# Properties

 Label 4840.2.a.j.1.1 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4840 = 2^{3} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4840.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: $$38.6475945783$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 4840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} +3.56155 q^{9} +O(q^{10})$$ $$q-2.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} +3.56155 q^{9} -2.00000 q^{13} -2.56155 q^{15} +0.561553 q^{17} -2.56155 q^{19} -6.56155 q^{21} +5.12311 q^{23} +1.00000 q^{25} -1.43845 q^{27} -9.68466 q^{29} +6.56155 q^{31} +2.56155 q^{35} -5.68466 q^{37} +5.12311 q^{39} -2.00000 q^{41} -10.2462 q^{43} +3.56155 q^{45} -13.1231 q^{47} -0.438447 q^{49} -1.43845 q^{51} +4.56155 q^{53} +6.56155 q^{57} +1.12311 q^{59} -2.31534 q^{61} +9.12311 q^{63} -2.00000 q^{65} +6.24621 q^{67} -13.1231 q^{69} +3.68466 q^{71} +2.00000 q^{73} -2.56155 q^{75} +15.3693 q^{79} -7.00000 q^{81} -5.12311 q^{83} +0.561553 q^{85} +24.8078 q^{87} -12.5616 q^{89} -5.12311 q^{91} -16.8078 q^{93} -2.56155 q^{95} +7.12311 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 + 2 * q^5 + q^7 + 3 * q^9 $$2 q - q^{3} + 2 q^{5} + q^{7} + 3 q^{9} - 4 q^{13} - q^{15} - 3 q^{17} - q^{19} - 9 q^{21} + 2 q^{23} + 2 q^{25} - 7 q^{27} - 7 q^{29} + 9 q^{31} + q^{35} + q^{37} + 2 q^{39} - 4 q^{41} - 4 q^{43} + 3 q^{45} - 18 q^{47} - 5 q^{49} - 7 q^{51} + 5 q^{53} + 9 q^{57} - 6 q^{59} - 17 q^{61} + 10 q^{63} - 4 q^{65} - 4 q^{67} - 18 q^{69} - 5 q^{71} + 4 q^{73} - q^{75} + 6 q^{79} - 14 q^{81} - 2 q^{83} - 3 q^{85} + 29 q^{87} - 21 q^{89} - 2 q^{91} - 13 q^{93} - q^{95} + 6 q^{97}+O(q^{100})$$ 2 * q - q^3 + 2 * q^5 + q^7 + 3 * q^9 - 4 * q^13 - q^15 - 3 * q^17 - q^19 - 9 * q^21 + 2 * q^23 + 2 * q^25 - 7 * q^27 - 7 * q^29 + 9 * q^31 + q^35 + q^37 + 2 * q^39 - 4 * q^41 - 4 * q^43 + 3 * q^45 - 18 * q^47 - 5 * q^49 - 7 * q^51 + 5 * q^53 + 9 * q^57 - 6 * q^59 - 17 * q^61 + 10 * q^63 - 4 * q^65 - 4 * q^67 - 18 * q^69 - 5 * q^71 + 4 * q^73 - q^75 + 6 * q^79 - 14 * q^81 - 2 * q^83 - 3 * q^85 + 29 * q^87 - 21 * q^89 - 2 * q^91 - 13 * q^93 - q^95 + 6 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.56155 −1.47891 −0.739457 0.673204i $$-0.764917\pi$$
−0.739457 + 0.673204i $$0.764917\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.56155 0.968176 0.484088 0.875019i $$-0.339151\pi$$
0.484088 + 0.875019i $$0.339151\pi$$
$$8$$ 0 0
$$9$$ 3.56155 1.18718
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ −2.56155 −0.661390
$$16$$ 0 0
$$17$$ 0.561553 0.136197 0.0680983 0.997679i $$-0.478307\pi$$
0.0680983 + 0.997679i $$0.478307\pi$$
$$18$$ 0 0
$$19$$ −2.56155 −0.587661 −0.293830 0.955858i $$-0.594930\pi$$
−0.293830 + 0.955858i $$0.594930\pi$$
$$20$$ 0 0
$$21$$ −6.56155 −1.43185
$$22$$ 0 0
$$23$$ 5.12311 1.06824 0.534121 0.845408i $$-0.320643\pi$$
0.534121 + 0.845408i $$0.320643\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.43845 −0.276829
$$28$$ 0 0
$$29$$ −9.68466 −1.79840 −0.899198 0.437542i $$-0.855849\pi$$
−0.899198 + 0.437542i $$0.855849\pi$$
$$30$$ 0 0
$$31$$ 6.56155 1.17849 0.589245 0.807955i $$-0.299425\pi$$
0.589245 + 0.807955i $$0.299425\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.56155 0.432981
$$36$$ 0 0
$$37$$ −5.68466 −0.934552 −0.467276 0.884111i $$-0.654765\pi$$
−0.467276 + 0.884111i $$0.654765\pi$$
$$38$$ 0 0
$$39$$ 5.12311 0.820353
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −10.2462 −1.56253 −0.781266 0.624198i $$-0.785426\pi$$
−0.781266 + 0.624198i $$0.785426\pi$$
$$44$$ 0 0
$$45$$ 3.56155 0.530925
$$46$$ 0 0
$$47$$ −13.1231 −1.91420 −0.957101 0.289755i $$-0.906426\pi$$
