# Properties

 Label 3450.2.d.a.2899.2 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 2899.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3450.2899 Dual form 3450.2.d.a.2899.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -6.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} -4.00000i q^{22} +1.00000i q^{23} +1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} +6.00000 q^{29} -8.00000 q^{31} +1.00000i q^{32} -4.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} -4.00000i q^{38} +6.00000 q^{39} +10.0000 q^{41} +4.00000i q^{43} +4.00000 q^{44} -1.00000 q^{46} +8.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} -6.00000 q^{51} +6.00000i q^{52} -14.0000i q^{53} +1.00000 q^{54} -4.00000i q^{57} +6.00000i q^{58} +10.0000 q^{61} -8.00000i q^{62} -1.00000 q^{64} +4.00000 q^{66} -4.00000i q^{67} -6.00000i q^{68} -1.00000 q^{69} +8.00000 q^{71} +1.00000i q^{72} +2.00000i q^{73} +6.00000 q^{74} +4.00000 q^{76} +6.00000i q^{78} +12.0000 q^{79} +1.00000 q^{81} +10.0000i q^{82} -16.0000i q^{83} -4.00000 q^{86} +6.00000i q^{87} +4.00000i q^{88} +2.00000 q^{89} -1.00000i q^{92} -8.00000i q^{93} -8.00000 q^{94} -1.00000 q^{96} +14.0000i q^{97} +7.00000i q^{98} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} - 8q^{11} + 2q^{16} - 8q^{19} + 2q^{24} + 12q^{26} + 12q^{29} - 16q^{31} - 12q^{34} + 2q^{36} + 12q^{39} + 20q^{41} + 8q^{44} - 2q^{46} + 14q^{49} - 12q^{51} + 2q^{54} + 20q^{61} - 2q^{64} + 8q^{66} - 2q^{69} + 16q^{71} + 12q^{74} + 8q^{76} + 24q^{79} + 2q^{81} - 8q^{86} + 4q^{89} - 16q^{94} - 2q^{96} + 8q^{99} + O(q^{100})$$

## 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
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ − 6.00000i − 1.66410i −0.554700 0.832050i $$-0.687167\pi$$
0.554700 0.832050i $$-0.312833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ − 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
$$22$$ − 4.00000i − 0.852803i
$$23$$ 1.00000i 0.208514i
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 6.00000 1.17670
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 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
$$33$$ − 4.00000i − 0.696311i
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 8.00000i 1.16692i 0.812142 + 0.583460i $$0.198301\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ 6.00000i 0.832050i
$$53$$ − 14.0000i − 1.92305i −0.274721 0.961524i $$-0.588586\pi$$
0.274721 0.961524i $$-0.411414\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 4.00000i − 0.529813i
$$58$$ 6.00000i 0.787839i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ − 8.00000i − 1.01600i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ − 6.00000i − 0.727607i
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 6.00000i 0.679366i
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.0000i 1.10432i
$$83$$ − 16.0000i − 1.75623i −0.478451 0.878114i $$-0.658802\pi$$
0.478451 0.878114i $$-0.341198\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 6.00000i 0.643268i
$$88$$ 4.00000i 0.426401i
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 1.00000i − 0.104257i
$$93$$ − 8.00000i − 0.829561i
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 14.0000i 1.42148i 0.703452 + 0.710742i $$0.251641\pi$$
−0.703452 + 0.710742i $$0.748359\pi$$
$$98$$ 7.00000i 0.707107i
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ − 16.0000i − 1.57653i −0.615338 0.788263i $$-0.710980\pi$$
0.615338 0.788263i $$-0.289020\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 14.0000 1.35980
$$107$$ − 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
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 6.00000i 0.554700i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000i 0.905357i
$$123$$ 10.0000i 0.901670i
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 12.0000i − 1.06483i −0.846484 0.532414i $$-0.821285\pi$$
0.846484 0.532414i $$-0.178715\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 4.00000i 0.348155i
