Properties

 Label 735.2.a.g.1.1 Level $735$ Weight $2$ Character 735.1 Self dual yes Analytic conductor $5.869$ 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] = [735,2,Mod(1,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("735.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 735.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: $$5.86900454856$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 735.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.73205 q^{2} -1.00000 q^{3} +5.46410 q^{4} +1.00000 q^{5} +2.73205 q^{6} -9.46410 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-2.73205 q^{2} -1.00000 q^{3} +5.46410 q^{4} +1.00000 q^{5} +2.73205 q^{6} -9.46410 q^{8} +1.00000 q^{9} -2.73205 q^{10} +0.732051 q^{11} -5.46410 q^{12} +2.26795 q^{13} -1.00000 q^{15} +14.9282 q^{16} +3.26795 q^{17} -2.73205 q^{18} +4.46410 q^{19} +5.46410 q^{20} -2.00000 q^{22} -4.73205 q^{23} +9.46410 q^{24} +1.00000 q^{25} -6.19615 q^{26} -1.00000 q^{27} -4.19615 q^{29} +2.73205 q^{30} -0.464102 q^{31} -21.8564 q^{32} -0.732051 q^{33} -8.92820 q^{34} +5.46410 q^{36} -3.19615 q^{37} -12.1962 q^{38} -2.26795 q^{39} -9.46410 q^{40} -0.732051 q^{41} +3.19615 q^{43} +4.00000 q^{44} +1.00000 q^{45} +12.9282 q^{46} +2.00000 q^{47} -14.9282 q^{48} -2.73205 q^{50} -3.26795 q^{51} +12.3923 q^{52} +12.3923 q^{53} +2.73205 q^{54} +0.732051 q^{55} -4.46410 q^{57} +11.4641 q^{58} -0.196152 q^{59} -5.46410 q^{60} +4.00000 q^{61} +1.26795 q^{62} +29.8564 q^{64} +2.26795 q^{65} +2.00000 q^{66} -14.6603 q^{67} +17.8564 q^{68} +4.73205 q^{69} +6.19615 q^{71} -9.46410 q^{72} +12.6603 q^{73} +8.73205 q^{74} -1.00000 q^{75} +24.3923 q^{76} +6.19615 q^{78} -7.39230 q^{79} +14.9282 q^{80} +1.00000 q^{81} +2.00000 q^{82} +15.1244 q^{83} +3.26795 q^{85} -8.73205 q^{86} +4.19615 q^{87} -6.92820 q^{88} +15.1244 q^{89} -2.73205 q^{90} -25.8564 q^{92} +0.464102 q^{93} -5.46410 q^{94} +4.46410 q^{95} +21.8564 q^{96} +14.9282 q^{97} +0.732051 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 12 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 + 2 * q^5 + 2 * q^6 - 12 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 12 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} + 8 q^{13} - 2 q^{15} + 16 q^{16} + 10 q^{17} - 2 q^{18} + 2 q^{19} + 4 q^{20} - 4 q^{22} - 6 q^{23} + 12 q^{24} + 2 q^{25} - 2 q^{26} - 2 q^{27} + 2 q^{29} + 2 q^{30} + 6 q^{31} - 16 q^{32} + 2 q^{33} - 4 q^{34} + 4 q^{36} + 4 q^{37} - 14 q^{38} - 8 q^{39} - 12 q^{40} + 2 q^{41} - 4 q^{43} + 8 q^{44} + 2 q^{45} + 12 q^{46} + 4 q^{47} - 16 q^{48} - 2 q^{50} - 10 q^{51} + 4 q^{52} + 4 q^{53} + 2 q^{54} - 2 q^{55} - 2 q^{57} + 16 q^{58} + 10 q^{59} - 4 q^{60} + 8 q^{61} + 6 q^{62} + 32 q^{64} + 8 q^{65} + 4 q^{66} - 12 q^{67} + 8 q^{68} + 6 q^{69} + 2 q^{71} - 12 q^{72} + 8 q^{73} + 14 q^{74} - 2 q^{75} + 28 q^{76} + 2 q^{78} + 6 q^{79} + 16 q^{80} + 2 q^{81} + 4 q^{82} + 6 q^{83} + 10 q^{85} - 14 q^{86} - 2 q^{87} + 6 q^{89} - 2 q^{90} - 24 q^{92} - 6 q^{93} - 4 q^{94} + 2 q^{95} + 16 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 + 2 * q^5 + 2 * q^6 - 12 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 - 4 * q^12 + 8 * q^13 - 2 * q^15 + 16 * q^16 + 10 * q^17 - 2 * q^18 + 2 * q^19 + 4 * q^20 - 4 * q^22 - 6 * q^23 + 12 * q^24 + 2 * q^25 - 2 * q^26 - 2 * q^27 + 2 * q^29 + 2 * q^30 + 6 * q^31 - 16 * q^32 + 2 * q^33 - 4 * q^34 + 4 * q^36 + 4 * q^37 - 14 * q^38 - 8 * q^39 - 12 * q^40 + 2 * q^41 - 4 * q^43 + 8 * q^44 + 2 * q^45 + 12 * q^46 + 4 * q^47 - 16 * q^48 - 2 * q^50 - 10 * q^51 + 4 * q^52 + 4 * q^53 + 2 * q^54 - 2 * q^55 - 2 * q^57 + 16 * q^58 + 10 * q^59 - 4 * q^60 + 8 * q^61 + 6 * q^62 + 32 * q^64 + 8 * q^65 + 4 * q^66 - 12 * q^67 + 8 * q^68 + 6 * q^69 + 2 * q^71 - 12 * q^72 + 8 * q^73 + 14 * q^74 - 2 * q^75 + 28 * q^76 + 2 * q^78 + 6 * q^79 + 16 * q^80 + 2 * q^81 + 4 * q^82 + 6 * q^83 + 10 * q^85 - 14 * q^86 - 2 * q^87 + 6 * q^89 - 2 * q^90 - 24 * q^92 - 6 * q^93 - 4 * q^94 + 2 * q^95 + 16 * q^96 + 16 * q^97 - 2 * q^99

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.73205 −1.93185 −0.965926 0.258819i $$-0.916667\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 5.46410 2.73205
$$5$$ 1.00000 0.447214
$$6$$ 2.73205 1.11536
$$7$$ 0 0
$$8$$ −9.46410 −3.34607
$$9$$ 1.00000 0.333333
$$10$$ −2.73205 −0.863950
$$11$$ 0.732051 0.220722 0.110361 0.993892i $$-0.464799\pi$$
0.110361 + 0.993892i $$0.464799\pi$$
$$12$$ −5.46410 −1.57735
$$13$$ 2.26795 0.629016 0.314508 0.949255i $$-0.398160\pi$$
0.314508 + 0.949255i $$0.398160\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 14.9282 3.73205
$$17$$ 3.26795 0.792594 0.396297 0.918122i $$-0.370295\pi$$
0.396297 + 0.918122i $$0.370295\pi$$
$$18$$ −2.73205 −0.643951
$$19$$ 4.46410 1.02414 0.512068 0.858945i $$-0.328880\pi$$
0.512068 + 0.858945i $$0.328880\pi$$
$$20$$ 5.46410 1.22181
$$21$$ 0 0
$$22$$ −2.00000 −0.426401
$$23$$ −4.73205 −0.986701 −0.493350 0.869831i $$-0.664228\pi$$
−0.493350 + 0.869831i $$0.664228\pi$$
$$24$$ 9.46410 1.93185
$$25$$ 1.00000 0.200000
$$26$$ −6.19615 −1.21517
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −4.19615 −0.779206 −0.389603 0.920983i $$-0.627388\pi$$
−0.389603 + 0.920983i $$0.627388\pi$$
