# Properties

 Label 882.2.a.n.1.2 Level $882$ Weight $2$ Character 882.1 Self dual yes Analytic conductor $7.043$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 882.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.04280545828$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 98) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 882.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} -1.00000 q^{8} -2.82843 q^{10} +2.00000 q^{11} +1.00000 q^{16} -1.41421 q^{17} +7.07107 q^{19} +2.82843 q^{20} -2.00000 q^{22} +4.00000 q^{23} +3.00000 q^{25} -2.00000 q^{29} -8.48528 q^{31} -1.00000 q^{32} +1.41421 q^{34} +10.0000 q^{37} -7.07107 q^{38} -2.82843 q^{40} -9.89949 q^{41} +2.00000 q^{43} +2.00000 q^{44} -4.00000 q^{46} +2.82843 q^{47} -3.00000 q^{50} +2.00000 q^{53} +5.65685 q^{55} +2.00000 q^{58} -1.41421 q^{59} -2.82843 q^{61} +8.48528 q^{62} +1.00000 q^{64} +12.0000 q^{67} -1.41421 q^{68} +12.0000 q^{71} +1.41421 q^{73} -10.0000 q^{74} +7.07107 q^{76} -4.00000 q^{79} +2.82843 q^{80} +9.89949 q^{82} +9.89949 q^{83} -4.00000 q^{85} -2.00000 q^{86} -2.00000 q^{88} -7.07107 q^{89} +4.00000 q^{92} -2.82843 q^{94} +20.0000 q^{95} -9.89949 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + O(q^{10})$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{11} + 2 q^{16} - 4 q^{22} + 8 q^{23} + 6 q^{25} - 4 q^{29} - 2 q^{32} + 20 q^{37} + 4 q^{43} + 4 q^{44} - 8 q^{46} - 6 q^{50} + 4 q^{53} + 4 q^{58} + 2 q^{64} + 24 q^{67} + 24 q^{71} - 20 q^{74} - 8 q^{79} - 8 q^{85} - 4 q^{86} - 4 q^{88} + 8 q^{92} + 40 q^{95} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 2.82843 1.26491 0.632456 0.774597i $$-0.282047\pi$$
0.632456 + 0.774597i $$0.282047\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −2.82843 −0.894427
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −1.41421 −0.342997 −0.171499 0.985184i $$-0.554861\pi$$
−0.171499 + 0.985184i $$0.554861\pi$$
$$18$$ 0 0
$$19$$ 7.07107 1.62221 0.811107 0.584898i $$-0.198865\pi$$
0.811107 + 0.584898i $$0.198865\pi$$
$$20$$ 2.82843 0.632456
$$21$$ 0 0
$$22$$ −2.00000 −0.426401
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 3.00000 0.600000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −8.48528 −1.52400 −0.762001 0.647576i $$-0.775783\pi$$
−0.762001 + 0.647576i $$0.775783\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 1.41421 0.242536
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ −7.07107 −1.14708
$$39$$ 0 0
$$40$$ −2.82843 −0.447214
$$41$$ −9.89949 −1.54604 −0.773021 0.634381i $$-0.781255\pi$$
−0.773021 + 0.634381i $$0.781255\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 2.82843 0.412568 0.206284 0.978492i $$-0.433863\pi$$
0.206284 + 0.978492i $$0.433863\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −3.00000 −0.424264
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ 5.65685 0.762770
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ −1.41421 −0.184115 −0.0920575 0.995754i $$-0.529344\pi$$
−0.0920575 + 0.995754i $$0.529344\pi$$
$$60$$ 0 0
$$61$$ −2.82843 −0.362143 −0.181071 0.983470i $$-0.557957\pi$$
−0.181071 + 0.983470i $$0.557957\pi$$
$$62$$ 8.48528 1.07763
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ −1.41421 −0.171499
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 1.41421 0.165521 0.0827606 0.996569i $$-0.473626\pi$$
0.0827606 + 0.996569i $$0.473626\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ 7.07107 0.811107
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 2.82843 0.316228
$$81$$ 0 0
$$82$$ 9.89949 1.09322
$$83$$ 9.89949 1.08661 0.543305 0.839535i $$-0.317173\pi$$
0.543305 + 0.839535i $$0.317173\pi$$
$$84$$ 0 0
$$85$$ −4.00000 −0.433861
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ −2.00000 −0.213201
$$89$$ −7.07107 −0.749532 −0.374766 0.927119i $$-0.622277\pi$$
−0.374766 + 0.927119i $$0.622277\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ −2.82843 −0.291730
$$95$$ 20.0000 2.05196