−0.957101 + 0.289755i $$0.906426\pi$$
$$48$$ 0 0
$$49$$ −0.438447 −0.0626353
$$50$$ 0 0
$$51$$ −1.43845 −0.201423
$$52$$ 0 0
$$53$$ 4.56155 0.626577 0.313289 0.949658i $$-0.398569\pi$$
0.313289 + 0.949658i $$0.398569\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 6.56155 0.869099
$$58$$ 0 0
$$59$$ 1.12311 0.146216 0.0731079 0.997324i $$-0.476708\pi$$
0.0731079 + 0.997324i $$0.476708\pi$$
$$60$$ 0 0
$$61$$ −2.31534 −0.296449 −0.148225 0.988954i $$-0.547356\pi$$
−0.148225 + 0.988954i $$0.547356\pi$$
$$62$$ 0 0
$$63$$ 9.12311 1.14940
$$64$$ 0 0
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ 6.24621 0.763096 0.381548 0.924349i $$-0.375391\pi$$
0.381548 + 0.924349i $$0.375391\pi$$
$$68$$ 0 0
$$69$$ −13.1231 −1.57984
$$70$$ 0 0
$$71$$ 3.68466 0.437289 0.218644 0.975805i $$-0.429837\pi$$
0.218644 + 0.975805i $$0.429837\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 0 0
$$75$$ −2.56155 −0.295783
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 15.3693 1.72918 0.864592 0.502475i $$-0.167577\pi$$
0.864592 + 0.502475i $$0.167577\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ −5.12311 −0.562334 −0.281167 0.959659i $$-0.590721\pi$$
−0.281167 + 0.959659i $$0.590721\pi$$
$$84$$ 0 0
$$85$$ 0.561553 0.0609090
$$86$$ 0 0
$$87$$ 24.8078 2.65967
$$88$$ 0 0
$$89$$ −12.5616 −1.33152 −0.665761 0.746165i $$-0.731893\pi$$
−0.665761 + 0.746165i $$0.731893\pi$$
$$90$$ 0 0
$$91$$ −5.12311 −0.537047
$$92$$ 0 0
$$93$$ −16.8078 −1.74288
$$94$$ 0 0
$$95$$ −2.56155 −0.262810
$$96$$ 0 0
$$97$$ 7.12311 0.723242 0.361621 0.932325i $$-0.382223\pi$$
0.361621 + 0.932325i $$0.382223\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 4.24621 0.422514 0.211257 0.977431i $$-0.432244\pi$$
0.211257 + 0.977431i $$0.432244\pi$$
$$102$$ 0 0
$$103$$ 10.2462 1.00959 0.504795 0.863239i $$-0.331568\pi$$
0.504795 + 0.863239i $$0.331568\pi$$
$$104$$ 0 0
$$105$$ −6.56155 −0.640342
$$106$$ 0 0
$$107$$ 16.0000 1.54678 0.773389 0.633932i $$-0.218560\pi$$
0.773389 + 0.633932i $$0.218560\pi$$
$$108$$ 0 0
$$109$$ −18.4924 −1.77125 −0.885626 0.464398i $$-0.846271\pi$$
−0.885626 + 0.464398i $$0.846271\pi$$
$$110$$ 0 0
$$111$$ 14.5616 1.38212
$$112$$ 0 0
$$113$$ −19.1231 −1.79895 −0.899475 0.436972i $$-0.856051\pi$$
−0.899475 + 0.436972i $$0.856051\pi$$
$$114$$ 0 0
$$115$$ 5.12311 0.477732
$$116$$ 0 0
$$117$$ −7.12311 −0.658531
$$118$$ 0 0
$$119$$ 1.43845 0.131862
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 5.12311 0.461935
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −14.2462 −1.26415 −0.632073 0.774909i $$-0.717796\pi$$
−0.632073 + 0.774909i $$0.717796\pi$$
$$128$$ 0 0
$$129$$ 26.2462 2.31085
$$130$$ 0 0
$$131$$ 7.68466 0.671412 0.335706 0.941967i $$-0.391025\pi$$
0.335706 + 0.941967i $$0.391025\pi$$
$$132$$ 0 0
$$133$$ −6.56155 −0.568959
$$134$$ 0 0
$$135$$ −1.43845 −0.123802
$$136$$ 0 0
$$137$$ 19.6155 1.67587 0.837934 0.545772i $$-0.183763\pi$$
0.837934 + 0.545772i $$0.183763\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 33.6155 2.83094
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −9.68466 −0.804267
$$146$$ 0 0
$$147$$ 1.12311 0.0926322
$$148$$ 0 0
$$149$$ −2.31534 −0.189680 −0.0948401 0.995493i $$-0.530234\pi$$
−0.0948401 + 0.995493i $$0.530234\pi$$
$$150$$ 0 0
$$151$$ −13.1231 −1.06794 −0.533972 0.845502i $$-0.679301\pi$$
−0.533972 + 0.845502i $$0.679301\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 6.56155 0.527037
$$156$$ 0 0
$$157$$ 14.8078 1.18179 0.590894 0.806749i $$-0.298775\pi$$
0.590894 + 0.806749i $$0.298775\pi$$
$$158$$ 0 0
$$159$$ −11.6847 −0.926654
$$160$$ 0 0
$$161$$ 13.1231 1.03425
$$162$$ 0 0
$$163$$ 3.19224 0.250035 0.125018 0.992155i $$-0.460101\pi$$
