# Properties

 Label 5070.2.a.i.1.1 Level $5070$ Weight $2$ Character 5070.1 Self dual yes Analytic conductor $40.484$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5070.1

## $q$-expansion

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

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ −5.00000 −1.88982 −0.944911 0.327327i $$-0.893852\pi$$
−0.944911 + 0.327327i $$0.893852\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ 5.00000 1.33631
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ −8.00000 −1.94029 −0.970143 0.242536i $$-0.922021\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −5.00000 −1.09109
$$22$$ 3.00000 0.639602
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ −5.00000 −0.944911
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 1.00000 0.182574
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −3.00000 −0.522233
$$34$$ 8.00000 1.37199
$$35$$ 5.00000 0.845154
$$36$$ 1.00000 0.166667
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ 5.00000 0.811107
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 5.00000 0.771517
$$43$$ 6.00000 0.914991 0.457496 0.889212i $$-0.348747\pi$$
0.457496 + 0.889212i $$0.348747\pi$$
$$44$$ −3.00000 −0.452267
$$45$$ −1.00000 −0.149071
$$46$$ 4.00000 0.589768
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 18.0000 2.57143
$$50$$ −1.00000 −0.141421
$$51$$ −8.00000 −1.12022
$$52$$ 0 0
$$53$$ 1.00000 0.137361 0.0686803 0.997639i $$-0.478121\pi$$
0.0686803 + 0.997639i $$0.478121\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 3.00000 0.404520
$$56$$ 5.00000 0.668153
$$57$$ −5.00000 −0.662266
$$58$$ 4.00000 0.525226
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 2.00000 0.254000
$$63$$ −5.00000 −0.629941
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 3.00000 0.369274
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ −8.00000 −0.970143
$$69$$ −4.00000 −0.481543
$$70$$ −5.00000 −0.597614
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 7.00000 0.813733
$$75$$ 1.00000 0.115470
$$76$$ −5.00000 −0.573539
$$77$$ 15.0000 1.70941
$$78$$ 0 0
$$79$$ −2.00000 −0.225018 −0.112509 0.993651i $$-0.535889\pi$$
−0.112509 + 0.993651i $$0.535889\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ −5.00000 −0.545545
$$85$$ 8.00000 0.867722
$$86$$ −6.00000 −0.646997
$$87$$ −4.00000 −0.428845
$$88$$ 3.00000 0.319801
$$89$$ −11.0000 −1.16600 −0.582999 0.812473i $$-0.698121\pi$$
−0.582999 + 0.812473i $$0.698121\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ −4.00000 −0.417029
$$93$$ −2.00000 −0.207390
$$94$$ 3.00000 0.309426
$$95$$ 5.00000 0.512989
$$96$$ −1.00000 −0.102062
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ −18.0000 −1.81827
$$99$$ −3.00000 −0.301511
$$100$$ 1.00000 0.100000
$$101$$ −8.00000 −0.796030 −0.398015 0.917379i $$-0.630301\pi$$
−0.398015 + 0.917379i $$0.630301\pi$$
$$102$$ 8.00000 0.792118
$$103$$ −7.00000 −0.689730 −0.344865 0.938652i $$-0.612075\pi$$
−0.344865 + 0.938652i $$0.612075\pi$$
$$104$$ 0 0
$$105$$ 5.00000 0.487950
$$106$$ −1.00000 −0.0971286
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ −3.00000 −0.286039
$$111$$ −7.00000 −0.664411
$$112$$ −5.00000 −0.472456
$$113$$ 8.00000 0.752577 0.376288 0.926503i $$-0.377200\pi$$
0.376288 + 0.926503i $$0.377200\pi$$
$$114$$ 5.00000 0.468293
$$115$$ 4.00000 0.373002
$$116$$ −4.00000 −0.371391
$$117$$ 0 0
$$118$$ −12.0000 −1.10469
$$119$$ 40.0000 3.66679
$$120$$ 1.00000 0.0912871
$$121$$ −2.00000 −0.181818
$$122$$ −2.00000 −0.181071
$$123$$ 6.00000 0.541002
$$124$$ −2.00000 −0.179605
$$125$$ −1.00000 −0.0894427
$$126$$ 5.00000 0.445435
$$127$$ −21.0000 −1.86345 −0.931724 0.363166i $$-0.881696\pi$$
−0.931724 + 0.363166i $$0.881696\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ −19.0000 −1.66004 −0.830019 0.557735i $$-0.811670\pi$$
−0.830019 + 0.557735i $$0.811670\pi$$
$$132$$ −3.00000 −0.261116
$$133$$ 25.0000 2.16777
$$134$$ −8.00000 −0.691095
$$135$$ −1.00000 −0.0860663
$$136$$ 8.00000 0.685994
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 4.00000 0.340503
$$139$$ 7.00000 0.593732 0.296866 0.954919i $$-0.404058\pi$$
0.296866 + 0.954919i $$0.404058\pi$$
$$140$$ 5.00000 0.422577
$$141$$ −3.00000 −0.252646
$$142$$ −2.00000 −0.167836
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 4.00000 0.332182
