# Properties

 Label 8670.2.a.be.1.2 Level $8670$ Weight $2$ Character 8670.1 Self dual yes Analytic conductor $69.230$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8670,2,Mod(1,8670)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8670.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$69.2302985525$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 510) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.44949$$ of defining polynomial Character $$\chi$$ $$=$$ 8670.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +4.89898 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +4.89898 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} +6.89898 q^{13} -4.89898 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +4.89898 q^{21} +4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -6.89898 q^{26} +1.00000 q^{27} +4.89898 q^{28} -6.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -4.89898 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} +6.89898 q^{39} +1.00000 q^{40} +2.89898 q^{41} -4.89898 q^{42} +8.89898 q^{43} -1.00000 q^{45} -4.00000 q^{46} +9.79796 q^{47} +1.00000 q^{48} +17.0000 q^{49} -1.00000 q^{50} +6.89898 q^{52} -7.79796 q^{53} -1.00000 q^{54} -4.89898 q^{56} +4.00000 q^{57} +6.00000 q^{58} +4.89898 q^{59} -1.00000 q^{60} -11.7980 q^{61} +4.00000 q^{62} +4.89898 q^{63} +1.00000 q^{64} -6.89898 q^{65} +0.898979 q^{67} +4.00000 q^{69} +4.89898 q^{70} +8.89898 q^{71} -1.00000 q^{72} +10.8990 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -6.89898 q^{78} +5.79796 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.89898 q^{82} +13.7980 q^{83} +4.89898 q^{84} -8.89898 q^{86} -6.00000 q^{87} -7.79796 q^{89} +1.00000 q^{90} +33.7980 q^{91} +4.00000 q^{92} -4.00000 q^{93} -9.79796 q^{94} -4.00000 q^{95} -1.00000 q^{96} +12.6969 q^{97} -17.0000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 4 q^{13} - 2 q^{15} + 2 q^{16} - 2 q^{18} + 8 q^{19} - 2 q^{20} + 8 q^{23} - 2 q^{24} + 2 q^{25} - 4 q^{26} + 2 q^{27} - 12 q^{29} + 2 q^{30} - 8 q^{31} - 2 q^{32} + 2 q^{36} - 12 q^{37} - 8 q^{38} + 4 q^{39} + 2 q^{40} - 4 q^{41} + 8 q^{43} - 2 q^{45} - 8 q^{46} + 2 q^{48} + 34 q^{49} - 2 q^{50} + 4 q^{52} + 4 q^{53} - 2 q^{54} + 8 q^{57} + 12 q^{58} - 2 q^{60} - 4 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} + 8 q^{69} + 8 q^{71} - 2 q^{72} + 12 q^{73} + 12 q^{74} + 2 q^{75} + 8 q^{76} - 4 q^{78} - 8 q^{79} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} - 8 q^{86} - 12 q^{87} + 4 q^{89} + 2 q^{90} + 48 q^{91} + 8 q^{92} - 8 q^{93} - 8 q^{95} - 2 q^{96} - 4 q^{97} - 34 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 4 * q^13 - 2 * q^15 + 2 * q^16 - 2 * q^18 + 8 * q^19 - 2 * q^20 + 8 * q^23 - 2 * q^24 + 2 * q^25 - 4 * q^26 + 2 * q^27 - 12 * q^29 + 2 * q^30 - 8 * q^31 - 2 * q^32 + 2 * q^36 - 12 * q^37 - 8 * q^38 + 4 * q^39 + 2 * q^40 - 4 * q^41 + 8 * q^43 - 2 * q^45 - 8 * q^46 + 2 * q^48 + 34 * q^49 - 2 * q^50 + 4 * q^52 + 4 * q^53 - 2 * q^54 + 8 * q^57 + 12 * q^58 - 2 * q^60 - 4 * q^61 + 8 * q^62 + 2 * q^64 - 4 * q^65 - 8 * q^67 + 8 * q^69 + 8 * q^71 - 2 * q^72 + 12 * q^73 + 12 * q^74 + 2 * q^75 + 8 * q^76 - 4 * q^78 - 8 * q^79 - 2 * q^80 + 2 * q^81 + 4 * q^82 + 8 * q^83 - 8 * q^86 - 12 * q^87 + 4 * q^89 + 2 * q^90 + 48 * q^91 + 8 * q^92 - 8 * q^93 - 8 * q^95 - 2 * q^96 - 4 * q^97 - 34 * q^98

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ 4.89898 1.85164 0.925820 0.377964i $$-0.123376\pi$$
0.925820 + 0.377964i $$0.123376\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 6.89898 1.91343 0.956716 0.291022i $$-0.0939953\pi$$
0.956716 + 0.291022i $$0.0939953\pi$$
$$14$$ −4.89898 −1.30931
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ 0 0
$$18$$ −1.00000 −0.235702
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 4.89898 1.06904
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ −6.89898 −1.35300
$$27$$ 1.00000 0.192450
$$28$$ 4.89898 0.925820
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 1.00000 0.182574
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −4.89898 −0.828079
$$36$$ 1.00000 0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 6.89898 1.10472
$$40$$ 1.00000 0.158114
$$41$$ 2.89898 0.452745 0.226372 0.974041i $$-0.427313\pi$$
0.226372 + 0.974041i $$0.427313\pi$$
$$42$$ −4.89898 −0.755929
$$43$$ 8.89898 1.35708 0.678541 0.734563i $$-0.262613\pi$$
0.678541 + 0.734563i $$0.262613\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ −4.00000 −0.589768
$$47$$ 9.79796 1.42918 0.714590 0.699544i $$-0.246613\pi$$
0.714590 + 0.699544i $$0.246613\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 17.0000 2.42857
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 6.89898 0.956716
$$53$$ −7.79796 −1.07113 −0.535566 0.844493i $$-0.679902\pi$$
−0.535566 + 0.844493i $$0.679902\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −4.89898 −0.654654
