# Properties

 Label 3675.2.a.bf.1.2 Level $3675$ Weight $2$ Character 3675.1 Self dual yes Analytic conductor $29.345$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3675 = 3 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3675.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.3450227428$$ Analytic rank: $$0$$ 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: no (minimal twist has level 147) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 3675.1

## $q$-expansion

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

## 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 $$-0.174443\pi$$
0.853553 + 0.521005i $$0.174443\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 3.82843 1.91421
$$5$$ 0 0
$$6$$ 2.41421 0.985599
$$7$$ 0 0
$$8$$ 4.41421 1.56066
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 3.82843 1.10517
$$13$$ 5.41421 1.50163 0.750816 0.660511i $$-0.229660\pi$$
0.750816 + 0.660511i $$0.229660\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ 6.24264 1.51406 0.757031 0.653379i $$-0.226649\pi$$
0.757031 + 0.653379i $$0.226649\pi$$
$$18$$ 2.41421 0.569036
$$19$$ −2.82843 −0.648886 −0.324443 0.945905i $$-0.605177\pi$$
−0.324443 + 0.945905i $$0.605177\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −4.82843 −1.02942
$$23$$ −3.65685 −0.762507 −0.381253 0.924471i $$-0.624507\pi$$
−0.381253 + 0.924471i $$0.624507\pi$$
$$24$$ 4.41421 0.901048
$$25$$ 0 0
$$26$$ 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 $$0.534694\pi$$
$$30$$ 0 0
$$31$$ 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 $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ −6.82843 −1.10772
$$39$$ 5.41421 0.866968
$$40$$ 0 0
$$41$$ 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 $$-0.358044\pi$$
0.431331 + 0.902194i $$0.358044\pi$$
$$44$$ −7.65685 −1.15431
$$45$$ 0 0
$$46$$ −8.82843 −1.30168
$$47$$ 2.82843 0.412568 0.206284 0.978492i $$-0.433863\pi$$
0.206284 + 0.978492i $$0.433863\pi$$
$$48$$ 3.00000 0.433013
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 6.24264 0.874145
$$52$$ 20.7279 2.87445
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 2.41421 0.328533
$$55$$ 0 0
$$56$$ 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 $$0.353384\pi$$
$$60$$ 0 0
$$61$$ −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
$$65$$ 0 0
$$66$$ −4.82843 −0.594338
$$67$$ −5.65685 −0.691095 −0.345547 0.938401i $$-0.612307\pi$$
−0.345547 + 0.938401i $$0.612307\pi$$
$$68$$ 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 $$-0.612203\pi$$
−0.345242 + 0.938514i $$0.612203\pi$$
$$74$$ 9.65685 1.12259
$$75$$ 0 0
$$76$$ −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 $$0.457920\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 5.41421 0.597900
$$83$$ −15.3137 −1.68090 −0.840449 0.541891i $$-0.817709\pi$$
−0.840449 + 0.541891i $$0.817709\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 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 $$0.401297\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −14.0000 −1.45960
$$93$$ 6.82843 0.708075
$$94$$ 6.82843 0.704298
$$95$$ 0 0
$$96$$ −1.58579 −0.161849
$$97$$ 5.41421 0.549730 0.274865 0.961483i $$-0.411367\pi$$
0.274865 + 0.961483i $$0.411367\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ −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 $$-0.710887\pi$$
−0.615106 + 0.788445i $$0.710887\pi$$