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 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
$$141$$ −8.00000 −0.673722
$$142$$ 8.00000i 0.671345i
$$143$$ 24.0000i 2.00698i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 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
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −6.00000 −0.480384
$$157$$ − 6.00000i − 0.478852i −0.970915 0.239426i $$-0.923041\pi$$
0.970915 0.239426i $$-0.0769593\pi$$
$$158$$ 12.0000i 0.954669i
$$159$$ 14.0000 1.11027
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ 16.0000i 1.23812i 0.785345 + 0.619059i $$0.212486\pi$$
−0.785345 + 0.619059i $$0.787514\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ − 4.00000i − 0.304997i
$$173$$ 18.0000i 1.36851i 0.729241 + 0.684257i $$0.239873\pi$$
−0.729241 + 0.684257i $$0.760127\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ 2.00000i 0.149906i
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ 10.0000i 0.739221i
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ − 24.0000i − 1.75505i
$$188$$ − 8.00000i − 0.583460i
$$189$$ 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
$$193$$ 10.0000i 0.719816i 0.932988 + 0.359908i $$0.117192\pi$$
−0.932988 + 0.359908i $$0.882808\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ − 26.0000i − 1.85242i −0.377004 0.926212i $$-0.623046\pi$$
0.377004 0.926212i $$-0.376954\pi$$
$$198$$ 4.00000i 0.284268i
$$199$$ 12.0000 0.850657 0.425329 0.905039i $$-0.360158\pi$$
0.425329 + 0.905039i $$0.360158\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 18.0000i 1.26648i
$$203$$ 0 0
$$204$$ 6.00000 0.420084
$$205$$ 0 0
$$206$$ 16.0000 1.11477
$$207$$ − 1.00000i − 0.0695048i
$$208$$ − 6.00000i − 0.416025i
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 14.0000i 0.961524i
$$213$$ 8.00000i 0.548151i
$$214$$ 8.00000 0.546869
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ − 2.00000i − 0.135457i
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 36.0000 2.42162
$$222$$ 6.00000i 0.402694i
$$223$$ − 20.0000i − 1.33930i −0.742677 0.669650i $$-0.766444\pi$$
0.742677 0.669650i $$-0.233556\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ − 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
$$232$$ − 6.00000i − 0.393919i
$$233$$ 22.0000i 1.44127i 0.693316 + 0.720634i $$0.256149\pi$$
−0.693316 + 0.720634i $$0.743851\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 12.0000i 0.779484i
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 5.00000i 0.321412i
$$243$$ 1.00000i 0.0641500i
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ −10.0000 −0.637577
$$247$$ 24.0000i 1.52708i
$$248$$ 8.00000i 0.508001i
$$249$$ 16.0000 1.01396
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ − 4.00000i − 0.251478i
$$254$$ 12.0000 0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 6.00000i − 0.374270i −0.982334 0.187135i $$-0.940080\pi$$
0.982334 0.187135i $$-0.0599201\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 8.00000i 0.494242i
$$263$$ − 32.0000i − 1.97320i −0.163144 0.986602i $$-0.552164\pi$$
0.163144 0.986602i $$-0.447836\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 2.00000i 0.122398i
$$268$$ 4.00000i 0.244339i
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 6.00000i 0.363803i
$$273$$ 0 0
$$274$$ 2.00000 0.120824
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ 30.0000i 1.80253i 0.433273 + 0.901263i $$0.357359\pi$$
−0.433273 + 0.901263i $$0.642641\pi$$
$$278$$ − 12.0000i − 0.719712i
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ − 8.00000i − 0.476393i
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ −24.0000 −1.41915
$$287$$ 0 0
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −14.0000 −0.820695
$$292$$ − 2.00000i − 0.117041i
$$293$$ − 14.0000i − 0.817889i −0.912559 0.408944i $$-0.865897\pi$$
0.912559 0.408944i $$-0.134103\pi$$
$$294$$ −7.00000 −0.408248
