# Properties

 Label 3450.2.d.u.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.u.2899.2

## $q$-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 $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 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 $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ −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 $$-0.893356\pi$$
0.944400 0.328798i $$-0.106644\pi$$
$$38$$ − 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 $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ −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 $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ −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 $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ −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 $$0.106810\pi$$
−0.944228 + 0.329293i $$0.893190\pi$$
$$84$$ −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 $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ − 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 $$0.163977\pi$$
−0.870219 + 0.492665i $$0.836023\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ − 6.00000i − 0.580042i −0.957020 0.290021i $$-0.906338\pi$$
0.957020 0.290021i $$-0.0936623\pi$$
$$108$$ 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 $$-0.728811\pi$$
0.658505 0.752577i $$-0.271189\pi$$
$$114$$ 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 $$-0.821285\pi$$
0.846484 0.532414i $$-0.178715\pi$$
$$128$$ 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 $$0.326049\pi$$
−0.519685 + 0.854358i $$0.673951\pi$$
$$138$$ 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 $$-0.948974\pi$$
0.987179 0.159617i $$-0.0510260\pi$$
$$158$$ − 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 $$-0.844270\pi$$
0.882690 0.469956i $$-0.155730\pi$$
$$164$$ −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 $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 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 $$0.224325\pi$$
−0.761781 + 0.647834i $$0.775675\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ − 22.0000i − 1.56744i −0.621117 0.783718i $$-0.713321\pi$$
0.621117 0.783718i $$-0.286679\pi$$
$$198$$ 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 $$0.386874\pi$$
−0.347960 + 0.937509i $$0.613126\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ −16.0000 −1.06430
$$227$$ 10.0000i 0.663723i 0.943328 + 0.331862i $$0.107677\pi$$
−0.943328 + 0.331862i $$0.892323\pi$$
$$228$$ − 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 $$-0.979132\pi$$
0.997852 0.0655122i $$-0.0208681\pi$$
$$234$$ −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 $$-0.940080\pi$$
0.982334 0.187135i $$-0.0599201\pi$$
$$258$$ 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 $$-0.960644\pi$$
0.992366 0.123325i $$-0.0393559\pi$$
$$264$$ −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 $$0.0971271\pi$$
−0.953807 + 0.300421i $$0.902873\pi$$
$$278$$ − 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 $$-0.687073\pi$$
0.554455 0.832214i $$-0.312927\pi$$
$$284$$ 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 $$-0.777845\pi$$
0.766179 0.642627i $$-0.222155\pi$$
$$294$$ 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 $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 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 $$0.169877\pi$$
−0.860938 + 0.508710i $$0.830123\pi$$
$$314$$ −4.00000 −0.225733
$$315$$ 0 0
$$316$$ −14.0000 −0.787562
$$317$$ 30.0000i 1.68497i 0.538721 + 0.842484i $$0.318908\pi$$
−0.538721 + 0.842484i $$0.681092\pi$$
$$318$$ 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 $$0.204518\pi$$
−0.800593 + 0.599208i $$0.795482\pi$$
$$338$$ − 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 $$-0.671133\pi$$
0.512101 0.858925i $$-0.328867\pi$$
$$348$$ − 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 $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 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 $$0.0166232\pi$$
−0.998637 + 0.0521996i $$0.983377\pi$$
$$368$$ 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 $$0.0330221\pi$$
−0.994624 + 0.103556i $$0.966978\pi$$
$$374$$ 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 $$0.0991866\pi$$
−0.951843 + 0.306586i $$0.900813\pi$$
$$384$$ −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 $$0.114267\pi$$
−0.936255 + 0.351320i $$0.885733\pi$$
$$398$$ − 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 $$0.214795\pi$$
−0.780833 + 0.624740i $$0.785205\pi$$
$$434$$ 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 $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ −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 $$0.138322\pi$$
−0.907060 + 0.421002i $$0.861678\pi$$
$$458$$ 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 $$-0.733125\pi$$
0.668644 0.743583i $$-0.266875\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −2.00000 −0.0926482
$$467$$ 30.0000i 1.38823i 0.719862 + 0.694117i $$0.244205\pi$$
−0.719862 + 0.694117i $$0.755795\pi$$
$$468$$ 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 $$-0.741826\pi$$
0.688718 0.725029i $$-0.258174\pi$$
$$488$$ 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 $$-0.785413\pi$$
