# Properties

 Label 5577.2.a.x.1.7 Level $5577$ Weight $2$ Character 5577.1 Self dual yes Analytic conductor $44.533$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5577 = 3 \cdot 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5577.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.5325692073$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3 x^{6} - 7 x^{5} + 21 x^{4} + 13 x^{3} - 33 x^{2} - 7 x + 7$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 429) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.7 Root $$-2.15754$$ of defining polynomial Character $$\chi$$ $$=$$ 5577.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.15754 q^{2} +1.00000 q^{3} +2.65498 q^{4} -0.710210 q^{5} +2.15754 q^{6} -2.30964 q^{7} +1.41315 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.15754 q^{2} +1.00000 q^{3} +2.65498 q^{4} -0.710210 q^{5} +2.15754 q^{6} -2.30964 q^{7} +1.41315 q^{8} +1.00000 q^{9} -1.53231 q^{10} -1.00000 q^{11} +2.65498 q^{12} -4.98315 q^{14} -0.710210 q^{15} -2.26104 q^{16} -6.68027 q^{17} +2.15754 q^{18} +0.242517 q^{19} -1.88559 q^{20} -2.30964 q^{21} -2.15754 q^{22} +9.53531 q^{23} +1.41315 q^{24} -4.49560 q^{25} +1.00000 q^{27} -6.13206 q^{28} -2.95273 q^{29} -1.53231 q^{30} +4.02017 q^{31} -7.70458 q^{32} -1.00000 q^{33} -14.4130 q^{34} +1.64033 q^{35} +2.65498 q^{36} -3.42304 q^{37} +0.523240 q^{38} -1.00363 q^{40} -9.46022 q^{41} -4.98315 q^{42} -11.8791 q^{43} -2.65498 q^{44} -0.710210 q^{45} +20.5728 q^{46} +12.9178 q^{47} -2.26104 q^{48} -1.66555 q^{49} -9.69944 q^{50} -6.68027 q^{51} +6.78954 q^{53} +2.15754 q^{54} +0.710210 q^{55} -3.26386 q^{56} +0.242517 q^{57} -6.37063 q^{58} -7.81320 q^{59} -1.88559 q^{60} +0.910585 q^{61} +8.67368 q^{62} -2.30964 q^{63} -12.1009 q^{64} -2.15754 q^{66} -9.32216 q^{67} -17.7360 q^{68} +9.53531 q^{69} +3.53908 q^{70} -12.9704 q^{71} +1.41315 q^{72} -8.51665 q^{73} -7.38534 q^{74} -4.49560 q^{75} +0.643877 q^{76} +2.30964 q^{77} +1.82360 q^{79} +1.60581 q^{80} +1.00000 q^{81} -20.4108 q^{82} -4.64671 q^{83} -6.13206 q^{84} +4.74440 q^{85} -25.6297 q^{86} -2.95273 q^{87} -1.41315 q^{88} -14.7714 q^{89} -1.53231 q^{90} +25.3161 q^{92} +4.02017 q^{93} +27.8707 q^{94} -0.172238 q^{95} -7.70458 q^{96} -1.86542 q^{97} -3.59349 q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q - 3q^{2} + 7q^{3} + 9q^{4} - 6q^{5} - 3q^{6} - 6q^{7} - 15q^{8} + 7q^{9} + O(q^{10})$$ $$7q - 3q^{2} + 7q^{3} + 9q^{4} - 6q^{5} - 3q^{6} - 6q^{7} - 15q^{8} + 7q^{9} - 7q^{11} + 9q^{12} + 8q^{14} - 6q^{15} + 17q^{16} - 2q^{17} - 3q^{18} - 8q^{19} + 2q^{20} - 6q^{21} + 3q^{22} + 4q^{23} - 15q^{24} + 13q^{25} + 7q^{27} - 12q^{28} - 12q^{29} + 10q^{31} - 33q^{32} - 7q^{33} - 28q^{34} - 4q^{35} + 9q^{36} - 6q^{37} + 16q^{38} - 10q^{40} - 2q^{41} + 8q^{42} - 16q^{43} - 9q^{44} - 6q^{45} + 26q^{46} - 18q^{47} + 17q^{48} + 23q^{49} - 39q^{50} - 2q^{51} + 10q^{53} - 3q^{54} + 6q^{55} + 16q^{56} - 8q^{57} - 10q^{58} - 2q^{59} + 2q^{60} - 10q^{61} - 36q^{62} - 6q^{63} + 29q^{64} + 3q^{66} - 8q^{67} - 10q^{68} + 4q^{69} + 20q^{70} - 36q^{71} - 15q^{72} - 20q^{73} + 13q^{75} - 10q^{76} + 6q^{77} + 6q^{79} + 20q^{80} + 7q^{81} - 10q^{82} - 30q^{83} - 12q^{84} + 40q^{85} - 6q^{86} - 12q^{87} + 15q^{88} - 34q^{89} - 12q^{92} + 10q^{93} + 32q^{94} + 18q^{95} - 33q^{96} - 16q^{97} - q^{98} - 7q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.15754 1.52561 0.762806 0.646628i $$-0.223821\pi$$
0.762806 + 0.646628i $$0.223821\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 2.65498 1.32749
$$5$$ −0.710210 −0.317616 −0.158808 0.987310i $$-0.550765\pi$$
−0.158808 + 0.987310i $$0.550765\pi$$
$$6$$ 2.15754 0.880812
$$7$$ −2.30964 −0.872963 −0.436482 0.899713i $$-0.643776\pi$$
−0.436482 + 0.899713i $$0.643776\pi$$
$$8$$ 1.41315 0.499622
$$9$$ 1.00000 0.333333
$$10$$ −1.53231 −0.484558
$$11$$ −1.00000 −0.301511
$$12$$ 2.65498 0.766427
$$13$$ 0 0
$$14$$ −4.98315 −1.33180
$$15$$ −0.710210 −0.183375
$$16$$ −2.26104 −0.565261
$$17$$ −6.68027 −1.62020 −0.810102 0.586289i $$-0.800588\pi$$
−0.810102 + 0.586289i $$0.800588\pi$$
$$18$$ 2.15754 0.508537
$$19$$ 0.242517 0.0556372 0.0278186 0.999613i $$-0.491144\pi$$
0.0278186 + 0.999613i $$0.491144\pi$$
$$20$$ −1.88559 −0.421631
$$21$$ −2.30964 −0.504005
$$22$$ −2.15754 −0.459989
$$23$$ 9.53531 1.98825 0.994125 0.108242i $$-0.0345220\pi$$
0.994125 + 0.108242i $$0.0345220\pi$$
$$24$$ 1.41315 0.288457
$$25$$ −4.49560 −0.899120
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ −6.13206 −1.15885
$$29$$ −2.95273 −0.548308 −0.274154 0.961686i $$-0.588398\pi$$
−0.274154 + 0.961686i $$0.588398\pi$$
$$30$$ −1.53231 −0.279760
$$31$$ 4.02017 0.722044 0.361022 0.932557i $$-0.382428\pi$$
0.361022 + 0.932557i $$0.382428\pi$$
$$32$$ −7.70458 −1.36199
$$33$$ −1.00000 −0.174078
$$34$$ −14.4130 −2.47180
$$35$$ 1.64033 0.277267
$$36$$ 2.65498 0.442497
$$37$$ −3.42304 −0.562744 −0.281372 0.959599i $$-0.590789\pi$$
−0.281372 + 0.959599i $$0.590789\pi$$
$$38$$ 0.523240 0.0848807
$$39$$ 0 0
$$40$$ −1.00363 −0.158688
$$41$$ −9.46022 −1.47744 −0.738719 0.674013i $$-0.764569\pi$$
−0.738719 + 0.674013i $$0.764569\pi$$
$$42$$ −4.98315 −0.768916
