# Properties

 Label 966.2.a.p.1.1 Level $966$ Weight $2$ Character 966.1 Self dual yes Analytic conductor $7.714$ 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] = [966,2,Mod(1,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, 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("966.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.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: $$7.71354883526$$ Analytic rank: $$0$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 966.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{12} -4.47214 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +4.47214 q^{17} +1.00000 q^{18} +2.47214 q^{19} +2.00000 q^{20} +1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -4.47214 q^{26} +1.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} -2.47214 q^{31} +1.00000 q^{32} +4.47214 q^{34} +2.00000 q^{35} +1.00000 q^{36} +6.94427 q^{37} +2.47214 q^{38} -4.47214 q^{39} +2.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} -4.94427 q^{43} +2.00000 q^{45} -1.00000 q^{46} -2.47214 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.47214 q^{51} -4.47214 q^{52} -10.9443 q^{53} +1.00000 q^{54} +1.00000 q^{56} +2.47214 q^{57} -2.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} -6.00000 q^{61} -2.47214 q^{62} +1.00000 q^{63} +1.00000 q^{64} -8.94427 q^{65} +4.94427 q^{67} +4.47214 q^{68} -1.00000 q^{69} +2.00000 q^{70} +4.94427 q^{71} +1.00000 q^{72} +14.9443 q^{73} +6.94427 q^{74} -1.00000 q^{75} +2.47214 q^{76} -4.47214 q^{78} -4.94427 q^{79} +2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +2.47214 q^{83} +1.00000 q^{84} +8.94427 q^{85} -4.94427 q^{86} -2.00000 q^{87} +9.41641 q^{89} +2.00000 q^{90} -4.47214 q^{91} -1.00000 q^{92} -2.47214 q^{93} -2.47214 q^{94} +4.94427 q^{95} +1.00000 q^{96} -0.472136 q^{97} +1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + 4 q^{10} + 2 q^{12} + 2 q^{14} + 4 q^{15} + 2 q^{16} + 2 q^{18} - 4 q^{19} + 4 q^{20} + 2 q^{21} - 2 q^{23} + 2 q^{24} - 2 q^{25} + 2 q^{27} + 2 q^{28} - 4 q^{29} + 4 q^{30} + 4 q^{31} + 2 q^{32} + 4 q^{35} + 2 q^{36} - 4 q^{37} - 4 q^{38} + 4 q^{40} - 12 q^{41} + 2 q^{42} + 8 q^{43} + 4 q^{45} - 2 q^{46} + 4 q^{47} + 2 q^{48} + 2 q^{49} - 2 q^{50} - 4 q^{53} + 2 q^{54} + 2 q^{56} - 4 q^{57} - 4 q^{58} + 8 q^{59} + 4 q^{60} - 12 q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} - 8 q^{67} - 2 q^{69} + 4 q^{70} - 8 q^{71} + 2 q^{72} + 12 q^{73} - 4 q^{74} - 2 q^{75} - 4 q^{76} + 8 q^{79} + 4 q^{80} + 2 q^{81} - 12 q^{82} - 4 q^{83} + 2 q^{84} + 8 q^{86} - 4 q^{87} - 8 q^{89} + 4 q^{90} - 2 q^{92} + 4 q^{93} + 4 q^{94} - 8 q^{95} + 2 q^{96} + 8 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9 + 4 * q^10 + 2 * q^12 + 2 * q^14 + 4 * q^15 + 2 * q^16 + 2 * q^18 - 4 * q^19 + 4 * q^20 + 2 * q^21 - 2 * q^23 + 2 * q^24 - 2 * q^25 + 2 * q^27 + 2 * q^28 - 4 * q^29 + 4 * q^30 + 4 * q^31 + 2 * q^32 + 4 * q^35 + 2 * q^36 - 4 * q^37 - 4 * q^38 + 4 * q^40 - 12 * q^41 + 2 * q^42 + 8 * q^43 + 4 * q^45 - 2 * q^46 + 4 * q^47 + 2 * q^48 + 2 * q^49 - 2 * q^50 - 4 * q^53 + 2 * q^54 + 2 * q^56 - 4 * q^57 - 4 * q^58 + 8 * q^59 + 4 * q^60 - 12 * q^61 + 4 * q^62 + 2 * q^63 + 2 * q^64 - 8 * q^67 - 2 * q^69 + 4 * q^70 - 8 * q^71 + 2 * q^72 + 12 * q^73 - 4 * q^74 - 2 * q^75 - 4 * q^76 + 8 * q^79 + 4 * q^80 + 2 * q^81 - 12 * q^82 - 4 * q^83 + 2 * q^84 + 8 * q^86 - 4 * q^87 - 8 * q^89 + 4 * q^90 - 2 * q^92 + 4 * q^93 + 4 * q^94 - 8 * q^95 + 2 * q^96 + 8 * q^97 + 2 * q^98

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 2.00000 0.632456
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −4.47214 −1.24035 −0.620174 0.784465i $$-0.712938\pi$$
−0.620174 + 0.784465i $$0.712938\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 2.00000 0.516398
$$16$$ 1.00000 0.250000
$$17$$ 4.47214 1.08465 0.542326 0.840168i $$-0.317544\pi$$
0.542326 + 0.840168i $$0.317544\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 2.47214 0.567147 0.283573 0.958951i $$-0.408480\pi$$
0.283573 + 0.958951i $$0.408480\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 1.00000 0.204124
$$25$$ −1.00000 −0.200000
$$26$$ −4.47214 −0.877058
$$27$$ 1.00000 0.192450
$$28$$ 1.00000 0.188982
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 2.00000 0.365148
$$31$$ −2.47214 −0.444009 −0.222004 0.975046i $$-0.571260\pi$$
−0.222004 + 0.975046i $$0.571260\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 4.47214 0.766965
$$35$$ 2.00000 0.338062
$$36$$ 1.00000 0.166667
$$37$$ 6.94427 1.14163 0.570816 0.821078i $$-0.306627\pi$$
0.570816 + 0.821078i $$0.306627\pi$$
$$38$$ 2.47214 0.401033
$$39$$ −4.47214 −0.716115
$$40$$ 2.00000 0.316228
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 1.00000 0.154303
$$43$$ −4.94427 −0.753994 −0.376997 0.926214i $$-0.623043\pi$$
−0.376997 + 0.926214i $$0.623043\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ −1.00000 −0.147442
$$47$$ −2.47214 −0.360598 −0.180299 0.983612i $$-0.557707\pi$$
