# Properties

 Label 231.2.a.b.1.2 Level $231$ Weight $2$ Character 231.1 Self dual yes Analytic conductor $1.845$ 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] = [231,2,Mod(1,231)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(231, 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("231.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 231.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: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 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 1.2 Root $$-1.79129$$ of defining polynomial Character $$\chi$$ $$=$$ 231.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.79129 q^{2} -1.00000 q^{3} +1.20871 q^{4} +3.00000 q^{5} -1.79129 q^{6} +1.00000 q^{7} -1.41742 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.79129 q^{2} -1.00000 q^{3} +1.20871 q^{4} +3.00000 q^{5} -1.79129 q^{6} +1.00000 q^{7} -1.41742 q^{8} +1.00000 q^{9} +5.37386 q^{10} -1.00000 q^{11} -1.20871 q^{12} +1.00000 q^{13} +1.79129 q^{14} -3.00000 q^{15} -4.95644 q^{16} +7.58258 q^{17} +1.79129 q^{18} -6.58258 q^{19} +3.62614 q^{20} -1.00000 q^{21} -1.79129 q^{22} -5.58258 q^{23} +1.41742 q^{24} +4.00000 q^{25} +1.79129 q^{26} -1.00000 q^{27} +1.20871 q^{28} -8.16515 q^{29} -5.37386 q^{30} +3.58258 q^{31} -6.04356 q^{32} +1.00000 q^{33} +13.5826 q^{34} +3.00000 q^{35} +1.20871 q^{36} +1.00000 q^{37} -11.7913 q^{38} -1.00000 q^{39} -4.25227 q^{40} -11.1652 q^{41} -1.79129 q^{42} +1.58258 q^{43} -1.20871 q^{44} +3.00000 q^{45} -10.0000 q^{46} +1.41742 q^{47} +4.95644 q^{48} +1.00000 q^{49} +7.16515 q^{50} -7.58258 q^{51} +1.20871 q^{52} -9.58258 q^{53} -1.79129 q^{54} -3.00000 q^{55} -1.41742 q^{56} +6.58258 q^{57} -14.6261 q^{58} +4.58258 q^{59} -3.62614 q^{60} +10.0000 q^{61} +6.41742 q^{62} +1.00000 q^{63} -0.912878 q^{64} +3.00000 q^{65} +1.79129 q^{66} +8.58258 q^{67} +9.16515 q^{68} +5.58258 q^{69} +5.37386 q^{70} +11.1652 q^{71} -1.41742 q^{72} +7.00000 q^{73} +1.79129 q^{74} -4.00000 q^{75} -7.95644 q^{76} -1.00000 q^{77} -1.79129 q^{78} +7.16515 q^{79} -14.8693 q^{80} +1.00000 q^{81} -20.0000 q^{82} -11.5826 q^{83} -1.20871 q^{84} +22.7477 q^{85} +2.83485 q^{86} +8.16515 q^{87} +1.41742 q^{88} +9.16515 q^{89} +5.37386 q^{90} +1.00000 q^{91} -6.74773 q^{92} -3.58258 q^{93} +2.53901 q^{94} -19.7477 q^{95} +6.04356 q^{96} -2.41742 q^{97} +1.79129 q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 2 q^{3} + 7 q^{4} + 6 q^{5} + q^{6} + 2 q^{7} - 12 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - q^2 - 2 * q^3 + 7 * q^4 + 6 * q^5 + q^6 + 2 * q^7 - 12 * q^8 + 2 * q^9 $$2 q - q^{2} - 2 q^{3} + 7 q^{4} + 6 q^{5} + q^{6} + 2 q^{7} - 12 q^{8} + 2 q^{9} - 3 q^{10} - 2 q^{11} - 7 q^{12} + 2 q^{13} - q^{14} - 6 q^{15} + 13 q^{16} + 6 q^{17} - q^{18} - 4 q^{19} + 21 q^{20} - 2 q^{21} + q^{22} - 2 q^{23} + 12 q^{24} + 8 q^{25} - q^{26} - 2 q^{27} + 7 q^{28} + 2 q^{29} + 3 q^{30} - 2 q^{31} - 35 q^{32} + 2 q^{33} + 18 q^{34} + 6 q^{35} + 7 q^{36} + 2 q^{37} - 19 q^{38} - 2 q^{39} - 36 q^{40} - 4 q^{41} + q^{42} - 6 q^{43} - 7 q^{44} + 6 q^{45} - 20 q^{46} + 12 q^{47} - 13 q^{48} + 2 q^{49} - 4 q^{50} - 6 q^{51} + 7 q^{52} - 10 q^{53} + q^{54} - 6 q^{55} - 12 q^{56} + 4 q^{57} - 43 q^{58} - 21 q^{60} + 20 q^{61} + 22 q^{62} + 2 q^{63} + 44 q^{64} + 6 q^{65} - q^{66} + 8 q^{67} + 2 q^{69} - 3 q^{70} + 4 q^{71} - 12 q^{72} + 14 q^{73} - q^{74} - 8 q^{75} + 7 q^{76} - 2 q^{77} + q^{78} - 4 q^{79} + 39 q^{80} + 2 q^{81} - 40 q^{82} - 14 q^{83} - 7 q^{84} + 18 q^{85} + 24 q^{86} - 2 q^{87} + 12 q^{88} - 3 q^{90} + 2 q^{91} + 14 q^{92} + 2 q^{93} - 27 q^{94} - 12 q^{95} + 35 q^{96} - 14 q^{97} - q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q - q^2 - 2 * q^3 + 7 * q^4 + 6 * q^5 + q^6 + 2 * q^7 - 12 * q^8 + 2 * q^9 - 3 * q^10 - 2 * q^11 - 7 * q^12 + 2 * q^13 - q^14 - 6 * q^15 + 13 * q^16 + 6 * q^17 - q^18 - 4 * q^19 + 21 * q^20 - 2 * q^21 + q^22 - 2 * q^23 + 12 * q^24 + 8 * q^25 - q^26 - 2 * q^27 + 7 * q^28 + 2 * q^29 + 3 * q^30 - 2 * q^31 - 35 * q^32 + 2 * q^33 + 18 * q^34 + 6 * q^35 + 7 * q^36 + 2 * q^37 - 19 * q^38 - 2 * q^39 - 36 * q^40 - 4 * q^41 + q^42 - 6 * q^43 - 7 * q^44 + 6 * q^45 - 20 * q^46 + 12 * q^47 - 13 * q^48 + 2 * q^49 - 4 * q^50 - 6 * q^51 + 7 * q^52 - 10 * q^53 + q^54 - 6 * q^55 - 12 * q^56 + 4 * q^57 - 43 * q^58 - 21 * q^60 + 20 * q^61 + 22 * q^62 + 2 * q^63 + 44 * q^64 + 6 * q^65 - q^66 + 8 * q^67 + 2 * q^69 - 3 * q^70 + 4 * q^71 - 12 * q^72 + 14 * q^73 - q^74 - 8 * q^75 + 7 * q^76 - 2 * q^77 + q^78 - 4 * q^79 + 39 * q^80 + 2 * q^81 - 40 * q^82 - 14 * q^83 - 7 * q^84 + 18 * q^85 + 24 * q^86 - 2 * q^87 + 12 * q^88 - 3 * q^90 + 2 * q^91 + 14 * q^92 + 2 * q^93 - 27 * q^94 - 12 * q^95 + 35 * q^96 - 14 * q^97 - q^98 - 2 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.79129 1.26663 0.633316 0.773893i $$-0.281693\pi$$
0.633316 + 0.773893i $$0.281693\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 1.20871 0.604356
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ −1.79129 −0.731290
$$7$$ 1.00000 0.377964
$$8$$ −1.41742 −0.501135
$$9$$ 1.00000 0.333333
$$10$$ 5.37386 1.69936
$$11$$ −1.00000 −0.301511
$$12$$ −1.20871 −0.348925
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 1.79129 0.478742
$$15$$ −3.00000 −0.774597
$$16$$ −4.95644 −1.23911
$$17$$ 7.58258 1.83904 0.919522 0.393038i $$-0.128576\pi$$
0.919522 + 0.393038i $$0.128576\pi$$
$$18$$ 1.79129 0.422211