$$43$$ −11.8791 −1.81155 −0.905776 0.423757i $$-0.860711\pi$$
−0.905776 + 0.423757i $$0.860711\pi$$
$$44$$ −2.65498 −0.400253
$$45$$ −0.710210 −0.105872
$$46$$ 20.5728 3.03330
$$47$$ 12.9178 1.88426 0.942130 0.335249i $$-0.108820\pi$$
0.942130 + 0.335249i $$0.108820\pi$$
$$48$$ −2.26104 −0.326353
$$49$$ −1.66555 −0.237935
$$50$$ −9.69944 −1.37171
$$51$$ −6.68027 −0.935425
$$52$$ 0 0
$$53$$ 6.78954 0.932615 0.466307 0.884623i $$-0.345584\pi$$
0.466307 + 0.884623i $$0.345584\pi$$
$$54$$ 2.15754 0.293604
$$55$$ 0.710210 0.0957647
$$56$$ −3.26386 −0.436152
$$57$$ 0.242517 0.0321221
$$58$$ −6.37063 −0.836504
$$59$$ −7.81320 −1.01719 −0.508596 0.861005i $$-0.669835\pi$$
−0.508596 + 0.861005i $$0.669835\pi$$
$$60$$ −1.88559 −0.243429
$$61$$ 0.910585 0.116588 0.0582942 0.998299i $$-0.481434\pi$$
0.0582942 + 0.998299i $$0.481434\pi$$
$$62$$ 8.67368 1.10156
$$63$$ −2.30964 −0.290988
$$64$$ −12.1009 −1.51261
$$65$$ 0 0
$$66$$ −2.15754 −0.265575
$$67$$ −9.32216 −1.13888 −0.569442 0.822032i $$-0.692841\pi$$
−0.569442 + 0.822032i $$0.692841\pi$$
$$68$$ −17.7360 −2.15080
$$69$$ 9.53531 1.14792
$$70$$ 3.53908 0.423001
$$71$$ −12.9704 −1.53930 −0.769650 0.638466i $$-0.779569\pi$$
−0.769650 + 0.638466i $$0.779569\pi$$
$$72$$ 1.41315 0.166541
$$73$$ −8.51665 −0.996798 −0.498399 0.866948i $$-0.666079\pi$$
−0.498399 + 0.866948i $$0.666079\pi$$
$$74$$ −7.38534 −0.858529
$$75$$ −4.49560 −0.519107
$$76$$ 0.643877 0.0738578
$$77$$ 2.30964 0.263208
$$78$$ 0 0
$$79$$ 1.82360 0.205172 0.102586 0.994724i $$-0.467288\pi$$
0.102586 + 0.994724i $$0.467288\pi$$
$$80$$ 1.60581 0.179536
$$81$$ 1.00000 0.111111
$$82$$ −20.4108 −2.25400
$$83$$ −4.64671 −0.510042 −0.255021 0.966935i $$-0.582082\pi$$
−0.255021 + 0.966935i $$0.582082\pi$$
$$84$$ −6.13206 −0.669062
$$85$$ 4.74440 0.514602
$$86$$ −25.6297 −2.76372
$$87$$ −2.95273 −0.316566
$$88$$ −1.41315 −0.150642
$$89$$ −14.7714 −1.56576 −0.782881 0.622171i $$-0.786251\pi$$
−0.782881 + 0.622171i $$0.786251\pi$$
$$90$$ −1.53231 −0.161519
$$91$$ 0 0
$$92$$ 25.3161 2.63938
$$93$$ 4.02017 0.416872
$$94$$ 27.8707 2.87465
$$95$$ −0.172238 −0.0176712
$$96$$ −7.70458 −0.786345
$$97$$ −1.86542 −0.189405 −0.0947025 0.995506i $$-0.530190\pi$$
−0.0947025 + 0.995506i $$0.530190\pi$$
$$98$$ −3.59349 −0.362997
$$99$$ −1.00000 −0.100504
$$100$$ −11.9357 −1.19357
$$101$$ −2.45086 −0.243870 −0.121935 0.992538i $$-0.538910\pi$$
−0.121935 + 0.992538i $$0.538910\pi$$
$$102$$ −14.4130 −1.42709
$$103$$ 10.2811 1.01302 0.506512 0.862233i $$-0.330934\pi$$
0.506512 + 0.862233i $$0.330934\pi$$
$$104$$ 0 0
$$105$$ 1.64033 0.160080
$$106$$ 14.6487 1.42281
$$107$$ 19.6690 1.90148 0.950739 0.309993i $$-0.100327\pi$$
0.950739 + 0.309993i $$0.100327\pi$$
$$108$$ 2.65498 0.255476
$$109$$ −1.27506 −0.122128 −0.0610642 0.998134i $$-0.519449\pi$$
−0.0610642 + 0.998134i $$0.519449\pi$$
$$110$$ 1.53231 0.146100
$$111$$ −3.42304 −0.324900
$$112$$ 5.22220 0.493452
$$113$$ −0.638739 −0.0600875 −0.0300438 0.999549i $$-0.509565\pi$$
−0.0300438 + 0.999549i $$0.509565\pi$$
$$114$$ 0.523240 0.0490059
$$115$$ −6.77207 −0.631499
$$116$$ −7.83943 −0.727873
$$117$$ 0 0
$$118$$ −16.8573 −1.55184
$$119$$ 15.4290 1.41438
$$120$$ −1.00363 −0.0916185
$$121$$ 1.00000 0.0909091
$$122$$ 1.96462 0.177869
$$123$$ −9.46022 −0.853000
$$124$$ 10.6735 0.958506
$$125$$ 6.74387 0.603190
$$126$$ −4.98315 −0.443934
$$127$$ 10.3681 0.920021 0.460010 0.887914i $$-0.347846\pi$$
0.460010 + 0.887914i $$0.347846\pi$$
$$128$$ −10.6989 −0.945660
$$129$$ −11.8791 −1.04590
$$130$$ 0 0
$$131$$ −5.68692 −0.496868 −0.248434 0.968649i $$-0.579916\pi$$
−0.248434 + 0.968649i $$0.579916\pi$$
$$132$$ −2.65498 −0.231086
$$133$$ −0.560127 −0.0485692
$$134$$ −20.1129 −1.73749
$$135$$ −0.710210 −0.0611251
$$136$$ −9.44019 −0.809490
$$137$$ −4.86363 −0.415528 −0.207764 0.978179i $$-0.566619\pi$$
−0.207764 + 0.978179i $$0.566619\pi$$
$$138$$ 20.5728 1.75127
$$139$$ −4.41515 −0.374488 −0.187244 0.982313i $$-0.559956\pi$$
−0.187244 + 0.982313i $$0.559956\pi$$
$$140$$ 4.35505 0.368069
$$141$$ 12.9178 1.08788
$$142$$ −27.9841 −2.34837
$$143$$ 0 0
$$144$$ −2.26104 −0.188420
$$145$$ 2.09706 0.174151
$$146$$ −18.3750 −1.52073
$$147$$ −1.66555 −0.137372
$$148$$ −9.08810 −0.747037
$$149$$ −9.50356 −0.778562 −0.389281 0.921119i $$-0.627276\pi$$
−0.389281 + 0.921119i $$0.627276\pi$$
$$150$$ −9.69944 −0.791956
$$151$$ −15.2126 −1.23798 −0.618991 0.785398i $$-0.712458\pi$$
−0.618991 + 0.785398i $$0.712458\pi$$
$$152$$ 0.342711 0.0277976
$$153$$ −6.68027 −0.540068
$$154$$ 4.98315 0.401554
$$155$$ −2.85516 −0.229332
$$156$$ 0 0
$$157$$ 20.1949 1.61173 0.805866 0.592098i $$-0.201700\pi$$
0.805866 + 0.592098i $$0.201700\pi$$
$$158$$ 3.93450 0.313012
$$159$$ 6.78954 0.538445
$$160$$ 5.47187 0.432589
$$161$$ −22.0232 −1.73567
$$162$$ 2.15754 0.169512
$$163$$ 16.4974 1.29217 0.646087 0.763264i $$-0.276404\pi$$
0.646087 + 0.763264i $$0.276404\pi$$
$$164$$ −25.1167 −1.96129
$$165$$ 0.710210 0.0552898
$$166$$ −10.0255 −0.778126
$$167$$ 5.50507 0.425995 0.212998 0.977053i $$-0.431677\pi$$
0.212998 + 0.977053i $$0.431677\pi$$
$$168$$ −3.26386 −0.251812
$$169$$ 0 0
$$170$$ 10.2362 0.785082
$$171$$ 0.242517 0.0185457
$$172$$ −31.5389 −2.40482
