# Properties

 Label 3450.2.d.t.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.t.2899.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} +2.00000i q^{13} +1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +4.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} +2.00000 q^{29} +1.00000i q^{32} -4.00000i q^{33} +6.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} +4.00000i q^{43} -4.00000 q^{44} -1.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} -6.00000 q^{51} -2.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +4.00000i q^{57} +2.00000i q^{58} +4.00000 q^{59} -10.0000 q^{61} -1.00000 q^{64} +4.00000 q^{66} -12.0000i q^{67} +6.00000i q^{68} +1.00000 q^{69} -8.00000 q^{71} +1.00000i q^{72} -10.0000i q^{73} +2.00000 q^{74} +4.00000 q^{76} +2.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +4.00000i q^{83} -4.00000 q^{86} -2.00000i q^{87} -4.00000i q^{88} -18.0000 q^{89} -1.00000i q^{92} +1.00000 q^{96} +2.00000i q^{97} +7.00000i q^{98} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + 8q^{11} + 2q^{16} - 8q^{19} - 2q^{24} - 4q^{26} + 4q^{29} + 12q^{34} + 2q^{36} + 4q^{39} + 20q^{41} - 8q^{44} - 2q^{46} + 14q^{49} - 12q^{51} - 2q^{54} + 8q^{59} - 20q^{61} - 2q^{64} + 8q^{66} + 2q^{69} - 16q^{71} + 4q^{74} + 8q^{76} + 16q^{79} + 2q^{81} - 8q^{86} - 36q^{89} + 2q^{96} - 8q^{99} + O(q^{100})$$

## 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 $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000i 0.852803i
$$23$$ 1.00000i 0.208514i
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 4.00000i − 0.696311i
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 2.00000 0.320256
$$40$$ 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
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ − 2.00000i − 0.277350i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 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
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ 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
$$73$$ − 10.0000i − 1.17041i −0.810885 0.585206i $$-0.801014\pi$$
0.810885 0.585206i $$-0.198986\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 2.00000i 0.226455i
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.0000i 1.10432i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ − 2.00000i − 0.214423i
$$88$$ − 4.00000i − 0.426401i
$$89$$ −18.0000 −1.90800 −0.953998 0.299813i $$-0.903076\pi$$
−0.953998 + 0.299813i $$0.903076\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 1.00000i − 0.104257i
$$93$$ 0 0
$$94$$ 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 $$0.967625\pi$$
$$98$$ 7.00000i 0.707107i
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ −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
$$121$$ 5.00000 0.454545
$$122$$ − 10.0000i − 0.905357i
$$123$$ − 10.0000i − 0.901670i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 16.0000i − 1.41977i −0.704317 0.709885i $$-0.748747\pi$$
0.704317 0.709885i $$-0.251253\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 20.0000 1.74741 0.873704 0.486458i $$-0.161711\pi$$
0.873704 + 0.486458i $$0.161711\pi$$
$$132$$ 4.00000i 0.348155i
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 2.00000i 0.170872i 0.996344 + 0.0854358i $$0.0272282\pi$$
−0.996344 + 0.0854358i $$0.972772\pi$$
$$138$$ 1.00000i 0.0851257i
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 8.00000i − 0.671345i
$$143$$ 8.00000i 0.668994i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 10.0000 0.827606
$$147$$ − 7.00000i − 0.577350i
$$148$$ 2.00000i 0.164399i
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ 6.00000i 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ − 10.0000i − 0.798087i −0.916932 0.399043i $$-0.869342\pi$$
0.916932 0.399043i $$-0.130658\pi$$
$$158$$ 8.00000i 0.636446i
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ − 20.0000i − 1.56652i −0.621694 0.783260i $$-0.713555\pi$$
0.621694 0.783260i $$-0.286445\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ − 4.00000i − 0.304997i
$$173$$ − 14.0000i − 1.06440i −0.846619 0.532200i $$-0.821365\pi$$
0.846619 0.532200i $$-0.178635\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ − 4.00000i − 0.300658i
$$178$$ − 18.0000i − 1.34916i
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 10.0000i 0.739221i
