# Properties

 Label 33.3.b.a.23.1 Level $33$ Weight $3$ Character 33.23 Analytic conductor $0.899$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,3,Mod(23,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("33.23");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 33.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$0.899184872389$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 3$$ x^2 - x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 23.1 Root $$0.500000 + 1.65831i$$ of defining polynomial Character $$\chi$$ $$=$$ 33.23 Dual form 33.3.b.a.23.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.31662i q^{2} +3.00000 q^{3} -7.00000 q^{4} +6.63325i q^{5} -9.94987i q^{6} -8.00000 q^{7} +9.94987i q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-3.31662i q^{2} +3.00000 q^{3} -7.00000 q^{4} +6.63325i q^{5} -9.94987i q^{6} -8.00000 q^{7} +9.94987i q^{8} +9.00000 q^{9} +22.0000 q^{10} -3.31662i q^{11} -21.0000 q^{12} +4.00000 q^{13} +26.5330i q^{14} +19.8997i q^{15} +5.00000 q^{16} -13.2665i q^{17} -29.8496i q^{18} -6.00000 q^{19} -46.4327i q^{20} -24.0000 q^{21} -11.0000 q^{22} -6.63325i q^{23} +29.8496i q^{24} -19.0000 q^{25} -13.2665i q^{26} +27.0000 q^{27} +56.0000 q^{28} +39.7995i q^{29} +66.0000 q^{30} -26.0000 q^{31} +23.2164i q^{32} -9.94987i q^{33} -44.0000 q^{34} -53.0660i q^{35} -63.0000 q^{36} +30.0000 q^{37} +19.8997i q^{38} +12.0000 q^{39} -66.0000 q^{40} -13.2665i q^{41} +79.5990i q^{42} +42.0000 q^{43} +23.2164i q^{44} +59.6992i q^{45} -22.0000 q^{46} -86.2322i q^{47} +15.0000 q^{48} +15.0000 q^{49} +63.0159i q^{50} -39.7995i q^{51} -28.0000 q^{52} +59.6992i q^{53} -89.5489i q^{54} +22.0000 q^{55} -79.5990i q^{56} -18.0000 q^{57} +132.000 q^{58} -66.3325i q^{59} -139.298i q^{60} +12.0000 q^{61} +86.2322i q^{62} -72.0000 q^{63} +97.0000 q^{64} +26.5330i q^{65} -33.0000 q^{66} +2.00000 q^{67} +92.8655i q^{68} -19.8997i q^{69} -176.000 q^{70} +59.6992i q^{71} +89.5489i q^{72} -74.0000 q^{73} -99.4987i q^{74} -57.0000 q^{75} +42.0000 q^{76} +26.5330i q^{77} -39.7995i q^{78} -40.0000 q^{79} +33.1662i q^{80} +81.0000 q^{81} -44.0000 q^{82} +39.7995i q^{83} +168.000 q^{84} +88.0000 q^{85} -139.298i q^{86} +119.398i q^{87} +33.0000 q^{88} +119.398i q^{89} +198.000 q^{90} -32.0000 q^{91} +46.4327i q^{92} -78.0000 q^{93} -286.000 q^{94} -39.7995i q^{95} +69.6491i q^{96} +62.0000 q^{97} -49.7494i q^{98} -29.8496i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 14 q^{4} - 16 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 14 * q^4 - 16 * q^7 + 18 * q^9 $$2 q + 6 q^{3} - 14 q^{4} - 16 q^{7} + 18 q^{9} + 44 q^{10} - 42 q^{12} + 8 q^{13} + 10 q^{16} - 12 q^{19} - 48 q^{21} - 22 q^{22} - 38 q^{25} + 54 q^{27} + 112 q^{28} + 132 q^{30} - 52 q^{31} - 88 q^{34} - 126 q^{36} + 60 q^{37} + 24 q^{39} - 132 q^{40} + 84 q^{43} - 44 q^{46} + 30 q^{48} + 30 q^{49} - 56 q^{52} + 44 q^{55} - 36 q^{57} + 264 q^{58} + 24 q^{61} - 144 q^{63} + 194 q^{64} - 66 q^{66} + 4 q^{67} - 352 q^{70} - 148 q^{73} - 114 q^{75} + 84 q^{76} - 80 q^{79} + 162 q^{81} - 88 q^{82} + 336 q^{84} + 176 q^{85} + 66 q^{88} + 396 q^{90} - 64 q^{91} - 156 q^{93} - 572 q^{94} + 124 q^{97}+O(q^{100})$$ 2 * q + 6 * q^3 - 14 * q^4 - 16 * q^7 + 18 * q^9 + 44 * q^10 - 42 * q^12 + 8 * q^13 + 10 * q^16 - 12 * q^19 - 48 * q^21 - 22 * q^22 - 38 * q^25 + 54 * q^27 + 112 * q^28 + 132 * q^30 - 52 * q^31 - 88 * q^34 - 126 * q^36 + 60 * q^37 + 24 * q^39 - 132 * q^40 + 84 * q^43 - 44 * q^46 + 30 * q^48 + 30 * q^49 - 56 * q^52 + 44 * q^55 - 36 * q^57 + 264 * q^58 + 24 * q^61 - 144 * q^63 + 194 * q^64 - 66 * q^66 + 4 * q^67 - 352 * q^70 - 148 * q^73 - 114 * q^75 + 84 * q^76 - 80 * q^79 + 162 * q^81 - 88 * q^82 + 336 * q^84 + 176 * q^85 + 66 * q^88 + 396 * q^90 - 64 * q^91 - 156 * q^93 - 572 * q^94 + 124 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 3.31662i − 1.65831i −0.559017 0.829156i $$-0.688821\pi$$
0.559017 0.829156i $$-0.311179\pi$$
$$3$$ 3.00000 1.00000
$$4$$ −7.00000 −1.75000
$$5$$ 6.63325i 1.32665i 0.748331 + 0.663325i $$0.230855\pi$$
−0.748331 + 0.663325i $$0.769145\pi$$
$$6$$ − 9.94987i − 1.65831i
$$7$$ −8.00000 −1.14286 −0.571429 0.820652i $$-0.693611\pi$$
−0.571429 + 0.820652i $$0.693611\pi$$
$$8$$ 9.94987i 1.24373i
$$9$$ 9.00000 1.00000
$$10$$ 22.0000 2.20000
$$11$$ − 3.31662i − 0.301511i
$$12$$ −21.0000 −1.75000
$$13$$ 4.00000 0.307692 0.153846 0.988095i $$-0.450834\pi$$
0.153846 + 0.988095i $$0.450834\pi$$
$$14$$ 26.5330i 1.89521i
$$15$$ 19.8997i 1.32665i
$$16$$ 5.00000 0.312500
$$17$$ − 13.2665i − 0.780382i −0.920734 0.390191i $$-0.872409\pi$$
0.920734 0.390191i $$-0.127591\pi$$
$$18$$ − 29.8496i − 1.65831i
$$19$$ −6.00000 −0.315789 −0.157895 0.987456i $$-0.550471\pi$$
−0.157895 + 0.987456i $$0.550471\pi$$
$$20$$ − 46.4327i − 2.32164i
$$21$$ −24.0000 −1.14286
$$22$$ −11.0000 −0.500000
$$23$$ − 6.63325i − 0.288402i −0.989548 0.144201i $$-0.953939\pi$$
0.989548 0.144201i $$-0.0460612\pi$$
$$24$$ 29.8496i 1.24373i
$$25$$ −19.0000 −0.760000
$$26$$ − 13.2665i − 0.510250i
$$27$$ 27.0000 1.00000
$$28$$ 56.0000 2.00000
$$29$$ 39.7995i 1.37240i 0.727415 + 0.686198i $$0.240722\pi$$
−0.727415 + 0.686198i $$0.759278\pi$$
$$30$$ 66.0000 2.20000
$$31$$ −26.0000 −0.838710 −0.419355 0.907822i $$-0.637744\pi$$
−0.419355 + 0.907822i $$0.637744\pi$$
$$32$$ 23.2164i 0.725512i
$$33$$ − 9.94987i − 0.301511i
$$34$$ −44.0000 −1.29412
$$35$$ − 53.0660i − 1.51617i
$$36$$ −63.0000 −1.75000
$$37$$ 30.0000 0.810811 0.405405 0.914137i $$-0.367130\pi$$
0.405405 + 0.914137i $$0.367130\pi$$
$$38$$ 19.8997i 0.523678i
$$39$$ 12.0000 0.307692
$$40$$ −66.0000 −1.65000
