# Properties

 Label 3450.2.d.j.2899.2 Level $3450$ Weight $2$ Character 3450.2899 Analytic conductor $27.548$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.5483886973$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 138) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2899.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3450.2899 Dual form 3450.2.d.j.2899.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} +2.00000 q^{21} -6.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} +6.00000i q^{33} +1.00000 q^{36} -2.00000 q^{39} +10.0000 q^{41} +2.00000i q^{42} -12.0000i q^{43} +6.00000 q^{44} +1.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +2.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -6.00000i q^{58} +12.0000 q^{59} +4.00000 q^{61} +8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} +12.0000i q^{67} -1.00000 q^{69} +1.00000i q^{72} -10.0000i q^{73} -12.0000i q^{77} -2.00000i q^{78} +6.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +14.0000i q^{83} -2.00000 q^{84} +12.0000 q^{86} +6.00000i q^{87} +6.00000i q^{88} +4.00000 q^{91} +1.00000i q^{92} -8.00000i q^{93} -8.00000 q^{94} +1.00000 q^{96} +6.00000i q^{97} +3.00000i q^{98} +6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 12 q^{11} - 4 q^{14} + 2 q^{16} + 4 q^{21} - 2 q^{24} + 4 q^{26} - 12 q^{29} + 16 q^{31} + 2 q^{36} - 4 q^{39} + 20 q^{41} + 12 q^{44} + 2 q^{46} + 6 q^{49} - 2 q^{54} + 4 q^{56} + 24 q^{59} + 8 q^{61} - 2 q^{64} - 12 q^{66} - 2 q^{69} + 12 q^{79} + 2 q^{81} - 4 q^{84} + 24 q^{86} + 8 q^{91} - 16 q^{94} + 2 q^{96} + 12 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ − 6.00000i − 1.27920i
$$23$$ − 1.00000i − 0.208514i
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ − 2.00000i − 0.377964i
$$29$$ −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
$$33$$ 6.00000i 1.04447i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 2.00000i 0.308607i
$$43$$ − 12.0000i − 1.82998i −0.403473 0.914991i $$-0.632197\pi$$
0.403473 0.914991i $$-0.367803\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ 8.00000i 1.16692i 0.812142 + 0.583460i $$0.198301\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000i 0.277350i
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ 0 0
$$58$$ − 6.00000i − 0.787839i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 8.00000i 1.01600i
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −6.00000 −0.738549
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ 0 0
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 10.0000i − 1.17041i −0.810885 0.585206i $$-0.801014\pi$$
0.810885 0.585206i $$-0.198986\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 12.0000i − 1.36753i
$$78$$ − 2.00000i − 0.226455i
$$79$$ 6.00000 0.675053 0.337526 0.941316i $$-0.390410\pi$$
0.337526 + 0.941316i $$0.390410\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.0000i 1.10432i
$$83$$ 14.0000i 1.53670i 0.640030 + 0.768350i $$0.278922\pi$$
−0.640030 + 0.768350i $$0.721078\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 12.0000 1.29399
$$87$$ 6.00000i 0.643268i
$$88$$ 6.00000i 0.639602i
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 1.00000i 0.104257i
$$93$$ − 8.00000i − 0.829561i
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 6.00000i 0.609208i 0.952479 + 0.304604i $$0.0985241\pi$$
−0.952479 + 0.304604i $$0.901476\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 14.0000i 1.37946i 0.724066 + 0.689730i $$0.242271\pi$$
−0.724066 + 0.689730i $$0.757729\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ − 14.0000i − 1.35343i −0.736245 0.676716i $$-0.763403\pi$$
0.736245 0.676716i $$-0.236597\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 16.0000 1.53252 0.766261 0.642529i $$-0.222115\pi$$
0.766261 + 0.642529i $$0.222115\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 2.00000i 0.188982i
$$113$$ − 8.00000i − 0.752577i −0.926503 0.376288i $$-0.877200\pi$$
