Properties

 Label 147.2.a.e.1.2 Level $147$ Weight $2$ Character 147.1 Self dual yes Analytic conductor $1.174$ 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] = [147,2,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, 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("147.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 147.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: $$1.17380090971$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 147.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} +3.41421 q^{5} +0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} +3.41421 q^{5} +0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} +1.41421 q^{10} -2.00000 q^{11} -1.82843 q^{12} +2.58579 q^{13} +3.41421 q^{15} +3.00000 q^{16} -2.24264 q^{17} +0.414214 q^{18} -2.82843 q^{19} -6.24264 q^{20} -0.828427 q^{22} -7.65685 q^{23} -1.58579 q^{24} +6.65685 q^{25} +1.07107 q^{26} +1.00000 q^{27} -6.82843 q^{29} +1.41421 q^{30} -1.17157 q^{31} +4.41421 q^{32} -2.00000 q^{33} -0.928932 q^{34} -1.82843 q^{36} -4.00000 q^{37} -1.17157 q^{38} +2.58579 q^{39} -5.41421 q^{40} +6.24264 q^{41} +5.65685 q^{43} +3.65685 q^{44} +3.41421 q^{45} -3.17157 q^{46} -2.82843 q^{47} +3.00000 q^{48} +2.75736 q^{50} -2.24264 q^{51} -4.72792 q^{52} -2.00000 q^{53} +0.414214 q^{54} -6.82843 q^{55} -2.82843 q^{57} -2.82843 q^{58} -1.17157 q^{59} -6.24264 q^{60} +12.2426 q^{61} -0.485281 q^{62} -4.17157 q^{64} +8.82843 q^{65} -0.828427 q^{66} -5.65685 q^{67} +4.10051 q^{68} -7.65685 q^{69} +9.31371 q^{71} -1.58579 q^{72} +13.8995 q^{73} -1.65685 q^{74} +6.65685 q^{75} +5.17157 q^{76} +1.07107 q^{78} +13.6569 q^{79} +10.2426 q^{80} +1.00000 q^{81} +2.58579 q^{82} +7.31371 q^{83} -7.65685 q^{85} +2.34315 q^{86} -6.82843 q^{87} +3.17157 q^{88} -14.2426 q^{89} +1.41421 q^{90} +14.0000 q^{92} -1.17157 q^{93} -1.17157 q^{94} -9.65685 q^{95} +4.41421 q^{96} +2.58579 q^{97} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 - 2 * q^6 - 6 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} + 8 q^{13} + 4 q^{15} + 6 q^{16} + 4 q^{17} - 2 q^{18} - 4 q^{20} + 4 q^{22} - 4 q^{23} - 6 q^{24} + 2 q^{25} - 12 q^{26} + 2 q^{27} - 8 q^{29} - 8 q^{31} + 6 q^{32} - 4 q^{33} - 16 q^{34} + 2 q^{36} - 8 q^{37} - 8 q^{38} + 8 q^{39} - 8 q^{40} + 4 q^{41} - 4 q^{44} + 4 q^{45} - 12 q^{46} + 6 q^{48} + 14 q^{50} + 4 q^{51} + 16 q^{52} - 4 q^{53} - 2 q^{54} - 8 q^{55} - 8 q^{59} - 4 q^{60} + 16 q^{61} + 16 q^{62} - 14 q^{64} + 12 q^{65} + 4 q^{66} + 28 q^{68} - 4 q^{69} - 4 q^{71} - 6 q^{72} + 8 q^{73} + 8 q^{74} + 2 q^{75} + 16 q^{76} - 12 q^{78} + 16 q^{79} + 12 q^{80} + 2 q^{81} + 8 q^{82} - 8 q^{83} - 4 q^{85} + 16 q^{86} - 8 q^{87} + 12 q^{88} - 20 q^{89} + 28 q^{92} - 8 q^{93} - 8 q^{94} - 8 q^{95} + 6 q^{96} + 8 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 - 2 * q^6 - 6 * q^8 + 2 * q^9 - 4 * q^11 + 2 * q^12 + 8 * q^13 + 4 * q^15 + 6 * q^16 + 4 * q^17 - 2 * q^18 - 4 * q^20 + 4 * q^22 - 4 * q^23 - 6 * q^24 + 2 * q^25 - 12 * q^26 + 2 * q^27 - 8 * q^29 - 8 * q^31 + 6 * q^32 - 4 * q^33 - 16 * q^34 + 2 * q^36 - 8 * q^37 - 8 * q^38 + 8 * q^39 - 8 * q^40 + 4 * q^41 - 4 * q^44 + 4 * q^45 - 12 * q^46 + 6 * q^48 + 14 * q^50 + 4 * q^51 + 16 * q^52 - 4 * q^53 - 2 * q^54 - 8 * q^55 - 8 * q^59 - 4 * q^60 + 16 * q^61 + 16 * q^62 - 14 * q^64 + 12 * q^65 + 4 * q^66 + 28 * q^68 - 4 * q^69 - 4 * q^71 - 6 * q^72 + 8 * q^73 + 8 * q^74 + 2 * q^75 + 16 * q^76 - 12 * q^78 + 16 * q^79 + 12 * q^80 + 2 * q^81 + 8 * q^82 - 8 * q^83 - 4 * q^85 + 16 * q^86 - 8 * q^87 + 12 * q^88 - 20 * q^89 + 28 * q^92 - 8 * q^93 - 8 * q^94 - 8 * q^95 + 6 * q^96 + 8 * q^97 - 4 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.414214 0.292893 0.146447 0.989219i $$-0.453216\pi$$
0.146447 + 0.989219i $$0.453216\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.82843 −0.914214
$$5$$ 3.41421 1.52688 0.763441 0.645877i $$-0.223508\pi$$
0.763441 + 0.645877i $$0.223508\pi$$
$$6$$ 0.414214 0.169102
$$7$$ 0 0
$$8$$ −1.58579 −0.560660
$$9$$ 1.00000 0.333333
$$10$$ 1.41421 0.447214
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ −1.82843 −0.527821
$$13$$ 2.58579 0.717168 0.358584 0.933497i $$-0.383260\pi$$
0.358584 + 0.933497i $$0.383260\pi$$
$$14$$ 0 0
$$15$$ 3.41421 0.881546
$$16$$ 3.00000 0.750000
$$17$$ −2.24264 −0.543920 −0.271960 0.962309i $$-0.587672\pi$$
−0.271960 + 0.962309i $$0.587672\pi$$
$$18$$ 0.414214 0.0976311
$$19$$ −2.82843 −0.648886 −0.324443 0.945905i $$-0.605177\pi$$
−0.324443 + 0.945905i $$0.605177\pi$$
$$20$$ −6.24264 −1.39590
$$21$$ 0 0
$$22$$ −0.828427 −0.176621
$$23$$ −7.65685 −1.59656 −0.798282 0.602284i $$-0.794258\pi$$
−0.798282 + 0.602284i $$0.794258\pi$$
$$24$$ −1.58579 −0.323697
$$25$$ 6.65685 1.33137
$$26$$ 1.07107 0.210054
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −6.82843 −1.26801 −0.634004 0.773330i $$-0.718590\pi$$
−0.634004 + 0.773330i $$0.718590\pi$$
$$30$$ 1.41421 0.258199
$$31$$ −1.17157 −0.210421 −0.105210 0.994450i $$-0.533552\pi$$
−0.105210 + 0.994450i $$0.533552\pi$$
$$32$$ 4.41421 0.780330
$$33$$ −2.00000 −0.348155
$$34$$ −0.928932 −0.159311
$$35$$ 0 0
$$36$$ −1.82843 −0.304738
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ −1.17157 −0.190054
