# Properties

 Label 3234.2.a.bg.1.3 Level 3234 Weight 2 Character 3234.1 Self dual yes Analytic conductor 25.824 Analytic rank 0 Dimension 3 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.8236200137$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 Defining polynomial: $$x^{3} - 6 x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 462) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-0.523976$$ of defining polynomial Character $$\chi$$ $$=$$ 3234.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.20147 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.20147 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.20147 q^{10} +1.00000 q^{11} -1.00000 q^{12} +6.40294 q^{13} -3.20147 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -1.15352 q^{19} +3.20147 q^{20} +1.00000 q^{22} +1.95205 q^{23} -1.00000 q^{24} +5.24943 q^{25} +6.40294 q^{26} -1.00000 q^{27} +7.24943 q^{29} -3.20147 q^{30} -10.4989 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +5.15352 q^{37} -1.15352 q^{38} -6.40294 q^{39} +3.20147 q^{40} -8.24943 q^{41} +5.15352 q^{43} +1.00000 q^{44} +3.20147 q^{45} +1.95205 q^{46} +6.04795 q^{47} -1.00000 q^{48} +5.24943 q^{50} +1.00000 q^{51} +6.40294 q^{52} -6.40294 q^{53} -1.00000 q^{54} +3.20147 q^{55} +1.15352 q^{57} +7.24943 q^{58} -11.5565 q^{59} -3.20147 q^{60} -9.70032 q^{61} -10.4989 q^{62} +1.00000 q^{64} +20.4989 q^{65} -1.00000 q^{66} +6.24943 q^{67} -1.00000 q^{68} -1.95205 q^{69} +5.24943 q^{71} +1.00000 q^{72} +2.09591 q^{73} +5.15352 q^{74} -5.24943 q^{75} -1.15352 q^{76} -6.40294 q^{78} -5.60442 q^{79} +3.20147 q^{80} +1.00000 q^{81} -8.24943 q^{82} -6.55646 q^{83} -3.20147 q^{85} +5.15352 q^{86} -7.24943 q^{87} +1.00000 q^{88} +18.4029 q^{89} +3.20147 q^{90} +1.95205 q^{92} +10.4989 q^{93} +6.04795 q^{94} -3.69296 q^{95} -1.00000 q^{96} +5.49885 q^{97} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 3q^{8} + 3q^{9} + 3q^{11} - 3q^{12} + 3q^{16} - 3q^{17} + 3q^{18} + 3q^{19} + 3q^{22} + 9q^{23} - 3q^{24} + 3q^{25} - 3q^{27} + 9q^{29} - 6q^{31} + 3q^{32} - 3q^{33} - 3q^{34} + 3q^{36} + 9q^{37} + 3q^{38} - 12q^{41} + 9q^{43} + 3q^{44} + 9q^{46} + 15q^{47} - 3q^{48} + 3q^{50} + 3q^{51} - 3q^{54} - 3q^{57} + 9q^{58} - 9q^{59} + 6q^{61} - 6q^{62} + 3q^{64} + 36q^{65} - 3q^{66} + 6q^{67} - 3q^{68} - 9q^{69} + 3q^{71} + 3q^{72} + 9q^{74} - 3q^{75} + 3q^{76} + 12q^{79} + 3q^{81} - 12q^{82} + 6q^{83} + 9q^{86} - 9q^{87} + 3q^{88} + 36q^{89} + 9q^{92} + 6q^{93} + 15q^{94} - 24q^{95} - 3q^{96} - 9q^{97} + 3q^{99} + O(q^{100})$$

## 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
$$5$$ 3.20147 1.43174 0.715871 0.698233i $$-0.246030\pi$$
0.715871 + 0.698233i $$0.246030\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 3.20147 1.01239
$$11$$ 1.00000 0.301511
$$12$$ −1.00000 −0.288675
$$13$$ 6.40294 1.77586 0.887929 0.459981i $$-0.152144\pi$$
0.887929 + 0.459981i $$0.152144\pi$$
$$14$$ 0 0
$$15$$ −3.20147 −0.826617
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536 −0.121268 0.992620i $$-0.538696\pi$$
−0.121268 + 0.992620i $$0.538696\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −1.15352 −0.264636 −0.132318 0.991207i $$-0.542242\pi$$
−0.132318 + 0.991207i $$0.542242\pi$$
$$20$$ 3.20147 0.715871
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ 1.95205 0.407030 0.203515 0.979072i $$-0.434763\pi$$
0.203515 + 0.979072i $$0.434763\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 5.24943 1.04989
$$26$$ 6.40294 1.25572
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 7.24943 1.34618 0.673092 0.739559i $$-0.264966\pi$$
0.673092 + 0.739559i $$0.264966\pi$$
$$30$$ −3.20147 −0.584506
$$31$$ −10.4989 −1.88565 −0.942825 0.333289i $$-0.891841\pi$$
−0.942825 + 0.333289i $$0.891841\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −1.00000 −0.174078
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 5.15352 0.847233 0.423617 0.905842i $$-0.360760\pi$$
0.423617 + 0.905842i $$0.360760\pi$$
$$38$$ −1.15352 −0.187126
$$39$$ −6.40294 −1.02529
$$40$$ 3.20147 0.506197
$$41$$ −8.24943 −1.28834 −0.644172 0.764881i $$-0.722798\pi$$
−0.644172 + 0.764881i $$0.722798\pi$$
$$42$$ 0 0
$$43$$ 5.15352 0.785904 0.392952 0.919559i $$-0.371454\pi$$
0.392952 + 0.919559i $$0.371454\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 3.20147 0.477247
$$46$$ 1.95205 0.287814
$$47$$ 6.04795 0.882185 0.441092 0.897462i $$-0.354591\pi$$
0.441092 + 0.897462i $$0.354591\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ 5.24943 0.742381
$$51$$ 1.00000 0.140028
$$52$$ 6.40294 0.887929
$$53$$ −6.40294 −0.879512 −0.439756 0.898117i $$-0.644935\pi$$
−0.439756 + 0.898117i $$0.644935\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 3.20147 0.431686
$$56$$ 0 0
$$57$$ 1.15352 0.152787
$$58$$ 7.24943 0.951896
$$59$$ −11.5565 −1.50452 −0.752262 0.658864i $$-0.771037\pi$$
−0.752262 + 0.658864i $$0.771037\pi$$
$$60$$ −3.20147 −0.413308
$$61$$ −9.70032 −1.24200 −0.621000 0.783811i $$-0.713273\pi$$
