# Properties

 Label 418.2.a.e.1.2 Level $418$ Weight $2$ Character 418.1 Self dual yes Analytic conductor $3.338$ 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] = [418,2,Mod(1,418)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(418, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("418.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.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: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 418.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +2.56155 q^{3} +1.00000 q^{4} +2.00000 q^{5} -2.56155 q^{6} -0.561553 q^{7} -1.00000 q^{8} +3.56155 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +2.56155 q^{3} +1.00000 q^{4} +2.00000 q^{5} -2.56155 q^{6} -0.561553 q^{7} -1.00000 q^{8} +3.56155 q^{9} -2.00000 q^{10} +1.00000 q^{11} +2.56155 q^{12} +0.561553 q^{13} +0.561553 q^{14} +5.12311 q^{15} +1.00000 q^{16} -0.561553 q^{17} -3.56155 q^{18} -1.00000 q^{19} +2.00000 q^{20} -1.43845 q^{21} -1.00000 q^{22} +1.43845 q^{23} -2.56155 q^{24} -1.00000 q^{25} -0.561553 q^{26} +1.43845 q^{27} -0.561553 q^{28} -5.68466 q^{29} -5.12311 q^{30} +2.00000 q^{31} -1.00000 q^{32} +2.56155 q^{33} +0.561553 q^{34} -1.12311 q^{35} +3.56155 q^{36} -5.12311 q^{37} +1.00000 q^{38} +1.43845 q^{39} -2.00000 q^{40} +2.00000 q^{41} +1.43845 q^{42} +1.00000 q^{44} +7.12311 q^{45} -1.43845 q^{46} +8.00000 q^{47} +2.56155 q^{48} -6.68466 q^{49} +1.00000 q^{50} -1.43845 q^{51} +0.561553 q^{52} +12.8078 q^{53} -1.43845 q^{54} +2.00000 q^{55} +0.561553 q^{56} -2.56155 q^{57} +5.68466 q^{58} -7.68466 q^{59} +5.12311 q^{60} +6.24621 q^{61} -2.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +1.12311 q^{65} -2.56155 q^{66} -7.68466 q^{67} -0.561553 q^{68} +3.68466 q^{69} +1.12311 q^{70} +6.00000 q^{71} -3.56155 q^{72} -9.68466 q^{73} +5.12311 q^{74} -2.56155 q^{75} -1.00000 q^{76} -0.561553 q^{77} -1.43845 q^{78} -4.00000 q^{79} +2.00000 q^{80} -7.00000 q^{81} -2.00000 q^{82} +14.2462 q^{83} -1.43845 q^{84} -1.12311 q^{85} -14.5616 q^{87} -1.00000 q^{88} -0.876894 q^{89} -7.12311 q^{90} -0.315342 q^{91} +1.43845 q^{92} +5.12311 q^{93} -8.00000 q^{94} -2.00000 q^{95} -2.56155 q^{96} -7.12311 q^{97} +6.68466 q^{98} +3.56155 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 3 q^{7} - 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 + 4 * q^5 - q^6 + 3 * q^7 - 2 * q^8 + 3 * q^9 $$2 q - 2 q^{2} + q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 3 q^{7} - 2 q^{8} + 3 q^{9} - 4 q^{10} + 2 q^{11} + q^{12} - 3 q^{13} - 3 q^{14} + 2 q^{15} + 2 q^{16} + 3 q^{17} - 3 q^{18} - 2 q^{19} + 4 q^{20} - 7 q^{21} - 2 q^{22} + 7 q^{23} - q^{24} - 2 q^{25} + 3 q^{26} + 7 q^{27} + 3 q^{28} + q^{29} - 2 q^{30} + 4 q^{31} - 2 q^{32} + q^{33} - 3 q^{34} + 6 q^{35} + 3 q^{36} - 2 q^{37} + 2 q^{38} + 7 q^{39} - 4 q^{40} + 4 q^{41} + 7 q^{42} + 2 q^{44} + 6 q^{45} - 7 q^{46} + 16 q^{47} + q^{48} - q^{49} + 2 q^{50} - 7 q^{51} - 3 q^{52} + 5 q^{53} - 7 q^{54} + 4 q^{55} - 3 q^{56} - q^{57} - q^{58} - 3 q^{59} + 2 q^{60} - 4 q^{61} - 4 q^{62} - 4 q^{63} + 2 q^{64} - 6 q^{65} - q^{66} - 3 q^{67} + 3 q^{68} - 5 q^{69} - 6 q^{70} + 12 q^{71} - 3 q^{72} - 7 q^{73} + 2 q^{74} - q^{75} - 2 q^{76} + 3 q^{77} - 7 q^{78} - 8 q^{79} + 4 q^{80} - 14 q^{81} - 4 q^{82} + 12 q^{83} - 7 q^{84} + 6 q^{85} - 25 q^{87} - 2 q^{88} - 10 q^{89} - 6 q^{90} - 13 q^{91} + 7 q^{92} + 2 q^{93} - 16 q^{94} - 4 q^{95} - q^{96} - 6 q^{97} + q^{98} + 3 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 + 4 * q^5 - q^6 + 3 * q^7 - 2 * q^8 + 3 * q^9 - 4 * q^10 + 2 * q^11 + q^12 - 3 * q^13 - 3 * q^14 + 2 * q^15 + 2 * q^16 + 3 * q^17 - 3 * q^18 - 2 * q^19 + 4 * q^20 - 7 * q^21 - 2 * q^22 + 7 * q^23 - q^24 - 2 * q^25 + 3 * q^26 + 7 * q^27 + 3 * q^28 + q^29 - 2 * q^30 + 4 * q^31 - 2 * q^32 + q^33 - 3 * q^34 + 6 * q^35 + 3 * q^36 - 2 * q^37 + 2 * q^38 + 7 * q^39 - 4 * q^40 + 4 * q^41 + 7 * q^42 + 2 * q^44 + 6 * q^45 - 7 * q^46 + 16 * q^47 + q^48 - q^49 + 2 * q^50 - 7 * q^51 - 3 * q^52 + 5 * q^53 - 7 * q^54 + 4 * q^55 - 3 * q^56 - q^57 - q^58 - 3 * q^59 + 2 * q^60 - 4 * q^61 - 4 * q^62 - 4 * q^63 + 2 * q^64 - 6 * q^65 - q^66 - 3 * q^67 + 3 * q^68 - 5 * q^69 - 6 * q^70 + 12 * q^71 - 3 * q^72 - 7 * q^73 + 2 * q^74 - q^75 - 2 * q^76 + 3 * q^77 - 7 * q^78 - 8 * q^79 + 4 * q^80 - 14 * q^81 - 4 * q^82 + 12 * q^83 - 7 * q^84 + 6 * q^85 - 25 * q^87 - 2 * q^88 - 10 * q^89 - 6 * q^90 - 13 * q^91 + 7 * q^92 + 2 * q^93 - 16 * q^94 - 4 * q^95 - q^96 - 6 * q^97 + q^98 + 3 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 2.56155 1.47891 0.739457 0.673204i $$-0.235083\pi$$
0.739457 + 0.673204i $$0.235083\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ −2.56155 −1.04575
$$7$$ −0.561553 −0.212247 −0.106124 0.994353i $$-0.533844\pi$$
−0.106124 + 0.994353i $$0.533844\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 3.56155 1.18718
$$10$$ −2.00000 −0.632456
$$11$$ 1.00000 0.301511
$$12$$ 2.56155 0.739457
$$13$$ 0.561553 0.155747 0.0778734 0.996963i $$-0.475187\pi$$
0.0778734 + 0.996963i $$0.475187\pi$$
$$14$$ 0.561553 0.150081
$$15$$ 5.12311 1.32278
$$16$$ 1.00000 0.250000
$$17$$ −0.561553 −0.136197 −0.0680983 0.997679i $$-0.521693\pi$$
−0.0680983 + 0.997679i $$0.521693\pi$$
$$18$$ −3.56155 −0.839466
$$19$$ −1.00000 −0.229416
$$20$$ 2.00000 0.447214
$$21$$ −1.43845 −0.313895
$$22$$ −1.00000 −0.213201
