# Properties

 Label 81.2.a.a.1.1 Level $81$ Weight $2$ Character 81.1 Self dual yes Analytic conductor $0.647$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 81.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.646788256372$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 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 $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 81.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.73205 q^{2} +1.00000 q^{4} +1.73205 q^{5} +2.00000 q^{7} +1.73205 q^{8} +O(q^{10})$$ $$q-1.73205 q^{2} +1.00000 q^{4} +1.73205 q^{5} +2.00000 q^{7} +1.73205 q^{8} -3.00000 q^{10} +3.46410 q^{11} -1.00000 q^{13} -3.46410 q^{14} -5.00000 q^{16} -5.19615 q^{17} +2.00000 q^{19} +1.73205 q^{20} -6.00000 q^{22} -3.46410 q^{23} -2.00000 q^{25} +1.73205 q^{26} +2.00000 q^{28} -1.73205 q^{29} +8.00000 q^{31} +5.19615 q^{32} +9.00000 q^{34} +3.46410 q^{35} -7.00000 q^{37} -3.46410 q^{38} +3.00000 q^{40} +6.92820 q^{41} +2.00000 q^{43} +3.46410 q^{44} +6.00000 q^{46} -6.92820 q^{47} -3.00000 q^{49} +3.46410 q^{50} -1.00000 q^{52} +6.00000 q^{55} +3.46410 q^{56} +3.00000 q^{58} -13.8564 q^{59} -7.00000 q^{61} -13.8564 q^{62} +1.00000 q^{64} -1.73205 q^{65} -10.0000 q^{67} -5.19615 q^{68} -6.00000 q^{70} +10.3923 q^{71} -7.00000 q^{73} +12.1244 q^{74} +2.00000 q^{76} +6.92820 q^{77} +2.00000 q^{79} -8.66025 q^{80} -12.0000 q^{82} +13.8564 q^{83} -9.00000 q^{85} -3.46410 q^{86} +6.00000 q^{88} +5.19615 q^{89} -2.00000 q^{91} -3.46410 q^{92} +12.0000 q^{94} +3.46410 q^{95} +2.00000 q^{97} +5.19615 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 4 q^{7}+O(q^{10})$$ 2 * q + 2 * q^4 + 4 * q^7 $$2 q + 2 q^{4} + 4 q^{7} - 6 q^{10} - 2 q^{13} - 10 q^{16} + 4 q^{19} - 12 q^{22} - 4 q^{25} + 4 q^{28} + 16 q^{31} + 18 q^{34} - 14 q^{37} + 6 q^{40} + 4 q^{43} + 12 q^{46} - 6 q^{49} - 2 q^{52} + 12 q^{55} + 6 q^{58} - 14 q^{61} + 2 q^{64} - 20 q^{67} - 12 q^{70} - 14 q^{73} + 4 q^{76} + 4 q^{79} - 24 q^{82} - 18 q^{85} + 12 q^{88} - 4 q^{91} + 24 q^{94} + 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^4 + 4 * q^7 - 6 * q^10 - 2 * q^13 - 10 * q^16 + 4 * q^19 - 12 * q^22 - 4 * q^25 + 4 * q^28 + 16 * q^31 + 18 * q^34 - 14 * q^37 + 6 * q^40 + 4 * q^43 + 12 * q^46 - 6 * q^49 - 2 * q^52 + 12 * q^55 + 6 * q^58 - 14 * q^61 + 2 * q^64 - 20 * q^67 - 12 * q^70 - 14 * q^73 + 4 * q^76 + 4 * q^79 - 24 * q^82 - 18 * q^85 + 12 * q^88 - 4 * q^91 + 24 * q^94 + 4 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.73205 −1.22474 −0.612372 0.790569i $$-0.709785\pi$$
−0.612372 + 0.790569i $$0.709785\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.73205 0.774597 0.387298 0.921954i $$-0.373408\pi$$
0.387298 + 0.921954i $$0.373408\pi$$
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 1.73205 0.612372
$$9$$ 0 0
$$10$$ −3.00000 −0.948683
$$11$$ 3.46410 1.04447 0.522233 0.852803i $$-0.325099\pi$$
0.522233 + 0.852803i $$0.325099\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ −3.46410 −0.925820
$$15$$ 0 0
$$16$$ −5.00000 −1.25000
$$17$$ −5.19615 −1.26025 −0.630126 0.776493i $$-0.716997\pi$$
−0.630126 + 0.776493i $$0.716997\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 1.73205 0.387298
$$21$$ 0 0
$$22$$ −6.00000 −1.27920
$$23$$ −3.46410 −0.722315 −0.361158 0.932505i $$-0.617618\pi$$
−0.361158 + 0.932505i $$0.617618\pi$$
$$24$$ 0 0
$$25$$ −2.00000 −0.400000
$$26$$ 1.73205 0.339683
$$27$$ 0 0
$$28$$ 2.00000 0.377964
$$29$$ −1.73205 −0.321634 −0.160817 0.986984i $$-0.551413\pi$$
−0.160817 + 0.986984i $$0.551413\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 5.19615 0.918559
$$33$$ 0 0
$$34$$ 9.00000 1.54349
$$35$$ 3.46410 0.585540
$$36$$ 0 0
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ −3.46410 −0.561951
$$39$$ 0 0
$$40$$ 3.00000 0.474342
$$41$$ 6.92820 1.08200 0.541002 0.841021i $$-0.318045\pi$$
0.541002 + 0.841021i $$0.318045\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 3.46410 0.522233
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ −6.92820 −1.01058 −0.505291 0.862949i $$-0.668615\pi$$
−0.505291 + 0.862949i $$0.668615\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 3.46410 0.489898
$$51$$ 0 0
$$52$$ −1.00000 −0.138675
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 6.00000 0.809040
$$56$$ 3.46410 0.462910
$$57$$ 0 0
$$58$$ 3.00000 0.393919
$$59$$ −13.8564 −1.80395 −0.901975 0.431788i $$-0.857883\pi$$
−0.901975 + 0.431788i $$0.857883\pi$$
$$60$$ 0 0
$$61$$ −7.00000 −0.896258 −0.448129 0.893969i $$-0.647910\pi$$
−0.448129 + 0.893969i $$0.647910\pi$$