$$19$$ −6.58258 −1.51015 −0.755073 0.655640i $$-0.772399\pi$$
−0.755073 + 0.655640i $$0.772399\pi$$
$$20$$ 3.62614 0.810829
$$21$$ −1.00000 −0.218218
$$22$$ −1.79129 −0.381904
$$23$$ −5.58258 −1.16405 −0.582024 0.813172i $$-0.697739\pi$$
−0.582024 + 0.813172i $$0.697739\pi$$
$$24$$ 1.41742 0.289331
$$25$$ 4.00000 0.800000
$$26$$ 1.79129 0.351300
$$27$$ −1.00000 −0.192450
$$28$$ 1.20871 0.228425
$$29$$ −8.16515 −1.51623 −0.758115 0.652121i $$-0.773880\pi$$
−0.758115 + 0.652121i $$0.773880\pi$$
$$30$$ −5.37386 −0.981129
$$31$$ 3.58258 0.643450 0.321725 0.946833i $$-0.395737\pi$$
0.321725 + 0.946833i $$0.395737\pi$$
$$32$$ −6.04356 −1.06836
$$33$$ 1.00000 0.174078
$$34$$ 13.5826 2.32939
$$35$$ 3.00000 0.507093
$$36$$ 1.20871 0.201452
$$37$$ 1.00000 0.164399 0.0821995 0.996616i $$-0.473806\pi$$
0.0821995 + 0.996616i $$0.473806\pi$$
$$38$$ −11.7913 −1.91280
$$39$$ −1.00000 −0.160128
$$40$$ −4.25227 −0.672343
$$41$$ −11.1652 −1.74370 −0.871852 0.489770i $$-0.837081\pi$$
−0.871852 + 0.489770i $$0.837081\pi$$
$$42$$ −1.79129 −0.276402
$$43$$ 1.58258 0.241341 0.120670 0.992693i $$-0.461496\pi$$
0.120670 + 0.992693i $$0.461496\pi$$
$$44$$ −1.20871 −0.182220
$$45$$ 3.00000 0.447214
$$46$$ −10.0000 −1.47442
$$47$$ 1.41742 0.206753 0.103376 0.994642i $$-0.467035\pi$$
0.103376 + 0.994642i $$0.467035\pi$$
$$48$$ 4.95644 0.715400
$$49$$ 1.00000 0.142857
$$50$$ 7.16515 1.01331
$$51$$ −7.58258 −1.06177
$$52$$ 1.20871 0.167618
$$53$$ −9.58258 −1.31627 −0.658134 0.752901i $$-0.728654\pi$$
−0.658134 + 0.752901i $$0.728654\pi$$
$$54$$ −1.79129 −0.243763
$$55$$ −3.00000 −0.404520
$$56$$ −1.41742 −0.189411
$$57$$ 6.58258 0.871883
$$58$$ −14.6261 −1.92051
$$59$$ 4.58258 0.596601 0.298300 0.954472i $$-0.403580\pi$$
0.298300 + 0.954472i $$0.403580\pi$$
$$60$$ −3.62614 −0.468132
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 6.41742 0.815014
$$63$$ 1.00000 0.125988
$$64$$ −0.912878 −0.114110
$$65$$ 3.00000 0.372104
$$66$$ 1.79129 0.220492
$$67$$ 8.58258 1.04853 0.524264 0.851556i $$-0.324340\pi$$
0.524264 + 0.851556i $$0.324340\pi$$
$$68$$ 9.16515 1.11144
$$69$$ 5.58258 0.672063
$$70$$ 5.37386 0.642300
$$71$$ 11.1652 1.32506 0.662530 0.749036i $$-0.269483\pi$$
0.662530 + 0.749036i $$0.269483\pi$$
$$72$$ −1.41742 −0.167045
$$73$$ 7.00000 0.819288 0.409644 0.912245i $$-0.365653\pi$$
0.409644 + 0.912245i $$0.365653\pi$$
$$74$$ 1.79129 0.208233
$$75$$ −4.00000 −0.461880
$$76$$ −7.95644 −0.912666
$$77$$ −1.00000 −0.113961
$$78$$ −1.79129 −0.202823
$$79$$ 7.16515 0.806143 0.403071 0.915169i $$-0.367943\pi$$
0.403071 + 0.915169i $$0.367943\pi$$
$$80$$ −14.8693 −1.66244
$$81$$ 1.00000 0.111111
$$82$$ −20.0000 −2.20863
$$83$$ −11.5826 −1.27135 −0.635676 0.771956i $$-0.719279\pi$$
−0.635676 + 0.771956i $$0.719279\pi$$
$$84$$ −1.20871 −0.131881
$$85$$ 22.7477 2.46734
$$86$$ 2.83485 0.305690
$$87$$ 8.16515 0.875396
$$88$$ 1.41742 0.151098
$$89$$ 9.16515 0.971504 0.485752 0.874097i $$-0.338546\pi$$
0.485752 + 0.874097i $$0.338546\pi$$
$$90$$ 5.37386 0.566455
$$91$$ 1.00000 0.104828
$$92$$ −6.74773 −0.703499
$$93$$ −3.58258 −0.371496
$$94$$ 2.53901 0.261879
$$95$$ −19.7477 −2.02607
$$96$$ 6.04356 0.616818
$$97$$ −2.41742 −0.245452 −0.122726 0.992441i $$-0.539164\pi$$
−0.122726 + 0.992441i $$0.539164\pi$$
$$98$$ 1.79129 0.180947
$$99$$ −1.00000 −0.100504
$$100$$ 4.83485 0.483485
$$101$$ 11.5826 1.15251 0.576255 0.817270i $$-0.304514\pi$$
0.576255 + 0.817270i $$0.304514\pi$$
$$102$$ −13.5826 −1.34488
$$103$$ −1.16515 −0.114806 −0.0574029 0.998351i $$-0.518282\pi$$
−0.0574029 + 0.998351i $$0.518282\pi$$
$$104$$ −1.41742 −0.138990
$$105$$ −3.00000 −0.292770
$$106$$ −17.1652 −1.66723
$$107$$ −12.5826 −1.21640 −0.608202 0.793782i $$-0.708109\pi$$
−0.608202 + 0.793782i $$0.708109\pi$$
$$108$$ −1.20871 −0.116308
$$109$$ 3.58258 0.343149 0.171574 0.985171i $$-0.445115\pi$$
0.171574 + 0.985171i $$0.445115\pi$$
$$110$$ −5.37386 −0.512378
$$111$$ −1.00000 −0.0949158
$$112$$ −4.95644 −0.468339
$$113$$ 9.16515 0.862185 0.431092 0.902308i $$-0.358128\pi$$
0.431092 + 0.902308i $$0.358128\pi$$
$$114$$ 11.7913 1.10436
$$115$$ −16.7477 −1.56173
$$116$$ −9.86932 −0.916343
$$117$$ 1.00000 0.0924500
$$118$$ 8.20871 0.755673
$$119$$ 7.58258 0.695094
$$120$$ 4.25227 0.388178
$$121$$ 1.00000 0.0909091
$$122$$ 17.9129 1.62176
$$123$$ 11.1652 1.00673
$$124$$ 4.33030 0.388873
$$125$$ −3.00000 −0.268328
$$126$$ 1.79129 0.159581
$$127$$ −11.5826 −1.02779 −0.513894 0.857854i $$-0.671797\pi$$
−0.513894 + 0.857854i $$0.671797\pi$$
$$128$$ 10.4519 0.923826
$$129$$ −1.58258 −0.139338
$$130$$ 5.37386 0.471319
$$131$$ −16.0000 −1.39793 −0.698963 0.715158i $$-0.746355\pi$$
−0.698963 + 0.715158i $$0.746355\pi$$
$$132$$ 1.20871 0.105205
$$133$$ −6.58258 −0.570782
$$134$$ 15.3739 1.32810
$$135$$ −3.00000 −0.258199
$$136$$ −10.7477 −0.921610
$$137$$ −11.5826 −0.989566 −0.494783 0.869016i $$-0.664752\pi$$
−0.494783 + 0.869016i $$0.664752\pi$$
$$138$$ 10.0000 0.851257
$$139$$ 11.1652 0.947016 0.473508 0.880790i $$-0.342988\pi$$
0.473508 + 0.880790i $$0.342988\pi$$
$$140$$ 3.62614 0.306464
$$141$$ −1.41742 −0.119369
$$142$$ 20.0000 1.67836
$$143$$ −1.00000 −0.0836242
$$144$$ −4.95644 −0.413037
$$145$$ −24.4955 −2.03424
$$146$$ 12.5390 1.03774
$$147$$ −1.00000 −0.0824786
$$148$$ 1.20871 0.0993555
$$149$$ 6.16515 0.505069 0.252534 0.967588i $$-0.418736\pi$$
0.252534 + 0.967588i $$0.418736\pi$$
$$150$$ −7.16515 −0.585032
