# Properties

 Label 3450.2.d.r.2899.1 Level $3450$ Weight $2$ Character 3450.2899 Analytic conductor $27.548$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.5483886973$$ 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 690) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2899.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3450.2899 Dual form 3450.2.d.r.2899.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -2.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} -1.00000i q^{27} -8.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} +4.00000i q^{38} -2.00000 q^{41} -2.00000i q^{43} -2.00000 q^{44} +1.00000 q^{46} -4.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} -2.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -2.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +2.00000 q^{66} +2.00000i q^{67} +6.00000i q^{68} -1.00000 q^{69} -10.0000 q^{71} -1.00000i q^{72} -10.0000i q^{73} +6.00000 q^{74} +4.00000 q^{76} +1.00000 q^{81} +2.00000i q^{82} -4.00000i q^{83} -2.00000 q^{86} +2.00000i q^{88} -4.00000 q^{89} -1.00000i q^{92} -8.00000i q^{93} -4.00000 q^{94} +1.00000 q^{96} -16.0000i q^{97} -7.00000i q^{98} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + 4q^{11} + 2q^{16} - 8q^{19} - 2q^{24} - 16q^{31} - 12q^{34} + 2q^{36} - 4q^{41} - 4q^{44} + 2q^{46} + 14q^{49} + 12q^{51} - 2q^{54} - 4q^{61} - 2q^{64} + 4q^{66} - 2q^{69} - 20q^{71} + 12q^{74} + 8q^{76} + 2q^{81} - 4q^{86} - 8q^{89} - 8q^{94} + 2q^{96} - 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 2.00000i − 0.426401i
$$23$$ 1.00000i 0.208514i
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 2.00000i 0.348155i
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ − 2.00000i − 0.304997i −0.988304 0.152499i $$-0.951268\pi$$
0.988304 0.152499i $$-0.0487319\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ − 4.00000i − 0.583460i −0.956501 0.291730i $$-0.905769\pi$$
0.956501 0.291730i $$-0.0942309\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 0 0
$$53$$ − 2.00000i − 0.274721i −0.990521 0.137361i $$-0.956138\pi$$
0.990521 0.137361i $$-0.0438619\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 4.00000i − 0.529813i
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 8.00000i 1.01600i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ 2.00000i 0.244339i 0.992509 + 0.122169i $$0.0389851\pi$$
−0.992509 + 0.122169i $$0.961015\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −10.0000 −1.18678 −0.593391 0.804914i $$-0.702211\pi$$
−0.593391 + 0.804914i $$0.702211\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 10.0000i − 1.17041i −0.810885 0.585206i $$-0.801014\pi$$
0.810885 0.585206i $$-0.198986\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.00000i 0.220863i
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ 2.00000i 0.213201i
$$89$$ −4.00000 −0.423999 −0.212000 0.977270i $$-0.567998\pi$$
−0.212000 + 0.977270i $$0.567998\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 1.00000i − 0.104257i
$$93$$ − 8.00000i − 0.829561i
$$94$$ −4.00000 −0.412568
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 16.0000i − 1.62455i −0.583272 0.812277i $$-0.698228\pi$$
0.583272 0.812277i $$-0.301772\pi$$
$$98$$ − 7.00000i − 0.707107i
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ − 16.0000i − 1.57653i −0.615338 0.788263i $$-0.710980\pi$$
0.615338 0.788263i $$-0.289020\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ − 8.00000i − 0.773389i −0.922208 0.386695i $$-0.873617\pi$$
0.922208 0.386695i $$-0.126383\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 0 0
$$113$$ 10.0000i 0.940721i 0.882474 + 0.470360i $$0.155876\pi$$
