# Properties

 Label 4650.2.d.y.3349.2 Level $4650$ Weight $2$ Character 4650.3349 Analytic conductor $37.130$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$37.1304369399$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3349.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4650.3349 Dual form 4650.2.d.y.3349.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} -1.00000i q^{18} +3.00000 q^{19} +3.00000 q^{21} +3.00000i q^{22} +5.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -3.00000i q^{28} -4.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} -3.00000i q^{33} -4.00000 q^{34} +1.00000 q^{36} +3.00000i q^{38} -2.00000 q^{39} +4.00000 q^{41} +3.00000i q^{42} +1.00000i q^{43} -3.00000 q^{44} -5.00000 q^{46} -10.0000i q^{47} -1.00000i q^{48} -2.00000 q^{49} +4.00000 q^{51} +2.00000i q^{52} +3.00000i q^{53} -1.00000 q^{54} +3.00000 q^{56} -3.00000i q^{57} -4.00000i q^{58} -6.00000 q^{59} -2.00000 q^{61} +1.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -2.00000i q^{67} -4.00000i q^{68} +5.00000 q^{69} +7.00000 q^{71} +1.00000i q^{72} +5.00000i q^{73} -3.00000 q^{76} +9.00000i q^{77} -2.00000i q^{78} +1.00000 q^{79} +1.00000 q^{81} +4.00000i q^{82} +12.0000i q^{83} -3.00000 q^{84} -1.00000 q^{86} +4.00000i q^{87} -3.00000i q^{88} -1.00000 q^{89} +6.00000 q^{91} -5.00000i q^{92} -1.00000i q^{93} +10.0000 q^{94} +1.00000 q^{96} +10.0000i q^{97} -2.00000i q^{98} -3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + 6q^{11} - 6q^{14} + 2q^{16} + 6q^{19} + 6q^{21} - 2q^{24} + 4q^{26} - 8q^{29} + 2q^{31} - 8q^{34} + 2q^{36} - 4q^{39} + 8q^{41} - 6q^{44} - 10q^{46} - 4q^{49} + 8q^{51} - 2q^{54} + 6q^{56} - 12q^{59} - 4q^{61} - 2q^{64} + 6q^{66} + 10q^{69} + 14q^{71} - 6q^{76} + 2q^{79} + 2q^{81} - 6q^{84} - 2q^{86} - 2q^{89} + 12q^{91} + 20q^{94} + 2q^{96} - 6q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$$.

 $$n$$ $$1801$$ $$2977$$ $$3101$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 3.00000i 1.13389i 0.823754 + 0.566947i $$0.191875\pi$$
−0.823754 + 0.566947i $$0.808125\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ −3.00000 −0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000i 0.970143i 0.874475 + 0.485071i $$0.161206\pi$$
−0.874475 + 0.485071i $$0.838794\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 3.00000 0.688247 0.344124 0.938924i $$-0.388176\pi$$
0.344124 + 0.938924i $$0.388176\pi$$
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ 3.00000i 0.639602i
$$23$$ 5.00000i 1.04257i 0.853382 + 0.521286i $$0.174548\pi$$
−0.853382 + 0.521286i $$0.825452\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ − 3.00000i − 0.566947i
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605
$$32$$ 1.00000i 0.176777i
$$33$$ − 3.00000i − 0.522233i
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 3.00000i 0.486664i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ 3.00000i 0.462910i
$$43$$ 1.00000i 0.152499i 0.997089 + 0.0762493i $$0.0242945\pi$$
−0.997089 + 0.0762493i $$0.975706\pi$$
$$44$$ −3.00000 −0.452267
$$45$$ 0 0
$$46$$ −5.00000 −0.737210
$$47$$ − 10.0000i − 1.45865i −0.684167 0.729325i $$-0.739834\pi$$
0.684167 0.729325i $$-0.260166\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ 2.00000i 0.277350i
$$53$$ 3.00000i 0.412082i 0.978543 + 0.206041i $$0.0660580\pi$$
−0.978543 + 0.206041i $$0.933942\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ − 3.00000i − 0.397360i
$$58$$ − 4.00000i − 0.525226i
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 1.00000i 0.127000i
$$63$$ − 3.00000i − 0.377964i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 3.00000 0.369274
$$67$$ − 2.00000i − 0.244339i −0.992509 0.122169i $$-0.961015\pi$$
0.992509 0.122169i $$-0.0389851\pi$$
$$68$$ − 4.00000i − 0.485071i
$$69$$ 5.00000 0.601929
$$70$$ 0 0
$$71$$ 7.00000 0.830747 0.415374 0.909651i $$-0.363651\pi$$
0.415374 + 0.909651i $$0.363651\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 5.00000i 0.585206i 0.956234 + 0.292603i $$0.0945214\pi$$
−0.956234 + 0.292603i $$0.905479\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ −3.00000 −0.344124
$$77$$ 9.00000i 1.02565i
$$78$$ − 2.00000i − 0.226455i
$$79$$ 1.00000 0.112509 0.0562544 0.998416i $$-0.482084\pi$$
0.0562544 + 0.998416i $$0.482084\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 4.00000i 0.441726i
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ −3.00000 −0.327327
$$85$$ 0 0
$$86$$ −1.00000 −0.107833
$$87$$ 4.00000i 0.428845i