$$151$$ 3.58258 0.291546 0.145773 0.989318i $$-0.453433\pi$$
0.145773 + 0.989318i $$0.453433\pi$$
$$152$$ 9.33030 0.756787
$$153$$ 7.58258 0.613015
$$154$$ −1.79129 −0.144346
$$155$$ 10.7477 0.863278
$$156$$ −1.20871 −0.0967744
$$157$$ 19.1652 1.52955 0.764773 0.644300i $$-0.222851\pi$$
0.764773 + 0.644300i $$0.222851\pi$$
$$158$$ 12.8348 1.02109
$$159$$ 9.58258 0.759948
$$160$$ −18.1307 −1.43336
$$161$$ −5.58258 −0.439969
$$162$$ 1.79129 0.140737
$$163$$ 8.58258 0.672239 0.336120 0.941819i $$-0.390885\pi$$
0.336120 + 0.941819i $$0.390885\pi$$
$$164$$ −13.4955 −1.05382
$$165$$ 3.00000 0.233550
$$166$$ −20.7477 −1.61034
$$167$$ −4.74773 −0.367390 −0.183695 0.982983i $$-0.558806\pi$$
−0.183695 + 0.982983i $$0.558806\pi$$
$$168$$ 1.41742 0.109357
$$169$$ −12.0000 −0.923077
$$170$$ 40.7477 3.12521
$$171$$ −6.58258 −0.503382
$$172$$ 1.91288 0.145856
$$173$$ −7.16515 −0.544756 −0.272378 0.962190i $$-0.587810\pi$$
−0.272378 + 0.962190i $$0.587810\pi$$
$$174$$ 14.6261 1.10880
$$175$$ 4.00000 0.302372
$$176$$ 4.95644 0.373606
$$177$$ −4.58258 −0.344447
$$178$$ 16.4174 1.23054
$$179$$ 14.3303 1.07110 0.535549 0.844504i $$-0.320105\pi$$
0.535549 + 0.844504i $$0.320105\pi$$
$$180$$ 3.62614 0.270276
$$181$$ −5.58258 −0.414950 −0.207475 0.978240i $$-0.566524\pi$$
−0.207475 + 0.978240i $$0.566524\pi$$
$$182$$ 1.79129 0.132779
$$183$$ −10.0000 −0.739221
$$184$$ 7.91288 0.583345
$$185$$ 3.00000 0.220564
$$186$$ −6.41742 −0.470548
$$187$$ −7.58258 −0.554493
$$188$$ 1.71326 0.124952
$$189$$ −1.00000 −0.0727393
$$190$$ −35.3739 −2.56629
$$191$$ 11.5826 0.838086 0.419043 0.907966i $$-0.362366\pi$$
0.419043 + 0.907966i $$0.362366\pi$$
$$192$$ 0.912878 0.0658813
$$193$$ −2.41742 −0.174010 −0.0870050 0.996208i $$-0.527730\pi$$
−0.0870050 + 0.996208i $$0.527730\pi$$
$$194$$ −4.33030 −0.310898
$$195$$ −3.00000 −0.214834
$$196$$ 1.20871 0.0863366
$$197$$ 5.16515 0.368002 0.184001 0.982926i $$-0.441095\pi$$
0.184001 + 0.982926i $$0.441095\pi$$
$$198$$ −1.79129 −0.127301
$$199$$ −9.58258 −0.679291 −0.339645 0.940554i $$-0.610307\pi$$
−0.339645 + 0.940554i $$0.610307\pi$$
$$200$$ −5.66970 −0.400908
$$201$$ −8.58258 −0.605368
$$202$$ 20.7477 1.45980
$$203$$ −8.16515 −0.573081
$$204$$ −9.16515 −0.641689
$$205$$ −33.4955 −2.33942
$$206$$ −2.08712 −0.145417
$$207$$ −5.58258 −0.388016
$$208$$ −4.95644 −0.343667
$$209$$ 6.58258 0.455326
$$210$$ −5.37386 −0.370832
$$211$$ −13.1652 −0.906326 −0.453163 0.891428i $$-0.649705\pi$$
−0.453163 + 0.891428i $$0.649705\pi$$
$$212$$ −11.5826 −0.795495
$$213$$ −11.1652 −0.765024
$$214$$ −22.5390 −1.54074
$$215$$ 4.74773 0.323792
$$216$$ 1.41742 0.0964435
$$217$$ 3.58258 0.243201
$$218$$ 6.41742 0.434643
$$219$$ −7.00000 −0.473016
$$220$$ −3.62614 −0.244474
$$221$$ 7.58258 0.510059
$$222$$ −1.79129 −0.120223
$$223$$ 6.00000 0.401790 0.200895 0.979613i $$-0.435615\pi$$
0.200895 + 0.979613i $$0.435615\pi$$
$$224$$ −6.04356 −0.403802
$$225$$ 4.00000 0.266667
$$226$$ 16.4174 1.09207
$$227$$ −22.0000 −1.46019 −0.730096 0.683345i $$-0.760525\pi$$
−0.730096 + 0.683345i $$0.760525\pi$$
$$228$$ 7.95644 0.526928
$$229$$ 0.747727 0.0494112 0.0247056 0.999695i $$-0.492135\pi$$
0.0247056 + 0.999695i $$0.492135\pi$$
$$230$$ −30.0000 −1.97814
$$231$$ 1.00000 0.0657952
$$232$$ 11.5735 0.759836
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ 1.79129 0.117100
$$235$$ 4.25227 0.277388
$$236$$ 5.53901 0.360559
$$237$$ −7.16515 −0.465427
$$238$$ 13.5826 0.880428
$$239$$ 16.5826 1.07264 0.536319 0.844015i $$-0.319814\pi$$
0.536319 + 0.844015i $$0.319814\pi$$
$$240$$ 14.8693 0.959810
$$241$$ −10.1652 −0.654795 −0.327397 0.944887i $$-0.606172\pi$$
−0.327397 + 0.944887i $$0.606172\pi$$
$$242$$ 1.79129 0.115148
$$243$$ −1.00000 −0.0641500
$$244$$ 12.0871 0.773799
$$245$$ 3.00000 0.191663
$$246$$ 20.0000 1.27515
$$247$$ −6.58258 −0.418839
$$248$$ −5.07803 −0.322455
$$249$$ 11.5826 0.734016
$$250$$ −5.37386 −0.339873
$$251$$ 7.41742 0.468184 0.234092 0.972214i $$-0.424788\pi$$
0.234092 + 0.972214i $$0.424788\pi$$
$$252$$ 1.20871 0.0761417
$$253$$ 5.58258 0.350974
$$254$$ −20.7477 −1.30183
$$255$$ −22.7477 −1.42452
$$256$$ 20.5481 1.28426
$$257$$ 19.0000 1.18519 0.592594 0.805502i $$-0.298104\pi$$
0.592594 + 0.805502i $$0.298104\pi$$
$$258$$ −2.83485 −0.176490
$$259$$ 1.00000 0.0621370
$$260$$ 3.62614 0.224883
$$261$$ −8.16515 −0.505410
$$262$$ −28.6606 −1.77066
$$263$$ −22.9129 −1.41287 −0.706434 0.707779i $$-0.749697\pi$$
−0.706434 + 0.707779i $$0.749697\pi$$
$$264$$ −1.41742 −0.0872364
$$265$$ −28.7477 −1.76596
$$266$$ −11.7913 −0.722970
$$267$$ −9.16515 −0.560898
$$268$$ 10.3739 0.633685
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ −5.37386 −0.327043
$$271$$ 5.41742 0.329085 0.164543 0.986370i $$-0.447385\pi$$
0.164543 + 0.986370i $$0.447385\pi$$
$$272$$ −37.5826 −2.27878
$$273$$ −1.00000 −0.0605228
$$274$$ −20.7477 −1.25342
$$275$$ −4.00000 −0.241209
$$276$$ 6.74773 0.406165
$$277$$ −19.1652 −1.15152 −0.575761 0.817618i $$-0.695294\pi$$
−0.575761 + 0.817618i $$0.695294\pi$$
$$278$$ 20.0000 1.19952
$$279$$ 3.58258 0.214483
$$280$$ −4.25227 −0.254122
$$281$$ −27.3303 −1.63039 −0.815195 0.579187i $$-0.803370\pi$$
−0.815195 + 0.579187i $$0.803370\pi$$
$$282$$ −2.53901 −0.151196
$$283$$ 27.7477 1.64943 0.824716 0.565548i $$-0.191335\pi$$
0.824716 + 0.565548i $$0.191335\pi$$
$$284$$ 13.4955 0.800808
$$285$$ 19.7477 1.16975
$$286$$ −1.79129 −0.105921
