# Properties

 Label 3450.2.d.s.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.s.2899.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} -4.00000 q^{21} -2.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} -8.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} -4.00000i q^{38} +4.00000 q^{39} +6.00000 q^{41} +4.00000i q^{42} -6.00000i q^{43} -2.00000 q^{44} +1.00000 q^{46} -4.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} +6.00000 q^{51} +4.00000i q^{52} +14.0000i q^{53} -1.00000 q^{54} -4.00000 q^{56} +4.00000i q^{57} +8.00000i q^{58} -4.00000 q^{59} +6.00000 q^{61} -8.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +14.0000i q^{67} +6.00000i q^{68} -1.00000 q^{69} +10.0000 q^{71} -1.00000i q^{72} -14.0000i q^{73} -10.0000 q^{74} -4.00000 q^{76} +8.00000i q^{77} -4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +4.00000i q^{83} +4.00000 q^{84} -6.00000 q^{86} -8.00000i q^{87} +2.00000i q^{88} +16.0000 q^{91} -1.00000i q^{92} +8.00000i q^{93} -4.00000 q^{94} +1.00000 q^{96} -8.00000i q^{97} +9.00000i q^{98} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + 4q^{11} + 8q^{14} + 2q^{16} + 8q^{19} - 8q^{21} - 2q^{24} - 8q^{26} - 16q^{29} + 16q^{31} - 12q^{34} + 2q^{36} + 8q^{39} + 12q^{41} - 4q^{44} + 2q^{46} - 18q^{49} + 12q^{51} - 2q^{54} - 8q^{56} - 8q^{59} + 12q^{61} - 2q^{64} + 4q^{66} - 2q^{69} + 20q^{71} - 20q^{74} - 8q^{76} + 16q^{79} + 2q^{81} + 8q^{84} - 12q^{86} + 32q^{91} - 8q^{94} + 2q^{96} - 4q^{99} + O(q^{100})$$

## 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
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ − 4.00000i − 1.10940i −0.832050 0.554700i $$-0.812833\pi$$
0.832050 0.554700i $$-0.187167\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 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 $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ − 2.00000i − 0.426401i
$$23$$ 1.00000i 0.208514i
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 4.00000i − 0.755929i
$$29$$ −8.00000 −1.48556 −0.742781 0.669534i $$-0.766494\pi$$
−0.742781 + 0.669534i $$0.766494\pi$$
$$30$$ 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
$$33$$ 2.00000i 0.348155i
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 10.0000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 4.00000i 0.617213i
$$43$$ − 6.00000i − 0.914991i −0.889212 0.457496i $$-0.848747\pi$$
0.889212 0.457496i $$-0.151253\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ − 4.00000i − 0.583460i −0.956501 0.291730i $$-0.905769\pi$$
0.956501 0.291730i $$-0.0942309\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 4.00000i 0.554700i
$$53$$ 14.0000i 1.92305i 0.274721 + 0.961524i $$0.411414\pi$$
−0.274721 + 0.961524i $$0.588586\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −4.00000 −0.534522
$$57$$ 4.00000i 0.529813i
$$58$$ 8.00000i 1.05045i
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ − 8.00000i − 1.01600i
$$63$$ − 4.00000i − 0.503953i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ 14.0000i 1.71037i 0.518321 + 0.855186i $$0.326557\pi$$
−0.518321 + 0.855186i $$0.673443\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ 10.0000 1.18678 0.593391 0.804914i $$-0.297789\pi$$
0.593391 + 0.804914i $$0.297789\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 14.0000i − 1.63858i −0.573382 0.819288i $$-0.694369\pi$$
0.573382 0.819288i $$-0.305631\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 8.00000i 0.911685i
$$78$$ − 4.00000i − 0.452911i
$$79$$ 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
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ −6.00000 −0.646997
$$87$$ − 8.00000i − 0.857690i
$$88$$ 2.00000i 0.213201i
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 16.0000 1.67726
$$92$$ − 1.00000i − 0.104257i
$$93$$ 8.00000i 0.829561i
$$94$$ −4.00000 −0.412568
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 8.00000i − 0.812277i −0.913812 0.406138i $$-0.866875\pi$$
0.913812 0.406138i $$-0.133125\pi$$