$$62$$ −13.8564 −1.75977
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −1.73205 −0.214834
$$66$$ 0 0
$$67$$ −10.0000 −1.22169 −0.610847 0.791748i $$-0.709171\pi$$
−0.610847 + 0.791748i $$0.709171\pi$$
$$68$$ −5.19615 −0.630126
$$69$$ 0 0
$$70$$ −6.00000 −0.717137
$$71$$ 10.3923 1.23334 0.616670 0.787222i $$-0.288481\pi$$
0.616670 + 0.787222i $$0.288481\pi$$
$$72$$ 0 0
$$73$$ −7.00000 −0.819288 −0.409644 0.912245i $$-0.634347\pi$$
−0.409644 + 0.912245i $$0.634347\pi$$
$$74$$ 12.1244 1.40943
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 6.92820 0.789542
$$78$$ 0 0
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ −8.66025 −0.968246
$$81$$ 0 0
$$82$$ −12.0000 −1.32518
$$83$$ 13.8564 1.52094 0.760469 0.649374i $$-0.224969\pi$$
0.760469 + 0.649374i $$0.224969\pi$$
$$84$$ 0 0
$$85$$ −9.00000 −0.976187
$$86$$ −3.46410 −0.373544
$$87$$ 0 0
$$88$$ 6.00000 0.639602
$$89$$ 5.19615 0.550791 0.275396 0.961331i $$-0.411191\pi$$
0.275396 + 0.961331i $$0.411191\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ −3.46410 −0.361158
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 3.46410 0.355409
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 5.19615 0.524891
$$99$$ 0 0
$$100$$ −2.00000 −0.200000
$$101$$ −6.92820 −0.689382 −0.344691 0.938716i $$-0.612016\pi$$
−0.344691 + 0.938716i $$0.612016\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ −1.73205 −0.169842
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ −10.3923 −0.990867
$$111$$ 0 0
$$112$$ −10.0000 −0.944911
$$113$$ 1.73205 0.162938 0.0814688 0.996676i $$-0.474039\pi$$
0.0814688 + 0.996676i $$0.474039\pi$$
$$114$$ 0 0
$$115$$ −6.00000 −0.559503
$$116$$ −1.73205 −0.160817
$$117$$ 0 0
$$118$$ 24.0000 2.20938
$$119$$ −10.3923 −0.952661
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 12.1244 1.09769
$$123$$ 0 0
$$124$$ 8.00000 0.718421
$$125$$ −12.1244 −1.08444
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ −12.1244 −1.07165
$$129$$ 0 0
$$130$$ 3.00000 0.263117
$$131$$ −3.46410 −0.302660 −0.151330 0.988483i $$-0.548356\pi$$
−0.151330 + 0.988483i $$0.548356\pi$$
$$132$$ 0 0
$$133$$ 4.00000 0.346844
$$134$$ 17.3205 1.49626
$$135$$ 0 0
$$136$$ −9.00000 −0.771744
$$137$$ −1.73205 −0.147979 −0.0739895 0.997259i $$-0.523573\pi$$
−0.0739895 + 0.997259i $$0.523573\pi$$
$$138$$ 0 0
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 3.46410 0.292770
$$141$$ 0 0
$$142$$ −18.0000 −1.51053
$$143$$ −3.46410 −0.289683
$$144$$ 0 0
$$145$$ −3.00000 −0.249136
$$146$$ 12.1244 1.00342
$$147$$ 0 0
$$148$$ −7.00000 −0.575396
$$149$$ −8.66025 −0.709476 −0.354738 0.934966i $$-0.615430\pi$$
−0.354738 + 0.934966i $$0.615430\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 3.46410 0.280976
$$153$$ 0 0
$$154$$ −12.0000 −0.966988
$$155$$ 13.8564 1.11297
$$156$$ 0 0
$$157$$ 17.0000 1.35675 0.678374 0.734717i $$-0.262685\pi$$
0.678374 + 0.734717i $$0.262685\pi$$
$$158$$ −3.46410 −0.275589
$$159$$ 0 0
$$160$$ 9.00000 0.711512
$$161$$ −6.92820 −0.546019
$$162$$ 0 0
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 6.92820 0.541002
$$165$$ 0 0
$$166$$ −24.0000 −1.86276
$$167$$ 17.3205 1.34030 0.670151 0.742225i $$-0.266230\pi$$
0.670151 + 0.742225i $$0.266230\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 15.5885 1.19558
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ 19.0526 1.44854 0.724270 0.689517i $$-0.242177\pi$$
0.724270 + 0.689517i $$0.242177\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ −17.3205 −1.30558
$$177$$ 0 0
$$178$$ −9.00000 −0.674579
$$179$$ −20.7846 −1.55351 −0.776757 0.629800i $$-0.783137\pi$$
−0.776757 + 0.629800i $$0.783137\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 3.46410 0.256776
$$183$$ 0 0
$$184$$ −6.00000 −0.442326
$$185$$ −12.1244 −0.891400
$$186$$ 0 0
$$187$$ −18.0000 −1.31629
$$188$$ −6.92820 −0.505291
$$189$$ 0 0
$$190$$ −6.00000 −0.435286
$$191$$ −17.3205 −1.25327 −0.626634 0.779314i $$-0.715568\pi$$
−0.626634 + 0.779314i $$0.715568\pi$$
$$192$$ 0 0
$$193$$ −1.00000 −0.0719816 −0.0359908 0.999352i $$-0.511459\pi$$
−0.0359908 + 0.999352i $$0.511459\pi$$
$$194$$ −3.46410 −0.248708
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 5.19615 0.370211 0.185105 0.982719i $$-0.440737\pi$$
0.185105 + 0.982719i $$0.440737\pi$$
$$198$$ 0 0
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ −3.46410 −0.244949
$$201$$ 0 0
$$202$$ 12.0000 0.844317
$$203$$ −3.46410 −0.243132
$$204$$ 0 0
$$205$$ 12.0000 0.838116
$$206$$ −13.8564 −0.965422
$$207$$ 0 0
$$208$$ 5.00000 0.346688
$$209$$ 6.92820 0.479234
$$210$$ 0 0
