# Properties

 Label 5047.2.a.a.1.2 Level $5047$ Weight $2$ Character 5047.1 Self dual yes Analytic conductor $40.300$ Analytic rank $1$ 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] = [5047,2,Mod(1,5047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5047, 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("5047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5047 = 7^{2} \cdot 103$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5047.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: $$40.3004979001$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 103) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 5047.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} +2.61803 q^{5} -0.381966 q^{6} +1.47214 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} +2.61803 q^{5} -0.381966 q^{6} +1.47214 q^{8} -2.00000 q^{9} -1.00000 q^{10} -0.381966 q^{11} -1.85410 q^{12} -1.85410 q^{13} +2.61803 q^{15} +3.14590 q^{16} +3.38197 q^{17} +0.763932 q^{18} +0.854102 q^{19} -4.85410 q^{20} +0.145898 q^{22} -4.47214 q^{23} +1.47214 q^{24} +1.85410 q^{25} +0.708204 q^{26} -5.00000 q^{27} -0.763932 q^{29} -1.00000 q^{30} -6.70820 q^{31} -4.14590 q^{32} -0.381966 q^{33} -1.29180 q^{34} +3.70820 q^{36} -6.70820 q^{37} -0.326238 q^{38} -1.85410 q^{39} +3.85410 q^{40} +8.94427 q^{41} +4.70820 q^{43} +0.708204 q^{44} -5.23607 q^{45} +1.70820 q^{46} +7.09017 q^{47} +3.14590 q^{48} -0.708204 q^{50} +3.38197 q^{51} +3.43769 q^{52} -10.0902 q^{53} +1.90983 q^{54} -1.00000 q^{55} +0.854102 q^{57} +0.291796 q^{58} -8.61803 q^{59} -4.85410 q^{60} -10.8541 q^{61} +2.56231 q^{62} -4.70820 q^{64} -4.85410 q^{65} +0.145898 q^{66} -12.4164 q^{67} -6.27051 q^{68} -4.47214 q^{69} +7.09017 q^{71} -2.94427 q^{72} +4.14590 q^{73} +2.56231 q^{74} +1.85410 q^{75} -1.58359 q^{76} +0.708204 q^{78} +13.5623 q^{79} +8.23607 q^{80} +1.00000 q^{81} -3.41641 q^{82} -9.32624 q^{83} +8.85410 q^{85} -1.79837 q^{86} -0.763932 q^{87} -0.562306 q^{88} +15.7082 q^{89} +2.00000 q^{90} +8.29180 q^{92} -6.70820 q^{93} -2.70820 q^{94} +2.23607 q^{95} -4.14590 q^{96} -11.7082 q^{97} +0.763932 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{8} - 4 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 - 6 * q^8 - 4 * q^9 $$2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{8} - 4 q^{9} - 2 q^{10} - 3 q^{11} + 3 q^{12} + 3 q^{13} + 3 q^{15} + 13 q^{16} + 9 q^{17} + 6 q^{18} - 5 q^{19} - 3 q^{20} + 7 q^{22} - 6 q^{24} - 3 q^{25} - 12 q^{26} - 10 q^{27} - 6 q^{29} - 2 q^{30} - 15 q^{32} - 3 q^{33} - 16 q^{34} - 6 q^{36} + 15 q^{38} + 3 q^{39} + q^{40} - 4 q^{43} - 12 q^{44} - 6 q^{45} - 10 q^{46} + 3 q^{47} + 13 q^{48} + 12 q^{50} + 9 q^{51} + 27 q^{52} - 9 q^{53} + 15 q^{54} - 2 q^{55} - 5 q^{57} + 14 q^{58} - 15 q^{59} - 3 q^{60} - 15 q^{61} - 15 q^{62} + 4 q^{64} - 3 q^{65} + 7 q^{66} + 2 q^{67} + 21 q^{68} + 3 q^{71} + 12 q^{72} + 15 q^{73} - 15 q^{74} - 3 q^{75} - 30 q^{76} - 12 q^{78} + 7 q^{79} + 12 q^{80} + 2 q^{81} + 20 q^{82} - 3 q^{83} + 11 q^{85} + 21 q^{86} - 6 q^{87} + 19 q^{88} + 18 q^{89} + 4 q^{90} + 30 q^{92} + 8 q^{94} - 15 q^{96} - 10 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 - 6 * q^8 - 4 * q^9 - 2 * q^10 - 3 * q^11 + 3 * q^12 + 3 * q^13 + 3 * q^15 + 13 * q^16 + 9 * q^17 + 6 * q^18 - 5 * q^19 - 3 * q^20 + 7 * q^22 - 6 * q^24 - 3 * q^25 - 12 * q^26 - 10 * q^27 - 6 * q^29 - 2 * q^30 - 15 * q^32 - 3 * q^33 - 16 * q^34 - 6 * q^36 + 15 * q^38 + 3 * q^39 + q^40 - 4 * q^43 - 12 * q^44 - 6 * q^45 - 10 * q^46 + 3 * q^47 + 13 * q^48 + 12 * q^50 + 9 * q^51 + 27 * q^52 - 9 * q^53 + 15 * q^54 - 2 * q^55 - 5 * q^57 + 14 * q^58 - 15 * q^59 - 3 * q^60 - 15 * q^61 - 15 * q^62 + 4 * q^64 - 3 * q^65 + 7 * q^66 + 2 * q^67 + 21 * q^68 + 3 * q^71 + 12 * q^72 + 15 * q^73 - 15 * q^74 - 3 * q^75 - 30 * q^76 - 12 * q^78 + 7 * q^79 + 12 * q^80 + 2 * q^81 + 20 * q^82 - 3 * q^83 + 11 * q^85 + 21 * q^86 - 6 * q^87 + 19 * q^88 + 18 * q^89 + 4 * q^90 + 30 * q^92 + 8 * q^94 - 15 * q^96 - 10 * q^97 + 6 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.381966 −0.270091 −0.135045 0.990839i $$-0.543118\pi$$
−0.135045 + 0.990839i $$0.543118\pi$$
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ −1.85410 −0.927051
$$5$$ 2.61803 1.17082 0.585410 0.810737i $$-0.300933\pi$$
0.585410 + 0.810737i $$0.300933\pi$$
$$6$$ −0.381966 −0.155937
$$7$$ 0 0
$$8$$ 1.47214 0.520479
$$9$$ −2.00000 −0.666667
$$10$$ −1.00000 −0.316228
$$11$$ −0.381966 −0.115167 −0.0575835 0.998341i $$-0.518340\pi$$
−0.0575835 + 0.998341i $$0.518340\pi$$
$$12$$ −1.85410 −0.535233
$$13$$ −1.85410 −0.514235 −0.257118 0.966380i $$-0.582773\pi$$
−0.257118 + 0.966380i $$0.582773\pi$$
$$14$$ 0 0
$$15$$ 2.61803 0.675973
$$16$$ 3.14590 0.786475
$$17$$ 3.38197 0.820247 0.410124 0.912030i $$-0.365486\pi$$
0.410124 + 0.912030i $$0.365486\pi$$
$$18$$ 0.763932 0.180061
$$19$$ 0.854102 0.195944 0.0979722 0.995189i $$-0.468764\pi$$
0.0979722 + 0.995189i $$0.468764\pi$$
$$20$$ −4.85410 −1.08541
$$21$$ 0 0
$$22$$ 0.145898 0.0311056
$$23$$ −4.47214 −0.932505 −0.466252 0.884652i $$-0.654396\pi$$
−0.466252 + 0.884652i $$0.654396\pi$$
$$24$$ 1.47214 0.300498
$$25$$ 1.85410 0.370820