−0.621000 + 0.783811i $$0.713273\pi$$
$$62$$ −10.4989 −1.33336
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 20.4989 2.54257
$$66$$ −1.00000 −0.123091
$$67$$ 6.24943 0.763489 0.381744 0.924268i $$-0.375323\pi$$
0.381744 + 0.924268i $$0.375323\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ −1.95205 −0.234999
$$70$$ 0 0
$$71$$ 5.24943 0.622992 0.311496 0.950247i $$-0.399170\pi$$
0.311496 + 0.950247i $$0.399170\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 2.09591 0.245307 0.122654 0.992450i $$-0.460860\pi$$
0.122654 + 0.992450i $$0.460860\pi$$
$$74$$ 5.15352 0.599084
$$75$$ −5.24943 −0.606151
$$76$$ −1.15352 −0.132318
$$77$$ 0 0
$$78$$ −6.40294 −0.724991
$$79$$ −5.60442 −0.630546 −0.315273 0.949001i $$-0.602096\pi$$
−0.315273 + 0.949001i $$0.602096\pi$$
$$80$$ 3.20147 0.357935
$$81$$ 1.00000 0.111111
$$82$$ −8.24943 −0.910997
$$83$$ −6.55646 −0.719665 −0.359833 0.933017i $$-0.617166\pi$$
−0.359833 + 0.933017i $$0.617166\pi$$
$$84$$ 0 0
$$85$$ −3.20147 −0.347248
$$86$$ 5.15352 0.555718
$$87$$ −7.24943 −0.777220
$$88$$ 1.00000 0.106600
$$89$$ 18.4029 1.95071 0.975354 0.220645i $$-0.0708163\pi$$
0.975354 + 0.220645i $$0.0708163\pi$$
$$90$$ 3.20147 0.337465
$$91$$ 0 0
$$92$$ 1.95205 0.203515
$$93$$ 10.4989 1.08868
$$94$$ 6.04795 0.623799
$$95$$ −3.69296 −0.378890
$$96$$ −1.00000 −0.102062
$$97$$ 5.49885 0.558324 0.279162 0.960244i $$-0.409943\pi$$
0.279162 + 0.960244i $$0.409943\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ 5.24943 0.524943
$$101$$ 19.9594 1.98604 0.993018 0.117965i $$-0.0376372\pi$$
0.993018 + 0.117965i $$0.0376372\pi$$
$$102$$ 1.00000 0.0990148
$$103$$ −0.307039 −0.0302535 −0.0151267 0.999886i $$-0.504815\pi$$
−0.0151267 + 0.999886i $$0.504815\pi$$
$$104$$ 6.40294 0.627860
$$105$$ 0 0
$$106$$ −6.40294 −0.621909
$$107$$ −9.05531 −0.875410 −0.437705 0.899119i $$-0.644209\pi$$
−0.437705 + 0.899119i $$0.644209\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −9.70032 −0.929122 −0.464561 0.885541i $$-0.653788\pi$$
−0.464561 + 0.885541i $$0.653788\pi$$
$$110$$ 3.20147 0.305248
$$111$$ −5.15352 −0.489150
$$112$$ 0 0
$$113$$ −12.4989 −1.17579 −0.587896 0.808936i $$-0.700044\pi$$
−0.587896 + 0.808936i $$0.700044\pi$$
$$114$$ 1.15352 0.108037
$$115$$ 6.24943 0.582762
$$116$$ 7.24943 0.673092
$$117$$ 6.40294 0.591952
$$118$$ −11.5565 −1.06386
$$119$$ 0 0
$$120$$ −3.20147 −0.292253
$$121$$ 1.00000 0.0909091
$$122$$ −9.70032 −0.878226
$$123$$ 8.24943 0.743826
$$124$$ −10.4989 −0.942825
$$125$$ 0.798528 0.0714225
$$126$$ 0 0
$$127$$ 8.35499 0.741386 0.370693 0.928756i $$-0.379120\pi$$
0.370693 + 0.928756i $$0.379120\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −5.15352 −0.453742
$$130$$ 20.4989 1.79787
$$131$$ −15.3047 −1.33718 −0.668591 0.743631i $$-0.733102\pi$$
−0.668591 + 0.743631i $$0.733102\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ 0 0
$$134$$ 6.24943 0.539868
$$135$$ −3.20147 −0.275539
$$136$$ −1.00000 −0.0857493
$$137$$ −6.40294 −0.547040 −0.273520 0.961866i $$-0.588188\pi$$
−0.273520 + 0.961866i $$0.588188\pi$$
$$138$$ −1.95205 −0.166169
$$139$$ −21.7483 −1.84466 −0.922332 0.386398i $$-0.873719\pi$$
−0.922332 + 0.386398i $$0.873719\pi$$
$$140$$ 0 0
$$141$$ −6.04795 −0.509330
$$142$$ 5.24943 0.440522
$$143$$ 6.40294 0.535441
$$144$$ 1.00000 0.0833333
$$145$$ 23.2088 1.92739
$$146$$ 2.09591 0.173458
$$147$$ 0 0
$$148$$ 5.15352 0.423617
$$149$$ 6.84648 0.560886 0.280443 0.959871i $$-0.409519\pi$$
0.280443 + 0.959871i $$0.409519\pi$$
$$150$$ −5.24943 −0.428614
$$151$$ 14.8538 1.20879 0.604394 0.796685i $$-0.293415\pi$$
0.604394 + 0.796685i $$0.293415\pi$$
$$152$$ −1.15352 −0.0935628
$$153$$ −1.00000 −0.0808452
$$154$$ 0 0
$$155$$ −33.6118 −2.69976
$$156$$ −6.40294 −0.512646
$$157$$ 9.46056 0.755035 0.377517 0.926002i $$-0.376778\pi$$
0.377517 + 0.926002i $$0.376778\pi$$
$$158$$ −5.60442 −0.445863
$$159$$ 6.40294 0.507787
$$160$$ 3.20147 0.253099
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 13.0553 1.02257 0.511286 0.859411i $$-0.329169\pi$$
0.511286 + 0.859411i $$0.329169\pi$$
$$164$$ −8.24943 −0.644172
$$165$$ −3.20147 −0.249234
$$166$$ −6.55646 −0.508880
$$167$$ 6.70998 0.519234 0.259617 0.965712i $$-0.416404\pi$$
0.259617 + 0.965712i $$0.416404\pi$$
$$168$$ 0 0
$$169$$ 27.9977 2.15367
$$170$$ −3.20147 −0.245542
$$171$$ −1.15352 −0.0882118
$$172$$ 5.15352 0.392952
$$173$$ 5.50115 0.418245 0.209122 0.977889i $$-0.432939\pi$$
0.209122 + 0.977889i $$0.432939\pi$$
$$174$$ −7.24943 −0.549578
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 11.5565 0.868637
$$178$$ 18.4029 1.37936
$$179$$ 1.65237 0.123504 0.0617520 0.998092i $$-0.480331\pi$$
0.0617520 + 0.998092i $$0.480331\pi$$
$$180$$ 3.20147 0.238624
$$181$$ −24.9018 −1.85094 −0.925468 0.378826i $$-0.876328\pi$$
−0.925468 + 0.378826i $$0.876328\pi$$
$$182$$ 0 0
$$183$$ 9.70032 0.717068
$$184$$ 1.95205 0.143907
$$185$$ 16.4989 1.21302
$$186$$ 10.4989 0.769813
$$187$$ −1.00000 −0.0731272
