# Properties

 Label 147.2.a.d.1.1 Level $147$ Weight $2$ Character 147.1 Self dual yes Analytic conductor $1.174$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 147.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.17380090971$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 147.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -0.585786 q^{5} +2.41421 q^{6} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -0.585786 q^{5} +2.41421 q^{6} -4.41421 q^{8} +1.00000 q^{9} +1.41421 q^{10} -2.00000 q^{11} -3.82843 q^{12} -5.41421 q^{13} +0.585786 q^{15} +3.00000 q^{16} -6.24264 q^{17} -2.41421 q^{18} -2.82843 q^{19} -2.24264 q^{20} +4.82843 q^{22} +3.65685 q^{23} +4.41421 q^{24} -4.65685 q^{25} +13.0711 q^{26} -1.00000 q^{27} -1.17157 q^{29} -1.41421 q^{30} +6.82843 q^{31} +1.58579 q^{32} +2.00000 q^{33} +15.0711 q^{34} +3.82843 q^{36} -4.00000 q^{37} +6.82843 q^{38} +5.41421 q^{39} +2.58579 q^{40} +2.24264 q^{41} -5.65685 q^{43} -7.65685 q^{44} -0.585786 q^{45} -8.82843 q^{46} -2.82843 q^{47} -3.00000 q^{48} +11.2426 q^{50} +6.24264 q^{51} -20.7279 q^{52} -2.00000 q^{53} +2.41421 q^{54} +1.17157 q^{55} +2.82843 q^{57} +2.82843 q^{58} +6.82843 q^{59} +2.24264 q^{60} -3.75736 q^{61} -16.4853 q^{62} -9.82843 q^{64} +3.17157 q^{65} -4.82843 q^{66} +5.65685 q^{67} -23.8995 q^{68} -3.65685 q^{69} -13.3137 q^{71} -4.41421 q^{72} +5.89949 q^{73} +9.65685 q^{74} +4.65685 q^{75} -10.8284 q^{76} -13.0711 q^{78} +2.34315 q^{79} -1.75736 q^{80} +1.00000 q^{81} -5.41421 q^{82} +15.3137 q^{83} +3.65685 q^{85} +13.6569 q^{86} +1.17157 q^{87} +8.82843 q^{88} +5.75736 q^{89} +1.41421 q^{90} +14.0000 q^{92} -6.82843 q^{93} +6.82843 q^{94} +1.65685 q^{95} -1.58579 q^{96} -5.41421 q^{97} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} - 6q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} - 6q^{8} + 2q^{9} - 4q^{11} - 2q^{12} - 8q^{13} + 4q^{15} + 6q^{16} - 4q^{17} - 2q^{18} + 4q^{20} + 4q^{22} - 4q^{23} + 6q^{24} + 2q^{25} + 12q^{26} - 2q^{27} - 8q^{29} + 8q^{31} + 6q^{32} + 4q^{33} + 16q^{34} + 2q^{36} - 8q^{37} + 8q^{38} + 8q^{39} + 8q^{40} - 4q^{41} - 4q^{44} - 4q^{45} - 12q^{46} - 6q^{48} + 14q^{50} + 4q^{51} - 16q^{52} - 4q^{53} + 2q^{54} + 8q^{55} + 8q^{59} - 4q^{60} - 16q^{61} - 16q^{62} - 14q^{64} + 12q^{65} - 4q^{66} - 28q^{68} + 4q^{69} - 4q^{71} - 6q^{72} - 8q^{73} + 8q^{74} - 2q^{75} - 16q^{76} - 12q^{78} + 16q^{79} - 12q^{80} + 2q^{81} - 8q^{82} + 8q^{83} - 4q^{85} + 16q^{86} + 8q^{87} + 12q^{88} + 20q^{89} + 28q^{92} - 8q^{93} + 8q^{94} - 8q^{95} - 6q^{96} - 8q^{97} - 4q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.41421 −1.70711 −0.853553 0.521005i $$-0.825557\pi$$
−0.853553 + 0.521005i $$0.825557\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 3.82843 1.91421
$$5$$ −0.585786 −0.261972 −0.130986 0.991384i $$-0.541814\pi$$
−0.130986 + 0.991384i $$0.541814\pi$$
$$6$$ 2.41421 0.985599
$$7$$ 0 0
$$8$$ −4.41421 −1.56066
$$9$$ 1.00000 0.333333
$$10$$ 1.41421 0.447214
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ −3.82843 −1.10517
$$13$$ −5.41421 −1.50163 −0.750816 0.660511i $$-0.770340\pi$$
−0.750816 + 0.660511i $$0.770340\pi$$
$$14$$ 0 0
$$15$$ 0.585786 0.151249
$$16$$ 3.00000 0.750000
$$17$$ −6.24264 −1.51406 −0.757031 0.653379i $$-0.773351\pi$$
−0.757031 + 0.653379i $$0.773351\pi$$
$$18$$ −2.41421 −0.569036
$$19$$ −2.82843 −0.648886 −0.324443 0.945905i $$-0.605177\pi$$
−0.324443 + 0.945905i $$0.605177\pi$$
$$20$$ −2.24264 −0.501470
$$21$$ 0 0
$$22$$ 4.82843 1.02942
$$23$$ 3.65685 0.762507 0.381253 0.924471i $$-0.375493\pi$$
0.381253 + 0.924471i $$0.375493\pi$$
$$24$$ 4.41421 0.901048
$$25$$ −4.65685 −0.931371
$$26$$ 13.0711 2.56345
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −1.17157 −0.217556 −0.108778 0.994066i $$-0.534694\pi$$
−0.108778 + 0.994066i $$0.534694\pi$$
$$30$$ −1.41421 −0.258199
$$31$$ 6.82843 1.22642 0.613211 0.789919i $$-0.289878\pi$$
0.613211 + 0.789919i $$0.289878\pi$$
$$32$$ 1.58579 0.280330
$$33$$ 2.00000 0.348155
$$34$$ 15.0711 2.58467
$$35$$ 0 0
$$36$$ 3.82843 0.638071
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 6.82843 1.10772
$$39$$ 5.41421 0.866968
$$40$$ 2.58579 0.408849
$$41$$ 2.24264 0.350242 0.175121 0.984547i $$-0.443968\pi$$
0.175121 + 0.984547i $$0.443968\pi$$
$$42$$ 0 0
$$43$$ −5.65685 −0.862662 −0.431331 0.902194i $$-0.641956\pi$$
−0.431331 + 0.902194i $$0.641956\pi$$
$$44$$ −7.65685 −1.15431
$$45$$ −0.585786 −0.0873239
$$46$$ −8.82843 −1.30168
$$47$$ −2.82843 −0.412568 −0.206284 0.978492i $$-0.566137\pi$$
−0.206284 + 0.978492i $$0.566137\pi$$
$$48$$ −3.00000 −0.433013
$$49$$ 0 0
$$50$$ 11.2426 1.58995
$$51$$ 6.24264 0.874145
$$52$$ −20.7279 −2.87445
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 2.41421 0.328533
$$55$$ 1.17157 0.157975
$$56$$ 0 0
$$57$$ 2.82843 0.374634
$$58$$ 2.82843 0.371391
$$59$$ 6.82843 0.888985 0.444493 0.895782i $$-0.353384\pi$$
0.444493 + 0.895782i $$0.353384\pi$$
$$60$$ 2.24264 0.289524
$$61$$ −3.75736 −0.481081 −0.240540 0.970639i $$-0.577325\pi$$