−0.882474 + 0.470360i $$0.844124\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 2.00000i 0.181071i
$$123$$ − 2.00000i − 0.180334i
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 6.00000i − 0.532414i −0.963916 0.266207i $$-0.914230\pi$$
0.963916 0.266207i $$-0.0857705\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 2.00000 0.176090
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ − 2.00000i − 0.174078i
$$133$$ 0 0
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ − 2.00000i − 0.170872i −0.996344 0.0854358i $$-0.972772\pi$$
0.996344 0.0854358i $$-0.0272282\pi$$
$$138$$ 1.00000i 0.0851257i
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ 10.0000i 0.839181i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −10.0000 −0.827606
$$147$$ 7.00000i 0.577350i
$$148$$ − 6.00000i − 0.493197i
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ −20.0000 −1.62758 −0.813788 0.581161i $$-0.802599\pi$$
−0.813788 + 0.581161i $$0.802599\pi$$
$$152$$ − 4.00000i − 0.324443i
$$153$$ 6.00000i 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 6.00000i 0.478852i 0.970915 + 0.239426i $$0.0769593\pi$$
−0.970915 + 0.239426i $$0.923041\pi$$
$$158$$ 0 0
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 8.00000i − 0.626608i −0.949653 0.313304i $$-0.898564\pi$$
0.949653 0.313304i $$-0.101436\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ − 8.00000i − 0.619059i −0.950890 0.309529i $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 2.00000i 0.152499i
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 0 0
$$178$$ 4.00000i 0.299813i
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 0 0
$$183$$ − 2.00000i − 0.147844i
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ − 12.0000i − 0.877527i
$$188$$ 4.00000i 0.291730i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ − 14.0000i − 1.00774i −0.863779 0.503871i $$-0.831909\pi$$
0.863779 0.503871i $$-0.168091\pi$$
$$194$$ −16.0000 −1.14873
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ − 14.0000i − 0.997459i −0.866758 0.498729i $$-0.833800\pi$$
0.866758 0.498729i $$-0.166200\pi$$
$$198$$ 2.00000i 0.142134i
$$199$$ −24.0000 −1.70131 −0.850657 0.525720i $$-0.823796\pi$$
−0.850657 + 0.525720i $$0.823796\pi$$
$$200$$ 0 0
$$201$$ −2.00000 −0.141069
$$202$$ 0 0
$$203$$ 0 0
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ −16.0000 −1.11477
$$207$$ − 1.00000i − 0.0695048i
$$208$$ 0 0
$$209$$ −8.00000 −0.553372
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 2.00000i 0.137361i
$$213$$ − 10.0000i − 0.685189i
$$214$$ −8.00000 −0.546869
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ 2.00000i 0.135457i
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 6.00000i 0.402694i
$$223$$ − 14.0000i − 0.937509i −0.883328 0.468755i $$-0.844703\pi$$
0.883328 0.468755i $$-0.155297\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 10.0000 0.665190
$$227$$ − 8.00000i − 0.530979i −0.964114 0.265489i $$-0.914466\pi$$
0.964114 0.265489i $$-0.0855335\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 22.0000i 1.44127i 0.693316 + 0.720634i $$0.256149\pi$$
−0.693316 + 0.720634i $$0.743851\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 26.0000 1.68180 0.840900 0.541190i $$-0.182026\pi$$
0.840900 + 0.541190i $$0.182026\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 7.00000i 0.449977i
$$243$$ 1.00000i 0.0641500i
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ −2.00000 −0.127515
$$247$$ 0 0
$$248$$ − 8.00000i − 0.508001i
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 0 0
$$253$$ 2.00000i 0.125739i
$$254$$ −6.00000 −0.376473
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 6.00000i − 0.374270i −0.982334 0.187135i $$-0.940080\pi$$