−0.180299 + 0.983612i $$0.557707\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 4.47214 0.626224
$$52$$ −4.47214 −0.620174
$$53$$ −10.9443 −1.50331 −0.751656 0.659556i $$-0.770744\pi$$
−0.751656 + 0.659556i $$0.770744\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 2.47214 0.327442
$$58$$ −2.00000 −0.262613
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 2.00000 0.258199
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ −2.47214 −0.313962
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ −8.94427 −1.10940
$$66$$ 0 0
$$67$$ 4.94427 0.604039 0.302019 0.953302i $$-0.402339\pi$$
0.302019 + 0.953302i $$0.402339\pi$$
$$68$$ 4.47214 0.542326
$$69$$ −1.00000 −0.120386
$$70$$ 2.00000 0.239046
$$71$$ 4.94427 0.586777 0.293389 0.955993i $$-0.405217\pi$$
0.293389 + 0.955993i $$0.405217\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 14.9443 1.74909 0.874547 0.484940i $$-0.161159\pi$$
0.874547 + 0.484940i $$0.161159\pi$$
$$74$$ 6.94427 0.807255
$$75$$ −1.00000 −0.115470
$$76$$ 2.47214 0.283573
$$77$$ 0 0
$$78$$ −4.47214 −0.506370
$$79$$ −4.94427 −0.556274 −0.278137 0.960541i $$-0.589717\pi$$
−0.278137 + 0.960541i $$0.589717\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 2.47214 0.271352 0.135676 0.990753i $$-0.456679\pi$$
0.135676 + 0.990753i $$0.456679\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 8.94427 0.970143
$$86$$ −4.94427 −0.533155
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ 9.41641 0.998137 0.499069 0.866562i $$-0.333676\pi$$
0.499069 + 0.866562i $$0.333676\pi$$
$$90$$ 2.00000 0.210819
$$91$$ −4.47214 −0.468807
$$92$$ −1.00000 −0.104257
$$93$$ −2.47214 −0.256349
$$94$$ −2.47214 −0.254981
$$95$$ 4.94427 0.507272
$$96$$ 1.00000 0.102062
$$97$$ −0.472136 −0.0479381 −0.0239691 0.999713i $$-0.507630\pi$$
−0.0239691 + 0.999713i $$0.507630\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ −9.41641 −0.936968 −0.468484 0.883472i $$-0.655200\pi$$
−0.468484 + 0.883472i $$0.655200\pi$$
$$102$$ 4.47214 0.442807
$$103$$ −4.94427 −0.487174 −0.243587 0.969879i $$-0.578324\pi$$
−0.243587 + 0.969879i $$0.578324\pi$$
$$104$$ −4.47214 −0.438529
$$105$$ 2.00000 0.195180
$$106$$ −10.9443 −1.06300
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −2.94427 −0.282010 −0.141005 0.990009i $$-0.545033\pi$$
−0.141005 + 0.990009i $$0.545033\pi$$
$$110$$ 0 0
$$111$$ 6.94427 0.659121
$$112$$ 1.00000 0.0944911
$$113$$ −18.9443 −1.78213 −0.891064 0.453878i $$-0.850040\pi$$
−0.891064 + 0.453878i $$0.850040\pi$$
$$114$$ 2.47214 0.231537
$$115$$ −2.00000 −0.186501
$$116$$ −2.00000 −0.185695
$$117$$ −4.47214 −0.413449
$$118$$ 4.00000 0.368230
$$119$$ 4.47214 0.409960
$$120$$ 2.00000 0.182574
$$121$$ −11.0000 −1.00000
$$122$$ −6.00000 −0.543214
$$123$$ −6.00000 −0.541002
$$124$$ −2.47214 −0.222004
$$125$$ −12.0000 −1.07331
$$126$$ 1.00000 0.0890871
$$127$$ 20.9443 1.85850 0.929252 0.369447i $$-0.120453\pi$$
0.929252 + 0.369447i $$0.120453\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −4.94427 −0.435319
$$130$$ −8.94427 −0.784465
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 2.47214 0.214361
$$134$$ 4.94427 0.427120
$$135$$ 2.00000 0.172133
$$136$$ 4.47214 0.383482
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ −1.00000 −0.0851257
$$139$$ 8.94427 0.758643 0.379322 0.925265i $$-0.376157\pi$$
0.379322 + 0.925265i $$0.376157\pi$$
$$140$$ 2.00000 0.169031
$$141$$ −2.47214 −0.208191
$$142$$ 4.94427 0.414914
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −4.00000 −0.332182
$$146$$ 14.9443 1.23680
$$147$$ 1.00000 0.0824786
$$148$$ 6.94427 0.570816
$$149$$ −1.05573 −0.0864886 −0.0432443 0.999065i $$-0.513769\pi$$
−0.0432443 + 0.999065i $$0.513769\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 4.94427 0.402359 0.201180 0.979554i $$-0.435523\pi$$
0.201180 + 0.979554i $$0.435523\pi$$
$$152$$ 2.47214 0.200517
$$153$$ 4.47214 0.361551
$$154$$ 0 0
$$155$$ −4.94427 −0.397133
$$156$$ −4.47214 −0.358057
$$157$$ −2.94427 −0.234978 −0.117489 0.993074i $$-0.537485\pi$$
−0.117489 + 0.993074i $$0.537485\pi$$
$$158$$ −4.94427 −0.393345
$$159$$ −10.9443 −0.867937
$$160$$ 2.00000 0.158114
$$161$$ −1.00000 −0.0788110
$$162$$ 1.00000 0.0785674
$$163$$ −8.94427 −0.700569 −0.350285 0.936643i $$-0.613915\pi$$
−0.350285 + 0.936643i $$0.613915\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 2.47214 0.191875
$$167$$ −15.4164 −1.19296 −0.596479 0.802629i $$-0.703434\pi$$
−0.596479 + 0.802629i $$0.703434\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 7.00000 0.538462
$$170$$ 8.94427 0.685994
$$171$$ 2.47214 0.189049
$$172$$ −4.94427 −0.376997
$$173$$ 0.472136 0.0358958 0.0179479 0.999839i $$-0.494287\pi$$
0.0179479 + 0.999839i $$0.494287\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ 4.00000 0.300658
$$178$$ 9.41641 0.705790
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ −4.47214 −0.331497
$$183$$ −6.00000 −0.443533
$$184$$ −1.00000 −0.0737210
$$185$$ 13.8885 1.02111
$$186$$ −2.47214 −0.181266
$$187$$ 0 0