0.926503 0.376288i $$-0.122800\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 2.00000i 0.184900i
$$118$$ 12.0000i 1.10469i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 4.00000i 0.362143i
$$123$$ − 10.0000i − 0.901670i
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ − 12.0000i − 1.06483i −0.846484 0.532414i $$-0.821285\pi$$
0.846484 0.532414i $$-0.178715\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −12.0000 −1.05654
$$130$$ 0 0
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ − 6.00000i − 0.522233i
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.0000i 1.02523i 0.858619 + 0.512615i $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$138$$ − 1.00000i − 0.0851257i
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 8.00000 0.673722
$$142$$ 0 0
$$143$$ 12.0000i 1.00349i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 10.0000 0.827606
$$147$$ − 3.00000i − 0.247436i
$$148$$ 0 0
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 12.0000 0.966988
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ − 16.0000i − 1.27694i −0.769647 0.638470i $$-0.779568\pi$$
0.769647 0.638470i $$-0.220432\pi$$
$$158$$ 6.00000i 0.477334i
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ 2.00000 0.157622
$$162$$ 1.00000i 0.0785674i
$$163$$ − 12.0000i − 0.939913i −0.882690 0.469956i $$-0.844270\pi$$
0.882690 0.469956i $$-0.155730\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ −14.0000 −1.08661
$$167$$ − 8.00000i − 0.619059i −0.950890 0.309529i $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ − 2.00000i − 0.154303i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 12.0000i 0.914991i
$$173$$ − 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
$$176$$ −6.00000 −0.452267
$$177$$ − 12.0000i − 0.901975i
$$178$$ 0 0
$$179$$ 16.0000 1.19590 0.597948 0.801535i $$-0.295983\pi$$
0.597948 + 0.801535i $$0.295983\pi$$
$$180$$ 0 0
$$181$$ 8.00000 0.594635 0.297318 0.954779i $$-0.403908\pi$$
0.297318 + 0.954779i $$0.403908\pi$$
$$182$$ 4.00000i 0.296500i
$$183$$ − 4.00000i − 0.295689i
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ 0 0
$$188$$ − 8.00000i − 0.583460i
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 6.00000i 0.426401i
$$199$$ −2.00000 −0.141776 −0.0708881 0.997484i $$-0.522583\pi$$
−0.0708881 + 0.997484i $$0.522583\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ − 6.00000i − 0.422159i
$$203$$ − 12.0000i − 0.842235i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ 1.00000i 0.0695048i
$$208$$ − 2.00000i − 0.138675i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ − 2.00000i − 0.137361i
$$213$$ 0 0
$$214$$ 14.0000 0.957020
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 16.0000i 1.08615i
$$218$$ 16.0000i 1.08366i
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 12.0000i 0.803579i 0.915732 + 0.401790i $$0.131612\pi$$
−0.915732 + 0.401790i $$0.868388\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ 8.00000 0.532152
$$227$$ 10.0000i 0.663723i 0.943328 + 0.331862i $$0.107677\pi$$
−0.943328 + 0.331862i $$0.892323\pi$$
$$228$$ 0 0
$$229$$ 24.0000 1.58596 0.792982 0.609245i $$-0.208527\pi$$
0.792982 + 0.609245i $$0.208527\pi$$
$$230$$ 0 0
$$231$$ −12.0000 −0.789542
$$232$$ 6.00000i 0.393919i
$$233$$ 10.0000i 0.655122i 0.944830 + 0.327561i $$0.106227\pi$$
−0.944830 + 0.327561i $$0.893773\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ − 6.00000i − 0.389742i
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ −6.00000 −0.386494 −0.193247 0.981150i $$-0.561902\pi$$
−0.193247 + 0.981150i $$0.561902\pi$$
$$242$$ 25.0000i 1.60706i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −4.00000 −0.256074
$$245$$ 0 0
$$246$$ 10.0000 0.637577
$$247$$ 0 0
$$248$$ − 8.00000i − 0.508001i
$$249$$ 14.0000 0.887214
$$250$$ 0 0
$$251$$ −6.00000 −0.378717 −0.189358 0.981908i $$-0.560641\pi$$
−0.189358 + 0.981908i $$0.560641\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 6.00000i 0.377217i