$$30$$ 2.73205 0.498802
$$31$$ −0.464102 −0.0833551 −0.0416776 0.999131i $$-0.513270\pi$$
−0.0416776 + 0.999131i $$0.513270\pi$$
$$32$$ −21.8564 −3.86370
$$33$$ −0.732051 −0.127434
$$34$$ −8.92820 −1.53117
$$35$$ 0 0
$$36$$ 5.46410 0.910684
$$37$$ −3.19615 −0.525444 −0.262722 0.964872i $$-0.584620\pi$$
−0.262722 + 0.964872i $$0.584620\pi$$
$$38$$ −12.1962 −1.97848
$$39$$ −2.26795 −0.363163
$$40$$ −9.46410 −1.49641
$$41$$ −0.732051 −0.114327 −0.0571636 0.998365i $$-0.518206\pi$$
−0.0571636 + 0.998365i $$0.518206\pi$$
$$42$$ 0 0
$$43$$ 3.19615 0.487409 0.243704 0.969850i $$-0.421637\pi$$
0.243704 + 0.969850i $$0.421637\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 1.00000 0.149071
$$46$$ 12.9282 1.90616
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ −14.9282 −2.15470
$$49$$ 0 0
$$50$$ −2.73205 −0.386370
$$51$$ −3.26795 −0.457604
$$52$$ 12.3923 1.71850
$$53$$ 12.3923 1.70221 0.851107 0.524992i $$-0.175932\pi$$
0.851107 + 0.524992i $$0.175932\pi$$
$$54$$ 2.73205 0.371785
$$55$$ 0.732051 0.0987097
$$56$$ 0 0
$$57$$ −4.46410 −0.591285
$$58$$ 11.4641 1.50531
$$59$$ −0.196152 −0.0255369 −0.0127684 0.999918i $$-0.504064\pi$$
−0.0127684 + 0.999918i $$0.504064\pi$$
$$60$$ −5.46410 −0.705412
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 1.26795 0.161030
$$63$$ 0 0
$$64$$ 29.8564 3.73205
$$65$$ 2.26795 0.281304
$$66$$ 2.00000 0.246183
$$67$$ −14.6603 −1.79104 −0.895518 0.445026i $$-0.853194\pi$$
−0.895518 + 0.445026i $$0.853194\pi$$
$$68$$ 17.8564 2.16541
$$69$$ 4.73205 0.569672
$$70$$ 0 0
$$71$$ 6.19615 0.735348 0.367674 0.929955i $$-0.380154\pi$$
0.367674 + 0.929955i $$0.380154\pi$$
$$72$$ −9.46410 −1.11536
$$73$$ 12.6603 1.48177 0.740885 0.671632i $$-0.234406\pi$$
0.740885 + 0.671632i $$0.234406\pi$$
$$74$$ 8.73205 1.01508
$$75$$ −1.00000 −0.115470
$$76$$ 24.3923 2.79799
$$77$$ 0 0
$$78$$ 6.19615 0.701576
$$79$$ −7.39230 −0.831699 −0.415850 0.909433i $$-0.636516\pi$$
−0.415850 + 0.909433i $$0.636516\pi$$
$$80$$ 14.9282 1.66902
$$81$$ 1.00000 0.111111
$$82$$ 2.00000 0.220863
$$83$$ 15.1244 1.66011 0.830057 0.557679i $$-0.188308\pi$$
0.830057 + 0.557679i $$0.188308\pi$$
$$84$$ 0 0
$$85$$ 3.26795 0.354459
$$86$$ −8.73205 −0.941601
$$87$$ 4.19615 0.449875
$$88$$ −6.92820 −0.738549
$$89$$ 15.1244 1.60318 0.801589 0.597875i $$-0.203988\pi$$
0.801589 + 0.597875i $$0.203988\pi$$
$$90$$ −2.73205 −0.287983
$$91$$ 0 0
$$92$$ −25.8564 −2.69572
$$93$$ 0.464102 0.0481251
$$94$$ −5.46410 −0.563579
$$95$$ 4.46410 0.458007
$$96$$ 21.8564 2.23071
$$97$$ 14.9282 1.51573 0.757865 0.652412i $$-0.226243\pi$$
0.757865 + 0.652412i $$0.226243\pi$$
$$98$$ 0 0
$$99$$ 0.732051 0.0735739
$$100$$ 5.46410 0.546410
$$101$$ −7.26795 −0.723188 −0.361594 0.932336i $$-0.617767\pi$$
−0.361594 + 0.932336i $$0.617767\pi$$
$$102$$ 8.92820 0.884024
$$103$$ −9.19615 −0.906124 −0.453062 0.891479i $$-0.649668\pi$$
−0.453062 + 0.891479i $$0.649668\pi$$
$$104$$ −21.4641 −2.10473
$$105$$ 0 0
$$106$$ −33.8564 −3.28842
$$107$$ 2.19615 0.212310 0.106155 0.994350i $$-0.466146\pi$$
0.106155 + 0.994350i $$0.466146\pi$$
$$108$$ −5.46410 −0.525783
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ 3.19615 0.303365
$$112$$ 0 0
$$113$$ 8.92820 0.839895 0.419947 0.907548i $$-0.362049\pi$$
0.419947 + 0.907548i $$0.362049\pi$$
$$114$$ 12.1962 1.14227
$$115$$ −4.73205 −0.441266
$$116$$ −22.9282 −2.12883
$$117$$ 2.26795 0.209672
$$118$$ 0.535898 0.0493334
$$119$$ 0 0
$$120$$ 9.46410 0.863950
$$121$$ −10.4641 −0.951282
$$122$$ −10.9282 −0.989393
$$123$$ 0.732051 0.0660068
$$124$$ −2.53590 −0.227730
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 4.80385 0.426273 0.213136 0.977022i $$-0.431632\pi$$
0.213136 + 0.977022i $$0.431632\pi$$
$$128$$ −37.8564 −3.34607
$$129$$ −3.19615 −0.281406
$$130$$ −6.19615 −0.543439
$$131$$ 15.4641 1.35110 0.675552 0.737312i $$-0.263905\pi$$
0.675552 + 0.737312i $$0.263905\pi$$
$$132$$ −4.00000 −0.348155
$$133$$ 0 0
$$134$$ 40.0526 3.46001
$$135$$ −1.00000 −0.0860663
$$136$$ −30.9282 −2.65207
$$137$$ 2.19615 0.187630 0.0938150 0.995590i $$-0.470094\pi$$
0.0938150 + 0.995590i $$0.470094\pi$$
$$138$$ −12.9282 −1.10052
$$139$$ 5.92820 0.502824 0.251412 0.967880i $$-0.419105\pi$$
0.251412 + 0.967880i $$0.419105\pi$$
$$140$$ 0 0
$$141$$ −2.00000 −0.168430
$$142$$ −16.9282 −1.42058
$$143$$ 1.66025 0.138837
$$144$$ 14.9282 1.24402
$$145$$ −4.19615 −0.348471
$$146$$ −34.5885 −2.86256
$$147$$ 0 0
$$148$$ −17.4641 −1.43554
$$149$$ 5.85641 0.479776 0.239888 0.970801i $$-0.422889\pi$$
0.239888 + 0.970801i $$0.422889\pi$$
$$150$$ 2.73205 0.223071
$$151$$ −8.92820 −0.726567 −0.363283 0.931679i $$-0.618344\pi$$
−0.363283 + 0.931679i $$0.618344\pi$$
$$152$$ −42.2487 −3.42682
$$153$$ 3.26795 0.264198
$$154$$ 0 0
$$155$$ −0.464102 −0.0372775
$$156$$ −12.3923 −0.992178
$$157$$ 6.39230 0.510161 0.255081 0.966920i $$-0.417898\pi$$
0.255081 + 0.966920i $$0.417898\pi$$
$$158$$ 20.1962 1.60672
$$159$$ −12.3923 −0.982774
$$160$$ −21.8564 −1.72790
$$161$$ 0 0
$$162$$ −2.73205 −0.214650
$$163$$ 21.8564 1.71193 0.855963 0.517037i $$-0.172965\pi$$
0.855963 + 0.517037i $$0.172965\pi$$
$$164$$ −4.00000 −0.312348
$$165$$ −0.732051 −0.0569901
$$166$$ −41.3205 −3.20709
