# Properties

 Label 2541.2.a.z.1.1 Level $2541$ Weight $2$ Character 2541.1 Self dual yes Analytic conductor $20.290$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2541 = 3 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2541.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$20.2899871536$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.79129$$ of defining polynomial Character $$\chi$$ $$=$$ 2541.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.79129 q^{2} -1.00000 q^{3} +1.20871 q^{4} +3.00000 q^{5} +1.79129 q^{6} -1.00000 q^{7} +1.41742 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.79129 q^{2} -1.00000 q^{3} +1.20871 q^{4} +3.00000 q^{5} +1.79129 q^{6} -1.00000 q^{7} +1.41742 q^{8} +1.00000 q^{9} -5.37386 q^{10} -1.20871 q^{12} -1.00000 q^{13} +1.79129 q^{14} -3.00000 q^{15} -4.95644 q^{16} -7.58258 q^{17} -1.79129 q^{18} +6.58258 q^{19} +3.62614 q^{20} +1.00000 q^{21} -5.58258 q^{23} -1.41742 q^{24} +4.00000 q^{25} +1.79129 q^{26} -1.00000 q^{27} -1.20871 q^{28} +8.16515 q^{29} +5.37386 q^{30} +3.58258 q^{31} +6.04356 q^{32} +13.5826 q^{34} -3.00000 q^{35} +1.20871 q^{36} +1.00000 q^{37} -11.7913 q^{38} +1.00000 q^{39} +4.25227 q^{40} +11.1652 q^{41} -1.79129 q^{42} -1.58258 q^{43} +3.00000 q^{45} +10.0000 q^{46} +1.41742 q^{47} +4.95644 q^{48} +1.00000 q^{49} -7.16515 q^{50} +7.58258 q^{51} -1.20871 q^{52} -9.58258 q^{53} +1.79129 q^{54} -1.41742 q^{56} -6.58258 q^{57} -14.6261 q^{58} +4.58258 q^{59} -3.62614 q^{60} -10.0000 q^{61} -6.41742 q^{62} -1.00000 q^{63} -0.912878 q^{64} -3.00000 q^{65} +8.58258 q^{67} -9.16515 q^{68} +5.58258 q^{69} +5.37386 q^{70} +11.1652 q^{71} +1.41742 q^{72} -7.00000 q^{73} -1.79129 q^{74} -4.00000 q^{75} +7.95644 q^{76} -1.79129 q^{78} -7.16515 q^{79} -14.8693 q^{80} +1.00000 q^{81} -20.0000 q^{82} +11.5826 q^{83} +1.20871 q^{84} -22.7477 q^{85} +2.83485 q^{86} -8.16515 q^{87} +9.16515 q^{89} -5.37386 q^{90} +1.00000 q^{91} -6.74773 q^{92} -3.58258 q^{93} -2.53901 q^{94} +19.7477 q^{95} -6.04356 q^{96} -2.41742 q^{97} -1.79129 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 2 q^{3} + 7 q^{4} + 6 q^{5} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + q^2 - 2 * q^3 + 7 * q^4 + 6 * q^5 - q^6 - 2 * q^7 + 12 * q^8 + 2 * q^9 $$2 q + q^{2} - 2 q^{3} + 7 q^{4} + 6 q^{5} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} + 3 q^{10} - 7 q^{12} - 2 q^{13} - q^{14} - 6 q^{15} + 13 q^{16} - 6 q^{17} + q^{18} + 4 q^{19} + 21 q^{20} + 2 q^{21} - 2 q^{23} - 12 q^{24} + 8 q^{25} - q^{26} - 2 q^{27} - 7 q^{28} - 2 q^{29} - 3 q^{30} - 2 q^{31} + 35 q^{32} + 18 q^{34} - 6 q^{35} + 7 q^{36} + 2 q^{37} - 19 q^{38} + 2 q^{39} + 36 q^{40} + 4 q^{41} + q^{42} + 6 q^{43} + 6 q^{45} + 20 q^{46} + 12 q^{47} - 13 q^{48} + 2 q^{49} + 4 q^{50} + 6 q^{51} - 7 q^{52} - 10 q^{53} - q^{54} - 12 q^{56} - 4 q^{57} - 43 q^{58} - 21 q^{60} - 20 q^{61} - 22 q^{62} - 2 q^{63} + 44 q^{64} - 6 q^{65} + 8 q^{67} + 2 q^{69} - 3 q^{70} + 4 q^{71} + 12 q^{72} - 14 q^{73} + q^{74} - 8 q^{75} - 7 q^{76} + q^{78} + 4 q^{79} + 39 q^{80} + 2 q^{81} - 40 q^{82} + 14 q^{83} + 7 q^{84} - 18 q^{85} + 24 q^{86} + 2 q^{87} + 3 q^{90} + 2 q^{91} + 14 q^{92} + 2 q^{93} + 27 q^{94} + 12 q^{95} - 35 q^{96} - 14 q^{97} + q^{98}+O(q^{100})$$ 2 * q + q^2 - 2 * q^3 + 7 * q^4 + 6 * q^5 - q^6 - 2 * q^7 + 12 * q^8 + 2 * q^9 + 3 * q^10 - 7 * q^12 - 2 * q^13 - q^14 - 6 * q^15 + 13 * q^16 - 6 * q^17 + q^18 + 4 * q^19 + 21 * q^20 + 2 * q^21 - 2 * q^23 - 12 * q^24 + 8 * q^25 - q^26 - 2 * q^27 - 7 * q^28 - 2 * q^29 - 3 * q^30 - 2 * q^31 + 35 * q^32 + 18 * q^34 - 6 * q^35 + 7 * q^36 + 2 * q^37 - 19 * q^38 + 2 * q^39 + 36 * q^40 + 4 * q^41 + q^42 + 6 * q^43 + 6 * q^45 + 20 * q^46 + 12 * q^47 - 13 * q^48 + 2 * q^49 + 4 * q^50 + 6 * q^51 - 7 * q^52 - 10 * q^53 - q^54 - 12 * q^56 - 4 * q^57 - 43 * q^58 - 21 * q^60 - 20 * q^61 - 22 * q^62 - 2 * q^63 + 44 * q^64 - 6 * q^65 + 8 * q^67 + 2 * q^69 - 3 * q^70 + 4 * q^71 + 12 * q^72 - 14 * q^73 + q^74 - 8 * q^75 - 7 * q^76 + q^78 + 4 * q^79 + 39 * q^80 + 2 * q^81 - 40 * q^82 + 14 * q^83 + 7 * q^84 - 18 * q^85 + 24 * q^86 + 2 * q^87 + 3 * q^90 + 2 * q^91 + 14 * q^92 + 2 * q^93 + 27 * q^94 + 12 * q^95 - 35 * q^96 - 14 * q^97 + q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.79129 −1.26663 −0.633316 0.773893i $$-0.718307\pi$$
−0.633316 + 0.773893i $$0.718307\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 1.20871 0.604356
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 1.79129 0.731290
$$7$$ −1.00000 −0.377964
$$8$$ 1.41742 0.501135
$$9$$ 1.00000 0.333333
$$10$$ −5.37386 −1.69936
$$11$$ 0 0
$$12$$ −1.20871 −0.348925
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 1.79129 0.478742
$$15$$ −3.00000 −0.774597
$$16$$ −4.95644 −1.23911
$$17$$ −7.58258 −1.83904 −0.919522 0.393038i $$-0.871424\pi$$
−0.919522 + 0.393038i $$0.871424\pi$$
$$18$$ −1.79129 −0.422211
$$19$$ 6.58258 1.51015 0.755073 0.655640i $$-0.227601\pi$$
0.755073 + 0.655640i $$0.227601\pi$$
$$20$$ 3.62614 0.810829
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ −5.58258 −1.16405 −0.582024 0.813172i $$-0.697739\pi$$
−0.582024 + 0.813172i $$0.697739\pi$$
$$24$$ −1.41742 −0.289331