$$23$$ 1.43845 0.299937 0.149968 0.988691i $$-0.452083\pi$$
0.149968 + 0.988691i $$0.452083\pi$$
$$24$$ −2.56155 −0.522875
$$25$$ −1.00000 −0.200000
$$26$$ −0.561553 −0.110130
$$27$$ 1.43845 0.276829
$$28$$ −0.561553 −0.106124
$$29$$ −5.68466 −1.05561 −0.527807 0.849364i $$-0.676986\pi$$
−0.527807 + 0.849364i $$0.676986\pi$$
$$30$$ −5.12311 −0.935347
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 2.56155 0.445909
$$34$$ 0.561553 0.0963055
$$35$$ −1.12311 −0.189839
$$36$$ 3.56155 0.593592
$$37$$ −5.12311 −0.842233 −0.421117 0.907006i $$-0.638362\pi$$
−0.421117 + 0.907006i $$0.638362\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 1.43845 0.230336
$$40$$ −2.00000 −0.316228
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 1.43845 0.221957
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 7.12311 1.06185
$$46$$ −1.43845 −0.212087
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 2.56155 0.369728
$$49$$ −6.68466 −0.954951
$$50$$ 1.00000 0.141421
$$51$$ −1.43845 −0.201423
$$52$$ 0.561553 0.0778734
$$53$$ 12.8078 1.75928 0.879641 0.475638i $$-0.157783\pi$$
0.879641 + 0.475638i $$0.157783\pi$$
$$54$$ −1.43845 −0.195748
$$55$$ 2.00000 0.269680
$$56$$ 0.561553 0.0750407
$$57$$ −2.56155 −0.339286
$$58$$ 5.68466 0.746432
$$59$$ −7.68466 −1.00046 −0.500229 0.865893i $$-0.666751\pi$$
−0.500229 + 0.865893i $$0.666751\pi$$
$$60$$ 5.12311 0.661390
$$61$$ 6.24621 0.799745 0.399873 0.916571i $$-0.369054\pi$$
0.399873 + 0.916571i $$0.369054\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ −2.00000 −0.251976
$$64$$ 1.00000 0.125000
$$65$$ 1.12311 0.139304
$$66$$ −2.56155 −0.315305
$$67$$ −7.68466 −0.938830 −0.469415 0.882978i $$-0.655535\pi$$
−0.469415 + 0.882978i $$0.655535\pi$$
$$68$$ −0.561553 −0.0680983
$$69$$ 3.68466 0.443581
$$70$$ 1.12311 0.134237
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ −3.56155 −0.419733
$$73$$ −9.68466 −1.13350 −0.566752 0.823889i $$-0.691800\pi$$
−0.566752 + 0.823889i $$0.691800\pi$$
$$74$$ 5.12311 0.595549
$$75$$ −2.56155 −0.295783
$$76$$ −1.00000 −0.114708
$$77$$ −0.561553 −0.0639949
$$78$$ −1.43845 −0.162872
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 2.00000 0.223607
$$81$$ −7.00000 −0.777778
$$82$$ −2.00000 −0.220863
$$83$$ 14.2462 1.56372 0.781862 0.623451i $$-0.214270\pi$$
0.781862 + 0.623451i $$0.214270\pi$$
$$84$$ −1.43845 −0.156947
$$85$$ −1.12311 −0.121818
$$86$$ 0 0
$$87$$ −14.5616 −1.56116
$$88$$ −1.00000 −0.106600
$$89$$ −0.876894 −0.0929506 −0.0464753 0.998919i $$-0.514799\pi$$
−0.0464753 + 0.998919i $$0.514799\pi$$
$$90$$ −7.12311 −0.750841
$$91$$ −0.315342 −0.0330568
$$92$$ 1.43845 0.149968
$$93$$ 5.12311 0.531241
$$94$$ −8.00000 −0.825137
$$95$$ −2.00000 −0.205196
$$96$$ −2.56155 −0.261437
$$97$$ −7.12311 −0.723242 −0.361621 0.932325i $$-0.617777\pi$$
−0.361621 + 0.932325i $$0.617777\pi$$
$$98$$ 6.68466 0.675252
$$99$$ 3.56155 0.357950
$$100$$ −1.00000 −0.100000
$$101$$ −6.87689 −0.684277 −0.342138 0.939650i $$-0.611151\pi$$
−0.342138 + 0.939650i $$0.611151\pi$$
$$102$$ 1.43845 0.142427
$$103$$ −13.3693 −1.31732 −0.658659 0.752442i $$-0.728876\pi$$
−0.658659 + 0.752442i $$0.728876\pi$$
$$104$$ −0.561553 −0.0550648
$$105$$ −2.87689 −0.280756
$$106$$ −12.8078 −1.24400
$$107$$ −9.93087 −0.960053 −0.480027 0.877254i $$-0.659373\pi$$
−0.480027 + 0.877254i $$0.659373\pi$$
$$108$$ 1.43845 0.138415
$$109$$ −6.31534 −0.604900 −0.302450 0.953165i $$-0.597805\pi$$
−0.302450 + 0.953165i $$0.597805\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ −13.1231 −1.24559
$$112$$ −0.561553 −0.0530618
$$113$$ −3.12311 −0.293797 −0.146899 0.989152i $$-0.546929\pi$$
−0.146899 + 0.989152i $$0.546929\pi$$
$$114$$ 2.56155 0.239911
$$115$$ 2.87689 0.268272
$$116$$ −5.68466 −0.527807
$$117$$ 2.00000 0.184900
$$118$$ 7.68466 0.707430
$$119$$ 0.315342 0.0289073
$$120$$ −5.12311 −0.467673
$$121$$ 1.00000 0.0909091
$$122$$ −6.24621 −0.565505
$$123$$ 5.12311 0.461935
$$124$$ 2.00000 0.179605
$$125$$ −12.0000 −1.07331
$$126$$ 2.00000 0.178174
$$127$$ 2.24621 0.199319 0.0996595 0.995022i $$-0.468225\pi$$
0.0996595 + 0.995022i $$0.468225\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ −1.12311 −0.0985029
$$131$$ 1.12311 0.0981262 0.0490631 0.998796i $$-0.484376\pi$$
0.0490631 + 0.998796i $$0.484376\pi$$
$$132$$ 2.56155 0.222955
$$133$$ 0.561553 0.0486928
$$134$$ 7.68466 0.663853
$$135$$ 2.87689 0.247604
$$136$$ 0.561553 0.0481528
$$137$$ 12.5616 1.07321 0.536603 0.843835i $$-0.319707\pi$$
0.536603 + 0.843835i $$0.319707\pi$$
$$138$$ −3.68466 −0.313659
$$139$$ 7.36932 0.625057 0.312529 0.949908i $$-0.398824\pi$$
0.312529 + 0.949908i $$0.398824\pi$$
$$140$$ −1.12311 −0.0949197
$$141$$ 20.4924 1.72577
$$142$$ −6.00000 −0.503509
$$143$$ 0.561553 0.0469594
$$144$$ 3.56155 0.296796
$$145$$ −11.3693 −0.944170
$$146$$ 9.68466 0.801508
$$147$$ −17.1231 −1.41229
$$148$$ −5.12311 −0.421117
$$149$$ −6.87689 −0.563377 −0.281689 0.959506i $$-0.590894\pi$$
−0.281689 + 0.959506i $$0.590894\pi$$
$$150$$ 2.56155 0.209150
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ −2.00000 −0.161690
$$154$$ 0.561553 0.0452512
$$155$$ 4.00000 0.321288
$$156$$ 1.43845 0.115168
$$157$$ −14.4924 −1.15662 −0.578311 0.815817i $$-0.696288\pi$$
−0.578311 + 0.815817i $$0.696288\pi$$
$$158$$ 4.00000 0.318223
$$159$$ 32.8078 2.60182
$$160$$ −2.00000 −0.158114
$$161$$ −0.807764 −0.0636607