$$167$$ −17.6603 −1.36659 −0.683296 0.730142i $$-0.739454\pi$$
−0.683296 + 0.730142i $$0.739454\pi$$
$$168$$ 0 0
$$169$$ −7.85641 −0.604339
$$170$$ −8.92820 −0.684762
$$171$$ 4.46410 0.341378
$$172$$ 17.4641 1.33163
$$173$$ 14.5359 1.10514 0.552572 0.833465i $$-0.313646\pi$$
0.552572 + 0.833465i $$0.313646\pi$$
$$174$$ −11.4641 −0.869091
$$175$$ 0 0
$$176$$ 10.9282 0.823744
$$177$$ 0.196152 0.0147437
$$178$$ −41.3205 −3.09710
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 5.46410 0.407270
$$181$$ −24.3205 −1.80773 −0.903865 0.427819i $$-0.859282\pi$$
−0.903865 + 0.427819i $$0.859282\pi$$
$$182$$ 0 0
$$183$$ −4.00000 −0.295689
$$184$$ 44.7846 3.30157
$$185$$ −3.19615 −0.234986
$$186$$ −1.26795 −0.0929705
$$187$$ 2.39230 0.174943
$$188$$ 10.9282 0.797021
$$189$$ 0 0
$$190$$ −12.1962 −0.884802
$$191$$ −8.92820 −0.646022 −0.323011 0.946395i $$-0.604695\pi$$
−0.323011 + 0.946395i $$0.604695\pi$$
$$192$$ −29.8564 −2.15470
$$193$$ 1.19615 0.0861009 0.0430505 0.999073i $$-0.486292\pi$$
0.0430505 + 0.999073i $$0.486292\pi$$
$$194$$ −40.7846 −2.92816
$$195$$ −2.26795 −0.162411
$$196$$ 0 0
$$197$$ −0.339746 −0.0242059 −0.0121029 0.999927i $$-0.503853\pi$$
−0.0121029 + 0.999927i $$0.503853\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ 22.0000 1.55954 0.779769 0.626067i $$-0.215336\pi$$
0.779769 + 0.626067i $$0.215336\pi$$
$$200$$ −9.46410 −0.669213
$$201$$ 14.6603 1.03405
$$202$$ 19.8564 1.39709
$$203$$ 0 0
$$204$$ −17.8564 −1.25020
$$205$$ −0.732051 −0.0511286
$$206$$ 25.1244 1.75050
$$207$$ −4.73205 −0.328900
$$208$$ 33.8564 2.34752
$$209$$ 3.26795 0.226049
$$210$$ 0 0
$$211$$ 7.07180 0.486843 0.243421 0.969921i $$-0.421730\pi$$
0.243421 + 0.969921i $$0.421730\pi$$
$$212$$ 67.7128 4.65054
$$213$$ −6.19615 −0.424553
$$214$$ −6.00000 −0.410152
$$215$$ 3.19615 0.217976
$$216$$ 9.46410 0.643951
$$217$$ 0 0
$$218$$ −30.0526 −2.03542
$$219$$ −12.6603 −0.855501
$$220$$ 4.00000 0.269680
$$221$$ 7.41154 0.498554
$$222$$ −8.73205 −0.586057
$$223$$ −20.3923 −1.36557 −0.682785 0.730619i $$-0.739231\pi$$
−0.682785 + 0.730619i $$0.739231\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ −24.3923 −1.62255
$$227$$ −1.66025 −0.110195 −0.0550975 0.998481i $$-0.517547\pi$$
−0.0550975 + 0.998481i $$0.517547\pi$$
$$228$$ −24.3923 −1.61542
$$229$$ −3.00000 −0.198246 −0.0991228 0.995075i $$-0.531604\pi$$
−0.0991228 + 0.995075i $$0.531604\pi$$
$$230$$ 12.9282 0.852460
$$231$$ 0 0
$$232$$ 39.7128 2.60727
$$233$$ 17.3205 1.13470 0.567352 0.823475i $$-0.307968\pi$$
0.567352 + 0.823475i $$0.307968\pi$$
$$234$$ −6.19615 −0.405055
$$235$$ 2.00000 0.130466
$$236$$ −1.07180 −0.0697680
$$237$$ 7.39230 0.480182
$$238$$ 0 0
$$239$$ 7.07180 0.457437 0.228718 0.973493i $$-0.426547\pi$$
0.228718 + 0.973493i $$0.426547\pi$$
$$240$$ −14.9282 −0.963611
$$241$$ 13.4641 0.867299 0.433650 0.901082i $$-0.357226\pi$$
0.433650 + 0.901082i $$0.357226\pi$$
$$242$$ 28.5885 1.83774
$$243$$ −1.00000 −0.0641500
$$244$$ 21.8564 1.39921
$$245$$ 0 0
$$246$$ −2.00000 −0.127515
$$247$$ 10.1244 0.644197
$$248$$ 4.39230 0.278912
$$249$$ −15.1244 −0.958467
$$250$$ −2.73205 −0.172790
$$251$$ −24.5885 −1.55201 −0.776005 0.630727i $$-0.782757\pi$$
−0.776005 + 0.630727i $$0.782757\pi$$
$$252$$ 0 0
$$253$$ −3.46410 −0.217786
$$254$$ −13.1244 −0.823495
$$255$$ −3.26795 −0.204647
$$256$$ 43.7128 2.73205
$$257$$ −5.66025 −0.353077 −0.176538 0.984294i $$-0.556490\pi$$
−0.176538 + 0.984294i $$0.556490\pi$$
$$258$$ 8.73205 0.543634
$$259$$ 0 0
$$260$$ 12.3923 0.768538
$$261$$ −4.19615 −0.259735
$$262$$ −42.2487 −2.61013
$$263$$ 8.39230 0.517492 0.258746 0.965945i $$-0.416691\pi$$
0.258746 + 0.965945i $$0.416691\pi$$
$$264$$ 6.92820 0.426401
$$265$$ 12.3923 0.761253
$$266$$ 0 0
$$267$$ −15.1244 −0.925596
$$268$$ −80.1051 −4.89320
$$269$$ −12.5359 −0.764327 −0.382164 0.924095i $$-0.624821\pi$$
−0.382164 + 0.924095i $$0.624821\pi$$
$$270$$ 2.73205 0.166267
$$271$$ 3.07180 0.186598 0.0932992 0.995638i $$-0.470259\pi$$
0.0932992 + 0.995638i $$0.470259\pi$$
$$272$$ 48.7846 2.95800
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0.732051 0.0441443
$$276$$ 25.8564 1.55637
$$277$$ −14.6603 −0.880849 −0.440425 0.897790i $$-0.645172\pi$$
−0.440425 + 0.897790i $$0.645172\pi$$
$$278$$ −16.1962 −0.971381
$$279$$ −0.464102 −0.0277850
$$280$$ 0 0
$$281$$ 13.8564 0.826604 0.413302 0.910594i $$-0.364375\pi$$
0.413302 + 0.910594i $$0.364375\pi$$
$$282$$ 5.46410 0.325383
$$283$$ −24.1244 −1.43404 −0.717022 0.697050i $$-0.754495\pi$$
−0.717022 + 0.697050i $$0.754495\pi$$
$$284$$ 33.8564 2.00901
$$285$$ −4.46410 −0.264431
$$286$$ −4.53590 −0.268213
$$287$$ 0 0
$$288$$ −21.8564 −1.28790
$$289$$ −6.32051 −0.371795
$$290$$ 11.4641 0.673195
$$291$$ −14.9282 −0.875107
$$292$$ 69.1769 4.04827
$$293$$ 18.9282 1.10580 0.552899 0.833248i $$-0.313522\pi$$
0.552899 + 0.833248i $$0.313522\pi$$
$$294$$ 0 0
$$295$$ −0.196152 −0.0114204
$$296$$ 30.2487 1.75817
$$297$$ −0.732051 −0.0424779
$$298$$ −16.0000 −0.926855
$$299$$ −10.7321 −0.620651
$$300$$ −5.46410 −0.315470
$$301$$ 0 0
$$302$$ 24.3923 1.40362
$$303$$ 7.26795 0.417533
$$304$$ 66.6410 3.82212
$$305$$ 4.00000 0.229039
$$306$$ −8.92820 −0.510391
$$307$$ −32.1244 −1.83343 −0.916717 0.399537i $$-0.869171\pi$$