$$25$$ 4.00000 0.800000
$$26$$ 1.79129 0.351300
$$27$$ −1.00000 −0.192450
$$28$$ −1.20871 −0.228425
$$29$$ 8.16515 1.51623 0.758115 0.652121i $$-0.226120\pi$$
0.758115 + 0.652121i $$0.226120\pi$$
$$30$$ 5.37386 0.981129
$$31$$ 3.58258 0.643450 0.321725 0.946833i $$-0.395737\pi$$
0.321725 + 0.946833i $$0.395737\pi$$
$$32$$ 6.04356 1.06836
$$33$$ 0 0
$$34$$ 13.5826 2.32939
$$35$$ −3.00000 −0.507093
$$36$$ 1.20871 0.201452
$$37$$ 1.00000 0.164399 0.0821995 0.996616i $$-0.473806\pi$$
0.0821995 + 0.996616i $$0.473806\pi$$
$$38$$ −11.7913 −1.91280
$$39$$ 1.00000 0.160128
$$40$$ 4.25227 0.672343
$$41$$ 11.1652 1.74370 0.871852 0.489770i $$-0.162919\pi$$
0.871852 + 0.489770i $$0.162919\pi$$
$$42$$ −1.79129 −0.276402
$$43$$ −1.58258 −0.241341 −0.120670 0.992693i $$-0.538504\pi$$
−0.120670 + 0.992693i $$0.538504\pi$$
$$44$$ 0 0
$$45$$ 3.00000 0.447214
$$46$$ 10.0000 1.47442
$$47$$ 1.41742 0.206753 0.103376 0.994642i $$-0.467035\pi$$
0.103376 + 0.994642i $$0.467035\pi$$
$$48$$ 4.95644 0.715400
$$49$$ 1.00000 0.142857
$$50$$ −7.16515 −1.01331
$$51$$ 7.58258 1.06177
$$52$$ −1.20871 −0.167618
$$53$$ −9.58258 −1.31627 −0.658134 0.752901i $$-0.728654\pi$$
−0.658134 + 0.752901i $$0.728654\pi$$
$$54$$ 1.79129 0.243763
$$55$$ 0 0
$$56$$ −1.41742 −0.189411
$$57$$ −6.58258 −0.871883
$$58$$ −14.6261 −1.92051
$$59$$ 4.58258 0.596601 0.298300 0.954472i $$-0.403580\pi$$
0.298300 + 0.954472i $$0.403580\pi$$
$$60$$ −3.62614 −0.468132
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −6.41742 −0.815014
$$63$$ −1.00000 −0.125988
$$64$$ −0.912878 −0.114110
$$65$$ −3.00000 −0.372104
$$66$$ 0 0
$$67$$ 8.58258 1.04853 0.524264 0.851556i $$-0.324340\pi$$
0.524264 + 0.851556i $$0.324340\pi$$
$$68$$ −9.16515 −1.11144
$$69$$ 5.58258 0.672063
$$70$$ 5.37386 0.642300
$$71$$ 11.1652 1.32506 0.662530 0.749036i $$-0.269483\pi$$
0.662530 + 0.749036i $$0.269483\pi$$
$$72$$ 1.41742 0.167045
$$73$$ −7.00000 −0.819288 −0.409644 0.912245i $$-0.634347\pi$$
−0.409644 + 0.912245i $$0.634347\pi$$
$$74$$ −1.79129 −0.208233
$$75$$ −4.00000 −0.461880
$$76$$ 7.95644 0.912666
$$77$$ 0 0
$$78$$ −1.79129 −0.202823
$$79$$ −7.16515 −0.806143 −0.403071 0.915169i $$-0.632057\pi$$
−0.403071 + 0.915169i $$0.632057\pi$$
$$80$$ −14.8693 −1.66244
$$81$$ 1.00000 0.111111
$$82$$ −20.0000 −2.20863
$$83$$ 11.5826 1.27135 0.635676 0.771956i $$-0.280721\pi$$
0.635676 + 0.771956i $$0.280721\pi$$
$$84$$ 1.20871 0.131881
$$85$$ −22.7477 −2.46734
$$86$$ 2.83485 0.305690
$$87$$ −8.16515 −0.875396
$$88$$ 0 0
$$89$$ 9.16515 0.971504 0.485752 0.874097i $$-0.338546\pi$$
0.485752 + 0.874097i $$0.338546\pi$$
$$90$$ −5.37386 −0.566455
$$91$$ 1.00000 0.104828
$$92$$ −6.74773 −0.703499
$$93$$ −3.58258 −0.371496
$$94$$ −2.53901 −0.261879
$$95$$ 19.7477 2.02607
$$96$$ −6.04356 −0.616818
$$97$$ −2.41742 −0.245452 −0.122726 0.992441i $$-0.539164\pi$$
−0.122726 + 0.992441i $$0.539164\pi$$
$$98$$ −1.79129 −0.180947
$$99$$ 0 0
$$100$$ 4.83485 0.483485
$$101$$ −11.5826 −1.15251 −0.576255 0.817270i $$-0.695486\pi$$
−0.576255 + 0.817270i $$0.695486\pi$$
$$102$$ −13.5826 −1.34488
$$103$$ −1.16515 −0.114806 −0.0574029 0.998351i $$-0.518282\pi$$
−0.0574029 + 0.998351i $$0.518282\pi$$
$$104$$ −1.41742 −0.138990
$$105$$ 3.00000 0.292770
$$106$$ 17.1652 1.66723
$$107$$ 12.5826 1.21640 0.608202 0.793782i $$-0.291891\pi$$
0.608202 + 0.793782i $$0.291891\pi$$
$$108$$ −1.20871 −0.116308
$$109$$ −3.58258 −0.343149 −0.171574 0.985171i $$-0.554885\pi$$
−0.171574 + 0.985171i $$0.554885\pi$$
$$110$$ 0 0
$$111$$ −1.00000 −0.0949158
$$112$$ 4.95644 0.468339
$$113$$ 9.16515 0.862185 0.431092 0.902308i $$-0.358128\pi$$
0.431092 + 0.902308i $$0.358128\pi$$
$$114$$ 11.7913 1.10436
$$115$$ −16.7477 −1.56173
$$116$$ 9.86932 0.916343
$$117$$ −1.00000 −0.0924500
$$118$$ −8.20871 −0.755673
$$119$$ 7.58258 0.695094
$$120$$ −4.25227 −0.388178
$$121$$ 0 0
$$122$$ 17.9129 1.62176
$$123$$ −11.1652 −1.00673
$$124$$ 4.33030 0.388873
$$125$$ −3.00000 −0.268328
$$126$$ 1.79129 0.159581
$$127$$ 11.5826 1.02779 0.513894 0.857854i $$-0.328203\pi$$
0.513894 + 0.857854i $$0.328203\pi$$
$$128$$ −10.4519 −0.923826
$$129$$ 1.58258 0.139338
$$130$$ 5.37386 0.471319
$$131$$ 16.0000 1.39793 0.698963 0.715158i $$-0.253645\pi$$
0.698963 + 0.715158i $$0.253645\pi$$
$$132$$ 0 0
$$133$$ −6.58258 −0.570782
$$134$$ −15.3739 −1.32810
$$135$$ −3.00000 −0.258199
$$136$$ −10.7477 −0.921610
$$137$$ −11.5826 −0.989566 −0.494783 0.869016i $$-0.664752\pi$$
−0.494783 + 0.869016i $$0.664752\pi$$
$$138$$ −10.0000 −0.851257
$$139$$ −11.1652 −0.947016 −0.473508 0.880790i $$-0.657012\pi$$
−0.473508 + 0.880790i $$0.657012\pi$$
$$140$$ −3.62614 −0.306464
$$141$$ −1.41742 −0.119369
$$142$$ −20.0000 −1.67836
$$143$$ 0 0
$$144$$ −4.95644 −0.413037
$$145$$ 24.4955 2.03424
$$146$$ 12.5390 1.03774
$$147$$ −1.00000 −0.0824786
$$148$$ 1.20871 0.0993555
$$149$$ −6.16515 −0.505069 −0.252534 0.967588i $$-0.581264\pi$$
−0.252534 + 0.967588i $$0.581264\pi$$
$$150$$ 7.16515 0.585032
$$151$$ −3.58258 −0.291546 −0.145773 0.989318i $$-0.546567\pi$$
−0.145773 + 0.989318i $$0.546567\pi$$
$$152$$ 9.33030 0.756787
$$153$$ −7.58258 −0.613015
$$154$$ 0 0
$$155$$ 10.7477 0.863278
$$156$$ 1.20871 0.0967744