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 24.0000i − 1.75505i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 2.00000i − 0.143963i −0.997406 0.0719816i $$-0.977068\pi$$
0.997406 0.0719816i $$-0.0229323\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ − 4.00000i − 0.284268i
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ 6.00000i 0.422159i
$$203$$ 0 0
$$204$$ 6.00000 0.420084
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 1.00000i − 0.0695048i
$$208$$ 2.00000i 0.138675i
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 8.00000i 0.548151i
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ 10.0000i 0.677285i
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ − 2.00000i − 0.134231i
$$223$$ − 16.0000i − 1.07144i −0.844396 0.535720i $$-0.820040\pi$$
0.844396 0.535720i $$-0.179960\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 2.00000i − 0.131306i
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ − 8.00000i − 0.519656i
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ 5.00000i 0.321412i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 10.0000 0.637577
$$247$$ − 8.00000i − 0.509028i
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 0 0
$$253$$ 4.00000i 0.251478i
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 14.0000i − 0.873296i −0.899632 0.436648i $$-0.856166\pi$$
0.899632 0.436648i $$-0.143834\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 20.0000i 1.23560i
$$263$$ 24.0000i 1.47990i 0.672660 + 0.739952i $$0.265152\pi$$
−0.672660 + 0.739952i $$0.734848\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 18.0000i 1.10158i
$$268$$ 12.0000i 0.733017i
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ − 6.00000i − 0.363803i
$$273$$ 0 0
$$274$$ −2.00000 −0.120824
$$275$$ 0 0
$$276$$ −1.00000 −0.0601929
$$277$$ − 10.0000i − 0.600842i −0.953807 0.300421i $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971271\pi$$
$$278$$ 20.0000i 1.19952i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ 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
$$286$$ −8.00000 −0.473050
$$287$$ 0 0
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ 10.0000i 0.585206i
$$293$$ − 22.0000i − 1.28525i −0.766179 0.642627i $$-0.777845\pi$$
0.766179 0.642627i $$-0.222155\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 4.00000i 0.232104i
$$298$$ 10.0000i 0.579284i
$$299$$ −2.00000 −0.115663
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 8.00000i 0.460348i
$$303$$ − 6.00000i − 0.344691i
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ 4.00000i 0.228292i 0.993464 + 0.114146i $$0.0364132\pi$$
−0.993464 + 0.114146i $$0.963587\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ − 2.00000i − 0.113228i
$$313$$ − 10.0000i − 0.565233i −0.959233 0.282617i $$-0.908798\pi$$
0.959233 0.282617i $$-0.0912024\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 30.0000i 1.68497i 0.538721 + 0.842484i $$0.318908\pi$$
−0.538721 + 0.842484i $$0.681092\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 24.0000i 1.33540i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 20.0000 1.10770
$$327$$ − 10.0000i − 0.553001i
$$328$$ − 10.0000i − 0.552158i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ − 4.00000i − 0.219529i
$$333$$ 2.00000i 0.109599i
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.00000i 0.108947i 0.998515 + 0.0544735i $$0.0173480\pi$$
−0.998515 + 0.0544735i $$0.982652\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.00000i 0.216295i
$$343$$ 0 0
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ 12.0000i 0.644194i 0.946707 + 0.322097i $$0.104388\pi$$
−0.946707 + 0.322097i $$0.895612\pi$$
$$348$$ 2.00000i 0.107211i
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 4.00000i 0.213201i
$$353$$ − 34.0000i − 1.80964i −0.425797 0.904819i $$-0.640006\pi$$
0.425797 0.904819i $$-0.359994\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ 18.0000 0.953998
$$357$$ 0 0
$$358$$ − 4.00000i − 0.211407i
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 2.00000i − 0.105118i
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 1.00000i 0.0521286i
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 18.0000i 0.932005i 0.884783 + 0.466002i $$0.154306\pi$$
−0.884783 + 0.466002i $$0.845694\pi$$
$$374$$ 24.0000 1.24101
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 4.00000i 0.206010i