$$173$$ −12.7126 −0.966524 −0.483262 0.875476i $$-0.660548\pi$$
−0.483262 + 0.875476i $$0.660548\pi$$
$$174$$ −6.37063 −0.482956
$$175$$ 10.3832 0.784899
$$176$$ 2.26104 0.170432
$$177$$ −7.81320 −0.587276
$$178$$ −31.8698 −2.38875
$$179$$ 24.4279 1.82583 0.912913 0.408154i $$-0.133828\pi$$
0.912913 + 0.408154i $$0.133828\pi$$
$$180$$ −1.88559 −0.140544
$$181$$ 0.931324 0.0692248 0.0346124 0.999401i $$-0.488980\pi$$
0.0346124 + 0.999401i $$0.488980\pi$$
$$182$$ 0 0
$$183$$ 0.910585 0.0673124
$$184$$ 13.4748 0.993374
$$185$$ 2.43108 0.178736
$$186$$ 8.67368 0.635985
$$187$$ 6.68027 0.488510
$$188$$ 34.2966 2.50134
$$189$$ −2.30964 −0.168002
$$190$$ −0.371610 −0.0269594
$$191$$ 4.68209 0.338784 0.169392 0.985549i $$-0.445820\pi$$
0.169392 + 0.985549i $$0.445820\pi$$
$$192$$ −12.1009 −0.873304
$$193$$ 10.2671 0.739039 0.369520 0.929223i $$-0.379522\pi$$
0.369520 + 0.929223i $$0.379522\pi$$
$$194$$ −4.02473 −0.288959
$$195$$ 0 0
$$196$$ −4.42200 −0.315857
$$197$$ −11.5842 −0.825341 −0.412671 0.910880i $$-0.635404\pi$$
−0.412671 + 0.910880i $$0.635404\pi$$
$$198$$ −2.15754 −0.153330
$$199$$ 17.2330 1.22162 0.610808 0.791779i $$-0.290845\pi$$
0.610808 + 0.791779i $$0.290845\pi$$
$$200$$ −6.35294 −0.449221
$$201$$ −9.32216 −0.657535
$$202$$ −5.28783 −0.372050
$$203$$ 6.81975 0.478652
$$204$$ −17.7360 −1.24177
$$205$$ 6.71875 0.469258
$$206$$ 22.1818 1.54548
$$207$$ 9.53531 0.662750
$$208$$ 0 0
$$209$$ −0.242517 −0.0167752
$$210$$ 3.53908 0.244220
$$211$$ −10.6905 −0.735967 −0.367983 0.929832i $$-0.619952\pi$$
−0.367983 + 0.929832i $$0.619952\pi$$
$$212$$ 18.0261 1.23804
$$213$$ −12.9704 −0.888716
$$214$$ 42.4367 2.90092
$$215$$ 8.43668 0.575377
$$216$$ 1.41315 0.0961523
$$217$$ −9.28516 −0.630318
$$218$$ −2.75099 −0.186320
$$219$$ −8.51665 −0.575502
$$220$$ 1.88559 0.127127
$$221$$ 0 0
$$222$$ −7.38534 −0.495672
$$223$$ 5.33992 0.357587 0.178794 0.983887i $$-0.442781\pi$$
0.178794 + 0.983887i $$0.442781\pi$$
$$224$$ 17.7948 1.18897
$$225$$ −4.49560 −0.299707
$$226$$ −1.37811 −0.0916702
$$227$$ 7.82431 0.519318 0.259659 0.965700i $$-0.416390\pi$$
0.259659 + 0.965700i $$0.416390\pi$$
$$228$$ 0.643877 0.0426418
$$229$$ 10.6805 0.705784 0.352892 0.935664i $$-0.385198\pi$$
0.352892 + 0.935664i $$0.385198\pi$$
$$230$$ −14.6110 −0.963422
$$231$$ 2.30964 0.151963
$$232$$ −4.17263 −0.273947
$$233$$ 15.4509 1.01222 0.506112 0.862468i $$-0.331082\pi$$
0.506112 + 0.862468i $$0.331082\pi$$
$$234$$ 0 0
$$235$$ −9.17437 −0.598470
$$236$$ −20.7439 −1.35031
$$237$$ 1.82360 0.118456
$$238$$ 33.2888 2.15779
$$239$$ 16.9365 1.09553 0.547765 0.836632i $$-0.315479\pi$$
0.547765 + 0.836632i $$0.315479\pi$$
$$240$$ 1.60581 0.103655
$$241$$ 0.0799646 0.00515098 0.00257549 0.999997i $$-0.499180\pi$$
0.00257549 + 0.999997i $$0.499180\pi$$
$$242$$ 2.15754 0.138692
$$243$$ 1.00000 0.0641500
$$244$$ 2.41758 0.154770
$$245$$ 1.18289 0.0755720
$$246$$ −20.4108 −1.30135
$$247$$ 0 0
$$248$$ 5.68108 0.360749
$$249$$ −4.64671 −0.294473
$$250$$ 14.5502 0.920234
$$251$$ −7.57174 −0.477924 −0.238962 0.971029i $$-0.576807\pi$$
−0.238962 + 0.971029i $$0.576807\pi$$
$$252$$ −6.13206 −0.386283
$$253$$ −9.53531 −0.599480
$$254$$ 22.3696 1.40359
$$255$$ 4.74440 0.297106
$$256$$ 1.11835 0.0698970
$$257$$ 23.0242 1.43621 0.718105 0.695935i $$-0.245010\pi$$
0.718105 + 0.695935i $$0.245010\pi$$
$$258$$ −25.6297 −1.59564
$$259$$ 7.90600 0.491255
$$260$$ 0 0
$$261$$ −2.95273 −0.182769
$$262$$ −12.2698 −0.758028
$$263$$ −5.95047 −0.366922 −0.183461 0.983027i $$-0.558730\pi$$
−0.183461 + 0.983027i $$0.558730\pi$$
$$264$$ −1.41315 −0.0869731
$$265$$ −4.82200 −0.296213
$$266$$ −1.20850 −0.0740977
$$267$$ −14.7714 −0.903994
$$268$$ −24.7502 −1.51186
$$269$$ −1.17571 −0.0716841 −0.0358420 0.999357i $$-0.511411\pi$$
−0.0358420 + 0.999357i $$0.511411\pi$$
$$270$$ −1.53231 −0.0932532
$$271$$ 7.12706 0.432938 0.216469 0.976290i $$-0.430546\pi$$
0.216469 + 0.976290i $$0.430546\pi$$
$$272$$ 15.1044 0.915837
$$273$$ 0 0
$$274$$ −10.4935 −0.633934
$$275$$ 4.49560 0.271095
$$276$$ 25.3161 1.52385
$$277$$ −11.9736 −0.719426 −0.359713 0.933063i $$-0.617125\pi$$
−0.359713 + 0.933063i $$0.617125\pi$$
$$278$$ −9.52586 −0.571323
$$279$$ 4.02017 0.240681
$$280$$ 2.31803 0.138529
$$281$$ −18.7901 −1.12092 −0.560461 0.828181i $$-0.689376\pi$$
−0.560461 + 0.828181i $$0.689376\pi$$
$$282$$ 27.8707 1.65968
$$283$$ 10.5237 0.625568 0.312784 0.949824i $$-0.398738\pi$$
0.312784 + 0.949824i $$0.398738\pi$$
$$284$$ −34.4361 −2.04341
$$285$$ −0.172238 −0.0102025
$$286$$ 0 0
$$287$$ 21.8497 1.28975
$$288$$ −7.70458 −0.453997
$$289$$ 27.6260 1.62506
$$290$$ 4.52448 0.265687
$$291$$ −1.86542 −0.109353
$$292$$ −22.6115 −1.32324
$$293$$ 14.0111 0.818540 0.409270 0.912413i $$-0.365783\pi$$
0.409270 + 0.912413i $$0.365783\pi$$
$$294$$ −3.59349 −0.209576
$$295$$ 5.54901 0.323076
$$296$$ −4.83725 −0.281159
$$297$$ −1.00000 −0.0580259
$$298$$ −20.5043 −1.18778
$$299$$ 0 0
$$300$$ −11.9357 −0.689110
$$301$$ 27.4366 1.58142
$$302$$ −32.8217 −1.88868
$$303$$ −2.45086 −0.140798
$$304$$ −0.548341 −0.0314495
$$305$$ −0.646706 −0.0370303
$$306$$ −14.4130 −0.823934
$$307$$ −2.52126 −0.143896 −0.0719481 0.997408i $$-0.522922\pi$$