$$188$$ −2.47214 −0.180299
$$189$$ 1.00000 0.0727393
$$190$$ 4.94427 0.358695
$$191$$ −3.05573 −0.221105 −0.110552 0.993870i $$-0.535262\pi$$
−0.110552 + 0.993870i $$0.535262\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 11.8885 0.855756 0.427878 0.903836i $$-0.359261\pi$$
0.427878 + 0.903836i $$0.359261\pi$$
$$194$$ −0.472136 −0.0338974
$$195$$ −8.94427 −0.640513
$$196$$ 1.00000 0.0714286
$$197$$ −14.9443 −1.06474 −0.532368 0.846513i $$-0.678698\pi$$
−0.532368 + 0.846513i $$0.678698\pi$$
$$198$$ 0 0
$$199$$ 3.05573 0.216615 0.108307 0.994117i $$-0.465457\pi$$
0.108307 + 0.994117i $$0.465457\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 4.94427 0.348742
$$202$$ −9.41641 −0.662536
$$203$$ −2.00000 −0.140372
$$204$$ 4.47214 0.313112
$$205$$ −12.0000 −0.838116
$$206$$ −4.94427 −0.344484
$$207$$ −1.00000 −0.0695048
$$208$$ −4.47214 −0.310087
$$209$$ 0 0
$$210$$ 2.00000 0.138013
$$211$$ −0.944272 −0.0650064 −0.0325032 0.999472i $$-0.510348\pi$$
−0.0325032 + 0.999472i $$0.510348\pi$$
$$212$$ −10.9443 −0.751656
$$213$$ 4.94427 0.338776
$$214$$ −8.00000 −0.546869
$$215$$ −9.88854 −0.674393
$$216$$ 1.00000 0.0680414
$$217$$ −2.47214 −0.167820
$$218$$ −2.94427 −0.199411
$$219$$ 14.9443 1.00984
$$220$$ 0 0
$$221$$ −20.0000 −1.34535
$$222$$ 6.94427 0.466069
$$223$$ 18.4721 1.23699 0.618493 0.785790i $$-0.287744\pi$$
0.618493 + 0.785790i $$0.287744\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −1.00000 −0.0666667
$$226$$ −18.9443 −1.26015
$$227$$ 5.52786 0.366897 0.183449 0.983029i $$-0.441274\pi$$
0.183449 + 0.983029i $$0.441274\pi$$
$$228$$ 2.47214 0.163721
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ −2.00000 −0.131876
$$231$$ 0 0
$$232$$ −2.00000 −0.131306
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ −4.47214 −0.292353
$$235$$ −4.94427 −0.322529
$$236$$ 4.00000 0.260378
$$237$$ −4.94427 −0.321165
$$238$$ 4.47214 0.289886
$$239$$ 20.9443 1.35477 0.677386 0.735628i $$-0.263113\pi$$
0.677386 + 0.735628i $$0.263113\pi$$
$$240$$ 2.00000 0.129099
$$241$$ 20.4721 1.31873 0.659363 0.751825i $$-0.270826\pi$$
0.659363 + 0.751825i $$0.270826\pi$$
$$242$$ −11.0000 −0.707107
$$243$$ 1.00000 0.0641500
$$244$$ −6.00000 −0.384111
$$245$$ 2.00000 0.127775
$$246$$ −6.00000 −0.382546
$$247$$ −11.0557 −0.703459
$$248$$ −2.47214 −0.156981
$$249$$ 2.47214 0.156665
$$250$$ −12.0000 −0.758947
$$251$$ −5.52786 −0.348916 −0.174458 0.984665i $$-0.555817\pi$$
−0.174458 + 0.984665i $$0.555817\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 0 0
$$254$$ 20.9443 1.31416
$$255$$ 8.94427 0.560112
$$256$$ 1.00000 0.0625000
$$257$$ −2.94427 −0.183659 −0.0918293 0.995775i $$-0.529271\pi$$
−0.0918293 + 0.995775i $$0.529271\pi$$
$$258$$ −4.94427 −0.307817
$$259$$ 6.94427 0.431496
$$260$$ −8.94427 −0.554700
$$261$$ −2.00000 −0.123797
$$262$$ 12.0000 0.741362
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ −21.8885 −1.34460
$$266$$ 2.47214 0.151576
$$267$$ 9.41641 0.576275
$$268$$ 4.94427 0.302019
$$269$$ 16.4721 1.00432 0.502162 0.864774i $$-0.332538\pi$$
0.502162 + 0.864774i $$0.332538\pi$$
$$270$$ 2.00000 0.121716
$$271$$ 15.4164 0.936480 0.468240 0.883601i $$-0.344888\pi$$
0.468240 + 0.883601i $$0.344888\pi$$
$$272$$ 4.47214 0.271163
$$273$$ −4.47214 −0.270666
$$274$$ −14.0000 −0.845771
$$275$$ 0 0
$$276$$ −1.00000 −0.0601929
$$277$$ −11.8885 −0.714313 −0.357157 0.934044i $$-0.616254\pi$$
−0.357157 + 0.934044i $$0.616254\pi$$
$$278$$ 8.94427 0.536442
$$279$$ −2.47214 −0.148003
$$280$$ 2.00000 0.119523
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ −2.47214 −0.147214
$$283$$ 18.4721 1.09805 0.549027 0.835804i $$-0.314998\pi$$
0.549027 + 0.835804i $$0.314998\pi$$
$$284$$ 4.94427 0.293389
$$285$$ 4.94427 0.292873
$$286$$ 0 0
$$287$$ −6.00000 −0.354169
$$288$$ 1.00000 0.0589256
$$289$$ 3.00000 0.176471
$$290$$ −4.00000 −0.234888
$$291$$ −0.472136 −0.0276771
$$292$$ 14.9443 0.874547
$$293$$ 27.8885 1.62927 0.814633 0.579977i $$-0.196938\pi$$
0.814633 + 0.579977i $$0.196938\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ 8.00000 0.465778
$$296$$ 6.94427 0.403628
$$297$$ 0 0
$$298$$ −1.05573 −0.0611567
$$299$$ 4.47214 0.258630
$$300$$ −1.00000 −0.0577350
$$301$$ −4.94427 −0.284983
$$302$$ 4.94427 0.284511
$$303$$ −9.41641 −0.540958
$$304$$ 2.47214 0.141787
$$305$$ −12.0000 −0.687118
$$306$$ 4.47214 0.255655
$$307$$ 24.9443 1.42364 0.711822 0.702360i $$-0.247870\pi$$
0.711822 + 0.702360i $$0.247870\pi$$
$$308$$ 0 0
$$309$$ −4.94427 −0.281270
$$310$$ −4.94427 −0.280816
$$311$$ −2.47214 −0.140182 −0.0700910 0.997541i $$-0.522329\pi$$
−0.0700910 + 0.997541i $$0.522329\pi$$
$$312$$ −4.47214 −0.253185
$$313$$ −13.4164 −0.758340 −0.379170 0.925327i $$-0.623790\pi$$
−0.379170 + 0.925327i $$0.623790\pi$$
$$314$$ −2.94427 −0.166155
$$315$$ 2.00000 0.112687
$$316$$ −4.94427 −0.278137
$$317$$ −13.0557 −0.733283 −0.366641 0.930362i $$-0.619492\pi$$
−0.366641 + 0.930362i $$0.619492\pi$$
$$318$$ −10.9443 −0.613724
$$319$$ 0 0
$$320$$ 2.00000 0.111803
