} -1.00000i q^{96} +18.0000i q^{98} +3.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{4} + 2q^{9} - 2q^{10} - 2q^{12} + 10q^{14} + 2q^{16} + 16q^{17} + 6q^{22} + 8q^{23} - 2q^{25} + 2q^{27} - 8q^{29} - 2q^{30} + 10q^{35} - 2q^{36} - 10q^{38} + 2q^{40} + 10q^{42} - 12q^{43} + 2q^{48} - 36q^{49} + 16q^{51} + 2q^{53} + 6q^{55} - 10q^{56} + 4q^{61} - 4q^{62} - 2q^{64} + 6q^{66} - 16q^{68} + 8q^{69} + 14q^{74} - 2q^{75} - 30q^{77} - 4q^{79} + 2q^{81} + 12q^{82} - 8q^{87} - 6q^{88} - 2q^{90} - 8q^{92} + 6q^{94} - 10q^{95} + O(q^{100})$$

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
$$7$$ 5.00000i 1.88982i 0.327327 + 0.944911i $$0.393852\pi$$
−0.327327 + 0.944911i $$0.606148\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 3.00000i 0.904534i 0.891883 + 0.452267i $$0.149385\pi$$
−0.891883 + 0.452267i $$0.850615\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 5.00000 1.33631
$$15$$ − 1.00000i − 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ 8.00000 1.94029 0.970143 0.242536i $$-0.0779791\pi$$
0.970143 + 0.242536i $$0.0779791\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ − 5.00000i − 1.14708i −0.819178 0.573539i $$-0.805570\pi$$
0.819178 0.573539i $$-0.194430\pi$$
$$20$$ 1.00000i 0.223607i
$$21$$ 5.00000i 1.09109i
$$22$$ 3.00000 0.639602
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ − 5.00000i − 0.944911i
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ − 2.00000i − 0.359211i −0.983739 0.179605i $$-0.942518\pi$$
0.983739 0.179605i $$-0.0574821\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 3.00000i 0.522233i
$$34$$ − 8.00000i − 1.37199i
$$35$$ 5.00000 0.845154
$$36$$ −1.00000 −0.166667
$$37$$ 7.00000i 1.15079i 0.817875 + 0.575396i $$0.195152\pi$$
−0.817875 + 0.575396i $$0.804848\pi$$
$$38$$ −5.00000 −0.811107
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 6.00000i 0.937043i 0.883452 + 0.468521i $$0.155213\pi$$
−0.883452 + 0.468521i $$0.844787\pi$$
$$42$$ 5.00000 0.771517
$$43$$ −6.00000 −0.914991 −0.457496 0.889212i $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ − 3.00000i − 0.452267i
$$45$$ − 1.00000i − 0.149071i
$$46$$ − 4.00000i − 0.589768i
$$47$$ 3.00000i 0.437595i 0.975770 + 0.218797i $$0.0702134\pi$$
−0.975770 + 0.218797i $$0.929787\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −18.0000 −2.57143
$$50$$ 1.00000i 0.141421i
$$51$$ 8.00000 1.12022
$$52$$ 0 0
$$53$$ 1.00000 0.137361 0.0686803 0.997639i $$-0.478121\pi$$
0.0686803 + 0.997639i $$0.478121\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 3.00000 0.404520
$$56$$ −5.00000 −0.668153
$$57$$ − 5.00000i − 0.662266i
$$58$$ 4.00000i 0.525226i
$$59$$ − 12.0000i − 1.56227i −0.624364 0.781133i $$-0.714642\pi$$
0.624364 0.781133i $$-0.285358\pi$$
$$60$$ 1.00000i 0.129099i
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ 5.00000i 0.629941i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 3.00000 0.369274
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ −8.00000 −0.970143
$$69$$ 4.00000 0.481543
$$70$$ − 5.00000i − 0.597614i
$$71$$ 2.00000i 0.237356i 0.992933 + 0.118678i $$0.0378657\pi$$
−0.992933 + 0.118678i $$0.962134\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 7.00000 0.813733
$$75$$ −1.00000 −0.115470
$$76$$ 5.00000i 0.573539i
$$77$$ −15.0000 −1.70941
$$78$$ 0 0
$$79$$ −2.00000 −0.225018 −0.112509 0.993651i $$-0.535889\pi$$
−0.112509 + 0.993651i $$0.535889\pi$$
$$80$$ − 1.00000i − 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ 8.00000i 0.878114i 0.898459 + 0.439057i $$0.144687\pi$$
−0.898459 + 0.439057i $$0.855313\pi$$
$$84$$ − 5.00000i − 0.545545i
$$85$$ − 8.00000i − 0.867722i
$$86$$ 6.00000i 0.646997i
$$87$$ −4.00000 −0.428845
$$88$$ −3.00000 −0.319801
$$89$$ 11.0000i 1.16600i 0.812473 + 0.582999i $$0.198121\pi$$
−0.812473 + 0.582999i $$0.801879\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −4.00000 −0.417029
$$93$$ − 2.00000i − 0.207390i
$$94$$ 3.00000 0.309426
$$95$$ −5.00000 −0.512989
$$96$$ − 1.00000i − 0.102062i
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 18.0000i 1.81827i
$$99$$ 3.00000i 0.301511i
$$100$$ 1.00000 0.100000
$$101$$ 8.00000 0.796030 0.398015 0.917379i $$-0.369699\pi$$
0.398015 + 0.917379i $$0.369699\pi$$
$$102$$ − 8.00000i − 0.792118i
$$103$$ 7.00000 0.689730 0.344865 0.938652i $$-0.387925\pi$$
0.344865 + 0.938652i $$0.387925\pi$$
$$104$$ 0 0
$$105$$ 5.00000 0.487950
$$106$$ − 1.00000i − 0.0971286i
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 14.0000i 1.34096i 0.741929 + 0.670478i $$0.233911\pi$$
−0.741929 + 0.670478i $$0.766089\pi$$
$$110$$ − 3.00000i − 0.286039i
$$111$$ 7.00000i 0.664411i
$$112$$ 5.00000i 0.472456i
$$113$$ 8.00000 0.752577 0.376288 0.926503i $$-0.377200\pi$$
0.376288 + 0.926503i $$0.377200\pi$$
$$114$$ −5.00000 −0.468293
$$115$$ − 4.00000i − 0.373002i
