Properties

 Label 1386.2.a.a.1.1 Level $1386$ Weight $2$ Character 1386.1 Self dual yes Analytic conductor $11.067$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1386.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$11.0672657201$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 462) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1386.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} -1.00000 q^{11} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -2.00000 q^{20} +1.00000 q^{22} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{28} +2.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} +2.00000 q^{35} -2.00000 q^{37} +2.00000 q^{40} +10.0000 q^{41} +4.00000 q^{43} -1.00000 q^{44} -4.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} +2.00000 q^{53} +2.00000 q^{55} +1.00000 q^{56} -2.00000 q^{58} +12.0000 q^{59} -2.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +12.0000 q^{67} +2.00000 q^{68} -2.00000 q^{70} -8.00000 q^{71} +6.00000 q^{73} +2.00000 q^{74} +1.00000 q^{77} -8.00000 q^{79} -2.00000 q^{80} -10.0000 q^{82} +8.00000 q^{83} -4.00000 q^{85} -4.00000 q^{86} +1.00000 q^{88} +14.0000 q^{89} +2.00000 q^{91} +4.00000 q^{94} -14.0000 q^{97} -1.00000 q^{98} +O(q^{100})$$

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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 2.00000 0.632456
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ −2.00000 −0.447214
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ −1.00000 −0.188982
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 2.00000 0.338062
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 2.00000 0.316228
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ −2.00000 −0.262613
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ −2.00000 −0.239046
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.00000 0.113961
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ −2.00000 −0.223607
$$81$$ 0 0
$$82$$ −10.0000 −1.10432
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ 0 0
$$85$$ −4.00000 −0.433861
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 1.00000 0.106600
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ 0 0
$$112$$ −1.00000 −0.0944911
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ 0 0
$$118$$ −12.0000 −1.10469
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 2.00000 0.181071
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ −4.00000 −0.350823
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ 2.00000 0.167248
$$144$$ 0 0
$$145$$ −4.00000 −0.332182
$$146$$ −6.00000 −0.496564
$$147$$ 0 0
$$148$$ −2.00000 −0.164399
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ −1.00000 −0.0805823
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 0 0
$$160$$ 2.00000 0.158114
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ −14.0000 −1.04934
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ −2.00000 −0.146254
$$188$$ −4.00000 −0.291730
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ −10.0000 −0.703598
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ −20.0000 −1.39686
$$206$$ −4.00000 −0.278693
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ −8.00000 −0.545595
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ −14.0000 −0.948200
$$219$$ 0 0
$$220$$ 2.00000 0.134840
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ 20.0000 1.33930 0.669650 0.742677i $$-0.266444\pi$$
0.669650 + 0.742677i $$0.266444\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −2.00000 −0.131306
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ 12.0000 0.781133
$$237$$ 0 0
$$238$$ 2.00000 0.129641
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 30.0000 1.93247 0.966235 0.257663i $$-0.0829523\pi$$