$$57$$ 4.00000 0.529813
$$58$$ 6.00000 0.787839
$$59$$ 4.89898 0.637793 0.318896 0.947790i $$-0.396688\pi$$
0.318896 + 0.947790i $$0.396688\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ −11.7980 −1.51057 −0.755287 0.655394i $$-0.772502\pi$$
−0.755287 + 0.655394i $$0.772502\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 4.89898 0.617213
$$64$$ 1.00000 0.125000
$$65$$ −6.89898 −0.855713
$$66$$ 0 0
$$67$$ 0.898979 0.109828 0.0549139 0.998491i $$-0.482512\pi$$
0.0549139 + 0.998491i $$0.482512\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 4.89898 0.585540
$$71$$ 8.89898 1.05611 0.528057 0.849209i $$-0.322921\pi$$
0.528057 + 0.849209i $$0.322921\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 10.8990 1.27563 0.637815 0.770190i $$-0.279839\pi$$
0.637815 + 0.770190i $$0.279839\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 1.00000 0.115470
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ −6.89898 −0.781156
$$79$$ 5.79796 0.652321 0.326161 0.945314i $$-0.394245\pi$$
0.326161 + 0.945314i $$0.394245\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −2.89898 −0.320139
$$83$$ 13.7980 1.51452 0.757261 0.653112i $$-0.226537\pi$$
0.757261 + 0.653112i $$0.226537\pi$$
$$84$$ 4.89898 0.534522
$$85$$ 0 0
$$86$$ −8.89898 −0.959602
$$87$$ −6.00000 −0.643268
$$88$$ 0 0
$$89$$ −7.79796 −0.826582 −0.413291 0.910599i $$-0.635621\pi$$
−0.413291 + 0.910599i $$0.635621\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 33.7980 3.54299
$$92$$ 4.00000 0.417029
$$93$$ −4.00000 −0.414781
$$94$$ −9.79796 −1.01058
$$95$$ −4.00000 −0.410391
$$96$$ −1.00000 −0.102062
$$97$$ 12.6969 1.28918 0.644589 0.764529i $$-0.277028\pi$$
0.644589 + 0.764529i $$0.277028\pi$$
$$98$$ −17.0000 −1.71726
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −18.8990 −1.88052 −0.940259 0.340459i $$-0.889418\pi$$
−0.940259 + 0.340459i $$0.889418\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ −6.89898 −0.676501
$$105$$ −4.89898 −0.478091
$$106$$ 7.79796 0.757405
$$107$$ −5.79796 −0.560510 −0.280255 0.959926i $$-0.590419\pi$$
−0.280255 + 0.959926i $$0.590419\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −11.7980 −1.13004 −0.565020 0.825077i $$-0.691131\pi$$
−0.565020 + 0.825077i $$0.691131\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 4.89898 0.462910
$$113$$ −7.79796 −0.733570 −0.366785 0.930306i $$-0.619542\pi$$
−0.366785 + 0.930306i $$0.619542\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ −4.00000 −0.373002
$$116$$ −6.00000 −0.557086
$$117$$ 6.89898 0.637811
$$118$$ −4.89898 −0.450988
$$119$$ 0 0
$$120$$ 1.00000 0.0912871
$$121$$ −11.0000 −1.00000
$$122$$ 11.7980 1.06814
$$123$$ 2.89898 0.261392
$$124$$ −4.00000 −0.359211
$$125$$ −1.00000 −0.0894427
$$126$$ −4.89898 −0.436436
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 8.89898 0.783511
$$130$$ 6.89898 0.605081
$$131$$ −9.79796 −0.856052 −0.428026 0.903767i $$-0.640791\pi$$
−0.428026 + 0.903767i $$0.640791\pi$$
$$132$$ 0 0
$$133$$ 19.5959 1.69918
$$134$$ −0.898979 −0.0776600
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ 14.0000 1.19610 0.598050 0.801459i $$-0.295942\pi$$
0.598050 + 0.801459i $$0.295942\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ −13.7980 −1.17033 −0.585164 0.810915i $$-0.698970\pi$$
−0.585164 + 0.810915i $$0.698970\pi$$
$$140$$ −4.89898 −0.414039
$$141$$ 9.79796 0.825137
$$142$$ −8.89898 −0.746786
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 6.00000 0.498273
$$146$$ −10.8990 −0.902006
$$147$$ 17.0000 1.40214
$$148$$ −6.00000 −0.493197
$$149$$ −18.8990 −1.54826 −0.774132 0.633024i $$-0.781814\pi$$
−0.774132 + 0.633024i $$0.781814\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −9.79796 −0.797347 −0.398673 0.917093i $$-0.630529\pi$$
−0.398673 + 0.917093i $$0.630529\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ 6.89898 0.552360
$$157$$ −1.10102 −0.0878710 −0.0439355 0.999034i $$-0.513990\pi$$
−0.0439355 + 0.999034i $$0.513990\pi$$
$$158$$ −5.79796 −0.461261
$$159$$ −7.79796 −0.618418
$$160$$ 1.00000 0.0790569
$$161$$ 19.5959 1.54437
$$162$$ −1.00000 −0.0785674
$$163$$ 2.20204 0.172477 0.0862386 0.996275i $$-0.472515\pi$$
0.0862386 + 0.996275i $$0.472515\pi$$
$$164$$ 2.89898 0.226372
$$165$$ 0 0
$$166$$ −13.7980 −1.07093
$$167$$ −2.20204 −0.170399 −0.0851995 0.996364i $$-0.527153\pi$$
−0.0851995 + 0.996364i $$0.527153\pi$$
$$168$$ −4.89898 −0.377964
$$169$$ 34.5959 2.66122
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 8.89898 0.678541
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 4.89898 0.370328
$$176$$ 0 0
$$177$$ 4.89898 0.368230
$$178$$ 7.79796 0.584482
$$179$$ −4.89898 −0.366167 −0.183083 0.983097i $$-0.558608\pi$$
−0.183083 + 0.983097i $$0.558608\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 4.20204 0.312335 0.156168 0.987731i $$-0.450086\pi$$
0.156168 + 0.987731i $$0.450086\pi$$
$$182$$ −33.7980 −2.50527