$$39$$ 2.58579 0.414057
$$40$$ −5.41421 −0.856062
$$41$$ 6.24264 0.974937 0.487468 0.873141i $$-0.337920\pi$$
0.487468 + 0.873141i $$0.337920\pi$$
$$42$$ 0 0
$$43$$ 5.65685 0.862662 0.431331 0.902194i $$-0.358044\pi$$
0.431331 + 0.902194i $$0.358044\pi$$
$$44$$ 3.65685 0.551292
$$45$$ 3.41421 0.508961
$$46$$ −3.17157 −0.467623
$$47$$ −2.82843 −0.412568 −0.206284 0.978492i $$-0.566137\pi$$
−0.206284 + 0.978492i $$0.566137\pi$$
$$48$$ 3.00000 0.433013
$$49$$ 0 0
$$50$$ 2.75736 0.389949
$$51$$ −2.24264 −0.314033
$$52$$ −4.72792 −0.655645
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0.414214 0.0563673
$$55$$ −6.82843 −0.920745
$$56$$ 0 0
$$57$$ −2.82843 −0.374634
$$58$$ −2.82843 −0.371391
$$59$$ −1.17157 −0.152526 −0.0762629 0.997088i $$-0.524299\pi$$
−0.0762629 + 0.997088i $$0.524299\pi$$
$$60$$ −6.24264 −0.805921
$$61$$ 12.2426 1.56751 0.783755 0.621070i $$-0.213302\pi$$
0.783755 + 0.621070i $$0.213302\pi$$
$$62$$ −0.485281 −0.0616308
$$63$$ 0 0
$$64$$ −4.17157 −0.521447
$$65$$ 8.82843 1.09503
$$66$$ −0.828427 −0.101972
$$67$$ −5.65685 −0.691095 −0.345547 0.938401i $$-0.612307\pi$$
−0.345547 + 0.938401i $$0.612307\pi$$
$$68$$ 4.10051 0.497259
$$69$$ −7.65685 −0.921777
$$70$$ 0 0
$$71$$ 9.31371 1.10533 0.552667 0.833402i $$-0.313610\pi$$
0.552667 + 0.833402i $$0.313610\pi$$
$$72$$ −1.58579 −0.186887
$$73$$ 13.8995 1.62681 0.813406 0.581696i $$-0.197611\pi$$
0.813406 + 0.581696i $$0.197611\pi$$
$$74$$ −1.65685 −0.192605
$$75$$ 6.65685 0.768667
$$76$$ 5.17157 0.593220
$$77$$ 0 0
$$78$$ 1.07107 0.121275
$$79$$ 13.6569 1.53652 0.768258 0.640140i $$-0.221124\pi$$
0.768258 + 0.640140i $$0.221124\pi$$
$$80$$ 10.2426 1.14516
$$81$$ 1.00000 0.111111
$$82$$ 2.58579 0.285552
$$83$$ 7.31371 0.802784 0.401392 0.915906i $$-0.368527\pi$$
0.401392 + 0.915906i $$0.368527\pi$$
$$84$$ 0 0
$$85$$ −7.65685 −0.830502
$$86$$ 2.34315 0.252668
$$87$$ −6.82843 −0.732084
$$88$$ 3.17157 0.338091
$$89$$ −14.2426 −1.50972 −0.754858 0.655888i $$-0.772294\pi$$
−0.754858 + 0.655888i $$0.772294\pi$$
$$90$$ 1.41421 0.149071
$$91$$ 0 0
$$92$$ 14.0000 1.45960
$$93$$ −1.17157 −0.121486
$$94$$ −1.17157 −0.120839
$$95$$ −9.65685 −0.990772
$$96$$ 4.41421 0.450524
$$97$$ 2.58579 0.262547 0.131273 0.991346i $$-0.458093\pi$$
0.131273 + 0.991346i $$0.458093\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ −12.1716 −1.21716
$$101$$ 2.92893 0.291440 0.145720 0.989326i $$-0.453450\pi$$
0.145720 + 0.989326i $$0.453450\pi$$
$$102$$ −0.928932 −0.0919780
$$103$$ 4.48528 0.441948 0.220974 0.975280i $$-0.429076\pi$$
0.220974 + 0.975280i $$0.429076\pi$$
$$104$$ −4.10051 −0.402088
$$105$$ 0 0
$$106$$ −0.828427 −0.0804640
$$107$$ −0.343146 −0.0331732 −0.0165866 0.999862i $$-0.505280\pi$$
−0.0165866 + 0.999862i $$0.505280\pi$$
$$108$$ −1.82843 −0.175940
$$109$$ −5.65685 −0.541828 −0.270914 0.962604i $$-0.587326\pi$$
−0.270914 + 0.962604i $$0.587326\pi$$
$$110$$ −2.82843 −0.269680
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ −5.31371 −0.499872 −0.249936 0.968262i $$-0.580410\pi$$
−0.249936 + 0.968262i $$0.580410\pi$$
$$114$$ −1.17157 −0.109728
$$115$$ −26.1421 −2.43777
$$116$$ 12.4853 1.15923
$$117$$ 2.58579 0.239056
$$118$$ −0.485281 −0.0446738
$$119$$ 0 0
$$120$$ −5.41421 −0.494248
$$121$$ −7.00000 −0.636364
$$122$$ 5.07107 0.459113
$$123$$ 6.24264 0.562880
$$124$$ 2.14214 0.192369
$$125$$ 5.65685 0.505964
$$126$$ 0 0
$$127$$ −1.65685 −0.147022 −0.0735110 0.997294i $$-0.523420\pi$$
−0.0735110 + 0.997294i $$0.523420\pi$$
$$128$$ −10.5563 −0.933058
$$129$$ 5.65685 0.498058
$$130$$ 3.65685 0.320727
$$131$$ −15.3137 −1.33796 −0.668982 0.743278i $$-0.733270\pi$$
−0.668982 + 0.743278i $$0.733270\pi$$
$$132$$ 3.65685 0.318288
$$133$$ 0 0
$$134$$ −2.34315 −0.202417
$$135$$ 3.41421 0.293849
$$136$$ 3.55635 0.304954
$$137$$ 14.1421 1.20824 0.604122 0.796892i $$-0.293524\pi$$
0.604122 + 0.796892i $$0.293524\pi$$
$$138$$ −3.17157 −0.269982
$$139$$ −17.6569 −1.49763 −0.748817 0.662776i $$-0.769378\pi$$
−0.748817 + 0.662776i $$0.769378\pi$$
$$140$$ 0 0
$$141$$ −2.82843 −0.238197
$$142$$ 3.85786 0.323745
$$143$$ −5.17157 −0.432469
$$144$$ 3.00000 0.250000
$$145$$ −23.3137 −1.93610
$$146$$ 5.75736 0.476482
$$147$$ 0 0
$$148$$ 7.31371 0.601183
$$149$$ 17.3137 1.41839 0.709197 0.705010i $$-0.249058\pi$$
0.709197 + 0.705010i $$0.249058\pi$$
$$150$$ 2.75736 0.225137
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 4.48528 0.363804
$$153$$ −2.24264 −0.181307
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ −4.72792 −0.378537
$$157$$ 11.7574 0.938339 0.469170 0.883108i $$-0.344553\pi$$
0.469170 + 0.883108i $$0.344553\pi$$
$$158$$ 5.65685 0.450035
$$159$$ −2.00000 −0.158610
$$160$$ 15.0711 1.19147
$$161$$ 0 0
$$162$$ 0.414214 0.0325437
$$163$$ −11.3137 −0.886158 −0.443079 0.896483i $$-0.646114\pi$$
−0.443079 + 0.896483i $$0.646114\pi$$
$$164$$ −11.4142 −0.891300
$$165$$ −6.82843 −0.531592
$$166$$ 3.02944 0.235130
$$167$$ −19.7990 −1.53209 −0.766046 0.642786i $$-0.777779\pi$$
−0.766046 + 0.642786i $$0.777779\pi$$
$$168$$ 0 0
$$169$$ −6.31371 −0.485670
$$170$$ −3.17157 −0.243249
$$171$$ −2.82843 −0.216295
$$172$$ −10.3431 −0.788657
$$173$$ 21.0711 1.60200 0.801002 0.598662i $$-0.204301\pi$$