$$378$$ 0 0
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 0 0
$$383$$ 32.0000i 1.63512i 0.575841 + 0.817562i $$0.304675\pi$$
−0.575841 + 0.817562i $$0.695325\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ − 4.00000i − 0.203331i
$$388$$ − 2.00000i − 0.101535i
$$389$$ −38.0000 −1.92668 −0.963338 0.268290i $$-0.913542\pi$$
−0.963338 + 0.268290i $$0.913542\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ − 7.00000i − 0.353553i
$$393$$ − 20.0000i − 1.00887i
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ − 2.00000i − 0.100377i −0.998740 0.0501886i $$-0.984018\pi$$
0.998740 0.0501886i $$-0.0159822\pi$$
$$398$$ 16.0000i 0.802008i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ − 12.0000i − 0.598506i
$$403$$ 0 0
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 8.00000i − 0.396545i
$$408$$ 6.00000i 0.297044i
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ 2.00000 0.0986527
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 1.00000 0.0491473
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ − 20.0000i − 0.979404i
$$418$$ − 16.0000i − 0.782586i
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 4.00000i 0.194717i
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ 12.0000i 0.580042i
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ −32.0000 −1.54139 −0.770693 0.637207i $$-0.780090\pi$$
−0.770693 + 0.637207i $$0.780090\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 34.0000i − 1.63394i −0.576683 0.816968i $$-0.695653\pi$$
0.576683 0.816968i $$-0.304347\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ − 4.00000i − 0.191346i
$$438$$ − 10.0000i − 0.477818i
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 12.0000i 0.570782i
$$443$$ 4.00000i 0.190046i 0.995475 + 0.0950229i $$0.0302924\pi$$
−0.995475 + 0.0950229i $$0.969708\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ 16.0000 0.757622
$$447$$ − 10.0000i − 0.472984i
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 40.0000 1.88353
$$452$$ − 6.00000i − 0.282216i
$$453$$ − 8.00000i − 0.375873i
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ 10.0000i 0.467780i 0.972263 + 0.233890i $$0.0751456\pi$$
−0.972263 + 0.233890i $$0.924854\pi$$
$$458$$ − 14.0000i − 0.654177i
$$459$$ 6.00000 0.280056
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ − 16.0000i − 0.743583i −0.928316 0.371792i $$-0.878744\pi$$
0.928316 0.371792i $$-0.121256\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 12.0000i 0.555294i 0.960683 + 0.277647i $$0.0895545\pi$$
−0.960683 + 0.277647i $$0.910445\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −10.0000 −0.460776
$$472$$ − 4.00000i − 0.184115i
$$473$$ 16.0000i 0.735681i
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ − 16.0000i − 0.731823i
$$479$$ −32.0000 −1.46212 −0.731059 0.682315i $$-0.760973\pi$$
−0.731059 + 0.682315i $$0.760973\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 18.0000i 0.819878i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 40.0000i 1.81257i 0.422664 + 0.906287i $$0.361095\pi$$
−0.422664 + 0.906287i $$0.638905\pi$$
$$488$$ 10.0000i 0.452679i
$$489$$ −20.0000 −0.904431
$$490$$ 0 0
$$491$$ −4.00000 −0.180517 −0.0902587 0.995918i $$-0.528769\pi$$
−0.0902587 + 0.995918i $$0.528769\pi$$
$$492$$ 10.0000i 0.450835i
$$493$$ − 12.0000i − 0.540453i
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 4.00000i 0.179244i
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ 20.0000i 0.892644i
$$503$$ 40.0000i 1.78351i 0.452517 + 0.891756i $$0.350526\pi$$
−0.452517 + 0.891756i $$0.649474\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −4.00000 −0.177822
$$507$$ − 9.00000i − 0.399704i
$$508$$ 16.0000i 0.709885i
$$509$$ 34.0000 1.50702 0.753512 0.657434i $$-0.228358\pi$$
0.753512 + 0.657434i $$0.228358\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ − 4.00000i − 0.176604i
$$514$$ 14.0000 0.617514
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ − 2.00000i − 0.0875376i
$$523$$ 36.0000i 1.57417i 0.616844 + 0.787085i $$0.288411\pi$$
−0.616844 + 0.787085i $$0.711589\pi$$
$$524$$ −20.0000 −0.873704
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 0 0
$$528$$ − 4.00000i − 0.174078i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ 20.0000i 0.866296i
$$534$$ −18.0000 −0.778936