$$183$$ −11.7980 −0.872130
$$184$$ −4.00000 −0.294884
$$185$$ 6.00000 0.441129
$$186$$ 4.00000 0.293294
$$187$$ 0 0
$$188$$ 9.79796 0.714590
$$189$$ 4.89898 0.356348
$$190$$ 4.00000 0.290191
$$191$$ −9.79796 −0.708955 −0.354478 0.935064i $$-0.615341\pi$$
−0.354478 + 0.935064i $$0.615341\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 1.10102 0.0792532 0.0396266 0.999215i $$-0.487383\pi$$
0.0396266 + 0.999215i $$0.487383\pi$$
$$194$$ −12.6969 −0.911587
$$195$$ −6.89898 −0.494046
$$196$$ 17.0000 1.21429
$$197$$ 13.5959 0.968669 0.484335 0.874883i $$-0.339062\pi$$
0.484335 + 0.874883i $$0.339062\pi$$
$$198$$ 0 0
$$199$$ −15.5959 −1.10557 −0.552783 0.833325i $$-0.686434\pi$$
−0.552783 + 0.833325i $$0.686434\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0.898979 0.0634091
$$202$$ 18.8990 1.32973
$$203$$ −29.3939 −2.06305
$$204$$ 0 0
$$205$$ −2.89898 −0.202474
$$206$$ −4.00000 −0.278693
$$207$$ 4.00000 0.278019
$$208$$ 6.89898 0.478358
$$209$$ 0 0
$$210$$ 4.89898 0.338062
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ −7.79796 −0.535566
$$213$$ 8.89898 0.609748
$$214$$ 5.79796 0.396340
$$215$$ −8.89898 −0.606905
$$216$$ −1.00000 −0.0680414
$$217$$ −19.5959 −1.33026
$$218$$ 11.7980 0.799059
$$219$$ 10.8990 0.736485
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 6.00000 0.402694
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ −4.89898 −0.327327
$$225$$ 1.00000 0.0666667
$$226$$ 7.79796 0.518713
$$227$$ 21.7980 1.44678 0.723391 0.690439i $$-0.242583\pi$$
0.723391 + 0.690439i $$0.242583\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −13.5959 −0.898444 −0.449222 0.893420i $$-0.648299\pi$$
−0.449222 + 0.893420i $$0.648299\pi$$
$$230$$ 4.00000 0.263752
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ −6.89898 −0.451000
$$235$$ −9.79796 −0.639148
$$236$$ 4.89898 0.318896
$$237$$ 5.79796 0.376618
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ −13.5959 −0.875790 −0.437895 0.899026i $$-0.644276\pi$$
−0.437895 + 0.899026i $$0.644276\pi$$
$$242$$ 11.0000 0.707107
$$243$$ 1.00000 0.0641500
$$244$$ −11.7980 −0.755287
$$245$$ −17.0000 −1.08609
$$246$$ −2.89898 −0.184832
$$247$$ 27.5959 1.75589
$$248$$ 4.00000 0.254000
$$249$$ 13.7980 0.874410
$$250$$ 1.00000 0.0632456
$$251$$ 4.89898 0.309221 0.154610 0.987976i $$-0.450588\pi$$
0.154610 + 0.987976i $$0.450588\pi$$
$$252$$ 4.89898 0.308607
$$253$$ 0 0
$$254$$ 12.0000 0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ −8.89898 −0.554026
$$259$$ −29.3939 −1.82645
$$260$$ −6.89898 −0.427857
$$261$$ −6.00000 −0.371391
$$262$$ 9.79796 0.605320
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ 0 0
$$265$$ 7.79796 0.479025
$$266$$ −19.5959 −1.20150
$$267$$ −7.79796 −0.477227
$$268$$ 0.898979 0.0549139
$$269$$ −9.59592 −0.585073 −0.292537 0.956254i $$-0.594499\pi$$
−0.292537 + 0.956254i $$0.594499\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ −1.79796 −0.109218 −0.0546091 0.998508i $$-0.517391\pi$$
−0.0546091 + 0.998508i $$0.517391\pi$$
$$272$$ 0 0
$$273$$ 33.7980 2.04555
$$274$$ −14.0000 −0.845771
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ −7.79796 −0.468534 −0.234267 0.972172i $$-0.575269\pi$$
−0.234267 + 0.972172i $$0.575269\pi$$
$$278$$ 13.7980 0.827547
$$279$$ −4.00000 −0.239474
$$280$$ 4.89898 0.292770
$$281$$ 27.7980 1.65829 0.829144 0.559036i $$-0.188829\pi$$
0.829144 + 0.559036i $$0.188829\pi$$
$$282$$ −9.79796 −0.583460
$$283$$ −23.5959 −1.40263 −0.701316 0.712851i $$-0.747404\pi$$
−0.701316 + 0.712851i $$0.747404\pi$$
$$284$$ 8.89898 0.528057
$$285$$ −4.00000 −0.236940
$$286$$ 0 0
$$287$$ 14.2020 0.838320
$$288$$ −1.00000 −0.0589256
$$289$$ 0 0
$$290$$ −6.00000 −0.352332
$$291$$ 12.6969 0.744308
$$292$$ 10.8990 0.637815
$$293$$ 13.5959 0.794282 0.397141 0.917758i $$-0.370002\pi$$
0.397141 + 0.917758i $$0.370002\pi$$
$$294$$ −17.0000 −0.991460
$$295$$ −4.89898 −0.285230
$$296$$ 6.00000 0.348743
$$297$$ 0 0
$$298$$ 18.8990 1.09479
$$299$$ 27.5959 1.59591
$$300$$ 1.00000 0.0577350
$$301$$ 43.5959 2.51283
$$302$$ 9.79796 0.563809
$$303$$ −18.8990 −1.08572
$$304$$ 4.00000 0.229416
$$305$$ 11.7980 0.675549
$$306$$ 0 0
$$307$$ 0.898979 0.0513075 0.0256537 0.999671i $$-0.491833\pi$$
0.0256537 + 0.999671i $$0.491833\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ −4.00000 −0.227185
$$311$$ 7.10102 0.402662 0.201331 0.979523i $$-0.435473\pi$$
0.201331 + 0.979523i $$0.435473\pi$$
$$312$$ −6.89898 −0.390578
$$313$$ −6.89898 −0.389953 −0.194977 0.980808i $$-0.562463\pi$$
−0.194977 + 0.980808i $$0.562463\pi$$
$$314$$ 1.10102 0.0621342
$$315$$ −4.89898 −0.276026
$$316$$ 5.79796 0.326161
$$317$$ −14.0000 −0.786318 −0.393159 0.919470i $$-0.628618\pi$$
−0.393159 + 0.919470i $$0.628618\pi$$
$$318$$ 7.79796 0.437288
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ −5.79796 −0.323611
$$322$$ −19.5959 −1.09204
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 6.89898 0.382687
