# Properties

 Label 3450.2.d.u.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.u.2899.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +6.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} +4.00000 q^{19} +2.00000 q^{21} +6.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} +2.00000 q^{29} -8.00000 q^{31} +1.00000i q^{32} -6.00000i q^{33} +1.00000 q^{36} +4.00000i q^{37} +4.00000i q^{38} -2.00000 q^{39} +2.00000 q^{41} +2.00000i q^{42} -8.00000i q^{43} -6.00000 q^{44} +1.00000 q^{46} -1.00000i q^{48} +3.00000 q^{49} +2.00000i q^{52} -2.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -4.00000i q^{57} +2.00000i q^{58} +4.00000 q^{59} -8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +6.00000 q^{66} -1.00000 q^{69} -8.00000 q^{71} +1.00000i q^{72} +6.00000i q^{73} -4.00000 q^{74} -4.00000 q^{76} +12.0000i q^{77} -2.00000i q^{78} +14.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -6.00000i q^{83} -2.00000 q^{84} +8.00000 q^{86} -2.00000i q^{87} -6.00000i q^{88} +16.0000 q^{89} +4.00000 q^{91} +1.00000i q^{92} +8.00000i q^{93} +1.00000 q^{96} -2.00000i q^{97} +3.00000i q^{98} -6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + 12q^{11} - 4q^{14} + 2q^{16} + 8q^{19} + 4q^{21} - 2q^{24} + 4q^{26} + 4q^{29} - 16q^{31} + 2q^{36} - 4q^{39} + 4q^{41} - 12q^{44} + 2q^{46} + 6q^{49} - 2q^{54} + 4q^{56} + 8q^{59} - 2q^{64} + 12q^{66} - 2q^{69} - 16q^{71} - 8q^{74} - 8q^{76} + 28q^{79} + 2q^{81} - 4q^{84} + 16q^{86} + 32q^{89} + 8q^{91} + 2q^{96} - 12q^{99} + O(q^{100})$$

## 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
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 6.00000i 1.27920i
$$23$$ − 1.00000i − 0.208514i
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ − 2.00000i − 0.377964i
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 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
$$33$$ − 6.00000i − 1.04447i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 4.00000i 0.657596i 0.944400 + 0.328798i $$0.106644\pi$$
−0.944400 + 0.328798i $$0.893356\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 2.00000i 0.308607i
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000i 0.277350i
$$53$$ − 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
$$56$$ 2.00000 0.267261
$$57$$ − 4.00000i − 0.529813i
$$58$$ 2.00000i 0.262613i
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ − 8.00000i − 1.01600i
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 6.00000 0.738549
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 0 0
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 6.00000i 0.702247i 0.936329 + 0.351123i $$0.114200\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 12.0000i 1.36753i
$$78$$ − 2.00000i − 0.226455i
$$79$$ 14.0000 1.57512 0.787562 0.616236i $$-0.211343\pi$$
0.787562 + 0.616236i $$0.211343\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.00000i 0.220863i
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ − 2.00000i − 0.214423i
$$88$$ − 6.00000i − 0.639602i
$$89$$ 16.0000 1.69600 0.847998 0.529999i $$-0.177808\pi$$
0.847998 + 0.529999i $$0.177808\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 1.00000i 0.104257i
$$93$$ 8.00000i 0.829561i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 2.00000i − 0.203069i −0.994832 0.101535i $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ − 10.0000i − 0.985329i −0.870219 0.492665i $$-0.836023\pi$$
0.870219 0.492665i $$-0.163977\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ 6.00000i 0.580042i 0.957020 + 0.290021i $$0.0936623\pi$$
−0.957020 + 0.290021i $$0.906338\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −12.0000 −1.14939 −0.574696 0.818367i $$-0.694880\pi$$
−0.574696 + 0.818367i $$0.694880\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 2.00000i 0.188982i
$$113$$ 16.0000i 1.50515i 0.658505 + 0.752577i $$0.271189\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ 2.00000i 0.184900i
