# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$27.5483886973$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 690) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

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

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} +1.00000i q^{12} +4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{21} -2.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} +1.00000i q^{27} +4.00000i q^{28} +4.00000 q^{29} +1.00000i q^{32} +2.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} +6.00000 q^{41} -4.00000i q^{42} +2.00000i q^{43} +2.00000 q^{44} -1.00000 q^{46} -12.0000i q^{47} -1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} +4.00000i q^{58} -12.0000 q^{59} -14.0000 q^{61} +4.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -2.00000i q^{67} +2.00000i q^{68} +1.00000 q^{69} -2.00000 q^{71} +1.00000i q^{72} +6.00000i q^{73} +10.0000 q^{74} +8.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +8.00000i q^{83} +4.00000 q^{84} -2.00000 q^{86} -4.00000i q^{87} +2.00000i q^{88} +8.00000 q^{89} -1.00000i q^{92} +12.0000 q^{94} +1.00000 q^{96} -9.00000i q^{98} +2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} - 4q^{11} + 8q^{14} + 2q^{16} - 8q^{21} - 2q^{24} + 8q^{29} + 4q^{34} + 2q^{36} + 12q^{41} + 4q^{44} - 2q^{46} - 18q^{49} - 4q^{51} - 2q^{54} - 8q^{56} - 24q^{59} - 28q^{61} - 2q^{64} - 4q^{66} + 2q^{69} - 4q^{71} + 20q^{74} - 16q^{79} + 2q^{81} + 8q^{84} - 4q^{86} + 16q^{89} + 24q^{94} + 2q^{96} + 4q^{99} + O(q^{100})$$

## 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 $$-0.727186\pi$$
0.654654 0.755929i $$-0.272814\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 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
$$21$$ −4.00000 −0.872872
$$22$$ − 2.00000i − 0.426401i
$$23$$ 1.00000i 0.208514i
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 4.00000i 0.755929i
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 2.00000i 0.348155i
$$34$$ 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
$$39$$ 0 0
$$40$$ 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
$$43$$ 2.00000i 0.304997i 0.988304 + 0.152499i $$0.0487319\pi$$
−0.988304 + 0.152499i $$0.951268\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ − 12.0000i − 1.75038i −0.483779 0.875190i $$-0.660736\pi$$
0.483779 0.875190i $$-0.339264\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −4.00000 −0.534522
$$57$$ 0 0
$$58$$ 4.00000i 0.525226i
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 0 0
$$63$$ 4.00000i 0.503953i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −2.00000 −0.246183
$$67$$ − 2.00000i − 0.244339i −0.992509 0.122169i $$-0.961015\pi$$
0.992509 0.122169i $$-0.0389851\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 6.00000i 0.702247i 0.936329 + 0.351123i $$0.114200\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 8.00000i 0.911685i
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000i 0.662589i
$$83$$ 8.00000i 0.878114i 0.898459 + 0.439057i $$0.144687\pi$$
−0.898459 + 0.439057i $$0.855313\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ −2.00000 −0.215666
$$87$$ − 4.00000i − 0.428845i
$$88$$ 2.00000i 0.213201i
$$89$$ 8.00000 0.847998 0.423999 0.905663i $$-0.360626\pi$$
0.423999 + 0.905663i $$0.360626\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 1.00000i − 0.104257i
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ − 9.00000i − 0.909137i
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ − 2.00000i − 0.198030i
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ − 4.00000i − 0.377964i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −4.00000 −0.371391
$$117$$ 0 0
$$118$$ − 12.0000i − 1.10469i
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ − 14.0000i − 1.26750i