$$254$$ 12.0000 0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000i 0.374270i 0.982334 + 0.187135i $$0.0599201\pi$$
−0.982334 + 0.187135i $$0.940080\pi$$
$$258$$ − 12.0000i − 0.747087i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ − 8.00000i − 0.494242i
$$263$$ 4.00000i 0.246651i 0.992366 + 0.123325i $$0.0393559\pi$$
−0.992366 + 0.123325i $$0.960644\pi$$
$$264$$ 6.00000 0.369274
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ − 12.0000i − 0.733017i
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 28.0000 1.70088 0.850439 0.526073i $$-0.176336\pi$$
0.850439 + 0.526073i $$0.176336\pi$$
$$272$$ 0 0
$$273$$ − 4.00000i − 0.242091i
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ − 26.0000i − 1.56219i −0.624413 0.781094i $$-0.714662\pi$$
0.624413 0.781094i $$-0.285338\pi$$
$$278$$ − 12.0000i − 0.719712i
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 16.0000 0.954480 0.477240 0.878773i $$-0.341637\pi$$
0.477240 + 0.878773i $$0.341637\pi$$
$$282$$ 8.00000i 0.476393i
$$283$$ 24.0000i 1.42665i 0.700832 + 0.713326i $$0.252812\pi$$
−0.700832 + 0.713326i $$0.747188\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −12.0000 −0.709575
$$287$$ 20.0000i 1.18056i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ 6.00000 0.351726
$$292$$ 10.0000i 0.585206i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 6.00000i − 0.348155i
$$298$$ 18.0000i 1.04271i
$$299$$ −2.00000 −0.115663
$$300$$ 0 0
$$301$$ 24.0000 1.38334
$$302$$ − 12.0000i − 0.690522i
$$303$$ 6.00000i 0.344691i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 20.0000i 1.14146i 0.821138 + 0.570730i $$0.193340\pi$$
−0.821138 + 0.570730i $$0.806660\pi$$
$$308$$ 12.0000i 0.683763i
$$309$$ 14.0000 0.796432
$$310$$ 0 0
$$311$$ 16.0000 0.907277 0.453638 0.891186i $$-0.350126\pi$$
0.453638 + 0.891186i $$0.350126\pi$$
$$312$$ 2.00000i 0.113228i
$$313$$ − 26.0000i − 1.46961i −0.678280 0.734803i $$-0.737274\pi$$
0.678280 0.734803i $$-0.262726\pi$$
$$314$$ 16.0000 0.902932
$$315$$ 0 0
$$316$$ −6.00000 −0.337526
$$317$$ − 22.0000i − 1.23564i −0.786318 0.617822i $$-0.788015\pi$$
0.786318 0.617822i $$-0.211985\pi$$
$$318$$ 2.00000i 0.112154i
$$319$$ 36.0000 2.01561
$$320$$ 0 0
$$321$$ −14.0000 −0.781404
$$322$$ 2.00000i 0.111456i
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 12.0000 0.664619
$$327$$ − 16.0000i − 0.884802i
$$328$$ − 10.0000i − 0.552158i
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ − 14.0000i − 0.768350i
$$333$$ 0 0
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 10.0000i 0.544735i 0.962193 + 0.272367i $$0.0878066\pi$$
−0.962193 + 0.272367i $$0.912193\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ −8.00000 −0.434500
$$340$$ 0 0
$$341$$ −48.0000 −2.59935
$$342$$ 0 0
$$343$$ 20.0000i 1.07990i
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 8.00000i 0.429463i 0.976673 + 0.214731i $$0.0688876\pi$$
−0.976673 + 0.214731i $$0.931112\pi$$
$$348$$ − 6.00000i − 0.321634i
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ − 6.00000i − 0.319801i
$$353$$ − 2.00000i − 0.106449i −0.998583 0.0532246i $$-0.983050\pi$$
0.998583 0.0532246i $$-0.0169499\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 16.0000i 0.845626i
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 8.00000i 0.420471i
$$363$$ − 25.0000i − 1.31216i
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 4.00000 0.209083
$$367$$ 14.0000i 0.730794i 0.930852 + 0.365397i $$0.119067\pi$$
−0.930852 + 0.365397i $$0.880933\pi$$
$$368$$ − 1.00000i − 0.0521286i
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ −4.00000 −0.207670
$$372$$ 8.00000i 0.414781i
$$373$$ − 32.0000i − 1.65690i −0.560065 0.828449i $$-0.689224\pi$$
0.560065 0.828449i $$-0.310776\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 12.0000i 0.618031i
$$378$$ − 2.00000i − 0.102869i
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 0 0
$$383$$ 4.00000i 0.204390i 0.994764 + 0.102195i $$0.0325866\pi$$
−0.994764 + 0.102195i $$0.967413\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 12.0000i 0.609994i