0.125018 + 0.992155i $$0.460101\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 10.5616 0.817277 0.408639 0.912696i $$-0.366004\pi$$
0.408639 + 0.912696i $$0.366004\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −9.12311 −0.697661
$$172$$ 0 0
$$173$$ −4.24621 −0.322833 −0.161417 0.986886i $$-0.551606\pi$$
−0.161417 + 0.986886i $$0.551606\pi$$
$$174$$ 0 0
$$175$$ 2.56155 0.193635
$$176$$ 0 0
$$177$$ −2.87689 −0.216241
$$178$$ 0 0
$$179$$ −6.24621 −0.466864 −0.233432 0.972373i $$-0.574996\pi$$
−0.233432 + 0.972373i $$0.574996\pi$$
$$180$$ 0 0
$$181$$ −4.24621 −0.315618 −0.157809 0.987470i $$-0.550443\pi$$
−0.157809 + 0.987470i $$0.550443\pi$$
$$182$$ 0 0
$$183$$ 5.93087 0.438423
$$184$$ 0 0
$$185$$ −5.68466 −0.417944
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −3.68466 −0.268019
$$190$$ 0 0
$$191$$ 10.2462 0.741390 0.370695 0.928755i $$-0.379120\pi$$
0.370695 + 0.928755i $$0.379120\pi$$
$$192$$ 0 0
$$193$$ −5.19224 −0.373745 −0.186873 0.982384i $$-0.559835\pi$$
−0.186873 + 0.982384i $$0.559835\pi$$
$$194$$ 0 0
$$195$$ 5.12311 0.366873
$$196$$ 0 0
$$197$$ 5.36932 0.382548 0.191274 0.981537i $$-0.438738\pi$$
0.191274 + 0.981537i $$0.438738\pi$$
$$198$$ 0 0
$$199$$ −0.807764 −0.0572609 −0.0286304 0.999590i $$-0.509115\pi$$
−0.0286304 + 0.999590i $$0.509115\pi$$
$$200$$ 0 0
$$201$$ −16.0000 −1.12855
$$202$$ 0 0
$$203$$ −24.8078 −1.74116
$$204$$ 0 0
$$205$$ −2.00000 −0.139686
$$206$$ 0 0
$$207$$ 18.2462 1.26820
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −8.31534 −0.572452 −0.286226 0.958162i $$-0.592401\pi$$
−0.286226 + 0.958162i $$0.592401\pi$$
$$212$$ 0 0
$$213$$ −9.43845 −0.646712
$$214$$ 0 0
$$215$$ −10.2462 −0.698786
$$216$$ 0 0
$$217$$ 16.8078 1.14099
$$218$$ 0 0
$$219$$ −5.12311 −0.346187
$$220$$ 0 0
$$221$$ −1.12311 −0.0755483
$$222$$ 0 0
$$223$$ 13.1231 0.878788 0.439394 0.898294i $$-0.355193\pi$$
0.439394 + 0.898294i $$0.355193\pi$$
$$224$$ 0 0
$$225$$ 3.56155 0.237437
$$226$$ 0 0
$$227$$ −2.87689 −0.190946 −0.0954731 0.995432i $$-0.530436\pi$$
−0.0954731 + 0.995432i $$0.530436\pi$$
$$228$$ 0 0
$$229$$ −24.7386 −1.63477 −0.817387 0.576088i $$-0.804578\pi$$
−0.817387 + 0.576088i $$0.804578\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −19.9309 −1.30571 −0.652857 0.757481i $$-0.726430\pi$$
−0.652857 + 0.757481i $$0.726430\pi$$
$$234$$ 0 0
$$235$$ −13.1231 −0.856057
$$236$$ 0 0
$$237$$ −39.3693 −2.55731
$$238$$ 0 0
$$239$$ −25.6155 −1.65693 −0.828465 0.560040i $$-0.810786\pi$$
−0.828465 + 0.560040i $$0.810786\pi$$
$$240$$ 0 0
$$241$$ 16.2462 1.04651 0.523255 0.852176i $$-0.324717\pi$$
0.523255 + 0.852176i $$0.324717\pi$$
$$242$$ 0 0
$$243$$ 22.2462 1.42710
$$244$$ 0 0
$$245$$ −0.438447 −0.0280114
$$246$$ 0 0
$$247$$ 5.12311 0.325975
$$248$$ 0 0
$$249$$ 13.1231 0.831643
$$250$$ 0 0
$$251$$ −29.6155 −1.86932 −0.934658 0.355549i $$-0.884294\pi$$
−0.934658 + 0.355549i $$0.884294\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −1.43845 −0.0900791
$$256$$ 0 0
$$257$$ −28.7386 −1.79267 −0.896333 0.443381i $$-0.853779\pi$$
−0.896333 + 0.443381i $$0.853779\pi$$
$$258$$ 0 0
$$259$$ −14.5616 −0.904811
$$260$$ 0 0
$$261$$ −34.4924 −2.13503
$$262$$ 0 0
$$263$$ −4.80776 −0.296459 −0.148230 0.988953i $$-0.547357\pi$$
−0.148230 + 0.988953i $$0.547357\pi$$
$$264$$ 0 0
$$265$$ 4.56155 0.280214
$$266$$ 0 0
$$267$$ 32.1771 1.96921
$$268$$ 0 0
$$269$$ −19.6155 −1.19598 −0.597990 0.801504i $$-0.704034\pi$$
−0.597990 + 0.801504i $$0.704034\pi$$
$$270$$ 0 0
$$271$$ −28.4924 −1.73079 −0.865396 0.501089i $$-0.832933\pi$$
−0.865396 + 0.501089i $$0.832933\pi$$
$$272$$ 0 0
$$273$$ 13.1231 0.794246