$$123$$ − 6.00000i − 0.541002i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −4.00000 −0.356348
$$127$$ − 6.00000i − 0.532414i −0.963916 0.266207i $$-0.914230\pi$$
0.963916 0.266207i $$-0.0857705\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 2.00000 0.176090
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ − 2.00000i − 0.174078i
$$133$$ 0 0
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ 2.00000i 0.170872i 0.996344 + 0.0854358i $$0.0272282\pi$$
−0.996344 + 0.0854358i $$0.972772\pi$$
$$138$$ 1.00000i 0.0851257i
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ − 2.00000i − 0.167836i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −6.00000 −0.496564
$$147$$ 9.00000i 0.742307i
$$148$$ 10.0000i 0.821995i
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ 2.00000i 0.161690i
$$154$$ −8.00000 −0.644658
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 22.0000i 1.75579i 0.478852 + 0.877896i $$0.341053\pi$$
−0.478852 + 0.877896i $$0.658947\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 1.00000i 0.0785674i
$$163$$ − 16.0000i − 1.25322i −0.779334 0.626608i $$-0.784443\pi$$
0.779334 0.626608i $$-0.215557\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 4.00000i 0.308607i
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 2.00000i − 0.152499i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 4.00000 0.303239
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ 12.0000i 0.901975i
$$178$$ 8.00000i 0.599625i
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ 14.0000i 1.03491i
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 4.00000i 0.292509i
$$188$$ 12.0000i 0.875190i
$$189$$ 4.00000 0.290957
$$190$$ 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 $$-0.831909\pi$$
0.863779 0.503871i $$-0.168091\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 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
$$201$$ −2.00000 −0.141069
$$202$$ − 12.0000i − 0.844317i
$$203$$ − 16.0000i − 1.12298i
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ − 1.00000i − 0.0695048i
$$208$$ 0 0
$$209$$ 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
$$213$$ 2.00000i 0.137038i
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ − 14.0000i − 0.948200i
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 0 0
$$222$$ − 10.0000i − 0.671156i
$$223$$ 2.00000i 0.133930i 0.997755 + 0.0669650i $$0.0213316\pi$$
−0.997755 + 0.0669650i $$0.978668\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 0 0
$$229$$ 26.0000 1.71813 0.859064 0.511868i $$-0.171046\pi$$
0.859064 + 0.511868i $$0.171046\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ − 4.00000i − 0.262613i
$$233$$ 10.0000i 0.655122i 0.944830 + 0.327561i $$0.106227\pi$$
−0.944830 + 0.327561i $$0.893773\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ 8.00000i 0.519656i
$$238$$ − 8.00000i − 0.518563i
$$239$$ −22.0000 −1.42306 −0.711531 0.702655i $$-0.751998\pi$$
−0.711531 + 0.702655i $$0.751998\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ − 7.00000i − 0.449977i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 8.00000 0.506979
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ − 4.00000i − 0.251976i
$$253$$ − 2.00000i − 0.125739i
$$254$$ 6.00000 0.376473
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 2.00000i − 0.124757i −0.998053 0.0623783i $$-0.980131\pi$$
0.998053 0.0623783i $$-0.0198685\pi$$
$$258$$ 2.00000i 0.124515i
$$259$$ −40.0000 −2.48548
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 0 0
$$263$$ − 32.0000i − 1.97320i −0.163144 0.986602i $$-0.552164\pi$$
0.163144 0.986602i $$-0.447836\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 8.00000i − 0.489592i
$$268$$ 2.00000i 0.122169i
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 0 0
$$274$$ −2.00000 −0.120824