$$287$$ −11.1652 −0.659058
$$288$$ −6.04356 −0.356120
$$289$$ 40.4955 2.38209
$$290$$ −43.8784 −2.57663
$$291$$ 2.41742 0.141712
$$292$$ 8.46099 0.495142
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ −1.79129 −0.104470
$$295$$ 13.7477 0.800424
$$296$$ −1.41742 −0.0823861
$$297$$ 1.00000 0.0580259
$$298$$ 11.0436 0.639736
$$299$$ −5.58258 −0.322849
$$300$$ −4.83485 −0.279140
$$301$$ 1.58258 0.0912181
$$302$$ 6.41742 0.369281
$$303$$ −11.5826 −0.665402
$$304$$ 32.6261 1.87124
$$305$$ 30.0000 1.71780
$$306$$ 13.5826 0.776464
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ −1.20871 −0.0688728
$$309$$ 1.16515 0.0662831
$$310$$ 19.2523 1.09346
$$311$$ 14.3303 0.812597 0.406298 0.913740i $$-0.366819\pi$$
0.406298 + 0.913740i $$0.366819\pi$$
$$312$$ 1.41742 0.0802458
$$313$$ 19.5826 1.10687 0.553436 0.832891i $$-0.313316\pi$$
0.553436 + 0.832891i $$0.313316\pi$$
$$314$$ 34.3303 1.93737
$$315$$ 3.00000 0.169031
$$316$$ 8.66061 0.487197
$$317$$ −22.4174 −1.25909 −0.629544 0.776965i $$-0.716758\pi$$
−0.629544 + 0.776965i $$0.716758\pi$$
$$318$$ 17.1652 0.962574
$$319$$ 8.16515 0.457161
$$320$$ −2.73864 −0.153094
$$321$$ 12.5826 0.702291
$$322$$ −10.0000 −0.557278
$$323$$ −49.9129 −2.77723
$$324$$ 1.20871 0.0671507
$$325$$ 4.00000 0.221880
$$326$$ 15.3739 0.851480
$$327$$ −3.58258 −0.198117
$$328$$ 15.8258 0.873831
$$329$$ 1.41742 0.0781451
$$330$$ 5.37386 0.295821
$$331$$ −3.16515 −0.173972 −0.0869862 0.996210i $$-0.527724\pi$$
−0.0869862 + 0.996210i $$0.527724\pi$$
$$332$$ −14.0000 −0.768350
$$333$$ 1.00000 0.0547997
$$334$$ −8.50455 −0.465348
$$335$$ 25.7477 1.40675
$$336$$ 4.95644 0.270396
$$337$$ 17.5826 0.957784 0.478892 0.877874i $$-0.341039\pi$$
0.478892 + 0.877874i $$0.341039\pi$$
$$338$$ −21.4955 −1.16920
$$339$$ −9.16515 −0.497783
$$340$$ 27.4955 1.49115
$$341$$ −3.58258 −0.194007
$$342$$ −11.7913 −0.637600
$$343$$ 1.00000 0.0539949
$$344$$ −2.24318 −0.120944
$$345$$ 16.7477 0.901667
$$346$$ −12.8348 −0.690006
$$347$$ 26.3303 1.41348 0.706742 0.707471i $$-0.250164\pi$$
0.706742 + 0.707471i $$0.250164\pi$$
$$348$$ 9.86932 0.529051
$$349$$ −15.0000 −0.802932 −0.401466 0.915874i $$-0.631499\pi$$
−0.401466 + 0.915874i $$0.631499\pi$$
$$350$$ 7.16515 0.382993
$$351$$ −1.00000 −0.0533761
$$352$$ 6.04356 0.322123
$$353$$ 24.1652 1.28618 0.643091 0.765790i $$-0.277652\pi$$
0.643091 + 0.765790i $$0.277652\pi$$
$$354$$ −8.20871 −0.436288
$$355$$ 33.4955 1.77775
$$356$$ 11.0780 0.587134
$$357$$ −7.58258 −0.401312
$$358$$ 25.6697 1.35669
$$359$$ −8.83485 −0.466285 −0.233143 0.972443i $$-0.574901\pi$$
−0.233143 + 0.972443i $$0.574901\pi$$
$$360$$ −4.25227 −0.224114
$$361$$ 24.3303 1.28054
$$362$$ −10.0000 −0.525588
$$363$$ −1.00000 −0.0524864
$$364$$ 1.20871 0.0633537
$$365$$ 21.0000 1.09919
$$366$$ −17.9129 −0.936321
$$367$$ 22.0000 1.14839 0.574195 0.818718i $$-0.305315\pi$$
0.574195 + 0.818718i $$0.305315\pi$$
$$368$$ 27.6697 1.44238
$$369$$ −11.1652 −0.581235
$$370$$ 5.37386 0.279374
$$371$$ −9.58258 −0.497503
$$372$$ −4.33030 −0.224516
$$373$$ −34.7477 −1.79917 −0.899585 0.436747i $$-0.856131\pi$$
−0.899585 + 0.436747i $$0.856131\pi$$
$$374$$ −13.5826 −0.702338
$$375$$ 3.00000 0.154919
$$376$$ −2.00909 −0.103611
$$377$$ −8.16515 −0.420527
$$378$$ −1.79129 −0.0921339
$$379$$ −12.5826 −0.646323 −0.323162 0.946344i $$-0.604746\pi$$
−0.323162 + 0.946344i $$0.604746\pi$$
$$380$$ −23.8693 −1.22447
$$381$$ 11.5826 0.593393
$$382$$ 20.7477 1.06155
$$383$$ 10.3303 0.527854 0.263927 0.964543i $$-0.414982\pi$$
0.263927 + 0.964543i $$0.414982\pi$$
$$384$$ −10.4519 −0.533371
$$385$$ −3.00000 −0.152894
$$386$$ −4.33030 −0.220407
$$387$$ 1.58258 0.0804468
$$388$$ −2.92197 −0.148341
$$389$$ −26.3303 −1.33500 −0.667500 0.744610i $$-0.732635\pi$$
−0.667500 + 0.744610i $$0.732635\pi$$
$$390$$ −5.37386 −0.272116
$$391$$ −42.3303 −2.14074
$$392$$ −1.41742 −0.0715907
$$393$$ 16.0000 0.807093
$$394$$ 9.25227 0.466123
$$395$$ 21.4955 1.08155
$$396$$ −1.20871 −0.0607401
$$397$$ −31.5826 −1.58508 −0.792542 0.609817i $$-0.791243\pi$$
−0.792542 + 0.609817i $$0.791243\pi$$
$$398$$ −17.1652 −0.860411
$$399$$ 6.58258 0.329541
$$400$$ −19.8258 −0.991288
$$401$$ 31.9129 1.59365 0.796827 0.604208i $$-0.206510\pi$$
0.796827 + 0.604208i $$0.206510\pi$$
$$402$$ −15.3739 −0.766779
$$403$$ 3.58258 0.178461
$$404$$ 14.0000 0.696526
$$405$$ 3.00000 0.149071
$$406$$ −14.6261 −0.725883
$$407$$ −1.00000 −0.0495682
$$408$$ 10.7477 0.532092
$$409$$ 8.33030 0.411907 0.205953 0.978562i $$-0.433970\pi$$
0.205953 + 0.978562i $$0.433970\pi$$
$$410$$ −60.0000 −2.96319
$$411$$ 11.5826 0.571326
$$412$$ −1.40833 −0.0693836
$$413$$ 4.58258 0.225494
$$414$$ −10.0000 −0.491473
$$415$$ −34.7477 −1.70570
$$416$$ −6.04356 −0.296310
$$417$$ −11.1652 −0.546760
$$418$$ 11.7913 0.576731
$$419$$ 2.58258 0.126167 0.0630835 0.998008i $$-0.479907\pi$$
0.0630835 + 0.998008i $$0.479907\pi$$
$$420$$ −3.62614 −0.176937
$$421$$ −33.6606 −1.64052 −0.820259 0.571993i $$-0.806171\pi$$
−0.820259 + 0.571993i $$0.806171\pi$$
$$422$$ −23.5826 −1.14798
$$423$$ 1.41742 0.0689175
$$424$$ 13.5826 0.659628
$$425$$ 30.3303 1.47124
$$426$$ −20.0000 −0.969003
$$427$$ 10.0000 0.483934
$$428$$ −15.2087 −0.735141
$$429$$ 1.00000 0.0482805
$$430$$ 8.50455 0.410126
$$431$$ 17.7477 0.854878 0.427439 0.904044i $$-0.359416\pi$$
0.427439 + 0.904044i $$0.359416\pi$$
$$432$$ 4.95644 0.238467