$$96$$ 0 0
$$97$$ −9.89949 −1.00514 −0.502571 0.864536i $$-0.667612\pi$$
−0.502571 + 0.864536i $$0.667612\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 3.00000 0.300000
$$101$$ 8.48528 0.844317 0.422159 0.906522i $$-0.361273\pi$$
0.422159 + 0.906522i $$0.361273\pi$$
$$102$$ 0 0
$$103$$ −2.82843 −0.278693 −0.139347 0.990244i $$-0.544500\pi$$
−0.139347 + 0.990244i $$0.544500\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ −5.65685 −0.539360
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 12.0000 1.12887 0.564433 0.825479i $$-0.309095\pi$$
0.564433 + 0.825479i $$0.309095\pi$$
$$114$$ 0 0
$$115$$ 11.3137 1.05501
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ 1.41421 0.130189
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 2.82843 0.256074
$$123$$ 0 0
$$124$$ −8.48528 −0.762001
$$125$$ −5.65685 −0.505964
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 12.7279 1.11204 0.556022 0.831168i $$-0.312327\pi$$
0.556022 + 0.831168i $$0.312327\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 1.41421 0.121268
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ −9.89949 −0.839664 −0.419832 0.907602i $$-0.637911\pi$$
−0.419832 + 0.907602i $$0.637911\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −12.0000 −1.00702
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −5.65685 −0.469776
$$146$$ −1.41421 −0.117041
$$147$$ 0 0
$$148$$ 10.0000 0.821995
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ −7.07107 −0.573539
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −24.0000 −1.92773
$$156$$ 0 0
$$157$$ 11.3137 0.902932 0.451466 0.892288i $$-0.350901\pi$$
0.451466 + 0.892288i $$0.350901\pi$$
$$158$$ 4.00000 0.318223
$$159$$ 0 0
$$160$$ −2.82843 −0.223607
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 10.0000 0.783260 0.391630 0.920123i $$-0.371911\pi$$
0.391630 + 0.920123i $$0.371911\pi$$
$$164$$ −9.89949 −0.773021
$$165$$ 0 0
$$166$$ −9.89949 −0.768350
$$167$$ −19.7990 −1.53209 −0.766046 0.642786i $$-0.777779\pi$$
−0.766046 + 0.642786i $$0.777779\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ −16.9706 −1.29025 −0.645124 0.764078i $$-0.723194\pi$$
−0.645124 + 0.764078i $$0.723194\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 0 0
$$178$$ 7.07107 0.529999
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ 28.2843 2.07950
$$186$$ 0 0
$$187$$ −2.82843 −0.206835
$$188$$ 2.82843 0.206284
$$189$$ 0 0
$$190$$ −20.0000 −1.45095
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 0 0
$$193$$ −16.0000 −1.15171 −0.575853 0.817554i $$-0.695330\pi$$
−0.575853 + 0.817554i $$0.695330\pi$$
$$194$$ 9.89949 0.710742
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ −8.48528 −0.601506 −0.300753 0.953702i $$-0.597238\pi$$
−0.300753 + 0.953702i $$0.597238\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ 0 0
$$202$$ −8.48528 −0.597022
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −28.0000 −1.95560
$$206$$ 2.82843 0.197066
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 14.1421 0.978232
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 0 0
$$214$$ −4.00000 −0.273434
$$215$$ 5.65685 0.385794
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 2.00000 0.135457
$$219$$ 0 0
$$220$$ 5.65685 0.381385
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −12.0000 −0.798228
$$227$$ −21.2132 −1.40797 −0.703985 0.710215i $$-0.748598\pi$$
−0.703985 + 0.710215i $$0.748598\pi$$
$$228$$ 0 0
$$229$$ 16.9706 1.12145 0.560723 0.828003i $$-0.310523\pi$$
0.560723 + 0.828003i $$0.310523\pi$$
$$230$$ −11.3137 −0.746004
$$231$$ 0 0
$$232$$ 2.00000 0.131306
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ −1.41421 −0.0920575
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ 21.2132 1.36646 0.683231 0.730202i $$-0.260574\pi$$
0.683231 + 0.730202i $$0.260574\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 0 0
$$244$$ −2.82843 −0.181071
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 8.48528 0.538816
