Properties

 Label 3450.2.d.e.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.e.2899.2

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -1.00000i q^{23} +1.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} -6.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} -2.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} +6.00000 q^{39} -6.00000 q^{41} -8.00000i q^{43} -1.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} +6.00000i q^{58} +4.00000 q^{59} -6.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} -8.00000i q^{67} +2.00000i q^{68} -1.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} +10.0000i q^{73} -10.0000 q^{74} -6.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -8.00000i q^{83} -8.00000 q^{86} +6.00000i q^{87} +6.00000 q^{89} +1.00000i q^{92} -8.00000i q^{93} -8.00000 q^{94} -1.00000 q^{96} -18.0000i q^{97} -7.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + 2q^{16} + 2q^{24} + 12q^{26} - 12q^{29} + 16q^{31} - 4q^{34} + 2q^{36} + 12q^{39} - 12q^{41} - 2q^{46} + 14q^{49} - 4q^{51} + 2q^{54} + 8q^{59} - 12q^{61} - 2q^{64} - 2q^{69} - 16q^{71} - 20q^{74} + 16q^{79} + 2q^{81} - 16q^{86} + 12q^{89} - 16q^{94} - 2q^{96} + O(q^{100})$$

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
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 6.00000i 1.66410i 0.554700 + 0.832050i $$0.312833\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 1.00000i − 0.208514i
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 6.00000 1.17670
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 10.0000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ 0 0
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ − 8.00000i − 1.16692i −0.812142 0.583460i $$-0.801699\pi$$
0.812142 0.583460i $$-0.198301\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ − 6.00000i − 0.832050i
$$53$$ − 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
$$57$$ 0 0
$$58$$ 6.00000i 0.787839i
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ − 8.00000i − 1.01600i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ −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 $$0.801014\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ − 6.00000i − 0.679366i
$$79$$ 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
$$82$$ 6.00000i 0.662589i
$$83$$ − 8.00000i − 0.878114i −0.898459 0.439057i $$-0.855313\pi$$
0.898459 0.439057i $$-0.144687\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 6.00000i 0.643268i
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 1.00000i 0.104257i
$$93$$ − 8.00000i − 0.829561i
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ − 18.0000i − 1.82762i −0.406138 0.913812i $$-0.633125\pi$$
0.406138 0.913812i $$-0.366875\pi$$
$$98$$ − 7.00000i − 0.707107i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 2.00000i 0.198030i
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ 0 0
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ − 6.00000i − 0.554700i
$$118$$ − 4.00000i − 0.368230i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 6.00000i 0.543214i
$$123$$ 6.00000i 0.541002i
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 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
$$141$$ −8.00000 −0.673722
$$142$$ 8.00000i 0.671345i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 10.0000 0.827606
$$147$$ − 7.00000i − 0.577350i
$$148$$ 10.0000i 0.821995i
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ 2.00000i 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −6.00000 −0.480384
$$157$$ − 18.0000i − 1.43656i −0.695756 0.718278i $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 12.0000i 0.939913i 0.882690 + 0.469956i $$0.155730\pi$$