$$116$$ 4.00000 0.371391
$$117$$ 0 0
$$118$$ −12.0000 −1.10469
$$119$$ 40.0000i 3.66679i
$$120$$ 1.00000 0.0912871
$$121$$ 2.00000 0.181818
$$122$$ − 2.00000i − 0.181071i
$$123$$ 6.00000i 0.541002i
$$124$$ 2.00000i 0.179605i
$$125$$ 1.00000i 0.0894427i
$$126$$ 5.00000 0.445435
$$127$$ 21.0000 1.86345 0.931724 0.363166i $$-0.118304\pi$$
0.931724 + 0.363166i $$0.118304\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −6.00000 −0.528271
$$130$$ 0 0
$$131$$ −19.0000 −1.66004 −0.830019 0.557735i $$-0.811670\pi$$
−0.830019 + 0.557735i $$0.811670\pi$$
$$132$$ − 3.00000i − 0.261116i
$$133$$ 25.0000 2.16777
$$134$$ 8.00000 0.691095
$$135$$ − 1.00000i − 0.0860663i
$$136$$ 8.00000i 0.685994i
$$137$$ 12.0000i 1.02523i 0.858619 + 0.512615i $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ 7.00000 0.593732 0.296866 0.954919i $$-0.404058\pi$$
0.296866 + 0.954919i $$0.404058\pi$$
$$140$$ −5.00000 −0.422577
$$141$$ 3.00000i 0.252646i
$$142$$ 2.00000 0.167836
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 4.00000i 0.332182i
$$146$$ 0 0
$$147$$ −18.0000 −1.48461
$$148$$ − 7.00000i − 0.575396i
$$149$$ − 2.00000i − 0.163846i −0.996639 0.0819232i $$-0.973894\pi$$
0.996639 0.0819232i $$-0.0261062\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ − 22.0000i − 1.79033i −0.445730 0.895167i $$-0.647056\pi$$
0.445730 0.895167i $$-0.352944\pi$$
$$152$$ 5.00000 0.405554
$$153$$ 8.00000 0.646762
$$154$$ 15.0000i 1.20873i
$$155$$ −2.00000 −0.160644
$$156$$ 0 0
$$157$$ 15.0000 1.19713 0.598565 0.801074i $$-0.295738\pi$$
0.598565 + 0.801074i $$0.295738\pi$$
$$158$$ 2.00000i 0.159111i
$$159$$ 1.00000 0.0793052
$$160$$ −1.00000 −0.0790569
$$161$$ 20.0000i 1.57622i
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 20.0000i 1.56652i 0.621694 + 0.783260i $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ − 6.00000i − 0.468521i
$$165$$ 3.00000 0.233550
$$166$$ 8.00000 0.620920
$$167$$ 23.0000i 1.77979i 0.456162 + 0.889897i $$0.349224\pi$$
−0.456162 + 0.889897i $$0.650776\pi$$
$$168$$ −5.00000 −0.385758
$$169$$ 0 0
$$170$$ −8.00000 −0.613572
$$171$$ − 5.00000i − 0.382360i
$$172$$ 6.00000 0.457496
$$173$$ −5.00000 −0.380143 −0.190071 0.981770i $$-0.560872\pi$$
−0.190071 + 0.981770i $$0.560872\pi$$
$$174$$ 4.00000i 0.303239i
$$175$$ − 5.00000i − 0.377964i
$$176$$ 3.00000i 0.226134i
$$177$$ − 12.0000i − 0.901975i
$$178$$ 11.0000 0.824485
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 1.00000i 0.0745356i
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 4.00000i 0.294884i
$$185$$ 7.00000 0.514650
$$186$$ −2.00000 −0.146647
$$187$$ 24.0000i 1.75505i
$$188$$ − 3.00000i − 0.218797i
$$189$$ 5.00000i 0.363696i
$$190$$ 5.00000i 0.362738i
$$191$$ 2.00000 0.144715 0.0723575 0.997379i $$-0.476948\pi$$
0.0723575 + 0.997379i $$0.476948\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ − 24.0000i − 1.72756i −0.503871 0.863779i $$-0.668091\pi$$
0.503871 0.863779i $$-0.331909\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 18.0000 1.28571
$$197$$ 3.00000i 0.213741i 0.994273 + 0.106871i $$0.0340831\pi$$
−0.994273 + 0.106871i $$0.965917\pi$$
$$198$$ 3.00000 0.213201
$$199$$ −22.0000 −1.55954 −0.779769 0.626067i $$-0.784664\pi$$
−0.779769 + 0.626067i $$0.784664\pi$$
$$200$$ − 1.00000i − 0.0707107i
$$201$$ 8.00000i 0.564276i
$$202$$ − 8.00000i − 0.562878i
$$203$$ − 20.0000i − 1.40372i
$$204$$ −8.00000 −0.560112
$$205$$ 6.00000 0.419058
$$206$$ − 7.00000i − 0.487713i
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ 15.0000 1.03757
$$210$$ − 5.00000i − 0.345033i
$$211$$ −15.0000 −1.03264 −0.516321 0.856395i $$-0.672699\pi$$
−0.516321 + 0.856395i $$0.672699\pi$$
$$212$$ −1.00000 −0.0686803
$$213$$ 2.00000i 0.137038i
$$214$$ 6.00000i 0.410152i
$$215$$ 6.00000i 0.409197i
$$216$$ 1.00000i 0.0680414i
$$217$$ 10.0000 0.678844
$$218$$ 14.0000 0.948200
$$219$$ 0 0
$$220$$ −3.00000 −0.202260
$$221$$ 0 0
$$222$$ 7.00000 0.469809
$$223$$ − 3.00000i − 0.200895i −0.994942 0.100447i $$-0.967973\pi$$
0.994942 0.100447i $$-0.0320274\pi$$
$$224$$ 5.00000 0.334077
$$225$$ −1.00000 −0.0666667
$$226$$ − 8.00000i − 0.532152i
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 5.00000i 0.331133i
$$229$$ − 14.0000i − 0.925146i −0.886581 0.462573i $$-0.846926\pi$$
0.886581 0.462573i $$-0.153074\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ −15.0000 −0.986928
$$232$$ − 4.00000i − 0.262613i
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ 0 0
$$235$$ 3.00000 0.195698
$$236$$ 12.0000i 0.781133i
$$237$$ −2.00000 −0.129914
$$238$$ 40.0000 2.59281
$$239$$ − 18.0000i − 1.16432i −0.813073 0.582162i $$-0.802207\pi$$
0.813073 0.582162i $$-0.197793\pi$$
$$240$$ − 1.00000i − 0.0645497i
$$241$$ 25.0000i 1.61039i 0.593009 + 0.805196i $$0.297940\pi$$
−0.593009 + 0.805196i $$0.702060\pi$$