$$211$$ −10.0000 −0.688428 −0.344214 0.938891i $$-0.611855\pi$$
−0.344214 + 0.938891i $$0.611855\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 3.46410 0.236250
$$216$$ 0 0
$$217$$ 16.0000 1.08615
$$218$$ −19.0526 −1.29040
$$219$$ 0 0
$$220$$ 6.00000 0.404520
$$221$$ 5.19615 0.349531
$$222$$ 0 0
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 10.3923 0.694365
$$225$$ 0 0
$$226$$ −3.00000 −0.199557
$$227$$ 3.46410 0.229920 0.114960 0.993370i $$-0.463326\pi$$
0.114960 + 0.993370i $$0.463326\pi$$
$$228$$ 0 0
$$229$$ −1.00000 −0.0660819 −0.0330409 0.999454i $$-0.510519\pi$$
−0.0330409 + 0.999454i $$0.510519\pi$$
$$230$$ 10.3923 0.685248
$$231$$ 0 0
$$232$$ −3.00000 −0.196960
$$233$$ 25.9808 1.70206 0.851028 0.525120i $$-0.175980\pi$$
0.851028 + 0.525120i $$0.175980\pi$$
$$234$$ 0 0
$$235$$ −12.0000 −0.782794
$$236$$ −13.8564 −0.901975
$$237$$ 0 0
$$238$$ 18.0000 1.16677
$$239$$ 27.7128 1.79259 0.896296 0.443455i $$-0.146248\pi$$
0.896296 + 0.443455i $$0.146248\pi$$
$$240$$ 0 0
$$241$$ 29.0000 1.86805 0.934027 0.357202i $$-0.116269\pi$$
0.934027 + 0.357202i $$0.116269\pi$$
$$242$$ −1.73205 −0.111340
$$243$$ 0 0
$$244$$ −7.00000 −0.448129
$$245$$ −5.19615 −0.331970
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 13.8564 0.879883
$$249$$ 0 0
$$250$$ 21.0000 1.32816
$$251$$ −10.3923 −0.655956 −0.327978 0.944685i $$-0.606367\pi$$
−0.327978 + 0.944685i $$0.606367\pi$$
$$252$$ 0 0
$$253$$ −12.0000 −0.754434
$$254$$ −3.46410 −0.217357
$$255$$ 0 0
$$256$$ 19.0000 1.18750
$$257$$ −8.66025 −0.540212 −0.270106 0.962831i $$-0.587059\pi$$
−0.270106 + 0.962831i $$0.587059\pi$$
$$258$$ 0 0
$$259$$ −14.0000 −0.869918
$$260$$ −1.73205 −0.107417
$$261$$ 0 0
$$262$$ 6.00000 0.370681
$$263$$ −6.92820 −0.427211 −0.213606 0.976920i $$-0.568521\pi$$
−0.213606 + 0.976920i $$0.568521\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −6.92820 −0.424795
$$267$$ 0 0
$$268$$ −10.0000 −0.610847
$$269$$ 15.5885 0.950445 0.475223 0.879866i $$-0.342368\pi$$
0.475223 + 0.879866i $$0.342368\pi$$
$$270$$ 0 0
$$271$$ 2.00000 0.121491 0.0607457 0.998153i $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ 25.9808 1.57532
$$273$$ 0 0
$$274$$ 3.00000 0.181237
$$275$$ −6.92820 −0.417786
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ −13.8564 −0.831052
$$279$$ 0 0
$$280$$ 6.00000 0.358569
$$281$$ −12.1244 −0.723278 −0.361639 0.932318i $$-0.617783\pi$$
−0.361639 + 0.932318i $$0.617783\pi$$
$$282$$ 0 0
$$283$$ −28.0000 −1.66443 −0.832214 0.554455i $$-0.812927\pi$$
−0.832214 + 0.554455i $$0.812927\pi$$
$$284$$ 10.3923 0.616670
$$285$$ 0 0
$$286$$ 6.00000 0.354787
$$287$$ 13.8564 0.817918
$$288$$ 0 0
$$289$$ 10.0000 0.588235
$$290$$ 5.19615 0.305129
$$291$$ 0 0
$$292$$ −7.00000 −0.409644
$$293$$ −19.0526 −1.11306 −0.556531 0.830827i $$-0.687868\pi$$
−0.556531 + 0.830827i $$0.687868\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ −12.1244 −0.704714
$$297$$ 0 0
$$298$$ 15.0000 0.868927
$$299$$ 3.46410 0.200334
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ −34.6410 −1.99337
$$303$$ 0 0
$$304$$ −10.0000 −0.573539
$$305$$ −12.1244 −0.694239
$$306$$ 0 0
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 6.92820 0.394771
$$309$$ 0 0
$$310$$ −24.0000 −1.36311
$$311$$ 6.92820 0.392862 0.196431 0.980518i $$-0.437065\pi$$
0.196431 + 0.980518i $$0.437065\pi$$
$$312$$ 0 0
$$313$$ −25.0000 −1.41308 −0.706542 0.707671i $$-0.749746\pi$$
−0.706542 + 0.707671i $$0.749746\pi$$
$$314$$ −29.4449 −1.66167
$$315$$ 0 0
$$316$$ 2.00000 0.112509
$$317$$ 8.66025 0.486408 0.243204 0.969975i $$-0.421801\pi$$
0.243204 + 0.969975i $$0.421801\pi$$
$$318$$ 0 0
$$319$$ −6.00000 −0.335936
$$320$$ 1.73205 0.0968246
$$321$$ 0 0
$$322$$ 12.0000 0.668734
$$323$$ −10.3923 −0.578243
$$324$$ 0 0
$$325$$ 2.00000 0.110940
$$326$$ 27.7128 1.53487
$$327$$ 0 0
$$328$$ 12.0000 0.662589
$$329$$ −13.8564 −0.763928
$$330$$ 0 0
$$331$$ 2.00000 0.109930 0.0549650 0.998488i $$-0.482495\pi$$
0.0549650 + 0.998488i $$0.482495\pi$$
$$332$$ 13.8564 0.760469
$$333$$ 0 0
$$334$$ −30.0000 −1.64153
$$335$$ −17.3205 −0.946320
$$336$$ 0 0
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ 20.7846 1.13053
$$339$$ 0 0
$$340$$ −9.00000 −0.488094
$$341$$ 27.7128 1.50073
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 3.46410 0.186772
$$345$$ 0 0
$$346$$ −33.0000 −1.77409
$$347$$ −3.46410 −0.185963 −0.0929814 0.995668i $$-0.529640\pi$$
−0.0929814 + 0.995668i $$0.529640\pi$$
$$348$$ 0 0
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 6.92820 0.370328
$$351$$ 0 0
$$352$$ 18.0000 0.959403