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 4.00000i 0.172613i
$$538$$ − 14.0000i − 0.603583i
$$539$$ 28.0000 1.20605
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ 0 0
$$543$$ 2.00000i 0.0858282i
$$544$$ 6.00000 0.257248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 4.00000i 0.171028i 0.996337 + 0.0855138i $$0.0272532\pi$$
−0.996337 + 0.0855138i $$0.972747\pi$$
$$548$$ − 2.00000i − 0.0854358i
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ − 1.00000i − 0.0425628i
$$553$$ 0 0
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ 14.0000i 0.593199i 0.955002 + 0.296600i $$0.0958526\pi$$
−0.955002 + 0.296600i $$0.904147\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ −24.0000 −1.01328
$$562$$ 18.0000i 0.759284i
$$563$$ 36.0000i 1.51722i 0.651546 + 0.758610i $$0.274121\pi$$
−0.651546 + 0.758610i $$0.725879\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ 8.00000i 0.335673i
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ − 8.00000i − 0.334497i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 2.00000i 0.0829027i
$$583$$ − 24.0000i − 0.993978i
$$584$$ −10.0000 −0.413803
$$585$$ 0 0
$$586$$ 22.0000 0.908812
$$587$$ 28.0000i 1.15568i 0.816149 + 0.577842i $$0.196105\pi$$
−0.816149 + 0.577842i $$0.803895\pi$$
$$588$$ 7.00000i 0.288675i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ − 2.00000i − 0.0821995i
$$593$$ 30.0000i 1.23195i 0.787765 + 0.615976i $$0.211238\pi$$
−0.787765 + 0.615976i $$0.788762\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ − 16.0000i − 0.654836i
$$598$$ − 2.00000i − 0.0817861i
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ 0 0
$$601$$ −38.0000 −1.55005 −0.775026 0.631929i $$-0.782263\pi$$
−0.775026 + 0.631929i $$0.782263\pi$$
$$602$$ 0 0
$$603$$ 12.0000i 0.488678i
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 6.00000 0.243733
$$607$$ − 16.0000i − 0.649420i −0.945814 0.324710i $$-0.894733\pi$$
0.945814 0.324710i $$-0.105267\pi$$
$$608$$ − 4.00000i − 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 6.00000i − 0.242536i
$$613$$ 18.0000i 0.727013i 0.931592 + 0.363507i $$0.118421\pi$$
−0.931592 + 0.363507i $$0.881579\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ 8.00000i 0.320771i
$$623$$ 0 0
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 16.0000i 0.638978i
$$628$$ 10.0000i 0.399043i
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ − 8.00000i − 0.318223i
$$633$$ − 4.00000i − 0.158986i
$$634$$ −30.0000 −1.19145
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ 14.0000i 0.554700i
$$638$$ 8.00000i 0.316723i
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ − 12.0000i − 0.473602i
$$643$$ − 20.0000i − 0.788723i −0.918955 0.394362i $$-0.870966\pi$$
0.918955 0.394362i $$-0.129034\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ 8.00000i 0.314512i 0.987558 + 0.157256i $$0.0502649\pi$$
−0.987558 + 0.157256i $$0.949735\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 20.0000i 0.783260i
$$653$$ − 14.0000i − 0.547862i −0.961749 0.273931i $$-0.911676\pi$$
0.961749 0.273931i $$-0.0883240\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ 10.0000i 0.390137i
$$658$$ 0 0
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ −50.0000 −1.94477 −0.972387 0.233373i $$-0.925024\pi$$
−0.972387 + 0.233373i $$0.925024\pi$$
$$662$$ 28.0000i 1.08825i
$$663$$ − 12.0000i − 0.466041i
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 2.00000i 0.0774403i
$$668$$ − 8.00000i − 0.309529i
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ 0 0
$$673$$ − 2.00000i − 0.0770943i −0.999257 0.0385472i $$-0.987727\pi$$
0.999257 0.0385472i $$-0.0122730\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 38.0000i 1.46046i 0.683202 + 0.730229i $$0.260587\pi$$
−0.683202 + 0.730229i $$0.739413\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 0 0
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 14.0000i 0.534133i
$$688$$ 4.00000i 0.152499i
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 14.0000i 0.532200i
$$693$$ 0 0
$$694$$ −12.0000 −0.455514
$$695$$ 0 0
$$696$$ −2.00000 −0.0758098
$$697$$ − 60.0000i − 2.27266i
$$698$$ − 14.0000i − 0.529908i
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ −50.0000 −1.88847 −0.944237 0.329267i $$-0.893198\pi$$
−0.944237 + 0.329267i $$0.893198\pi$$
$$702$$ − 2.00000i − 0.0754851i
$$703$$ 8.00000i 0.301726i