$$118$$ 4.00000i 0.368230i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 0 0
$$123$$ − 2.00000i − 0.180334i
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ 12.0000i 1.06483i 0.846484 + 0.532414i $$0.178715\pi$$
−0.846484 + 0.532414i $$0.821285\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 6.00000i 0.522233i
$$133$$ 8.00000i 0.693688i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 20.0000i − 1.70872i −0.519685 0.854358i $$-0.673951\pi$$
0.519685 0.854358i $$-0.326049\pi$$
$$138$$ − 1.00000i − 0.0851257i
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 8.00000i − 0.671345i
$$143$$ − 12.0000i − 1.00349i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −6.00000 −0.496564
$$147$$ − 3.00000i − 0.247436i
$$148$$ − 4.00000i − 0.328798i
$$149$$ −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 $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ − 4.00000i − 0.324443i
$$153$$ 0 0
$$154$$ −12.0000 −0.966988
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 4.00000i 0.319235i 0.987179 + 0.159617i $$0.0510260\pi$$
−0.987179 + 0.159617i $$0.948974\pi$$
$$158$$ 14.0000i 1.11378i
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ 2.00000 0.157622
$$162$$ 1.00000i 0.0785674i
$$163$$ 12.0000i 0.939913i 0.882690 + 0.469956i $$0.155730\pi$$
−0.882690 + 0.469956i $$0.844270\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ − 2.00000i − 0.154303i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 8.00000i 0.609994i
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ 6.00000 0.452267
$$177$$ − 4.00000i − 0.300658i
$$178$$ 16.0000i 1.19925i
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 4.00000 0.297318 0.148659 0.988889i $$-0.452504\pi$$
0.148659 + 0.988889i $$0.452504\pi$$
$$182$$ 4.00000i 0.296500i
$$183$$ 0 0
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 18.0000i − 1.29567i −0.761781 0.647834i $$-0.775675\pi$$
0.761781 0.647834i $$-0.224325\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 22.0000i 1.56744i 0.621117 + 0.783718i $$0.286679\pi$$
−0.621117 + 0.783718i $$0.713321\pi$$
$$198$$ − 6.00000i − 0.426401i
$$199$$ 22.0000 1.55954 0.779769 0.626067i $$-0.215336\pi$$
0.779769 + 0.626067i $$0.215336\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 2.00000i 0.140720i
$$203$$ 4.00000i 0.280745i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 10.0000 0.696733
$$207$$ 1.00000i 0.0695048i
$$208$$ − 2.00000i − 0.138675i
$$209$$ 24.0000 1.66011
$$210$$ 0 0
$$211$$ 28.0000 1.92760 0.963800 0.266627i $$-0.0859092\pi$$
0.963800 + 0.266627i $$0.0859092\pi$$
$$212$$ 2.00000i 0.137361i
$$213$$ 8.00000i 0.548151i
$$214$$ −6.00000 −0.410152
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ − 16.0000i − 1.08615i
$$218$$ − 12.0000i − 0.812743i
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 4.00000i 0.268462i
$$223$$ − 28.0000i − 1.87502i −0.347960 0.937509i $$-0.613126\pi$$
0.347960 0.937509i $$-0.386874\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ −16.0000 −1.06430
$$227$$ − 10.0000i − 0.663723i −0.943328 0.331862i $$-0.892323\pi$$
0.943328 0.331862i $$-0.107677\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ −4.00000 −0.264327 −0.132164 0.991228i $$-0.542192\pi$$
−0.132164 + 0.991228i $$0.542192\pi$$
$$230$$ 0 0
$$231$$ 12.0000 0.789542
$$232$$ − 2.00000i − 0.131306i
$$233$$ 2.00000i 0.131024i 0.997852 + 0.0655122i $$0.0208681\pi$$
−0.997852 + 0.0655122i $$0.979132\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ − 14.0000i − 0.909398i
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ 25.0000i 1.60706i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 2.00000 0.127515
$$247$$ − 8.00000i − 0.509028i
$$248$$ 8.00000i 0.508001i
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ − 6.00000i − 0.377217i
$$254$$ −12.0000 −0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000i 0.374270i 0.982334 + 0.187135i $$0.0599201\pi$$