$$388$$ − 6.00000i − 0.304604i
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 3.00000i − 0.151523i
$$393$$ 8.00000i 0.403547i
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ −6.00000 −0.301511
$$397$$ − 6.00000i − 0.301131i −0.988600 0.150566i $$-0.951890\pi$$
0.988600 0.150566i $$-0.0481095\pi$$
$$398$$ − 2.00000i − 0.100251i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −32.0000 −1.59800 −0.799002 0.601329i $$-0.794638\pi$$
−0.799002 + 0.601329i $$0.794638\pi$$
$$402$$ 12.0000i 0.598506i
$$403$$ − 16.0000i − 0.797017i
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 12.0000 0.595550
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ − 14.0000i − 0.689730i
$$413$$ 24.0000i 1.18096i
$$414$$ −1.00000 −0.0491473
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 12.0000i 0.587643i
$$418$$ 0 0
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ 4.00000 0.194948 0.0974740 0.995238i $$-0.468924\pi$$
0.0974740 + 0.995238i $$0.468924\pi$$
$$422$$ 20.0000i 0.973585i
$$423$$ − 8.00000i − 0.388973i
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 8.00000i 0.387147i
$$428$$ 14.0000i 0.676716i
$$429$$ 12.0000 0.579365
$$430$$ 0 0
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 2.00000i − 0.0961139i −0.998845 0.0480569i $$-0.984697\pi$$
0.998845 0.0480569i $$-0.0153029\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 0 0
$$436$$ −16.0000 −0.766261
$$437$$ 0 0
$$438$$ − 10.0000i − 0.477818i
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ 36.0000i 1.71041i 0.518289 + 0.855206i $$0.326569\pi$$
−0.518289 + 0.855206i $$0.673431\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −12.0000 −0.568216
$$447$$ − 18.0000i − 0.851371i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 0 0
$$451$$ −60.0000 −2.82529
$$452$$ 8.00000i 0.376288i
$$453$$ 12.0000i 0.563809i
$$454$$ −10.0000 −0.469323
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 26.0000i − 1.21623i −0.793849 0.608114i $$-0.791926\pi$$
0.793849 0.608114i $$-0.208074\pi$$
$$458$$ 24.0000i 1.12145i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ − 12.0000i − 0.558291i
$$463$$ − 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
$$466$$ −10.0000 −0.463241
$$467$$ − 18.0000i − 0.832941i −0.909149 0.416470i $$-0.863267\pi$$
0.909149 0.416470i $$-0.136733\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ −24.0000 −1.10822
$$470$$ 0 0
$$471$$ −16.0000 −0.737241
$$472$$ − 12.0000i − 0.552345i
$$473$$ 72.0000i 3.31056i
$$474$$ 6.00000 0.275589
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 2.00000i − 0.0915737i
$$478$$ 16.0000i 0.731823i
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ − 6.00000i − 0.273293i
$$483$$ − 2.00000i − 0.0910032i
$$484$$ −25.0000 −1.13636
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 8.00000i − 0.362515i −0.983436 0.181257i $$-0.941983\pi$$
0.983436 0.181257i $$-0.0580167\pi$$
$$488$$ − 4.00000i − 0.181071i
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ −32.0000 −1.44414 −0.722070 0.691820i $$-0.756809\pi$$
−0.722070 + 0.691820i $$0.756809\pi$$
$$492$$ 10.0000i 0.450835i
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ 14.0000i 0.627355i
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 0 0
$$501$$ −8.00000 −0.357414
$$502$$ − 6.00000i − 0.267793i
$$503$$ − 12.0000i − 0.535054i −0.963550 0.267527i $$-0.913794\pi$$
0.963550 0.267527i $$-0.0862064\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ −6.00000 −0.266733
$$507$$ − 9.00000i − 0.399704i
$$508$$ 12.0000i 0.532414i
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ 20.0000 0.884748
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ 0 0
$$516$$ 12.0000 0.528271
$$517$$ − 48.0000i − 2.11104i
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ 4.00000 0.175243 0.0876216 0.996154i $$-0.472073\pi$$
0.0876216 + 0.996154i $$0.472073\pi$$
$$522$$ 6.00000i 0.262613i