$$321$$ −8.00000 −0.446516
$$322$$ −1.00000 −0.0557278
$$323$$ 11.0557 0.615157
$$324$$ 1.00000 0.0555556
$$325$$ 4.47214 0.248069
$$326$$ −8.94427 −0.495377
$$327$$ −2.94427 −0.162819
$$328$$ −6.00000 −0.331295
$$329$$ −2.47214 −0.136293
$$330$$ 0 0
$$331$$ −13.8885 −0.763383 −0.381692 0.924290i $$-0.624658\pi$$
−0.381692 + 0.924290i $$0.624658\pi$$
$$332$$ 2.47214 0.135676
$$333$$ 6.94427 0.380544
$$334$$ −15.4164 −0.843548
$$335$$ 9.88854 0.540269
$$336$$ 1.00000 0.0545545
$$337$$ −2.94427 −0.160385 −0.0801924 0.996779i $$-0.525553\pi$$
−0.0801924 + 0.996779i $$0.525553\pi$$
$$338$$ 7.00000 0.380750
$$339$$ −18.9443 −1.02891
$$340$$ 8.94427 0.485071
$$341$$ 0 0
$$342$$ 2.47214 0.133678
$$343$$ 1.00000 0.0539949
$$344$$ −4.94427 −0.266577
$$345$$ −2.00000 −0.107676
$$346$$ 0.472136 0.0253822
$$347$$ −0.944272 −0.0506912 −0.0253456 0.999679i $$-0.508069\pi$$
−0.0253456 + 0.999679i $$0.508069\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ 24.4721 1.30996 0.654982 0.755645i $$-0.272676\pi$$
0.654982 + 0.755645i $$0.272676\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ −4.47214 −0.238705
$$352$$ 0 0
$$353$$ 2.00000 0.106449 0.0532246 0.998583i $$-0.483050\pi$$
0.0532246 + 0.998583i $$0.483050\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 9.88854 0.524829
$$356$$ 9.41641 0.499069
$$357$$ 4.47214 0.236691
$$358$$ −20.0000 −1.05703
$$359$$ −30.8328 −1.62729 −0.813647 0.581359i $$-0.802521\pi$$
−0.813647 + 0.581359i $$0.802521\pi$$
$$360$$ 2.00000 0.105409
$$361$$ −12.8885 −0.678344
$$362$$ 2.00000 0.105118
$$363$$ −11.0000 −0.577350
$$364$$ −4.47214 −0.234404
$$365$$ 29.8885 1.56444
$$366$$ −6.00000 −0.313625
$$367$$ 30.8328 1.60946 0.804730 0.593641i $$-0.202310\pi$$
0.804730 + 0.593641i $$0.202310\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ −6.00000 −0.312348
$$370$$ 13.8885 0.722031
$$371$$ −10.9443 −0.568198
$$372$$ −2.47214 −0.128174
$$373$$ 6.94427 0.359561 0.179780 0.983707i $$-0.442461\pi$$
0.179780 + 0.983707i $$0.442461\pi$$
$$374$$ 0 0
$$375$$ −12.0000 −0.619677
$$376$$ −2.47214 −0.127491
$$377$$ 8.94427 0.460653
$$378$$ 1.00000 0.0514344
$$379$$ 24.0000 1.23280 0.616399 0.787434i $$-0.288591\pi$$
0.616399 + 0.787434i $$0.288591\pi$$
$$380$$ 4.94427 0.253636
$$381$$ 20.9443 1.07301
$$382$$ −3.05573 −0.156345
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 11.8885 0.605111
$$387$$ −4.94427 −0.251331
$$388$$ −0.472136 −0.0239691
$$389$$ −18.9443 −0.960513 −0.480256 0.877128i $$-0.659456\pi$$
−0.480256 + 0.877128i $$0.659456\pi$$
$$390$$ −8.94427 −0.452911
$$391$$ −4.47214 −0.226166
$$392$$ 1.00000 0.0505076
$$393$$ 12.0000 0.605320
$$394$$ −14.9443 −0.752882
$$395$$ −9.88854 −0.497547
$$396$$ 0 0
$$397$$ −36.4721 −1.83048 −0.915242 0.402905i $$-0.868001\pi$$
−0.915242 + 0.402905i $$0.868001\pi$$
$$398$$ 3.05573 0.153170
$$399$$ 2.47214 0.123762
$$400$$ −1.00000 −0.0500000
$$401$$ −22.0000 −1.09863 −0.549314 0.835616i $$-0.685111\pi$$
−0.549314 + 0.835616i $$0.685111\pi$$
$$402$$ 4.94427 0.246598
$$403$$ 11.0557 0.550725
$$404$$ −9.41641 −0.468484
$$405$$ 2.00000 0.0993808
$$406$$ −2.00000 −0.0992583
$$407$$ 0 0
$$408$$ 4.47214 0.221404
$$409$$ −15.8885 −0.785638 −0.392819 0.919616i $$-0.628500\pi$$
−0.392819 + 0.919616i $$0.628500\pi$$
$$410$$ −12.0000 −0.592638
$$411$$ −14.0000 −0.690569
$$412$$ −4.94427 −0.243587
$$413$$ 4.00000 0.196827
$$414$$ −1.00000 −0.0491473
$$415$$ 4.94427 0.242705
$$416$$ −4.47214 −0.219265
$$417$$ 8.94427 0.438003
$$418$$ 0 0
$$419$$ −7.41641 −0.362315 −0.181158 0.983454i $$-0.557984\pi$$
−0.181158 + 0.983454i $$0.557984\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ 32.8328 1.60017 0.800087 0.599884i $$-0.204787\pi$$
0.800087 + 0.599884i $$0.204787\pi$$
$$422$$ −0.944272 −0.0459664
$$423$$ −2.47214 −0.120199
$$424$$ −10.9443 −0.531501
$$425$$ −4.47214 −0.216930
$$426$$ 4.94427 0.239551
$$427$$ −6.00000 −0.290360
$$428$$ −8.00000 −0.386695
$$429$$ 0 0
$$430$$ −9.88854 −0.476868
$$431$$ 17.8885 0.861661 0.430830 0.902433i $$-0.358221\pi$$
0.430830 + 0.902433i $$0.358221\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 25.4164 1.22143 0.610717 0.791849i $$-0.290881\pi$$
0.610717 + 0.791849i $$0.290881\pi$$
$$434$$ −2.47214 −0.118666
$$435$$ −4.00000 −0.191785
$$436$$ −2.94427 −0.141005
$$437$$ −2.47214 −0.118258
$$438$$ 14.9443 0.714065
$$439$$ 28.3607 1.35358 0.676791 0.736175i $$-0.263370\pi$$
0.676791 + 0.736175i $$0.263370\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ −20.0000 −0.951303
$$443$$ 34.8328 1.65496 0.827479 0.561497i $$-0.189775\pi$$
0.827479 + 0.561497i $$0.189775\pi$$
$$444$$ 6.94427 0.329561
$$445$$ 18.8328 0.892761
$$446$$ 18.4721 0.874681
$$447$$ −1.05573 −0.0499342
$$448$$ 1.00000 0.0472456
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 0 0
$$452$$ −18.9443 −0.891064
$$453$$ 4.94427 0.232302
$$454$$ 5.52786 0.259436
$$455$$ −8.94427 −0.419314
$$456$$ 2.47214 0.115768
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 4.47214 0.208741