$$188$$ 6.04795 0.441092
$$189$$ 0 0
$$190$$ −3.69296 −0.267916
$$191$$ −4.80589 −0.347742 −0.173871 0.984768i $$-0.555628\pi$$
−0.173871 + 0.984768i $$0.555628\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 8.70998 0.626958 0.313479 0.949595i $$-0.398505\pi$$
0.313479 + 0.949595i $$0.398505\pi$$
$$194$$ 5.49885 0.394794
$$195$$ −20.4989 −1.46795
$$196$$ 0 0
$$197$$ 0.654669 0.0466433 0.0233216 0.999728i $$-0.492576\pi$$
0.0233216 + 0.999728i $$0.492576\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ 7.69296 0.545340 0.272670 0.962108i $$-0.412093\pi$$
0.272670 + 0.962108i $$0.412093\pi$$
$$200$$ 5.24943 0.371190
$$201$$ −6.24943 −0.440800
$$202$$ 19.9594 1.40434
$$203$$ 0 0
$$204$$ 1.00000 0.0700140
$$205$$ −26.4103 −1.84458
$$206$$ −0.307039 −0.0213924
$$207$$ 1.95205 0.135677
$$208$$ 6.40294 0.443964
$$209$$ −1.15352 −0.0797906
$$210$$ 0 0
$$211$$ 14.5948 1.00474 0.502372 0.864651i $$-0.332461\pi$$
0.502372 + 0.864651i $$0.332461\pi$$
$$212$$ −6.40294 −0.439756
$$213$$ −5.24943 −0.359685
$$214$$ −9.05531 −0.619009
$$215$$ 16.4989 1.12521
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ −9.70032 −0.656989
$$219$$ −2.09591 −0.141628
$$220$$ 3.20147 0.215843
$$221$$ −6.40294 −0.430709
$$222$$ −5.15352 −0.345882
$$223$$ 14.9018 0.997898 0.498949 0.866631i $$-0.333719\pi$$
0.498949 + 0.866631i $$0.333719\pi$$
$$224$$ 0 0
$$225$$ 5.24943 0.349962
$$226$$ −12.4989 −0.831411
$$227$$ 0.249425 0.0165549 0.00827746 0.999966i $$-0.497365\pi$$
0.00827746 + 0.999966i $$0.497365\pi$$
$$228$$ 1.15352 0.0763937
$$229$$ 4.09591 0.270665 0.135333 0.990800i $$-0.456790\pi$$
0.135333 + 0.990800i $$0.456790\pi$$
$$230$$ 6.24943 0.412075
$$231$$ 0 0
$$232$$ 7.24943 0.475948
$$233$$ 11.3070 0.740749 0.370374 0.928883i $$-0.379229\pi$$
0.370374 + 0.928883i $$0.379229\pi$$
$$234$$ 6.40294 0.418574
$$235$$ 19.3624 1.26306
$$236$$ −11.5565 −0.752262
$$237$$ 5.60442 0.364046
$$238$$ 0 0
$$239$$ 6.59476 0.426579 0.213290 0.976989i $$-0.431582\pi$$
0.213290 + 0.976989i $$0.431582\pi$$
$$240$$ −3.20147 −0.206654
$$241$$ −17.1129 −1.10234 −0.551170 0.834393i $$-0.685819\pi$$
−0.551170 + 0.834393i $$0.685819\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ −9.70032 −0.621000
$$245$$ 0 0
$$246$$ 8.24943 0.525964
$$247$$ −7.38592 −0.469955
$$248$$ −10.4989 −0.666678
$$249$$ 6.55646 0.415499
$$250$$ 0.798528 0.0505033
$$251$$ −7.95941 −0.502393 −0.251197 0.967936i $$-0.580824\pi$$
−0.251197 + 0.967936i $$0.580824\pi$$
$$252$$ 0 0
$$253$$ 1.95205 0.122724
$$254$$ 8.35499 0.524239
$$255$$ 3.20147 0.200484
$$256$$ 1.00000 0.0625000
$$257$$ 22.4029 1.39746 0.698729 0.715387i $$-0.253749\pi$$
0.698729 + 0.715387i $$0.253749\pi$$
$$258$$ −5.15352 −0.320844
$$259$$ 0 0
$$260$$ 20.4989 1.27128
$$261$$ 7.24943 0.448728
$$262$$ −15.3047 −0.945530
$$263$$ −25.0170 −1.54262 −0.771308 0.636462i $$-0.780397\pi$$
−0.771308 + 0.636462i $$0.780397\pi$$
$$264$$ −1.00000 −0.0615457
$$265$$ −20.4989 −1.25923
$$266$$ 0 0
$$267$$ −18.4029 −1.12624
$$268$$ 6.24943 0.381744
$$269$$ −8.10327 −0.494065 −0.247032 0.969007i $$-0.579455\pi$$
−0.247032 + 0.969007i $$0.579455\pi$$
$$270$$ −3.20147 −0.194835
$$271$$ 4.80589 0.291937 0.145968 0.989289i $$-0.453370\pi$$
0.145968 + 0.989289i $$0.453370\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ 0 0
$$274$$ −6.40294 −0.386816
$$275$$ 5.24943 0.316552
$$276$$ −1.95205 −0.117499
$$277$$ −11.2088 −0.673474 −0.336737 0.941599i $$-0.609323\pi$$
−0.336737 + 0.941599i $$0.609323\pi$$
$$278$$ −21.7483 −1.30437
$$279$$ −10.4989 −0.628550
$$280$$ 0 0
$$281$$ −17.4989 −1.04389 −0.521947 0.852978i $$-0.674794\pi$$
−0.521947 + 0.852978i $$0.674794\pi$$
$$282$$ −6.04795 −0.360150
$$283$$ −27.4006 −1.62880 −0.814400 0.580304i $$-0.802934\pi$$
−0.814400 + 0.580304i $$0.802934\pi$$
$$284$$ 5.24943 0.311496
$$285$$ 3.69296 0.218752
$$286$$ 6.40294 0.378614
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ −16.0000 −0.941176
$$290$$ 23.2088 1.36287
$$291$$ −5.49885 −0.322348
$$292$$ 2.09591 0.122654
$$293$$ 19.7483 1.15371 0.576853 0.816848i $$-0.304280\pi$$
0.576853 + 0.816848i $$0.304280\pi$$
$$294$$ 0 0
$$295$$ −36.9977 −2.15409
$$296$$ 5.15352 0.299542
$$297$$ −1.00000 −0.0580259
$$298$$ 6.84648 0.396606
$$299$$ 12.4989 0.722827
$$300$$ −5.24943 −0.303076
$$301$$ 0 0
$$302$$ 14.8538 0.854743
$$303$$ −19.9594 −1.14664
$$304$$ −1.15352 −0.0661589
$$305$$ −31.0553 −1.77822
$$306$$ −1.00000 −0.0571662
$$307$$ −4.30704 −0.245816 −0.122908 0.992418i $$-0.539222\pi$$
−0.122908 + 0.992418i $$0.539222\pi$$
$$308$$ 0 0
$$309$$ 0.307039 0.0174668
$$310$$ −33.6118 −1.90902
$$311$$ 14.7579 0.836846 0.418423 0.908252i $$-0.362583\pi$$
0.418423 + 0.908252i $$0.362583\pi$$
$$312$$ −6.40294 −0.362495
$$313$$ −20.7506 −1.17289 −0.586446 0.809988i $$-0.699473\pi$$
−0.586446 + 0.809988i $$0.699473\pi$$
$$314$$ 9.46056 0.533890
$$315$$ 0 0
$$316$$ −5.60442 −0.315273
$$317$$ 16.1992 0.909836 0.454918 0.890533i $$-0.349669\pi$$
0.454918 + 0.890533i $$0.349669\pi$$