$$242$$ − 2.00000i − 0.128565i
$$243$$ 1.00000 0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 18.0000i 1.14998i
$$246$$ 6.00000 0.382546
$$247$$ 0 0
$$248$$ 2.00000 0.127000
$$249$$ 8.00000i 0.506979i
$$250$$ 1.00000 0.0632456
$$251$$ −15.0000 −0.946792 −0.473396 0.880850i $$-0.656972\pi$$
−0.473396 + 0.880850i $$0.656972\pi$$
$$252$$ − 5.00000i − 0.314970i
$$253$$ 12.0000i 0.754434i
$$254$$ − 21.0000i − 1.31766i
$$255$$ − 8.00000i − 0.500979i
$$256$$ 1.00000 0.0625000
$$257$$ −28.0000 −1.74659 −0.873296 0.487190i $$-0.838022\pi$$
−0.873296 + 0.487190i $$0.838022\pi$$
$$258$$ 6.00000i 0.373544i
$$259$$ −35.0000 −2.17479
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 19.0000i 1.17382i
$$263$$ −15.0000 −0.924940 −0.462470 0.886635i $$-0.653037\pi$$
−0.462470 + 0.886635i $$0.653037\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ − 1.00000i − 0.0614295i
$$266$$ − 25.0000i − 1.53285i
$$267$$ 11.0000i 0.673189i
$$268$$ − 8.00000i − 0.488678i
$$269$$ 4.00000 0.243884 0.121942 0.992537i $$-0.461088\pi$$
0.121942 + 0.992537i $$0.461088\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ − 4.00000i − 0.242983i −0.992592 0.121491i $$-0.961232\pi$$
0.992592 0.121491i $$-0.0387677\pi$$
$$272$$ 8.00000 0.485071
$$273$$ 0 0
$$274$$ 12.0000 0.724947
$$275$$ − 3.00000i − 0.180907i
$$276$$ −4.00000 −0.240772
$$277$$ 15.0000 0.901263 0.450631 0.892710i $$-0.351199\pi$$
0.450631 + 0.892710i $$0.351199\pi$$
$$278$$ − 7.00000i − 0.419832i
$$279$$ − 2.00000i − 0.119737i
$$280$$ 5.00000i 0.298807i
$$281$$ − 18.0000i − 1.07379i −0.843649 0.536895i $$-0.819597\pi$$
0.843649 0.536895i $$-0.180403\pi$$
$$282$$ 3.00000 0.178647
$$283$$ 10.0000 0.594438 0.297219 0.954809i $$-0.403941\pi$$
0.297219 + 0.954809i $$0.403941\pi$$
$$284$$ − 2.00000i − 0.118678i
$$285$$ −5.00000 −0.296174
$$286$$ 0 0
$$287$$ −30.0000 −1.77084
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 47.0000 2.76471
$$290$$ 4.00000 0.234888
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 9.00000i 0.525786i 0.964825 + 0.262893i $$0.0846766\pi$$
−0.964825 + 0.262893i $$0.915323\pi$$
$$294$$ 18.0000i 1.04978i
$$295$$ −12.0000 −0.698667
$$296$$ −7.00000 −0.406867
$$297$$ 3.00000i 0.174078i
$$298$$ −2.00000 −0.115857
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ − 30.0000i − 1.72917i
$$302$$ −22.0000 −1.26596
$$303$$ 8.00000 0.459588
$$304$$ − 5.00000i − 0.286770i
$$305$$ − 2.00000i − 0.114520i
$$306$$ − 8.00000i − 0.457330i
$$307$$ 6.00000i 0.342438i 0.985233 + 0.171219i $$0.0547706\pi$$
−0.985233 + 0.171219i $$0.945229\pi$$
$$308$$ 15.0000 0.854704
$$309$$ 7.00000 0.398216
$$310$$ 2.00000i 0.113592i
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ − 15.0000i − 0.846499i
$$315$$ 5.00000 0.281718
$$316$$ 2.00000 0.112509
$$317$$ − 23.0000i − 1.29181i −0.763418 0.645904i $$-0.776480\pi$$
0.763418 0.645904i $$-0.223520\pi$$
$$318$$ − 1.00000i − 0.0560772i
$$319$$ − 12.0000i − 0.671871i
$$320$$ 1.00000i 0.0559017i
$$321$$ −6.00000 −0.334887
$$322$$ 20.0000 1.11456
$$323$$ − 40.0000i − 2.22566i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 20.0000 1.10770
$$327$$ 14.0000i 0.774202i
$$328$$ −6.00000 −0.331295
$$329$$ −15.0000 −0.826977
$$330$$ − 3.00000i − 0.165145i
$$331$$ 4.00000i 0.219860i 0.993939 + 0.109930i $$0.0350627\pi$$
−0.993939 + 0.109930i $$0.964937\pi$$
$$332$$ − 8.00000i − 0.439057i
$$333$$ 7.00000i 0.383598i
$$334$$ 23.0000 1.25850
$$335$$ 8.00000 0.437087
$$336$$ 5.00000i 0.272772i
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ 8.00000 0.434500
$$340$$ 8.00000i 0.433861i
$$341$$ 6.00000 0.324918
$$342$$ −5.00000 −0.270369
$$343$$ − 55.0000i − 2.96972i
$$344$$ − 6.00000i − 0.323498i
$$345$$ − 4.00000i − 0.215353i
$$346$$ 5.00000i 0.268802i
$$347$$ −16.0000 −0.858925 −0.429463 0.903085i $$-0.641297\pi$$
−0.429463 + 0.903085i $$0.641297\pi$$
$$348$$ 4.00000 0.214423
$$349$$ − 8.00000i − 0.428230i −0.976808 0.214115i $$-0.931313\pi$$
0.976808 0.214115i $$-0.0686868\pi$$
$$350$$ −5.00000 −0.267261
$$351$$ 0 0
$$352$$ 3.00000 0.159901
$$353$$ 16.0000i 0.851594i 0.904819 + 0.425797i $$0.140006\pi$$
−0.904819 + 0.425797i $$0.859994\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 2.00000 0.106149
$$356$$ − 11.0000i − 0.582999i
$$357$$ 40.0000i 2.11702i
$$358$$ 4.00000i 0.211407i
$$359$$ 18.0000i 0.950004i 0.879985 + 0.475002i $$0.157553\pi$$
−0.879985 + 0.475002i $$0.842447\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −6.00000 −0.315789
$$362$$ − 2.00000i − 0.105118i
$$363$$ 2.00000 0.104973
$$364$$ 0 0
$$365$$ 0 0
$$366$$ − 2.00000i − 0.104542i
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 6.00000i 0.312348i
$$370$$ − 7.00000i − 0.363913i
$$371$$ 5.00000i 0.259587i
$$372$$ 2.00000i 0.103695i
$$373$$ 38.0000 1.96757 0.983783 0.179364i $$-0.0574041\pi$$