$$275$$ 0 0
$$276$$ −1.00000 −0.0601929
$$277$$ − 8.00000i − 0.480673i −0.970690 0.240337i $$-0.922742\pi$$
0.970690 0.240337i $$-0.0772579\pi$$
$$278$$ − 20.0000i − 1.19952i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 32.0000 1.90896 0.954480 0.298275i $$-0.0964112\pi$$
0.954480 + 0.298275i $$0.0964112\pi$$
$$282$$ − 12.0000i − 0.714590i
$$283$$ − 10.0000i − 0.594438i −0.954809 0.297219i $$-0.903941\pi$$
0.954809 0.297219i $$-0.0960592\pi$$
$$284$$ 2.00000 0.118678
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 24.0000i − 1.41668i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 6.00000i − 0.351123i
$$293$$ 14.0000i 0.817889i 0.912559 + 0.408944i $$0.134103\pi$$
−0.912559 + 0.408944i $$0.865897\pi$$
$$294$$ −9.00000 −0.524891
$$295$$ 0 0
$$296$$ −10.0000 −0.581238
$$297$$ − 2.00000i − 0.116052i
$$298$$ 14.0000i 0.810998i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ − 12.0000i − 0.690522i
$$303$$ 12.0000i 0.689382i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ − 4.00000i − 0.228292i −0.993464 0.114146i $$-0.963587\pi$$
0.993464 0.114146i $$-0.0364132\pi$$
$$308$$ − 8.00000i − 0.455842i
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ −2.00000 −0.113410 −0.0567048 0.998391i $$-0.518059\pi$$
−0.0567048 + 0.998391i $$0.518059\pi$$
$$312$$ 0 0
$$313$$ 20.0000i 1.13047i 0.824931 + 0.565233i $$0.191214\pi$$
−0.824931 + 0.565233i $$0.808786\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ 10.0000i 0.561656i 0.959758 + 0.280828i $$0.0906090\pi$$
−0.959758 + 0.280828i $$0.909391\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 4.00000i 0.222911i
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 16.0000 0.886158
$$327$$ 14.0000i 0.774202i
$$328$$ − 6.00000i − 0.331295i
$$329$$ −48.0000 −2.64633
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ − 8.00000i − 0.439057i
$$333$$ 10.0000i 0.547997i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ 4.00000i 0.217894i 0.994048 + 0.108947i $$0.0347479\pi$$
−0.994048 + 0.108947i $$0.965252\pi$$
$$338$$ 13.0000i 0.707107i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 8.00000i 0.431959i
$$344$$ 2.00000 0.107833
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ − 4.00000i − 0.214731i −0.994220 0.107366i $$-0.965758\pi$$
0.994220 0.107366i $$-0.0342415\pi$$
$$348$$ 4.00000i 0.214423i
$$349$$ 30.0000 1.60586 0.802932 0.596071i $$-0.203272\pi$$
0.802932 + 0.596071i $$0.203272\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 2.00000i − 0.106600i
$$353$$ − 34.0000i − 1.80964i −0.425797 0.904819i $$-0.640006\pi$$
0.425797 0.904819i $$-0.359994\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ −8.00000 −0.423999
$$357$$ 8.00000i 0.423405i
$$358$$ − 4.00000i − 0.211407i
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ − 6.00000i − 0.315353i
$$363$$ 7.00000i 0.367405i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −14.0000 −0.731792
$$367$$ − 4.00000i − 0.208798i −0.994535 0.104399i $$-0.966708\pi$$
0.994535 0.104399i $$-0.0332919\pi$$
$$368$$ 1.00000i 0.0521286i
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 24.0000 1.24602
$$372$$ 0 0
$$373$$ − 22.0000i − 1.13912i −0.821951 0.569558i $$-0.807114\pi$$
0.821951 0.569558i $$-0.192886\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 0 0
$$378$$ 4.00000i 0.205738i
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ −6.00000 −0.307389
$$382$$ 0 0
$$383$$ − 24.0000i − 1.22634i −0.789950 0.613171i $$-0.789894\pi$$
0.789950 0.613171i $$-0.210106\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ − 2.00000i − 0.101666i
$$388$$ 0 0
$$389$$ −22.0000 −1.11544 −0.557722 0.830028i $$-0.688325\pi$$
−0.557722 + 0.830028i $$0.688325\pi$$
$$390$$ 0 0
$$391$$ 2.00000 0.101144
$$392$$ 9.00000i 0.454569i
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ −2.00000 −0.100504