−0.882690 + 0.469956i $$0.844270\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ 16.0000i 1.23812i 0.785345 + 0.619059i $$0.212486\pi$$
−0.785345 + 0.619059i $$0.787514\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 8.00000i 0.609994i
$$173$$ − 18.0000i − 1.36851i −0.729241 0.684257i $$-0.760127\pi$$
0.729241 0.684257i $$-0.239873\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 4.00000i − 0.300658i
$$178$$ − 6.00000i − 0.449719i
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ 6.00000i 0.443533i
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 0 0
$$188$$ 8.00000i 0.583460i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −24.0000 −1.73658 −0.868290 0.496058i $$-0.834780\pi$$
−0.868290 + 0.496058i $$0.834780\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 18.0000i 1.29567i 0.761781 + 0.647834i $$0.224325\pi$$
−0.761781 + 0.647834i $$0.775675\pi$$
$$194$$ −18.0000 −1.29232
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ − 22.0000i − 1.56744i −0.621117 0.783718i $$-0.713321\pi$$
0.621117 0.783718i $$-0.286679\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ − 6.00000i − 0.422159i
$$203$$ 0 0
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 1.00000i 0.0695048i
$$208$$ 6.00000i 0.416025i
$$209$$ 0 0
$$210$$ 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
$$214$$ 0 0
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ − 6.00000i − 0.406371i
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 10.0000i 0.671156i
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ −18.0000 −1.18947 −0.594737 0.803921i $$-0.702744\pi$$
−0.594737 + 0.803921i $$0.702744\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ − 22.0000i − 1.44127i −0.693316 0.720634i $$-0.743851\pi$$
0.693316 0.720634i $$-0.256149\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ − 8.00000i − 0.519656i
$$238$$ 0 0
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 11.0000i 0.707107i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 6.00000 0.384111
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ 0 0
$$248$$ 8.00000i 0.508001i
$$249$$ −8.00000 −0.506979
$$250$$ 0 0
$$251$$ 8.00000 0.504956 0.252478 0.967603i $$-0.418755\pi$$
0.252478 + 0.967603i $$0.418755\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ 8.00000i 0.498058i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ − 4.00000i − 0.247121i
$$263$$ − 8.00000i − 0.493301i −0.969104 0.246651i $$-0.920670\pi$$
0.969104 0.246651i $$-0.0793300\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 6.00000i − 0.367194i
$$268$$ 8.00000i 0.488678i
$$269$$ −22.0000 −1.34136 −0.670682 0.741745i $$-0.733998\pi$$
−0.670682 + 0.741745i $$0.733998\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 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
$$281$$ −22.0000 −1.31241 −0.656205 0.754583i $$-0.727839\pi$$
−0.656205 + 0.754583i $$0.727839\pi$$
$$282$$ 8.00000i 0.476393i
$$283$$ 24.0000i 1.42665i 0.700832 + 0.713326i $$0.252812\pi$$
−0.700832 + 0.713326i $$0.747188\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 1.00000i 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −18.0000 −1.05518
$$292$$ − 10.0000i − 0.585206i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ −7.00000 −0.408248
$$295$$ 0 0
$$296$$ 10.0000 0.581238
$$297$$ 0 0
$$298$$ − 6.00000i − 0.347571i
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 16.0000i 0.920697i
$$303$$ − 6.00000i − 0.344691i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ 4.00000i 0.228292i 0.993464 + 0.114146i $$0.0364132\pi$$
−0.993464 + 0.114146i $$0.963587\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 6.00000i 0.339683i
$$313$$ − 6.00000i − 0.339140i −0.985518 0.169570i $$-0.945762\pi$$
0.985518 0.169570i $$-0.0542379\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 2.00000i 0.112331i 0.998421 + 0.0561656i $$0.0178875\pi$$