$$249$$ 0 0
$$250$$ 5.65685 0.357771
$$251$$ 9.89949 0.624851 0.312425 0.949942i $$-0.398859\pi$$
0.312425 + 0.949942i $$0.398859\pi$$
$$252$$ 0 0
$$253$$ 8.00000 0.502956
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 12.7279 0.793946 0.396973 0.917830i $$-0.370061\pi$$
0.396973 + 0.917830i $$0.370061\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −12.7279 −0.786334
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ 5.65685 0.347498
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 12.0000 0.733017
$$269$$ −11.3137 −0.689809 −0.344904 0.938638i $$-0.612089\pi$$
−0.344904 + 0.938638i $$0.612089\pi$$
$$270$$ 0 0
$$271$$ −22.6274 −1.37452 −0.687259 0.726413i $$-0.741186\pi$$
−0.687259 + 0.726413i $$0.741186\pi$$
$$272$$ −1.41421 −0.0857493
$$273$$ 0 0
$$274$$ 12.0000 0.724947
$$275$$ 6.00000 0.361814
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 9.89949 0.593732
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −16.0000 −0.954480 −0.477240 0.878773i $$-0.658363\pi$$
−0.477240 + 0.878773i $$0.658363\pi$$
$$282$$ 0 0
$$283$$ 1.41421 0.0840663 0.0420331 0.999116i $$-0.486616\pi$$
0.0420331 + 0.999116i $$0.486616\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −15.0000 −0.882353
$$290$$ 5.65685 0.332182
$$291$$ 0 0
$$292$$ 1.41421 0.0827606
$$293$$ −19.7990 −1.15667 −0.578335 0.815800i $$-0.696297\pi$$
−0.578335 + 0.815800i $$0.696297\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ −10.0000 −0.581238
$$297$$ 0 0
$$298$$ 10.0000 0.579284
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 16.0000 0.920697
$$303$$ 0 0
$$304$$ 7.07107 0.405554
$$305$$ −8.00000 −0.458079
$$306$$ 0 0
$$307$$ 9.89949 0.564994 0.282497 0.959268i $$-0.408837\pi$$
0.282497 + 0.959268i $$0.408837\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 24.0000 1.36311
$$311$$ −11.3137 −0.641542 −0.320771 0.947157i $$-0.603942\pi$$
−0.320771 + 0.947157i $$0.603942\pi$$
$$312$$ 0 0
$$313$$ −12.7279 −0.719425 −0.359712 0.933063i $$-0.617125\pi$$
−0.359712 + 0.933063i $$0.617125\pi$$
$$314$$ −11.3137 −0.638470
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ −10.0000 −0.561656 −0.280828 0.959758i $$-0.590609\pi$$
−0.280828 + 0.959758i $$0.590609\pi$$
$$318$$ 0 0
$$319$$ −4.00000 −0.223957
$$320$$ 2.82843 0.158114
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −10.0000 −0.556415
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −10.0000 −0.553849
$$327$$ 0 0
$$328$$ 9.89949 0.546608
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 10.0000 0.549650 0.274825 0.961494i $$-0.411380\pi$$
0.274825 + 0.961494i $$0.411380\pi$$
$$332$$ 9.89949 0.543305
$$333$$ 0 0
$$334$$ 19.7990 1.08335
$$335$$ 33.9411 1.85440
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 13.0000 0.707107
$$339$$ 0 0
$$340$$ −4.00000 −0.216930
$$341$$ −16.9706 −0.919007
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −2.00000 −0.107833
$$345$$ 0 0
$$346$$ 16.9706 0.912343
$$347$$ 30.0000 1.61048 0.805242 0.592946i $$-0.202035\pi$$
0.805242 + 0.592946i $$0.202035\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −2.00000 −0.106600
$$353$$ −1.41421 −0.0752710 −0.0376355 0.999292i $$-0.511983\pi$$
−0.0376355 + 0.999292i $$0.511983\pi$$
$$354$$ 0 0
$$355$$ 33.9411 1.80141
$$356$$ −7.07107 −0.374766
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 0 0
$$361$$ 31.0000 1.63158
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 4.00000 0.209370
$$366$$ 0 0
$$367$$ −28.2843 −1.47643 −0.738213 0.674567i $$-0.764330\pi$$
−0.738213 + 0.674567i $$0.764330\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ −28.2843 −1.47043
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ 2.82843 0.146254
$$375$$ 0 0
$$376$$ −2.82843 −0.145865
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −26.0000 −1.33553 −0.667765 0.744372i $$-0.732749\pi$$
−0.667765 + 0.744372i $$0.732749\pi$$
$$380$$ 20.0000 1.02598
$$381$$ 0 0