$$318$$ 6.40294 0.359059
$$319$$ 7.24943 0.405890
$$320$$ 3.20147 0.178968
$$321$$ 9.05531 0.505418
$$322$$ 0 0
$$323$$ 1.15352 0.0641835
$$324$$ 1.00000 0.0555556
$$325$$ 33.6118 1.86445
$$326$$ 13.0553 0.723067
$$327$$ 9.70032 0.536429
$$328$$ −8.24943 −0.455498
$$329$$ 0 0
$$330$$ −3.20147 −0.176235
$$331$$ 21.2471 1.16785 0.583924 0.811808i $$-0.301517\pi$$
0.583924 + 0.811808i $$0.301517\pi$$
$$332$$ −6.55646 −0.359833
$$333$$ 5.15352 0.282411
$$334$$ 6.70998 0.367154
$$335$$ 20.0074 1.09312
$$336$$ 0 0
$$337$$ −25.5159 −1.38994 −0.694969 0.719040i $$-0.744582\pi$$
−0.694969 + 0.719040i $$0.744582\pi$$
$$338$$ 27.9977 1.52287
$$339$$ 12.4989 0.678844
$$340$$ −3.20147 −0.173624
$$341$$ −10.4989 −0.568545
$$342$$ −1.15352 −0.0623752
$$343$$ 0 0
$$344$$ 5.15352 0.277859
$$345$$ −6.24943 −0.336458
$$346$$ 5.50115 0.295744
$$347$$ 3.05531 0.164018 0.0820089 0.996632i $$-0.473866\pi$$
0.0820089 + 0.996632i $$0.473866\pi$$
$$348$$ −7.24943 −0.388610
$$349$$ 4.00736 0.214509 0.107255 0.994232i $$-0.465794\pi$$
0.107255 + 0.994232i $$0.465794\pi$$
$$350$$ 0 0
$$351$$ −6.40294 −0.341764
$$352$$ 1.00000 0.0533002
$$353$$ 1.09821 0.0584516 0.0292258 0.999573i $$-0.490696\pi$$
0.0292258 + 0.999573i $$0.490696\pi$$
$$354$$ 11.5565 0.614219
$$355$$ 16.8059 0.891964
$$356$$ 18.4029 0.975354
$$357$$ 0 0
$$358$$ 1.65237 0.0873305
$$359$$ 15.6118 0.823958 0.411979 0.911193i $$-0.364838\pi$$
0.411979 + 0.911193i $$0.364838\pi$$
$$360$$ 3.20147 0.168732
$$361$$ −17.6694 −0.929968
$$362$$ −24.9018 −1.30881
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ 6.70998 0.351217
$$366$$ 9.70032 0.507044
$$367$$ −0.791166 −0.0412985 −0.0206493 0.999787i $$-0.506573\pi$$
−0.0206493 + 0.999787i $$0.506573\pi$$
$$368$$ 1.95205 0.101757
$$369$$ −8.24943 −0.429448
$$370$$ 16.4989 0.857734
$$371$$ 0 0
$$372$$ 10.4989 0.544340
$$373$$ 15.9115 0.823864 0.411932 0.911215i $$-0.364854\pi$$
0.411932 + 0.911215i $$0.364854\pi$$
$$374$$ −1.00000 −0.0517088
$$375$$ −0.798528 −0.0412358
$$376$$ 6.04795 0.311899
$$377$$ 46.4177 2.39063
$$378$$ 0 0
$$379$$ −17.0553 −0.876073 −0.438036 0.898957i $$-0.644326\pi$$
−0.438036 + 0.898957i $$0.644326\pi$$
$$380$$ −3.69296 −0.189445
$$381$$ −8.35499 −0.428039
$$382$$ −4.80589 −0.245891
$$383$$ −13.4412 −0.686815 −0.343408 0.939186i $$-0.611581\pi$$
−0.343408 + 0.939186i $$0.611581\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 8.70998 0.443327
$$387$$ 5.15352 0.261968
$$388$$ 5.49885 0.279162
$$389$$ −7.20147 −0.365129 −0.182565 0.983194i $$-0.558440\pi$$
−0.182565 + 0.983194i $$0.558440\pi$$
$$390$$ −20.4989 −1.03800
$$391$$ −1.95205 −0.0987193
$$392$$ 0 0
$$393$$ 15.3047 0.772022
$$394$$ 0.654669 0.0329818
$$395$$ −17.9424 −0.902779
$$396$$ 1.00000 0.0502519
$$397$$ −12.9424 −0.649560 −0.324780 0.945790i $$-0.605290\pi$$
−0.324780 + 0.945790i $$0.605290\pi$$
$$398$$ 7.69296 0.385613
$$399$$ 0 0
$$400$$ 5.24943 0.262471
$$401$$ −16.9018 −0.844035 −0.422018 0.906588i $$-0.638678\pi$$
−0.422018 + 0.906588i $$0.638678\pi$$
$$402$$ −6.24943 −0.311693
$$403$$ −67.2236 −3.34864
$$404$$ 19.9594 0.993018
$$405$$ 3.20147 0.159082
$$406$$ 0 0
$$407$$ 5.15352 0.255450
$$408$$ 1.00000 0.0495074
$$409$$ 37.0936 1.83416 0.917080 0.398702i $$-0.130539\pi$$
0.917080 + 0.398702i $$0.130539\pi$$
$$410$$ −26.4103 −1.30431
$$411$$ 6.40294 0.315834
$$412$$ −0.307039 −0.0151267
$$413$$ 0 0
$$414$$ 1.95205 0.0959379
$$415$$ −20.9903 −1.03038
$$416$$ 6.40294 0.313930
$$417$$ 21.7483 1.06502
$$418$$ −1.15352 −0.0564205
$$419$$ −22.4583 −1.09716 −0.548579 0.836099i $$-0.684831\pi$$
−0.548579 + 0.836099i $$0.684831\pi$$
$$420$$ 0 0
$$421$$ −23.3453 −1.13778 −0.568891 0.822413i $$-0.692627\pi$$
−0.568891 + 0.822413i $$0.692627\pi$$
$$422$$ 14.5948 0.710462
$$423$$ 6.04795 0.294062
$$424$$ −6.40294 −0.310954
$$425$$ −5.24943 −0.254635
$$426$$ −5.24943 −0.254335
$$427$$ 0 0
$$428$$ −9.05531 −0.437705
$$429$$ −6.40294 −0.309137
$$430$$ 16.4989 0.795645
$$431$$ 31.8229 1.53286 0.766428 0.642330i $$-0.222032\pi$$
0.766428 + 0.642330i $$0.222032\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −30.3047 −1.45635 −0.728176 0.685390i $$-0.759632\pi$$
−0.728176 + 0.685390i $$0.759632\pi$$
$$434$$ 0 0
$$435$$ −23.2088 −1.11278
$$436$$ −9.70032 −0.464561
$$437$$ −2.25172 −0.107715
$$438$$ −2.09591 −0.100146
$$439$$ 10.6620 0.508871 0.254435 0.967090i $$-0.418110\pi$$
0.254435 + 0.967090i $$0.418110\pi$$
$$440$$ 3.20147 0.152624
$$441$$ 0 0
$$442$$ −6.40294 −0.304557
$$443$$ −36.0553 −1.71304 −0.856520 0.516114i $$-0.827378\pi$$
−0.856520 + 0.516114i $$0.827378\pi$$
$$444$$ −5.15352 −0.244575
$$445$$ 58.9165 2.79291
$$446$$ 14.9018 0.705620
$$447$$ −6.84648 −0.323827
$$448$$ 0 0
$$449$$ 23.4006 1.10434 0.552172 0.833730i $$-0.313799\pi$$
0.552172 + 0.833730i $$0.313799\pi$$
$$450$$ 5.24943 0.247460
$$451$$ −8.24943 −0.388450
$$452$$ −12.4989 −0.587896
$$453$$ −14.8538 −0.697894
$$454$$ 0.249425 0.0117061
$$455$$ 0 0
$$456$$ 1.15352 0.0540185