$$162$$ 7.00000 0.549972
$$163$$ 11.3693 0.890514 0.445257 0.895403i $$-0.353112\pi$$
0.445257 + 0.895403i $$0.353112\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 5.12311 0.398833
$$166$$ −14.2462 −1.10572
$$167$$ −0.630683 −0.0488037 −0.0244019 0.999702i $$-0.507768\pi$$
−0.0244019 + 0.999702i $$0.507768\pi$$
$$168$$ 1.43845 0.110979
$$169$$ −12.6847 −0.975743
$$170$$ 1.12311 0.0861383
$$171$$ −3.56155 −0.272359
$$172$$ 0 0
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 14.5616 1.10391
$$175$$ 0.561553 0.0424494
$$176$$ 1.00000 0.0753778
$$177$$ −19.6847 −1.47959
$$178$$ 0.876894 0.0657260
$$179$$ 5.75379 0.430058 0.215029 0.976608i $$-0.431015\pi$$
0.215029 + 0.976608i $$0.431015\pi$$
$$180$$ 7.12311 0.530925
$$181$$ −10.8769 −0.808473 −0.404237 0.914654i $$-0.632463\pi$$
−0.404237 + 0.914654i $$0.632463\pi$$
$$182$$ 0.315342 0.0233747
$$183$$ 16.0000 1.18275
$$184$$ −1.43845 −0.106044
$$185$$ −10.2462 −0.753316
$$186$$ −5.12311 −0.375644
$$187$$ −0.561553 −0.0410648
$$188$$ 8.00000 0.583460
$$189$$ −0.807764 −0.0587562
$$190$$ 2.00000 0.145095
$$191$$ −8.31534 −0.601677 −0.300838 0.953675i $$-0.597267\pi$$
−0.300838 + 0.953675i $$0.597267\pi$$
$$192$$ 2.56155 0.184864
$$193$$ 7.12311 0.512732 0.256366 0.966580i $$-0.417475\pi$$
0.256366 + 0.966580i $$0.417475\pi$$
$$194$$ 7.12311 0.511409
$$195$$ 2.87689 0.206019
$$196$$ −6.68466 −0.477476
$$197$$ 18.2462 1.29999 0.649994 0.759939i $$-0.274771\pi$$
0.649994 + 0.759939i $$0.274771\pi$$
$$198$$ −3.56155 −0.253109
$$199$$ 11.6847 0.828303 0.414152 0.910208i $$-0.364078\pi$$
0.414152 + 0.910208i $$0.364078\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −19.6847 −1.38845
$$202$$ 6.87689 0.483857
$$203$$ 3.19224 0.224051
$$204$$ −1.43845 −0.100711
$$205$$ 4.00000 0.279372
$$206$$ 13.3693 0.931484
$$207$$ 5.12311 0.356080
$$208$$ 0.561553 0.0389367
$$209$$ −1.00000 −0.0691714
$$210$$ 2.87689 0.198525
$$211$$ −0.315342 −0.0217090 −0.0108545 0.999941i $$-0.503455\pi$$
−0.0108545 + 0.999941i $$0.503455\pi$$
$$212$$ 12.8078 0.879641
$$213$$ 15.3693 1.05309
$$214$$ 9.93087 0.678860
$$215$$ 0 0
$$216$$ −1.43845 −0.0978739
$$217$$ −1.12311 −0.0762414
$$218$$ 6.31534 0.427729
$$219$$ −24.8078 −1.67635
$$220$$ 2.00000 0.134840
$$221$$ −0.315342 −0.0212122
$$222$$ 13.1231 0.880765
$$223$$ 24.7386 1.65662 0.828311 0.560269i $$-0.189302\pi$$
0.828311 + 0.560269i $$0.189302\pi$$
$$224$$ 0.561553 0.0375203
$$225$$ −3.56155 −0.237437
$$226$$ 3.12311 0.207746
$$227$$ 28.1771 1.87018 0.935089 0.354412i $$-0.115319\pi$$
0.935089 + 0.354412i $$0.115319\pi$$
$$228$$ −2.56155 −0.169643
$$229$$ 12.8769 0.850929 0.425465 0.904975i $$-0.360111\pi$$
0.425465 + 0.904975i $$0.360111\pi$$
$$230$$ −2.87689 −0.189697
$$231$$ −1.43845 −0.0946429
$$232$$ 5.68466 0.373216
$$233$$ −28.7386 −1.88273 −0.941365 0.337389i $$-0.890456\pi$$
−0.941365 + 0.337389i $$0.890456\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 16.0000 1.04372
$$236$$ −7.68466 −0.500229
$$237$$ −10.2462 −0.665563
$$238$$ −0.315342 −0.0204406
$$239$$ 23.9309 1.54796 0.773980 0.633210i $$-0.218263\pi$$
0.773980 + 0.633210i $$0.218263\pi$$
$$240$$ 5.12311 0.330695
$$241$$ −19.6155 −1.26355 −0.631774 0.775153i $$-0.717673\pi$$
−0.631774 + 0.775153i $$0.717673\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ −22.2462 −1.42710
$$244$$ 6.24621 0.399873
$$245$$ −13.3693 −0.854134
$$246$$ −5.12311 −0.326637
$$247$$ −0.561553 −0.0357307
$$248$$ −2.00000 −0.127000
$$249$$ 36.4924 2.31261
$$250$$ 12.0000 0.758947
$$251$$ 17.1231 1.08080 0.540400 0.841408i $$-0.318273\pi$$
0.540400 + 0.841408i $$0.318273\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 1.43845 0.0904344
$$254$$ −2.24621 −0.140940
$$255$$ −2.87689 −0.180158
$$256$$ 1.00000 0.0625000
$$257$$ −22.0000 −1.37232 −0.686161 0.727450i $$-0.740706\pi$$
−0.686161 + 0.727450i $$0.740706\pi$$
$$258$$ 0 0
$$259$$ 2.87689 0.178762
$$260$$ 1.12311 0.0696521
$$261$$ −20.2462 −1.25321
$$262$$ −1.12311 −0.0693857
$$263$$ 8.87689 0.547373 0.273686 0.961819i $$-0.411757\pi$$
0.273686 + 0.961819i $$0.411757\pi$$
$$264$$ −2.56155 −0.157653
$$265$$ 25.6155 1.57355
$$266$$ −0.561553 −0.0344310
$$267$$ −2.24621 −0.137466
$$268$$ −7.68466 −0.469415
$$269$$ 2.87689 0.175407 0.0877037 0.996147i $$-0.472047\pi$$
0.0877037 + 0.996147i $$0.472047\pi$$
$$270$$ −2.87689 −0.175082
$$271$$ 25.6847 1.56023 0.780116 0.625635i $$-0.215160\pi$$
0.780116 + 0.625635i $$0.215160\pi$$
$$272$$ −0.561553 −0.0340491
$$273$$ −0.807764 −0.0488881
$$274$$ −12.5616 −0.758871
$$275$$ −1.00000 −0.0603023
$$276$$ 3.68466 0.221790
$$277$$ 8.49242 0.510260 0.255130 0.966907i $$-0.417882\pi$$
0.255130 + 0.966907i $$0.417882\pi$$
$$278$$ −7.36932 −0.441982
$$279$$ 7.12311 0.426449
$$280$$ 1.12311 0.0671184
$$281$$ −25.3693 −1.51341 −0.756703 0.653758i $$-0.773191\pi$$
−0.756703 + 0.653758i $$0.773191\pi$$
$$282$$ −20.4924 −1.22031
$$283$$ 11.3693 0.675836 0.337918 0.941176i $$-0.390277\pi$$
0.337918 + 0.941176i $$0.390277\pi$$
$$284$$ 6.00000 0.356034
$$285$$ −5.12311 −0.303467
$$286$$ −0.561553 −0.0332053
$$287$$ −1.12311 −0.0662948
$$288$$ −3.56155 −0.209867
$$289$$ −16.6847 −0.981450
$$290$$ 11.3693 0.667629
$$291$$ −18.2462 −1.06961
$$292$$ −9.68466 −0.566752
$$293$$ 3.93087 0.229644 0.114822 0.993386i $$-0.463370\pi$$
0.114822 + 0.993386i $$0.463370\pi$$
$$294$$ 17.1231 0.998640