$$460$$ −2.00000 −0.0932505
$$461$$ 6.58359 0.306628 0.153314 0.988177i $$-0.451005\pi$$
0.153314 + 0.988177i $$0.451005\pi$$
$$462$$ 0 0
$$463$$ 6.11146 0.284023 0.142012 0.989865i $$-0.454643\pi$$
0.142012 + 0.989865i $$0.454643\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ −4.94427 −0.229285
$$466$$ −6.00000 −0.277945
$$467$$ 33.3050 1.54117 0.770585 0.637338i $$-0.219964\pi$$
0.770585 + 0.637338i $$0.219964\pi$$
$$468$$ −4.47214 −0.206725
$$469$$ 4.94427 0.228305
$$470$$ −4.94427 −0.228062
$$471$$ −2.94427 −0.135665
$$472$$ 4.00000 0.184115
$$473$$ 0 0
$$474$$ −4.94427 −0.227098
$$475$$ −2.47214 −0.113429
$$476$$ 4.47214 0.204980
$$477$$ −10.9443 −0.501104
$$478$$ 20.9443 0.957969
$$479$$ −4.94427 −0.225910 −0.112955 0.993600i $$-0.536032\pi$$
−0.112955 + 0.993600i $$0.536032\pi$$
$$480$$ 2.00000 0.0912871
$$481$$ −31.0557 −1.41602
$$482$$ 20.4721 0.932480
$$483$$ −1.00000 −0.0455016
$$484$$ −11.0000 −0.500000
$$485$$ −0.944272 −0.0428772
$$486$$ 1.00000 0.0453609
$$487$$ 24.0000 1.08754 0.543772 0.839233i $$-0.316996\pi$$
0.543772 + 0.839233i $$0.316996\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ −8.94427 −0.404474
$$490$$ 2.00000 0.0903508
$$491$$ 40.9443 1.84779 0.923895 0.382647i $$-0.124987\pi$$
0.923895 + 0.382647i $$0.124987\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ −8.94427 −0.402830
$$494$$ −11.0557 −0.497421
$$495$$ 0 0
$$496$$ −2.47214 −0.111002
$$497$$ 4.94427 0.221781
$$498$$ 2.47214 0.110779
$$499$$ 32.9443 1.47479 0.737394 0.675463i $$-0.236056\pi$$
0.737394 + 0.675463i $$0.236056\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ −15.4164 −0.688754
$$502$$ −5.52786 −0.246721
$$503$$ −9.88854 −0.440908 −0.220454 0.975397i $$-0.570754\pi$$
−0.220454 + 0.975397i $$0.570754\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ −18.8328 −0.838049
$$506$$ 0 0
$$507$$ 7.00000 0.310881
$$508$$ 20.9443 0.929252
$$509$$ 6.58359 0.291813 0.145906 0.989298i $$-0.453390\pi$$
0.145906 + 0.989298i $$0.453390\pi$$
$$510$$ 8.94427 0.396059
$$511$$ 14.9443 0.661096
$$512$$ 1.00000 0.0441942
$$513$$ 2.47214 0.109147
$$514$$ −2.94427 −0.129866
$$515$$ −9.88854 −0.435741
$$516$$ −4.94427 −0.217659
$$517$$ 0 0
$$518$$ 6.94427 0.305114
$$519$$ 0.472136 0.0207245
$$520$$ −8.94427 −0.392232
$$521$$ 33.4164 1.46400 0.732000 0.681305i $$-0.238587\pi$$
0.732000 + 0.681305i $$0.238587\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ −25.3050 −1.10651 −0.553254 0.833013i $$-0.686614\pi$$
−0.553254 + 0.833013i $$0.686614\pi$$
$$524$$ 12.0000 0.524222
$$525$$ −1.00000 −0.0436436
$$526$$ −24.0000 −1.04645
$$527$$ −11.0557 −0.481595
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ −21.8885 −0.950778
$$531$$ 4.00000 0.173585
$$532$$ 2.47214 0.107181
$$533$$ 26.8328 1.16226
$$534$$ 9.41641 0.407488
$$535$$ −16.0000 −0.691740
$$536$$ 4.94427 0.213560
$$537$$ −20.0000 −0.863064
$$538$$ 16.4721 0.710164
$$539$$ 0 0
$$540$$ 2.00000 0.0860663
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 15.4164 0.662191
$$543$$ 2.00000 0.0858282
$$544$$ 4.47214 0.191741
$$545$$ −5.88854 −0.252238
$$546$$ −4.47214 −0.191390
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ −14.0000 −0.598050
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ −4.94427 −0.210633
$$552$$ −1.00000 −0.0425628
$$553$$ −4.94427 −0.210252
$$554$$ −11.8885 −0.505096
$$555$$ 13.8885 0.589536
$$556$$ 8.94427 0.379322
$$557$$ −17.0557 −0.722674 −0.361337 0.932435i $$-0.617680\pi$$
−0.361337 + 0.932435i $$0.617680\pi$$
$$558$$ −2.47214 −0.104654
$$559$$ 22.1115 0.935215
$$560$$ 2.00000 0.0845154
$$561$$ 0 0
$$562$$ 10.0000 0.421825
$$563$$ 26.4721 1.11567 0.557834 0.829953i $$-0.311633\pi$$
0.557834 + 0.829953i $$0.311633\pi$$
$$564$$ −2.47214 −0.104096
$$565$$ −37.8885 −1.59398
$$566$$ 18.4721 0.776442
$$567$$ 1.00000 0.0419961
$$568$$ 4.94427 0.207457
$$569$$ −7.88854 −0.330705 −0.165352 0.986235i $$-0.552876\pi$$
−0.165352 + 0.986235i $$0.552876\pi$$
$$570$$ 4.94427 0.207093
$$571$$ −12.9443 −0.541701 −0.270850 0.962621i $$-0.587305\pi$$
−0.270850 + 0.962621i $$0.587305\pi$$
$$572$$ 0 0
$$573$$ −3.05573 −0.127655
$$574$$ −6.00000 −0.250435
$$575$$ 1.00000 0.0417029
$$576$$ 1.00000 0.0416667
$$577$$ −2.94427 −0.122572 −0.0612858 0.998120i $$-0.519520\pi$$
−0.0612858 + 0.998120i $$0.519520\pi$$
$$578$$ 3.00000 0.124784
$$579$$ 11.8885 0.494071
$$580$$ −4.00000 −0.166091
$$581$$ 2.47214 0.102561
$$582$$ −0.472136 −0.0195707
$$583$$ 0 0
$$584$$ 14.9443 0.618398
$$585$$ −8.94427 −0.369800
$$586$$ 27.8885 1.15207
$$587$$ −0.944272 −0.0389743 −0.0194871 0.999810i $$-0.506203\pi$$
−0.0194871 + 0.999810i $$0.506203\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ −6.11146 −0.251818
$$590$$ 8.00000 0.329355
$$591$$ −14.9443 −0.614725
$$592$$ 6.94427 0.285408
$$593$$ −28.8328 −1.18402 −0.592011 0.805930i $$-0.701666\pi$$
−0.592011 + 0.805930i $$0.701666\pi$$
$$594$$ 0 0
$$595$$ 8.94427 0.366679
$$596$$ −1.05573 −0.0432443
$$597$$ 3.05573 0.125063