−0.240540 + 0.970639i $$0.577325\pi$$
$$62$$ −16.4853 −2.09363
$$63$$ 0 0
$$64$$ −9.82843 −1.22855
$$65$$ 3.17157 0.393385
$$66$$ −4.82843 −0.594338
$$67$$ 5.65685 0.691095 0.345547 0.938401i $$-0.387693\pi$$
0.345547 + 0.938401i $$0.387693\pi$$
$$68$$ −23.8995 −2.89824
$$69$$ −3.65685 −0.440234
$$70$$ 0 0
$$71$$ −13.3137 −1.58005 −0.790023 0.613077i $$-0.789932\pi$$
−0.790023 + 0.613077i $$0.789932\pi$$
$$72$$ −4.41421 −0.520220
$$73$$ 5.89949 0.690484 0.345242 0.938514i $$-0.387797\pi$$
0.345242 + 0.938514i $$0.387797\pi$$
$$74$$ 9.65685 1.12259
$$75$$ 4.65685 0.537727
$$76$$ −10.8284 −1.24211
$$77$$ 0 0
$$78$$ −13.0711 −1.48001
$$79$$ 2.34315 0.263624 0.131812 0.991275i $$-0.457920\pi$$
0.131812 + 0.991275i $$0.457920\pi$$
$$80$$ −1.75736 −0.196479
$$81$$ 1.00000 0.111111
$$82$$ −5.41421 −0.597900
$$83$$ 15.3137 1.68090 0.840449 0.541891i $$-0.182291\pi$$
0.840449 + 0.541891i $$0.182291\pi$$
$$84$$ 0 0
$$85$$ 3.65685 0.396642
$$86$$ 13.6569 1.47266
$$87$$ 1.17157 0.125606
$$88$$ 8.82843 0.941113
$$89$$ 5.75736 0.610279 0.305139 0.952308i $$-0.401297\pi$$
0.305139 + 0.952308i $$0.401297\pi$$
$$90$$ 1.41421 0.149071
$$91$$ 0 0
$$92$$ 14.0000 1.45960
$$93$$ −6.82843 −0.708075
$$94$$ 6.82843 0.704298
$$95$$ 1.65685 0.169990
$$96$$ −1.58579 −0.161849
$$97$$ −5.41421 −0.549730 −0.274865 0.961483i $$-0.588633\pi$$
−0.274865 + 0.961483i $$0.588633\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ −17.8284 −1.78284
$$101$$ −17.0711 −1.69863 −0.849317 0.527883i $$-0.822986\pi$$
−0.849317 + 0.527883i $$0.822986\pi$$
$$102$$ −15.0711 −1.49226
$$103$$ 12.4853 1.23021 0.615106 0.788445i $$-0.289113\pi$$
0.615106 + 0.788445i $$0.289113\pi$$
$$104$$ 23.8995 2.34354
$$105$$ 0 0
$$106$$ 4.82843 0.468978
$$107$$ −11.6569 −1.12691 −0.563455 0.826147i $$-0.690528\pi$$
−0.563455 + 0.826147i $$0.690528\pi$$
$$108$$ −3.82843 −0.368391
$$109$$ 5.65685 0.541828 0.270914 0.962604i $$-0.412674\pi$$
0.270914 + 0.962604i $$0.412674\pi$$
$$110$$ −2.82843 −0.269680
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ 17.3137 1.62874 0.814368 0.580348i $$-0.197084\pi$$
0.814368 + 0.580348i $$0.197084\pi$$
$$114$$ −6.82843 −0.639541
$$115$$ −2.14214 −0.199755
$$116$$ −4.48528 −0.416448
$$117$$ −5.41421 −0.500544
$$118$$ −16.4853 −1.51759
$$119$$ 0 0
$$120$$ −2.58579 −0.236049
$$121$$ −7.00000 −0.636364
$$122$$ 9.07107 0.821256
$$123$$ −2.24264 −0.202212
$$124$$ 26.1421 2.34763
$$125$$ 5.65685 0.505964
$$126$$ 0 0
$$127$$ 9.65685 0.856907 0.428454 0.903564i $$-0.359059\pi$$
0.428454 + 0.903564i $$0.359059\pi$$
$$128$$ 20.5563 1.81694
$$129$$ 5.65685 0.498058
$$130$$ −7.65685 −0.671551
$$131$$ −7.31371 −0.639002 −0.319501 0.947586i $$-0.603515\pi$$
−0.319501 + 0.947586i $$0.603515\pi$$
$$132$$ 7.65685 0.666444
$$133$$ 0 0
$$134$$ −13.6569 −1.17977
$$135$$ 0.585786 0.0504165
$$136$$ 27.5563 2.36294
$$137$$ −14.1421 −1.20824 −0.604122 0.796892i $$-0.706476\pi$$
−0.604122 + 0.796892i $$0.706476\pi$$
$$138$$ 8.82843 0.751526
$$139$$ 6.34315 0.538019 0.269009 0.963138i $$-0.413304\pi$$
0.269009 + 0.963138i $$0.413304\pi$$
$$140$$ 0 0
$$141$$ 2.82843 0.238197
$$142$$ 32.1421 2.69731
$$143$$ 10.8284 0.905519
$$144$$ 3.00000 0.250000
$$145$$ 0.686292 0.0569934
$$146$$ −14.2426 −1.17873
$$147$$ 0 0
$$148$$ −15.3137 −1.25878
$$149$$ −5.31371 −0.435316 −0.217658 0.976025i $$-0.569842\pi$$
−0.217658 + 0.976025i $$0.569842\pi$$
$$150$$ −11.2426 −0.917958
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 12.4853 1.01269
$$153$$ −6.24264 −0.504688
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 20.7279 1.65956
$$157$$ −20.2426 −1.61554 −0.807769 0.589499i $$-0.799325\pi$$
−0.807769 + 0.589499i $$0.799325\pi$$
$$158$$ −5.65685 −0.450035
$$159$$ 2.00000 0.158610
$$160$$ −0.928932 −0.0734385
$$161$$ 0 0
$$162$$ −2.41421 −0.189679
$$163$$ 11.3137 0.886158 0.443079 0.896483i $$-0.353886\pi$$
0.443079 + 0.896483i $$0.353886\pi$$
$$164$$ 8.58579 0.670437
$$165$$ −1.17157 −0.0912068
$$166$$ −36.9706 −2.86947
$$167$$ −19.7990 −1.53209 −0.766046 0.642786i $$-0.777779\pi$$
−0.766046 + 0.642786i $$0.777779\pi$$
$$168$$ 0 0
$$169$$ 16.3137 1.25490
$$170$$ −8.82843 −0.677109
$$171$$ −2.82843 −0.216295
$$172$$ −21.6569 −1.65132
$$173$$ −6.92893 −0.526797 −0.263398 0.964687i $$-0.584843\pi$$
−0.263398 + 0.964687i $$0.584843\pi$$
$$174$$ −2.82843 −0.214423
$$175$$ 0 0
$$176$$ −6.00000 −0.452267
$$177$$ −6.82843 −0.513256
$$178$$ −13.8995 −1.04181
$$179$$ −8.34315 −0.623596 −0.311798 0.950148i $$-0.600931\pi$$
−0.311798 + 0.950148i $$0.600931\pi$$
$$180$$ −2.24264 −0.167157
$$181$$ 5.41421 0.402435 0.201218 0.979547i $$-0.435510\pi$$
0.201218 + 0.979547i $$0.435510\pi$$
$$182$$ 0 0
$$183$$ 3.75736 0.277752
$$184$$ −16.1421 −1.19001
$$185$$ 2.34315 0.172272
$$186$$ 16.4853 1.20876
$$187$$ 12.4853 0.913014
$$188$$ −10.8284 −0.789744
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 9.82843 0.709306
$$193$$ −17.3137 −1.24627 −0.623134 0.782115i $$-0.714141\pi$$
−0.623134 + 0.782115i $$0.714141\pi$$
$$194$$ 13.0711 0.938448