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ 4.00000i 0.232104i
$$298$$ − 10.0000i − 0.579284i
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 16.0000i 0.920697i
$$303$$ 18.0000i 1.03407i
$$304$$ −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
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ − 6.00000i − 0.339683i
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ 6.00000 0.338600
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 14.0000i 0.785081i
$$319$$ −24.0000 −1.34374
$$320$$ 0 0
$$321$$ 8.00000 0.446516
$$322$$ 0 0
$$323$$ − 24.0000i − 1.33540i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ − 2.00000i − 0.110600i
$$328$$ − 10.0000i − 0.552158i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 16.0000i 0.878114i
$$333$$ 6.00000i 0.328798i
$$334$$ −16.0000 −0.875481
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ − 23.0000i − 1.25104i
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ 32.0000 1.73290
$$342$$ 4.00000i 0.216295i
$$343$$ 0 0
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ 28.0000i 1.50312i 0.659665 + 0.751559i $$0.270698\pi$$
−0.659665 + 0.751559i $$0.729302\pi$$
$$348$$ − 6.00000i − 0.321634i
$$349$$ 34.0000 1.81998 0.909989 0.414632i $$-0.136090\pi$$
0.909989 + 0.414632i $$0.136090\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ − 4.00000i − 0.213201i
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 6.00000i − 0.315353i
$$363$$ 5.00000i 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ 32.0000i 1.67039i 0.549957 + 0.835193i $$0.314644\pi$$
−0.549957 + 0.835193i $$0.685356\pi$$
$$368$$ 1.00000i 0.0521286i
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 8.00000i 0.414781i
$$373$$ − 18.0000i − 0.932005i −0.884783 0.466002i $$-0.845694\pi$$
0.884783 0.466002i $$-0.154306\pi$$
$$374$$ 24.0000 1.24101
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ − 36.0000i − 1.85409i
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ 12.0000 0.614779
$$382$$ − 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
$$386$$ −10.0000 −0.508987
$$387$$ − 4.00000i − 0.203331i
$$388$$ − 14.0000i − 0.710742i
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ −6.00000 −0.303433
$$392$$ − 7.00000i − 0.353553i
$$393$$ 8.00000i 0.403547i
$$394$$ 26.0000 1.30986
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ − 2.00000i − 0.100377i −0.998740 0.0501886i $$-0.984018\pi$$
0.998740 0.0501886i $$-0.0159822\pi$$
$$398$$ 12.0000i 0.601506i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 4.00000i 0.199502i
$$403$$ 48.0000i 2.39105i
$$404$$ −18.0000 −0.895533
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 24.0000i 1.18964i
$$408$$ 6.00000i 0.297044i
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ 2.00000 0.0986527
$$412$$ 16.0000i 0.788263i
$$413$$ 0 0
$$414$$ 1.00000 0.0491473
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ − 12.0000i − 0.587643i
$$418$$ 16.0000i 0.782586i
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −38.0000 −1.85201 −0.926003 0.377515i $$-0.876779\pi$$
−0.926003 + 0.377515i $$0.876779\pi$$
$$422$$ − 20.0000i − 0.973585i
$$423$$ − 8.00000i − 0.388973i
$$424$$ −14.0000 −0.679900
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ 8.00000i 0.386695i
$$429$$ −24.0000 −1.15873
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 2.00000i 0.0961139i 0.998845 + 0.0480569i $$0.0153029\pi$$
−0.998845 + 0.0480569i $$0.984697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ − 4.00000i − 0.191346i
$$438$$ − 2.00000i − 0.0955637i
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 36.0000i 1.71235i
$$443$$ 12.0000i 0.570137i 0.958507 + 0.285069i $$0.0920164\pi$$
−0.958507 + 0.285069i $$0.907984\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ 0 0
$$446$$ 20.0000 0.947027
$$447$$ − 10.0000i − 0.472984i
$$448$$ 0 0
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ 14.0000i 0.658505i
$$453$$ 16.0000i 0.751746i