$$433$$ −11.1652 −0.536563 −0.268281 0.963341i $$-0.586456\pi$$
−0.268281 + 0.963341i $$0.586456\pi$$
$$434$$ 6.41742 0.308046
$$435$$ 24.4955 1.17447
$$436$$ 4.33030 0.207384
$$437$$ 36.7477 1.75788
$$438$$ −12.5390 −0.599137
$$439$$ 17.4174 0.831288 0.415644 0.909527i $$-0.363556\pi$$
0.415644 + 0.909527i $$0.363556\pi$$
$$440$$ 4.25227 0.202719
$$441$$ 1.00000 0.0476190
$$442$$ 13.5826 0.646057
$$443$$ −23.1652 −1.10061 −0.550305 0.834964i $$-0.685488\pi$$
−0.550305 + 0.834964i $$0.685488\pi$$
$$444$$ −1.20871 −0.0573629
$$445$$ 27.4955 1.30341
$$446$$ 10.7477 0.508920
$$447$$ −6.16515 −0.291602
$$448$$ −0.912878 −0.0431295
$$449$$ −18.3303 −0.865060 −0.432530 0.901619i $$-0.642379\pi$$
−0.432530 + 0.901619i $$0.642379\pi$$
$$450$$ 7.16515 0.337768
$$451$$ 11.1652 0.525746
$$452$$ 11.0780 0.521067
$$453$$ −3.58258 −0.168324
$$454$$ −39.4083 −1.84952
$$455$$ 3.00000 0.140642
$$456$$ −9.33030 −0.436931
$$457$$ −19.9129 −0.931485 −0.465743 0.884920i $$-0.654213\pi$$
−0.465743 + 0.884920i $$0.654213\pi$$
$$458$$ 1.33939 0.0625858
$$459$$ −7.58258 −0.353924
$$460$$ −20.2432 −0.943843
$$461$$ 18.3303 0.853727 0.426864 0.904316i $$-0.359618\pi$$
0.426864 + 0.904316i $$0.359618\pi$$
$$462$$ 1.79129 0.0833383
$$463$$ 8.58258 0.398866 0.199433 0.979911i $$-0.436090\pi$$
0.199433 + 0.979911i $$0.436090\pi$$
$$464$$ 40.4701 1.87878
$$465$$ −10.7477 −0.498414
$$466$$ −25.0780 −1.16172
$$467$$ −38.5826 −1.78539 −0.892694 0.450663i $$-0.851188\pi$$
−0.892694 + 0.450663i $$0.851188\pi$$
$$468$$ 1.20871 0.0558727
$$469$$ 8.58258 0.396307
$$470$$ 7.61704 0.351348
$$471$$ −19.1652 −0.883084
$$472$$ −6.49545 −0.298978
$$473$$ −1.58258 −0.0727669
$$474$$ −12.8348 −0.589524
$$475$$ −26.3303 −1.20812
$$476$$ 9.16515 0.420084
$$477$$ −9.58258 −0.438756
$$478$$ 29.7042 1.35864
$$479$$ −15.5826 −0.711986 −0.355993 0.934489i $$-0.615857\pi$$
−0.355993 + 0.934489i $$0.615857\pi$$
$$480$$ 18.1307 0.827549
$$481$$ 1.00000 0.0455961
$$482$$ −18.2087 −0.829384
$$483$$ 5.58258 0.254016
$$484$$ 1.20871 0.0549415
$$485$$ −7.25227 −0.329309
$$486$$ −1.79129 −0.0812545
$$487$$ 10.3303 0.468111 0.234055 0.972223i $$-0.424800\pi$$
0.234055 + 0.972223i $$0.424800\pi$$
$$488$$ −14.1742 −0.641638
$$489$$ −8.58258 −0.388117
$$490$$ 5.37386 0.242766
$$491$$ −22.9129 −1.03404 −0.517022 0.855972i $$-0.672959\pi$$
−0.517022 + 0.855972i $$0.672959\pi$$
$$492$$ 13.4955 0.608422
$$493$$ −61.9129 −2.78842
$$494$$ −11.7913 −0.530515
$$495$$ −3.00000 −0.134840
$$496$$ −17.7568 −0.797305
$$497$$ 11.1652 0.500825
$$498$$ 20.7477 0.929728
$$499$$ 41.7477 1.86888 0.934442 0.356114i $$-0.115899\pi$$
0.934442 + 0.356114i $$0.115899\pi$$
$$500$$ −3.62614 −0.162166
$$501$$ 4.74773 0.212113
$$502$$ 13.2867 0.593016
$$503$$ −0.747727 −0.0333395 −0.0166698 0.999861i $$-0.505306\pi$$
−0.0166698 + 0.999861i $$0.505306\pi$$
$$504$$ −1.41742 −0.0631371
$$505$$ 34.7477 1.54625
$$506$$ 10.0000 0.444554
$$507$$ 12.0000 0.532939
$$508$$ −14.0000 −0.621150
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ −40.7477 −1.80434
$$511$$ 7.00000 0.309662
$$512$$ 15.9038 0.702855
$$513$$ 6.58258 0.290628
$$514$$ 34.0345 1.50120
$$515$$ −3.49545 −0.154028
$$516$$ −1.91288 −0.0842098
$$517$$ −1.41742 −0.0623382
$$518$$ 1.79129 0.0787047
$$519$$ 7.16515 0.314515
$$520$$ −4.25227 −0.186475
$$521$$ 15.8348 0.693737 0.346869 0.937914i $$-0.387245\pi$$
0.346869 + 0.937914i $$0.387245\pi$$
$$522$$ −14.6261 −0.640169
$$523$$ 15.4174 0.674157 0.337078 0.941477i $$-0.390561\pi$$
0.337078 + 0.941477i $$0.390561\pi$$
$$524$$ −19.3394 −0.844845
$$525$$ −4.00000 −0.174574
$$526$$ −41.0436 −1.78958
$$527$$ 27.1652 1.18333
$$528$$ −4.95644 −0.215701
$$529$$ 8.16515 0.355007
$$530$$ −51.4955 −2.23682
$$531$$ 4.58258 0.198867
$$532$$ −7.95644 −0.344955
$$533$$ −11.1652 −0.483616
$$534$$ −16.4174 −0.710451
$$535$$ −37.7477 −1.63198
$$536$$ −12.1652 −0.525455
$$537$$ −14.3303 −0.618398
$$538$$ 17.9129 0.772279
$$539$$ −1.00000 −0.0430730
$$540$$ −3.62614 −0.156044
$$541$$ 18.3303 0.788081 0.394041 0.919093i $$-0.371077\pi$$
0.394041 + 0.919093i $$0.371077\pi$$
$$542$$ 9.70417 0.416830
$$543$$ 5.58258 0.239571
$$544$$ −45.8258 −1.96476
$$545$$ 10.7477 0.460382
$$546$$ −1.79129 −0.0766600
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ −14.0000 −0.598050
$$549$$ 10.0000 0.426790
$$550$$ −7.16515 −0.305523
$$551$$ 53.7477 2.28973
$$552$$ −7.91288 −0.336794
$$553$$ 7.16515 0.304693
$$554$$ −34.3303 −1.45855
$$555$$ −3.00000 −0.127343
$$556$$ 13.4955 0.572335
$$557$$ 9.33030 0.395338 0.197669 0.980269i $$-0.436663\pi$$
0.197669 + 0.980269i $$0.436663\pi$$
$$558$$ 6.41742 0.271671
$$559$$ 1.58258 0.0669358
$$560$$ −14.8693 −0.628343
$$561$$ 7.58258 0.320137
$$562$$ −48.9564 −2.06510
$$563$$ −37.5826 −1.58392 −0.791958 0.610575i $$-0.790938\pi$$
−0.791958 + 0.610575i $$0.790938\pi$$
$$564$$ −1.71326 −0.0721412
$$565$$ 27.4955 1.15674
$$566$$ 49.7042 2.08922
$$567$$ 1.00000 0.0419961
$$568$$ −15.8258 −0.664034
$$569$$ −26.6606 −1.11767 −0.558835 0.829279i $$-0.688752\pi$$
−0.558835 + 0.829279i $$0.688752\pi$$
$$570$$ 35.3739 1.48165
$$571$$ 28.8348 1.20670 0.603350 0.797476i $$-0.293832\pi$$
0.603350 + 0.797476i $$0.293832\pi$$
$$572$$ −1.20871 −0.0505388
$$573$$ −11.5826 −0.483869
$$574$$ −20.0000 −0.834784
$$575$$ −22.3303 −0.931238
$$576$$ −0.912878 −0.0380366
$$577$$ −21.9129 −0.912245 −0.456123 0.889917i $$-0.650762\pi$$
−0.456123 + 0.889917i $$0.650762\pi$$