$$26$$ 0.708204 0.138890
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ −0.763932 −0.141859 −0.0709293 0.997481i $$-0.522596\pi$$
−0.0709293 + 0.997481i $$0.522596\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ −6.70820 −1.20483 −0.602414 0.798183i $$-0.705795\pi$$
−0.602414 + 0.798183i $$0.705795\pi$$
$$32$$ −4.14590 −0.732898
$$33$$ −0.381966 −0.0664917
$$34$$ −1.29180 −0.221541
$$35$$ 0 0
$$36$$ 3.70820 0.618034
$$37$$ −6.70820 −1.10282 −0.551411 0.834234i $$-0.685910\pi$$
−0.551411 + 0.834234i $$0.685910\pi$$
$$38$$ −0.326238 −0.0529228
$$39$$ −1.85410 −0.296894
$$40$$ 3.85410 0.609387
$$41$$ 8.94427 1.39686 0.698430 0.715678i $$-0.253882\pi$$
0.698430 + 0.715678i $$0.253882\pi$$
$$42$$ 0 0
$$43$$ 4.70820 0.717994 0.358997 0.933339i $$-0.383119\pi$$
0.358997 + 0.933339i $$0.383119\pi$$
$$44$$ 0.708204 0.106766
$$45$$ −5.23607 −0.780547
$$46$$ 1.70820 0.251861
$$47$$ 7.09017 1.03421 0.517104 0.855923i $$-0.327010\pi$$
0.517104 + 0.855923i $$0.327010\pi$$
$$48$$ 3.14590 0.454071
$$49$$ 0 0
$$50$$ −0.708204 −0.100155
$$51$$ 3.38197 0.473570
$$52$$ 3.43769 0.476722
$$53$$ −10.0902 −1.38599 −0.692996 0.720942i $$-0.743710\pi$$
−0.692996 + 0.720942i $$0.743710\pi$$
$$54$$ 1.90983 0.259895
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 0.854102 0.113129
$$58$$ 0.291796 0.0383147
$$59$$ −8.61803 −1.12197 −0.560986 0.827825i $$-0.689578\pi$$
−0.560986 + 0.827825i $$0.689578\pi$$
$$60$$ −4.85410 −0.626662
$$61$$ −10.8541 −1.38973 −0.694863 0.719142i $$-0.744535\pi$$
−0.694863 + 0.719142i $$0.744535\pi$$
$$62$$ 2.56231 0.325413
$$63$$ 0 0
$$64$$ −4.70820 −0.588525
$$65$$ −4.85410 −0.602077
$$66$$ 0.145898 0.0179588
$$67$$ −12.4164 −1.51691 −0.758453 0.651728i $$-0.774044\pi$$
−0.758453 + 0.651728i $$0.774044\pi$$
$$68$$ −6.27051 −0.760411
$$69$$ −4.47214 −0.538382
$$70$$ 0 0
$$71$$ 7.09017 0.841448 0.420724 0.907189i $$-0.361776\pi$$
0.420724 + 0.907189i $$0.361776\pi$$
$$72$$ −2.94427 −0.346986
$$73$$ 4.14590 0.485241 0.242620 0.970121i $$-0.421993\pi$$
0.242620 + 0.970121i $$0.421993\pi$$
$$74$$ 2.56231 0.297862
$$75$$ 1.85410 0.214093
$$76$$ −1.58359 −0.181650
$$77$$ 0 0
$$78$$ 0.708204 0.0801883
$$79$$ 13.5623 1.52588 0.762939 0.646470i $$-0.223755\pi$$
0.762939 + 0.646470i $$0.223755\pi$$
$$80$$ 8.23607 0.920820
$$81$$ 1.00000 0.111111
$$82$$ −3.41641 −0.377279
$$83$$ −9.32624 −1.02369 −0.511844 0.859079i $$-0.671037\pi$$
−0.511844 + 0.859079i $$0.671037\pi$$
$$84$$ 0 0
$$85$$ 8.85410 0.960362
$$86$$ −1.79837 −0.193924
$$87$$ −0.763932 −0.0819021
$$88$$ −0.562306 −0.0599420
$$89$$ 15.7082 1.66507 0.832533 0.553975i $$-0.186890\pi$$
0.832533 + 0.553975i $$0.186890\pi$$
$$90$$ 2.00000 0.210819
$$91$$ 0 0
$$92$$ 8.29180 0.864479
$$93$$ −6.70820 −0.695608
$$94$$ −2.70820 −0.279330
$$95$$ 2.23607 0.229416
$$96$$ −4.14590 −0.423139
$$97$$ −11.7082 −1.18879 −0.594394 0.804174i $$-0.702608\pi$$
−0.594394 + 0.804174i $$0.702608\pi$$
$$98$$ 0 0
$$99$$ 0.763932 0.0767781
$$100$$ −3.43769 −0.343769
$$101$$ −13.0902 −1.30252 −0.651260 0.758854i $$-0.725759\pi$$
−0.651260 + 0.758854i $$0.725759\pi$$
$$102$$ −1.29180 −0.127907
$$103$$ 1.00000 0.0985329
$$104$$ −2.72949 −0.267649
$$105$$ 0 0
$$106$$ 3.85410 0.374343
$$107$$ 4.09017 0.395412 0.197706 0.980261i $$-0.436651\pi$$
0.197706 + 0.980261i $$0.436651\pi$$
$$108$$ 9.27051 0.892055
$$109$$ 10.5623 1.01169 0.505843 0.862626i $$-0.331182\pi$$
0.505843 + 0.862626i $$0.331182\pi$$
$$110$$ 0.381966 0.0364190
$$111$$ −6.70820 −0.636715
$$112$$ 0 0
$$113$$ −15.0000 −1.41108 −0.705541 0.708669i $$-0.749296\pi$$
−0.705541 + 0.708669i $$0.749296\pi$$
$$114$$ −0.326238 −0.0305550
$$115$$ −11.7082 −1.09180
$$116$$ 1.41641 0.131510
$$117$$ 3.70820 0.342824
$$118$$ 3.29180 0.303034
$$119$$ 0 0
$$120$$ 3.85410 0.351830
$$121$$ −10.8541 −0.986737
$$122$$ 4.14590 0.375352
$$123$$ 8.94427 0.806478
$$124$$ 12.4377 1.11694
$$125$$ −8.23607 −0.736656
$$126$$ 0 0
$$127$$ −18.2705 −1.62125 −0.810623 0.585569i $$-0.800871\pi$$
−0.810623 + 0.585569i $$0.800871\pi$$
$$128$$ 10.0902 0.891853
$$129$$ 4.70820 0.414534
$$130$$ 1.85410 0.162615
$$131$$ −2.23607 −0.195366 −0.0976831 0.995218i $$-0.531143\pi$$
−0.0976831 + 0.995218i $$0.531143\pi$$
$$132$$ 0.708204 0.0616412
$$133$$ 0 0
$$134$$ 4.74265 0.409702
$$135$$ −13.0902 −1.12662
$$136$$ 4.97871 0.426921
$$137$$ 0.708204 0.0605059 0.0302530 0.999542i $$-0.490369\pi$$
0.0302530 + 0.999542i $$0.490369\pi$$
$$138$$ 1.70820 0.145412
$$139$$ 17.8541 1.51437 0.757183 0.653203i $$-0.226575\pi$$
0.757183 + 0.653203i $$0.226575\pi$$
$$140$$ 0 0
$$141$$ 7.09017 0.597100
$$142$$ −2.70820 −0.227267
$$143$$ 0.708204 0.0592230
$$144$$ −6.29180 −0.524316
$$145$$ −2.00000 −0.166091
$$146$$ −1.58359 −0.131059
$$147$$ 0 0
$$148$$ 12.4377 1.02237
$$149$$ 1.47214 0.120602 0.0603010 0.998180i $$-0.480794\pi$$
0.0603010 + 0.998180i $$0.480794\pi$$
$$150$$ −0.708204 −0.0578246
$$151$$ −19.0000 −1.54620 −0.773099 0.634285i $$-0.781294\pi$$
−0.773099 + 0.634285i $$0.781294\pi$$
$$152$$ 1.25735 0.101985
$$153$$ −6.76393 −0.546831
$$154$$ 0 0
$$155$$ −17.5623 −1.41064
$$156$$ 3.43769 0.275236
$$157$$ −3.29180 −0.262714 −0.131357 0.991335i $$-0.541933\pi$$