$$295$$ −15.3693 −0.894836
$$296$$ 5.12311 0.297774
$$297$$ 1.43845 0.0834672
$$298$$ 6.87689 0.398368
$$299$$ 0.807764 0.0467142
$$300$$ −2.56155 −0.147891
$$301$$ 0 0
$$302$$ −4.00000 −0.230174
$$303$$ −17.6155 −1.01199
$$304$$ −1.00000 −0.0573539
$$305$$ 12.4924 0.715314
$$306$$ 2.00000 0.114332
$$307$$ −22.2462 −1.26966 −0.634829 0.772653i $$-0.718930\pi$$
−0.634829 + 0.772653i $$0.718930\pi$$
$$308$$ −0.561553 −0.0319974
$$309$$ −34.2462 −1.94820
$$310$$ −4.00000 −0.227185
$$311$$ −4.80776 −0.272623 −0.136312 0.990666i $$-0.543525\pi$$
−0.136312 + 0.990666i $$0.543525\pi$$
$$312$$ −1.43845 −0.0814360
$$313$$ −0.561553 −0.0317408 −0.0158704 0.999874i $$-0.505052\pi$$
−0.0158704 + 0.999874i $$0.505052\pi$$
$$314$$ 14.4924 0.817855
$$315$$ −4.00000 −0.225374
$$316$$ −4.00000 −0.225018
$$317$$ −7.05398 −0.396191 −0.198095 0.980183i $$-0.563476\pi$$
−0.198095 + 0.980183i $$0.563476\pi$$
$$318$$ −32.8078 −1.83977
$$319$$ −5.68466 −0.318280
$$320$$ 2.00000 0.111803
$$321$$ −25.4384 −1.41984
$$322$$ 0.807764 0.0450149
$$323$$ 0.561553 0.0312456
$$324$$ −7.00000 −0.388889
$$325$$ −0.561553 −0.0311493
$$326$$ −11.3693 −0.629688
$$327$$ −16.1771 −0.894595
$$328$$ −2.00000 −0.110432
$$329$$ −4.49242 −0.247675
$$330$$ −5.12311 −0.282018
$$331$$ −14.5616 −0.800375 −0.400188 0.916433i $$-0.631055\pi$$
−0.400188 + 0.916433i $$0.631055\pi$$
$$332$$ 14.2462 0.781862
$$333$$ −18.2462 −0.999886
$$334$$ 0.630683 0.0345094
$$335$$ −15.3693 −0.839715
$$336$$ −1.43845 −0.0784737
$$337$$ 11.1231 0.605914 0.302957 0.953004i $$-0.402026\pi$$
0.302957 + 0.953004i $$0.402026\pi$$
$$338$$ 12.6847 0.689954
$$339$$ −8.00000 −0.434500
$$340$$ −1.12311 −0.0609090
$$341$$ 2.00000 0.108306
$$342$$ 3.56155 0.192587
$$343$$ 7.68466 0.414933
$$344$$ 0 0
$$345$$ 7.36932 0.396751
$$346$$ −18.0000 −0.967686
$$347$$ 9.61553 0.516189 0.258094 0.966120i $$-0.416905\pi$$
0.258094 + 0.966120i $$0.416905\pi$$
$$348$$ −14.5616 −0.780581
$$349$$ 21.6155 1.15705 0.578526 0.815664i $$-0.303628\pi$$
0.578526 + 0.815664i $$0.303628\pi$$
$$350$$ −0.561553 −0.0300163
$$351$$ 0.807764 0.0431153
$$352$$ −1.00000 −0.0533002
$$353$$ 22.1771 1.18037 0.590183 0.807269i $$-0.299055\pi$$
0.590183 + 0.807269i $$0.299055\pi$$
$$354$$ 19.6847 1.04623
$$355$$ 12.0000 0.636894
$$356$$ −0.876894 −0.0464753
$$357$$ 0.807764 0.0427514
$$358$$ −5.75379 −0.304097
$$359$$ −15.9309 −0.840799 −0.420400 0.907339i $$-0.638110\pi$$
−0.420400 + 0.907339i $$0.638110\pi$$
$$360$$ −7.12311 −0.375421
$$361$$ 1.00000 0.0526316
$$362$$ 10.8769 0.571677
$$363$$ 2.56155 0.134447
$$364$$ −0.315342 −0.0165284
$$365$$ −19.3693 −1.01384
$$366$$ −16.0000 −0.836333
$$367$$ 30.2462 1.57884 0.789420 0.613854i $$-0.210382\pi$$
0.789420 + 0.613854i $$0.210382\pi$$
$$368$$ 1.43845 0.0749842
$$369$$ 7.12311 0.370814
$$370$$ 10.2462 0.532675
$$371$$ −7.19224 −0.373402
$$372$$ 5.12311 0.265621
$$373$$ 3.43845 0.178036 0.0890180 0.996030i $$-0.471627\pi$$
0.0890180 + 0.996030i $$0.471627\pi$$
$$374$$ 0.561553 0.0290372
$$375$$ −30.7386 −1.58734
$$376$$ −8.00000 −0.412568
$$377$$ −3.19224 −0.164409
$$378$$ 0.807764 0.0415469
$$379$$ 6.56155 0.337044 0.168522 0.985698i $$-0.446101\pi$$
0.168522 + 0.985698i $$0.446101\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ 5.75379 0.294776
$$382$$ 8.31534 0.425450
$$383$$ −2.00000 −0.102195 −0.0510976 0.998694i $$-0.516272\pi$$
−0.0510976 + 0.998694i $$0.516272\pi$$
$$384$$ −2.56155 −0.130719
$$385$$ −1.12311 −0.0572388
$$386$$ −7.12311 −0.362557
$$387$$ 0 0
$$388$$ −7.12311 −0.361621
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ −2.87689 −0.145677
$$391$$ −0.807764 −0.0408504
$$392$$ 6.68466 0.337626
$$393$$ 2.87689 0.145120
$$394$$ −18.2462 −0.919231
$$395$$ −8.00000 −0.402524
$$396$$ 3.56155 0.178975
$$397$$ 38.9848 1.95659 0.978297 0.207209i $$-0.0664381\pi$$
0.978297 + 0.207209i $$0.0664381\pi$$
$$398$$ −11.6847 −0.585699
$$399$$ 1.43845 0.0720124
$$400$$ −1.00000 −0.0500000
$$401$$ −7.75379 −0.387206 −0.193603 0.981080i $$-0.562017\pi$$
−0.193603 + 0.981080i $$0.562017\pi$$
$$402$$ 19.6847 0.981782
$$403$$ 1.12311 0.0559459
$$404$$ −6.87689 −0.342138
$$405$$ −14.0000 −0.695666
$$406$$ −3.19224 −0.158428
$$407$$ −5.12311 −0.253943
$$408$$ 1.43845 0.0712137
$$409$$ 16.2462 0.803323 0.401662 0.915788i $$-0.368433\pi$$
0.401662 + 0.915788i $$0.368433\pi$$
$$410$$ −4.00000 −0.197546
$$411$$ 32.1771 1.58718
$$412$$ −13.3693 −0.658659
$$413$$ 4.31534 0.212344
$$414$$ −5.12311 −0.251787
$$415$$ 28.4924 1.39864
$$416$$ −0.561553 −0.0275324
$$417$$ 18.8769 0.924405
$$418$$ 1.00000 0.0489116
$$419$$ −14.8769 −0.726784 −0.363392 0.931636i $$-0.618381\pi$$
−0.363392 + 0.931636i $$0.618381\pi$$
$$420$$ −2.87689 −0.140378
$$421$$ −29.9309 −1.45874 −0.729371 0.684119i $$-0.760187\pi$$
−0.729371 + 0.684119i $$0.760187\pi$$
$$422$$ 0.315342 0.0153506
$$423$$ 28.4924 1.38535
$$424$$ −12.8078 −0.622000
$$425$$ 0.561553 0.0272393
$$426$$ −15.3693 −0.744646
$$427$$ −3.50758 −0.169744
$$428$$ −9.93087 −0.480027
$$429$$ 1.43845 0.0694489
$$430$$ 0 0
$$431$$ 31.8617 1.53473 0.767363 0.641213i $$-0.221568\pi$$
0.767363 + 0.641213i $$0.221568\pi$$
$$432$$ 1.43845 0.0692073
$$433$$ 6.63068 0.318650 0.159325 0.987226i $$-0.449068\pi$$
0.159325 + 0.987226i $$0.449068\pi$$
$$434$$ 1.12311 0.0539108
$$435$$ −29.1231 −1.39635
$$436$$ −6.31534 −0.302450