−0.0719481 + 0.997408i $$0.522922\pi$$
$$308$$ 6.13206 0.349406
$$309$$ 10.2811 0.584870
$$310$$ −6.16013 −0.349872
$$311$$ −22.9380 −1.30069 −0.650347 0.759638i $$-0.725376\pi$$
−0.650347 + 0.759638i $$0.725376\pi$$
$$312$$ 0 0
$$313$$ −21.6566 −1.22410 −0.612051 0.790818i $$-0.709655\pi$$
−0.612051 + 0.790818i $$0.709655\pi$$
$$314$$ 43.5714 2.45888
$$315$$ 1.64033 0.0924222
$$316$$ 4.84163 0.272363
$$317$$ −26.6050 −1.49429 −0.747143 0.664664i $$-0.768575\pi$$
−0.747143 + 0.664664i $$0.768575\pi$$
$$318$$ 14.6487 0.821458
$$319$$ 2.95273 0.165321
$$320$$ 8.59415 0.480428
$$321$$ 19.6690 1.09782
$$322$$ −47.5159 −2.64796
$$323$$ −1.62008 −0.0901435
$$324$$ 2.65498 0.147499
$$325$$ 0 0
$$326$$ 35.5937 1.97135
$$327$$ −1.27506 −0.0705108
$$328$$ −13.3687 −0.738161
$$329$$ −29.8356 −1.64489
$$330$$ 1.53231 0.0843507
$$331$$ 24.7670 1.36132 0.680659 0.732600i $$-0.261693\pi$$
0.680659 + 0.732600i $$0.261693\pi$$
$$332$$ −12.3369 −0.677076
$$333$$ −3.42304 −0.187581
$$334$$ 11.8774 0.649903
$$335$$ 6.62069 0.361727
$$336$$ 5.22220 0.284894
$$337$$ −28.2890 −1.54100 −0.770499 0.637441i $$-0.779993\pi$$
−0.770499 + 0.637441i $$0.779993\pi$$
$$338$$ 0 0
$$339$$ −0.638739 −0.0346916
$$340$$ 12.5963 0.683129
$$341$$ −4.02017 −0.217704
$$342$$ 0.523240 0.0282936
$$343$$ 20.0143 1.08067
$$344$$ −16.7869 −0.905091
$$345$$ −6.77207 −0.364596
$$346$$ −27.4280 −1.47454
$$347$$ −18.0814 −0.970661 −0.485331 0.874331i $$-0.661301\pi$$
−0.485331 + 0.874331i $$0.661301\pi$$
$$348$$ −7.83943 −0.420238
$$349$$ −21.0326 −1.12585 −0.562924 0.826508i $$-0.690324\pi$$
−0.562924 + 0.826508i $$0.690324\pi$$
$$350$$ 22.4022 1.19745
$$351$$ 0 0
$$352$$ 7.70458 0.410655
$$353$$ −2.20127 −0.117162 −0.0585810 0.998283i $$-0.518658\pi$$
−0.0585810 + 0.998283i $$0.518658\pi$$
$$354$$ −16.8573 −0.895955
$$355$$ 9.21169 0.488906
$$356$$ −39.2177 −2.07853
$$357$$ 15.4290 0.816591
$$358$$ 52.7041 2.78550
$$359$$ 7.76540 0.409842 0.204921 0.978778i $$-0.434306\pi$$
0.204921 + 0.978778i $$0.434306\pi$$
$$360$$ −1.00363 −0.0528959
$$361$$ −18.9412 −0.996905
$$362$$ 2.00937 0.105610
$$363$$ 1.00000 0.0524864
$$364$$ 0 0
$$365$$ 6.04861 0.316599
$$366$$ 1.96462 0.102692
$$367$$ −11.9604 −0.624326 −0.312163 0.950028i $$-0.601054\pi$$
−0.312163 + 0.950028i $$0.601054\pi$$
$$368$$ −21.5597 −1.12388
$$369$$ −9.46022 −0.492480
$$370$$ 5.24514 0.272682
$$371$$ −15.6814 −0.814138
$$372$$ 10.6735 0.553394
$$373$$ 14.1243 0.731330 0.365665 0.930747i $$-0.380842\pi$$
0.365665 + 0.930747i $$0.380842\pi$$
$$374$$ 14.4130 0.745276
$$375$$ 6.74387 0.348252
$$376$$ 18.2548 0.941418
$$377$$ 0 0
$$378$$ −4.98315 −0.256305
$$379$$ 3.34432 0.171786 0.0858931 0.996304i $$-0.472626\pi$$
0.0858931 + 0.996304i $$0.472626\pi$$
$$380$$ −0.457288 −0.0234584
$$381$$ 10.3681 0.531174
$$382$$ 10.1018 0.516853
$$383$$ 5.63621 0.287997 0.143998 0.989578i $$-0.454004\pi$$
0.143998 + 0.989578i $$0.454004\pi$$
$$384$$ −10.6989 −0.545977
$$385$$ −1.64033 −0.0835991
$$386$$ 22.1516 1.12749
$$387$$ −11.8791 −0.603850
$$388$$ −4.95266 −0.251433
$$389$$ 14.3490 0.727523 0.363761 0.931492i $$-0.381492\pi$$
0.363761 + 0.931492i $$0.381492\pi$$
$$390$$ 0 0
$$391$$ −63.6984 −3.22137
$$392$$ −2.35366 −0.118878
$$393$$ −5.68692 −0.286867
$$394$$ −24.9934 −1.25915
$$395$$ −1.29514 −0.0651657
$$396$$ −2.65498 −0.133418
$$397$$ 14.8769 0.746648 0.373324 0.927701i $$-0.378218\pi$$
0.373324 + 0.927701i $$0.378218\pi$$
$$398$$ 37.1809 1.86371
$$399$$ −0.560127 −0.0280414
$$400$$ 10.1647 0.508237
$$401$$ −19.6380 −0.980675 −0.490337 0.871533i $$-0.663126\pi$$
−0.490337 + 0.871533i $$0.663126\pi$$
$$402$$ −20.1129 −1.00314
$$403$$ 0 0
$$404$$ −6.50698 −0.323734
$$405$$ −0.710210 −0.0352906
$$406$$ 14.7139 0.730237
$$407$$ 3.42304 0.169674
$$408$$ −9.44019 −0.467359
$$409$$ −23.1731 −1.14584 −0.572918 0.819613i $$-0.694189\pi$$
−0.572918 + 0.819613i $$0.694189\pi$$
$$410$$ 14.4960 0.715905
$$411$$ −4.86363 −0.239905
$$412$$ 27.2961 1.34478
$$413$$ 18.0457 0.887971
$$414$$ 20.5728 1.01110
$$415$$ 3.30014 0.161997
$$416$$ 0 0
$$417$$ −4.41515 −0.216211
$$418$$ −0.523240 −0.0255925
$$419$$ −3.70294 −0.180900 −0.0904502 0.995901i $$-0.528831\pi$$
−0.0904502 + 0.995901i $$0.528831\pi$$
$$420$$ 4.35505 0.212505
$$421$$ −23.7239 −1.15623 −0.578116 0.815955i $$-0.696212\pi$$
−0.578116 + 0.815955i $$0.696212\pi$$
$$422$$ −23.0653 −1.12280
$$423$$ 12.9178 0.628086
$$424$$ 9.59460 0.465955
$$425$$ 30.0318 1.45676
$$426$$ −27.9841 −1.35583
$$427$$ −2.10313 −0.101777
$$428$$ 52.2209 2.52419
$$429$$ 0 0
$$430$$ 18.2025 0.877802
$$431$$ 31.0695 1.49656 0.748282 0.663381i $$-0.230879\pi$$
0.748282 + 0.663381i $$0.230879\pi$$
$$432$$ −2.26104 −0.108784
$$433$$ 23.0541 1.10791 0.553954 0.832547i $$-0.313118\pi$$
0.553954 + 0.832547i $$0.313118\pi$$
$$434$$ −20.0331 −0.961620
$$435$$ 2.09706 0.100546
$$436$$ −3.38525 −0.162124
$$437$$ 2.31247 0.110621
$$438$$ −18.3750 −0.877992
$$439$$ 2.39273 0.114199 0.0570994 0.998368i $$-0.481815\pi$$
0.0570994 + 0.998368i $$0.481815\pi$$
$$440$$ 1.00363 0.0478462
$$441$$ −1.66555 −0.0793118
$$442$$ 0 0
$$443$$ −17.1624 −0.815411 −0.407705 0.913113i $$-0.633671\pi$$
−0.407705 + 0.913113i $$0.633671\pi$$