$$98$$ 9.00000i 0.909137i
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ −8.00000 −0.796030 −0.398015 0.917379i $$-0.630301\pi$$
−0.398015 + 0.917379i $$0.630301\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ 14.0000 1.35980
$$107$$ 8.00000i 0.773389i 0.922208 + 0.386695i $$0.126383\pi$$
−0.922208 + 0.386695i $$0.873617\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 4.00000i 0.377964i
$$113$$ 10.0000i 0.940721i 0.882474 + 0.470360i $$0.155876\pi$$
−0.882474 + 0.470360i $$0.844124\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 8.00000 0.742781
$$117$$ 4.00000i 0.369800i
$$118$$ 4.00000i 0.368230i
$$119$$ 24.0000 2.20008
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ − 6.00000i − 0.543214i
$$123$$ 6.00000i 0.541002i
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ −4.00000 −0.356348
$$127$$ 2.00000i 0.177471i 0.996055 + 0.0887357i $$0.0282826\pi$$
−0.996055 + 0.0887357i $$0.971717\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ 16.0000 1.39793 0.698963 0.715158i $$-0.253645\pi$$
0.698963 + 0.715158i $$0.253645\pi$$
$$132$$ − 2.00000i − 0.174078i
$$133$$ 16.0000i 1.38738i
$$134$$ 14.0000 1.20942
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 1.00000i 0.0851257i
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ − 10.0000i − 0.839181i
$$143$$ − 8.00000i − 0.668994i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −14.0000 −1.15865
$$147$$ − 9.00000i − 0.742307i
$$148$$ 10.0000i 0.821995i
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ 6.00000i 0.485071i
$$154$$ 8.00000 0.644658
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ −14.0000 −1.11027
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 8.00000i 0.626608i 0.949653 + 0.313304i $$0.101436\pi$$
−0.949653 + 0.313304i $$0.898564\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ − 8.00000i − 0.619059i −0.950890 0.309529i $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ − 4.00000i − 0.308607i
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 6.00000i 0.457496i
$$173$$ − 10.0000i − 0.760286i −0.924928 0.380143i $$-0.875875\pi$$
0.924928 0.380143i $$-0.124125\pi$$
$$174$$ −8.00000 −0.606478
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ − 4.00000i − 0.300658i
$$178$$ 0 0
$$179$$ 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 $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ − 16.0000i − 1.18600i
$$183$$ 6.00000i 0.443533i
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ − 12.0000i − 0.877527i
$$188$$ 4.00000i 0.291730i
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 22.0000i 1.58359i 0.610784 + 0.791797i $$0.290854\pi$$
−0.610784 + 0.791797i $$0.709146\pi$$
$$194$$ −8.00000 −0.574367
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ − 22.0000i − 1.56744i −0.621117 0.783718i $$-0.713321\pi$$
0.621117 0.783718i $$-0.286679\pi$$
$$198$$ 2.00000i 0.142134i
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ −14.0000 −0.987484
$$202$$ 8.00000i 0.562878i
$$203$$ − 32.0000i − 2.24596i
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ − 1.00000i − 0.0695048i
$$208$$ − 4.00000i − 0.277350i
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ − 14.0000i − 0.961524i
$$213$$ 10.0000i 0.685189i
$$214$$ 8.00000 0.546869
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 32.0000i 2.17230i
$$218$$ − 14.0000i − 0.948200i
$$219$$ 14.0000 0.946032
$$220$$ 0 0
$$221$$ −24.0000 −1.61441
$$222$$ − 10.0000i − 0.671156i
$$223$$ − 14.0000i − 0.937509i −0.883328 0.468755i $$-0.844703\pi$$
0.883328 0.468755i $$-0.155297\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ 10.0000 0.665190
$$227$$ 8.00000i 0.530979i 0.964114 + 0.265489i $$0.0855335\pi$$
−0.964114 + 0.265489i $$0.914466\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ −8.00000 −0.526361
$$232$$ − 8.00000i − 0.525226i
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 8.00000i 0.519656i
$$238$$ − 24.0000i − 1.55569i
$$239$$ −26.0000 −1.68180 −0.840900 0.541190i $$-0.817974\pi$$