0.982334 0.187135i $$-0.0599201\pi$$
$$258$$ − 2.00000i − 0.124515i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 4.00000i 0.247121i
$$263$$ 16.0000i 0.986602i 0.869859 + 0.493301i $$0.164210\pi$$
−0.869859 + 0.493301i $$0.835790\pi$$
$$264$$ −2.00000 −0.123091
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 4.00000i − 0.244796i
$$268$$ − 2.00000i − 0.122169i
$$269$$ −28.0000 −1.70719 −0.853595 0.520937i $$-0.825583\pi$$
−0.853595 + 0.520937i $$0.825583\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ − 6.00000i − 0.363803i
$$273$$ 0 0
$$274$$ −2.00000 −0.120824
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ − 24.0000i − 1.44202i −0.692925 0.721010i $$-0.743678\pi$$
0.692925 0.721010i $$-0.256322\pi$$
$$278$$ 12.0000i 0.719712i
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ − 4.00000i − 0.238197i
$$283$$ 10.0000i 0.594438i 0.954809 + 0.297219i $$0.0960592\pi$$
−0.954809 + 0.297219i $$0.903941\pi$$
$$284$$ 10.0000 0.593391
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 1.00000i 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 16.0000 0.937937
$$292$$ 10.0000i 0.585206i
$$293$$ 22.0000i 1.28525i 0.766179 + 0.642627i $$0.222155\pi$$
−0.766179 + 0.642627i $$0.777845\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ − 2.00000i − 0.116052i
$$298$$ 10.0000i 0.579284i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 20.0000i 1.15087i
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 0 0
$$309$$ 16.0000 0.910208
$$310$$ 0 0
$$311$$ −2.00000 −0.113410 −0.0567048 0.998391i $$-0.518059\pi$$
−0.0567048 + 0.998391i $$0.518059\pi$$
$$312$$ 0 0
$$313$$ − 20.0000i − 1.13047i −0.824931 0.565233i $$-0.808786\pi$$
0.824931 0.565233i $$-0.191214\pi$$
$$314$$ 6.00000 0.338600
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ − 2.00000i − 0.112154i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 8.00000 0.446516
$$322$$ 0 0
$$323$$ 24.0000i 1.33540i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −8.00000 −0.443079
$$327$$ − 2.00000i − 0.110600i
$$328$$ − 2.00000i − 0.110432i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 4.00000i 0.219529i
$$333$$ − 6.00000i − 0.328798i
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 28.0000i − 1.52526i −0.646837 0.762629i $$-0.723908\pi$$
0.646837 0.762629i $$-0.276092\pi$$
$$338$$ − 13.0000i − 0.707107i
$$339$$ −10.0000 −0.543125
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ − 4.00000i − 0.216295i
$$343$$ 0 0
$$344$$ 2.00000 0.107833
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ 4.00000i 0.214731i 0.994220 + 0.107366i $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 2.00000i − 0.106600i
$$353$$ − 6.00000i − 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 4.00000 0.212000
$$357$$ 0 0
$$358$$ − 24.0000i − 1.26844i
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 6.00000i − 0.315353i
$$363$$ − 7.00000i − 0.367405i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ − 16.0000i − 0.835193i −0.908633 0.417597i $$-0.862873\pi$$
0.908633 0.417597i $$-0.137127\pi$$
$$368$$ 1.00000i 0.0521286i
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 8.00000i 0.414781i
$$373$$ − 6.00000i − 0.310668i −0.987862 0.155334i $$-0.950355\pi$$
0.987862 0.155334i $$-0.0496454\pi$$
$$374$$ −12.0000 −0.620505
$$375$$ 0 0
$$376$$ 4.00000 0.206284
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ 6.00000 0.307389
$$382$$ 8.00000i 0.409316i
$$383$$ 16.0000i 0.817562i 0.912633 + 0.408781i $$0.134046\pi$$
−0.912633 + 0.408781i $$0.865954\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 2.00000i 0.101666i
$$388$$ 16.0000i 0.812277i
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ 7.00000i 0.353553i