$$104$$ 23.8995 2.34354
$$105$$ 0 0
$$106$$ 4.82843 0.468978
$$107$$ 11.6569 1.12691 0.563455 0.826147i $$-0.309472\pi$$
0.563455 + 0.826147i $$0.309472\pi$$
$$108$$ 3.82843 0.368391
$$109$$ 5.65685 0.541828 0.270914 0.962604i $$-0.412674\pi$$
0.270914 + 0.962604i $$0.412674\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ −17.3137 −1.62874 −0.814368 0.580348i $$-0.802916\pi$$
−0.814368 + 0.580348i $$0.802916\pi$$
$$114$$ −6.82843 −0.639541
$$115$$ 0 0
$$116$$ −4.48528 −0.416448
$$117$$ 5.41421 0.500544
$$118$$ 16.4853 1.51759
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −9.07107 −0.821256
$$123$$ 2.24264 0.202212
$$124$$ 26.1421 2.34763
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −9.65685 −0.856907 −0.428454 0.903564i $$-0.640941\pi$$
−0.428454 + 0.903564i $$0.640941\pi$$
$$128$$ −20.5563 −1.81694
$$129$$ 5.65685 0.498058
$$130$$ 0 0
$$131$$ −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 0
$$136$$ 27.5563 2.36294
$$137$$ 14.1421 1.20824 0.604122 0.796892i $$-0.293524\pi$$
0.604122 + 0.796892i $$0.293524\pi$$
$$138$$ −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 0
$$146$$ −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 $$0.569842\pi$$
$$150$$ 0 0
$$151$$ 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
$$155$$ 0 0
$$156$$ 20.7279 1.65956
$$157$$ 20.2426 1.61554 0.807769 0.589499i $$-0.200675\pi$$
0.807769 + 0.589499i $$0.200675\pi$$
$$158$$ 5.65685 0.450035
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 2.41421 0.189679
$$163$$ −11.3137 −0.886158 −0.443079 0.896483i $$-0.646114\pi$$
−0.443079 + 0.896483i $$0.646114\pi$$
$$164$$ 8.58579 0.670437
$$165$$ 0 0
$$166$$ −36.9706 −2.86947
$$167$$ 19.7990 1.53209 0.766046 0.642786i $$-0.222221\pi$$
0.766046 + 0.642786i $$0.222221\pi$$
$$168$$ 0 0
$$169$$ 16.3137 1.25490
$$170$$ 0 0
$$171$$ −2.82843 −0.216295
$$172$$ 21.6569 1.65132
$$173$$ 6.92893 0.526797 0.263398 0.964687i $$-0.415157\pi$$
0.263398 + 0.964687i $$0.415157\pi$$
$$174$$ −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 $$0.600931\pi$$
$$180$$ 0 0
$$181$$ 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
$$185$$ 0 0
$$186$$ 16.4853 1.20876
$$187$$ −12.4853 −0.913014
$$188$$ 10.8284 0.789744
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −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 $$-0.285859\pi$$
0.623134 + 0.782115i $$0.285859\pi$$
$$194$$ 13.0711 0.938448
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ −4.82843 −0.343141
$$199$$ −10.3431 −0.733206 −0.366603 0.930377i $$-0.619479\pi$$
−0.366603 + 0.930377i $$0.619479\pi$$
$$200$$ 0 0
$$201$$ −5.65685 −0.399004
$$202$$ −41.2132 −2.89975
$$203$$ 0 0
$$204$$ 23.8995 1.67330
$$205$$ 0 0
$$206$$ −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
$$215$$ 0 0
$$216$$ 4.41421 0.300349
$$217$$ 0 0
$$218$$ 13.6569 0.924959
$$219$$ −5.89949 −0.398651
$$220$$ 0 0
$$221$$ 33.7990 2.27357
$$222$$ 9.65685 0.648126
$$223$$ −8.97056 −0.600713 −0.300357 0.953827i $$-0.597106\pi$$
−0.300357 + 0.953827i $$0.597106\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −41.7990 −2.78043
$$227$$ −15.7990 −1.04862 −0.524308 0.851529i $$-0.675676\pi$$
−0.524308 + 0.851529i $$0.675676\pi$$
$$228$$ −10.8284 −0.717130
$$229$$ −8.24264 −0.544689 −0.272345 0.962200i $$-0.587799\pi$$
−0.272345 + 0.962200i $$0.587799\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −5.17157 −0.339530