$$88$$ − 3.00000i − 0.319801i
$$89$$ −1.00000 −0.106000 −0.0529999 0.998595i $$-0.516878\pi$$
−0.0529999 + 0.998595i $$0.516878\pi$$
$$90$$ 0 0
$$91$$ 6.00000 0.628971
$$92$$ − 5.00000i − 0.521286i
$$93$$ − 1.00000i − 0.103695i
$$94$$ 10.0000 1.03142
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ − 2.00000i − 0.202031i
$$99$$ −3.00000 −0.301511
$$100$$ 0 0
$$101$$ −1.00000 −0.0995037 −0.0497519 0.998762i $$-0.515843\pi$$
−0.0497519 + 0.998762i $$0.515843\pi$$
$$102$$ 4.00000i 0.396059i
$$103$$ 16.0000i 1.57653i 0.615338 + 0.788263i $$0.289020\pi$$
−0.615338 + 0.788263i $$0.710980\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −3.00000 −0.291386
$$107$$ − 9.00000i − 0.870063i −0.900415 0.435031i $$-0.856737\pi$$
0.900415 0.435031i $$-0.143263\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −20.0000 −1.91565 −0.957826 0.287348i $$-0.907226\pi$$
−0.957826 + 0.287348i $$0.907226\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 3.00000i 0.283473i
$$113$$ 9.00000i 0.846649i 0.905978 + 0.423324i $$0.139137\pi$$
−0.905978 + 0.423324i $$0.860863\pi$$
$$114$$ 3.00000 0.280976
$$115$$ 0 0
$$116$$ 4.00000 0.371391
$$117$$ 2.00000i 0.184900i
$$118$$ − 6.00000i − 0.552345i
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ − 2.00000i − 0.181071i
$$123$$ − 4.00000i − 0.360668i
$$124$$ −1.00000 −0.0898027
$$125$$ 0 0
$$126$$ 3.00000 0.267261
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 3.00000i 0.261116i
$$133$$ 9.00000i 0.780399i
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ 4.00000 0.342997
$$137$$ 10.0000i 0.854358i 0.904167 + 0.427179i $$0.140493\pi$$
−0.904167 + 0.427179i $$0.859507\pi$$
$$138$$ 5.00000i 0.425628i
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ 0 0
$$141$$ −10.0000 −0.842152
$$142$$ 7.00000i 0.587427i
$$143$$ − 6.00000i − 0.501745i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −5.00000 −0.413803
$$147$$ 2.00000i 0.164957i
$$148$$ 0 0
$$149$$ 11.0000 0.901155 0.450578 0.892737i $$-0.351218\pi$$
0.450578 + 0.892737i $$0.351218\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ − 3.00000i − 0.243332i
$$153$$ − 4.00000i − 0.323381i
$$154$$ −9.00000 −0.725241
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 5.00000i 0.399043i 0.979893 + 0.199522i $$0.0639388\pi$$
−0.979893 + 0.199522i $$0.936061\pi$$
$$158$$ 1.00000i 0.0795557i
$$159$$ 3.00000 0.237915
$$160$$ 0 0
$$161$$ −15.0000 −1.18217
$$162$$ 1.00000i 0.0785674i
$$163$$ 14.0000i 1.09656i 0.836293 + 0.548282i $$0.184718\pi$$
−0.836293 + 0.548282i $$0.815282\pi$$
$$164$$ −4.00000 −0.312348
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 19.0000i 1.47026i 0.677924 + 0.735132i $$0.262880\pi$$
−0.677924 + 0.735132i $$0.737120\pi$$
$$168$$ − 3.00000i − 0.231455i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −3.00000 −0.229416
$$172$$ − 1.00000i − 0.0762493i
$$173$$ − 22.0000i − 1.67263i −0.548250 0.836315i $$-0.684706\pi$$
0.548250 0.836315i $$-0.315294\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ 6.00000i 0.450988i
$$178$$ − 1.00000i − 0.0749532i
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ 5.00000 0.371647 0.185824 0.982583i $$-0.440505\pi$$
0.185824 + 0.982583i $$0.440505\pi$$
$$182$$ 6.00000i 0.444750i
$$183$$ 2.00000i 0.147844i
$$184$$ 5.00000 0.368605
$$185$$ 0 0
$$186$$ 1.00000 0.0733236
$$187$$ 12.0000i 0.877527i
$$188$$ 10.0000i 0.729325i
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 6.00000i − 0.431889i −0.976406 0.215945i $$-0.930717\pi$$
0.976406 0.215945i $$-0.0692831\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ − 3.00000i − 0.213201i
$$199$$ −7.00000 −0.496217 −0.248108 0.968732i $$-0.579809\pi$$
−0.248108 + 0.968732i $$0.579809\pi$$
$$200$$ 0 0
$$201$$ −2.00000 −0.141069
$$202$$ − 1.00000i − 0.0703598i
$$203$$ − 12.0000i − 0.842235i
$$204$$ −4.00000 −0.280056
$$205$$ 0 0
$$206$$ −16.0000 −1.11477
$$207$$ − 5.00000i − 0.347524i
$$208$$ − 2.00000i − 0.138675i
$$209$$ 9.00000 0.622543
$$210$$ 0 0
$$211$$ −19.0000 −1.30801 −0.654007 0.756489i $$-0.726913\pi$$
−0.654007 + 0.756489i $$0.726913\pi$$
$$212$$ − 3.00000i − 0.206041i
$$213$$ − 7.00000i − 0.479632i
$$214$$ 9.00000 0.615227
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 3.00000i 0.203653i
$$218$$ − 20.0000i − 1.35457i
$$219$$ 5.00000 0.337869
$$220$$ 0 0
$$221$$ 8.00000 0.538138
$$222$$ 0 0
$$223$$ − 6.00000i − 0.401790i −0.979613 0.200895i $$-0.935615\pi$$
0.979613 0.200895i $$-0.0643850\pi$$
$$224$$ −3.00000 −0.200446
$$225$$ 0 0