$$444$$ −9.08810 −0.431302
$$445$$ 10.4908 0.497311
$$446$$ 11.5211 0.545539
$$447$$ −9.50356 −0.449503
$$448$$ 27.9487 1.32045
$$449$$ −29.0646 −1.37164 −0.685822 0.727769i $$-0.740557\pi$$
−0.685822 + 0.727769i $$0.740557\pi$$
$$450$$ −9.69944 −0.457236
$$451$$ 9.46022 0.445465
$$452$$ −1.69584 −0.0797656
$$453$$ −15.2126 −0.714749
$$454$$ 16.8813 0.792277
$$455$$ 0 0
$$456$$ 0.342711 0.0160489
$$457$$ 31.9111 1.49274 0.746369 0.665532i $$-0.231795\pi$$
0.746369 + 0.665532i $$0.231795\pi$$
$$458$$ 23.0435 1.07675
$$459$$ −6.68027 −0.311808
$$460$$ −17.9797 −0.838308
$$461$$ 0.706685 0.0329136 0.0164568 0.999865i $$-0.494761\pi$$
0.0164568 + 0.999865i $$0.494761\pi$$
$$462$$ 4.98315 0.231837
$$463$$ −25.7813 −1.19816 −0.599079 0.800690i $$-0.704467\pi$$
−0.599079 + 0.800690i $$0.704467\pi$$
$$464$$ 6.67624 0.309937
$$465$$ −2.85516 −0.132405
$$466$$ 33.3360 1.54426
$$467$$ 37.3541 1.72854 0.864270 0.503028i $$-0.167781\pi$$
0.864270 + 0.503028i $$0.167781\pi$$
$$468$$ 0 0
$$469$$ 21.5309 0.994203
$$470$$ −19.7941 −0.913033
$$471$$ 20.1949 0.930533
$$472$$ −11.0412 −0.508212
$$473$$ 11.8791 0.546203
$$474$$ 3.93450 0.180718
$$475$$ −1.09026 −0.0500245
$$476$$ 40.9638 1.87757
$$477$$ 6.78954 0.310872
$$478$$ 36.5411 1.67135
$$479$$ −1.74064 −0.0795318 −0.0397659 0.999209i $$-0.512661\pi$$
−0.0397659 + 0.999209i $$0.512661\pi$$
$$480$$ 5.47187 0.249756
$$481$$ 0 0
$$482$$ 0.172527 0.00785839
$$483$$ −22.0232 −1.00209
$$484$$ 2.65498 0.120681
$$485$$ 1.32484 0.0601580
$$486$$ 2.15754 0.0978680
$$487$$ −5.69783 −0.258193 −0.129097 0.991632i $$-0.541208\pi$$
−0.129097 + 0.991632i $$0.541208\pi$$
$$488$$ 1.28679 0.0582502
$$489$$ 16.4974 0.746037
$$490$$ 2.55213 0.115293
$$491$$ −27.7388 −1.25183 −0.625917 0.779890i $$-0.715275\pi$$
−0.625917 + 0.779890i $$0.715275\pi$$
$$492$$ −25.1167 −1.13235
$$493$$ 19.7250 0.888370
$$494$$ 0 0
$$495$$ 0.710210 0.0319216
$$496$$ −9.08977 −0.408143
$$497$$ 29.9569 1.34375
$$498$$ −10.0255 −0.449251
$$499$$ −8.27073 −0.370249 −0.185124 0.982715i $$-0.559269\pi$$
−0.185124 + 0.982715i $$0.559269\pi$$
$$500$$ 17.9048 0.800729
$$501$$ 5.50507 0.245948
$$502$$ −16.3363 −0.729126
$$503$$ −12.7099 −0.566706 −0.283353 0.959016i $$-0.591447\pi$$
−0.283353 + 0.959016i $$0.591447\pi$$
$$504$$ −3.26386 −0.145384
$$505$$ 1.74062 0.0774568
$$506$$ −20.5728 −0.914573
$$507$$ 0 0
$$508$$ 27.5271 1.22132
$$509$$ −5.55978 −0.246433 −0.123216 0.992380i $$-0.539321\pi$$
−0.123216 + 0.992380i $$0.539321\pi$$
$$510$$ 10.2362 0.453268
$$511$$ 19.6704 0.870168
$$512$$ 23.8107 1.05230
$$513$$ 0.242517 0.0107074
$$514$$ 49.6756 2.19110
$$515$$ −7.30172 −0.321752
$$516$$ −31.5389 −1.38842
$$517$$ −12.9178 −0.568126
$$518$$ 17.0575 0.749464
$$519$$ −12.7126 −0.558023
$$520$$ 0 0
$$521$$ −4.41425 −0.193392 −0.0966958 0.995314i $$-0.530827\pi$$
−0.0966958 + 0.995314i $$0.530827\pi$$
$$522$$ −6.37063 −0.278835
$$523$$ 15.5162 0.678474 0.339237 0.940701i $$-0.389831\pi$$
0.339237 + 0.940701i $$0.389831\pi$$
$$524$$ −15.0986 −0.659587
$$525$$ 10.3832 0.453162
$$526$$ −12.8384 −0.559780
$$527$$ −26.8558 −1.16986
$$528$$ 2.26104 0.0983992
$$529$$ 67.9221 2.95314
$$530$$ −10.4037 −0.451906
$$531$$ −7.81320 −0.339064
$$532$$ −1.48713 −0.0644751
$$533$$ 0 0
$$534$$ −31.8698 −1.37914
$$535$$ −13.9691 −0.603939
$$536$$ −13.1736 −0.569011
$$537$$ 24.4279 1.05414
$$538$$ −2.53663 −0.109362
$$539$$ 1.66555 0.0717402
$$540$$ −1.88559 −0.0811430
$$541$$ 27.9827 1.20307 0.601536 0.798846i $$-0.294556\pi$$
0.601536 + 0.798846i $$0.294556\pi$$
$$542$$ 15.3769 0.660495
$$543$$ 0.931324 0.0399669
$$544$$ 51.4687 2.20670
$$545$$ 0.905559 0.0387899
$$546$$ 0 0
$$547$$ −24.1470 −1.03245 −0.516226 0.856453i $$-0.672663\pi$$
−0.516226 + 0.856453i $$0.672663\pi$$
$$548$$ −12.9128 −0.551609
$$549$$ 0.910585 0.0388628
$$550$$ 9.69944 0.413586
$$551$$ −0.716086 −0.0305063
$$552$$ 13.4748 0.573524
$$553$$ −4.21188 −0.179107
$$554$$ −25.8336 −1.09756
$$555$$ 2.43108 0.103193
$$556$$ −11.7221 −0.497129
$$557$$ −19.1326 −0.810675 −0.405337 0.914167i $$-0.632846\pi$$
−0.405337 + 0.914167i $$0.632846\pi$$
$$558$$ 8.67368 0.367186
$$559$$ 0 0
$$560$$ −3.70886 −0.156728
$$561$$ 6.68027 0.282041
$$562$$ −40.5404 −1.71009
$$563$$ −3.91199 −0.164871 −0.0824353 0.996596i $$-0.526270\pi$$
−0.0824353 + 0.996596i $$0.526270\pi$$
$$564$$ 34.2966 1.44415
$$565$$ 0.453639 0.0190847
$$566$$ 22.7053 0.954373
$$567$$ −2.30964 −0.0969959
$$568$$ −18.3290 −0.769069
$$569$$ −15.8002 −0.662377 −0.331189 0.943565i $$-0.607450\pi$$
−0.331189 + 0.943565i $$0.607450\pi$$
$$570$$ −0.371610 −0.0155650
$$571$$ 20.2458 0.847262 0.423631 0.905835i $$-0.360755\pi$$
0.423631 + 0.905835i $$0.360755\pi$$
$$572$$ 0 0
$$573$$ 4.68209 0.195597
$$574$$ 47.1417 1.96766
$$575$$ −42.8669 −1.78768
$$576$$ −12.1009 −0.504202
$$577$$ 7.07073 0.294358 0.147179 0.989110i $$-0.452981\pi$$
0.147179 + 0.989110i $$0.452981\pi$$
$$578$$ 59.6042 2.47921
$$579$$ 10.2671 0.426685
$$580$$ 5.56764 0.231184
$$581$$ 10.7322 0.445248
$$582$$ −4.02473 −0.166830
$$583$$ −6.78954 −0.281194
$$584$$ −12.0353 −0.498023
$$585$$ 0 0
$$586$$ 30.2296 1.24877
$$587$$ −27.1767 −1.12170 −0.560851 0.827917i $$-0.689526\pi$$