$$41$$ − 13.2665i − 0.323573i −0.986826 0.161787i $$-0.948274\pi$$
0.986826 0.161787i $$-0.0517256\pi$$
$$42$$ 79.5990i 1.89521i
$$43$$ 42.0000 0.976744 0.488372 0.872635i $$-0.337591\pi$$
0.488372 + 0.872635i $$0.337591\pi$$
$$44$$ 23.2164i 0.527645i
$$45$$ 59.6992i 1.32665i
$$46$$ −22.0000 −0.478261
$$47$$ − 86.2322i − 1.83473i −0.398049 0.917364i $$-0.630312\pi$$
0.398049 0.917364i $$-0.369688\pi$$
$$48$$ 15.0000 0.312500
$$49$$ 15.0000 0.306122
$$50$$ 63.0159i 1.26032i
$$51$$ − 39.7995i − 0.780382i
$$52$$ −28.0000 −0.538462
$$53$$ 59.6992i 1.12640i 0.826320 + 0.563200i $$0.190430\pi$$
−0.826320 + 0.563200i $$0.809570\pi$$
$$54$$ − 89.5489i − 1.65831i
$$55$$ 22.0000 0.400000
$$56$$ − 79.5990i − 1.42141i
$$57$$ −18.0000 −0.315789
$$58$$ 132.000 2.27586
$$59$$ − 66.3325i − 1.12428i −0.827042 0.562140i $$-0.809978\pi$$
0.827042 0.562140i $$-0.190022\pi$$
$$60$$ − 139.298i − 2.32164i
$$61$$ 12.0000 0.196721 0.0983607 0.995151i $$-0.468640\pi$$
0.0983607 + 0.995151i $$0.468640\pi$$
$$62$$ 86.2322i 1.39084i
$$63$$ −72.0000 −1.14286
$$64$$ 97.0000 1.51562
$$65$$ 26.5330i 0.408200i
$$66$$ −33.0000 −0.500000
$$67$$ 2.00000 0.0298507 0.0149254 0.999889i $$-0.495249\pi$$
0.0149254 + 0.999889i $$0.495249\pi$$
$$68$$ 92.8655i 1.36567i
$$69$$ − 19.8997i − 0.288402i
$$70$$ −176.000 −2.51429
$$71$$ 59.6992i 0.840834i 0.907331 + 0.420417i $$0.138116\pi$$
−0.907331 + 0.420417i $$0.861884\pi$$
$$72$$ 89.5489i 1.24373i
$$73$$ −74.0000 −1.01370 −0.506849 0.862035i $$-0.669190\pi$$
−0.506849 + 0.862035i $$0.669190\pi$$
$$74$$ − 99.4987i − 1.34458i
$$75$$ −57.0000 −0.760000
$$76$$ 42.0000 0.552632
$$77$$ 26.5330i 0.344584i
$$78$$ − 39.7995i − 0.510250i
$$79$$ −40.0000 −0.506329 −0.253165 0.967423i $$-0.581471\pi$$
−0.253165 + 0.967423i $$0.581471\pi$$
$$80$$ 33.1662i 0.414578i
$$81$$ 81.0000 1.00000
$$82$$ −44.0000 −0.536585
$$83$$ 39.7995i 0.479512i 0.970833 + 0.239756i $$0.0770674\pi$$
−0.970833 + 0.239756i $$0.922933\pi$$
$$84$$ 168.000 2.00000
$$85$$ 88.0000 1.03529
$$86$$ − 139.298i − 1.61975i
$$87$$ 119.398i 1.37240i
$$88$$ 33.0000 0.375000
$$89$$ 119.398i 1.34156i 0.741658 + 0.670778i $$0.234040\pi$$
−0.741658 + 0.670778i $$0.765960\pi$$
$$90$$ 198.000 2.20000
$$91$$ −32.0000 −0.351648
$$92$$ 46.4327i 0.504704i
$$93$$ −78.0000 −0.838710
$$94$$ −286.000 −3.04255
$$95$$ − 39.7995i − 0.418942i
$$96$$ 69.6491i 0.725512i
$$97$$ 62.0000 0.639175 0.319588 0.947557i $$-0.396456\pi$$
0.319588 + 0.947557i $$0.396456\pi$$
$$98$$ − 49.7494i − 0.507647i
$$99$$ − 29.8496i − 0.301511i
$$100$$ 133.000 1.33000
$$101$$ − 106.132i − 1.05081i −0.850852 0.525406i $$-0.823913\pi$$
0.850852 0.525406i $$-0.176087\pi$$
$$102$$ −132.000 −1.29412
$$103$$ 74.0000 0.718447 0.359223 0.933252i $$-0.383042\pi$$
0.359223 + 0.933252i $$0.383042\pi$$
$$104$$ 39.7995i 0.382687i
$$105$$ − 159.198i − 1.51617i
$$106$$ 198.000 1.86792
$$107$$ 39.7995i 0.371958i 0.982554 + 0.185979i $$0.0595456\pi$$
−0.982554 + 0.185979i $$0.940454\pi$$
$$108$$ −189.000 −1.75000
$$109$$ −200.000 −1.83486 −0.917431 0.397894i $$-0.869741\pi$$
−0.917431 + 0.397894i $$0.869741\pi$$
$$110$$ − 72.9657i − 0.663325i
$$111$$ 90.0000 0.810811
$$112$$ −40.0000 −0.357143
$$113$$ 39.7995i 0.352208i 0.984372 + 0.176104i $$0.0563495\pi$$
−0.984372 + 0.176104i $$0.943651\pi$$
$$114$$ 59.6992i 0.523678i
$$115$$ 44.0000 0.382609
$$116$$ − 278.596i − 2.40169i
$$117$$ 36.0000 0.307692
$$118$$ −220.000 −1.86441
$$119$$ 106.132i 0.891865i
$$120$$ −198.000 −1.65000
$$121$$ −11.0000 −0.0909091
$$122$$ − 39.7995i − 0.326225i
$$123$$ − 39.7995i − 0.323573i
$$124$$ 182.000 1.46774
$$125$$ 39.7995i 0.318396i
$$126$$ 238.797i 1.89521i
$$127$$ 188.000 1.48031 0.740157 0.672434i $$-0.234751\pi$$
0.740157 + 0.672434i $$0.234751\pi$$
$$128$$ − 228.847i − 1.78787i
$$129$$ 126.000 0.976744
$$130$$ 88.0000 0.676923
$$131$$ − 39.7995i − 0.303813i −0.988395 0.151906i $$-0.951459\pi$$
0.988395 0.151906i $$-0.0485413\pi$$
$$132$$ 69.6491i 0.527645i
$$133$$ 48.0000 0.360902
$$134$$ − 6.63325i − 0.0495019i
$$135$$ 179.098i 1.32665i
$$136$$ 132.000 0.970588
$$137$$ − 106.132i − 0.774686i −0.921936 0.387343i $$-0.873393\pi$$
0.921936 0.387343i $$-0.126607\pi$$
$$138$$ −66.0000 −0.478261
$$139$$ −74.0000 −0.532374 −0.266187 0.963921i $$-0.585764\pi$$
−0.266187 + 0.963921i $$0.585764\pi$$
$$140$$ 371.462i 2.65330i
$$141$$ − 258.697i − 1.83473i
$$142$$ 198.000 1.39437
$$143$$ − 13.2665i − 0.0927727i
$$144$$ 45.0000 0.312500
$$145$$ −264.000 −1.82069
$$146$$ 245.430i 1.68103i
$$147$$ 45.0000 0.306122
$$148$$ −210.000 −1.41892
$$149$$ 92.8655i 0.623258i 0.950204 + 0.311629i $$0.100875\pi$$
−0.950204 + 0.311629i $$0.899125\pi$$
$$150$$ 189.048i 1.26032i
$$151$$ −160.000 −1.05960 −0.529801 0.848122i $$-0.677734\pi$$
−0.529801 + 0.848122i $$0.677734\pi$$
$$152$$ − 59.6992i − 0.392758i
$$153$$ − 119.398i − 0.780382i
$$154$$ 88.0000 0.571429
$$155$$ − 172.464i − 1.11267i
$$156$$ −84.0000 −0.538462
$$157$$ 182.000 1.15924 0.579618 0.814888i $$-0.303202\pi$$
0.579618 + 0.814888i $$0.303202\pi$$
$$158$$ 132.665i 0.839652i
$$159$$ 179.098i 1.12640i
$$160$$ −154.000 −0.962500
$$161$$ 53.0660i 0.329602i
$$162$$ − 268.647i − 1.65831i
$$163$$ −290.000 −1.77914 −0.889571 0.456798i $$-0.848996\pi$$
−0.889571 + 0.456798i $$0.848996\pi$$
$$164$$ 92.8655i 0.566253i
$$165$$ 66.0000 0.400000
$$166$$ 132.000 0.795181
$$167$$ 238.797i 1.42992i 0.699164 + 0.714961i $$0.253556\pi$$
−0.699164 + 0.714961i $$0.746444\pi$$
$$168$$ − 238.797i − 1.42141i
$$169$$ −153.000 −0.905325
$$170$$ − 291.863i − 1.71684i
$$171$$ −54.0000 −0.315789
$$172$$ −294.000 −1.70930
$$173$$ − 198.997i − 1.15027i −0.818057 0.575137i $$-0.804949\pi$$