$$157$$ 19.1652 1.52955 0.764773 0.644300i $$-0.222851\pi$$
0.764773 + 0.644300i $$0.222851\pi$$
$$158$$ 12.8348 1.02109
$$159$$ 9.58258 0.759948
$$160$$ 18.1307 1.43336
$$161$$ 5.58258 0.439969
$$162$$ −1.79129 −0.140737
$$163$$ 8.58258 0.672239 0.336120 0.941819i $$-0.390885\pi$$
0.336120 + 0.941819i $$0.390885\pi$$
$$164$$ 13.4955 1.05382
$$165$$ 0 0
$$166$$ −20.7477 −1.61034
$$167$$ 4.74773 0.367390 0.183695 0.982983i $$-0.441194\pi$$
0.183695 + 0.982983i $$0.441194\pi$$
$$168$$ 1.41742 0.109357
$$169$$ −12.0000 −0.923077
$$170$$ 40.7477 3.12521
$$171$$ 6.58258 0.503382
$$172$$ −1.91288 −0.145856
$$173$$ 7.16515 0.544756 0.272378 0.962190i $$-0.412190\pi$$
0.272378 + 0.962190i $$0.412190\pi$$
$$174$$ 14.6261 1.10880
$$175$$ −4.00000 −0.302372
$$176$$ 0 0
$$177$$ −4.58258 −0.344447
$$178$$ −16.4174 −1.23054
$$179$$ 14.3303 1.07110 0.535549 0.844504i $$-0.320105\pi$$
0.535549 + 0.844504i $$0.320105\pi$$
$$180$$ 3.62614 0.270276
$$181$$ −5.58258 −0.414950 −0.207475 0.978240i $$-0.566524\pi$$
−0.207475 + 0.978240i $$0.566524\pi$$
$$182$$ −1.79129 −0.132779
$$183$$ 10.0000 0.739221
$$184$$ −7.91288 −0.583345
$$185$$ 3.00000 0.220564
$$186$$ 6.41742 0.470548
$$187$$ 0 0
$$188$$ 1.71326 0.124952
$$189$$ 1.00000 0.0727393
$$190$$ −35.3739 −2.56629
$$191$$ 11.5826 0.838086 0.419043 0.907966i $$-0.362366\pi$$
0.419043 + 0.907966i $$0.362366\pi$$
$$192$$ 0.912878 0.0658813
$$193$$ 2.41742 0.174010 0.0870050 0.996208i $$-0.472270\pi$$
0.0870050 + 0.996208i $$0.472270\pi$$
$$194$$ 4.33030 0.310898
$$195$$ 3.00000 0.214834
$$196$$ 1.20871 0.0863366
$$197$$ −5.16515 −0.368002 −0.184001 0.982926i $$-0.558905\pi$$
−0.184001 + 0.982926i $$0.558905\pi$$
$$198$$ 0 0
$$199$$ −9.58258 −0.679291 −0.339645 0.940554i $$-0.610307\pi$$
−0.339645 + 0.940554i $$0.610307\pi$$
$$200$$ 5.66970 0.400908
$$201$$ −8.58258 −0.605368
$$202$$ 20.7477 1.45980
$$203$$ −8.16515 −0.573081
$$204$$ 9.16515 0.641689
$$205$$ 33.4955 2.33942
$$206$$ 2.08712 0.145417
$$207$$ −5.58258 −0.388016
$$208$$ 4.95644 0.343667
$$209$$ 0 0
$$210$$ −5.37386 −0.370832
$$211$$ 13.1652 0.906326 0.453163 0.891428i $$-0.350295\pi$$
0.453163 + 0.891428i $$0.350295\pi$$
$$212$$ −11.5826 −0.795495
$$213$$ −11.1652 −0.765024
$$214$$ −22.5390 −1.54074
$$215$$ −4.74773 −0.323792
$$216$$ −1.41742 −0.0964435
$$217$$ −3.58258 −0.243201
$$218$$ 6.41742 0.434643
$$219$$ 7.00000 0.473016
$$220$$ 0 0
$$221$$ 7.58258 0.510059
$$222$$ 1.79129 0.120223
$$223$$ 6.00000 0.401790 0.200895 0.979613i $$-0.435615\pi$$
0.200895 + 0.979613i $$0.435615\pi$$
$$224$$ −6.04356 −0.403802
$$225$$ 4.00000 0.266667
$$226$$ −16.4174 −1.09207
$$227$$ 22.0000 1.46019 0.730096 0.683345i $$-0.239475\pi$$
0.730096 + 0.683345i $$0.239475\pi$$
$$228$$ −7.95644 −0.526928
$$229$$ 0.747727 0.0494112 0.0247056 0.999695i $$-0.492135\pi$$
0.0247056 + 0.999695i $$0.492135\pi$$
$$230$$ 30.0000 1.97814
$$231$$ 0 0
$$232$$ 11.5735 0.759836
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ 1.79129 0.117100
$$235$$ 4.25227 0.277388
$$236$$ 5.53901 0.360559
$$237$$ 7.16515 0.465427
$$238$$ −13.5826 −0.880428
$$239$$ −16.5826 −1.07264 −0.536319 0.844015i $$-0.680186\pi$$
−0.536319 + 0.844015i $$0.680186\pi$$
$$240$$ 14.8693 0.959810
$$241$$ 10.1652 0.654795 0.327397 0.944887i $$-0.393828\pi$$
0.327397 + 0.944887i $$0.393828\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ −12.0871 −0.773799
$$245$$ 3.00000 0.191663
$$246$$ 20.0000 1.27515
$$247$$ −6.58258 −0.418839
$$248$$ 5.07803 0.322455
$$249$$ −11.5826 −0.734016
$$250$$ 5.37386 0.339873
$$251$$ 7.41742 0.468184 0.234092 0.972214i $$-0.424788\pi$$
0.234092 + 0.972214i $$0.424788\pi$$
$$252$$ −1.20871 −0.0761417
$$253$$ 0 0
$$254$$ −20.7477 −1.30183
$$255$$ 22.7477 1.42452
$$256$$ 20.5481 1.28426
$$257$$ 19.0000 1.18519 0.592594 0.805502i $$-0.298104\pi$$
0.592594 + 0.805502i $$0.298104\pi$$
$$258$$ −2.83485 −0.176490
$$259$$ −1.00000 −0.0621370
$$260$$ −3.62614 −0.224883
$$261$$ 8.16515 0.505410
$$262$$ −28.6606 −1.77066
$$263$$ 22.9129 1.41287 0.706434 0.707779i $$-0.250303\pi$$
0.706434 + 0.707779i $$0.250303\pi$$
$$264$$ 0 0
$$265$$ −28.7477 −1.76596
$$266$$ 11.7913 0.722970
$$267$$ −9.16515 −0.560898
$$268$$ 10.3739 0.633685
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 5.37386 0.327043
$$271$$ −5.41742 −0.329085 −0.164543 0.986370i $$-0.552615\pi$$
−0.164543 + 0.986370i $$0.552615\pi$$
$$272$$ 37.5826 2.27878
$$273$$ −1.00000 −0.0605228
$$274$$ 20.7477 1.25342
$$275$$ 0 0
$$276$$ 6.74773 0.406165
$$277$$ 19.1652 1.15152 0.575761 0.817618i $$-0.304706\pi$$
0.575761 + 0.817618i $$0.304706\pi$$
$$278$$ 20.0000 1.19952
$$279$$ 3.58258 0.214483
$$280$$ −4.25227 −0.254122
$$281$$ 27.3303 1.63039 0.815195 0.579187i $$-0.196630\pi$$
0.815195 + 0.579187i $$0.196630\pi$$
$$282$$ 2.53901 0.151196
$$283$$ −27.7477 −1.64943 −0.824716 0.565548i $$-0.808665\pi$$
−0.824716 + 0.565548i $$0.808665\pi$$
$$284$$ 13.4955 0.800808
$$285$$ −19.7477 −1.16975
$$286$$ 0 0
$$287$$ −11.1652 −0.659058
$$288$$ 6.04356 0.356120
$$289$$ 40.4955 2.38209
$$290$$ −43.8784 −2.57663
$$291$$ 2.41742 0.141712
$$292$$ −8.46099 −0.495142
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 1.79129 0.104470