0.966235 + 0.257663i $$0.0829523\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 0 0
$$244$$ −2.00000 −0.128037
$$245$$ −2.00000 −0.127775
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −4.00000 −0.254000
$$249$$ 0 0
$$250$$ −12.0000 −0.758947
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 4.00000 0.248069
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −32.0000 −1.97320 −0.986602 0.163144i $$-0.947836\pi$$
−0.986602 + 0.163144i $$0.947836\pi$$
$$264$$ 0 0
$$265$$ −4.00000 −0.245718
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 12.0000 0.733017
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ 10.0000 0.604122
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ −2.00000 −0.119523
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ 16.0000 0.951101 0.475551 0.879688i $$-0.342249\pi$$
0.475551 + 0.879688i $$0.342249\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ −10.0000 −0.590281
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 4.00000 0.234888
$$291$$ 0 0
$$292$$ 6.00000 0.351123
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ −8.00000 −0.460348
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ 8.00000 0.456584 0.228292 0.973593i $$-0.426686\pi$$
0.228292 + 0.973593i $$0.426686\pi$$
$$308$$ 1.00000 0.0569803
$$309$$ 0 0
$$310$$ 8.00000 0.454369
$$311$$ −20.0000 −1.13410 −0.567048 0.823685i $$-0.691915\pi$$
−0.567048 + 0.823685i $$0.691915\pi$$
$$312$$ 0 0
$$313$$ −30.0000 −1.69570 −0.847850 0.530236i $$-0.822103\pi$$
−0.847850 + 0.530236i $$0.822103\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −22.0000 −1.23564 −0.617822 0.786318i $$-0.711985\pi$$
−0.617822 + 0.786318i $$0.711985\pi$$
$$318$$ 0 0
$$319$$ −2.00000 −0.111979
$$320$$ −2.00000 −0.111803
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 2.00000 0.110940
$$326$$ −4.00000 −0.221540
$$327$$ 0 0
$$328$$ −10.0000 −0.552158
$$329$$ 4.00000 0.220527
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 8.00000 0.439057
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −24.0000 −1.31126
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 0 0
$$340$$ −4.00000 −0.216930
$$341$$ −4.00000 −0.216612
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ 0 0
$$352$$ 1.00000 0.0533002
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ 14.0000 0.741999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ −2.00000 −0.105118
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ −12.0000 −0.628109
$$366$$ 0 0
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ −4.00000 −0.207950
$$371$$ −2.00000 −0.103835
$$372$$ 0 0
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 2.00000 0.103418
$$375$$ 0 0
$$376$$ 4.00000 0.206284
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −8.00000 −0.409316
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ 0 0
$$385$$ −2.00000 −0.101929
$$386$$ −2.00000 −0.101797
$$387$$ 0 0
$$388$$ −14.0000 −0.710742
$$389$$ −22.0000 −1.11544 −0.557722 0.830028i $$-0.688325\pi$$
−0.557722 + 0.830028i $$0.688325\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −1.00000 −0.0505076
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 16.0000 0.805047
$$396$$ 0 0
$$397$$ −30.0000 −1.50566 −0.752828 0.658217i $$-0.771311\pi$$
−0.752828 + 0.658217i $$0.771311\pi$$
$$398$$ −20.0000 −1.00251
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ −8.00000 −0.398508
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ 2.00000 0.0992583
$$407$$ 2.00000 0.0991363
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 20.0000 0.987730
$$411$$ 0 0
$$412$$ 4.00000 0.197066
$$413$$ −12.0000 −0.590481
$$414$$ 0 0
$$415$$ −16.0000 −0.785409
$$416$$ 2.00000 0.0980581
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ 0 0