0.818057 0.575137i $$-0.195051\pi$$
$$174$$ 396.000 2.27586
$$175$$ 152.000 0.868571
$$176$$ − 16.5831i − 0.0942223i
$$177$$ − 198.997i − 1.12428i
$$178$$ 396.000 2.22472
$$179$$ − 198.997i − 1.11172i −0.831277 0.555859i $$-0.812389\pi$$
0.831277 0.555859i $$-0.187611\pi$$
$$180$$ − 417.895i − 2.32164i
$$181$$ 10.0000 0.0552486 0.0276243 0.999618i $$-0.491206\pi$$
0.0276243 + 0.999618i $$0.491206\pi$$
$$182$$ 106.132i 0.583143i
$$183$$ 36.0000 0.196721
$$184$$ 66.0000 0.358696
$$185$$ 198.997i 1.07566i
$$186$$ 258.697i 1.39084i
$$187$$ −44.0000 −0.235294
$$188$$ 603.626i 3.21078i
$$189$$ −216.000 −1.14286
$$190$$ −132.000 −0.694737
$$191$$ 112.765i 0.590394i 0.955436 + 0.295197i $$0.0953853\pi$$
−0.955436 + 0.295197i $$0.904615\pi$$
$$192$$ 291.000 1.51562
$$193$$ 298.000 1.54404 0.772021 0.635597i $$-0.219246\pi$$
0.772021 + 0.635597i $$0.219246\pi$$
$$194$$ − 205.631i − 1.05995i
$$195$$ 79.5990i 0.408200i
$$196$$ −105.000 −0.535714
$$197$$ − 132.665i − 0.673426i −0.941607 0.336713i $$-0.890685\pi$$
0.941607 0.336713i $$-0.109315\pi$$
$$198$$ −99.0000 −0.500000
$$199$$ −42.0000 −0.211055 −0.105528 0.994416i $$-0.533653\pi$$
−0.105528 + 0.994416i $$0.533653\pi$$
$$200$$ − 189.048i − 0.945238i
$$201$$ 6.00000 0.0298507
$$202$$ −352.000 −1.74257
$$203$$ − 318.396i − 1.56845i
$$204$$ 278.596i 1.36567i
$$205$$ 88.0000 0.429268
$$206$$ − 245.430i − 1.19141i
$$207$$ − 59.6992i − 0.288402i
$$208$$ 20.0000 0.0961538
$$209$$ 19.8997i 0.0952141i
$$210$$ −528.000 −2.51429
$$211$$ 246.000 1.16588 0.582938 0.812516i $$-0.301903\pi$$
0.582938 + 0.812516i $$0.301903\pi$$
$$212$$ − 417.895i − 1.97120i
$$213$$ 179.098i 0.840834i
$$214$$ 132.000 0.616822
$$215$$ 278.596i 1.29580i
$$216$$ 268.647i 1.24373i
$$217$$ 208.000 0.958525
$$218$$ 663.325i 3.04278i
$$219$$ −222.000 −1.01370
$$220$$ −154.000 −0.700000
$$221$$ − 53.0660i − 0.240118i
$$222$$ − 298.496i − 1.34458i
$$223$$ −302.000 −1.35426 −0.677130 0.735863i $$-0.736777\pi$$
−0.677130 + 0.735863i $$0.736777\pi$$
$$224$$ − 185.731i − 0.829156i
$$225$$ −171.000 −0.760000
$$226$$ 132.000 0.584071
$$227$$ 198.997i 0.876641i 0.898819 + 0.438320i $$0.144426\pi$$
−0.898819 + 0.438320i $$0.855574\pi$$
$$228$$ 126.000 0.552632
$$229$$ 150.000 0.655022 0.327511 0.944847i $$-0.393790\pi$$
0.327511 + 0.944847i $$0.393790\pi$$
$$230$$ − 145.931i − 0.634485i
$$231$$ 79.5990i 0.344584i
$$232$$ −396.000 −1.70690
$$233$$ 437.794i 1.87895i 0.342623 + 0.939473i $$0.388685\pi$$
−0.342623 + 0.939473i $$0.611315\pi$$
$$234$$ − 119.398i − 0.510250i
$$235$$ 572.000 2.43404
$$236$$ 464.327i 1.96749i
$$237$$ −120.000 −0.506329
$$238$$ 352.000 1.47899
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 99.4987i 0.414578i
$$241$$ 130.000 0.539419 0.269710 0.962942i $$-0.413072\pi$$
0.269710 + 0.962942i $$0.413072\pi$$
$$242$$ 36.4829i 0.150756i
$$243$$ 243.000 1.00000
$$244$$ −84.0000 −0.344262
$$245$$ 99.4987i 0.406117i
$$246$$ −132.000 −0.536585
$$247$$ −24.0000 −0.0971660
$$248$$ − 258.697i − 1.04313i
$$249$$ 119.398i 0.479512i
$$250$$ 132.000 0.528000
$$251$$ − 291.863i − 1.16280i −0.813618 0.581400i $$-0.802505\pi$$
0.813618 0.581400i $$-0.197495\pi$$
$$252$$ 504.000 2.00000
$$253$$ −22.0000 −0.0869565
$$254$$ − 623.525i − 2.45482i
$$255$$ 264.000 1.03529
$$256$$ −371.000 −1.44922
$$257$$ − 490.860i − 1.90996i −0.296665 0.954981i $$-0.595875\pi$$
0.296665 0.954981i $$-0.404125\pi$$
$$258$$ − 417.895i − 1.61975i
$$259$$ −240.000 −0.926641
$$260$$ − 185.731i − 0.714350i
$$261$$ 358.195i 1.37240i
$$262$$ −132.000 −0.503817
$$263$$ − 225.530i − 0.857530i −0.903416 0.428765i $$-0.858949\pi$$
0.903416 0.428765i $$-0.141051\pi$$
$$264$$ 99.0000 0.375000
$$265$$ −396.000 −1.49434
$$266$$ − 159.198i − 0.598489i
$$267$$ 358.195i 1.34156i
$$268$$ −14.0000 −0.0522388
$$269$$ − 72.9657i − 0.271248i −0.990760 0.135624i $$-0.956696\pi$$
0.990760 0.135624i $$-0.0433039\pi$$
$$270$$ 594.000 2.20000
$$271$$ −448.000 −1.65314 −0.826568 0.562836i $$-0.809710\pi$$
−0.826568 + 0.562836i $$0.809710\pi$$
$$272$$ − 66.3325i − 0.243869i
$$273$$ −96.0000 −0.351648
$$274$$ −352.000 −1.28467
$$275$$ 63.0159i 0.229149i
$$276$$ 139.298i 0.504704i
$$277$$ −260.000 −0.938628 −0.469314 0.883031i $$-0.655499\pi$$
−0.469314 + 0.883031i $$0.655499\pi$$
$$278$$ 245.430i 0.882843i
$$279$$ −234.000 −0.838710
$$280$$ 528.000 1.88571
$$281$$ 198.997i 0.708176i 0.935212 + 0.354088i $$0.115209\pi$$
−0.935212 + 0.354088i $$0.884791\pi$$
$$282$$ −858.000 −3.04255
$$283$$ −50.0000 −0.176678 −0.0883392 0.996090i $$-0.528156\pi$$
−0.0883392 + 0.996090i $$0.528156\pi$$
$$284$$ − 417.895i − 1.47146i
$$285$$ − 119.398i − 0.418942i
$$286$$ −44.0000 −0.153846
$$287$$ 106.132i 0.369798i
$$288$$ 208.947i 0.725512i
$$289$$ 113.000 0.391003
$$290$$ 875.589i 3.01927i
$$291$$ 186.000 0.639175
$$292$$ 518.000 1.77397
$$293$$ − 477.594i − 1.63001i −0.579451 0.815007i $$-0.696733\pi$$
0.579451 0.815007i $$-0.303267\pi$$
$$294$$ − 149.248i − 0.507647i
$$295$$ 440.000 1.49153
$$296$$ 298.496i 1.00843i
$$297$$ − 89.5489i − 0.301511i
$$298$$ 308.000 1.03356
$$299$$ − 26.5330i − 0.0887391i
$$300$$ 399.000 1.33000
$$301$$ −336.000 −1.11628
$$302$$ 530.660i 1.75715i
$$303$$ − 318.396i − 1.05081i
$$304$$ −30.0000 −0.0986842
$$305$$ 79.5990i 0.260980i
$$306$$ −396.000 −1.29412
$$307$$ 86.0000 0.280130 0.140065 0.990142i $$-0.455269\pi$$
0.140065 + 0.990142i $$0.455269\pi$$
$$308$$ − 185.731i − 0.603023i
$$309$$ 222.000 0.718447
$$310$$ −572.000 −1.84516
$$311$$ − 19.8997i − 0.0639863i −0.999488 0.0319932i $$-0.989815\pi$$
0.999488 0.0319932i $$-0.0101855\pi$$
$$312$$ 119.398i 0.382687i