$$393$$ − 4.00000i − 0.201773i
$$394$$ −14.0000 −0.705310
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ − 8.00000i − 0.401508i −0.979642 0.200754i $$-0.935661\pi$$
0.979642 0.200754i $$-0.0643393\pi$$
$$398$$ 24.0000i 1.20301i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 2.00000i 0.0997509i
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 12.0000i 0.594818i
$$408$$ 6.00000i 0.297044i
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ 0 0
$$411$$ 2.00000 0.0986527
$$412$$ 16.0000i 0.788263i
$$413$$ 0 0
$$414$$ −1.00000 −0.0491473
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 12.0000i − 0.587643i
$$418$$ 8.00000i 0.391293i
$$419$$ 6.00000 0.293119 0.146560 0.989202i $$-0.453180\pi$$
0.146560 + 0.989202i $$0.453180\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 20.0000i 0.973585i
$$423$$ 4.00000i 0.194487i
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ −10.0000 −0.484502
$$427$$ 0 0
$$428$$ 8.00000i 0.386695i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −40.0000 −1.92673 −0.963366 0.268190i $$-0.913575\pi$$
−0.963366 + 0.268190i $$0.913575\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 32.0000i 1.53782i 0.639356 + 0.768911i $$0.279201\pi$$
−0.639356 + 0.768911i $$0.720799\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ − 4.00000i − 0.191346i
$$438$$ − 10.0000i − 0.477818i
$$439$$ 12.0000 0.572729 0.286364 0.958121i $$-0.407553\pi$$
0.286364 + 0.958121i $$0.407553\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 0 0
$$443$$ 36.0000i 1.71041i 0.518289 + 0.855206i $$0.326569\pi$$
−0.518289 + 0.855206i $$0.673431\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 0 0
$$446$$ −14.0000 −0.662919
$$447$$ − 10.0000i − 0.472984i
$$448$$ 0 0
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ 0 0
$$451$$ −4.00000 −0.188353
$$452$$ − 10.0000i − 0.470360i
$$453$$ − 20.0000i − 0.939682i
$$454$$ −8.00000 −0.375459
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ 8.00000i 0.374224i 0.982339 + 0.187112i $$0.0599128\pi$$
−0.982339 + 0.187112i $$0.940087\pi$$
$$458$$ − 14.0000i − 0.654177i
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ 32.0000 1.49039 0.745194 0.666847i $$-0.232357\pi$$
0.745194 + 0.666847i $$0.232357\pi$$
$$462$$ 0 0
$$463$$ 34.0000i 1.58011i 0.613033 + 0.790057i $$0.289949\pi$$
−0.613033 + 0.790057i $$0.710051\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 22.0000 1.01913
$$467$$ 4.00000i 0.185098i 0.995708 + 0.0925490i $$0.0295015\pi$$
−0.995708 + 0.0925490i $$0.970499\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −6.00000 −0.276465
$$472$$ 0 0
$$473$$ − 4.00000i − 0.183920i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 2.00000i 0.0915737i
$$478$$ − 26.0000i − 1.18921i
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ − 2.00000i − 0.0910975i
$$483$$ 0 0
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 2.00000i − 0.0906287i −0.998973 0.0453143i $$-0.985571\pi$$
0.998973 0.0453143i $$-0.0144289\pi$$
$$488$$ − 2.00000i − 0.0905357i
$$489$$ 8.00000 0.361773
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 2.00000i 0.0901670i
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ − 4.00000i − 0.179244i
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ 18.0000i 0.803379i
$$503$$ − 24.0000i − 1.07011i −0.844818 0.535054i $$-0.820291\pi$$
0.844818 0.535054i $$-0.179709\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 2.00000 0.0889108
$$507$$ 13.0000i 0.577350i
$$508$$ 6.00000i 0.266207i
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 4.00000i 0.176604i
$$514$$ −6.00000 −0.264649
$$515$$ 0 0
$$516$$ −2.00000 −0.0880451
$$517$$ − 8.00000i − 0.351840i