−0.131357 + 0.991335i $$0.541933\pi$$
$$158$$ −5.18034 −0.412126
$$159$$ −10.0902 −0.800203
$$160$$ −10.8541 −0.858092
$$161$$ 0 0
$$162$$ −0.381966 −0.0300101
$$163$$ −10.7082 −0.838731 −0.419366 0.907817i $$-0.637747\pi$$
−0.419366 + 0.907817i $$0.637747\pi$$
$$164$$ −16.5836 −1.29496
$$165$$ −1.00000 −0.0778499
$$166$$ 3.56231 0.276489
$$167$$ −9.00000 −0.696441 −0.348220 0.937413i $$-0.613214\pi$$
−0.348220 + 0.937413i $$0.613214\pi$$
$$168$$ 0 0
$$169$$ −9.56231 −0.735562
$$170$$ −3.38197 −0.259385
$$171$$ −1.70820 −0.130630
$$172$$ −8.72949 −0.665617
$$173$$ 13.0344 0.990990 0.495495 0.868611i $$-0.334987\pi$$
0.495495 + 0.868611i $$0.334987\pi$$
$$174$$ 0.291796 0.0221210
$$175$$ 0 0
$$176$$ −1.20163 −0.0905760
$$177$$ −8.61803 −0.647771
$$178$$ −6.00000 −0.449719
$$179$$ −1.14590 −0.0856484 −0.0428242 0.999083i $$-0.513636\pi$$
−0.0428242 + 0.999083i $$0.513636\pi$$
$$180$$ 9.70820 0.723607
$$181$$ 2.85410 0.212144 0.106072 0.994358i $$-0.466173\pi$$
0.106072 + 0.994358i $$0.466173\pi$$
$$182$$ 0 0
$$183$$ −10.8541 −0.802358
$$184$$ −6.58359 −0.485349
$$185$$ −17.5623 −1.29121
$$186$$ 2.56231 0.187877
$$187$$ −1.29180 −0.0944655
$$188$$ −13.1459 −0.958763
$$189$$ 0 0
$$190$$ −0.854102 −0.0619631
$$191$$ 3.38197 0.244710 0.122355 0.992486i $$-0.460955\pi$$
0.122355 + 0.992486i $$0.460955\pi$$
$$192$$ −4.70820 −0.339785
$$193$$ −20.1246 −1.44860 −0.724301 0.689484i $$-0.757837\pi$$
−0.724301 + 0.689484i $$0.757837\pi$$
$$194$$ 4.47214 0.321081
$$195$$ −4.85410 −0.347609
$$196$$ 0 0
$$197$$ 10.4164 0.742138 0.371069 0.928605i $$-0.378991\pi$$
0.371069 + 0.928605i $$0.378991\pi$$
$$198$$ −0.291796 −0.0207370
$$199$$ −3.41641 −0.242183 −0.121091 0.992641i $$-0.538639\pi$$
−0.121091 + 0.992641i $$0.538639\pi$$
$$200$$ 2.72949 0.193004
$$201$$ −12.4164 −0.875786
$$202$$ 5.00000 0.351799
$$203$$ 0 0
$$204$$ −6.27051 −0.439024
$$205$$ 23.4164 1.63547
$$206$$ −0.381966 −0.0266128
$$207$$ 8.94427 0.621670
$$208$$ −5.83282 −0.404433
$$209$$ −0.326238 −0.0225663
$$210$$ 0 0
$$211$$ 8.14590 0.560787 0.280393 0.959885i $$-0.409535\pi$$
0.280393 + 0.959885i $$0.409535\pi$$
$$212$$ 18.7082 1.28488
$$213$$ 7.09017 0.485810
$$214$$ −1.56231 −0.106797
$$215$$ 12.3262 0.840642
$$216$$ −7.36068 −0.500831
$$217$$ 0 0
$$218$$ −4.03444 −0.273247
$$219$$ 4.14590 0.280154
$$220$$ 1.85410 0.125004
$$221$$ −6.27051 −0.421800
$$222$$ 2.56231 0.171971
$$223$$ 5.70820 0.382250 0.191125 0.981566i $$-0.438786\pi$$
0.191125 + 0.981566i $$0.438786\pi$$
$$224$$ 0 0
$$225$$ −3.70820 −0.247214
$$226$$ 5.72949 0.381120
$$227$$ 14.9443 0.991886 0.495943 0.868355i $$-0.334822\pi$$
0.495943 + 0.868355i $$0.334822\pi$$
$$228$$ −1.58359 −0.104876
$$229$$ 6.70820 0.443291 0.221645 0.975127i $$-0.428857\pi$$
0.221645 + 0.975127i $$0.428857\pi$$
$$230$$ 4.47214 0.294884
$$231$$ 0 0
$$232$$ −1.12461 −0.0738344
$$233$$ 8.88854 0.582308 0.291154 0.956676i $$-0.405961\pi$$
0.291154 + 0.956676i $$0.405961\pi$$
$$234$$ −1.41641 −0.0925935
$$235$$ 18.5623 1.21087
$$236$$ 15.9787 1.04013
$$237$$ 13.5623 0.880966
$$238$$ 0 0
$$239$$ 24.3262 1.57353 0.786767 0.617250i $$-0.211753\pi$$
0.786767 + 0.617250i $$0.211753\pi$$
$$240$$ 8.23607 0.531636
$$241$$ −25.2705 −1.62782 −0.813908 0.580993i $$-0.802664\pi$$
−0.813908 + 0.580993i $$0.802664\pi$$
$$242$$ 4.14590 0.266508
$$243$$ 16.0000 1.02640
$$244$$ 20.1246 1.28835
$$245$$ 0 0
$$246$$ −3.41641 −0.217822
$$247$$ −1.58359 −0.100762
$$248$$ −9.87539 −0.627088
$$249$$ −9.32624 −0.591026
$$250$$ 3.14590 0.198964
$$251$$ −6.76393 −0.426936 −0.213468 0.976950i $$-0.568476\pi$$
−0.213468 + 0.976950i $$0.568476\pi$$
$$252$$ 0 0
$$253$$ 1.70820 0.107394
$$254$$ 6.97871 0.437883
$$255$$ 8.85410 0.554465
$$256$$ 5.56231 0.347644
$$257$$ 4.52786 0.282440 0.141220 0.989978i $$-0.454897\pi$$
0.141220 + 0.989978i $$0.454897\pi$$
$$258$$ −1.79837 −0.111962
$$259$$ 0 0
$$260$$ 9.00000 0.558156
$$261$$ 1.52786 0.0945724
$$262$$ 0.854102 0.0527666
$$263$$ −18.3820 −1.13348 −0.566740 0.823896i $$-0.691796\pi$$
−0.566740 + 0.823896i $$0.691796\pi$$
$$264$$ −0.562306 −0.0346075
$$265$$ −26.4164 −1.62275
$$266$$ 0 0
$$267$$ 15.7082 0.961326
$$268$$ 23.0213 1.40625
$$269$$ 3.32624 0.202804 0.101402 0.994846i $$-0.467667\pi$$
0.101402 + 0.994846i $$0.467667\pi$$
$$270$$ 5.00000 0.304290
$$271$$ −1.00000 −0.0607457 −0.0303728 0.999539i $$-0.509669\pi$$
−0.0303728 + 0.999539i $$0.509669\pi$$
$$272$$ 10.6393 0.645104
$$273$$ 0 0
$$274$$ −0.270510 −0.0163421
$$275$$ −0.708204 −0.0427063
$$276$$ 8.29180 0.499107
$$277$$ −8.70820 −0.523225 −0.261613 0.965173i $$-0.584254\pi$$
−0.261613 + 0.965173i $$0.584254\pi$$
$$278$$ −6.81966 −0.409016
$$279$$ 13.4164 0.803219
$$280$$ 0 0
$$281$$ −22.5279 −1.34390 −0.671950 0.740597i $$-0.734543\pi$$
−0.671950 + 0.740597i $$0.734543\pi$$
$$282$$ −2.70820 −0.161271
$$283$$ 6.29180 0.374008 0.187004 0.982359i $$-0.440122\pi$$
0.187004 + 0.982359i $$0.440122\pi$$
$$284$$ −13.1459 −0.780066
$$285$$ 2.23607 0.132453
$$286$$ −0.270510 −0.0159956
$$287$$ 0 0
$$288$$ 8.29180 0.488599
$$289$$ −5.56231 −0.327194
$$290$$ 0.763932 0.0448596
$$291$$ −11.7082 −0.686347