$$233$$ −22.1421 −1.45058 −0.725290 0.688444i $$-0.758294\pi$$
−0.725290 + 0.688444i $$0.758294\pi$$
$$234$$ 13.0711 0.854482
$$235$$ 0 0
$$236$$ 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 $$0.544861\pi$$
$$240$$ 0 0
$$241$$ 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
$$250$$ 0 0
$$251$$ −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
$$255$$ 0 0
$$256$$ −29.9706 −1.87316
$$257$$ −19.2132 −1.19849 −0.599243 0.800567i $$-0.704532\pi$$
−0.599243 + 0.800567i $$0.704532\pi$$
$$258$$ 13.6569 0.850239
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −1.17157 −0.0725185
$$262$$ −17.6569 −1.09084
$$263$$ 17.3137 1.06761 0.533805 0.845608i $$-0.320762\pi$$
0.533805 + 0.845608i $$0.320762\pi$$
$$264$$ −8.82843 −0.543352
$$265$$ 0 0
$$266$$ 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 $$0.606053\pi$$
$$270$$ 0 0
$$271$$ −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
$$275$$ 0 0
$$276$$ −14.0000 −0.842701
$$277$$ −13.3137 −0.799943 −0.399972 0.916528i $$-0.630980\pi$$
−0.399972 + 0.916528i $$0.630980\pi$$
$$278$$ 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 $$-0.418846\pi$$
0.252199 + 0.967675i $$0.418846\pi$$
$$284$$ −50.9706 −3.02455
$$285$$ 0 0
$$286$$ −26.1421 −1.54582
$$287$$ 0 0
$$288$$ −1.58579 −0.0934434
$$289$$ 21.9706 1.29239
$$290$$ 0 0
$$291$$ 5.41421 0.317387
$$292$$ −22.5858 −1.32173
$$293$$ 19.4142 1.13419 0.567095 0.823652i $$-0.308067\pi$$
0.567095 + 0.823652i $$0.308067\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 17.6569 1.02628
$$297$$ −2.00000 −0.116052
$$298$$ −12.8284 −0.743131
$$299$$ −19.7990 −1.14501
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 28.9706 1.66707
$$303$$ −17.0711 −0.980707
$$304$$ −8.48528 −0.486664
$$305$$ 0 0
$$306$$ 15.0711 0.861556
$$307$$ 1.85786 0.106034 0.0530170 0.998594i $$-0.483116\pi$$
0.0530170 + 0.998594i $$0.483116\pi$$
$$308$$ 0 0
$$309$$ −12.4853 −0.710263
$$310$$ 0 0
$$311$$ 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 $$-0.668828\pi$$
−0.505870 + 0.862610i $$0.668828\pi$$
$$314$$ 48.8701 2.75790
$$315$$ 0 0
$$316$$ 8.97056 0.504634
$$317$$ −10.0000 −0.561656 −0.280828 0.959758i $$-0.590609\pi$$
−0.280828 + 0.959758i $$0.590609\pi$$
$$318$$ 4.82843 0.270765
$$319$$ 2.34315 0.131191
$$320$$ 0 0
$$321$$ 11.6569 0.650622
$$322$$ 0 0
$$323$$ −17.6569 −0.982454
$$324$$ 3.82843 0.212690
$$325$$ 0 0
$$326$$ −27.3137 −1.51277
$$327$$ 5.65685 0.312825
$$328$$ 9.89949 0.546608
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −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
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 18.3431 0.999215 0.499607 0.866252i $$-0.333478\pi$$
0.499607 + 0.866252i $$0.333478\pi$$
$$338$$ 39.3848 2.14225
$$339$$ −17.3137 −0.940352
$$340$$ 0 0
$$341$$ −13.6569 −0.739560
$$342$$ −6.82843 −0.369239
$$343$$ 0 0
$$344$$ 24.9706 1.34632
$$345$$ 0 0
$$346$$ 16.7279 0.899299
$$347$$ −10.6863 −0.573670 −0.286835 0.957980i $$-0.592603\pi$$
−0.286835 + 0.957980i $$0.592603\pi$$
$$348$$ −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 $$-0.407842\pi$$
0.285495 + 0.958380i $$0.407842\pi$$
$$354$$ 16.4853 0.876183
$$355$$ 0 0
$$356$$ 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 $$0.599530\pi$$
$$360$$ 0 0
$$361$$ −11.0000 −0.578947
$$362$$ 13.0711 0.687000
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −9.07107 −0.474152
