Properties

 Label 3450.2.d.t.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.t.2899.2

$q$-expansion

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

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 $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 4.00000i − 0.852803i
$$23$$ − 1.00000i − 0.208514i
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −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 $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 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 $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ −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 $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ −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 $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ − 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 $$0.198986\pi$$
−0.810885 + 0.585206i $$0.801014\pi$$
$$74$$ 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 $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 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 $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ − 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 $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 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 $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ −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 $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ 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 $$-0.972772\pi$$
0.996344 0.0854358i $$-0.0272282\pi$$
$$138$$ − 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 $$0.130658\pi$$
−0.916932 + 0.399043i $$0.869342\pi$$
$$158$$ − 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 $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ − 8.00000i − 0.619059i −0.950890 0.309529i $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ 0 0
$$169$$ 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 $$0.178635\pi$$
−0.846619 + 0.532200i $$0.821365\pi$$
$$174$$ 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 $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 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 $$0.179960\pi$$
−0.844396 + 0.535720i $$0.820040\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ 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 $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ 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 $$0.143834\pi$$
−0.899632 + 0.436648i $$0.856166\pi$$
$$258$$ − 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 $$-0.734848\pi$$
0.672660 0.739952i $$-0.265152\pi$$
$$264$$ −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 $$0.0971271\pi$$
−0.953807 + 0.300421i $$0.902873\pi$$
$$278$$ − 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 $$-0.962067\pi$$
0.992908 0.118888i $$-0.0379328\pi$$
$$284$$ 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 $$0.222155\pi$$
−0.766179 + 0.642627i $$0.777845\pi$$
$$294$$ 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 $$-0.963587\pi$$
0.993464 0.114146i $$-0.0364132\pi$$
$$308$$ 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 $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ − 30.0000i − 1.68497i −0.538721 0.842484i $$-0.681092\pi$$
0.538721 0.842484i $$-0.318908\pi$$
$$318$$ 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 $$-0.982652\pi$$
0.998515 0.0544735i $$-0.0173480\pi$$
$$338$$ − 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 $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ − 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 $$0.359994\pi$$
−0.425797 + 0.904819i $$0.640006\pi$$
$$354$$ 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 $$0.0669552\pi$$
−0.977959 + 0.208798i $$0.933045\pi$$
$$368$$ − 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 $$-0.845694\pi$$
0.884783 0.466002i $$-0.154306\pi$$
$$374$$ 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 $$-0.695325\pi$$
0.575841 0.817562i $$-0.304675\pi$$
$$384$$ −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 $$0.0159822\pi$$
−0.998740 + 0.0501886i $$0.984018\pi$$
$$398$$ − 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 $$0.304347\pi$$
−0.576683 + 0.816968i $$0.695653\pi$$
$$434$$ 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 $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ 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 $$-0.924854\pi$$
0.972263 0.233890i $$-0.0751456\pi$$
$$458$$ 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 $$0.121256\pi$$
−0.928316 + 0.371792i $$0.878744\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ − 12.0000i − 0.555294i −0.960683 0.277647i $$-0.910445\pi$$
0.960683 0.277647i $$-0.0895545\pi$$
$$468$$ − 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 $$-0.638905\pi$$
0.422664 0.906287i $$-0.361095\pi$$
$$488$$ − 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 $$-0.649474\pi$$
0.452517 0.891756i $$-0.350526\pi$$