$$295$$ 13.7477 0.800424
$$296$$ 1.41742 0.0823861
$$297$$ 0 0
$$298$$ 11.0436 0.639736
$$299$$ 5.58258 0.322849
$$300$$ −4.83485 −0.279140
$$301$$ 1.58258 0.0912181
$$302$$ 6.41742 0.369281
$$303$$ 11.5826 0.665402
$$304$$ −32.6261 −1.87124
$$305$$ −30.0000 −1.71780
$$306$$ 13.5826 0.776464
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 1.16515 0.0662831
$$310$$ −19.2523 −1.09346
$$311$$ 14.3303 0.812597 0.406298 0.913740i $$-0.366819\pi$$
0.406298 + 0.913740i $$0.366819\pi$$
$$312$$ 1.41742 0.0802458
$$313$$ 19.5826 1.10687 0.553436 0.832891i $$-0.313316\pi$$
0.553436 + 0.832891i $$0.313316\pi$$
$$314$$ −34.3303 −1.93737
$$315$$ −3.00000 −0.169031
$$316$$ −8.66061 −0.487197
$$317$$ −22.4174 −1.25909 −0.629544 0.776965i $$-0.716758\pi$$
−0.629544 + 0.776965i $$0.716758\pi$$
$$318$$ −17.1652 −0.962574
$$319$$ 0 0
$$320$$ −2.73864 −0.153094
$$321$$ −12.5826 −0.702291
$$322$$ −10.0000 −0.557278
$$323$$ −49.9129 −2.77723
$$324$$ 1.20871 0.0671507
$$325$$ −4.00000 −0.221880
$$326$$ −15.3739 −0.851480
$$327$$ 3.58258 0.198117
$$328$$ 15.8258 0.873831
$$329$$ −1.41742 −0.0781451
$$330$$ 0 0
$$331$$ −3.16515 −0.173972 −0.0869862 0.996210i $$-0.527724\pi$$
−0.0869862 + 0.996210i $$0.527724\pi$$
$$332$$ 14.0000 0.768350
$$333$$ 1.00000 0.0547997
$$334$$ −8.50455 −0.465348
$$335$$ 25.7477 1.40675
$$336$$ −4.95644 −0.270396
$$337$$ −17.5826 −0.957784 −0.478892 0.877874i $$-0.658961\pi$$
−0.478892 + 0.877874i $$0.658961\pi$$
$$338$$ 21.4955 1.16920
$$339$$ −9.16515 −0.497783
$$340$$ −27.4955 −1.49115
$$341$$ 0 0
$$342$$ −11.7913 −0.637600
$$343$$ −1.00000 −0.0539949
$$344$$ −2.24318 −0.120944
$$345$$ 16.7477 0.901667
$$346$$ −12.8348 −0.690006
$$347$$ −26.3303 −1.41348 −0.706742 0.707471i $$-0.749836\pi$$
−0.706742 + 0.707471i $$0.749836\pi$$
$$348$$ −9.86932 −0.529051
$$349$$ 15.0000 0.802932 0.401466 0.915874i $$-0.368501\pi$$
0.401466 + 0.915874i $$0.368501\pi$$
$$350$$ 7.16515 0.382993
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ 24.1652 1.28618 0.643091 0.765790i $$-0.277652\pi$$
0.643091 + 0.765790i $$0.277652\pi$$
$$354$$ 8.20871 0.436288
$$355$$ 33.4955 1.77775
$$356$$ 11.0780 0.587134
$$357$$ −7.58258 −0.401312
$$358$$ −25.6697 −1.35669
$$359$$ 8.83485 0.466285 0.233143 0.972443i $$-0.425099\pi$$
0.233143 + 0.972443i $$0.425099\pi$$
$$360$$ 4.25227 0.224114
$$361$$ 24.3303 1.28054
$$362$$ 10.0000 0.525588
$$363$$ 0 0
$$364$$ 1.20871 0.0633537
$$365$$ −21.0000 −1.09919
$$366$$ −17.9129 −0.936321
$$367$$ 22.0000 1.14839 0.574195 0.818718i $$-0.305315\pi$$
0.574195 + 0.818718i $$0.305315\pi$$
$$368$$ 27.6697 1.44238
$$369$$ 11.1652 0.581235
$$370$$ −5.37386 −0.279374
$$371$$ 9.58258 0.497503
$$372$$ −4.33030 −0.224516
$$373$$ 34.7477 1.79917 0.899585 0.436747i $$-0.143869\pi$$
0.899585 + 0.436747i $$0.143869\pi$$
$$374$$ 0 0
$$375$$ 3.00000 0.154919
$$376$$ 2.00909 0.103611
$$377$$ −8.16515 −0.420527
$$378$$ −1.79129 −0.0921339
$$379$$ −12.5826 −0.646323 −0.323162 0.946344i $$-0.604746\pi$$
−0.323162 + 0.946344i $$0.604746\pi$$
$$380$$ 23.8693 1.22447
$$381$$ −11.5826 −0.593393
$$382$$ −20.7477 −1.06155
$$383$$ 10.3303 0.527854 0.263927 0.964543i $$-0.414982\pi$$
0.263927 + 0.964543i $$0.414982\pi$$
$$384$$ 10.4519 0.533371
$$385$$ 0 0
$$386$$ −4.33030 −0.220407
$$387$$ −1.58258 −0.0804468
$$388$$ −2.92197 −0.148341
$$389$$ −26.3303 −1.33500 −0.667500 0.744610i $$-0.732635\pi$$
−0.667500 + 0.744610i $$0.732635\pi$$
$$390$$ −5.37386 −0.272116
$$391$$ 42.3303 2.14074
$$392$$ 1.41742 0.0715907
$$393$$ −16.0000 −0.807093
$$394$$ 9.25227 0.466123
$$395$$ −21.4955 −1.08155
$$396$$ 0 0
$$397$$ −31.5826 −1.58508 −0.792542 0.609817i $$-0.791243\pi$$
−0.792542 + 0.609817i $$0.791243\pi$$
$$398$$ 17.1652 0.860411
$$399$$ 6.58258 0.329541
$$400$$ −19.8258 −0.991288
$$401$$ 31.9129 1.59365 0.796827 0.604208i $$-0.206510\pi$$
0.796827 + 0.604208i $$0.206510\pi$$
$$402$$ 15.3739 0.766779
$$403$$ −3.58258 −0.178461
$$404$$ −14.0000 −0.696526
$$405$$ 3.00000 0.149071
$$406$$ 14.6261 0.725883
$$407$$ 0 0
$$408$$ 10.7477 0.532092
$$409$$ −8.33030 −0.411907 −0.205953 0.978562i $$-0.566030\pi$$
−0.205953 + 0.978562i $$0.566030\pi$$
$$410$$ −60.0000 −2.96319
$$411$$ 11.5826 0.571326
$$412$$ −1.40833 −0.0693836
$$413$$ −4.58258 −0.225494
$$414$$ 10.0000 0.491473
$$415$$ 34.7477 1.70570
$$416$$ −6.04356 −0.296310
$$417$$ 11.1652 0.546760
$$418$$ 0 0
$$419$$ 2.58258 0.126167 0.0630835 0.998008i $$-0.479907\pi$$
0.0630835 + 0.998008i $$0.479907\pi$$
$$420$$ 3.62614 0.176937
$$421$$ −33.6606 −1.64052 −0.820259 0.571993i $$-0.806171\pi$$
−0.820259 + 0.571993i $$0.806171\pi$$
$$422$$ −23.5826 −1.14798
$$423$$ 1.41742 0.0689175
$$424$$ −13.5826 −0.659628
$$425$$ −30.3303 −1.47124
$$426$$ 20.0000 0.969003
$$427$$ 10.0000 0.483934
$$428$$ 15.2087 0.735141
$$429$$ 0 0
$$430$$ 8.50455 0.410126
$$431$$ −17.7477 −0.854878 −0.427439 0.904044i $$-0.640584\pi$$
−0.427439 + 0.904044i $$0.640584\pi$$
$$432$$ 4.95644 0.238467
$$433$$ −11.1652 −0.536563 −0.268281 0.963341i $$-0.586456\pi$$
−0.268281 + 0.963341i $$0.586456\pi$$
$$434$$ 6.41742 0.308046
$$435$$ −24.4955 −1.17447
$$436$$ −4.33030 −0.207384