$$367$$ 19.3137 1.00817 0.504084 0.863655i $$-0.331830\pi$$
0.504084 + 0.863655i $$0.331830\pi$$
$$368$$ −10.9706 −0.571880
$$369$$ 2.24264 0.116747
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 26.1421 1.35541
$$373$$ 33.3137 1.72492 0.862459 0.506127i $$-0.168923\pi$$
0.862459 + 0.506127i $$0.168923\pi$$
$$374$$ −30.1421 −1.55861
$$375$$ 0 0
$$376$$ 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 $$0.202573\pi$$
$$380$$ 0 0
$$381$$ −9.65685 −0.494736
$$382$$ −43.4558 −2.22339
$$383$$ −29.6569 −1.51539 −0.757697 0.652606i $$-0.773676\pi$$
−0.757697 + 0.652606i $$0.773676\pi$$
$$384$$ −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 $$0.417229\pi$$
$$390$$ 0 0
$$391$$ −22.8284 −1.15448
$$392$$ 0 0
$$393$$ −7.31371 −0.368928
$$394$$ −4.82843 −0.243253
$$395$$ 0 0
$$396$$ −7.65685 −0.384771
$$397$$ 34.3848 1.72572 0.862861 0.505441i $$-0.168670\pi$$
0.862861 + 0.505441i $$0.168670\pi$$
$$398$$ −24.9706 −1.25166
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 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 0
$$406$$ 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 $$0.348027\pi$$
$$410$$ 0 0
$$411$$ 14.1421 0.697580
$$412$$ −47.7990 −2.35489
$$413$$ 0 0
$$414$$ −8.82843 −0.433894
$$415$$ 0 0
$$416$$ −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
$$425$$ 0 0
$$426$$ −32.1421 −1.55729
$$427$$ 0 0
$$428$$ 44.6274 2.15715
$$429$$ −10.8284 −0.522801
$$430$$ 0 0
$$431$$ 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 $$-0.591167\pi$$
−0.282511 + 0.959264i $$0.591167\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 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 $$0.819044\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 81.5980 3.88122
$$443$$ −1.02944 −0.0489100 −0.0244550 0.999701i $$-0.507785\pi$$
−0.0244550 + 0.999701i $$0.507785\pi$$
$$444$$ 15.3137 0.726756
$$445$$ 0 0
$$446$$ −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 $$0.366037\pi$$
$$450$$ 0 0
$$451$$ −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 $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ −19.8995 −0.929842
$$459$$ 6.24264 0.291382
$$460$$ 0 0
$$461$$ 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 $$-0.642490\pi$$
−0.432845 + 0.901468i $$0.642490\pi$$
$$464$$ −3.51472 −0.163167
$$465$$ 0 0
$$466$$ −53.4558 −2.47629
$$467$$ 39.7990 1.84168 0.920839 0.389943i $$-0.127505\pi$$
0.920839 + 0.389943i $$0.127505\pi$$
$$468$$ 20.7279 0.958149
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 20.2426 0.932732
$$472$$ 30.1421 1.38740
$$473$$ −11.3137 −0.520205
$$474$$ 5.65685 0.259828
$$475$$ 0 0
$$476$$ 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 0
$$481$$ 21.6569 0.987468
$$482$$ 18.7279 0.853033
$$483$$ 0 0
$$484$$ −26.7990 −1.21814
$$485$$ 0 0
$$486$$ 2.41421 0.109511
$$487$$ 18.6274 0.844089 0.422044 0.906575i $$-0.361313\pi$$
0.422044 + 0.906575i $$0.361313\pi$$
$$488$$ −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
$$495$$ 0 0
$$496$$ 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 $$0.642298\pi$$
$$500$$ 0 0
$$501$$ 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
$$505$$ 0 0
$$506$$ 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 $$0.308342\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −31.2426 −1.38074
$$513$$ −2.82843 −0.124878
$$514$$ −46.3848 −2.04594
$$515$$ 0 0
$$516$$ 21.6569 0.953390