−0.998421 + 0.0561656i $$0.982113\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 12.0000 0.664619
$$327$$ − 6.00000i − 0.331801i
$$328$$ − 6.00000i − 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 8.00000i 0.439057i
$$333$$ 10.0000i 0.547997i
$$334$$ 16.0000 0.875481
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 22.0000i 1.19842i 0.800593 + 0.599208i $$0.204518\pi$$
−0.800593 + 0.599208i $$0.795482\pi$$
$$338$$ 23.0000i 1.25104i
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ − 36.0000i − 1.93258i −0.257454 0.966291i $$-0.582883\pi$$
0.257454 0.966291i $$-0.417117\pi$$
$$348$$ − 6.00000i − 0.321634i
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ 0 0
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ − 4.00000i − 0.211407i
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ − 2.00000i − 0.105118i
$$363$$ 11.0000i 0.577350i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 6.00000 0.313625
$$367$$ 24.0000i 1.25279i 0.779506 + 0.626395i $$0.215470\pi$$
−0.779506 + 0.626395i $$0.784530\pi$$
$$368$$ − 1.00000i − 0.0521286i
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 8.00000i 0.414781i
$$373$$ 26.0000i 1.34623i 0.739538 + 0.673114i $$0.235044\pi$$
−0.739538 + 0.673114i $$0.764956\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ − 36.0000i − 1.85409i
$$378$$ 0 0
$$379$$ 24.0000 1.23280 0.616399 0.787434i $$-0.288591\pi$$
0.616399 + 0.787434i $$0.288591\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 24.0000i 1.22795i
$$383$$ − 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
$$386$$ 18.0000 0.916176
$$387$$ 8.00000i 0.406663i
$$388$$ 18.0000i 0.913812i
$$389$$ −2.00000 −0.101404 −0.0507020 0.998714i $$-0.516146\pi$$
−0.0507020 + 0.998714i $$0.516146\pi$$
$$390$$ 0 0
$$391$$ −2.00000 −0.101144
$$392$$ 7.00000i 0.353553i
$$393$$ − 4.00000i − 0.201773i
$$394$$ −22.0000 −1.10834
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 6.00000i − 0.301131i −0.988600 0.150566i $$-0.951890\pi$$
0.988600 0.150566i $$-0.0481095\pi$$
$$398$$ − 16.0000i − 0.802008i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 26.0000 1.29838 0.649189 0.760627i $$-0.275108\pi$$
0.649189 + 0.760627i $$0.275108\pi$$
$$402$$ 8.00000i 0.399004i
$$403$$ 48.0000i 2.39105i
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ − 2.00000i − 0.0990148i
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ 8.00000i 0.394132i
$$413$$ 0 0
$$414$$ 1.00000 0.0491473
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ − 4.00000i − 0.195881i
$$418$$ 0 0
$$419$$ −16.0000 −0.781651 −0.390826 0.920465i $$-0.627810\pi$$
−0.390826 + 0.920465i $$0.627810\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ 8.00000i 0.388973i
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 8.00000 0.387601
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 14.0000i − 0.672797i −0.941720 0.336399i $$-0.890791\pi$$
0.941720 0.336399i $$-0.109209\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −6.00000 −0.287348
$$437$$ 0 0
$$438$$ − 10.0000i − 0.477818i
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ − 12.0000i − 0.570782i
$$443$$ 20.0000i 0.950229i 0.879924 + 0.475114i $$0.157593\pi$$
−0.879924 + 0.475114i $$0.842407\pi$$
$$444$$ 10.0000 0.474579
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 6.00000i − 0.283790i
$$448$$ 0 0
$$449$$ 14.0000 0.660701 0.330350 0.943858i $$-0.392833\pi$$
0.330350 + 0.943858i $$0.392833\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 6.00000i 0.282216i
$$453$$ 16.0000i 0.751746i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 38.0000i 1.77757i 0.458329 + 0.888783i $$0.348448\pi$$
−0.458329 + 0.888783i $$0.651552\pi$$
$$458$$ 18.0000i 0.841085i