$$397$$ − 32.0000i − 1.60603i −0.595956 0.803017i $$-0.703227\pi$$
0.595956 0.803017i $$-0.296773\pi$$
$$398$$ − 16.0000i − 0.802008i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ − 2.00000i − 0.0997509i
$$403$$ 0 0
$$404$$ 12.0000 0.597022
$$405$$ 0 0
$$406$$ 16.0000 0.794067
$$407$$ 20.0000i 0.991363i
$$408$$ 2.00000i 0.0990148i
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ 2.00000 0.0986527
$$412$$ − 4.00000i − 0.197066i
$$413$$ 48.0000i 2.36193i
$$414$$ 1.00000 0.0491473
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 20.0000i 0.979404i
$$418$$ 0 0
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 4.00000i 0.194717i
$$423$$ 12.0000i 0.583460i
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ −2.00000 −0.0969003
$$427$$ 56.0000i 2.71003i
$$428$$ − 12.0000i − 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 24.0000i 1.15337i 0.816968 + 0.576683i $$0.195653\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 0 0
$$438$$ 6.00000i 0.286691i
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ − 20.0000i − 0.950229i −0.879924 0.475114i $$-0.842407\pi$$
0.879924 0.475114i $$-0.157593\pi$$
$$444$$ 10.0000 0.474579
$$445$$ 0 0
$$446$$ −2.00000 −0.0947027
$$447$$ − 14.0000i − 0.662177i
$$448$$ 4.00000i 0.188982i
$$449$$ 34.0000 1.60456 0.802280 0.596948i $$-0.203620\pi$$
0.802280 + 0.596948i $$0.203620\pi$$
$$450$$ 0 0
$$451$$ −12.0000 −0.565058
$$452$$ − 6.00000i − 0.282216i
$$453$$ 12.0000i 0.563809i
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 24.0000i 1.12267i 0.827588 + 0.561336i $$0.189713\pi$$
−0.827588 + 0.561336i $$0.810287\pi$$
$$458$$ 26.0000i 1.21490i
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 20.0000 0.931493 0.465746 0.884918i $$-0.345786\pi$$
0.465746 + 0.884918i $$0.345786\pi$$
$$462$$ 8.00000i 0.372194i
$$463$$ 18.0000i 0.836531i 0.908325 + 0.418265i $$0.137362\pi$$
−0.908325 + 0.418265i $$0.862638\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ −10.0000 −0.463241
$$467$$ − 24.0000i − 1.11059i −0.831654 0.555294i $$-0.812606\pi$$
0.831654 0.555294i $$-0.187394\pi$$
$$468$$ 0 0
$$469$$ −8.00000 −0.369406
$$470$$ 0 0
$$471$$ 22.0000 1.01371
$$472$$ 12.0000i 0.552345i
$$473$$ − 4.00000i − 0.183920i
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ 8.00000 0.366679
$$477$$ − 6.00000i − 0.274721i
$$478$$ − 22.0000i − 1.00626i
$$479$$ 40.0000 1.82765 0.913823 0.406112i $$-0.133116\pi$$
0.913823 + 0.406112i $$0.133116\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ − 14.0000i − 0.637683i
$$483$$ − 4.00000i − 0.182006i
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 18.0000i − 0.815658i −0.913058 0.407829i $$-0.866286\pi$$
0.913058 0.407829i $$-0.133714\pi$$
$$488$$ 14.0000i 0.633750i
$$489$$ −16.0000 −0.723545
$$490$$ 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
$$493$$ − 8.00000i − 0.360302i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8.00000i 0.358849i
$$498$$ 8.00000i 0.358489i
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 18.0000i 0.803379i
$$503$$ − 8.00000i − 0.356702i −0.983967 0.178351i $$-0.942924\pi$$
0.983967 0.178351i $$-0.0570763\pi$$
$$504$$ 4.00000 0.178174
$$505$$ 0 0
$$506$$ 2.00000 0.0889108
$$507$$ − 13.0000i − 0.577350i
$$508$$ 6.00000i 0.266207i
$$509$$ −28.0000 −1.24108 −0.620539 0.784176i $$-0.713086\pi$$
−0.620539 + 0.784176i $$0.713086\pi$$
$$510$$ 0 0
$$511$$ 24.0000 1.06170
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ 2.00000 0.0882162
$$515$$ 0 0
$$516$$ −2.00000 −0.0880451
$$517$$ 24.0000i 1.05552i
$$518$$ − 40.0000i − 1.75750i
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ 32.0000 1.40195 0.700973 0.713188i $$-0.252749\pi$$
0.700973 + 0.713188i $$0.252749\pi$$
$$522$$ − 4.00000i − 0.175075i
$$523$$ 22.0000i 0.961993i 0.876723 + 0.480996i $$0.159725\pi$$