0.801002 + 0.598662i $$0.204301\pi$$
$$174$$ −2.82843 −0.214423
$$175$$ 0 0
$$176$$ −6.00000 −0.452267
$$177$$ −1.17157 −0.0880608
$$178$$ −5.89949 −0.442186
$$179$$ −19.6569 −1.46922 −0.734611 0.678488i $$-0.762635\pi$$
−0.734611 + 0.678488i $$0.762635\pi$$
$$180$$ −6.24264 −0.465299
$$181$$ −2.58579 −0.192200 −0.0961000 0.995372i $$-0.530637\pi$$
−0.0961000 + 0.995372i $$0.530637\pi$$
$$182$$ 0 0
$$183$$ 12.2426 0.905002
$$184$$ 12.1421 0.895130
$$185$$ −13.6569 −1.00407
$$186$$ −0.485281 −0.0355826
$$187$$ 4.48528 0.327996
$$188$$ 5.17157 0.377176
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ −4.17157 −0.301057
$$193$$ 5.31371 0.382489 0.191245 0.981542i $$-0.438748\pi$$
0.191245 + 0.981542i $$0.438748\pi$$
$$194$$ 1.07107 0.0768982
$$195$$ 8.82843 0.632217
$$196$$ 0 0
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ −0.828427 −0.0588738
$$199$$ 21.6569 1.53521 0.767607 0.640921i $$-0.221447\pi$$
0.767607 + 0.640921i $$0.221447\pi$$
$$200$$ −10.5563 −0.746447
$$201$$ −5.65685 −0.399004
$$202$$ 1.21320 0.0853607
$$203$$ 0 0
$$204$$ 4.10051 0.287093
$$205$$ 21.3137 1.48861
$$206$$ 1.85786 0.129444
$$207$$ −7.65685 −0.532188
$$208$$ 7.75736 0.537876
$$209$$ 5.65685 0.391293
$$210$$ 0 0
$$211$$ 12.9706 0.892930 0.446465 0.894801i $$-0.352683\pi$$
0.446465 + 0.894801i $$0.352683\pi$$
$$212$$ 3.65685 0.251154
$$213$$ 9.31371 0.638165
$$214$$ −0.142136 −0.00971619
$$215$$ 19.3137 1.31718
$$216$$ −1.58579 −0.107899
$$217$$ 0 0
$$218$$ −2.34315 −0.158698
$$219$$ 13.8995 0.939241
$$220$$ 12.4853 0.841757
$$221$$ −5.79899 −0.390082
$$222$$ −1.65685 −0.111201
$$223$$ 24.9706 1.67215 0.836076 0.548613i $$-0.184844\pi$$
0.836076 + 0.548613i $$0.184844\pi$$
$$224$$ 0 0
$$225$$ 6.65685 0.443790
$$226$$ −2.20101 −0.146409
$$227$$ 23.7990 1.57959 0.789797 0.613368i $$-0.210186\pi$$
0.789797 + 0.613368i $$0.210186\pi$$
$$228$$ 5.17157 0.342496
$$229$$ −0.242641 −0.0160341 −0.00801707 0.999968i $$-0.502552\pi$$
−0.00801707 + 0.999968i $$0.502552\pi$$
$$230$$ −10.8284 −0.714005
$$231$$ 0 0
$$232$$ 10.8284 0.710921
$$233$$ −6.14214 −0.402385 −0.201192 0.979552i $$-0.564482\pi$$
−0.201192 + 0.979552i $$0.564482\pi$$
$$234$$ 1.07107 0.0700179
$$235$$ −9.65685 −0.629944
$$236$$ 2.14214 0.139441
$$237$$ 13.6569 0.887108
$$238$$ 0 0
$$239$$ −15.6569 −1.01276 −0.506379 0.862311i $$-0.669016\pi$$
−0.506379 + 0.862311i $$0.669016\pi$$
$$240$$ 10.2426 0.661160
$$241$$ −16.2426 −1.04628 −0.523140 0.852247i $$-0.675240\pi$$
−0.523140 + 0.852247i $$0.675240\pi$$
$$242$$ −2.89949 −0.186387
$$243$$ 1.00000 0.0641500
$$244$$ −22.3848 −1.43304
$$245$$ 0 0
$$246$$ 2.58579 0.164864
$$247$$ −7.31371 −0.465360
$$248$$ 1.85786 0.117975
$$249$$ 7.31371 0.463487
$$250$$ 2.34315 0.148194
$$251$$ −12.4853 −0.788064 −0.394032 0.919097i $$-0.628920\pi$$
−0.394032 + 0.919097i $$0.628920\pi$$
$$252$$ 0 0
$$253$$ 15.3137 0.962765
$$254$$ −0.686292 −0.0430618
$$255$$ −7.65685 −0.479491
$$256$$ 3.97056 0.248160
$$257$$ 23.2132 1.44800 0.724000 0.689800i $$-0.242302\pi$$
0.724000 + 0.689800i $$0.242302\pi$$
$$258$$ 2.34315 0.145878
$$259$$ 0 0
$$260$$ −16.1421 −1.00109
$$261$$ −6.82843 −0.422669
$$262$$ −6.34315 −0.391881
$$263$$ 5.31371 0.327657 0.163829 0.986489i $$-0.447616\pi$$
0.163829 + 0.986489i $$0.447616\pi$$
$$264$$ 3.17157 0.195197
$$265$$ −6.82843 −0.419467
$$266$$ 0 0
$$267$$ −14.2426 −0.871635
$$268$$ 10.3431 0.631808
$$269$$ −14.7279 −0.897977 −0.448989 0.893537i $$-0.648216\pi$$
−0.448989 + 0.893537i $$0.648216\pi$$
$$270$$ 1.41421 0.0860663
$$271$$ −10.1421 −0.616091 −0.308045 0.951372i $$-0.599675\pi$$
−0.308045 + 0.951372i $$0.599675\pi$$
$$272$$ −6.72792 −0.407940
$$273$$ 0 0
$$274$$ 5.85786 0.353887
$$275$$ −13.3137 −0.802847
$$276$$ 14.0000 0.842701
$$277$$ −9.31371 −0.559607 −0.279803 0.960057i $$-0.590269\pi$$
−0.279803 + 0.960057i $$0.590269\pi$$
$$278$$ −7.31371 −0.438647
$$279$$ −1.17157 −0.0701402
$$280$$ 0 0
$$281$$ 0.485281 0.0289495 0.0144747 0.999895i $$-0.495392\pi$$
0.0144747 + 0.999895i $$0.495392\pi$$
$$282$$ −1.17157 −0.0697661
$$283$$ −8.48528 −0.504398 −0.252199 0.967675i $$-0.581154\pi$$
−0.252199 + 0.967675i $$0.581154\pi$$
$$284$$ −17.0294 −1.01051
$$285$$ −9.65685 −0.572023
$$286$$ −2.14214 −0.126667
$$287$$ 0 0
$$288$$ 4.41421 0.260110
$$289$$ −11.9706 −0.704151
$$290$$ −9.65685 −0.567070
$$291$$ 2.58579 0.151581
$$292$$ −25.4142 −1.48725
$$293$$ 16.5858 0.968952 0.484476 0.874805i $$-0.339010\pi$$
0.484476 + 0.874805i $$0.339010\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 6.34315 0.368688
$$297$$ −2.00000 −0.116052
$$298$$ 7.17157 0.415438
$$299$$ −19.7990 −1.14501
$$300$$ −12.1716 −0.702726
$$301$$ 0 0
$$302$$ 4.97056 0.286024
$$303$$ 2.92893 0.168263
$$304$$ −8.48528 −0.486664
$$305$$ 41.7990 2.39340
$$306$$ −0.928932 −0.0531035
$$307$$ 30.1421 1.72030 0.860151 0.510039i $$-0.170369\pi$$
0.860151 + 0.510039i $$0.170369\pi$$
$$308$$ 0 0
$$309$$ 4.48528 0.255159
$$310$$ −1.65685 −0.0941030
$$311$$ 6.14214 0.348289 0.174144 0.984720i $$-0.444284\pi$$
0.174144 + 0.984720i $$0.444284\pi$$
$$312$$ −4.10051 −0.232145
$$313$$ 1.89949 0.107366 0.0536829 0.998558i $$-0.482904\pi$$
0.0536829 + 0.998558i $$0.482904\pi$$
$$314$$ 4.87006 0.274833