−0.840900 + 0.541190i $$0.817974\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 7.00000i 0.449977i
$$243$$ 1.00000i 0.0641500i
$$244$$ −6.00000 −0.384111
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ − 16.0000i − 1.01806i
$$248$$ 8.00000i 0.508001i
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ −2.00000 −0.126239 −0.0631194 0.998006i $$-0.520105\pi$$
−0.0631194 + 0.998006i $$0.520105\pi$$
$$252$$ 4.00000i 0.251976i
$$253$$ 2.00000i 0.125739i
$$254$$ 2.00000 0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ − 6.00000i − 0.373544i
$$259$$ 40.0000 2.48548
$$260$$ 0 0
$$261$$ 8.00000 0.495188
$$262$$ − 16.0000i − 0.988483i
$$263$$ − 16.0000i − 0.986602i −0.869859 0.493301i $$-0.835790\pi$$
0.869859 0.493301i $$-0.164210\pi$$
$$264$$ −2.00000 −0.123091
$$265$$ 0 0
$$266$$ 16.0000 0.981023
$$267$$ 0 0
$$268$$ − 14.0000i − 0.855186i
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ − 6.00000i − 0.363803i
$$273$$ 16.0000i 0.968364i
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ − 12.0000i − 0.721010i −0.932757 0.360505i $$-0.882604\pi$$
0.932757 0.360505i $$-0.117396\pi$$
$$278$$ − 20.0000i − 1.19952i
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 8.00000 0.477240 0.238620 0.971113i $$-0.423305\pi$$
0.238620 + 0.971113i $$0.423305\pi$$
$$282$$ − 4.00000i − 0.238197i
$$283$$ 22.0000i 1.30776i 0.756596 + 0.653882i $$0.226861\pi$$
−0.756596 + 0.653882i $$0.773139\pi$$
$$284$$ −10.0000 −0.593391
$$285$$ 0 0
$$286$$ −8.00000 −0.473050
$$287$$ 24.0000i 1.41668i
$$288$$ 1.00000i 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ 14.0000i 0.819288i
$$293$$ − 26.0000i − 1.51894i −0.650545 0.759468i $$-0.725459\pi$$
0.650545 0.759468i $$-0.274541\pi$$
$$294$$ −9.00000 −0.524891
$$295$$ 0 0
$$296$$ 10.0000 0.581238
$$297$$ − 2.00000i − 0.116052i
$$298$$ − 10.0000i − 0.579284i
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ 24.0000 1.38334
$$302$$ − 20.0000i − 1.15087i
$$303$$ − 8.00000i − 0.459588i
$$304$$ 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 $$0.963587\pi$$
$$308$$ − 8.00000i − 0.455842i
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ −14.0000 −0.793867 −0.396934 0.917847i $$-0.629926\pi$$
−0.396934 + 0.917847i $$0.629926\pi$$
$$312$$ 4.00000i 0.226455i
$$313$$ 4.00000i 0.226093i 0.993590 + 0.113047i $$0.0360610\pi$$
−0.993590 + 0.113047i $$0.963939\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 14.0000i 0.785081i
$$319$$ −16.0000 −0.895828
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ 4.00000i 0.222911i
$$323$$ − 24.0000i − 1.33540i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 8.00000 0.443079
$$327$$ 14.0000i 0.774202i
$$328$$ 6.00000i 0.331295i
$$329$$ 16.0000 0.882109
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ − 4.00000i − 0.219529i
$$333$$ 10.0000i 0.547997i
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ − 20.0000i − 1.08947i −0.838608 0.544735i $$-0.816630\pi$$
0.838608 0.544735i $$-0.183370\pi$$
$$338$$ 3.00000i 0.163178i
$$339$$ −10.0000 −0.543125
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ 4.00000i 0.216295i
$$343$$ − 8.00000i − 0.431959i
$$344$$ 6.00000 0.323498
$$345$$ 0 0
$$346$$ −10.0000 −0.537603
$$347$$ 12.0000i 0.644194i 0.946707 + 0.322097i $$0.104388\pi$$
−0.946707 + 0.322097i $$0.895612\pi$$
$$348$$ 8.00000i 0.428845i
$$349$$ 30.0000 1.60586 0.802932 0.596071i $$-0.203272\pi$$
0.802932 + 0.596071i $$0.203272\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ − 2.00000i − 0.106600i
$$353$$ − 14.0000i − 0.745145i −0.928003 0.372572i $$-0.878476\pi$$
0.928003 0.372572i $$-0.121524\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 24.0000i 1.27021i
$$358$$ − 4.00000i − 0.211407i
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 2.00000i 0.105118i
$$363$$ − 7.00000i − 0.367405i
$$364$$ −16.0000 −0.838628
$$365$$ 0 0
$$366$$ 6.00000 0.313625
$$367$$ − 12.0000i − 0.626395i −0.949688 0.313197i $$-0.898600\pi$$