−0.560851 + 0.827917i $$0.689526\pi$$
$$588$$ −4.42200 −0.182360
$$589$$ 0.974959 0.0401725
$$590$$ 11.9722 0.492889
$$591$$ −11.5842 −0.476511
$$592$$ 7.73963 0.318097
$$593$$ −20.0431 −0.823070 −0.411535 0.911394i $$-0.635007\pi$$
−0.411535 + 0.911394i $$0.635007\pi$$
$$594$$ −2.15754 −0.0885249
$$595$$ −10.9579 −0.449228
$$596$$ −25.2318 −1.03353
$$597$$ 17.2330 0.705300
$$598$$ 0 0
$$599$$ −24.5855 −1.00454 −0.502268 0.864712i $$-0.667501\pi$$
−0.502268 + 0.864712i $$0.667501\pi$$
$$600$$ −6.35294 −0.259358
$$601$$ −34.1098 −1.39137 −0.695683 0.718349i $$-0.744898\pi$$
−0.695683 + 0.718349i $$0.744898\pi$$
$$602$$ 59.1955 2.41263
$$603$$ −9.32216 −0.379628
$$604$$ −40.3891 −1.64341
$$605$$ −0.710210 −0.0288741
$$606$$ −5.28783 −0.214803
$$607$$ 20.9453 0.850143 0.425071 0.905160i $$-0.360249\pi$$
0.425071 + 0.905160i $$0.360249\pi$$
$$608$$ −1.86849 −0.0757773
$$609$$ 6.81975 0.276350
$$610$$ −1.39530 −0.0564938
$$611$$ 0 0
$$612$$ −17.7360 −0.716935
$$613$$ 22.2792 0.899848 0.449924 0.893067i $$-0.351451\pi$$
0.449924 + 0.893067i $$0.351451\pi$$
$$614$$ −5.43973 −0.219530
$$615$$ 6.71875 0.270926
$$616$$ 3.26386 0.131505
$$617$$ −27.4366 −1.10456 −0.552279 0.833660i $$-0.686242\pi$$
−0.552279 + 0.833660i $$0.686242\pi$$
$$618$$ 22.1818 0.892284
$$619$$ 47.9288 1.92642 0.963211 0.268747i $$-0.0866096\pi$$
0.963211 + 0.268747i $$0.0866096\pi$$
$$620$$ −7.58040 −0.304436
$$621$$ 9.53531 0.382639
$$622$$ −49.4896 −1.98435
$$623$$ 34.1166 1.36685
$$624$$ 0 0
$$625$$ 17.6884 0.707538
$$626$$ −46.7250 −1.86751
$$627$$ −0.242517 −0.00968519
$$628$$ 53.6171 2.13956
$$629$$ 22.8668 0.911760
$$630$$ 3.53908 0.141000
$$631$$ −23.8277 −0.948567 −0.474284 0.880372i $$-0.657293\pi$$
−0.474284 + 0.880372i $$0.657293\pi$$
$$632$$ 2.57702 0.102508
$$633$$ −10.6905 −0.424911
$$634$$ −57.4013 −2.27970
$$635$$ −7.36353 −0.292213
$$636$$ 18.0261 0.714781
$$637$$ 0 0
$$638$$ 6.37063 0.252216
$$639$$ −12.9704 −0.513100
$$640$$ 7.59848 0.300356
$$641$$ −8.30355 −0.327970 −0.163985 0.986463i $$-0.552435\pi$$
−0.163985 + 0.986463i $$0.552435\pi$$
$$642$$ 42.4367 1.67484
$$643$$ −29.9185 −1.17987 −0.589934 0.807451i $$-0.700846\pi$$
−0.589934 + 0.807451i $$0.700846\pi$$
$$644$$ −58.4710 −2.30408
$$645$$ 8.43668 0.332194
$$646$$ −3.49538 −0.137524
$$647$$ 5.87308 0.230895 0.115447 0.993314i $$-0.463170\pi$$
0.115447 + 0.993314i $$0.463170\pi$$
$$648$$ 1.41315 0.0555136
$$649$$ 7.81320 0.306695
$$650$$ 0 0
$$651$$ −9.28516 −0.363914
$$652$$ 43.8002 1.71535
$$653$$ −5.43262 −0.212595 −0.106297 0.994334i $$-0.533900\pi$$
−0.106297 + 0.994334i $$0.533900\pi$$
$$654$$ −2.75099 −0.107572
$$655$$ 4.03891 0.157813
$$656$$ 21.3900 0.835138
$$657$$ −8.51665 −0.332266
$$658$$ −64.3715 −2.50946
$$659$$ 26.5954 1.03601 0.518004 0.855378i $$-0.326675\pi$$
0.518004 + 0.855378i $$0.326675\pi$$
$$660$$ 1.88559 0.0733966
$$661$$ 27.7763 1.08037 0.540187 0.841545i $$-0.318354\pi$$
0.540187 + 0.841545i $$0.318354\pi$$
$$662$$ 53.4359 2.07684
$$663$$ 0 0
$$664$$ −6.56647 −0.254828
$$665$$ 0.397808 0.0154263
$$666$$ −7.38534 −0.286176
$$667$$ −28.1552 −1.09017
$$668$$ 14.6158 0.565504
$$669$$ 5.33992 0.206453
$$670$$ 14.2844 0.551855
$$671$$ −0.910585 −0.0351527
$$672$$ 17.7948 0.686450
$$673$$ 3.00054 0.115663 0.0578313 0.998326i $$-0.481581\pi$$
0.0578313 + 0.998326i $$0.481581\pi$$
$$674$$ −61.0346 −2.35096
$$675$$ −4.49560 −0.173036
$$676$$ 0 0
$$677$$ −25.1724 −0.967452 −0.483726 0.875219i $$-0.660717\pi$$
−0.483726 + 0.875219i $$0.660717\pi$$
$$678$$ −1.37811 −0.0529258
$$679$$ 4.30846 0.165344
$$680$$ 6.70452 0.257107
$$681$$ 7.82431 0.299828
$$682$$ −8.67368 −0.332132
$$683$$ −39.1652 −1.49861 −0.749307 0.662223i $$-0.769613\pi$$
−0.749307 + 0.662223i $$0.769613\pi$$
$$684$$ 0.643877 0.0246193
$$685$$ 3.45420 0.131978
$$686$$ 43.1817 1.64869
$$687$$ 10.6805 0.407485
$$688$$ 26.8592 1.02400
$$689$$ 0 0
$$690$$ −14.6110 −0.556232
$$691$$ −33.3472 −1.26859 −0.634294 0.773092i $$-0.718709\pi$$
−0.634294 + 0.773092i $$0.718709\pi$$
$$692$$ −33.7518 −1.28305
$$693$$ 2.30964 0.0877361
$$694$$ −39.0114 −1.48085
$$695$$ 3.13568 0.118943
$$696$$ −4.17263 −0.158163
$$697$$ 63.1969 2.39375
$$698$$ −45.3787 −1.71761
$$699$$ 15.4509 0.584408
$$700$$ 27.5673 1.04195
$$701$$ 8.45704 0.319418 0.159709 0.987164i $$-0.448944\pi$$
0.159709 + 0.987164i $$0.448944\pi$$
$$702$$ 0 0
$$703$$ −0.830144 −0.0313095
$$704$$ 12.1009 0.456068
$$705$$ −9.17437 −0.345527
$$706$$ −4.74934 −0.178744
$$707$$ 5.66061 0.212889
$$708$$ −20.7439 −0.779603
$$709$$ 7.03783 0.264311 0.132156 0.991229i $$-0.457810\pi$$
0.132156 + 0.991229i $$0.457810\pi$$
$$710$$ 19.8746 0.745880
$$711$$ 1.82360 0.0683905
$$712$$ −20.8741 −0.782290
$$713$$ 38.3336 1.43560
$$714$$ 33.2888 1.24580
$$715$$ 0 0
$$716$$ 64.8555 2.42377
$$717$$ 16.9365 0.632504
$$718$$ 16.7542 0.625260
$$719$$ −5.52947 −0.206215 −0.103107 0.994670i $$-0.532879\pi$$
−0.103107 + 0.994670i $$0.532879\pi$$
$$720$$ 1.60581 0.0598452
$$721$$ −23.7456 −0.884333
$$722$$ −40.8664 −1.52089
$$723$$ 0.0799646 0.00297392
$$724$$ 2.47265 0.0918952
$$725$$ 13.2743 0.492995
$$726$$ 2.15754 0.0800738
$$727$$ −38.7533 −1.43728 −0.718640 0.695383i $$-0.755235\pi$$