$$523$$ − 4.00000i − 0.174908i −0.996169 0.0874539i $$-0.972127\pi$$
0.996169 0.0874539i $$-0.0278730\pi$$
$$524$$ 8.00000 0.349482
$$525$$ 0 0
$$526$$ −4.00000 −0.174408
$$527$$ 0 0
$$528$$ 6.00000i 0.261116i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ − 20.0000i − 0.866296i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ − 16.0000i − 0.690451i
$$538$$ − 18.0000i − 0.776035i
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ 26.0000 1.11783 0.558914 0.829226i $$-0.311218\pi$$
0.558914 + 0.829226i $$0.311218\pi$$
$$542$$ 28.0000i 1.20270i
$$543$$ − 8.00000i − 0.343313i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 4.00000 0.171184
$$547$$ − 28.0000i − 1.19719i −0.801050 0.598597i $$-0.795725\pi$$
0.801050 0.598597i $$-0.204275\pi$$
$$548$$ − 12.0000i − 0.512615i
$$549$$ −4.00000 −0.170716
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 1.00000i 0.0425628i
$$553$$ 12.0000i 0.510292i
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ 12.0000 0.508913
$$557$$ − 6.00000i − 0.254228i −0.991888 0.127114i $$-0.959429\pi$$
0.991888 0.127114i $$-0.0405714\pi$$
$$558$$ − 8.00000i − 0.338667i
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 16.0000i 0.674919i
$$563$$ 14.0000i 0.590030i 0.955493 + 0.295015i $$0.0953246\pi$$
−0.955493 + 0.295015i $$0.904675\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ −24.0000 −1.00880
$$567$$ 2.00000i 0.0839921i
$$568$$ 0 0
$$569$$ −16.0000 −0.670755 −0.335377 0.942084i $$-0.608864\pi$$
−0.335377 + 0.942084i $$0.608864\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ − 12.0000i − 0.501745i
$$573$$ 0 0
$$574$$ −20.0000 −0.834784
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ 17.0000i 0.707107i
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ −28.0000 −1.16164
$$582$$ 6.00000i 0.248708i
$$583$$ − 12.0000i − 0.496989i
$$584$$ −10.0000 −0.413803
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ − 16.0000i − 0.660391i −0.943913 0.330195i $$-0.892885\pi$$
0.943913 0.330195i $$-0.107115\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 0 0
$$593$$ − 6.00000i − 0.246390i −0.992382 0.123195i $$-0.960686\pi$$
0.992382 0.123195i $$-0.0393141\pi$$
$$594$$ 6.00000 0.246183
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ 2.00000i 0.0818546i
$$598$$ − 2.00000i − 0.0817861i
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 24.0000i 0.978167i
$$603$$ − 12.0000i − 0.488678i
$$604$$ 12.0000 0.488273
$$605$$ 0 0
$$606$$ −6.00000 −0.243733
$$607$$ 4.00000i 0.162355i 0.996700 + 0.0811775i $$0.0258681\pi$$
−0.996700 + 0.0811775i $$0.974132\pi$$
$$608$$ 0 0
$$609$$ −12.0000 −0.486265
$$610$$ 0 0
$$611$$ 16.0000 0.647291
$$612$$ 0 0
$$613$$ − 4.00000i − 0.161558i −0.996732 0.0807792i $$-0.974259\pi$$
0.996732 0.0807792i $$-0.0257409\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ −12.0000 −0.483494
$$617$$ 32.0000i 1.28827i 0.764911 + 0.644136i $$0.222783\pi$$
−0.764911 + 0.644136i $$0.777217\pi$$
$$618$$ 14.0000i 0.563163i
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 16.0000i 0.641542i
$$623$$ 0 0
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ 26.0000 1.03917
$$627$$ 0 0
$$628$$ 16.0000i 0.638470i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −2.00000 −0.0796187 −0.0398094 0.999207i $$-0.512675\pi$$
−0.0398094 + 0.999207i $$0.512675\pi$$
$$632$$ − 6.00000i − 0.238667i
$$633$$ − 20.0000i − 0.794929i
$$634$$ 22.0000 0.873732
$$635$$ 0 0
$$636$$ −2.00000 −0.0793052
$$637$$ − 6.00000i − 0.237729i
$$638$$ 36.0000i 1.42525i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 16.0000 0.631962 0.315981 0.948766i $$-0.397666\pi$$
0.315981 + 0.948766i $$0.397666\pi$$
$$642$$ − 14.0000i − 0.552536i
$$643$$ 44.0000i 1.73519i 0.497271 + 0.867595i $$0.334335\pi$$
−0.497271 + 0.867595i $$0.665665\pi$$
$$644$$ −2.00000 −0.0788110
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 40.0000i 1.57256i 0.617869 + 0.786281i $$0.287996\pi$$
−0.617869 + 0.786281i $$0.712004\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −72.0000 −2.82625