$$195$$ −3.17157 −0.227121
$$196$$ 0 0
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 4.82843 0.343141
$$199$$ −10.3431 −0.733206 −0.366603 0.930377i $$-0.619479\pi$$
−0.366603 + 0.930377i $$0.619479\pi$$
$$200$$ 20.5563 1.45355
$$201$$ −5.65685 −0.399004
$$202$$ 41.2132 2.89975
$$203$$ 0 0
$$204$$ 23.8995 1.67330
$$205$$ −1.31371 −0.0917534
$$206$$ −30.1421 −2.10010
$$207$$ 3.65685 0.254169
$$208$$ −16.2426 −1.12622
$$209$$ 5.65685 0.391293
$$210$$ 0 0
$$211$$ −20.9706 −1.44367 −0.721837 0.692064i $$-0.756702\pi$$
−0.721837 + 0.692064i $$0.756702\pi$$
$$212$$ −7.65685 −0.525875
$$213$$ 13.3137 0.912240
$$214$$ 28.1421 1.92376
$$215$$ 3.31371 0.225993
$$216$$ 4.41421 0.300349
$$217$$ 0 0
$$218$$ −13.6569 −0.924959
$$219$$ −5.89949 −0.398651
$$220$$ 4.48528 0.302398
$$221$$ 33.7990 2.27357
$$222$$ −9.65685 −0.648126
$$223$$ 8.97056 0.600713 0.300357 0.953827i $$-0.402894\pi$$
0.300357 + 0.953827i $$0.402894\pi$$
$$224$$ 0 0
$$225$$ −4.65685 −0.310457
$$226$$ −41.7990 −2.78043
$$227$$ 15.7990 1.04862 0.524308 0.851529i $$-0.324324\pi$$
0.524308 + 0.851529i $$0.324324\pi$$
$$228$$ 10.8284 0.717130
$$229$$ −8.24264 −0.544689 −0.272345 0.962200i $$-0.587799\pi$$
−0.272345 + 0.962200i $$0.587799\pi$$
$$230$$ 5.17157 0.341003
$$231$$ 0 0
$$232$$ 5.17157 0.339530
$$233$$ 22.1421 1.45058 0.725290 0.688444i $$-0.241706\pi$$
0.725290 + 0.688444i $$0.241706\pi$$
$$234$$ 13.0711 0.854482
$$235$$ 1.65685 0.108081
$$236$$ 26.1421 1.70171
$$237$$ −2.34315 −0.152204
$$238$$ 0 0
$$239$$ −4.34315 −0.280935 −0.140467 0.990085i $$-0.544861\pi$$
−0.140467 + 0.990085i $$0.544861\pi$$
$$240$$ 1.75736 0.113437
$$241$$ 7.75736 0.499695 0.249848 0.968285i $$-0.419619\pi$$
0.249848 + 0.968285i $$0.419619\pi$$
$$242$$ 16.8995 1.08634
$$243$$ −1.00000 −0.0641500
$$244$$ −14.3848 −0.920891
$$245$$ 0 0
$$246$$ 5.41421 0.345198
$$247$$ 15.3137 0.974388
$$248$$ −30.1421 −1.91403
$$249$$ −15.3137 −0.970467
$$250$$ −13.6569 −0.863735
$$251$$ −4.48528 −0.283108 −0.141554 0.989931i $$-0.545210\pi$$
−0.141554 + 0.989931i $$0.545210\pi$$
$$252$$ 0 0
$$253$$ −7.31371 −0.459809
$$254$$ −23.3137 −1.46283
$$255$$ −3.65685 −0.229001
$$256$$ −29.9706 −1.87316
$$257$$ 19.2132 1.19849 0.599243 0.800567i $$-0.295468\pi$$
0.599243 + 0.800567i $$0.295468\pi$$
$$258$$ −13.6569 −0.850239
$$259$$ 0 0
$$260$$ 12.1421 0.753023
$$261$$ −1.17157 −0.0725185
$$262$$ 17.6569 1.09084
$$263$$ −17.3137 −1.06761 −0.533805 0.845608i $$-0.679238\pi$$
−0.533805 + 0.845608i $$0.679238\pi$$
$$264$$ −8.82843 −0.543352
$$265$$ 1.17157 0.0719691
$$266$$ 0 0
$$267$$ −5.75736 −0.352345
$$268$$ 21.6569 1.32290
$$269$$ −10.7279 −0.654093 −0.327046 0.945008i $$-0.606053\pi$$
−0.327046 + 0.945008i $$0.606053\pi$$
$$270$$ −1.41421 −0.0860663
$$271$$ −18.1421 −1.10206 −0.551028 0.834487i $$-0.685764\pi$$
−0.551028 + 0.834487i $$0.685764\pi$$
$$272$$ −18.7279 −1.13555
$$273$$ 0 0
$$274$$ 34.1421 2.06260
$$275$$ 9.31371 0.561638
$$276$$ −14.0000 −0.842701
$$277$$ 13.3137 0.799943 0.399972 0.916528i $$-0.369020\pi$$
0.399972 + 0.916528i $$0.369020\pi$$
$$278$$ −15.3137 −0.918455
$$279$$ 6.82843 0.408807
$$280$$ 0 0
$$281$$ −16.4853 −0.983429 −0.491715 0.870756i $$-0.663630\pi$$
−0.491715 + 0.870756i $$0.663630\pi$$
$$282$$ −6.82843 −0.406627
$$283$$ −8.48528 −0.504398 −0.252199 0.967675i $$-0.581154\pi$$
−0.252199 + 0.967675i $$0.581154\pi$$
$$284$$ −50.9706 −3.02455
$$285$$ −1.65685 −0.0981436
$$286$$ −26.1421 −1.54582
$$287$$ 0 0
$$288$$ 1.58579 0.0934434
$$289$$ 21.9706 1.29239
$$290$$ −1.65685 −0.0972938
$$291$$ 5.41421 0.317387
$$292$$ 22.5858 1.32173
$$293$$ −19.4142 −1.13419 −0.567095 0.823652i $$-0.691933\pi$$
−0.567095 + 0.823652i $$0.691933\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 17.6569 1.02628
$$297$$ 2.00000 0.116052
$$298$$ 12.8284 0.743131
$$299$$ −19.7990 −1.14501
$$300$$ 17.8284 1.02932
$$301$$ 0 0
$$302$$ −28.9706 −1.66707
$$303$$ 17.0711 0.980707
$$304$$ −8.48528 −0.486664
$$305$$ 2.20101 0.126029
$$306$$ 15.0711 0.861556
$$307$$ −1.85786 −0.106034 −0.0530170 0.998594i $$-0.516884\pi$$
−0.0530170 + 0.998594i $$0.516884\pi$$
$$308$$ 0 0
$$309$$ −12.4853 −0.710263
$$310$$ 9.65685 0.548472
$$311$$ 22.1421 1.25557 0.627783 0.778389i $$-0.283963\pi$$
0.627783 + 0.778389i $$0.283963\pi$$
$$312$$ −23.8995 −1.35304
$$313$$ 17.8995 1.01174 0.505870 0.862610i $$-0.331172\pi$$
0.505870 + 0.862610i $$0.331172\pi$$
$$314$$ 48.8701 2.75790
$$315$$ 0 0
$$316$$ 8.97056 0.504634
$$317$$ 10.0000 0.561656 0.280828 0.959758i $$-0.409391\pi$$
0.280828 + 0.959758i $$0.409391\pi$$
$$318$$ −4.82843 −0.270765
$$319$$ 2.34315 0.131191
$$320$$ 5.75736 0.321846
$$321$$ 11.6569 0.650622
$$322$$ 0 0
$$323$$ 17.6569 0.982454
$$324$$ 3.82843 0.212690
$$325$$ 25.2132 1.39858
$$326$$ −27.3137 −1.51277
$$327$$ −5.65685 −0.312825
$$328$$ −9.89949 −0.546608
$$329$$ 0 0
$$330$$ 2.82843 0.155700
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 58.6274 3.21760
$$333$$ −4.00000 −0.219199
$$334$$ 47.7990 2.61544
$$335$$ −3.31371 −0.181047
$$336$$ 0 0