$$517$$ −5.65685 −0.248788
$$518$$ 0 0
$$519$$ 6.92893 0.304146
$$520$$ 0 0
$$521$$ 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 $$-0.601532\pi$$
−0.313591 + 0.949558i $$0.601532\pi$$
$$524$$ −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
$$530$$ 0 0
$$531$$ 6.82843 0.296328
$$532$$ 0 0
$$533$$ 12.1421 0.525934
$$534$$ 13.8995 0.601490
$$535$$ 0 0
$$536$$ −24.9706 −1.07856
$$537$$ −8.34315 −0.360033
$$538$$ −25.8995 −1.11661
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −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
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 3.02944 0.129529 0.0647647 0.997901i $$-0.479370\pi$$
0.0647647 + 0.997901i $$0.479370\pi$$
$$548$$ 54.1421 2.31284
$$549$$ −3.75736 −0.160360
$$550$$ 0 0
$$551$$ 3.31371 0.141169
$$552$$ −16.1421 −0.687055
$$553$$ 0 0
$$554$$ −32.1421 −1.36559
$$555$$ 0 0
$$556$$ 24.2843 1.02988
$$557$$ −26.0000 −1.10166 −0.550828 0.834619i $$-0.685688\pi$$
−0.550828 + 0.834619i $$0.685688\pi$$
$$558$$ 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 $$-0.545962\pi$$
−0.143892 + 0.989593i $$0.545962\pi$$
$$564$$ 10.8284 0.455959
$$565$$ 0 0
$$566$$ 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 $$0.496762\pi$$
$$570$$ 0 0
$$571$$ 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
$$575$$ 0 0
$$576$$ −9.82843 −0.409518
$$577$$ −14.1005 −0.587012 −0.293506 0.955957i $$-0.594822\pi$$
−0.293506 + 0.955957i $$0.594822\pi$$
$$578$$ 53.0416 2.20624
$$579$$ 17.3137 0.719533
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 13.0711 0.541813
$$583$$ −4.00000 −0.165663
$$584$$ −26.0416 −1.07761
$$585$$ 0 0
$$586$$ 46.8701 1.93618
$$587$$ −17.1716 −0.708747 −0.354373 0.935104i $$-0.615306\pi$$
−0.354373 + 0.935104i $$0.615306\pi$$
$$588$$ 0 0
$$589$$ −19.3137 −0.795807
$$590$$ 0 0
$$591$$ −2.00000 −0.0822690
$$592$$ 12.0000 0.493197
$$593$$ −21.0711 −0.865285 −0.432643 0.901566i $$-0.642419\pi$$
−0.432643 + 0.901566i $$0.642419\pi$$
$$594$$ −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 $$0.513009\pi$$
$$600$$ 0 0
$$601$$ 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
$$605$$ 0 0
$$606$$ −41.2132 −1.67417
$$607$$ −29.6569 −1.20373 −0.601867 0.798596i $$-0.705576\pi$$
−0.601867 + 0.798596i $$0.705576\pi$$
$$608$$ 4.48528 0.181902
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 15.3137 0.619526
$$612$$ 23.8995 0.966080
$$613$$ −27.3137 −1.10319 −0.551595 0.834112i $$-0.685981\pi$$
−0.551595 + 0.834112i $$0.685981\pi$$
$$614$$ 4.48528 0.181011
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 7.51472 0.302531 0.151266 0.988493i $$-0.451665\pi$$
0.151266 + 0.988493i $$0.451665\pi$$
$$618$$ −30.1421 −1.21249
$$619$$ 4.97056 0.199784 0.0998919 0.994998i $$-0.468150\pi$$
0.0998919 + 0.994998i $$0.468150\pi$$
$$620$$ 0 0
$$621$$ −3.65685 −0.146745
$$622$$ 53.4558 2.14338
$$623$$ 0 0
$$624$$ 16.2426 0.650226
$$625$$ 0 0
$$626$$ −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
$$635$$ 0 0
$$636$$ 7.65685 0.303614
$$637$$ 0 0
$$638$$ 5.65685 0.223957
$$639$$ −13.3137 −0.526682
$$640$$ 0 0
$$641$$ −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 $$-0.0339675\pi$$
0.994312 + 0.106510i $$0.0339675\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −42.6274 −1.67715
$$647$$ 21.1716 0.832340 0.416170 0.909287i $$-0.363372\pi$$
0.416170 + 0.909287i $$0.363372\pi$$
$$648$$ 4.41421 0.173407