$$326$$ −2.20204 −0.121960
$$327$$ −11.7980 −0.652429
$$328$$ −2.89898 −0.160069
$$329$$ 48.0000 2.64633
$$330$$ 0 0
$$331$$ −5.79796 −0.318685 −0.159342 0.987223i $$-0.550937\pi$$
−0.159342 + 0.987223i $$0.550937\pi$$
$$332$$ 13.7980 0.757261
$$333$$ −6.00000 −0.328798
$$334$$ 2.20204 0.120490
$$335$$ −0.898979 −0.0491165
$$336$$ 4.89898 0.267261
$$337$$ −32.6969 −1.78112 −0.890558 0.454870i $$-0.849686\pi$$
−0.890558 + 0.454870i $$0.849686\pi$$
$$338$$ −34.5959 −1.88177
$$339$$ −7.79796 −0.423527
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −4.00000 −0.216295
$$343$$ 48.9898 2.64520
$$344$$ −8.89898 −0.479801
$$345$$ −4.00000 −0.215353
$$346$$ 6.00000 0.322562
$$347$$ −21.7980 −1.17018 −0.585088 0.810970i $$-0.698940\pi$$
−0.585088 + 0.810970i $$0.698940\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ 4.20204 0.224930 0.112465 0.993656i $$-0.464125\pi$$
0.112465 + 0.993656i $$0.464125\pi$$
$$350$$ −4.89898 −0.261861
$$351$$ 6.89898 0.368240
$$352$$ 0 0
$$353$$ −26.0000 −1.38384 −0.691920 0.721974i $$-0.743235\pi$$
−0.691920 + 0.721974i $$0.743235\pi$$
$$354$$ −4.89898 −0.260378
$$355$$ −8.89898 −0.472309
$$356$$ −7.79796 −0.413291
$$357$$ 0 0
$$358$$ 4.89898 0.258919
$$359$$ 37.3939 1.97357 0.986787 0.162025i $$-0.0518025\pi$$
0.986787 + 0.162025i $$0.0518025\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −3.00000 −0.157895
$$362$$ −4.20204 −0.220854
$$363$$ −11.0000 −0.577350
$$364$$ 33.7980 1.77149
$$365$$ −10.8990 −0.570479
$$366$$ 11.7980 0.616689
$$367$$ 27.1010 1.41466 0.707331 0.706883i $$-0.249899\pi$$
0.707331 + 0.706883i $$0.249899\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 2.89898 0.150915
$$370$$ −6.00000 −0.311925
$$371$$ −38.2020 −1.98335
$$372$$ −4.00000 −0.207390
$$373$$ 24.6969 1.27876 0.639380 0.768891i $$-0.279191\pi$$
0.639380 + 0.768891i $$0.279191\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ −9.79796 −0.505291
$$377$$ −41.3939 −2.13189
$$378$$ −4.89898 −0.251976
$$379$$ 29.7980 1.53062 0.765309 0.643663i $$-0.222586\pi$$
0.765309 + 0.643663i $$0.222586\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ −12.0000 −0.614779
$$382$$ 9.79796 0.501307
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −1.10102 −0.0560405
$$387$$ 8.89898 0.452361
$$388$$ 12.6969 0.644589
$$389$$ −38.4949 −1.95177 −0.975884 0.218288i $$-0.929953\pi$$
−0.975884 + 0.218288i $$0.929953\pi$$
$$390$$ 6.89898 0.349343
$$391$$ 0 0
$$392$$ −17.0000 −0.858630
$$393$$ −9.79796 −0.494242
$$394$$ −13.5959 −0.684952
$$395$$ −5.79796 −0.291727
$$396$$ 0 0
$$397$$ −1.59592 −0.0800968 −0.0400484 0.999198i $$-0.512751\pi$$
−0.0400484 + 0.999198i $$0.512751\pi$$
$$398$$ 15.5959 0.781753
$$399$$ 19.5959 0.981023
$$400$$ 1.00000 0.0500000
$$401$$ −22.8990 −1.14352 −0.571760 0.820421i $$-0.693739\pi$$
−0.571760 + 0.820421i $$0.693739\pi$$
$$402$$ −0.898979 −0.0448370
$$403$$ −27.5959 −1.37465
$$404$$ −18.8990 −0.940259
$$405$$ −1.00000 −0.0496904
$$406$$ 29.3939 1.45879
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −17.5959 −0.870062 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$410$$ 2.89898 0.143170
$$411$$ 14.0000 0.690569
$$412$$ 4.00000 0.197066
$$413$$ 24.0000 1.18096
$$414$$ −4.00000 −0.196589
$$415$$ −13.7980 −0.677315
$$416$$ −6.89898 −0.338250
$$417$$ −13.7980 −0.675689
$$418$$ 0 0
$$419$$ 17.7980 0.869487 0.434744 0.900554i $$-0.356839\pi$$
0.434744 + 0.900554i $$0.356839\pi$$
$$420$$ −4.89898 −0.239046
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ 9.79796 0.476393
$$424$$ 7.79796 0.378702
$$425$$ 0 0
$$426$$ −8.89898 −0.431157
$$427$$ −57.7980 −2.79704
$$428$$ −5.79796 −0.280255
$$429$$ 0 0
$$430$$ 8.89898 0.429147
$$431$$ 23.1010 1.11274 0.556369 0.830936i $$-0.312194\pi$$
0.556369 + 0.830936i $$0.312194\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −35.3939 −1.70092 −0.850461 0.526039i $$-0.823677\pi$$
−0.850461 + 0.526039i $$0.823677\pi$$
$$434$$ 19.5959 0.940634
$$435$$ 6.00000 0.287678
$$436$$ −11.7980 −0.565020
$$437$$ 16.0000 0.765384
$$438$$ −10.8990 −0.520773
$$439$$ −5.79796 −0.276721 −0.138361 0.990382i $$-0.544183\pi$$
−0.138361 + 0.990382i $$0.544183\pi$$
$$440$$ 0 0
$$441$$ 17.0000 0.809524
$$442$$ 0 0
$$443$$ 37.7980 1.79584 0.897918 0.440164i $$-0.145080\pi$$
0.897918 + 0.440164i $$0.145080\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ 7.79796 0.369659
$$446$$ 4.00000 0.189405
$$447$$ −18.8990 −0.893891
$$448$$ 4.89898 0.231455
$$449$$ −6.89898 −0.325583 −0.162791 0.986660i $$-0.552050\pi$$
−0.162791 + 0.986660i $$0.552050\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 0 0
$$452$$ −7.79796 −0.366785
$$453$$ −9.79796 −0.460348
$$454$$ −21.7980 −1.02303
$$455$$ −33.7980 −1.58447
$$456$$ −4.00000 −0.187317
$$457$$ 16.2020 0.757900 0.378950 0.925417i $$-0.376285\pi$$
0.378950 + 0.925417i $$0.376285\pi$$
$$458$$ 13.5959 0.635296
$$459$$ 0 0