$$598$$ 4.47214 0.182879
$$599$$ 33.8885 1.38465 0.692324 0.721587i $$-0.256587\pi$$
0.692324 + 0.721587i $$0.256587\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ 40.8328 1.66561 0.832803 0.553570i $$-0.186735\pi$$
0.832803 + 0.553570i $$0.186735\pi$$
$$602$$ −4.94427 −0.201513
$$603$$ 4.94427 0.201346
$$604$$ 4.94427 0.201180
$$605$$ −22.0000 −0.894427
$$606$$ −9.41641 −0.382515
$$607$$ −28.3607 −1.15112 −0.575562 0.817758i $$-0.695217\pi$$
−0.575562 + 0.817758i $$0.695217\pi$$
$$608$$ 2.47214 0.100258
$$609$$ −2.00000 −0.0810441
$$610$$ −12.0000 −0.485866
$$611$$ 11.0557 0.447267
$$612$$ 4.47214 0.180775
$$613$$ 40.8328 1.64922 0.824611 0.565700i $$-0.191394\pi$$
0.824611 + 0.565700i $$0.191394\pi$$
$$614$$ 24.9443 1.00667
$$615$$ −12.0000 −0.483887
$$616$$ 0 0
$$617$$ 26.0000 1.04672 0.523360 0.852111i $$-0.324678\pi$$
0.523360 + 0.852111i $$0.324678\pi$$
$$618$$ −4.94427 −0.198888
$$619$$ −34.4721 −1.38555 −0.692776 0.721153i $$-0.743613\pi$$
−0.692776 + 0.721153i $$0.743613\pi$$
$$620$$ −4.94427 −0.198567
$$621$$ −1.00000 −0.0401286
$$622$$ −2.47214 −0.0991236
$$623$$ 9.41641 0.377260
$$624$$ −4.47214 −0.179029
$$625$$ −19.0000 −0.760000
$$626$$ −13.4164 −0.536228
$$627$$ 0 0
$$628$$ −2.94427 −0.117489
$$629$$ 31.0557 1.23827
$$630$$ 2.00000 0.0796819
$$631$$ −11.0557 −0.440122 −0.220061 0.975486i $$-0.570626\pi$$
−0.220061 + 0.975486i $$0.570626\pi$$
$$632$$ −4.94427 −0.196673
$$633$$ −0.944272 −0.0375314
$$634$$ −13.0557 −0.518509
$$635$$ 41.8885 1.66230
$$636$$ −10.9443 −0.433969
$$637$$ −4.47214 −0.177192
$$638$$ 0 0
$$639$$ 4.94427 0.195592
$$640$$ 2.00000 0.0790569
$$641$$ 10.0000 0.394976 0.197488 0.980305i $$-0.436722\pi$$
0.197488 + 0.980305i $$0.436722\pi$$
$$642$$ −8.00000 −0.315735
$$643$$ −10.4721 −0.412981 −0.206490 0.978449i $$-0.566204\pi$$
−0.206490 + 0.978449i $$0.566204\pi$$
$$644$$ −1.00000 −0.0394055
$$645$$ −9.88854 −0.389361
$$646$$ 11.0557 0.434982
$$647$$ 12.3607 0.485948 0.242974 0.970033i $$-0.421877\pi$$
0.242974 + 0.970033i $$0.421877\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ 4.47214 0.175412
$$651$$ −2.47214 −0.0968906
$$652$$ −8.94427 −0.350285
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ −2.94427 −0.115130
$$655$$ 24.0000 0.937758
$$656$$ −6.00000 −0.234261
$$657$$ 14.9443 0.583032
$$658$$ −2.47214 −0.0963739
$$659$$ 46.8328 1.82435 0.912174 0.409804i $$-0.134403\pi$$
0.912174 + 0.409804i $$0.134403\pi$$
$$660$$ 0 0
$$661$$ −30.0000 −1.16686 −0.583432 0.812162i $$-0.698291\pi$$
−0.583432 + 0.812162i $$0.698291\pi$$
$$662$$ −13.8885 −0.539794
$$663$$ −20.0000 −0.776736
$$664$$ 2.47214 0.0959375
$$665$$ 4.94427 0.191731
$$666$$ 6.94427 0.269085
$$667$$ 2.00000 0.0774403
$$668$$ −15.4164 −0.596479
$$669$$ 18.4721 0.714174
$$670$$ 9.88854 0.382028
$$671$$ 0 0
$$672$$ 1.00000 0.0385758
$$673$$ −30.0000 −1.15642 −0.578208 0.815890i $$-0.696248\pi$$
−0.578208 + 0.815890i $$0.696248\pi$$
$$674$$ −2.94427 −0.113409
$$675$$ −1.00000 −0.0384900
$$676$$ 7.00000 0.269231
$$677$$ −4.11146 −0.158016 −0.0790080 0.996874i $$-0.525175\pi$$
−0.0790080 + 0.996874i $$0.525175\pi$$
$$678$$ −18.9443 −0.727550
$$679$$ −0.472136 −0.0181189
$$680$$ 8.94427 0.342997
$$681$$ 5.52786 0.211828
$$682$$ 0 0
$$683$$ 16.9443 0.648355 0.324177 0.945996i $$-0.394913\pi$$
0.324177 + 0.945996i $$0.394913\pi$$
$$684$$ 2.47214 0.0945245
$$685$$ −28.0000 −1.06983
$$686$$ 1.00000 0.0381802
$$687$$ −14.0000 −0.534133
$$688$$ −4.94427 −0.188499
$$689$$ 48.9443 1.86463
$$690$$ −2.00000 −0.0761387
$$691$$ −15.0557 −0.572747 −0.286373 0.958118i $$-0.592450\pi$$
−0.286373 + 0.958118i $$0.592450\pi$$
$$692$$ 0.472136 0.0179479
$$693$$ 0 0
$$694$$ −0.944272 −0.0358441
$$695$$ 17.8885 0.678551
$$696$$ −2.00000 −0.0758098
$$697$$ −26.8328 −1.01637
$$698$$ 24.4721 0.926284
$$699$$ −6.00000 −0.226941
$$700$$ −1.00000 −0.0377964
$$701$$ −20.8328 −0.786845 −0.393422 0.919358i $$-0.628709\pi$$
−0.393422 + 0.919358i $$0.628709\pi$$
$$702$$ −4.47214 −0.168790
$$703$$ 17.1672 0.647473
$$704$$ 0 0
$$705$$ −4.94427 −0.186212
$$706$$ 2.00000 0.0752710
$$707$$ −9.41641 −0.354140
$$708$$ 4.00000 0.150329
$$709$$ 6.94427 0.260798 0.130399 0.991462i $$-0.458374\pi$$
0.130399 + 0.991462i $$0.458374\pi$$
$$710$$ 9.88854 0.371110
$$711$$ −4.94427 −0.185425
$$712$$ 9.41641 0.352895
$$713$$ 2.47214 0.0925822
$$714$$ 4.47214 0.167365
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ 20.9443 0.782178
$$718$$ −30.8328 −1.15067
$$719$$ −5.52786 −0.206155 −0.103077 0.994673i $$-0.532869\pi$$
−0.103077 + 0.994673i $$0.532869\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ −4.94427 −0.184134
$$722$$ −12.8885 −0.479662
$$723$$ 20.4721 0.761367
$$724$$ 2.00000 0.0743294
$$725$$ 2.00000 0.0742781
$$726$$ −11.0000 −0.408248
$$727$$ 3.05573 0.113331 0.0566653 0.998393i $$-0.481953\pi$$
0.0566653 + 0.998393i $$0.481953\pi$$
$$728$$ −4.47214 −0.165748
$$729$$ 1.00000 0.0370370
$$730$$ 29.8885 1.10622
$$731$$ −22.1115 −0.817822
$$732$$ −6.00000 −0.221766
