# Properties

 Label 3969.2.a.p.1.3 Level $3969$ Weight $2$ Character 3969.1 Self dual yes Analytic conductor $31.693$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, 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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3969 = 3^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3969.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: $$31.6926245622$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.321.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 1$$ x^3 - x^2 - 4*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.46050$$ of defining polynomial Character $$\chi$$ $$=$$ 3969.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.46050 q^{2} +4.05408 q^{4} -2.59358 q^{5} +5.05408 q^{8} +O(q^{10})$$ $$q+2.46050 q^{2} +4.05408 q^{4} -2.59358 q^{5} +5.05408 q^{8} -6.38151 q^{10} -4.51459 q^{11} -1.00000 q^{13} +4.32743 q^{16} -0.945916 q^{17} +4.05408 q^{19} -10.5146 q^{20} -11.1082 q^{22} +0.273346 q^{23} +1.72665 q^{25} -2.46050 q^{26} -2.46050 q^{29} -2.32743 q^{31} +0.539495 q^{32} -2.32743 q^{34} +1.78074 q^{37} +9.97509 q^{38} -13.1082 q^{40} -6.40642 q^{41} -10.4356 q^{43} -18.3025 q^{44} +0.672570 q^{46} -12.1623 q^{47} +4.24844 q^{50} -4.05408 q^{52} +6.27335 q^{53} +11.7089 q^{55} -6.05408 q^{58} -2.72665 q^{59} +2.27335 q^{61} -5.72665 q^{62} -7.32743 q^{64} +2.59358 q^{65} -15.8171 q^{67} -3.83482 q^{68} -3.27335 q^{71} +1.50739 q^{73} +4.38151 q^{74} +16.4356 q^{76} +14.7089 q^{79} -11.2235 q^{80} -15.7630 q^{82} -0.945916 q^{83} +2.45331 q^{85} -25.6768 q^{86} -22.8171 q^{88} -14.3566 q^{89} +1.10817 q^{92} -29.9253 q^{94} -10.5146 q^{95} +11.4897 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 3 q^{4} - 5 q^{5} + 6 q^{8}+O(q^{10})$$ 3 * q + q^2 + 3 * q^4 - 5 * q^5 + 6 * q^8 $$3 q + q^{2} + 3 q^{4} - 5 q^{5} + 6 q^{8} + 2 q^{11} - 3 q^{13} + 3 q^{16} - 12 q^{17} + 3 q^{19} - 16 q^{20} - 15 q^{22} + 6 q^{25} - q^{26} - q^{29} + 3 q^{31} + 8 q^{32} + 3 q^{34} - 3 q^{37} + 8 q^{38} - 21 q^{40} - 22 q^{41} - 3 q^{43} - 23 q^{44} + 12 q^{46} - 9 q^{47} - 10 q^{50} - 3 q^{52} + 18 q^{53} + 6 q^{55} - 9 q^{58} - 9 q^{59} + 6 q^{61} - 18 q^{62} - 12 q^{64} + 5 q^{65} + 6 q^{68} - 9 q^{71} - 3 q^{73} - 6 q^{74} + 21 q^{76} + 15 q^{79} + 11 q^{80} - 9 q^{82} - 12 q^{83} + 9 q^{85} - 34 q^{86} - 21 q^{88} - 2 q^{89} - 15 q^{92} - 24 q^{94} - 16 q^{95} - 3 q^{97}+O(q^{100})$$ 3 * q + q^2 + 3 * q^4 - 5 * q^5 + 6 * q^8 + 2 * q^11 - 3 * q^13 + 3 * q^16 - 12 * q^17 + 3 * q^19 - 16 * q^20 - 15 * q^22 + 6 * q^25 - q^26 - q^29 + 3 * q^31 + 8 * q^32 + 3 * q^34 - 3 * q^37 + 8 * q^38 - 21 * q^40 - 22 * q^41 - 3 * q^43 - 23 * q^44 + 12 * q^46 - 9 * q^47 - 10 * q^50 - 3 * q^52 + 18 * q^53 + 6 * q^55 - 9 * q^58 - 9 * q^59 + 6 * q^61 - 18 * q^62 - 12 * q^64 + 5 * q^65 + 6 * q^68 - 9 * q^71 - 3 * q^73 - 6 * q^74 + 21 * q^76 + 15 * q^79 + 11 * q^80 - 9 * q^82 - 12 * q^83 + 9 * q^85 - 34 * q^86 - 21 * q^88 - 2 * q^89 - 15 * q^92 - 24 * q^94 - 16 * q^95 - 3 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.46050 1.73984 0.869920 0.493193i $$-0.164170\pi$$
0.869920 + 0.493193i $$0.164170\pi$$
$$3$$ 0 0
$$4$$ 4.05408 2.02704
$$5$$ −2.59358 −1.15988 −0.579942 0.814658i $$-0.696925\pi$$
−0.579942 + 0.814658i $$0.696925\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 5.05408 1.78689
$$9$$ 0 0
$$10$$ −6.38151 −2.01801
$$11$$ −4.51459 −1.36120 −0.680600 0.732655i $$-0.738281\pi$$
−0.680600 + 0.732655i $$0.738281\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 4.32743 1.08186
$$17$$ −0.945916 −0.229418 −0.114709 0.993399i $$-0.536594\pi$$
−0.114709 + 0.993399i $$0.536594\pi$$
$$18$$ 0 0
$$19$$ 4.05408 0.930071 0.465035 0.885292i $$-0.346042\pi$$
0.465035 + 0.885292i $$0.346042\pi$$
$$20$$ −10.5146 −2.35113
$$21$$ 0 0
$$22$$ −11.1082 −2.36827
$$23$$ 0.273346 0.0569966 0.0284983 0.999594i $$-0.490927\pi$$
0.0284983 + 0.999594i $$0.490927\pi$$
$$24$$ 0 0
$$25$$ 1.72665 0.345331
$$26$$ −2.46050 −0.482545
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.46050 −0.456904 −0.228452 0.973555i $$-0.573366\pi$$
−0.228452 + 0.973555i $$0.573366\pi$$
$$30$$ 0 0
$$31$$ −2.32743 −0.418019 −0.209009 0.977914i $$-0.567024\pi$$
−0.209009 + 0.977914i $$0.567024\pi$$
$$32$$ 0.539495 0.0953702
$$33$$ 0 0
$$34$$ −2.32743 −0.399151
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 1.78074 0.292752 0.146376 0.989229i $$-0.453239\pi$$
0.146376 + 0.989229i $$0.453239\pi$$
$$38$$ 9.97509 1.61817
$$39$$ 0 0
$$40$$ −13.1082 −2.07258
$$41$$ −6.40642 −1.00051 −0.500257 0.865877i $$-0.666761\pi$$
−0.500257 + 0.865877i $$0.666761\pi$$
$$42$$ 0 0
$$43$$ −10.4356 −1.59141 −0.795707 0.605682i $$-0.792900\pi$$
−0.795707 + 0.605682i $$0.792900\pi$$
$$44$$ −18.3025 −2.75921
$$45$$ 0 0
$$46$$ 0.672570 0.0991650
$$47$$ −12.1623 −1.77405 −0.887023 0.461724i $$-0.847231\pi$$
−0.887023 + 0.461724i $$0.847231\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 4.24844 0.600820