$$437$$ −36.7477 −1.75788
$$438$$ −12.5390 −0.599137
$$439$$ −17.4174 −0.831288 −0.415644 0.909527i $$-0.636444\pi$$
−0.415644 + 0.909527i $$0.636444\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ −13.5826 −0.646057
$$443$$ −23.1652 −1.10061 −0.550305 0.834964i $$-0.685488\pi$$
−0.550305 + 0.834964i $$0.685488\pi$$
$$444$$ −1.20871 −0.0573629
$$445$$ 27.4955 1.30341
$$446$$ −10.7477 −0.508920
$$447$$ 6.16515 0.291602
$$448$$ 0.912878 0.0431295
$$449$$ −18.3303 −0.865060 −0.432530 0.901619i $$-0.642379\pi$$
−0.432530 + 0.901619i $$0.642379\pi$$
$$450$$ −7.16515 −0.337768
$$451$$ 0 0
$$452$$ 11.0780 0.521067
$$453$$ 3.58258 0.168324
$$454$$ −39.4083 −1.84952
$$455$$ 3.00000 0.140642
$$456$$ −9.33030 −0.436931
$$457$$ 19.9129 0.931485 0.465743 0.884920i $$-0.345787\pi$$
0.465743 + 0.884920i $$0.345787\pi$$
$$458$$ −1.33939 −0.0625858
$$459$$ 7.58258 0.353924
$$460$$ −20.2432 −0.943843
$$461$$ −18.3303 −0.853727 −0.426864 0.904316i $$-0.640382\pi$$
−0.426864 + 0.904316i $$0.640382\pi$$
$$462$$ 0 0
$$463$$ 8.58258 0.398866 0.199433 0.979911i $$-0.436090\pi$$
0.199433 + 0.979911i $$0.436090\pi$$
$$464$$ −40.4701 −1.87878
$$465$$ −10.7477 −0.498414
$$466$$ −25.0780 −1.16172
$$467$$ −38.5826 −1.78539 −0.892694 0.450663i $$-0.851188\pi$$
−0.892694 + 0.450663i $$0.851188\pi$$
$$468$$ −1.20871 −0.0558727
$$469$$ −8.58258 −0.396307
$$470$$ −7.61704 −0.351348
$$471$$ −19.1652 −0.883084
$$472$$ 6.49545 0.298978
$$473$$ 0 0
$$474$$ −12.8348 −0.589524
$$475$$ 26.3303 1.20812
$$476$$ 9.16515 0.420084
$$477$$ −9.58258 −0.438756
$$478$$ 29.7042 1.35864
$$479$$ 15.5826 0.711986 0.355993 0.934489i $$-0.384143\pi$$
0.355993 + 0.934489i $$0.384143\pi$$
$$480$$ −18.1307 −0.827549
$$481$$ −1.00000 −0.0455961
$$482$$ −18.2087 −0.829384
$$483$$ −5.58258 −0.254016
$$484$$ 0 0
$$485$$ −7.25227 −0.329309
$$486$$ 1.79129 0.0812545
$$487$$ 10.3303 0.468111 0.234055 0.972223i $$-0.424800\pi$$
0.234055 + 0.972223i $$0.424800\pi$$
$$488$$ −14.1742 −0.641638
$$489$$ −8.58258 −0.388117
$$490$$ −5.37386 −0.242766
$$491$$ 22.9129 1.03404 0.517022 0.855972i $$-0.327041\pi$$
0.517022 + 0.855972i $$0.327041\pi$$
$$492$$ −13.4955 −0.608422
$$493$$ −61.9129 −2.78842
$$494$$ 11.7913 0.530515
$$495$$ 0 0
$$496$$ −17.7568 −0.797305
$$497$$ −11.1652 −0.500825
$$498$$ 20.7477 0.929728
$$499$$ 41.7477 1.86888 0.934442 0.356114i $$-0.115899\pi$$
0.934442 + 0.356114i $$0.115899\pi$$
$$500$$ −3.62614 −0.162166
$$501$$ −4.74773 −0.212113
$$502$$ −13.2867 −0.593016
$$503$$ 0.747727 0.0333395 0.0166698 0.999861i $$-0.494694\pi$$
0.0166698 + 0.999861i $$0.494694\pi$$
$$504$$ −1.41742 −0.0631371
$$505$$ −34.7477 −1.54625
$$506$$ 0 0
$$507$$ 12.0000 0.532939
$$508$$ 14.0000 0.621150
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ −40.7477 −1.80434
$$511$$ 7.00000 0.309662
$$512$$ −15.9038 −0.702855
$$513$$ −6.58258 −0.290628
$$514$$ −34.0345 −1.50120
$$515$$ −3.49545 −0.154028
$$516$$ 1.91288 0.0842098
$$517$$ 0 0
$$518$$ 1.79129 0.0787047
$$519$$ −7.16515 −0.314515
$$520$$ −4.25227 −0.186475
$$521$$ 15.8348 0.693737 0.346869 0.937914i $$-0.387245\pi$$
0.346869 + 0.937914i $$0.387245\pi$$
$$522$$ −14.6261 −0.640169
$$523$$ −15.4174 −0.674157 −0.337078 0.941477i $$-0.609439\pi$$
−0.337078 + 0.941477i $$0.609439\pi$$
$$524$$ 19.3394 0.844845
$$525$$ 4.00000 0.174574
$$526$$ −41.0436 −1.78958
$$527$$ −27.1652 −1.18333
$$528$$ 0 0
$$529$$ 8.16515 0.355007
$$530$$ 51.4955 2.23682
$$531$$ 4.58258 0.198867
$$532$$ −7.95644 −0.344955
$$533$$ −11.1652 −0.483616
$$534$$ 16.4174 0.710451
$$535$$ 37.7477 1.63198
$$536$$ 12.1652 0.525455
$$537$$ −14.3303 −0.618398
$$538$$ −17.9129 −0.772279
$$539$$ 0 0
$$540$$ −3.62614 −0.156044
$$541$$ −18.3303 −0.788081 −0.394041 0.919093i $$-0.628923\pi$$
−0.394041 + 0.919093i $$0.628923\pi$$
$$542$$ 9.70417 0.416830
$$543$$ 5.58258 0.239571
$$544$$ −45.8258 −1.96476
$$545$$ −10.7477 −0.460382
$$546$$ 1.79129 0.0766600
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ −14.0000 −0.598050
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 53.7477 2.28973
$$552$$ 7.91288 0.336794
$$553$$ 7.16515 0.304693
$$554$$ −34.3303 −1.45855
$$555$$ −3.00000 −0.127343
$$556$$ −13.4955 −0.572335
$$557$$ −9.33030 −0.395338 −0.197669 0.980269i $$-0.563337\pi$$
−0.197669 + 0.980269i $$0.563337\pi$$
$$558$$ −6.41742 −0.271671
$$559$$ 1.58258 0.0669358
$$560$$ 14.8693 0.628343
$$561$$ 0 0
$$562$$ −48.9564 −2.06510
$$563$$ 37.5826 1.58392 0.791958 0.610575i $$-0.209062\pi$$
0.791958 + 0.610575i $$0.209062\pi$$
$$564$$ −1.71326 −0.0721412
$$565$$ 27.4955 1.15674
$$566$$ 49.7042 2.08922
$$567$$ −1.00000 −0.0419961
$$568$$ 15.8258 0.664034
$$569$$ 26.6606 1.11767 0.558835 0.829279i $$-0.311248\pi$$
0.558835 + 0.829279i $$0.311248\pi$$
$$570$$ 35.3739 1.48165
$$571$$ −28.8348 −1.20670 −0.603350 0.797476i $$-0.706168\pi$$
−0.603350 + 0.797476i $$0.706168\pi$$
$$572$$ 0 0
$$573$$ −11.5826 −0.483869
$$574$$ 20.0000 0.834784
$$575$$ −22.3303 −0.931238
$$576$$ −0.912878 −0.0380366
$$577$$ −21.9129 −0.912245 −0.456123 0.889917i $$-0.650762\pi$$
−0.456123 + 0.889917i $$0.650762\pi$$
$$578$$ −72.5390 −3.01723
$$579$$ −2.41742 −0.100465
$$580$$ 29.6080 1.22940