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ − 40.0000i − 1.85896i −0.368875 0.929479i $$-0.620257\pi$$
0.368875 0.929479i $$-0.379743\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ − 8.00000i − 0.370196i −0.982720 0.185098i $$-0.940740\pi$$
0.982720 0.185098i $$-0.0592602\pi$$
$$468$$ 6.00000i 0.277350i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ 4.00000i 0.184115i
$$473$$ 0 0
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ 8.00000i 0.365911i
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 60.0000 2.73576
$$482$$ − 26.0000i − 1.18427i
$$483$$ 0 0
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 16.0000i − 0.725029i −0.931978 0.362515i $$-0.881918\pi$$
0.931978 0.362515i $$-0.118082\pi$$
$$488$$ − 6.00000i − 0.271607i
$$489$$ 12.0000 0.542659
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ 12.0000i 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ 8.00000i 0.358489i
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 0 0
$$501$$ 16.0000 0.714827
$$502$$ − 8.00000i − 0.357057i
$$503$$ 16.0000i 0.713405i 0.934218 + 0.356702i $$0.116099\pi$$
−0.934218 + 0.356702i $$0.883901\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 23.0000i 1.02147i
$$508$$ − 8.00000i − 0.354943i
$$509$$ 26.0000 1.15243 0.576215 0.817298i $$-0.304529\pi$$
0.576215 + 0.817298i $$0.304529\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ − 6.00000i − 0.262613i
$$523$$ − 16.0000i − 0.699631i −0.936819 0.349816i $$-0.886244\pi$$
0.936819 0.349816i $$-0.113756\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ − 16.0000i − 0.696971i
$$528$$ 0 0
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ − 36.0000i − 1.55933i
$$534$$ −6.00000 −0.259645
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ − 4.00000i − 0.172613i
$$538$$ 22.0000i 0.948487i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ − 16.0000i − 0.687259i
$$543$$ − 2.00000i − 0.0858282i
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 4.00000i − 0.171028i −0.996337 0.0855138i $$-0.972747\pi$$
0.996337 0.0855138i $$-0.0272532\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ 0 0
$$552$$ − 1.00000i − 0.0425628i
$$553$$ 0 0
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ − 10.0000i − 0.423714i −0.977301 0.211857i $$-0.932049\pi$$
0.977301 0.211857i $$-0.0679510\pi$$
$$558$$ 8.00000i 0.338667i
$$559$$ 48.0000 2.03018
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 22.0000i 0.928014i
$$563$$ 32.0000i 1.34864i 0.738440 + 0.674320i $$0.235563\pi$$
−0.738440 + 0.674320i $$0.764437\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 0 0
$$566$$ 24.0000 1.00880
$$567$$ 0 0
$$568$$ − 8.00000i − 0.335673i
$$569$$ 14.0000 0.586911 0.293455 0.955973i $$-0.405195\pi$$
0.293455 + 0.955973i $$0.405195\pi$$
$$570$$ 0 0
$$571$$ 8.00000 0.334790 0.167395 0.985890i $$-0.446465\pi$$
0.167395 + 0.985890i $$0.446465\pi$$
$$572$$ 0 0
$$573$$ 24.0000i 1.00261i
$$574$$ 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 $$-0.0132552\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ 18.0000 0.748054
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 18.0000i 0.746124i
$$583$$ 0 0
$$584$$ −10.0000 −0.413803
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ 7.00000i 0.288675i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −22.0000 −0.904959
$$592$$ − 10.0000i − 0.410997i
$$593$$ − 30.0000i − 1.23195i −0.787765 0.615976i $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ − 16.0000i − 0.654836i
$$598$$ − 6.00000i − 0.245358i
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ 8.00000i 0.325785i
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ −6.00000 −0.243733
$$607$$ 16.0000i 0.649420i 0.945814 + 0.324710i $$0.105267\pi$$
−0.945814 + 0.324710i $$0.894733\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 48.0000 1.94187