0.949688 0.313197i $$-0.101400\pi$$
$$368$$ 1.00000i 0.0521286i
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −56.0000 −2.90738
$$372$$ − 8.00000i − 0.414781i
$$373$$ 18.0000i 0.932005i 0.884783 + 0.466002i $$0.154306\pi$$
−0.884783 + 0.466002i $$0.845694\pi$$
$$374$$ −12.0000 −0.620505
$$375$$ 0 0
$$376$$ 4.00000 0.206284
$$377$$ 32.0000i 1.64808i
$$378$$ − 4.00000i − 0.205738i
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ −2.00000 −0.102463
$$382$$ 16.0000i 0.818631i
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 22.0000 1.11977
$$387$$ 6.00000i 0.304997i
$$388$$ 8.00000i 0.406138i
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ − 9.00000i − 0.454569i
$$393$$ 16.0000i 0.807093i
$$394$$ −22.0000 −1.10834
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ − 28.0000i − 1.40528i −0.711546 0.702640i $$-0.752005\pi$$
0.711546 0.702640i $$-0.247995\pi$$
$$398$$ 16.0000i 0.802008i
$$399$$ −16.0000 −0.801002
$$400$$ 0 0
$$401$$ 20.0000 0.998752 0.499376 0.866385i $$-0.333563\pi$$
0.499376 + 0.866385i $$0.333563\pi$$
$$402$$ 14.0000i 0.698257i
$$403$$ − 32.0000i − 1.59403i
$$404$$ 8.00000 0.398015
$$405$$ 0 0
$$406$$ −32.0000 −1.58813
$$407$$ − 20.0000i − 0.991363i
$$408$$ 6.00000i 0.297044i
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ 4.00000i 0.197066i
$$413$$ − 16.0000i − 0.787309i
$$414$$ −1.00000 −0.0491473
$$415$$ 0 0
$$416$$ −4.00000 −0.196116
$$417$$ 20.0000i 0.979404i
$$418$$ − 8.00000i − 0.391293i
$$419$$ 30.0000 1.46560 0.732798 0.680446i $$-0.238214\pi$$
0.732798 + 0.680446i $$0.238214\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 20.0000i 0.973585i
$$423$$ 4.00000i 0.194487i
$$424$$ −14.0000 −0.679900
$$425$$ 0 0
$$426$$ 10.0000 0.484502
$$427$$ 24.0000i 1.16144i
$$428$$ − 8.00000i − 0.386695i
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ − 32.0000i − 1.53782i −0.639356 0.768911i $$-0.720799\pi$$
0.639356 0.768911i $$-0.279201\pi$$
$$434$$ 32.0000 1.53605
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ 4.00000i 0.191346i
$$438$$ − 14.0000i − 0.668946i
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 24.0000i 1.14156i
$$443$$ 28.0000i 1.33032i 0.746701 + 0.665160i $$0.231637\pi$$
−0.746701 + 0.665160i $$0.768363\pi$$
$$444$$ −10.0000 −0.474579
$$445$$ 0 0
$$446$$ −14.0000 −0.662919
$$447$$ 10.0000i 0.472984i
$$448$$ − 4.00000i − 0.188982i
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ 12.0000 0.565058
$$452$$ − 10.0000i − 0.470360i
$$453$$ 20.0000i 0.939682i
$$454$$ 8.00000 0.375459
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ 16.0000i 0.748448i 0.927338 + 0.374224i $$0.122091\pi$$
−0.927338 + 0.374224i $$0.877909\pi$$
$$458$$ − 22.0000i − 1.02799i
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 8.00000i 0.372194i
$$463$$ 10.0000i 0.464739i 0.972628 + 0.232370i $$0.0746479\pi$$
−0.972628 + 0.232370i $$0.925352\pi$$
$$464$$ −8.00000 −0.371391
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ − 36.0000i − 1.66588i −0.553362 0.832941i $$-0.686655\pi$$
0.553362 0.832941i $$-0.313345\pi$$
$$468$$ − 4.00000i − 0.184900i
$$469$$ −56.0000 −2.58584
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ − 4.00000i − 0.184115i
$$473$$ − 12.0000i − 0.551761i
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ −24.0000 −1.10004
$$477$$ − 14.0000i − 0.641016i
$$478$$ 26.0000i 1.18921i
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ −40.0000 −1.82384
$$482$$ − 10.0000i − 0.455488i
$$483$$ − 4.00000i − 0.182006i
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 10.0000i − 0.453143i −0.973995 0.226572i $$-0.927248\pi$$
0.973995 0.226572i $$-0.0727517\pi$$
$$488$$ 6.00000i 0.271607i
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ −24.0000 −1.08310 −0.541552 0.840667i $$-0.682163\pi$$
−0.541552 + 0.840667i $$0.682163\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ 48.0000i 2.16181i
$$494$$ −16.0000 −0.719874
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 40.0000i 1.79425i