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ 12.0000i 0.469956i
$$653$$ − 22.0000i − 0.860927i −0.902608 0.430463i $$-0.858350\pi$$
0.902608 0.430463i $$-0.141650\pi$$
$$654$$ 16.0000 0.625650
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ 10.0000i 0.390137i
$$658$$ − 16.0000i − 0.623745i
$$659$$ −42.0000 −1.63609 −0.818044 0.575156i $$-0.804941\pi$$
−0.818044 + 0.575156i $$0.804941\pi$$
$$660$$ 0 0
$$661$$ −4.00000 −0.155582 −0.0777910 0.996970i $$-0.524787\pi$$
−0.0777910 + 0.996970i $$0.524787\pi$$
$$662$$ 4.00000i 0.155464i
$$663$$ 0 0
$$664$$ 14.0000 0.543305
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 6.00000i 0.232321i
$$668$$ 8.00000i 0.309529i
$$669$$ 12.0000 0.463947
$$670$$ 0 0
$$671$$ −24.0000 −0.926510
$$672$$ 2.00000i 0.0771517i
$$673$$ − 46.0000i − 1.77317i −0.462566 0.886585i $$-0.653071\pi$$
0.462566 0.886585i $$-0.346929\pi$$
$$674$$ −10.0000 −0.385186
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 6.00000i − 0.230599i −0.993331 0.115299i $$-0.963217\pi$$
0.993331 0.115299i $$-0.0367827\pi$$
$$678$$ − 8.00000i − 0.307238i
$$679$$ −12.0000 −0.460518
$$680$$ 0 0
$$681$$ 10.0000 0.383201
$$682$$ − 48.0000i − 1.83801i
$$683$$ − 24.0000i − 0.918334i −0.888350 0.459167i $$-0.848148\pi$$
0.888350 0.459167i $$-0.151852\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ − 24.0000i − 0.915657i
$$688$$ − 12.0000i − 0.457496i
$$689$$ 4.00000 0.152388
$$690$$ 0 0
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ 18.0000i 0.684257i
$$693$$ 12.0000i 0.455842i
$$694$$ −8.00000 −0.303676
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ 0 0
$$698$$ − 26.0000i − 0.984115i
$$699$$ 10.0000 0.378235
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ 0 0
$$704$$ 6.00000 0.226134
$$705$$ 0 0
$$706$$ 2.00000 0.0752710
$$707$$ − 12.0000i − 0.451306i
$$708$$ 12.0000i 0.450988i
$$709$$ −4.00000 −0.150223 −0.0751116 0.997175i $$-0.523931\pi$$
−0.0751116 + 0.997175i $$0.523931\pi$$
$$710$$ 0 0
$$711$$ −6.00000 −0.225018
$$712$$ 0 0
$$713$$ − 8.00000i − 0.299602i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −16.0000 −0.597948
$$717$$ − 16.0000i − 0.597531i
$$718$$ − 24.0000i − 0.895672i
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ 0 0
$$721$$ −28.0000 −1.04277
$$722$$ − 19.0000i − 0.707107i
$$723$$ 6.00000i 0.223142i
$$724$$ −8.00000 −0.297318
$$725$$ 0 0
$$726$$ 25.0000 0.927837
$$727$$ 18.0000i 0.667583i 0.942647 + 0.333792i $$0.108328\pi$$
−0.942647 + 0.333792i $$0.891672\pi$$
$$728$$ − 4.00000i − 0.148250i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 4.00000i 0.147844i
$$733$$ 36.0000i 1.32969i 0.746981 + 0.664845i $$0.231502\pi$$
−0.746981 + 0.664845i $$0.768498\pi$$
$$734$$ −14.0000 −0.516749
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ − 72.0000i − 2.65215i
$$738$$ − 10.0000i − 0.368105i
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 4.00000i − 0.146845i
$$743$$ 4.00000i 0.146746i 0.997305 + 0.0733729i $$0.0233763\pi$$
−0.997305 + 0.0733729i $$0.976624\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ 32.0000 1.17160
$$747$$ − 14.0000i − 0.512233i
$$748$$ 0 0
$$749$$ 28.0000 1.02310
$$750$$ 0 0
$$751$$ −38.0000 −1.38664 −0.693320 0.720630i $$-0.743853\pi$$
−0.693320 + 0.720630i $$0.743853\pi$$
$$752$$ 8.00000i 0.291730i
$$753$$ 6.00000i 0.218652i
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ 32.0000i 1.16306i 0.813525 + 0.581530i $$0.197546\pi$$
−0.813525 + 0.581530i $$0.802454\pi$$
$$758$$ 0 0
$$759$$ 6.00000 0.217786
$$760$$ 0 0
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ − 12.0000i − 0.434714i
$$763$$ 32.0000i 1.15848i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −4.00000 −0.144526
$$767$$ − 24.0000i − 0.866590i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ − 14.0000i − 0.503871i
$$773$$ − 30.0000i − 1.07903i −0.841978 0.539513i $$-0.818609\pi$$
0.841978 0.539513i $$-0.181391\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 0 0