0.983783 + 0.179364i $$0.0574041\pi$$
$$374$$ 24.0000 1.24101
$$375$$ 1.00000i 0.0516398i
$$376$$ −3.00000 −0.154713
$$377$$ 0 0
$$378$$ 5.00000 0.257172
$$379$$ − 25.0000i − 1.28416i −0.766636 0.642082i $$-0.778071\pi$$
0.766636 0.642082i $$-0.221929\pi$$
$$380$$ 5.00000 0.256495
$$381$$ 21.0000 1.07586
$$382$$ − 2.00000i − 0.102329i
$$383$$ − 28.0000i − 1.43073i −0.698749 0.715367i $$-0.746260\pi$$
0.698749 0.715367i $$-0.253740\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 15.0000i 0.764471i
$$386$$ −24.0000 −1.22157
$$387$$ −6.00000 −0.304997
$$388$$ 0 0
$$389$$ 32.0000 1.62246 0.811232 0.584724i $$-0.198797\pi$$
0.811232 + 0.584724i $$0.198797\pi$$
$$390$$ 0 0
$$391$$ 32.0000 1.61831
$$392$$ − 18.0000i − 0.909137i
$$393$$ −19.0000 −0.958423
$$394$$ 3.00000 0.151138
$$395$$ 2.00000i 0.100631i
$$396$$ − 3.00000i − 0.150756i
$$397$$ − 25.0000i − 1.25471i −0.778732 0.627357i $$-0.784137\pi$$
0.778732 0.627357i $$-0.215863\pi$$
$$398$$ 22.0000i 1.10276i
$$399$$ 25.0000 1.25157
$$400$$ −1.00000 −0.0500000
$$401$$ 19.0000i 0.948815i 0.880305 + 0.474407i $$0.157338\pi$$
−0.880305 + 0.474407i $$0.842662\pi$$
$$402$$ 8.00000 0.399004
$$403$$ 0 0
$$404$$ −8.00000 −0.398015
$$405$$ − 1.00000i − 0.0496904i
$$406$$ −20.0000 −0.992583
$$407$$ −21.0000 −1.04093
$$408$$ 8.00000i 0.396059i
$$409$$ 25.0000i 1.23617i 0.786111 + 0.618085i $$0.212091\pi$$
−0.786111 + 0.618085i $$0.787909\pi$$
$$410$$ − 6.00000i − 0.296319i
$$411$$ 12.0000i 0.591916i
$$412$$ −7.00000 −0.344865
$$413$$ 60.0000 2.95241
$$414$$ − 4.00000i − 0.196589i
$$415$$ 8.00000 0.392705
$$416$$ 0 0
$$417$$ 7.00000 0.342791
$$418$$ − 15.0000i − 0.733674i
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ −5.00000 −0.243975
$$421$$ 12.0000i 0.584844i 0.956289 + 0.292422i $$0.0944612\pi$$
−0.956289 + 0.292422i $$0.905539\pi$$
$$422$$ 15.0000i 0.730189i
$$423$$ 3.00000i 0.145865i
$$424$$ 1.00000i 0.0485643i
$$425$$ −8.00000 −0.388057
$$426$$ 2.00000 0.0969003
$$427$$ 10.0000i 0.483934i
$$428$$ 6.00000 0.290021
$$429$$ 0 0
$$430$$ 6.00000 0.289346
$$431$$ 12.0000i 0.578020i 0.957326 + 0.289010i $$0.0933260\pi$$
−0.957326 + 0.289010i $$0.906674\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ − 10.0000i − 0.480015i
$$435$$ 4.00000i 0.191785i
$$436$$ − 14.0000i − 0.670478i
$$437$$ − 20.0000i − 0.956730i
$$438$$ 0 0
$$439$$ −10.0000 −0.477274 −0.238637 0.971109i $$-0.576701\pi$$
−0.238637 + 0.971109i $$0.576701\pi$$
$$440$$ 3.00000i 0.143019i
$$441$$ −18.0000 −0.857143
$$442$$ 0 0
$$443$$ 6.00000 0.285069 0.142534 0.989790i $$-0.454475\pi$$
0.142534 + 0.989790i $$0.454475\pi$$
$$444$$ − 7.00000i − 0.332205i
$$445$$ 11.0000 0.521450
$$446$$ −3.00000 −0.142054
$$447$$ − 2.00000i − 0.0945968i
$$448$$ − 5.00000i − 0.236228i
$$449$$ 27.0000i 1.27421i 0.770778 + 0.637104i $$0.219868\pi$$
−0.770778 + 0.637104i $$0.780132\pi$$
$$450$$ 1.00000i 0.0471405i
$$451$$ −18.0000 −0.847587
$$452$$ −8.00000 −0.376288
$$453$$ − 22.0000i − 1.03365i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 5.00000 0.234146
$$457$$ 30.0000i 1.40334i 0.712502 + 0.701670i $$0.247562\pi$$
−0.712502 + 0.701670i $$0.752438\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 8.00000 0.373408
$$460$$ 4.00000i 0.186501i
$$461$$ 8.00000i 0.372597i 0.982493 + 0.186299i $$0.0596492\pi$$
−0.982493 + 0.186299i $$0.940351\pi$$
$$462$$ 15.0000i 0.697863i
$$463$$ − 8.00000i − 0.371792i −0.982569 0.185896i $$-0.940481\pi$$
0.982569 0.185896i $$-0.0595187\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ −2.00000 −0.0927478
$$466$$ − 14.0000i − 0.648537i
$$467$$ 24.0000 1.11059 0.555294 0.831654i $$-0.312606\pi$$
0.555294 + 0.831654i $$0.312606\pi$$
$$468$$ 0 0
$$469$$ −40.0000 −1.84703
$$470$$ − 3.00000i − 0.138380i
$$471$$ 15.0000 0.691164
$$472$$ 12.0000 0.552345
$$473$$ − 18.0000i − 0.827641i
$$474$$ 2.00000i 0.0918630i
$$475$$ 5.00000i 0.229416i
$$476$$ − 40.0000i − 1.83340i
$$477$$ 1.00000 0.0457869
$$478$$ −18.0000 −0.823301
$$479$$ 28.0000i 1.27935i 0.768644 + 0.639676i $$0.220932\pi$$
−0.768644 + 0.639676i $$0.779068\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ 25.0000 1.13872
$$483$$ 20.0000i 0.910032i
$$484$$ −2.00000 −0.0909091
$$485$$ 0 0
$$486$$ − 1.00000i − 0.0453609i
$$487$$ 37.0000i 1.67663i 0.545186 + 0.838315i $$0.316459\pi$$
−0.545186 + 0.838315i $$0.683541\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ 20.0000i 0.904431i
$$490$$ 18.0000 0.813157
$$491$$ −21.0000 −0.947717 −0.473858 0.880601i $$-0.657139\pi$$
−0.473858 + 0.880601i $$0.657139\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ −32.0000 −1.44121
$$494$$ 0 0
$$495$$ 3.00000 0.134840
$$496$$ − 2.00000i − 0.0898027i
$$497$$ −10.0000 −0.448561
$$498$$ 8.00000 0.358489
$$499$$ 4.00000i 0.179065i 0.995984 + 0.0895323i $$0.0285372\pi$$