−0.916717 + 0.399537i $$0.869171\pi$$
$$308$$ 0 0
$$309$$ 9.19615 0.523151
$$310$$ 1.26795 0.0720147
$$311$$ 9.12436 0.517395 0.258697 0.965958i $$-0.416707\pi$$
0.258697 + 0.965958i $$0.416707\pi$$
$$312$$ 21.4641 1.21517
$$313$$ −12.6603 −0.715600 −0.357800 0.933798i $$-0.616473\pi$$
−0.357800 + 0.933798i $$0.616473\pi$$
$$314$$ −17.4641 −0.985556
$$315$$ 0 0
$$316$$ −40.3923 −2.27224
$$317$$ −28.4449 −1.59762 −0.798811 0.601582i $$-0.794537\pi$$
−0.798811 + 0.601582i $$0.794537\pi$$
$$318$$ 33.8564 1.89857
$$319$$ −3.07180 −0.171988
$$320$$ 29.8564 1.66902
$$321$$ −2.19615 −0.122577
$$322$$ 0 0
$$323$$ 14.5885 0.811723
$$324$$ 5.46410 0.303561
$$325$$ 2.26795 0.125803
$$326$$ −59.7128 −3.30719
$$327$$ −11.0000 −0.608301
$$328$$ 6.92820 0.382546
$$329$$ 0 0
$$330$$ 2.00000 0.110096
$$331$$ 8.07180 0.443666 0.221833 0.975085i $$-0.428796\pi$$
0.221833 + 0.975085i $$0.428796\pi$$
$$332$$ 82.6410 4.53551
$$333$$ −3.19615 −0.175148
$$334$$ 48.2487 2.64005
$$335$$ −14.6603 −0.800975
$$336$$ 0 0
$$337$$ 17.9808 0.979475 0.489737 0.871870i $$-0.337093\pi$$
0.489737 + 0.871870i $$0.337093\pi$$
$$338$$ 21.4641 1.16749
$$339$$ −8.92820 −0.484913
$$340$$ 17.8564 0.968400
$$341$$ −0.339746 −0.0183983
$$342$$ −12.1962 −0.659492
$$343$$ 0 0
$$344$$ −30.2487 −1.63090
$$345$$ 4.73205 0.254765
$$346$$ −39.7128 −2.13497
$$347$$ −21.0718 −1.13119 −0.565597 0.824682i $$-0.691354\pi$$
−0.565597 + 0.824682i $$0.691354\pi$$
$$348$$ 22.9282 1.22908
$$349$$ 22.0000 1.17763 0.588817 0.808267i $$-0.299594\pi$$
0.588817 + 0.808267i $$0.299594\pi$$
$$350$$ 0 0
$$351$$ −2.26795 −0.121054
$$352$$ −16.0000 −0.852803
$$353$$ −3.12436 −0.166293 −0.0831463 0.996537i $$-0.526497\pi$$
−0.0831463 + 0.996537i $$0.526497\pi$$
$$354$$ −0.535898 −0.0284827
$$355$$ 6.19615 0.328858
$$356$$ 82.6410 4.37997
$$357$$ 0 0
$$358$$ 27.3205 1.44393
$$359$$ −1.26795 −0.0669198 −0.0334599 0.999440i $$-0.510653\pi$$
−0.0334599 + 0.999440i $$0.510653\pi$$
$$360$$ −9.46410 −0.498802
$$361$$ 0.928203 0.0488528
$$362$$ 66.4449 3.49226
$$363$$ 10.4641 0.549223
$$364$$ 0 0
$$365$$ 12.6603 0.662668
$$366$$ 10.9282 0.571226
$$367$$ 11.1962 0.584434 0.292217 0.956352i $$-0.405607\pi$$
0.292217 + 0.956352i $$0.405607\pi$$
$$368$$ −70.6410 −3.68242
$$369$$ −0.732051 −0.0381090
$$370$$ 8.73205 0.453958
$$371$$ 0 0
$$372$$ 2.53590 0.131480
$$373$$ 26.5167 1.37298 0.686490 0.727139i $$-0.259150\pi$$
0.686490 + 0.727139i $$0.259150\pi$$
$$374$$ −6.53590 −0.337963
$$375$$ −1.00000 −0.0516398
$$376$$ −18.9282 −0.976148
$$377$$ −9.51666 −0.490133
$$378$$ 0 0
$$379$$ 6.32051 0.324663 0.162331 0.986736i $$-0.448099\pi$$
0.162331 + 0.986736i $$0.448099\pi$$
$$380$$ 24.3923 1.25130
$$381$$ −4.80385 −0.246109
$$382$$ 24.3923 1.24802
$$383$$ −23.3205 −1.19162 −0.595811 0.803125i $$-0.703169\pi$$
−0.595811 + 0.803125i $$0.703169\pi$$
$$384$$ 37.8564 1.93185
$$385$$ 0 0
$$386$$ −3.26795 −0.166334
$$387$$ 3.19615 0.162470
$$388$$ 81.5692 4.14105
$$389$$ 5.41154 0.274376 0.137188 0.990545i $$-0.456194\pi$$
0.137188 + 0.990545i $$0.456194\pi$$
$$390$$ 6.19615 0.313754
$$391$$ −15.4641 −0.782053
$$392$$ 0 0
$$393$$ −15.4641 −0.780061
$$394$$ 0.928203 0.0467622
$$395$$ −7.39230 −0.371947
$$396$$ 4.00000 0.201008
$$397$$ −31.1962 −1.56569 −0.782845 0.622217i $$-0.786232\pi$$
−0.782845 + 0.622217i $$0.786232\pi$$
$$398$$ −60.1051 −3.01280
$$399$$ 0 0
$$400$$ 14.9282 0.746410
$$401$$ −16.3923 −0.818593 −0.409296 0.912402i $$-0.634226\pi$$
−0.409296 + 0.912402i $$0.634226\pi$$
$$402$$ −40.0526 −1.99764
$$403$$ −1.05256 −0.0524317
$$404$$ −39.7128 −1.97579
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ −2.33975 −0.115977
$$408$$ 30.9282 1.53117
$$409$$ 3.14359 0.155441 0.0777203 0.996975i $$-0.475236\pi$$
0.0777203 + 0.996975i $$0.475236\pi$$
$$410$$ 2.00000 0.0987730
$$411$$ −2.19615 −0.108328
$$412$$ −50.2487 −2.47558
$$413$$ 0 0
$$414$$ 12.9282 0.635387
$$415$$ 15.1244 0.742425
$$416$$ −49.5692 −2.43033
$$417$$ −5.92820 −0.290305
$$418$$ −8.92820 −0.436693
$$419$$ −35.4641 −1.73253 −0.866267 0.499581i $$-0.833487\pi$$
−0.866267 + 0.499581i $$0.833487\pi$$
$$420$$ 0 0
$$421$$ 0.0717968 0.00349916 0.00174958 0.999998i $$-0.499443\pi$$
0.00174958 + 0.999998i $$0.499443\pi$$
$$422$$ −19.3205 −0.940508
$$423$$ 2.00000 0.0972433
$$424$$ −117.282 −5.69572
$$425$$ 3.26795 0.158519
$$426$$ 16.9282 0.820174
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ −1.66025 −0.0801578
$$430$$ −8.73205 −0.421097
$$431$$ 17.3205 0.834300 0.417150 0.908838i $$-0.363029\pi$$
0.417150 + 0.908838i $$0.363029\pi$$
$$432$$ −14.9282 −0.718234
$$433$$ 15.1962 0.730280 0.365140 0.930953i $$-0.381021\pi$$
0.365140 + 0.930953i $$0.381021\pi$$
$$434$$ 0 0
$$435$$ 4.19615 0.201190
$$436$$ 60.1051 2.87851
$$437$$ −21.1244 −1.01051
$$438$$ 34.5885 1.65270
$$439$$ 0.535898 0.0255770 0.0127885 0.999918i $$-0.495929\pi$$
0.0127885 + 0.999918i $$0.495929\pi$$
$$440$$ −6.92820 −0.330289
$$441$$ 0 0
$$442$$ −20.2487 −0.963133
$$443$$ 9.46410 0.449653 0.224827 0.974399i $$-0.427818\pi$$
0.224827 + 0.974399i $$0.427818\pi$$
$$444$$ 17.4641 0.828810
$$445$$ 15.1244 0.716963
$$446$$ 55.7128 2.63808
$$447$$ −5.85641 −0.276999
$$448$$ 0 0