$$778$$ − 6.00000i − 0.215110i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ − 6.00000i − 0.214423i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ −8.00000 −0.285351
$$787$$ 28.0000i 0.998092i 0.866575 + 0.499046i $$0.166316\pi$$
−0.866575 + 0.499046i $$0.833684\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 4.00000 0.142404
$$790$$ 0 0
$$791$$ 16.0000 0.568895
$$792$$ − 6.00000i − 0.213201i
$$793$$ − 8.00000i − 0.284088i
$$794$$ 6.00000 0.212932
$$795$$ 0 0
$$796$$ 2.00000 0.0708881
$$797$$ 18.0000i 0.637593i 0.947823 + 0.318796i $$0.103279\pi$$
−0.947823 + 0.318796i $$0.896721\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 32.0000i − 1.12996i
$$803$$ 60.0000i 2.11735i
$$804$$ −12.0000 −0.423207
$$805$$ 0 0
$$806$$ 16.0000 0.563576
$$807$$ 18.0000i 0.633630i
$$808$$ 6.00000i 0.211079i
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 12.0000i 0.421117i
$$813$$ − 28.0000i − 0.982003i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 10.0000i 0.349642i
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ −6.00000 −0.209401 −0.104701 0.994504i $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ 12.0000i 0.418548i
$$823$$ − 24.0000i − 0.836587i −0.908312 0.418294i $$-0.862628\pi$$
0.908312 0.418294i $$-0.137372\pi$$
$$824$$ 14.0000 0.487713
$$825$$ 0 0
$$826$$ −24.0000 −0.835067
$$827$$ − 18.0000i − 0.625921i −0.949766 0.312961i $$-0.898679\pi$$
0.949766 0.312961i $$-0.101321\pi$$
$$828$$ − 1.00000i − 0.0347524i
$$829$$ −42.0000 −1.45872 −0.729360 0.684130i $$-0.760182\pi$$
−0.729360 + 0.684130i $$0.760182\pi$$
$$830$$ 0 0
$$831$$ −26.0000 −0.901930
$$832$$ 2.00000i 0.0693375i
$$833$$ 0 0
$$834$$ −12.0000 −0.415526
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 8.00000i 0.276520i
$$838$$ 26.0000i 0.898155i
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 4.00000i 0.137849i
$$843$$ − 16.0000i − 0.551069i
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ 50.0000i 1.71802i
$$848$$ 2.00000i 0.0686803i
$$849$$ 24.0000 0.823678
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 6.00000i 0.205436i 0.994711 + 0.102718i $$0.0327539\pi$$
−0.994711 + 0.102718i $$0.967246\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ 0 0
$$856$$ −14.0000 −0.478510
$$857$$ − 2.00000i − 0.0683187i −0.999416 0.0341593i $$-0.989125\pi$$
0.999416 0.0341593i $$-0.0108754\pi$$
$$858$$ 12.0000i 0.409673i
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ 20.0000 0.681598
$$862$$ − 12.0000i − 0.408722i
$$863$$ 32.0000i 1.08929i 0.838666 + 0.544646i $$0.183336\pi$$
−0.838666 + 0.544646i $$0.816664\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 2.00000 0.0679628
$$867$$ − 17.0000i − 0.577350i
$$868$$ − 16.0000i − 0.543075i
$$869$$ −36.0000 −1.22122
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ − 16.0000i − 0.541828i
$$873$$ − 6.00000i − 0.203069i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 10.0000 0.337869
$$877$$ − 10.0000i − 0.337676i −0.985644 0.168838i $$-0.945999\pi$$
0.985644 0.168838i $$-0.0540015\pi$$
$$878$$ 16.0000i 0.539974i
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −32.0000 −1.07811 −0.539054 0.842271i $$-0.681218\pi$$
−0.539054 + 0.842271i $$0.681218\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ 36.0000i 1.21150i 0.795656 + 0.605748i $$0.207126\pi$$
−0.795656 + 0.605748i $$0.792874\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ 16.0000i 0.537227i 0.963248 + 0.268614i $$0.0865655\pi$$
−0.963248 + 0.268614i $$0.913434\pi$$
$$888$$ 0 0
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ −6.00000 −0.201008
$$892$$ − 12.0000i − 0.401790i
$$893$$ 0 0
$$894$$ 18.0000 0.602010
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ 2.00000i 0.0667781i
$$898$$ − 2.00000i − 0.0667409i
$$899$$ −48.0000 −1.60089
$$900$$ 0 0
$$901$$ 0 0
$$902$$ − 60.0000i − 1.99778i
$$903$$ − 24.0000i − 0.798670i
$$904$$ −8.00000 −0.266076
$$905$$ 0 0
$$906$$ −12.0000 −0.398673