$$337$$ −18.3431 −0.999215 −0.499607 0.866252i $$-0.666522\pi$$
−0.499607 + 0.866252i $$0.666522\pi$$
$$338$$ −39.3848 −2.14225
$$339$$ −17.3137 −0.940352
$$340$$ 14.0000 0.759257
$$341$$ −13.6569 −0.739560
$$342$$ 6.82843 0.369239
$$343$$ 0 0
$$344$$ 24.9706 1.34632
$$345$$ 2.14214 0.115329
$$346$$ 16.7279 0.899299
$$347$$ 10.6863 0.573670 0.286835 0.957980i $$-0.407397\pi$$
0.286835 + 0.957980i $$0.407397\pi$$
$$348$$ 4.48528 0.240436
$$349$$ −9.89949 −0.529908 −0.264954 0.964261i $$-0.585357\pi$$
−0.264954 + 0.964261i $$0.585357\pi$$
$$350$$ 0 0
$$351$$ 5.41421 0.288989
$$352$$ −3.17157 −0.169045
$$353$$ −10.7279 −0.570990 −0.285495 0.958380i $$-0.592158\pi$$
−0.285495 + 0.958380i $$0.592158\pi$$
$$354$$ 16.4853 0.876183
$$355$$ 7.79899 0.413927
$$356$$ 22.0416 1.16820
$$357$$ 0 0
$$358$$ 20.1421 1.06454
$$359$$ −11.6569 −0.615225 −0.307613 0.951512i $$-0.599530\pi$$
−0.307613 + 0.951512i $$0.599530\pi$$
$$360$$ 2.58579 0.136283
$$361$$ −11.0000 −0.578947
$$362$$ −13.0711 −0.687000
$$363$$ 7.00000 0.367405
$$364$$ 0 0
$$365$$ −3.45584 −0.180887
$$366$$ −9.07107 −0.474152
$$367$$ −19.3137 −1.00817 −0.504084 0.863655i $$-0.668170\pi$$
−0.504084 + 0.863655i $$0.668170\pi$$
$$368$$ 10.9706 0.571880
$$369$$ 2.24264 0.116747
$$370$$ −5.65685 −0.294086
$$371$$ 0 0
$$372$$ −26.1421 −1.35541
$$373$$ −33.3137 −1.72492 −0.862459 0.506127i $$-0.831077\pi$$
−0.862459 + 0.506127i $$0.831077\pi$$
$$374$$ −30.1421 −1.55861
$$375$$ −5.65685 −0.292119
$$376$$ 12.4853 0.643879
$$377$$ 6.34315 0.326689
$$378$$ 0 0
$$379$$ 31.3137 1.60848 0.804239 0.594307i $$-0.202573\pi$$
0.804239 + 0.594307i $$0.202573\pi$$
$$380$$ 6.34315 0.325397
$$381$$ −9.65685 −0.494736
$$382$$ 43.4558 2.22339
$$383$$ 29.6569 1.51539 0.757697 0.652606i $$-0.226324\pi$$
0.757697 + 0.652606i $$0.226324\pi$$
$$384$$ −20.5563 −1.04901
$$385$$ 0 0
$$386$$ 41.7990 2.12751
$$387$$ −5.65685 −0.287554
$$388$$ −20.7279 −1.05230
$$389$$ 10.1421 0.514227 0.257113 0.966381i $$-0.417229\pi$$
0.257113 + 0.966381i $$0.417229\pi$$
$$390$$ 7.65685 0.387720
$$391$$ −22.8284 −1.15448
$$392$$ 0 0
$$393$$ 7.31371 0.368928
$$394$$ −4.82843 −0.243253
$$395$$ −1.37258 −0.0690621
$$396$$ −7.65685 −0.384771
$$397$$ −34.3848 −1.72572 −0.862861 0.505441i $$-0.831330\pi$$
−0.862861 + 0.505441i $$0.831330\pi$$
$$398$$ 24.9706 1.25166
$$399$$ 0 0
$$400$$ −13.9706 −0.698528
$$401$$ 22.1421 1.10573 0.552863 0.833272i $$-0.313535\pi$$
0.552863 + 0.833272i $$0.313535\pi$$
$$402$$ 13.6569 0.681142
$$403$$ −36.9706 −1.84163
$$404$$ −65.3553 −3.25155
$$405$$ −0.585786 −0.0291080
$$406$$ 0 0
$$407$$ 8.00000 0.396545
$$408$$ −27.5563 −1.36424
$$409$$ 18.5858 0.919008 0.459504 0.888176i $$-0.348027\pi$$
0.459504 + 0.888176i $$0.348027\pi$$
$$410$$ 3.17157 0.156633
$$411$$ 14.1421 0.697580
$$412$$ 47.7990 2.35489
$$413$$ 0 0
$$414$$ −8.82843 −0.433894
$$415$$ −8.97056 −0.440348
$$416$$ −8.58579 −0.420953
$$417$$ −6.34315 −0.310625
$$418$$ −13.6569 −0.667979
$$419$$ −38.8284 −1.89689 −0.948446 0.316938i $$-0.897345\pi$$
−0.948446 + 0.316938i $$0.897345\pi$$
$$420$$ 0 0
$$421$$ −28.6274 −1.39521 −0.697607 0.716480i $$-0.745752\pi$$
−0.697607 + 0.716480i $$0.745752\pi$$
$$422$$ 50.6274 2.46450
$$423$$ −2.82843 −0.137523
$$424$$ 8.82843 0.428746
$$425$$ 29.0711 1.41015
$$426$$ −32.1421 −1.55729
$$427$$ 0 0
$$428$$ −44.6274 −2.15715
$$429$$ −10.8284 −0.522801
$$430$$ −8.00000 −0.385794
$$431$$ 6.97056 0.335760 0.167880 0.985807i $$-0.446308\pi$$
0.167880 + 0.985807i $$0.446308\pi$$
$$432$$ −3.00000 −0.144338
$$433$$ 11.7574 0.565023 0.282511 0.959264i $$-0.408833\pi$$
0.282511 + 0.959264i $$0.408833\pi$$
$$434$$ 0 0
$$435$$ −0.686292 −0.0329052
$$436$$ 21.6569 1.03718
$$437$$ −10.3431 −0.494780
$$438$$ 14.2426 0.680540
$$439$$ −35.3137 −1.68543 −0.842716 0.538359i $$-0.819044\pi$$
−0.842716 + 0.538359i $$0.819044\pi$$
$$440$$ −5.17157 −0.246545
$$441$$ 0 0
$$442$$ −81.5980 −3.88122
$$443$$ 1.02944 0.0489100 0.0244550 0.999701i $$-0.492215\pi$$
0.0244550 + 0.999701i $$0.492215\pi$$
$$444$$ 15.3137 0.726756
$$445$$ −3.37258 −0.159876
$$446$$ −21.6569 −1.02548
$$447$$ 5.31371 0.251330
$$448$$ 0 0
$$449$$ 17.3137 0.817084 0.408542 0.912739i $$-0.366037\pi$$
0.408542 + 0.912739i $$0.366037\pi$$
$$450$$ 11.2426 0.529983
$$451$$ −4.48528 −0.211204
$$452$$ 66.2843 3.11775
$$453$$ −12.0000 −0.563809
$$454$$ −38.1421 −1.79010
$$455$$ 0 0
$$456$$ −12.4853 −0.584677
$$457$$ −18.0000 −0.842004 −0.421002 0.907060i $$-0.638322\pi$$
−0.421002 + 0.907060i $$0.638322\pi$$
$$458$$ 19.8995 0.929842
$$459$$ 6.24264 0.291382
$$460$$ −8.20101 −0.382374
$$461$$ 19.4142 0.904210 0.452105 0.891965i $$-0.350673\pi$$
0.452105 + 0.891965i $$0.350673\pi$$
$$462$$ 0 0
$$463$$ 18.6274 0.865689 0.432845 0.901468i $$-0.357510\pi$$
0.432845 + 0.901468i $$0.357510\pi$$
$$464$$ −3.51472 −0.163167
$$465$$ 4.00000 0.185496
$$466$$ −53.4558 −2.47629
$$467$$ −39.7990 −1.84168 −0.920839 0.389943i $$-0.872495\pi$$
−0.920839 + 0.389943i $$0.872495\pi$$
$$468$$ −20.7279 −0.958149
$$469$$ 0 0
$$470$$ −4.00000 −0.184506