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −7.12311 −0.427986 −0.213993 0.976835i $$-0.568647\pi$$
−0.213993 + 0.976835i $$0.568647\pi$$
$$278$$ 0 0
$$279$$ 23.3693 1.39908
$$280$$ 0 0
$$281$$ 8.24621 0.491928 0.245964 0.969279i $$-0.420896\pi$$
0.245964 + 0.969279i $$0.420896\pi$$
$$282$$ 0 0
$$283$$ 23.3693 1.38916 0.694581 0.719415i $$-0.255590\pi$$
0.694581 + 0.719415i $$0.255590\pi$$
$$284$$ 0 0
$$285$$ 6.56155 0.388673
$$286$$ 0 0
$$287$$ −5.12311 −0.302407
$$288$$ 0 0
$$289$$ −16.6847 −0.981450
$$290$$ 0 0
$$291$$ −18.2462 −1.06961
$$292$$ 0 0
$$293$$ −14.4924 −0.846656 −0.423328 0.905976i $$-0.639138\pi$$
−0.423328 + 0.905976i $$0.639138\pi$$
$$294$$ 0 0
$$295$$ 1.12311 0.0653897
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −10.2462 −0.592554
$$300$$ 0 0
$$301$$ −26.2462 −1.51281
$$302$$ 0 0
$$303$$ −10.8769 −0.624861
$$304$$ 0 0
$$305$$ −2.31534 −0.132576
$$306$$ 0 0
$$307$$ 25.6155 1.46196 0.730978 0.682401i $$-0.239064\pi$$
0.730978 + 0.682401i $$0.239064\pi$$
$$308$$ 0 0
$$309$$ −26.2462 −1.49309
$$310$$ 0 0
$$311$$ −25.4384 −1.44248 −0.721241 0.692684i $$-0.756428\pi$$
−0.721241 + 0.692684i $$0.756428\pi$$
$$312$$ 0 0
$$313$$ 30.4924 1.72353 0.861767 0.507305i $$-0.169358\pi$$
0.861767 + 0.507305i $$0.169358\pi$$
$$314$$ 0 0
$$315$$ 9.12311 0.514029
$$316$$ 0 0
$$317$$ 13.1922 0.740950 0.370475 0.928842i $$-0.379195\pi$$
0.370475 + 0.928842i $$0.379195\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −40.9848 −2.28755
$$322$$ 0 0
$$323$$ −1.43845 −0.0800373
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ 0 0
$$327$$ 47.3693 2.61953
$$328$$ 0 0
$$329$$ −33.6155 −1.85328
$$330$$ 0 0
$$331$$ −8.49242 −0.466786 −0.233393 0.972383i $$-0.574983\pi$$
−0.233393 + 0.972383i $$0.574983\pi$$
$$332$$ 0 0
$$333$$ −20.2462 −1.10949
$$334$$ 0 0
$$335$$ 6.24621 0.341267
$$336$$ 0 0
$$337$$ −2.31534 −0.126125 −0.0630623 0.998010i $$-0.520087\pi$$
−0.0630623 + 0.998010i $$0.520087\pi$$
$$338$$ 0 0
$$339$$ 48.9848 2.66049
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −19.0540 −1.02882
$$344$$ 0 0
$$345$$ −13.1231 −0.706524
$$346$$ 0 0
$$347$$ 18.8769 1.01336 0.506682 0.862133i $$-0.330872\pi$$
0.506682 + 0.862133i $$0.330872\pi$$
$$348$$ 0 0
$$349$$ −10.4924 −0.561646 −0.280823 0.959760i $$-0.590607\pi$$
−0.280823 + 0.959760i $$0.590607\pi$$
$$350$$ 0 0
$$351$$ 2.87689 0.153557
$$352$$ 0 0
$$353$$ −34.4924 −1.83585 −0.917923 0.396758i $$-0.870135\pi$$
−0.917923 + 0.396758i $$0.870135\pi$$
$$354$$ 0 0
$$355$$ 3.68466 0.195561
$$356$$ 0 0
$$357$$ −3.68466 −0.195013
$$358$$ 0 0
$$359$$ −26.2462 −1.38522 −0.692611 0.721311i $$-0.743540\pi$$
−0.692611 + 0.721311i $$0.743540\pi$$
$$360$$ 0 0
$$361$$ −12.4384 −0.654655
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 0 0
$$367$$ −14.7386 −0.769350 −0.384675 0.923052i $$-0.625687\pi$$
−0.384675 + 0.923052i $$0.625687\pi$$
$$368$$ 0 0
$$369$$ −7.12311 −0.370814
$$370$$ 0 0
$$371$$ 11.6847 0.606637
$$372$$ 0 0
$$373$$ −2.00000 −0.103556 −0.0517780 0.998659i $$-0.516489\pi$$
−0.0517780 + 0.998659i $$0.516489\pi$$
$$374$$ 0 0
$$375$$ −2.56155 −0.132278
$$376$$ 0 0
$$377$$ 19.3693 0.997571
$$378$$ 0 0
$$379$$ −25.1231 −1.29049 −0.645244 0.763977i $$-0.723244\pi$$
−0.645244 + 0.763977i $$0.723244\pi$$
$$380$$ 0 0
$$381$$ 36.4924 1.86956
$$382$$ 0 0
$$383$$ −31.3693 −1.60290 −0.801449 0.598064i $$-0.795937\pi$$
−0.801449 + 0.598064i $$0.795937\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −36.4924 −1.85501
$$388$$ 0 0
$$389$$ 32.2462 1.63495 0.817474 0.575966i $$-0.195374\pi$$
0.817474 + 0.575966i $$0.195374\pi$$
$$390$$ 0 0
$$391$$ 2.87689 0.145491
$$392$$ 0 0
$$393$$ −19.6847 −0.992960
$$394$$ 0 0
$$395$$ 15.3693 0.773314