$$457$$ −18.6141 −0.870730 −0.435365 0.900254i $$-0.643381\pi$$
−0.435365 + 0.900254i $$0.643381\pi$$
$$458$$ 4.09591 0.191389
$$459$$ 1.00000 0.0466760
$$460$$ 6.24943 0.291381
$$461$$ 31.4412 1.46436 0.732182 0.681109i $$-0.238502\pi$$
0.732182 + 0.681109i $$0.238502\pi$$
$$462$$ 0 0
$$463$$ −22.0959 −1.02688 −0.513442 0.858124i $$-0.671630\pi$$
−0.513442 + 0.858124i $$0.671630\pi$$
$$464$$ 7.24943 0.336546
$$465$$ 33.6118 1.55871
$$466$$ 11.3070 0.523788
$$467$$ −21.0576 −0.974430 −0.487215 0.873282i $$-0.661987\pi$$
−0.487215 + 0.873282i $$0.661987\pi$$
$$468$$ 6.40294 0.295976
$$469$$ 0 0
$$470$$ 19.3624 0.893119
$$471$$ −9.46056 −0.435920
$$472$$ −11.5565 −0.531929
$$473$$ 5.15352 0.236959
$$474$$ 5.60442 0.257419
$$475$$ −6.05531 −0.277837
$$476$$ 0 0
$$477$$ −6.40294 −0.293171
$$478$$ 6.59476 0.301637
$$479$$ 21.2088 0.969056 0.484528 0.874776i $$-0.338991\pi$$
0.484528 + 0.874776i $$0.338991\pi$$
$$480$$ −3.20147 −0.146127
$$481$$ 32.9977 1.50457
$$482$$ −17.1129 −0.779473
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 17.6044 0.799375
$$486$$ −1.00000 −0.0453609
$$487$$ −38.7100 −1.75412 −0.877058 0.480384i $$-0.840497\pi$$
−0.877058 + 0.480384i $$0.840497\pi$$
$$488$$ −9.70032 −0.439113
$$489$$ −13.0553 −0.590382
$$490$$ 0 0
$$491$$ 19.1705 0.865154 0.432577 0.901597i $$-0.357604\pi$$
0.432577 + 0.901597i $$0.357604\pi$$
$$492$$ 8.24943 0.371913
$$493$$ −7.24943 −0.326498
$$494$$ −7.38592 −0.332308
$$495$$ 3.20147 0.143895
$$496$$ −10.4989 −0.471412
$$497$$ 0 0
$$498$$ 6.55646 0.293802
$$499$$ −27.9188 −1.24982 −0.624909 0.780698i $$-0.714864\pi$$
−0.624909 + 0.780698i $$0.714864\pi$$
$$500$$ 0.798528 0.0357112
$$501$$ −6.70998 −0.299780
$$502$$ −7.95941 −0.355246
$$503$$ 8.70998 0.388359 0.194179 0.980966i $$-0.437796\pi$$
0.194179 + 0.980966i $$0.437796\pi$$
$$504$$ 0 0
$$505$$ 63.8995 2.84349
$$506$$ 1.95205 0.0867791
$$507$$ −27.9977 −1.24342
$$508$$ 8.35499 0.370693
$$509$$ 1.78887 0.0792901 0.0396451 0.999214i $$-0.487377\pi$$
0.0396451 + 0.999214i $$0.487377\pi$$
$$510$$ 3.20147 0.141764
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 1.15352 0.0509291
$$514$$ 22.4029 0.988152
$$515$$ −0.982977 −0.0433151
$$516$$ −5.15352 −0.226871
$$517$$ 6.04795 0.265989
$$518$$ 0 0
$$519$$ −5.50115 −0.241474
$$520$$ 20.4989 0.898934
$$521$$ −23.6118 −1.03445 −0.517225 0.855849i $$-0.673035\pi$$
−0.517225 + 0.855849i $$0.673035\pi$$
$$522$$ 7.24943 0.317299
$$523$$ −43.4006 −1.89778 −0.948889 0.315610i $$-0.897791\pi$$
−0.948889 + 0.315610i $$0.897791\pi$$
$$524$$ −15.3047 −0.668591
$$525$$ 0 0
$$526$$ −25.0170 −1.09079
$$527$$ 10.4989 0.457337
$$528$$ −1.00000 −0.0435194
$$529$$ −19.1895 −0.834327
$$530$$ −20.4989 −0.890413
$$531$$ −11.5565 −0.501508
$$532$$ 0 0
$$533$$ −52.8206 −2.28791
$$534$$ −18.4029 −0.796373
$$535$$ −28.9903 −1.25336
$$536$$ 6.24943 0.269934
$$537$$ −1.65237 −0.0713050
$$538$$ −8.10327 −0.349357
$$539$$ 0 0
$$540$$ −3.20147 −0.137769
$$541$$ 16.9903 0.730472 0.365236 0.930915i $$-0.380988\pi$$
0.365236 + 0.930915i $$0.380988\pi$$
$$542$$ 4.80589 0.206431
$$543$$ 24.9018 1.06864
$$544$$ −1.00000 −0.0428746
$$545$$ −31.0553 −1.33026
$$546$$ 0 0
$$547$$ −1.34533 −0.0575222 −0.0287611 0.999586i $$-0.509156\pi$$
−0.0287611 + 0.999586i $$0.509156\pi$$
$$548$$ −6.40294 −0.273520
$$549$$ −9.70032 −0.414000
$$550$$ 5.24943 0.223836
$$551$$ −8.36235 −0.356248
$$552$$ −1.95205 −0.0830846
$$553$$ 0 0
$$554$$ −11.2088 −0.476218
$$555$$ −16.4989 −0.700337
$$556$$ −21.7483 −0.922332
$$557$$ −16.6547 −0.705681 −0.352840 0.935683i $$-0.614784\pi$$
−0.352840 + 0.935683i $$0.614784\pi$$
$$558$$ −10.4989 −0.444452
$$559$$ 32.9977 1.39565
$$560$$ 0 0
$$561$$ 1.00000 0.0422200
$$562$$ −17.4989 −0.738144
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ −6.04795 −0.254665
$$565$$ −40.0147 −1.68343
$$566$$ −27.4006 −1.15174
$$567$$ 0 0
$$568$$ 5.24943 0.220261
$$569$$ −10.5542 −0.442454 −0.221227 0.975222i $$-0.571006\pi$$
−0.221227 + 0.975222i $$0.571006\pi$$
$$570$$ 3.69296 0.154681
$$571$$ 4.75057 0.198805 0.0994027 0.995047i $$-0.468307\pi$$
0.0994027 + 0.995047i $$0.468307\pi$$
$$572$$ 6.40294 0.267721
$$573$$ 4.80589 0.200769
$$574$$ 0 0
$$575$$ 10.2471 0.427335
$$576$$ 1.00000 0.0416667
$$577$$ −37.5542 −1.56340 −0.781700 0.623654i $$-0.785647\pi$$
−0.781700 + 0.623654i $$0.785647\pi$$
$$578$$ −16.0000 −0.665512
$$579$$ −8.70998 −0.361975
$$580$$ 23.2088 0.963694
$$581$$ 0 0
$$582$$ −5.49885 −0.227935
$$583$$ −6.40294 −0.265183
$$584$$ 2.09591 0.0867292
$$585$$ 20.4989 0.847523
$$586$$ 19.7483 0.815794
$$587$$ 17.0982 0.705718 0.352859 0.935676i $$-0.385209\pi$$
0.352859 + 0.935676i $$0.385209\pi$$
$$588$$ 0 0
$$589$$ 12.1106 0.499010
$$590$$ −36.9977 −1.52317
$$591$$ −0.654669 −0.0269295
$$592$$ 5.15352 0.211808
$$593$$ 32.2471 1.32423 0.662115 0.749402i $$-0.269659\pi$$
0.662115 + 0.749402i $$0.269659\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ 0 0
$$596$$ 6.84648 0.280443
$$597$$ −7.69296 −0.314852