$$504$$ 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 $$-0.711589\pi$$
0.616844 0.787085i $$-0.288411\pi$$
$$524$$ −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 $$-0.972747\pi$$
0.996337 0.0855138i $$-0.0272532\pi$$
$$548$$ 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 $$-0.904147\pi$$
0.955002 0.296600i $$-0.0958526\pi$$
$$558$$ 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 $$-0.725879\pi$$
0.651546 0.758610i $$-0.274121\pi$$
$$564$$ 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 $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ 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 $$-0.803895\pi$$
0.816149 0.577842i $$-0.196105\pi$$
$$588$$ − 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 $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ −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 $$0.105267\pi$$
−0.945814 + 0.324710i $$0.894733\pi$$
$$608$$ 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 $$-0.881579\pi$$
0.931592 0.363507i $$-0.118421\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 18.0000i − 0.724653i −0.932051 0.362326i $$-0.881983\pi$$
0.932051 0.362326i $$-0.118017\pi$$
$$618$$ 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 $$0.129034\pi$$
−0.918955 + 0.394362i $$0.870966\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ − 8.00000i − 0.314512i −0.987558 0.157256i $$-0.949735\pi$$
0.987558 0.157256i $$-0.0502649\pi$$
$$648$$ 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 $$0.0883240\pi$$
−0.961749 + 0.273931i $$0.911676\pi$$
$$654$$ 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 $$0.0122730\pi$$
−0.999257 + 0.0385472i $$0.987727\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 38.0000i − 1.46046i −0.683202 0.730229i $$-0.739413\pi$$
0.683202 0.730229i $$-0.260587\pi$$
$$678$$ − 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 $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ −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 $$-0.904113\pi$$
0.954970 0.296704i $$-0.0958873\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 10.0000i 0.369611i
$$733$$ 6.00000i 0.221615i 0.993842 + 0.110808i $$0.0353437\pi$$
−0.993842 + 0.110808i $$0.964656\pi$$
$$734$$ 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 $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 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 $$0.106075\pi$$
−0.944986 + 0.327111i $$0.893925\pi$$
$$758$$ 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 $$0.423318\pi$$
−0.238581 + 0.971123i $$0.576682\pi$$
$$774$$ 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 $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ 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 $$0.205697\pi$$
−0.798367 + 0.602171i $$0.794303\pi$$
$$798$$ 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 $$-0.754448\pi$$
0.716919 0.697156i $$-0.245552\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ − 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 $$0.0327539\pi$$
−0.994711 + 0.102718i $$0.967246\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ − 42.0000i − 1.43469i −0.696717 0.717346i $$-0.745357\pi$$
0.696717 0.717346i $$-0.254643\pi$$
$$858$$ − 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 $$-0.816664\pi$$
0.838666 0.544646i $$-0.183336\pi$$
$$864$$ −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 $$0.319918\pi$$
−0.536044 + 0.844190i $$0.680082\pi$$
$$878$$ − 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 $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 24.0000i 0.805841i 0.915235 + 0.402921i $$0.132005\pi$$
−0.915235 + 0.402921i $$0.867995\pi$$
$$888$$ − 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 $$-0.739288\pi$$
0.682915 0.730498i $$-0.260712\pi$$
$$908$$ − 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 $$0.117003\pi$$
−0.933201 + 0.359354i $$0.882997\pi$$
$$938$$ 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 $$-0.979298\pi$$
0.997886 0.0649913i $$-0.0207020\pi$$
$$948$$ − 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 $$0.0103129\pi$$
−0.999475 + 0.0323932i $$0.989687\pi$$
$$954$$ −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 $$0.126107\pi$$
−0.922543 + 0.385894i $$0.873893\pi$$
$$968$$ 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 $$-0.765500\pi$$
0.740688 0.671850i $$-0.234500\pi$$
$$978$$ 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 $$0.125020\pi$$
−0.923856 + 0.382741i $$0.874980\pi$$
$$984$$ −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 $$-0.794476\pi$$
0.798695 0.601736i $$-0.205524\pi$$
$$998$$ 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.2899.1 2
5.2 odd 4 690.2.a.k.1.1 1
5.3 odd 4 3450.2.a.d.1.1 1
5.4 even 2 inner 3450.2.d.t.2899.2 2
15.2 even 4 2070.2.a.b.1.1 1
20.7 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 5.2 odd 4
2070.2.a.b.1.1 1 15.2 even 4
3450.2.a.d.1.1 1 5.3 odd 4
3450.2.d.t.2899.1 2 1.1 even 1 trivial
3450.2.d.t.2899.2 2 5.4 even 2 inner
5520.2.a.i.1.1 1 20.7 even 4