$$146$$ 0 0
$$147$$ 18.0000 1.48461
$$148$$ −7.00000 −0.575396
$$149$$ −2.00000 −0.163846 −0.0819232 0.996639i $$-0.526106\pi$$
−0.0819232 + 0.996639i $$0.526106\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 22.0000 1.79033 0.895167 0.445730i $$-0.147056\pi$$
0.895167 + 0.445730i $$0.147056\pi$$
$$152$$ 5.00000 0.405554
$$153$$ −8.00000 −0.646762
$$154$$ −15.0000 −1.20873
$$155$$ 2.00000 0.160644
$$156$$ 0 0
$$157$$ 15.0000 1.19713 0.598565 0.801074i $$-0.295738\pi$$
0.598565 + 0.801074i $$0.295738\pi$$
$$158$$ 2.00000 0.159111
$$159$$ 1.00000 0.0793052
$$160$$ 1.00000 0.0790569
$$161$$ 20.0000 1.57622
$$162$$ −1.00000 −0.0785674
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 3.00000 0.233550
$$166$$ −8.00000 −0.620920
$$167$$ −23.0000 −1.77979 −0.889897 0.456162i $$-0.849224\pi$$
−0.889897 + 0.456162i $$0.849224\pi$$
$$168$$ 5.00000 0.385758
$$169$$ 0 0
$$170$$ −8.00000 −0.613572
$$171$$ −5.00000 −0.382360
$$172$$ 6.00000 0.457496
$$173$$ 5.00000 0.380143 0.190071 0.981770i $$-0.439128\pi$$
0.190071 + 0.981770i $$0.439128\pi$$
$$174$$ 4.00000 0.303239
$$175$$ −5.00000 −0.377964
$$176$$ −3.00000 −0.226134
$$177$$ 12.0000 0.901975
$$178$$ 11.0000 0.824485
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 4.00000 0.294884
$$185$$ 7.00000 0.514650
$$186$$ 2.00000 0.146647
$$187$$ 24.0000 1.75505
$$188$$ −3.00000 −0.218797
$$189$$ −5.00000 −0.363696
$$190$$ −5.00000 −0.362738
$$191$$ 2.00000 0.144715 0.0723575 0.997379i $$-0.476948\pi$$
0.0723575 + 0.997379i $$0.476948\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 24.0000 1.72756 0.863779 0.503871i $$-0.168091\pi$$
0.863779 + 0.503871i $$0.168091\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 18.0000 1.28571
$$197$$ 3.00000 0.213741 0.106871 0.994273i $$-0.465917\pi$$
0.106871 + 0.994273i $$0.465917\pi$$
$$198$$ 3.00000 0.213201
$$199$$ 22.0000 1.55954 0.779769 0.626067i $$-0.215336\pi$$
0.779769 + 0.626067i $$0.215336\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 8.00000 0.564276
$$202$$ 8.00000 0.562878
$$203$$ 20.0000 1.40372
$$204$$ −8.00000 −0.560112
$$205$$ −6.00000 −0.419058
$$206$$ 7.00000 0.487713
$$207$$ −4.00000 −0.278019
$$208$$ 0 0
$$209$$ 15.0000 1.03757
$$210$$ −5.00000 −0.345033
$$211$$ −15.0000 −1.03264 −0.516321 0.856395i $$-0.672699\pi$$
−0.516321 + 0.856395i $$0.672699\pi$$
$$212$$ 1.00000 0.0686803
$$213$$ 2.00000 0.137038
$$214$$ 6.00000 0.410152
$$215$$ −6.00000 −0.409197
$$216$$ −1.00000 −0.0680414
$$217$$ 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
$$223$$ −3.00000 −0.200895 −0.100447 0.994942i $$-0.532027\pi$$
−0.100447 + 0.994942i $$0.532027\pi$$
$$224$$ 5.00000 0.334077
$$225$$ 1.00000 0.0666667
$$226$$ −8.00000 −0.532152
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ −5.00000 −0.331133
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 15.0000 0.986928
$$232$$ 4.00000 0.262613
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ 0 0
$$235$$ 3.00000 0.195698
$$236$$ 12.0000 0.781133
$$237$$ −2.00000 −0.129914
$$238$$ −40.0000 −2.59281
$$239$$ −18.0000 −1.16432 −0.582162 0.813073i $$-0.697793\pi$$
−0.582162 + 0.813073i $$0.697793\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ −25.0000 −1.61039 −0.805196 0.593009i $$-0.797940\pi$$
−0.805196 + 0.593009i $$0.797940\pi$$
$$242$$ 2.00000 0.128565
$$243$$ 1.00000 0.0641500
$$244$$ 2.00000 0.128037
$$245$$ −18.0000 −1.14998
$$246$$ −6.00000 −0.382546
$$247$$ 0 0
$$248$$ 2.00000 0.127000
$$249$$ 8.00000 0.506979
$$250$$ 1.00000 0.0632456
$$251$$ 15.0000 0.946792 0.473396 0.880850i $$-0.343028\pi$$
0.473396 + 0.880850i $$0.343028\pi$$
$$252$$ −5.00000 −0.314970
$$253$$ 12.0000 0.754434
$$254$$ 21.0000 1.31766
$$255$$ 8.00000 0.500979
$$256$$ 1.00000 0.0625000
$$257$$ 28.0000 1.74659 0.873296 0.487190i $$-0.161978\pi$$
0.873296 + 0.487190i $$0.161978\pi$$
$$258$$ −6.00000 −0.373544
$$259$$ 35.0000 2.17479
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 19.0000 1.17382
$$263$$ −15.0000 −0.924940 −0.462470 0.886635i $$-0.653037\pi$$
−0.462470 + 0.886635i $$0.653037\pi$$
$$264$$ 3.00000 0.184637
$$265$$ −1.00000 −0.0614295
$$266$$ −25.0000 −1.53285
$$267$$ −11.0000 −0.673189
$$268$$ 8.00000 0.488678
$$269$$ 4.00000 0.243884 0.121942 0.992537i $$-0.461088\pi$$