−0.982334 + 0.187135i $$0.940080\pi$$
$$258$$ − 8.00000i − 0.498058i
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 8.00000i 0.494242i
$$263$$ 4.00000i 0.246651i 0.992366 + 0.123325i $$0.0393559\pi$$
−0.992366 + 0.123325i $$0.960644\pi$$
$$264$$ −6.00000 −0.369274
$$265$$ 0 0
$$266$$ −8.00000 −0.490511
$$267$$ − 16.0000i − 0.979184i
$$268$$ 0 0
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ 0 0
$$273$$ − 4.00000i − 0.242091i
$$274$$ 20.0000 1.20824
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ − 10.0000i − 0.600842i −0.953807 0.300421i $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971271\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −16.0000 −0.954480 −0.477240 0.878773i $$-0.658363\pi$$
−0.477240 + 0.878773i $$0.658363\pi$$
$$282$$ 0 0
$$283$$ 28.0000i 1.66443i 0.554455 + 0.832214i $$0.312927\pi$$
−0.554455 + 0.832214i $$0.687073\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ 12.0000 0.709575
$$287$$ 4.00000i 0.236113i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ − 6.00000i − 0.351123i
$$293$$ 22.0000i 1.28525i 0.766179 + 0.642627i $$0.222155\pi$$
−0.766179 + 0.642627i $$0.777845\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ 4.00000 0.232495
$$297$$ 6.00000i 0.348155i
$$298$$ − 10.0000i − 0.579284i
$$299$$ −2.00000 −0.115663
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ 20.0000i 1.15087i
$$303$$ − 2.00000i − 0.114897i
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ − 12.0000i − 0.683763i
$$309$$ −10.0000 −0.568880
$$310$$ 0 0
$$311$$ −32.0000 −1.81455 −0.907277 0.420534i $$-0.861843\pi$$
−0.907277 + 0.420534i $$0.861843\pi$$
$$312$$ 2.00000i 0.113228i
$$313$$ − 18.0000i − 1.01742i −0.860938 0.508710i $$-0.830123\pi$$
0.860938 0.508710i $$-0.169877\pi$$
$$314$$ −4.00000 −0.225733
$$315$$ 0 0
$$316$$ −14.0000 −0.787562
$$317$$ − 30.0000i − 1.68497i −0.538721 0.842484i $$-0.681092\pi$$
0.538721 0.842484i $$-0.318908\pi$$
$$318$$ − 2.00000i − 0.112154i
$$319$$ 12.0000 0.671871
$$320$$ 0 0
$$321$$ 6.00000 0.334887
$$322$$ 2.00000i 0.111456i
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ 12.0000i 0.663602i
$$328$$ − 2.00000i − 0.110432i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 6.00000i 0.329293i
$$333$$ − 4.00000i − 0.219199i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ − 22.0000i − 1.19842i −0.800593 0.599208i $$-0.795482\pi$$
0.800593 0.599208i $$-0.204518\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 16.0000 0.869001
$$340$$ 0 0
$$341$$ −48.0000 −2.59935
$$342$$ − 4.00000i − 0.216295i
$$343$$ 20.0000i 1.07990i
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 32.0000i 1.71785i 0.512101 + 0.858925i $$0.328867\pi$$
−0.512101 + 0.858925i $$0.671133\pi$$
$$348$$ 2.00000i 0.107211i
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 6.00000i 0.319801i
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ −16.0000 −0.847998
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −32.0000 −1.68890 −0.844448 0.535638i $$-0.820071\pi$$
−0.844448 + 0.535638i $$0.820071\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 4.00000i 0.210235i
$$363$$ − 25.0000i − 1.31216i
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 2.00000i − 0.104399i −0.998637 0.0521996i $$-0.983377\pi$$
0.998637 0.0521996i $$-0.0166232\pi$$
$$368$$ − 1.00000i − 0.0521286i
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ 4.00000 0.207670
$$372$$ − 8.00000i − 0.414781i
$$373$$ − 4.00000i − 0.207112i −0.994624 0.103556i $$-0.966978\pi$$
0.994624 0.103556i $$-0.0330221\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 4.00000i − 0.206010i
$$378$$ − 2.00000i − 0.102869i
$$379$$ 36.0000 1.84920 0.924598 0.380945i $$-0.124401\pi$$
0.924598 + 0.380945i $$0.124401\pi$$
$$380$$ 0 0
$$381$$ 12.0000 0.614779
$$382$$ 8.00000i 0.409316i
$$383$$ − 12.0000i − 0.613171i −0.951843 0.306586i $$-0.900813\pi$$
0.951843 0.306586i $$-0.0991866\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 18.0000 0.916176