$$226$$ −9.00000 −0.598671
$$227$$ 27.0000i 1.79205i 0.444001 + 0.896026i $$0.353559\pi$$
−0.444001 + 0.896026i $$0.646441\pi$$
$$228$$ 3.00000i 0.198680i
$$229$$ −13.0000 −0.859064 −0.429532 0.903052i $$-0.641321\pi$$
−0.429532 + 0.903052i $$0.641321\pi$$
$$230$$ 0 0
$$231$$ 9.00000 0.592157
$$232$$ 4.00000i 0.262613i
$$233$$ 15.0000i 0.982683i 0.870967 + 0.491341i $$0.163493\pi$$
−0.870967 + 0.491341i $$0.836507\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 6.00000 0.390567
$$237$$ − 1.00000i − 0.0649570i
$$238$$ − 12.0000i − 0.777844i
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ − 2.00000i − 0.128565i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 4.00000 0.255031
$$247$$ − 6.00000i − 0.381771i
$$248$$ − 1.00000i − 0.0635001i
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 3.00000i 0.188982i
$$253$$ 15.0000i 0.943042i
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 23.0000i 1.43470i 0.696713 + 0.717350i $$0.254645\pi$$
−0.696713 + 0.717350i $$0.745355\pi$$
$$258$$ 1.00000i 0.0622573i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 4.00000 0.247594
$$262$$ 6.00000i 0.370681i
$$263$$ 24.0000i 1.47990i 0.672660 + 0.739952i $$0.265152\pi$$
−0.672660 + 0.739952i $$0.734848\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ 0 0
$$266$$ −9.00000 −0.551825
$$267$$ 1.00000i 0.0611990i
$$268$$ 2.00000i 0.122169i
$$269$$ −2.00000 −0.121942 −0.0609711 0.998140i $$-0.519420\pi$$
−0.0609711 + 0.998140i $$0.519420\pi$$
$$270$$ 0 0
$$271$$ 13.0000 0.789694 0.394847 0.918747i $$-0.370798\pi$$
0.394847 + 0.918747i $$0.370798\pi$$
$$272$$ 4.00000i 0.242536i
$$273$$ − 6.00000i − 0.363137i
$$274$$ −10.0000 −0.604122
$$275$$ 0 0
$$276$$ −5.00000 −0.300965
$$277$$ − 12.0000i − 0.721010i −0.932757 0.360505i $$-0.882604\pi$$
0.932757 0.360505i $$-0.117396\pi$$
$$278$$ 14.0000i 0.839664i
$$279$$ −1.00000 −0.0598684
$$280$$ 0 0
$$281$$ 16.0000 0.954480 0.477240 0.878773i $$-0.341637\pi$$
0.477240 + 0.878773i $$0.341637\pi$$
$$282$$ − 10.0000i − 0.595491i
$$283$$ − 24.0000i − 1.42665i −0.700832 0.713326i $$-0.747188\pi$$
0.700832 0.713326i $$-0.252812\pi$$
$$284$$ −7.00000 −0.415374
$$285$$ 0 0
$$286$$ 6.00000 0.354787
$$287$$ 12.0000i 0.708338i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 10.0000 0.586210
$$292$$ − 5.00000i − 0.292603i
$$293$$ 30.0000i 1.75262i 0.481749 + 0.876309i $$0.340002\pi$$
−0.481749 + 0.876309i $$0.659998\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 3.00000i 0.174078i
$$298$$ 11.0000i 0.637213i
$$299$$ 10.0000 0.578315
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ 0 0
$$303$$ 1.00000i 0.0574485i
$$304$$ 3.00000 0.172062
$$305$$ 0 0
$$306$$ 4.00000 0.228665
$$307$$ − 16.0000i − 0.913168i −0.889680 0.456584i $$-0.849073\pi$$
0.889680 0.456584i $$-0.150927\pi$$
$$308$$ − 9.00000i − 0.512823i
$$309$$ 16.0000 0.910208
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 2.00000i 0.113228i
$$313$$ 30.0000i 1.69570i 0.530236 + 0.847850i $$0.322103\pi$$
−0.530236 + 0.847850i $$0.677897\pi$$
$$314$$ −5.00000 −0.282166
$$315$$ 0 0
$$316$$ −1.00000 −0.0562544
$$317$$ 8.00000i 0.449325i 0.974437 + 0.224662i $$0.0721279\pi$$
−0.974437 + 0.224662i $$0.927872\pi$$
$$318$$ 3.00000i 0.168232i
$$319$$ −12.0000 −0.671871
$$320$$ 0 0
$$321$$ −9.00000 −0.502331
$$322$$ − 15.0000i − 0.835917i
$$323$$ 12.0000i 0.667698i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −14.0000 −0.775388
$$327$$ 20.0000i 1.10600i
$$328$$ − 4.00000i − 0.220863i
$$329$$ 30.0000 1.65395
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 0 0
$$334$$ −19.0000 −1.03963
$$335$$ 0 0
$$336$$ 3.00000 0.163663
$$337$$ 2.00000i 0.108947i 0.998515 + 0.0544735i $$0.0173480\pi$$
−0.998515 + 0.0544735i $$0.982652\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 9.00000 0.488813
$$340$$ 0 0
$$341$$ 3.00000 0.162459
$$342$$ − 3.00000i − 0.162221i
$$343$$ 15.0000i 0.809924i
$$344$$ 1.00000 0.0539164
$$345$$ 0 0
$$346$$ 22.0000 1.18273
$$347$$ 32.0000i 1.71785i 0.512101 + 0.858925i $$0.328867\pi$$
−0.512101 + 0.858925i $$0.671133\pi$$
$$348$$ − 4.00000i − 0.214423i
$$349$$ 24.0000 1.28469 0.642345 0.766415i $$-0.277962\pi$$
0.642345 + 0.766415i $$0.277962\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 3.00000i 0.159901i
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ −6.00000 −0.318896
$$355$$ 0 0
$$356$$ 1.00000 0.0529999
$$357$$ 12.0000i 0.635107i
$$358$$ − 4.00000i − 0.211407i
$$359$$ −15.0000 −0.791670 −0.395835 0.918322i $$-0.629545\pi$$