$$612$$ − 2.00000i − 0.0808452i
$$613$$ − 46.0000i − 1.85792i −0.370177 0.928961i $$-0.620703\pi$$
0.370177 0.928961i $$-0.379297\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 2.00000i − 0.0805170i −0.999189 0.0402585i $$-0.987182\pi$$
0.999189 0.0402585i $$-0.0128181\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ 16.0000 0.643094 0.321547 0.946894i $$-0.395797\pi$$
0.321547 + 0.946894i $$0.395797\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 6.00000 0.240192
$$625$$ 0 0
$$626$$ −6.00000 −0.239808
$$627$$ 0 0
$$628$$ 18.0000i 0.718278i
$$629$$ −20.0000 −0.797452
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ − 4.00000i − 0.158986i
$$634$$ 2.00000 0.0794301
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ 42.0000i 1.66410i
$$638$$ 0 0
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ −22.0000 −0.868948 −0.434474 0.900684i $$-0.643066\pi$$
−0.434474 + 0.900684i $$0.643066\pi$$
$$642$$ 0 0
$$643$$ − 16.0000i − 0.630978i −0.948929 0.315489i $$-0.897831\pi$$
0.948929 0.315489i $$-0.102169\pi$$
$$644$$ 0 0
$$645$$ 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
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 12.0000i − 0.469956i
$$653$$ − 42.0000i − 1.64359i −0.569785 0.821794i $$-0.692974\pi$$
0.569785 0.821794i $$-0.307026\pi$$
$$654$$ −6.00000 −0.234619
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ − 10.0000i − 0.390137i
$$658$$ 0 0
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ 0 0
$$661$$ 2.00000 0.0777910 0.0388955 0.999243i $$-0.487616\pi$$
0.0388955 + 0.999243i $$0.487616\pi$$
$$662$$ 20.0000i 0.777322i
$$663$$ − 12.0000i − 0.466041i
$$664$$ 8.00000 0.310460
$$665$$ 0 0
$$666$$ 10.0000 0.387492
$$667$$ 6.00000i 0.232321i
$$668$$ − 16.0000i − 0.619059i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ − 30.0000i − 1.15642i −0.815890 0.578208i $$-0.803752\pi$$
0.815890 0.578208i $$-0.196248\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 6.00000i 0.230599i 0.993331 + 0.115299i $$0.0367827\pi$$
−0.993331 + 0.115299i $$0.963217\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 4.00000i − 0.153056i −0.997067 0.0765279i $$-0.975617\pi$$
0.997067 0.0765279i $$-0.0243834\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 18.0000i 0.686743i
$$688$$ − 8.00000i − 0.304997i
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ −12.0000 −0.456502 −0.228251 0.973602i $$-0.573301\pi$$
−0.228251 + 0.973602i $$0.573301\pi$$
$$692$$ 18.0000i 0.684257i
$$693$$ 0 0
$$694$$ −36.0000 −1.36654
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ 12.0000i 0.454532i
$$698$$ − 26.0000i − 0.984115i
$$699$$ −22.0000 −0.832116
$$700$$ 0 0
$$701$$ 42.0000 1.58632 0.793159 0.609015i $$-0.208435\pi$$
0.793159 + 0.609015i $$0.208435\pi$$
$$702$$ 6.00000i 0.226455i
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 0 0
$$708$$ 4.00000i 0.150329i
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 6.00000i 0.224860i
$$713$$ − 8.00000i − 0.299602i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −4.00000 −0.149487
$$717$$ 8.00000i 0.298765i
$$718$$ 16.0000i 0.597115i
$$719$$ −40.0000 −1.49175 −0.745874 0.666087i $$-0.767968\pi$$
−0.745874 + 0.666087i $$0.767968\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 19.0000i 0.707107i
$$723$$ − 26.0000i − 0.966950i
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ 11.0000 0.408248
$$727$$ 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
$$731$$ −16.0000 −0.591781
$$732$$ − 6.00000i − 0.221766i
$$733$$ 2.00000i 0.0738717i 0.999318 + 0.0369358i $$0.0117597\pi$$
−0.999318 + 0.0369358i $$0.988240\pi$$
$$734$$ 24.0000 0.885856
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ 0 0
$$738$$ − 6.00000i − 0.220863i