$$578$$ 72.5390 3.01723
$$579$$ 2.41742 0.100465
$$580$$ −29.6080 −1.22940
$$581$$ −11.5826 −0.480526
$$582$$ 4.33030 0.179497
$$583$$ 9.58258 0.396870
$$584$$ −9.92197 −0.410574
$$585$$ 3.00000 0.124035
$$586$$ 0 0
$$587$$ 37.7477 1.55802 0.779008 0.627014i $$-0.215723\pi$$
0.779008 + 0.627014i $$0.215723\pi$$
$$588$$ −1.20871 −0.0498464
$$589$$ −23.5826 −0.971703
$$590$$ 24.6261 1.01384
$$591$$ −5.16515 −0.212466
$$592$$ −4.95644 −0.203708
$$593$$ 16.0000 0.657041 0.328521 0.944497i $$-0.393450\pi$$
0.328521 + 0.944497i $$0.393450\pi$$
$$594$$ 1.79129 0.0734974
$$595$$ 22.7477 0.932566
$$596$$ 7.45189 0.305241
$$597$$ 9.58258 0.392189
$$598$$ −10.0000 −0.408930
$$599$$ 7.16515 0.292760 0.146380 0.989228i $$-0.453238\pi$$
0.146380 + 0.989228i $$0.453238\pi$$
$$600$$ 5.66970 0.231464
$$601$$ 24.4955 0.999190 0.499595 0.866259i $$-0.333482\pi$$
0.499595 + 0.866259i $$0.333482\pi$$
$$602$$ 2.83485 0.115540
$$603$$ 8.58258 0.349510
$$604$$ 4.33030 0.176198
$$605$$ 3.00000 0.121967
$$606$$ −20.7477 −0.842819
$$607$$ −21.7477 −0.882713 −0.441357 0.897332i $$-0.645503\pi$$
−0.441357 + 0.897332i $$0.645503\pi$$
$$608$$ 39.7822 1.61338
$$609$$ 8.16515 0.330869
$$610$$ 53.7386 2.17581
$$611$$ 1.41742 0.0573428
$$612$$ 9.16515 0.370479
$$613$$ −26.7477 −1.08033 −0.540165 0.841559i $$-0.681638\pi$$
−0.540165 + 0.841559i $$0.681638\pi$$
$$614$$ 0 0
$$615$$ 33.4955 1.35067
$$616$$ 1.41742 0.0571097
$$617$$ 2.83485 0.114127 0.0570634 0.998371i $$-0.481826\pi$$
0.0570634 + 0.998371i $$0.481826\pi$$
$$618$$ 2.08712 0.0839563
$$619$$ −29.0780 −1.16874 −0.584372 0.811486i $$-0.698659\pi$$
−0.584372 + 0.811486i $$0.698659\pi$$
$$620$$ 12.9909 0.521727
$$621$$ 5.58258 0.224021
$$622$$ 25.6697 1.02926
$$623$$ 9.16515 0.367194
$$624$$ 4.95644 0.198416
$$625$$ −29.0000 −1.16000
$$626$$ 35.0780 1.40200
$$627$$ −6.58258 −0.262883
$$628$$ 23.1652 0.924390
$$629$$ 7.58258 0.302337
$$630$$ 5.37386 0.214100
$$631$$ 23.1652 0.922190 0.461095 0.887351i $$-0.347457\pi$$
0.461095 + 0.887351i $$0.347457\pi$$
$$632$$ −10.1561 −0.403986
$$633$$ 13.1652 0.523268
$$634$$ −40.1561 −1.59480
$$635$$ −34.7477 −1.37892
$$636$$ 11.5826 0.459279
$$637$$ 1.00000 0.0396214
$$638$$ 14.6261 0.579054
$$639$$ 11.1652 0.441687
$$640$$ 31.3557 1.23944
$$641$$ 43.5826 1.72141 0.860704 0.509106i $$-0.170024\pi$$
0.860704 + 0.509106i $$0.170024\pi$$
$$642$$ 22.5390 0.889544
$$643$$ 38.2432 1.50816 0.754082 0.656780i $$-0.228082\pi$$
0.754082 + 0.656780i $$0.228082\pi$$
$$644$$ −6.74773 −0.265898
$$645$$ −4.74773 −0.186942
$$646$$ −89.4083 −3.51772
$$647$$ −10.9129 −0.429030 −0.214515 0.976721i $$-0.568817\pi$$
−0.214515 + 0.976721i $$0.568817\pi$$
$$648$$ −1.41742 −0.0556817
$$649$$ −4.58258 −0.179882
$$650$$ 7.16515 0.281040
$$651$$ −3.58258 −0.140412
$$652$$ 10.3739 0.406272
$$653$$ −30.3303 −1.18692 −0.593458 0.804865i $$-0.702238\pi$$
−0.593458 + 0.804865i $$0.702238\pi$$
$$654$$ −6.41742 −0.250941
$$655$$ −48.0000 −1.87552
$$656$$ 55.3394 2.16064
$$657$$ 7.00000 0.273096
$$658$$ 2.53901 0.0989811
$$659$$ −28.5826 −1.11342 −0.556710 0.830707i $$-0.687936\pi$$
−0.556710 + 0.830707i $$0.687936\pi$$
$$660$$ 3.62614 0.141147
$$661$$ −39.0780 −1.51996 −0.759980 0.649947i $$-0.774791\pi$$
−0.759980 + 0.649947i $$0.774791\pi$$
$$662$$ −5.66970 −0.220359
$$663$$ −7.58258 −0.294483
$$664$$ 16.4174 0.637120
$$665$$ −19.7477 −0.765784
$$666$$ 1.79129 0.0694110
$$667$$ 45.5826 1.76496
$$668$$ −5.73864 −0.222034
$$669$$ −6.00000 −0.231973
$$670$$ 46.1216 1.78183
$$671$$ −10.0000 −0.386046
$$672$$ 6.04356 0.233135
$$673$$ 11.2523 0.433743 0.216872 0.976200i $$-0.430415\pi$$
0.216872 + 0.976200i $$0.430415\pi$$
$$674$$ 31.4955 1.21316
$$675$$ −4.00000 −0.153960
$$676$$ −14.5045 −0.557867
$$677$$ −45.1652 −1.73584 −0.867919 0.496706i $$-0.834543\pi$$
−0.867919 + 0.496706i $$0.834543\pi$$
$$678$$ −16.4174 −0.630507
$$679$$ −2.41742 −0.0927722
$$680$$ −32.2432 −1.23647
$$681$$ 22.0000 0.843042
$$682$$ −6.41742 −0.245736
$$683$$ −33.0780 −1.26570 −0.632848 0.774276i $$-0.718114\pi$$
−0.632848 + 0.774276i $$0.718114\pi$$
$$684$$ −7.95644 −0.304222
$$685$$ −34.7477 −1.32764
$$686$$ 1.79129 0.0683917
$$687$$ −0.747727 −0.0285276
$$688$$ −7.84394 −0.299047
$$689$$ −9.58258 −0.365067
$$690$$ 30.0000 1.14208
$$691$$ 10.0000 0.380418 0.190209 0.981744i $$-0.439083\pi$$
0.190209 + 0.981744i $$0.439083\pi$$
$$692$$ −8.66061 −0.329227
$$693$$ −1.00000 −0.0379869
$$694$$ 47.1652 1.79036
$$695$$ 33.4955 1.27055
$$696$$ −11.5735 −0.438692
$$697$$ −84.6606 −3.20675
$$698$$ −26.8693 −1.01702
$$699$$ 14.0000 0.529529
$$700$$ 4.83485 0.182740
$$701$$ 10.0000 0.377695 0.188847 0.982006i $$-0.439525\pi$$
0.188847 + 0.982006i $$0.439525\pi$$
$$702$$ −1.79129 −0.0676078
$$703$$ −6.58258 −0.248267
$$704$$ 0.912878 0.0344054
$$705$$ −4.25227 −0.160150
$$706$$ 43.2867 1.62912
$$707$$ 11.5826 0.435608
$$708$$ −5.53901 −0.208169
$$709$$ 27.6606 1.03882 0.519408 0.854526i $$-0.326153\pi$$
0.519408 + 0.854526i $$0.326153\pi$$
$$710$$ 60.0000 2.25176
$$711$$ 7.16515 0.268714
$$712$$ −12.9909 −0.486855
$$713$$ −20.0000 −0.749006
$$714$$ −13.5826 −0.508315
$$715$$ −3.00000 −0.112194
$$716$$ 17.3212 0.647324
$$717$$ −16.5826 −0.619288
$$718$$ −15.8258 −0.590612
$$719$$ −14.0780 −0.525022 −0.262511 0.964929i $$-0.584551\pi$$
−0.262511 + 0.964929i $$0.584551\pi$$
$$720$$ −14.8693 −0.554147
$$721$$ −1.16515 −0.0433925
$$722$$ 43.5826 1.62198