−0.876723 + 0.480996i $$0.840275\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ 0 0
$$528$$ 2.00000i 0.0870388i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 8.00000 0.346194
$$535$$ 0 0
$$536$$ −2.00000 −0.0863868
$$537$$ 4.00000i 0.172613i
$$538$$ − 24.0000i − 1.03471i
$$539$$ 18.0000 0.775315
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ 8.00000i 0.343629i
$$543$$ 6.00000i 0.257485i
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 12.0000i − 0.513083i −0.966533 0.256541i $$-0.917417\pi$$
0.966533 0.256541i $$-0.0825830\pi$$
$$548$$ − 2.00000i − 0.0854358i
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ 0 0
$$552$$ − 1.00000i − 0.0425628i
$$553$$ 32.0000i 1.36078i
$$554$$ 8.00000 0.339887
$$555$$ 0 0
$$556$$ 20.0000 0.848189
$$557$$ − 26.0000i − 1.10166i −0.834619 0.550828i $$-0.814312\pi$$
0.834619 0.550828i $$-0.185688\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 4.00000 0.168880
$$562$$ 32.0000i 1.34984i
$$563$$ 8.00000i 0.337160i 0.985688 + 0.168580i $$0.0539181\pi$$
−0.985688 + 0.168580i $$0.946082\pi$$
$$564$$ 12.0000 0.505291
$$565$$ 0 0
$$566$$ 10.0000 0.420331
$$567$$ − 4.00000i − 0.167984i
$$568$$ 2.00000i 0.0839181i
$$569$$ 12.0000 0.503066 0.251533 0.967849i $$-0.419065\pi$$
0.251533 + 0.967849i $$0.419065\pi$$
$$570$$ 0 0
$$571$$ −36.0000 −1.50655 −0.753277 0.657704i $$-0.771528\pi$$
−0.753277 + 0.657704i $$0.771528\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 24.0000 1.00174
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 10.0000i 0.416305i 0.978096 + 0.208153i $$0.0667451\pi$$
−0.978096 + 0.208153i $$0.933255\pi$$
$$578$$ 13.0000i 0.540729i
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ 32.0000 1.32758
$$582$$ 0 0
$$583$$ − 12.0000i − 0.496989i
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ 4.00000i 0.165098i 0.996587 + 0.0825488i $$0.0263060\pi$$
−0.996587 + 0.0825488i $$0.973694\pi$$
$$588$$ − 9.00000i − 0.371154i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ − 10.0000i − 0.410997i
$$593$$ − 30.0000i − 1.23195i −0.787765 0.615976i $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ −14.0000 −0.573462
$$597$$ 16.0000i 0.654836i
$$598$$ 0 0
$$599$$ 30.0000 1.22577 0.612883 0.790173i $$-0.290010\pi$$
0.612883 + 0.790173i $$0.290010\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 8.00000i 0.326056i
$$603$$ 2.00000i 0.0814463i
$$604$$ 12.0000 0.488273
$$605$$ 0 0
$$606$$ −12.0000 −0.487467
$$607$$ 14.0000i 0.568242i 0.958788 + 0.284121i $$0.0917018\pi$$
−0.958788 + 0.284121i $$0.908298\pi$$
$$608$$ 0 0
$$609$$ −16.0000 −0.648353
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 2.00000i − 0.0808452i
$$613$$ 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
$$617$$ 14.0000i 0.563619i 0.959470 + 0.281809i $$0.0909346\pi$$
−0.959470 + 0.281809i $$0.909065\pi$$
$$618$$ 4.00000i 0.160904i
$$619$$ 24.0000 0.964641 0.482321 0.875995i $$-0.339794\pi$$
0.482321 + 0.875995i $$0.339794\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ − 2.00000i − 0.0801927i
$$623$$ − 32.0000i − 1.28205i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −20.0000 −0.799361
$$627$$ 0 0
$$628$$ − 22.0000i − 0.877896i
$$629$$ −20.0000 −0.797452
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ − 4.00000i − 0.158986i
$$634$$ −10.0000 −0.397151
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 0 0
$$638$$ − 8.00000i − 0.316723i
$$639$$ 2.00000 0.0791188
$$640$$ 0 0
$$641$$ −8.00000 −0.315981 −0.157991 0.987441i $$-0.550502\pi$$
−0.157991 + 0.987441i $$0.550502\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ 14.0000i 0.552106i 0.961142 + 0.276053i $$0.0890266\pi$$
−0.961142 + 0.276053i $$0.910973\pi$$
$$644$$ −4.00000 −0.157622
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 12.0000i − 0.471769i −0.971781 0.235884i $$-0.924201\pi$$