$$313$$ 98.0000 0.313099 0.156550 0.987670i $$-0.449963\pi$$
0.156550 + 0.987670i $$0.449963\pi$$
$$314$$ − 603.626i − 1.92237i
$$315$$ − 477.594i − 1.51617i
$$316$$ 280.000 0.886076
$$317$$ − 311.763i − 0.983479i −0.870742 0.491739i $$-0.836361\pi$$
0.870742 0.491739i $$-0.163639\pi$$
$$318$$ 594.000 1.86792
$$319$$ 132.000 0.413793
$$320$$ 643.425i 2.01070i
$$321$$ 119.398i 0.371958i
$$322$$ 176.000 0.546584
$$323$$ 79.5990i 0.246437i
$$324$$ −567.000 −1.75000
$$325$$ −76.0000 −0.233846
$$326$$ 961.821i 2.95037i
$$327$$ −600.000 −1.83486
$$328$$ 132.000 0.402439
$$329$$ 689.858i 2.09683i
$$330$$ − 218.897i − 0.663325i
$$331$$ −218.000 −0.658610 −0.329305 0.944224i $$-0.606814\pi$$
−0.329305 + 0.944224i $$0.606814\pi$$
$$332$$ − 278.596i − 0.839146i
$$333$$ 270.000 0.810811
$$334$$ 792.000 2.37126
$$335$$ 13.2665i 0.0396015i
$$336$$ −120.000 −0.357143
$$337$$ 278.000 0.824926 0.412463 0.910974i $$-0.364669\pi$$
0.412463 + 0.910974i $$0.364669\pi$$
$$338$$ 507.444i 1.50131i
$$339$$ 119.398i 0.352208i
$$340$$ −616.000 −1.81176
$$341$$ 86.2322i 0.252880i
$$342$$ 179.098i 0.523678i
$$343$$ 272.000 0.793003
$$344$$ 417.895i 1.21481i
$$345$$ 132.000 0.382609
$$346$$ −660.000 −1.90751
$$347$$ − 331.662i − 0.955800i −0.878414 0.477900i $$-0.841398\pi$$
0.878414 0.477900i $$-0.158602\pi$$
$$348$$ − 835.789i − 2.40169i
$$349$$ 324.000 0.928367 0.464183 0.885739i $$-0.346348\pi$$
0.464183 + 0.885739i $$0.346348\pi$$
$$350$$ − 504.127i − 1.44036i
$$351$$ 108.000 0.307692
$$352$$ 77.0000 0.218750
$$353$$ − 397.995i − 1.12746i −0.825958 0.563732i $$-0.809365\pi$$
0.825958 0.563732i $$-0.190635\pi$$
$$354$$ −660.000 −1.86441
$$355$$ −396.000 −1.11549
$$356$$ − 835.789i − 2.34772i
$$357$$ 318.396i 0.891865i
$$358$$ −660.000 −1.84358
$$359$$ 305.129i 0.849943i 0.905207 + 0.424971i $$0.139716\pi$$
−0.905207 + 0.424971i $$0.860284\pi$$
$$360$$ −594.000 −1.65000
$$361$$ −325.000 −0.900277
$$362$$ − 33.1662i − 0.0916195i
$$363$$ −33.0000 −0.0909091
$$364$$ 224.000 0.615385
$$365$$ − 490.860i − 1.34482i
$$366$$ − 119.398i − 0.326225i
$$367$$ −278.000 −0.757493 −0.378747 0.925500i $$-0.623645\pi$$
−0.378747 + 0.925500i $$0.623645\pi$$
$$368$$ − 33.1662i − 0.0901257i
$$369$$ − 119.398i − 0.323573i
$$370$$ 660.000 1.78378
$$371$$ − 477.594i − 1.28732i
$$372$$ 546.000 1.46774
$$373$$ −68.0000 −0.182306 −0.0911528 0.995837i $$-0.529055\pi$$
−0.0911528 + 0.995837i $$0.529055\pi$$
$$374$$ 145.931i 0.390191i
$$375$$ 119.398i 0.318396i
$$376$$ 858.000 2.28191
$$377$$ 159.198i 0.422276i
$$378$$ 716.391i 1.89521i
$$379$$ 670.000 1.76781 0.883905 0.467666i $$-0.154905\pi$$
0.883905 + 0.467666i $$0.154905\pi$$
$$380$$ 278.596i 0.733149i
$$381$$ 564.000 1.48031
$$382$$ 374.000 0.979058
$$383$$ 33.1662i 0.0865959i 0.999062 + 0.0432980i $$0.0137865\pi$$
−0.999062 + 0.0432980i $$0.986214\pi$$
$$384$$ − 686.541i − 1.78787i
$$385$$ −176.000 −0.457143
$$386$$ − 988.354i − 2.56050i
$$387$$ 378.000 0.976744
$$388$$ −434.000 −1.11856
$$389$$ 417.895i 1.07428i 0.843493 + 0.537140i $$0.180495\pi$$
−0.843493 + 0.537140i $$0.819505\pi$$
$$390$$ 264.000 0.676923
$$391$$ −88.0000 −0.225064
$$392$$ 149.248i 0.380735i
$$393$$ − 119.398i − 0.303813i
$$394$$ −440.000 −1.11675
$$395$$ − 265.330i − 0.671721i
$$396$$ 208.947i 0.527645i
$$397$$ −86.0000 −0.216625 −0.108312 0.994117i $$-0.534545\pi$$
−0.108312 + 0.994117i $$0.534545\pi$$
$$398$$ 139.298i 0.349996i
$$399$$ 144.000 0.360902
$$400$$ −95.0000 −0.237500
$$401$$ 252.063i 0.628587i 0.949326 + 0.314294i $$0.101768\pi$$
−0.949326 + 0.314294i $$0.898232\pi$$
$$402$$ − 19.8997i − 0.0495019i
$$403$$ −104.000 −0.258065
$$404$$ 742.924i 1.83892i
$$405$$ 537.293i 1.32665i
$$406$$ −1056.00 −2.60099
$$407$$ − 99.4987i − 0.244469i
$$408$$ 396.000 0.970588
$$409$$ 510.000 1.24694 0.623472 0.781846i $$-0.285722\pi$$
0.623472 + 0.781846i $$0.285722\pi$$
$$410$$ − 291.863i − 0.711861i
$$411$$ − 318.396i − 0.774686i
$$412$$ −518.000 −1.25728
$$413$$ 530.660i 1.28489i
$$414$$ −198.000 −0.478261
$$415$$ −264.000 −0.636145
$$416$$ 92.8655i 0.223234i
$$417$$ −222.000 −0.532374
$$418$$ 66.0000 0.157895
$$419$$ 530.660i 1.26649i 0.773951 + 0.633246i $$0.218278\pi$$
−0.773951 + 0.633246i $$0.781722\pi$$
$$420$$ 1114.39i 2.65330i
$$421$$ −170.000 −0.403800 −0.201900 0.979406i $$-0.564712\pi$$
−0.201900 + 0.979406i $$0.564712\pi$$
$$422$$ − 815.890i − 1.93339i
$$423$$ − 776.090i − 1.83473i
$$424$$ −594.000 −1.40094
$$425$$ 252.063i 0.593091i
$$426$$ 594.000 1.39437
$$427$$ −96.0000 −0.224824
$$428$$ − 278.596i − 0.650926i
$$429$$ − 39.7995i − 0.0927727i
$$430$$ 924.000 2.14884
$$431$$ − 278.596i − 0.646396i −0.946331 0.323198i $$-0.895242\pi$$
0.946331 0.323198i $$-0.104758\pi$$
$$432$$ 135.000 0.312500
$$433$$ −542.000 −1.25173 −0.625866 0.779931i $$-0.715254\pi$$
−0.625866 + 0.779931i $$0.715254\pi$$
$$434$$ − 689.858i − 1.58953i
$$435$$ −792.000 −1.82069
$$436$$ 1400.00 3.21101
$$437$$ 39.7995i 0.0910744i
$$438$$ 736.291i 1.68103i
$$439$$ 328.000 0.747153 0.373576 0.927599i $$-0.378131\pi$$
0.373576 + 0.927599i $$0.378131\pi$$
$$440$$ 218.897i 0.497494i
$$441$$ 135.000 0.306122
$$442$$ −176.000 −0.398190
$$443$$ − 132.665i − 0.299470i −0.988726 0.149735i $$-0.952158\pi$$
0.988726 0.149735i $$-0.0478420\pi$$
$$444$$ −630.000 −1.41892
$$445$$ −792.000 −1.77978
$$446$$ 1001.62i 2.24579i
$$447$$ 278.596i 0.623258i
$$448$$ −776.000 −1.73214
$$449$$ 451.061i 1.00459i 0.864696 + 0.502295i $$0.167511\pi$$
−0.864696 + 0.502295i $$0.832489\pi$$
$$450$$ 567.143i 1.26032i
$$451$$ −44.0000 −0.0975610
$$452$$ − 278.596i − 0.616364i
$$453$$ −480.000 −1.05960
$$454$$ 660.000 1.45374