$$723$$ 10.1652 0.378046
$$724$$ −6.74773 −0.250777
$$725$$ −32.6606 −1.21298
$$726$$ −1.79129 −0.0664809
$$727$$ −15.9129 −0.590176 −0.295088 0.955470i $$-0.595349\pi$$
−0.295088 + 0.955470i $$0.595349\pi$$
$$728$$ −1.41742 −0.0525332
$$729$$ 1.00000 0.0370370
$$730$$ 37.6170 1.39227
$$731$$ 12.0000 0.443836
$$732$$ −12.0871 −0.446753
$$733$$ −34.0000 −1.25582 −0.627909 0.778287i $$-0.716089\pi$$
−0.627909 + 0.778287i $$0.716089\pi$$
$$734$$ 39.4083 1.45459
$$735$$ −3.00000 −0.110657
$$736$$ 33.7386 1.24362
$$737$$ −8.58258 −0.316143
$$738$$ −20.0000 −0.736210
$$739$$ −31.9129 −1.17393 −0.586967 0.809611i $$-0.699678\pi$$
−0.586967 + 0.809611i $$0.699678\pi$$
$$740$$ 3.62614 0.133299
$$741$$ 6.58258 0.241817
$$742$$ −17.1652 −0.630153
$$743$$ −53.2432 −1.95330 −0.976651 0.214830i $$-0.931080\pi$$
−0.976651 + 0.214830i $$0.931080\pi$$
$$744$$ 5.07803 0.186170
$$745$$ 18.4955 0.677621
$$746$$ −62.2432 −2.27888
$$747$$ −11.5826 −0.423784
$$748$$ −9.16515 −0.335111
$$749$$ −12.5826 −0.459757
$$750$$ 5.37386 0.196226
$$751$$ −8.91288 −0.325236 −0.162618 0.986689i $$-0.551994\pi$$
−0.162618 + 0.986689i $$0.551994\pi$$
$$752$$ −7.02538 −0.256189
$$753$$ −7.41742 −0.270306
$$754$$ −14.6261 −0.532652
$$755$$ 10.7477 0.391150
$$756$$ −1.20871 −0.0439604
$$757$$ 27.3303 0.993337 0.496668 0.867940i $$-0.334557\pi$$
0.496668 + 0.867940i $$0.334557\pi$$
$$758$$ −22.5390 −0.818654
$$759$$ −5.58258 −0.202635
$$760$$ 27.9909 1.01534
$$761$$ 42.3303 1.53447 0.767236 0.641365i $$-0.221631\pi$$
0.767236 + 0.641365i $$0.221631\pi$$
$$762$$ 20.7477 0.751611
$$763$$ 3.58258 0.129698
$$764$$ 14.0000 0.506502
$$765$$ 22.7477 0.822446
$$766$$ 18.5045 0.668596
$$767$$ 4.58258 0.165467
$$768$$ −20.5481 −0.741466
$$769$$ 6.49545 0.234232 0.117116 0.993118i $$-0.462635\pi$$
0.117116 + 0.993118i $$0.462635\pi$$
$$770$$ −5.37386 −0.193661
$$771$$ −19.0000 −0.684268
$$772$$ −2.92197 −0.105164
$$773$$ 6.16515 0.221745 0.110873 0.993835i $$-0.464635\pi$$
0.110873 + 0.993835i $$0.464635\pi$$
$$774$$ 2.83485 0.101897
$$775$$ 14.3303 0.514760
$$776$$ 3.42652 0.123005
$$777$$ −1.00000 −0.0358748
$$778$$ −47.1652 −1.69095
$$779$$ 73.4955 2.63325
$$780$$ −3.62614 −0.129837
$$781$$ −11.1652 −0.399521
$$782$$ −75.8258 −2.71152
$$783$$ 8.16515 0.291799
$$784$$ −4.95644 −0.177016
$$785$$ 57.4955 2.05210
$$786$$ 28.6606 1.02229
$$787$$ 38.5826 1.37532 0.687660 0.726033i $$-0.258638\pi$$
0.687660 + 0.726033i $$0.258638\pi$$
$$788$$ 6.24318 0.222404
$$789$$ 22.9129 0.815720
$$790$$ 38.5045 1.36993
$$791$$ 9.16515 0.325875
$$792$$ 1.41742 0.0503660
$$793$$ 10.0000 0.355110
$$794$$ −56.5735 −2.00772
$$795$$ 28.7477 1.01958
$$796$$ −11.5826 −0.410534
$$797$$ −52.4955 −1.85948 −0.929742 0.368211i $$-0.879970\pi$$
−0.929742 + 0.368211i $$0.879970\pi$$
$$798$$ 11.7913 0.417407
$$799$$ 10.7477 0.380227
$$800$$ −24.1742 −0.854689
$$801$$ 9.16515 0.323835
$$802$$ 57.1652 2.01857
$$803$$ −7.00000 −0.247025
$$804$$ −10.3739 −0.365858
$$805$$ −16.7477 −0.590280
$$806$$ 6.41742 0.226044
$$807$$ −10.0000 −0.352017
$$808$$ −16.4174 −0.577563
$$809$$ 9.33030 0.328036 0.164018 0.986457i $$-0.447554\pi$$
0.164018 + 0.986457i $$0.447554\pi$$
$$810$$ 5.37386 0.188818
$$811$$ −2.25227 −0.0790880 −0.0395440 0.999218i $$-0.512591\pi$$
−0.0395440 + 0.999218i $$0.512591\pi$$
$$812$$ −9.86932 −0.346345
$$813$$ −5.41742 −0.189997
$$814$$ −1.79129 −0.0627846
$$815$$ 25.7477 0.901904
$$816$$ 37.5826 1.31565
$$817$$ −10.4174 −0.364460
$$818$$ 14.9220 0.521734
$$819$$ 1.00000 0.0349428
$$820$$ −40.4864 −1.41385
$$821$$ −47.0000 −1.64031 −0.820156 0.572140i $$-0.806113\pi$$
−0.820156 + 0.572140i $$0.806113\pi$$
$$822$$ 20.7477 0.723660
$$823$$ −30.5826 −1.06604 −0.533021 0.846102i $$-0.678943\pi$$
−0.533021 + 0.846102i $$0.678943\pi$$
$$824$$ 1.65151 0.0575332
$$825$$ 4.00000 0.139262
$$826$$ 8.20871 0.285618
$$827$$ 8.91288 0.309931 0.154966 0.987920i $$-0.450473\pi$$
0.154966 + 0.987920i $$0.450473\pi$$
$$828$$ −6.74773 −0.234500
$$829$$ −40.0000 −1.38926 −0.694629 0.719368i $$-0.744431\pi$$
−0.694629 + 0.719368i $$0.744431\pi$$
$$830$$ −62.2432 −2.16049
$$831$$ 19.1652 0.664832
$$832$$ −0.912878 −0.0316484
$$833$$ 7.58258 0.262721
$$834$$ −20.0000 −0.692543
$$835$$ −14.2432 −0.492906
$$836$$ 7.95644 0.275179
$$837$$ −3.58258 −0.123832
$$838$$ 4.62614 0.159807
$$839$$ 7.08712 0.244675 0.122337 0.992489i $$-0.460961\pi$$
0.122337 + 0.992489i $$0.460961\pi$$
$$840$$ 4.25227 0.146717
$$841$$ 37.6697 1.29896
$$842$$ −60.2958 −2.07793
$$843$$ 27.3303 0.941306
$$844$$ −15.9129 −0.547744
$$845$$ −36.0000 −1.23844
$$846$$ 2.53901 0.0872931
$$847$$ 1.00000 0.0343604
$$848$$ 47.4955 1.63100
$$849$$ −27.7477 −0.952300
$$850$$ 54.3303 1.86351
$$851$$ −5.58258 −0.191368
$$852$$ −13.4955 −0.462347
$$853$$ 17.1652 0.587724 0.293862 0.955848i $$-0.405059\pi$$
0.293862 + 0.955848i $$0.405059\pi$$
$$854$$ 17.9129 0.612966
$$855$$ −19.7477 −0.675358
$$856$$ 17.8348 0.609583
$$857$$ −7.66970 −0.261992 −0.130996 0.991383i $$-0.541817\pi$$
−0.130996 + 0.991383i $$0.541817\pi$$
$$858$$ 1.79129 0.0611536
$$859$$ 12.0000 0.409435 0.204717 0.978821i $$-0.434372\pi$$
0.204717 + 0.978821i $$0.434372\pi$$
$$860$$ 5.73864 0.195686
$$861$$ 11.1652 0.380507
$$862$$ 31.7913 1.08282
$$863$$ 14.4174 0.490775 0.245387 0.969425i $$-0.421085\pi$$
0.245387 + 0.969425i $$0.421085\pi$$
$$864$$ 6.04356 0.205606
$$865$$ −21.4955 −0.730867
$$866$$ −20.0000 −0.679628