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ 34.0000 1.27961
$$707$$ 0 0
$$708$$ 4.00000i 0.150329i
$$709$$ 34.0000 1.27690 0.638448 0.769665i $$-0.279577\pi$$
0.638448 + 0.769665i $$0.279577\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 18.0000i 0.674579i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 16.0000i 0.597531i
$$718$$ − 8.00000i − 0.298557i
$$719$$ 32.0000 1.19340 0.596699 0.802465i $$-0.296479\pi$$
0.596699 + 0.802465i $$0.296479\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 3.00000i − 0.111648i
$$723$$ − 18.0000i − 0.669427i
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ 16.0000i 0.593407i 0.954970 + 0.296704i $$0.0958873\pi$$
−0.954970 + 0.296704i $$0.904113\pi$$
$$728$$ 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 $$-0.964656\pi$$
0.993842 0.110808i $$-0.0353437\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ − 48.0000i − 1.76810i
$$738$$ − 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
$$741$$ −8.00000 −0.293887
$$742$$ 0 0
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −18.0000 −0.659027
$$747$$ − 4.00000i − 0.146352i
$$748$$ 24.0000i 0.877527i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 0 0
$$753$$ − 20.0000i − 0.728841i
$$754$$ −4.00000 −0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 18.0000i − 0.654221i −0.944986 0.327111i $$-0.893925\pi$$
0.944986 0.327111i $$-0.106075\pi$$
$$758$$ − 12.0000i − 0.435860i
$$759$$ 4.00000 0.145191
$$760$$ 0 0
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ − 16.0000i − 0.579619i
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −32.0000 −1.15621
$$767$$ 8.00000i 0.288863i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ −14.0000 −0.504198
$$772$$ 2.00000i 0.0719816i
$$773$$ − 54.0000i − 1.94225i −0.238581 0.971123i $$-0.576682\pi$$
0.238581 0.971123i $$-0.423318\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ 0 0
$$778$$ − 38.0000i − 1.36237i
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ 6.00000i 0.214560i
$$783$$ 2.00000i 0.0714742i
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ 20.0000 0.713376
$$787$$ 4.00000i 0.142585i 0.997455 + 0.0712923i $$0.0227123\pi$$
−0.997455 + 0.0712923i $$0.977288\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 4.00000i 0.142134i
$$793$$ − 20.0000i − 0.710221i
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ − 34.0000i − 1.20434i −0.798367 0.602171i $$-0.794303\pi$$
0.798367 0.602171i $$-0.205697\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 18.0000 0.635999
$$802$$ 10.0000i 0.353112i
$$803$$ − 40.0000i − 1.41157i
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 14.0000i 0.492823i
$$808$$ − 6.00000i − 0.211079i
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ −6.00000 −0.210042
$$817$$ − 16.0000i − 0.559769i
$$818$$ − 26.0000i − 0.909069i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −42.0000 −1.46581 −0.732905 0.680331i $$-0.761836\pi$$
−0.732905 + 0.680331i $$0.761836\pi$$
$$822$$ 2.00000i 0.0697580i
$$823$$ 40.0000i 1.39431i 0.716919 + 0.697156i $$0.245552\pi$$
−0.716919 + 0.697156i $$0.754448\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 1.00000i 0.0347524i
$$829$$ −46.0000 −1.59765 −0.798823 0.601566i $$-0.794544\pi$$
−0.798823 + 0.601566i $$0.794544\pi$$
$$830$$ 0 0
$$831$$ −10.0000 −0.346896
$$832$$ − 2.00000i − 0.0693375i
$$833$$ − 42.0000i − 1.45521i
$$834$$ 20.0000 0.692543
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ − 12.0000i − 0.414533i
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ − 2.00000i − 0.0689246i
$$843$$ − 18.0000i − 0.619953i
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ − 6.00000i − 0.206041i
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ 2.00000 0.0685591
$$852$$ − 8.00000i − 0.274075i
$$853$$ − 6.00000i − 0.205436i −0.994711 0.102718i $$-0.967246\pi$$
0.994711 0.102718i $$-0.0327539\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 42.0000i 1.43469i 0.696717 + 0.717346i $$0.254643\pi$$
−0.696717 + 0.717346i $$0.745357\pi$$
$$858$$ 8.00000i 0.273115i
$$859$$ −44.0000 −1.50126 −0.750630 0.660722i $$-0.770250\pi$$
−0.750630 + 0.660722i $$0.770250\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 32.0000i − 1.08992i
$$863$$ 32.0000i 1.08929i 0.838666 + 0.544646i $$0.183336\pi$$