$$455$$ − 212.264i − 0.466514i
$$456$$ − 179.098i − 0.392758i
$$457$$ 342.000 0.748359 0.374179 0.927356i $$-0.377924\pi$$
0.374179 + 0.927356i $$0.377924\pi$$
$$458$$ − 497.494i − 1.08623i
$$459$$ − 358.195i − 0.780382i
$$460$$ −308.000 −0.669565
$$461$$ 79.5990i 0.172666i 0.996266 + 0.0863330i $$0.0275149\pi$$
−0.996266 + 0.0863330i $$0.972485\pi$$
$$462$$ 264.000 0.571429
$$463$$ −86.0000 −0.185745 −0.0928726 0.995678i $$-0.529605\pi$$
−0.0928726 + 0.995678i $$0.529605\pi$$
$$464$$ 198.997i 0.428874i
$$465$$ − 517.393i − 1.11267i
$$466$$ 1452.00 3.11588
$$467$$ 596.992i 1.27836i 0.769059 + 0.639178i $$0.220725\pi$$
−0.769059 + 0.639178i $$0.779275\pi$$
$$468$$ −252.000 −0.538462
$$469$$ −16.0000 −0.0341151
$$470$$ − 1897.11i − 4.03640i
$$471$$ 546.000 1.15924
$$472$$ 660.000 1.39831
$$473$$ − 139.298i − 0.294499i
$$474$$ 397.995i 0.839652i
$$475$$ 114.000 0.240000
$$476$$ − 742.924i − 1.56076i
$$477$$ 537.293i 1.12640i
$$478$$ 0 0
$$479$$ − 225.530i − 0.470836i −0.971894 0.235418i $$-0.924354\pi$$
0.971894 0.235418i $$-0.0756459\pi$$
$$480$$ −462.000 −0.962500
$$481$$ 120.000 0.249480
$$482$$ − 431.161i − 0.894525i
$$483$$ 159.198i 0.329602i
$$484$$ 77.0000 0.159091
$$485$$ 411.261i 0.847962i
$$486$$ − 805.940i − 1.65831i
$$487$$ 446.000 0.915811 0.457906 0.889001i $$-0.348600\pi$$
0.457906 + 0.889001i $$0.348600\pi$$
$$488$$ 119.398i 0.244669i
$$489$$ −870.000 −1.77914
$$490$$ 330.000 0.673469
$$491$$ 703.124i 1.43203i 0.698087 + 0.716013i $$0.254035\pi$$
−0.698087 + 0.716013i $$0.745965\pi$$
$$492$$ 278.596i 0.566253i
$$493$$ 528.000 1.07099
$$494$$ 79.5990i 0.161132i
$$495$$ 198.000 0.400000
$$496$$ −130.000 −0.262097
$$497$$ − 477.594i − 0.960954i
$$498$$ 396.000 0.795181
$$499$$ −58.0000 −0.116232 −0.0581162 0.998310i $$-0.518509\pi$$
−0.0581162 + 0.998310i $$0.518509\pi$$
$$500$$ − 278.596i − 0.557193i
$$501$$ 716.391i 1.42992i
$$502$$ −968.000 −1.92829
$$503$$ 504.127i 1.00224i 0.865378 + 0.501120i $$0.167079\pi$$
−0.865378 + 0.501120i $$0.832921\pi$$
$$504$$ − 716.391i − 1.42141i
$$505$$ 704.000 1.39406
$$506$$ 72.9657i 0.144201i
$$507$$ −459.000 −0.905325
$$508$$ −1316.00 −2.59055
$$509$$ − 736.291i − 1.44654i −0.690563 0.723272i $$-0.742637\pi$$
0.690563 0.723272i $$-0.257363\pi$$
$$510$$ − 875.589i − 1.71684i
$$511$$ 592.000 1.15851
$$512$$ 315.079i 0.615389i
$$513$$ −162.000 −0.315789
$$514$$ −1628.00 −3.16732
$$515$$ 490.860i 0.953127i
$$516$$ −882.000 −1.70930
$$517$$ −286.000 −0.553191
$$518$$ 795.990i 1.53666i
$$519$$ − 596.992i − 1.15027i
$$520$$ −264.000 −0.507692
$$521$$ − 278.596i − 0.534734i −0.963595 0.267367i $$-0.913846\pi$$
0.963595 0.267367i $$-0.0861536\pi$$
$$522$$ 1188.00 2.27586
$$523$$ −142.000 −0.271511 −0.135755 0.990742i $$-0.543346\pi$$
−0.135755 + 0.990742i $$0.543346\pi$$
$$524$$ 278.596i 0.531673i
$$525$$ 456.000 0.868571
$$526$$ −748.000 −1.42205
$$527$$ 344.929i 0.654514i
$$528$$ − 49.7494i − 0.0942223i
$$529$$ 485.000 0.916824
$$530$$ 1313.38i 2.47808i
$$531$$ − 596.992i − 1.12428i
$$532$$ −336.000 −0.631579
$$533$$ − 53.0660i − 0.0995610i
$$534$$ 1188.00 2.22472
$$535$$ −264.000 −0.493458
$$536$$ 19.8997i 0.0371264i
$$537$$ − 596.992i − 1.11172i
$$538$$ −242.000 −0.449814
$$539$$ − 49.7494i − 0.0922994i
$$540$$ − 1253.68i − 2.32164i
$$541$$ 400.000 0.739372 0.369686 0.929157i $$-0.379465\pi$$
0.369686 + 0.929157i $$0.379465\pi$$
$$542$$ 1485.85i 2.74142i
$$543$$ 30.0000 0.0552486
$$544$$ 308.000 0.566176
$$545$$ − 1326.65i − 2.43422i
$$546$$ 318.396i 0.583143i
$$547$$ 170.000 0.310786 0.155393 0.987853i $$-0.450336\pi$$
0.155393 + 0.987853i $$0.450336\pi$$
$$548$$ 742.924i 1.35570i
$$549$$ 108.000 0.196721
$$550$$ 209.000 0.380000
$$551$$ − 238.797i − 0.433388i
$$552$$ 198.000 0.358696
$$553$$ 320.000 0.578662
$$554$$ 862.322i 1.55654i
$$555$$ 596.992i 1.07566i
$$556$$ 518.000 0.931655
$$557$$ − 849.056i − 1.52434i −0.647378 0.762169i $$-0.724135\pi$$
0.647378 0.762169i $$-0.275865\pi$$
$$558$$ 776.090i 1.39084i
$$559$$ 168.000 0.300537
$$560$$ − 265.330i − 0.473804i
$$561$$ −132.000 −0.235294
$$562$$ 660.000 1.17438
$$563$$ 703.124i 1.24889i 0.781069 + 0.624444i $$0.214675\pi$$
−0.781069 + 0.624444i $$0.785325\pi$$
$$564$$ 1810.88i 3.21078i
$$565$$ −264.000 −0.467257
$$566$$ 165.831i 0.292988i
$$567$$ −648.000 −1.14286
$$568$$ −594.000 −1.04577
$$569$$ 119.398i 0.209839i 0.994481 + 0.104920i $$0.0334585\pi$$
−0.994481 + 0.104920i $$0.966541\pi$$
$$570$$ −396.000 −0.694737
$$571$$ −706.000 −1.23643 −0.618214 0.786010i $$-0.712143\pi$$
−0.618214 + 0.786010i $$0.712143\pi$$
$$572$$ 92.8655i 0.162352i
$$573$$ 338.296i 0.590394i
$$574$$ 352.000 0.613240
$$575$$ 126.032i 0.219186i
$$576$$ 873.000 1.51562
$$577$$ −738.000 −1.27903 −0.639515 0.768779i $$-0.720865\pi$$
−0.639515 + 0.768779i $$0.720865\pi$$
$$578$$ − 374.779i − 0.648406i
$$579$$ 894.000 1.54404
$$580$$ 1848.00 3.18621
$$581$$ − 318.396i − 0.548014i
$$582$$ − 616.892i − 1.05995i
$$583$$ 198.000 0.339623
$$584$$ − 736.291i − 1.26077i
$$585$$ 238.797i 0.408200i
$$586$$ −1584.00 −2.70307
$$587$$ 941.921i 1.60464i 0.596897 + 0.802318i $$0.296400\pi$$
−0.596897 + 0.802318i $$0.703600\pi$$
$$588$$ −315.000 −0.535714
$$589$$ 156.000 0.264856
$$590$$ − 1459.31i − 2.47342i
$$591$$ − 397.995i − 0.673426i
$$592$$ 150.000 0.253378
$$593$$ − 543.926i − 0.917245i −0.888631 0.458623i $$-0.848343\pi$$
0.888631 0.458623i $$-0.151657\pi$$
$$594$$ −297.000 −0.500000
$$595$$ −704.000 −1.18319
$$596$$ − 650.058i − 1.09070i
$$597$$ −126.000 −0.211055
$$598$$ −88.0000 −0.147157
$$599$$ 46.4327i 0.0775171i 0.999249 + 0.0387586i $$0.0123403\pi$$
−0.999249 + 0.0387586i $$0.987660\pi$$