$$867$$ −40.4955 −1.37530
$$868$$ 4.33030 0.146980
$$869$$ −7.16515 −0.243061
$$870$$ 43.8784 1.48762
$$871$$ 8.58258 0.290809
$$872$$ −5.07803 −0.171964
$$873$$ −2.41742 −0.0818174
$$874$$ 65.8258 2.22659
$$875$$ −3.00000 −0.101419
$$876$$ −8.46099 −0.285870
$$877$$ 9.49545 0.320639 0.160319 0.987065i $$-0.448748\pi$$
0.160319 + 0.987065i $$0.448748\pi$$
$$878$$ 31.1996 1.05294
$$879$$ 0 0
$$880$$ 14.8693 0.501245
$$881$$ −37.6606 −1.26882 −0.634409 0.772998i $$-0.718756\pi$$
−0.634409 + 0.772998i $$0.718756\pi$$
$$882$$ 1.79129 0.0603158
$$883$$ −55.7477 −1.87606 −0.938030 0.346554i $$-0.887352\pi$$
−0.938030 + 0.346554i $$0.887352\pi$$
$$884$$ 9.16515 0.308257
$$885$$ −13.7477 −0.462125
$$886$$ −41.4955 −1.39407
$$887$$ 38.7477 1.30102 0.650511 0.759497i $$-0.274555\pi$$
0.650511 + 0.759497i $$0.274555\pi$$
$$888$$ 1.41742 0.0475656
$$889$$ −11.5826 −0.388467
$$890$$ 49.2523 1.65094
$$891$$ −1.00000 −0.0335013
$$892$$ 7.25227 0.242824
$$893$$ −9.33030 −0.312227
$$894$$ −11.0436 −0.369352
$$895$$ 42.9909 1.43703
$$896$$ 10.4519 0.349173
$$897$$ 5.58258 0.186397
$$898$$ −32.8348 −1.09571
$$899$$ −29.2523 −0.975618
$$900$$ 4.83485 0.161162
$$901$$ −72.6606 −2.42068
$$902$$ 20.0000 0.665927
$$903$$ −1.58258 −0.0526648
$$904$$ −12.9909 −0.432071
$$905$$ −16.7477 −0.556713
$$906$$ −6.41742 −0.213205
$$907$$ 42.3303 1.40555 0.702777 0.711410i $$-0.251943\pi$$
0.702777 + 0.711410i $$0.251943\pi$$
$$908$$ −26.5917 −0.882475
$$909$$ 11.5826 0.384170
$$910$$ 5.37386 0.178142
$$911$$ −3.49545 −0.115810 −0.0579048 0.998322i $$-0.518442\pi$$
−0.0579048 + 0.998322i $$0.518442\pi$$
$$912$$ −32.6261 −1.08036
$$913$$ 11.5826 0.383327
$$914$$ −35.6697 −1.17985
$$915$$ −30.0000 −0.991769
$$916$$ 0.903787 0.0298620
$$917$$ −16.0000 −0.528367
$$918$$ −13.5826 −0.448292
$$919$$ −8.08712 −0.266770 −0.133385 0.991064i $$-0.542585\pi$$
−0.133385 + 0.991064i $$0.542585\pi$$
$$920$$ 23.7386 0.782640
$$921$$ 0 0
$$922$$ 32.8348 1.08136
$$923$$ 11.1652 0.367505
$$924$$ 1.20871 0.0397637
$$925$$ 4.00000 0.131519
$$926$$ 15.3739 0.505217
$$927$$ −1.16515 −0.0382686
$$928$$ 49.3466 1.61988
$$929$$ −21.3303 −0.699825 −0.349912 0.936782i $$-0.613789\pi$$
−0.349912 + 0.936782i $$0.613789\pi$$
$$930$$ −19.2523 −0.631307
$$931$$ −6.58258 −0.215735
$$932$$ −16.9220 −0.554298
$$933$$ −14.3303 −0.469153
$$934$$ −69.1125 −2.26143
$$935$$ −22.7477 −0.743930
$$936$$ −1.41742 −0.0463300
$$937$$ 10.0000 0.326686 0.163343 0.986569i $$-0.447772\pi$$
0.163343 + 0.986569i $$0.447772\pi$$
$$938$$ 15.3739 0.501974
$$939$$ −19.5826 −0.639053
$$940$$ 5.13977 0.167641
$$941$$ 35.1652 1.14635 0.573176 0.819433i $$-0.305711\pi$$
0.573176 + 0.819433i $$0.305711\pi$$
$$942$$ −34.3303 −1.11854
$$943$$ 62.3303 2.02975
$$944$$ −22.7133 −0.739254
$$945$$ −3.00000 −0.0975900
$$946$$ −2.83485 −0.0921689
$$947$$ 45.1652 1.46767 0.733835 0.679328i $$-0.237728\pi$$
0.733835 + 0.679328i $$0.237728\pi$$
$$948$$ −8.66061 −0.281283
$$949$$ 7.00000 0.227230
$$950$$ −47.1652 −1.53024
$$951$$ 22.4174 0.726935
$$952$$ −10.7477 −0.348336
$$953$$ −28.1652 −0.912359 −0.456179 0.889888i $$-0.650782\pi$$
−0.456179 + 0.889888i $$0.650782\pi$$
$$954$$ −17.1652 −0.555742
$$955$$ 34.7477 1.12441
$$956$$ 20.0436 0.648255
$$957$$ −8.16515 −0.263942
$$958$$ −27.9129 −0.901824
$$959$$ −11.5826 −0.374021
$$960$$ 2.73864 0.0883891
$$961$$ −18.1652 −0.585973
$$962$$ 1.79129 0.0577534
$$963$$ −12.5826 −0.405468
$$964$$ −12.2867 −0.395729
$$965$$ −7.25227 −0.233459
$$966$$ 10.0000 0.321745
$$967$$ −13.1652 −0.423363 −0.211681 0.977339i $$-0.567894\pi$$
−0.211681 + 0.977339i $$0.567894\pi$$
$$968$$ −1.41742 −0.0455577
$$969$$ 49.9129 1.60343
$$970$$ −12.9909 −0.417113
$$971$$ −12.5826 −0.403794 −0.201897 0.979407i $$-0.564711\pi$$
−0.201897 + 0.979407i $$0.564711\pi$$
$$972$$ −1.20871 −0.0387695
$$973$$ 11.1652 0.357938
$$974$$ 18.5045 0.592924
$$975$$ −4.00000 −0.128103
$$976$$ −49.5644 −1.58652
$$977$$ 23.2523 0.743906 0.371953 0.928252i $$-0.378688\pi$$
0.371953 + 0.928252i $$0.378688\pi$$
$$978$$ −15.3739 −0.491602
$$979$$ −9.16515 −0.292920
$$980$$ 3.62614 0.115833
$$981$$ 3.58258 0.114383
$$982$$ −41.0436 −1.30975
$$983$$ −4.83485 −0.154208 −0.0771039 0.997023i $$-0.524567\pi$$
−0.0771039 + 0.997023i $$0.524567\pi$$
$$984$$ −15.8258 −0.504507
$$985$$ 15.4955 0.493726
$$986$$ −110.904 −3.53190
$$987$$ −1.41742 −0.0451171
$$988$$ −7.95644 −0.253128
$$989$$ −8.83485 −0.280932
$$990$$ −5.37386 −0.170793
$$991$$ −20.2523 −0.643335 −0.321667 0.946853i $$-0.604243\pi$$
−0.321667 + 0.946853i $$0.604243\pi$$
$$992$$ −21.6515 −0.687436
$$993$$ 3.16515 0.100443
$$994$$ 20.0000 0.634361
$$995$$ −28.7477 −0.911364
$$996$$ 14.0000 0.443607
$$997$$ 10.6606 0.337625 0.168812 0.985648i $$-0.446007\pi$$
0.168812 + 0.985648i $$0.446007\pi$$
$$998$$ 74.7822 2.36719
$$999$$ −1.00000 −0.0316386
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 231.2.a.b.1.2 2
3.2 odd 2 693.2.a.j.1.1 2
4.3 odd 2 3696.2.a.bl.1.2 2
5.4 even 2 5775.2.a.bn.1.1 2
7.6 odd 2 1617.2.a.o.1.2 2
11.10 odd 2 2541.2.a.z.1.1 2
21.20 even 2 4851.2.a.ba.1.1 2
33.32 even 2 7623.2.a.bf.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.b.1.2 2 1.1 even 1 trivial
693.2.a.j.1.1 2 3.2 odd 2
1617.2.a.o.1.2 2 7.6 odd 2
2541.2.a.z.1.1 2 11.10 odd 2
3696.2.a.bl.1.2 2 4.3 odd 2
4851.2.a.ba.1.1 2 21.20 even 2
5775.2.a.bn.1.1 2 5.4 even 2
7623.2.a.bf.1.2 2 33.32 even 2