$$598$$ 12.4989 0.511116
$$599$$ 3.39328 0.138646 0.0693229 0.997594i $$-0.477916\pi$$
0.0693229 + 0.997594i $$0.477916\pi$$
$$600$$ −5.24943 −0.214307
$$601$$ −30.4989 −1.24407 −0.622037 0.782988i $$-0.713695\pi$$
−0.622037 + 0.782988i $$0.713695\pi$$
$$602$$ 0 0
$$603$$ 6.24943 0.254496
$$604$$ 14.8538 0.604394
$$605$$ 3.20147 0.130158
$$606$$ −19.9594 −0.810796
$$607$$ −12.9903 −0.527262 −0.263631 0.964624i $$-0.584920\pi$$
−0.263631 + 0.964624i $$0.584920\pi$$
$$608$$ −1.15352 −0.0467814
$$609$$ 0 0
$$610$$ −31.0553 −1.25739
$$611$$ 38.7247 1.56663
$$612$$ −1.00000 −0.0404226
$$613$$ 20.1992 0.815837 0.407918 0.913018i $$-0.366255\pi$$
0.407918 + 0.913018i $$0.366255\pi$$
$$614$$ −4.30704 −0.173818
$$615$$ 26.4103 1.06497
$$616$$ 0 0
$$617$$ 17.0936 0.688163 0.344081 0.938940i $$-0.388190\pi$$
0.344081 + 0.938940i $$0.388190\pi$$
$$618$$ 0.307039 0.0123509
$$619$$ −41.6694 −1.67483 −0.837417 0.546564i $$-0.815935\pi$$
−0.837417 + 0.546564i $$0.815935\pi$$
$$620$$ −33.6118 −1.34988
$$621$$ −1.95205 −0.0783330
$$622$$ 14.7579 0.591739
$$623$$ 0 0
$$624$$ −6.40294 −0.256323
$$625$$ −23.6907 −0.947626
$$626$$ −20.7506 −0.829360
$$627$$ 1.15352 0.0460671
$$628$$ 9.46056 0.377517
$$629$$ −5.15352 −0.205484
$$630$$ 0 0
$$631$$ −23.8995 −0.951424 −0.475712 0.879601i $$-0.657810\pi$$
−0.475712 + 0.879601i $$0.657810\pi$$
$$632$$ −5.60442 −0.222932
$$633$$ −14.5948 −0.580089
$$634$$ 16.1992 0.643351
$$635$$ 26.7483 1.06147
$$636$$ 6.40294 0.253893
$$637$$ 0 0
$$638$$ 7.24943 0.287007
$$639$$ 5.24943 0.207664
$$640$$ 3.20147 0.126549
$$641$$ −14.3070 −0.565094 −0.282547 0.959253i $$-0.591179\pi$$
−0.282547 + 0.959253i $$0.591179\pi$$
$$642$$ 9.05531 0.357385
$$643$$ 35.4966 1.39985 0.699924 0.714218i $$-0.253217\pi$$
0.699924 + 0.714218i $$0.253217\pi$$
$$644$$ 0 0
$$645$$ −16.4989 −0.649642
$$646$$ 1.15352 0.0453846
$$647$$ −23.8156 −0.936286 −0.468143 0.883653i $$-0.655077\pi$$
−0.468143 + 0.883653i $$0.655077\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −11.5565 −0.453631
$$650$$ 33.6118 1.31836
$$651$$ 0 0
$$652$$ 13.0553 0.511286
$$653$$ −13.7962 −0.539888 −0.269944 0.962876i $$-0.587005\pi$$
−0.269944 + 0.962876i $$0.587005\pi$$
$$654$$ 9.70032 0.379313
$$655$$ −48.9977 −1.91450
$$656$$ −8.24943 −0.322086
$$657$$ 2.09591 0.0817691
$$658$$ 0 0
$$659$$ 18.9447 0.737980 0.368990 0.929433i $$-0.379704\pi$$
0.368990 + 0.929433i $$0.379704\pi$$
$$660$$ −3.20147 −0.124617
$$661$$ −3.95941 −0.154003 −0.0770016 0.997031i $$-0.524535\pi$$
−0.0770016 + 0.997031i $$0.524535\pi$$
$$662$$ 21.2471 0.825793
$$663$$ 6.40294 0.248670
$$664$$ −6.55646 −0.254440
$$665$$ 0 0
$$666$$ 5.15352 0.199695
$$667$$ 14.1512 0.547937
$$668$$ 6.70998 0.259617
$$669$$ −14.9018 −0.576137
$$670$$ 20.0074 0.772952
$$671$$ −9.70032 −0.374477
$$672$$ 0 0
$$673$$ −8.69066 −0.335000 −0.167500 0.985872i $$-0.553569\pi$$
−0.167500 + 0.985872i $$0.553569\pi$$
$$674$$ −25.5159 −0.982835
$$675$$ −5.24943 −0.202050
$$676$$ 27.9977 1.07683
$$677$$ 31.4606 1.20913 0.604564 0.796557i $$-0.293347\pi$$
0.604564 + 0.796557i $$0.293347\pi$$
$$678$$ 12.4989 0.480015
$$679$$ 0 0
$$680$$ −3.20147 −0.122771
$$681$$ −0.249425 −0.00955799
$$682$$ −10.4989 −0.402022
$$683$$ 24.7460 0.946878 0.473439 0.880826i $$-0.343012\pi$$
0.473439 + 0.880826i $$0.343012\pi$$
$$684$$ −1.15352 −0.0441059
$$685$$ −20.4989 −0.783221
$$686$$ 0 0
$$687$$ −4.09591 −0.156269
$$688$$ 5.15352 0.196476
$$689$$ −40.9977 −1.56189
$$690$$ −6.24943 −0.237912
$$691$$ −25.5542 −0.972126 −0.486063 0.873924i $$-0.661567\pi$$
−0.486063 + 0.873924i $$0.661567\pi$$
$$692$$ 5.50115 0.209122
$$693$$ 0 0
$$694$$ 3.05531 0.115978
$$695$$ −69.6265 −2.64108
$$696$$ −7.24943 −0.274789
$$697$$ 8.24943 0.312469
$$698$$ 4.00736 0.151681
$$699$$ −11.3070 −0.427671
$$700$$ 0 0
$$701$$ −14.1512 −0.534484 −0.267242 0.963629i $$-0.586112\pi$$
−0.267242 + 0.963629i $$0.586112\pi$$
$$702$$ −6.40294 −0.241664
$$703$$ −5.94469 −0.224208
$$704$$ 1.00000 0.0376889
$$705$$ −19.3624 −0.729228
$$706$$ 1.09821 0.0413315
$$707$$ 0 0
$$708$$ 11.5565 0.434319
$$709$$ −10.7506 −0.403746 −0.201873 0.979412i $$-0.564703\pi$$
−0.201873 + 0.979412i $$0.564703\pi$$
$$710$$ 16.8059 0.630714
$$711$$ −5.60442 −0.210182
$$712$$ 18.4029 0.689680
$$713$$ −20.4943 −0.767516
$$714$$ 0 0
$$715$$ 20.4989 0.766614
$$716$$ 1.65237 0.0617520
$$717$$ −6.59476 −0.246286
$$718$$ 15.6118 0.582626
$$719$$ −2.75794 −0.102854 −0.0514268 0.998677i $$-0.516377\pi$$
−0.0514268 + 0.998677i $$0.516377\pi$$
$$720$$ 3.20147 0.119312
$$721$$ 0 0
$$722$$ −17.6694 −0.657587
$$723$$ 17.1129 0.636437
$$724$$ −24.9018 −0.925468
$$725$$ 38.0553 1.41334
$$726$$ −1.00000 −0.0371135
$$727$$ −14.6141 −0.542006 −0.271003 0.962578i $$-0.587355\pi$$
−0.271003 + 0.962578i $$0.587355\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 6.70998 0.248348
$$731$$ −5.15352 −0.190610
$$732$$ 9.70032 0.358534
$$733$$ −7.10557 −0.262450 −0.131225 0.991353i $$-0.541891\pi$$