$$600$$ − 567.143i − 0.945238i
$$601$$ 542.000 0.901830 0.450915 0.892567i $$-0.351098\pi$$
0.450915 + 0.892567i $$0.351098\pi$$
$$602$$ 1114.39i 1.85114i
$$603$$ 18.0000 0.0298507
$$604$$ 1120.00 1.85430
$$605$$ − 72.9657i − 0.120605i
$$606$$ −1056.00 −1.74257
$$607$$ −700.000 −1.15321 −0.576606 0.817022i $$-0.695623\pi$$
−0.576606 + 0.817022i $$0.695623\pi$$
$$608$$ − 139.298i − 0.229109i
$$609$$ − 955.188i − 1.56845i
$$610$$ 264.000 0.432787
$$611$$ − 344.929i − 0.564532i
$$612$$ 835.789i 1.36567i
$$613$$ 764.000 1.24633 0.623165 0.782091i $$-0.285847\pi$$
0.623165 + 0.782091i $$0.285847\pi$$
$$614$$ − 285.230i − 0.464544i
$$615$$ 264.000 0.429268
$$616$$ −264.000 −0.428571
$$617$$ − 39.7995i − 0.0645049i −0.999480 0.0322524i $$-0.989732\pi$$
0.999480 0.0322524i $$-0.0102680\pi$$
$$618$$ − 736.291i − 1.19141i
$$619$$ −742.000 −1.19871 −0.599354 0.800484i $$-0.704576\pi$$
−0.599354 + 0.800484i $$0.704576\pi$$
$$620$$ 1207.25i 1.94718i
$$621$$ − 179.098i − 0.288402i
$$622$$ −66.0000 −0.106109
$$623$$ − 955.188i − 1.53321i
$$624$$ 60.0000 0.0961538
$$625$$ −739.000 −1.18240
$$626$$ − 325.029i − 0.519216i
$$627$$ 59.6992i 0.0952141i
$$628$$ −1274.00 −2.02866
$$629$$ − 397.995i − 0.632742i
$$630$$ −1584.00 −2.51429
$$631$$ −410.000 −0.649762 −0.324881 0.945755i $$-0.605324\pi$$
−0.324881 + 0.945755i $$0.605324\pi$$
$$632$$ − 397.995i − 0.629739i
$$633$$ 738.000 1.16588
$$634$$ −1034.00 −1.63091
$$635$$ 1247.05i 1.96386i
$$636$$ − 1253.68i − 1.97120i
$$637$$ 60.0000 0.0941915
$$638$$ − 437.794i − 0.686198i
$$639$$ 537.293i 0.840834i
$$640$$ 1518.00 2.37188
$$641$$ 756.190i 1.17970i 0.807511 + 0.589852i $$0.200814\pi$$
−0.807511 + 0.589852i $$0.799186\pi$$
$$642$$ 396.000 0.616822
$$643$$ 890.000 1.38414 0.692068 0.721832i $$-0.256700\pi$$
0.692068 + 0.721832i $$0.256700\pi$$
$$644$$ − 371.462i − 0.576804i
$$645$$ 835.789i 1.29580i
$$646$$ 264.000 0.408669
$$647$$ − 484.227i − 0.748419i −0.927344 0.374210i $$-0.877914\pi$$
0.927344 0.374210i $$-0.122086\pi$$
$$648$$ 805.940i 1.24373i
$$649$$ −220.000 −0.338983
$$650$$ 252.063i 0.387790i
$$651$$ 624.000 0.958525
$$652$$ 2030.00 3.11350
$$653$$ − 391.362i − 0.599329i −0.954045 0.299664i $$-0.903125\pi$$
0.954045 0.299664i $$-0.0968747\pi$$
$$654$$ 1989.97i 3.04278i
$$655$$ 264.000 0.403053
$$656$$ − 66.3325i − 0.101117i
$$657$$ −666.000 −1.01370
$$658$$ 2288.00 3.47720
$$659$$ 384.728i 0.583806i 0.956448 + 0.291903i $$0.0942885\pi$$
−0.956448 + 0.291903i $$0.905711\pi$$
$$660$$ −462.000 −0.700000
$$661$$ −746.000 −1.12859 −0.564297 0.825572i $$-0.690853\pi$$
−0.564297 + 0.825572i $$0.690853\pi$$
$$662$$ 723.024i 1.09218i
$$663$$ − 159.198i − 0.240118i
$$664$$ −396.000 −0.596386
$$665$$ 318.396i 0.478791i
$$666$$ − 895.489i − 1.34458i
$$667$$ 264.000 0.395802
$$668$$ − 1671.58i − 2.50236i
$$669$$ −906.000 −1.35426
$$670$$ 44.0000 0.0656716
$$671$$ − 39.7995i − 0.0593137i
$$672$$ − 557.193i − 0.829156i
$$673$$ −634.000 −0.942051 −0.471025 0.882120i $$-0.656116\pi$$
−0.471025 + 0.882120i $$0.656116\pi$$
$$674$$ − 922.022i − 1.36798i
$$675$$ −513.000 −0.760000
$$676$$ 1071.00 1.58432
$$677$$ − 795.990i − 1.17576i −0.808948 0.587880i $$-0.799963\pi$$
0.808948 0.587880i $$-0.200037\pi$$
$$678$$ 396.000 0.584071
$$679$$ −496.000 −0.730486
$$680$$ 875.589i 1.28763i
$$681$$ 596.992i 0.876641i
$$682$$ 286.000 0.419355
$$683$$ 451.061i 0.660411i 0.943909 + 0.330206i $$0.107118\pi$$
−0.943909 + 0.330206i $$0.892882\pi$$
$$684$$ 378.000 0.552632
$$685$$ 704.000 1.02774
$$686$$ − 902.122i − 1.31505i
$$687$$ 450.000 0.655022
$$688$$ 210.000 0.305233
$$689$$ 238.797i 0.346585i
$$690$$ − 437.794i − 0.634485i
$$691$$ 458.000 0.662808 0.331404 0.943489i $$-0.392478\pi$$
0.331404 + 0.943489i $$0.392478\pi$$
$$692$$ 1392.98i 2.01298i
$$693$$ 238.797i 0.344584i
$$694$$ −1100.00 −1.58501
$$695$$ − 490.860i − 0.706274i
$$696$$ −1188.00 −1.70690
$$697$$ −176.000 −0.252511
$$698$$ − 1074.59i − 1.53952i
$$699$$ 1313.38i 1.87895i
$$700$$ −1064.00 −1.52000
$$701$$ − 504.127i − 0.719154i −0.933115 0.359577i $$-0.882921\pi$$
0.933115 0.359577i $$-0.117079\pi$$
$$702$$ − 358.195i − 0.510250i
$$703$$ −180.000 −0.256046
$$704$$ − 321.713i − 0.456978i
$$705$$ 1716.00 2.43404
$$706$$ −1320.00 −1.86969
$$707$$ 849.056i 1.20093i
$$708$$ 1392.98i 1.96749i
$$709$$ −562.000 −0.792666 −0.396333 0.918107i $$-0.629717\pi$$
−0.396333 + 0.918107i $$0.629717\pi$$
$$710$$ 1313.38i 1.84984i
$$711$$ −360.000 −0.506329
$$712$$ −1188.00 −1.66854
$$713$$ 172.464i 0.241886i
$$714$$ 1056.00 1.47899
$$715$$ 88.0000 0.123077
$$716$$ 1392.98i 1.94551i
$$717$$ 0 0
$$718$$ 1012.00 1.40947
$$719$$ − 338.296i − 0.470509i −0.971934 0.235254i $$-0.924408\pi$$
0.971934 0.235254i $$-0.0755923\pi$$
$$720$$ 298.496i 0.414578i
$$721$$ −592.000 −0.821082
$$722$$ 1077.90i 1.49294i
$$723$$ 390.000 0.539419
$$724$$ −70.0000 −0.0966851
$$725$$ − 756.190i − 1.04302i
$$726$$ 109.449i 0.150756i
$$727$$ −42.0000 −0.0577717 −0.0288858 0.999583i $$-0.509196\pi$$
−0.0288858 + 0.999583i $$0.509196\pi$$
$$728$$ − 318.396i − 0.437357i
$$729$$ 729.000 1.00000
$$730$$ −1628.00 −2.23014
$$731$$ − 557.193i − 0.762234i
$$732$$ −252.000 −0.344262
$$733$$ −624.000 −0.851296 −0.425648 0.904889i $$-0.639954\pi$$
−0.425648 + 0.904889i $$0.639954\pi$$
$$734$$ 922.022i 1.25616i
$$735$$ 298.496i 0.406117i
$$736$$ 154.000 0.209239
$$737$$ − 6.63325i − 0.00900034i
$$738$$ −396.000 −0.536585
$$739$$ 686.000 0.928281 0.464141 0.885761i $$-0.346363\pi$$
0.464141 + 0.885761i $$0.346363\pi$$
$$740$$ − 1392.98i − 1.88241i
$$741$$ −72.0000 −0.0971660
$$742$$ −1584.00 −2.13477
$$743$$ 862.322i 1.16060i 0.814404 + 0.580298i $$0.197064\pi$$