$$460$$ −4.00000 −0.186501
$$461$$ −28.6969 −1.33655 −0.668275 0.743914i $$-0.732967\pi$$
−0.668275 + 0.743914i $$0.732967\pi$$
$$462$$ 0 0
$$463$$ 7.59592 0.353012 0.176506 0.984300i $$-0.443520\pi$$
0.176506 + 0.984300i $$0.443520\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 4.00000 0.185496
$$466$$ 14.0000 0.648537
$$467$$ −7.59592 −0.351497 −0.175749 0.984435i $$-0.556235\pi$$
−0.175749 + 0.984435i $$0.556235\pi$$
$$468$$ 6.89898 0.318905
$$469$$ 4.40408 0.203362
$$470$$ 9.79796 0.451946
$$471$$ −1.10102 −0.0507323
$$472$$ −4.89898 −0.225494
$$473$$ 0 0
$$474$$ −5.79796 −0.266309
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ −7.79796 −0.357044
$$478$$ 0 0
$$479$$ 32.8990 1.50319 0.751596 0.659623i $$-0.229284\pi$$
0.751596 + 0.659623i $$0.229284\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ −41.3939 −1.88740
$$482$$ 13.5959 0.619277
$$483$$ 19.5959 0.891645
$$484$$ −11.0000 −0.500000
$$485$$ −12.6969 −0.576538
$$486$$ −1.00000 −0.0453609
$$487$$ 20.8990 0.947023 0.473512 0.880788i $$-0.342986\pi$$
0.473512 + 0.880788i $$0.342986\pi$$
$$488$$ 11.7980 0.534069
$$489$$ 2.20204 0.0995797
$$490$$ 17.0000 0.767982
$$491$$ −1.30306 −0.0588063 −0.0294032 0.999568i $$-0.509361\pi$$
−0.0294032 + 0.999568i $$0.509361\pi$$
$$492$$ 2.89898 0.130696
$$493$$ 0 0
$$494$$ −27.5959 −1.24160
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 43.5959 1.95554
$$498$$ −13.7980 −0.618301
$$499$$ 39.5959 1.77256 0.886278 0.463153i $$-0.153282\pi$$
0.886278 + 0.463153i $$0.153282\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −2.20204 −0.0983799
$$502$$ −4.89898 −0.218652
$$503$$ −4.00000 −0.178351 −0.0891756 0.996016i $$-0.528423\pi$$
−0.0891756 + 0.996016i $$0.528423\pi$$
$$504$$ −4.89898 −0.218218
$$505$$ 18.8990 0.840994
$$506$$ 0 0
$$507$$ 34.5959 1.53646
$$508$$ −12.0000 −0.532414
$$509$$ −15.3031 −0.678296 −0.339148 0.940733i $$-0.610139\pi$$
−0.339148 + 0.940733i $$0.610139\pi$$
$$510$$ 0 0
$$511$$ 53.3939 2.36201
$$512$$ −1.00000 −0.0441942
$$513$$ 4.00000 0.176604
$$514$$ 2.00000 0.0882162
$$515$$ −4.00000 −0.176261
$$516$$ 8.89898 0.391756
$$517$$ 0 0
$$518$$ 29.3939 1.29149
$$519$$ −6.00000 −0.263371
$$520$$ 6.89898 0.302540
$$521$$ 40.2929 1.76526 0.882631 0.470066i $$-0.155770\pi$$
0.882631 + 0.470066i $$0.155770\pi$$
$$522$$ 6.00000 0.262613
$$523$$ 18.6969 0.817560 0.408780 0.912633i $$-0.365954\pi$$
0.408780 + 0.912633i $$0.365954\pi$$
$$524$$ −9.79796 −0.428026
$$525$$ 4.89898 0.213809
$$526$$ −8.00000 −0.348817
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −7.79796 −0.338722
$$531$$ 4.89898 0.212598
$$532$$ 19.5959 0.849591
$$533$$ 20.0000 0.866296
$$534$$ 7.79796 0.337451
$$535$$ 5.79796 0.250668
$$536$$ −0.898979 −0.0388300
$$537$$ −4.89898 −0.211407
$$538$$ 9.59592 0.413709
$$539$$ 0 0
$$540$$ −1.00000 −0.0430331
$$541$$ 7.79796 0.335260 0.167630 0.985850i $$-0.446389\pi$$
0.167630 + 0.985850i $$0.446389\pi$$
$$542$$ 1.79796 0.0772290
$$543$$ 4.20204 0.180327
$$544$$ 0 0
$$545$$ 11.7980 0.505369
$$546$$ −33.7980 −1.44642
$$547$$ −0.404082 −0.0172773 −0.00863865 0.999963i $$-0.502750\pi$$
−0.00863865 + 0.999963i $$0.502750\pi$$
$$548$$ 14.0000 0.598050
$$549$$ −11.7980 −0.503525
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ −4.00000 −0.170251
$$553$$ 28.4041 1.20786
$$554$$ 7.79796 0.331304
$$555$$ 6.00000 0.254686
$$556$$ −13.7980 −0.585164
$$557$$ 19.7980 0.838866 0.419433 0.907786i $$-0.362229\pi$$
0.419433 + 0.907786i $$0.362229\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 61.3939 2.59668
$$560$$ −4.89898 −0.207020
$$561$$ 0 0
$$562$$ −27.7980 −1.17259
$$563$$ 21.7980 0.918674 0.459337 0.888262i $$-0.348087\pi$$
0.459337 + 0.888262i $$0.348087\pi$$
$$564$$ 9.79796 0.412568
$$565$$ 7.79796 0.328063
$$566$$ 23.5959 0.991810
$$567$$ 4.89898 0.205738
$$568$$ −8.89898 −0.373393
$$569$$ 34.0000 1.42535 0.712677 0.701492i $$-0.247483\pi$$
0.712677 + 0.701492i $$0.247483\pi$$
$$570$$ 4.00000 0.167542
$$571$$ −29.7980 −1.24701 −0.623503 0.781821i $$-0.714291\pi$$
−0.623503 + 0.781821i $$0.714291\pi$$
$$572$$ 0 0
$$573$$ −9.79796 −0.409316
$$574$$ −14.2020 −0.592782
$$575$$ 4.00000 0.166812
$$576$$ 1.00000 0.0416667
$$577$$ −27.3939 −1.14042 −0.570211 0.821498i $$-0.693139\pi$$
−0.570211 + 0.821498i $$0.693139\pi$$
$$578$$ 0 0
$$579$$ 1.10102 0.0457569
$$580$$ 6.00000 0.249136
$$581$$ 67.5959 2.80435
$$582$$ −12.6969 −0.526305
$$583$$ 0 0
$$584$$ −10.8990 −0.451003
$$585$$ −6.89898 −0.285238
$$586$$ −13.5959 −0.561642
$$587$$ −41.3939 −1.70851 −0.854254 0.519856i $$-0.825986\pi$$
−0.854254 + 0.519856i $$0.825986\pi$$
$$588$$ 17.0000 0.701068
$$589$$ −16.0000 −0.659269
$$590$$ 4.89898 0.201688
$$591$$ 13.5959 0.559261
$$592$$ −6.00000 −0.246598
$$593$$ 1.59592 0.0655365 0.0327682 0.999463i $$-0.489568\pi$$
0.0327682 + 0.999463i $$0.489568\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −18.8990 −0.774132