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ 36.0000 1.57719 0.788594 0.614914i $$-0.210809\pi$$
0.788594 + 0.614914i $$0.210809\pi$$
$$522$$ 0 0
$$523$$ 34.0000i 1.48672i 0.668894 + 0.743358i $$0.266768\pi$$
−0.668894 + 0.743358i $$0.733232\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ 16.0000 0.697633
$$527$$ 48.0000i 2.09091i
$$528$$ 2.00000i 0.0870388i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −4.00000 −0.173097
$$535$$ 0 0
$$536$$ −2.00000 −0.0863868
$$537$$ 24.0000i 1.03568i
$$538$$ 28.0000i 1.20717i
$$539$$ 14.0000 0.603023
$$540$$ 0 0
$$541$$ 26.0000 1.11783 0.558914 0.829226i $$-0.311218\pi$$
0.558914 + 0.829226i $$0.311218\pi$$
$$542$$ 8.00000i 0.343629i
$$543$$ 6.00000i 0.257485i
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 20.0000i − 0.855138i −0.903983 0.427569i $$-0.859370\pi$$
0.903983 0.427569i $$-0.140630\pi$$
$$548$$ 2.00000i 0.0854358i
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ 0 0
$$552$$ − 1.00000i − 0.0425628i
$$553$$ 0 0
$$554$$ −24.0000 −1.01966
$$555$$ 0 0
$$556$$ 12.0000 0.508913
$$557$$ 30.0000i 1.27114i 0.772043 + 0.635570i $$0.219235\pi$$
−0.772043 + 0.635570i $$0.780765\pi$$
$$558$$ − 8.00000i − 0.338667i
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 12.0000 0.506640
$$562$$ − 12.0000i − 0.506189i
$$563$$ 4.00000i 0.168580i 0.996441 + 0.0842900i $$0.0268622\pi$$
−0.996441 + 0.0842900i $$0.973138\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 0 0
$$566$$ 10.0000 0.420331
$$567$$ 0 0
$$568$$ − 10.0000i − 0.419591i
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ −16.0000 −0.669579 −0.334790 0.942293i $$-0.608665\pi$$
−0.334790 + 0.942293i $$0.608665\pi$$
$$572$$ 0 0
$$573$$ − 8.00000i − 0.334205i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ 19.0000i 0.790296i
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 16.0000i − 0.663221i
$$583$$ − 4.00000i − 0.165663i
$$584$$ 10.0000 0.413803
$$585$$ 0 0
$$586$$ 22.0000 0.908812
$$587$$ 28.0000i 1.15568i 0.816149 + 0.577842i $$0.196105\pi$$
−0.816149 + 0.577842i $$0.803895\pi$$
$$588$$ − 7.00000i − 0.288675i
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ 14.0000 0.575883
$$592$$ 6.00000i 0.246598i
$$593$$ − 42.0000i − 1.72473i −0.506284 0.862367i $$-0.668981\pi$$
0.506284 0.862367i $$-0.331019\pi$$
$$594$$ −2.00000 −0.0820610
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ − 24.0000i − 0.982255i
$$598$$ 0 0
$$599$$ −26.0000 −1.06233 −0.531166 0.847268i $$-0.678246\pi$$
−0.531166 + 0.847268i $$0.678246\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 0 0
$$603$$ − 2.00000i − 0.0814463i
$$604$$ 20.0000 0.813788
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 22.0000i 0.892952i 0.894795 + 0.446476i $$0.147321\pi$$
−0.894795 + 0.446476i $$0.852679\pi$$
$$608$$ 4.00000i 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 6.00000i − 0.242536i
$$613$$ − 10.0000i − 0.403896i −0.979396 0.201948i $$-0.935273\pi$$
0.979396 0.201948i $$-0.0647272\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 42.0000i 1.69086i 0.534089 + 0.845428i $$0.320655\pi$$
−0.534089 + 0.845428i $$0.679345\pi$$
$$618$$ − 16.0000i − 0.643614i
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 2.00000i 0.0801927i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −20.0000 −0.799361
$$627$$ − 8.00000i − 0.319489i
$$628$$ − 6.00000i − 0.239426i
$$629$$ 36.0000 1.43541
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 0 0
$$633$$ − 20.0000i − 0.794929i
$$634$$ 6.00000 0.238290
$$635$$ 0 0
$$636$$ −2.00000 −0.0793052
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 10.0000 0.395594
$$640$$ 0 0
$$641$$ 4.00000 0.157991 0.0789953 0.996875i $$-0.474829\pi$$