$$382$$ −4.00000 −0.204658
$$383$$ −36.7696 −1.87884 −0.939418 0.342773i $$-0.888634\pi$$
−0.939418 + 0.342773i $$0.888634\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 16.0000 0.814379
$$387$$ 0 0
$$388$$ −9.89949 −0.502571
$$389$$ −26.0000 −1.31825 −0.659126 0.752032i $$-0.729074\pi$$
−0.659126 + 0.752032i $$0.729074\pi$$
$$390$$ 0 0
$$391$$ −5.65685 −0.286079
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 2.00000 0.100759
$$395$$ −11.3137 −0.569254
$$396$$ 0 0
$$397$$ −22.6274 −1.13564 −0.567819 0.823154i $$-0.692213\pi$$
−0.567819 + 0.823154i $$0.692213\pi$$
$$398$$ 8.48528 0.425329
$$399$$ 0 0
$$400$$ 3.00000 0.150000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 8.48528 0.422159
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 20.0000 0.991363
$$408$$ 0 0
$$409$$ −38.1838 −1.88807 −0.944033 0.329851i $$-0.893001\pi$$
−0.944033 + 0.329851i $$0.893001\pi$$
$$410$$ 28.0000 1.38282
$$411$$ 0 0
$$412$$ −2.82843 −0.139347
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 28.0000 1.37447
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −14.1421 −0.691714
$$419$$ −9.89949 −0.483622 −0.241811 0.970323i $$-0.577741\pi$$
−0.241811 + 0.970323i $$0.577741\pi$$
$$420$$ 0 0
$$421$$ 30.0000 1.46211 0.731055 0.682318i $$-0.239028\pi$$
0.731055 + 0.682318i $$0.239028\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 0 0
$$424$$ −2.00000 −0.0971286
$$425$$ −4.24264 −0.205798
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 4.00000 0.193347
$$429$$ 0 0
$$430$$ −5.65685 −0.272798
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ 29.6985 1.42722 0.713609 0.700544i $$-0.247059\pi$$
0.713609 + 0.700544i $$0.247059\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 28.2843 1.35302
$$438$$ 0 0
$$439$$ 16.9706 0.809961 0.404980 0.914325i $$-0.367278\pi$$
0.404980 + 0.914325i $$0.367278\pi$$
$$440$$ −5.65685 −0.269680
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 0 0
$$445$$ −20.0000 −0.948091
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ −19.7990 −0.932298
$$452$$ 12.0000 0.564433
$$453$$ 0 0
$$454$$ 21.2132 0.995585
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 24.0000 1.12267 0.561336 0.827588i $$-0.310287\pi$$
0.561336 + 0.827588i $$0.310287\pi$$
$$458$$ −16.9706 −0.792982
$$459$$ 0 0
$$460$$ 11.3137 0.527504
$$461$$ 39.5980 1.84426 0.922131 0.386878i $$-0.126447\pi$$
0.922131 + 0.386878i $$0.126447\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 24.0000 1.11178
$$467$$ 32.5269 1.50517 0.752583 0.658497i $$-0.228808\pi$$
0.752583 + 0.658497i $$0.228808\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −8.00000 −0.369012
$$471$$ 0 0
$$472$$ 1.41421 0.0650945
$$473$$ 4.00000 0.183920
$$474$$ 0 0
$$475$$ 21.2132 0.973329
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −12.0000 −0.548867
$$479$$ −31.1127 −1.42158 −0.710788 0.703407i $$-0.751661\pi$$
−0.710788 + 0.703407i $$0.751661\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −21.2132 −0.966235
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ −28.0000 −1.27141
$$486$$ 0 0
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ 2.82843 0.128037
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 2.82843 0.127386
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.48528 −0.381000
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ −5.65685 −0.252982
$$501$$ 0 0
$$502$$ −9.89949 −0.441836
$$503$$ 39.5980 1.76559 0.882793 0.469762i $$-0.155660\pi$$
0.882793 + 0.469762i $$0.155660\pi$$
$$504$$ 0 0
$$505$$ 24.0000 1.06799
$$506$$ −8.00000 −0.355643
$$507$$ 0 0
$$508$$ 16.0000 0.709885
$$509$$ 22.6274 1.00294 0.501471 0.865174i $$-0.332792\pi$$
0.501471 + 0.865174i $$0.332792\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −12.7279 −0.561405
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ 5.65685 0.248788
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −1.41421 −0.0619578 −0.0309789 0.999520i $$-0.509862\pi$$