$$424$$ −2.00000 −0.0971286
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ 2.00000 0.0967868
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 8.00000 0.385794
$$431$$ 40.0000 1.92673 0.963366 0.268190i $$-0.0864254\pi$$
0.963366 + 0.268190i $$0.0864254\pi$$
$$432$$ 0 0
$$433$$ 18.0000 0.865025 0.432512 0.901628i $$-0.357627\pi$$
0.432512 + 0.901628i $$0.357627\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ −2.00000 −0.0953463
$$441$$ 0 0
$$442$$ 4.00000 0.190261
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 0 0
$$445$$ −28.0000 −1.32733
$$446$$ −20.0000 −0.947027
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ −10.0000 −0.470882
$$452$$ −18.0000 −0.846649
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −4.00000 −0.187523
$$456$$ 0 0
$$457$$ −38.0000 −1.77757 −0.888783 0.458329i $$-0.848448\pi$$
−0.888783 + 0.458329i $$0.848448\pi$$
$$458$$ 14.0000 0.654177
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ −4.00000 −0.185098 −0.0925490 0.995708i $$-0.529501\pi$$
−0.0925490 + 0.995708i $$0.529501\pi$$
$$468$$ 0 0
$$469$$ −12.0000 −0.554109
$$470$$ −8.00000 −0.369012
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ −4.00000 −0.183920
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ 0 0
$$478$$ 16.0000 0.731823
$$479$$ 32.0000 1.46212 0.731059 0.682315i $$-0.239027\pi$$
0.731059 + 0.682315i $$0.239027\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ −30.0000 −1.36646
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 28.0000 1.27141
$$486$$ 0 0
$$487$$ −32.0000 −1.45006 −0.725029 0.688718i $$-0.758174\pi$$
−0.725029 + 0.688718i $$0.758174\pi$$
$$488$$ 2.00000 0.0905357
$$489$$ 0 0
$$490$$ 2.00000 0.0903508
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 0 0
$$493$$ 4.00000 0.180151
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 8.00000 0.358849
$$498$$ 0 0
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ 12.0000 0.536656
$$501$$ 0 0
$$502$$ 4.00000 0.178529
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ −20.0000 −0.889988
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ −10.0000 −0.443242 −0.221621 0.975133i $$-0.571135\pi$$
−0.221621 + 0.975133i $$0.571135\pi$$
$$510$$ 0 0
$$511$$ −6.00000 −0.265424
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ 4.00000 0.175920
$$518$$ −2.00000 −0.0878750
$$519$$ 0 0
$$520$$ −4.00000 −0.175412
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 0 0
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ 8.00000 0.348485
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 4.00000 0.173749
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −20.0000 −0.866296
$$534$$ 0 0
$$535$$ −24.0000 −1.03761
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ 10.0000 0.431131
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 8.00000 0.343629
$$543$$ 0 0
$$544$$ −2.00000 −0.0857493
$$545$$ −28.0000 −1.19939
$$546$$ 0 0
$$547$$ −44.0000 −1.88130 −0.940652 0.339372i $$-0.889785\pi$$
−0.940652 + 0.339372i $$0.889785\pi$$
$$548$$ −10.0000 −0.427179
$$549$$ 0 0
$$550$$ −1.00000 −0.0426401
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −14.0000 −0.593199 −0.296600 0.955002i $$-0.595853\pi$$
−0.296600 + 0.955002i $$0.595853\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 2.00000 0.0845154
$$561$$ 0 0
$$562$$ 10.0000 0.421825
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 36.0000 1.51453
$$566$$ −16.0000 −0.672530
$$567$$ 0 0
$$568$$ 8.00000 0.335673
$$569$$ 22.0000 0.922288 0.461144 0.887325i $$-0.347439\pi$$
0.461144 + 0.887325i $$0.347439\pi$$
$$570$$ 0 0
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ 0 0
$$574$$ 10.0000 0.417392
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 10.0000 0.416305 0.208153 0.978096i $$-0.433255\pi$$