0.781241 0.624229i $$-0.214587\pi$$
$$504$$ −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 $$-0.661164\pi$$
0.484955 0.874539i $$-0.338836\pi$$
$$524$$ −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 $$-0.917417\pi$$
0.966533 0.256541i $$-0.0825830\pi$$
$$548$$ − 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 $$0.0679510\pi$$
−0.977301 + 0.211857i $$0.932049\pi$$
$$558$$ − 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 $$-0.876163\pi$$
0.925272 0.379305i $$-0.123837\pi$$
$$564$$ 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 $$-0.905874\pi$$
0.956597 0.291414i $$-0.0941257\pi$$
$$578$$ − 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 $$0.164939\pi$$
−0.868726 + 0.495293i $$0.835061\pi$$
$$588$$ − 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 $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ −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 $$0.0258681\pi$$
−0.996700 + 0.0811775i $$0.974132\pi$$
$$608$$ − 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 $$0.104731\pi$$
−0.946359 + 0.323117i $$0.895269\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 12.0000 0.483494
$$617$$ − 40.0000i − 1.61034i −0.593045 0.805170i $$-0.702074\pi$$
0.593045 0.805170i $$-0.297926\pi$$
$$618$$ 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 $$0.102169\pi$$
−0.948929 + 0.315489i $$0.897831\pi$$
$$644$$ −2.00000 −0.0788110
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.0000i 0.943537i 0.881722 + 0.471769i $$0.156384\pi$$
−0.881722 + 0.471769i $$0.843616\pi$$
$$648$$ 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 $$-0.768320\pi$$
0.746611 0.665261i $$-0.231680\pi$$
$$654$$ −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 $$0.196248\pi$$
−0.815890 + 0.578208i $$0.803752\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 18.0000i 0.691796i 0.938272 + 0.345898i $$0.112426\pi$$
−0.938272 + 0.345898i $$0.887574\pi$$
$$678$$ − 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 $$-0.900972\pi$$
0.951996 0.306111i $$-0.0990280\pi$$
$$684$$ 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 $$-0.988192\pi$$
0.999312 0.0370879i $$-0.0118082\pi$$
$$728$$ 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 $$0.346843\pi$$
−0.462805 + 0.886460i $$0.653157\pi$$
$$734$$ 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 $$-0.597084\pi$$
0.300291 0.953847i $$-0.402916\pi$$
$$744$$ 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 $$0.118403\pi$$
−0.931611 + 0.363456i $$0.881597\pi$$
$$758$$ − 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 $$0.272517\pi$$
−0.655359 + 0.755318i $$0.727483\pi$$
$$774$$ −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 $$0.0455413\pi$$
−0.989783 + 0.142585i $$0.954459\pi$$
$$788$$ 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 $$-0.821693\pi$$
0.847167 0.531327i $$-0.178307\pi$$
$$798$$ − 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 $$0.188326\pi$$
−0.830026 + 0.557725i $$0.811674\pi$$
$$824$$ −10.0000 −0.348367
$$825$$ 0 0
$$826$$ −8.00000 −0.278356
$$827$$ 30.0000i 1.04320i 0.853189 + 0.521601i $$0.174665\pi$$
−0.853189 + 0.521601i $$0.825335\pi$$
$$828$$ 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 $$0.0997113\pi$$
−0.951336 + 0.308154i $$0.900289\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 6.00000 0.205076
$$857$$ 10.0000i 0.341593i 0.985306 + 0.170797i $$0.0546341\pi$$
−0.985306 + 0.170797i $$0.945366\pi$$
$$858$$ 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 $$0.304348\pi$$
−0.576681 + 0.816970i $$0.695652\pi$$
$$864$$ −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 $$-0.967699\pi$$
0.994856 0.101303i $$-0.0323011\pi$$
$$878$$ 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 $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ − 40.0000i − 1.34307i −0.740973 0.671534i $$-0.765636\pi$$
0.740973 0.671534i $$-0.234364\pi$$
$$888$$ 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 $$-0.739288\pi$$
0.682915 0.730498i $$-0.260712\pi$$
$$908$$ − 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 $$0.0104006\pi$$
−0.999466 + 0.0326686i $$0.989599\pi$$
$$938$$ 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 $$0.0837111\pi$$
−0.965618 + 0.259965i $$0.916289\pi$$
$$948$$ − 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 $$-0.872918\pi$$
0.921357 0.388718i $$-0.127082\pi$$
$$954$$ −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 $$0.172032\pi$$
−0.857475 + 0.514525i $$0.827968\pi$$
$$968$$ 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 $$0.147828\pi$$
−0.894084 + 0.447900i $$0.852172\pi$$
$$978$$ − 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 $$-0.852659\pi$$
0.894768 0.446531i $$-0.147341\pi$$
$$984$$ −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 $$0.180969\pi$$
−0.842692 + 0.538395i $$0.819031\pi$$
$$998$$ 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.2899.1 2
5.2 odd 4 3450.2.a.ba.1.1 1
5.3 odd 4 690.2.a.a.1.1 1
5.4 even 2 inner 3450.2.d.u.2899.2 2
15.8 even 4 2070.2.a.q.1.1 1
20.3 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 5.3 odd 4
2070.2.a.q.1.1 1 15.8 even 4
3450.2.a.ba.1.1 1 5.2 odd 4
3450.2.d.u.2899.1 2 1.1 even 1 trivial
3450.2.d.u.2899.2 2 5.4 even 2 inner
5520.2.a.y.1.1 1 20.3 even 4