−0.0309789 + 0.999520i $$0.509862\pi$$
$$522$$ 0 0
$$523$$ −12.7279 −0.556553 −0.278277 0.960501i $$-0.589763\pi$$
−0.278277 + 0.960501i $$0.589763\pi$$
$$524$$ 12.7279 0.556022
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −5.65685 −0.245718
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 11.3137 0.489134
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ 11.3137 0.487769
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 10.0000 0.429934 0.214967 0.976621i $$-0.431036\pi$$
0.214967 + 0.976621i $$0.431036\pi$$
$$542$$ 22.6274 0.971931
$$543$$ 0 0
$$544$$ 1.41421 0.0606339
$$545$$ −5.65685 −0.242313
$$546$$ 0 0
$$547$$ −26.0000 −1.11168 −0.555840 0.831289i $$-0.687603\pi$$
−0.555840 + 0.831289i $$0.687603\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ 0 0
$$550$$ −6.00000 −0.255841
$$551$$ −14.1421 −0.602475
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −9.89949 −0.419832
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 16.0000 0.674919
$$563$$ −1.41421 −0.0596020 −0.0298010 0.999556i $$-0.509487\pi$$
−0.0298010 + 0.999556i $$0.509487\pi$$
$$564$$ 0 0
$$565$$ 33.9411 1.42791
$$566$$ −1.41421 −0.0594438
$$567$$ 0 0
$$568$$ −12.0000 −0.503509
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ −2.00000 −0.0836974 −0.0418487 0.999124i $$-0.513325\pi$$
−0.0418487 + 0.999124i $$0.513325\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 12.0000 0.500435
$$576$$ 0 0
$$577$$ 21.2132 0.883117 0.441559 0.897232i $$-0.354426\pi$$
0.441559 + 0.897232i $$0.354426\pi$$
$$578$$ 15.0000 0.623918
$$579$$ 0 0
$$580$$ −5.65685 −0.234888
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 4.00000 0.165663
$$584$$ −1.41421 −0.0585206
$$585$$ 0 0
$$586$$ 19.7990 0.817889
$$587$$ −29.6985 −1.22579 −0.612894 0.790165i $$-0.709995\pi$$
−0.612894 + 0.790165i $$0.709995\pi$$
$$588$$ 0 0
$$589$$ −60.0000 −2.47226
$$590$$ 4.00000 0.164677
$$591$$ 0 0
$$592$$ 10.0000 0.410997
$$593$$ −7.07107 −0.290374 −0.145187 0.989404i $$-0.546378\pi$$
−0.145187 + 0.989404i $$0.546378\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 16.0000 0.653742 0.326871 0.945069i $$-0.394006\pi$$
0.326871 + 0.945069i $$0.394006\pi$$
$$600$$ 0 0
$$601$$ −29.6985 −1.21143 −0.605713 0.795683i $$-0.707112\pi$$
−0.605713 + 0.795683i $$0.707112\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −16.0000 −0.651031
$$605$$ −19.7990 −0.804943
$$606$$ 0 0
$$607$$ 16.9706 0.688814 0.344407 0.938820i $$-0.388080\pi$$
0.344407 + 0.938820i $$0.388080\pi$$
$$608$$ −7.07107 −0.286770
$$609$$ 0 0
$$610$$ 8.00000 0.323911
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −30.0000 −1.21169 −0.605844 0.795583i $$-0.707165\pi$$
−0.605844 + 0.795583i $$0.707165\pi$$
$$614$$ −9.89949 −0.399511
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 26.0000 1.04672 0.523360 0.852111i $$-0.324678\pi$$
0.523360 + 0.852111i $$0.324678\pi$$
$$618$$ 0 0
$$619$$ −18.3848 −0.738947 −0.369473 0.929241i $$-0.620462\pi$$
−0.369473 + 0.929241i $$0.620462\pi$$
$$620$$ −24.0000 −0.963863
$$621$$ 0 0
$$622$$ 11.3137 0.453638
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −31.0000 −1.24000
$$626$$ 12.7279 0.508710
$$627$$ 0 0
$$628$$ 11.3137 0.451466
$$629$$ −14.1421 −0.563884
$$630$$ 0 0
$$631$$ 44.0000 1.75161 0.875806 0.482663i $$-0.160330\pi$$
0.875806 + 0.482663i $$0.160330\pi$$
$$632$$ 4.00000 0.159111
$$633$$ 0 0
$$634$$ 10.0000 0.397151
$$635$$ 45.2548 1.79588
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 4.00000 0.158362
$$639$$ 0 0
$$640$$ −2.82843 −0.111803
$$641$$ −26.0000 −1.02694 −0.513469 0.858108i $$-0.671640\pi$$
−0.513469 + 0.858108i $$0.671640\pi$$
$$642$$ 0 0
$$643$$ −9.89949 −0.390398 −0.195199 0.980764i $$-0.562535\pi$$
−0.195199 + 0.980764i $$0.562535\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 10.0000 0.393445
$$647$$ 8.48528 0.333591 0.166795 0.985992i $$-0.446658\pi$$
0.166795 + 0.985992i $$0.446658\pi$$
$$648$$ 0 0
$$649$$ −2.82843 −0.111025