$$437$$ −1.43845 −0.0688103
$$438$$ 24.8078 1.18536
$$439$$ −9.61553 −0.458924 −0.229462 0.973318i $$-0.573697\pi$$
−0.229462 + 0.973318i $$0.573697\pi$$
$$440$$ −2.00000 −0.0953463
$$441$$ −23.8078 −1.13370
$$442$$ 0.315342 0.0149993
$$443$$ 39.8617 1.89389 0.946944 0.321398i $$-0.104153\pi$$
0.946944 + 0.321398i $$0.104153\pi$$
$$444$$ −13.1231 −0.622795
$$445$$ −1.75379 −0.0831376
$$446$$ −24.7386 −1.17141
$$447$$ −17.6155 −0.833186
$$448$$ −0.561553 −0.0265309
$$449$$ 23.6155 1.11449 0.557243 0.830350i $$-0.311859\pi$$
0.557243 + 0.830350i $$0.311859\pi$$
$$450$$ 3.56155 0.167893
$$451$$ 2.00000 0.0941763
$$452$$ −3.12311 −0.146899
$$453$$ 10.2462 0.481409
$$454$$ −28.1771 −1.32242
$$455$$ −0.630683 −0.0295669
$$456$$ 2.56155 0.119956
$$457$$ 36.5616 1.71028 0.855139 0.518399i $$-0.173472\pi$$
0.855139 + 0.518399i $$0.173472\pi$$
$$458$$ −12.8769 −0.601698
$$459$$ −0.807764 −0.0377032
$$460$$ 2.87689 0.134136
$$461$$ 26.7386 1.24534 0.622671 0.782484i $$-0.286047\pi$$
0.622671 + 0.782484i $$0.286047\pi$$
$$462$$ 1.43845 0.0669226
$$463$$ 3.50758 0.163011 0.0815055 0.996673i $$-0.474027\pi$$
0.0815055 + 0.996673i $$0.474027\pi$$
$$464$$ −5.68466 −0.263904
$$465$$ 10.2462 0.475157
$$466$$ 28.7386 1.33129
$$467$$ 9.75379 0.451352 0.225676 0.974202i $$-0.427541\pi$$
0.225676 + 0.974202i $$0.427541\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 4.31534 0.199264
$$470$$ −16.0000 −0.738025
$$471$$ −37.1231 −1.71054
$$472$$ 7.68466 0.353715
$$473$$ 0 0
$$474$$ 10.2462 0.470624
$$475$$ 1.00000 0.0458831
$$476$$ 0.315342 0.0144537
$$477$$ 45.6155 2.08859
$$478$$ −23.9309 −1.09457
$$479$$ −13.3693 −0.610860 −0.305430 0.952215i $$-0.598800\pi$$
−0.305430 + 0.952215i $$0.598800\pi$$
$$480$$ −5.12311 −0.233837
$$481$$ −2.87689 −0.131175
$$482$$ 19.6155 0.893463
$$483$$ −2.06913 −0.0941487
$$484$$ 1.00000 0.0454545
$$485$$ −14.2462 −0.646887
$$486$$ 22.2462 1.00911
$$487$$ 31.1231 1.41032 0.705161 0.709047i $$-0.250875\pi$$
0.705161 + 0.709047i $$0.250875\pi$$
$$488$$ −6.24621 −0.282753
$$489$$ 29.1231 1.31699
$$490$$ 13.3693 0.603964
$$491$$ −39.3693 −1.77671 −0.888356 0.459155i $$-0.848152\pi$$
−0.888356 + 0.459155i $$0.848152\pi$$
$$492$$ 5.12311 0.230967
$$493$$ 3.19224 0.143771
$$494$$ 0.561553 0.0252655
$$495$$ 7.12311 0.320160
$$496$$ 2.00000 0.0898027
$$497$$ −3.36932 −0.151135
$$498$$ −36.4924 −1.63526
$$499$$ 9.75379 0.436640 0.218320 0.975877i $$-0.429942\pi$$
0.218320 + 0.975877i $$0.429942\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ −1.61553 −0.0721765
$$502$$ −17.1231 −0.764242
$$503$$ 21.6847 0.966871 0.483436 0.875380i $$-0.339389\pi$$
0.483436 + 0.875380i $$0.339389\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ −13.7538 −0.612036
$$506$$ −1.43845 −0.0639468
$$507$$ −32.4924 −1.44304
$$508$$ 2.24621 0.0996595
$$509$$ −31.3693 −1.39042 −0.695210 0.718806i $$-0.744689\pi$$
−0.695210 + 0.718806i $$0.744689\pi$$
$$510$$ 2.87689 0.127391
$$511$$ 5.43845 0.240583
$$512$$ −1.00000 −0.0441942
$$513$$ −1.43845 −0.0635090
$$514$$ 22.0000 0.970378
$$515$$ −26.7386 −1.17824
$$516$$ 0 0
$$517$$ 8.00000 0.351840
$$518$$ −2.87689 −0.126403
$$519$$ 46.1080 2.02391
$$520$$ −1.12311 −0.0492514
$$521$$ −7.12311 −0.312069 −0.156034 0.987752i $$-0.549871\pi$$
−0.156034 + 0.987752i $$0.549871\pi$$
$$522$$ 20.2462 0.886153
$$523$$ 31.0540 1.35790 0.678948 0.734187i $$-0.262436\pi$$
0.678948 + 0.734187i $$0.262436\pi$$
$$524$$ 1.12311 0.0490631
$$525$$ 1.43845 0.0627790
$$526$$ −8.87689 −0.387051
$$527$$ −1.12311 −0.0489232
$$528$$ 2.56155 0.111477
$$529$$ −20.9309 −0.910038
$$530$$ −25.6155 −1.11267
$$531$$ −27.3693 −1.18773
$$532$$ 0.561553 0.0243464
$$533$$ 1.12311 0.0486471
$$534$$ 2.24621 0.0972031
$$535$$ −19.8617 −0.858698
$$536$$ 7.68466 0.331927
$$537$$ 14.7386 0.636019
$$538$$ −2.87689 −0.124032
$$539$$ −6.68466 −0.287929
$$540$$ 2.87689 0.123802
$$541$$ −39.2311 −1.68667 −0.843337 0.537384i $$-0.819412\pi$$
−0.843337 + 0.537384i $$0.819412\pi$$
$$542$$ −25.6847 −1.10325
$$543$$ −27.8617 −1.19566
$$544$$ 0.561553 0.0240764
$$545$$ −12.6307 −0.541039
$$546$$ 0.807764 0.0345691
$$547$$ −42.7386 −1.82737 −0.913686 0.406421i $$-0.866777\pi$$
−0.913686 + 0.406421i $$0.866777\pi$$
$$548$$ 12.5616 0.536603
$$549$$ 22.2462 0.949445
$$550$$ 1.00000 0.0426401
$$551$$ 5.68466 0.242175
$$552$$ −3.68466 −0.156829
$$553$$ 2.24621 0.0955186
$$554$$ −8.49242 −0.360808
$$555$$ −26.2462 −1.11409
$$556$$ 7.36932 0.312529
$$557$$ 17.1231 0.725529 0.362765 0.931881i $$-0.381833\pi$$
0.362765 + 0.931881i $$0.381833\pi$$
$$558$$ −7.12311 −0.301545
$$559$$ 0 0
$$560$$ −1.12311 −0.0474599
$$561$$ −1.43845 −0.0607313
$$562$$ 25.3693 1.07014
$$563$$ −42.7386 −1.80122 −0.900609 0.434630i $$-0.856879\pi$$
−0.900609 + 0.434630i $$0.856879\pi$$
$$564$$ 20.4924 0.862887
$$565$$ −6.24621 −0.262780
$$566$$ −11.3693 −0.477888
$$567$$ 3.93087 0.165081
$$568$$ −6.00000 −0.251754
$$569$$ −21.3693 −0.895848 −0.447924 0.894072i $$-0.647837\pi$$
−0.447924 + 0.894072i $$0.647837\pi$$
$$570$$ 5.12311 0.214583
$$571$$ −5.61553 −0.235003 −0.117501 0.993073i $$-0.537488\pi$$
−0.117501 + 0.993073i $$0.537488\pi$$
$$572$$ 0.561553 0.0234797
$$573$$ −21.3002 −0.889828
$$574$$ 1.12311 0.0468775
$$575$$ −1.43845 −0.0599874
$$576$$ 3.56155 0.148398
$$577$$ 41.0540 1.70910 0.854550 0.519370i $$-0.173833\pi$$
0.854550 + 0.519370i $$0.173833\pi$$