$$396$$ 0 0
$$397$$ −18.0000 −0.903394 −0.451697 0.892171i $$-0.649181\pi$$
−0.451697 + 0.892171i $$0.649181\pi$$
$$398$$ 0 0
$$399$$ 16.8078 0.841441
$$400$$ 0 0
$$401$$ −7.43845 −0.371458 −0.185729 0.982601i $$-0.559465\pi$$
−0.185729 + 0.982601i $$0.559465\pi$$
$$402$$ 0 0
$$403$$ −13.1231 −0.653708
$$404$$ 0 0
$$405$$ −7.00000 −0.347833
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −7.12311 −0.352215 −0.176107 0.984371i $$-0.556351\pi$$
−0.176107 + 0.984371i $$0.556351\pi$$
$$410$$ 0 0
$$411$$ −50.2462 −2.47846
$$412$$ 0 0
$$413$$ 2.87689 0.141563
$$414$$ 0 0
$$415$$ −5.12311 −0.251483
$$416$$ 0 0
$$417$$ 30.7386 1.50528
$$418$$ 0 0
$$419$$ −26.7386 −1.30627 −0.653134 0.757242i $$-0.726546\pi$$
−0.653134 + 0.757242i $$0.726546\pi$$
$$420$$ 0 0
$$421$$ 7.61553 0.371158 0.185579 0.982629i $$-0.440584\pi$$
0.185579 + 0.982629i $$0.440584\pi$$
$$422$$ 0 0
$$423$$ −46.7386 −2.27251
$$424$$ 0 0
$$425$$ 0.561553 0.0272393
$$426$$ 0 0
$$427$$ −5.93087 −0.287015
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 0 0
$$433$$ 33.3693 1.60363 0.801814 0.597574i $$-0.203869\pi$$
0.801814 + 0.597574i $$0.203869\pi$$
$$434$$ 0 0
$$435$$ 24.8078 1.18944
$$436$$ 0 0
$$437$$ −13.1231 −0.627763
$$438$$ 0 0
$$439$$ 34.2462 1.63448 0.817241 0.576296i $$-0.195502\pi$$
0.817241 + 0.576296i $$0.195502\pi$$
$$440$$ 0 0
$$441$$ −1.56155 −0.0743597
$$442$$ 0 0
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ 0 0
$$445$$ −12.5616 −0.595475
$$446$$ 0 0
$$447$$ 5.93087 0.280521
$$448$$ 0 0
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 33.6155 1.57940
$$454$$ 0 0
$$455$$ −5.12311 −0.240175
$$456$$ 0 0
$$457$$ −28.5616 −1.33605 −0.668027 0.744137i $$-0.732861\pi$$
−0.668027 + 0.744137i $$0.732861\pi$$
$$458$$ 0 0
$$459$$ −0.807764 −0.0377032
$$460$$ 0 0
$$461$$ −20.5616 −0.957647 −0.478823 0.877911i $$-0.658937\pi$$
−0.478823 + 0.877911i $$0.658937\pi$$
$$462$$ 0 0
$$463$$ −7.36932 −0.342481 −0.171241 0.985229i $$-0.554778\pi$$
−0.171241 + 0.985229i $$0.554778\pi$$
$$464$$ 0 0
$$465$$ −16.8078 −0.779441
$$466$$ 0 0
$$467$$ 9.93087 0.459546 0.229773 0.973244i $$-0.426202\pi$$
0.229773 + 0.973244i $$0.426202\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ −37.9309 −1.74776
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −2.56155 −0.117532
$$476$$ 0 0
$$477$$ 16.2462 0.743863
$$478$$ 0 0
$$479$$ 32.9848 1.50712 0.753558 0.657381i $$-0.228336\pi$$
0.753558 + 0.657381i $$0.228336\pi$$
$$480$$ 0 0
$$481$$ 11.3693 0.518396
$$482$$ 0 0
$$483$$ −33.6155 −1.52956
$$484$$ 0 0
$$485$$ 7.12311 0.323444
$$486$$ 0 0
$$487$$ −2.24621 −0.101786 −0.0508928 0.998704i $$-0.516207\pi$$
−0.0508928 + 0.998704i $$0.516207\pi$$
$$488$$ 0 0
$$489$$ −8.17708 −0.369780
$$490$$ 0 0
$$491$$ −7.05398 −0.318341 −0.159171 0.987251i $$-0.550882\pi$$
−0.159171 + 0.987251i $$0.550882\pi$$
$$492$$ 0 0
$$493$$ −5.43845 −0.244935
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 9.43845 0.423372
$$498$$ 0 0
$$499$$ 16.4924 0.738302 0.369151 0.929369i $$-0.379648\pi$$
0.369151 + 0.929369i $$0.379648\pi$$
$$500$$ 0 0
$$501$$ −27.0540 −1.20868
$$502$$ 0 0
$$503$$ 14.2462 0.635207 0.317604 0.948224i $$-0.397122\pi$$
0.317604 + 0.948224i $$0.397122\pi$$
$$504$$ 0 0
$$505$$ 4.24621 0.188954
$$506$$ 0 0
$$507$$ 23.0540 1.02386
$$508$$ 0 0
$$509$$ 21.3693 0.947178 0.473589 0.880746i $$-0.342958\pi$$
0.473589 + 0.880746i $$0.342958\pi$$
$$510$$ 0 0
$$511$$ 5.12311 0.226633
$$512$$ 0 0
$$513$$ 3.68466 0.162682
$$514$$ 0 0
$$515$$ 10.2462 0.451502
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 10.8769 0.477443
$$520$$ 0 0
$$521$$ 8.73863 0.382846 0.191423 0.981508i $$-0.438690\pi$$