−0.395835 + 0.918322i $$0.629545\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ 5.00000i 0.262794i
$$363$$ 2.00000i 0.104973i
$$364$$ −6.00000 −0.314485
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ − 2.00000i − 0.104399i −0.998637 0.0521996i $$-0.983377\pi$$
0.998637 0.0521996i $$-0.0166232\pi$$
$$368$$ 5.00000i 0.260643i
$$369$$ −4.00000 −0.208232
$$370$$ 0 0
$$371$$ −9.00000 −0.467257
$$372$$ 1.00000i 0.0518476i
$$373$$ − 9.00000i − 0.466002i −0.972476 0.233001i $$-0.925145\pi$$
0.972476 0.233001i $$-0.0748546\pi$$
$$374$$ −12.0000 −0.620505
$$375$$ 0 0
$$376$$ −10.0000 −0.515711
$$377$$ 8.00000i 0.412021i
$$378$$ − 3.00000i − 0.154303i
$$379$$ −15.0000 −0.770498 −0.385249 0.922813i $$-0.625884\pi$$
−0.385249 + 0.922813i $$0.625884\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 16.0000i 0.818631i
$$383$$ 12.0000i 0.613171i 0.951843 + 0.306586i $$0.0991866\pi$$
−0.951843 + 0.306586i $$0.900813\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 6.00000 0.305392
$$387$$ − 1.00000i − 0.0508329i
$$388$$ − 10.0000i − 0.507673i
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ −20.0000 −1.01144
$$392$$ 2.00000i 0.101015i
$$393$$ − 6.00000i − 0.302660i
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 3.00000 0.150756
$$397$$ − 35.0000i − 1.75660i −0.478110 0.878300i $$-0.658678\pi$$
0.478110 0.878300i $$-0.341322\pi$$
$$398$$ − 7.00000i − 0.350878i
$$399$$ 9.00000 0.450564
$$400$$ 0 0
$$401$$ 25.0000 1.24844 0.624220 0.781248i $$-0.285417\pi$$
0.624220 + 0.781248i $$0.285417\pi$$
$$402$$ − 2.00000i − 0.0997509i
$$403$$ − 2.00000i − 0.0996271i
$$404$$ 1.00000 0.0497519
$$405$$ 0 0
$$406$$ 12.0000 0.595550
$$407$$ 0 0
$$408$$ − 4.00000i − 0.198030i
$$409$$ −8.00000 −0.395575 −0.197787 0.980245i $$-0.563376\pi$$
−0.197787 + 0.980245i $$0.563376\pi$$
$$410$$ 0 0
$$411$$ 10.0000 0.493264
$$412$$ − 16.0000i − 0.788263i
$$413$$ − 18.0000i − 0.885722i
$$414$$ 5.00000 0.245737
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ − 14.0000i − 0.685583i
$$418$$ 9.00000i 0.440204i
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ − 19.0000i − 0.924906i
$$423$$ 10.0000i 0.486217i
$$424$$ 3.00000 0.145693
$$425$$ 0 0
$$426$$ 7.00000 0.339151
$$427$$ − 6.00000i − 0.290360i
$$428$$ 9.00000i 0.435031i
$$429$$ −6.00000 −0.289683
$$430$$ 0 0
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 1.00000i − 0.0480569i −0.999711 0.0240285i $$-0.992351\pi$$
0.999711 0.0240285i $$-0.00764923\pi$$
$$434$$ −3.00000 −0.144005
$$435$$ 0 0
$$436$$ 20.0000 0.957826
$$437$$ 15.0000i 0.717547i
$$438$$ 5.00000i 0.238909i
$$439$$ −18.0000 −0.859093 −0.429547 0.903045i $$-0.641327\pi$$
−0.429547 + 0.903045i $$0.641327\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 8.00000i 0.380521i
$$443$$ 15.0000i 0.712672i 0.934358 + 0.356336i $$0.115974\pi$$
−0.934358 + 0.356336i $$0.884026\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 6.00000 0.284108
$$447$$ − 11.0000i − 0.520282i
$$448$$ − 3.00000i − 0.141737i
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 0 0
$$451$$ 12.0000 0.565058
$$452$$ − 9.00000i − 0.423324i
$$453$$ 0 0
$$454$$ −27.0000 −1.26717
$$455$$ 0 0
$$456$$ −3.00000 −0.140488
$$457$$ − 10.0000i − 0.467780i −0.972263 0.233890i $$-0.924854\pi$$
0.972263 0.233890i $$-0.0751456\pi$$
$$458$$ − 13.0000i − 0.607450i
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 9.00000i 0.418718i
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 0 0
$$466$$ −15.0000 −0.694862
$$467$$ − 24.0000i − 1.11059i −0.831654 0.555294i $$-0.812606\pi$$
0.831654 0.555294i $$-0.187394\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ 6.00000 0.277054
$$470$$ 0 0
$$471$$ 5.00000 0.230388
$$472$$ 6.00000i 0.276172i
$$473$$ 3.00000i 0.137940i
$$474$$ 1.00000 0.0459315
$$475$$ 0 0
$$476$$ 12.0000 0.550019
$$477$$ − 3.00000i − 0.137361i
$$478$$ − 12.0000i − 0.548867i
$$479$$ −3.00000 −0.137073 −0.0685367 0.997649i $$-0.521833\pi$$
−0.0685367 + 0.997649i $$0.521833\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ − 18.0000i − 0.819878i
$$483$$ 15.0000i 0.682524i
$$484$$ 2.00000 0.0909091
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 32.0000i − 1.45006i −0.688718 0.725029i $$-0.741826\pi$$
0.688718 0.725029i $$-0.258174\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ 14.0000 0.633102
$$490$$ 0 0
$$491$$ −37.0000 −1.66979 −0.834893 0.550412i $$-0.814471\pi$$
−0.834893 + 0.550412i $$0.814471\pi$$
$$492$$ 4.00000i 0.180334i
$$493$$ − 16.0000i − 0.720604i