−0.131225 + 0.991353i $$0.541891\pi$$
$$734$$ −0.791166 −0.0292025
$$735$$ 0 0
$$736$$ 1.95205 0.0719534
$$737$$ 6.24943 0.230201
$$738$$ −8.24943 −0.303666
$$739$$ −41.4966 −1.52648 −0.763238 0.646118i $$-0.776391\pi$$
−0.763238 + 0.646118i $$0.776391\pi$$
$$740$$ 16.4989 0.606510
$$741$$ 7.38592 0.271329
$$742$$ 0 0
$$743$$ −31.9041 −1.17045 −0.585224 0.810872i $$-0.698993\pi$$
−0.585224 + 0.810872i $$0.698993\pi$$
$$744$$ 10.4989 0.384907
$$745$$ 21.9188 0.803043
$$746$$ 15.9115 0.582560
$$747$$ −6.55646 −0.239888
$$748$$ −1.00000 −0.0365636
$$749$$ 0 0
$$750$$ −0.798528 −0.0291581
$$751$$ 13.5159 0.493201 0.246601 0.969117i $$-0.420686\pi$$
0.246601 + 0.969117i $$0.420686\pi$$
$$752$$ 6.04795 0.220546
$$753$$ 7.95941 0.290057
$$754$$ 46.4177 1.69043
$$755$$ 47.5542 1.73067
$$756$$ 0 0
$$757$$ −18.5542 −0.674363 −0.337181 0.941440i $$-0.609474\pi$$
−0.337181 + 0.941440i $$0.609474\pi$$
$$758$$ −17.0553 −0.619477
$$759$$ −1.95205 −0.0708548
$$760$$ −3.69296 −0.133958
$$761$$ −15.7506 −0.570958 −0.285479 0.958385i $$-0.592153\pi$$
−0.285479 + 0.958385i $$0.592153\pi$$
$$762$$ −8.35499 −0.302669
$$763$$ 0 0
$$764$$ −4.80589 −0.173871
$$765$$ −3.20147 −0.115749
$$766$$ −13.4412 −0.485652
$$767$$ −73.9954 −2.67182
$$768$$ −1.00000 −0.0360844
$$769$$ −41.2854 −1.48879 −0.744395 0.667739i $$-0.767262\pi$$
−0.744395 + 0.667739i $$0.767262\pi$$
$$770$$ 0 0
$$771$$ −22.4029 −0.806822
$$772$$ 8.70998 0.313479
$$773$$ −4.90916 −0.176570 −0.0882850 0.996095i $$-0.528139\pi$$
−0.0882850 + 0.996095i $$0.528139\pi$$
$$774$$ 5.15352 0.185239
$$775$$ −55.1129 −1.97971
$$776$$ 5.49885 0.197397
$$777$$ 0 0
$$778$$ −7.20147 −0.258185
$$779$$ 9.51587 0.340942
$$780$$ −20.4989 −0.733977
$$781$$ 5.24943 0.187839
$$782$$ −1.95205 −0.0698051
$$783$$ −7.24943 −0.259073
$$784$$ 0 0
$$785$$ 30.2877 1.08101
$$786$$ 15.3047 0.545902
$$787$$ −0.769897 −0.0274439 −0.0137219 0.999906i $$-0.504368\pi$$
−0.0137219 + 0.999906i $$0.504368\pi$$
$$788$$ 0.654669 0.0233216
$$789$$ 25.0170 0.890630
$$790$$ −17.9424 −0.638361
$$791$$ 0 0
$$792$$ 1.00000 0.0355335
$$793$$ −62.1106 −2.20561
$$794$$ −12.9424 −0.459308
$$795$$ 20.4989 0.727019
$$796$$ 7.69296 0.272670
$$797$$ −28.9092 −1.02401 −0.512007 0.858981i $$-0.671098\pi$$
−0.512007 + 0.858981i $$0.671098\pi$$
$$798$$ 0 0
$$799$$ −6.04795 −0.213961
$$800$$ 5.24943 0.185595
$$801$$ 18.4029 0.650236
$$802$$ −16.9018 −0.596823
$$803$$ 2.09591 0.0739629
$$804$$ −6.24943 −0.220400
$$805$$ 0 0
$$806$$ −67.2236 −2.36785
$$807$$ 8.10327 0.285249
$$808$$ 19.9594 0.702170
$$809$$ −20.2494 −0.711932 −0.355966 0.934499i $$-0.615848\pi$$
−0.355966 + 0.934499i $$0.615848\pi$$
$$810$$ 3.20147 0.112488
$$811$$ 48.4989 1.70302 0.851512 0.524334i $$-0.175686\pi$$
0.851512 + 0.524334i $$0.175686\pi$$
$$812$$ 0 0
$$813$$ −4.80589 −0.168550
$$814$$ 5.15352 0.180631
$$815$$ 41.7962 1.46406
$$816$$ 1.00000 0.0350070
$$817$$ −5.94469 −0.207978
$$818$$ 37.0936 1.29695
$$819$$ 0 0
$$820$$ −26.4103 −0.922288
$$821$$ 31.3047 1.09254 0.546271 0.837608i $$-0.316047\pi$$
0.546271 + 0.837608i $$0.316047\pi$$
$$822$$ 6.40294 0.223328
$$823$$ 0.287717 0.0100292 0.00501459 0.999987i $$-0.498404\pi$$
0.00501459 + 0.999987i $$0.498404\pi$$
$$824$$ −0.307039 −0.0106962
$$825$$ −5.24943 −0.182762
$$826$$ 0 0
$$827$$ 37.2471 1.29521 0.647605 0.761976i $$-0.275771\pi$$
0.647605 + 0.761976i $$0.275771\pi$$
$$828$$ 1.95205 0.0678383
$$829$$ −49.1535 −1.70717 −0.853586 0.520952i $$-0.825577\pi$$
−0.853586 + 0.520952i $$0.825577\pi$$
$$830$$ −20.9903 −0.728585
$$831$$ 11.2088 0.388830
$$832$$ 6.40294 0.221982
$$833$$ 0 0
$$834$$ 21.7483 0.753081
$$835$$ 21.4818 0.743409
$$836$$ −1.15352 −0.0398953
$$837$$ 10.4989 0.362893
$$838$$ −22.4583 −0.775808
$$839$$ 2.81785 0.0972830 0.0486415 0.998816i $$-0.484511\pi$$
0.0486415 + 0.998816i $$0.484511\pi$$
$$840$$ 0 0
$$841$$ 23.5542 0.812213
$$842$$ −23.3453 −0.804533
$$843$$ 17.4989 0.602692
$$844$$ 14.5948 0.502372
$$845$$ 89.6339 3.08350
$$846$$ 6.04795 0.207933
$$847$$ 0 0
$$848$$ −6.40294 −0.219878
$$849$$ 27.4006 0.940388
$$850$$ −5.24943 −0.180054
$$851$$ 10.0599 0.344849
$$852$$ −5.24943 −0.179842
$$853$$ −2.39558 −0.0820232 −0.0410116 0.999159i $$-0.513058\pi$$
−0.0410116 + 0.999159i $$0.513058\pi$$
$$854$$ 0 0
$$855$$ −3.69296 −0.126297
$$856$$ −9.05531 −0.309504
$$857$$ −0.808189 −0.0276072 −0.0138036 0.999905i $$-0.504394\pi$$
−0.0138036 + 0.999905i $$0.504394\pi$$
$$858$$ −6.40294 −0.218593
$$859$$ −6.55646 −0.223704 −0.111852 0.993725i $$-0.535678\pi$$
−0.111852 + 0.993725i $$0.535678\pi$$
$$860$$ 16.4989 0.562606
$$861$$ 0 0
$$862$$ 31.8229 1.08389
$$863$$ −20.6067 −0.701461 −0.350730 0.936476i $$-0.614067\pi$$
−0.350730 + 0.936476i $$0.614067\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 17.6118 0.598818
$$866$$ −30.3047 −1.02980
$$867$$ 16.0000 0.543388
$$868$$ 0 0
$$869$$ −5.60442 −0.190117
$$870$$ −23.2088 −0.786853
$$871$$ 40.0147 1.35585
$$872$$ −9.70032 −0.328494
$$873$$ 5.49885 0.186108