$$387$$ 8.00000i 0.406663i
$$388$$ 2.00000i 0.101535i
$$389$$ −10.0000 −0.507020 −0.253510 0.967333i $$-0.581585\pi$$
−0.253510 + 0.967333i $$0.581585\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 3.00000i − 0.151523i
$$393$$ − 8.00000i − 0.403547i
$$394$$ −22.0000 −1.10834
$$395$$ 0 0
$$396$$ 6.00000 0.301511
$$397$$ − 14.0000i − 0.702640i −0.936255 0.351320i $$-0.885733\pi$$
0.936255 0.351320i $$-0.114267\pi$$
$$398$$ 22.0000i 1.10276i
$$399$$ 8.00000 0.400501
$$400$$ 0 0
$$401$$ 8.00000 0.399501 0.199750 0.979847i $$-0.435987\pi$$
0.199750 + 0.979847i $$0.435987\pi$$
$$402$$ 0 0
$$403$$ 16.0000i 0.797017i
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ −4.00000 −0.198517
$$407$$ 24.0000i 1.18964i
$$408$$ 0 0
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ −20.0000 −0.986527
$$412$$ 10.0000i 0.492665i
$$413$$ 8.00000i 0.393654i
$$414$$ −1.00000 −0.0491473
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ − 4.00000i − 0.195881i
$$418$$ 24.0000i 1.17388i
$$419$$ −26.0000 −1.27018 −0.635092 0.772437i $$-0.719038\pi$$
−0.635092 + 0.772437i $$0.719038\pi$$
$$420$$ 0 0
$$421$$ 24.0000 1.16969 0.584844 0.811146i $$-0.301156\pi$$
0.584844 + 0.811146i $$0.301156\pi$$
$$422$$ 28.0000i 1.36302i
$$423$$ 0 0
$$424$$ −2.00000 −0.0971286
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ − 6.00000i − 0.290021i
$$429$$ −12.0000 −0.579365
$$430$$ 0 0
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 26.0000i − 1.24948i −0.780833 0.624740i $$-0.785205\pi$$
0.780833 0.624740i $$-0.214795\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ 12.0000 0.574696
$$437$$ − 4.00000i − 0.191346i
$$438$$ 6.00000i 0.286691i
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ 4.00000i 0.190046i 0.995475 + 0.0950229i $$0.0302924\pi$$
−0.995475 + 0.0950229i $$0.969708\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ 28.0000 1.32584
$$447$$ 10.0000i 0.472984i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 0 0
$$451$$ 12.0000 0.565058
$$452$$ − 16.0000i − 0.752577i
$$453$$ − 20.0000i − 0.939682i
$$454$$ 10.0000 0.469323
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ − 18.0000i − 0.842004i −0.907060 0.421002i $$-0.861678\pi$$
0.907060 0.421002i $$-0.138322\pi$$
$$458$$ − 4.00000i − 0.186908i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ 12.0000i 0.558291i
$$463$$ 32.0000i 1.48717i 0.668644 + 0.743583i $$0.266875\pi$$
−0.668644 + 0.743583i $$0.733125\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −2.00000 −0.0926482
$$467$$ − 30.0000i − 1.38823i −0.719862 0.694117i $$-0.755795\pi$$
0.719862 0.694117i $$-0.244205\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 4.00000 0.184310
$$472$$ − 4.00000i − 0.184115i
$$473$$ − 48.0000i − 2.20704i
$$474$$ 14.0000 0.643041
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 2.00000i 0.0915737i
$$478$$ 0 0
$$479$$ 40.0000 1.82765 0.913823 0.406112i $$-0.133116\pi$$
0.913823 + 0.406112i $$0.133116\pi$$
$$480$$ 0 0
$$481$$ 8.00000 0.364769
$$482$$ 18.0000i 0.819878i
$$483$$ − 2.00000i − 0.0910032i
$$484$$ −25.0000 −1.13636
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 32.0000i 1.45006i 0.688718 + 0.725029i $$0.258174\pi$$
−0.688718 + 0.725029i $$0.741826\pi$$
$$488$$ 0 0
$$489$$ 12.0000 0.542659
$$490$$ 0 0
$$491$$ −32.0000 −1.44414 −0.722070 0.691820i $$-0.756809\pi$$
−0.722070 + 0.691820i $$0.756809\pi$$
$$492$$ 2.00000i 0.0901670i
$$493$$ 0 0
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ − 16.0000i − 0.717698i
$$498$$ − 6.00000i − 0.268866i
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 18.0000i − 0.803379i
$$503$$ 28.0000i 1.24846i 0.781241 + 0.624229i $$0.214587\pi$$
−0.781241 + 0.624229i $$0.785413\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 6.00000 0.266733
$$507$$ − 9.00000i − 0.399704i
$$508$$ − 12.0000i − 0.532414i
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ −12.0000 −0.530849