0.208153 + 0.978096i $$0.433255\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 0 0
$$580$$ −4.00000 −0.166091
$$581$$ −8.00000 −0.331896
$$582$$ 0 0
$$583$$ −2.00000 −0.0828315
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 24.0000 0.988064
$$591$$ 0 0
$$592$$ −2.00000 −0.0821995
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ 0 0
$$595$$ 4.00000 0.163984
$$596$$ 10.0000 0.409616
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −18.0000 −0.734235 −0.367118 0.930175i $$-0.619655\pi$$
−0.367118 + 0.930175i $$0.619655\pi$$
$$602$$ 4.00000 0.163028
$$603$$ 0 0
$$604$$ 8.00000 0.325515
$$605$$ −2.00000 −0.0813116
$$606$$ 0 0
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ −4.00000 −0.161955
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ 38.0000 1.53481 0.767403 0.641165i $$-0.221549\pi$$
0.767403 + 0.641165i $$0.221549\pi$$
$$614$$ −8.00000 −0.322854
$$615$$ 0 0
$$616$$ −1.00000 −0.0402911
$$617$$ −26.0000 −1.04672 −0.523360 0.852111i $$-0.675322\pi$$
−0.523360 + 0.852111i $$0.675322\pi$$
$$618$$ 0 0
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ −8.00000 −0.321288
$$621$$ 0 0
$$622$$ 20.0000 0.801927
$$623$$ −14.0000 −0.560898
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 30.0000 1.19904
$$627$$ 0 0
$$628$$ 18.0000 0.718278
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 8.00000 0.318223
$$633$$ 0 0
$$634$$ 22.0000 0.873732
$$635$$ −16.0000 −0.634941
$$636$$ 0 0
$$637$$ −2.00000 −0.0792429
$$638$$ 2.00000 0.0791808
$$639$$ 0 0
$$640$$ 2.00000 0.0790569
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ 44.0000 1.73519 0.867595 0.497271i $$-0.165665\pi$$
0.867595 + 0.497271i $$0.165665\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 28.0000 1.10079 0.550397 0.834903i $$-0.314476\pi$$
0.550397 + 0.834903i $$0.314476\pi$$
$$648$$ 0 0
$$649$$ −12.0000 −0.471041
$$650$$ −2.00000 −0.0784465
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ 0 0
$$658$$ −4.00000 −0.155936
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ 18.0000 0.700119 0.350059 0.936727i $$-0.386161\pi$$
0.350059 + 0.936727i $$0.386161\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ −8.00000 −0.310460
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 24.0000 0.927201
$$671$$ 2.00000 0.0772091
$$672$$ 0 0
$$673$$ 26.0000 1.00223 0.501113 0.865382i $$-0.332924\pi$$
0.501113 + 0.865382i $$0.332924\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −14.0000 −0.538064 −0.269032 0.963131i $$-0.586704\pi$$
−0.269032 + 0.963131i $$0.586704\pi$$
$$678$$ 0 0
$$679$$ 14.0000 0.537271
$$680$$ 4.00000 0.153393
$$681$$ 0 0
$$682$$ 4.00000 0.153168
$$683$$ 28.0000 1.07139 0.535695 0.844411i $$-0.320050\pi$$
0.535695 + 0.844411i $$0.320050\pi$$
$$684$$ 0 0
$$685$$ 20.0000 0.764161
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ −12.0000 −0.455514
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 20.0000 0.757554
$$698$$ 26.0000 0.984115
$$699$$ 0 0
$$700$$ 1.00000 0.0377964
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ −10.0000 −0.376089
$$708$$ 0 0
$$709$$ −2.00000 −0.0751116 −0.0375558 0.999295i $$-0.511957\pi$$
−0.0375558 + 0.999295i $$0.511957\pi$$
$$710$$ −16.0000 −0.600469
$$711$$ 0 0
$$712$$ −14.0000 −0.524672
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −4.00000 −0.149592
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ −32.0000 −1.19423
$$719$$ −28.0000 −1.04422 −0.522112 0.852877i $$-0.674856\pi$$
−0.522112 + 0.852877i $$0.674856\pi$$
$$720$$ 0 0
$$721$$ −4.00000 −0.148968
$$722$$ 19.0000 0.707107
$$723$$ 0 0
$$724$$ 2.00000 0.0743294
$$725$$ −2.00000 −0.0742781
$$726$$ 0 0
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ −2.00000 −0.0741249
$$729$$ 0 0
$$730$$ 12.0000 0.444140