0.191423 + 0.981508i $$0.438690\pi$$
$$522$$ 0 0
$$523$$ 3.50758 0.153376 0.0766878 0.997055i $$-0.475566\pi$$
0.0766878 + 0.997055i $$0.475566\pi$$
$$524$$ 0 0
$$525$$ −6.56155 −0.286370
$$526$$ 0 0
$$527$$ 3.68466 0.160506
$$528$$ 0 0
$$529$$ 3.24621 0.141140
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ 4.00000 0.173259
$$534$$ 0 0
$$535$$ 16.0000 0.691740
$$536$$ 0 0
$$537$$ 16.0000 0.690451
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 33.5464 1.44227 0.721136 0.692793i $$-0.243620\pi$$
0.721136 + 0.692793i $$0.243620\pi$$
$$542$$ 0 0
$$543$$ 10.8769 0.466772
$$544$$ 0 0
$$545$$ −18.4924 −0.792128
$$546$$ 0 0
$$547$$ 16.0000 0.684111 0.342055 0.939680i $$-0.388877\pi$$
0.342055 + 0.939680i $$0.388877\pi$$
$$548$$ 0 0
$$549$$ −8.24621 −0.351940
$$550$$ 0 0
$$551$$ 24.8078 1.05685
$$552$$ 0 0
$$553$$ 39.3693 1.67415
$$554$$ 0 0
$$555$$ 14.5616 0.618103
$$556$$ 0 0
$$557$$ −46.4924 −1.96995 −0.984974 0.172705i $$-0.944749\pi$$
−0.984974 + 0.172705i $$0.944749\pi$$
$$558$$ 0 0
$$559$$ 20.4924 0.866737
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 2.87689 0.121247 0.0606233 0.998161i $$-0.480691\pi$$
0.0606233 + 0.998161i $$0.480691\pi$$
$$564$$ 0 0
$$565$$ −19.1231 −0.804515
$$566$$ 0 0
$$567$$ −17.9309 −0.753026
$$568$$ 0 0
$$569$$ 28.1080 1.17835 0.589173 0.808007i $$-0.299454\pi$$
0.589173 + 0.808007i $$0.299454\pi$$
$$570$$ 0 0
$$571$$ −30.4233 −1.27318 −0.636588 0.771204i $$-0.719655\pi$$
−0.636588 + 0.771204i $$0.719655\pi$$
$$572$$ 0 0
$$573$$ −26.2462 −1.09645
$$574$$ 0 0
$$575$$ 5.12311 0.213648
$$576$$ 0 0
$$577$$ 24.7386 1.02988 0.514941 0.857225i $$-0.327814\pi$$
0.514941 + 0.857225i $$0.327814\pi$$
$$578$$ 0 0
$$579$$ 13.3002 0.552737
$$580$$ 0 0
$$581$$ −13.1231 −0.544438
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −7.12311 −0.294504
$$586$$ 0 0
$$587$$ 32.3153 1.33380 0.666898 0.745149i $$-0.267621\pi$$
0.666898 + 0.745149i $$0.267621\pi$$
$$588$$ 0 0
$$589$$ −16.8078 −0.692552
$$590$$ 0 0
$$591$$ −13.7538 −0.565755
$$592$$ 0 0
$$593$$ −1.50758 −0.0619088 −0.0309544 0.999521i $$-0.509855\pi$$
−0.0309544 + 0.999521i $$0.509855\pi$$
$$594$$ 0 0
$$595$$ 1.43845 0.0589706
$$596$$ 0 0
$$597$$ 2.06913 0.0846839
$$598$$ 0 0
$$599$$ −17.4384 −0.712516 −0.356258 0.934388i $$-0.615948\pi$$
−0.356258 + 0.934388i $$0.615948\pi$$
$$600$$ 0 0
$$601$$ 30.0000 1.22373 0.611863 0.790964i $$-0.290420\pi$$
0.611863 + 0.790964i $$0.290420\pi$$
$$602$$ 0 0
$$603$$ 22.2462 0.905936
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −20.8078 −0.844561 −0.422281 0.906465i $$-0.638770\pi$$
−0.422281 + 0.906465i $$0.638770\pi$$
$$608$$ 0 0
$$609$$ 63.5464 2.57503
$$610$$ 0 0
$$611$$ 26.2462 1.06181
$$612$$ 0 0
$$613$$ −16.7386 −0.676067 −0.338034 0.941134i $$-0.609762\pi$$
−0.338034 + 0.941134i $$0.609762\pi$$
$$614$$ 0 0
$$615$$ 5.12311 0.206584
$$616$$ 0 0
$$617$$ −6.63068 −0.266941 −0.133471 0.991053i $$-0.542612\pi$$
−0.133471 + 0.991053i $$0.542612\pi$$
$$618$$ 0 0
$$619$$ −8.49242 −0.341339 −0.170670 0.985328i $$-0.554593\pi$$
−0.170670 + 0.985328i $$0.554593\pi$$
$$620$$ 0 0
$$621$$ −7.36932 −0.295720
$$622$$ 0 0
$$623$$ −32.1771 −1.28915
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −3.19224 −0.127283
$$630$$ 0 0
$$631$$ −16.8078 −0.669107 −0.334553 0.942377i $$-0.608585\pi$$
−0.334553 + 0.942377i $$0.608585\pi$$
$$632$$ 0 0
$$633$$ 21.3002 0.846606
$$634$$ 0 0
$$635$$ −14.2462 −0.565344
$$636$$ 0 0
$$637$$ 0.876894 0.0347438
$$638$$ 0 0
$$639$$ 13.1231 0.519142
$$640$$ 0 0
$$641$$ −38.1771 −1.50790 −0.753952 0.656929i $$-0.771855\pi$$
−0.753952 + 0.656929i $$0.771855\pi$$
$$642$$ 0 0
$$643$$ 23.6847 0.934032 0.467016 0.884249i $$-0.345329\pi$$