−0.814404 + 0.580298i $$0.802936\pi$$
$$744$$ − 776.090i − 1.04313i
$$745$$ −616.000 −0.826846
$$746$$ 225.530i 0.302320i
$$747$$ 358.195i 0.479512i
$$748$$ 308.000 0.411765
$$749$$ − 318.396i − 0.425095i
$$750$$ 396.000 0.528000
$$751$$ 94.0000 0.125166 0.0625832 0.998040i $$-0.480066\pi$$
0.0625832 + 0.998040i $$0.480066\pi$$
$$752$$ − 431.161i − 0.573353i
$$753$$ − 875.589i − 1.16280i
$$754$$ 528.000 0.700265
$$755$$ − 1061.32i − 1.40572i
$$756$$ 1512.00 2.00000
$$757$$ 1118.00 1.47688 0.738441 0.674318i $$-0.235562\pi$$
0.738441 + 0.674318i $$0.235562\pi$$
$$758$$ − 2222.14i − 2.93158i
$$759$$ −66.0000 −0.0869565
$$760$$ 396.000 0.521053
$$761$$ 1154.19i 1.51667i 0.651865 + 0.758335i $$0.273987\pi$$
−0.651865 + 0.758335i $$0.726013\pi$$
$$762$$ − 1870.58i − 2.45482i
$$763$$ 1600.00 2.09699
$$764$$ − 789.357i − 1.03319i
$$765$$ 792.000 1.03529
$$766$$ 110.000 0.143603
$$767$$ − 265.330i − 0.345932i
$$768$$ −1113.00 −1.44922
$$769$$ 1274.00 1.65670 0.828349 0.560213i $$-0.189281\pi$$
0.828349 + 0.560213i $$0.189281\pi$$
$$770$$ 583.726i 0.758086i
$$771$$ − 1472.58i − 1.90996i
$$772$$ −2086.00 −2.70207
$$773$$ 935.288i 1.20995i 0.796246 + 0.604973i $$0.206816\pi$$
−0.796246 + 0.604973i $$0.793184\pi$$
$$774$$ − 1253.68i − 1.61975i
$$775$$ 494.000 0.637419
$$776$$ 616.892i 0.794964i
$$777$$ −720.000 −0.926641
$$778$$ 1386.00 1.78149
$$779$$ 79.5990i 0.102181i
$$780$$ − 557.193i − 0.714350i
$$781$$ 198.000 0.253521
$$782$$ 291.863i 0.373226i
$$783$$ 1074.59i 1.37240i
$$784$$ 75.0000 0.0956633
$$785$$ 1207.25i 1.53790i
$$786$$ −396.000 −0.503817
$$787$$ 298.000 0.378653 0.189327 0.981914i $$-0.439370\pi$$
0.189327 + 0.981914i $$0.439370\pi$$
$$788$$ 928.655i 1.17850i
$$789$$ − 676.591i − 0.857530i
$$790$$ −880.000 −1.11392
$$791$$ − 318.396i − 0.402523i
$$792$$ 297.000 0.375000
$$793$$ 48.0000 0.0605296
$$794$$ 285.230i 0.359231i
$$795$$ −1188.00 −1.49434
$$796$$ 294.000 0.369347
$$797$$ − 524.027i − 0.657499i −0.944417 0.328750i $$-0.893373\pi$$
0.944417 0.328750i $$-0.106627\pi$$
$$798$$ − 477.594i − 0.598489i
$$799$$ −1144.00 −1.43179
$$800$$ − 441.111i − 0.551389i
$$801$$ 1074.59i 1.34156i
$$802$$ 836.000 1.04239
$$803$$ 245.430i 0.305642i
$$804$$ −42.0000 −0.0522388
$$805$$ −352.000 −0.437267
$$806$$ 344.929i 0.427952i
$$807$$ − 218.897i − 0.271248i
$$808$$ 1056.00 1.30693
$$809$$ 915.388i 1.13151i 0.824575 + 0.565753i $$0.191414\pi$$
−0.824575 + 0.565753i $$0.808586\pi$$
$$810$$ 1782.00 2.20000
$$811$$ −182.000 −0.224414 −0.112207 0.993685i $$-0.535792\pi$$
−0.112207 + 0.993685i $$0.535792\pi$$
$$812$$ 2228.77i 2.74479i
$$813$$ −1344.00 −1.65314
$$814$$ −330.000 −0.405405
$$815$$ − 1923.64i − 2.36030i
$$816$$ − 198.997i − 0.243869i
$$817$$ −252.000 −0.308446
$$818$$ − 1691.48i − 2.06782i
$$819$$ −288.000 −0.351648
$$820$$ −616.000 −0.751220
$$821$$ 822.523i 1.00185i 0.865489 + 0.500927i $$0.167008\pi$$
−0.865489 + 0.500927i $$0.832992\pi$$
$$822$$ −1056.00 −1.28467
$$823$$ −246.000 −0.298906 −0.149453 0.988769i $$-0.547751\pi$$
−0.149453 + 0.988769i $$0.547751\pi$$
$$824$$ 736.291i 0.893557i
$$825$$ 189.048i 0.229149i
$$826$$ 1760.00 2.13075
$$827$$ − 543.926i − 0.657710i −0.944380 0.328855i $$-0.893337\pi$$
0.944380 0.328855i $$-0.106663\pi$$
$$828$$ 417.895i 0.504704i
$$829$$ 250.000 0.301568 0.150784 0.988567i $$-0.451820\pi$$
0.150784 + 0.988567i $$0.451820\pi$$
$$830$$ 875.589i 1.05493i
$$831$$ −780.000 −0.938628
$$832$$ 388.000 0.466346
$$833$$ − 198.997i − 0.238893i
$$834$$ 736.291i 0.882843i
$$835$$ −1584.00 −1.89701
$$836$$ − 139.298i − 0.166625i
$$837$$ −702.000 −0.838710
$$838$$ 1760.00 2.10024
$$839$$ − 470.961i − 0.561336i −0.959805 0.280668i $$-0.909444\pi$$
0.959805 0.280668i $$-0.0905559\pi$$
$$840$$ 1584.00 1.88571
$$841$$ −743.000 −0.883472
$$842$$ 563.826i 0.669627i
$$843$$ 596.992i 0.708176i
$$844$$ −1722.00 −2.04028
$$845$$ − 1014.89i − 1.20105i
$$846$$ −2574.00 −3.04255
$$847$$ 88.0000 0.103896
$$848$$ 298.496i 0.352000i
$$849$$ −150.000 −0.176678
$$850$$ 836.000 0.983529
$$851$$ − 198.997i − 0.233840i
$$852$$ − 1253.68i − 1.47146i
$$853$$ 56.0000 0.0656506 0.0328253 0.999461i $$-0.489549\pi$$
0.0328253 + 0.999461i $$0.489549\pi$$
$$854$$ 318.396i 0.372829i
$$855$$ − 358.195i − 0.418942i
$$856$$ −396.000 −0.462617
$$857$$ 252.063i 0.294123i 0.989127 + 0.147062i $$0.0469815\pi$$
−0.989127 + 0.147062i $$0.953018\pi$$
$$858$$ −132.000 −0.153846
$$859$$ 1278.00 1.48778 0.743888 0.668304i $$-0.232979\pi$$
0.743888 + 0.668304i $$0.232979\pi$$
$$860$$ − 1950.18i − 2.26765i
$$861$$ 318.396i 0.369798i
$$862$$ −924.000 −1.07193
$$863$$ 1134.29i 1.31435i 0.753737 + 0.657176i $$0.228249\pi$$
−0.753737 + 0.657176i $$0.771751\pi$$
$$864$$ 626.842i 0.725512i
$$865$$ 1320.00 1.52601
$$866$$ 1797.61i 2.07576i
$$867$$ 339.000 0.391003
$$868$$ −1456.00 −1.67742
$$869$$ 132.665i 0.152664i
$$870$$ 2626.77i 3.01927i
$$871$$ 8.00000 0.00918485
$$872$$ − 1989.97i − 2.28208i
$$873$$ 558.000 0.639175
$$874$$ 132.000 0.151030
$$875$$ − 318.396i − 0.363881i
$$876$$ 1554.00 1.77397
$$877$$ 456.000 0.519954 0.259977 0.965615i $$-0.416285\pi$$
0.259977 + 0.965615i $$0.416285\pi$$
$$878$$ − 1087.85i − 1.23901i
$$879$$ − 1432.78i − 1.63001i
$$880$$ 110.000 0.125000
$$881$$ 610.259i 0.692689i 0.938107 + 0.346344i $$0.112577\pi$$
−0.938107 + 0.346344i $$0.887423\pi$$
$$882$$ − 447.744i − 0.507647i
$$883$$ −1094.00 −1.23896 −0.619479 0.785013i $$-0.712656\pi$$
−0.619479 + 0.785013i $$0.712656\pi$$
$$884$$ 371.462i 0.420206i
$$885$$ 1320.00 1.49153
$$886$$ −440.000 −0.496614
$$887$$ − 145.931i − 0.164523i −0.996611 0.0822613i $$-0.973786\pi$$