$$512$$ 1.00000i 0.0441942i
$$513$$ 4.00000i 0.176604i
$$514$$ −6.00000 −0.264649
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ − 8.00000i − 0.351500i
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −4.00000 −0.175243 −0.0876216 0.996154i $$-0.527927\pi$$
−0.0876216 + 0.996154i $$0.527927\pi$$
$$522$$ − 2.00000i − 0.0875376i
$$523$$ 40.0000i 1.74908i 0.484955 + 0.874539i $$0.338836\pi$$
−0.484955 + 0.874539i $$0.661164\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ −4.00000 −0.174408
$$527$$ 0 0
$$528$$ − 6.00000i − 0.261116i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ − 8.00000i − 0.346844i
$$533$$ − 4.00000i − 0.173259i
$$534$$ 16.0000 0.692388
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 30.0000i 1.29339i
$$539$$ 18.0000 0.775315
$$540$$ 0 0
$$541$$ −6.00000 −0.257960 −0.128980 0.991647i $$-0.541170\pi$$
−0.128980 + 0.991647i $$0.541170\pi$$
$$542$$ − 4.00000i − 0.171815i
$$543$$ − 4.00000i − 0.171656i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 4.00000 0.171184
$$547$$ 12.0000i 0.513083i 0.966533 + 0.256541i $$0.0825830\pi$$
−0.966533 + 0.256541i $$0.917417\pi$$
$$548$$ 20.0000i 0.854358i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 1.00000i 0.0425628i
$$553$$ 28.0000i 1.19068i
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ − 10.0000i − 0.423714i −0.977301 0.211857i $$-0.932049\pi$$
0.977301 0.211857i $$-0.0679510\pi$$
$$558$$ 8.00000i 0.338667i
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 16.0000i − 0.674919i
$$563$$ 18.0000i 0.758610i 0.925272 + 0.379305i $$0.123837\pi$$
−0.925272 + 0.379305i $$0.876163\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −28.0000 −1.17693
$$567$$ 2.00000i 0.0839921i
$$568$$ 8.00000i 0.335673i
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ −32.0000 −1.33916 −0.669579 0.742741i $$-0.733526\pi$$
−0.669579 + 0.742741i $$0.733526\pi$$
$$572$$ 12.0000i 0.501745i
$$573$$ − 8.00000i − 0.334205i
$$574$$ −4.00000 −0.166957
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 14.0000i 0.582828i 0.956597 + 0.291414i $$0.0941257\pi$$
−0.956597 + 0.291414i $$0.905874\pi$$
$$578$$ 17.0000i 0.707107i
$$579$$ −18.0000 −0.748054
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ − 2.00000i − 0.0829027i
$$583$$ − 12.0000i − 0.496989i
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ −22.0000 −0.908812
$$587$$ − 24.0000i − 0.990586i −0.868726 0.495293i $$-0.835061\pi$$
0.868726 0.495293i $$-0.164939\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 22.0000 0.904959
$$592$$ 4.00000i 0.164399i
$$593$$ − 6.00000i − 0.246390i −0.992382 0.123195i $$-0.960686\pi$$
0.992382 0.123195i $$-0.0393141\pi$$
$$594$$ −6.00000 −0.246183
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ − 22.0000i − 0.900400i
$$598$$ − 2.00000i − 0.0817861i
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ 0 0
$$601$$ −38.0000 −1.55005 −0.775026 0.631929i $$-0.782263\pi$$
−0.775026 + 0.631929i $$0.782263\pi$$
$$602$$ 16.0000i 0.652111i
$$603$$ 0 0
$$604$$ −20.0000 −0.813788
$$605$$ 0 0
$$606$$ 2.00000 0.0812444
$$607$$ − 4.00000i − 0.162355i −0.996700 0.0811775i $$-0.974132\pi$$
0.996700 0.0811775i $$-0.0258681\pi$$
$$608$$ 4.00000i 0.162221i
$$609$$ 4.00000 0.162088
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 16.0000i − 0.646234i −0.946359 0.323117i $$-0.895269\pi$$
0.946359 0.323117i $$-0.104731\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 12.0000 0.483494
$$617$$ 40.0000i 1.61034i 0.593045 + 0.805170i $$0.297926\pi$$
−0.593045 + 0.805170i $$0.702074\pi$$
$$618$$ − 10.0000i − 0.402259i
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ − 32.0000i − 1.28308i
$$623$$ 32.0000i 1.28205i
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ 18.0000 0.719425
$$627$$ − 24.0000i − 0.958468i
$$628$$ − 4.00000i − 0.159617i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −34.0000 −1.35352 −0.676759 0.736204i $$-0.736616\pi$$