0.121942 + 0.992537i $$0.461088\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 4.00000 0.242983 0.121491 0.992592i $$-0.461232\pi$$
0.121491 + 0.992592i $$0.461232\pi$$
$$272$$ −8.00000 −0.485071
$$273$$ 0 0
$$274$$ 12.0000 0.724947
$$275$$ −3.00000 −0.180907
$$276$$ −4.00000 −0.240772
$$277$$ −15.0000 −0.901263 −0.450631 0.892710i $$-0.648801\pi$$
−0.450631 + 0.892710i $$0.648801\pi$$
$$278$$ −7.00000 −0.419832
$$279$$ −2.00000 −0.119737
$$280$$ −5.00000 −0.298807
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 3.00000 0.178647
$$283$$ −10.0000 −0.594438 −0.297219 0.954809i $$-0.596059\pi$$
−0.297219 + 0.954809i $$0.596059\pi$$
$$284$$ 2.00000 0.118678
$$285$$ 5.00000 0.296174
$$286$$ 0 0
$$287$$ −30.0000 −1.77084
$$288$$ −1.00000 −0.0589256
$$289$$ 47.0000 2.76471
$$290$$ −4.00000 −0.234888
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −9.00000 −0.525786 −0.262893 0.964825i $$-0.584677\pi$$
−0.262893 + 0.964825i $$0.584677\pi$$
$$294$$ −18.0000 −1.04978
$$295$$ −12.0000 −0.698667
$$296$$ 7.00000 0.406867
$$297$$ −3.00000 −0.174078
$$298$$ 2.00000 0.115857
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ −30.0000 −1.72917
$$302$$ −22.0000 −1.26596
$$303$$ −8.00000 −0.459588
$$304$$ −5.00000 −0.286770
$$305$$ −2.00000 −0.114520
$$306$$ 8.00000 0.457330
$$307$$ −6.00000 −0.342438 −0.171219 0.985233i $$-0.554771\pi$$
−0.171219 + 0.985233i $$0.554771\pi$$
$$308$$ 15.0000 0.854704
$$309$$ −7.00000 −0.398216
$$310$$ −2.00000 −0.113592
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ −15.0000 −0.846499
$$315$$ 5.00000 0.281718
$$316$$ −2.00000 −0.112509
$$317$$ −23.0000 −1.29181 −0.645904 0.763418i $$-0.723520\pi$$
−0.645904 + 0.763418i $$0.723520\pi$$
$$318$$ −1.00000 −0.0560772
$$319$$ 12.0000 0.671871
$$320$$ −1.00000 −0.0559017
$$321$$ −6.00000 −0.334887
$$322$$ −20.0000 −1.11456
$$323$$ 40.0000 2.22566
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 20.0000 1.10770
$$327$$ 14.0000 0.774202
$$328$$ −6.00000 −0.331295
$$329$$ 15.0000 0.826977
$$330$$ −3.00000 −0.165145
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 8.00000 0.439057
$$333$$ −7.00000 −0.383598
$$334$$ 23.0000 1.25850
$$335$$ −8.00000 −0.437087
$$336$$ −5.00000 −0.272772
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ 8.00000 0.434500
$$340$$ 8.00000 0.433861
$$341$$ 6.00000 0.324918
$$342$$ 5.00000 0.270369
$$343$$ −55.0000 −2.96972
$$344$$ −6.00000 −0.323498
$$345$$ 4.00000 0.215353
$$346$$ −5.00000 −0.268802
$$347$$ −16.0000 −0.858925 −0.429463 0.903085i $$-0.641297\pi$$
−0.429463 + 0.903085i $$0.641297\pi$$
$$348$$ −4.00000 −0.214423
$$349$$ 8.00000 0.428230 0.214115 0.976808i $$-0.431313\pi$$
0.214115 + 0.976808i $$0.431313\pi$$
$$350$$ 5.00000 0.267261
$$351$$ 0 0
$$352$$ 3.00000 0.159901
$$353$$ 16.0000 0.851594 0.425797 0.904819i $$-0.359994\pi$$
0.425797 + 0.904819i $$0.359994\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ −2.00000 −0.106149
$$356$$ −11.0000 −0.582999
$$357$$ 40.0000 2.11702
$$358$$ −4.00000 −0.211407
$$359$$ −18.0000 −0.950004 −0.475002 0.879985i $$-0.657553\pi$$
−0.475002 + 0.879985i $$0.657553\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ 6.00000 0.315789
$$362$$ 2.00000 0.105118
$$363$$ −2.00000 −0.104973
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 6.00000 0.312348
$$370$$ −7.00000 −0.363913
$$371$$ −5.00000 −0.259587
$$372$$ −2.00000 −0.103695
$$373$$ 38.0000 1.96757 0.983783 0.179364i $$-0.0574041\pi$$
0.983783 + 0.179364i $$0.0574041\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ −1.00000 −0.0516398
$$376$$ 3.00000 0.154713
$$377$$ 0 0
$$378$$ 5.00000 0.257172
$$379$$ −25.0000 −1.28416 −0.642082 0.766636i $$-0.721929\pi$$
−0.642082 + 0.766636i $$0.721929\pi$$
$$380$$ 5.00000 0.256495
$$381$$ −21.0000 −1.07586
$$382$$ −2.00000 −0.102329
$$383$$ −28.0000 −1.43073 −0.715367 0.698749i $$-0.753740\pi$$
−0.715367 + 0.698749i $$0.753740\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −15.0000 −0.764471
$$386$$ −24.0000 −1.22157
$$387$$ 6.00000 0.304997
$$388$$ 0 0
$$389$$ −32.0000 −1.62246 −0.811232 0.584724i $$-0.801203\pi$$
−0.811232 + 0.584724i $$0.801203\pi$$
$$390$$ 0 0
$$391$$ 32.0000 1.61831
$$392$$ −18.0000 −0.909137
$$393$$ −19.0000 −0.958423
$$394$$ −3.00000 −0.151138
$$395$$ 2.00000 0.100631
$$396$$ −3.00000 −0.150756
$$397$$ 25.0000 1.25471 0.627357 0.778732i $$-0.284137\pi$$