$$471$$ 20.2426 0.932732
$$472$$ −30.1421 −1.38740
$$473$$ 11.3137 0.520205
$$474$$ 5.65685 0.259828
$$475$$ 13.1716 0.604353
$$476$$ 0 0
$$477$$ −2.00000 −0.0915737
$$478$$ 10.4853 0.479586
$$479$$ −30.1421 −1.37723 −0.688615 0.725127i $$-0.741781\pi$$
−0.688615 + 0.725127i $$0.741781\pi$$
$$480$$ 0.928932 0.0423998
$$481$$ 21.6569 0.987468
$$482$$ −18.7279 −0.853033
$$483$$ 0 0
$$484$$ −26.7990 −1.21814
$$485$$ 3.17157 0.144014
$$486$$ 2.41421 0.109511
$$487$$ −18.6274 −0.844089 −0.422044 0.906575i $$-0.638687\pi$$
−0.422044 + 0.906575i $$0.638687\pi$$
$$488$$ 16.5858 0.750803
$$489$$ −11.3137 −0.511624
$$490$$ 0 0
$$491$$ 38.9706 1.75872 0.879358 0.476160i $$-0.157972\pi$$
0.879358 + 0.476160i $$0.157972\pi$$
$$492$$ −8.58579 −0.387077
$$493$$ 7.31371 0.329393
$$494$$ −36.9706 −1.66338
$$495$$ 1.17157 0.0526583
$$496$$ 20.4853 0.919816
$$497$$ 0 0
$$498$$ 36.9706 1.65669
$$499$$ −19.3137 −0.864600 −0.432300 0.901730i $$-0.642298\pi$$
−0.432300 + 0.901730i $$0.642298\pi$$
$$500$$ 21.6569 0.968524
$$501$$ 19.7990 0.884554
$$502$$ 10.8284 0.483296
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 10.0000 0.444994
$$506$$ 17.6569 0.784943
$$507$$ −16.3137 −0.724517
$$508$$ 36.9706 1.64030
$$509$$ 25.5563 1.13277 0.566383 0.824142i $$-0.308342\pi$$
0.566383 + 0.824142i $$0.308342\pi$$
$$510$$ 8.82843 0.390929
$$511$$ 0 0
$$512$$ 31.2426 1.38074
$$513$$ 2.82843 0.124878
$$514$$ −46.3848 −2.04594
$$515$$ −7.31371 −0.322281
$$516$$ 21.6569 0.953390
$$517$$ 5.65685 0.248788
$$518$$ 0 0
$$519$$ 6.92893 0.304146
$$520$$ −14.0000 −0.613941
$$521$$ 32.5858 1.42761 0.713805 0.700345i $$-0.246970\pi$$
0.713805 + 0.700345i $$0.246970\pi$$
$$522$$ 2.82843 0.123797
$$523$$ 14.3431 0.627182 0.313591 0.949558i $$-0.398468\pi$$
0.313591 + 0.949558i $$0.398468\pi$$
$$524$$ −28.0000 −1.22319
$$525$$ 0 0
$$526$$ 41.7990 1.82252
$$527$$ −42.6274 −1.85688
$$528$$ 6.00000 0.261116
$$529$$ −9.62742 −0.418583
$$530$$ −2.82843 −0.122859
$$531$$ 6.82843 0.296328
$$532$$ 0 0
$$533$$ −12.1421 −0.525934
$$534$$ 13.8995 0.601490
$$535$$ 6.82843 0.295219
$$536$$ −24.9706 −1.07856
$$537$$ 8.34315 0.360033
$$538$$ 25.8995 1.11661
$$539$$ 0 0
$$540$$ 2.24264 0.0965079
$$541$$ −5.31371 −0.228454 −0.114227 0.993455i $$-0.536439\pi$$
−0.114227 + 0.993455i $$0.536439\pi$$
$$542$$ 43.7990 1.88133
$$543$$ −5.41421 −0.232346
$$544$$ −9.89949 −0.424437
$$545$$ −3.31371 −0.141944
$$546$$ 0 0
$$547$$ −3.02944 −0.129529 −0.0647647 0.997901i $$-0.520630\pi$$
−0.0647647 + 0.997901i $$0.520630\pi$$
$$548$$ −54.1421 −2.31284
$$549$$ −3.75736 −0.160360
$$550$$ −22.4853 −0.958776
$$551$$ 3.31371 0.141169
$$552$$ 16.1421 0.687055
$$553$$ 0 0
$$554$$ −32.1421 −1.36559
$$555$$ −2.34315 −0.0994610
$$556$$ 24.2843 1.02988
$$557$$ 26.0000 1.10166 0.550828 0.834619i $$-0.314312\pi$$
0.550828 + 0.834619i $$0.314312\pi$$
$$558$$ −16.4853 −0.697878
$$559$$ 30.6274 1.29540
$$560$$ 0 0
$$561$$ −12.4853 −0.527129
$$562$$ 39.7990 1.67882
$$563$$ 6.82843 0.287784 0.143892 0.989593i $$-0.454038\pi$$
0.143892 + 0.989593i $$0.454038\pi$$
$$564$$ 10.8284 0.455959
$$565$$ −10.1421 −0.426683
$$566$$ 20.4853 0.861061
$$567$$ 0 0
$$568$$ 58.7696 2.46592
$$569$$ 0.485281 0.0203441 0.0101720 0.999948i $$-0.496762\pi$$
0.0101720 + 0.999948i $$0.496762\pi$$
$$570$$ 4.00000 0.167542
$$571$$ 33.6569 1.40850 0.704248 0.709954i $$-0.251284\pi$$
0.704248 + 0.709954i $$0.251284\pi$$
$$572$$ 41.4558 1.73336
$$573$$ 18.0000 0.751961
$$574$$ 0 0
$$575$$ −17.0294 −0.710177
$$576$$ −9.82843 −0.409518
$$577$$ 14.1005 0.587012 0.293506 0.955957i $$-0.405178\pi$$
0.293506 + 0.955957i $$0.405178\pi$$
$$578$$ −53.0416 −2.20624
$$579$$ 17.3137 0.719533
$$580$$ 2.62742 0.109098
$$581$$ 0 0
$$582$$ −13.0711 −0.541813
$$583$$ 4.00000 0.165663
$$584$$ −26.0416 −1.07761
$$585$$ 3.17157 0.131128
$$586$$ 46.8701 1.93618
$$587$$ 17.1716 0.708747 0.354373 0.935104i $$-0.384694\pi$$
0.354373 + 0.935104i $$0.384694\pi$$
$$588$$ 0 0
$$589$$ −19.3137 −0.795807
$$590$$ 9.65685 0.397566
$$591$$ −2.00000 −0.0822690
$$592$$ −12.0000 −0.493197
$$593$$ 21.0711 0.865285 0.432643 0.901566i $$-0.357581\pi$$
0.432643 + 0.901566i $$0.357581\pi$$
$$594$$ −4.82843 −0.198113
$$595$$ 0 0
$$596$$ −20.3431 −0.833288
$$597$$ 10.3431 0.423317
$$598$$ 47.7990 1.95465
$$599$$ −2.00000 −0.0817178 −0.0408589 0.999165i $$-0.513009\pi$$
−0.0408589 + 0.999165i $$0.513009\pi$$
$$600$$ −20.5563 −0.839209
$$601$$ 0.928932 0.0378919 0.0189460 0.999821i $$-0.493969\pi$$
0.0189460 + 0.999821i $$0.493969\pi$$
$$602$$ 0 0
$$603$$ 5.65685 0.230365
$$604$$ 45.9411 1.86932
$$605$$ 4.10051 0.166709
$$606$$ −41.2132 −1.67417
$$607$$ 29.6569 1.20373 0.601867 0.798596i $$-0.294424\pi$$
0.601867 + 0.798596i $$0.294424\pi$$
$$608$$ −4.48528 −0.181902
$$609$$ 0 0
$$610$$ −5.31371 −0.215146
$$611$$ 15.3137 0.619526
$$612$$ −23.8995 −0.966080
$$613$$ 27.3137 1.10319 0.551595 0.834112i $$-0.314019\pi$$
0.551595 + 0.834112i $$0.314019\pi$$
$$614$$ 4.48528 0.181011
$$615$$ 1.31371 0.0529738
$$616$$ 0 0
$$617$$ −7.51472 −0.302531 −0.151266 0.988493i $$-0.548335\pi$$