$$449$$ −35.8564 −1.69217 −0.846084 0.533049i $$-0.821046\pi$$
−0.846084 + 0.533049i $$0.821046\pi$$
$$450$$ −2.73205 −0.128790
$$451$$ −0.535898 −0.0252345
$$452$$ 48.7846 2.29464
$$453$$ 8.92820 0.419484
$$454$$ 4.53590 0.212880
$$455$$ 0 0
$$456$$ 42.2487 1.97848
$$457$$ −16.6603 −0.779334 −0.389667 0.920956i $$-0.627410\pi$$
−0.389667 + 0.920956i $$0.627410\pi$$
$$458$$ 8.19615 0.382981
$$459$$ −3.26795 −0.152535
$$460$$ −25.8564 −1.20556
$$461$$ 16.9808 0.790873 0.395436 0.918493i $$-0.370593\pi$$
0.395436 + 0.918493i $$0.370593\pi$$
$$462$$ 0 0
$$463$$ 25.7321 1.19587 0.597935 0.801545i $$-0.295988\pi$$
0.597935 + 0.801545i $$0.295988\pi$$
$$464$$ −62.6410 −2.90804
$$465$$ 0.464102 0.0215222
$$466$$ −47.3205 −2.19208
$$467$$ −0.143594 −0.00664472 −0.00332236 0.999994i $$-0.501058\pi$$
−0.00332236 + 0.999994i $$0.501058\pi$$
$$468$$ 12.3923 0.572834
$$469$$ 0 0
$$470$$ −5.46410 −0.252040
$$471$$ −6.39230 −0.294542
$$472$$ 1.85641 0.0854480
$$473$$ 2.33975 0.107582
$$474$$ −20.1962 −0.927640
$$475$$ 4.46410 0.204827
$$476$$ 0 0
$$477$$ 12.3923 0.567405
$$478$$ −19.3205 −0.883699
$$479$$ 8.78461 0.401379 0.200690 0.979655i $$-0.435682\pi$$
0.200690 + 0.979655i $$0.435682\pi$$
$$480$$ 21.8564 0.997604
$$481$$ −7.24871 −0.330513
$$482$$ −36.7846 −1.67549
$$483$$ 0 0
$$484$$ −57.1769 −2.59895
$$485$$ 14.9282 0.677855
$$486$$ 2.73205 0.123928
$$487$$ −0.411543 −0.0186488 −0.00932439 0.999957i $$-0.502968\pi$$
−0.00932439 + 0.999957i $$0.502968\pi$$
$$488$$ −37.8564 −1.71368
$$489$$ −21.8564 −0.988381
$$490$$ 0 0
$$491$$ −38.2487 −1.72614 −0.863070 0.505084i $$-0.831461\pi$$
−0.863070 + 0.505084i $$0.831461\pi$$
$$492$$ 4.00000 0.180334
$$493$$ −13.7128 −0.617594
$$494$$ −27.6603 −1.24449
$$495$$ 0.732051 0.0329032
$$496$$ −6.92820 −0.311086
$$497$$ 0 0
$$498$$ 41.3205 1.85162
$$499$$ −13.5359 −0.605950 −0.302975 0.952998i $$-0.597980\pi$$
−0.302975 + 0.952998i $$0.597980\pi$$
$$500$$ 5.46410 0.244362
$$501$$ 17.6603 0.789002
$$502$$ 67.1769 2.99825
$$503$$ 14.3923 0.641721 0.320861 0.947126i $$-0.396028\pi$$
0.320861 + 0.947126i $$0.396028\pi$$
$$504$$ 0 0
$$505$$ −7.26795 −0.323419
$$506$$ 9.46410 0.420731
$$507$$ 7.85641 0.348915
$$508$$ 26.2487 1.16460
$$509$$ 4.53590 0.201050 0.100525 0.994935i $$-0.467948\pi$$
0.100525 + 0.994935i $$0.467948\pi$$
$$510$$ 8.92820 0.395347
$$511$$ 0 0
$$512$$ −43.7128 −1.93185
$$513$$ −4.46410 −0.197095
$$514$$ 15.4641 0.682092
$$515$$ −9.19615 −0.405231
$$516$$ −17.4641 −0.768814
$$517$$ 1.46410 0.0643911
$$518$$ 0 0
$$519$$ −14.5359 −0.638055
$$520$$ −21.4641 −0.941263
$$521$$ −5.46410 −0.239387 −0.119693 0.992811i $$-0.538191\pi$$
−0.119693 + 0.992811i $$0.538191\pi$$
$$522$$ 11.4641 0.501770
$$523$$ −27.7321 −1.21264 −0.606319 0.795222i $$-0.707355\pi$$
−0.606319 + 0.795222i $$0.707355\pi$$
$$524$$ 84.4974 3.69129
$$525$$ 0 0
$$526$$ −22.9282 −0.999717
$$527$$ −1.51666 −0.0660668
$$528$$ −10.9282 −0.475589
$$529$$ −0.607695 −0.0264215
$$530$$ −33.8564 −1.47063
$$531$$ −0.196152 −0.00851229
$$532$$ 0 0
$$533$$ −1.66025 −0.0719136
$$534$$ 41.3205 1.78811
$$535$$ 2.19615 0.0949479
$$536$$ 138.746 5.99292
$$537$$ 10.0000 0.431532
$$538$$ 34.2487 1.47657
$$539$$ 0 0
$$540$$ −5.46410 −0.235137
$$541$$ 5.78461 0.248700 0.124350 0.992238i $$-0.460315\pi$$
0.124350 + 0.992238i $$0.460315\pi$$
$$542$$ −8.39230 −0.360480
$$543$$ 24.3205 1.04369
$$544$$ −71.4256 −3.06235
$$545$$ 11.0000 0.471188
$$546$$ 0 0
$$547$$ −26.2487 −1.12231 −0.561157 0.827709i $$-0.689644\pi$$
−0.561157 + 0.827709i $$0.689644\pi$$
$$548$$ 12.0000 0.512615
$$549$$ 4.00000 0.170716
$$550$$ −2.00000 −0.0852803
$$551$$ −18.7321 −0.798012
$$552$$ −44.7846 −1.90616
$$553$$ 0 0
$$554$$ 40.0526 1.70167
$$555$$ 3.19615 0.135669
$$556$$ 32.3923 1.37374
$$557$$ 14.7846 0.626444 0.313222 0.949680i $$-0.398592\pi$$
0.313222 + 0.949680i $$0.398592\pi$$
$$558$$ 1.26795 0.0536766
$$559$$ 7.24871 0.306588
$$560$$ 0 0
$$561$$ −2.39230 −0.101003
$$562$$ −37.8564 −1.59688
$$563$$ −18.0000 −0.758610 −0.379305 0.925272i $$-0.623837\pi$$
−0.379305 + 0.925272i $$0.623837\pi$$
$$564$$ −10.9282 −0.460160
$$565$$ 8.92820 0.375612
$$566$$ 65.9090 2.77036
$$567$$ 0 0
$$568$$ −58.6410 −2.46052
$$569$$ 32.4449 1.36016 0.680080 0.733138i $$-0.261945\pi$$
0.680080 + 0.733138i $$0.261945\pi$$
$$570$$ 12.1962 0.510841
$$571$$ 18.6077 0.778708 0.389354 0.921088i $$-0.372698\pi$$
0.389354 + 0.921088i $$0.372698\pi$$
$$572$$ 9.07180 0.379311
$$573$$ 8.92820 0.372981
$$574$$ 0 0
$$575$$ −4.73205 −0.197340
$$576$$ 29.8564 1.24402
$$577$$ 28.6603 1.19314 0.596571 0.802560i $$-0.296529\pi$$
0.596571 + 0.802560i $$0.296529\pi$$
$$578$$ 17.2679 0.718252
$$579$$ −1.19615 −0.0497104
$$580$$ −22.9282 −0.952042
$$581$$ 0 0
$$582$$ 40.7846 1.69058
$$583$$ 9.07180 0.375715
$$584$$ −119.818 −4.95810
$$585$$ 2.26795 0.0937682
$$586$$ −51.7128 −2.13624
$$587$$ 40.7321 1.68119 0.840596 0.541663i $$-0.182205\pi$$
0.840596 + 0.541663i $$0.182205\pi$$
$$588$$ 0 0
$$589$$ −2.07180 −0.0853669
$$590$$ 0.535898 0.0220626
$$591$$ 0.339746 0.0139753
$$592$$ −47.7128 −1.96098
$$593$$ 27.9090 1.14608 0.573042 0.819526i $$-0.305763\pi$$
0.573042 + 0.819526i $$0.305763\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ 32.0000 1.31077