0.627357 + 0.778732i $$0.284137\pi$$
$$398$$ −22.0000 −1.10276
$$399$$ 25.0000 1.25157
$$400$$ 1.00000 0.0500000
$$401$$ −19.0000 −0.948815 −0.474407 0.880305i $$-0.657338\pi$$
−0.474407 + 0.880305i $$0.657338\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ 0 0
$$404$$ −8.00000 −0.398015
$$405$$ −1.00000 −0.0496904
$$406$$ −20.0000 −0.992583
$$407$$ 21.0000 1.04093
$$408$$ 8.00000 0.396059
$$409$$ 25.0000 1.23617 0.618085 0.786111i $$-0.287909\pi$$
0.618085 + 0.786111i $$0.287909\pi$$
$$410$$ 6.00000 0.296319
$$411$$ −12.0000 −0.591916
$$412$$ −7.00000 −0.344865
$$413$$ −60.0000 −2.95241
$$414$$ 4.00000 0.196589
$$415$$ −8.00000 −0.392705
$$416$$ 0 0
$$417$$ 7.00000 0.342791
$$418$$ −15.0000 −0.733674
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 5.00000 0.243975
$$421$$ 12.0000 0.584844 0.292422 0.956289i $$-0.405539\pi$$
0.292422 + 0.956289i $$0.405539\pi$$
$$422$$ 15.0000 0.730189
$$423$$ −3.00000 −0.145865
$$424$$ −1.00000 −0.0485643
$$425$$ −8.00000 −0.388057
$$426$$ −2.00000 −0.0969003
$$427$$ −10.0000 −0.483934
$$428$$ −6.00000 −0.290021
$$429$$ 0 0
$$430$$ 6.00000 0.289346
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 16.0000 0.768911 0.384455 0.923144i $$-0.374389\pi$$
0.384455 + 0.923144i $$0.374389\pi$$
$$434$$ −10.0000 −0.480015
$$435$$ 4.00000 0.191785
$$436$$ 14.0000 0.670478
$$437$$ 20.0000 0.956730
$$438$$ 0 0
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ −3.00000 −0.143019
$$441$$ 18.0000 0.857143
$$442$$ 0 0
$$443$$ 6.00000 0.285069 0.142534 0.989790i $$-0.454475\pi$$
0.142534 + 0.989790i $$0.454475\pi$$
$$444$$ −7.00000 −0.332205
$$445$$ 11.0000 0.521450
$$446$$ 3.00000 0.142054
$$447$$ −2.00000 −0.0945968
$$448$$ −5.00000 −0.236228
$$449$$ −27.0000 −1.27421 −0.637104 0.770778i $$-0.719868\pi$$
−0.637104 + 0.770778i $$0.719868\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ −18.0000 −0.847587
$$452$$ 8.00000 0.376288
$$453$$ 22.0000 1.03365
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 5.00000 0.234146
$$457$$ 30.0000 1.40334 0.701670 0.712502i $$-0.252438\pi$$
0.701670 + 0.712502i $$0.252438\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ −8.00000 −0.373408
$$460$$ 4.00000 0.186501
$$461$$ 8.00000 0.372597 0.186299 0.982493i $$-0.440351\pi$$
0.186299 + 0.982493i $$0.440351\pi$$
$$462$$ −15.0000 −0.697863
$$463$$ 8.00000 0.371792 0.185896 0.982569i $$-0.440481\pi$$
0.185896 + 0.982569i $$0.440481\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 2.00000 0.0927478
$$466$$ 14.0000 0.648537
$$467$$ −24.0000 −1.11059 −0.555294 0.831654i $$-0.687394\pi$$
−0.555294 + 0.831654i $$0.687394\pi$$
$$468$$ 0 0
$$469$$ −40.0000 −1.84703
$$470$$ −3.00000 −0.138380
$$471$$ 15.0000 0.691164
$$472$$ −12.0000 −0.552345
$$473$$ −18.0000 −0.827641
$$474$$ 2.00000 0.0918630
$$475$$ −5.00000 −0.229416
$$476$$ 40.0000 1.83340
$$477$$ 1.00000 0.0457869
$$478$$ 18.0000 0.823301
$$479$$ −28.0000 −1.27935 −0.639676 0.768644i $$-0.720932\pi$$
−0.639676 + 0.768644i $$0.720932\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ 25.0000 1.13872
$$483$$ 20.0000 0.910032
$$484$$ −2.00000 −0.0909091
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 37.0000 1.67663 0.838315 0.545186i $$-0.183541\pi$$
0.838315 + 0.545186i $$0.183541\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ −20.0000 −0.904431
$$490$$ 18.0000 0.813157
$$491$$ 21.0000 0.947717 0.473858 0.880601i $$-0.342861\pi$$
0.473858 + 0.880601i $$0.342861\pi$$
$$492$$ 6.00000 0.270501
$$493$$ 32.0000 1.44121
$$494$$ 0 0
$$495$$ 3.00000 0.134840
$$496$$ −2.00000 −0.0898027
$$497$$ −10.0000 −0.448561
$$498$$ −8.00000 −0.358489
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −23.0000 −1.02756
$$502$$ −15.0000 −0.669483
$$503$$ −11.0000 −0.490466 −0.245233 0.969464i $$-0.578864\pi$$
−0.245233 + 0.969464i $$0.578864\pi$$
$$504$$ 5.00000 0.222718
$$505$$ 8.00000 0.355995
$$506$$ −12.0000 −0.533465
$$507$$ 0 0
$$508$$ −21.0000 −0.931724
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ −8.00000 −0.354246
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ −5.00000 −0.220755
$$514$$ −28.0000 −1.23503
$$515$$ 7.00000 0.308457
$$516$$ 6.00000 0.264135
$$517$$ 9.00000 0.395820
$$518$$ −35.0000 −1.53781
$$519$$ 5.00000 0.219476
$$520$$ 0 0