$$874$$ −2.25172 −0.0761657
$$875$$ 0 0
$$876$$ −2.09591 −0.0708141
$$877$$ 26.6021 0.898290 0.449145 0.893459i $$-0.351729\pi$$
0.449145 + 0.893459i $$0.351729\pi$$
$$878$$ 10.6620 0.359826
$$879$$ −19.7483 −0.666093
$$880$$ 3.20147 0.107922
$$881$$ 8.49885 0.286334 0.143167 0.989699i $$-0.454271\pi$$
0.143167 + 0.989699i $$0.454271\pi$$
$$882$$ 0 0
$$883$$ 46.2448 1.55626 0.778131 0.628102i $$-0.216168\pi$$
0.778131 + 0.628102i $$0.216168\pi$$
$$884$$ −6.40294 −0.215354
$$885$$ 36.9977 1.24366
$$886$$ −36.0553 −1.21130
$$887$$ −18.2877 −0.614041 −0.307021 0.951703i $$-0.599332\pi$$
−0.307021 + 0.951703i $$0.599332\pi$$
$$888$$ −5.15352 −0.172941
$$889$$ 0 0
$$890$$ 58.9165 1.97489
$$891$$ 1.00000 0.0335013
$$892$$ 14.9018 0.498949
$$893$$ −6.97643 −0.233457
$$894$$ −6.84648 −0.228981
$$895$$ 5.29002 0.176826
$$896$$ 0 0
$$897$$ −12.4989 −0.417324
$$898$$ 23.4006 0.780890
$$899$$ −76.1106 −2.53843
$$900$$ 5.24943 0.174981
$$901$$ 6.40294 0.213313
$$902$$ −8.24943 −0.274676
$$903$$ 0 0
$$904$$ −12.4989 −0.415706
$$905$$ −79.7224 −2.65006
$$906$$ −14.8538 −0.493486
$$907$$ −31.6694 −1.05156 −0.525782 0.850619i $$-0.676227\pi$$
−0.525782 + 0.850619i $$0.676227\pi$$
$$908$$ 0.249425 0.00827746
$$909$$ 19.9594 0.662012
$$910$$ 0 0
$$911$$ 58.7727 1.94723 0.973613 0.228207i $$-0.0732864\pi$$
0.973613 + 0.228207i $$0.0732864\pi$$
$$912$$ 1.15352 0.0381968
$$913$$ −6.55646 −0.216987
$$914$$ −18.6141 −0.615699
$$915$$ 31.0553 1.02666
$$916$$ 4.09591 0.135333
$$917$$ 0 0
$$918$$ 1.00000 0.0330049
$$919$$ −32.9691 −1.08755 −0.543775 0.839231i $$-0.683005\pi$$
−0.543775 + 0.839231i $$0.683005\pi$$
$$920$$ 6.24943 0.206037
$$921$$ 4.30704 0.141922
$$922$$ 31.4412 1.03546
$$923$$ 33.6118 1.10635
$$924$$ 0 0
$$925$$ 27.0530 0.889498
$$926$$ −22.0959 −0.726117
$$927$$ −0.307039 −0.0100845
$$928$$ 7.24943 0.237974
$$929$$ −41.8995 −1.37468 −0.687339 0.726337i $$-0.741221\pi$$
−0.687339 + 0.726337i $$0.741221\pi$$
$$930$$ 33.6118 1.10217
$$931$$ 0 0
$$932$$ 11.3070 0.370374
$$933$$ −14.7579 −0.483153
$$934$$ −21.0576 −0.689026
$$935$$ −3.20147 −0.104699
$$936$$ 6.40294 0.209287
$$937$$ 33.4966 1.09428 0.547142 0.837040i $$-0.315716\pi$$
0.547142 + 0.837040i $$0.315716\pi$$
$$938$$ 0 0
$$939$$ 20.7506 0.677169
$$940$$ 19.3624 0.631530
$$941$$ 45.7483 1.49135 0.745676 0.666309i $$-0.232127\pi$$
0.745676 + 0.666309i $$0.232127\pi$$
$$942$$ −9.46056 −0.308242
$$943$$ −16.1033 −0.524395
$$944$$ −11.5565 −0.376131
$$945$$ 0 0
$$946$$ 5.15352 0.167555
$$947$$ −45.0530 −1.46403 −0.732013 0.681291i $$-0.761419\pi$$
−0.732013 + 0.681291i $$0.761419\pi$$
$$948$$ 5.60442 0.182023
$$949$$ 13.4200 0.435631
$$950$$ −6.05531 −0.196460
$$951$$ −16.1992 −0.525294
$$952$$ 0 0
$$953$$ 37.1705 1.20407 0.602036 0.798469i $$-0.294356\pi$$
0.602036 + 0.798469i $$0.294356\pi$$
$$954$$ −6.40294 −0.207303
$$955$$ −15.3859 −0.497877
$$956$$ 6.59476 0.213290
$$957$$ −7.24943 −0.234341
$$958$$ 21.2088 0.685226
$$959$$ 0 0
$$960$$ −3.20147 −0.103327
$$961$$ 79.2259 2.55567
$$962$$ 32.9977 1.06389
$$963$$ −9.05531 −0.291803
$$964$$ −17.1129 −0.551170
$$965$$ 27.8848 0.897643
$$966$$ 0 0
$$967$$ 42.7579 1.37500 0.687501 0.726183i $$-0.258708\pi$$
0.687501 + 0.726183i $$0.258708\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ −1.15352 −0.0370564
$$970$$ 17.6044 0.565244
$$971$$ −0.690661 −0.0221644 −0.0110822 0.999939i $$-0.503528\pi$$
−0.0110822 + 0.999939i $$0.503528\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ −38.7100 −1.24035
$$975$$ −33.6118 −1.07644
$$976$$ −9.70032 −0.310500
$$977$$ −33.1895 −1.06183 −0.530913 0.847426i $$-0.678151\pi$$
−0.530913 + 0.847426i $$0.678151\pi$$
$$978$$ −13.0553 −0.417463
$$979$$ 18.4029 0.588161
$$980$$ 0 0
$$981$$ −9.70032 −0.309707
$$982$$ 19.1705 0.611757
$$983$$ 42.6620 1.36071 0.680354 0.732884i $$-0.261826\pi$$
0.680354 + 0.732884i $$0.261826\pi$$
$$984$$ 8.24943 0.262982
$$985$$ 2.09591 0.0667811
$$986$$ −7.24943 −0.230869
$$987$$ 0 0
$$988$$ −7.38592 −0.234977
$$989$$ 10.0599 0.319887
$$990$$ 3.20147 0.101749
$$991$$ −24.0147 −0.762853 −0.381426 0.924399i $$-0.624567\pi$$
−0.381426 + 0.924399i $$0.624567\pi$$
$$992$$ −10.4989 −0.333339
$$993$$ −21.2471 −0.674257
$$994$$ 0 0
$$995$$ 24.6288 0.780785
$$996$$ 6.55646 0.207750
$$997$$ −33.5971 −1.06403 −0.532015 0.846735i $$-0.678565\pi$$
−0.532015 + 0.846735i $$0.678565\pi$$
$$998$$ −27.9188 −0.883755
$$999$$ −5.15352 −0.163050
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 3234.2.a.bg.1.3 3
3.2 odd 2 9702.2.a.du.1.1 3
7.3 odd 6 462.2.i.f.331.3 yes 6
7.5 odd 6 462.2.i.f.67.3 6
7.6 odd 2 3234.2.a.bi.1.1 3
21.5 even 6 1386.2.k.w.991.1 6
21.17 even 6 1386.2.k.w.793.1 6
21.20 even 2 9702.2.a.dt.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.f.67.3 6 7.5 odd 6
462.2.i.f.331.3 yes 6 7.3 odd 6
1386.2.k.w.793.1 6 21.17 even 6
1386.2.k.w.991.1 6 21.5 even 6
3234.2.a.bg.1.3 3 1.1 even 1 trivial
3234.2.a.bi.1.1 3 7.6 odd 2
9702.2.a.dt.1.3 3 21.20 even 2
9702.2.a.du.1.1 3 3.2 odd 2