$$454$$ 8.00000 0.375459
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ − 10.0000i − 0.467780i −0.972263 0.233890i $$-0.924854\pi$$
0.972263 0.233890i $$-0.0751456\pi$$
$$458$$ 14.0000i 0.654177i
$$459$$ 6.00000 0.280056
$$460$$ 0 0
$$461$$ 2.00000 0.0931493 0.0465746 0.998915i $$-0.485169\pi$$
0.0465746 + 0.998915i $$0.485169\pi$$
$$462$$ 0 0
$$463$$ − 20.0000i − 0.929479i −0.885448 0.464739i $$-0.846148\pi$$
0.885448 0.464739i $$-0.153852\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ − 8.00000i − 0.370196i −0.982720 0.185098i $$-0.940740\pi$$
0.982720 0.185098i $$-0.0592602\pi$$
$$468$$ − 6.00000i − 0.277350i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 6.00000 0.276465
$$472$$ 0 0
$$473$$ − 16.0000i − 0.735681i
$$474$$ −12.0000 −0.551178
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 14.0000i 0.641016i
$$478$$ 8.00000i 0.365911i
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ −36.0000 −1.64146
$$482$$ 2.00000i 0.0910975i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 20.0000i − 0.906287i −0.891438 0.453143i $$-0.850303\pi$$
0.891438 0.453143i $$-0.149697\pi$$
$$488$$ − 10.0000i − 0.452679i
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ − 10.0000i − 0.450835i
$$493$$ 36.0000i 1.62136i
$$494$$ −24.0000 −1.07981
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ 16.0000i 0.716977i
$$499$$ −12.0000 −0.537194 −0.268597 0.963253i $$-0.586560\pi$$
−0.268597 + 0.963253i $$0.586560\pi$$
$$500$$ 0 0
$$501$$ −16.0000 −0.714827
$$502$$ − 12.0000i − 0.535586i
$$503$$ − 24.0000i − 1.07011i −0.844818 0.535054i $$-0.820291\pi$$
0.844818 0.535054i $$-0.179709\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 4.00000 0.177822
$$507$$ − 23.0000i − 1.02147i
$$508$$ 12.0000i 0.532414i
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ 4.00000i 0.176604i
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ − 32.0000i − 1.40736i
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −42.0000 −1.84005 −0.920027 0.391856i $$-0.871833\pi$$
−0.920027 + 0.391856i $$0.871833\pi$$
$$522$$ − 6.00000i − 0.262613i
$$523$$ − 20.0000i − 0.874539i −0.899331 0.437269i $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ − 48.0000i − 2.09091i
$$528$$ − 4.00000i − 0.174078i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 60.0000i − 2.59889i
$$534$$ −2.00000 −0.0865485
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ 0 0
$$538$$ − 10.0000i − 0.431131i
$$539$$ −28.0000 −1.20605
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ − 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
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ 1.00000i 0.0425628i
$$553$$ 0 0
$$554$$ −30.0000 −1.27458
$$555$$ 0 0
$$556$$ 12.0000 0.508913
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ 8.00000i 0.338667i
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 24.0000 1.01328
$$562$$ 30.0000i 1.26547i
$$563$$ − 8.00000i − 0.337160i −0.985688 0.168580i $$-0.946082\pi$$
0.985688 0.168580i $$-0.0539181\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ − 8.00000i − 0.335673i
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ − 24.0000i − 1.00349i
$$573$$ − 8.00000i − 0.334205i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 34.0000i − 1.41544i −0.706494 0.707719i $$-0.749724\pi$$
0.706494 0.707719i $$-0.250276\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ −10.0000 −0.415586
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 14.0000i − 0.580319i
$$583$$ 56.0000i 2.31928i
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ 4.00000i 0.165098i 0.996587 + 0.0825488i $$0.0263060\pi$$
−0.996587 + 0.0825488i $$0.973694\pi$$
$$588$$ − 7.00000i − 0.288675i
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ 26.0000 1.06950
$$592$$ − 6.00000i − 0.246598i
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 12.0000i 0.491127i
$$598$$ 6.00000i 0.245358i
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ 4.00000i 0.162893i