$$292$$ −7.68692 −0.449843
$$293$$ −9.65248 −0.563904 −0.281952 0.959429i $$-0.590982\pi$$
−0.281952 + 0.959429i $$0.590982\pi$$
$$294$$ 0 0
$$295$$ −22.5623 −1.31363
$$296$$ −9.87539 −0.573995
$$297$$ 1.90983 0.110820
$$298$$ −0.562306 −0.0325735
$$299$$ 8.29180 0.479527
$$300$$ −3.43769 −0.198475
$$301$$ 0 0
$$302$$ 7.25735 0.417614
$$303$$ −13.0902 −0.752011
$$304$$ 2.68692 0.154105
$$305$$ −28.4164 −1.62712
$$306$$ 2.58359 0.147694
$$307$$ −3.85410 −0.219965 −0.109983 0.993934i $$-0.535080\pi$$
−0.109983 + 0.993934i $$0.535080\pi$$
$$308$$ 0 0
$$309$$ 1.00000 0.0568880
$$310$$ 6.70820 0.381000
$$311$$ 32.8885 1.86494 0.932469 0.361250i $$-0.117650\pi$$
0.932469 + 0.361250i $$0.117650\pi$$
$$312$$ −2.72949 −0.154527
$$313$$ −29.7082 −1.67921 −0.839603 0.543200i $$-0.817213\pi$$
−0.839603 + 0.543200i $$0.817213\pi$$
$$314$$ 1.25735 0.0709566
$$315$$ 0 0
$$316$$ −25.1459 −1.41457
$$317$$ 1.58359 0.0889434 0.0444717 0.999011i $$-0.485840\pi$$
0.0444717 + 0.999011i $$0.485840\pi$$
$$318$$ 3.85410 0.216127
$$319$$ 0.291796 0.0163374
$$320$$ −12.3262 −0.689058
$$321$$ 4.09017 0.228291
$$322$$ 0 0
$$323$$ 2.88854 0.160723
$$324$$ −1.85410 −0.103006
$$325$$ −3.43769 −0.190689
$$326$$ 4.09017 0.226534
$$327$$ 10.5623 0.584097
$$328$$ 13.1672 0.727036
$$329$$ 0 0
$$330$$ 0.381966 0.0210265
$$331$$ −22.8541 −1.25618 −0.628088 0.778143i $$-0.716162\pi$$
−0.628088 + 0.778143i $$0.716162\pi$$
$$332$$ 17.2918 0.949011
$$333$$ 13.4164 0.735215
$$334$$ 3.43769 0.188102
$$335$$ −32.5066 −1.77602
$$336$$ 0 0
$$337$$ −22.5623 −1.22905 −0.614524 0.788898i $$-0.710652\pi$$
−0.614524 + 0.788898i $$0.710652\pi$$
$$338$$ 3.65248 0.198668
$$339$$ −15.0000 −0.814688
$$340$$ −16.4164 −0.890305
$$341$$ 2.56231 0.138757
$$342$$ 0.652476 0.0352819
$$343$$ 0 0
$$344$$ 6.93112 0.373701
$$345$$ −11.7082 −0.630349
$$346$$ −4.97871 −0.267657
$$347$$ 7.47214 0.401125 0.200563 0.979681i $$-0.435723\pi$$
0.200563 + 0.979681i $$0.435723\pi$$
$$348$$ 1.41641 0.0759274
$$349$$ 11.4164 0.611106 0.305553 0.952175i $$-0.401159\pi$$
0.305553 + 0.952175i $$0.401159\pi$$
$$350$$ 0 0
$$351$$ 9.27051 0.494823
$$352$$ 1.58359 0.0844057
$$353$$ −4.03444 −0.214732 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$354$$ 3.29180 0.174957
$$355$$ 18.5623 0.985185
$$356$$ −29.1246 −1.54360
$$357$$ 0 0
$$358$$ 0.437694 0.0231329
$$359$$ 30.3262 1.60056 0.800279 0.599628i $$-0.204685\pi$$
0.800279 + 0.599628i $$0.204685\pi$$
$$360$$ −7.70820 −0.406258
$$361$$ −18.2705 −0.961606
$$362$$ −1.09017 −0.0572981
$$363$$ −10.8541 −0.569693
$$364$$ 0 0
$$365$$ 10.8541 0.568130
$$366$$ 4.14590 0.216710
$$367$$ −36.5623 −1.90854 −0.954268 0.298951i $$-0.903363\pi$$
−0.954268 + 0.298951i $$0.903363\pi$$
$$368$$ −14.0689 −0.733391
$$369$$ −17.8885 −0.931240
$$370$$ 6.70820 0.348743
$$371$$ 0 0
$$372$$ 12.4377 0.644864
$$373$$ 37.6869 1.95135 0.975677 0.219212i $$-0.0703485\pi$$
0.975677 + 0.219212i $$0.0703485\pi$$
$$374$$ 0.493422 0.0255143
$$375$$ −8.23607 −0.425309
$$376$$ 10.4377 0.538283
$$377$$ 1.41641 0.0729487
$$378$$ 0 0
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ −4.14590 −0.212680
$$381$$ −18.2705 −0.936027
$$382$$ −1.29180 −0.0660940
$$383$$ −23.1803 −1.18446 −0.592230 0.805769i $$-0.701752\pi$$
−0.592230 + 0.805769i $$0.701752\pi$$
$$384$$ 10.0902 0.514912
$$385$$ 0 0
$$386$$ 7.68692 0.391254
$$387$$ −9.41641 −0.478663
$$388$$ 21.7082 1.10207
$$389$$ −19.4164 −0.984451 −0.492225 0.870468i $$-0.663816\pi$$
−0.492225 + 0.870468i $$0.663816\pi$$
$$390$$ 1.85410 0.0938861
$$391$$ −15.1246 −0.764884
$$392$$ 0 0
$$393$$ −2.23607 −0.112795
$$394$$ −3.97871 −0.200445
$$395$$ 35.5066 1.78653
$$396$$ −1.41641 −0.0711772
$$397$$ −20.0000 −1.00377 −0.501886 0.864934i $$-0.667360\pi$$
−0.501886 + 0.864934i $$0.667360\pi$$
$$398$$ 1.30495 0.0654113
$$399$$ 0 0
$$400$$ 5.83282 0.291641
$$401$$ 11.8885 0.593686 0.296843 0.954926i $$-0.404066\pi$$
0.296843 + 0.954926i $$0.404066\pi$$
$$402$$ 4.74265 0.236542
$$403$$ 12.4377 0.619566
$$404$$ 24.2705 1.20750
$$405$$ 2.61803 0.130091
$$406$$ 0 0
$$407$$ 2.56231 0.127009
$$408$$ 4.97871 0.246483
$$409$$ −23.2918 −1.15171 −0.575853 0.817554i $$-0.695330\pi$$
−0.575853 + 0.817554i $$0.695330\pi$$
$$410$$ −8.94427 −0.441726
$$411$$ 0.708204 0.0349331
$$412$$ −1.85410 −0.0913450
$$413$$ 0 0
$$414$$ −3.41641 −0.167907
$$415$$ −24.4164 −1.19855
$$416$$ 7.68692 0.376882
$$417$$ 17.8541 0.874319
$$418$$ 0.124612 0.00609496
$$419$$ −7.09017 −0.346377 −0.173189 0.984889i $$-0.555407\pi$$
−0.173189 + 0.984889i $$0.555407\pi$$
$$420$$ 0 0
$$421$$ 3.00000 0.146211 0.0731055 0.997324i $$-0.476709\pi$$
0.0731055 + 0.997324i $$0.476709\pi$$
$$422$$ −3.11146 −0.151463
$$423$$ −14.1803 −0.689472
$$424$$ −14.8541 −0.721379
$$425$$ 6.27051 0.304164
$$426$$ −2.70820 −0.131213
$$427$$ 0 0
$$428$$ −7.58359 −0.366567
$$429$$ 0.708204 0.0341924
$$430$$ −4.70820 −0.227050
$$431$$ −10.3607 −0.499056 −0.249528 0.968368i $$-0.580276\pi$$
−0.249528 + 0.968368i $$0.580276\pi$$
$$432$$ −15.7295 −0.756785
$$433$$ 12.4164 0.596694 0.298347 0.954457i $$-0.403565\pi$$
0.298347 + 0.954457i $$0.403565\pi$$
$$434$$ 0 0
$$435$$ −2.00000 −0.0958927