−0.676759 + 0.736204i $$0.736616\pi$$
$$632$$ − 14.0000i − 0.556890i
$$633$$ − 28.0000i − 1.11290i
$$634$$ 30.0000 1.19145
$$635$$ 0 0
$$636$$ 2.00000 0.0793052
$$637$$ − 6.00000i − 0.237729i
$$638$$ 12.0000i 0.475085i
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ 6.00000i 0.236801i
$$643$$ − 16.0000i − 0.630978i −0.948929 0.315489i $$-0.897831\pi$$
0.948929 0.315489i $$-0.102169\pi$$
$$644$$ −2.00000 −0.0788110
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ −16.0000 −0.627089
$$652$$ − 12.0000i − 0.469956i
$$653$$ 34.0000i 1.33052i 0.746611 + 0.665261i $$0.231680\pi$$
−0.746611 + 0.665261i $$0.768320\pi$$
$$654$$ −12.0000 −0.469237
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ − 6.00000i − 0.234082i
$$658$$ 0 0
$$659$$ −30.0000 −1.16863 −0.584317 0.811525i $$-0.698638\pi$$
−0.584317 + 0.811525i $$0.698638\pi$$
$$660$$ 0 0
$$661$$ 40.0000 1.55582 0.777910 0.628376i $$-0.216280\pi$$
0.777910 + 0.628376i $$0.216280\pi$$
$$662$$ − 20.0000i − 0.777322i
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ − 2.00000i − 0.0774403i
$$668$$ 0 0
$$669$$ −28.0000 −1.08254
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 2.00000i 0.0771517i
$$673$$ − 30.0000i − 1.15642i −0.815890 0.578208i $$-0.803752\pi$$
0.815890 0.578208i $$-0.196248\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 18.0000i − 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 16.0000i 0.614476i
$$679$$ 4.00000 0.153506
$$680$$ 0 0
$$681$$ −10.0000 −0.383201
$$682$$ − 48.0000i − 1.83801i
$$683$$ 16.0000i 0.612223i 0.951996 + 0.306111i $$0.0990280\pi$$
−0.951996 + 0.306111i $$0.900972\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ 4.00000i 0.152610i
$$688$$ − 8.00000i − 0.304997i
$$689$$ −4.00000 −0.152388
$$690$$ 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
$$693$$ − 12.0000i − 0.455842i
$$694$$ −32.0000 −1.21470
$$695$$ 0 0
$$696$$ −2.00000 −0.0758098
$$697$$ 0 0
$$698$$ − 10.0000i − 0.378506i
$$699$$ 2.00000 0.0756469
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ 16.0000i 0.603451i
$$704$$ −6.00000 −0.226134
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 4.00000i 0.150435i
$$708$$ 4.00000i 0.150329i
$$709$$ −24.0000 −0.901339 −0.450669 0.892691i $$-0.648815\pi$$
−0.450669 + 0.892691i $$0.648815\pi$$
$$710$$ 0 0
$$711$$ −14.0000 −0.525041
$$712$$ − 16.0000i − 0.599625i
$$713$$ 8.00000i 0.299602i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ − 32.0000i − 1.19423i
$$719$$ −32.0000 −1.19340 −0.596699 0.802465i $$-0.703521\pi$$
−0.596699 + 0.802465i $$0.703521\pi$$
$$720$$ 0 0
$$721$$ 20.0000 0.744839
$$722$$ − 3.00000i − 0.111648i
$$723$$ − 18.0000i − 0.669427i
$$724$$ −4.00000 −0.148659
$$725$$ 0 0
$$726$$ 25.0000 0.927837
$$727$$ 2.00000i 0.0741759i 0.999312 + 0.0370879i $$0.0118082\pi$$
−0.999312 + 0.0370879i $$0.988192\pi$$
$$728$$ − 4.00000i − 0.148250i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ − 48.0000i − 1.77292i −0.462805 0.886460i $$-0.653157\pi$$
0.462805 0.886460i $$-0.346843\pi$$
$$734$$ 2.00000 0.0738213
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ 0 0
$$738$$ − 2.00000i − 0.0736210i
$$739$$ −12.0000 −0.441427 −0.220714 0.975339i $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ 4.00000i 0.146845i
$$743$$ 52.0000i 1.90769i 0.300291 + 0.953847i $$0.402916\pi$$
−0.300291 + 0.953847i $$0.597084\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ 4.00000 0.146450
$$747$$ 6.00000i 0.219529i
$$748$$ 0 0
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ 26.0000 0.948753 0.474377 0.880322i $$-0.342673\pi$$
0.474377 + 0.880322i $$0.342673\pi$$
$$752$$ 0 0
$$753$$ 18.0000i 0.655956i
$$754$$ 4.00000 0.145671
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ − 20.0000i − 0.726912i −0.931611 0.363456i $$-0.881597\pi$$
0.931611 0.363456i $$-0.118403\pi$$
$$758$$ 36.0000i 1.30758i
$$759$$ −6.00000 −0.217786
$$760$$ 0 0
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ 12.0000i 0.434714i