$$498$$ 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
$$502$$ 2.00000i 0.0892644i
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 4.00000 0.178174
$$505$$ 0 0
$$506$$ 2.00000 0.0889108
$$507$$ − 3.00000i − 0.133235i
$$508$$ − 2.00000i − 0.0887357i
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ 0 0
$$511$$ 56.0000 2.47729
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 4.00000i − 0.176604i
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ −6.00000 −0.264135
$$517$$ − 8.00000i − 0.351840i
$$518$$ − 40.0000i − 1.75750i
$$519$$ 10.0000 0.438951
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ − 8.00000i − 0.350150i
$$523$$ 14.0000i 0.612177i 0.952003 + 0.306089i $$0.0990204\pi$$
−0.952003 + 0.306089i $$0.900980\pi$$
$$524$$ −16.0000 −0.698963
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ − 48.0000i − 2.09091i
$$528$$ 2.00000i 0.0870388i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ − 16.0000i − 0.693688i
$$533$$ − 24.0000i − 1.03956i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −14.0000 −0.604708
$$537$$ 4.00000i 0.172613i
$$538$$ 12.0000i 0.517357i
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ 10.0000 0.429934 0.214967 0.976621i $$-0.431036\pi$$
0.214967 + 0.976621i $$0.431036\pi$$
$$542$$ 8.00000i 0.343629i
$$543$$ − 2.00000i − 0.0858282i
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 16.0000 0.684737
$$547$$ 28.0000i 1.19719i 0.801050 + 0.598597i $$0.204275\pi$$
−0.801050 + 0.598597i $$0.795725\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ −32.0000 −1.36325
$$552$$ − 1.00000i − 0.0425628i
$$553$$ 32.0000i 1.36078i
$$554$$ −12.0000 −0.509831
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ − 2.00000i − 0.0847427i −0.999102 0.0423714i $$-0.986509\pi$$
0.999102 0.0423714i $$-0.0134913\pi$$
$$558$$ 8.00000i 0.338667i
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 12.0000 0.506640
$$562$$ − 8.00000i − 0.337460i
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 0 0
$$566$$ 22.0000 0.924729
$$567$$ 4.00000i 0.167984i
$$568$$ 10.0000i 0.419591i
$$569$$ 44.0000 1.84458 0.922288 0.386503i $$-0.126317\pi$$
0.922288 + 0.386503i $$0.126317\pi$$
$$570$$ 0 0
$$571$$ −24.0000 −1.00437 −0.502184 0.864761i $$-0.667470\pi$$
−0.502184 + 0.864761i $$0.667470\pi$$
$$572$$ 8.00000i 0.334497i
$$573$$ − 16.0000i − 0.668410i
$$574$$ 24.0000 1.00174
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 34.0000i − 1.41544i −0.706494 0.707719i $$-0.749724\pi$$
0.706494 0.707719i $$-0.250276\pi$$
$$578$$ 19.0000i 0.790296i
$$579$$ −22.0000 −0.914289
$$580$$ 0 0
$$581$$ −16.0000 −0.663792
$$582$$ − 8.00000i − 0.331611i
$$583$$ 28.0000i 1.15964i
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ −26.0000 −1.07405
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 9.00000i 0.371154i
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ 22.0000 0.904959
$$592$$ − 10.0000i − 0.410997i
$$593$$ − 18.0000i − 0.739171i −0.929197 0.369586i $$-0.879500\pi$$
0.929197 0.369586i $$-0.120500\pi$$
$$594$$ −2.00000 −0.0820610
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ − 16.0000i − 0.654836i
$$598$$ − 4.00000i − 0.163572i
$$599$$ −14.0000 −0.572024 −0.286012 0.958226i $$-0.592330\pi$$
−0.286012 + 0.958226i $$0.592330\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ − 24.0000i − 0.978167i
$$603$$ − 14.0000i − 0.570124i
$$604$$ −20.0000 −0.813788
$$605$$ 0 0
$$606$$ −8.00000 −0.324978
$$607$$ 30.0000i 1.21766i 0.793300 + 0.608831i $$0.208361\pi$$
−0.793300 + 0.608831i $$0.791639\pi$$
$$608$$ − 4.00000i − 0.162221i
$$609$$ 32.0000 1.29671
$$610$$ 0 0
$$611$$ −16.0000 −0.647291
$$612$$ − 6.00000i − 0.242536i
$$613$$ 14.0000i 0.565455i 0.959200 + 0.282727i $$0.0912392\pi$$
−0.959200 + 0.282727i $$0.908761\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ −8.00000 −0.322329
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ − 4.00000i − 0.160904i
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 14.0000i 0.561349i