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ 46.0000 1.69905 0.849524 0.527549i $$-0.176889\pi$$
0.849524 + 0.527549i $$0.176889\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −12.0000 −0.442026
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 4.00000 0.147043
$$741$$ 0 0
$$742$$ 2.00000 0.0734223
$$743$$ 40.0000 1.46746 0.733729 0.679442i $$-0.237778\pi$$
0.733729 + 0.679442i $$0.237778\pi$$
$$744$$ 0 0
$$745$$ −20.0000 −0.732743
$$746$$ −22.0000 −0.805477
$$747$$ 0 0
$$748$$ −2.00000 −0.0731272
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ −4.00000 −0.145865
$$753$$ 0 0
$$754$$ 4.00000 0.145671
$$755$$ −16.0000 −0.582300
$$756$$ 0 0
$$757$$ −18.0000 −0.654221 −0.327111 0.944986i $$-0.606075\pi$$
−0.327111 + 0.944986i $$0.606075\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 2.00000 0.0724999 0.0362500 0.999343i $$-0.488459\pi$$
0.0362500 + 0.999343i $$0.488459\pi$$
$$762$$ 0 0
$$763$$ −14.0000 −0.506834
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 12.0000 0.433578
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ 2.00000 0.0720750
$$771$$ 0 0
$$772$$ 2.00000 0.0719816
$$773$$ −10.0000 −0.359675 −0.179838 0.983696i $$-0.557557\pi$$
−0.179838 + 0.983696i $$0.557557\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 14.0000 0.502571
$$777$$ 0 0
$$778$$ 22.0000 0.788738
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 8.00000 0.286263
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ −36.0000 −1.28490
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 0 0
$$790$$ −16.0000 −0.569254
$$791$$ 18.0000 0.640006
$$792$$ 0 0
$$793$$ 4.00000 0.142044
$$794$$ 30.0000 1.06466
$$795$$ 0 0
$$796$$ 20.0000 0.708881
$$797$$ −34.0000 −1.20434 −0.602171 0.798367i $$-0.705697\pi$$
−0.602171 + 0.798367i $$0.705697\pi$$
$$798$$ 0 0
$$799$$ −8.00000 −0.283020
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ 18.0000 0.635602
$$803$$ −6.00000 −0.211735
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 0 0
$$808$$ −10.0000 −0.351799
$$809$$ 14.0000 0.492214 0.246107 0.969243i $$-0.420849\pi$$
0.246107 + 0.969243i $$0.420849\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ −2.00000 −0.0701862
$$813$$ 0 0
$$814$$ −2.00000 −0.0701000
$$815$$ −8.00000 −0.280228
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −14.0000 −0.489499
$$819$$ 0 0
$$820$$ −20.0000 −0.698430
$$821$$ −22.0000 −0.767805 −0.383903 0.923374i $$-0.625420\pi$$
−0.383903 + 0.923374i $$0.625420\pi$$
$$822$$ 0 0
$$823$$ −48.0000 −1.67317 −0.836587 0.547833i $$-0.815453\pi$$
−0.836587 + 0.547833i $$0.815453\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 0 0
$$829$$ −6.00000 −0.208389 −0.104194 0.994557i $$-0.533226\pi$$
−0.104194 + 0.994557i $$0.533226\pi$$
$$830$$ 16.0000 0.555368
$$831$$ 0 0
$$832$$ −2.00000 −0.0693375
$$833$$ 2.00000 0.0692959
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −12.0000 −0.414533
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −6.00000 −0.206774
$$843$$ 0 0
$$844$$ 12.0000 0.413057
$$845$$ 18.0000 0.619219
$$846$$ 0 0
$$847$$ −1.00000 −0.0343604
$$848$$ 2.00000 0.0686803
$$849$$ 0 0
$$850$$ 2.00000 0.0685994
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 6.00000 0.205436 0.102718 0.994711i $$-0.467246\pi$$
0.102718 + 0.994711i $$0.467246\pi$$
$$854$$ −2.00000 −0.0684386
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 34.0000 1.16142 0.580709 0.814111i $$-0.302775\pi$$
0.580709 + 0.814111i $$0.302775\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 0 0
$$862$$ −40.0000 −1.36241
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ 12.0000 0.408012
$$866$$ −18.0000 −0.611665
$$867$$ 0 0
$$868$$ −4.00000 −0.135769
$$869$$ 8.00000 0.271381
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ −14.0000 −0.474100