$$597$$ −15.5959 −0.638298
$$598$$ −27.5959 −1.12848
$$599$$ 21.3939 0.874130 0.437065 0.899430i $$-0.356018\pi$$
0.437065 + 0.899430i $$0.356018\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −16.2020 −0.660895 −0.330448 0.943824i $$-0.607200\pi$$
−0.330448 + 0.943824i $$0.607200\pi$$
$$602$$ −43.5959 −1.77684
$$603$$ 0.898979 0.0366093
$$604$$ −9.79796 −0.398673
$$605$$ 11.0000 0.447214
$$606$$ 18.8990 0.767719
$$607$$ 32.4949 1.31893 0.659464 0.751736i $$-0.270783\pi$$
0.659464 + 0.751736i $$0.270783\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ −29.3939 −1.19110
$$610$$ −11.7980 −0.477685
$$611$$ 67.5959 2.73464
$$612$$ 0 0
$$613$$ −14.4949 −0.585443 −0.292722 0.956198i $$-0.594561\pi$$
−0.292722 + 0.956198i $$0.594561\pi$$
$$614$$ −0.898979 −0.0362799
$$615$$ −2.89898 −0.116898
$$616$$ 0 0
$$617$$ 37.5959 1.51355 0.756777 0.653673i $$-0.226773\pi$$
0.756777 + 0.653673i $$0.226773\pi$$
$$618$$ −4.00000 −0.160904
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 4.00000 0.160514
$$622$$ −7.10102 −0.284725
$$623$$ −38.2020 −1.53053
$$624$$ 6.89898 0.276180
$$625$$ 1.00000 0.0400000
$$626$$ 6.89898 0.275739
$$627$$ 0 0
$$628$$ −1.10102 −0.0439355
$$629$$ 0 0
$$630$$ 4.89898 0.195180
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ −5.79796 −0.230630
$$633$$ 12.0000 0.476957
$$634$$ 14.0000 0.556011
$$635$$ 12.0000 0.476205
$$636$$ −7.79796 −0.309209
$$637$$ 117.283 4.64691
$$638$$ 0 0
$$639$$ 8.89898 0.352038
$$640$$ 1.00000 0.0395285
$$641$$ 20.6969 0.817480 0.408740 0.912651i $$-0.365968\pi$$
0.408740 + 0.912651i $$0.365968\pi$$
$$642$$ 5.79796 0.228827
$$643$$ −29.7980 −1.17512 −0.587558 0.809182i $$-0.699911\pi$$
−0.587558 + 0.809182i $$0.699911\pi$$
$$644$$ 19.5959 0.772187
$$645$$ −8.89898 −0.350397
$$646$$ 0 0
$$647$$ −32.0000 −1.25805 −0.629025 0.777385i $$-0.716546\pi$$
−0.629025 + 0.777385i $$0.716546\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ −6.89898 −0.270600
$$651$$ −19.5959 −0.768025
$$652$$ 2.20204 0.0862386
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 11.7980 0.461337
$$655$$ 9.79796 0.382838
$$656$$ 2.89898 0.113186
$$657$$ 10.8990 0.425210
$$658$$ −48.0000 −1.87123
$$659$$ 30.6969 1.19578 0.597891 0.801577i $$-0.296005\pi$$
0.597891 + 0.801577i $$0.296005\pi$$
$$660$$ 0 0
$$661$$ −19.7980 −0.770051 −0.385026 0.922906i $$-0.625807\pi$$
−0.385026 + 0.922906i $$0.625807\pi$$
$$662$$ 5.79796 0.225344
$$663$$ 0 0
$$664$$ −13.7980 −0.535465
$$665$$ −19.5959 −0.759897
$$666$$ 6.00000 0.232495
$$667$$ −24.0000 −0.929284
$$668$$ −2.20204 −0.0851995
$$669$$ −4.00000 −0.154649
$$670$$ 0.898979 0.0347306
$$671$$ 0 0
$$672$$ −4.89898 −0.188982
$$673$$ 33.1010 1.27595 0.637975 0.770057i $$-0.279772\pi$$
0.637975 + 0.770057i $$0.279772\pi$$
$$674$$ 32.6969 1.25944
$$675$$ 1.00000 0.0384900
$$676$$ 34.5959 1.33061
$$677$$ 13.5959 0.522534 0.261267 0.965267i $$-0.415860\pi$$
0.261267 + 0.965267i $$0.415860\pi$$
$$678$$ 7.79796 0.299479
$$679$$ 62.2020 2.38710
$$680$$ 0 0
$$681$$ 21.7980 0.835300
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 4.00000 0.152944
$$685$$ −14.0000 −0.534913
$$686$$ −48.9898 −1.87044
$$687$$ −13.5959 −0.518717
$$688$$ 8.89898 0.339270
$$689$$ −53.7980 −2.04954
$$690$$ 4.00000 0.152277
$$691$$ 27.1918 1.03443 0.517213 0.855857i $$-0.326969\pi$$
0.517213 + 0.855857i $$0.326969\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ 21.7980 0.827439
$$695$$ 13.7980 0.523386
$$696$$ 6.00000 0.227429
$$697$$ 0 0
$$698$$ −4.20204 −0.159050
$$699$$ −14.0000 −0.529529
$$700$$ 4.89898 0.185164
$$701$$ 14.8990 0.562727 0.281363 0.959601i $$-0.409213\pi$$
0.281363 + 0.959601i $$0.409213\pi$$
$$702$$ −6.89898 −0.260385
$$703$$ −24.0000 −0.905177
$$704$$ 0 0
$$705$$ −9.79796 −0.369012
$$706$$ 26.0000 0.978523
$$707$$ −92.5857 −3.48204
$$708$$ 4.89898 0.184115
$$709$$ 31.7980 1.19420 0.597099 0.802168i $$-0.296320\pi$$
0.597099 + 0.802168i $$0.296320\pi$$
$$710$$ 8.89898 0.333973
$$711$$ 5.79796 0.217440
$$712$$ 7.79796 0.292241
$$713$$ −16.0000 −0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −4.89898 −0.183083
$$717$$ 0 0
$$718$$ −37.3939 −1.39553
$$719$$ 12.4949 0.465981 0.232991 0.972479i $$-0.425149\pi$$
0.232991 + 0.972479i $$0.425149\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 19.5959 0.729790
$$722$$ 3.00000 0.111648
$$723$$ −13.5959 −0.505638
$$724$$ 4.20204 0.156168
$$725$$ −6.00000 −0.222834
$$726$$ 11.0000 0.408248
$$727$$ 0.404082 0.0149866 0.00749329 0.999972i $$-0.497615\pi$$
0.00749329 + 0.999972i $$0.497615\pi$$
$$728$$ −33.7980 −1.25264
$$729$$ 1.00000 0.0370370
$$730$$ 10.8990 0.403389
$$731$$ 0 0
$$732$$ −11.7980 −0.436065
$$733$$ −46.4949 −1.71733 −0.858664 0.512539i $$-0.828705\pi$$
−0.858664 + 0.512539i $$0.828705\pi$$
$$734$$ −27.1010 −1.00032
$$735$$ −17.0000 −0.627054
$$736$$ −4.00000 −0.147442
$$737$$ 0 0
$$738$$ −2.89898 −0.106713