0.996611 0.0822613i $$-0.0262142\pi$$
$$888$$ 895.489i 1.00843i
$$889$$ −1504.00 −1.69179
$$890$$ 2626.77i 2.95142i
$$891$$ − 268.647i − 0.301511i
$$892$$ 2114.00 2.36996
$$893$$ 517.393i 0.579388i
$$894$$ 924.000 1.03356
$$895$$ 1320.00 1.47486
$$896$$ 1830.78i 2.04328i
$$897$$ − 79.5990i − 0.0887391i
$$898$$ 1496.00 1.66592
$$899$$ − 1034.79i − 1.15104i
$$900$$ 1197.00 1.33000
$$901$$ 792.000 0.879023
$$902$$ 145.931i 0.161787i
$$903$$ −1008.00 −1.11628
$$904$$ −396.000 −0.438053
$$905$$ 66.3325i 0.0732956i
$$906$$ 1591.98i 1.75715i
$$907$$ 186.000 0.205072 0.102536 0.994729i $$-0.467304\pi$$
0.102536 + 0.994729i $$0.467304\pi$$
$$908$$ − 1392.98i − 1.53412i
$$909$$ − 955.188i − 1.05081i
$$910$$ −704.000 −0.773626
$$911$$ − 855.689i − 0.939286i −0.882857 0.469643i $$-0.844383\pi$$
0.882857 0.469643i $$-0.155617\pi$$
$$912$$ −90.0000 −0.0986842
$$913$$ 132.000 0.144578
$$914$$ − 1134.29i − 1.24101i
$$915$$ 238.797i 0.260980i
$$916$$ −1050.00 −1.14629
$$917$$ 318.396i 0.347215i
$$918$$ −1188.00 −1.29412
$$919$$ −428.000 −0.465724 −0.232862 0.972510i $$-0.574809\pi$$
−0.232862 + 0.972510i $$0.574809\pi$$
$$920$$ 437.794i 0.475864i
$$921$$ 258.000 0.280130
$$922$$ 264.000 0.286334
$$923$$ 238.797i 0.258718i
$$924$$ − 557.193i − 0.603023i
$$925$$ −570.000 −0.616216
$$926$$ 285.230i 0.308023i
$$927$$ 666.000 0.718447
$$928$$ −924.000 −0.995690
$$929$$ − 636.792i − 0.685460i −0.939434 0.342730i $$-0.888648\pi$$
0.939434 0.342730i $$-0.111352\pi$$
$$930$$ −1716.00 −1.84516
$$931$$ −90.0000 −0.0966702
$$932$$ − 3064.56i − 3.28816i
$$933$$ − 59.6992i − 0.0639863i
$$934$$ 1980.00 2.11991
$$935$$ − 291.863i − 0.312153i
$$936$$ 358.195i 0.382687i
$$937$$ 290.000 0.309498 0.154749 0.987954i $$-0.450543\pi$$
0.154749 + 0.987954i $$0.450543\pi$$
$$938$$ 53.0660i 0.0565736i
$$939$$ 294.000 0.313099
$$940$$ −4004.00 −4.25957
$$941$$ − 875.589i − 0.930488i −0.885183 0.465244i $$-0.845967\pi$$
0.885183 0.465244i $$-0.154033\pi$$
$$942$$ − 1810.88i − 1.92237i
$$943$$ −88.0000 −0.0933192
$$944$$ − 331.662i − 0.351337i
$$945$$ − 1432.78i − 1.51617i
$$946$$ −462.000 −0.488372
$$947$$ 79.5990i 0.0840538i 0.999116 + 0.0420269i $$0.0133815\pi$$
−0.999116 + 0.0420269i $$0.986618\pi$$
$$948$$ 840.000 0.886076
$$949$$ −296.000 −0.311907
$$950$$ − 378.095i − 0.397995i
$$951$$ − 935.288i − 0.983479i
$$952$$ −1056.00 −1.10924
$$953$$ 782.723i 0.821326i 0.911787 + 0.410663i $$0.134703\pi$$
−0.911787 + 0.410663i $$0.865297\pi$$
$$954$$ 1782.00 1.86792
$$955$$ −748.000 −0.783246
$$956$$ 0 0
$$957$$ 396.000 0.413793
$$958$$ −748.000 −0.780793
$$959$$ 849.056i 0.885356i
$$960$$ 1930.28i 2.01070i
$$961$$ −285.000 −0.296566
$$962$$ − 397.995i − 0.413716i
$$963$$ 358.195i 0.371958i
$$964$$ −910.000 −0.943983
$$965$$ 1976.71i 2.04840i
$$966$$ 528.000 0.546584
$$967$$ 460.000 0.475698 0.237849 0.971302i $$-0.423558\pi$$
0.237849 + 0.971302i $$0.423558\pi$$
$$968$$ − 109.449i − 0.113067i
$$969$$ 238.797i 0.246437i
$$970$$ 1364.00 1.40619
$$971$$ 1167.45i 1.20232i 0.799129 + 0.601160i $$0.205294\pi$$
−0.799129 + 0.601160i $$0.794706\pi$$
$$972$$ −1701.00 −1.75000
$$973$$ 592.000 0.608428
$$974$$ − 1479.21i − 1.51870i
$$975$$ −228.000 −0.233846
$$976$$ 60.0000 0.0614754
$$977$$ − 1313.38i − 1.34430i −0.740414 0.672151i $$-0.765370\pi$$
0.740414 0.672151i $$-0.234630\pi$$
$$978$$ 2885.46i 2.95037i
$$979$$ 396.000 0.404494
$$980$$ − 696.491i − 0.710705i
$$981$$ −1800.00 −1.83486
$$982$$ 2332.00 2.37475
$$983$$ − 417.895i − 0.425122i −0.977148 0.212561i $$-0.931820\pi$$
0.977148 0.212561i $$-0.0681804\pi$$
$$984$$ 396.000 0.402439
$$985$$ 880.000 0.893401
$$986$$ − 1751.18i − 1.77604i
$$987$$ 2069.57i 2.09683i
$$988$$ 168.000 0.170040
$$989$$ − 278.596i − 0.281695i
$$990$$ − 656.692i − 0.663325i
$$991$$ 838.000 0.845610 0.422805 0.906221i $$-0.361045\pi$$
0.422805 + 0.906221i $$0.361045\pi$$
$$992$$ − 603.626i − 0.608494i
$$993$$ −654.000 −0.658610
$$994$$ −1584.00 −1.59356
$$995$$ − 278.596i − 0.279996i
$$996$$ − 835.789i − 0.839146i
$$997$$ −52.0000 −0.0521565 −0.0260782 0.999660i $$-0.508302\pi$$
−0.0260782 + 0.999660i $$0.508302\pi$$
$$998$$ 192.364i 0.192750i
$$999$$ 810.000 0.810811
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 33.3.b.a.23.1 2
3.2 odd 2 inner 33.3.b.a.23.2 yes 2
4.3 odd 2 528.3.i.a.353.2 2
11.2 odd 10 363.3.h.d.323.2 8
11.3 even 5 363.3.h.e.251.1 8
11.4 even 5 363.3.h.e.269.2 8
11.5 even 5 363.3.h.e.245.2 8
11.6 odd 10 363.3.h.d.245.1 8
11.7 odd 10 363.3.h.d.269.1 8
11.8 odd 10 363.3.h.d.251.2 8
11.9 even 5 363.3.h.e.323.1 8
11.10 odd 2 363.3.b.d.122.2 2
12.11 even 2 528.3.i.a.353.1 2
33.2 even 10 363.3.h.d.323.1 8
33.5 odd 10 363.3.h.e.245.1 8
33.8 even 10 363.3.h.d.251.1 8
33.14 odd 10 363.3.h.e.251.2 8
33.17 even 10 363.3.h.d.245.2 8
33.20 odd 10 363.3.h.e.323.2 8
33.26 odd 10 363.3.h.e.269.1 8
33.29 even 10 363.3.h.d.269.2 8
33.32 even 2 363.3.b.d.122.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.3.b.a.23.1 2 1.1 even 1 trivial
33.3.b.a.23.2 yes 2 3.2 odd 2 inner
363.3.b.d.122.1 2 33.32 even 2
363.3.b.d.122.2 2 11.10 odd 2
363.3.h.d.245.1 8 11.6 odd 10
363.3.h.d.245.2 8 33.17 even 10
363.3.h.d.251.1 8 33.8 even 10
363.3.h.d.251.2 8 11.8 odd 10
363.3.h.d.269.1 8 11.7 odd 10
363.3.h.d.269.2 8 33.29 even 10
363.3.h.d.323.1 8 33.2 even 10
363.3.h.d.323.2 8 11.2 odd 10
363.3.h.e.245.1 8 33.5 odd 10
363.3.h.e.245.2 8 11.5 even 5
363.3.h.e.251.1 8 11.3 even 5
363.3.h.e.251.2 8 33.14 odd 10
363.3.h.e.269.1 8 33.26 odd 10
363.3.h.e.269.2 8 11.4 even 5
363.3.h.e.323.1 8 11.9 even 5
363.3.h.e.323.2 8 33.20 odd 10
528.3.i.a.353.1 2 12.11 even 2
528.3.i.a.353.2 2 4.3 odd 2