$$763$$ − 24.0000i − 0.868858i
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ 12.0000 0.433578
$$767$$ − 8.00000i − 0.288863i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ 18.0000i 0.647834i
$$773$$ − 42.0000i − 1.51064i −0.655359 0.755318i $$-0.727483\pi$$
0.655359 0.755318i $$-0.272517\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ 8.00000i 0.286998i
$$778$$ − 10.0000i − 0.358517i
$$779$$ 8.00000 0.286630
$$780$$ 0 0
$$781$$ −48.0000 −1.71758
$$782$$ 0 0
$$783$$ 2.00000i 0.0714742i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 8.00000 0.285351
$$787$$ − 8.00000i − 0.285169i −0.989783 0.142585i $$-0.954459\pi$$
0.989783 0.142585i $$-0.0455413\pi$$
$$788$$ − 22.0000i − 0.783718i
$$789$$ 4.00000 0.142404
$$790$$ 0 0
$$791$$ −32.0000 −1.13779
$$792$$ 6.00000i 0.213201i
$$793$$ 0 0
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ −22.0000 −0.779769
$$797$$ 30.0000i 1.06265i 0.847167 + 0.531327i $$0.178307\pi$$
−0.847167 + 0.531327i $$0.821693\pi$$
$$798$$ 8.00000i 0.283197i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −16.0000 −0.565332
$$802$$ 8.00000i 0.282490i
$$803$$ 36.0000i 1.27041i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ − 30.0000i − 1.05605i
$$808$$ − 2.00000i − 0.0703598i
$$809$$ 50.0000 1.75791 0.878953 0.476908i $$-0.158243\pi$$
0.878953 + 0.476908i $$0.158243\pi$$
$$810$$ 0 0
$$811$$ 36.0000 1.26413 0.632065 0.774915i $$-0.282207\pi$$
0.632065 + 0.774915i $$0.282207\pi$$
$$812$$ − 4.00000i − 0.140372i
$$813$$ 4.00000i 0.140286i
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 32.0000i − 1.11954i
$$818$$ 10.0000i 0.349642i
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ − 20.0000i − 0.697580i
$$823$$ − 32.0000i − 1.11545i −0.830026 0.557725i $$-0.811674\pi$$
0.830026 0.557725i $$-0.188326\pi$$
$$824$$ −10.0000 −0.348367
$$825$$ 0 0
$$826$$ −8.00000 −0.278356
$$827$$ − 30.0000i − 1.04320i −0.853189 0.521601i $$-0.825335\pi$$
0.853189 0.521601i $$-0.174665\pi$$
$$828$$ − 1.00000i − 0.0347524i
$$829$$ −26.0000 −0.903017 −0.451509 0.892267i $$-0.649114\pi$$
−0.451509 + 0.892267i $$0.649114\pi$$
$$830$$ 0 0
$$831$$ −10.0000 −0.346896
$$832$$ 2.00000i 0.0693375i
$$833$$ 0 0
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ −24.0000 −0.830057
$$837$$ − 8.00000i − 0.276520i
$$838$$ − 26.0000i − 0.898155i
$$839$$ 44.0000 1.51905 0.759524 0.650479i $$-0.225432\pi$$
0.759524 + 0.650479i $$0.225432\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 24.0000i 0.827095i
$$843$$ 16.0000i 0.551069i
$$844$$ −28.0000 −0.963800
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 50.0000i 1.71802i
$$848$$ − 2.00000i − 0.0686803i
$$849$$ 28.0000 0.960958
$$850$$ 0 0
$$851$$ 4.00000 0.137118
$$852$$ − 8.00000i − 0.274075i
$$853$$ − 18.0000i − 0.616308i −0.951336 0.308154i $$-0.900289\pi$$
0.951336 0.308154i $$-0.0997113\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 6.00000 0.205076
$$857$$ − 10.0000i − 0.341593i −0.985306 0.170797i $$-0.945366\pi$$
0.985306 0.170797i $$-0.0546341\pi$$
$$858$$ − 12.0000i − 0.409673i
$$859$$ 36.0000 1.22830 0.614152 0.789188i $$-0.289498\pi$$
0.614152 + 0.789188i $$0.289498\pi$$
$$860$$ 0 0
$$861$$ 4.00000 0.136320
$$862$$ − 36.0000i − 1.22616i
$$863$$ − 48.0000i − 1.63394i −0.576681 0.816970i $$-0.695652\pi$$
0.576681 0.816970i $$-0.304348\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 26.0000 0.883516
$$867$$ − 17.0000i − 0.577350i
$$868$$ 16.0000i 0.543075i
$$869$$ 84.0000 2.84950
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 12.0000i 0.406371i
$$873$$ 2.00000i 0.0676897i
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ −6.00000 −0.202721
$$877$$ 6.00000i 0.202606i 0.994856 + 0.101303i $$0.0323011\pi$$
−0.994856 + 0.101303i $$0.967699\pi$$
$$878$$ 0 0
$$879$$ 22.0000 0.742042
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ − 20.0000i − 0.673054i −0.941674 0.336527i $$-0.890748\pi$$