$$436$$ −19.5836 −0.937884
$$437$$ −3.81966 −0.182719
$$438$$ −1.58359 −0.0756670
$$439$$ −9.43769 −0.450437 −0.225218 0.974308i $$-0.572310\pi$$
−0.225218 + 0.974308i $$0.572310\pi$$
$$440$$ −1.47214 −0.0701813
$$441$$ 0 0
$$442$$ 2.39512 0.113924
$$443$$ −35.5623 −1.68962 −0.844808 0.535069i $$-0.820285\pi$$
−0.844808 + 0.535069i $$0.820285\pi$$
$$444$$ 12.4377 0.590267
$$445$$ 41.1246 1.94949
$$446$$ −2.18034 −0.103242
$$447$$ 1.47214 0.0696296
$$448$$ 0 0
$$449$$ 31.3607 1.48000 0.740001 0.672606i $$-0.234825\pi$$
0.740001 + 0.672606i $$0.234825\pi$$
$$450$$ 1.41641 0.0667701
$$451$$ −3.41641 −0.160872
$$452$$ 27.8115 1.30814
$$453$$ −19.0000 −0.892698
$$454$$ −5.70820 −0.267899
$$455$$ 0 0
$$456$$ 1.25735 0.0588810
$$457$$ −3.14590 −0.147159 −0.0735795 0.997289i $$-0.523442\pi$$
−0.0735795 + 0.997289i $$0.523442\pi$$
$$458$$ −2.56231 −0.119729
$$459$$ −16.9098 −0.789283
$$460$$ 21.7082 1.01215
$$461$$ 39.2148 1.82641 0.913207 0.407495i $$-0.133598\pi$$
0.913207 + 0.407495i $$0.133598\pi$$
$$462$$ 0 0
$$463$$ 5.41641 0.251722 0.125861 0.992048i $$-0.459831\pi$$
0.125861 + 0.992048i $$0.459831\pi$$
$$464$$ −2.40325 −0.111568
$$465$$ −17.5623 −0.814432
$$466$$ −3.39512 −0.157276
$$467$$ −27.6525 −1.27960 −0.639802 0.768540i $$-0.720984\pi$$
−0.639802 + 0.768540i $$0.720984\pi$$
$$468$$ −6.87539 −0.317815
$$469$$ 0 0
$$470$$ −7.09017 −0.327045
$$471$$ −3.29180 −0.151678
$$472$$ −12.6869 −0.583963
$$473$$ −1.79837 −0.0826893
$$474$$ −5.18034 −0.237941
$$475$$ 1.58359 0.0726602
$$476$$ 0 0
$$477$$ 20.1803 0.923994
$$478$$ −9.29180 −0.424997
$$479$$ 14.1803 0.647916 0.323958 0.946071i $$-0.394986\pi$$
0.323958 + 0.946071i $$0.394986\pi$$
$$480$$ −10.8541 −0.495420
$$481$$ 12.4377 0.567110
$$482$$ 9.65248 0.439658
$$483$$ 0 0
$$484$$ 20.1246 0.914755
$$485$$ −30.6525 −1.39186
$$486$$ −6.11146 −0.277221
$$487$$ −23.0000 −1.04223 −0.521115 0.853487i $$-0.674484\pi$$
−0.521115 + 0.853487i $$0.674484\pi$$
$$488$$ −15.9787 −0.723322
$$489$$ −10.7082 −0.484242
$$490$$ 0 0
$$491$$ −35.7771 −1.61460 −0.807299 0.590143i $$-0.799071\pi$$
−0.807299 + 0.590143i $$0.799071\pi$$
$$492$$ −16.5836 −0.747646
$$493$$ −2.58359 −0.116359
$$494$$ 0.604878 0.0272148
$$495$$ 2.00000 0.0898933
$$496$$ −21.1033 −0.947567
$$497$$ 0 0
$$498$$ 3.56231 0.159631
$$499$$ −20.2705 −0.907433 −0.453716 0.891146i $$-0.649902\pi$$
−0.453716 + 0.891146i $$0.649902\pi$$
$$500$$ 15.2705 0.682918
$$501$$ −9.00000 −0.402090
$$502$$ 2.58359 0.115311
$$503$$ 31.3607 1.39830 0.699152 0.714973i $$-0.253561\pi$$
0.699152 + 0.714973i $$0.253561\pi$$
$$504$$ 0 0
$$505$$ −34.2705 −1.52502
$$506$$ −0.652476 −0.0290061
$$507$$ −9.56231 −0.424677
$$508$$ 33.8754 1.50298
$$509$$ −24.3820 −1.08071 −0.540356 0.841437i $$-0.681710\pi$$
−0.540356 + 0.841437i $$0.681710\pi$$
$$510$$ −3.38197 −0.149756
$$511$$ 0 0
$$512$$ −22.3050 −0.985749
$$513$$ −4.27051 −0.188548
$$514$$ −1.72949 −0.0762845
$$515$$ 2.61803 0.115364
$$516$$ −8.72949 −0.384294
$$517$$ −2.70820 −0.119107
$$518$$ 0 0
$$519$$ 13.0344 0.572148
$$520$$ −7.14590 −0.313368
$$521$$ 5.18034 0.226955 0.113477 0.993541i $$-0.463801\pi$$
0.113477 + 0.993541i $$0.463801\pi$$
$$522$$ −0.583592 −0.0255431
$$523$$ 37.4164 1.63611 0.818053 0.575143i $$-0.195054\pi$$
0.818053 + 0.575143i $$0.195054\pi$$
$$524$$ 4.14590 0.181114
$$525$$ 0 0
$$526$$ 7.02129 0.306143
$$527$$ −22.6869 −0.988258
$$528$$ −1.20163 −0.0522941
$$529$$ −3.00000 −0.130435
$$530$$ 10.0902 0.438289
$$531$$ 17.2361 0.747982
$$532$$ 0 0
$$533$$ −16.5836 −0.718315
$$534$$ −6.00000 −0.259645
$$535$$ 10.7082 0.462956
$$536$$ −18.2786 −0.789517
$$537$$ −1.14590 −0.0494492
$$538$$ −1.27051 −0.0547756
$$539$$ 0 0
$$540$$ 24.2705 1.04444
$$541$$ 15.8541 0.681621 0.340811 0.940132i $$-0.389299\pi$$
0.340811 + 0.940132i $$0.389299\pi$$
$$542$$ 0.381966 0.0164068
$$543$$ 2.85410 0.122481
$$544$$ −14.0213 −0.601158
$$545$$ 27.6525 1.18450
$$546$$ 0 0
$$547$$ 31.2705 1.33703 0.668515 0.743698i $$-0.266930\pi$$
0.668515 + 0.743698i $$0.266930\pi$$
$$548$$ −1.31308 −0.0560921
$$549$$ 21.7082 0.926484
$$550$$ 0.270510 0.0115346
$$551$$ −0.652476 −0.0277964
$$552$$ −6.58359 −0.280216
$$553$$ 0 0
$$554$$ 3.32624 0.141318
$$555$$ −17.5623 −0.745478
$$556$$ −33.1033 −1.40389
$$557$$ 28.6869 1.21550 0.607752 0.794127i $$-0.292072\pi$$
0.607752 + 0.794127i $$0.292072\pi$$
$$558$$ −5.12461 −0.216942
$$559$$ −8.72949 −0.369218
$$560$$ 0 0
$$561$$ −1.29180 −0.0545397
$$562$$ 8.60488 0.362975
$$563$$ 25.7984 1.08727 0.543636 0.839321i $$-0.317047\pi$$
0.543636 + 0.839321i $$0.317047\pi$$
$$564$$ −13.1459 −0.553542
$$565$$ −39.2705 −1.65212
$$566$$ −2.40325 −0.101016
$$567$$ 0 0
$$568$$ 10.4377 0.437956
$$569$$ 42.1591 1.76740 0.883700 0.468054i $$-0.155045\pi$$
0.883700 + 0.468054i $$0.155045\pi$$
$$570$$ −0.854102 −0.0357744
$$571$$ −2.43769 −0.102014 −0.0510072 0.998698i $$-0.516243\pi$$
−0.0510072 + 0.998698i $$0.516243\pi$$
$$572$$ −1.31308 −0.0549027
$$573$$ 3.38197 0.141284
$$574$$ 0 0
$$575$$ −8.29180 −0.345792
$$576$$ 9.41641 0.392350
$$577$$ −16.8328 −0.700759 −0.350380 0.936608i $$-0.613947\pi$$
−0.350380 + 0.936608i $$0.613947\pi$$
$$578$$ 2.12461 0.0883722