$$315$$ 0 0
$$316$$ −24.9706 −1.40470
$$317$$ 10.0000 0.561656 0.280828 0.959758i $$-0.409391\pi$$
0.280828 + 0.959758i $$0.409391\pi$$
$$318$$ −0.828427 −0.0464559
$$319$$ 13.6569 0.764637
$$320$$ −14.2426 −0.796188
$$321$$ −0.343146 −0.0191525
$$322$$ 0 0
$$323$$ 6.34315 0.352942
$$324$$ −1.82843 −0.101579
$$325$$ 17.2132 0.954817
$$326$$ −4.68629 −0.259550
$$327$$ −5.65685 −0.312825
$$328$$ −9.89949 −0.546608
$$329$$ 0 0
$$330$$ −2.82843 −0.155700
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ −13.3726 −0.733916
$$333$$ −4.00000 −0.219199
$$334$$ −8.20101 −0.448739
$$335$$ −19.3137 −1.05522
$$336$$ 0 0
$$337$$ −29.6569 −1.61551 −0.807756 0.589517i $$-0.799318\pi$$
−0.807756 + 0.589517i $$0.799318\pi$$
$$338$$ −2.61522 −0.142249
$$339$$ −5.31371 −0.288601
$$340$$ 14.0000 0.759257
$$341$$ 2.34315 0.126888
$$342$$ −1.17157 −0.0633514
$$343$$ 0 0
$$344$$ −8.97056 −0.483660
$$345$$ −26.1421 −1.40745
$$346$$ 8.72792 0.469216
$$347$$ 33.3137 1.78837 0.894187 0.447694i $$-0.147755\pi$$
0.894187 + 0.447694i $$0.147755\pi$$
$$348$$ 12.4853 0.669281
$$349$$ −9.89949 −0.529908 −0.264954 0.964261i $$-0.585357\pi$$
−0.264954 + 0.964261i $$0.585357\pi$$
$$350$$ 0 0
$$351$$ 2.58579 0.138019
$$352$$ −8.82843 −0.470557
$$353$$ −14.7279 −0.783888 −0.391944 0.919989i $$-0.628197\pi$$
−0.391944 + 0.919989i $$0.628197\pi$$
$$354$$ −0.485281 −0.0257924
$$355$$ 31.7990 1.68772
$$356$$ 26.0416 1.38020
$$357$$ 0 0
$$358$$ −8.14214 −0.430325
$$359$$ −0.343146 −0.0181105 −0.00905527 0.999959i $$-0.502882\pi$$
−0.00905527 + 0.999959i $$0.502882\pi$$
$$360$$ −5.41421 −0.285354
$$361$$ −11.0000 −0.578947
$$362$$ −1.07107 −0.0562941
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ 47.4558 2.48395
$$366$$ 5.07107 0.265069
$$367$$ −3.31371 −0.172974 −0.0864871 0.996253i $$-0.527564\pi$$
−0.0864871 + 0.996253i $$0.527564\pi$$
$$368$$ −22.9706 −1.19742
$$369$$ 6.24264 0.324979
$$370$$ −5.65685 −0.294086
$$371$$ 0 0
$$372$$ 2.14214 0.111065
$$373$$ −10.6863 −0.553315 −0.276658 0.960969i $$-0.589227\pi$$
−0.276658 + 0.960969i $$0.589227\pi$$
$$374$$ 1.85786 0.0960679
$$375$$ 5.65685 0.292119
$$376$$ 4.48528 0.231311
$$377$$ −17.6569 −0.909374
$$378$$ 0 0
$$379$$ 8.68629 0.446185 0.223092 0.974797i $$-0.428385\pi$$
0.223092 + 0.974797i $$0.428385\pi$$
$$380$$ 17.6569 0.905778
$$381$$ −1.65685 −0.0848832
$$382$$ −7.45584 −0.381474
$$383$$ −18.3431 −0.937291 −0.468645 0.883386i $$-0.655258\pi$$
−0.468645 + 0.883386i $$0.655258\pi$$
$$384$$ −10.5563 −0.538701
$$385$$ 0 0
$$386$$ 2.20101 0.112028
$$387$$ 5.65685 0.287554
$$388$$ −4.72792 −0.240024
$$389$$ −18.1421 −0.919843 −0.459921 0.887960i $$-0.652122\pi$$
−0.459921 + 0.887960i $$0.652122\pi$$
$$390$$ 3.65685 0.185172
$$391$$ 17.1716 0.868404
$$392$$ 0 0
$$393$$ −15.3137 −0.772474
$$394$$ 0.828427 0.0417356
$$395$$ 46.6274 2.34608
$$396$$ 3.65685 0.183764
$$397$$ −2.38478 −0.119688 −0.0598442 0.998208i $$-0.519060\pi$$
−0.0598442 + 0.998208i $$0.519060\pi$$
$$398$$ 8.97056 0.449654
$$399$$ 0 0
$$400$$ 19.9706 0.998528
$$401$$ −6.14214 −0.306724 −0.153362 0.988170i $$-0.549010\pi$$
−0.153362 + 0.988170i $$0.549010\pi$$
$$402$$ −2.34315 −0.116865
$$403$$ −3.02944 −0.150907
$$404$$ −5.35534 −0.266438
$$405$$ 3.41421 0.169654
$$406$$ 0 0
$$407$$ 8.00000 0.396545
$$408$$ 3.55635 0.176066
$$409$$ −21.4142 −1.05886 −0.529432 0.848352i $$-0.677595\pi$$
−0.529432 + 0.848352i $$0.677595\pi$$
$$410$$ 8.82843 0.436005
$$411$$ 14.1421 0.697580
$$412$$ −8.20101 −0.404035
$$413$$ 0 0
$$414$$ −3.17157 −0.155874
$$415$$ 24.9706 1.22576
$$416$$ 11.4142 0.559628
$$417$$ −17.6569 −0.864660
$$418$$ 2.34315 0.114607
$$419$$ 33.1716 1.62054 0.810269 0.586059i $$-0.199321\pi$$
0.810269 + 0.586059i $$0.199321\pi$$
$$420$$ 0 0
$$421$$ 16.6274 0.810371 0.405185 0.914235i $$-0.367207\pi$$
0.405185 + 0.914235i $$0.367207\pi$$
$$422$$ 5.37258 0.261533
$$423$$ −2.82843 −0.137523
$$424$$ 3.17157 0.154025
$$425$$ −14.9289 −0.724160
$$426$$ 3.85786 0.186914
$$427$$ 0 0
$$428$$ 0.627417 0.0303273
$$429$$ −5.17157 −0.249686
$$430$$ 8.00000 0.385794
$$431$$ −26.9706 −1.29913 −0.649563 0.760308i $$-0.725048\pi$$
−0.649563 + 0.760308i $$0.725048\pi$$
$$432$$ 3.00000 0.144338
$$433$$ −20.2426 −0.972799 −0.486400 0.873736i $$-0.661690\pi$$
−0.486400 + 0.873736i $$0.661690\pi$$
$$434$$ 0 0
$$435$$ −23.3137 −1.11781
$$436$$ 10.3431 0.495347
$$437$$ 21.6569 1.03599
$$438$$ 5.75736 0.275097
$$439$$ 12.6863 0.605484 0.302742 0.953073i $$-0.402098\pi$$
0.302742 + 0.953073i $$0.402098\pi$$
$$440$$ 10.8284 0.516225
$$441$$ 0 0
$$442$$ −2.40202 −0.114252
$$443$$ 34.9706 1.66150 0.830751 0.556645i $$-0.187911\pi$$
0.830751 + 0.556645i $$0.187911\pi$$
$$444$$ 7.31371 0.347093
$$445$$ −48.6274 −2.30516
$$446$$ 10.3431 0.489762
$$447$$ 17.3137 0.818910
$$448$$ 0 0
$$449$$ −5.31371 −0.250769 −0.125385 0.992108i $$-0.540017\pi$$
−0.125385 + 0.992108i $$0.540017\pi$$
$$450$$ 2.75736 0.129983
$$451$$ −12.4853 −0.587909
$$452$$ 9.71573 0.456989
$$453$$ 12.0000 0.563809
$$454$$ 9.85786 0.462652
$$455$$ 0 0
$$456$$ 4.48528 0.210043
$$457$$ −18.0000 −0.842004 −0.421002 0.907060i $$-0.638322\pi$$
−0.421002 + 0.907060i $$0.638322\pi$$
$$458$$ −0.100505 −0.00469629