$$733$$ −50.9443 −1.88167 −0.940835 0.338866i $$-0.889957\pi$$
−0.940835 + 0.338866i $$0.889957\pi$$
$$734$$ 30.8328 1.13806
$$735$$ 2.00000 0.0737711
$$736$$ −1.00000 −0.0368605
$$737$$ 0 0
$$738$$ −6.00000 −0.220863
$$739$$ −8.94427 −0.329020 −0.164510 0.986375i $$-0.552604\pi$$
−0.164510 + 0.986375i $$0.552604\pi$$
$$740$$ 13.8885 0.510553
$$741$$ −11.0557 −0.406142
$$742$$ −10.9443 −0.401777
$$743$$ −22.8328 −0.837655 −0.418827 0.908066i $$-0.637559\pi$$
−0.418827 + 0.908066i $$0.637559\pi$$
$$744$$ −2.47214 −0.0906329
$$745$$ −2.11146 −0.0773578
$$746$$ 6.94427 0.254248
$$747$$ 2.47214 0.0904507
$$748$$ 0 0
$$749$$ −8.00000 −0.292314
$$750$$ −12.0000 −0.438178
$$751$$ 17.8885 0.652762 0.326381 0.945238i $$-0.394171\pi$$
0.326381 + 0.945238i $$0.394171\pi$$
$$752$$ −2.47214 −0.0901495
$$753$$ −5.52786 −0.201447
$$754$$ 8.94427 0.325731
$$755$$ 9.88854 0.359881
$$756$$ 1.00000 0.0363696
$$757$$ −42.9443 −1.56084 −0.780418 0.625258i $$-0.784994\pi$$
−0.780418 + 0.625258i $$0.784994\pi$$
$$758$$ 24.0000 0.871719
$$759$$ 0 0
$$760$$ 4.94427 0.179348
$$761$$ 21.0557 0.763270 0.381635 0.924313i $$-0.375361\pi$$
0.381635 + 0.924313i $$0.375361\pi$$
$$762$$ 20.9443 0.758731
$$763$$ −2.94427 −0.106590
$$764$$ −3.05573 −0.110552
$$765$$ 8.94427 0.323381
$$766$$ 16.0000 0.578103
$$767$$ −17.8885 −0.645918
$$768$$ 1.00000 0.0360844
$$769$$ 30.3607 1.09483 0.547417 0.836860i $$-0.315611\pi$$
0.547417 + 0.836860i $$0.315611\pi$$
$$770$$ 0 0
$$771$$ −2.94427 −0.106035
$$772$$ 11.8885 0.427878
$$773$$ 5.05573 0.181842 0.0909210 0.995858i $$-0.471019\pi$$
0.0909210 + 0.995858i $$0.471019\pi$$
$$774$$ −4.94427 −0.177718
$$775$$ 2.47214 0.0888017
$$776$$ −0.472136 −0.0169487
$$777$$ 6.94427 0.249124
$$778$$ −18.9443 −0.679185
$$779$$ −14.8328 −0.531441
$$780$$ −8.94427 −0.320256
$$781$$ 0 0
$$782$$ −4.47214 −0.159923
$$783$$ −2.00000 −0.0714742
$$784$$ 1.00000 0.0357143
$$785$$ −5.88854 −0.210171
$$786$$ 12.0000 0.428026
$$787$$ −0.583592 −0.0208028 −0.0104014 0.999946i $$-0.503311\pi$$
−0.0104014 + 0.999946i $$0.503311\pi$$
$$788$$ −14.9443 −0.532368
$$789$$ −24.0000 −0.854423
$$790$$ −9.88854 −0.351819
$$791$$ −18.9443 −0.673581
$$792$$ 0 0
$$793$$ 26.8328 0.952861
$$794$$ −36.4721 −1.29435
$$795$$ −21.8885 −0.776307
$$796$$ 3.05573 0.108307
$$797$$ −28.8328 −1.02131 −0.510655 0.859785i $$-0.670597\pi$$
−0.510655 + 0.859785i $$0.670597\pi$$
$$798$$ 2.47214 0.0875127
$$799$$ −11.0557 −0.391124
$$800$$ −1.00000 −0.0353553
$$801$$ 9.41641 0.332712
$$802$$ −22.0000 −0.776847
$$803$$ 0 0
$$804$$ 4.94427 0.174371
$$805$$ −2.00000 −0.0704907
$$806$$ 11.0557 0.389421
$$807$$ 16.4721 0.579847
$$808$$ −9.41641 −0.331268
$$809$$ 35.8885 1.26177 0.630887 0.775875i $$-0.282691\pi$$
0.630887 + 0.775875i $$0.282691\pi$$
$$810$$ 2.00000 0.0702728
$$811$$ −37.8885 −1.33045 −0.665223 0.746644i $$-0.731664\pi$$
−0.665223 + 0.746644i $$0.731664\pi$$
$$812$$ −2.00000 −0.0701862
$$813$$ 15.4164 0.540677
$$814$$ 0 0
$$815$$ −17.8885 −0.626608
$$816$$ 4.47214 0.156556
$$817$$ −12.2229 −0.427626
$$818$$ −15.8885 −0.555530
$$819$$ −4.47214 −0.156269
$$820$$ −12.0000 −0.419058
$$821$$ −16.8328 −0.587469 −0.293735 0.955887i $$-0.594898\pi$$
−0.293735 + 0.955887i $$0.594898\pi$$
$$822$$ −14.0000 −0.488306
$$823$$ 20.9443 0.730071 0.365036 0.930994i $$-0.381057\pi$$
0.365036 + 0.930994i $$0.381057\pi$$
$$824$$ −4.94427 −0.172242
$$825$$ 0 0
$$826$$ 4.00000 0.139178
$$827$$ 33.8885 1.17842 0.589210 0.807980i $$-0.299439\pi$$
0.589210 + 0.807980i $$0.299439\pi$$
$$828$$ −1.00000 −0.0347524
$$829$$ 13.4164 0.465971 0.232986 0.972480i $$-0.425151\pi$$
0.232986 + 0.972480i $$0.425151\pi$$
$$830$$ 4.94427 0.171618
$$831$$ −11.8885 −0.412409
$$832$$ −4.47214 −0.155043
$$833$$ 4.47214 0.154950
$$834$$ 8.94427 0.309715
$$835$$ −30.8328 −1.06701
$$836$$ 0 0
$$837$$ −2.47214 −0.0854495
$$838$$ −7.41641 −0.256196
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 2.00000 0.0690066
$$841$$ −25.0000 −0.862069
$$842$$ 32.8328 1.13149
$$843$$ 10.0000 0.344418
$$844$$ −0.944272 −0.0325032
$$845$$ 14.0000 0.481615
$$846$$ −2.47214 −0.0849938
$$847$$ −11.0000 −0.377964
$$848$$ −10.9443 −0.375828
$$849$$ 18.4721 0.633962
$$850$$ −4.47214 −0.153393
$$851$$ −6.94427 −0.238047
$$852$$ 4.94427 0.169388
$$853$$ −6.36068 −0.217786 −0.108893 0.994054i $$-0.534731\pi$$
−0.108893 + 0.994054i $$0.534731\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ 4.94427 0.169091
$$856$$ −8.00000 −0.273434
$$857$$ −47.8885 −1.63584 −0.817921 0.575331i $$-0.804873\pi$$
−0.817921 + 0.575331i $$0.804873\pi$$
$$858$$ 0 0
$$859$$ 32.9443 1.12404 0.562022 0.827122i $$-0.310024\pi$$
0.562022 + 0.827122i $$0.310024\pi$$
$$860$$ −9.88854 −0.337197
$$861$$ −6.00000 −0.204479
$$862$$ 17.8885 0.609286
$$863$$ 20.9443 0.712951 0.356476 0.934305i $$-0.383978\pi$$
0.356476 + 0.934305i $$0.383978\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0.944272 0.0321062
$$866$$ 25.4164 0.863685
$$867$$ 3.00000 0.101885
$$868$$ −2.47214 −0.0839098