$$494$$ 6.00000 0.269953
$$495$$ 0 0
$$496$$ 1.00000 0.0449013
$$497$$ 21.0000i 0.941979i
$$498$$ 12.0000i 0.537733i
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 19.0000 0.848857
$$502$$ − 12.0000i − 0.535586i
$$503$$ 20.0000i 0.891756i 0.895094 + 0.445878i $$0.147108\pi$$
−0.895094 + 0.445878i $$0.852892\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 0 0
$$506$$ −15.0000 −0.666831
$$507$$ − 9.00000i − 0.399704i
$$508$$ 8.00000i 0.354943i
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ −15.0000 −0.663561
$$512$$ 1.00000i 0.0441942i
$$513$$ 3.00000i 0.132453i
$$514$$ −23.0000 −1.01449
$$515$$ 0 0
$$516$$ −1.00000 −0.0440225
$$517$$ − 30.0000i − 1.31940i
$$518$$ 0 0
$$519$$ −22.0000 −0.965693
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 4.00000i 0.175075i
$$523$$ − 43.0000i − 1.88026i −0.340818 0.940129i $$-0.610704\pi$$
0.340818 0.940129i $$-0.389296\pi$$
$$524$$ −6.00000 −0.262111
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 4.00000i 0.174243i
$$528$$ − 3.00000i − 0.130558i
$$529$$ −2.00000 −0.0869565
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ − 9.00000i − 0.390199i
$$533$$ − 8.00000i − 0.346518i
$$534$$ −1.00000 −0.0432742
$$535$$ 0 0
$$536$$ −2.00000 −0.0863868
$$537$$ 4.00000i 0.172613i
$$538$$ − 2.00000i − 0.0862261i
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ 13.0000i 0.558398i
$$543$$ − 5.00000i − 0.214571i
$$544$$ −4.00000 −0.171499
$$545$$ 0 0
$$546$$ 6.00000 0.256776
$$547$$ − 22.0000i − 0.940652i −0.882493 0.470326i $$-0.844136\pi$$
0.882493 0.470326i $$-0.155864\pi$$
$$548$$ − 10.0000i − 0.427179i
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ −12.0000 −0.511217
$$552$$ − 5.00000i − 0.212814i
$$553$$ 3.00000i 0.127573i
$$554$$ 12.0000 0.509831
$$555$$ 0 0
$$556$$ −14.0000 −0.593732
$$557$$ 1.00000i 0.0423714i 0.999776 + 0.0211857i $$0.00674412\pi$$
−0.999776 + 0.0211857i $$0.993256\pi$$
$$558$$ − 1.00000i − 0.0423334i
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ 12.0000 0.506640
$$562$$ 16.0000i 0.674919i
$$563$$ − 4.00000i − 0.168580i −0.996441 0.0842900i $$-0.973138\pi$$
0.996441 0.0842900i $$-0.0268622\pi$$
$$564$$ 10.0000 0.421076
$$565$$ 0 0
$$566$$ 24.0000 1.00880
$$567$$ 3.00000i 0.125988i
$$568$$ − 7.00000i − 0.293713i
$$569$$ 23.0000 0.964210 0.482105 0.876113i $$-0.339872\pi$$
0.482105 + 0.876113i $$0.339872\pi$$
$$570$$ 0 0
$$571$$ 22.0000 0.920671 0.460336 0.887745i $$-0.347729\pi$$
0.460336 + 0.887745i $$0.347729\pi$$
$$572$$ 6.00000i 0.250873i
$$573$$ − 16.0000i − 0.668410i
$$574$$ −12.0000 −0.500870
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 4.00000i − 0.166522i −0.996528 0.0832611i $$-0.973466\pi$$
0.996528 0.0832611i $$-0.0265335\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ −6.00000 −0.249351
$$580$$ 0 0
$$581$$ −36.0000 −1.49353
$$582$$ 10.0000i 0.414513i
$$583$$ 9.00000i 0.372742i
$$584$$ 5.00000 0.206901
$$585$$ 0 0
$$586$$ −30.0000 −1.23929
$$587$$ − 44.0000i − 1.81607i −0.418890 0.908037i $$-0.637581\pi$$
0.418890 0.908037i $$-0.362419\pi$$
$$588$$ − 2.00000i − 0.0824786i
$$589$$ 3.00000 0.123613
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 0 0
$$593$$ 22.0000i 0.903432i 0.892162 + 0.451716i $$0.149188\pi$$
−0.892162 + 0.451716i $$0.850812\pi$$
$$594$$ −3.00000 −0.123091
$$595$$ 0 0
$$596$$ −11.0000 −0.450578
$$597$$ 7.00000i 0.286491i
$$598$$ 10.0000i 0.408930i
$$599$$ 27.0000 1.10319 0.551595 0.834112i $$-0.314019\pi$$
0.551595 + 0.834112i $$0.314019\pi$$
$$600$$ 0 0
$$601$$ 42.0000 1.71322 0.856608 0.515968i $$-0.172568\pi$$
0.856608 + 0.515968i $$0.172568\pi$$
$$602$$ − 3.00000i − 0.122271i
$$603$$ 2.00000i 0.0814463i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ −1.00000 −0.0406222
$$607$$ − 29.0000i − 1.17707i −0.808470 0.588537i $$-0.799704\pi$$
0.808470 0.588537i $$-0.200296\pi$$
$$608$$ 3.00000i 0.121666i
$$609$$ −12.0000 −0.486265
$$610$$ 0 0
$$611$$ −20.0000 −0.809113
$$612$$ 4.00000i 0.161690i
$$613$$ − 22.0000i − 0.888572i −0.895885 0.444286i $$-0.853457\pi$$
0.895885 0.444286i $$-0.146543\pi$$
$$614$$ 16.0000 0.645707
$$615$$ 0 0
$$616$$ 9.00000 0.362620
$$617$$ 27.0000i 1.08698i 0.839416 + 0.543490i $$0.182897\pi$$
−0.839416 + 0.543490i $$0.817103\pi$$
$$618$$ 16.0000i 0.643614i
$$619$$ 38.0000 1.52735 0.763674 0.645601i $$-0.223393\pi$$
0.763674 + 0.645601i $$0.223393\pi$$
$$620$$ 0 0
$$621$$ −5.00000 −0.200643
$$622$$ 0 0
$$623$$ − 3.00000i − 0.120192i
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ −30.0000 −1.19904