−0.838666 + 0.544646i $$0.816664\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 34.0000 1.15537
$$867$$ 19.0000i 0.645274i
$$868$$ 0 0
$$869$$ 32.0000 1.08553
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ − 10.0000i − 0.338643i
$$873$$ − 2.00000i − 0.0676897i
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 10.0000 0.337869
$$877$$ − 50.0000i − 1.68838i −0.536044 0.844190i $$-0.680082\pi$$
0.536044 0.844190i $$-0.319918\pi$$
$$878$$ 8.00000i 0.269987i
$$879$$ −22.0000 −0.742042
$$880$$ 0 0
$$881$$ 10.0000 0.336909 0.168454 0.985709i $$-0.446122\pi$$
0.168454 + 0.985709i $$0.446122\pi$$
$$882$$ − 7.00000i − 0.235702i
$$883$$ − 20.0000i − 0.673054i −0.941674 0.336527i $$-0.890748\pi$$
0.941674 0.336527i $$-0.109252\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ − 24.0000i − 0.805841i −0.915235 0.402921i $$-0.867995\pi$$
0.915235 0.402921i $$-0.132005\pi$$
$$888$$ 2.00000i 0.0671156i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 16.0000i 0.535720i
$$893$$ 0 0
$$894$$ 10.0000 0.334450
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 2.00000i 0.0667781i
$$898$$ 30.0000i 1.00111i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 40.0000i 1.33185i
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 8.00000 0.265782
$$907$$ 44.0000i 1.46100i 0.682915 + 0.730498i $$0.260712\pi$$
−0.682915 + 0.730498i $$0.739288\pi$$
$$908$$ 20.0000i 0.663723i
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ 4.00000i 0.132453i
$$913$$ 16.0000i 0.529523i
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 0 0
$$918$$ 6.00000i 0.198030i
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ − 18.0000i − 0.592798i
$$923$$ − 16.0000i − 0.526646i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 16.0000 0.525793
$$927$$ 0 0
$$928$$ 2.00000i 0.0656532i
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ − 6.00000i − 0.196537i
$$933$$ − 8.00000i − 0.261908i
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ − 22.0000i − 0.718709i −0.933201 0.359354i $$-0.882997\pi$$
0.933201 0.359354i $$-0.117003\pi$$
$$938$$ 0 0
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ 14.0000 0.456387 0.228193 0.973616i $$-0.426718\pi$$
0.228193 + 0.973616i $$0.426718\pi$$
$$942$$ − 10.0000i − 0.325818i
$$943$$ 10.0000i 0.325645i
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ 4.00000i 0.129983i 0.997886 + 0.0649913i $$0.0207020\pi$$
−0.997886 + 0.0649913i $$0.979298\pi$$
$$948$$ 8.00000i 0.259828i
$$949$$ 20.0000 0.649227
$$950$$ 0 0
$$951$$ 30.0000 0.972817
$$952$$ 0 0
$$953$$ − 2.00000i − 0.0647864i −0.999475 0.0323932i $$-0.989687\pi$$
0.999475 0.0323932i $$-0.0103129\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ − 8.00000i − 0.258603i
$$958$$ − 32.0000i − 1.03387i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 4.00000i 0.128965i
$$963$$ 12.0000i 0.386695i
$$964$$ −18.0000 −0.579741
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 24.0000i − 0.771788i −0.922543 0.385894i $$-0.873893\pi$$
0.922543 0.385894i $$-0.126107\pi$$
$$968$$ − 5.00000i − 0.160706i
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ 52.0000 1.66876 0.834380 0.551190i $$-0.185826\pi$$
0.834380 + 0.551190i $$0.185826\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 0 0
$$974$$ −40.0000 −1.28168
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ − 20.0000i − 0.639529i
$$979$$ −72.0000 −2.30113
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ − 4.00000i − 0.127645i
$$983$$ − 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
$$986$$ 12.0000 0.382158
$$987$$ 0 0
$$988$$ 8.00000i 0.254514i
$$989$$ −4.00000 −0.127193
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 0 0
$$993$$ − 28.0000i − 0.888553i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −4.00000 −0.126745
$$997$$ 38.0000i 1.20347i 0.798695 + 0.601736i $$0.205524\pi$$
−0.798695 + 0.601736i $$0.794476\pi$$
$$998$$ − 20.0000i − 0.633089i
$$999$$ 2.00000 0.0632772
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.t.2899.2 2
5.2 odd 4 3450.2.a.d.1.1 1
5.3 odd 4 690.2.a.k.1.1 1
5.4 even 2 inner 3450.2.d.t.2899.1 2
15.8 even 4 2070.2.a.b.1.1 1
20.3 even 4 5520.2.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.k.1.1 1 5.3 odd 4
2070.2.a.b.1.1 1 15.8 even 4
3450.2.a.d.1.1 1 5.2 odd 4
3450.2.d.t.2899.1 2 5.4 even 2 inner
3450.2.d.t.2899.2 2 1.1 even 1 trivial
5520.2.a.i.1.1 1 20.3 even 4