$$581$$ −11.5826 −0.480526
$$582$$ −4.33030 −0.179497
$$583$$ 0 0
$$584$$ −9.92197 −0.410574
$$585$$ −3.00000 −0.124035
$$586$$ 0 0
$$587$$ 37.7477 1.55802 0.779008 0.627014i $$-0.215723\pi$$
0.779008 + 0.627014i $$0.215723\pi$$
$$588$$ −1.20871 −0.0498464
$$589$$ 23.5826 0.971703
$$590$$ −24.6261 −1.01384
$$591$$ 5.16515 0.212466
$$592$$ −4.95644 −0.203708
$$593$$ −16.0000 −0.657041 −0.328521 0.944497i $$-0.606550\pi$$
−0.328521 + 0.944497i $$0.606550\pi$$
$$594$$ 0 0
$$595$$ 22.7477 0.932566
$$596$$ −7.45189 −0.305241
$$597$$ 9.58258 0.392189
$$598$$ −10.0000 −0.408930
$$599$$ 7.16515 0.292760 0.146380 0.989228i $$-0.453238\pi$$
0.146380 + 0.989228i $$0.453238\pi$$
$$600$$ −5.66970 −0.231464
$$601$$ −24.4955 −0.999190 −0.499595 0.866259i $$-0.666518\pi$$
−0.499595 + 0.866259i $$0.666518\pi$$
$$602$$ −2.83485 −0.115540
$$603$$ 8.58258 0.349510
$$604$$ −4.33030 −0.176198
$$605$$ 0 0
$$606$$ −20.7477 −0.842819
$$607$$ 21.7477 0.882713 0.441357 0.897332i $$-0.354497\pi$$
0.441357 + 0.897332i $$0.354497\pi$$
$$608$$ 39.7822 1.61338
$$609$$ 8.16515 0.330869
$$610$$ 53.7386 2.17581
$$611$$ −1.41742 −0.0573428
$$612$$ −9.16515 −0.370479
$$613$$ 26.7477 1.08033 0.540165 0.841559i $$-0.318362\pi$$
0.540165 + 0.841559i $$0.318362\pi$$
$$614$$ 0 0
$$615$$ −33.4955 −1.35067
$$616$$ 0 0
$$617$$ 2.83485 0.114127 0.0570634 0.998371i $$-0.481826\pi$$
0.0570634 + 0.998371i $$0.481826\pi$$
$$618$$ −2.08712 −0.0839563
$$619$$ −29.0780 −1.16874 −0.584372 0.811486i $$-0.698659\pi$$
−0.584372 + 0.811486i $$0.698659\pi$$
$$620$$ 12.9909 0.521727
$$621$$ 5.58258 0.224021
$$622$$ −25.6697 −1.02926
$$623$$ −9.16515 −0.367194
$$624$$ −4.95644 −0.198416
$$625$$ −29.0000 −1.16000
$$626$$ −35.0780 −1.40200
$$627$$ 0 0
$$628$$ 23.1652 0.924390
$$629$$ −7.58258 −0.302337
$$630$$ 5.37386 0.214100
$$631$$ 23.1652 0.922190 0.461095 0.887351i $$-0.347457\pi$$
0.461095 + 0.887351i $$0.347457\pi$$
$$632$$ −10.1561 −0.403986
$$633$$ −13.1652 −0.523268
$$634$$ 40.1561 1.59480
$$635$$ 34.7477 1.37892
$$636$$ 11.5826 0.459279
$$637$$ −1.00000 −0.0396214
$$638$$ 0 0
$$639$$ 11.1652 0.441687
$$640$$ −31.3557 −1.23944
$$641$$ 43.5826 1.72141 0.860704 0.509106i $$-0.170024\pi$$
0.860704 + 0.509106i $$0.170024\pi$$
$$642$$ 22.5390 0.889544
$$643$$ 38.2432 1.50816 0.754082 0.656780i $$-0.228082\pi$$
0.754082 + 0.656780i $$0.228082\pi$$
$$644$$ 6.74773 0.265898
$$645$$ 4.74773 0.186942
$$646$$ 89.4083 3.51772
$$647$$ −10.9129 −0.429030 −0.214515 0.976721i $$-0.568817\pi$$
−0.214515 + 0.976721i $$0.568817\pi$$
$$648$$ 1.41742 0.0556817
$$649$$ 0 0
$$650$$ 7.16515 0.281040
$$651$$ 3.58258 0.140412
$$652$$ 10.3739 0.406272
$$653$$ −30.3303 −1.18692 −0.593458 0.804865i $$-0.702238\pi$$
−0.593458 + 0.804865i $$0.702238\pi$$
$$654$$ −6.41742 −0.250941
$$655$$ 48.0000 1.87552
$$656$$ −55.3394 −2.16064
$$657$$ −7.00000 −0.273096
$$658$$ 2.53901 0.0989811
$$659$$ 28.5826 1.11342 0.556710 0.830707i $$-0.312064\pi$$
0.556710 + 0.830707i $$0.312064\pi$$
$$660$$ 0 0
$$661$$ −39.0780 −1.51996 −0.759980 0.649947i $$-0.774791\pi$$
−0.759980 + 0.649947i $$0.774791\pi$$
$$662$$ 5.66970 0.220359
$$663$$ −7.58258 −0.294483
$$664$$ 16.4174 0.637120
$$665$$ −19.7477 −0.765784
$$666$$ −1.79129 −0.0694110
$$667$$ −45.5826 −1.76496
$$668$$ 5.73864 0.222034
$$669$$ −6.00000 −0.231973
$$670$$ −46.1216 −1.78183
$$671$$ 0 0
$$672$$ 6.04356 0.233135
$$673$$ −11.2523 −0.433743 −0.216872 0.976200i $$-0.569585\pi$$
−0.216872 + 0.976200i $$0.569585\pi$$
$$674$$ 31.4955 1.21316
$$675$$ −4.00000 −0.153960
$$676$$ −14.5045 −0.557867
$$677$$ 45.1652 1.73584 0.867919 0.496706i $$-0.165457\pi$$
0.867919 + 0.496706i $$0.165457\pi$$
$$678$$ 16.4174 0.630507
$$679$$ 2.41742 0.0927722
$$680$$ −32.2432 −1.23647
$$681$$ −22.0000 −0.843042
$$682$$ 0 0
$$683$$ −33.0780 −1.26570 −0.632848 0.774276i $$-0.718114\pi$$
−0.632848 + 0.774276i $$0.718114\pi$$
$$684$$ 7.95644 0.304222
$$685$$ −34.7477 −1.32764
$$686$$ 1.79129 0.0683917
$$687$$ −0.747727 −0.0285276
$$688$$ 7.84394 0.299047
$$689$$ 9.58258 0.365067
$$690$$ −30.0000 −1.14208
$$691$$ 10.0000 0.380418 0.190209 0.981744i $$-0.439083\pi$$
0.190209 + 0.981744i $$0.439083\pi$$
$$692$$ 8.66061 0.329227
$$693$$ 0 0
$$694$$ 47.1652 1.79036
$$695$$ −33.4955 −1.27055
$$696$$ −11.5735 −0.438692
$$697$$ −84.6606 −3.20675
$$698$$ −26.8693 −1.01702
$$699$$ −14.0000 −0.529529
$$700$$ −4.83485 −0.182740
$$701$$ −10.0000 −0.377695 −0.188847 0.982006i $$-0.560475\pi$$
−0.188847 + 0.982006i $$0.560475\pi$$
$$702$$ −1.79129 −0.0676078
$$703$$ 6.58258 0.248267
$$704$$ 0 0
$$705$$ −4.25227 −0.160150
$$706$$ −43.2867 −1.62912
$$707$$ 11.5826 0.435608
$$708$$ −5.53901 −0.208169
$$709$$ 27.6606 1.03882 0.519408 0.854526i $$-0.326153\pi$$
0.519408 + 0.854526i $$0.326153\pi$$
$$710$$ −60.0000 −2.25176
$$711$$ −7.16515 −0.268714
$$712$$ 12.9909 0.486855
$$713$$ −20.0000 −0.749006
$$714$$ 13.5826 0.508315
$$715$$ 0 0
$$716$$ 17.3212 0.647324
$$717$$ 16.5826 0.619288
$$718$$ −15.8258 −0.590612
$$719$$ −14.0780 −0.525022 −0.262511 0.964929i $$-0.584551\pi$$
−0.262511 + 0.964929i $$0.584551\pi$$
$$720$$ −14.8693 −0.554147
$$721$$ 1.16515 0.0433925
$$722$$ −43.5826 −1.62198
$$723$$ −10.1652 −0.378046