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ −18.0000 −0.731200
$$607$$ 28.0000i 1.13648i 0.822861 + 0.568242i $$0.192376\pi$$
−0.822861 + 0.568242i $$0.807624\pi$$
$$608$$ − 4.00000i − 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 48.0000 1.94187
$$612$$ 6.00000i 0.242536i
$$613$$ 14.0000i 0.565455i 0.959200 + 0.282727i $$0.0912392\pi$$
−0.959200 + 0.282727i $$0.908761\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 18.0000i − 0.724653i −0.932051 0.362326i $$-0.881983\pi$$
0.932051 0.362326i $$-0.118017\pi$$
$$618$$ 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
$$622$$ − 8.00000i − 0.320771i
$$623$$ 0 0
$$624$$ 6.00000 0.240192
$$625$$ 0 0
$$626$$ −10.0000 −0.399680
$$627$$ 16.0000i 0.638978i
$$628$$ 6.00000i 0.239426i
$$629$$ 36.0000 1.43541
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ − 12.0000i − 0.477334i
$$633$$ − 20.0000i − 0.794929i
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ −14.0000 −0.555136
$$637$$ − 42.0000i − 1.66410i
$$638$$ − 24.0000i − 0.950169i
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 8.00000i 0.315735i
$$643$$ − 44.0000i − 1.73519i −0.497271 0.867595i $$-0.665665\pi$$
0.497271 0.867595i $$-0.334335\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 4.00000i − 0.156652i
$$653$$ − 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
$$656$$ 10.0000 0.390434
$$657$$ − 2.00000i − 0.0780274i
$$658$$ 0 0
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ 34.0000 1.32245 0.661223 0.750189i $$-0.270038\pi$$
0.661223 + 0.750189i $$0.270038\pi$$
$$662$$ − 20.0000i − 0.777322i
$$663$$ 36.0000i 1.39812i
$$664$$ −16.0000 −0.620920
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ 6.00000i 0.232321i
$$668$$ − 16.0000i − 0.619059i
$$669$$ 20.0000 0.773245
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ 0 0
$$673$$ − 22.0000i − 0.848038i −0.905653 0.424019i $$-0.860619\pi$$
0.905653 0.424019i $$-0.139381\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 6.00000i 0.230599i 0.993331 + 0.115299i $$0.0367827\pi$$
−0.993331 + 0.115299i $$0.963217\pi$$
$$678$$ 14.0000i 0.537667i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 8.00000 0.306561
$$682$$ 32.0000i 1.22534i
$$683$$ 44.0000i 1.68361i 0.539779 + 0.841807i $$0.318508\pi$$
−0.539779 + 0.841807i $$0.681492\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 14.0000i 0.534133i
$$688$$ 4.00000i 0.152499i
$$689$$ −84.0000 −3.20015
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ − 18.0000i − 0.684257i
$$693$$ 0 0
$$694$$ −28.0000 −1.06287
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ 60.0000i 2.27266i
$$698$$ 34.0000i 1.28692i
$$699$$ −22.0000 −0.832116
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ − 6.00000i − 0.226455i
$$703$$ 24.0000i 0.905177i
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ − 2.00000i − 0.0749532i
$$713$$ − 8.00000i − 0.299602i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 8.00000i 0.298765i
$$718$$ 8.00000i 0.298557i
$$719$$ 40.0000 1.49175 0.745874 0.666087i $$-0.232032\pi$$
0.745874 + 0.666087i $$0.232032\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 3.00000i − 0.111648i
$$723$$ 2.00000i 0.0743808i
$$724$$ 6.00000 0.222988
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ − 8.00000i − 0.296704i −0.988935 0.148352i $$-0.952603\pi$$
0.988935 0.148352i $$-0.0473968\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ − 10.0000i − 0.369611i
$$733$$ 6.00000i 0.221615i 0.993842 + 0.110808i $$0.0353437\pi$$
−0.993842 + 0.110808i $$0.964656\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ 16.0000i 0.589368i
$$738$$ − 10.0000i − 0.368105i
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 0 0
$$741$$ −24.0000 −0.881662
$$742$$ 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
$$746$$ 18.0000 0.659027
$$747$$ 16.0000i 0.585409i
$$748$$ 24.0000i 0.877527i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −28.0000 −1.02173 −0.510867 0.859660i $$-0.670676\pi$$