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −12.0000 −0.405674
$$876$$ 0 0
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ −16.0000 −0.539974
$$879$$ 0 0
$$880$$ 2.00000 0.0674200
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 0 0
$$883$$ −28.0000 −0.942275 −0.471138 0.882060i $$-0.656156\pi$$
−0.471138 + 0.882060i $$0.656156\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 28.0000 0.938562
$$891$$ 0 0
$$892$$ 20.0000 0.669650
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ −30.0000 −1.00111
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ 4.00000 0.133259
$$902$$ 10.0000 0.332964
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ −4.00000 −0.132964
$$906$$ 0 0
$$907$$ −12.0000 −0.398453 −0.199227 0.979953i $$-0.563843\pi$$
−0.199227 + 0.979953i $$0.563843\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 4.00000 0.132599
$$911$$ 40.0000 1.32526 0.662630 0.748947i $$-0.269440\pi$$
0.662630 + 0.748947i $$0.269440\pi$$
$$912$$ 0 0
$$913$$ −8.00000 −0.264761
$$914$$ 38.0000 1.25693
$$915$$ 0 0
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −18.0000 −0.592798
$$923$$ 16.0000 0.526646
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ −16.0000 −0.525793
$$927$$ 0 0
$$928$$ −2.00000 −0.0656532
$$929$$ 54.0000 1.77168 0.885841 0.463988i $$-0.153582\pi$$
0.885841 + 0.463988i $$0.153582\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000 0.196537
$$933$$ 0 0
$$934$$ 4.00000 0.130884
$$935$$ 4.00000 0.130814
$$936$$ 0 0
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 12.0000 0.391814
$$939$$ 0 0
$$940$$ 8.00000 0.260931
$$941$$ −6.00000 −0.195594 −0.0977972 0.995206i $$-0.531180\pi$$
−0.0977972 + 0.995206i $$0.531180\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ −4.00000 −0.129983 −0.0649913 0.997886i $$-0.520702\pi$$
−0.0649913 + 0.997886i $$0.520702\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 2.00000 0.0648204
$$953$$ −18.0000 −0.583077 −0.291539 0.956559i $$-0.594167\pi$$
−0.291539 + 0.956559i $$0.594167\pi$$
$$954$$ 0 0
$$955$$ −16.0000 −0.517748
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ −32.0000 −1.03387
$$959$$ 10.0000 0.322917
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −4.00000 −0.128965
$$963$$ 0 0
$$964$$ 30.0000 0.966235
$$965$$ −4.00000 −0.128765
$$966$$ 0 0
$$967$$ −40.0000 −1.28631 −0.643157 0.765735i $$-0.722376\pi$$
−0.643157 + 0.765735i $$0.722376\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 0 0
$$970$$ −28.0000 −0.899026
$$971$$ 52.0000 1.66876 0.834380 0.551190i $$-0.185826\pi$$
0.834380 + 0.551190i $$0.185826\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 32.0000 1.02535
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ −2.00000 −0.0639857 −0.0319928 0.999488i $$-0.510185\pi$$
−0.0319928 + 0.999488i $$0.510185\pi$$
$$978$$ 0 0
$$979$$ −14.0000 −0.447442
$$980$$ −2.00000 −0.0638877
$$981$$ 0 0
$$982$$ −36.0000 −1.14881
$$983$$ 4.00000 0.127580 0.0637901 0.997963i $$-0.479681\pi$$
0.0637901 + 0.997963i $$0.479681\pi$$
$$984$$ 0 0
$$985$$ 12.0000 0.382352
$$986$$ −4.00000 −0.127386
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 0 0
$$994$$ −8.00000 −0.253745
$$995$$ −40.0000 −1.26809
$$996$$ 0 0
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ −28.0000 −0.886325
$$999$$ 0 0
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 1386.2.a.a.1.1 1
3.2 odd 2 462.2.a.g.1.1 1
7.6 odd 2 9702.2.a.r.1.1 1
12.11 even 2 3696.2.a.m.1.1 1
21.20 even 2 3234.2.a.p.1.1 1
33.32 even 2 5082.2.a.n.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.g.1.1 1 3.2 odd 2
1386.2.a.a.1.1 1 1.1 even 1 trivial
3234.2.a.p.1.1 1 21.20 even 2
3696.2.a.m.1.1 1 12.11 even 2
5082.2.a.n.1.1 1 33.32 even 2
9702.2.a.r.1.1 1 7.6 odd 2