0.0789953 + 0.996875i $$0.474829\pi$$
$$642$$ − 8.00000i − 0.315735i
$$643$$ − 14.0000i − 0.552106i −0.961142 0.276053i $$-0.910973\pi$$
0.961142 0.276053i $$-0.0890266\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ 36.0000i 1.41531i 0.706560 + 0.707653i $$0.250246\pi$$
−0.706560 + 0.707653i $$0.749754\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 8.00000i 0.313304i
$$653$$ − 14.0000i − 0.547862i −0.961749 0.273931i $$-0.911676\pi$$
0.961749 0.273931i $$-0.0883240\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ 10.0000i 0.390137i
$$658$$ 0 0
$$659$$ −18.0000 −0.701180 −0.350590 0.936529i $$-0.614019\pi$$
−0.350590 + 0.936529i $$0.614019\pi$$
$$660$$ 0 0
$$661$$ 10.0000 0.388955 0.194477 0.980907i $$-0.437699\pi$$
0.194477 + 0.980907i $$0.437699\pi$$
$$662$$ − 4.00000i − 0.155464i
$$663$$ 0 0
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ 0 0
$$668$$ 8.00000i 0.309529i
$$669$$ 14.0000 0.541271
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ 0 0
$$673$$ − 10.0000i − 0.385472i −0.981251 0.192736i $$-0.938264\pi$$
0.981251 0.192736i $$-0.0617360\pi$$
$$674$$ −28.0000 −1.07852
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ 6.00000i 0.230599i 0.993331 + 0.115299i $$0.0367827\pi$$
−0.993331 + 0.115299i $$0.963217\pi$$
$$678$$ 10.0000i 0.384048i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 8.00000 0.306561
$$682$$ 16.0000i 0.612672i
$$683$$ 20.0000i 0.765279i 0.923898 + 0.382639i $$0.124985\pi$$
−0.923898 + 0.382639i $$0.875015\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 14.0000i 0.534133i
$$688$$ − 2.00000i − 0.0762493i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ − 6.00000i − 0.228086i
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 12.0000i 0.454532i
$$698$$ 26.0000i 0.984115i
$$699$$ −22.0000 −0.832116
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 0 0
$$703$$ − 24.0000i − 0.905177i
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 4.00000i − 0.149906i
$$713$$ − 8.00000i − 0.299602i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −24.0000 −0.896922
$$717$$ 26.0000i 0.970988i
$$718$$ − 32.0000i − 1.19423i
$$719$$ 22.0000 0.820462 0.410231 0.911982i $$-0.365448\pi$$
0.410231 + 0.911982i $$0.365448\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 3.00000i 0.111648i
$$723$$ 2.00000i 0.0743808i
$$724$$ −6.00000 −0.222988
$$725$$ 0 0
$$726$$ −7.00000 −0.259794
$$727$$ 28.0000i 1.03846i 0.854634 + 0.519231i $$0.173782\pi$$
−0.854634 + 0.519231i $$0.826218\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −12.0000 −0.443836
$$732$$ 2.00000i 0.0739221i
$$733$$ − 30.0000i − 1.10808i −0.832492 0.554038i $$-0.813086\pi$$
0.832492 0.554038i $$-0.186914\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ 4.00000i 0.147342i
$$738$$ − 2.00000i − 0.0736210i
$$739$$ −36.0000 −1.32428 −0.662141 0.749380i $$-0.730352\pi$$
−0.662141 + 0.749380i $$0.730352\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ −6.00000 −0.219676
$$747$$ 4.00000i 0.146352i
$$748$$ 12.0000i 0.438763i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ − 4.00000i − 0.145865i
$$753$$ − 18.0000i − 0.655956i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 42.0000i 1.52652i 0.646094 + 0.763258i $$0.276401\pi$$
−0.646094 + 0.763258i $$0.723599\pi$$
$$758$$ − 8.00000i − 0.290573i
$$759$$ −2.00000 −0.0725954
$$760$$ 0 0
$$761$$ −14.0000 −0.507500 −0.253750 0.967270i $$-0.581664\pi$$
−0.253750 + 0.967270i $$0.581664\pi$$
$$762$$ − 6.00000i − 0.217357i
$$763$$ 0 0
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ 0 0
$$768$$ 1.00000i 0.0360844i
$$769$$ 42.0000 1.51456 0.757279 0.653091i $$-0.226528\pi$$