$$907$$ − 16.0000i − 0.531271i −0.964073 0.265636i $$-0.914418\pi$$
0.964073 0.265636i $$-0.0855818\pi$$
$$908$$ − 10.0000i − 0.331862i
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ −20.0000 −0.662630 −0.331315 0.943520i $$-0.607492\pi$$
−0.331315 + 0.943520i $$0.607492\pi$$
$$912$$ 0 0
$$913$$ − 84.0000i − 2.77999i
$$914$$ 26.0000 0.860004
$$915$$ 0 0
$$916$$ −24.0000 −0.792982
$$917$$ − 16.0000i − 0.528367i
$$918$$ 0 0
$$919$$ −10.0000 −0.329870 −0.164935 0.986304i $$-0.552741\pi$$
−0.164935 + 0.986304i $$0.552741\pi$$
$$920$$ 0 0
$$921$$ 20.0000 0.659022
$$922$$ − 30.0000i − 0.987997i
$$923$$ 0 0
$$924$$ 12.0000 0.394771
$$925$$ 0 0
$$926$$ 40.0000 1.31448
$$927$$ − 14.0000i − 0.459820i
$$928$$ − 6.00000i − 0.196960i
$$929$$ 54.0000 1.77168 0.885841 0.463988i $$-0.153582\pi$$
0.885841 + 0.463988i $$0.153582\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 10.0000i − 0.327561i
$$933$$ − 16.0000i − 0.523816i
$$934$$ 18.0000 0.588978
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ − 2.00000i − 0.0653372i −0.999466 0.0326686i $$-0.989599\pi$$
0.999466 0.0326686i $$-0.0104006\pi$$
$$938$$ − 24.0000i − 0.783628i
$$939$$ −26.0000 −0.848478
$$940$$ 0 0
$$941$$ 54.0000 1.76035 0.880175 0.474650i $$-0.157425\pi$$
0.880175 + 0.474650i $$0.157425\pi$$
$$942$$ − 16.0000i − 0.521308i
$$943$$ − 10.0000i − 0.325645i
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ −72.0000 −2.34092
$$947$$ − 8.00000i − 0.259965i −0.991516 0.129983i $$-0.958508\pi$$
0.991516 0.129983i $$-0.0414921\pi$$
$$948$$ 6.00000i 0.194871i
$$949$$ −20.0000 −0.649227
$$950$$ 0 0
$$951$$ −22.0000 −0.713399
$$952$$ 0 0
$$953$$ − 24.0000i − 0.777436i −0.921357 0.388718i $$-0.872918\pi$$
0.921357 0.388718i $$-0.127082\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ 0 0
$$956$$ −16.0000 −0.517477
$$957$$ − 36.0000i − 1.16371i
$$958$$ − 24.0000i − 0.775405i
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 14.0000i 0.451144i
$$964$$ 6.00000 0.193247
$$965$$ 0 0
$$966$$ 2.00000 0.0643489
$$967$$ 8.00000i 0.257263i 0.991692 + 0.128631i $$0.0410584\pi$$
−0.991692 + 0.128631i $$0.958942\pi$$
$$968$$ − 25.0000i − 0.803530i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −10.0000 −0.320915 −0.160458 0.987043i $$-0.551297\pi$$
−0.160458 + 0.987043i $$0.551297\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ − 24.0000i − 0.769405i
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ 4.00000 0.128037
$$977$$ − 12.0000i − 0.383914i −0.981403 0.191957i $$-0.938517\pi$$
0.981403 0.191957i $$-0.0614834\pi$$
$$978$$ − 12.0000i − 0.383718i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ − 32.0000i − 1.02116i
$$983$$ 52.0000i 1.65854i 0.558846 + 0.829271i $$0.311244\pi$$
−0.558846 + 0.829271i $$0.688756\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 16.0000i 0.509286i
$$988$$ 0 0
$$989$$ −12.0000 −0.381578
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 8.00000i 0.254000i
$$993$$ − 4.00000i − 0.126936i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −14.0000 −0.443607
$$997$$ − 10.0000i − 0.316703i −0.987383 0.158352i $$-0.949382\pi$$
0.987383 0.158352i $$-0.0506179\pi$$
$$998$$ 20.0000i 0.633089i
$$999$$ 0 0
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.j.2899.2 2
5.2 odd 4 138.2.a.a.1.1 1
5.3 odd 4 3450.2.a.y.1.1 1
5.4 even 2 inner 3450.2.d.j.2899.1 2
15.2 even 4 414.2.a.d.1.1 1
20.7 even 4 1104.2.a.e.1.1 1
35.27 even 4 6762.2.a.q.1.1 1
40.27 even 4 4416.2.a.m.1.1 1
40.37 odd 4 4416.2.a.z.1.1 1
60.47 odd 4 3312.2.a.n.1.1 1
115.22 even 4 3174.2.a.b.1.1 1
345.137 odd 4 9522.2.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.a.1.1 1 5.2 odd 4
414.2.a.d.1.1 1 15.2 even 4
1104.2.a.e.1.1 1 20.7 even 4
3174.2.a.b.1.1 1 115.22 even 4
3312.2.a.n.1.1 1 60.47 odd 4
3450.2.a.y.1.1 1 5.3 odd 4
3450.2.d.j.2899.1 2 5.4 even 2 inner
3450.2.d.j.2899.2 2 1.1 even 1 trivial
4416.2.a.m.1.1 1 40.27 even 4
4416.2.a.z.1.1 1 40.37 odd 4
6762.2.a.q.1.1 1 35.27 even 4
9522.2.a.i.1.1 1 345.137 odd 4