−0.995984 + 0.0895323i $$0.971463\pi$$
$$500$$ − 1.00000i − 0.0447214i
$$501$$ 23.0000i 1.02756i
$$502$$ 15.0000i 0.669483i
$$503$$ −11.0000 −0.490466 −0.245233 0.969464i $$-0.578864\pi$$
−0.245233 + 0.969464i $$0.578864\pi$$
$$504$$ −5.00000 −0.222718
$$505$$ − 8.00000i − 0.355995i
$$506$$ 12.0000 0.533465
$$507$$ 0 0
$$508$$ −21.0000 −0.931724
$$509$$ 10.0000i 0.443242i 0.975133 + 0.221621i $$0.0711348\pi$$
−0.975133 + 0.221621i $$0.928865\pi$$
$$510$$ −8.00000 −0.354246
$$511$$ 0 0
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 5.00000i − 0.220755i
$$514$$ 28.0000i 1.23503i
$$515$$ − 7.00000i − 0.308457i
$$516$$ 6.00000 0.264135
$$517$$ −9.00000 −0.395820
$$518$$ 35.0000i 1.53781i
$$519$$ −5.00000 −0.219476
$$520$$ 0 0
$$521$$ 5.00000 0.219054 0.109527 0.993984i $$-0.465066\pi$$
0.109527 + 0.993984i $$0.465066\pi$$
$$522$$ 4.00000i 0.175075i
$$523$$ −10.0000 −0.437269 −0.218635 0.975807i $$-0.570160\pi$$
−0.218635 + 0.975807i $$0.570160\pi$$
$$524$$ 19.0000 0.830019
$$525$$ − 5.00000i − 0.218218i
$$526$$ 15.0000i 0.654031i
$$527$$ − 16.0000i − 0.696971i
$$528$$ 3.00000i 0.130558i
$$529$$ −7.00000 −0.304348
$$530$$ −1.00000 −0.0434372
$$531$$ − 12.0000i − 0.520756i
$$532$$ −25.0000 −1.08389
$$533$$ 0 0
$$534$$ 11.0000 0.476017
$$535$$ 6.00000i 0.259403i
$$536$$ −8.00000 −0.345547
$$537$$ −4.00000 −0.172613
$$538$$ − 4.00000i − 0.172452i
$$539$$ − 54.0000i − 2.32594i
$$540$$ 1.00000i 0.0430331i
$$541$$ − 34.0000i − 1.46177i −0.682498 0.730887i $$-0.739107\pi$$
0.682498 0.730887i $$-0.260893\pi$$
$$542$$ −4.00000 −0.171815
$$543$$ 2.00000 0.0858282
$$544$$ − 8.00000i − 0.342997i
$$545$$ 14.0000 0.599694
$$546$$ 0 0
$$547$$ 6.00000 0.256541 0.128271 0.991739i $$-0.459057\pi$$
0.128271 + 0.991739i $$0.459057\pi$$
$$548$$ − 12.0000i − 0.512615i
$$549$$ 2.00000 0.0853579
$$550$$ −3.00000 −0.127920
$$551$$ 20.0000i 0.852029i
$$552$$ 4.00000i 0.170251i
$$553$$ − 10.0000i − 0.425243i
$$554$$ − 15.0000i − 0.637289i
$$555$$ 7.00000 0.297133
$$556$$ −7.00000 −0.296866
$$557$$ − 15.0000i − 0.635570i −0.948163 0.317785i $$-0.897061\pi$$
0.948163 0.317785i $$-0.102939\pi$$
$$558$$ −2.00000 −0.0846668
$$559$$ 0 0
$$560$$ 5.00000 0.211289
$$561$$ 24.0000i 1.01328i
$$562$$ −18.0000 −0.759284
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ − 3.00000i − 0.126323i
$$565$$ − 8.00000i − 0.336563i
$$566$$ − 10.0000i − 0.420331i
$$567$$ 5.00000i 0.209980i
$$568$$ −2.00000 −0.0839181
$$569$$ 15.0000 0.628833 0.314416 0.949285i $$-0.398191\pi$$
0.314416 + 0.949285i $$0.398191\pi$$
$$570$$ 5.00000i 0.209427i
$$571$$ 33.0000 1.38101 0.690504 0.723329i $$-0.257389\pi$$
0.690504 + 0.723329i $$0.257389\pi$$
$$572$$ 0 0
$$573$$ 2.00000 0.0835512
$$574$$ 30.0000i 1.25218i
$$575$$ −4.00000 −0.166812
$$576$$ −1.00000 −0.0416667
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ − 47.0000i − 1.95494i
$$579$$ − 24.0000i − 0.997406i
$$580$$ − 4.00000i − 0.166091i
$$581$$ −40.0000 −1.65948
$$582$$ 0 0
$$583$$ 3.00000i 0.124247i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 9.00000 0.371787
$$587$$ 18.0000i 0.742940i 0.928445 + 0.371470i $$0.121146\pi$$
−0.928445 + 0.371470i $$0.878854\pi$$
$$588$$ 18.0000 0.742307
$$589$$ −10.0000 −0.412043
$$590$$ 12.0000i 0.494032i
$$591$$ 3.00000i 0.123404i
$$592$$ 7.00000i 0.287698i
$$593$$ − 20.0000i − 0.821302i −0.911793 0.410651i $$-0.865302\pi$$
0.911793 0.410651i $$-0.134698\pi$$
$$594$$ 3.00000 0.123091
$$595$$ 40.0000 1.63984
$$596$$ 2.00000i 0.0819232i
$$597$$ −22.0000 −0.900400
$$598$$ 0 0
$$599$$ 34.0000 1.38920 0.694601 0.719395i $$-0.255581\pi$$
0.694601 + 0.719395i $$0.255581\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ 37.0000 1.50926 0.754631 0.656150i $$-0.227816\pi$$
0.754631 + 0.656150i $$0.227816\pi$$
$$602$$ −30.0000 −1.22271
$$603$$ 8.00000i 0.325785i
$$604$$ 22.0000i 0.895167i
$$605$$ − 2.00000i − 0.0813116i
$$606$$ − 8.00000i − 0.324978i
$$607$$ −29.0000 −1.17707 −0.588537 0.808470i $$-0.700296\pi$$
−0.588537 + 0.808470i $$0.700296\pi$$
$$608$$ −5.00000 −0.202777
$$609$$ − 20.0000i − 0.810441i
$$610$$ −2.00000 −0.0809776
$$611$$ 0 0
$$612$$ −8.00000 −0.323381
$$613$$ 25.0000i 1.00974i 0.863195 + 0.504870i $$0.168460\pi$$
−0.863195 + 0.504870i $$0.831540\pi$$
$$614$$ 6.00000 0.242140
$$615$$ 6.00000 0.241943
$$616$$ − 15.0000i − 0.604367i
$$617$$ − 14.0000i − 0.563619i −0.959470 0.281809i $$-0.909065\pi$$
0.959470 0.281809i $$-0.0909346\pi$$
$$618$$ − 7.00000i − 0.281581i
$$619$$ − 17.0000i − 0.683288i −0.939829 0.341644i $$-0.889016\pi$$
0.939829 0.341644i $$-0.110984\pi$$
$$620$$ 2.00000 0.0803219
$$621$$ 4.00000 0.160514
$$622$$ − 12.0000i − 0.481156i
$$623$$ −55.0000 −2.20353
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 6.00000i 0.239808i
$$627$$ 15.0000 0.599042
$$628$$ −15.0000 −0.598565
$$629$$ 56.0000i 2.23287i