$$578$$ 16.6847 0.693990
$$579$$ 18.2462 0.758287
$$580$$ −11.3693 −0.472085
$$581$$ −8.00000 −0.331896
$$582$$ 18.2462 0.756330
$$583$$ 12.8078 0.530443
$$584$$ 9.68466 0.400754
$$585$$ 4.00000 0.165380
$$586$$ −3.93087 −0.162383
$$587$$ −40.4924 −1.67130 −0.835651 0.549261i $$-0.814909\pi$$
−0.835651 + 0.549261i $$0.814909\pi$$
$$588$$ −17.1231 −0.706145
$$589$$ −2.00000 −0.0824086
$$590$$ 15.3693 0.632745
$$591$$ 46.7386 1.92257
$$592$$ −5.12311 −0.210558
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ −1.43845 −0.0590202
$$595$$ 0.630683 0.0258555
$$596$$ −6.87689 −0.281689
$$597$$ 29.9309 1.22499
$$598$$ −0.807764 −0.0330319
$$599$$ −17.8617 −0.729811 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$600$$ 2.56155 0.104575
$$601$$ 21.3693 0.871673 0.435836 0.900026i $$-0.356453\pi$$
0.435836 + 0.900026i $$0.356453\pi$$
$$602$$ 0 0
$$603$$ −27.3693 −1.11456
$$604$$ 4.00000 0.162758
$$605$$ 2.00000 0.0813116
$$606$$ 17.6155 0.715582
$$607$$ −13.6155 −0.552637 −0.276319 0.961066i $$-0.589115\pi$$
−0.276319 + 0.961066i $$0.589115\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 8.17708 0.331352
$$610$$ −12.4924 −0.505803
$$611$$ 4.49242 0.181744
$$612$$ −2.00000 −0.0808452
$$613$$ 18.2462 0.736958 0.368479 0.929636i $$-0.379879\pi$$
0.368479 + 0.929636i $$0.379879\pi$$
$$614$$ 22.2462 0.897784
$$615$$ 10.2462 0.413167
$$616$$ 0.561553 0.0226256
$$617$$ 44.7386 1.80111 0.900555 0.434743i $$-0.143161\pi$$
0.900555 + 0.434743i $$0.143161\pi$$
$$618$$ 34.2462 1.37758
$$619$$ −9.75379 −0.392038 −0.196019 0.980600i $$-0.562801\pi$$
−0.196019 + 0.980600i $$0.562801\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 2.06913 0.0830313
$$622$$ 4.80776 0.192774
$$623$$ 0.492423 0.0197285
$$624$$ 1.43845 0.0575840
$$625$$ −19.0000 −0.760000
$$626$$ 0.561553 0.0224442
$$627$$ −2.56155 −0.102299
$$628$$ −14.4924 −0.578311
$$629$$ 2.87689 0.114709
$$630$$ 4.00000 0.159364
$$631$$ 3.50758 0.139634 0.0698172 0.997560i $$-0.477758\pi$$
0.0698172 + 0.997560i $$0.477758\pi$$
$$632$$ 4.00000 0.159111
$$633$$ −0.807764 −0.0321057
$$634$$ 7.05398 0.280149
$$635$$ 4.49242 0.178276
$$636$$ 32.8078 1.30091
$$637$$ −3.75379 −0.148731
$$638$$ 5.68466 0.225058
$$639$$ 21.3693 0.845357
$$640$$ −2.00000 −0.0790569
$$641$$ 31.6155 1.24874 0.624369 0.781129i $$-0.285356\pi$$
0.624369 + 0.781129i $$0.285356\pi$$
$$642$$ 25.4384 1.00398
$$643$$ −12.6307 −0.498106 −0.249053 0.968490i $$-0.580119\pi$$
−0.249053 + 0.968490i $$0.580119\pi$$
$$644$$ −0.807764 −0.0318304
$$645$$ 0 0
$$646$$ −0.561553 −0.0220940
$$647$$ −47.5464 −1.86924 −0.934621 0.355646i $$-0.884261\pi$$
−0.934621 + 0.355646i $$0.884261\pi$$
$$648$$ 7.00000 0.274986
$$649$$ −7.68466 −0.301649
$$650$$ 0.561553 0.0220259
$$651$$ −2.87689 −0.112754
$$652$$ 11.3693 0.445257
$$653$$ −15.6155 −0.611083 −0.305541 0.952179i $$-0.598837\pi$$
−0.305541 + 0.952179i $$0.598837\pi$$
$$654$$ 16.1771 0.632574
$$655$$ 2.24621 0.0877667
$$656$$ 2.00000 0.0780869
$$657$$ −34.4924 −1.34568
$$658$$ 4.49242 0.175133
$$659$$ −14.4233 −0.561852 −0.280926 0.959729i $$-0.590641\pi$$
−0.280926 + 0.959729i $$0.590641\pi$$
$$660$$ 5.12311 0.199417
$$661$$ −26.5616 −1.03312 −0.516562 0.856250i $$-0.672789\pi$$
−0.516562 + 0.856250i $$0.672789\pi$$
$$662$$ 14.5616 0.565951
$$663$$ −0.807764 −0.0313710
$$664$$ −14.2462 −0.552860
$$665$$ 1.12311 0.0435522
$$666$$ 18.2462 0.707026
$$667$$ −8.17708 −0.316618
$$668$$ −0.630683 −0.0244019
$$669$$ 63.3693 2.45000
$$670$$ 15.3693 0.593769
$$671$$ 6.24621 0.241132
$$672$$ 1.43845 0.0554893
$$673$$ −27.1231 −1.04552 −0.522759 0.852480i $$-0.675097\pi$$
−0.522759 + 0.852480i $$0.675097\pi$$
$$674$$ −11.1231 −0.428446
$$675$$ −1.43845 −0.0553659
$$676$$ −12.6847 −0.487871
$$677$$ 6.17708 0.237405 0.118702 0.992930i $$-0.462127\pi$$
0.118702 + 0.992930i $$0.462127\pi$$
$$678$$ 8.00000 0.307238
$$679$$ 4.00000 0.153506
$$680$$ 1.12311 0.0430691
$$681$$ 72.1771 2.76583
$$682$$ −2.00000 −0.0765840
$$683$$ 28.4924 1.09023 0.545116 0.838361i $$-0.316486\pi$$
0.545116 + 0.838361i $$0.316486\pi$$
$$684$$ −3.56155 −0.136179
$$685$$ 25.1231 0.959905
$$686$$ −7.68466 −0.293402
$$687$$ 32.9848 1.25845
$$688$$ 0 0
$$689$$ 7.19224 0.274002
$$690$$ −7.36932 −0.280545
$$691$$ −2.38447 −0.0907096 −0.0453548 0.998971i $$-0.514442\pi$$
−0.0453548 + 0.998971i $$0.514442\pi$$
$$692$$ 18.0000 0.684257
$$693$$ −2.00000 −0.0759737
$$694$$ −9.61553 −0.365000
$$695$$ 14.7386 0.559068
$$696$$ 14.5616 0.551954
$$697$$ −1.12311 −0.0425407
$$698$$ −21.6155 −0.818160
$$699$$ −73.6155 −2.78439
$$700$$ 0.561553 0.0212247
$$701$$ −40.9848 −1.54798 −0.773988 0.633200i $$-0.781741\pi$$
−0.773988 + 0.633200i $$0.781741\pi$$
$$702$$ −0.807764 −0.0304871
$$703$$ 5.12311 0.193222
$$704$$ 1.00000 0.0376889
$$705$$ 40.9848 1.54358
$$706$$ −22.1771 −0.834645
$$707$$ 3.86174 0.145236
$$708$$ −19.6847 −0.739795
$$709$$ −42.4924 −1.59584 −0.797918 0.602766i $$-0.794065\pi$$
−0.797918 + 0.602766i $$0.794065\pi$$
$$710$$ −12.0000 −0.450352
$$711$$ −14.2462 −0.534275
$$712$$ 0.876894 0.0328630
$$713$$ 2.87689 0.107741
$$714$$ −0.807764 −0.0302298
$$715$$ 1.12311 0.0420018
$$716$$ 5.75379 0.215029
$$717$$ 61.3002 2.28930
$$718$$ 15.9309 0.594535
$$719$$ −10.5616 −0.393879 −0.196940 0.980416i $$-0.563100\pi$$
−0.196940 + 0.980416i $$0.563100\pi$$
$$720$$ 7.12311 0.265462
$$721$$ 7.50758 0.279597