$$353$$ 13.8564 0.737502 0.368751 0.929528i $$-0.379785\pi$$
0.368751 + 0.929528i $$0.379785\pi$$
$$354$$ 0 0
$$355$$ 18.0000 0.955341
$$356$$ 5.19615 0.275396
$$357$$ 0 0
$$358$$ 36.0000 1.90266
$$359$$ −10.3923 −0.548485 −0.274242 0.961661i $$-0.588427\pi$$
−0.274242 + 0.961661i $$0.588427\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −3.46410 −0.182069
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ −12.1244 −0.634618
$$366$$ 0 0
$$367$$ 20.0000 1.04399 0.521996 0.852948i $$-0.325188\pi$$
0.521996 + 0.852948i $$0.325188\pi$$
$$368$$ 17.3205 0.902894
$$369$$ 0 0
$$370$$ 21.0000 1.09174
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ 31.1769 1.61212
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 1.73205 0.0892052
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 3.46410 0.177705
$$381$$ 0 0
$$382$$ 30.0000 1.53493
$$383$$ 17.3205 0.885037 0.442518 0.896759i $$-0.354085\pi$$
0.442518 + 0.896759i $$0.354085\pi$$
$$384$$ 0 0
$$385$$ 12.0000 0.611577
$$386$$ 1.73205 0.0881591
$$387$$ 0 0
$$388$$ 2.00000 0.101535
$$389$$ −27.7128 −1.40510 −0.702548 0.711637i $$-0.747954\pi$$
−0.702548 + 0.711637i $$0.747954\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ −5.19615 −0.262445
$$393$$ 0 0
$$394$$ −9.00000 −0.453413
$$395$$ 3.46410 0.174298
$$396$$ 0 0
$$397$$ 29.0000 1.45547 0.727734 0.685859i $$-0.240573\pi$$
0.727734 + 0.685859i $$0.240573\pi$$
$$398$$ −34.6410 −1.73640
$$399$$ 0 0
$$400$$ 10.0000 0.500000
$$401$$ 12.1244 0.605461 0.302731 0.953076i $$-0.402102\pi$$
0.302731 + 0.953076i $$0.402102\pi$$
$$402$$ 0 0
$$403$$ −8.00000 −0.398508
$$404$$ −6.92820 −0.344691
$$405$$ 0 0
$$406$$ 6.00000 0.297775
$$407$$ −24.2487 −1.20196
$$408$$ 0 0
$$409$$ −19.0000 −0.939490 −0.469745 0.882802i $$-0.655654\pi$$
−0.469745 + 0.882802i $$0.655654\pi$$
$$410$$ −20.7846 −1.02648
$$411$$ 0 0
$$412$$ 8.00000 0.394132
$$413$$ −27.7128 −1.36366
$$414$$ 0 0
$$415$$ 24.0000 1.17811
$$416$$ −5.19615 −0.254762
$$417$$ 0 0
$$418$$ −12.0000 −0.586939
$$419$$ 6.92820 0.338465 0.169232 0.985576i $$-0.445871\pi$$
0.169232 + 0.985576i $$0.445871\pi$$
$$420$$ 0 0
$$421$$ −25.0000 −1.21843 −0.609213 0.793007i $$-0.708514\pi$$
−0.609213 + 0.793007i $$0.708514\pi$$
$$422$$ 17.3205 0.843149
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 10.3923 0.504101
$$426$$ 0 0
$$427$$ −14.0000 −0.677507
$$428$$ 0 0
$$429$$ 0 0
$$430$$ −6.00000 −0.289346
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 11.0000 0.528626 0.264313 0.964437i $$-0.414855\pi$$
0.264313 + 0.964437i $$0.414855\pi$$
$$434$$ −27.7128 −1.33026
$$435$$ 0 0
$$436$$ 11.0000 0.526804
$$437$$ −6.92820 −0.331421
$$438$$ 0 0
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 10.3923 0.495434
$$441$$ 0 0
$$442$$ −9.00000 −0.428086
$$443$$ 34.6410 1.64584 0.822922 0.568154i $$-0.192342\pi$$
0.822922 + 0.568154i $$0.192342\pi$$
$$444$$ 0 0
$$445$$ 9.00000 0.426641
$$446$$ −3.46410 −0.164030
$$447$$ 0 0
$$448$$ 2.00000 0.0944911
$$449$$ −20.7846 −0.980886 −0.490443 0.871473i $$-0.663165\pi$$
−0.490443 + 0.871473i $$0.663165\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 1.73205 0.0814688
$$453$$ 0 0
$$454$$ −6.00000 −0.281594
$$455$$ −3.46410 −0.162400
$$456$$ 0 0
$$457$$ 29.0000 1.35656 0.678281 0.734802i $$-0.262725\pi$$
0.678281 + 0.734802i $$0.262725\pi$$
$$458$$ 1.73205 0.0809334
$$459$$ 0 0
$$460$$ −6.00000 −0.279751
$$461$$ 13.8564 0.645357 0.322679 0.946509i $$-0.395417\pi$$
0.322679 + 0.946509i $$0.395417\pi$$
$$462$$ 0 0
$$463$$ 8.00000 0.371792 0.185896 0.982569i $$-0.440481\pi$$
0.185896 + 0.982569i $$0.440481\pi$$
$$464$$ 8.66025 0.402042
$$465$$ 0 0
$$466$$ −45.0000 −2.08458
$$467$$ 20.7846 0.961797 0.480899 0.876776i $$-0.340311\pi$$
0.480899 + 0.876776i $$0.340311\pi$$
$$468$$ 0 0
$$469$$ −20.0000 −0.923514
$$470$$ 20.7846 0.958723
$$471$$ 0 0
$$472$$ −24.0000 −1.10469
$$473$$ 6.92820 0.318559
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ −10.3923 −0.476331
$$477$$ 0 0
$$478$$ −48.0000 −2.19547
$$479$$ 24.2487 1.10795 0.553976 0.832533i $$-0.313110\pi$$
0.553976 + 0.832533i $$0.313110\pi$$
$$480$$ 0 0
$$481$$ 7.00000 0.319173
$$482$$ −50.2295 −2.28789
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 3.46410 0.157297
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ −12.1244 −0.548844
$$489$$ 0 0
$$490$$ 9.00000 0.406579
$$491$$ 17.3205 0.781664 0.390832 0.920462i $$-0.372187\pi$$
0.390832 + 0.920462i $$0.372187\pi$$
$$492$$ 0 0
$$493$$ 9.00000 0.405340
$$494$$ 3.46410 0.155857
$$495$$ 0 0
$$496$$ −40.0000 −1.79605
$$497$$ 20.7846 0.932317