$$459$$ −2.24264 −0.104678
$$460$$ 47.7990 2.22864
$$461$$ −16.5858 −0.772477 −0.386239 0.922399i $$-0.626226\pi$$
−0.386239 + 0.922399i $$0.626226\pi$$
$$462$$ 0 0
$$463$$ −26.6274 −1.23748 −0.618741 0.785595i $$-0.712357\pi$$
−0.618741 + 0.785595i $$0.712357\pi$$
$$464$$ −20.4853 −0.951005
$$465$$ −4.00000 −0.185496
$$466$$ −2.54416 −0.117856
$$467$$ 0.201010 0.00930164 0.00465082 0.999989i $$-0.498520\pi$$
0.00465082 + 0.999989i $$0.498520\pi$$
$$468$$ −4.72792 −0.218548
$$469$$ 0 0
$$470$$ −4.00000 −0.184506
$$471$$ 11.7574 0.541751
$$472$$ 1.85786 0.0855151
$$473$$ −11.3137 −0.520205
$$474$$ 5.65685 0.259828
$$475$$ −18.8284 −0.863907
$$476$$ 0 0
$$477$$ −2.00000 −0.0915737
$$478$$ −6.48528 −0.296630
$$479$$ 1.85786 0.0848880 0.0424440 0.999099i $$-0.486486\pi$$
0.0424440 + 0.999099i $$0.486486\pi$$
$$480$$ 15.0711 0.687897
$$481$$ −10.3431 −0.471607
$$482$$ −6.72792 −0.306448
$$483$$ 0 0
$$484$$ 12.7990 0.581772
$$485$$ 8.82843 0.400878
$$486$$ 0.414214 0.0187891
$$487$$ 26.6274 1.20660 0.603302 0.797513i $$-0.293851\pi$$
0.603302 + 0.797513i $$0.293851\pi$$
$$488$$ −19.4142 −0.878840
$$489$$ −11.3137 −0.511624
$$490$$ 0 0
$$491$$ 5.02944 0.226975 0.113488 0.993539i $$-0.463798\pi$$
0.113488 + 0.993539i $$0.463798\pi$$
$$492$$ −11.4142 −0.514592
$$493$$ 15.3137 0.689695
$$494$$ −3.02944 −0.136301
$$495$$ −6.82843 −0.306915
$$496$$ −3.51472 −0.157816
$$497$$ 0 0
$$498$$ 3.02944 0.135752
$$499$$ 3.31371 0.148342 0.0741710 0.997246i $$-0.476369\pi$$
0.0741710 + 0.997246i $$0.476369\pi$$
$$500$$ −10.3431 −0.462560
$$501$$ −19.7990 −0.884554
$$502$$ −5.17157 −0.230819
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 10.0000 0.444994
$$506$$ 6.34315 0.281987
$$507$$ −6.31371 −0.280402
$$508$$ 3.02944 0.134410
$$509$$ 5.55635 0.246281 0.123140 0.992389i $$-0.460703\pi$$
0.123140 + 0.992389i $$0.460703\pi$$
$$510$$ −3.17157 −0.140440
$$511$$ 0 0
$$512$$ 22.7574 1.00574
$$513$$ −2.82843 −0.124878
$$514$$ 9.61522 0.424109
$$515$$ 15.3137 0.674803
$$516$$ −10.3431 −0.455332
$$517$$ 5.65685 0.248788
$$518$$ 0 0
$$519$$ 21.0711 0.924917
$$520$$ −14.0000 −0.613941
$$521$$ −35.4142 −1.55152 −0.775762 0.631025i $$-0.782634\pi$$
−0.775762 + 0.631025i $$0.782634\pi$$
$$522$$ −2.82843 −0.123797
$$523$$ −25.6569 −1.12190 −0.560948 0.827851i $$-0.689563\pi$$
−0.560948 + 0.827851i $$0.689563\pi$$
$$524$$ 28.0000 1.22319
$$525$$ 0 0
$$526$$ 2.20101 0.0959686
$$527$$ 2.62742 0.114452
$$528$$ −6.00000 −0.261116
$$529$$ 35.6274 1.54902
$$530$$ −2.82843 −0.122859
$$531$$ −1.17157 −0.0508419
$$532$$ 0 0
$$533$$ 16.1421 0.699194
$$534$$ −5.89949 −0.255296
$$535$$ −1.17157 −0.0506515
$$536$$ 8.97056 0.387469
$$537$$ −19.6569 −0.848256
$$538$$ −6.10051 −0.263011
$$539$$ 0 0
$$540$$ −6.24264 −0.268640
$$541$$ 17.3137 0.744374 0.372187 0.928158i $$-0.378608\pi$$
0.372187 + 0.928158i $$0.378608\pi$$
$$542$$ −4.20101 −0.180449
$$543$$ −2.58579 −0.110967
$$544$$ −9.89949 −0.424437
$$545$$ −19.3137 −0.827308
$$546$$ 0 0
$$547$$ −36.9706 −1.58075 −0.790374 0.612625i $$-0.790114\pi$$
−0.790374 + 0.612625i $$0.790114\pi$$
$$548$$ −25.8579 −1.10459
$$549$$ 12.2426 0.522503
$$550$$ −5.51472 −0.235148
$$551$$ 19.3137 0.822792
$$552$$ 12.1421 0.516804
$$553$$ 0 0
$$554$$ −3.85786 −0.163905
$$555$$ −13.6569 −0.579701
$$556$$ 32.2843 1.36916
$$557$$ 26.0000 1.10166 0.550828 0.834619i $$-0.314312\pi$$
0.550828 + 0.834619i $$0.314312\pi$$
$$558$$ −0.485281 −0.0205436
$$559$$ 14.6274 0.618674
$$560$$ 0 0
$$561$$ 4.48528 0.189369
$$562$$ 0.201010 0.00847910
$$563$$ −1.17157 −0.0493759 −0.0246880 0.999695i $$-0.507859\pi$$
−0.0246880 + 0.999695i $$0.507859\pi$$
$$564$$ 5.17157 0.217763
$$565$$ −18.1421 −0.763245
$$566$$ −3.51472 −0.147735
$$567$$ 0 0
$$568$$ −14.7696 −0.619717
$$569$$ −16.4853 −0.691099 −0.345549 0.938401i $$-0.612307\pi$$
−0.345549 + 0.938401i $$0.612307\pi$$
$$570$$ −4.00000 −0.167542
$$571$$ 22.3431 0.935032 0.467516 0.883985i $$-0.345149\pi$$
0.467516 + 0.883985i $$0.345149\pi$$
$$572$$ 9.45584 0.395369
$$573$$ −18.0000 −0.751961
$$574$$ 0 0
$$575$$ −50.9706 −2.12562
$$576$$ −4.17157 −0.173816
$$577$$ −33.8995 −1.41125 −0.705627 0.708583i $$-0.749335\pi$$
−0.705627 + 0.708583i $$0.749335\pi$$
$$578$$ −4.95837 −0.206241
$$579$$ 5.31371 0.220830
$$580$$ 42.6274 1.77001
$$581$$ 0 0
$$582$$ 1.07107 0.0443972
$$583$$ 4.00000 0.165663
$$584$$ −22.0416 −0.912089
$$585$$ 8.82843 0.365011
$$586$$ 6.87006 0.283799
$$587$$ −22.8284 −0.942230 −0.471115 0.882072i $$-0.656148\pi$$
−0.471115 + 0.882072i $$0.656148\pi$$
$$588$$ 0 0
$$589$$ 3.31371 0.136539
$$590$$ −1.65685 −0.0682116
$$591$$ 2.00000 0.0822690
$$592$$ −12.0000 −0.493197
$$593$$ −6.92893 −0.284537 −0.142269 0.989828i $$-0.545440\pi$$
−0.142269 + 0.989828i $$0.545440\pi$$
$$594$$ −0.828427 −0.0339908
$$595$$ 0 0
$$596$$ −31.6569 −1.29672
$$597$$ 21.6569 0.886356
$$598$$ −8.20101 −0.335364
$$599$$ −2.00000 −0.0817178 −0.0408589 0.999165i $$-0.513009\pi$$
−0.0408589 + 0.999165i $$0.513009\pi$$
$$600$$ −10.5563 −0.430961
$$601$$ −15.0711 −0.614762 −0.307381 0.951587i $$-0.599453\pi$$
−0.307381 + 0.951587i $$0.599453\pi$$
$$602$$ 0 0
$$603$$ −5.65685 −0.230365
$$604$$ −21.9411 −0.892772
$$605$$ −23.8995 −0.971653