$$739$$ −9.39388 −0.345559 −0.172780 0.984960i $$-0.555275\pi$$
−0.172780 + 0.984960i $$0.555275\pi$$
$$740$$ 6.00000 0.220564
$$741$$ 27.5959 1.01376
$$742$$ 38.2020 1.40244
$$743$$ 9.39388 0.344628 0.172314 0.985042i $$-0.444876\pi$$
0.172314 + 0.985042i $$0.444876\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 18.8990 0.692405
$$746$$ −24.6969 −0.904219
$$747$$ 13.7980 0.504841
$$748$$ 0 0
$$749$$ −28.4041 −1.03786
$$750$$ 1.00000 0.0365148
$$751$$ −21.7980 −0.795419 −0.397709 0.917511i $$-0.630195\pi$$
−0.397709 + 0.917511i $$0.630195\pi$$
$$752$$ 9.79796 0.357295
$$753$$ 4.89898 0.178529
$$754$$ 41.3939 1.50748
$$755$$ 9.79796 0.356584
$$756$$ 4.89898 0.178174
$$757$$ −4.69694 −0.170713 −0.0853566 0.996350i $$-0.527203\pi$$
−0.0853566 + 0.996350i $$0.527203\pi$$
$$758$$ −29.7980 −1.08231
$$759$$ 0 0
$$760$$ 4.00000 0.145095
$$761$$ −1.59592 −0.0578520 −0.0289260 0.999582i $$-0.509209\pi$$
−0.0289260 + 0.999582i $$0.509209\pi$$
$$762$$ 12.0000 0.434714
$$763$$ −57.7980 −2.09243
$$764$$ −9.79796 −0.354478
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ 33.7980 1.22037
$$768$$ 1.00000 0.0360844
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ −2.00000 −0.0720282
$$772$$ 1.10102 0.0396266
$$773$$ −35.3939 −1.27303 −0.636515 0.771265i $$-0.719625\pi$$
−0.636515 + 0.771265i $$0.719625\pi$$
$$774$$ −8.89898 −0.319867
$$775$$ −4.00000 −0.143684
$$776$$ −12.6969 −0.455794
$$777$$ −29.3939 −1.05450
$$778$$ 38.4949 1.38011
$$779$$ 11.5959 0.415467
$$780$$ −6.89898 −0.247023
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −6.00000 −0.214423
$$784$$ 17.0000 0.607143
$$785$$ 1.10102 0.0392971
$$786$$ 9.79796 0.349482
$$787$$ 45.7980 1.63252 0.816260 0.577684i $$-0.196043\pi$$
0.816260 + 0.577684i $$0.196043\pi$$
$$788$$ 13.5959 0.484335
$$789$$ 8.00000 0.284808
$$790$$ 5.79796 0.206282
$$791$$ −38.2020 −1.35831
$$792$$ 0 0
$$793$$ −81.3939 −2.89038
$$794$$ 1.59592 0.0566370
$$795$$ 7.79796 0.276565
$$796$$ −15.5959 −0.552783
$$797$$ 26.0000 0.920967 0.460484 0.887668i $$-0.347676\pi$$
0.460484 + 0.887668i $$0.347676\pi$$
$$798$$ −19.5959 −0.693688
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ −7.79796 −0.275527
$$802$$ 22.8990 0.808591
$$803$$ 0 0
$$804$$ 0.898979 0.0317046
$$805$$ −19.5959 −0.690665
$$806$$ 27.5959 0.972025
$$807$$ −9.59592 −0.337792
$$808$$ 18.8990 0.664864
$$809$$ −32.6969 −1.14956 −0.574782 0.818307i $$-0.694913\pi$$
−0.574782 + 0.818307i $$0.694913\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −25.3939 −0.891700 −0.445850 0.895108i $$-0.647098\pi$$
−0.445850 + 0.895108i $$0.647098\pi$$
$$812$$ −29.3939 −1.03152
$$813$$ −1.79796 −0.0630572
$$814$$ 0 0
$$815$$ −2.20204 −0.0771341
$$816$$ 0 0
$$817$$ 35.5959 1.24534
$$818$$ 17.5959 0.615227
$$819$$ 33.7980 1.18100
$$820$$ −2.89898 −0.101237
$$821$$ −15.7980 −0.551353 −0.275676 0.961251i $$-0.588902\pi$$
−0.275676 + 0.961251i $$0.588902\pi$$
$$822$$ −14.0000 −0.488306
$$823$$ −20.8990 −0.728493 −0.364246 0.931303i $$-0.618673\pi$$
−0.364246 + 0.931303i $$0.618673\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ −24.0000 −0.835067
$$827$$ 4.00000 0.139094 0.0695468 0.997579i $$-0.477845\pi$$
0.0695468 + 0.997579i $$0.477845\pi$$
$$828$$ 4.00000 0.139010
$$829$$ 3.39388 0.117874 0.0589371 0.998262i $$-0.481229\pi$$
0.0589371 + 0.998262i $$0.481229\pi$$
$$830$$ 13.7980 0.478934
$$831$$ −7.79796 −0.270508
$$832$$ 6.89898 0.239179
$$833$$ 0 0
$$834$$ 13.7980 0.477784
$$835$$ 2.20204 0.0762048
$$836$$ 0 0
$$837$$ −4.00000 −0.138260
$$838$$ −17.7980 −0.614820
$$839$$ 7.10102 0.245154 0.122577 0.992459i $$-0.460884\pi$$
0.122577 + 0.992459i $$0.460884\pi$$
$$840$$ 4.89898 0.169031
$$841$$ 7.00000 0.241379
$$842$$ −14.0000 −0.482472
$$843$$ 27.7980 0.957413
$$844$$ 12.0000 0.413057
$$845$$ −34.5959 −1.19014
$$846$$ −9.79796 −0.336861
$$847$$ −53.8888 −1.85164
$$848$$ −7.79796 −0.267783
$$849$$ −23.5959 −0.809810
$$850$$ 0 0
$$851$$ −24.0000 −0.822709
$$852$$ 8.89898 0.304874
$$853$$ −11.3939 −0.390119 −0.195059 0.980791i $$-0.562490\pi$$
−0.195059 + 0.980791i $$0.562490\pi$$
$$854$$ 57.7980 1.97781
$$855$$ −4.00000 −0.136797
$$856$$ 5.79796 0.198170
$$857$$ 10.0000 0.341593 0.170797 0.985306i $$-0.445366\pi$$
0.170797 + 0.985306i $$0.445366\pi$$
$$858$$ 0 0
$$859$$ 5.79796 0.197824 0.0989119 0.995096i $$-0.468464\pi$$
0.0989119 + 0.995096i $$0.468464\pi$$
$$860$$ −8.89898 −0.303453
$$861$$ 14.2020 0.484004
$$862$$ −23.1010 −0.786824
$$863$$ −45.3939 −1.54523 −0.772613 0.634878i $$-0.781051\pi$$
−0.772613 + 0.634878i $$0.781051\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 6.00000 0.204006
$$866$$ 35.3939 1.20273
$$867$$ 0 0
$$868$$ −19.5959 −0.665129
$$869$$ 0 0
$$870$$ −6.00000 −0.203419
$$871$$ 6.20204 0.210148
$$872$$ 11.7980 0.399529
$$873$$ 12.6969 0.429726
$$874$$ −16.0000 −0.541208
$$875$$ −4.89898 −0.165616
$$876$$ 10.8990 0.368242