$$649$$ −13.6569 −0.536078
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −43.3137 −1.69630
$$653$$ −19.5147 −0.763670 −0.381835 0.924231i $$-0.624708\pi$$
−0.381835 + 0.924231i $$0.624708\pi$$
$$654$$ 13.6569 0.534025
$$655$$ 0 0
$$656$$ 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 $$0.583497\pi$$
$$660$$ 0 0
$$661$$ −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
$$670$$ 0 0
$$671$$ 7.51472 0.290102
$$672$$ 0 0
$$673$$ −0.686292 −0.0264546 −0.0132273 0.999913i $$-0.504211\pi$$
−0.0132273 + 0.999913i $$0.504211\pi$$
$$674$$ 44.2843 1.70577
$$675$$ 0 0
$$676$$ 62.4558 2.40215
$$677$$ 28.5858 1.09864 0.549321 0.835612i $$-0.314887\pi$$
0.549321 + 0.835612i $$0.314887\pi$$
$$678$$ −41.7990 −1.60528
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −15.7990 −0.605419
$$682$$ −32.9706 −1.26251
$$683$$ 8.34315 0.319242 0.159621 0.987178i $$-0.448973\pi$$
0.159621 + 0.987178i $$0.448973\pi$$
$$684$$ −10.8284 −0.414035
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −8.24264 −0.314476
$$688$$ 16.9706 0.646997
$$689$$ 10.8284 0.412530
$$690$$ 0 0
$$691$$ 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
$$695$$ 0 0
$$696$$ −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
$$705$$ 0 0
$$706$$ 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 $$0.375616\pi$$
$$710$$ 0 0
$$711$$ 2.34315 0.0878748
$$712$$ 25.4142 0.952438
$$713$$ −24.9706 −0.935155
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −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 $$0.339285\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −26.5563 −0.988325
$$723$$ 7.75736 0.288499
$$724$$ 20.7279 0.770347
$$725$$ 0 0
$$726$$ −16.8995 −0.627199
$$727$$ −4.48528 −0.166350 −0.0831749 0.996535i $$-0.526506\pi$$
−0.0831749 + 0.996535i $$0.526506\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 35.3137 1.30612
$$732$$ −14.3848 −0.531677
$$733$$ −9.69848 −0.358222 −0.179111 0.983829i $$-0.557322\pi$$
−0.179111 + 0.983829i $$0.557322\pi$$
$$734$$ 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 $$0.332460\pi$$
$$740$$ 0 0
$$741$$ −15.3137 −0.562563
$$742$$ 0 0
$$743$$ 17.0294 0.624749 0.312375 0.949959i $$-0.398876\pi$$
0.312375 + 0.949959i $$0.398876\pi$$
$$744$$ 30.1421 1.10506
$$745$$ 0 0
$$746$$ 80.4264 2.94462
$$747$$ −15.3137 −0.560299
$$748$$ −47.7990 −1.74770
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 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
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −37.6569 −1.36866 −0.684331 0.729172i $$-0.739906\pi$$
−0.684331 + 0.729172i $$0.739906\pi$$
$$758$$ 75.5980 2.74584
$$759$$ 7.31371 0.265471
$$760$$ 0 0
$$761$$ −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
$$765$$ 0 0
$$766$$ −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 $$-0.373282\pi$$
0.387664 + 0.921801i $$0.373282\pi$$
$$774$$ 13.6569 0.490885
$$775$$ 0 0
$$776$$ 23.8995 0.857942
$$777$$ 0 0
$$778$$ 24.4853 0.877840
$$779$$ −6.34315 −0.227267
$$780$$ 0 0
$$781$$ 26.6274 0.952804
$$782$$ −55.1127 −1.97083
$$783$$ −1.17157 −0.0418686
$$784$$ 0 0
$$785$$ 0 0
$$786$$ −17.6569 −0.629799
$$787$$ 47.3137 1.68655 0.843276 0.537481i $$-0.180624\pi$$
0.843276 + 0.537481i $$0.180624\pi$$
$$788$$ −7.65685 −0.272764
$$789$$ 17.3137 0.616384
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −8.82843 −0.313704
$$793$$ −20.3431 −0.722406
$$794$$ 83.0122 2.94599