$$606$$ 1.21320 0.0492830
$$607$$ −18.3431 −0.744525 −0.372263 0.928127i $$-0.621418\pi$$
−0.372263 + 0.928127i $$0.621418\pi$$
$$608$$ −12.4853 −0.506345
$$609$$ 0 0
$$610$$ 17.3137 0.701012
$$611$$ −7.31371 −0.295881
$$612$$ 4.10051 0.165753
$$613$$ 4.68629 0.189278 0.0946388 0.995512i $$-0.469830\pi$$
0.0946388 + 0.995512i $$0.469830\pi$$
$$614$$ 12.4853 0.503865
$$615$$ 21.3137 0.859452
$$616$$ 0 0
$$617$$ −24.4853 −0.985740 −0.492870 0.870103i $$-0.664052\pi$$
−0.492870 + 0.870103i $$0.664052\pi$$
$$618$$ 1.85786 0.0747343
$$619$$ 28.9706 1.16443 0.582213 0.813037i $$-0.302187\pi$$
0.582213 + 0.813037i $$0.302187\pi$$
$$620$$ 7.31371 0.293726
$$621$$ −7.65685 −0.307259
$$622$$ 2.54416 0.102011
$$623$$ 0 0
$$624$$ 7.75736 0.310543
$$625$$ −13.9706 −0.558823
$$626$$ 0.786797 0.0314467
$$627$$ 5.65685 0.225913
$$628$$ −21.4975 −0.857843
$$629$$ 8.97056 0.357680
$$630$$ 0 0
$$631$$ 23.3137 0.928104 0.464052 0.885808i $$-0.346395\pi$$
0.464052 + 0.885808i $$0.346395\pi$$
$$632$$ −21.6569 −0.861463
$$633$$ 12.9706 0.515534
$$634$$ 4.14214 0.164505
$$635$$ −5.65685 −0.224485
$$636$$ 3.65685 0.145004
$$637$$ 0 0
$$638$$ 5.65685 0.223957
$$639$$ 9.31371 0.368445
$$640$$ −36.0416 −1.42467
$$641$$ −10.8284 −0.427697 −0.213849 0.976867i $$-0.568600\pi$$
−0.213849 + 0.976867i $$0.568600\pi$$
$$642$$ −0.142136 −0.00560965
$$643$$ −34.4264 −1.35764 −0.678822 0.734302i $$-0.737509\pi$$
−0.678822 + 0.734302i $$0.737509\pi$$
$$644$$ 0 0
$$645$$ 19.3137 0.760477
$$646$$ 2.62742 0.103374
$$647$$ 26.8284 1.05473 0.527367 0.849638i $$-0.323179\pi$$
0.527367 + 0.849638i $$0.323179\pi$$
$$648$$ −1.58579 −0.0622956
$$649$$ 2.34315 0.0919765
$$650$$ 7.12994 0.279659
$$651$$ 0 0
$$652$$ 20.6863 0.810138
$$653$$ 36.4853 1.42778 0.713890 0.700258i $$-0.246932\pi$$
0.713890 + 0.700258i $$0.246932\pi$$
$$654$$ −2.34315 −0.0916242
$$655$$ −52.2843 −2.04292
$$656$$ 18.7279 0.731203
$$657$$ 13.8995 0.542271
$$658$$ 0 0
$$659$$ 9.31371 0.362811 0.181405 0.983408i $$-0.441935\pi$$
0.181405 + 0.983408i $$0.441935\pi$$
$$660$$ 12.4853 0.485989
$$661$$ −23.5563 −0.916236 −0.458118 0.888891i $$-0.651476\pi$$
−0.458118 + 0.888891i $$0.651476\pi$$
$$662$$ −1.65685 −0.0643955
$$663$$ −5.79899 −0.225214
$$664$$ −11.5980 −0.450089
$$665$$ 0 0
$$666$$ −1.65685 −0.0642018
$$667$$ 52.2843 2.02446
$$668$$ 36.2010 1.40066
$$669$$ 24.9706 0.965418
$$670$$ −8.00000 −0.309067
$$671$$ −24.4853 −0.945244
$$672$$ 0 0
$$673$$ 23.3137 0.898677 0.449339 0.893361i $$-0.351660\pi$$
0.449339 + 0.893361i $$0.351660\pi$$
$$674$$ −12.2843 −0.473172
$$675$$ 6.65685 0.256222
$$676$$ 11.5442 0.444006
$$677$$ 31.4142 1.20735 0.603673 0.797232i $$-0.293703\pi$$
0.603673 + 0.797232i $$0.293703\pi$$
$$678$$ −2.20101 −0.0845293
$$679$$ 0 0
$$680$$ 12.1421 0.465630
$$681$$ 23.7990 0.911979
$$682$$ 0.970563 0.0371648
$$683$$ −19.6569 −0.752149 −0.376074 0.926590i $$-0.622726\pi$$
−0.376074 + 0.926590i $$0.622726\pi$$
$$684$$ 5.17157 0.197740
$$685$$ 48.2843 1.84485
$$686$$ 0 0
$$687$$ −0.242641 −0.00925732
$$688$$ 16.9706 0.646997
$$689$$ −5.17157 −0.197021
$$690$$ −10.8284 −0.412231
$$691$$ −0.686292 −0.0261078 −0.0130539 0.999915i $$-0.504155\pi$$
−0.0130539 + 0.999915i $$0.504155\pi$$
$$692$$ −38.5269 −1.46457
$$693$$ 0 0
$$694$$ 13.7990 0.523802
$$695$$ −60.2843 −2.28671
$$696$$ 10.8284 0.410450
$$697$$ −14.0000 −0.530288
$$698$$ −4.10051 −0.155206
$$699$$ −6.14214 −0.232317
$$700$$ 0 0
$$701$$ −17.1716 −0.648561 −0.324281 0.945961i $$-0.605122\pi$$
−0.324281 + 0.945961i $$0.605122\pi$$
$$702$$ 1.07107 0.0404248
$$703$$ 11.3137 0.426705
$$704$$ 8.34315 0.314444
$$705$$ −9.65685 −0.363698
$$706$$ −6.10051 −0.229596
$$707$$ 0 0
$$708$$ 2.14214 0.0805064
$$709$$ −36.2843 −1.36268 −0.681342 0.731965i $$-0.738603\pi$$
−0.681342 + 0.731965i $$0.738603\pi$$
$$710$$ 13.1716 0.494320
$$711$$ 13.6569 0.512172
$$712$$ 22.5858 0.846438
$$713$$ 8.97056 0.335950
$$714$$ 0 0
$$715$$ −17.6569 −0.660329
$$716$$ 35.9411 1.34318
$$717$$ −15.6569 −0.584716
$$718$$ −0.142136 −0.00530445
$$719$$ 41.9411 1.56414 0.782070 0.623191i $$-0.214164\pi$$
0.782070 + 0.623191i $$0.214164\pi$$
$$720$$ 10.2426 0.381721
$$721$$ 0 0
$$722$$ −4.55635 −0.169570
$$723$$ −16.2426 −0.604070
$$724$$ 4.72792 0.175712
$$725$$ −45.4558 −1.68819
$$726$$ −2.89949 −0.107610
$$727$$ 12.4853 0.463053 0.231527 0.972829i $$-0.425628\pi$$
0.231527 + 0.972829i $$0.425628\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 19.6569 0.727533
$$731$$ −12.6863 −0.469219
$$732$$ −22.3848 −0.827365
$$733$$ 49.6985 1.83566 0.917828 0.396979i $$-0.129941\pi$$
0.917828 + 0.396979i $$0.129941\pi$$
$$734$$ −1.37258 −0.0506630
$$735$$ 0 0
$$736$$ −33.7990 −1.24585
$$737$$ 11.3137 0.416746
$$738$$ 2.58579 0.0951841
$$739$$ 4.68629 0.172388 0.0861940 0.996278i $$-0.472530\pi$$
0.0861940 + 0.996278i $$0.472530\pi$$
$$740$$ 24.9706 0.917936
$$741$$ −7.31371 −0.268676
$$742$$ 0 0
$$743$$ −50.9706 −1.86993 −0.934964 0.354742i $$-0.884569\pi$$
−0.934964 + 0.354742i $$0.884569\pi$$
$$744$$ 1.85786 0.0681126
$$745$$ 59.1127 2.16572
$$746$$ −4.42641 −0.162062
$$747$$ 7.31371 0.267595
$$748$$ −8.20101 −0.299859
$$749$$ 0 0
$$750$$ 2.34315 0.0855596
$$751$$ 13.6569 0.498346 0.249173 0.968459i $$-0.419841\pi$$