$$521$$ 5.00000 0.219054 0.109527 0.993984i $$-0.465066\pi$$
0.109527 + 0.993984i $$0.465066\pi$$
$$522$$ 4.00000 0.175075
$$523$$ −10.0000 −0.437269 −0.218635 0.975807i $$-0.570160\pi$$
−0.218635 + 0.975807i $$0.570160\pi$$
$$524$$ −19.0000 −0.830019
$$525$$ −5.00000 −0.218218
$$526$$ 15.0000 0.654031
$$527$$ 16.0000 0.696971
$$528$$ −3.00000 −0.130558
$$529$$ −7.00000 −0.304348
$$530$$ 1.00000 0.0434372
$$531$$ 12.0000 0.520756
$$532$$ 25.0000 1.08389
$$533$$ 0 0
$$534$$ 11.0000 0.476017
$$535$$ 6.00000 0.259403
$$536$$ −8.00000 −0.345547
$$537$$ 4.00000 0.172613
$$538$$ −4.00000 −0.172452
$$539$$ −54.0000 −2.32594
$$540$$ −1.00000 −0.0430331
$$541$$ 34.0000 1.46177 0.730887 0.682498i $$-0.239107\pi$$
0.730887 + 0.682498i $$0.239107\pi$$
$$542$$ −4.00000 −0.171815
$$543$$ −2.00000 −0.0858282
$$544$$ 8.00000 0.342997
$$545$$ −14.0000 −0.599694
$$546$$ 0 0
$$547$$ 6.00000 0.256541 0.128271 0.991739i $$-0.459057\pi$$
0.128271 + 0.991739i $$0.459057\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ 2.00000 0.0853579
$$550$$ 3.00000 0.127920
$$551$$ 20.0000 0.852029
$$552$$ 4.00000 0.170251
$$553$$ 10.0000 0.425243
$$554$$ 15.0000 0.637289
$$555$$ 7.00000 0.297133
$$556$$ 7.00000 0.296866
$$557$$ 15.0000 0.635570 0.317785 0.948163i $$-0.397061\pi$$
0.317785 + 0.948163i $$0.397061\pi$$
$$558$$ 2.00000 0.0846668
$$559$$ 0 0
$$560$$ 5.00000 0.211289
$$561$$ 24.0000 1.01328
$$562$$ −18.0000 −0.759284
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ −3.00000 −0.126323
$$565$$ −8.00000 −0.336563
$$566$$ 10.0000 0.420331
$$567$$ −5.00000 −0.209980
$$568$$ −2.00000 −0.0839181
$$569$$ −15.0000 −0.628833 −0.314416 0.949285i $$-0.601809\pi$$
−0.314416 + 0.949285i $$0.601809\pi$$
$$570$$ −5.00000 −0.209427
$$571$$ −33.0000 −1.38101 −0.690504 0.723329i $$-0.742611\pi$$
−0.690504 + 0.723329i $$0.742611\pi$$
$$572$$ 0 0
$$573$$ 2.00000 0.0835512
$$574$$ 30.0000 1.25218
$$575$$ −4.00000 −0.166812
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −47.0000 −1.95494
$$579$$ 24.0000 0.997406
$$580$$ 4.00000 0.166091
$$581$$ −40.0000 −1.65948
$$582$$ 0 0
$$583$$ −3.00000 −0.124247
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 9.00000 0.371787
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ 18.0000 0.742307
$$589$$ 10.0000 0.412043
$$590$$ 12.0000 0.494032
$$591$$ 3.00000 0.123404
$$592$$ −7.00000 −0.287698
$$593$$ 20.0000 0.821302 0.410651 0.911793i $$-0.365302\pi$$
0.410651 + 0.911793i $$0.365302\pi$$
$$594$$ 3.00000 0.123091
$$595$$ −40.0000 −1.63984
$$596$$ −2.00000 −0.0819232
$$597$$ 22.0000 0.900400
$$598$$ 0 0
$$599$$ 34.0000 1.38920 0.694601 0.719395i $$-0.255581\pi$$
0.694601 + 0.719395i $$0.255581\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ 37.0000 1.50926 0.754631 0.656150i $$-0.227816\pi$$
0.754631 + 0.656150i $$0.227816\pi$$
$$602$$ 30.0000 1.22271
$$603$$ 8.00000 0.325785
$$604$$ 22.0000 0.895167
$$605$$ 2.00000 0.0813116
$$606$$ 8.00000 0.324978
$$607$$ −29.0000 −1.17707 −0.588537 0.808470i $$-0.700296\pi$$
−0.588537 + 0.808470i $$0.700296\pi$$
$$608$$ 5.00000 0.202777
$$609$$ 20.0000 0.810441
$$610$$ 2.00000 0.0809776
$$611$$ 0 0
$$612$$ −8.00000 −0.323381
$$613$$ 25.0000 1.00974 0.504870 0.863195i $$-0.331540\pi$$
0.504870 + 0.863195i $$0.331540\pi$$
$$614$$ 6.00000 0.242140
$$615$$ −6.00000 −0.241943
$$616$$ −15.0000 −0.604367
$$617$$ −14.0000 −0.563619 −0.281809 0.959470i $$-0.590935\pi$$
−0.281809 + 0.959470i $$0.590935\pi$$
$$618$$ 7.00000 0.281581
$$619$$ 17.0000 0.683288 0.341644 0.939829i $$-0.389016\pi$$
0.341644 + 0.939829i $$0.389016\pi$$
$$620$$ 2.00000 0.0803219
$$621$$ −4.00000 −0.160514
$$622$$ 12.0000 0.481156
$$623$$ 55.0000 2.20353
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 6.00000 0.239808
$$627$$ 15.0000 0.599042
$$628$$ 15.0000 0.598565
$$629$$ 56.0000 2.23287
$$630$$ −5.00000 −0.199205
$$631$$ 12.0000 0.477712 0.238856 0.971055i $$-0.423228\pi$$
0.238856 + 0.971055i $$0.423228\pi$$
$$632$$ 2.00000 0.0795557
$$633$$ −15.0000 −0.596196
$$634$$ 23.0000 0.913447
$$635$$ 21.0000 0.833360
$$636$$ 1.00000 0.0396526
$$637$$ 0 0
$$638$$ −12.0000 −0.475085
$$639$$ 2.00000 0.0791188
$$640$$ 1.00000 0.0395285
$$641$$ 27.0000 1.06644 0.533218 0.845978i $$-0.320983\pi$$
0.533218 + 0.845978i $$0.320983\pi$$
$$642$$ 6.00000 0.236801
$$643$$ −44.0000 −1.73519 −0.867595 0.497271i $$-0.834335\pi$$