0.467016 + 0.884249i $$0.345329\pi$$
$$644$$ 0 0
$$645$$ 26.2462 1.03344
$$646$$ 0 0
$$647$$ −13.1231 −0.515923 −0.257961 0.966155i $$-0.583051\pi$$
−0.257961 + 0.966155i $$0.583051\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −43.0540 −1.68742
$$652$$ 0 0
$$653$$ −10.8078 −0.422940 −0.211470 0.977384i $$-0.567825\pi$$
−0.211470 + 0.977384i $$0.567825\pi$$
$$654$$ 0 0
$$655$$ 7.68466 0.300264
$$656$$ 0 0
$$657$$ 7.12311 0.277899
$$658$$ 0 0
$$659$$ −19.1922 −0.747623 −0.373812 0.927505i $$-0.621949\pi$$
−0.373812 + 0.927505i $$0.621949\pi$$
$$660$$ 0 0
$$661$$ 16.2462 0.631904 0.315952 0.948775i $$-0.397676\pi$$
0.315952 + 0.948775i $$0.397676\pi$$
$$662$$ 0 0
$$663$$ 2.87689 0.111729
$$664$$ 0 0
$$665$$ −6.56155 −0.254446
$$666$$ 0 0
$$667$$ −49.6155 −1.92112
$$668$$ 0 0
$$669$$ −33.6155 −1.29965
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 35.7926 1.37970 0.689852 0.723951i $$-0.257676\pi$$
0.689852 + 0.723951i $$0.257676\pi$$
$$674$$ 0 0
$$675$$ −1.43845 −0.0553659
$$676$$ 0 0
$$677$$ −27.6155 −1.06135 −0.530675 0.847575i $$-0.678062\pi$$
−0.530675 + 0.847575i $$0.678062\pi$$
$$678$$ 0 0
$$679$$ 18.2462 0.700225
$$680$$ 0 0
$$681$$ 7.36932 0.282393
$$682$$ 0 0
$$683$$ −25.9309 −0.992217 −0.496109 0.868260i $$-0.665238\pi$$
−0.496109 + 0.868260i $$0.665238\pi$$
$$684$$ 0 0
$$685$$ 19.6155 0.749471
$$686$$ 0 0
$$687$$ 63.3693 2.41769
$$688$$ 0 0
$$689$$ −9.12311 −0.347563
$$690$$ 0 0
$$691$$ 3.36932 0.128175 0.0640874 0.997944i $$-0.479586\pi$$
0.0640874 + 0.997944i $$0.479586\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −12.0000 −0.455186
$$696$$ 0 0
$$697$$ −1.12311 −0.0425407
$$698$$ 0 0
$$699$$ 51.0540 1.93104
$$700$$ 0 0
$$701$$ −34.3153 −1.29607 −0.648036 0.761609i $$-0.724409\pi$$
−0.648036 + 0.761609i $$0.724409\pi$$
$$702$$ 0 0
$$703$$ 14.5616 0.549199
$$704$$ 0 0
$$705$$ 33.6155 1.26603
$$706$$ 0 0
$$707$$ 10.8769 0.409068
$$708$$ 0 0
$$709$$ 1.50758 0.0566183 0.0283091 0.999599i $$-0.490988\pi$$
0.0283091 + 0.999599i $$0.490988\pi$$
$$710$$ 0 0
$$711$$ 54.7386 2.05286
$$712$$ 0 0
$$713$$ 33.6155 1.25891
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 65.6155 2.45046
$$718$$ 0 0
$$719$$ 7.82292 0.291746 0.145873 0.989303i $$-0.453401\pi$$
0.145873 + 0.989303i $$0.453401\pi$$
$$720$$ 0 0
$$721$$ 26.2462 0.977460
$$722$$ 0 0
$$723$$ −41.6155 −1.54770
$$724$$ 0 0
$$725$$ −9.68466 −0.359679
$$726$$ 0 0
$$727$$ 17.6155 0.653324 0.326662 0.945141i $$-0.394076\pi$$
0.326662 + 0.945141i $$0.394076\pi$$
$$728$$ 0 0
$$729$$ −35.9848 −1.33277
$$730$$ 0 0
$$731$$ −5.75379 −0.212812
$$732$$ 0 0
$$733$$ 19.1231 0.706328 0.353164 0.935561i $$-0.385106\pi$$
0.353164 + 0.935561i $$0.385106\pi$$
$$734$$ 0 0
$$735$$ 1.12311 0.0414264
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 42.7386 1.57217 0.786083 0.618121i $$-0.212106\pi$$
0.786083 + 0.618121i $$0.212106\pi$$
$$740$$ 0 0
$$741$$ −13.1231 −0.482089
$$742$$ 0 0
$$743$$ 2.56155 0.0939743 0.0469871 0.998895i $$-0.485038\pi$$
0.0469871 + 0.998895i $$0.485038\pi$$
$$744$$ 0 0
$$745$$ −2.31534 −0.0848276
$$746$$ 0 0
$$747$$ −18.2462 −0.667594
$$748$$ 0 0
$$749$$ 40.9848 1.49755
$$750$$ 0 0
$$751$$ −16.1771 −0.590310 −0.295155 0.955449i $$-0.595371\pi$$
−0.295155 + 0.955449i $$0.595371\pi$$
$$752$$ 0 0
$$753$$ 75.8617 2.76456
$$754$$ 0 0
$$755$$ −13.1231 −0.477599
$$756$$ 0 0
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 36.7386 1.33177 0.665887 0.746052i $$-0.268053\pi$$
0.665887 + 0.746052i $$0.268053\pi$$
$$762$$ 0 0
$$763$$ −47.3693 −1.71488
$$764$$ 0 0
$$765$$ 2.00000 0.0723102
$$766$$ 0 0
$$767$$ −2.24621 −0.0811060
$$768$$ 0 0
$$769$$ −24.7386 −0.892098 −0.446049 0.895009i $$-0.647169\pi$$