−0.151266 + 0.988493i $$0.548335\pi$$
$$618$$ 30.1421 1.21249
$$619$$ 4.97056 0.199784 0.0998919 0.994998i $$-0.468150\pi$$
0.0998919 + 0.994998i $$0.468150\pi$$
$$620$$ −15.3137 −0.615013
$$621$$ −3.65685 −0.146745
$$622$$ −53.4558 −2.14338
$$623$$ 0 0
$$624$$ 16.2426 0.650226
$$625$$ 19.9706 0.798823
$$626$$ −43.2132 −1.72715
$$627$$ −5.65685 −0.225913
$$628$$ −77.4975 −3.09249
$$629$$ 24.9706 0.995642
$$630$$ 0 0
$$631$$ 0.686292 0.0273208 0.0136604 0.999907i $$-0.495652\pi$$
0.0136604 + 0.999907i $$0.495652\pi$$
$$632$$ −10.3431 −0.411428
$$633$$ 20.9706 0.833505
$$634$$ −24.1421 −0.958807
$$635$$ −5.65685 −0.224485
$$636$$ 7.65685 0.303614
$$637$$ 0 0
$$638$$ −5.65685 −0.223957
$$639$$ −13.3137 −0.526682
$$640$$ −12.0416 −0.475987
$$641$$ −5.17157 −0.204265 −0.102132 0.994771i $$-0.532567\pi$$
−0.102132 + 0.994771i $$0.532567\pi$$
$$642$$ −28.1421 −1.11068
$$643$$ −50.4264 −1.98862 −0.994312 0.106510i $$-0.966033\pi$$
−0.994312 + 0.106510i $$0.966033\pi$$
$$644$$ 0 0
$$645$$ −3.31371 −0.130477
$$646$$ −42.6274 −1.67715
$$647$$ −21.1716 −0.832340 −0.416170 0.909287i $$-0.636628\pi$$
−0.416170 + 0.909287i $$0.636628\pi$$
$$648$$ −4.41421 −0.173407
$$649$$ −13.6569 −0.536078
$$650$$ −60.8701 −2.38752
$$651$$ 0 0
$$652$$ 43.3137 1.69630
$$653$$ 19.5147 0.763670 0.381835 0.924231i $$-0.375292\pi$$
0.381835 + 0.924231i $$0.375292\pi$$
$$654$$ 13.6569 0.534025
$$655$$ 4.28427 0.167400
$$656$$ 6.72792 0.262681
$$657$$ 5.89949 0.230161
$$658$$ 0 0
$$659$$ −13.3137 −0.518628 −0.259314 0.965793i $$-0.583497\pi$$
−0.259314 + 0.965793i $$0.583497\pi$$
$$660$$ −4.48528 −0.174589
$$661$$ −7.55635 −0.293908 −0.146954 0.989143i $$-0.546947\pi$$
−0.146954 + 0.989143i $$0.546947\pi$$
$$662$$ 9.65685 0.375324
$$663$$ −33.7990 −1.31264
$$664$$ −67.5980 −2.62331
$$665$$ 0 0
$$666$$ 9.65685 0.374196
$$667$$ −4.28427 −0.165888
$$668$$ −75.7990 −2.93275
$$669$$ −8.97056 −0.346822
$$670$$ 8.00000 0.309067
$$671$$ 7.51472 0.290102
$$672$$ 0 0
$$673$$ 0.686292 0.0264546 0.0132273 0.999913i $$-0.495789\pi$$
0.0132273 + 0.999913i $$0.495789\pi$$
$$674$$ 44.2843 1.70577
$$675$$ 4.65685 0.179242
$$676$$ 62.4558 2.40215
$$677$$ −28.5858 −1.09864 −0.549321 0.835612i $$-0.685113\pi$$
−0.549321 + 0.835612i $$0.685113\pi$$
$$678$$ 41.7990 1.60528
$$679$$ 0 0
$$680$$ −16.1421 −0.619023
$$681$$ −15.7990 −0.605419
$$682$$ 32.9706 1.26251
$$683$$ −8.34315 −0.319242 −0.159621 0.987178i $$-0.551027\pi$$
−0.159621 + 0.987178i $$0.551027\pi$$
$$684$$ −10.8284 −0.414035
$$685$$ 8.28427 0.316526
$$686$$ 0 0
$$687$$ 8.24264 0.314476
$$688$$ −16.9706 −0.646997
$$689$$ 10.8284 0.412530
$$690$$ −5.17157 −0.196878
$$691$$ 23.3137 0.886895 0.443448 0.896300i $$-0.353755\pi$$
0.443448 + 0.896300i $$0.353755\pi$$
$$692$$ −26.5269 −1.00840
$$693$$ 0 0
$$694$$ −25.7990 −0.979316
$$695$$ −3.71573 −0.140946
$$696$$ −5.17157 −0.196028
$$697$$ −14.0000 −0.530288
$$698$$ 23.8995 0.904609
$$699$$ −22.1421 −0.837492
$$700$$ 0 0
$$701$$ −22.8284 −0.862218 −0.431109 0.902300i $$-0.641878\pi$$
−0.431109 + 0.902300i $$0.641878\pi$$
$$702$$ −13.0711 −0.493336
$$703$$ 11.3137 0.426705
$$704$$ 19.6569 0.740846
$$705$$ −1.65685 −0.0624007
$$706$$ 25.8995 0.974740
$$707$$ 0 0
$$708$$ −26.1421 −0.982482
$$709$$ 20.2843 0.761792 0.380896 0.924618i $$-0.375616\pi$$
0.380896 + 0.924618i $$0.375616\pi$$
$$710$$ −18.8284 −0.706618
$$711$$ 2.34315 0.0878748
$$712$$ −25.4142 −0.952438
$$713$$ 24.9706 0.935155
$$714$$ 0 0
$$715$$ −6.34315 −0.237220
$$716$$ −31.9411 −1.19370
$$717$$ 4.34315 0.162198
$$718$$ 28.1421 1.05026
$$719$$ 25.9411 0.967441 0.483720 0.875223i $$-0.339285\pi$$
0.483720 + 0.875223i $$0.339285\pi$$
$$720$$ −1.75736 −0.0654929
$$721$$ 0 0
$$722$$ 26.5563 0.988325
$$723$$ −7.75736 −0.288499
$$724$$ 20.7279 0.770347
$$725$$ 5.45584 0.202625
$$726$$ −16.8995 −0.627199
$$727$$ 4.48528 0.166350 0.0831749 0.996535i $$-0.473494\pi$$
0.0831749 + 0.996535i $$0.473494\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 8.34315 0.308794
$$731$$ 35.3137 1.30612
$$732$$ 14.3848 0.531677
$$733$$ 9.69848 0.358222 0.179111 0.983829i $$-0.442678\pi$$
0.179111 + 0.983829i $$0.442678\pi$$
$$734$$ 46.6274 1.72105
$$735$$ 0 0
$$736$$ 5.79899 0.213754
$$737$$ −11.3137 −0.416746
$$738$$ −5.41421 −0.199300
$$739$$ 27.3137 1.00475 0.502376 0.864650i $$-0.332460\pi$$
0.502376 + 0.864650i $$0.332460\pi$$
$$740$$ 8.97056 0.329764
$$741$$ −15.3137 −0.562563
$$742$$ 0 0
$$743$$ −17.0294 −0.624749 −0.312375 0.949959i $$-0.601124\pi$$
−0.312375 + 0.949959i $$0.601124\pi$$
$$744$$ 30.1421 1.10506
$$745$$ 3.11270 0.114040
$$746$$ 80.4264 2.94462
$$747$$ 15.3137 0.560299
$$748$$ 47.7990 1.74770
$$749$$ 0 0
$$750$$ 13.6569 0.498678
$$751$$ 2.34315 0.0855026 0.0427513 0.999086i $$-0.486388\pi$$
0.0427513 + 0.999086i $$0.486388\pi$$
$$752$$ −8.48528 −0.309426
$$753$$ 4.48528 0.163453
$$754$$ −15.3137 −0.557692
$$755$$ −7.02944 −0.255827
$$756$$ 0 0
$$757$$ 37.6569 1.36866 0.684331 0.729172i $$-0.260094\pi$$
0.684331 + 0.729172i $$0.260094\pi$$