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 8.00000i − 0.293492i −0.989174 0.146746i $$-0.953120\pi$$
0.989174 0.146746i $$-0.0468799\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ 26.0000 0.951928
$$747$$ 8.00000i 0.292705i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ − 8.00000i − 0.291730i
$$753$$ − 8.00000i − 0.291536i
$$754$$ −36.0000 −1.31104
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 34.0000i − 1.23575i −0.786276 0.617876i $$-0.787994\pi$$
0.786276 0.617876i $$-0.212006\pi$$
$$758$$ − 24.0000i − 0.871719i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ − 8.00000i − 0.289809i
$$763$$ 0 0
$$764$$ 24.0000 0.868290
$$765$$ 0 0
$$766$$ −32.0000 −1.15621
$$767$$ 24.0000i 0.866590i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 6.00000 0.216366 0.108183 0.994131i $$-0.465497\pi$$
0.108183 + 0.994131i $$0.465497\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ − 18.0000i − 0.647834i
$$773$$ 2.00000i 0.0719350i 0.999353 + 0.0359675i $$0.0114513\pi$$
−0.999353 + 0.0359675i $$0.988549\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ 18.0000 0.646162
$$777$$ 0 0
$$778$$ 2.00000i 0.0717035i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 2.00000i 0.0715199i
$$783$$ − 6.00000i − 0.214423i
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ −4.00000 −0.142675
$$787$$ 32.0000i 1.14068i 0.821410 + 0.570338i $$0.193188\pi$$
−0.821410 + 0.570338i $$0.806812\pi$$
$$788$$ 22.0000i 0.783718i
$$789$$ −8.00000 −0.284808
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ − 36.0000i − 1.27840i
$$794$$ −6.00000 −0.212932
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ 54.0000i 1.91278i 0.292096 + 0.956389i $$0.405647\pi$$
−0.292096 + 0.956389i $$0.594353\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ − 26.0000i − 0.918092i
$$803$$ 0 0
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ 48.0000 1.69073
$$807$$ 22.0000i 0.774437i
$$808$$ 6.00000i 0.211079i
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 0 0
$$813$$ − 16.0000i − 0.561144i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ 0 0
$$818$$ 10.0000i 0.349642i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ − 6.00000i − 0.209274i
$$823$$ 56.0000i 1.95204i 0.217687 + 0.976019i $$0.430149\pi$$
−0.217687 + 0.976019i $$0.569851\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 48.0000i 1.66912i 0.550914 + 0.834562i $$0.314279\pi$$
−0.550914 + 0.834562i $$0.685721\pi$$
$$828$$ − 1.00000i − 0.0347524i
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ − 6.00000i − 0.208013i
$$833$$ − 14.0000i − 0.485071i
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 8.00000i 0.276520i
$$838$$ 16.0000i 0.552711i
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 22.0000i 0.758170i
$$843$$ 22.0000i 0.757720i
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ 0 0
$$848$$ − 6.00000i − 0.206041i
$$849$$ 24.0000 0.823678
$$850$$ 0 0
$$851$$ −10.0000 −0.342796
$$852$$ − 8.00000i − 0.274075i
$$853$$ − 50.0000i − 1.71197i −0.517003 0.855984i $$-0.672952\pi$$
0.517003 0.855984i $$-0.327048\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 10.0000i − 0.341593i −0.985306 0.170797i $$-0.945366\pi$$
0.985306 0.170797i $$-0.0546341\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 8.00000i 0.272481i
$$863$$ 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
$$866$$ −14.0000 −0.475739
$$867$$ − 13.0000i − 0.441503i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 48.0000 1.62642
$$872$$ 6.00000i 0.203186i
$$873$$ 18.0000i 0.609208i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −10.0000 −0.337869
$$877$$ − 14.0000i − 0.472746i −0.971662 0.236373i $$-0.924041\pi$$
0.971662 0.236373i $$-0.0759588\pi$$
$$878$$ 8.00000i 0.269987i