−0.510867 + 0.859660i $$0.670676\pi$$
$$752$$ 8.00000i 0.291730i
$$753$$ − 12.0000i − 0.437304i
$$754$$ 36.0000 1.31104
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 54.0000i − 1.96266i −0.192323 0.981332i $$-0.561602\pi$$
0.192323 0.981332i $$-0.438398\pi$$
$$758$$ − 4.00000i − 0.145287i
$$759$$ 4.00000 0.145191
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 12.0000i 0.434714i
$$763$$ 0 0
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ 0 0
$$768$$ 1.00000i 0.0360844i
$$769$$ 6.00000 0.216366 0.108183 0.994131i $$-0.465497\pi$$
0.108183 + 0.994131i $$0.465497\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ − 10.0000i − 0.359908i
$$773$$ 2.00000i 0.0719350i 0.999353 + 0.0359675i $$0.0114513\pi$$
−0.999353 + 0.0359675i $$0.988549\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ 14.0000 0.502571
$$777$$ 0 0
$$778$$ 6.00000i 0.215110i
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ − 6.00000i − 0.214560i
$$783$$ − 6.00000i − 0.214423i
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ −8.00000 −0.285351
$$787$$ − 36.0000i − 1.28326i −0.767014 0.641631i $$-0.778258\pi$$
0.767014 0.641631i $$-0.221742\pi$$
$$788$$ 26.0000i 0.926212i
$$789$$ 32.0000 1.13923
$$790$$ 0 0
$$791$$ 0 0
$$792$$ − 4.00000i − 0.142134i
$$793$$ − 60.0000i − 2.13066i
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ −12.0000 −0.425329
$$797$$ − 18.0000i − 0.637593i −0.947823 0.318796i $$-0.896721\pi$$
0.947823 0.318796i $$-0.103279\pi$$
$$798$$ 0 0
$$799$$ −48.0000 −1.69812
$$800$$ 0 0
$$801$$ −2.00000 −0.0706665
$$802$$ 6.00000i 0.211867i
$$803$$ − 8.00000i − 0.282314i
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ −48.0000 −1.69073
$$807$$ − 10.0000i − 0.352017i
$$808$$ − 18.0000i − 0.633238i
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 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
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ −6.00000 −0.210042
$$817$$ − 16.0000i − 0.559769i
$$818$$ − 10.0000i − 0.349642i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −46.0000 −1.60541 −0.802706 0.596376i $$-0.796607\pi$$
−0.802706 + 0.596376i $$0.796607\pi$$
$$822$$ 2.00000i 0.0697580i
$$823$$ 20.0000i 0.697156i 0.937280 + 0.348578i $$0.113335\pi$$
−0.937280 + 0.348578i $$0.886665\pi$$
$$824$$ −16.0000 −0.557386
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 8.00000i 0.278187i 0.990279 + 0.139094i $$0.0444189\pi$$
−0.990279 + 0.139094i $$0.955581\pi$$
$$828$$ 1.00000i 0.0347524i
$$829$$ 18.0000 0.625166 0.312583 0.949890i $$-0.398806\pi$$
0.312583 + 0.949890i $$0.398806\pi$$
$$830$$ 0 0
$$831$$ −30.0000 −1.04069
$$832$$ 6.00000i 0.208013i
$$833$$ 42.0000i 1.45521i
$$834$$ 12.0000 0.415526
$$835$$ 0 0
$$836$$ −16.0000 −0.553372
$$837$$ 8.00000i 0.276520i
$$838$$ 12.0000i 0.414533i
$$839$$ 48.0000 1.65714 0.828572 0.559883i $$-0.189154\pi$$
0.828572 + 0.559883i $$0.189154\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 38.0000i − 1.30957i
$$843$$ 30.0000i 1.03325i
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ 0 0
$$848$$ − 14.0000i − 0.480762i
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ 6.00000 0.205677
$$852$$ − 8.00000i − 0.274075i
$$853$$ − 22.0000i − 0.753266i −0.926363 0.376633i $$-0.877082\pi$$
0.926363 0.376633i $$-0.122918\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −8.00000 −0.273434
$$857$$ 42.0000i 1.43469i 0.696717 + 0.717346i $$0.254643\pi$$
−0.696717 + 0.717346i $$0.745357\pi$$
$$858$$ − 24.0000i − 0.819346i
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 8.00000i 0.272481i
$$863$$ 24.0000i 0.816970i 0.912765 + 0.408485i $$0.133943\pi$$
−0.912765 + 0.408485i $$0.866057\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ − 19.0000i − 0.645274i
$$868$$ 0 0
$$869$$ −48.0000 −1.62829
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 2.00000i 0.0677285i
$$873$$ − 14.0000i − 0.473828i