$$51$$ 0 0
$$52$$ −4.05408 −0.562200
$$53$$ 6.27335 0.861710 0.430855 0.902421i $$-0.358212\pi$$
0.430855 + 0.902421i $$0.358212\pi$$
$$54$$ 0 0
$$55$$ 11.7089 1.57883
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −6.05408 −0.794940
$$59$$ −2.72665 −0.354980 −0.177490 0.984123i $$-0.556798\pi$$
−0.177490 + 0.984123i $$0.556798\pi$$
$$60$$ 0 0
$$61$$ 2.27335 0.291072 0.145536 0.989353i $$-0.453509\pi$$
0.145536 + 0.989353i $$0.453509\pi$$
$$62$$ −5.72665 −0.727286
$$63$$ 0 0
$$64$$ −7.32743 −0.915929
$$65$$ 2.59358 0.321694
$$66$$ 0 0
$$67$$ −15.8171 −1.93237 −0.966184 0.257854i $$-0.916985\pi$$
−0.966184 + 0.257854i $$0.916985\pi$$
$$68$$ −3.83482 −0.465041
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −3.27335 −0.388475 −0.194237 0.980955i $$-0.562223\pi$$
−0.194237 + 0.980955i $$0.562223\pi$$
$$72$$ 0 0
$$73$$ 1.50739 0.176427 0.0882134 0.996102i $$-0.471884\pi$$
0.0882134 + 0.996102i $$0.471884\pi$$
$$74$$ 4.38151 0.509341
$$75$$ 0 0
$$76$$ 16.4356 1.88529
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 14.7089 1.65489 0.827443 0.561550i $$-0.189795\pi$$
0.827443 + 0.561550i $$0.189795\pi$$
$$80$$ −11.2235 −1.25483
$$81$$ 0 0
$$82$$ −15.7630 −1.74074
$$83$$ −0.945916 −0.103828 −0.0519139 0.998652i $$-0.516532\pi$$
−0.0519139 + 0.998652i $$0.516532\pi$$
$$84$$ 0 0
$$85$$ 2.45331 0.266099
$$86$$ −25.6768 −2.76881
$$87$$ 0 0
$$88$$ −22.8171 −2.43231
$$89$$ −14.3566 −1.52180 −0.760899 0.648871i $$-0.775242\pi$$
−0.760899 + 0.648871i $$0.775242\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 1.10817 0.115535
$$93$$ 0 0
$$94$$ −29.9253 −3.08656
$$95$$ −10.5146 −1.07877
$$96$$ 0 0
$$97$$ 11.4897 1.16660 0.583300 0.812257i $$-0.301761\pi$$
0.583300 + 0.812257i $$0.301761\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 7.00000 0.700000
$$101$$ −3.67977 −0.366150 −0.183075 0.983099i $$-0.558605\pi$$
−0.183075 + 0.983099i $$0.558605\pi$$
$$102$$ 0 0
$$103$$ 9.72665 0.958396 0.479198 0.877707i $$-0.340928\pi$$
0.479198 + 0.877707i $$0.340928\pi$$
$$104$$ −5.05408 −0.495594
$$105$$ 0 0
$$106$$ 15.4356 1.49924
$$107$$ 1.37432 0.132860 0.0664301 0.997791i $$-0.478839\pi$$
0.0664301 + 0.997791i $$0.478839\pi$$
$$108$$ 0 0
$$109$$ −3.39922 −0.325587 −0.162793 0.986660i $$-0.552050\pi$$
−0.162793 + 0.986660i $$0.552050\pi$$
$$110$$ 28.8099 2.74692
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −10.3887 −0.977288 −0.488644 0.872483i $$-0.662508\pi$$
−0.488644 + 0.872483i $$0.662508\pi$$
$$114$$ 0 0
$$115$$ −0.708945 −0.0661095
$$116$$ −9.97509 −0.926164
$$117$$ 0 0
$$118$$ −6.70895 −0.617608
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 9.38151 0.852865
$$122$$ 5.59358 0.506419
$$123$$ 0 0
$$124$$ −9.43560 −0.847342
$$125$$ 8.48968 0.759340
$$126$$ 0 0
$$127$$ 0.672570 0.0596809 0.0298405 0.999555i $$-0.490500\pi$$
0.0298405 + 0.999555i $$0.490500\pi$$
$$128$$ −19.1082 −1.68894
$$129$$ 0 0
$$130$$ 6.38151 0.559696
$$131$$ 7.91381 0.691433 0.345717 0.938339i $$-0.387636\pi$$
0.345717 + 0.938339i $$0.387636\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −38.9181 −3.36201
$$135$$ 0 0
$$136$$ −4.78074 −0.409945
$$137$$ 3.67257 0.313769 0.156884 0.987617i $$-0.449855\pi$$
0.156884 + 0.987617i $$0.449855\pi$$
$$138$$ 0 0
$$139$$ 2.05408 0.174225 0.0871126 0.996198i $$-0.472236\pi$$
0.0871126 + 0.996198i $$0.472236\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −8.05408 −0.675884
$$143$$ 4.51459 0.377529
$$144$$ 0 0
$$145$$ 6.38151 0.529956
$$146$$ 3.70895 0.306954
$$147$$ 0 0
$$148$$ 7.21926 0.593420
$$149$$ 13.5438 1.10955 0.554774 0.832001i $$-0.312805\pi$$
0.554774 + 0.832001i $$0.312805\pi$$
$$150$$ 0 0
$$151$$ 9.92821 0.807946 0.403973 0.914771i $$-0.367629\pi$$
0.403973 + 0.914771i $$0.367629\pi$$
$$152$$ 20.4897 1.66193
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 6.03638 0.484853
$$156$$ 0 0
$$157$$ −6.05408 −0.483169 −0.241584 0.970380i $$-0.577667\pi$$
−0.241584 + 0.970380i $$0.577667\pi$$
$$158$$ 36.1914 2.87924
$$159$$ 0 0
$$160$$ −1.39922 −0.110618
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 17.8171 1.39554 0.697772 0.716320i $$-0.254175\pi$$
0.697772 + 0.716320i $$0.254175\pi$$
$$164$$ −25.9722 −2.02809
$$165$$ 0 0
$$166$$ −2.32743 −0.180644
$$167$$ −8.46770 −0.655250 −0.327625 0.944808i $$-0.606248\pi$$
−0.327625 + 0.944808i $$0.606248\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 6.03638 0.462969
$$171$$ 0 0
$$172$$ −42.3068 −3.22586
$$173$$ 17.3566 1.31960 0.659799 0.751442i $$-0.270641\pi$$
0.659799 + 0.751442i $$0.270641\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −19.5366 −1.47262
$$177$$ 0 0
$$178$$ −35.3245 −2.64768
$$179$$ 11.3494 0.848295 0.424147 0.905593i $$-0.360574\pi$$
0.424147 + 0.905593i $$0.360574\pi$$
$$180$$ 0 0
$$181$$ −21.8889 −1.62699 −0.813495 0.581572i $$-0.802438\pi$$
−0.813495 + 0.581572i $$0.802438\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 1.38151 0.101847