−0.867595 + 0.497271i $$0.834335\pi$$
$$644$$ 20.0000 0.788110
$$645$$ −6.00000 −0.236250
$$646$$ −40.0000 −1.57378
$$647$$ 3.00000 0.117942 0.0589711 0.998260i $$-0.481218\pi$$
0.0589711 + 0.998260i $$0.481218\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −36.0000 −1.41312
$$650$$ 0 0
$$651$$ 10.0000 0.391931
$$652$$ −20.0000 −0.783260
$$653$$ 27.0000 1.05659 0.528296 0.849060i $$-0.322831\pi$$
0.528296 + 0.849060i $$0.322831\pi$$
$$654$$ −14.0000 −0.547443
$$655$$ 19.0000 0.742391
$$656$$ 6.00000 0.234261
$$657$$ 0 0
$$658$$ −15.0000 −0.584761
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 3.00000 0.116775
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ 0 0
$$664$$ −8.00000 −0.310460
$$665$$ −25.0000 −0.969458
$$666$$ 7.00000 0.271244
$$667$$ 16.0000 0.619522
$$668$$ −23.0000 −0.889897
$$669$$ −3.00000 −0.115987
$$670$$ 8.00000 0.309067
$$671$$ −6.00000 −0.231627
$$672$$ 5.00000 0.192879
$$673$$ −32.0000 −1.23351 −0.616755 0.787155i $$-0.711553\pi$$
−0.616755 + 0.787155i $$0.711553\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −22.0000 −0.845529 −0.422764 0.906240i $$-0.638940\pi$$
−0.422764 + 0.906240i $$0.638940\pi$$
$$678$$ −8.00000 −0.307238
$$679$$ 0 0
$$680$$ −8.00000 −0.306786
$$681$$ 0 0
$$682$$ −6.00000 −0.229752
$$683$$ 30.0000 1.14792 0.573959 0.818884i $$-0.305407\pi$$
0.573959 + 0.818884i $$0.305407\pi$$
$$684$$ −5.00000 −0.191180
$$685$$ 12.0000 0.458496
$$686$$ 55.0000 2.09991
$$687$$ 14.0000 0.534133
$$688$$ 6.00000 0.228748
$$689$$ 0 0
$$690$$ −4.00000 −0.152277
$$691$$ 17.0000 0.646710 0.323355 0.946278i $$-0.395189\pi$$
0.323355 + 0.946278i $$0.395189\pi$$
$$692$$ 5.00000 0.190071
$$693$$ 15.0000 0.569803
$$694$$ 16.0000 0.607352
$$695$$ −7.00000 −0.265525
$$696$$ 4.00000 0.151620
$$697$$ −48.0000 −1.81813
$$698$$ −8.00000 −0.302804
$$699$$ −14.0000 −0.529529
$$700$$ −5.00000 −0.188982
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 35.0000 1.32005
$$704$$ −3.00000 −0.113067
$$705$$ 3.00000 0.112987
$$706$$ −16.0000 −0.602168
$$707$$ 40.0000 1.50435
$$708$$ 12.0000 0.450988
$$709$$ 32.0000 1.20179 0.600893 0.799330i $$-0.294812\pi$$
0.600893 + 0.799330i $$0.294812\pi$$
$$710$$ 2.00000 0.0750587
$$711$$ −2.00000 −0.0750059
$$712$$ 11.0000 0.412242
$$713$$ 8.00000 0.299602
$$714$$ −40.0000 −1.49696
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ −18.0000 −0.672222
$$718$$ 18.0000 0.671754
$$719$$ −20.0000 −0.745874 −0.372937 0.927857i $$-0.621649\pi$$
−0.372937 + 0.927857i $$0.621649\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 35.0000 1.30347
$$722$$ −6.00000 −0.223297
$$723$$ −25.0000 −0.929760
$$724$$ −2.00000 −0.0743294
$$725$$ −4.00000 −0.148556
$$726$$ 2.00000 0.0742270
$$727$$ −11.0000 −0.407967 −0.203984 0.978974i $$-0.565389\pi$$
−0.203984 + 0.978974i $$0.565389\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −48.0000 −1.77534
$$732$$ 2.00000 0.0739221
$$733$$ −43.0000 −1.58824 −0.794121 0.607760i $$-0.792068\pi$$
−0.794121 + 0.607760i $$0.792068\pi$$
$$734$$ 8.00000 0.295285
$$735$$ −18.0000 −0.663940
$$736$$ 4.00000 0.147442
$$737$$ −24.0000 −0.884051
$$738$$ −6.00000 −0.220863
$$739$$ 19.0000 0.698926 0.349463 0.936950i $$-0.386364\pi$$
0.349463 + 0.936950i $$0.386364\pi$$
$$740$$ 7.00000 0.257325
$$741$$ 0 0
$$742$$ 5.00000 0.183556
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 2.00000 0.0733236
$$745$$ 2.00000 0.0732743
$$746$$ −38.0000 −1.39128
$$747$$ 8.00000 0.292705
$$748$$ 24.0000 0.877527
$$749$$ 30.0000 1.09618
$$750$$ 1.00000 0.0365148
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ −3.00000 −0.109399
$$753$$ 15.0000 0.546630
$$754$$ 0 0
$$755$$ −22.0000 −0.800662
$$756$$ −5.00000 −0.181848
$$757$$ 17.0000 0.617876 0.308938 0.951082i $$-0.400027\pi$$
0.308938 + 0.951082i $$0.400027\pi$$
$$758$$ 25.0000 0.908041
$$759$$ 12.0000 0.435572
$$760$$ −5.00000 −0.181369
$$761$$ 9.00000 0.326250 0.163125 0.986605i $$-0.447843\pi$$
0.163125 + 0.986605i $$0.447843\pi$$
$$762$$ 21.0000 0.760750
$$763$$ −70.0000 −2.53417
$$764$$ 2.00000 0.0723575
$$765$$ 8.00000 0.289241
$$766$$ 28.0000 1.01168
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ −34.0000 −1.22607 −0.613036 0.790055i $$-0.710052\pi$$
−0.613036 + 0.790055i $$0.710052\pi$$
$$770$$ 15.0000 0.540562
$$771$$ 28.0000 1.00840