0.249173 + 0.968459i $$0.419841\pi$$
$$752$$ −8.48528 −0.309426
$$753$$ −12.4853 −0.454989
$$754$$ −7.31371 −0.266350
$$755$$ 40.9706 1.49107
$$756$$ 0 0
$$757$$ 26.3431 0.957458 0.478729 0.877963i $$-0.341098\pi$$
0.478729 + 0.877963i $$0.341098\pi$$
$$758$$ 3.59798 0.130685
$$759$$ 15.3137 0.555852
$$760$$ 15.3137 0.555487
$$761$$ −18.5269 −0.671600 −0.335800 0.941933i $$-0.609007\pi$$
−0.335800 + 0.941933i $$0.609007\pi$$
$$762$$ −0.686292 −0.0248617
$$763$$ 0 0
$$764$$ 32.9117 1.19070
$$765$$ −7.65685 −0.276834
$$766$$ −7.59798 −0.274526
$$767$$ −3.02944 −0.109387
$$768$$ 3.97056 0.143275
$$769$$ 29.6985 1.07095 0.535477 0.844550i $$-0.320132\pi$$
0.535477 + 0.844550i $$0.320132\pi$$
$$770$$ 0 0
$$771$$ 23.2132 0.836003
$$772$$ −9.71573 −0.349677
$$773$$ −9.55635 −0.343718 −0.171859 0.985122i $$-0.554977\pi$$
−0.171859 + 0.985122i $$0.554977\pi$$
$$774$$ 2.34315 0.0842226
$$775$$ −7.79899 −0.280148
$$776$$ −4.10051 −0.147200
$$777$$ 0 0
$$778$$ −7.51472 −0.269416
$$779$$ −17.6569 −0.632622
$$780$$ −16.1421 −0.577981
$$781$$ −18.6274 −0.666541
$$782$$ 7.11270 0.254350
$$783$$ −6.82843 −0.244028
$$784$$ 0 0
$$785$$ 40.1421 1.43273
$$786$$ −6.34315 −0.226253
$$787$$ 24.6863 0.879971 0.439986 0.898005i $$-0.354984\pi$$
0.439986 + 0.898005i $$0.354984\pi$$
$$788$$ −3.65685 −0.130270
$$789$$ 5.31371 0.189173
$$790$$ 19.3137 0.687151
$$791$$ 0 0
$$792$$ 3.17157 0.112697
$$793$$ 31.6569 1.12417
$$794$$ −0.987807 −0.0350559
$$795$$ −6.82843 −0.242179
$$796$$ −39.5980 −1.40351
$$797$$ −8.38478 −0.297004 −0.148502 0.988912i $$-0.547445\pi$$
−0.148502 + 0.988912i $$0.547445\pi$$
$$798$$ 0 0
$$799$$ 6.34315 0.224404
$$800$$ 29.3848 1.03891
$$801$$ −14.2426 −0.503239
$$802$$ −2.54416 −0.0898373
$$803$$ −27.7990 −0.981005
$$804$$ 10.3431 0.364775
$$805$$ 0 0
$$806$$ −1.25483 −0.0441996
$$807$$ −14.7279 −0.518447
$$808$$ −4.64466 −0.163399
$$809$$ 19.9411 0.701093 0.350546 0.936545i $$-0.385996\pi$$
0.350546 + 0.936545i $$0.385996\pi$$
$$810$$ 1.41421 0.0496904
$$811$$ −17.6569 −0.620016 −0.310008 0.950734i $$-0.600332\pi$$
−0.310008 + 0.950734i $$0.600332\pi$$
$$812$$ 0 0
$$813$$ −10.1421 −0.355700
$$814$$ 3.31371 0.116145
$$815$$ −38.6274 −1.35306
$$816$$ −6.72792 −0.235524
$$817$$ −16.0000 −0.559769
$$818$$ −8.87006 −0.310134
$$819$$ 0 0
$$820$$ −38.9706 −1.36091
$$821$$ −10.6863 −0.372954 −0.186477 0.982459i $$-0.559707\pi$$
−0.186477 + 0.982459i $$0.559707\pi$$
$$822$$ 5.85786 0.204316
$$823$$ 8.97056 0.312694 0.156347 0.987702i $$-0.450028\pi$$
0.156347 + 0.987702i $$0.450028\pi$$
$$824$$ −7.11270 −0.247783
$$825$$ −13.3137 −0.463524
$$826$$ 0 0
$$827$$ 47.6569 1.65719 0.828596 0.559848i $$-0.189140\pi$$
0.828596 + 0.559848i $$0.189140\pi$$
$$828$$ 14.0000 0.486534
$$829$$ −0.727922 −0.0252818 −0.0126409 0.999920i $$-0.504024\pi$$
−0.0126409 + 0.999920i $$0.504024\pi$$
$$830$$ 10.3431 0.359016
$$831$$ −9.31371 −0.323089
$$832$$ −10.7868 −0.373965
$$833$$ 0 0
$$834$$ −7.31371 −0.253253
$$835$$ −67.5980 −2.33932
$$836$$ −10.3431 −0.357725
$$837$$ −1.17157 −0.0404955
$$838$$ 13.7401 0.474644
$$839$$ 50.8284 1.75479 0.877396 0.479767i $$-0.159279\pi$$
0.877396 + 0.479767i $$0.159279\pi$$
$$840$$ 0 0
$$841$$ 17.6274 0.607842
$$842$$ 6.88730 0.237352
$$843$$ 0.485281 0.0167140
$$844$$ −23.7157 −0.816329
$$845$$ −21.5563 −0.741561
$$846$$ −1.17157 −0.0402795
$$847$$ 0 0
$$848$$ −6.00000 −0.206041
$$849$$ −8.48528 −0.291214
$$850$$ −6.18377 −0.212101
$$851$$ 30.6274 1.04989
$$852$$ −17.0294 −0.583419
$$853$$ −49.4975 −1.69476 −0.847381 0.530986i $$-0.821822\pi$$
−0.847381 + 0.530986i $$0.821822\pi$$
$$854$$ 0 0
$$855$$ −9.65685 −0.330257
$$856$$ 0.544156 0.0185989
$$857$$ 15.4142 0.526540 0.263270 0.964722i $$-0.415199\pi$$
0.263270 + 0.964722i $$0.415199\pi$$
$$858$$ −2.14214 −0.0731313
$$859$$ 57.4558 1.96037 0.980184 0.198089i $$-0.0634735\pi$$
0.980184 + 0.198089i $$0.0634735\pi$$
$$860$$ −35.3137 −1.20419
$$861$$ 0 0
$$862$$ −11.1716 −0.380505
$$863$$ 17.3137 0.589365 0.294683 0.955595i $$-0.404786\pi$$
0.294683 + 0.955595i $$0.404786\pi$$
$$864$$ 4.41421 0.150175
$$865$$ 71.9411 2.44607
$$866$$ −8.38478 −0.284926
$$867$$ −11.9706 −0.406542
$$868$$ 0 0
$$869$$ −27.3137 −0.926554
$$870$$ −9.65685 −0.327398
$$871$$ −14.6274 −0.495631
$$872$$ 8.97056 0.303782
$$873$$ 2.58579 0.0875156
$$874$$ 8.97056 0.303434
$$875$$ 0 0
$$876$$ −25.4142 −0.858667
$$877$$ −11.3137 −0.382037 −0.191018 0.981586i $$-0.561179\pi$$
−0.191018 + 0.981586i $$0.561179\pi$$
$$878$$ 5.25483 0.177342
$$879$$ 16.5858 0.559425
$$880$$ −20.4853 −0.690559
$$881$$ 21.7574 0.733024 0.366512 0.930413i $$-0.380552\pi$$
0.366512 + 0.930413i $$0.380552\pi$$
$$882$$ 0 0
$$883$$ −4.68629 −0.157706 −0.0788531 0.996886i $$-0.525126\pi$$
−0.0788531 + 0.996886i $$0.525126\pi$$
$$884$$ 10.6030 0.356619
$$885$$ −4.00000 −0.134459
$$886$$ 14.4853 0.486643
$$887$$ −2.82843 −0.0949693 −0.0474846 0.998872i $$-0.515121\pi$$
−0.0474846 + 0.998872i $$0.515121\pi$$
$$888$$ 6.34315 0.212862
$$889$$ 0 0
$$890$$ −20.1421 −0.675166
$$891$$ −2.00000 −0.0670025
$$892$$ −45.6569 −1.52870
$$893$$ 8.00000 0.267710
$$894$$ 7.17157 0.239853
$$895$$ −67.1127 −2.24333
$$896$$ 0 0
$$897$$ −19.7990 −0.661069