$$185$$ −4.61849 −0.339558
$$186$$ 0 0
$$187$$ 4.27042 0.312284
$$188$$ −49.3068 −3.59607
$$189$$ 0 0
$$190$$ −25.8712 −1.87689
$$191$$ 0.701748 0.0507767 0.0253883 0.999678i $$-0.491918\pi$$
0.0253883 + 0.999678i $$0.491918\pi$$
$$192$$ 0 0
$$193$$ 12.1445 0.874183 0.437092 0.899417i $$-0.356009\pi$$
0.437092 + 0.899417i $$0.356009\pi$$
$$194$$ 28.2704 2.02970
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 16.4107 1.16921 0.584607 0.811317i $$-0.301249\pi$$
0.584607 + 0.811317i $$0.301249\pi$$
$$198$$ 0 0
$$199$$ −22.7060 −1.60959 −0.804794 0.593555i $$-0.797724\pi$$
−0.804794 + 0.593555i $$0.797724\pi$$
$$200$$ 8.72665 0.617068
$$201$$ 0 0
$$202$$ −9.05408 −0.637043
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 16.6156 1.16048
$$206$$ 23.9325 1.66745
$$207$$ 0 0
$$208$$ −4.32743 −0.300053
$$209$$ −18.3025 −1.26601
$$210$$ 0 0
$$211$$ 4.56148 0.314025 0.157012 0.987597i $$-0.449814\pi$$
0.157012 + 0.987597i $$0.449814\pi$$
$$212$$ 25.4327 1.74672
$$213$$ 0 0
$$214$$ 3.38151 0.231156
$$215$$ 27.0656 1.84586
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −8.36381 −0.566468
$$219$$ 0 0
$$220$$ 47.4690 3.20036
$$221$$ 0.945916 0.0636292
$$222$$ 0 0
$$223$$ −13.3245 −0.892275 −0.446137 0.894964i $$-0.647201\pi$$
−0.446137 + 0.894964i $$0.647201\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −25.5615 −1.70032
$$227$$ 1.38151 0.0916943 0.0458472 0.998948i $$-0.485401\pi$$
0.0458472 + 0.998948i $$0.485401\pi$$
$$228$$ 0 0
$$229$$ 17.9794 1.18811 0.594055 0.804424i $$-0.297526\pi$$
0.594055 + 0.804424i $$0.297526\pi$$
$$230$$ −1.74436 −0.115020
$$231$$ 0 0
$$232$$ −12.4356 −0.816437
$$233$$ 18.9823 1.24357 0.621786 0.783187i $$-0.286408\pi$$
0.621786 + 0.783187i $$0.286408\pi$$
$$234$$ 0 0
$$235$$ 31.5438 2.05769
$$236$$ −11.0541 −0.719560
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −4.89183 −0.316426 −0.158213 0.987405i $$-0.550573\pi$$
−0.158213 + 0.987405i $$0.550573\pi$$
$$240$$ 0 0
$$241$$ 26.1593 1.68507 0.842535 0.538641i $$-0.181062\pi$$
0.842535 + 0.538641i $$0.181062\pi$$
$$242$$ 23.0833 1.48385
$$243$$ 0 0
$$244$$ 9.21634 0.590016
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4.05408 −0.257955
$$248$$ −11.7630 −0.746953
$$249$$ 0 0
$$250$$ 20.8889 1.32113
$$251$$ 18.4576 1.16503 0.582516 0.812819i $$-0.302068\pi$$
0.582516 + 0.812819i $$0.302068\pi$$
$$252$$ 0 0
$$253$$ −1.23405 −0.0775838
$$254$$ 1.65486 0.103835
$$255$$ 0 0
$$256$$ −32.3609 −2.02256
$$257$$ −11.7339 −0.731938 −0.365969 0.930627i $$-0.619262\pi$$
−0.365969 + 0.930627i $$0.619262\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 10.5146 0.652087
$$261$$ 0 0
$$262$$ 19.4720 1.20298
$$263$$ 7.52179 0.463813 0.231907 0.972738i $$-0.425504\pi$$
0.231907 + 0.972738i $$0.425504\pi$$
$$264$$ 0 0
$$265$$ −16.2704 −0.999484
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −64.1239 −3.91699
$$269$$ −18.8348 −1.14838 −0.574190 0.818722i $$-0.694683\pi$$
−0.574190 + 0.818722i $$0.694683\pi$$
$$270$$ 0 0
$$271$$ 23.9823 1.45682 0.728410 0.685141i $$-0.240260\pi$$
0.728410 + 0.685141i $$0.240260\pi$$
$$272$$ −4.09338 −0.248198
$$273$$ 0 0
$$274$$ 9.03638 0.545907
$$275$$ −7.79513 −0.470064
$$276$$ 0 0
$$277$$ 7.16225 0.430338 0.215169 0.976577i $$-0.430970\pi$$
0.215169 + 0.976577i $$0.430970\pi$$
$$278$$ 5.05408 0.303124
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −14.8817 −0.887768 −0.443884 0.896084i $$-0.646400\pi$$
−0.443884 + 0.896084i $$0.646400\pi$$
$$282$$ 0 0
$$283$$ −19.9971 −1.18870 −0.594351 0.804205i $$-0.702591\pi$$
−0.594351 + 0.804205i $$0.702591\pi$$
$$284$$ −13.2704 −0.787455
$$285$$ 0 0
$$286$$ 11.1082 0.656840
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −16.1052 −0.947367
$$290$$ 15.7017 0.922038
$$291$$ 0 0
$$292$$ 6.11109 0.357625
$$293$$ 15.0656 0.880139 0.440070 0.897964i $$-0.354954\pi$$
0.440070 + 0.897964i $$0.354954\pi$$
$$294$$ 0 0
$$295$$ 7.07179 0.411736
$$296$$ 9.00000 0.523114
$$297$$ 0 0
$$298$$ 33.3245 1.93044
$$299$$ −0.273346 −0.0158080
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 24.4284 1.40570
$$303$$ 0 0
$$304$$ 17.5438 1.00620
$$305$$ −5.89610 −0.337610
$$306$$ 0 0
$$307$$ 27.2704 1.55641 0.778203 0.628013i $$-0.216132\pi$$
0.778203 + 0.628013i $$0.216132\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 14.8525 0.843567
$$311$$ −15.9823 −0.906273 −0.453136 0.891441i $$-0.649695\pi$$
−0.453136 + 0.891441i $$0.649695\pi$$
$$312$$ 0 0
$$313$$ −11.5979 −0.655549 −0.327775 0.944756i $$-0.606299\pi$$
−0.327775 + 0.944756i $$0.606299\pi$$
$$314$$ −14.8961 −0.840636
$$315$$ 0 0
$$316$$ 59.6313 3.35452
$$317$$ 2.01771 0.113326 0.0566629 0.998393i $$-0.481954\pi$$
0.0566629 + 0.998393i $$0.481954\pi$$
$$318$$ 0 0
$$319$$ 11.1082 0.621938
$$320$$ 19.0043 1.06237
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −3.83482 −0.213375
$$324$$ 0 0
$$325$$ −1.72665 −0.0957775
$$326$$ 43.8391 2.42802