$$758$$ −75.5980 −2.74584
$$759$$ 7.31371 0.265471
$$760$$ −7.31371 −0.265296
$$761$$ −46.5269 −1.68660 −0.843300 0.537444i $$-0.819390\pi$$
−0.843300 + 0.537444i $$0.819390\pi$$
$$762$$ 23.3137 0.844567
$$763$$ 0 0
$$764$$ −68.9117 −2.49314
$$765$$ 3.65685 0.132214
$$766$$ −71.5980 −2.58694
$$767$$ −36.9706 −1.33493
$$768$$ 29.9706 1.08147
$$769$$ 29.6985 1.07095 0.535477 0.844550i $$-0.320132\pi$$
0.535477 + 0.844550i $$0.320132\pi$$
$$770$$ 0 0
$$771$$ −19.2132 −0.691947
$$772$$ −66.2843 −2.38562
$$773$$ −21.5563 −0.775328 −0.387664 0.921801i $$-0.626718\pi$$
−0.387664 + 0.921801i $$0.626718\pi$$
$$774$$ 13.6569 0.490885
$$775$$ −31.7990 −1.14225
$$776$$ 23.8995 0.857942
$$777$$ 0 0
$$778$$ −24.4853 −0.877840
$$779$$ −6.34315 −0.227267
$$780$$ −12.1421 −0.434758
$$781$$ 26.6274 0.952804
$$782$$ 55.1127 1.97083
$$783$$ 1.17157 0.0418686
$$784$$ 0 0
$$785$$ 11.8579 0.423225
$$786$$ −17.6569 −0.629799
$$787$$ −47.3137 −1.68655 −0.843276 0.537481i $$-0.819376\pi$$
−0.843276 + 0.537481i $$0.819376\pi$$
$$788$$ 7.65685 0.272764
$$789$$ 17.3137 0.616384
$$790$$ 3.31371 0.117896
$$791$$ 0 0
$$792$$ 8.82843 0.313704
$$793$$ 20.3431 0.722406
$$794$$ 83.0122 2.94599
$$795$$ −1.17157 −0.0415514
$$796$$ −39.5980 −1.40351
$$797$$ −28.3848 −1.00544 −0.502720 0.864449i $$-0.667667\pi$$
−0.502720 + 0.864449i $$0.667667\pi$$
$$798$$ 0 0
$$799$$ 17.6569 0.624655
$$800$$ −7.38478 −0.261091
$$801$$ 5.75736 0.203426
$$802$$ −53.4558 −1.88759
$$803$$ −11.7990 −0.416377
$$804$$ −21.6569 −0.763778
$$805$$ 0 0
$$806$$ 89.2548 3.14387
$$807$$ 10.7279 0.377641
$$808$$ 75.3553 2.65099
$$809$$ −47.9411 −1.68552 −0.842760 0.538289i $$-0.819071\pi$$
−0.842760 + 0.538289i $$0.819071\pi$$
$$810$$ 1.41421 0.0496904
$$811$$ 6.34315 0.222738 0.111369 0.993779i $$-0.464476\pi$$
0.111369 + 0.993779i $$0.464476\pi$$
$$812$$ 0 0
$$813$$ 18.1421 0.636272
$$814$$ −19.3137 −0.676945
$$815$$ −6.62742 −0.232148
$$816$$ 18.7279 0.655608
$$817$$ 16.0000 0.559769
$$818$$ −44.8701 −1.56884
$$819$$ 0 0
$$820$$ −5.02944 −0.175636
$$821$$ −33.3137 −1.16266 −0.581328 0.813669i $$-0.697467\pi$$
−0.581328 + 0.813669i $$0.697467\pi$$
$$822$$ −34.1421 −1.19084
$$823$$ −24.9706 −0.870419 −0.435210 0.900329i $$-0.643326\pi$$
−0.435210 + 0.900329i $$0.643326\pi$$
$$824$$ −55.1127 −1.91994
$$825$$ −9.31371 −0.324262
$$826$$ 0 0
$$827$$ 36.3431 1.26378 0.631888 0.775060i $$-0.282280\pi$$
0.631888 + 0.775060i $$0.282280\pi$$
$$828$$ 14.0000 0.486534
$$829$$ −24.7279 −0.858836 −0.429418 0.903106i $$-0.641281\pi$$
−0.429418 + 0.903106i $$0.641281\pi$$
$$830$$ 21.6569 0.751720
$$831$$ −13.3137 −0.461847
$$832$$ 53.2132 1.84484
$$833$$ 0 0
$$834$$ 15.3137 0.530270
$$835$$ 11.5980 0.401365
$$836$$ 21.6569 0.749018
$$837$$ −6.82843 −0.236025
$$838$$ 93.7401 3.23820
$$839$$ −45.1716 −1.55950 −0.779748 0.626094i $$-0.784653\pi$$
−0.779748 + 0.626094i $$0.784653\pi$$
$$840$$ 0 0
$$841$$ −27.6274 −0.952670
$$842$$ 69.1127 2.38178
$$843$$ 16.4853 0.567783
$$844$$ −80.2843 −2.76350
$$845$$ −9.55635 −0.328748
$$846$$ 6.82843 0.234766
$$847$$ 0 0
$$848$$ −6.00000 −0.206041
$$849$$ 8.48528 0.291214
$$850$$ −70.1838 −2.40728
$$851$$ −14.6274 −0.501421
$$852$$ 50.9706 1.74622
$$853$$ −49.4975 −1.69476 −0.847381 0.530986i $$-0.821822\pi$$
−0.847381 + 0.530986i $$0.821822\pi$$
$$854$$ 0 0
$$855$$ 1.65685 0.0566632
$$856$$ 51.4558 1.75872
$$857$$ −12.5858 −0.429922 −0.214961 0.976623i $$-0.568962\pi$$
−0.214961 + 0.976623i $$0.568962\pi$$
$$858$$ 26.1421 0.892478
$$859$$ −6.54416 −0.223284 −0.111642 0.993749i $$-0.535611\pi$$
−0.111642 + 0.993749i $$0.535611\pi$$
$$860$$ 12.6863 0.432599
$$861$$ 0 0
$$862$$ −16.8284 −0.573179
$$863$$ −5.31371 −0.180881 −0.0904404 0.995902i $$-0.528827\pi$$
−0.0904404 + 0.995902i $$0.528827\pi$$
$$864$$ −1.58579 −0.0539496
$$865$$ 4.05887 0.138006
$$866$$ −28.3848 −0.964554
$$867$$ −21.9706 −0.746159
$$868$$ 0 0
$$869$$ −4.68629 −0.158972
$$870$$ 1.65685 0.0561726
$$871$$ −30.6274 −1.03777
$$872$$ −24.9706 −0.845610
$$873$$ −5.41421 −0.183243
$$874$$ 24.9706 0.844642
$$875$$ 0 0
$$876$$ −22.5858 −0.763103
$$877$$ 11.3137 0.382037 0.191018 0.981586i $$-0.438821\pi$$
0.191018 + 0.981586i $$0.438821\pi$$
$$878$$ 85.2548 2.87721
$$879$$ 19.4142 0.654825
$$880$$ 3.51472 0.118481
$$881$$ −30.2426 −1.01890 −0.509450 0.860500i $$-0.670151\pi$$
−0.509450 + 0.860500i $$0.670151\pi$$
$$882$$ 0 0
$$883$$ −27.3137 −0.919179 −0.459590 0.888131i $$-0.652004\pi$$
−0.459590 + 0.888131i $$0.652004\pi$$
$$884$$ 129.397 4.35209
$$885$$ 4.00000 0.134459
$$886$$ −2.48528 −0.0834947
$$887$$ −2.82843 −0.0949693 −0.0474846 0.998872i $$-0.515121\pi$$
−0.0474846 + 0.998872i $$0.515121\pi$$
$$888$$ −17.6569 −0.592525
$$889$$ 0 0
$$890$$ 8.14214 0.272925
$$891$$ −2.00000 −0.0670025
$$892$$ 34.3431 1.14989
$$893$$ 8.00000 0.267710
$$894$$ −12.8284 −0.429047
$$895$$ 4.88730 0.163364
$$896$$ 0 0
$$897$$ 19.7990 0.661069
$$898$$ −41.7990 −1.39485
$$899$$ −8.00000 −0.266815
$$900$$ −17.8284 −0.594281
$$901$$ 12.4853 0.415945
$$902$$ 10.8284 0.360547
$$903$$ 0 0