0.971781 0.235884i $$-0.0757987\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 16.0000i 0.626608i
$$653$$ − 26.0000i − 1.01746i −0.860927 0.508729i $$-0.830115\pi$$
0.860927 0.508729i $$-0.169885\pi$$
$$654$$ −14.0000 −0.547443
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ − 6.00000i − 0.234082i
$$658$$ − 48.0000i − 1.87123i
$$659$$ 18.0000 0.701180 0.350590 0.936529i $$-0.385981\pi$$
0.350590 + 0.936529i $$0.385981\pi$$
$$660$$ 0 0
$$661$$ 6.00000 0.233373 0.116686 0.993169i $$-0.462773\pi$$
0.116686 + 0.993169i $$0.462773\pi$$
$$662$$ − 28.0000i − 1.08825i
$$663$$ 0 0
$$664$$ 8.00000 0.310460
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ 4.00000i 0.154881i
$$668$$ 0 0
$$669$$ 2.00000 0.0773245
$$670$$ 0 0
$$671$$ 28.0000 1.08093
$$672$$ − 4.00000i − 0.154303i
$$673$$ − 18.0000i − 0.693849i −0.937893 0.346925i $$-0.887226\pi$$
0.937893 0.346925i $$-0.112774\pi$$
$$674$$ −4.00000 −0.154074
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ 22.0000i 0.845529i 0.906240 + 0.422764i $$0.138940\pi$$
−0.906240 + 0.422764i $$0.861060\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ − 36.0000i − 1.37750i −0.724998 0.688751i $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −8.00000 −0.305441
$$687$$ − 26.0000i − 0.991962i
$$688$$ 2.00000i 0.0762493i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ − 8.00000i − 0.303895i
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ −4.00000 −0.151620
$$697$$ − 12.0000i − 0.454532i
$$698$$ 30.0000i 1.13552i
$$699$$ 10.0000 0.378235
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ 34.0000 1.27961
$$707$$ 48.0000i 1.80523i
$$708$$ − 12.0000i − 0.450988i
$$709$$ −22.0000 −0.826227 −0.413114 0.910679i $$-0.635559\pi$$
−0.413114 + 0.910679i $$0.635559\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ − 8.00000i − 0.299813i
$$713$$ 0 0
$$714$$ −8.00000 −0.299392
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 22.0000i 0.821605i
$$718$$ 0 0
$$719$$ −18.0000 −0.671287 −0.335643 0.941989i $$-0.608954\pi$$
−0.335643 + 0.941989i $$0.608954\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ − 19.0000i − 0.707107i
$$723$$ 14.0000i 0.520666i
$$724$$ 6.00000 0.222988
$$725$$ 0 0
$$726$$ −7.00000 −0.259794
$$727$$ − 8.00000i − 0.296704i −0.988935 0.148352i $$-0.952603\pi$$
0.988935 0.148352i $$-0.0473968\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ − 14.0000i − 0.517455i
$$733$$ − 30.0000i − 1.10808i −0.832492 0.554038i $$-0.813086\pi$$
0.832492 0.554038i $$-0.186914\pi$$
$$734$$ 4.00000 0.147643
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ 4.00000i 0.147342i
$$738$$ − 6.00000i − 0.220863i
$$739$$ −36.0000 −1.32428 −0.662141 0.749380i $$-0.730352\pi$$
−0.662141 + 0.749380i $$0.730352\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 24.0000i 0.881068i
$$743$$ 16.0000i 0.586983i 0.955962 + 0.293492i $$0.0948173\pi$$
−0.955962 + 0.293492i $$0.905183\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 22.0000 0.805477
$$747$$ − 8.00000i − 0.292705i
$$748$$ − 4.00000i − 0.146254i
$$749$$ 48.0000 1.75388
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ − 12.0000i − 0.437595i
$$753$$ − 18.0000i − 0.655956i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ − 22.0000i − 0.799604i −0.916602 0.399802i $$-0.869079\pi$$
0.916602 0.399802i $$-0.130921\pi$$
$$758$$ − 4.00000i − 0.145287i
$$759$$ −2.00000 −0.0725954
$$760$$ 0 0
$$761$$ 2.00000 0.0724999 0.0362500 0.999343i $$-0.488459\pi$$
0.0362500 + 0.999343i $$0.488459\pi$$
$$762$$ − 6.00000i − 0.217357i
$$763$$ 56.0000i 2.02734i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 0 0
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −6.00000 −0.216366 −0.108183 0.994131i $$-0.534503\pi$$