$$327$$ 0 0
$$328$$ −32.3786 −1.78781
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −19.7089 −1.08330 −0.541651 0.840604i $$-0.682200\pi$$
−0.541651 + 0.840604i $$0.682200\pi$$
$$332$$ −3.83482 −0.210463
$$333$$ 0 0
$$334$$ −20.8348 −1.14003
$$335$$ 41.0229 2.24132
$$336$$ 0 0
$$337$$ −29.0512 −1.58252 −0.791259 0.611481i $$-0.790574\pi$$
−0.791259 + 0.611481i $$0.790574\pi$$
$$338$$ −29.5261 −1.60601
$$339$$ 0 0
$$340$$ 9.94592 0.539393
$$341$$ 10.5074 0.569007
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −52.7424 −2.84368
$$345$$ 0 0
$$346$$ 42.7060 2.29589
$$347$$ −29.0833 −1.56127 −0.780636 0.624986i $$-0.785105\pi$$
−0.780636 + 0.624986i $$0.785105\pi$$
$$348$$ 0 0
$$349$$ −24.7630 −1.32553 −0.662767 0.748825i $$-0.730618\pi$$
−0.662767 + 0.748825i $$0.730618\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −2.43560 −0.129818
$$353$$ −33.3025 −1.77251 −0.886257 0.463193i $$-0.846704\pi$$
−0.886257 + 0.463193i $$0.846704\pi$$
$$354$$ 0 0
$$355$$ 8.48968 0.450586
$$356$$ −58.2029 −3.08475
$$357$$ 0 0
$$358$$ 27.9253 1.47590
$$359$$ −25.5366 −1.34777 −0.673884 0.738837i $$-0.735375\pi$$
−0.673884 + 0.738837i $$0.735375\pi$$
$$360$$ 0 0
$$361$$ −2.56440 −0.134968
$$362$$ −53.8578 −2.83070
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −3.90954 −0.204635
$$366$$ 0 0
$$367$$ −27.4504 −1.43290 −0.716449 0.697639i $$-0.754234\pi$$
−0.716449 + 0.697639i $$0.754234\pi$$
$$368$$ 1.18289 0.0616622
$$369$$ 0 0
$$370$$ −11.3638 −0.590776
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 16.3274 0.845402 0.422701 0.906269i $$-0.361082\pi$$
0.422701 + 0.906269i $$0.361082\pi$$
$$374$$ 10.5074 0.543324
$$375$$ 0 0
$$376$$ −61.4690 −3.17002
$$377$$ 2.46050 0.126722
$$378$$ 0 0
$$379$$ 12.0364 0.618267 0.309134 0.951019i $$-0.399961\pi$$
0.309134 + 0.951019i $$0.399961\pi$$
$$380$$ −42.6270 −2.18672
$$381$$ 0 0
$$382$$ 1.72665 0.0883433
$$383$$ −12.4356 −0.635429 −0.317715 0.948186i $$-0.602915\pi$$
−0.317715 + 0.948186i $$0.602915\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 29.8817 1.52094
$$387$$ 0 0
$$388$$ 46.5801 2.36475
$$389$$ −20.6008 −1.04450 −0.522250 0.852792i $$-0.674907\pi$$
−0.522250 + 0.852792i $$0.674907\pi$$
$$390$$ 0 0
$$391$$ −0.258562 −0.0130761
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 40.3786 2.03424
$$395$$ −38.1488 −1.91948
$$396$$ 0 0
$$397$$ 23.6372 1.18631 0.593157 0.805087i $$-0.297881\pi$$
0.593157 + 0.805087i $$0.297881\pi$$
$$398$$ −55.8683 −2.80042
$$399$$ 0 0
$$400$$ 7.47197 0.373599
$$401$$ 2.56440 0.128060 0.0640300 0.997948i $$-0.479605\pi$$
0.0640300 + 0.997948i $$0.479605\pi$$
$$402$$ 0 0
$$403$$ 2.32743 0.115938
$$404$$ −14.9181 −0.742202
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −8.03930 −0.398493
$$408$$ 0 0
$$409$$ 34.3245 1.69724 0.848619 0.529005i $$-0.177435\pi$$
0.848619 + 0.529005i $$0.177435\pi$$
$$410$$ 40.8827 2.01905
$$411$$ 0 0
$$412$$ 39.4327 1.94271
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 2.45331 0.120428
$$416$$ −0.539495 −0.0264509
$$417$$ 0 0
$$418$$ −45.0335 −2.20266
$$419$$ 4.05701 0.198198 0.0990989 0.995078i $$-0.468404\pi$$
0.0990989 + 0.995078i $$0.468404\pi$$
$$420$$ 0 0
$$421$$ −21.0689 −1.02683 −0.513417 0.858139i $$-0.671621\pi$$
−0.513417 + 0.858139i $$0.671621\pi$$
$$422$$ 11.2235 0.546353
$$423$$ 0 0
$$424$$ 31.7060 1.53978
$$425$$ −1.63327 −0.0792252
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 5.57160 0.269313
$$429$$ 0 0
$$430$$ 66.5949 3.21149
$$431$$ −22.6185 −1.08949 −0.544747 0.838600i $$-0.683374\pi$$
−0.544747 + 0.838600i $$0.683374\pi$$
$$432$$ 0 0
$$433$$ −2.41789 −0.116196 −0.0580982 0.998311i $$-0.518504\pi$$
−0.0580982 + 0.998311i $$0.518504\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −13.7807 −0.659978
$$437$$ 1.10817 0.0530109
$$438$$ 0 0
$$439$$ 23.4897 1.12110 0.560551 0.828120i $$-0.310589\pi$$
0.560551 + 0.828120i $$0.310589\pi$$
$$440$$ 59.1780 2.82120
$$441$$ 0 0
$$442$$ 2.32743 0.110705
$$443$$ 13.4179 0.637503 0.318752 0.947838i $$-0.396736\pi$$
0.318752 + 0.947838i $$0.396736\pi$$
$$444$$ 0 0
$$445$$ 37.2350 1.76511
$$446$$ −32.7850 −1.55242
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 9.16225 0.432393 0.216197 0.976350i $$-0.430635\pi$$
0.216197 + 0.976350i $$0.430635\pi$$
$$450$$ 0 0
$$451$$ 28.9224 1.36190
$$452$$ −42.1167 −1.98100
$$453$$ 0 0
$$454$$ 3.39922 0.159533
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 8.81711 0.412447 0.206224 0.978505i $$-0.433883\pi$$
0.206224 + 0.978505i $$0.433883\pi$$
$$458$$ 44.2383 2.06712
$$459$$ 0 0
$$460$$ −2.87412 −0.134007
$$461$$ −5.65913 −0.263572 −0.131786 0.991278i $$-0.542071\pi$$
−0.131786 + 0.991278i $$0.542071\pi$$
$$462$$ 0 0
$$463$$ 15.7267 0.730880 0.365440 0.930835i $$-0.380919\pi$$
0.365440 + 0.930835i $$0.380919\pi$$
$$464$$ −10.6477 −0.494305
$$465$$ 0 0
$$466$$ 46.7060 2.16361
$$467$$ −21.9971 −1.01790 −0.508952 0.860795i $$-0.669967\pi$$