$$724$$ −6.74773 −0.250777
$$725$$ 32.6606 1.21298
$$726$$ 0 0
$$727$$ −15.9129 −0.590176 −0.295088 0.955470i $$-0.595349\pi$$
−0.295088 + 0.955470i $$0.595349\pi$$
$$728$$ 1.41742 0.0525332
$$729$$ 1.00000 0.0370370
$$730$$ 37.6170 1.39227
$$731$$ 12.0000 0.443836
$$732$$ 12.0871 0.446753
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ −39.4083 −1.45459
$$735$$ −3.00000 −0.110657
$$736$$ −33.7386 −1.24362
$$737$$ 0 0
$$738$$ −20.0000 −0.736210
$$739$$ 31.9129 1.17393 0.586967 0.809611i $$-0.300322\pi$$
0.586967 + 0.809611i $$0.300322\pi$$
$$740$$ 3.62614 0.133299
$$741$$ 6.58258 0.241817
$$742$$ −17.1652 −0.630153
$$743$$ 53.2432 1.95330 0.976651 0.214830i $$-0.0689198\pi$$
0.976651 + 0.214830i $$0.0689198\pi$$
$$744$$ −5.07803 −0.186170
$$745$$ −18.4955 −0.677621
$$746$$ −62.2432 −2.27888
$$747$$ 11.5826 0.423784
$$748$$ 0 0
$$749$$ −12.5826 −0.459757
$$750$$ −5.37386 −0.196226
$$751$$ −8.91288 −0.325236 −0.162618 0.986689i $$-0.551994\pi$$
−0.162618 + 0.986689i $$0.551994\pi$$
$$752$$ −7.02538 −0.256189
$$753$$ −7.41742 −0.270306
$$754$$ 14.6261 0.532652
$$755$$ −10.7477 −0.391150
$$756$$ 1.20871 0.0439604
$$757$$ 27.3303 0.993337 0.496668 0.867940i $$-0.334557\pi$$
0.496668 + 0.867940i $$0.334557\pi$$
$$758$$ 22.5390 0.818654
$$759$$ 0 0
$$760$$ 27.9909 1.01534
$$761$$ −42.3303 −1.53447 −0.767236 0.641365i $$-0.778369\pi$$
−0.767236 + 0.641365i $$0.778369\pi$$
$$762$$ 20.7477 0.751611
$$763$$ 3.58258 0.129698
$$764$$ 14.0000 0.506502
$$765$$ −22.7477 −0.822446
$$766$$ −18.5045 −0.668596
$$767$$ −4.58258 −0.165467
$$768$$ −20.5481 −0.741466
$$769$$ −6.49545 −0.234232 −0.117116 0.993118i $$-0.537365\pi$$
−0.117116 + 0.993118i $$0.537365\pi$$
$$770$$ 0 0
$$771$$ −19.0000 −0.684268
$$772$$ 2.92197 0.105164
$$773$$ 6.16515 0.221745 0.110873 0.993835i $$-0.464635\pi$$
0.110873 + 0.993835i $$0.464635\pi$$
$$774$$ 2.83485 0.101897
$$775$$ 14.3303 0.514760
$$776$$ −3.42652 −0.123005
$$777$$ 1.00000 0.0358748
$$778$$ 47.1652 1.69095
$$779$$ 73.4955 2.63325
$$780$$ 3.62614 0.129837
$$781$$ 0 0
$$782$$ −75.8258 −2.71152
$$783$$ −8.16515 −0.291799
$$784$$ −4.95644 −0.177016
$$785$$ 57.4955 2.05210
$$786$$ 28.6606 1.02229
$$787$$ −38.5826 −1.37532 −0.687660 0.726033i $$-0.741362\pi$$
−0.687660 + 0.726033i $$0.741362\pi$$
$$788$$ −6.24318 −0.222404
$$789$$ −22.9129 −0.815720
$$790$$ 38.5045 1.36993
$$791$$ −9.16515 −0.325875
$$792$$ 0 0
$$793$$ 10.0000 0.355110
$$794$$ 56.5735 2.00772
$$795$$ 28.7477 1.01958
$$796$$ −11.5826 −0.410534
$$797$$ −52.4955 −1.85948 −0.929742 0.368211i $$-0.879970\pi$$
−0.929742 + 0.368211i $$0.879970\pi$$
$$798$$ −11.7913 −0.417407
$$799$$ −10.7477 −0.380227
$$800$$ 24.1742 0.854689
$$801$$ 9.16515 0.323835
$$802$$ −57.1652 −2.01857
$$803$$ 0 0
$$804$$ −10.3739 −0.365858
$$805$$ 16.7477 0.590280
$$806$$ 6.41742 0.226044
$$807$$ −10.0000 −0.352017
$$808$$ −16.4174 −0.577563
$$809$$ −9.33030 −0.328036 −0.164018 0.986457i $$-0.552446\pi$$
−0.164018 + 0.986457i $$0.552446\pi$$
$$810$$ −5.37386 −0.188818
$$811$$ 2.25227 0.0790880 0.0395440 0.999218i $$-0.487409\pi$$
0.0395440 + 0.999218i $$0.487409\pi$$
$$812$$ −9.86932 −0.346345
$$813$$ 5.41742 0.189997
$$814$$ 0 0
$$815$$ 25.7477 0.901904
$$816$$ −37.5826 −1.31565
$$817$$ −10.4174 −0.364460
$$818$$ 14.9220 0.521734
$$819$$ 1.00000 0.0349428
$$820$$ 40.4864 1.41385
$$821$$ 47.0000 1.64031 0.820156 0.572140i $$-0.193887\pi$$
0.820156 + 0.572140i $$0.193887\pi$$
$$822$$ −20.7477 −0.723660
$$823$$ −30.5826 −1.06604 −0.533021 0.846102i $$-0.678943\pi$$
−0.533021 + 0.846102i $$0.678943\pi$$
$$824$$ −1.65151 −0.0575332
$$825$$ 0 0
$$826$$ 8.20871 0.285618
$$827$$ −8.91288 −0.309931 −0.154966 0.987920i $$-0.549527\pi$$
−0.154966 + 0.987920i $$0.549527\pi$$
$$828$$ −6.74773 −0.234500
$$829$$ −40.0000 −1.38926 −0.694629 0.719368i $$-0.744431\pi$$
−0.694629 + 0.719368i $$0.744431\pi$$
$$830$$ −62.2432 −2.16049
$$831$$ −19.1652 −0.664832
$$832$$ 0.912878 0.0316484
$$833$$ −7.58258 −0.262721
$$834$$ −20.0000 −0.692543
$$835$$ 14.2432 0.492906
$$836$$ 0 0
$$837$$ −3.58258 −0.123832
$$838$$ −4.62614 −0.159807
$$839$$ 7.08712 0.244675 0.122337 0.992489i $$-0.460961\pi$$
0.122337 + 0.992489i $$0.460961\pi$$
$$840$$ 4.25227 0.146717
$$841$$ 37.6697 1.29896
$$842$$ 60.2958 2.07793
$$843$$ −27.3303 −0.941306
$$844$$ 15.9129 0.547744
$$845$$ −36.0000 −1.23844
$$846$$ −2.53901 −0.0872931
$$847$$ 0 0
$$848$$ 47.4955 1.63100
$$849$$ 27.7477 0.952300
$$850$$ 54.3303 1.86351
$$851$$ −5.58258 −0.191368
$$852$$ −13.4955 −0.462347
$$853$$ −17.1652 −0.587724 −0.293862 0.955848i $$-0.594941\pi$$
−0.293862 + 0.955848i $$0.594941\pi$$
$$854$$ −17.9129 −0.612966
$$855$$ 19.7477 0.675358
$$856$$ 17.8348 0.609583
$$857$$ 7.66970 0.261992 0.130996 0.991383i $$-0.458183\pi$$
0.130996 + 0.991383i $$0.458183\pi$$
$$858$$ 0 0
$$859$$ 12.0000 0.409435 0.204717 0.978821i $$-0.434372\pi$$
0.204717 + 0.978821i $$0.434372\pi$$
$$860$$ −5.73864 −0.195686
$$861$$ 11.1652 0.380507
$$862$$ 31.7913 1.08282
$$863$$ 14.4174 0.490775 0.245387 0.969425i $$-0.421085\pi$$
0.245387 + 0.969425i $$0.421085\pi$$
$$864$$ −6.04356 −0.205606
$$865$$ 21.4955 0.730867
$$866$$ 20.0000 0.679628