$$597$$ −22.0000 −0.900400
$$598$$ 29.3205 1.19900
$$599$$ −38.2487 −1.56280 −0.781400 0.624030i $$-0.785494\pi$$
−0.781400 + 0.624030i $$0.785494\pi$$
$$600$$ 9.46410 0.386370
$$601$$ −0.0717968 −0.00292865 −0.00146433 0.999999i $$-0.500466\pi$$
−0.00146433 + 0.999999i $$0.500466\pi$$
$$602$$ 0 0
$$603$$ −14.6603 −0.597012
$$604$$ −48.7846 −1.98502
$$605$$ −10.4641 −0.425426
$$606$$ −19.8564 −0.806611
$$607$$ 3.19615 0.129728 0.0648639 0.997894i $$-0.479339\pi$$
0.0648639 + 0.997894i $$0.479339\pi$$
$$608$$ −97.5692 −3.95695
$$609$$ 0 0
$$610$$ −10.9282 −0.442470
$$611$$ 4.53590 0.183503
$$612$$ 17.8564 0.721802
$$613$$ −26.9282 −1.08762 −0.543810 0.839208i $$-0.683019\pi$$
−0.543810 + 0.839208i $$0.683019\pi$$
$$614$$ 87.7654 3.54192
$$615$$ 0.732051 0.0295191
$$616$$ 0 0
$$617$$ −36.2487 −1.45932 −0.729659 0.683811i $$-0.760321\pi$$
−0.729659 + 0.683811i $$0.760321\pi$$
$$618$$ −25.1244 −1.01065
$$619$$ −30.0718 −1.20869 −0.604344 0.796724i $$-0.706565\pi$$
−0.604344 + 0.796724i $$0.706565\pi$$
$$620$$ −2.53590 −0.101844
$$621$$ 4.73205 0.189891
$$622$$ −24.9282 −0.999530
$$623$$ 0 0
$$624$$ −33.8564 −1.35534
$$625$$ 1.00000 0.0400000
$$626$$ 34.5885 1.38243
$$627$$ −3.26795 −0.130509
$$628$$ 34.9282 1.39379
$$629$$ −10.4449 −0.416464
$$630$$ 0 0
$$631$$ 48.7846 1.94208 0.971042 0.238908i $$-0.0767893\pi$$
0.971042 + 0.238908i $$0.0767893\pi$$
$$632$$ 69.9615 2.78292
$$633$$ −7.07180 −0.281079
$$634$$ 77.7128 3.08637
$$635$$ 4.80385 0.190635
$$636$$ −67.7128 −2.68499
$$637$$ 0 0
$$638$$ 8.39230 0.332255
$$639$$ 6.19615 0.245116
$$640$$ −37.8564 −1.49641
$$641$$ 3.80385 0.150243 0.0751215 0.997174i $$-0.476066\pi$$
0.0751215 + 0.997174i $$0.476066\pi$$
$$642$$ 6.00000 0.236801
$$643$$ 4.51666 0.178120 0.0890599 0.996026i $$-0.471614\pi$$
0.0890599 + 0.996026i $$0.471614\pi$$
$$644$$ 0 0
$$645$$ −3.19615 −0.125848
$$646$$ −39.8564 −1.56813
$$647$$ −27.9090 −1.09721 −0.548607 0.836080i $$-0.684842\pi$$
−0.548607 + 0.836080i $$0.684842\pi$$
$$648$$ −9.46410 −0.371785
$$649$$ −0.143594 −0.00563654
$$650$$ −6.19615 −0.243033
$$651$$ 0 0
$$652$$ 119.426 4.67707
$$653$$ −44.5885 −1.74488 −0.872441 0.488720i $$-0.837464\pi$$
−0.872441 + 0.488720i $$0.837464\pi$$
$$654$$ 30.0526 1.17515
$$655$$ 15.4641 0.604232
$$656$$ −10.9282 −0.426675
$$657$$ 12.6603 0.493924
$$658$$ 0 0
$$659$$ 2.92820 0.114067 0.0570333 0.998372i $$-0.481836\pi$$
0.0570333 + 0.998372i $$0.481836\pi$$
$$660$$ −4.00000 −0.155700
$$661$$ 10.4641 0.407006 0.203503 0.979074i $$-0.434767\pi$$
0.203503 + 0.979074i $$0.434767\pi$$
$$662$$ −22.0526 −0.857097
$$663$$ −7.41154 −0.287840
$$664$$ −143.138 −5.55485
$$665$$ 0 0
$$666$$ 8.73205 0.338360
$$667$$ 19.8564 0.768843
$$668$$ −96.4974 −3.73360
$$669$$ 20.3923 0.788412
$$670$$ 40.0526 1.54737
$$671$$ 2.92820 0.113042
$$672$$ 0 0
$$673$$ −27.3397 −1.05387 −0.526935 0.849906i $$-0.676659\pi$$
−0.526935 + 0.849906i $$0.676659\pi$$
$$674$$ −49.1244 −1.89220
$$675$$ −1.00000 −0.0384900
$$676$$ −42.9282 −1.65108
$$677$$ −33.1244 −1.27307 −0.636536 0.771247i $$-0.719633\pi$$
−0.636536 + 0.771247i $$0.719633\pi$$
$$678$$ 24.3923 0.936781
$$679$$ 0 0
$$680$$ −30.9282 −1.18604
$$681$$ 1.66025 0.0636211
$$682$$ 0.928203 0.0355427
$$683$$ −28.0526 −1.07340 −0.536701 0.843773i $$-0.680330\pi$$
−0.536701 + 0.843773i $$0.680330\pi$$
$$684$$ 24.3923 0.932663
$$685$$ 2.19615 0.0839107
$$686$$ 0 0
$$687$$ 3.00000 0.114457
$$688$$ 47.7128 1.81903
$$689$$ 28.1051 1.07072
$$690$$ −12.9282 −0.492168
$$691$$ 8.85641 0.336914 0.168457 0.985709i $$-0.446122\pi$$
0.168457 + 0.985709i $$0.446122\pi$$
$$692$$ 79.4256 3.01931
$$693$$ 0 0
$$694$$ 57.5692 2.18530
$$695$$ 5.92820 0.224870
$$696$$ −39.7128 −1.50531
$$697$$ −2.39230 −0.0906150
$$698$$ −60.1051 −2.27501
$$699$$ −17.3205 −0.655122
$$700$$ 0 0
$$701$$ −8.58846 −0.324382 −0.162191 0.986759i $$-0.551856\pi$$
−0.162191 + 0.986759i $$0.551856\pi$$
$$702$$ 6.19615 0.233859
$$703$$ −14.2679 −0.538126
$$704$$ 21.8564 0.823744
$$705$$ −2.00000 −0.0753244
$$706$$ 8.53590 0.321253
$$707$$ 0 0
$$708$$ 1.07180 0.0402806
$$709$$ −1.07180 −0.0402522 −0.0201261 0.999797i $$-0.506407\pi$$
−0.0201261 + 0.999797i $$0.506407\pi$$
$$710$$ −16.9282 −0.635304
$$711$$ −7.39230 −0.277233
$$712$$ −143.138 −5.36434
$$713$$ 2.19615 0.0822466
$$714$$ 0 0
$$715$$ 1.66025 0.0620900
$$716$$ −54.6410 −2.04203
$$717$$ −7.07180 −0.264101
$$718$$ 3.46410 0.129279
$$719$$ −20.5359 −0.765860 −0.382930 0.923777i $$-0.625085\pi$$
−0.382930 + 0.923777i $$0.625085\pi$$
$$720$$ 14.9282 0.556341
$$721$$ 0 0
$$722$$ −2.53590 −0.0943764
$$723$$ −13.4641 −0.500735
$$724$$ −132.890 −4.93881
$$725$$ −4.19615 −0.155841
$$726$$ −28.5885 −1.06102
$$727$$ −13.3397 −0.494744 −0.247372 0.968921i $$-0.579567\pi$$
−0.247372 + 0.968921i $$0.579567\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −34.5885 −1.28018
$$731$$ 10.4449 0.386317
$$732$$ −21.8564 −0.807836
$$733$$ −1.33975 −0.0494846 −0.0247423 0.999694i $$-0.507877\pi$$
−0.0247423 + 0.999694i $$0.507877\pi$$
$$734$$ −30.5885 −1.12904
$$735$$ 0 0
$$736$$ 103.426 3.81232
$$737$$ −10.7321 −0.395320
$$738$$ 2.00000 0.0736210
$$739$$ 27.7846 1.02207 0.511037 0.859559i $$-0.329262\pi$$