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ 50.0000 1.68454 0.842271 0.539054i $$-0.181218\pi$$
0.842271 + 0.539054i $$0.181218\pi$$
$$882$$ 7.00000i 0.235702i
$$883$$ 36.0000i 1.21150i 0.795656 + 0.605748i $$0.207126\pi$$
−0.795656 + 0.605748i $$0.792874\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ 20.0000 0.671913
$$887$$ − 16.0000i − 0.537227i −0.963248 0.268614i $$-0.913434\pi$$
0.963248 0.268614i $$-0.0865655\pi$$
$$888$$ − 10.0000i − 0.335578i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 6.00000i − 0.200334i
$$898$$ − 14.0000i − 0.467186i
$$899$$ −48.0000 −1.60089
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 16.0000 0.531564
$$907$$ − 8.00000i − 0.265636i −0.991140 0.132818i $$-0.957597\pi$$
0.991140 0.132818i $$-0.0424025\pi$$
$$908$$ 0 0
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 40.0000 1.32526 0.662630 0.748947i $$-0.269440\pi$$
0.662630 + 0.748947i $$0.269440\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 38.0000 1.25693
$$915$$ 0 0
$$916$$ 18.0000 0.594737
$$917$$ 0 0
$$918$$ − 2.00000i − 0.0660098i
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ − 6.00000i − 0.197599i
$$923$$ − 48.0000i − 1.57994i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −40.0000 −1.31448
$$927$$ 8.00000i 0.262754i
$$928$$ 6.00000i 0.196960i
$$929$$ 14.0000 0.459325 0.229663 0.973270i $$-0.426238\pi$$
0.229663 + 0.973270i $$0.426238\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 22.0000i 0.720634i
$$933$$ 0 0
$$934$$ −8.00000 −0.261768
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ − 2.00000i − 0.0653372i −0.999466 0.0326686i $$-0.989599\pi$$
0.999466 0.0326686i $$-0.0104006\pi$$
$$938$$ 0 0
$$939$$ −6.00000 −0.195803
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 18.0000i 0.586472i
$$943$$ 6.00000i 0.195387i
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 44.0000i − 1.42981i −0.699223 0.714904i $$-0.746470\pi$$
0.699223 0.714904i $$-0.253530\pi$$
$$948$$ 8.00000i 0.259828i
$$949$$ −60.0000 −1.94768
$$950$$ 0 0
$$951$$ 2.00000 0.0648544
$$952$$ 0 0
$$953$$ − 6.00000i − 0.194359i −0.995267 0.0971795i $$-0.969018\pi$$
0.995267 0.0971795i $$-0.0309821\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 8.00000 0.258738
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 60.0000i − 1.93448i
$$963$$ 0 0
$$964$$ −26.0000 −0.837404
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 56.0000i 1.80084i 0.435023 + 0.900419i $$0.356740\pi$$
−0.435023 + 0.900419i $$0.643260\pi$$
$$968$$ − 11.0000i − 0.353553i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −32.0000 −1.02693 −0.513464 0.858111i $$-0.671638\pi$$
−0.513464 + 0.858111i $$0.671638\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 0 0
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ −6.00000 −0.192055
$$977$$ − 50.0000i − 1.59964i −0.600239 0.799821i $$-0.704928\pi$$
0.600239 0.799821i $$-0.295072\pi$$
$$978$$ − 12.0000i − 0.383718i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −6.00000 −0.191565
$$982$$ − 36.0000i − 1.14881i
$$983$$ − 48.0000i − 1.53096i −0.643458 0.765481i $$-0.722501\pi$$
0.643458 0.765481i $$-0.277499\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 12.0000 0.382158
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ − 8.00000i − 0.254000i
$$993$$ 20.0000i 0.634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 8.00000 0.253490
$$997$$ − 38.0000i − 1.20347i −0.798695 0.601736i $$-0.794476\pi$$
0.798695 0.601736i $$-0.205524\pi$$
$$998$$ − 20.0000i − 0.633089i
$$999$$ 10.0000 0.316386
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.e.2899.1 2
5.2 odd 4 690.2.a.h.1.1 1
5.3 odd 4 3450.2.a.j.1.1 1
5.4 even 2 inner 3450.2.d.e.2899.2 2
15.2 even 4 2070.2.a.h.1.1 1
20.7 even 4 5520.2.a.x.1.1 1

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