$$795$$ 0 0
$$796$$ −39.5980 −1.40351
$$797$$ 28.3848 1.00544 0.502720 0.864449i $$-0.332333\pi$$
0.502720 + 0.864449i $$0.332333\pi$$
$$798$$ 0 0
$$799$$ 17.6569 0.624655
$$800$$ 0 0
$$801$$ 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 $$0.819071\pi$$
$$810$$ 0 0
$$811$$ 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
$$815$$ 0 0
$$816$$ 18.7279 0.655608
$$817$$ −16.0000 −0.559769
$$818$$ 44.8701 1.56884
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −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 $$-0.356674\pi$$
0.435210 + 0.900329i $$0.356674\pi$$
$$824$$ −55.1127 −1.91994
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −36.3431 −1.26378 −0.631888 0.775060i $$-0.717720\pi$$
−0.631888 + 0.775060i $$0.717720\pi$$
$$828$$ −14.0000 −0.486534
$$829$$ −24.7279 −0.858836 −0.429418 0.903106i $$-0.641281\pi$$
−0.429418 + 0.903106i $$0.641281\pi$$
$$830$$ 0 0
$$831$$ −13.3137 −0.461847
$$832$$ −53.2132 −1.84484
$$833$$ 0 0
$$834$$ 15.3137 0.530270
$$835$$ 0 0
$$836$$ 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
$$845$$ 0 0
$$846$$ 6.82843 0.234766
$$847$$ 0 0
$$848$$ 6.00000 0.206041
$$849$$ 8.48528 0.291214
$$850$$ 0 0
$$851$$ −14.6274 −0.501421
$$852$$ −50.9706 −1.74622
$$853$$ 49.4975 1.69476 0.847381 0.530986i $$-0.178178\pi$$
0.847381 + 0.530986i $$0.178178\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 51.4558 1.75872
$$857$$ 12.5858 0.429922 0.214961 0.976623i $$-0.431038\pi$$
0.214961 + 0.976623i $$0.431038\pi$$
$$858$$ −26.1421 −0.892478
$$859$$ −6.54416 −0.223284 −0.111642 0.993749i $$-0.535611\pi$$
−0.111642 + 0.993749i $$0.535611\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 16.8284 0.573179
$$863$$ 5.31371 0.180881 0.0904404 0.995902i $$-0.471173\pi$$
0.0904404 + 0.995902i $$0.471173\pi$$
$$864$$ −1.58579 −0.0539496
$$865$$ 0 0
$$866$$ −28.3848 −0.964554
$$867$$ 21.9706 0.746159
$$868$$ 0 0
$$869$$ −4.68629 −0.158972
$$870$$ 0 0
$$871$$ −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 $$-0.561179\pi$$
−0.191018 + 0.981586i $$0.561179\pi$$
$$878$$ −85.2548 −2.87721
$$879$$ 19.4142 0.654825
$$880$$ 0 0
$$881$$ −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 $$-0.347996\pi$$
0.459590 + 0.888131i $$0.347996\pi$$
$$884$$ 129.397 4.35209
$$885$$ 0 0
$$886$$ −2.48528 −0.0834947
$$887$$ 2.82843 0.0949693 0.0474846 0.998872i $$-0.484879\pi$$
0.0474846 + 0.998872i $$0.484879\pi$$
$$888$$ 17.6569 0.592525
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ −34.3431 −1.14989
$$893$$ −8.00000 −0.267710
$$894$$ −12.8284 −0.429047
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −19.7990 −0.661069
$$898$$ 41.7990 1.39485
$$899$$ −8.00000 −0.266815
$$900$$ 0 0
$$901$$ 12.4853 0.415945
$$902$$ −10.8284 −0.360547
$$903$$ 0 0
$$904$$ −76.4264 −2.54190
$$905$$ 0 0
$$906$$ 28.9706 0.962482
$$907$$ 16.0000 0.531271 0.265636 0.964073i $$-0.414418\pi$$
0.265636 + 0.964073i $$0.414418\pi$$
$$908$$ −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
$$915$$ 0 0
$$916$$ −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 $$0.206748\pi$$
$$920$$ 0 0
$$921$$ 1.85786 0.0612187
$$922$$ 46.8701 1.54358
$$923$$ −72.0833 −2.37265
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −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 $$0.516786\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −84.7696 −2.77672
$$933$$ 22.1421 0.724901