0.511037 + 0.859559i $$0.329262\pi$$
$$740$$ −17.4641 −0.641993
$$741$$ −10.1244 −0.371927
$$742$$ 0 0
$$743$$ −15.9090 −0.583643 −0.291822 0.956473i $$-0.594261\pi$$
−0.291822 + 0.956473i $$0.594261\pi$$
$$744$$ −4.39230 −0.161030
$$745$$ 5.85641 0.214562
$$746$$ −72.4449 −2.65239
$$747$$ 15.1244 0.553371
$$748$$ 13.0718 0.477952
$$749$$ 0 0
$$750$$ 2.73205 0.0997604
$$751$$ 18.0718 0.659449 0.329725 0.944077i $$-0.393044\pi$$
0.329725 + 0.944077i $$0.393044\pi$$
$$752$$ 29.8564 1.08875
$$753$$ 24.5885 0.896053
$$754$$ 26.0000 0.946864
$$755$$ −8.92820 −0.324931
$$756$$ 0 0
$$757$$ −27.8564 −1.01246 −0.506229 0.862399i $$-0.668961\pi$$
−0.506229 + 0.862399i $$0.668961\pi$$
$$758$$ −17.2679 −0.627200
$$759$$ 3.46410 0.125739
$$760$$ −42.2487 −1.53252
$$761$$ 46.7321 1.69404 0.847018 0.531565i $$-0.178396\pi$$
0.847018 + 0.531565i $$0.178396\pi$$
$$762$$ 13.1244 0.475445
$$763$$ 0 0
$$764$$ −48.7846 −1.76497
$$765$$ 3.26795 0.118153
$$766$$ 63.7128 2.30204
$$767$$ −0.444864 −0.0160631
$$768$$ −43.7128 −1.57735
$$769$$ 52.3205 1.88673 0.943363 0.331763i $$-0.107643\pi$$
0.943363 + 0.331763i $$0.107643\pi$$
$$770$$ 0 0
$$771$$ 5.66025 0.203849
$$772$$ 6.53590 0.235232
$$773$$ 43.5167 1.56519 0.782593 0.622534i $$-0.213897\pi$$
0.782593 + 0.622534i $$0.213897\pi$$
$$774$$ −8.73205 −0.313867
$$775$$ −0.464102 −0.0166710
$$776$$ −141.282 −5.07173
$$777$$ 0 0
$$778$$ −14.7846 −0.530054
$$779$$ −3.26795 −0.117086
$$780$$ −12.3923 −0.443716
$$781$$ 4.53590 0.162307
$$782$$ 42.2487 1.51081
$$783$$ 4.19615 0.149958
$$784$$ 0 0
$$785$$ 6.39230 0.228151
$$786$$ 42.2487 1.50696
$$787$$ 13.4641 0.479943 0.239972 0.970780i $$-0.422862\pi$$
0.239972 + 0.970780i $$0.422862\pi$$
$$788$$ −1.85641 −0.0661317
$$789$$ −8.39230 −0.298774
$$790$$ 20.1962 0.718547
$$791$$ 0 0
$$792$$ −6.92820 −0.246183
$$793$$ 9.07180 0.322149
$$794$$ 85.2295 3.02468
$$795$$ −12.3923 −0.439510
$$796$$ 120.210 4.26074
$$797$$ −3.94744 −0.139826 −0.0699128 0.997553i $$-0.522272\pi$$
−0.0699128 + 0.997553i $$0.522272\pi$$
$$798$$ 0 0
$$799$$ 6.53590 0.231223
$$800$$ −21.8564 −0.772741
$$801$$ 15.1244 0.534393
$$802$$ 44.7846 1.58140
$$803$$ 9.26795 0.327059
$$804$$ 80.1051 2.82509
$$805$$ 0 0
$$806$$ 2.87564 0.101290
$$807$$ 12.5359 0.441285
$$808$$ 68.7846 2.41983
$$809$$ 25.7128 0.904014 0.452007 0.892014i $$-0.350708\pi$$
0.452007 + 0.892014i $$0.350708\pi$$
$$810$$ −2.73205 −0.0959945
$$811$$ −3.46410 −0.121641 −0.0608205 0.998149i $$-0.519372\pi$$
−0.0608205 + 0.998149i $$0.519372\pi$$
$$812$$ 0 0
$$813$$ −3.07180 −0.107733
$$814$$ 6.39230 0.224050
$$815$$ 21.8564 0.765597
$$816$$ −48.7846 −1.70780
$$817$$ 14.2679 0.499172
$$818$$ −8.58846 −0.300288
$$819$$ 0 0
$$820$$ −4.00000 −0.139686
$$821$$ −25.5167 −0.890538 −0.445269 0.895397i $$-0.646892\pi$$
−0.445269 + 0.895397i $$0.646892\pi$$
$$822$$ 6.00000 0.209274
$$823$$ 39.1769 1.36562 0.682811 0.730595i $$-0.260757\pi$$
0.682811 + 0.730595i $$0.260757\pi$$
$$824$$ 87.0333 3.03195
$$825$$ −0.732051 −0.0254867
$$826$$ 0 0
$$827$$ −3.75129 −0.130445 −0.0652225 0.997871i $$-0.520776\pi$$
−0.0652225 + 0.997871i $$0.520776\pi$$
$$828$$ −25.8564 −0.898572
$$829$$ −4.60770 −0.160032 −0.0800159 0.996794i $$-0.525497\pi$$
−0.0800159 + 0.996794i $$0.525497\pi$$
$$830$$ −41.3205 −1.43426
$$831$$ 14.6603 0.508559
$$832$$ 67.7128 2.34752
$$833$$ 0 0
$$834$$ 16.1962 0.560827
$$835$$ −17.6603 −0.611158
$$836$$ 17.8564 0.617577
$$837$$ 0.464102 0.0160417
$$838$$ 96.8897 3.34700
$$839$$ 18.4449 0.636787 0.318394 0.947959i $$-0.396857\pi$$
0.318394 + 0.947959i $$0.396857\pi$$
$$840$$ 0 0
$$841$$ −11.3923 −0.392838
$$842$$ −0.196152 −0.00675986
$$843$$ −13.8564 −0.477240
$$844$$ 38.6410 1.33008
$$845$$ −7.85641 −0.270269
$$846$$ −5.46410 −0.187860
$$847$$ 0 0
$$848$$ 184.995 6.35275
$$849$$ 24.1244 0.827946
$$850$$ −8.92820 −0.306235
$$851$$ 15.1244 0.518456
$$852$$ −33.8564 −1.15990
$$853$$ −31.9808 −1.09500 −0.547500 0.836806i $$-0.684420\pi$$
−0.547500 + 0.836806i $$0.684420\pi$$
$$854$$ 0 0
$$855$$ 4.46410 0.152669
$$856$$ −20.7846 −0.710403
$$857$$ 29.1244 0.994869 0.497435 0.867502i $$-0.334275\pi$$
0.497435 + 0.867502i $$0.334275\pi$$
$$858$$ 4.53590 0.154853
$$859$$ 7.46410 0.254672 0.127336 0.991860i $$-0.459357\pi$$
0.127336 + 0.991860i $$0.459357\pi$$
$$860$$ 17.4641 0.595521
$$861$$ 0 0
$$862$$ −47.3205 −1.61174
$$863$$ 14.3923 0.489920 0.244960 0.969533i $$-0.421225\pi$$
0.244960 + 0.969533i $$0.421225\pi$$
$$864$$ 21.8564 0.743570
$$865$$ 14.5359 0.494235
$$866$$ −41.5167 −1.41079
$$867$$ 6.32051 0.214656
$$868$$ 0 0
$$869$$ −5.41154 −0.183574
$$870$$ −11.4641 −0.388669
$$871$$ −33.2487 −1.12659
$$872$$ −104.105 −3.52544
$$873$$ 14.9282 0.505243
$$874$$ 57.7128 1.95217
$$875$$ 0 0
$$876$$ −69.1769 −2.33727
$$877$$ −4.14359 −0.139919 −0.0699596 0.997550i $$-0.522287\pi$$
−0.0699596 + 0.997550i $$0.522287\pi$$
$$878$$ −1.46410 −0.0494110
$$879$$ −18.9282 −0.638432
$$880$$ 10.9282 0.368390
$$881$$ 9.85641 0.332071 0.166035 0.986120i $$-0.446903\pi$$
0.166035 + 0.986120i $$0.446903\pi$$
$$882$$ 0 0
$$883$$ 53.5885 1.80340 0.901698 0.432367i $$-0.142322\pi$$
0.901698 + 0.432367i $$0.142322\pi$$
$$884$$ 40.4974 1.36208