−0.108183 + 0.994131i $$0.534503\pi$$
$$770$$ 0 0
$$771$$ −2.00000 −0.0720282
$$772$$ 14.0000i 0.503871i
$$773$$ 6.00000i 0.215805i 0.994161 + 0.107903i $$0.0344134\pi$$
−0.994161 + 0.107903i $$0.965587\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 40.0000i 1.43499i
$$778$$ − 22.0000i − 0.788738i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 4.00000 0.143131
$$782$$ 2.00000i 0.0715199i
$$783$$ 4.00000i 0.142948i
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 34.0000i 1.21197i 0.795476 + 0.605985i $$0.207221\pi$$
−0.795476 + 0.605985i $$0.792779\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ −32.0000 −1.13923
$$790$$ 0 0
$$791$$ 24.0000 0.853342
$$792$$ − 2.00000i − 0.0710669i
$$793$$ 0 0
$$794$$ 32.0000 1.13564
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ − 50.0000i − 1.77109i −0.464553 0.885545i $$-0.653785\pi$$
0.464553 0.885545i $$-0.346215\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ −8.00000 −0.282666
$$802$$ − 12.0000i − 0.423735i
$$803$$ − 12.0000i − 0.423471i
$$804$$ 2.00000 0.0705346
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 24.0000i 0.844840i
$$808$$ 12.0000i 0.422159i
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ 52.0000 1.82597 0.912983 0.407997i $$-0.133772\pi$$
0.912983 + 0.407997i $$0.133772\pi$$
$$812$$ 16.0000i 0.561490i
$$813$$ − 8.00000i − 0.280572i
$$814$$ −20.0000 −0.701000
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ 0 0
$$818$$ 10.0000i 0.349642i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −16.0000 −0.558404 −0.279202 0.960232i $$-0.590070\pi$$
−0.279202 + 0.960232i $$0.590070\pi$$
$$822$$ 2.00000i 0.0697580i
$$823$$ − 10.0000i − 0.348578i −0.984695 0.174289i $$-0.944237\pi$$
0.984695 0.174289i $$-0.0557627\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ −48.0000 −1.67013
$$827$$ − 44.0000i − 1.53003i −0.644013 0.765015i $$-0.722732\pi$$
0.644013 0.765015i $$-0.277268\pi$$
$$828$$ 1.00000i 0.0347524i
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ 0 0
$$831$$ −8.00000 −0.277517
$$832$$ 0 0
$$833$$ 18.0000i 0.623663i
$$834$$ −20.0000 −0.692543
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 26.0000i 0.898155i
$$839$$ 20.0000 0.690477 0.345238 0.938515i $$-0.387798\pi$$
0.345238 + 0.938515i $$0.387798\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ − 6.00000i − 0.206774i
$$843$$ − 32.0000i − 1.10214i
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ 28.0000i 0.962091i
$$848$$ 6.00000i 0.206041i
$$849$$ −10.0000 −0.343199
$$850$$ 0 0
$$851$$ 10.0000 0.342796
$$852$$ − 2.00000i − 0.0685189i
$$853$$ 56.0000i 1.91740i 0.284413 + 0.958702i $$0.408201\pi$$
−0.284413 + 0.958702i $$0.591799\pi$$
$$854$$ −56.0000 −1.91628
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 42.0000i 1.43469i 0.696717 + 0.717346i $$0.254643\pi$$
−0.696717 + 0.717346i $$0.745357\pi$$
$$858$$ 0 0
$$859$$ −36.0000 −1.22830 −0.614152 0.789188i $$-0.710502\pi$$
−0.614152 + 0.789188i $$0.710502\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ 24.0000i 0.817443i
$$863$$ 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
$$866$$ −24.0000 −0.815553
$$867$$ − 13.0000i − 0.441503i
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 14.0000i 0.474100i
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −6.00000 −0.202721
$$877$$ 56.0000i 1.89099i 0.325643 + 0.945493i $$0.394419\pi$$
−0.325643 + 0.945493i $$0.605581\pi$$
$$878$$ − 20.0000i − 0.674967i
$$879$$ 14.0000 0.472208
$$880$$ 0 0
$$881$$ 40.0000 1.34763 0.673817 0.738898i $$-0.264654\pi$$
0.673817 + 0.738898i $$0.264654\pi$$
$$882$$ 9.00000i 0.303046i
$$883$$ − 48.0000i − 1.61533i −0.589643 0.807664i $$-0.700731\pi$$
0.589643 0.807664i $$-0.299269\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 20.0000 0.671913
$$887$$ 36.0000i 1.20876i 0.796696 + 0.604381i $$0.206579\pi$$