$$623$$ 0 0
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ 4.00000 0.159872
$$627$$ 8.00000i 0.319489i
$$628$$ 2.00000i 0.0798087i
$$629$$ −60.0000 −2.39236
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ − 20.0000i − 0.794929i
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ 14.0000 0.555136
$$637$$ 36.0000i 1.42637i
$$638$$ 16.0000i 0.633446i
$$639$$ −10.0000 −0.395594
$$640$$ 0 0
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ 8.00000i 0.315735i
$$643$$ − 34.0000i − 1.34083i −0.741987 0.670415i $$-0.766116\pi$$
0.741987 0.670415i $$-0.233884\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ − 28.0000i − 1.10079i −0.834903 0.550397i $$-0.814476\pi$$
0.834903 0.550397i $$-0.185524\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ −8.00000 −0.314027
$$650$$ 0 0
$$651$$ −32.0000 −1.25418
$$652$$ − 8.00000i − 0.313304i
$$653$$ 42.0000i 1.64359i 0.569785 + 0.821794i $$0.307026\pi$$
−0.569785 + 0.821794i $$0.692974\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 14.0000i 0.546192i
$$658$$ − 16.0000i − 0.623745i
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ − 12.0000i − 0.466393i
$$663$$ − 24.0000i − 0.932083i
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ 10.0000 0.387492
$$667$$ − 8.00000i − 0.309761i
$$668$$ 8.00000i 0.309529i
$$669$$ 14.0000 0.541271
$$670$$ 0 0
$$671$$ 12.0000 0.463255
$$672$$ 4.00000i 0.154303i
$$673$$ 34.0000i 1.31060i 0.755367 + 0.655302i $$0.227459\pi$$
−0.755367 + 0.655302i $$0.772541\pi$$
$$674$$ −20.0000 −0.770371
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ − 26.0000i − 0.999261i −0.866239 0.499631i $$-0.833469\pi$$
0.866239 0.499631i $$-0.166531\pi$$
$$678$$ 10.0000i 0.384048i
$$679$$ 32.0000 1.22805
$$680$$ 0 0
$$681$$ −8.00000 −0.306561
$$682$$ − 16.0000i − 0.612672i
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ −8.00000 −0.305441
$$687$$ 22.0000i 0.839352i
$$688$$ − 6.00000i − 0.228748i
$$689$$ 56.0000 2.13343
$$690$$ 0 0
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ 10.0000i 0.380143i
$$693$$ − 8.00000i − 0.303895i
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ 8.00000 0.303239
$$697$$ − 36.0000i − 1.36360i
$$698$$ − 30.0000i − 1.13552i
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 4.00000i 0.150970i
$$703$$ − 40.0000i − 1.50863i
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ −14.0000 −0.526897
$$707$$ − 32.0000i − 1.20348i
$$708$$ 4.00000i 0.150329i
$$709$$ −34.0000 −1.27690 −0.638448 0.769665i $$-0.720423\pi$$
−0.638448 + 0.769665i $$0.720423\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 8.00000i 0.299602i
$$714$$ 24.0000 0.898177
$$715$$ 0 0
$$716$$ −4.00000 −0.149487
$$717$$ − 26.0000i − 0.970988i
$$718$$ − 16.0000i − 0.597115i
$$719$$ −46.0000 −1.71551 −0.857755 0.514058i $$-0.828142\pi$$
−0.857755 + 0.514058i $$0.828142\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 3.00000i 0.111648i
$$723$$ 10.0000i 0.371904i
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ −7.00000 −0.259794
$$727$$ 32.0000i 1.18681i 0.804902 + 0.593407i $$0.202218\pi$$
−0.804902 + 0.593407i $$0.797782\pi$$
$$728$$ 16.0000i 0.592999i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −36.0000 −1.33151
$$732$$ − 6.00000i − 0.221766i
$$733$$ − 14.0000i − 0.517102i −0.965998 0.258551i $$-0.916755\pi$$
0.965998 0.258551i $$-0.0832450\pi$$
$$734$$ −12.0000 −0.442928
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ 28.0000i 1.03139i
$$738$$ 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
$$741$$ 16.0000 0.587775
$$742$$ 56.0000i 2.05582i
$$743$$ − 32.0000i − 1.17397i −0.809599 0.586983i $$-0.800316\pi$$
0.809599 0.586983i $$-0.199684\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ 18.0000 0.659027
$$747$$ − 4.00000i − 0.146352i
$$748$$ 12.0000i 0.438763i
$$749$$ −32.0000 −1.16925
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ − 4.00000i − 0.145865i
$$753$$ − 2.00000i − 0.0728841i
$$754$$ 32.0000 1.16537