0.757279 + 0.653091i $$0.226528\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ 14.0000i 0.503871i
$$773$$ 38.0000i 1.36677i 0.730061 + 0.683383i $$0.239492\pi$$
−0.730061 + 0.683383i $$0.760508\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ 0 0
$$776$$ 16.0000 0.574367
$$777$$ 0 0
$$778$$ − 18.0000i − 0.645331i
$$779$$ 8.00000 0.286630
$$780$$ 0 0
$$781$$ −20.0000 −0.715656
$$782$$ − 6.00000i − 0.214560i
$$783$$ 0 0
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ −4.00000 −0.142675
$$787$$ − 18.0000i − 0.641631i −0.947142 0.320815i $$-0.896043\pi$$
0.947142 0.320815i $$-0.103957\pi$$
$$788$$ 14.0000i 0.498729i
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ 0 0
$$792$$ − 2.00000i − 0.0710669i
$$793$$ 0 0
$$794$$ −8.00000 −0.283909
$$795$$ 0 0
$$796$$ 24.0000 0.850657
$$797$$ − 42.0000i − 1.48772i −0.668338 0.743858i $$-0.732994\pi$$
0.668338 0.743858i $$-0.267006\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ 4.00000 0.141333
$$802$$ 0 0
$$803$$ − 20.0000i − 0.705785i
$$804$$ 2.00000 0.0705346
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 28.0000i − 0.985647i
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 0 0
$$813$$ − 8.00000i − 0.280572i
$$814$$ 12.0000 0.420600
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ 8.00000i 0.279885i
$$818$$ 22.0000i 0.769212i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 44.0000 1.53561 0.767805 0.640683i $$-0.221349\pi$$
0.767805 + 0.640683i $$0.221349\pi$$
$$822$$ − 2.00000i − 0.0697580i
$$823$$ − 10.0000i − 0.348578i −0.984695 0.174289i $$-0.944237\pi$$
0.984695 0.174289i $$-0.0557627\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 32.0000i 1.11275i 0.830932 + 0.556375i $$0.187808\pi$$
−0.830932 + 0.556375i $$0.812192\pi$$
$$828$$ 1.00000i 0.0347524i
$$829$$ 6.00000 0.208389 0.104194 0.994557i $$-0.466774\pi$$
0.104194 + 0.994557i $$0.466774\pi$$
$$830$$ 0 0
$$831$$ 24.0000 0.832551
$$832$$ 0 0
$$833$$ − 42.0000i − 1.45521i
$$834$$ −12.0000 −0.415526
$$835$$ 0 0
$$836$$ 8.00000 0.276686
$$837$$ 8.00000i 0.276520i
$$838$$ − 6.00000i − 0.207267i
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ − 22.0000i − 0.758170i
$$843$$ 12.0000i 0.413302i
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 4.00000 0.137523
$$847$$ 0 0
$$848$$ − 2.00000i − 0.0686803i
$$849$$ −10.0000 −0.343199
$$850$$ 0 0
$$851$$ −6.00000 −0.205677
$$852$$ 10.0000i 0.342594i
$$853$$ − 16.0000i − 0.547830i −0.961754 0.273915i $$-0.911681\pi$$
0.961754 0.273915i $$-0.0883186\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 8.00000 0.273434
$$857$$ − 18.0000i − 0.614868i −0.951569 0.307434i $$-0.900530\pi$$
0.951569 0.307434i $$-0.0994704\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 40.0000i 1.36241i
$$863$$ − 12.0000i − 0.408485i −0.978920 0.204242i $$-0.934527\pi$$
0.978920 0.204242i $$-0.0654731\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 32.0000 1.08740
$$867$$ − 19.0000i − 0.645274i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ − 2.00000i − 0.0677285i
$$873$$ 16.0000i 0.541518i
$$874$$ −4.00000 −0.135302
$$875$$ 0 0
$$876$$ −10.0000 −0.337869
$$877$$ − 32.0000i − 1.08056i −0.841484 0.540282i $$-0.818318\pi$$
0.841484 0.540282i $$-0.181682\pi$$
$$878$$ − 12.0000i − 0.404980i
$$879$$ −22.0000 −0.742042
$$880$$ 0 0
$$881$$ 36.0000 1.21287 0.606435 0.795133i $$-0.292599\pi$$
0.606435 + 0.795133i $$0.292599\pi$$
$$882$$ 7.00000i 0.235702i
$$883$$ 8.00000i 0.269221i 0.990899 + 0.134611i $$0.0429784\pi$$
−0.990899 + 0.134611i $$0.957022\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 36.0000 1.20944
$$887$$ 28.0000i 0.940148i 0.882627 + 0.470074i $$0.155773\pi$$
−0.882627 + 0.470074i $$0.844227\pi$$