$$772$$ 24.0000 0.863779
$$773$$ 1.00000 0.0359675 0.0179838 0.999838i $$-0.494275\pi$$
0.0179838 + 0.999838i $$0.494275\pi$$
$$774$$ −6.00000 −0.215666
$$775$$ −2.00000 −0.0718421
$$776$$ 0 0
$$777$$ 35.0000 1.25562
$$778$$ 32.0000 1.14726
$$779$$ −30.0000 −1.07486
$$780$$ 0 0
$$781$$ −6.00000 −0.214697
$$782$$ −32.0000 −1.14432
$$783$$ −4.00000 −0.142948
$$784$$ 18.0000 0.642857
$$785$$ −15.0000 −0.535373
$$786$$ 19.0000 0.677708
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ 3.00000 0.106871
$$789$$ −15.0000 −0.534014
$$790$$ −2.00000 −0.0711568
$$791$$ −40.0000 −1.42224
$$792$$ 3.00000 0.106600
$$793$$ 0 0
$$794$$ −25.0000 −0.887217
$$795$$ −1.00000 −0.0354663
$$796$$ 22.0000 0.779769
$$797$$ −10.0000 −0.354218 −0.177109 0.984191i $$-0.556675\pi$$
−0.177109 + 0.984191i $$0.556675\pi$$
$$798$$ −25.0000 −0.884990
$$799$$ 24.0000 0.849059
$$800$$ −1.00000 −0.0353553
$$801$$ −11.0000 −0.388666
$$802$$ 19.0000 0.670913
$$803$$ 0 0
$$804$$ 8.00000 0.282138
$$805$$ −20.0000 −0.704907
$$806$$ 0 0
$$807$$ 4.00000 0.140807
$$808$$ 8.00000 0.281439
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 25.0000 0.877869 0.438934 0.898519i $$-0.355356\pi$$
0.438934 + 0.898519i $$0.355356\pi$$
$$812$$ 20.0000 0.701862
$$813$$ 4.00000 0.140286
$$814$$ −21.0000 −0.736050
$$815$$ 20.0000 0.700569
$$816$$ −8.00000 −0.280056
$$817$$ −30.0000 −1.04957
$$818$$ −25.0000 −0.874105
$$819$$ 0 0
$$820$$ −6.00000 −0.209529
$$821$$ −22.0000 −0.767805 −0.383903 0.923374i $$-0.625420\pi$$
−0.383903 + 0.923374i $$0.625420\pi$$
$$822$$ 12.0000 0.418548
$$823$$ −35.0000 −1.22002 −0.610012 0.792392i $$-0.708835\pi$$
−0.610012 + 0.792392i $$0.708835\pi$$
$$824$$ 7.00000 0.243857
$$825$$ −3.00000 −0.104447
$$826$$ 60.0000 2.08767
$$827$$ 6.00000 0.208640 0.104320 0.994544i $$-0.466733\pi$$
0.104320 + 0.994544i $$0.466733\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ −16.0000 −0.555703 −0.277851 0.960624i $$-0.589622\pi$$
−0.277851 + 0.960624i $$0.589622\pi$$
$$830$$ 8.00000 0.277684
$$831$$ −15.0000 −0.520344
$$832$$ 0 0
$$833$$ −144.000 −4.98930
$$834$$ −7.00000 −0.242390
$$835$$ 23.0000 0.795948
$$836$$ 15.0000 0.518786
$$837$$ −2.00000 −0.0691301
$$838$$ 12.0000 0.414533
$$839$$ −14.0000 −0.483334 −0.241667 0.970359i $$-0.577694\pi$$
−0.241667 + 0.970359i $$0.577694\pi$$
$$840$$ −5.00000 −0.172516
$$841$$ −13.0000 −0.448276
$$842$$ −12.0000 −0.413547
$$843$$ 18.0000 0.619953
$$844$$ −15.0000 −0.516321
$$845$$ 0 0
$$846$$ 3.00000 0.103142
$$847$$ 10.0000 0.343604
$$848$$ 1.00000 0.0343401
$$849$$ −10.0000 −0.343199
$$850$$ 8.00000 0.274398
$$851$$ 28.0000 0.959828
$$852$$ 2.00000 0.0685189
$$853$$ −26.0000 −0.890223 −0.445112 0.895475i $$-0.646836\pi$$
−0.445112 + 0.895475i $$0.646836\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 5.00000 0.170996
$$856$$ 6.00000 0.205076
$$857$$ 38.0000 1.29806 0.649028 0.760765i $$-0.275176\pi$$
0.649028 + 0.760765i $$0.275176\pi$$
$$858$$ 0 0
$$859$$ 3.00000 0.102359 0.0511793 0.998689i $$-0.483702\pi$$
0.0511793 + 0.998689i $$0.483702\pi$$
$$860$$ −6.00000 −0.204598
$$861$$ −30.0000 −1.02240
$$862$$ −12.0000 −0.408722
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −5.00000 −0.170005
$$866$$ −16.0000 −0.543702
$$867$$ 47.0000 1.59620
$$868$$ 10.0000 0.339422
$$869$$ 6.00000 0.203536
$$870$$ −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
$$877$$ −54.0000 −1.82345 −0.911725 0.410801i $$-0.865249\pi$$
−0.911725 + 0.410801i $$0.865249\pi$$
$$878$$ −10.0000 −0.337484
$$879$$ −9.00000 −0.303562
$$880$$ 3.00000 0.101130
$$881$$ −15.0000 −0.505363 −0.252681 0.967550i $$-0.581312\pi$$
−0.252681 + 0.967550i $$0.581312\pi$$
$$882$$ −18.0000 −0.606092
$$883$$ 30.0000 1.00958 0.504790 0.863242i $$-0.331570\pi$$
0.504790 + 0.863242i $$0.331570\pi$$
$$884$$ 0 0
$$885$$ −12.0000 −0.403376
$$886$$ −6.00000 −0.201574
$$887$$ 41.0000 1.37665 0.688323 0.725405i $$-0.258347\pi$$
0.688323 + 0.725405i $$0.258347\pi$$
$$888$$ 7.00000 0.234905
$$889$$ 105.000 3.52159
$$890$$ −11.0000 −0.368721
$$891$$ −3.00000 −0.100504
$$892$$ −3.00000 −0.100447
$$893$$ 15.0000 0.501956
$$894$$ 2.00000 0.0668900
$$895$$ −4.00000 −0.133705
$$896$$ 5.00000 0.167038
$$897$$ 0 0
$$898$$ 27.0000 0.901002
$$899$$ 8.00000 0.266815
$$900$$ 1.00000 0.0333333
$$901$$ −8.00000 −0.266519