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ 34.0000i 1.23575i 0.786276 + 0.617876i $$0.212006\pi$$
−0.786276 + 0.617876i $$0.787994\pi$$
$$758$$ − 16.0000i − 0.581146i
$$759$$ −2.00000 −0.0725954
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 2.00000i 0.0724524i
$$763$$ 56.0000i 2.02734i
$$764$$ 16.0000 0.578860
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 16.0000i 0.577727i
$$768$$ 1.00000i 0.0360844i
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ − 22.0000i − 0.791797i
$$773$$ − 42.0000i − 1.51064i −0.655359 0.755318i $$-0.727483\pi$$
0.655359 0.755318i $$-0.272517\pi$$
$$774$$ 6.00000 0.215666
$$775$$ 0 0
$$776$$ 8.00000 0.287183
$$777$$ 40.0000i 1.43499i
$$778$$ − 14.0000i − 0.501924i
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 20.0000 0.715656
$$782$$ − 6.00000i − 0.214560i
$$783$$ 8.00000i 0.285897i
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ 16.0000 0.570701
$$787$$ 50.0000i 1.78231i 0.453701 + 0.891154i $$0.350103\pi$$
−0.453701 + 0.891154i $$0.649897\pi$$
$$788$$ 22.0000i 0.783718i
$$789$$ 16.0000 0.569615
$$790$$ 0 0
$$791$$ −40.0000 −1.42224
$$792$$ − 2.00000i − 0.0710669i
$$793$$ − 24.0000i − 0.852265i
$$794$$ −28.0000 −0.993683
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ 54.0000i 1.91278i 0.292096 + 0.956389i $$0.405647\pi$$
−0.292096 + 0.956389i $$0.594353\pi$$
$$798$$ 16.0000i 0.566394i
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 20.0000i − 0.706225i
$$803$$ − 28.0000i − 0.988099i
$$804$$ 14.0000 0.493742
$$805$$ 0 0
$$806$$ −32.0000 −1.12715
$$807$$ − 12.0000i − 0.422420i
$$808$$ − 8.00000i − 0.281439i
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ −12.0000 −0.421377 −0.210688 0.977553i $$-0.567571\pi$$
−0.210688 + 0.977553i $$0.567571\pi$$
$$812$$ 32.0000i 1.12298i
$$813$$ − 8.00000i − 0.280572i
$$814$$ −20.0000 −0.701000
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ − 24.0000i − 0.839654i
$$818$$ − 10.0000i − 0.349642i
$$819$$ −16.0000 −0.559085
$$820$$ 0 0
$$821$$ 12.0000 0.418803 0.209401 0.977830i $$-0.432848\pi$$
0.209401 + 0.977830i $$0.432848\pi$$
$$822$$ 6.00000i 0.209274i
$$823$$ − 42.0000i − 1.46403i −0.681290 0.732014i $$-0.738581\pi$$
0.681290 0.732014i $$-0.261419\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ −16.0000 −0.556711
$$827$$ − 48.0000i − 1.66912i −0.550914 0.834562i $$-0.685721\pi$$
0.550914 0.834562i $$-0.314279\pi$$
$$828$$ 1.00000i 0.0347524i
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ 0 0
$$831$$ 12.0000 0.416275
$$832$$ 4.00000i 0.138675i
$$833$$ 54.0000i 1.87099i
$$834$$ 20.0000 0.692543
$$835$$ 0 0
$$836$$ −8.00000 −0.276686
$$837$$ − 8.00000i − 0.276520i
$$838$$ − 30.0000i − 1.03633i
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ − 6.00000i − 0.206774i
$$843$$ 8.00000i 0.275535i
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 4.00000 0.137523
$$847$$ − 28.0000i − 0.962091i
$$848$$ 14.0000i 0.480762i
$$849$$ −22.0000 −0.755038
$$850$$ 0 0
$$851$$ 10.0000 0.342796
$$852$$ − 10.0000i − 0.342594i
$$853$$ 4.00000i 0.136957i 0.997653 + 0.0684787i $$0.0218145\pi$$
−0.997653 + 0.0684787i $$0.978185\pi$$
$$854$$ 24.0000 0.821263
$$855$$ 0 0
$$856$$ −8.00000 −0.273434
$$857$$ 6.00000i 0.204956i 0.994735 + 0.102478i $$0.0326771\pi$$
−0.994735 + 0.102478i $$0.967323\pi$$
$$858$$ − 8.00000i − 0.273115i
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ − 8.00000i − 0.272481i
$$863$$ 36.0000i 1.22545i 0.790295 + 0.612727i $$0.209928\pi$$
−0.790295 + 0.612727i $$0.790072\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −32.0000 −1.08740
$$867$$ − 19.0000i − 0.645274i
$$868$$ − 32.0000i − 1.08615i
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 56.0000 1.89749
$$872$$ 14.0000i 0.474100i
$$873$$ 8.00000i 0.270759i
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ −14.0000 −0.473016
$$877$$ 20.0000i 0.675352i 0.941262 + 0.337676i $$0.109641\pi$$
−0.941262 + 0.337676i $$0.890359\pi$$
$$878$$ − 20.0000i − 0.674967i
$$879$$ 26.0000 0.876958