$$877$$ 31.3939 1.06010 0.530048 0.847968i $$-0.322174\pi$$
0.530048 + 0.847968i $$0.322174\pi$$
$$878$$ 5.79796 0.195672
$$879$$ 13.5959 0.458579
$$880$$ 0 0
$$881$$ −34.4949 −1.16216 −0.581081 0.813846i $$-0.697370\pi$$
−0.581081 + 0.813846i $$0.697370\pi$$
$$882$$ −17.0000 −0.572420
$$883$$ 26.6969 0.898424 0.449212 0.893425i $$-0.351705\pi$$
0.449212 + 0.893425i $$0.351705\pi$$
$$884$$ 0 0
$$885$$ −4.89898 −0.164677
$$886$$ −37.7980 −1.26985
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ 6.00000 0.201347
$$889$$ −58.7878 −1.97168
$$890$$ −7.79796 −0.261388
$$891$$ 0 0
$$892$$ −4.00000 −0.133930
$$893$$ 39.1918 1.31150
$$894$$ 18.8990 0.632076
$$895$$ 4.89898 0.163755
$$896$$ −4.89898 −0.163663
$$897$$ 27.5959 0.921401
$$898$$ 6.89898 0.230222
$$899$$ 24.0000 0.800445
$$900$$ 1.00000 0.0333333
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 43.5959 1.45078
$$904$$ 7.79796 0.259356
$$905$$ −4.20204 −0.139681
$$906$$ 9.79796 0.325515
$$907$$ 33.3939 1.10883 0.554413 0.832242i $$-0.312943\pi$$
0.554413 + 0.832242i $$0.312943\pi$$
$$908$$ 21.7980 0.723391
$$909$$ −18.8990 −0.626840
$$910$$ 33.7980 1.12039
$$911$$ −23.1010 −0.765371 −0.382685 0.923879i $$-0.625001\pi$$
−0.382685 + 0.923879i $$0.625001\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 0 0
$$914$$ −16.2020 −0.535916
$$915$$ 11.7980 0.390028
$$916$$ −13.5959 −0.449222
$$917$$ −48.0000 −1.58510
$$918$$ 0 0
$$919$$ 1.79796 0.0593092 0.0296546 0.999560i $$-0.490559\pi$$
0.0296546 + 0.999560i $$0.490559\pi$$
$$920$$ 4.00000 0.131876
$$921$$ 0.898979 0.0296224
$$922$$ 28.6969 0.945083
$$923$$ 61.3939 2.02080
$$924$$ 0 0
$$925$$ −6.00000 −0.197279
$$926$$ −7.59592 −0.249617
$$927$$ 4.00000 0.131377
$$928$$ 6.00000 0.196960
$$929$$ −36.2929 −1.19073 −0.595365 0.803455i $$-0.702993\pi$$
−0.595365 + 0.803455i $$0.702993\pi$$
$$930$$ −4.00000 −0.131165
$$931$$ 68.0000 2.22861
$$932$$ −14.0000 −0.458585
$$933$$ 7.10102 0.232477
$$934$$ 7.59592 0.248546
$$935$$ 0 0
$$936$$ −6.89898 −0.225500
$$937$$ 39.3939 1.28694 0.643471 0.765471i $$-0.277494\pi$$
0.643471 + 0.765471i $$0.277494\pi$$
$$938$$ −4.40408 −0.143798
$$939$$ −6.89898 −0.225140
$$940$$ −9.79796 −0.319574
$$941$$ 16.2020 0.528171 0.264086 0.964499i $$-0.414930\pi$$
0.264086 + 0.964499i $$0.414930\pi$$
$$942$$ 1.10102 0.0358732
$$943$$ 11.5959 0.377615
$$944$$ 4.89898 0.159448
$$945$$ −4.89898 −0.159364
$$946$$ 0 0
$$947$$ −21.7980 −0.708338 −0.354169 0.935181i $$-0.615236\pi$$
−0.354169 + 0.935181i $$0.615236\pi$$
$$948$$ 5.79796 0.188309
$$949$$ 75.1918 2.44083
$$950$$ −4.00000 −0.129777
$$951$$ −14.0000 −0.453981
$$952$$ 0 0
$$953$$ −41.1918 −1.33433 −0.667167 0.744908i $$-0.732493\pi$$
−0.667167 + 0.744908i $$0.732493\pi$$
$$954$$ 7.79796 0.252468
$$955$$ 9.79796 0.317055
$$956$$ 0 0
$$957$$ 0 0
$$958$$ −32.8990 −1.06292
$$959$$ 68.5857 2.21475
$$960$$ −1.00000 −0.0322749
$$961$$ −15.0000 −0.483871
$$962$$ 41.3939 1.33459
$$963$$ −5.79796 −0.186837
$$964$$ −13.5959 −0.437895
$$965$$ −1.10102 −0.0354431
$$966$$ −19.5959 −0.630488
$$967$$ −41.3939 −1.33114 −0.665569 0.746337i $$-0.731811\pi$$
−0.665569 + 0.746337i $$0.731811\pi$$
$$968$$ 11.0000 0.353553
$$969$$ 0 0
$$970$$ 12.6969 0.407674
$$971$$ 38.6969 1.24184 0.620922 0.783872i $$-0.286758\pi$$
0.620922 + 0.783872i $$0.286758\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −67.5959 −2.16703
$$974$$ −20.8990 −0.669646
$$975$$ 6.89898 0.220944
$$976$$ −11.7980 −0.377643
$$977$$ −29.5959 −0.946857 −0.473429 0.880832i $$-0.656984\pi$$
−0.473429 + 0.880832i $$0.656984\pi$$
$$978$$ −2.20204 −0.0704135
$$979$$ 0 0
$$980$$ −17.0000 −0.543045
$$981$$ −11.7980 −0.376680
$$982$$ 1.30306 0.0415824
$$983$$ −9.39388 −0.299618 −0.149809 0.988715i $$-0.547866\pi$$
−0.149809 + 0.988715i $$0.547866\pi$$
$$984$$ −2.89898 −0.0924161
$$985$$ −13.5959 −0.433202
$$986$$ 0 0
$$987$$ 48.0000 1.52786
$$988$$ 27.5959 0.877943
$$989$$ 35.5959 1.13188
$$990$$ 0 0
$$991$$ 15.5959 0.495421 0.247710 0.968834i $$-0.420322\pi$$
0.247710 + 0.968834i $$0.420322\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −5.79796 −0.183993
$$994$$ −43.5959 −1.38278
$$995$$ 15.5959 0.494424
$$996$$ 13.7980 0.437205
$$997$$ 21.5959 0.683950 0.341975 0.939709i $$-0.388904\pi$$
0.341975 + 0.939709i $$0.388904\pi$$
$$998$$ −39.5959 −1.25339
$$999$$ −6.00000 −0.189832
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 8670.2.a.be.1.2 2
17.16 even 2 510.2.a.h.1.1 2
51.50 odd 2 1530.2.a.s.1.1 2
68.67 odd 2 4080.2.a.bq.1.2 2
85.33 odd 4 2550.2.d.u.2449.4 4
85.67 odd 4 2550.2.d.u.2449.1 4
85.84 even 2 2550.2.a.bl.1.2 2
255.254 odd 2 7650.2.a.cu.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.h.1.1 2 17.16 even 2
1530.2.a.s.1.1 2 51.50 odd 2
2550.2.a.bl.1.2 2 85.84 even 2
2550.2.d.u.2449.1 4 85.67 odd 4
2550.2.d.u.2449.4 4 85.33 odd 4
4080.2.a.bq.1.2 2 68.67 odd 2
7650.2.a.cu.1.2 2 255.254 odd 2
8670.2.a.be.1.2 2 1.1 even 1 trivial