$$630$$ − 5.00000i − 0.199205i
$$631$$ − 12.0000i − 0.477712i −0.971055 0.238856i $$-0.923228\pi$$
0.971055 0.238856i $$-0.0767725\pi$$
$$632$$ − 2.00000i − 0.0795557i
$$633$$ −15.0000 −0.596196
$$634$$ −23.0000 −0.913447
$$635$$ − 21.0000i − 0.833360i
$$636$$ −1.00000 −0.0396526
$$637$$ 0 0
$$638$$ −12.0000 −0.475085
$$639$$ 2.00000i 0.0791188i
$$640$$ 1.00000 0.0395285
$$641$$ −27.0000 −1.06644 −0.533218 0.845978i $$-0.679017\pi$$
−0.533218 + 0.845978i $$0.679017\pi$$
$$642$$ 6.00000i 0.236801i
$$643$$ − 44.0000i − 1.73519i −0.497271 0.867595i $$-0.665665\pi$$
0.497271 0.867595i $$-0.334335\pi$$
$$644$$ − 20.0000i − 0.788110i
$$645$$ 6.00000i 0.236250i
$$646$$ −40.0000 −1.57378
$$647$$ −3.00000 −0.117942 −0.0589711 0.998260i $$-0.518782\pi$$
−0.0589711 + 0.998260i $$0.518782\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 36.0000 1.41312
$$650$$ 0 0
$$651$$ 10.0000 0.391931
$$652$$ − 20.0000i − 0.783260i
$$653$$ 27.0000 1.05659 0.528296 0.849060i $$-0.322831\pi$$
0.528296 + 0.849060i $$0.322831\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 19.0000i 0.742391i
$$656$$ 6.00000i 0.234261i
$$657$$ 0 0
$$658$$ 15.0000i 0.584761i
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ −3.00000 −0.116775
$$661$$ 10.0000i 0.388955i 0.980907 + 0.194477i $$0.0623011\pi$$
−0.980907 + 0.194477i $$0.937699\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ −8.00000 −0.310460
$$665$$ − 25.0000i − 0.969458i
$$666$$ 7.00000 0.271244
$$667$$ −16.0000 −0.619522
$$668$$ − 23.0000i − 0.889897i
$$669$$ − 3.00000i − 0.115987i
$$670$$ − 8.00000i − 0.309067i
$$671$$ 6.00000i 0.231627i
$$672$$ 5.00000 0.192879
$$673$$ 32.0000 1.23351 0.616755 0.787155i $$-0.288447\pi$$
0.616755 + 0.787155i $$0.288447\pi$$
$$674$$ 14.0000i 0.539260i
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −22.0000 −0.845529 −0.422764 0.906240i $$-0.638940\pi$$
−0.422764 + 0.906240i $$0.638940\pi$$
$$678$$ − 8.00000i − 0.307238i
$$679$$ 0 0
$$680$$ 8.00000 0.306786
$$681$$ 0 0
$$682$$ − 6.00000i − 0.229752i
$$683$$ − 30.0000i − 1.14792i −0.818884 0.573959i $$-0.805407\pi$$
0.818884 0.573959i $$-0.194593\pi$$
$$684$$ 5.00000i 0.191180i
$$685$$ 12.0000 0.458496
$$686$$ −55.0000 −2.09991
$$687$$ − 14.0000i − 0.534133i
$$688$$ −6.00000 −0.228748
$$689$$ 0 0
$$690$$ −4.00000 −0.152277
$$691$$ 17.0000i 0.646710i 0.946278 + 0.323355i $$0.104811\pi$$
−0.946278 + 0.323355i $$0.895189\pi$$
$$692$$ 5.00000 0.190071
$$693$$ −15.0000 −0.569803
$$694$$ 16.0000i 0.607352i
$$695$$ − 7.00000i − 0.265525i
$$696$$ − 4.00000i − 0.151620i
$$697$$ 48.0000i 1.81813i
$$698$$ −8.00000 −0.302804
$$699$$ 14.0000 0.529529
$$700$$ 5.00000i 0.188982i
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 35.0000 1.32005
$$704$$ − 3.00000i − 0.113067i
$$705$$ 3.00000 0.112987
$$706$$ 16.0000 0.602168
$$707$$ 40.0000i 1.50435i
$$708$$ 12.0000i 0.450988i
$$709$$ − 32.0000i − 1.20179i −0.799330 0.600893i $$-0.794812\pi$$
0.799330 0.600893i $$-0.205188\pi$$
$$710$$ − 2.00000i − 0.0750587i
$$711$$ −2.00000 −0.0750059
$$712$$ −11.0000 −0.412242
$$713$$ − 8.00000i − 0.299602i
$$714$$ 40.0000 1.49696
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ − 18.0000i − 0.672222i
$$718$$ 18.0000 0.671754
$$719$$ 20.0000 0.745874 0.372937 0.927857i $$-0.378351\pi$$
0.372937 + 0.927857i $$0.378351\pi$$
$$720$$ − 1.00000i − 0.0372678i
$$721$$ 35.0000i 1.30347i
$$722$$ 6.00000i 0.223297i
$$723$$ 25.0000i 0.929760i
$$724$$ −2.00000 −0.0743294
$$725$$ 4.00000 0.148556
$$726$$ − 2.00000i − 0.0742270i
$$727$$ 11.0000 0.407967 0.203984 0.978974i $$-0.434611\pi$$
0.203984 + 0.978974i $$0.434611\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −48.0000 −1.77534
$$732$$ −2.00000 −0.0739221
$$733$$ − 43.0000i − 1.58824i −0.607760 0.794121i $$-0.707932\pi$$
0.607760 0.794121i $$-0.292068\pi$$
$$734$$ 8.00000i 0.295285i
$$735$$ 18.0000i 0.663940i
$$736$$ − 4.00000i − 0.147442i
$$737$$ −24.0000 −0.884051
$$738$$ 6.00000 0.220863
$$739$$ − 19.0000i − 0.698926i −0.936950 0.349463i $$-0.886364\pi$$
0.936950 0.349463i $$-0.113636\pi$$
$$740$$ −7.00000 −0.257325
$$741$$ 0 0
$$742$$ 5.00000 0.183556
$$743$$ − 16.0000i − 0.586983i −0.955962 0.293492i $$-0.905183\pi$$
0.955962 0.293492i $$-0.0948173\pi$$
$$744$$ 2.00000 0.0733236
$$745$$ −2.00000 −0.0732743
$$746$$ − 38.0000i − 1.39128i
$$747$$ 8.00000i 0.292705i
$$748$$ − 24.0000i − 0.877527i
$$749$$ − 30.0000i − 1.09618i
$$750$$ 1.00000 0.0365148
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 3.00000i 0.109399i
$$753$$ −15.0000 −0.546630
$$754$$ 0 0
$$755$$ −22.0000 −0.800662
$$756$$ − 5.00000i − 0.181848i
$$757$$ 17.0000 0.617876 0.308938 0.951082i $$-0.400027\pi$$
0.308938 + 0.951082i $$0.400027\pi$$
$$758$$ −25.0000 −0.908041
$$759$$ 12.0000i 0.435572i
$$760$$ − 5.00000i − 0.181369i