−0.508952 + 0.860795i $$0.669967\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 77.6136 3.58005
$$471$$ 0 0
$$472$$ −13.7807 −0.634310
$$473$$ 47.1124 2.16623
$$474$$ 0 0
$$475$$ 7.00000 0.321182
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −12.0364 −0.550531
$$479$$ 24.9751 1.14114 0.570571 0.821249i $$-0.306722\pi$$
0.570571 + 0.821249i $$0.306722\pi$$
$$480$$ 0 0
$$481$$ −1.78074 −0.0811947
$$482$$ 64.3652 2.93175
$$483$$ 0 0
$$484$$ 38.0335 1.72879
$$485$$ −29.7994 −1.35312
$$486$$ 0 0
$$487$$ −17.5979 −0.797435 −0.398717 0.917074i $$-0.630545\pi$$
−0.398717 + 0.917074i $$0.630545\pi$$
$$488$$ 11.4897 0.520114
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −13.7951 −0.622566 −0.311283 0.950317i $$-0.600759\pi$$
−0.311283 + 0.950317i $$0.600759\pi$$
$$492$$ 0 0
$$493$$ 2.32743 0.104822
$$494$$ −9.97509 −0.448801
$$495$$ 0 0
$$496$$ −10.0718 −0.452237
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 13.0875 0.585879 0.292939 0.956131i $$-0.405367\pi$$
0.292939 + 0.956131i $$0.405367\pi$$
$$500$$ 34.4179 1.53921
$$501$$ 0 0
$$502$$ 45.4150 2.02697
$$503$$ −22.3068 −0.994611 −0.497305 0.867576i $$-0.665677\pi$$
−0.497305 + 0.867576i $$0.665677\pi$$
$$504$$ 0 0
$$505$$ 9.54377 0.424692
$$506$$ −3.03638 −0.134983
$$507$$ 0 0
$$508$$ 2.72665 0.120976
$$509$$ −15.8932 −0.704453 −0.352226 0.935915i $$-0.614575\pi$$
−0.352226 + 0.935915i $$0.614575\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −41.4078 −1.82998
$$513$$ 0 0
$$514$$ −28.8712 −1.27345
$$515$$ −25.2268 −1.11163
$$516$$ 0 0
$$517$$ 54.9076 2.41483
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 13.1082 0.574831
$$521$$ 4.41789 0.193551 0.0967756 0.995306i $$-0.469147\pi$$
0.0967756 + 0.995306i $$0.469147\pi$$
$$522$$ 0 0
$$523$$ −25.2733 −1.10513 −0.552563 0.833471i $$-0.686350\pi$$
−0.552563 + 0.833471i $$0.686350\pi$$
$$524$$ 32.0833 1.40156
$$525$$ 0 0
$$526$$ 18.5074 0.806961
$$527$$ 2.20155 0.0959012
$$528$$ 0 0
$$529$$ −22.9253 −0.996751
$$530$$ −40.0335 −1.73894
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 6.40642 0.277493
$$534$$ 0 0
$$535$$ −3.56440 −0.154103
$$536$$ −79.9410 −3.45293
$$537$$ 0 0
$$538$$ −46.3432 −1.99800
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −3.43852 −0.147834 −0.0739168 0.997264i $$-0.523550\pi$$
−0.0739168 + 0.997264i $$0.523550\pi$$
$$542$$ 59.0085 2.53463
$$543$$ 0 0
$$544$$ −0.510317 −0.0218797
$$545$$ 8.81616 0.377643
$$546$$ 0 0
$$547$$ −6.92821 −0.296229 −0.148114 0.988970i $$-0.547320\pi$$
−0.148114 + 0.988970i $$0.547320\pi$$
$$548$$ 14.8889 0.636023
$$549$$ 0 0
$$550$$ −19.1800 −0.817836
$$551$$ −9.97509 −0.424953
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 17.6228 0.748719
$$555$$ 0 0
$$556$$ 8.32743 0.353162
$$557$$ −33.5835 −1.42298 −0.711488 0.702698i $$-0.751979\pi$$
−0.711488 + 0.702698i $$0.751979\pi$$
$$558$$ 0 0
$$559$$ 10.4356 0.441379
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −36.6165 −1.54457
$$563$$ 42.4792 1.79028 0.895142 0.445781i $$-0.147074\pi$$
0.895142 + 0.445781i $$0.147074\pi$$
$$564$$ 0 0
$$565$$ 26.9439 1.13354
$$566$$ −49.2029 −2.06815
$$567$$ 0 0
$$568$$ −16.5438 −0.694161
$$569$$ −10.4035 −0.436137 −0.218069 0.975933i $$-0.569976\pi$$
−0.218069 + 0.975933i $$0.569976\pi$$
$$570$$ 0 0
$$571$$ 17.8496 0.746983 0.373491 0.927634i $$-0.378161\pi$$
0.373491 + 0.927634i $$0.378161\pi$$
$$572$$ 18.3025 0.765267
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0.471974 0.0196827
$$576$$ 0 0
$$577$$ −11.9430 −0.497193 −0.248597 0.968607i $$-0.579969\pi$$
−0.248597 + 0.968607i $$0.579969\pi$$
$$578$$ −39.6270 −1.64827
$$579$$ 0 0
$$580$$ 25.8712 1.07424
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −28.3216 −1.17296
$$584$$ 7.61849 0.315255
$$585$$ 0 0
$$586$$ 37.0689 1.53130
$$587$$ 23.8597 0.984796 0.492398 0.870370i $$-0.336120\pi$$
0.492398 + 0.870370i $$0.336120\pi$$
$$588$$ 0 0
$$589$$ −9.43560 −0.388787
$$590$$ 17.4002 0.716354
$$591$$ 0 0
$$592$$ 7.70602 0.316715
$$593$$ 19.5801 0.804060 0.402030 0.915626i $$-0.368305\pi$$
0.402030 + 0.915626i $$0.368305\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 54.9076 2.24910
$$597$$ 0 0
$$598$$ −0.672570 −0.0275034
$$599$$ −18.5467 −0.757797 −0.378899 0.925438i $$-0.623697\pi$$
−0.378899 + 0.925438i $$0.623697\pi$$
$$600$$ 0 0
$$601$$ 18.1986 0.742338 0.371169 0.928565i $$-0.378957\pi$$
0.371169 + 0.928565i $$0.378957\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 40.2498 1.63774
$$605$$ −24.3317 −0.989224
$$606$$ 0 0
$$607$$ 22.3097 0.905524 0.452762 0.891631i $$-0.350439\pi$$
0.452762 + 0.891631i $$0.350439\pi$$
$$608$$ 2.18716 0.0887010
$$609$$ 0 0
$$610$$ −14.5074 −0.587387
$$611$$ 12.1623 0.492032
$$612$$ 0 0
$$613$$ 10.2370 0.413467 0.206734 0.978397i $$-0.433717\pi$$
0.206734 + 0.978397i $$0.433717\pi$$
$$614$$ 67.0990 2.70790
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 11.3274 0.456025 0.228013 0.973658i $$-0.426777\pi$$