$$722$$ −1.00000 −0.0372161
$$723$$ −50.2462 −1.86868
$$724$$ −10.8769 −0.404237
$$725$$ 5.68466 0.211123
$$726$$ −2.56155 −0.0950681
$$727$$ 22.4233 0.831634 0.415817 0.909448i $$-0.363496\pi$$
0.415817 + 0.909448i $$0.363496\pi$$
$$728$$ 0.315342 0.0116873
$$729$$ −35.9848 −1.33277
$$730$$ 19.3693 0.716891
$$731$$ 0 0
$$732$$ 16.0000 0.591377
$$733$$ 26.1080 0.964319 0.482160 0.876083i $$-0.339853\pi$$
0.482160 + 0.876083i $$0.339853\pi$$
$$734$$ −30.2462 −1.11641
$$735$$ −34.2462 −1.26319
$$736$$ −1.43845 −0.0530219
$$737$$ −7.68466 −0.283068
$$738$$ −7.12311 −0.262205
$$739$$ −44.9848 −1.65479 −0.827397 0.561617i $$-0.810179\pi$$
−0.827397 + 0.561617i $$0.810179\pi$$
$$740$$ −10.2462 −0.376658
$$741$$ −1.43845 −0.0528427
$$742$$ 7.19224 0.264035
$$743$$ −42.2462 −1.54986 −0.774932 0.632045i $$-0.782216\pi$$
−0.774932 + 0.632045i $$0.782216\pi$$
$$744$$ −5.12311 −0.187822
$$745$$ −13.7538 −0.503900
$$746$$ −3.43845 −0.125890
$$747$$ 50.7386 1.85643
$$748$$ −0.561553 −0.0205324
$$749$$ 5.57671 0.203768
$$750$$ 30.7386 1.12242
$$751$$ 41.2311 1.50454 0.752271 0.658853i $$-0.228958\pi$$
0.752271 + 0.658853i $$0.228958\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 43.8617 1.59841
$$754$$ 3.19224 0.116254
$$755$$ 8.00000 0.291150
$$756$$ −0.807764 −0.0293781
$$757$$ −42.4924 −1.54441 −0.772207 0.635371i $$-0.780847\pi$$
−0.772207 + 0.635371i $$0.780847\pi$$
$$758$$ −6.56155 −0.238326
$$759$$ 3.68466 0.133745
$$760$$ 2.00000 0.0725476
$$761$$ −42.8078 −1.55178 −0.775890 0.630868i $$-0.782699\pi$$
−0.775890 + 0.630868i $$0.782699\pi$$
$$762$$ −5.75379 −0.208438
$$763$$ 3.54640 0.128388
$$764$$ −8.31534 −0.300838
$$765$$ −4.00000 −0.144620
$$766$$ 2.00000 0.0722629
$$767$$ −4.31534 −0.155818
$$768$$ 2.56155 0.0924321
$$769$$ −13.6847 −0.493481 −0.246741 0.969082i $$-0.579360\pi$$
−0.246741 + 0.969082i $$0.579360\pi$$
$$770$$ 1.12311 0.0404739
$$771$$ −56.3542 −2.02955
$$772$$ 7.12311 0.256366
$$773$$ 37.9309 1.36428 0.682139 0.731222i $$-0.261050\pi$$
0.682139 + 0.731222i $$0.261050\pi$$
$$774$$ 0 0
$$775$$ −2.00000 −0.0718421
$$776$$ 7.12311 0.255705
$$777$$ 7.36932 0.264373
$$778$$ −18.0000 −0.645331
$$779$$ −2.00000 −0.0716574
$$780$$ 2.87689 0.103009
$$781$$ 6.00000 0.214697
$$782$$ 0.807764 0.0288856
$$783$$ −8.17708 −0.292225
$$784$$ −6.68466 −0.238738
$$785$$ −28.9848 −1.03451
$$786$$ −2.87689 −0.102615
$$787$$ 15.6847 0.559098 0.279549 0.960131i $$-0.409815\pi$$
0.279549 + 0.960131i $$0.409815\pi$$
$$788$$ 18.2462 0.649994
$$789$$ 22.7386 0.809517
$$790$$ 8.00000 0.284627
$$791$$ 1.75379 0.0623575
$$792$$ −3.56155 −0.126554
$$793$$ 3.50758 0.124558
$$794$$ −38.9848 −1.38352
$$795$$ 65.6155 2.32714
$$796$$ 11.6847 0.414152
$$797$$ −15.1922 −0.538137 −0.269068 0.963121i $$-0.586716\pi$$
−0.269068 + 0.963121i $$0.586716\pi$$
$$798$$ −1.43845 −0.0509205
$$799$$ −4.49242 −0.158930
$$800$$ 1.00000 0.0353553
$$801$$ −3.12311 −0.110350
$$802$$ 7.75379 0.273796
$$803$$ −9.68466 −0.341764
$$804$$ −19.6847 −0.694224
$$805$$ −1.61553 −0.0569399
$$806$$ −1.12311 −0.0395597
$$807$$ 7.36932 0.259412
$$808$$ 6.87689 0.241928
$$809$$ 8.06913 0.283696 0.141848 0.989888i $$-0.454696\pi$$
0.141848 + 0.989888i $$0.454696\pi$$
$$810$$ 14.0000 0.491910
$$811$$ 8.31534 0.291991 0.145996 0.989285i $$-0.453361\pi$$
0.145996 + 0.989285i $$0.453361\pi$$
$$812$$ 3.19224 0.112026
$$813$$ 65.7926 2.30745
$$814$$ 5.12311 0.179565
$$815$$ 22.7386 0.796500
$$816$$ −1.43845 −0.0503557
$$817$$ 0 0
$$818$$ −16.2462 −0.568035
$$819$$ −1.12311 −0.0392445
$$820$$ 4.00000 0.139686
$$821$$ −20.9848 −0.732376 −0.366188 0.930541i $$-0.619337\pi$$
−0.366188 + 0.930541i $$0.619337\pi$$
$$822$$ −32.1771 −1.12230
$$823$$ −13.9309 −0.485600 −0.242800 0.970076i $$-0.578066\pi$$
−0.242800 + 0.970076i $$0.578066\pi$$
$$824$$ 13.3693 0.465742
$$825$$ −2.56155 −0.0891818
$$826$$ −4.31534 −0.150150
$$827$$ 49.9309 1.73627 0.868133 0.496331i $$-0.165320\pi$$
0.868133 + 0.496331i $$0.165320\pi$$
$$828$$ 5.12311 0.178040
$$829$$ 12.1771 0.422928 0.211464 0.977386i $$-0.432177\pi$$
0.211464 + 0.977386i $$0.432177\pi$$
$$830$$ −28.4924 −0.988986
$$831$$ 21.7538 0.754631
$$832$$ 0.561553 0.0194683
$$833$$ 3.75379 0.130061
$$834$$ −18.8769 −0.653653
$$835$$ −1.26137 −0.0436514
$$836$$ −1.00000 −0.0345857
$$837$$ 2.87689 0.0994400
$$838$$ 14.8769 0.513914
$$839$$ −7.12311 −0.245917 −0.122958 0.992412i $$-0.539238\pi$$
−0.122958 + 0.992412i $$0.539238\pi$$
$$840$$ 2.87689 0.0992623
$$841$$ 3.31534 0.114322
$$842$$ 29.9309 1.03149
$$843$$ −64.9848 −2.23820
$$844$$ −0.315342 −0.0108545
$$845$$ −25.3693 −0.872731
$$846$$ −28.4924 −0.979590
$$847$$ −0.561553 −0.0192952
$$848$$ 12.8078 0.439820
$$849$$ 29.1231 0.999502
$$850$$ −0.561553 −0.0192611
$$851$$ −7.36932 −0.252617
$$852$$ 15.3693 0.526544
$$853$$ −52.3542 −1.79257 −0.896286 0.443476i $$-0.853745\pi$$
−0.896286 + 0.443476i $$0.853745\pi$$
$$854$$ 3.50758 0.120027
$$855$$ −7.12311 −0.243605
$$856$$ 9.93087 0.339430
$$857$$ −10.0000 −0.341593 −0.170797 0.985306i $$-0.554634\pi$$
−0.170797 + 0.985306i $$0.554634\pi$$
$$858$$ −1.43845 −0.0491078
$$859$$ 15.8617 0.541196 0.270598 0.962692i $$-0.412779\pi$$
0.270598 + 0.962692i $$0.412779\pi$$
$$860$$ 0 0
$$861$$ −2.87689 −0.0980443
$$862$$ −31.8617 −1.08522
$$863$$ 0.246211 0.00838113 0.00419056 0.999991i $$-0.498666\pi$$
0.00419056 + 0.999991i $$0.498666\pi$$