$$761$$ − 9.00000i − 0.326250i −0.986605 0.163125i $$-0.947843\pi$$
0.986605 0.163125i $$-0.0521573\pi$$
$$762$$ − 21.0000i − 0.760750i
$$763$$ −70.0000 −2.53417
$$764$$ −2.00000 −0.0723575
$$765$$ − 8.00000i − 0.289241i
$$766$$ −28.0000 −1.01168
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ − 34.0000i − 1.22607i −0.790055 0.613036i $$-0.789948\pi$$
0.790055 0.613036i $$-0.210052\pi$$
$$770$$ 15.0000 0.540562
$$771$$ −28.0000 −1.00840
$$772$$ 24.0000i 0.863779i
$$773$$ 1.00000i 0.0359675i 0.999838 + 0.0179838i $$0.00572471\pi$$
−0.999838 + 0.0179838i $$0.994275\pi$$
$$774$$ 6.00000i 0.215666i
$$775$$ 2.00000i 0.0718421i
$$776$$ 0 0
$$777$$ −35.0000 −1.25562
$$778$$ − 32.0000i − 1.14726i
$$779$$ 30.0000 1.07486
$$780$$ 0 0
$$781$$ −6.00000 −0.214697
$$782$$ − 32.0000i − 1.14432i
$$783$$ −4.00000 −0.142948
$$784$$ −18.0000 −0.642857
$$785$$ − 15.0000i − 0.535373i
$$786$$ 19.0000i 0.677708i
$$787$$ 28.0000i 0.998092i 0.866575 + 0.499046i $$0.166316\pi$$
−0.866575 + 0.499046i $$0.833684\pi$$
$$788$$ − 3.00000i − 0.106871i
$$789$$ −15.0000 −0.534014
$$790$$ 2.00000 0.0711568
$$791$$ 40.0000i 1.42224i
$$792$$ −3.00000 −0.106600
$$793$$ 0 0
$$794$$ −25.0000 −0.887217
$$795$$ − 1.00000i − 0.0354663i
$$796$$ 22.0000 0.779769
$$797$$ 10.0000 0.354218 0.177109 0.984191i $$-0.443325\pi$$
0.177109 + 0.984191i $$0.443325\pi$$
$$798$$ − 25.0000i − 0.884990i
$$799$$ 24.0000i 0.849059i
$$800$$ 1.00000i 0.0353553i
$$801$$ 11.0000i 0.388666i
$$802$$ 19.0000 0.670913
$$803$$ 0 0
$$804$$ − 8.00000i − 0.282138i
$$805$$ 20.0000 0.704907
$$806$$ 0 0
$$807$$ 4.00000 0.140807
$$808$$ 8.00000i 0.281439i
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 25.0000i 0.877869i 0.898519 + 0.438934i $$0.144644\pi$$
−0.898519 + 0.438934i $$0.855356\pi$$
$$812$$ 20.0000i 0.701862i
$$813$$ − 4.00000i − 0.140286i
$$814$$ 21.0000i 0.736050i
$$815$$ 20.0000 0.700569
$$816$$ 8.00000 0.280056
$$817$$ 30.0000i 1.04957i
$$818$$ 25.0000 0.874105
$$819$$ 0 0
$$820$$ −6.00000 −0.209529
$$821$$ − 22.0000i − 0.767805i −0.923374 0.383903i $$-0.874580\pi$$
0.923374 0.383903i $$-0.125420\pi$$
$$822$$ 12.0000 0.418548
$$823$$ 35.0000 1.22002 0.610012 0.792392i $$-0.291165\pi$$
0.610012 + 0.792392i $$0.291165\pi$$
$$824$$ 7.00000i 0.243857i
$$825$$ − 3.00000i − 0.104447i
$$826$$ − 60.0000i − 2.08767i
$$827$$ − 6.00000i − 0.208640i −0.994544 0.104320i $$-0.966733\pi$$
0.994544 0.104320i $$-0.0332667\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ 16.0000 0.555703 0.277851 0.960624i $$-0.410378\pi$$
0.277851 + 0.960624i $$0.410378\pi$$
$$830$$ − 8.00000i − 0.277684i
$$831$$ 15.0000 0.520344
$$832$$ 0 0
$$833$$ −144.000 −4.98930
$$834$$ − 7.00000i − 0.242390i
$$835$$ 23.0000 0.795948
$$836$$ −15.0000 −0.518786
$$837$$ − 2.00000i − 0.0691301i
$$838$$ 12.0000i 0.414533i
$$839$$ 14.0000i 0.483334i 0.970359 + 0.241667i $$0.0776941\pi$$
−0.970359 + 0.241667i $$0.922306\pi$$
$$840$$ 5.00000i 0.172516i
$$841$$ −13.0000 −0.448276
$$842$$ 12.0000 0.413547
$$843$$ − 18.0000i − 0.619953i
$$844$$ 15.0000 0.516321
$$845$$ 0 0
$$846$$ 3.00000 0.103142
$$847$$ 10.0000i 0.343604i
$$848$$ 1.00000 0.0343401
$$849$$ 10.0000 0.343199
$$850$$ 8.00000i 0.274398i
$$851$$ 28.0000i 0.959828i
$$852$$ − 2.00000i − 0.0685189i
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ 10.0000 0.342193
$$855$$ −5.00000 −0.170996
$$856$$ − 6.00000i − 0.205076i
$$857$$ −38.0000 −1.29806 −0.649028 0.760765i $$-0.724824\pi$$
−0.649028 + 0.760765i $$0.724824\pi$$
$$858$$ 0 0
$$859$$ 3.00000 0.102359 0.0511793 0.998689i $$-0.483702\pi$$
0.0511793 + 0.998689i $$0.483702\pi$$
$$860$$ − 6.00000i − 0.204598i
$$861$$ −30.0000 −1.02240
$$862$$ 12.0000 0.408722
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 5.00000i 0.170005i
$$866$$ 16.0000i 0.543702i
$$867$$ 47.0000 1.59620
$$868$$ −10.0000 −0.339422
$$869$$ − 6.00000i − 0.203536i
$$870$$ 4.00000 0.135613
$$871$$ 0 0
$$872$$ −14.0000 −0.474100
$$873$$ 0 0
$$874$$ −20.0000 −0.676510
$$875$$ −5.00000 −0.169031
$$876$$ 0 0
$$877$$ − 54.0000i − 1.82345i −0.410801 0.911725i $$-0.634751\pi$$
0.410801 0.911725i $$-0.365249\pi$$
$$878$$ 10.0000i 0.337484i
$$879$$ 9.00000i 0.303562i
$$880$$ 3.00000 0.101130
$$881$$ 15.0000 0.505363 0.252681 0.967550i $$-0.418688\pi$$
0.252681 + 0.967550i $$0.418688\pi$$
$$882$$ 18.0000i 0.606092i
$$883$$ −30.0000 −1.00958 −0.504790 0.863242i $$-0.668430\pi$$
−0.504790 + 0.863242i $$0.668430\pi$$
$$884$$ 0 0
$$885$$ −12.0000 −0.403376
$$886$$ − 6.00000i − 0.201574i
$$887$$ 41.0000 1.37665 0.688323 0.725405i $$-0.258347\pi$$
0.688323 + 0.725405i $$0.258347\pi$$
$$888$$ −7.00000 −0.234905
$$889$$ 105.000i 3.52159i
$$890$$ − 11.0000i − 0.368721i
$$891$$ 3.00000i 0.100504i
$$892$$ 3.00000i 0.100447i
$$893$$ 15.0000 0.501956
$$894$$ −2.00000 −0.0668900
$$895$$ 4.00000i 0.133705i