$$888$$ − 6.00000i − 0.201347i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 14.0000i 0.468755i
$$893$$ 16.0000i 0.535420i
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ − 2.00000i − 0.0667409i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ 4.00000i 0.133185i
$$903$$ 0 0
$$904$$ −10.0000 −0.332595
$$905$$ 0 0
$$906$$ −20.0000 −0.664455
$$907$$ − 38.0000i − 1.26177i −0.775877 0.630885i $$-0.782692\pi$$
0.775877 0.630885i $$-0.217308\pi$$
$$908$$ 8.00000i 0.265489i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 20.0000 0.662630 0.331315 0.943520i $$-0.392508\pi$$
0.331315 + 0.943520i $$0.392508\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ − 8.00000i − 0.264761i
$$914$$ 8.00000 0.264616
$$915$$ 0 0
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ 6.00000i 0.198030i
$$919$$ −48.0000 −1.58337 −0.791687 0.610927i $$-0.790797\pi$$
−0.791687 + 0.610927i $$0.790797\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ − 32.0000i − 1.05386i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 34.0000 1.11731
$$927$$ 16.0000i 0.525509i
$$928$$ 0 0
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ − 22.0000i − 0.720634i
$$933$$ − 2.00000i − 0.0654771i
$$934$$ 4.00000 0.130884
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 40.0000i − 1.30674i −0.757037 0.653372i $$-0.773354\pi$$
0.757037 0.653372i $$-0.226646\pi$$
$$938$$ 0 0
$$939$$ 20.0000 0.652675
$$940$$ 0 0
$$941$$ −38.0000 −1.23876 −0.619382 0.785090i $$-0.712617\pi$$
−0.619382 + 0.785090i $$0.712617\pi$$
$$942$$ 6.00000i 0.195491i
$$943$$ − 2.00000i − 0.0651290i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ − 52.0000i − 1.68977i −0.534946 0.844886i $$-0.679668\pi$$
0.534946 0.844886i $$-0.320332\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ 0 0
$$953$$ − 22.0000i − 0.712650i −0.934362 0.356325i $$-0.884030\pi$$
0.934362 0.356325i $$-0.115970\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ 0 0
$$956$$ −26.0000 −0.840900
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 8.00000i 0.257796i
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 38.0000i 1.22200i 0.791632 + 0.610999i $$0.209232\pi$$
−0.791632 + 0.610999i $$0.790768\pi$$
$$968$$ − 7.00000i − 0.224989i
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ 6.00000 0.192549 0.0962746 0.995355i $$-0.469307\pi$$
0.0962746 + 0.995355i $$0.469307\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 0 0
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ − 38.0000i − 1.21573i −0.794041 0.607864i $$-0.792027\pi$$
0.794041 0.607864i $$-0.207973\pi$$
$$978$$ − 8.00000i − 0.255812i
$$979$$ −8.00000 −0.255681
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 12.0000i 0.382935i
$$983$$ 24.0000i 0.765481i 0.923856 + 0.382741i $$0.125020\pi$$
−0.923856 + 0.382741i $$0.874980\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 2.00000 0.0635963
$$990$$ 0 0
$$991$$ −36.0000 −1.14358 −0.571789 0.820401i $$-0.693750\pi$$
−0.571789 + 0.820401i $$0.693750\pi$$
$$992$$ 8.00000i 0.254000i
$$993$$ 4.00000i 0.126936i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −4.00000 −0.126745
$$997$$ 16.0000i 0.506725i 0.967371 + 0.253363i $$0.0815366\pi$$
−0.967371 + 0.253363i $$0.918463\pi$$
$$998$$ − 36.0000i − 1.13956i
$$999$$ 6.00000 0.189832
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 3450.2.d.r.2899.1 2
5.2 odd 4 690.2.a.i.1.1 1
5.3 odd 4 3450.2.a.c.1.1 1
5.4 even 2 inner 3450.2.d.r.2899.2 2
15.2 even 4 2070.2.a.g.1.1 1
20.7 even 4 5520.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.i.1.1 1 5.2 odd 4
2070.2.a.g.1.1 1 15.2 even 4
3450.2.a.c.1.1 1 5.3 odd 4
3450.2.d.r.2899.1 2 1.1 even 1 trivial
3450.2.d.r.2899.2 2 5.4 even 2 inner
5520.2.a.d.1.1 1 20.7 even 4