$$498$$ 0 0
$$499$$ −10.0000 −0.447661 −0.223831 0.974628i $$-0.571856\pi$$
−0.223831 + 0.974628i $$0.571856\pi$$
$$500$$ −12.1244 −0.542218
$$501$$ 0 0
$$502$$ 18.0000 0.803379
$$503$$ −20.7846 −0.926740 −0.463370 0.886165i $$-0.653360\pi$$
−0.463370 + 0.886165i $$0.653360\pi$$
$$504$$ 0 0
$$505$$ −12.0000 −0.533993
$$506$$ 20.7846 0.923989
$$507$$ 0 0
$$508$$ 2.00000 0.0887357
$$509$$ 27.7128 1.22835 0.614174 0.789170i $$-0.289489\pi$$
0.614174 + 0.789170i $$0.289489\pi$$
$$510$$ 0 0
$$511$$ −14.0000 −0.619324
$$512$$ −8.66025 −0.382733
$$513$$ 0 0
$$514$$ 15.0000 0.661622
$$515$$ 13.8564 0.610586
$$516$$ 0 0
$$517$$ −24.0000 −1.05552
$$518$$ 24.2487 1.06543
$$519$$ 0 0
$$520$$ −3.00000 −0.131559
$$521$$ 20.7846 0.910590 0.455295 0.890341i $$-0.349534\pi$$
0.455295 + 0.890341i $$0.349534\pi$$
$$522$$ 0 0
$$523$$ 38.0000 1.66162 0.830812 0.556553i $$-0.187876\pi$$
0.830812 + 0.556553i $$0.187876\pi$$
$$524$$ −3.46410 −0.151330
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ −41.5692 −1.81078
$$528$$ 0 0
$$529$$ −11.0000 −0.478261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 4.00000 0.173422
$$533$$ −6.92820 −0.300094
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −17.3205 −0.748132
$$537$$ 0 0
$$538$$ −27.0000 −1.16405
$$539$$ −10.3923 −0.447628
$$540$$ 0 0
$$541$$ 11.0000 0.472927 0.236463 0.971640i $$-0.424012\pi$$
0.236463 + 0.971640i $$0.424012\pi$$
$$542$$ −3.46410 −0.148796
$$543$$ 0 0
$$544$$ −27.0000 −1.15762
$$545$$ 19.0526 0.816122
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ −1.73205 −0.0739895
$$549$$ 0 0
$$550$$ 12.0000 0.511682
$$551$$ −3.46410 −0.147576
$$552$$ 0 0
$$553$$ 4.00000 0.170097
$$554$$ −3.46410 −0.147176
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ −36.3731 −1.54118 −0.770588 0.637333i $$-0.780037\pi$$
−0.770588 + 0.637333i $$0.780037\pi$$
$$558$$ 0 0
$$559$$ −2.00000 −0.0845910
$$560$$ −17.3205 −0.731925
$$561$$ 0 0
$$562$$ 21.0000 0.885832
$$563$$ −34.6410 −1.45994 −0.729972 0.683477i $$-0.760467\pi$$
−0.729972 + 0.683477i $$0.760467\pi$$
$$564$$ 0 0
$$565$$ 3.00000 0.126211
$$566$$ 48.4974 2.03850
$$567$$ 0 0
$$568$$ 18.0000 0.755263
$$569$$ −32.9090 −1.37962 −0.689808 0.723993i $$-0.742305\pi$$
−0.689808 + 0.723993i $$0.742305\pi$$
$$570$$ 0 0
$$571$$ 8.00000 0.334790 0.167395 0.985890i $$-0.446465\pi$$
0.167395 + 0.985890i $$0.446465\pi$$
$$572$$ −3.46410 −0.144841
$$573$$ 0 0
$$574$$ −24.0000 −1.00174
$$575$$ 6.92820 0.288926
$$576$$ 0 0
$$577$$ 11.0000 0.457936 0.228968 0.973434i $$-0.426465\pi$$
0.228968 + 0.973434i $$0.426465\pi$$
$$578$$ −17.3205 −0.720438
$$579$$ 0 0
$$580$$ −3.00000 −0.124568
$$581$$ 27.7128 1.14972
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −12.1244 −0.501709
$$585$$ 0 0
$$586$$ 33.0000 1.36322
$$587$$ −38.1051 −1.57277 −0.786383 0.617739i $$-0.788049\pi$$
−0.786383 + 0.617739i $$0.788049\pi$$
$$588$$ 0 0
$$589$$ 16.0000 0.659269
$$590$$ 41.5692 1.71138
$$591$$ 0 0
$$592$$ 35.0000 1.43849
$$593$$ −15.5885 −0.640141 −0.320071 0.947394i $$-0.603707\pi$$
−0.320071 + 0.947394i $$0.603707\pi$$
$$594$$ 0 0
$$595$$ −18.0000 −0.737928
$$596$$ −8.66025 −0.354738
$$597$$ 0 0
$$598$$ −6.00000 −0.245358
$$599$$ −13.8564 −0.566157 −0.283079 0.959097i $$-0.591356\pi$$
−0.283079 + 0.959097i $$0.591356\pi$$
$$600$$ 0 0
$$601$$ −25.0000 −1.01977 −0.509886 0.860242i $$-0.670312\pi$$
−0.509886 + 0.860242i $$0.670312\pi$$
$$602$$ −6.92820 −0.282372
$$603$$ 0 0
$$604$$ 20.0000 0.813788
$$605$$ 1.73205 0.0704179
$$606$$ 0 0
$$607$$ 26.0000 1.05531 0.527654 0.849460i $$-0.323072\pi$$
0.527654 + 0.849460i $$0.323072\pi$$
$$608$$ 10.3923 0.421464
$$609$$ 0 0
$$610$$ 21.0000 0.850265
$$611$$ 6.92820 0.280285
$$612$$ 0 0
$$613$$ −34.0000 −1.37325 −0.686624 0.727013i $$-0.740908\pi$$
−0.686624 + 0.727013i $$0.740908\pi$$
$$614$$ 27.7128 1.11840
$$615$$ 0 0
$$616$$ 12.0000 0.483494
$$617$$ 12.1244 0.488108 0.244054 0.969762i $$-0.421523\pi$$
0.244054 + 0.969762i $$0.421523\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 13.8564 0.556487
$$621$$ 0 0
$$622$$ −12.0000 −0.481156
$$623$$ 10.3923 0.416359
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ 43.3013 1.73067
$$627$$ 0 0
$$628$$ 17.0000 0.678374
$$629$$ 36.3731 1.45029
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ 3.46410 0.137795
$$633$$ 0 0
$$634$$ −15.0000 −0.595726
$$635$$ 3.46410 0.137469
$$636$$ 0 0
$$637$$ 3.00000 0.118864
$$638$$ 10.3923 0.411435
$$639$$ 0 0
$$640$$ −21.0000 −0.830098
$$641$$ −22.5167 −0.889355 −0.444677 0.895691i $$-0.646682\pi$$
−0.444677 + 0.895691i $$0.646682\pi$$