$$864$$ −1.43845 −0.0489370
$$865$$ 36.0000 1.22404
$$866$$ −6.63068 −0.225320
$$867$$ −42.7386 −1.45148
$$868$$ −1.12311 −0.0381207
$$869$$ −4.00000 −0.135691
$$870$$ 29.1231 0.987366
$$871$$ −4.31534 −0.146220
$$872$$ 6.31534 0.213864
$$873$$ −25.3693 −0.858621
$$874$$ 1.43845 0.0486562
$$875$$ 6.73863 0.227807
$$876$$ −24.8078 −0.838177
$$877$$ −44.5616 −1.50474 −0.752368 0.658743i $$-0.771089\pi$$
−0.752368 + 0.658743i $$0.771089\pi$$
$$878$$ 9.61553 0.324508
$$879$$ 10.0691 0.339623
$$880$$ 2.00000 0.0674200
$$881$$ 20.7386 0.698702 0.349351 0.936992i $$-0.386402\pi$$
0.349351 + 0.936992i $$0.386402\pi$$
$$882$$ 23.8078 0.801649
$$883$$ −5.26137 −0.177059 −0.0885295 0.996074i $$-0.528217\pi$$
−0.0885295 + 0.996074i $$0.528217\pi$$
$$884$$ −0.315342 −0.0106061
$$885$$ −39.3693 −1.32339
$$886$$ −39.8617 −1.33918
$$887$$ −32.0000 −1.07445 −0.537227 0.843437i $$-0.680528\pi$$
−0.537227 + 0.843437i $$0.680528\pi$$
$$888$$ 13.1231 0.440383
$$889$$ −1.26137 −0.0423049
$$890$$ 1.75379 0.0587871
$$891$$ −7.00000 −0.234509
$$892$$ 24.7386 0.828311
$$893$$ −8.00000 −0.267710
$$894$$ 17.6155 0.589151
$$895$$ 11.5076 0.384656
$$896$$ 0.561553 0.0187602
$$897$$ 2.06913 0.0690863
$$898$$ −23.6155 −0.788060
$$899$$ −11.3693 −0.379188
$$900$$ −3.56155 −0.118718
$$901$$ −7.19224 −0.239608
$$902$$ −2.00000 −0.0665927
$$903$$ 0 0
$$904$$ 3.12311 0.103873
$$905$$ −21.7538 −0.723120
$$906$$ −10.2462 −0.340408
$$907$$ −29.7926 −0.989247 −0.494624 0.869107i $$-0.664694\pi$$
−0.494624 + 0.869107i $$0.664694\pi$$
$$908$$ 28.1771 0.935089
$$909$$ −24.4924 −0.812362
$$910$$ 0.630683 0.0209069
$$911$$ 18.9848 0.628996 0.314498 0.949258i $$-0.398164\pi$$
0.314498 + 0.949258i $$0.398164\pi$$
$$912$$ −2.56155 −0.0848215
$$913$$ 14.2462 0.471481
$$914$$ −36.5616 −1.20935
$$915$$ 32.0000 1.05789
$$916$$ 12.8769 0.425465
$$917$$ −0.630683 −0.0208270
$$918$$ 0.807764 0.0266602
$$919$$ 32.5616 1.07411 0.537053 0.843548i $$-0.319537\pi$$
0.537053 + 0.843548i $$0.319537\pi$$
$$920$$ −2.87689 −0.0948484
$$921$$ −56.9848 −1.87771
$$922$$ −26.7386 −0.880590
$$923$$ 3.36932 0.110902
$$924$$ −1.43845 −0.0473214
$$925$$ 5.12311 0.168447
$$926$$ −3.50758 −0.115266
$$927$$ −47.6155 −1.56390
$$928$$ 5.68466 0.186608
$$929$$ 11.9309 0.391439 0.195720 0.980660i $$-0.437296\pi$$
0.195720 + 0.980660i $$0.437296\pi$$
$$930$$ −10.2462 −0.335987
$$931$$ 6.68466 0.219081
$$932$$ −28.7386 −0.941365
$$933$$ −12.3153 −0.403186
$$934$$ −9.75379 −0.319154
$$935$$ −1.12311 −0.0367295
$$936$$ −2.00000 −0.0653720
$$937$$ −0.699813 −0.0228619 −0.0114310 0.999935i $$-0.503639\pi$$
−0.0114310 + 0.999935i $$0.503639\pi$$
$$938$$ −4.31534 −0.140901
$$939$$ −1.43845 −0.0469419
$$940$$ 16.0000 0.521862
$$941$$ 24.5616 0.800684 0.400342 0.916366i $$-0.368891\pi$$
0.400342 + 0.916366i $$0.368891\pi$$
$$942$$ 37.1231 1.20954
$$943$$ 2.87689 0.0936846
$$944$$ −7.68466 −0.250114
$$945$$ −1.61553 −0.0525531
$$946$$ 0 0
$$947$$ −36.9848 −1.20185 −0.600923 0.799307i $$-0.705200\pi$$
−0.600923 + 0.799307i $$0.705200\pi$$
$$948$$ −10.2462 −0.332781
$$949$$ −5.43845 −0.176539
$$950$$ −1.00000 −0.0324443
$$951$$ −18.0691 −0.585932
$$952$$ −0.315342 −0.0102203
$$953$$ −0.876894 −0.0284054 −0.0142027 0.999899i $$-0.504521\pi$$
−0.0142027 + 0.999899i $$0.504521\pi$$
$$954$$ −45.6155 −1.47686
$$955$$ −16.6307 −0.538156
$$956$$ 23.9309 0.773980
$$957$$ −14.5616 −0.470708
$$958$$ 13.3693 0.431943
$$959$$ −7.05398 −0.227785
$$960$$ 5.12311 0.165348
$$961$$ −27.0000 −0.870968
$$962$$ 2.87689 0.0927548
$$963$$ −35.3693 −1.13976
$$964$$ −19.6155 −0.631774
$$965$$ 14.2462 0.458602
$$966$$ 2.06913 0.0665732
$$967$$ 53.8617 1.73208 0.866038 0.499978i $$-0.166658\pi$$
0.866038 + 0.499978i $$0.166658\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 1.43845 0.0462096
$$970$$ 14.2462 0.457418
$$971$$ 2.24621 0.0720843 0.0360422 0.999350i $$-0.488525\pi$$
0.0360422 + 0.999350i $$0.488525\pi$$
$$972$$ −22.2462 −0.713548
$$973$$ −4.13826 −0.132667
$$974$$ −31.1231 −0.997249
$$975$$ −1.43845 −0.0460672
$$976$$ 6.24621 0.199936
$$977$$ 30.6307 0.979962 0.489981 0.871733i $$-0.337004\pi$$
0.489981 + 0.871733i $$0.337004\pi$$
$$978$$ −29.1231 −0.931254
$$979$$ −0.876894 −0.0280257
$$980$$ −13.3693 −0.427067
$$981$$ −22.4924 −0.718128
$$982$$ 39.3693 1.25633
$$983$$ −31.7538 −1.01279 −0.506394 0.862302i $$-0.669022\pi$$
−0.506394 + 0.862302i $$0.669022\pi$$
$$984$$ −5.12311 −0.163319
$$985$$ 36.4924 1.16275
$$986$$ −3.19224 −0.101662
$$987$$ −11.5076 −0.366290
$$988$$ −0.561553 −0.0178654
$$989$$ 0 0
$$990$$ −7.12311 −0.226387
$$991$$ 12.8769 0.409048 0.204524 0.978862i $$-0.434435\pi$$
0.204524 + 0.978862i $$0.434435\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ −37.3002 −1.18369
$$994$$ 3.36932 0.106868
$$995$$ 23.3693 0.740857
$$996$$ 36.4924 1.15631
$$997$$ −28.6307 −0.906743 −0.453371 0.891322i $$-0.649779\pi$$
−0.453371 + 0.891322i $$0.649779\pi$$
$$998$$ −9.75379 −0.308751
$$999$$ −7.36932 −0.233155
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 418.2.a.e.1.2 2
3.2 odd 2 3762.2.a.y.1.1 2
4.3 odd 2 3344.2.a.k.1.1 2
11.10 odd 2 4598.2.a.bj.1.2 2
19.18 odd 2 7942.2.a.x.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.e.1.2 2 1.1 even 1 trivial
3344.2.a.k.1.1 2 4.3 odd 2
3762.2.a.y.1.1 2 3.2 odd 2
4598.2.a.bj.1.2 2 11.10 odd 2
7942.2.a.x.1.1 2 19.18 odd 2