−0.796696 + 0.604381i $$0.793421\pi$$
$$888$$ 10.0000i 0.335578i
$$889$$ −24.0000 −0.804934
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ − 2.00000i − 0.0669650i
$$893$$ 0 0
$$894$$ 14.0000 0.468230
$$895$$ 0 0
$$896$$ −4.00000 −0.133631
$$897$$ 0 0
$$898$$ 34.0000i 1.13459i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ − 12.0000i − 0.399556i
$$903$$ − 8.00000i − 0.266223i
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ −12.0000 −0.398673
$$907$$ − 42.0000i − 1.39459i −0.716786 0.697294i $$-0.754387\pi$$
0.716786 0.697294i $$-0.245613\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ 12.0000 0.398015
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ − 16.0000i − 0.529523i
$$914$$ −24.0000 −0.793849
$$915$$ 0 0
$$916$$ −26.0000 −0.859064
$$917$$ 0 0
$$918$$ 2.00000i 0.0660098i
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 20.0000i 0.658665i
$$923$$ 0 0
$$924$$ −8.00000 −0.263181
$$925$$ 0 0
$$926$$ −18.0000 −0.591517
$$927$$ − 4.00000i − 0.131377i
$$928$$ 4.00000i 0.131306i
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 10.0000i − 0.327561i
$$933$$ 2.00000i 0.0654771i
$$934$$ 24.0000 0.785304
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 48.0000i − 1.56809i −0.620703 0.784046i $$-0.713153\pi$$
0.620703 0.784046i $$-0.286847\pi$$
$$938$$ − 8.00000i − 0.261209i
$$939$$ 20.0000 0.652675
$$940$$ 0 0
$$941$$ 26.0000 0.847576 0.423788 0.905761i $$-0.360700\pi$$
0.423788 + 0.905761i $$0.360700\pi$$
$$942$$ 22.0000i 0.716799i
$$943$$ 6.00000i 0.195387i
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ 52.0000i 1.68977i 0.534946 + 0.844886i $$0.320332\pi$$
−0.534946 + 0.844886i $$0.679668\pi$$
$$948$$ − 8.00000i − 0.259828i
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 10.0000 0.324272
$$952$$ 8.00000i 0.259281i
$$953$$ 14.0000i 0.453504i 0.973952 + 0.226752i $$0.0728108\pi$$
−0.973952 + 0.226752i $$0.927189\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 22.0000 0.711531
$$957$$ 8.00000i 0.258603i
$$958$$ 40.0000i 1.29234i
$$959$$ 8.00000 0.258333
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 14.0000 0.450910
$$965$$ 0 0
$$966$$ 4.00000 0.128698
$$967$$ − 58.0000i − 1.86515i −0.360971 0.932577i $$-0.617555\pi$$
0.360971 0.932577i $$-0.382445\pi$$
$$968$$ 7.00000i 0.224989i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 50.0000 1.60458 0.802288 0.596937i $$-0.203616\pi$$
0.802288 + 0.596937i $$0.203616\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 80.0000i 2.56468i
$$974$$ 18.0000 0.576757
$$975$$ 0 0
$$976$$ −14.0000 −0.448129
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ − 16.0000i − 0.511624i
$$979$$ −16.0000 −0.511362
$$980$$ 0 0
$$981$$ 14.0000 0.446986
$$982$$ − 24.0000i − 0.765871i
$$983$$ 24.0000i 0.765481i 0.923856 + 0.382741i $$0.125020\pi$$
−0.923856 + 0.382741i $$0.874980\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 8.00000 0.254772
$$987$$ 48.0000i 1.52786i
$$988$$ 0 0
$$989$$ −2.00000 −0.0635963
$$990$$ 0 0
$$991$$ 52.0000 1.65183 0.825917 0.563791i $$-0.190658\pi$$
0.825917 + 0.563791i $$0.190658\pi$$
$$992$$ 0 0
$$993$$ 28.0000i 0.888553i
$$994$$ −8.00000 −0.253745
$$995$$ 0 0
$$996$$ −8.00000 −0.253490
$$997$$ − 40.0000i − 1.26681i −0.773819 0.633406i $$-0.781656\pi$$
0.773819 0.633406i $$-0.218344\pi$$
$$998$$ 28.0000i 0.886325i
$$999$$ 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.n.2899.2 2
5.2 odd 4 690.2.a.b.1.1 1
5.3 odd 4 3450.2.a.t.1.1 1
5.4 even 2 inner 3450.2.d.n.2899.1 2
15.2 even 4 2070.2.a.s.1.1 1
20.7 even 4 5520.2.a.r.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.b.1.1 1 5.2 odd 4
2070.2.a.s.1.1 1 15.2 even 4
3450.2.a.t.1.1 1 5.3 odd 4
3450.2.d.n.2899.1 2 5.4 even 2 inner
3450.2.d.n.2899.2 2 1.1 even 1 trivial
5520.2.a.r.1.1 1 20.7 even 4