$$642$$ 0 0
$$643$$ 8.00000 0.315489 0.157745 0.987480i $$-0.449578\pi$$
0.157745 + 0.987480i $$0.449578\pi$$
$$644$$ −6.92820 −0.273009
$$645$$ 0 0
$$646$$ 18.0000 0.708201
$$647$$ 31.1769 1.22569 0.612845 0.790203i $$-0.290025\pi$$
0.612845 + 0.790203i $$0.290025\pi$$
$$648$$ 0 0
$$649$$ −48.0000 −1.88416
$$650$$ −3.46410 −0.135873
$$651$$ 0 0
$$652$$ −16.0000 −0.626608
$$653$$ −13.8564 −0.542243 −0.271122 0.962545i $$-0.587395\pi$$
−0.271122 + 0.962545i $$0.587395\pi$$
$$654$$ 0 0
$$655$$ −6.00000 −0.234439
$$656$$ −34.6410 −1.35250
$$657$$ 0 0
$$658$$ 24.0000 0.935617
$$659$$ 3.46410 0.134942 0.0674711 0.997721i $$-0.478507\pi$$
0.0674711 + 0.997721i $$0.478507\pi$$
$$660$$ 0 0
$$661$$ 17.0000 0.661223 0.330612 0.943767i $$-0.392745\pi$$
0.330612 + 0.943767i $$0.392745\pi$$
$$662$$ −3.46410 −0.134636
$$663$$ 0 0
$$664$$ 24.0000 0.931381
$$665$$ 6.92820 0.268664
$$666$$ 0 0
$$667$$ 6.00000 0.232321
$$668$$ 17.3205 0.670151
$$669$$ 0 0
$$670$$ 30.0000 1.15900
$$671$$ −24.2487 −0.936111
$$672$$ 0 0
$$673$$ −25.0000 −0.963679 −0.481840 0.876259i $$-0.660031\pi$$
−0.481840 + 0.876259i $$0.660031\pi$$
$$674$$ −45.0333 −1.73462
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ 13.8564 0.532545 0.266272 0.963898i $$-0.414208\pi$$
0.266272 + 0.963898i $$0.414208\pi$$
$$678$$ 0 0
$$679$$ 4.00000 0.153506
$$680$$ −15.5885 −0.597790
$$681$$ 0 0
$$682$$ −48.0000 −1.83801
$$683$$ 20.7846 0.795301 0.397650 0.917537i $$-0.369826\pi$$
0.397650 + 0.917537i $$0.369826\pi$$
$$684$$ 0 0
$$685$$ −3.00000 −0.114624
$$686$$ 34.6410 1.32260
$$687$$ 0 0
$$688$$ −10.0000 −0.381246
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 2.00000 0.0760836 0.0380418 0.999276i $$-0.487888\pi$$
0.0380418 + 0.999276i $$0.487888\pi$$
$$692$$ 19.0526 0.724270
$$693$$ 0 0
$$694$$ 6.00000 0.227757
$$695$$ 13.8564 0.525603
$$696$$ 0 0
$$697$$ −36.0000 −1.36360
$$698$$ −3.46410 −0.131118
$$699$$ 0 0
$$700$$ −4.00000 −0.151186
$$701$$ −46.7654 −1.76630 −0.883152 0.469087i $$-0.844583\pi$$
−0.883152 + 0.469087i $$0.844583\pi$$
$$702$$ 0 0
$$703$$ −14.0000 −0.528020
$$704$$ 3.46410 0.130558
$$705$$ 0 0
$$706$$ −24.0000 −0.903252
$$707$$ −13.8564 −0.521124
$$708$$ 0 0
$$709$$ −25.0000 −0.938895 −0.469447 0.882960i $$-0.655547\pi$$
−0.469447 + 0.882960i $$0.655547\pi$$
$$710$$ −31.1769 −1.17005
$$711$$ 0 0
$$712$$ 9.00000 0.337289
$$713$$ −27.7128 −1.03785
$$714$$ 0 0
$$715$$ −6.00000 −0.224387
$$716$$ −20.7846 −0.776757
$$717$$ 0 0
$$718$$ 18.0000 0.671754
$$719$$ 10.3923 0.387568 0.193784 0.981044i $$-0.437924\pi$$
0.193784 + 0.981044i $$0.437924\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 25.9808 0.966904
$$723$$ 0 0
$$724$$ 2.00000 0.0743294
$$725$$ 3.46410 0.128654
$$726$$ 0 0
$$727$$ −34.0000 −1.26099 −0.630495 0.776193i $$-0.717148\pi$$
−0.630495 + 0.776193i $$0.717148\pi$$
$$728$$ −3.46410 −0.128388
$$729$$ 0 0
$$730$$ 21.0000 0.777245
$$731$$ −10.3923 −0.384373
$$732$$ 0 0
$$733$$ −46.0000 −1.69905 −0.849524 0.527549i $$-0.823111\pi$$
−0.849524 + 0.527549i $$0.823111\pi$$
$$734$$ −34.6410 −1.27862
$$735$$ 0 0
$$736$$ −18.0000 −0.663489
$$737$$ −34.6410 −1.27602
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ −12.1244 −0.445700
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 6.92820 0.254171 0.127086 0.991892i $$-0.459438\pi$$
0.127086 + 0.991892i $$0.459438\pi$$
$$744$$ 0 0
$$745$$ −15.0000 −0.549557
$$746$$ 17.3205 0.634149
$$747$$ 0 0
$$748$$ −18.0000 −0.658145
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −10.0000 −0.364905 −0.182453 0.983215i $$-0.558404\pi$$
−0.182453 + 0.983215i $$0.558404\pi$$
$$752$$ 34.6410 1.26323
$$753$$ 0 0
$$754$$ −3.00000 −0.109254
$$755$$ 34.6410 1.26072
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 27.7128 1.00657
$$759$$ 0 0
$$760$$ 6.00000 0.217643
$$761$$ −29.4449 −1.06738 −0.533688 0.845682i $$-0.679194\pi$$
−0.533688 + 0.845682i $$0.679194\pi$$
$$762$$ 0 0
$$763$$ 22.0000 0.796453
$$764$$ −17.3205 −0.626634
$$765$$ 0 0
$$766$$ −30.0000 −1.08394
$$767$$ 13.8564 0.500326
$$768$$ 0 0
$$769$$ −1.00000 −0.0360609 −0.0180305 0.999837i $$-0.505740\pi$$
−0.0180305 + 0.999837i $$0.505740\pi$$
$$770$$ −20.7846 −0.749025
$$771$$ 0 0
$$772$$ −1.00000 −0.0359908
$$773$$ 25.9808 0.934463 0.467232 0.884135i $$-0.345251\pi$$
0.467232 + 0.884135i $$0.345251\pi$$
$$774$$ 0 0
$$775$$ −16.0000 −0.574737
$$776$$ 3.46410 0.124354
$$777$$ 0 0
$$778$$ 48.0000 1.72088
$$779$$ 13.8564 0.496457
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ −31.1769 −1.11488
$$783$$ 0 0
$$784$$ 15.0000 0.535714