$$867$$ −40.4955 −1.37530
$$868$$ −4.33030 −0.146980
$$869$$ 0 0
$$870$$ 43.8784 1.48762
$$871$$ −8.58258 −0.290809
$$872$$ −5.07803 −0.171964
$$873$$ −2.41742 −0.0818174
$$874$$ 65.8258 2.22659
$$875$$ 3.00000 0.101419
$$876$$ 8.46099 0.285870
$$877$$ −9.49545 −0.320639 −0.160319 0.987065i $$-0.551252\pi$$
−0.160319 + 0.987065i $$0.551252\pi$$
$$878$$ 31.1996 1.05294
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −37.6606 −1.26882 −0.634409 0.772998i $$-0.718756\pi$$
−0.634409 + 0.772998i $$0.718756\pi$$
$$882$$ −1.79129 −0.0603158
$$883$$ −55.7477 −1.87606 −0.938030 0.346554i $$-0.887352\pi$$
−0.938030 + 0.346554i $$0.887352\pi$$
$$884$$ 9.16515 0.308257
$$885$$ −13.7477 −0.462125
$$886$$ 41.4955 1.39407
$$887$$ −38.7477 −1.30102 −0.650511 0.759497i $$-0.725445\pi$$
−0.650511 + 0.759497i $$0.725445\pi$$
$$888$$ −1.41742 −0.0475656
$$889$$ −11.5826 −0.388467
$$890$$ −49.2523 −1.65094
$$891$$ 0 0
$$892$$ 7.25227 0.242824
$$893$$ 9.33030 0.312227
$$894$$ −11.0436 −0.369352
$$895$$ 42.9909 1.43703
$$896$$ 10.4519 0.349173
$$897$$ −5.58258 −0.186397
$$898$$ 32.8348 1.09571
$$899$$ 29.2523 0.975618
$$900$$ 4.83485 0.161162
$$901$$ 72.6606 2.42068
$$902$$ 0 0
$$903$$ −1.58258 −0.0526648
$$904$$ 12.9909 0.432071
$$905$$ −16.7477 −0.556713
$$906$$ −6.41742 −0.213205
$$907$$ 42.3303 1.40555 0.702777 0.711410i $$-0.251943\pi$$
0.702777 + 0.711410i $$0.251943\pi$$
$$908$$ 26.5917 0.882475
$$909$$ −11.5826 −0.384170
$$910$$ −5.37386 −0.178142
$$911$$ −3.49545 −0.115810 −0.0579048 0.998322i $$-0.518442\pi$$
−0.0579048 + 0.998322i $$0.518442\pi$$
$$912$$ 32.6261 1.08036
$$913$$ 0 0
$$914$$ −35.6697 −1.17985
$$915$$ 30.0000 0.991769
$$916$$ 0.903787 0.0298620
$$917$$ −16.0000 −0.528367
$$918$$ −13.5826 −0.448292
$$919$$ 8.08712 0.266770 0.133385 0.991064i $$-0.457415\pi$$
0.133385 + 0.991064i $$0.457415\pi$$
$$920$$ −23.7386 −0.782640
$$921$$ 0 0
$$922$$ 32.8348 1.08136
$$923$$ −11.1652 −0.367505
$$924$$ 0 0
$$925$$ 4.00000 0.131519
$$926$$ −15.3739 −0.505217
$$927$$ −1.16515 −0.0382686
$$928$$ 49.3466 1.61988
$$929$$ −21.3303 −0.699825 −0.349912 0.936782i $$-0.613789\pi$$
−0.349912 + 0.936782i $$0.613789\pi$$
$$930$$ 19.2523 0.631307
$$931$$ 6.58258 0.215735
$$932$$ 16.9220 0.554298
$$933$$ −14.3303 −0.469153
$$934$$ 69.1125 2.26143
$$935$$ 0 0
$$936$$ −1.41742 −0.0463300
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 15.3739 0.501974
$$939$$ −19.5826 −0.639053
$$940$$ 5.13977 0.167641
$$941$$ −35.1652 −1.14635 −0.573176 0.819433i $$-0.694289\pi$$
−0.573176 + 0.819433i $$0.694289\pi$$
$$942$$ 34.3303 1.11854
$$943$$ −62.3303 −2.02975
$$944$$ −22.7133 −0.739254
$$945$$ 3.00000 0.0975900
$$946$$ 0 0
$$947$$ 45.1652 1.46767 0.733835 0.679328i $$-0.237728\pi$$
0.733835 + 0.679328i $$0.237728\pi$$
$$948$$ 8.66061 0.281283
$$949$$ 7.00000 0.227230
$$950$$ −47.1652 −1.53024
$$951$$ 22.4174 0.726935
$$952$$ 10.7477 0.348336
$$953$$ 28.1652 0.912359 0.456179 0.889888i $$-0.349218\pi$$
0.456179 + 0.889888i $$0.349218\pi$$
$$954$$ 17.1652 0.555742
$$955$$ 34.7477 1.12441
$$956$$ −20.0436 −0.648255
$$957$$ 0 0
$$958$$ −27.9129 −0.901824
$$959$$ 11.5826 0.374021
$$960$$ 2.73864 0.0883891
$$961$$ −18.1652 −0.585973
$$962$$ 1.79129 0.0577534
$$963$$ 12.5826 0.405468
$$964$$ 12.2867 0.395729
$$965$$ 7.25227 0.233459
$$966$$ 10.0000 0.321745
$$967$$ 13.1652 0.423363 0.211681 0.977339i $$-0.432106\pi$$
0.211681 + 0.977339i $$0.432106\pi$$
$$968$$ 0 0
$$969$$ 49.9129 1.60343
$$970$$ 12.9909 0.417113
$$971$$ −12.5826 −0.403794 −0.201897 0.979407i $$-0.564711\pi$$
−0.201897 + 0.979407i $$0.564711\pi$$
$$972$$ −1.20871 −0.0387695
$$973$$ 11.1652 0.357938
$$974$$ −18.5045 −0.592924
$$975$$ 4.00000 0.128103
$$976$$ 49.5644 1.58652
$$977$$ 23.2523 0.743906 0.371953 0.928252i $$-0.378688\pi$$
0.371953 + 0.928252i $$0.378688\pi$$
$$978$$ 15.3739 0.491602
$$979$$ 0 0
$$980$$ 3.62614 0.115833
$$981$$ −3.58258 −0.114383
$$982$$ −41.0436 −1.30975
$$983$$ −4.83485 −0.154208 −0.0771039 0.997023i $$-0.524567\pi$$
−0.0771039 + 0.997023i $$0.524567\pi$$
$$984$$ −15.8258 −0.504507
$$985$$ −15.4955 −0.493726
$$986$$ 110.904 3.53190
$$987$$ 1.41742 0.0451171
$$988$$ −7.95644 −0.253128
$$989$$ 8.83485 0.280932
$$990$$ 0 0
$$991$$ −20.2523 −0.643335 −0.321667 0.946853i $$-0.604243\pi$$
−0.321667 + 0.946853i $$0.604243\pi$$
$$992$$ 21.6515 0.687436
$$993$$ 3.16515 0.100443
$$994$$ 20.0000 0.634361
$$995$$ −28.7477 −0.911364
$$996$$ −14.0000 −0.443607
$$997$$ −10.6606 −0.337625 −0.168812 0.985648i $$-0.553993\pi$$
−0.168812 + 0.985648i $$0.553993\pi$$
$$998$$ −74.7822 −2.36719
$$999$$ −1.00000 −0.0316386
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 2541.2.a.z.1.1 2
3.2 odd 2 7623.2.a.bf.1.2 2
11.10 odd 2 231.2.a.b.1.2 2
33.32 even 2 693.2.a.j.1.1 2
44.43 even 2 3696.2.a.bl.1.2 2
55.54 odd 2 5775.2.a.bn.1.1 2
77.76 even 2 1617.2.a.o.1.2 2
231.230 odd 2 4851.2.a.ba.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.b.1.2 2 11.10 odd 2
693.2.a.j.1.1 2 33.32 even 2
1617.2.a.o.1.2 2 77.76 even 2
2541.2.a.z.1.1 2 1.1 even 1 trivial
3696.2.a.bl.1.2 2 44.43 even 2
4851.2.a.ba.1.1 2 231.230 odd 2
5775.2.a.bn.1.1 2 55.54 odd 2
7623.2.a.bf.1.2 2 3.2 odd 2