$$579$$ −20.1246 −0.836350
$$580$$ 3.70820 0.153975
$$581$$ 0 0
$$582$$ 4.47214 0.185376
$$583$$ 3.85410 0.159621
$$584$$ 6.10333 0.252557
$$585$$ 9.70820 0.401385
$$586$$ 3.68692 0.152305
$$587$$ −11.0689 −0.456862 −0.228431 0.973560i $$-0.573359\pi$$
−0.228431 + 0.973560i $$0.573359\pi$$
$$588$$ 0 0
$$589$$ −5.72949 −0.236080
$$590$$ 8.61803 0.354799
$$591$$ 10.4164 0.428474
$$592$$ −21.1033 −0.867341
$$593$$ 20.1803 0.828707 0.414354 0.910116i $$-0.364008\pi$$
0.414354 + 0.910116i $$0.364008\pi$$
$$594$$ −0.729490 −0.0299313
$$595$$ 0 0
$$596$$ −2.72949 −0.111804
$$597$$ −3.41641 −0.139824
$$598$$ −3.16718 −0.129516
$$599$$ 20.4508 0.835599 0.417800 0.908539i $$-0.362801\pi$$
0.417800 + 0.908539i $$0.362801\pi$$
$$600$$ 2.72949 0.111431
$$601$$ 16.5623 0.675591 0.337795 0.941220i $$-0.390319\pi$$
0.337795 + 0.941220i $$0.390319\pi$$
$$602$$ 0 0
$$603$$ 24.8328 1.01127
$$604$$ 35.2279 1.43340
$$605$$ −28.4164 −1.15529
$$606$$ 5.00000 0.203111
$$607$$ 7.70820 0.312866 0.156433 0.987689i $$-0.450000\pi$$
0.156433 + 0.987689i $$0.450000\pi$$
$$608$$ −3.54102 −0.143607
$$609$$ 0 0
$$610$$ 10.8541 0.439470
$$611$$ −13.1459 −0.531826
$$612$$ 12.5410 0.506941
$$613$$ −2.41641 −0.0975978 −0.0487989 0.998809i $$-0.515539\pi$$
−0.0487989 + 0.998809i $$0.515539\pi$$
$$614$$ 1.47214 0.0594106
$$615$$ 23.4164 0.944241
$$616$$ 0 0
$$617$$ −30.2705 −1.21864 −0.609322 0.792923i $$-0.708558\pi$$
−0.609322 + 0.792923i $$0.708558\pi$$
$$618$$ −0.381966 −0.0153649
$$619$$ −31.6869 −1.27360 −0.636802 0.771027i $$-0.719743\pi$$
−0.636802 + 0.771027i $$0.719743\pi$$
$$620$$ 32.5623 1.30773
$$621$$ 22.3607 0.897303
$$622$$ −12.5623 −0.503703
$$623$$ 0 0
$$624$$ −5.83282 −0.233500
$$625$$ −30.8328 −1.23331
$$626$$ 11.3475 0.453538
$$627$$ −0.326238 −0.0130287
$$628$$ 6.10333 0.243549
$$629$$ −22.6869 −0.904587
$$630$$ 0 0
$$631$$ 8.72949 0.347516 0.173758 0.984788i $$-0.444409\pi$$
0.173758 + 0.984788i $$0.444409\pi$$
$$632$$ 19.9656 0.794187
$$633$$ 8.14590 0.323770
$$634$$ −0.604878 −0.0240228
$$635$$ −47.8328 −1.89819
$$636$$ 18.7082 0.741829
$$637$$ 0 0
$$638$$ −0.111456 −0.00441259
$$639$$ −14.1803 −0.560966
$$640$$ 26.4164 1.04420
$$641$$ −15.0000 −0.592464 −0.296232 0.955116i $$-0.595730\pi$$
−0.296232 + 0.955116i $$0.595730\pi$$
$$642$$ −1.56231 −0.0616593
$$643$$ −5.00000 −0.197181 −0.0985904 0.995128i $$-0.531433\pi$$
−0.0985904 + 0.995128i $$0.531433\pi$$
$$644$$ 0 0
$$645$$ 12.3262 0.485345
$$646$$ −1.10333 −0.0434098
$$647$$ −43.7426 −1.71970 −0.859850 0.510546i $$-0.829443\pi$$
−0.859850 + 0.510546i $$0.829443\pi$$
$$648$$ 1.47214 0.0578310
$$649$$ 3.29180 0.129214
$$650$$ 1.31308 0.0515033
$$651$$ 0 0
$$652$$ 19.8541 0.777547
$$653$$ −0.763932 −0.0298950 −0.0149475 0.999888i $$-0.504758\pi$$
−0.0149475 + 0.999888i $$0.504758\pi$$
$$654$$ −4.03444 −0.157759
$$655$$ −5.85410 −0.228739
$$656$$ 28.1378 1.09860
$$657$$ −8.29180 −0.323494
$$658$$ 0 0
$$659$$ −12.5967 −0.490700 −0.245350 0.969435i $$-0.578903\pi$$
−0.245350 + 0.969435i $$0.578903\pi$$
$$660$$ 1.85410 0.0721708
$$661$$ 11.4377 0.444875 0.222437 0.974947i $$-0.428599\pi$$
0.222437 + 0.974947i $$0.428599\pi$$
$$662$$ 8.72949 0.339281
$$663$$ −6.27051 −0.243526
$$664$$ −13.7295 −0.532808
$$665$$ 0 0
$$666$$ −5.12461 −0.198575
$$667$$ 3.41641 0.132284
$$668$$ 16.6869 0.645636
$$669$$ 5.70820 0.220692
$$670$$ 12.4164 0.479688
$$671$$ 4.14590 0.160051
$$672$$ 0 0
$$673$$ −37.7082 −1.45354 −0.726772 0.686879i $$-0.758980\pi$$
−0.726772 + 0.686879i $$0.758980\pi$$
$$674$$ 8.61803 0.331954
$$675$$ −9.27051 −0.356822
$$676$$ 17.7295 0.681903
$$677$$ −10.9656 −0.421441 −0.210720 0.977546i $$-0.567581\pi$$
−0.210720 + 0.977546i $$0.567581\pi$$
$$678$$ 5.72949 0.220040
$$679$$ 0 0
$$680$$ 13.0344 0.499848
$$681$$ 14.9443 0.572666
$$682$$ −0.978714 −0.0374769
$$683$$ 42.7639 1.63632 0.818158 0.574993i $$-0.194995\pi$$
0.818158 + 0.574993i $$0.194995\pi$$
$$684$$ 3.16718 0.121100
$$685$$ 1.85410 0.0708416
$$686$$ 0 0
$$687$$ 6.70820 0.255934
$$688$$ 14.8115 0.564684
$$689$$ 18.7082 0.712726
$$690$$ 4.47214 0.170251
$$691$$ −1.14590 −0.0435920 −0.0217960 0.999762i $$-0.506938\pi$$
−0.0217960 + 0.999762i $$0.506938\pi$$
$$692$$ −24.1672 −0.918698
$$693$$ 0 0
$$694$$ −2.85410 −0.108340
$$695$$ 46.7426 1.77305
$$696$$ −1.12461 −0.0426283
$$697$$ 30.2492 1.14577
$$698$$ −4.36068 −0.165054
$$699$$ 8.88854 0.336196
$$700$$ 0 0
$$701$$ −1.20163 −0.0453848 −0.0226924 0.999742i $$-0.507224\pi$$
−0.0226924 + 0.999742i $$0.507224\pi$$
$$702$$ −3.54102 −0.133647
$$703$$ −5.72949 −0.216092
$$704$$ 1.79837 0.0677788
$$705$$ 18.5623 0.699097
$$706$$ 1.54102 0.0579970
$$707$$ 0 0
$$708$$ 15.9787 0.600517
$$709$$ −47.9787 −1.80188 −0.900939 0.433945i $$-0.857121\pi$$
−0.900939 + 0.433945i $$0.857121\pi$$
$$710$$ −7.09017 −0.266089
$$711$$ −27.1246 −1.01725
$$712$$ 23.1246 0.866631
$$713$$ 30.0000 1.12351
$$714$$ 0 0
$$715$$ 1.85410 0.0693395
$$716$$ 2.12461 0.0794005
$$717$$ 24.3262 0.908480
$$718$$ −11.5836 −0.432296
$$719$$ −23.6738 −0.882882 −0.441441 0.897290i $$-0.645533\pi$$
−0.441441 + 0.897290i $$0.645533\pi$$
$$720$$ −16.4721 −0.613880
$$721$$ 0 0
$$722$$ 6.97871 0.259721