−0.718640 + 0.695383i $$0.755235\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 13.0501 0.483006
$$731$$ 79.3558 2.93508
$$732$$ 2.41758 0.0893565
$$733$$ −9.60595 −0.354804 −0.177402 0.984138i $$-0.556769\pi$$
−0.177402 + 0.984138i $$0.556769\pi$$
$$734$$ −25.8050 −0.952480
$$735$$ 1.18289 0.0436315
$$736$$ −73.4655 −2.70798
$$737$$ 9.32216 0.343386
$$738$$ −20.4108 −0.751332
$$739$$ −21.6659 −0.796995 −0.398497 0.917169i $$-0.630468\pi$$
−0.398497 + 0.917169i $$0.630468\pi$$
$$740$$ 6.45446 0.237271
$$741$$ 0 0
$$742$$ −33.8333 −1.24206
$$743$$ −12.3149 −0.451790 −0.225895 0.974152i $$-0.572531\pi$$
−0.225895 + 0.974152i $$0.572531\pi$$
$$744$$ 5.68108 0.208279
$$745$$ 6.74952 0.247283
$$746$$ 30.4738 1.11572
$$747$$ −4.64671 −0.170014
$$748$$ 17.7360 0.648492
$$749$$ −45.4285 −1.65992
$$750$$ 14.5502 0.531297
$$751$$ 2.50813 0.0915231 0.0457616 0.998952i $$-0.485429\pi$$
0.0457616 + 0.998952i $$0.485429\pi$$
$$752$$ −29.2078 −1.06510
$$753$$ −7.57174 −0.275930
$$754$$ 0 0
$$755$$ 10.8041 0.393202
$$756$$ −6.13206 −0.223021
$$757$$ 2.89889 0.105362 0.0526809 0.998611i $$-0.483223\pi$$
0.0526809 + 0.998611i $$0.483223\pi$$
$$758$$ 7.21551 0.262079
$$759$$ −9.53531 −0.346110
$$760$$ −0.243397 −0.00882894
$$761$$ 33.6764 1.22077 0.610384 0.792105i $$-0.291015\pi$$
0.610384 + 0.792105i $$0.291015\pi$$
$$762$$ 22.3696 0.810365
$$763$$ 2.94493 0.106614
$$764$$ 12.4309 0.449733
$$765$$ 4.74440 0.171534
$$766$$ 12.1604 0.439371
$$767$$ 0 0
$$768$$ 1.11835 0.0403551
$$769$$ −25.1224 −0.905935 −0.452968 0.891527i $$-0.649635\pi$$
−0.452968 + 0.891527i $$0.649635\pi$$
$$770$$ −3.53908 −0.127540
$$771$$ 23.0242 0.829196
$$772$$ 27.2588 0.981067
$$773$$ −31.6336 −1.13778 −0.568891 0.822413i $$-0.692627\pi$$
−0.568891 + 0.822413i $$0.692627\pi$$
$$774$$ −25.6297 −0.921241
$$775$$ −18.0731 −0.649204
$$776$$ −2.63611 −0.0946310
$$777$$ 7.90600 0.283626
$$778$$ 30.9585 1.10992
$$779$$ −2.29426 −0.0822005
$$780$$ 0 0
$$781$$ 12.9704 0.464117
$$782$$ −137.432 −4.91456
$$783$$ −2.95273 −0.105522
$$784$$ 3.76587 0.134495
$$785$$ −14.3426 −0.511911
$$786$$ −12.2698 −0.437648
$$787$$ −23.2041 −0.827137 −0.413569 0.910473i $$-0.635718\pi$$
−0.413569 + 0.910473i $$0.635718\pi$$
$$788$$ −30.7559 −1.09563
$$789$$ −5.95047 −0.211842
$$790$$ −2.79432 −0.0994175
$$791$$ 1.47526 0.0524542
$$792$$ −1.41315 −0.0502139
$$793$$ 0 0
$$794$$ 32.0974 1.13909
$$795$$ −4.82200 −0.171019
$$796$$ 45.7533 1.62168
$$797$$ 16.5706 0.586960 0.293480 0.955965i $$-0.405187\pi$$
0.293480 + 0.955965i $$0.405187\pi$$
$$798$$ −1.20850 −0.0427803
$$799$$ −86.2946 −3.05288
$$800$$ 34.6367 1.22459
$$801$$ −14.7714 −0.521921
$$802$$ −42.3698 −1.49613
$$803$$ 8.51665 0.300546
$$804$$ −24.7502 −0.872871
$$805$$ 15.6411 0.551275
$$806$$ 0 0
$$807$$ −1.17571 −0.0413868
$$808$$ −3.46342 −0.121843
$$809$$ 8.97637 0.315592 0.157796 0.987472i $$-0.449561\pi$$
0.157796 + 0.987472i $$0.449561\pi$$
$$810$$ −1.53231 −0.0538398
$$811$$ −2.09499 −0.0735651 −0.0367826 0.999323i $$-0.511711\pi$$
−0.0367826 + 0.999323i $$0.511711\pi$$
$$812$$ 18.1063 0.635406
$$813$$ 7.12706 0.249957
$$814$$ 7.38534 0.258856
$$815$$ −11.7166 −0.410414
$$816$$ 15.1044 0.528759
$$817$$ −2.88089 −0.100790
$$818$$ −49.9969 −1.74810
$$819$$ 0 0
$$820$$ 17.8381 0.622935
$$821$$ 32.8066 1.14496 0.572480 0.819919i $$-0.305982\pi$$
0.572480 + 0.819919i $$0.305982\pi$$
$$822$$ −10.4935 −0.366002
$$823$$ −6.37447 −0.222200 −0.111100 0.993809i $$-0.535437\pi$$
−0.111100 + 0.993809i $$0.535437\pi$$
$$824$$ 14.5287 0.506130
$$825$$ 4.49560 0.156517
$$826$$ 38.9343 1.35470
$$827$$ 14.7476 0.512825 0.256413 0.966567i $$-0.417459\pi$$
0.256413 + 0.966567i $$0.417459\pi$$
$$828$$ 25.3161 0.879794
$$829$$ −25.4837 −0.885085 −0.442542 0.896748i $$-0.645923\pi$$
−0.442542 + 0.896748i $$0.645923\pi$$
$$830$$ 7.12018 0.247145
$$831$$ −11.9736 −0.415361
$$832$$ 0 0
$$833$$ 11.1263 0.385504
$$834$$ −9.52586 −0.329854
$$835$$ −3.90976 −0.135303
$$836$$ −0.643877 −0.0222690
$$837$$ 4.02017 0.138957
$$838$$ −7.98924 −0.275984
$$839$$ 41.7953 1.44293 0.721467 0.692449i $$-0.243468\pi$$
0.721467 + 0.692449i $$0.243468\pi$$
$$840$$ 2.31803 0.0799795
$$841$$ −20.2814 −0.699359
$$842$$ −51.1852 −1.76396
$$843$$ −18.7901 −0.647165
$$844$$ −28.3831 −0.976988
$$845$$ 0 0
$$846$$ 27.8707 0.958216
$$847$$ −2.30964 −0.0793603
$$848$$ −15.3514 −0.527170
$$849$$ 10.5237 0.361172
$$850$$ 64.7949 2.22245
$$851$$ −32.6397 −1.11888
$$852$$ −34.4361 −1.17976
$$853$$ 25.2825 0.865655 0.432827 0.901477i $$-0.357516\pi$$
0.432827 + 0.901477i $$0.357516\pi$$
$$854$$ −4.53758 −0.155273
$$855$$ −0.172238 −0.00589041
$$856$$ 27.7952 0.950021
$$857$$ −26.8626 −0.917609 −0.458804 0.888537i $$-0.651722\pi$$
−0.458804 + 0.888537i $$0.651722\pi$$
$$858$$ 0 0
$$859$$ 6.11713 0.208714 0.104357 0.994540i $$-0.466722\pi$$
0.104357 + 0.994540i $$0.466722\pi$$
$$860$$ 22.3992 0.763807
$$861$$ 21.8497 0.744637
$$862$$ 67.0336 2.28318
$$863$$ −32.1517 −1.09446 −0.547228 0.836984i $$-0.684317\pi$$
−0.547228 + 0.836984i $$0.684317\pi$$
$$864$$ −7.70458 −0.262115
$$865$$ 9.02865 0.306983
$$866$$ 49.7401 1.69024
$$867$$ 27.6260 0.938228