$$869$$ 0 0
$$870$$ −4.00000 −0.135613
$$871$$ −22.1115 −0.749218
$$872$$ −2.94427 −0.0997056
$$873$$ −0.472136 −0.0159794
$$874$$ −2.47214 −0.0836212
$$875$$ −12.0000 −0.405674
$$876$$ 14.9443 0.504920
$$877$$ −34.7214 −1.17246 −0.586229 0.810146i $$-0.699388\pi$$
−0.586229 + 0.810146i $$0.699388\pi$$
$$878$$ 28.3607 0.957127
$$879$$ 27.8885 0.940657
$$880$$ 0 0
$$881$$ 22.3607 0.753350 0.376675 0.926345i $$-0.377067\pi$$
0.376675 + 0.926345i $$0.377067\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ 2.83282 0.0953318 0.0476659 0.998863i $$-0.484822\pi$$
0.0476659 + 0.998863i $$0.484822\pi$$
$$884$$ −20.0000 −0.672673
$$885$$ 8.00000 0.268917
$$886$$ 34.8328 1.17023
$$887$$ −5.52786 −0.185608 −0.0928038 0.995684i $$-0.529583\pi$$
−0.0928038 + 0.995684i $$0.529583\pi$$
$$888$$ 6.94427 0.233035
$$889$$ 20.9443 0.702448
$$890$$ 18.8328 0.631277
$$891$$ 0 0
$$892$$ 18.4721 0.618493
$$893$$ −6.11146 −0.204512
$$894$$ −1.05573 −0.0353088
$$895$$ −40.0000 −1.33705
$$896$$ 1.00000 0.0334077
$$897$$ 4.47214 0.149320
$$898$$ 18.0000 0.600668
$$899$$ 4.94427 0.164901
$$900$$ −1.00000 −0.0333333
$$901$$ −48.9443 −1.63057
$$902$$ 0 0
$$903$$ −4.94427 −0.164535
$$904$$ −18.9443 −0.630077
$$905$$ 4.00000 0.132964
$$906$$ 4.94427 0.164262
$$907$$ −41.8885 −1.39089 −0.695443 0.718581i $$-0.744792\pi$$
−0.695443 + 0.718581i $$0.744792\pi$$
$$908$$ 5.52786 0.183449
$$909$$ −9.41641 −0.312323
$$910$$ −8.94427 −0.296500
$$911$$ −48.7214 −1.61421 −0.807105 0.590407i $$-0.798967\pi$$
−0.807105 + 0.590407i $$0.798967\pi$$
$$912$$ 2.47214 0.0818606
$$913$$ 0 0
$$914$$ 2.00000 0.0661541
$$915$$ −12.0000 −0.396708
$$916$$ −14.0000 −0.462573
$$917$$ 12.0000 0.396275
$$918$$ 4.47214 0.147602
$$919$$ 35.0557 1.15638 0.578191 0.815902i $$-0.303759\pi$$
0.578191 + 0.815902i $$0.303759\pi$$
$$920$$ −2.00000 −0.0659380
$$921$$ 24.9443 0.821942
$$922$$ 6.58359 0.216819
$$923$$ −22.1115 −0.727807
$$924$$ 0 0
$$925$$ −6.94427 −0.228326
$$926$$ 6.11146 0.200835
$$927$$ −4.94427 −0.162391
$$928$$ −2.00000 −0.0656532
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ −4.94427 −0.162129
$$931$$ 2.47214 0.0810210
$$932$$ −6.00000 −0.196537
$$933$$ −2.47214 −0.0809341
$$934$$ 33.3050 1.08977
$$935$$ 0 0
$$936$$ −4.47214 −0.146176
$$937$$ 40.2492 1.31488 0.657442 0.753505i $$-0.271638\pi$$
0.657442 + 0.753505i $$0.271638\pi$$
$$938$$ 4.94427 0.161436
$$939$$ −13.4164 −0.437828
$$940$$ −4.94427 −0.161264
$$941$$ −6.00000 −0.195594 −0.0977972 0.995206i $$-0.531180\pi$$
−0.0977972 + 0.995206i $$0.531180\pi$$
$$942$$ −2.94427 −0.0959296
$$943$$ 6.00000 0.195387
$$944$$ 4.00000 0.130189
$$945$$ 2.00000 0.0650600
$$946$$ 0 0
$$947$$ 0.944272 0.0306847 0.0153424 0.999882i $$-0.495116\pi$$
0.0153424 + 0.999882i $$0.495116\pi$$
$$948$$ −4.94427 −0.160582
$$949$$ −66.8328 −2.16949
$$950$$ −2.47214 −0.0802067
$$951$$ −13.0557 −0.423361
$$952$$ 4.47214 0.144943
$$953$$ −9.05573 −0.293344 −0.146672 0.989185i $$-0.546856\pi$$
−0.146672 + 0.989185i $$0.546856\pi$$
$$954$$ −10.9443 −0.354334
$$955$$ −6.11146 −0.197762
$$956$$ 20.9443 0.677386
$$957$$ 0 0
$$958$$ −4.94427 −0.159742
$$959$$ −14.0000 −0.452084
$$960$$ 2.00000 0.0645497
$$961$$ −24.8885 −0.802856
$$962$$ −31.0557 −1.00128
$$963$$ −8.00000 −0.257796
$$964$$ 20.4721 0.659363
$$965$$ 23.7771 0.765412
$$966$$ −1.00000 −0.0321745
$$967$$ −16.0000 −0.514525 −0.257263 0.966342i $$-0.582821\pi$$
−0.257263 + 0.966342i $$0.582821\pi$$
$$968$$ −11.0000 −0.353553
$$969$$ 11.0557 0.355161
$$970$$ −0.944272 −0.0303187
$$971$$ −23.4164 −0.751468 −0.375734 0.926727i $$-0.622609\pi$$
−0.375734 + 0.926727i $$0.622609\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 8.94427 0.286740
$$974$$ 24.0000 0.769010
$$975$$ 4.47214 0.143223
$$976$$ −6.00000 −0.192055
$$977$$ −6.00000 −0.191957 −0.0959785 0.995383i $$-0.530598\pi$$
−0.0959785 + 0.995383i $$0.530598\pi$$
$$978$$ −8.94427 −0.286006
$$979$$ 0 0
$$980$$ 2.00000 0.0638877
$$981$$ −2.94427 −0.0940034
$$982$$ 40.9443 1.30658
$$983$$ −22.1115 −0.705246 −0.352623 0.935765i $$-0.614710\pi$$
−0.352623 + 0.935765i $$0.614710\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ −29.8885 −0.952328
$$986$$ −8.94427 −0.284844
$$987$$ −2.47214 −0.0786890
$$988$$ −11.0557 −0.351730
$$989$$ 4.94427 0.157219
$$990$$ 0 0
$$991$$ 60.9443 1.93596 0.967979 0.251030i $$-0.0807693\pi$$
0.967979 + 0.251030i $$0.0807693\pi$$
$$992$$ −2.47214 −0.0784904
$$993$$ −13.8885 −0.440740
$$994$$ 4.94427 0.156823
$$995$$ 6.11146 0.193746
$$996$$ 2.47214 0.0783326
$$997$$ −46.3607 −1.46826 −0.734129 0.679010i $$-0.762409\pi$$
−0.734129 + 0.679010i $$0.762409\pi$$
$$998$$ 32.9443 1.04283
$$999$$ 6.94427 0.219707
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 966.2.a.p.1.1 2
3.2 odd 2 2898.2.a.v.1.1 2
4.3 odd 2 7728.2.a.bd.1.1 2
7.6 odd 2 6762.2.a.bz.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.p.1.1 2 1.1 even 1 trivial
2898.2.a.v.1.1 2 3.2 odd 2
6762.2.a.bz.1.2 2 7.6 odd 2
7728.2.a.bd.1.1 2 4.3 odd 2