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ − 2.00000i − 0.0675352i −0.999430 0.0337676i $$-0.989249\pi$$
0.999430 0.0337676i $$-0.0107506\pi$$
$$878$$ 24.0000i 0.809961i
$$879$$ 14.0000 0.472208
$$880$$ 0 0
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ − 7.00000i − 0.235702i
$$883$$ 44.0000i 1.48072i 0.672212 + 0.740359i $$0.265344\pi$$
−0.672212 + 0.740359i $$0.734656\pi$$
$$884$$ −36.0000 −1.21081
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ − 8.00000i − 0.268614i −0.990940 0.134307i $$-0.957119\pi$$
0.990940 0.134307i $$-0.0428808\pi$$
$$888$$ − 6.00000i − 0.201347i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 20.0000i 0.669650i
$$893$$ − 32.0000i − 1.07084i
$$894$$ 10.0000 0.334450
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 6.00000i 0.200334i
$$898$$ − 10.0000i − 0.333704i
$$899$$ −48.0000 −1.60089
$$900$$ 0 0
$$901$$ 84.0000 2.79845
$$902$$ − 40.0000i − 1.33185i
$$903$$ 0 0
$$904$$ −14.0000 −0.465633
$$905$$ 0 0
$$906$$ −16.0000 −0.531564
$$907$$ 4.00000i 0.132818i 0.997792 + 0.0664089i $$0.0211542\pi$$
−0.997792 + 0.0664089i $$0.978846\pi$$
$$908$$ 8.00000i 0.265489i
$$909$$ −18.0000 −0.597022
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ 64.0000i 2.11809i
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ 6.00000i 0.198030i
$$919$$ −12.0000 −0.395843 −0.197922 0.980218i $$-0.563419\pi$$
−0.197922 + 0.980218i $$0.563419\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ 2.00000i 0.0658665i
$$923$$ − 48.0000i − 1.57994i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 20.0000 0.657241
$$927$$ 16.0000i 0.525509i
$$928$$ 6.00000i 0.196960i
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ − 22.0000i − 0.720634i
$$933$$ − 8.00000i − 0.261908i
$$934$$ 8.00000 0.261768
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ − 10.0000i − 0.326686i −0.986569 0.163343i $$-0.947772\pi$$
0.986569 0.163343i $$-0.0522277\pi$$
$$938$$ 0 0
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ −14.0000 −0.456387 −0.228193 0.973616i $$-0.573282\pi$$
−0.228193 + 0.973616i $$0.573282\pi$$
$$942$$ 6.00000i 0.195491i
$$943$$ 10.0000i 0.325645i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ 20.0000i 0.649913i 0.945729 + 0.324956i $$0.105350\pi$$
−0.945729 + 0.324956i $$0.894650\pi$$
$$948$$ − 12.0000i − 0.389742i
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ − 22.0000i − 0.712650i −0.934362 0.356325i $$-0.884030\pi$$
0.934362 0.356325i $$-0.115970\pi$$
$$954$$ −14.0000 −0.453267
$$955$$ 0 0
$$956$$ −8.00000 −0.258738
$$957$$ − 24.0000i − 0.775810i
$$958$$ − 24.0000i − 0.775405i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 36.0000i − 1.16069i
$$963$$ 8.00000i 0.257796i
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 28.0000i − 0.900419i −0.892923 0.450210i $$-0.851349\pi$$
0.892923 0.450210i $$-0.148651\pi$$
$$968$$ − 5.00000i − 0.160706i
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 0 0
$$974$$ 20.0000 0.640841
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ − 2.00000i − 0.0639857i −0.999488 0.0319928i $$-0.989815\pi$$
0.999488 0.0319928i $$-0.0101854\pi$$
$$978$$ − 4.00000i − 0.127906i
$$979$$ −8.00000 −0.255681
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ 10.0000 0.318788
$$985$$ 0 0
$$986$$ −36.0000 −1.14647
$$987$$ 0 0
$$988$$ − 24.0000i − 0.763542i
$$989$$ −4.00000 −0.127193
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ − 8.00000i − 0.254000i
$$993$$ − 20.0000i − 0.634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −16.0000 −0.506979
$$997$$ 22.0000i 0.696747i 0.937356 + 0.348373i $$0.113266\pi$$
−0.937356 + 0.348373i $$0.886734\pi$$
$$998$$ − 12.0000i − 0.379853i
$$999$$ −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.a.2899.2 2
5.2 odd 4 690.2.a.e.1.1 1
5.3 odd 4 3450.2.a.p.1.1 1
5.4 even 2 inner 3450.2.d.a.2899.1 2
15.2 even 4 2070.2.a.r.1.1 1
20.7 even 4 5520.2.a.f.1.1 1

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