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 10.0000 0.391630
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 0 0
$$655$$ 36.0000 1.40664
$$656$$ −9.89949 −0.386510
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −30.0000 −1.16863 −0.584317 0.811525i $$-0.698638\pi$$
−0.584317 + 0.811525i $$0.698638\pi$$
$$660$$ 0 0
$$661$$ −8.48528 −0.330039 −0.165020 0.986290i $$-0.552769\pi$$
−0.165020 + 0.986290i $$0.552769\pi$$
$$662$$ −10.0000 −0.388661
$$663$$ 0 0
$$664$$ −9.89949 −0.384175
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −8.00000 −0.309761
$$668$$ −19.7990 −0.766046
$$669$$ 0 0
$$670$$ −33.9411 −1.31126
$$671$$ −5.65685 −0.218380
$$672$$ 0 0
$$673$$ −12.0000 −0.462566 −0.231283 0.972887i $$-0.574292\pi$$
−0.231283 + 0.972887i $$0.574292\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ −16.9706 −0.652232 −0.326116 0.945330i $$-0.605740\pi$$
−0.326116 + 0.945330i $$0.605740\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 4.00000 0.153393
$$681$$ 0 0
$$682$$ 16.9706 0.649836
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ −33.9411 −1.29682
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 2.00000 0.0762493
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −12.7279 −0.484193 −0.242096 0.970252i $$-0.577835\pi$$
−0.242096 + 0.970252i $$0.577835\pi$$
$$692$$ −16.9706 −0.645124
$$693$$ 0 0
$$694$$ −30.0000 −1.13878
$$695$$ −28.0000 −1.06210
$$696$$ 0 0
$$697$$ 14.0000 0.530288
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ 70.7107 2.66690
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ 1.41421 0.0532246
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ −33.9411 −1.27379
$$711$$ 0 0
$$712$$ 7.07107 0.264999
$$713$$ −33.9411 −1.27111
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ −32.0000 −1.19423
$$719$$ 2.82843 0.105483 0.0527413 0.998608i $$-0.483204\pi$$
0.0527413 + 0.998608i $$0.483204\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −31.0000 −1.15370
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ 19.7990 0.734304 0.367152 0.930161i $$-0.380333\pi$$
0.367152 + 0.930161i $$0.380333\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −4.00000 −0.148047
$$731$$ −2.82843 −0.104613
$$732$$ 0 0
$$733$$ −42.4264 −1.56706 −0.783528 0.621357i $$-0.786582\pi$$
−0.783528 + 0.621357i $$0.786582\pi$$
$$734$$ 28.2843 1.04399
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 24.0000 0.884051
$$738$$ 0 0
$$739$$ −30.0000 −1.10357 −0.551784 0.833987i $$-0.686053\pi$$
−0.551784 + 0.833987i $$0.686053\pi$$
$$740$$ 28.2843 1.03975
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 0 0
$$745$$ −28.2843 −1.03626
$$746$$ −10.0000 −0.366126
$$747$$ 0 0
$$748$$ −2.82843 −0.103418
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ 2.82843 0.103142
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −45.2548 −1.64699
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 26.0000 0.944363
$$759$$ 0 0
$$760$$ −20.0000 −0.725476
$$761$$ −7.07107 −0.256326 −0.128163 0.991753i $$-0.540908\pi$$
−0.128163 + 0.991753i $$0.540908\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 4.00000 0.144715
$$765$$ 0 0
$$766$$ 36.7696 1.32854
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −29.6985 −1.07095 −0.535477 0.844550i $$-0.679868\pi$$
−0.535477 + 0.844550i $$0.679868\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −16.0000 −0.575853
$$773$$ 48.0833 1.72943 0.864717 0.502259i $$-0.167498\pi$$
0.864717 + 0.502259i $$0.167498\pi$$
$$774$$ 0 0
$$775$$ −25.4558 −0.914401
$$776$$ 9.89949 0.355371
$$777$$ 0 0
$$778$$ 26.0000 0.932145
$$779$$ −70.0000 −2.50801
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ 5.65685 0.202289
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 32.0000 1.14213
$$786$$ 0 0
$$787$$ 1.41421 0.0504113 0.0252056 0.999682i $$-0.491976\pi$$
0.0252056 + 0.999682i $$0.491976\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ 0 0