$$627$$ − 9.00000i − 0.359425i
$$628$$ − 5.00000i − 0.199522i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 17.0000 0.676759 0.338380 0.941010i $$-0.390121\pi$$
0.338380 + 0.941010i $$0.390121\pi$$
$$632$$ − 1.00000i − 0.0397779i
$$633$$ 19.0000i 0.755182i
$$634$$ −8.00000 −0.317721
$$635$$ 0 0
$$636$$ −3.00000 −0.118958
$$637$$ 4.00000i 0.158486i
$$638$$ − 12.0000i − 0.475085i
$$639$$ −7.00000 −0.276916
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ − 9.00000i − 0.355202i
$$643$$ 19.0000i 0.749287i 0.927169 + 0.374643i $$0.122235\pi$$
−0.927169 + 0.374643i $$0.877765\pi$$
$$644$$ 15.0000 0.591083
$$645$$ 0 0
$$646$$ −12.0000 −0.472134
$$647$$ − 3.00000i − 0.117942i −0.998260 0.0589711i $$-0.981218\pi$$
0.998260 0.0589711i $$-0.0187820\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −18.0000 −0.706562
$$650$$ 0 0
$$651$$ 3.00000 0.117579
$$652$$ − 14.0000i − 0.548282i
$$653$$ − 26.0000i − 1.01746i −0.860927 0.508729i $$-0.830115\pi$$
0.860927 0.508729i $$-0.169885\pi$$
$$654$$ −20.0000 −0.782062
$$655$$ 0 0
$$656$$ 4.00000 0.156174
$$657$$ − 5.00000i − 0.195069i
$$658$$ 30.0000i 1.16952i
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ 0 0
$$661$$ −14.0000 −0.544537 −0.272268 0.962221i $$-0.587774\pi$$
−0.272268 + 0.962221i $$0.587774\pi$$
$$662$$ 12.0000i 0.466393i
$$663$$ − 8.00000i − 0.310694i
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 20.0000i − 0.774403i
$$668$$ − 19.0000i − 0.735132i
$$669$$ −6.00000 −0.231973
$$670$$ 0 0
$$671$$ −6.00000 −0.231627
$$672$$ 3.00000i 0.115728i
$$673$$ 26.0000i 1.00223i 0.865382 + 0.501113i $$0.167076\pi$$
−0.865382 + 0.501113i $$0.832924\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 27.0000i − 1.03769i −0.854867 0.518847i $$-0.826361\pi$$
0.854867 0.518847i $$-0.173639\pi$$
$$678$$ 9.00000i 0.345643i
$$679$$ −30.0000 −1.15129
$$680$$ 0 0
$$681$$ 27.0000 1.03464
$$682$$ 3.00000i 0.114876i
$$683$$ − 19.0000i − 0.727015i −0.931591 0.363507i $$-0.881579\pi$$
0.931591 0.363507i $$-0.118421\pi$$
$$684$$ 3.00000 0.114708
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ 13.0000i 0.495981i
$$688$$ 1.00000i 0.0381246i
$$689$$ 6.00000 0.228582
$$690$$ 0 0
$$691$$ −31.0000 −1.17930 −0.589648 0.807661i $$-0.700733\pi$$
−0.589648 + 0.807661i $$0.700733\pi$$
$$692$$ 22.0000i 0.836315i
$$693$$ − 9.00000i − 0.341882i
$$694$$ −32.0000 −1.21470
$$695$$ 0 0
$$696$$ 4.00000 0.151620
$$697$$ 16.0000i 0.606043i
$$698$$ 24.0000i 0.908413i
$$699$$ 15.0000 0.567352
$$700$$ 0 0
$$701$$ −21.0000 −0.793159 −0.396580 0.918000i $$-0.629803\pi$$
−0.396580 + 0.918000i $$0.629803\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ 0 0
$$704$$ −3.00000 −0.113067
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ − 3.00000i − 0.112827i
$$708$$ − 6.00000i − 0.225494i
$$709$$ 25.0000 0.938895 0.469447 0.882960i $$-0.344453\pi$$
0.469447 + 0.882960i $$0.344453\pi$$
$$710$$ 0 0
$$711$$ −1.00000 −0.0375029
$$712$$ 1.00000i 0.0374766i
$$713$$ 5.00000i 0.187251i
$$714$$ −12.0000 −0.449089
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 12.0000i 0.448148i
$$718$$ − 15.0000i − 0.559795i
$$719$$ −26.0000 −0.969636 −0.484818 0.874615i $$-0.661114\pi$$
−0.484818 + 0.874615i $$0.661114\pi$$
$$720$$ 0 0
$$721$$ −48.0000 −1.78761
$$722$$ − 10.0000i − 0.372161i
$$723$$ 18.0000i 0.669427i
$$724$$ −5.00000 −0.185824
$$725$$ 0 0
$$726$$ −2.00000 −0.0742270
$$727$$ − 45.0000i − 1.66896i −0.551040 0.834479i $$-0.685769\pi$$
0.551040 0.834479i $$-0.314231\pi$$
$$728$$ − 6.00000i − 0.222375i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −4.00000 −0.147945
$$732$$ − 2.00000i − 0.0739221i
$$733$$ − 18.0000i − 0.664845i −0.943131 0.332423i $$-0.892134\pi$$
0.943131 0.332423i $$-0.107866\pi$$
$$734$$ 2.00000 0.0738213
$$735$$ 0 0
$$736$$ −5.00000 −0.184302
$$737$$ − 6.00000i − 0.221013i
$$738$$ − 4.00000i − 0.147242i
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ −6.00000 −0.220416
$$742$$ − 9.00000i − 0.330400i
$$743$$ − 37.0000i − 1.35740i −0.734416 0.678699i $$-0.762544\pi$$
0.734416 0.678699i $$-0.237456\pi$$
$$744$$ −1.00000 −0.0366618
$$745$$ 0 0
$$746$$ 9.00000 0.329513
$$747$$ − 12.0000i − 0.439057i
$$748$$ − 12.0000i − 0.438763i
$$749$$ 27.0000 0.986559
$$750$$ 0 0
$$751$$ −50.0000 −1.82453 −0.912263 0.409605i $$-0.865667\pi$$
−0.912263 + 0.409605i $$0.865667\pi$$
$$752$$ − 10.0000i − 0.364662i
$$753$$ 12.0000i 0.437304i
$$754$$ −8.00000 −0.291343
$$755$$ 0 0
$$756$$ 3.00000 0.109109