$$723$$ −25.2705 −0.939820
$$724$$ −5.29180 −0.196668
$$725$$ −1.41641 −0.0526041
$$726$$ 4.14590 0.153869
$$727$$ 38.2705 1.41937 0.709687 0.704517i $$-0.248836\pi$$
0.709687 + 0.704517i $$0.248836\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ −4.14590 −0.153447
$$731$$ 15.9230 0.588933
$$732$$ 20.1246 0.743827
$$733$$ −1.29180 −0.0477136 −0.0238568 0.999715i $$-0.507595\pi$$
−0.0238568 + 0.999715i $$0.507595\pi$$
$$734$$ 13.9656 0.515478
$$735$$ 0 0
$$736$$ 18.5410 0.683431
$$737$$ 4.74265 0.174698
$$738$$ 6.83282 0.251519
$$739$$ 36.8328 1.35492 0.677459 0.735561i $$-0.263081\pi$$
0.677459 + 0.735561i $$0.263081\pi$$
$$740$$ 32.5623 1.19701
$$741$$ −1.58359 −0.0581747
$$742$$ 0 0
$$743$$ 17.2918 0.634374 0.317187 0.948363i $$-0.397262\pi$$
0.317187 + 0.948363i $$0.397262\pi$$
$$744$$ −9.87539 −0.362049
$$745$$ 3.85410 0.141203
$$746$$ −14.3951 −0.527043
$$747$$ 18.6525 0.682458
$$748$$ 2.39512 0.0875743
$$749$$ 0 0
$$750$$ 3.14590 0.114872
$$751$$ −3.87539 −0.141415 −0.0707075 0.997497i $$-0.522526\pi$$
−0.0707075 + 0.997497i $$0.522526\pi$$
$$752$$ 22.3050 0.813378
$$753$$ −6.76393 −0.246491
$$754$$ −0.541020 −0.0197028
$$755$$ −49.7426 −1.81032
$$756$$ 0 0
$$757$$ −34.7082 −1.26149 −0.630746 0.775990i $$-0.717251\pi$$
−0.630746 + 0.775990i $$0.717251\pi$$
$$758$$ −1.90983 −0.0693682
$$759$$ 1.70820 0.0620039
$$760$$ 3.29180 0.119406
$$761$$ −1.47214 −0.0533649 −0.0266824 0.999644i $$-0.508494\pi$$
−0.0266824 + 0.999644i $$0.508494\pi$$
$$762$$ 6.97871 0.252812
$$763$$ 0 0
$$764$$ −6.27051 −0.226859
$$765$$ −17.7082 −0.640241
$$766$$ 8.85410 0.319912
$$767$$ 15.9787 0.576958
$$768$$ 5.56231 0.200712
$$769$$ 42.3951 1.52881 0.764404 0.644738i $$-0.223034\pi$$
0.764404 + 0.644738i $$0.223034\pi$$
$$770$$ 0 0
$$771$$ 4.52786 0.163067
$$772$$ 37.3131 1.34293
$$773$$ 25.4721 0.916169 0.458085 0.888909i $$-0.348536\pi$$
0.458085 + 0.888909i $$0.348536\pi$$
$$774$$ 3.59675 0.129282
$$775$$ −12.4377 −0.446775
$$776$$ −17.2361 −0.618739
$$777$$ 0 0
$$778$$ 7.41641 0.265891
$$779$$ 7.63932 0.273707
$$780$$ 9.00000 0.322252
$$781$$ −2.70820 −0.0969072
$$782$$ 5.77709 0.206588
$$783$$ 3.81966 0.136504
$$784$$ 0 0
$$785$$ −8.61803 −0.307591
$$786$$ 0.854102 0.0304648
$$787$$ −16.5836 −0.591141 −0.295571 0.955321i $$-0.595510\pi$$
−0.295571 + 0.955321i $$0.595510\pi$$
$$788$$ −19.3131 −0.688000
$$789$$ −18.3820 −0.654415
$$790$$ −13.5623 −0.482525
$$791$$ 0 0
$$792$$ 1.12461 0.0399613
$$793$$ 20.1246 0.714646
$$794$$ 7.63932 0.271109
$$795$$ −26.4164 −0.936893
$$796$$ 6.33437 0.224516
$$797$$ 9.87539 0.349804 0.174902 0.984586i $$-0.444039\pi$$
0.174902 + 0.984586i $$0.444039\pi$$
$$798$$ 0 0
$$799$$ 23.9787 0.848306
$$800$$ −7.68692 −0.271774
$$801$$ −31.4164 −1.11004
$$802$$ −4.54102 −0.160349
$$803$$ −1.58359 −0.0558838
$$804$$ 23.0213 0.811898
$$805$$ 0 0
$$806$$ −4.75078 −0.167339
$$807$$ 3.32624 0.117089
$$808$$ −19.2705 −0.677934
$$809$$ −14.9443 −0.525413 −0.262706 0.964876i $$-0.584615\pi$$
−0.262706 + 0.964876i $$0.584615\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 45.5410 1.59916 0.799581 0.600559i $$-0.205055\pi$$
0.799581 + 0.600559i $$0.205055\pi$$
$$812$$ 0 0
$$813$$ −1.00000 −0.0350715
$$814$$ −0.978714 −0.0343039
$$815$$ −28.0344 −0.982004
$$816$$ 10.6393 0.372451
$$817$$ 4.02129 0.140687
$$818$$ 8.89667 0.311065
$$819$$ 0 0
$$820$$ −43.4164 −1.51617
$$821$$ 33.0000 1.15171 0.575854 0.817553i $$-0.304670\pi$$
0.575854 + 0.817553i $$0.304670\pi$$
$$822$$ −0.270510 −0.00943511
$$823$$ 23.5623 0.821330 0.410665 0.911786i $$-0.365297\pi$$
0.410665 + 0.911786i $$0.365297\pi$$
$$824$$ 1.47214 0.0512843
$$825$$ −0.708204 −0.0246565
$$826$$ 0 0
$$827$$ −6.70820 −0.233267 −0.116634 0.993175i $$-0.537210\pi$$
−0.116634 + 0.993175i $$0.537210\pi$$
$$828$$ −16.5836 −0.576320
$$829$$ 19.2705 0.669292 0.334646 0.942344i $$-0.391383\pi$$
0.334646 + 0.942344i $$0.391383\pi$$
$$830$$ 9.32624 0.323718
$$831$$ −8.70820 −0.302084
$$832$$ 8.72949 0.302641
$$833$$ 0 0
$$834$$ −6.81966 −0.236146
$$835$$ −23.5623 −0.815407
$$836$$ 0.604878 0.0209202
$$837$$ 33.5410 1.15935
$$838$$ 2.70820 0.0935534
$$839$$ −21.3820 −0.738187 −0.369094 0.929392i $$-0.620332\pi$$
−0.369094 + 0.929392i $$0.620332\pi$$
$$840$$ 0 0
$$841$$ −28.4164 −0.979876
$$842$$ −1.14590 −0.0394903
$$843$$ −22.5279 −0.775901
$$844$$ −15.1033 −0.519878
$$845$$ −25.0344 −0.861211
$$846$$ 5.41641 0.186220
$$847$$ 0 0
$$848$$ −31.7426 −1.09005
$$849$$ 6.29180 0.215934
$$850$$ −2.39512 −0.0821520
$$851$$ 30.0000 1.02839
$$852$$ −13.1459 −0.450371
$$853$$ 10.7295 0.367371 0.183685 0.982985i $$-0.441197\pi$$
0.183685 + 0.982985i $$0.441197\pi$$
$$854$$ 0 0
$$855$$ −4.47214 −0.152944
$$856$$ 6.02129 0.205803
$$857$$ −3.76393 −0.128573 −0.0642867 0.997931i $$-0.520477\pi$$
−0.0642867 + 0.997931i $$0.520477\pi$$
$$858$$ −0.270510 −0.00923505
$$859$$ 9.56231 0.326262 0.163131 0.986604i $$-0.447841\pi$$
0.163131 + 0.986604i $$0.447841\pi$$
$$860$$ −22.8541 −0.779318
$$861$$ 0 0
$$862$$ 3.95743 0.134791
$$863$$ −8.29180 −0.282256 −0.141128 0.989991i $$-0.545073\pi$$
−0.141128 + 0.989991i $$0.545073\pi$$
$$864$$ 20.7295 0.705232