$$880$$ 0 0
$$881$$ −48.0000 −1.61716 −0.808581 0.588386i $$-0.799764\pi$$
−0.808581 + 0.588386i $$0.799764\pi$$
$$882$$ − 9.00000i − 0.303046i
$$883$$ − 8.00000i − 0.269221i −0.990899 0.134611i $$-0.957022\pi$$
0.990899 0.134611i $$-0.0429784\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ 28.0000 0.940678
$$887$$ 12.0000i 0.402921i 0.979497 + 0.201460i $$0.0645687\pi$$
−0.979497 + 0.201460i $$0.935431\pi$$
$$888$$ 10.0000i 0.335578i
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 14.0000i 0.468755i
$$893$$ − 16.0000i − 0.535420i
$$894$$ 10.0000 0.334450
$$895$$ 0 0
$$896$$ −4.00000 −0.133631
$$897$$ 4.00000i 0.133556i
$$898$$ 30.0000i 1.00111i
$$899$$ −64.0000 −2.13452
$$900$$ 0 0
$$901$$ 84.0000 2.79845
$$902$$ − 12.0000i − 0.399556i
$$903$$ 24.0000i 0.798670i
$$904$$ −10.0000 −0.332595
$$905$$ 0 0
$$906$$ 20.0000 0.664455
$$907$$ 46.0000i 1.52740i 0.645568 + 0.763702i $$0.276621\pi$$
−0.645568 + 0.763702i $$0.723379\pi$$
$$908$$ − 8.00000i − 0.265489i
$$909$$ 8.00000 0.265343
$$910$$ 0 0
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ 4.00000i 0.132453i
$$913$$ 8.00000i 0.264761i
$$914$$ 16.0000 0.529233
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ 64.0000i 2.11347i
$$918$$ 6.00000i 0.198030i
$$919$$ 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
$$922$$ 0 0
$$923$$ − 40.0000i − 1.31662i
$$924$$ 8.00000 0.263181
$$925$$ 0 0
$$926$$ 10.0000 0.328620
$$927$$ 4.00000i 0.131377i
$$928$$ 8.00000i 0.262613i
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ −36.0000 −1.17985
$$932$$ − 6.00000i − 0.196537i
$$933$$ − 14.0000i − 0.458339i
$$934$$ −36.0000 −1.17796
$$935$$ 0 0
$$936$$ −4.00000 −0.130744
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ 56.0000i 1.82846i
$$939$$ −4.00000 −0.130535
$$940$$ 0 0
$$941$$ 38.0000 1.23876 0.619382 0.785090i $$-0.287383\pi$$
0.619382 + 0.785090i $$0.287383\pi$$
$$942$$ − 2.00000i − 0.0651635i
$$943$$ 6.00000i 0.195387i
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ − 20.0000i − 0.649913i −0.945729 0.324956i $$-0.894650\pi$$
0.945729 0.324956i $$-0.105350\pi$$
$$948$$ − 8.00000i − 0.259828i
$$949$$ −56.0000 −1.81784
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 24.0000i 0.777844i
$$953$$ − 38.0000i − 1.23094i −0.788160 0.615470i $$-0.788966\pi$$
0.788160 0.615470i $$-0.211034\pi$$
$$954$$ −14.0000 −0.453267
$$955$$ 0 0
$$956$$ 26.0000 0.840900
$$957$$ − 16.0000i − 0.517207i
$$958$$ 24.0000i 0.775405i
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 40.0000i 1.28965i
$$963$$ − 8.00000i − 0.257796i
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ −4.00000 −0.128698
$$967$$ 22.0000i 0.707472i 0.935345 + 0.353736i $$0.115089\pi$$
−0.935345 + 0.353736i $$0.884911\pi$$
$$968$$ − 7.00000i − 0.224989i
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ 6.00000 0.192549 0.0962746 0.995355i $$-0.469307\pi$$
0.0962746 + 0.995355i $$0.469307\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 80.0000i 2.56468i
$$974$$ −10.0000 −0.320421
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ 10.0000i 0.319928i 0.987123 + 0.159964i $$0.0511379\pi$$
−0.987123 + 0.159964i $$0.948862\pi$$
$$978$$ 8.00000i 0.255812i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ 24.0000i 0.765871i
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 48.0000 1.52863
$$987$$ 16.0000i 0.509286i
$$988$$ 16.0000i 0.509028i
$$989$$ 6.00000 0.190789
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ − 8.00000i − 0.254000i
$$993$$ 12.0000i 0.380808i
$$994$$ 40.0000 1.26872
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ 4.00000i 0.126681i 0.997992 + 0.0633406i $$0.0201755\pi$$
−0.997992 + 0.0633406i $$0.979825\pi$$
$$998$$ 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.s.2899.1 2
5.2 odd 4 3450.2.a.u.1.1 1
5.3 odd 4 690.2.a.d.1.1 1
5.4 even 2 inner 3450.2.d.s.2899.2 2
15.8 even 4 2070.2.a.o.1.1 1
20.3 even 4 5520.2.a.bb.1.1 1

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