$$868$$ −24.6519 −0.836740
$$869$$ −1.82360 −0.0618616
$$870$$ 4.52448 0.153394
$$871$$ 0 0
$$872$$ −1.80184 −0.0610180
$$873$$ −1.86542 −0.0631350
$$874$$ 4.98925 0.168764
$$875$$ −15.5759 −0.526563
$$876$$ −22.6115 −0.763973
$$877$$ 17.9062 0.604648 0.302324 0.953205i $$-0.402237\pi$$
0.302324 + 0.953205i $$0.402237\pi$$
$$878$$ 5.16241 0.174223
$$879$$ 14.0111 0.472584
$$880$$ −1.60581 −0.0541320
$$881$$ 40.5751 1.36701 0.683504 0.729947i $$-0.260455\pi$$
0.683504 + 0.729947i $$0.260455\pi$$
$$882$$ −3.59349 −0.120999
$$883$$ −24.6601 −0.829879 −0.414940 0.909849i $$-0.636197\pi$$
−0.414940 + 0.909849i $$0.636197\pi$$
$$884$$ 0 0
$$885$$ 5.54901 0.186528
$$886$$ −37.0286 −1.24400
$$887$$ 27.2662 0.915510 0.457755 0.889078i $$-0.348654\pi$$
0.457755 + 0.889078i $$0.348654\pi$$
$$888$$ −4.83725 −0.162327
$$889$$ −23.9466 −0.803144
$$890$$ 22.6343 0.758703
$$891$$ −1.00000 −0.0335013
$$892$$ 14.1774 0.474693
$$893$$ 3.13279 0.104835
$$894$$ −20.5043 −0.685767
$$895$$ −17.3489 −0.579911
$$896$$ 24.7107 0.825526
$$897$$ 0 0
$$898$$ −62.7081 −2.09260
$$899$$ −11.8705 −0.395902
$$900$$ −11.9357 −0.397858
$$901$$ −45.3560 −1.51103
$$902$$ 20.4108 0.679606
$$903$$ 27.4366 0.913032
$$904$$ −0.902631 −0.0300211
$$905$$ −0.661436 −0.0219869
$$906$$ −32.8217 −1.09043
$$907$$ 0.121526 0.00403521 0.00201760 0.999998i $$-0.499358\pi$$
0.00201760 + 0.999998i $$0.499358\pi$$
$$908$$ 20.7734 0.689389
$$909$$ −2.45086 −0.0812898
$$910$$ 0 0
$$911$$ −40.6736 −1.34758 −0.673788 0.738925i $$-0.735334\pi$$
−0.673788 + 0.738925i $$0.735334\pi$$
$$912$$ −0.548341 −0.0181574
$$913$$ 4.64671 0.153784
$$914$$ 68.8495 2.27734
$$915$$ −0.646706 −0.0213795
$$916$$ 28.3564 0.936921
$$917$$ 13.1347 0.433748
$$918$$ −14.4130 −0.475698
$$919$$ −26.0713 −0.860012 −0.430006 0.902826i $$-0.641489\pi$$
−0.430006 + 0.902826i $$0.641489\pi$$
$$920$$ −9.56992 −0.315511
$$921$$ −2.52126 −0.0830785
$$922$$ 1.52470 0.0502134
$$923$$ 0 0
$$924$$ 6.13206 0.201730
$$925$$ 15.3886 0.505975
$$926$$ −55.6242 −1.82792
$$927$$ 10.2811 0.337675
$$928$$ 22.7495 0.746790
$$929$$ 26.9849 0.885346 0.442673 0.896683i $$-0.354030\pi$$
0.442673 + 0.896683i $$0.354030\pi$$
$$930$$ −6.16013 −0.201999
$$931$$ −0.403923 −0.0132381
$$932$$ 41.0219 1.34372
$$933$$ −22.9380 −0.750956
$$934$$ 80.5929 2.63708
$$935$$ −4.74440 −0.155158
$$936$$ 0 0
$$937$$ 31.9915 1.04512 0.522559 0.852603i $$-0.324977\pi$$
0.522559 + 0.852603i $$0.324977\pi$$
$$938$$ 46.4537 1.51677
$$939$$ −21.6566 −0.706736
$$940$$ −24.3578 −0.794463
$$941$$ 31.7257 1.03423 0.517115 0.855916i $$-0.327006\pi$$
0.517115 + 0.855916i $$0.327006\pi$$
$$942$$ 43.5714 1.41963
$$943$$ −90.2062 −2.93752
$$944$$ 17.6660 0.574979
$$945$$ 1.64033 0.0533600
$$946$$ 25.6297 0.833294
$$947$$ 46.5444 1.51249 0.756245 0.654289i $$-0.227032\pi$$
0.756245 + 0.654289i $$0.227032\pi$$
$$948$$ 4.84163 0.157249
$$949$$ 0 0
$$950$$ −2.35228 −0.0763180
$$951$$ −26.6050 −0.862726
$$952$$ 21.8035 0.706655
$$953$$ 37.1054 1.20196 0.600981 0.799263i $$-0.294777\pi$$
0.600981 + 0.799263i $$0.294777\pi$$
$$954$$ 14.6487 0.474269
$$955$$ −3.32527 −0.107603
$$956$$ 44.9660 1.45430
$$957$$ 2.95273 0.0954481
$$958$$ −3.75550 −0.121335
$$959$$ 11.2332 0.362740
$$960$$ 8.59415 0.277375
$$961$$ −14.8382 −0.478653
$$962$$ 0 0
$$963$$ 19.6690 0.633826
$$964$$ 0.212305 0.00683787
$$965$$ −7.29177 −0.234730
$$966$$ −47.5159 −1.52880
$$967$$ −33.1854 −1.06717 −0.533586 0.845746i $$-0.679156\pi$$
−0.533586 + 0.845746i $$0.679156\pi$$
$$968$$ 1.41315 0.0454202
$$969$$ −1.62008 −0.0520444
$$970$$ 2.85840 0.0917777
$$971$$ 23.0990 0.741283 0.370642 0.928776i $$-0.379138\pi$$
0.370642 + 0.928776i $$0.379138\pi$$
$$972$$ 2.65498 0.0851585
$$973$$ 10.1974 0.326914
$$974$$ −12.2933 −0.393903
$$975$$ 0 0
$$976$$ −2.05887 −0.0659028
$$977$$ −32.4063 −1.03677 −0.518385 0.855147i $$-0.673467\pi$$
−0.518385 + 0.855147i $$0.673467\pi$$
$$978$$ 35.5937 1.13816
$$979$$ 14.7714 0.472095
$$980$$ 3.14055 0.100321
$$981$$ −1.27506 −0.0407095
$$982$$ −59.8475 −1.90981
$$983$$ 1.17252 0.0373976 0.0186988 0.999825i $$-0.494048\pi$$
0.0186988 + 0.999825i $$0.494048\pi$$
$$984$$ −13.3687 −0.426178
$$985$$ 8.22723 0.262141
$$986$$ 42.5575 1.35531
$$987$$ −29.8356 −0.949677
$$988$$ 0 0
$$989$$ −113.271 −3.60182
$$990$$ 1.53231 0.0486999
$$991$$ −4.78159 −0.151892 −0.0759462 0.997112i $$-0.524198\pi$$
−0.0759462 + 0.997112i $$0.524198\pi$$
$$992$$ −30.9737 −0.983416
$$993$$ 24.7670 0.785958
$$994$$ 64.6333 2.05004
$$995$$ −12.2391 −0.388004
$$996$$ −12.3369 −0.390910
$$997$$ 55.7577 1.76586 0.882932 0.469502i $$-0.155566\pi$$
0.882932 + 0.469502i $$0.155566\pi$$
$$998$$ −17.8444 −0.564856
$$999$$ −3.42304 −0.108300
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 5577.2.a.x.1.7 7
13.5 odd 4 429.2.b.b.298.3 14
13.8 odd 4 429.2.b.b.298.12 yes 14
13.12 even 2 5577.2.a.y.1.1 7
39.5 even 4 1287.2.b.c.298.12 14
39.8 even 4 1287.2.b.c.298.3 14

By twisted newform
Twist Min Dim Char Parity Ord Type
429.2.b.b.298.3 14 13.5 odd 4
429.2.b.b.298.12 yes 14 13.8 odd 4
1287.2.b.c.298.3 14 39.8 even 4
1287.2.b.c.298.12 14 39.5 even 4
5577.2.a.x.1.7 7 1.1 even 1 trivial
5577.2.a.y.1.1 7 13.12 even 2