$$865$$ 34.1246 1.16027
$$866$$ −4.74265 −0.161162
$$867$$ −5.56231 −0.188906
$$868$$ 0 0
$$869$$ −5.18034 −0.175731
$$870$$ 0.763932 0.0258997
$$871$$ 23.0213 0.780047
$$872$$ 15.5492 0.526561
$$873$$ 23.4164 0.792525
$$874$$ 1.45898 0.0493507
$$875$$ 0 0
$$876$$ −7.68692 −0.259717
$$877$$ −13.0000 −0.438979 −0.219489 0.975615i $$-0.570439\pi$$
−0.219489 + 0.975615i $$0.570439\pi$$
$$878$$ 3.60488 0.121659
$$879$$ −9.65248 −0.325570
$$880$$ −3.14590 −0.106048
$$881$$ 5.88854 0.198390 0.0991950 0.995068i $$-0.468373\pi$$
0.0991950 + 0.995068i $$0.468373\pi$$
$$882$$ 0 0
$$883$$ 46.1246 1.55222 0.776108 0.630600i $$-0.217191\pi$$
0.776108 + 0.630600i $$0.217191\pi$$
$$884$$ 11.6262 0.391030
$$885$$ −22.5623 −0.758424
$$886$$ 13.5836 0.456350
$$887$$ 58.1935 1.95395 0.976973 0.213362i $$-0.0684414\pi$$
0.976973 + 0.213362i $$0.0684414\pi$$
$$888$$ −9.87539 −0.331396
$$889$$ 0 0
$$890$$ −15.7082 −0.526540
$$891$$ −0.381966 −0.0127963
$$892$$ −10.5836 −0.354365
$$893$$ 6.05573 0.202647
$$894$$ −0.562306 −0.0188063
$$895$$ −3.00000 −0.100279
$$896$$ 0 0
$$897$$ 8.29180 0.276855
$$898$$ −11.9787 −0.399735
$$899$$ 5.12461 0.170915
$$900$$ 6.87539 0.229180
$$901$$ −34.1246 −1.13686
$$902$$ 1.30495 0.0434501
$$903$$ 0 0
$$904$$ −22.0820 −0.734438
$$905$$ 7.47214 0.248382
$$906$$ 7.25735 0.241109
$$907$$ 33.1246 1.09988 0.549942 0.835203i $$-0.314650\pi$$
0.549942 + 0.835203i $$0.314650\pi$$
$$908$$ −27.7082 −0.919529
$$909$$ 26.1803 0.868347
$$910$$ 0 0
$$911$$ 19.0344 0.630639 0.315320 0.948986i $$-0.397888\pi$$
0.315320 + 0.948986i $$0.397888\pi$$
$$912$$ 2.68692 0.0889727
$$913$$ 3.56231 0.117895
$$914$$ 1.20163 0.0397463
$$915$$ −28.4164 −0.939417
$$916$$ −12.4377 −0.410953
$$917$$ 0 0
$$918$$ 6.45898 0.213178
$$919$$ 18.9787 0.626050 0.313025 0.949745i $$-0.398658\pi$$
0.313025 + 0.949745i $$0.398658\pi$$
$$920$$ −17.2361 −0.568256
$$921$$ −3.85410 −0.126997
$$922$$ −14.9787 −0.493298
$$923$$ −13.1459 −0.432703
$$924$$ 0 0
$$925$$ −12.4377 −0.408949
$$926$$ −2.06888 −0.0679877
$$927$$ −2.00000 −0.0656886
$$928$$ 3.16718 0.103968
$$929$$ 2.94427 0.0965984 0.0482992 0.998833i $$-0.484620\pi$$
0.0482992 + 0.998833i $$0.484620\pi$$
$$930$$ 6.70820 0.219971
$$931$$ 0 0
$$932$$ −16.4803 −0.539829
$$933$$ 32.8885 1.07672
$$934$$ 10.5623 0.345609
$$935$$ −3.38197 −0.110602
$$936$$ 5.45898 0.178432
$$937$$ 11.0000 0.359354 0.179677 0.983726i $$-0.442495\pi$$
0.179677 + 0.983726i $$0.442495\pi$$
$$938$$ 0 0
$$939$$ −29.7082 −0.969491
$$940$$ −34.4164 −1.12254
$$941$$ −50.3951 −1.64283 −0.821417 0.570328i $$-0.806816\pi$$
−0.821417 + 0.570328i $$0.806816\pi$$
$$942$$ 1.25735 0.0409668
$$943$$ −40.0000 −1.30258
$$944$$ −27.1115 −0.882403
$$945$$ 0 0
$$946$$ 0.686918 0.0223336
$$947$$ 35.0132 1.13777 0.568887 0.822415i $$-0.307374\pi$$
0.568887 + 0.822415i $$0.307374\pi$$
$$948$$ −25.1459 −0.816701
$$949$$ −7.68692 −0.249528
$$950$$ −0.604878 −0.0196248
$$951$$ 1.58359 0.0513515
$$952$$ 0 0
$$953$$ −31.3607 −1.01587 −0.507936 0.861395i $$-0.669591\pi$$
−0.507936 + 0.861395i $$0.669591\pi$$
$$954$$ −7.70820 −0.249562
$$955$$ 8.85410 0.286512
$$956$$ −45.1033 −1.45875
$$957$$ 0.291796 0.00943243
$$958$$ −5.41641 −0.174996
$$959$$ 0 0
$$960$$ −12.3262 −0.397828
$$961$$ 14.0000 0.451613
$$962$$ −4.75078 −0.153171
$$963$$ −8.18034 −0.263608
$$964$$ 46.8541 1.50907
$$965$$ −52.6869 −1.69605
$$966$$ 0 0
$$967$$ 12.4164 0.399285 0.199642 0.979869i $$-0.436022\pi$$
0.199642 + 0.979869i $$0.436022\pi$$
$$968$$ −15.9787 −0.513575
$$969$$ 2.88854 0.0927934
$$970$$ 11.7082 0.375928
$$971$$ −23.0132 −0.738527 −0.369264 0.929325i $$-0.620390\pi$$
−0.369264 + 0.929325i $$0.620390\pi$$
$$972$$ −29.6656 −0.951526
$$973$$ 0 0
$$974$$ 8.78522 0.281497
$$975$$ −3.43769 −0.110094
$$976$$ −34.1459 −1.09298
$$977$$ 4.25735 0.136205 0.0681024 0.997678i $$-0.478306\pi$$
0.0681024 + 0.997678i $$0.478306\pi$$
$$978$$ 4.09017 0.130789
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ −21.1246 −0.674457
$$982$$ 13.6656 0.436088
$$983$$ 12.6525 0.403551 0.201776 0.979432i $$-0.435329\pi$$
0.201776 + 0.979432i $$0.435329\pi$$
$$984$$ 13.1672 0.419755
$$985$$ 27.2705 0.868911
$$986$$ 0.986844 0.0314275
$$987$$ 0 0
$$988$$ 2.93614 0.0934111
$$989$$ −21.0557 −0.669533
$$990$$ −0.763932 −0.0242794
$$991$$ −9.27051 −0.294487 −0.147244 0.989100i $$-0.547040\pi$$
−0.147244 + 0.989100i $$0.547040\pi$$
$$992$$ 27.8115 0.883017
$$993$$ −22.8541 −0.725253
$$994$$ 0 0
$$995$$ −8.94427 −0.283552
$$996$$ 17.2918 0.547912
$$997$$ 52.7082 1.66929 0.834643 0.550792i $$-0.185674\pi$$
0.834643 + 0.550792i $$0.185674\pi$$
$$998$$ 7.74265 0.245089
$$999$$ 33.5410 1.06119
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 5047.2.a.a.1.2 2
7.6 odd 2 103.2.a.a.1.2 2
21.20 even 2 927.2.a.b.1.1 2
28.27 even 2 1648.2.a.f.1.1 2
35.34 odd 2 2575.2.a.g.1.1 2
56.13 odd 2 6592.2.a.t.1.2 2
56.27 even 2 6592.2.a.h.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
103.2.a.a.1.2 2 7.6 odd 2
927.2.a.b.1.1 2 21.20 even 2
1648.2.a.f.1.1 2 28.27 even 2
2575.2.a.g.1.1 2 35.34 odd 2
5047.2.a.a.1.2 2 1.1 even 1 trivial
6592.2.a.h.1.2 2 56.27 even 2
6592.2.a.t.1.2 2 56.13 odd 2