$$934$$ 96.0833 3.14394
$$935$$ 0 0
$$936$$ 23.8995 0.781179
$$937$$ 33.4142 1.09159 0.545797 0.837917i $$-0.316227\pi$$
0.545797 + 0.837917i $$0.316227\pi$$
$$938$$ 0 0
$$939$$ −17.8995 −0.584128
$$940$$ 0 0
$$941$$ 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 $$-0.833465\pi$$
−0.866231 + 0.499643i $$0.833465\pi$$
$$948$$ 8.97056 0.291350
$$949$$ −31.9411 −1.03685
$$950$$ 0 0
$$951$$ −10.0000 −0.324272
$$952$$ 0 0
$$953$$ −2.00000 −0.0647864 −0.0323932 0.999475i $$-0.510313\pi$$
−0.0323932 + 0.999475i $$0.510313\pi$$
$$954$$ 4.82843 0.156326
$$955$$ 0 0
$$956$$ −16.6274 −0.537769
$$957$$ 2.34315 0.0757431
$$958$$ −72.7696 −2.35108
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 15.6274 0.504110
$$962$$ 52.2843 1.68571
$$963$$ 11.6569 0.375637
$$964$$ 29.6985 0.956524
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −22.3431 −0.718507 −0.359254 0.933240i $$-0.616969\pi$$
−0.359254 + 0.933240i $$0.616969\pi$$
$$968$$ −30.8995 −0.993147
$$969$$ −17.6569 −0.567220
$$970$$ 0 0
$$971$$ 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
$$975$$ 0 0
$$976$$ −11.2721 −0.360810
$$977$$ 26.8284 0.858317 0.429159 0.903229i $$-0.358810\pi$$
0.429159 + 0.903229i $$0.358810\pi$$
$$978$$ −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 $$-0.297499\pi$$
0.594122 + 0.804375i $$0.297499\pi$$
$$984$$ 9.89949 0.315584
$$985$$ 0 0
$$986$$ −17.6569 −0.562309
$$987$$ 0 0
$$988$$ −58.6274 −1.86519
$$989$$ −20.6863 −0.657786
$$990$$ 0 0
$$991$$ 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
$$995$$ 0 0
$$996$$ −58.6274 −1.85768
$$997$$ −10.3848 −0.328889 −0.164445 0.986386i $$-0.552583\pi$$
−0.164445 + 0.986386i $$0.552583\pi$$
$$998$$ −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 3675.2.a.bf.1.2 2
5.4 even 2 147.2.a.d.1.1 2
7.6 odd 2 3675.2.a.bd.1.2 2
15.14 odd 2 441.2.a.j.1.2 2
20.19 odd 2 2352.2.a.be.1.2 2
35.4 even 6 147.2.e.e.79.2 4
35.9 even 6 147.2.e.e.67.2 4
35.19 odd 6 147.2.e.d.67.2 4
35.24 odd 6 147.2.e.d.79.2 4
35.34 odd 2 147.2.a.e.1.1 yes 2
40.19 odd 2 9408.2.a.dq.1.1 2
40.29 even 2 9408.2.a.ef.1.1 2
60.59 even 2 7056.2.a.cv.1.1 2
105.44 odd 6 441.2.e.f.361.1 4
105.59 even 6 441.2.e.g.226.1 4
105.74 odd 6 441.2.e.f.226.1 4
105.89 even 6 441.2.e.g.361.1 4
105.104 even 2 441.2.a.i.1.2 2
140.19 even 6 2352.2.q.bd.1537.2 4
140.39 odd 6 2352.2.q.bb.961.1 4
140.59 even 6 2352.2.q.bd.961.2 4
140.79 odd 6 2352.2.q.bb.1537.1 4
140.139 even 2 2352.2.a.bc.1.1 2
280.69 odd 2 9408.2.a.di.1.2 2
280.139 even 2 9408.2.a.dt.1.2 2
420.419 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 5.4 even 2
147.2.a.e.1.1 yes 2 35.34 odd 2
147.2.e.d.67.2 4 35.19 odd 6
147.2.e.d.79.2 4 35.24 odd 6
147.2.e.e.67.2 4 35.9 even 6
147.2.e.e.79.2 4 35.4 even 6
441.2.a.i.1.2 2 105.104 even 2
441.2.a.j.1.2 2 15.14 odd 2
441.2.e.f.226.1 4 105.74 odd 6
441.2.e.f.361.1 4 105.44 odd 6
441.2.e.g.226.1 4 105.59 even 6
441.2.e.g.361.1 4 105.89 even 6
2352.2.a.bc.1.1 2 140.139 even 2
2352.2.a.be.1.2 2 20.19 odd 2
2352.2.q.bb.961.1 4 140.39 odd 6
2352.2.q.bb.1537.1 4 140.79 odd 6
2352.2.q.bd.961.2 4 140.59 even 6
2352.2.q.bd.1537.2 4 140.19 even 6
3675.2.a.bd.1.2 2 7.6 odd 2
3675.2.a.bf.1.2 2 1.1 even 1 trivial
7056.2.a.cf.1.2 2 420.419 odd 2
7056.2.a.cv.1.1 2 60.59 even 2
9408.2.a.di.1.2 2 280.69 odd 2
9408.2.a.dq.1.1 2 40.19 odd 2
9408.2.a.dt.1.2 2 280.139 even 2
9408.2.a.ef.1.1 2 40.29 even 2