$$896$$ −5.00000 −0.167038
$$897$$ 0 0
$$898$$ 27.0000 0.901002
$$899$$ 8.00000i 0.266815i
$$900$$ 1.00000 0.0333333
$$901$$ 8.00000 0.266519
$$902$$ 18.0000i 0.599334i
$$903$$ − 30.0000i − 0.998337i
$$904$$ 8.00000i 0.266076i
$$905$$ − 2.00000i − 0.0664822i
$$906$$ −22.0000 −0.730901
$$907$$ −10.0000 −0.332045 −0.166022 0.986122i $$-0.553092\pi$$
−0.166022 + 0.986122i $$0.553092\pi$$
$$908$$ 0 0
$$909$$ 8.00000 0.265343
$$910$$ 0 0
$$911$$ −20.0000 −0.662630 −0.331315 0.943520i $$-0.607492\pi$$
−0.331315 + 0.943520i $$0.607492\pi$$
$$912$$ − 5.00000i − 0.165567i
$$913$$ −24.0000 −0.794284
$$914$$ 30.0000 0.992312
$$915$$ − 2.00000i − 0.0661180i
$$916$$ 14.0000i 0.462573i
$$917$$ − 95.0000i − 3.13718i
$$918$$ − 8.00000i − 0.264039i
$$919$$ −2.00000 −0.0659739 −0.0329870 0.999456i $$-0.510502\pi$$
−0.0329870 + 0.999456i $$0.510502\pi$$
$$920$$ 4.00000 0.131876
$$921$$ 6.00000i 0.197707i
$$922$$ 8.00000 0.263466
$$923$$ 0 0
$$924$$ 15.0000 0.493464
$$925$$ − 7.00000i − 0.230159i
$$926$$ −8.00000 −0.262896
$$927$$ 7.00000 0.229910
$$928$$ 4.00000i 0.131306i
$$929$$ 2.00000i 0.0656179i 0.999462 + 0.0328089i $$0.0104453\pi$$
−0.999462 + 0.0328089i $$0.989555\pi$$
$$930$$ 2.00000i 0.0655826i
$$931$$ 90.0000i 2.94963i
$$932$$ −14.0000 −0.458585
$$933$$ 12.0000 0.392862
$$934$$ − 24.0000i − 0.785304i
$$935$$ 24.0000 0.784884
$$936$$ 0 0
$$937$$ 30.0000 0.980057 0.490029 0.871706i $$-0.336986\pi$$
0.490029 + 0.871706i $$0.336986\pi$$
$$938$$ 40.0000i 1.30605i
$$939$$ −6.00000 −0.195803
$$940$$ −3.00000 −0.0978492
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ − 15.0000i − 0.488726i
$$943$$ 24.0000i 0.781548i
$$944$$ − 12.0000i − 0.390567i
$$945$$ 5.00000 0.162650
$$946$$ −18.0000 −0.585230
$$947$$ − 58.0000i − 1.88475i −0.334563 0.942373i $$-0.608589\pi$$
0.334563 0.942373i $$-0.391411\pi$$
$$948$$ 2.00000 0.0649570
$$949$$ 0 0
$$950$$ 5.00000 0.162221
$$951$$ − 23.0000i − 0.745826i
$$952$$ −40.0000 −1.29641
$$953$$ 54.0000 1.74923 0.874616 0.484817i $$-0.161114\pi$$
0.874616 + 0.484817i $$0.161114\pi$$
$$954$$ − 1.00000i − 0.0323762i
$$955$$ − 2.00000i − 0.0647185i
$$956$$ 18.0000i 0.582162i
$$957$$ − 12.0000i − 0.387905i
$$958$$ 28.0000 0.904639
$$959$$ −60.0000 −1.93750
$$960$$ 1.00000i 0.0322749i
$$961$$ 27.0000 0.870968
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ − 25.0000i − 0.805196i
$$965$$ −24.0000 −0.772587
$$966$$ 20.0000 0.643489
$$967$$ − 31.0000i − 0.996893i −0.866921 0.498446i $$-0.833904\pi$$
0.866921 0.498446i $$-0.166096\pi$$
$$968$$ 2.00000i 0.0642824i
$$969$$ − 40.0000i − 1.28499i
$$970$$ 0 0
$$971$$ 15.0000 0.481373 0.240686 0.970603i $$-0.422627\pi$$
0.240686 + 0.970603i $$0.422627\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 35.0000i 1.12205i
$$974$$ 37.0000 1.18556
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 20.0000i 0.639857i 0.947442 + 0.319928i $$0.103659\pi$$
−0.947442 + 0.319928i $$0.896341\pi$$
$$978$$ 20.0000 0.639529
$$979$$ −33.0000 −1.05468
$$980$$ − 18.0000i − 0.574989i
$$981$$ 14.0000i 0.446986i
$$982$$ 21.0000i 0.670137i
$$983$$ − 23.0000i − 0.733586i −0.930303 0.366793i $$-0.880456\pi$$
0.930303 0.366793i $$-0.119544\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 3.00000 0.0955879
$$986$$ 32.0000i 1.01909i
$$987$$ −15.0000 −0.477455
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ − 3.00000i − 0.0953463i
$$991$$ 10.0000 0.317660 0.158830 0.987306i $$-0.449228\pi$$
0.158830 + 0.987306i $$0.449228\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 4.00000i 0.126936i
$$994$$ 10.0000i 0.317181i
$$995$$ 22.0000i 0.697447i
$$996$$ − 8.00000i − 0.253490i
$$997$$ −1.00000 −0.0316703 −0.0158352 0.999875i $$-0.505041\pi$$
−0.0158352 + 0.999875i $$0.505041\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 7.00000i 0.221470i
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.l.1351.1 2
13.2 odd 12 390.2.i.d.61.1 2
13.5 odd 4 5070.2.a.i.1.1 1
13.6 odd 12 390.2.i.d.211.1 yes 2
13.8 odd 4 5070.2.a.x.1.1 1
13.12 even 2 inner 5070.2.b.l.1351.2 2
39.2 even 12 1170.2.i.g.451.1 2
39.32 even 12 1170.2.i.g.991.1 2
65.2 even 12 1950.2.z.j.1699.2 4
65.19 odd 12 1950.2.i.i.601.1 2
65.28 even 12 1950.2.z.j.1699.1 4
65.32 even 12 1950.2.z.j.1849.1 4
65.54 odd 12 1950.2.i.i.451.1 2
65.58 even 12 1950.2.z.j.1849.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.d.61.1 2 13.2 odd 12
390.2.i.d.211.1 yes 2 13.6 odd 12
1170.2.i.g.451.1 2 39.2 even 12
1170.2.i.g.991.1 2 39.32 even 12
1950.2.i.i.451.1 2 65.54 odd 12
1950.2.i.i.601.1 2 65.19 odd 12
1950.2.z.j.1699.1 4 65.28 even 12
1950.2.z.j.1699.2 4 65.2 even 12
1950.2.z.j.1849.1 4 65.32 even 12
1950.2.z.j.1849.2 4 65.58 even 12
5070.2.a.i.1.1 1 13.5 odd 4
5070.2.a.x.1.1 1 13.8 odd 4
5070.2.b.l.1351.1 2 1.1 even 1 trivial
5070.2.b.l.1351.2 2 13.12 even 2 inner