$$904$$ −76.4264 −2.54190
$$905$$ −3.17157 −0.105427
$$906$$ 28.9706 0.962482
$$907$$ −16.0000 −0.531271 −0.265636 0.964073i $$-0.585582\pi$$
−0.265636 + 0.964073i $$0.585582\pi$$
$$908$$ 60.4853 2.00727
$$909$$ −17.0711 −0.566212
$$910$$ 0 0
$$911$$ −34.9706 −1.15863 −0.579313 0.815105i $$-0.696679\pi$$
−0.579313 + 0.815105i $$0.696679\pi$$
$$912$$ 8.48528 0.280976
$$913$$ −30.6274 −1.01362
$$914$$ 43.4558 1.43739
$$915$$ −2.20101 −0.0727632
$$916$$ −31.5563 −1.04265
$$917$$ 0 0
$$918$$ −15.0711 −0.497419
$$919$$ 48.2843 1.59275 0.796376 0.604802i $$-0.206748\pi$$
0.796376 + 0.604802i $$0.206748\pi$$
$$920$$ 9.45584 0.311750
$$921$$ 1.85786 0.0612187
$$922$$ −46.8701 −1.54358
$$923$$ 72.0833 2.37265
$$924$$ 0 0
$$925$$ 18.6274 0.612466
$$926$$ −44.9706 −1.47782
$$927$$ 12.4853 0.410070
$$928$$ −1.85786 −0.0609874
$$929$$ −3.21320 −0.105422 −0.0527109 0.998610i $$-0.516786\pi$$
−0.0527109 + 0.998610i $$0.516786\pi$$
$$930$$ −9.65685 −0.316661
$$931$$ 0 0
$$932$$ 84.7696 2.77672
$$933$$ −22.1421 −0.724901
$$934$$ 96.0833 3.14394
$$935$$ −7.31371 −0.239184
$$936$$ 23.8995 0.781179
$$937$$ −33.4142 −1.09159 −0.545797 0.837917i $$-0.683773\pi$$
−0.545797 + 0.837917i $$0.683773\pi$$
$$938$$ 0 0
$$939$$ −17.8995 −0.584128
$$940$$ 6.34315 0.206891
$$941$$ 7.21320 0.235144 0.117572 0.993064i $$-0.462489\pi$$
0.117572 + 0.993064i $$0.462489\pi$$
$$942$$ −48.8701 −1.59227
$$943$$ 8.20101 0.267062
$$944$$ 20.4853 0.666739
$$945$$ 0 0
$$946$$ −27.3137 −0.888045
$$947$$ 53.3137 1.73246 0.866231 0.499643i $$-0.166535\pi$$
0.866231 + 0.499643i $$0.166535\pi$$
$$948$$ −8.97056 −0.291350
$$949$$ −31.9411 −1.03685
$$950$$ −31.7990 −1.03170
$$951$$ −10.0000 −0.324272
$$952$$ 0 0
$$953$$ 2.00000 0.0647864 0.0323932 0.999475i $$-0.489687\pi$$
0.0323932 + 0.999475i $$0.489687\pi$$
$$954$$ 4.82843 0.156326
$$955$$ 10.5442 0.341201
$$956$$ −16.6274 −0.537769
$$957$$ −2.34315 −0.0757431
$$958$$ 72.7696 2.35108
$$959$$ 0 0
$$960$$ −5.75736 −0.185818
$$961$$ 15.6274 0.504110
$$962$$ −52.2843 −1.68571
$$963$$ −11.6569 −0.375637
$$964$$ 29.6985 0.956524
$$965$$ 10.1421 0.326487
$$966$$ 0 0
$$967$$ 22.3431 0.718507 0.359254 0.933240i $$-0.383031\pi$$
0.359254 + 0.933240i $$0.383031\pi$$
$$968$$ 30.8995 0.993147
$$969$$ −17.6569 −0.567220
$$970$$ −7.65685 −0.245847
$$971$$ 5.37258 0.172414 0.0862072 0.996277i $$-0.472525\pi$$
0.0862072 + 0.996277i $$0.472525\pi$$
$$972$$ −3.82843 −0.122797
$$973$$ 0 0
$$974$$ 44.9706 1.44095
$$975$$ −25.2132 −0.807469
$$976$$ −11.2721 −0.360810
$$977$$ −26.8284 −0.858317 −0.429159 0.903229i $$-0.641190\pi$$
−0.429159 + 0.903229i $$0.641190\pi$$
$$978$$ 27.3137 0.873396
$$979$$ −11.5147 −0.368012
$$980$$ 0 0
$$981$$ 5.65685 0.180609
$$982$$ −94.0833 −3.00232
$$983$$ −37.2548 −1.18824 −0.594122 0.804375i $$-0.702501\pi$$
−0.594122 + 0.804375i $$0.702501\pi$$
$$984$$ 9.89949 0.315584
$$985$$ −1.17157 −0.0373294
$$986$$ −17.6569 −0.562309
$$987$$ 0 0
$$988$$ 58.6274 1.86519
$$989$$ −20.6863 −0.657786
$$990$$ −2.82843 −0.0898933
$$991$$ 20.9706 0.666152 0.333076 0.942900i $$-0.391913\pi$$
0.333076 + 0.942900i $$0.391913\pi$$
$$992$$ 10.8284 0.343803
$$993$$ 4.00000 0.126936
$$994$$ 0 0
$$995$$ 6.05887 0.192079
$$996$$ −58.6274 −1.85768
$$997$$ 10.3848 0.328889 0.164445 0.986386i $$-0.447417\pi$$
0.164445 + 0.986386i $$0.447417\pi$$
$$998$$ 46.6274 1.47597
$$999$$ 4.00000 0.126554
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 147.2.a.d.1.1 2
3.2 odd 2 441.2.a.j.1.2 2
4.3 odd 2 2352.2.a.be.1.2 2
5.4 even 2 3675.2.a.bf.1.2 2
7.2 even 3 147.2.e.e.67.2 4
7.3 odd 6 147.2.e.d.79.2 4
7.4 even 3 147.2.e.e.79.2 4
7.5 odd 6 147.2.e.d.67.2 4
7.6 odd 2 147.2.a.e.1.1 yes 2
8.3 odd 2 9408.2.a.dq.1.1 2
8.5 even 2 9408.2.a.ef.1.1 2
12.11 even 2 7056.2.a.cv.1.1 2
21.2 odd 6 441.2.e.f.361.1 4
21.5 even 6 441.2.e.g.361.1 4
21.11 odd 6 441.2.e.f.226.1 4
21.17 even 6 441.2.e.g.226.1 4
21.20 even 2 441.2.a.i.1.2 2
28.3 even 6 2352.2.q.bd.961.2 4
28.11 odd 6 2352.2.q.bb.961.1 4
28.19 even 6 2352.2.q.bd.1537.2 4
28.23 odd 6 2352.2.q.bb.1537.1 4
28.27 even 2 2352.2.a.bc.1.1 2
35.34 odd 2 3675.2.a.bd.1.2 2
56.13 odd 2 9408.2.a.di.1.2 2
56.27 even 2 9408.2.a.dt.1.2 2
84.83 odd 2 7056.2.a.cf.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.a.d.1.1 2 1.1 even 1 trivial
147.2.a.e.1.1 yes 2 7.6 odd 2
147.2.e.d.67.2 4 7.5 odd 6
147.2.e.d.79.2 4 7.3 odd 6
147.2.e.e.67.2 4 7.2 even 3
147.2.e.e.79.2 4 7.4 even 3
441.2.a.i.1.2 2 21.20 even 2
441.2.a.j.1.2 2 3.2 odd 2
441.2.e.f.226.1 4 21.11 odd 6
441.2.e.f.361.1 4 21.2 odd 6
441.2.e.g.226.1 4 21.17 even 6
441.2.e.g.361.1 4 21.5 even 6
2352.2.a.bc.1.1 2 28.27 even 2
2352.2.a.be.1.2 2 4.3 odd 2
2352.2.q.bb.961.1 4 28.11 odd 6
2352.2.q.bb.1537.1 4 28.23 odd 6
2352.2.q.bd.961.2 4 28.3 even 6
2352.2.q.bd.1537.2 4 28.19 even 6
3675.2.a.bd.1.2 2 35.34 odd 2
3675.2.a.bf.1.2 2 5.4 even 2
7056.2.a.cf.1.2 2 84.83 odd 2
7056.2.a.cv.1.1 2 12.11 even 2
9408.2.a.di.1.2 2 56.13 odd 2
9408.2.a.dq.1.1 2 8.3 odd 2
9408.2.a.dt.1.2 2 56.27 even 2
9408.2.a.ef.1.1 2 8.5 even 2