$$790$$ 11.3137 0.402524
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 22.6274 0.803017
$$795$$ 0 0
$$796$$ −8.48528 −0.300753
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ −4.00000 −0.141510
$$800$$ −3.00000 −0.106066
$$801$$ 0 0
$$802$$ −18.0000 −0.635602
$$803$$ 2.82843 0.0998130
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −8.48528 −0.298511
$$809$$ 16.0000 0.562530 0.281265 0.959630i $$-0.409246\pi$$
0.281265 + 0.959630i $$0.409246\pi$$
$$810$$ 0 0
$$811$$ 29.6985 1.04285 0.521427 0.853296i $$-0.325400\pi$$
0.521427 + 0.853296i $$0.325400\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −20.0000 −0.701000
$$815$$ 28.2843 0.990755
$$816$$ 0 0
$$817$$ 14.1421 0.494771
$$818$$ 38.1838 1.33506
$$819$$ 0 0
$$820$$ −28.0000 −0.977802
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ 2.82843 0.0985329
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ 31.1127 1.08059 0.540294 0.841476i $$-0.318313\pi$$
0.540294 + 0.841476i $$0.318313\pi$$
$$830$$ −28.0000 −0.971894
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −56.0000 −1.93796
$$836$$ 14.1421 0.489116
$$837$$ 0 0
$$838$$ 9.89949 0.341972
$$839$$ 19.7990 0.683537 0.341769 0.939784i $$-0.388974\pi$$
0.341769 + 0.939784i $$0.388974\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −30.0000 −1.03387
$$843$$ 0 0
$$844$$ −12.0000 −0.413057
$$845$$ −36.7696 −1.26491
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 2.00000 0.0686803
$$849$$ 0 0
$$850$$ 4.24264 0.145521
$$851$$ 40.0000 1.37118
$$852$$ 0 0
$$853$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ 18.3848 0.628012 0.314006 0.949421i $$-0.398329\pi$$
0.314006 + 0.949421i $$0.398329\pi$$
$$858$$ 0 0
$$859$$ 26.8701 0.916795 0.458397 0.888747i $$-0.348424\pi$$
0.458397 + 0.888747i $$0.348424\pi$$
$$860$$ 5.65685 0.192897
$$861$$ 0 0
$$862$$ 12.0000 0.408722
$$863$$ 4.00000 0.136162 0.0680808 0.997680i $$-0.478312\pi$$
0.0680808 + 0.997680i $$0.478312\pi$$
$$864$$ 0 0
$$865$$ −48.0000 −1.63205
$$866$$ −29.6985 −1.00920
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −8.00000 −0.271381
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 2.00000 0.0677285
$$873$$ 0 0
$$874$$ −28.2843 −0.956730
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −46.0000 −1.55331 −0.776655 0.629926i $$-0.783085\pi$$
−0.776655 + 0.629926i $$0.783085\pi$$
$$878$$ −16.9706 −0.572729
$$879$$ 0 0
$$880$$ 5.65685 0.190693
$$881$$ 29.6985 1.00057 0.500284 0.865862i $$-0.333229\pi$$
0.500284 + 0.865862i $$0.333229\pi$$
$$882$$ 0 0
$$883$$ 44.0000 1.48072 0.740359 0.672212i $$-0.234656\pi$$
0.740359 + 0.672212i $$0.234656\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ −36.7696 −1.23460 −0.617300 0.786728i $$-0.711774\pi$$
−0.617300 + 0.786728i $$0.711774\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 20.0000 0.670402
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 20.0000 0.669274
$$894$$ 0 0
$$895$$ −33.9411 −1.13453
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 30.0000 1.00111
$$899$$ 16.9706 0.566000
$$900$$ 0 0
$$901$$ −2.82843 −0.0942286
$$902$$ 19.7990 0.659234
$$903$$ 0 0
$$904$$ −12.0000 −0.399114
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ −21.2132 −0.703985
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 40.0000 1.32526 0.662630 0.748947i $$-0.269440\pi$$
0.662630 + 0.748947i $$0.269440\pi$$
$$912$$ 0 0
$$913$$ 19.7990 0.655251
$$914$$ −24.0000 −0.793849
$$915$$ 0 0
$$916$$ 16.9706 0.560723
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ −11.3137 −0.373002
$$921$$ 0 0
$$922$$ −39.5980 −1.30409
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 30.0000 0.986394
$$926$$ −16.0000 −0.525793
$$927$$ 0 0
$$928$$ 2.00000 0.0656532
$$929$$ 32.5269 1.06717 0.533587 0.845745i $$-0.320844\pi$$
0.533587 + 0.845745i $$0.320844\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −24.0000 −0.786146
$$933$$ 0 0
$$934$$ −32.5269 −1.06431
$$935$$ −8.00000 −0.261628