$$757$$ − 50.0000i − 1.81728i −0.417579 0.908640i $$-0.637121\pi$$
0.417579 0.908640i $$-0.362879\pi$$
$$758$$ − 15.0000i − 0.544825i
$$759$$ 15.0000 0.544466
$$760$$ 0 0
$$761$$ 45.0000 1.63125 0.815624 0.578582i $$-0.196394\pi$$
0.815624 + 0.578582i $$0.196394\pi$$
$$762$$ − 8.00000i − 0.289809i
$$763$$ − 60.0000i − 2.17215i
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ −12.0000 −0.433578
$$767$$ 12.0000i 0.433295i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −9.00000 −0.324548 −0.162274 0.986746i $$-0.551883\pi$$
−0.162274 + 0.986746i $$0.551883\pi$$
$$770$$ 0 0
$$771$$ 23.0000 0.828325
$$772$$ 6.00000i 0.215945i
$$773$$ − 31.0000i − 1.11499i −0.830179 0.557496i $$-0.811762\pi$$
0.830179 0.557496i $$-0.188238\pi$$
$$774$$ 1.00000 0.0359443
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ 0 0
$$778$$ 26.0000i 0.932145i
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ 21.0000 0.751439
$$782$$ − 20.0000i − 0.715199i
$$783$$ − 4.00000i − 0.142948i
$$784$$ −2.00000 −0.0714286
$$785$$ 0 0
$$786$$ 6.00000 0.214013
$$787$$ 35.0000i 1.24762i 0.781578 + 0.623808i $$0.214415\pi$$
−0.781578 + 0.623808i $$0.785585\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ −27.0000 −0.960009
$$792$$ 3.00000i 0.106600i
$$793$$ 4.00000i 0.142044i
$$794$$ 35.0000 1.24210
$$795$$ 0 0
$$796$$ 7.00000 0.248108
$$797$$ − 2.00000i − 0.0708436i −0.999372 0.0354218i $$-0.988723\pi$$
0.999372 0.0354218i $$-0.0112775\pi$$
$$798$$ 9.00000i 0.318597i
$$799$$ 40.0000 1.41510
$$800$$ 0 0
$$801$$ 1.00000 0.0353333
$$802$$ 25.0000i 0.882781i
$$803$$ 15.0000i 0.529339i
$$804$$ 2.00000 0.0705346
$$805$$ 0 0
$$806$$ 2.00000 0.0704470
$$807$$ 2.00000i 0.0704033i
$$808$$ 1.00000i 0.0351799i
$$809$$ −17.0000 −0.597688 −0.298844 0.954302i $$-0.596601\pi$$
−0.298844 + 0.954302i $$0.596601\pi$$
$$810$$ 0 0
$$811$$ −27.0000 −0.948098 −0.474049 0.880498i $$-0.657208\pi$$
−0.474049 + 0.880498i $$0.657208\pi$$
$$812$$ 12.0000i 0.421117i
$$813$$ − 13.0000i − 0.455930i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 4.00000 0.140028
$$817$$ 3.00000i 0.104957i
$$818$$ − 8.00000i − 0.279713i
$$819$$ −6.00000 −0.209657
$$820$$ 0 0
$$821$$ 40.0000 1.39601 0.698005 0.716093i $$-0.254071\pi$$
0.698005 + 0.716093i $$0.254071\pi$$
$$822$$ 10.0000i 0.348790i
$$823$$ − 18.0000i − 0.627441i −0.949515 0.313720i $$-0.898425\pi$$
0.949515 0.313720i $$-0.101575\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 0 0
$$826$$ 18.0000 0.626300
$$827$$ − 20.0000i − 0.695468i −0.937593 0.347734i $$-0.886951\pi$$
0.937593 0.347734i $$-0.113049\pi$$
$$828$$ 5.00000i 0.173762i
$$829$$ 35.0000 1.21560 0.607800 0.794090i $$-0.292052\pi$$
0.607800 + 0.794090i $$0.292052\pi$$
$$830$$ 0 0
$$831$$ −12.0000 −0.416275
$$832$$ 2.00000i 0.0693375i
$$833$$ − 8.00000i − 0.277184i
$$834$$ 14.0000 0.484780
$$835$$ 0 0
$$836$$ −9.00000 −0.311272
$$837$$ 1.00000i 0.0345651i
$$838$$ − 12.0000i − 0.414533i
$$839$$ −51.0000 −1.76072 −0.880358 0.474310i $$-0.842698\pi$$
−0.880358 + 0.474310i $$0.842698\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ − 22.0000i − 0.758170i
$$843$$ − 16.0000i − 0.551069i
$$844$$ 19.0000 0.654007
$$845$$ 0 0
$$846$$ −10.0000 −0.343807
$$847$$ − 6.00000i − 0.206162i
$$848$$ 3.00000i 0.103020i
$$849$$ −24.0000 −0.823678
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 7.00000i 0.239816i
$$853$$ − 41.0000i − 1.40381i −0.712269 0.701907i $$-0.752332\pi$$
0.712269 0.701907i $$-0.247668\pi$$
$$854$$ 6.00000 0.205316
$$855$$ 0 0
$$856$$ −9.00000 −0.307614
$$857$$ − 42.0000i − 1.43469i −0.696717 0.717346i $$-0.745357\pi$$
0.696717 0.717346i $$-0.254643\pi$$
$$858$$ − 6.00000i − 0.204837i
$$859$$ −32.0000 −1.09183 −0.545913 0.837842i $$-0.683817\pi$$
−0.545913 + 0.837842i $$0.683817\pi$$
$$860$$ 0 0
$$861$$ 12.0000 0.408959
$$862$$ 16.0000i 0.544962i
$$863$$ 43.0000i 1.46374i 0.681446 + 0.731869i $$0.261351\pi$$
−0.681446 + 0.731869i $$0.738649\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 1.00000 0.0339814
$$867$$ − 1.00000i − 0.0339618i
$$868$$ − 3.00000i − 0.101827i
$$869$$ 3.00000 0.101768
$$870$$ 0 0
$$871$$ −4.00000 −0.135535
$$872$$ 20.0000i 0.677285i
$$873$$ − 10.0000i − 0.338449i
$$874$$ −15.0000 −0.507383
$$875$$ 0 0
$$876$$ −5.00000 −0.168934
$$877$$ − 46.0000i − 1.55331i −0.629926 0.776655i $$-0.716915\pi$$
0.629926 0.776655i $$-0.283085\pi$$
$$878$$ − 18.0000i − 0.607471i
$$879$$ 30.0000 1.01187
$$880$$ 0 0
$$881$$ 26.0000 0.875962 0.437981 0.898984i $$-0.355694\pi$$
0.437981 + 0.898984i $$0.355694\pi$$