$$885$$ 0.196152 0.00659359
$$886$$ −25.8564 −0.868663
$$887$$ −25.2679 −0.848415 −0.424207 0.905565i $$-0.639447\pi$$
−0.424207 + 0.905565i $$0.639447\pi$$
$$888$$ −30.2487 −1.01508
$$889$$ 0 0
$$890$$ −41.3205 −1.38507
$$891$$ 0.732051 0.0245246
$$892$$ −111.426 −3.73081
$$893$$ 8.92820 0.298771
$$894$$ 16.0000 0.535120
$$895$$ −10.0000 −0.334263
$$896$$ 0 0
$$897$$ 10.7321 0.358333
$$898$$ 97.9615 3.26902
$$899$$ 1.94744 0.0649508
$$900$$ 5.46410 0.182137
$$901$$ 40.4974 1.34916
$$902$$ 1.46410 0.0487493
$$903$$ 0 0
$$904$$ −84.4974 −2.81034
$$905$$ −24.3205 −0.808441
$$906$$ −24.3923 −0.810380
$$907$$ 33.5885 1.11529 0.557643 0.830081i $$-0.311706\pi$$
0.557643 + 0.830081i $$0.311706\pi$$
$$908$$ −9.07180 −0.301058
$$909$$ −7.26795 −0.241063
$$910$$ 0 0
$$911$$ −14.7321 −0.488095 −0.244047 0.969763i $$-0.578475\pi$$
−0.244047 + 0.969763i $$0.578475\pi$$
$$912$$ −66.6410 −2.20670
$$913$$ 11.0718 0.366423
$$914$$ 45.5167 1.50556
$$915$$ −4.00000 −0.132236
$$916$$ −16.3923 −0.541617
$$917$$ 0 0
$$918$$ 8.92820 0.294675
$$919$$ −30.8564 −1.01786 −0.508929 0.860808i $$-0.669959\pi$$
−0.508929 + 0.860808i $$0.669959\pi$$
$$920$$ 44.7846 1.47650
$$921$$ 32.1244 1.05853
$$922$$ −46.3923 −1.52785
$$923$$ 14.0526 0.462546
$$924$$ 0 0
$$925$$ −3.19615 −0.105089
$$926$$ −70.3013 −2.31024
$$927$$ −9.19615 −0.302041
$$928$$ 91.7128 3.01062
$$929$$ −52.4449 −1.72066 −0.860330 0.509737i $$-0.829743\pi$$
−0.860330 + 0.509737i $$0.829743\pi$$
$$930$$ −1.26795 −0.0415777
$$931$$ 0 0
$$932$$ 94.6410 3.10007
$$933$$ −9.12436 −0.298718
$$934$$ 0.392305 0.0128366
$$935$$ 2.39230 0.0782367
$$936$$ −21.4641 −0.701576
$$937$$ 31.7321 1.03664 0.518320 0.855186i $$-0.326557\pi$$
0.518320 + 0.855186i $$0.326557\pi$$
$$938$$ 0 0
$$939$$ 12.6603 0.413152
$$940$$ 10.9282 0.356439
$$941$$ −30.0526 −0.979685 −0.489843 0.871811i $$-0.662946\pi$$
−0.489843 + 0.871811i $$0.662946\pi$$
$$942$$ 17.4641 0.569011
$$943$$ 3.46410 0.112807
$$944$$ −2.92820 −0.0953049
$$945$$ 0 0
$$946$$ −6.39230 −0.207832
$$947$$ −5.66025 −0.183934 −0.0919668 0.995762i $$-0.529315\pi$$
−0.0919668 + 0.995762i $$0.529315\pi$$
$$948$$ 40.3923 1.31188
$$949$$ 28.7128 0.932057
$$950$$ −12.1962 −0.395695
$$951$$ 28.4449 0.922388
$$952$$ 0 0
$$953$$ −36.1051 −1.16956 −0.584780 0.811192i $$-0.698819\pi$$
−0.584780 + 0.811192i $$0.698819\pi$$
$$954$$ −33.8564 −1.09614
$$955$$ −8.92820 −0.288910
$$956$$ 38.6410 1.24974
$$957$$ 3.07180 0.0992971
$$958$$ −24.0000 −0.775405
$$959$$ 0 0
$$960$$ −29.8564 −0.963611
$$961$$ −30.7846 −0.993052
$$962$$ 19.8038 0.638502
$$963$$ 2.19615 0.0707700
$$964$$ 73.5692 2.36951
$$965$$ 1.19615 0.0385055
$$966$$ 0 0
$$967$$ −10.1244 −0.325577 −0.162789 0.986661i $$-0.552049\pi$$
−0.162789 + 0.986661i $$0.552049\pi$$
$$968$$ 99.0333 3.18305
$$969$$ −14.5885 −0.468649
$$970$$ −40.7846 −1.30951
$$971$$ −24.0000 −0.770197 −0.385098 0.922876i $$-0.625832\pi$$
−0.385098 + 0.922876i $$0.625832\pi$$
$$972$$ −5.46410 −0.175261
$$973$$ 0 0
$$974$$ 1.12436 0.0360267
$$975$$ −2.26795 −0.0726325
$$976$$ 59.7128 1.91136
$$977$$ −16.5885 −0.530712 −0.265356 0.964151i $$-0.585489\pi$$
−0.265356 + 0.964151i $$0.585489\pi$$
$$978$$ 59.7128 1.90941
$$979$$ 11.0718 0.353856
$$980$$ 0 0
$$981$$ 11.0000 0.351203
$$982$$ 104.497 3.33465
$$983$$ 9.80385 0.312694 0.156347 0.987702i $$-0.450028\pi$$
0.156347 + 0.987702i $$0.450028\pi$$
$$984$$ −6.92820 −0.220863
$$985$$ −0.339746 −0.0108252
$$986$$ 37.4641 1.19310
$$987$$ 0 0
$$988$$ 55.3205 1.75998
$$989$$ −15.1244 −0.480927
$$990$$ −2.00000 −0.0635642
$$991$$ −21.1051 −0.670426 −0.335213 0.942142i $$-0.608808\pi$$
−0.335213 + 0.942142i $$0.608808\pi$$
$$992$$ 10.1436 0.322059
$$993$$ −8.07180 −0.256151
$$994$$ 0 0
$$995$$ 22.0000 0.697447
$$996$$ −82.6410 −2.61858
$$997$$ −55.9808 −1.77293 −0.886464 0.462797i $$-0.846846\pi$$
−0.886464 + 0.462797i $$0.846846\pi$$
$$998$$ 36.9808 1.17061
$$999$$ 3.19615 0.101122
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 735.2.a.g.1.1 2
3.2 odd 2 2205.2.a.z.1.2 2
5.4 even 2 3675.2.a.bg.1.2 2
7.2 even 3 105.2.i.d.46.2 yes 4
7.3 odd 6 735.2.i.l.226.2 4
7.4 even 3 105.2.i.d.16.2 4
7.5 odd 6 735.2.i.l.361.2 4
7.6 odd 2 735.2.a.h.1.1 2
21.2 odd 6 315.2.j.c.46.1 4
21.11 odd 6 315.2.j.c.226.1 4
21.20 even 2 2205.2.a.ba.1.2 2
28.11 odd 6 1680.2.bg.o.961.1 4
28.23 odd 6 1680.2.bg.o.1201.1 4
35.2 odd 12 525.2.r.a.424.1 4
35.4 even 6 525.2.i.f.226.1 4
35.9 even 6 525.2.i.f.151.1 4
35.18 odd 12 525.2.r.a.499.1 4
35.23 odd 12 525.2.r.f.424.2 4
35.32 odd 12 525.2.r.f.499.2 4
35.34 odd 2 3675.2.a.be.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.i.d.16.2 4 7.4 even 3
105.2.i.d.46.2 yes 4 7.2 even 3
315.2.j.c.46.1 4 21.2 odd 6
315.2.j.c.226.1 4 21.11 odd 6
525.2.i.f.151.1 4 35.9 even 6
525.2.i.f.226.1 4 35.4 even 6
525.2.r.a.424.1 4 35.2 odd 12
525.2.r.a.499.1 4 35.18 odd 12
525.2.r.f.424.2 4 35.23 odd 12
525.2.r.f.499.2 4 35.32 odd 12
735.2.a.g.1.1 2 1.1 even 1 trivial
735.2.a.h.1.1 2 7.6 odd 2
735.2.i.l.226.2 4 7.3 odd 6
735.2.i.l.361.2 4 7.5 odd 6
1680.2.bg.o.961.1 4 28.11 odd 6
1680.2.bg.o.1201.1 4 28.23 odd 6
2205.2.a.z.1.2 2 3.2 odd 2
2205.2.a.ba.1.2 2 21.20 even 2
3675.2.a.be.1.2 2 35.34 odd 2
3675.2.a.bg.1.2 2 5.4 even 2