$$898$$ −2.20101 −0.0734487
$$899$$ 8.00000 0.266815
$$900$$ −12.1716 −0.405719
$$901$$ 4.48528 0.149426
$$902$$ −5.17157 −0.172195
$$903$$ 0 0
$$904$$ 8.42641 0.280258
$$905$$ −8.82843 −0.293467
$$906$$ 4.97056 0.165136
$$907$$ −16.0000 −0.531271 −0.265636 0.964073i $$-0.585582\pi$$
−0.265636 + 0.964073i $$0.585582\pi$$
$$908$$ −43.5147 −1.44409
$$909$$ 2.92893 0.0971465
$$910$$ 0 0
$$911$$ −1.02944 −0.0341068 −0.0170534 0.999855i $$-0.505429\pi$$
−0.0170534 + 0.999855i $$0.505429\pi$$
$$912$$ −8.48528 −0.280976
$$913$$ −14.6274 −0.484097
$$914$$ −7.45584 −0.246617
$$915$$ 41.7990 1.38183
$$916$$ 0.443651 0.0146586
$$917$$ 0 0
$$918$$ −0.928932 −0.0306593
$$919$$ −8.28427 −0.273273 −0.136636 0.990621i $$-0.543629\pi$$
−0.136636 + 0.990621i $$0.543629\pi$$
$$920$$ 41.4558 1.36676
$$921$$ 30.1421 0.993217
$$922$$ −6.87006 −0.226253
$$923$$ 24.0833 0.792710
$$924$$ 0 0
$$925$$ −26.6274 −0.875504
$$926$$ −11.0294 −0.362450
$$927$$ 4.48528 0.147316
$$928$$ −30.1421 −0.989464
$$929$$ −39.2132 −1.28654 −0.643272 0.765638i $$-0.722423\pi$$
−0.643272 + 0.765638i $$0.722423\pi$$
$$930$$ −1.65685 −0.0543304
$$931$$ 0 0
$$932$$ 11.2304 0.367866
$$933$$ 6.14214 0.201084
$$934$$ 0.0832611 0.00272439
$$935$$ 15.3137 0.500812
$$936$$ −4.10051 −0.134029
$$937$$ 30.5858 0.999194 0.499597 0.866258i $$-0.333481\pi$$
0.499597 + 0.866258i $$0.333481\pi$$
$$938$$ 0 0
$$939$$ 1.89949 0.0619877
$$940$$ 17.6569 0.575903
$$941$$ 35.2132 1.14792 0.573959 0.818884i $$-0.305407\pi$$
0.573959 + 0.818884i $$0.305407\pi$$
$$942$$ 4.87006 0.158675
$$943$$ −47.7990 −1.55655
$$944$$ −3.51472 −0.114394
$$945$$ 0 0
$$946$$ −4.68629 −0.152364
$$947$$ 30.6863 0.997170 0.498585 0.866841i $$-0.333853\pi$$
0.498585 + 0.866841i $$0.333853\pi$$
$$948$$ −24.9706 −0.811006
$$949$$ 35.9411 1.16670
$$950$$ −7.79899 −0.253033
$$951$$ 10.0000 0.324272
$$952$$ 0 0
$$953$$ 2.00000 0.0647864 0.0323932 0.999475i $$-0.489687\pi$$
0.0323932 + 0.999475i $$0.489687\pi$$
$$954$$ −0.828427 −0.0268213
$$955$$ −61.4558 −1.98866
$$956$$ 28.6274 0.925877
$$957$$ 13.6569 0.441463
$$958$$ 0.769553 0.0248631
$$959$$ 0 0
$$960$$ −14.2426 −0.459679
$$961$$ −29.6274 −0.955723
$$962$$ −4.28427 −0.138130
$$963$$ −0.343146 −0.0110577
$$964$$ 29.6985 0.956524
$$965$$ 18.1421 0.584016
$$966$$ 0 0
$$967$$ 33.6569 1.08233 0.541166 0.840916i $$-0.317983\pi$$
0.541166 + 0.840916i $$0.317983\pi$$
$$968$$ 11.1005 0.356784
$$969$$ 6.34315 0.203771
$$970$$ 3.65685 0.117415
$$971$$ −50.6274 −1.62471 −0.812356 0.583162i $$-0.801815\pi$$
−0.812356 + 0.583162i $$0.801815\pi$$
$$972$$ −1.82843 −0.0586468
$$973$$ 0 0
$$974$$ 11.0294 0.353406
$$975$$ 17.2132 0.551264
$$976$$ 36.7279 1.17563
$$977$$ −21.1716 −0.677339 −0.338669 0.940905i $$-0.609977\pi$$
−0.338669 + 0.940905i $$0.609977\pi$$
$$978$$ −4.68629 −0.149851
$$979$$ 28.4853 0.910394
$$980$$ 0 0
$$981$$ −5.65685 −0.180609
$$982$$ 2.08326 0.0664795
$$983$$ −53.2548 −1.69857 −0.849283 0.527938i $$-0.822965\pi$$
−0.849283 + 0.527938i $$0.822965\pi$$
$$984$$ −9.89949 −0.315584
$$985$$ 6.82843 0.217572
$$986$$ 6.34315 0.202007
$$987$$ 0 0
$$988$$ 13.3726 0.425439
$$989$$ −43.3137 −1.37730
$$990$$ −2.82843 −0.0898933
$$991$$ −12.9706 −0.412024 −0.206012 0.978550i $$-0.566049\pi$$
−0.206012 + 0.978550i $$0.566049\pi$$
$$992$$ −5.17157 −0.164198
$$993$$ −4.00000 −0.126936
$$994$$ 0 0
$$995$$ 73.9411 2.34409
$$996$$ −13.3726 −0.423727
$$997$$ 26.3848 0.835614 0.417807 0.908536i $$-0.362799\pi$$
0.417807 + 0.908536i $$0.362799\pi$$
$$998$$ 1.37258 0.0434484
$$999$$ −4.00000 −0.126554
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 147.2.a.e.1.2 yes 2
3.2 odd 2 441.2.a.i.1.1 2
4.3 odd 2 2352.2.a.bc.1.2 2
5.4 even 2 3675.2.a.bd.1.1 2
7.2 even 3 147.2.e.d.67.1 4
7.3 odd 6 147.2.e.e.79.1 4
7.4 even 3 147.2.e.d.79.1 4
7.5 odd 6 147.2.e.e.67.1 4
7.6 odd 2 147.2.a.d.1.2 2
8.3 odd 2 9408.2.a.dt.1.1 2
8.5 even 2 9408.2.a.di.1.1 2
12.11 even 2 7056.2.a.cf.1.1 2
21.2 odd 6 441.2.e.g.361.2 4
21.5 even 6 441.2.e.f.361.2 4
21.11 odd 6 441.2.e.g.226.2 4
21.17 even 6 441.2.e.f.226.2 4
21.20 even 2 441.2.a.j.1.1 2
28.3 even 6 2352.2.q.bb.961.2 4
28.11 odd 6 2352.2.q.bd.961.1 4
28.19 even 6 2352.2.q.bb.1537.2 4
28.23 odd 6 2352.2.q.bd.1537.1 4
28.27 even 2 2352.2.a.be.1.1 2
35.34 odd 2 3675.2.a.bf.1.1 2
56.13 odd 2 9408.2.a.ef.1.2 2
56.27 even 2 9408.2.a.dq.1.2 2
84.83 odd 2 7056.2.a.cv.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.a.d.1.2 2 7.6 odd 2
147.2.a.e.1.2 yes 2 1.1 even 1 trivial
147.2.e.d.67.1 4 7.2 even 3
147.2.e.d.79.1 4 7.4 even 3
147.2.e.e.67.1 4 7.5 odd 6
147.2.e.e.79.1 4 7.3 odd 6
441.2.a.i.1.1 2 3.2 odd 2
441.2.a.j.1.1 2 21.20 even 2
441.2.e.f.226.2 4 21.17 even 6
441.2.e.f.361.2 4 21.5 even 6
441.2.e.g.226.2 4 21.11 odd 6
441.2.e.g.361.2 4 21.2 odd 6
2352.2.a.bc.1.2 2 4.3 odd 2
2352.2.a.be.1.1 2 28.27 even 2
2352.2.q.bb.961.2 4 28.3 even 6
2352.2.q.bb.1537.2 4 28.19 even 6
2352.2.q.bd.961.1 4 28.11 odd 6
2352.2.q.bd.1537.1 4 28.23 odd 6
3675.2.a.bd.1.1 2 5.4 even 2
3675.2.a.bf.1.1 2 35.34 odd 2
7056.2.a.cf.1.1 2 12.11 even 2
7056.2.a.cv.1.2 2 84.83 odd 2
9408.2.a.di.1.1 2 8.5 even 2
9408.2.a.dq.1.2 2 56.27 even 2
9408.2.a.dt.1.1 2 8.3 odd 2
9408.2.a.ef.1.2 2 56.13 odd 2