$$785$$ 29.4449 1.05093
$$786$$ 0 0
$$787$$ 26.0000 0.926800 0.463400 0.886149i $$-0.346629\pi$$
0.463400 + 0.886149i $$0.346629\pi$$
$$788$$ 5.19615 0.185105
$$789$$ 0 0
$$790$$ −6.00000 −0.213470
$$791$$ 3.46410 0.123169
$$792$$ 0 0
$$793$$ 7.00000 0.248577
$$794$$ −50.2295 −1.78258
$$795$$ 0 0
$$796$$ 20.0000 0.708881
$$797$$ 53.6936 1.90192 0.950962 0.309308i $$-0.100097\pi$$
0.950962 + 0.309308i $$0.100097\pi$$
$$798$$ 0 0
$$799$$ 36.0000 1.27359
$$800$$ −10.3923 −0.367423
$$801$$ 0 0
$$802$$ −21.0000 −0.741536
$$803$$ −24.2487 −0.855718
$$804$$ 0 0
$$805$$ −12.0000 −0.422944
$$806$$ 13.8564 0.488071
$$807$$ 0 0
$$808$$ −12.0000 −0.422159
$$809$$ 46.7654 1.64418 0.822091 0.569355i $$-0.192807\pi$$
0.822091 + 0.569355i $$0.192807\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ −3.46410 −0.121566
$$813$$ 0 0
$$814$$ 42.0000 1.47210
$$815$$ −27.7128 −0.970737
$$816$$ 0 0
$$817$$ 4.00000 0.139942
$$818$$ 32.9090 1.15063
$$819$$ 0 0
$$820$$ 12.0000 0.419058
$$821$$ −12.1244 −0.423143 −0.211571 0.977363i $$-0.567858\pi$$
−0.211571 + 0.977363i $$0.567858\pi$$
$$822$$ 0 0
$$823$$ −28.0000 −0.976019 −0.488009 0.872838i $$-0.662277\pi$$
−0.488009 + 0.872838i $$0.662277\pi$$
$$824$$ 13.8564 0.482711
$$825$$ 0 0
$$826$$ 48.0000 1.67013
$$827$$ 10.3923 0.361376 0.180688 0.983540i $$-0.442168\pi$$
0.180688 + 0.983540i $$0.442168\pi$$
$$828$$ 0 0
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ −41.5692 −1.44289
$$831$$ 0 0
$$832$$ −1.00000 −0.0346688
$$833$$ 15.5885 0.540108
$$834$$ 0 0
$$835$$ 30.0000 1.03819
$$836$$ 6.92820 0.239617
$$837$$ 0 0
$$838$$ −12.0000 −0.414533
$$839$$ 45.0333 1.55472 0.777361 0.629054i $$-0.216558\pi$$
0.777361 + 0.629054i $$0.216558\pi$$
$$840$$ 0 0
$$841$$ −26.0000 −0.896552
$$842$$ 43.3013 1.49226
$$843$$ 0 0
$$844$$ −10.0000 −0.344214
$$845$$ −20.7846 −0.715012
$$846$$ 0 0
$$847$$ 2.00000 0.0687208
$$848$$ 0 0
$$849$$ 0 0
$$850$$ −18.0000 −0.617395
$$851$$ 24.2487 0.831235
$$852$$ 0 0
$$853$$ −34.0000 −1.16414 −0.582069 0.813139i $$-0.697757\pi$$
−0.582069 + 0.813139i $$0.697757\pi$$
$$854$$ 24.2487 0.829774
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −22.5167 −0.769154 −0.384577 0.923093i $$-0.625653\pi$$
−0.384577 + 0.923093i $$0.625653\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 3.46410 0.118125
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −31.1769 −1.06127 −0.530637 0.847599i $$-0.678047\pi$$
−0.530637 + 0.847599i $$0.678047\pi$$
$$864$$ 0 0
$$865$$ 33.0000 1.12203
$$866$$ −19.0526 −0.647432
$$867$$ 0 0
$$868$$ 16.0000 0.543075
$$869$$ 6.92820 0.235023
$$870$$ 0 0
$$871$$ 10.0000 0.338837
$$872$$ 19.0526 0.645201
$$873$$ 0 0
$$874$$ 12.0000 0.405906
$$875$$ −24.2487 −0.819756
$$876$$ 0 0
$$877$$ 53.0000 1.78968 0.894841 0.446384i $$-0.147289\pi$$
0.894841 + 0.446384i $$0.147289\pi$$
$$878$$ −34.6410 −1.16908
$$879$$ 0 0
$$880$$ −30.0000 −1.01130
$$881$$ −20.7846 −0.700251 −0.350126 0.936703i $$-0.613861\pi$$
−0.350126 + 0.936703i $$0.613861\pi$$
$$882$$ 0 0
$$883$$ 56.0000 1.88455 0.942275 0.334840i $$-0.108682\pi$$
0.942275 + 0.334840i $$0.108682\pi$$
$$884$$ 5.19615 0.174766
$$885$$ 0 0
$$886$$ −60.0000 −2.01574
$$887$$ −3.46410 −0.116313 −0.0581566 0.998307i $$-0.518522\pi$$
−0.0581566 + 0.998307i $$0.518522\pi$$
$$888$$ 0 0
$$889$$ 4.00000 0.134156
$$890$$ −15.5885 −0.522526
$$891$$ 0 0
$$892$$ 2.00000 0.0669650
$$893$$ −13.8564 −0.463687
$$894$$ 0 0
$$895$$ −36.0000 −1.20335
$$896$$ −24.2487 −0.810093
$$897$$ 0 0
$$898$$ 36.0000 1.20134
$$899$$ −13.8564 −0.462137
$$900$$ 0 0
$$901$$ 0 0
$$902$$ −41.5692 −1.38410
$$903$$ 0 0
$$904$$ 3.00000 0.0997785
$$905$$ 3.46410 0.115151
$$906$$ 0 0
$$907$$ −52.0000 −1.72663 −0.863316 0.504664i $$-0.831616\pi$$
−0.863316 + 0.504664i $$0.831616\pi$$
$$908$$ 3.46410 0.114960
$$909$$ 0 0
$$910$$ 6.00000 0.198898
$$911$$ 24.2487 0.803396 0.401698 0.915772i $$-0.368420\pi$$
0.401698 + 0.915772i $$0.368420\pi$$
$$912$$ 0 0
$$913$$ 48.0000 1.58857
$$914$$ −50.2295 −1.66144
$$915$$ 0 0
$$916$$ −1.00000 −0.0330409
$$917$$ −6.92820 −0.228789
$$918$$ 0 0
$$919$$ 2.00000 0.0659739 0.0329870 0.999456i $$-0.489498\pi$$
0.0329870 + 0.999456i $$0.489498\pi$$
$$920$$ −10.3923 −0.342624
$$921$$ 0 0
$$922$$ −24.0000 −0.790398
$$923$$ −10.3923 −0.342067
$$924$$ 0 0
$$925$$ 14.0000 0.460317
$$926$$ −13.8564 −0.455350
$$927$$ 0 0
$$928$$ −9.00000 −0.295439
$$929$$ 50.2295 1.64798 0.823988 0.566608i $$-0.191744\pi$$
0.823988 + 0.566608i $$0.191744\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 25.9808 0.851028
$$933$$ 0 0