0.941674 0.336527i $$-0.109252\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 40.0000i 1.34307i 0.740973 + 0.671534i $$0.234364\pi$$
−0.740973 + 0.671534i $$0.765636\pi$$
$$888$$ − 4.00000i − 0.134231i
$$889$$ −24.0000 −0.804934
$$890$$ 0 0
$$891$$ 6.00000 0.201008
$$892$$ 28.0000i 0.937509i
$$893$$ 0 0
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ 2.00000i 0.0667781i
$$898$$ − 10.0000i − 0.333704i
$$899$$ −16.0000 −0.533630
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 12.0000i 0.399556i
$$903$$ − 16.0000i − 0.532447i
$$904$$ 16.0000 0.532152
$$905$$ 0 0
$$906$$ 20.0000 0.664455
$$907$$ 44.0000i 1.46100i 0.682915 + 0.730498i $$0.260712\pi$$
−0.682915 + 0.730498i $$0.739288\pi$$
$$908$$ 10.0000i 0.331862i
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ − 36.0000i − 1.19143i
$$914$$ 18.0000 0.595387
$$915$$ 0 0
$$916$$ 4.00000 0.132164
$$917$$ 16.0000i 0.528367i
$$918$$ 0 0
$$919$$ 30.0000 0.989609 0.494804 0.869004i $$-0.335240\pi$$
0.494804 + 0.869004i $$0.335240\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ − 14.0000i − 0.461065i
$$923$$ 16.0000i 0.526646i
$$924$$ −12.0000 −0.394771
$$925$$ 0 0
$$926$$ −32.0000 −1.05159
$$927$$ 10.0000i 0.328443i
$$928$$ 2.00000i 0.0656532i
$$929$$ −2.00000 −0.0656179 −0.0328089 0.999462i $$-0.510445\pi$$
−0.0328089 + 0.999462i $$0.510445\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ − 2.00000i − 0.0655122i
$$933$$ 32.0000i 1.04763i
$$934$$ 30.0000 0.981630
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ − 2.00000i − 0.0653372i −0.999466 0.0326686i $$-0.989599\pi$$
0.999466 0.0326686i $$-0.0104006\pi$$
$$938$$ 0 0
$$939$$ −18.0000 −0.587408
$$940$$ 0 0
$$941$$ −30.0000 −0.977972 −0.488986 0.872292i $$-0.662633\pi$$
−0.488986 + 0.872292i $$0.662633\pi$$
$$942$$ 4.00000i 0.130327i
$$943$$ − 2.00000i − 0.0651290i
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 48.0000 1.56061
$$947$$ − 16.0000i − 0.519930i −0.965618 0.259965i $$-0.916289\pi$$
0.965618 0.259965i $$-0.0837111\pi$$
$$948$$ 14.0000i 0.454699i
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ −30.0000 −0.972817
$$952$$ 0 0
$$953$$ 24.0000i 0.777436i 0.921357 + 0.388718i $$0.127082\pi$$
−0.921357 + 0.388718i $$0.872918\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 12.0000i − 0.387905i
$$958$$ 40.0000i 1.29234i
$$959$$ 40.0000 1.29167
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 8.00000i 0.257930i
$$963$$ − 6.00000i − 0.193347i
$$964$$ −18.0000 −0.579741
$$965$$ 0 0
$$966$$ 2.00000 0.0643489
$$967$$ − 32.0000i − 1.02905i −0.857475 0.514525i $$-0.827968\pi$$
0.857475 0.514525i $$-0.172032\pi$$
$$968$$ − 25.0000i − 0.803530i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 50.0000 1.60458 0.802288 0.596937i $$-0.203616\pi$$
0.802288 + 0.596937i $$0.203616\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 8.00000i 0.256468i
$$974$$ −32.0000 −1.02535
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 28.0000i − 0.895799i −0.894084 0.447900i $$-0.852172\pi$$
0.894084 0.447900i $$-0.147828\pi$$
$$978$$ 12.0000i 0.383718i
$$979$$ 96.0000 3.06817
$$980$$ 0 0
$$981$$ 12.0000 0.383131
$$982$$ − 32.0000i − 1.02116i
$$983$$ 28.0000i 0.893061i 0.894768 + 0.446531i $$0.147341\pi$$
−0.894768 + 0.446531i $$0.852659\pi$$
$$984$$ −2.00000 −0.0637577
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 8.00000i 0.254514i
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ − 8.00000i − 0.254000i
$$993$$ 20.0000i 0.634681i
$$994$$ 16.0000 0.507489
$$995$$ 0 0
$$996$$ 6.00000 0.190117
$$997$$ − 34.0000i − 1.07679i −0.842692 0.538395i $$-0.819031\pi$$
0.842692 0.538395i $$-0.180969\pi$$
$$998$$ − 36.0000i − 1.13956i
$$999$$ −4.00000 −0.126554
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.u.2899.2 2
5.2 odd 4 690.2.a.a.1.1 1
5.3 odd 4 3450.2.a.ba.1.1 1
5.4 even 2 inner 3450.2.d.u.2899.1 2
15.2 even 4 2070.2.a.q.1.1 1
20.7 even 4 5520.2.a.y.1.1 1

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