0.228013 + 0.973658i $$0.426777\pi$$
$$618$$ 0 0
$$619$$ −8.63327 −0.347000 −0.173500 0.984834i $$-0.555508\pi$$
−0.173500 + 0.984834i $$0.555508\pi$$
$$620$$ 24.4720 0.982818
$$621$$ 0 0
$$622$$ −39.3245 −1.57677
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −30.6519 −1.22608
$$626$$ −28.5366 −1.14055
$$627$$ 0 0
$$628$$ −24.5438 −0.979403
$$629$$ −1.68443 −0.0671626
$$630$$ 0 0
$$631$$ −14.8535 −0.591308 −0.295654 0.955295i $$-0.595538\pi$$
−0.295654 + 0.955295i $$0.595538\pi$$
$$632$$ 74.3402 2.95710
$$633$$ 0 0
$$634$$ 4.96458 0.197169
$$635$$ −1.74436 −0.0692229
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 27.3317 1.08207
$$639$$ 0 0
$$640$$ 49.5586 1.95897
$$641$$ 34.1593 1.34921 0.674606 0.738178i $$-0.264313\pi$$
0.674606 + 0.738178i $$0.264313\pi$$
$$642$$ 0 0
$$643$$ 10.8348 0.427284 0.213642 0.976912i $$-0.431467\pi$$
0.213642 + 0.976912i $$0.431467\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −9.43560 −0.371239
$$647$$ 32.9692 1.29615 0.648077 0.761575i $$-0.275573\pi$$
0.648077 + 0.761575i $$0.275573\pi$$
$$648$$ 0 0
$$649$$ 12.3097 0.483199
$$650$$ −4.24844 −0.166638
$$651$$ 0 0
$$652$$ 72.2321 2.82883
$$653$$ 3.93113 0.153837 0.0769185 0.997037i $$-0.475492\pi$$
0.0769185 + 0.997037i $$0.475492\pi$$
$$654$$ 0 0
$$655$$ −20.5251 −0.801982
$$656$$ −27.7233 −1.08241
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −16.8171 −0.655102 −0.327551 0.944834i $$-0.606223\pi$$
−0.327551 + 0.944834i $$0.606223\pi$$
$$660$$ 0 0
$$661$$ 17.0216 0.662063 0.331032 0.943620i $$-0.392603\pi$$
0.331032 + 0.943620i $$0.392603\pi$$
$$662$$ −48.4940 −1.88477
$$663$$ 0 0
$$664$$ −4.78074 −0.185529
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −0.672570 −0.0260420
$$668$$ −34.3288 −1.32822
$$669$$ 0 0
$$670$$ 100.937 3.89954
$$671$$ −10.2632 −0.396207
$$672$$ 0 0
$$673$$ 28.7453 1.10805 0.554025 0.832500i $$-0.313091\pi$$
0.554025 + 0.832500i $$0.313091\pi$$
$$674$$ −71.4805 −2.75333
$$675$$ 0 0
$$676$$ −48.6490 −1.87112
$$677$$ −6.03638 −0.231997 −0.115998 0.993249i $$-0.537007\pi$$
−0.115998 + 0.993249i $$0.537007\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 12.3992 0.475489
$$681$$ 0 0
$$682$$ 25.8535 0.989981
$$683$$ −20.5113 −0.784842 −0.392421 0.919786i $$-0.628362\pi$$
−0.392421 + 0.919786i $$0.628362\pi$$
$$684$$ 0 0
$$685$$ −9.52510 −0.363935
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −45.1593 −1.72168
$$689$$ −6.27335 −0.238995
$$690$$ 0 0
$$691$$ 15.0029 0.570738 0.285369 0.958418i $$-0.407884\pi$$
0.285369 + 0.958418i $$0.407884\pi$$
$$692$$ 70.3652 2.67488
$$693$$ 0 0
$$694$$ −71.5595 −2.71636
$$695$$ −5.32743 −0.202081
$$696$$ 0 0
$$697$$ 6.05993 0.229536
$$698$$ −60.9296 −2.30622
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −38.5113 −1.45455 −0.727275 0.686346i $$-0.759214\pi$$
−0.727275 + 0.686346i $$0.759214\pi$$
$$702$$ 0 0
$$703$$ 7.21926 0.272280
$$704$$ 33.0803 1.24676
$$705$$ 0 0
$$706$$ −81.9410 −3.08389
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 7.64008 0.286929 0.143465 0.989655i $$-0.454176\pi$$
0.143465 + 0.989655i $$0.454176\pi$$
$$710$$ 20.8889 0.783947
$$711$$ 0 0
$$712$$ −72.5595 −2.71928
$$713$$ −0.636194 −0.0238257
$$714$$ 0 0
$$715$$ −11.7089 −0.437890
$$716$$ 46.0115 1.71953
$$717$$ 0 0
$$718$$ −62.8329 −2.34490
$$719$$ −30.0364 −1.12017 −0.560084 0.828436i $$-0.689231\pi$$
−0.560084 + 0.828436i $$0.689231\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −6.30972 −0.234824
$$723$$ 0 0
$$724$$ −88.7395 −3.29798
$$725$$ −4.24844 −0.157783
$$726$$ 0 0
$$727$$ −3.45623 −0.128185 −0.0640923 0.997944i $$-0.520415\pi$$
−0.0640923 + 0.997944i $$0.520415\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −9.61944 −0.356032
$$731$$ 9.87120 0.365099
$$732$$ 0 0
$$733$$ −38.5261 −1.42299 −0.711496 0.702690i $$-0.751982\pi$$
−0.711496 + 0.702690i $$0.751982\pi$$
$$734$$ −67.5418 −2.49301
$$735$$ 0 0
$$736$$ 0.147469 0.00543578
$$737$$ 71.4078 2.63034
$$738$$ 0 0
$$739$$ 45.1239 1.65991 0.829955 0.557830i $$-0.188366\pi$$
0.829955 + 0.557830i $$0.188366\pi$$
$$740$$ −18.7237 −0.688298
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −9.48676 −0.348035 −0.174018 0.984743i $$-0.555675\pi$$
−0.174018 + 0.984743i $$0.555675\pi$$
$$744$$ 0 0
$$745$$ −35.1268 −1.28695
$$746$$ 40.1737 1.47086
$$747$$ 0 0
$$748$$ 17.3126 0.633013
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −9.83190 −0.358771 −0.179386 0.983779i $$-0.557411\pi$$
−0.179386 + 0.983779i $$0.557411\pi$$
$$752$$ −52.6313 −1.91927
$$753$$ 0 0
$$754$$ 6.05408 0.220477
$$755$$ −25.7496 −0.937124
$$756$$ 0 0
$$757$$ −41.8171 −1.51987 −0.759934 0.650000i $$-0.774769\pi$$
−0.759934 + 0.650000i $$0.774769\pi$$
$$758$$ 29.6156 1.07569
$$759$$ 0 0
$$760$$ −53.1416 −1.92765
$$761$$ 22.9794 0.833001 0.416501 0.909135i $$-0.363256\pi$$
0.416501 + 0.909135i $$0.363256\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 2.84494 0.102926