$$936$$ 0 0
$$937$$ −9.89949 −0.323402 −0.161701 0.986840i $$-0.551698\pi$$
−0.161701 + 0.986840i $$0.551698\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 8.00000 0.260931
$$941$$ −31.1127 −1.01424 −0.507122 0.861874i $$-0.669291\pi$$
−0.507122 + 0.861874i $$0.669291\pi$$
$$942$$ 0 0
$$943$$ −39.5980 −1.28949
$$944$$ −1.41421 −0.0460287
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ 18.0000 0.584921 0.292461 0.956278i $$-0.405526\pi$$
0.292461 + 0.956278i $$0.405526\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −21.2132 −0.688247
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 26.0000 0.842223 0.421111 0.907009i $$-0.361640\pi$$
0.421111 + 0.907009i $$0.361640\pi$$
$$954$$ 0 0
$$955$$ 11.3137 0.366103
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ 31.1127 1.00521
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 41.0000 1.32258
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 21.2132 0.683231
$$965$$ −45.2548 −1.45680
$$966$$ 0 0
$$967$$ −12.0000 −0.385894 −0.192947 0.981209i $$-0.561805\pi$$
−0.192947 + 0.981209i $$0.561805\pi$$
$$968$$ 7.00000 0.224989
$$969$$ 0 0
$$970$$ 28.0000 0.899026
$$971$$ 32.5269 1.04384 0.521919 0.852995i $$-0.325216\pi$$
0.521919 + 0.852995i $$0.325216\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −12.0000 −0.384505
$$975$$ 0 0
$$976$$ −2.82843 −0.0905357
$$977$$ −12.0000 −0.383914 −0.191957 0.981403i $$-0.561483\pi$$
−0.191957 + 0.981403i $$0.561483\pi$$
$$978$$ 0 0
$$979$$ −14.1421 −0.451985
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −12.0000 −0.382935
$$983$$ 48.0833 1.53362 0.766809 0.641875i $$-0.221843\pi$$
0.766809 + 0.641875i $$0.221843\pi$$
$$984$$ 0 0
$$985$$ −5.65685 −0.180242
$$986$$ −2.82843 −0.0900755
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 8.48528 0.269408
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −24.0000 −0.760851
$$996$$ 0 0
$$997$$ 31.1127 0.985349 0.492675 0.870214i $$-0.336019\pi$$
0.492675 + 0.870214i $$0.336019\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 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 882.2.a.n.1.2 2
3.2 odd 2 98.2.a.b.1.2 yes 2
4.3 odd 2 7056.2.a.cl.1.2 2
7.2 even 3 882.2.g.l.361.1 4
7.3 odd 6 882.2.g.l.667.2 4
7.4 even 3 882.2.g.l.667.1 4
7.5 odd 6 882.2.g.l.361.2 4
7.6 odd 2 inner 882.2.a.n.1.1 2
12.11 even 2 784.2.a.l.1.1 2
15.2 even 4 2450.2.c.v.99.3 4
15.8 even 4 2450.2.c.v.99.2 4
15.14 odd 2 2450.2.a.bj.1.1 2
21.2 odd 6 98.2.c.c.67.1 4
21.5 even 6 98.2.c.c.67.2 4
21.11 odd 6 98.2.c.c.79.1 4
21.17 even 6 98.2.c.c.79.2 4
21.20 even 2 98.2.a.b.1.1 2
24.5 odd 2 3136.2.a.bn.1.1 2
24.11 even 2 3136.2.a.bm.1.2 2
28.27 even 2 7056.2.a.cl.1.1 2
84.11 even 6 784.2.i.m.177.2 4
84.23 even 6 784.2.i.m.753.2 4
84.47 odd 6 784.2.i.m.753.1 4
84.59 odd 6 784.2.i.m.177.1 4
84.83 odd 2 784.2.a.l.1.2 2
105.62 odd 4 2450.2.c.v.99.4 4
105.83 odd 4 2450.2.c.v.99.1 4
105.104 even 2 2450.2.a.bj.1.2 2
168.83 odd 2 3136.2.a.bm.1.1 2
168.125 even 2 3136.2.a.bn.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
98.2.a.b.1.1 2 21.20 even 2
98.2.a.b.1.2 yes 2 3.2 odd 2
98.2.c.c.67.1 4 21.2 odd 6
98.2.c.c.67.2 4 21.5 even 6
98.2.c.c.79.1 4 21.11 odd 6
98.2.c.c.79.2 4 21.17 even 6
784.2.a.l.1.1 2 12.11 even 2
784.2.a.l.1.2 2 84.83 odd 2
784.2.i.m.177.1 4 84.59 odd 6
784.2.i.m.177.2 4 84.11 even 6
784.2.i.m.753.1 4 84.47 odd 6
784.2.i.m.753.2 4 84.23 even 6
882.2.a.n.1.1 2 7.6 odd 2 inner
882.2.a.n.1.2 2 1.1 even 1 trivial
882.2.g.l.361.1 4 7.2 even 3
882.2.g.l.361.2 4 7.5 odd 6
882.2.g.l.667.1 4 7.4 even 3
882.2.g.l.667.2 4 7.3 odd 6
2450.2.a.bj.1.1 2 15.14 odd 2
2450.2.a.bj.1.2 2 105.104 even 2
2450.2.c.v.99.1 4 105.83 odd 4
2450.2.c.v.99.2 4 15.8 even 4
2450.2.c.v.99.3 4 15.2 even 4
2450.2.c.v.99.4 4 105.62 odd 4
3136.2.a.bm.1.1 2 168.83 odd 2
3136.2.a.bm.1.2 2 24.11 even 2
3136.2.a.bn.1.1 2 24.5 odd 2
3136.2.a.bn.1.2 2 168.125 even 2
7056.2.a.cl.1.1 2 28.27 even 2
7056.2.a.cl.1.2 2 4.3 odd 2