$$882$$ 2.00000i 0.0673435i
$$883$$ 11.0000i 0.370179i 0.982722 + 0.185090i $$0.0592576\pi$$
−0.982722 + 0.185090i $$0.940742\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ 0 0
$$886$$ −15.0000 −0.503935
$$887$$ 46.0000i 1.54453i 0.635301 + 0.772264i $$0.280876\pi$$
−0.635301 + 0.772264i $$0.719124\pi$$
$$888$$ 0 0
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ 3.00000 0.100504
$$892$$ 6.00000i 0.200895i
$$893$$ − 30.0000i − 1.00391i
$$894$$ 11.0000 0.367895
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ − 10.0000i − 0.333890i
$$898$$ − 2.00000i − 0.0667409i
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ 12.0000i 0.399556i
$$903$$ 3.00000i 0.0998337i
$$904$$ 9.00000 0.299336
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 52.0000i 1.72663i 0.504664 + 0.863316i $$0.331616\pi$$
−0.504664 + 0.863316i $$0.668384\pi$$
$$908$$ − 27.0000i − 0.896026i
$$909$$ 1.00000 0.0331679
$$910$$ 0 0
$$911$$ 20.0000 0.662630 0.331315 0.943520i $$-0.392508\pi$$
0.331315 + 0.943520i $$0.392508\pi$$
$$912$$ − 3.00000i − 0.0993399i
$$913$$ 36.0000i 1.19143i
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ 13.0000 0.429532
$$917$$ 18.0000i 0.594412i
$$918$$ − 4.00000i − 0.132020i
$$919$$ 38.0000 1.25350 0.626752 0.779219i $$-0.284384\pi$$
0.626752 + 0.779219i $$0.284384\pi$$
$$920$$ 0 0
$$921$$ −16.0000 −0.527218
$$922$$ 0 0
$$923$$ − 14.0000i − 0.460816i
$$924$$ −9.00000 −0.296078
$$925$$ 0 0
$$926$$ −4.00000 −0.131448
$$927$$ − 16.0000i − 0.525509i
$$928$$ − 4.00000i − 0.131306i
$$929$$ 41.0000 1.34517 0.672583 0.740022i $$-0.265185\pi$$
0.672583 + 0.740022i $$0.265185\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ − 15.0000i − 0.491341i
$$933$$ 0 0
$$934$$ 24.0000 0.785304
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ 52.0000i 1.69877i 0.527777 + 0.849383i $$0.323026\pi$$
−0.527777 + 0.849383i $$0.676974\pi$$
$$938$$ 6.00000i 0.195907i
$$939$$ 30.0000 0.979013
$$940$$ 0 0
$$941$$ 40.0000 1.30396 0.651981 0.758235i $$-0.273938\pi$$
0.651981 + 0.758235i $$0.273938\pi$$
$$942$$ 5.00000i 0.162909i
$$943$$ 20.0000i 0.651290i
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ −3.00000 −0.0975384
$$947$$ − 30.0000i − 0.974869i −0.873160 0.487435i $$-0.837933\pi$$
0.873160 0.487435i $$-0.162067\pi$$
$$948$$ 1.00000i 0.0324785i
$$949$$ 10.0000 0.324614
$$950$$ 0 0
$$951$$ 8.00000 0.259418
$$952$$ 12.0000i 0.388922i
$$953$$ − 52.0000i − 1.68445i −0.539130 0.842223i $$-0.681247\pi$$
0.539130 0.842223i $$-0.318753\pi$$
$$954$$ 3.00000 0.0971286
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 12.0000i 0.387905i
$$958$$ − 3.00000i − 0.0969256i
$$959$$ −30.0000 −0.968751
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 0 0
$$963$$ 9.00000i 0.290021i
$$964$$ 18.0000 0.579741
$$965$$ 0 0
$$966$$ −15.0000 −0.482617
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ 2.00000i 0.0642824i
$$969$$ 12.0000 0.385496
$$970$$ 0 0
$$971$$ −54.0000 −1.73294 −0.866471 0.499227i $$-0.833617\pi$$
−0.866471 + 0.499227i $$0.833617\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 42.0000i 1.34646i
$$974$$ 32.0000 1.02535
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ − 38.0000i − 1.21573i −0.794041 0.607864i $$-0.792027\pi$$
0.794041 0.607864i $$-0.207973\pi$$
$$978$$ 14.0000i 0.447671i
$$979$$ −3.00000 −0.0958804
$$980$$ 0 0
$$981$$ 20.0000 0.638551
$$982$$ − 37.0000i − 1.18072i
$$983$$ − 56.0000i − 1.78612i −0.449935 0.893061i $$-0.648553\pi$$
0.449935 0.893061i $$-0.351447\pi$$
$$984$$ −4.00000 −0.127515
$$985$$ 0 0
$$986$$ 16.0000 0.509544
$$987$$ − 30.0000i − 0.954911i
$$988$$ 6.00000i 0.190885i
$$989$$ −5.00000 −0.158991
$$990$$ 0 0
$$991$$ −25.0000 −0.794151 −0.397076 0.917786i $$-0.629975\pi$$
−0.397076 + 0.917786i $$0.629975\pi$$
$$992$$ 1.00000i 0.0317500i
$$993$$ − 12.0000i − 0.380808i
$$994$$ −21.0000 −0.666080
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ 42.0000i 1.33015i 0.746775 + 0.665077i $$0.231601\pi$$
−0.746775 + 0.665077i $$0.768399\pi$$
$$998$$ − 4.00000i − 0.126618i
$$999$$ 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 4650.2.d.y.3349.2 2
5.2 odd 4 930.2.a.a.1.1 1
5.3 odd 4 4650.2.a.bv.1.1 1
5.4 even 2 inner 4650.2.d.y.3349.1 2
15.2 even 4 2790.2.a.y.1.1 1
20.7 even 4 7440.2.a.v.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.a.1.1 1 5.2 odd 4
2790.2.a.y.1.1 1 15.2 even 4
4650.2.a.bv.1.1 1 5.3 odd 4
4650.2.d.y.3349.1 2 5.4 even 2 inner
4650.2.d.y.3349.2 2 1.1 even 1 trivial
7440.2.a.v.1.1 1 20.7 even 4