$$765$$ 0 0
$$766$$ −30.5979 −1.10555
$$767$$ 2.72665 0.0984538
$$768$$ 0 0
$$769$$ −6.08658 −0.219488 −0.109744 0.993960i $$-0.535003\pi$$
−0.109744 + 0.993960i $$0.535003\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 49.2350 1.77201
$$773$$ −41.8214 −1.50421 −0.752105 0.659043i $$-0.770962\pi$$
−0.752105 + 0.659043i $$0.770962\pi$$
$$774$$ 0 0
$$775$$ −4.01867 −0.144355
$$776$$ 58.0698 2.08459
$$777$$ 0 0
$$778$$ −50.6883 −1.81726
$$779$$ −25.9722 −0.930550
$$780$$ 0 0
$$781$$ 14.7778 0.528792
$$782$$ −0.636194 −0.0227503
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 15.7017 0.560419
$$786$$ 0 0
$$787$$ −32.2920 −1.15109 −0.575543 0.817772i $$-0.695209\pi$$
−0.575543 + 0.817772i $$0.695209\pi$$
$$788$$ 66.5303 2.37004
$$789$$ 0 0
$$790$$ −93.8653 −3.33958
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −2.27335 −0.0807289
$$794$$ 58.1593 2.06400
$$795$$ 0 0
$$796$$ −92.0521 −3.26270
$$797$$ 46.5657 1.64944 0.824722 0.565539i $$-0.191332\pi$$
0.824722 + 0.565539i $$0.191332\pi$$
$$798$$ 0 0
$$799$$ 11.5045 0.406999
$$800$$ 0.931521 0.0329343
$$801$$ 0 0
$$802$$ 6.30972 0.222804
$$803$$ −6.80525 −0.240152
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 5.72665 0.201713
$$807$$ 0 0
$$808$$ −18.5979 −0.654270
$$809$$ 10.8023 0.379790 0.189895 0.981804i $$-0.439185\pi$$
0.189895 + 0.981804i $$0.439185\pi$$
$$810$$ 0 0
$$811$$ −5.58307 −0.196048 −0.0980240 0.995184i $$-0.531252\pi$$
−0.0980240 + 0.995184i $$0.531252\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −19.7807 −0.693315
$$815$$ −46.2101 −1.61867
$$816$$ 0 0
$$817$$ −42.3068 −1.48013
$$818$$ 84.4556 2.95292
$$819$$ 0 0
$$820$$ 67.3609 2.35234
$$821$$ 31.7879 1.10941 0.554703 0.832048i $$-0.312832\pi$$
0.554703 + 0.832048i $$0.312832\pi$$
$$822$$ 0 0
$$823$$ −36.0000 −1.25488 −0.627441 0.778664i $$-0.715897\pi$$
−0.627441 + 0.778664i $$0.715897\pi$$
$$824$$ 49.1593 1.71255
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −15.9224 −0.553675 −0.276837 0.960917i $$-0.589286\pi$$
−0.276837 + 0.960917i $$0.589286\pi$$
$$828$$ 0 0
$$829$$ −35.4720 −1.23199 −0.615996 0.787749i $$-0.711246\pi$$
−0.615996 + 0.787749i $$0.711246\pi$$
$$830$$ 6.03638 0.209526
$$831$$ 0 0
$$832$$ 7.32743 0.254033
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 21.9617 0.760014
$$836$$ −74.2000 −2.56626
$$837$$ 0 0
$$838$$ 9.98229 0.344833
$$839$$ −54.6782 −1.88770 −0.943850 0.330373i $$-0.892825\pi$$
−0.943850 + 0.330373i $$0.892825\pi$$
$$840$$ 0 0
$$841$$ −22.9459 −0.791238
$$842$$ −51.8401 −1.78653
$$843$$ 0 0
$$844$$ 18.4926 0.636542
$$845$$ 31.1230 1.07066
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 27.1475 0.932248
$$849$$ 0 0
$$850$$ −4.01867 −0.137839
$$851$$ 0.486758 0.0166858
$$852$$ 0 0
$$853$$ 2.19767 0.0752468 0.0376234 0.999292i $$-0.488021\pi$$
0.0376234 + 0.999292i $$0.488021\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 6.94592 0.237407
$$857$$ −15.7765 −0.538914 −0.269457 0.963012i $$-0.586844\pi$$
−0.269457 + 0.963012i $$0.586844\pi$$
$$858$$ 0 0
$$859$$ −5.57626 −0.190260 −0.0951298 0.995465i $$-0.530327\pi$$
−0.0951298 + 0.995465i $$0.530327\pi$$
$$860$$ 109.726 3.74163
$$861$$ 0 0
$$862$$ −55.6529 −1.89555
$$863$$ −23.1268 −0.787247 −0.393623 0.919272i $$-0.628779\pi$$
−0.393623 + 0.919272i $$0.628779\pi$$
$$864$$ 0 0
$$865$$ −45.0157 −1.53058
$$866$$ −5.94923 −0.202163
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −66.4048 −2.25263
$$870$$ 0 0
$$871$$ 15.8171 0.535942
$$872$$ −17.1800 −0.581787
$$873$$ 0 0
$$874$$ 2.72665 0.0922304
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 3.92528 0.132547 0.0662737 0.997801i $$-0.478889\pi$$
0.0662737 + 0.997801i $$0.478889\pi$$
$$878$$ 57.7965 1.95054
$$879$$ 0 0
$$880$$ 50.6696 1.70807
$$881$$ 27.1986 0.916345 0.458173 0.888863i $$-0.348504\pi$$
0.458173 + 0.888863i $$0.348504\pi$$
$$882$$ 0 0
$$883$$ 8.21341 0.276403 0.138202 0.990404i $$-0.455868\pi$$
0.138202 + 0.990404i $$0.455868\pi$$
$$884$$ 3.83482 0.128979
$$885$$ 0 0
$$886$$ 33.0148 1.10915
$$887$$ −6.48114 −0.217615 −0.108808 0.994063i $$-0.534703\pi$$
−0.108808 + 0.994063i $$0.534703\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 91.6169 3.07101
$$891$$ 0 0
$$892$$ −54.0187 −1.80868
$$893$$ −49.3068 −1.64999
$$894$$ 0 0
$$895$$ −29.4356 −0.983924
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 22.5438 0.752295
$$899$$ 5.72665 0.190995
$$900$$ 0 0
$$901$$ −5.93406 −0.197692
$$902$$ 71.1636 2.36949
$$903$$ 0 0
$$904$$ −52.5054 −1.74630
$$905$$ 56.7706 1.88712
$$906$$ 0 0
$$907$$ 10.1288 0.336321 0.168161 0.985760i $$-0.446217\pi$$
0.168161 + 0.985760i $$0.446217\pi$$
$$908$$ 5.60078 0.185868
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −45.9224 −1.52148 −0.760738 0.649059i $$-0.775163\pi$$
−0.760738 + 0.649059i $$0.775163\pi$$
$$912$$ 0 0
$$913$$ 4.27042 0.141330
$$914$$ 21.6946 0.717592
$$915$$ 0 0
$$916$$ 72.8899 2.40835