$$902$$ 18.0000 0.599334
$$903$$ −30.0000 −0.998337
$$904$$ −8.00000 −0.266076
$$905$$ 2.00000 0.0664822
$$906$$ −22.0000 −0.730901
$$907$$ 10.0000 0.332045 0.166022 0.986122i $$-0.446908\pi$$
0.166022 + 0.986122i $$0.446908\pi$$
$$908$$ 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 $$0.607492\pi$$
$$912$$ −5.00000 −0.165567
$$913$$ −24.0000 −0.794284
$$914$$ −30.0000 −0.992312
$$915$$ −2.00000 −0.0661180
$$916$$ 14.0000 0.462573
$$917$$ 95.0000 3.13718
$$918$$ 8.00000 0.264039
$$919$$ −2.00000 −0.0659739 −0.0329870 0.999456i $$-0.510502\pi$$
−0.0329870 + 0.999456i $$0.510502\pi$$
$$920$$ −4.00000 −0.131876
$$921$$ −6.00000 −0.197707
$$922$$ −8.00000 −0.263466
$$923$$ 0 0
$$924$$ 15.0000 0.493464
$$925$$ −7.00000 −0.230159
$$926$$ −8.00000 −0.262896
$$927$$ −7.00000 −0.229910
$$928$$ 4.00000 0.131306
$$929$$ 2.00000 0.0656179 0.0328089 0.999462i $$-0.489555\pi$$
0.0328089 + 0.999462i $$0.489555\pi$$
$$930$$ −2.00000 −0.0655826
$$931$$ −90.0000 −2.94963
$$932$$ −14.0000 −0.458585
$$933$$ −12.0000 −0.392862
$$934$$ 24.0000 0.785304
$$935$$ −24.0000 −0.784884
$$936$$ 0 0
$$937$$ 30.0000 0.980057 0.490029 0.871706i $$-0.336986\pi$$
0.490029 + 0.871706i $$0.336986\pi$$
$$938$$ 40.0000 1.30605
$$939$$ −6.00000 −0.195803
$$940$$ 3.00000 0.0978492
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ −15.0000 −0.488726
$$943$$ −24.0000 −0.781548
$$944$$ 12.0000 0.390567
$$945$$ 5.00000 0.162650
$$946$$ 18.0000 0.585230
$$947$$ 58.0000 1.88475 0.942373 0.334563i $$-0.108589\pi$$
0.942373 + 0.334563i $$0.108589\pi$$
$$948$$ −2.00000 −0.0649570
$$949$$ 0 0
$$950$$ 5.00000 0.162221
$$951$$ −23.0000 −0.745826
$$952$$ −40.0000 −1.29641
$$953$$ −54.0000 −1.74923 −0.874616 0.484817i $$-0.838886\pi$$
−0.874616 + 0.484817i $$0.838886\pi$$
$$954$$ −1.00000 −0.0323762
$$955$$ −2.00000 −0.0647185
$$956$$ −18.0000 −0.582162
$$957$$ 12.0000 0.387905
$$958$$ 28.0000 0.904639
$$959$$ 60.0000 1.93750
$$960$$ −1.00000 −0.0322749
$$961$$ −27.0000 −0.870968
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ −25.0000 −0.805196
$$965$$ −24.0000 −0.772587
$$966$$ −20.0000 −0.643489
$$967$$ −31.0000 −0.996893 −0.498446 0.866921i $$-0.666096\pi$$
−0.498446 + 0.866921i $$0.666096\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ 40.0000 1.28499
$$970$$ 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
$$973$$ −35.0000 −1.12205
$$974$$ −37.0000 −1.18556
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 20.0000 0.639857 0.319928 0.947442i $$-0.396341\pi$$
0.319928 + 0.947442i $$0.396341\pi$$
$$978$$ 20.0000 0.639529
$$979$$ 33.0000 1.05468
$$980$$ −18.0000 −0.574989
$$981$$ 14.0000 0.446986
$$982$$ −21.0000 −0.670137
$$983$$ 23.0000 0.733586 0.366793 0.930303i $$-0.380456\pi$$
0.366793 + 0.930303i $$0.380456\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ −3.00000 −0.0955879
$$986$$ −32.0000 −1.01909
$$987$$ 15.0000 0.477455
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ −3.00000 −0.0953463
$$991$$ 10.0000 0.317660 0.158830 0.987306i $$-0.449228\pi$$
0.158830 + 0.987306i $$0.449228\pi$$
$$992$$ 2.00000 0.0635001
$$993$$ 4.00000 0.126936
$$994$$ 10.0000 0.317181
$$995$$ −22.0000 −0.697447
$$996$$ 8.00000 0.253490
$$997$$ −1.00000 −0.0316703 −0.0158352 0.999875i $$-0.505041\pi$$
−0.0158352 + 0.999875i $$0.505041\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ −7.00000 −0.221470
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.a.i.1.1 1
13.3 even 3 390.2.i.d.61.1 2
13.5 odd 4 5070.2.b.l.1351.2 2
13.8 odd 4 5070.2.b.l.1351.1 2
13.9 even 3 390.2.i.d.211.1 yes 2
13.12 even 2 5070.2.a.x.1.1 1
39.29 odd 6 1170.2.i.g.451.1 2
39.35 odd 6 1170.2.i.g.991.1 2
65.3 odd 12 1950.2.z.j.1699.1 4
65.9 even 6 1950.2.i.i.601.1 2
65.22 odd 12 1950.2.z.j.1849.1 4
65.29 even 6 1950.2.i.i.451.1 2
65.42 odd 12 1950.2.z.j.1699.2 4
65.48 odd 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 13.3 even 3
390.2.i.d.211.1 yes 2 13.9 even 3
1170.2.i.g.451.1 2 39.29 odd 6
1170.2.i.g.991.1 2 39.35 odd 6
1950.2.i.i.451.1 2 65.29 even 6
1950.2.i.i.601.1 2 65.9 even 6
1950.2.z.j.1699.1 4 65.3 odd 12
1950.2.z.j.1699.2 4 65.42 odd 12
1950.2.z.j.1849.1 4 65.22 odd 12
1950.2.z.j.1849.2 4 65.48 odd 12
5070.2.a.i.1.1 1 1.1 even 1 trivial
5070.2.a.x.1.1 1 13.12 even 2
5070.2.b.l.1351.1 2 13.8 odd 4
5070.2.b.l.1351.2 2 13.5 odd 4