$$934$$ −36.0000 −1.17796
$$935$$ −31.1769 −1.01959
$$936$$ 0 0
$$937$$ −25.0000 −0.816714 −0.408357 0.912822i $$-0.633898\pi$$
−0.408357 + 0.912822i $$0.633898\pi$$
$$938$$ 34.6410 1.13107
$$939$$ 0 0
$$940$$ −12.0000 −0.391397
$$941$$ −50.2295 −1.63743 −0.818717 0.574197i $$-0.805314\pi$$
−0.818717 + 0.574197i $$0.805314\pi$$
$$942$$ 0 0
$$943$$ −24.0000 −0.781548
$$944$$ 69.2820 2.25494
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ −17.3205 −0.562841 −0.281420 0.959585i $$-0.590806\pi$$
−0.281420 + 0.959585i $$0.590806\pi$$
$$948$$ 0 0
$$949$$ 7.00000 0.227230
$$950$$ 6.92820 0.224781
$$951$$ 0 0
$$952$$ −18.0000 −0.583383
$$953$$ 5.19615 0.168320 0.0841599 0.996452i $$-0.473179\pi$$
0.0841599 + 0.996452i $$0.473179\pi$$
$$954$$ 0 0
$$955$$ −30.0000 −0.970777
$$956$$ 27.7128 0.896296
$$957$$ 0 0
$$958$$ −42.0000 −1.35696
$$959$$ −3.46410 −0.111862
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −12.1244 −0.390905
$$963$$ 0 0
$$964$$ 29.0000 0.934027
$$965$$ −1.73205 −0.0557567
$$966$$ 0 0
$$967$$ −46.0000 −1.47926 −0.739630 0.673014i $$-0.765000\pi$$
−0.739630 + 0.673014i $$0.765000\pi$$
$$968$$ 1.73205 0.0556702
$$969$$ 0 0
$$970$$ −6.00000 −0.192648
$$971$$ −31.1769 −1.00051 −0.500257 0.865877i $$-0.666761\pi$$
−0.500257 + 0.865877i $$0.666761\pi$$
$$972$$ 0 0
$$973$$ 16.0000 0.512936
$$974$$ 27.7128 0.887976
$$975$$ 0 0
$$976$$ 35.0000 1.12032
$$977$$ 48.4974 1.55157 0.775785 0.630997i $$-0.217354\pi$$
0.775785 + 0.630997i $$0.217354\pi$$
$$978$$ 0 0
$$979$$ 18.0000 0.575282
$$980$$ −5.19615 −0.165985
$$981$$ 0 0
$$982$$ −30.0000 −0.957338
$$983$$ 34.6410 1.10488 0.552438 0.833554i $$-0.313697\pi$$
0.552438 + 0.833554i $$0.313697\pi$$
$$984$$ 0 0
$$985$$ 9.00000 0.286764
$$986$$ −15.5885 −0.496438
$$987$$ 0 0
$$988$$ −2.00000 −0.0636285
$$989$$ −6.92820 −0.220304
$$990$$ 0 0
$$991$$ −34.0000 −1.08005 −0.540023 0.841650i $$-0.681584\pi$$
−0.540023 + 0.841650i $$0.681584\pi$$
$$992$$ 41.5692 1.31982
$$993$$ 0 0
$$994$$ −36.0000 −1.14185
$$995$$ 34.6410 1.09819
$$996$$ 0 0
$$997$$ −7.00000 −0.221692 −0.110846 0.993838i $$-0.535356\pi$$
−0.110846 + 0.993838i $$0.535356\pi$$
$$998$$ 17.3205 0.548271
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 81.2.a.a.1.1 2
3.2 odd 2 inner 81.2.a.a.1.2 yes 2
4.3 odd 2 1296.2.a.o.1.2 2
5.2 odd 4 2025.2.b.k.649.2 4
5.3 odd 4 2025.2.b.k.649.3 4
5.4 even 2 2025.2.a.j.1.2 2
7.6 odd 2 3969.2.a.i.1.1 2
8.3 odd 2 5184.2.a.bq.1.1 2
8.5 even 2 5184.2.a.br.1.1 2
9.2 odd 6 81.2.c.b.28.1 4
9.4 even 3 81.2.c.b.55.2 4
9.5 odd 6 81.2.c.b.55.1 4
9.7 even 3 81.2.c.b.28.2 4
11.10 odd 2 9801.2.a.v.1.2 2
12.11 even 2 1296.2.a.o.1.1 2
15.2 even 4 2025.2.b.k.649.4 4
15.8 even 4 2025.2.b.k.649.1 4
15.14 odd 2 2025.2.a.j.1.1 2
21.20 even 2 3969.2.a.i.1.2 2
24.5 odd 2 5184.2.a.br.1.2 2
24.11 even 2 5184.2.a.bq.1.2 2
27.2 odd 18 729.2.e.o.568.1 12
27.4 even 9 729.2.e.o.406.1 12
27.5 odd 18 729.2.e.o.649.2 12
27.7 even 9 729.2.e.o.325.1 12
27.11 odd 18 729.2.e.o.82.2 12
27.13 even 9 729.2.e.o.163.2 12
27.14 odd 18 729.2.e.o.163.1 12
27.16 even 9 729.2.e.o.82.1 12
27.20 odd 18 729.2.e.o.325.2 12
27.22 even 9 729.2.e.o.649.1 12
27.23 odd 18 729.2.e.o.406.2 12
27.25 even 9 729.2.e.o.568.2 12
33.32 even 2 9801.2.a.v.1.1 2
36.7 odd 6 1296.2.i.s.433.1 4
36.11 even 6 1296.2.i.s.433.2 4
36.23 even 6 1296.2.i.s.865.2 4
36.31 odd 6 1296.2.i.s.865.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
81.2.a.a.1.1 2 1.1 even 1 trivial
81.2.a.a.1.2 yes 2 3.2 odd 2 inner
81.2.c.b.28.1 4 9.2 odd 6
81.2.c.b.28.2 4 9.7 even 3
81.2.c.b.55.1 4 9.5 odd 6
81.2.c.b.55.2 4 9.4 even 3
729.2.e.o.82.1 12 27.16 even 9
729.2.e.o.82.2 12 27.11 odd 18
729.2.e.o.163.1 12 27.14 odd 18
729.2.e.o.163.2 12 27.13 even 9
729.2.e.o.325.1 12 27.7 even 9
729.2.e.o.325.2 12 27.20 odd 18
729.2.e.o.406.1 12 27.4 even 9
729.2.e.o.406.2 12 27.23 odd 18
729.2.e.o.568.1 12 27.2 odd 18
729.2.e.o.568.2 12 27.25 even 9
729.2.e.o.649.1 12 27.22 even 9
729.2.e.o.649.2 12 27.5 odd 18
1296.2.a.o.1.1 2 12.11 even 2
1296.2.a.o.1.2 2 4.3 odd 2
1296.2.i.s.433.1 4 36.7 odd 6
1296.2.i.s.433.2 4 36.11 even 6
1296.2.i.s.865.1 4 36.31 odd 6
1296.2.i.s.865.2 4 36.23 even 6
2025.2.a.j.1.1 2 15.14 odd 2
2025.2.a.j.1.2 2 5.4 even 2
2025.2.b.k.649.1 4 15.8 even 4
2025.2.b.k.649.2 4 5.2 odd 4
2025.2.b.k.649.3 4 5.3 odd 4
2025.2.b.k.649.4 4 15.2 even 4
3969.2.a.i.1.1 2 7.6 odd 2
3969.2.a.i.1.2 2 21.20 even 2
5184.2.a.bq.1.1 2 8.3 odd 2
5184.2.a.bq.1.2 2 24.11 even 2
5184.2.a.br.1.1 2 8.5 even 2
5184.2.a.br.1.2 2 24.5 odd 2
9801.2.a.v.1.1 2 33.32 even 2
9801.2.a.v.1.2 2 11.10 odd 2