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −4.92432 −0.162438 −0.0812192 0.996696i $$-0.525881\pi$$
−0.0812192 + 0.996696i $$0.525881\pi$$
$$920$$ −3.58307 −0.118130
$$921$$ 0 0
$$922$$ −13.9243 −0.458573
$$923$$ 3.27335 0.107744
$$924$$ 0 0
$$925$$ 3.07472 0.101096
$$926$$ 38.6955 1.27161
$$927$$ 0 0
$$928$$ −1.32743 −0.0435750
$$929$$ 0.00758649 0.000248905 0 0.000124452 1.00000i $$-0.499960\pi$$
0.000124452 1.00000i $$0.499960\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 76.9558 2.52077
$$933$$ 0 0
$$934$$ −54.1239 −1.77099
$$935$$ −11.0757 −0.362213
$$936$$ 0 0
$$937$$ −21.1623 −0.691341 −0.345670 0.938356i $$-0.612348\pi$$
−0.345670 + 0.938356i $$0.612348\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 127.881 4.17102
$$941$$ 4.55816 0.148592 0.0742959 0.997236i $$-0.476329\pi$$
0.0742959 + 0.997236i $$0.476329\pi$$
$$942$$ 0 0
$$943$$ −1.75117 −0.0570260
$$944$$ −11.7994 −0.384038
$$945$$ 0 0
$$946$$ 115.920 3.76890
$$947$$ −13.7352 −0.446334 −0.223167 0.974780i $$-0.571640\pi$$
−0.223167 + 0.974780i $$0.571640\pi$$
$$948$$ 0 0
$$949$$ −1.50739 −0.0489320
$$950$$ 17.2235 0.558805
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −8.80699 −0.285286 −0.142643 0.989774i $$-0.545560\pi$$
−0.142643 + 0.989774i $$0.545560\pi$$
$$954$$ 0 0
$$955$$ −1.82004 −0.0588951
$$956$$ −19.8319 −0.641409
$$957$$ 0 0
$$958$$ 61.4513 1.98540
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −25.5831 −0.825260
$$962$$ −4.38151 −0.141266
$$963$$ 0 0
$$964$$ 106.052 3.41571
$$965$$ −31.4978 −1.01395
$$966$$ 0 0
$$967$$ 38.3284 1.23256 0.616279 0.787528i $$-0.288639\pi$$
0.616279 + 0.787528i $$0.288639\pi$$
$$968$$ 47.4150 1.52397
$$969$$ 0 0
$$970$$ −73.3216 −2.35421
$$971$$ −31.0187 −0.995436 −0.497718 0.867339i $$-0.665829\pi$$
−0.497718 + 0.867339i $$0.665829\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −43.2996 −1.38741
$$975$$ 0 0
$$976$$ 9.83775 0.314899
$$977$$ 52.7424 1.68738 0.843689 0.536832i $$-0.180379\pi$$
0.843689 + 0.536832i $$0.180379\pi$$
$$978$$ 0 0
$$979$$ 64.8142 2.07147
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −33.9430 −1.08316
$$983$$ −18.3029 −0.583772 −0.291886 0.956453i $$-0.594283\pi$$
−0.291886 + 0.956453i $$0.594283\pi$$
$$984$$ 0 0
$$985$$ −42.5624 −1.35615
$$986$$ 5.72665 0.182374
$$987$$ 0 0
$$988$$ −16.4356 −0.522886
$$989$$ −2.85253 −0.0907052
$$990$$ 0 0
$$991$$ −12.6008 −0.400277 −0.200138 0.979768i $$-0.564139\pi$$
−0.200138 + 0.979768i $$0.564139\pi$$
$$992$$ −1.25564 −0.0398665
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 58.8899 1.86693
$$996$$ 0 0
$$997$$ −11.7424 −0.371885 −0.185943 0.982561i $$-0.559534\pi$$
−0.185943 + 0.982561i $$0.559534\pi$$
$$998$$ 32.2019 1.01933
$$999$$ 0 0
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 3969.2.a.p.1.3 3
3.2 odd 2 3969.2.a.m.1.1 3
7.6 odd 2 567.2.a.g.1.3 3
9.2 odd 6 441.2.f.d.148.3 6
9.4 even 3 1323.2.f.c.883.1 6
9.5 odd 6 441.2.f.d.295.3 6
9.7 even 3 1323.2.f.c.442.1 6
21.20 even 2 567.2.a.d.1.1 3
28.27 even 2 9072.2.a.cd.1.2 3
63.2 odd 6 441.2.g.d.67.3 6
63.4 even 3 1323.2.g.b.667.1 6
63.5 even 6 441.2.h.c.214.1 6
63.11 odd 6 441.2.h.b.373.1 6
63.13 odd 6 189.2.f.a.127.1 6
63.16 even 3 1323.2.g.b.361.1 6
63.20 even 6 63.2.f.b.22.3 6
63.23 odd 6 441.2.h.b.214.1 6
63.25 even 3 1323.2.h.e.226.3 6
63.31 odd 6 1323.2.g.c.667.1 6
63.32 odd 6 441.2.g.d.79.3 6
63.34 odd 6 189.2.f.a.64.1 6
63.38 even 6 441.2.h.c.373.1 6
63.40 odd 6 1323.2.h.d.802.3 6
63.41 even 6 63.2.f.b.43.3 yes 6
63.47 even 6 441.2.g.e.67.3 6
63.52 odd 6 1323.2.h.d.226.3 6
63.58 even 3 1323.2.h.e.802.3 6
63.59 even 6 441.2.g.e.79.3 6
63.61 odd 6 1323.2.g.c.361.1 6
84.83 odd 2 9072.2.a.bq.1.2 3
252.83 odd 6 1008.2.r.k.337.3 6
252.139 even 6 3024.2.r.g.2017.2 6
252.167 odd 6 1008.2.r.k.673.3 6
252.223 even 6 3024.2.r.g.1009.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.3 6 63.20 even 6
63.2.f.b.43.3 yes 6 63.41 even 6
189.2.f.a.64.1 6 63.34 odd 6
189.2.f.a.127.1 6 63.13 odd 6
441.2.f.d.148.3 6 9.2 odd 6
441.2.f.d.295.3 6 9.5 odd 6
441.2.g.d.67.3 6 63.2 odd 6
441.2.g.d.79.3 6 63.32 odd 6
441.2.g.e.67.3 6 63.47 even 6
441.2.g.e.79.3 6 63.59 even 6
441.2.h.b.214.1 6 63.23 odd 6
441.2.h.b.373.1 6 63.11 odd 6
441.2.h.c.214.1 6 63.5 even 6
441.2.h.c.373.1 6 63.38 even 6
567.2.a.d.1.1 3 21.20 even 2
567.2.a.g.1.3 3 7.6 odd 2
1008.2.r.k.337.3 6 252.83 odd 6
1008.2.r.k.673.3 6 252.167 odd 6
1323.2.f.c.442.1 6 9.7 even 3
1323.2.f.c.883.1 6 9.4 even 3
1323.2.g.b.361.1 6 63.16 even 3
1323.2.g.b.667.1 6 63.4 even 3
1323.2.g.c.361.1 6 63.61 odd 6
1323.2.g.c.667.1 6 63.31 odd 6
1323.2.h.d.226.3 6 63.52 odd 6
1323.2.h.d.802.3 6 63.40 odd 6
1323.2.h.e.226.3 6 63.25 even 3
1323.2.h.e.802.3 6 63.58 even 3
3024.2.r.g.1009.2 6 252.223 even 6
3024.2.r.g.2017.2 6 252.139 even 6
3969.2.a.m.1.1 3 3.2 odd 2
3969.2.a.p.1.3 3 1.1 even 1 trivial
9072.2.a.bq.1.2 3 84.83 odd 2
9072.2.a.cd.1.2 3 28.27 even 2