−0.446049 + 0.895009i $$0.647169\pi$$
$$770$$ 0 0
$$771$$ 73.6155 2.65120
$$772$$ 0 0
$$773$$ −37.6847 −1.35542 −0.677711 0.735328i $$-0.737028\pi$$
−0.677711 + 0.735328i $$0.737028\pi$$
$$774$$ 0 0
$$775$$ 6.56155 0.235698
$$776$$ 0 0
$$777$$ 37.3002 1.33814
$$778$$ 0 0
$$779$$ 5.12311 0.183554
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 13.9309 0.497849
$$784$$ 0 0
$$785$$ 14.8078 0.528512
$$786$$ 0 0
$$787$$ 54.7386 1.95122 0.975611 0.219508i $$-0.0704451\pi$$
0.975611 + 0.219508i $$0.0704451\pi$$
$$788$$ 0 0
$$789$$ 12.3153 0.438438
$$790$$ 0 0
$$791$$ −48.9848 −1.74170
$$792$$ 0 0
$$793$$ 4.63068 0.164440
$$794$$ 0 0
$$795$$ −11.6847 −0.414412
$$796$$ 0 0
$$797$$ −23.7538 −0.841402 −0.420701 0.907199i $$-0.638216\pi$$
−0.420701 + 0.907199i $$0.638216\pi$$
$$798$$ 0 0
$$799$$ −7.36932 −0.260708
$$800$$ 0 0
$$801$$ −44.7386 −1.58076
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 13.1231 0.462529
$$806$$ 0 0
$$807$$ 50.2462 1.76875
$$808$$ 0 0
$$809$$ −6.49242 −0.228261 −0.114131 0.993466i $$-0.536408\pi$$
−0.114131 + 0.993466i $$0.536408\pi$$
$$810$$ 0 0
$$811$$ −27.1922 −0.954849 −0.477424 0.878673i $$-0.658430\pi$$
−0.477424 + 0.878673i $$0.658430\pi$$
$$812$$ 0 0
$$813$$ 72.9848 2.55969
$$814$$ 0 0
$$815$$ 3.19224 0.111819
$$816$$ 0 0
$$817$$ 26.2462 0.918239
$$818$$ 0 0
$$819$$ −18.2462 −0.637574
$$820$$ 0 0
$$821$$ 2.00000 0.0698005 0.0349002 0.999391i $$-0.488889\pi$$
0.0349002 + 0.999391i $$0.488889\pi$$
$$822$$ 0 0
$$823$$ −52.4924 −1.82977 −0.914885 0.403714i $$-0.867719\pi$$
−0.914885 + 0.403714i $$0.867719\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 4.49242 0.156217 0.0781084 0.996945i $$-0.475112\pi$$
0.0781084 + 0.996945i $$0.475112\pi$$
$$828$$ 0 0
$$829$$ 5.36932 0.186484 0.0932420 0.995643i $$-0.470277\pi$$
0.0932420 + 0.995643i $$0.470277\pi$$
$$830$$ 0 0
$$831$$ 18.2462 0.632954
$$832$$ 0 0
$$833$$ −0.246211 −0.00853071
$$834$$ 0 0
$$835$$ 10.5616 0.365498
$$836$$ 0 0
$$837$$ −9.43845 −0.326240
$$838$$ 0 0
$$839$$ 11.5076 0.397286 0.198643 0.980072i $$-0.436347\pi$$
0.198643 + 0.980072i $$0.436347\pi$$
$$840$$ 0 0
$$841$$ 64.7926 2.23423
$$842$$ 0 0
$$843$$ −21.1231 −0.727518
$$844$$ 0 0
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −59.8617 −2.05445
$$850$$ 0 0
$$851$$ −29.1231 −0.998327
$$852$$ 0 0
$$853$$ 3.75379 0.128527 0.0642636 0.997933i $$-0.479530\pi$$
0.0642636 + 0.997933i $$0.479530\pi$$
$$854$$ 0 0
$$855$$ −9.12311 −0.312004
$$856$$ 0 0
$$857$$ −32.4233 −1.10756 −0.553779 0.832664i $$-0.686815\pi$$
−0.553779 + 0.832664i $$0.686815\pi$$
$$858$$ 0 0
$$859$$ −38.8769 −1.32646 −0.663231 0.748415i $$-0.730815\pi$$
−0.663231 + 0.748415i $$0.730815\pi$$
$$860$$ 0 0
$$861$$ 13.1231 0.447234
$$862$$ 0 0
$$863$$ 9.61553 0.327316 0.163658 0.986517i $$-0.447671\pi$$
0.163658 + 0.986517i $$0.447671\pi$$
$$864$$ 0 0
$$865$$ −4.24621 −0.144376
$$866$$ 0 0
$$867$$ 42.7386 1.45148
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −12.4924 −0.423290
$$872$$ 0 0
$$873$$ 25.3693 0.858621
$$874$$ 0 0
$$875$$ 2.56155 0.0865963
$$876$$ 0 0
$$877$$ 32.2462 1.08888 0.544439 0.838801i $$-0.316743\pi$$
0.544439 + 0.838801i $$0.316743\pi$$
$$878$$ 0 0
$$879$$ 37.1231 1.25213
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 0 0
$$883$$ 6.06913 0.204242 0.102121 0.994772i $$-0.467437\pi$$
0.102121 + 0.994772i $$0.467437\pi$$
$$884$$ 0 0
$$885$$ −2.87689 −0.0967057
$$886$$ 0 0
$$887$$ 30.2462 1.01557 0.507784 0.861484i $$-0.330465\pi$$
0.507784 + 0.861484i $$0.330465\pi$$
$$888$$ 0 0
$$889$$ −36.4924 −1.22392
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 33.6155 1.12490
$$894$$ 0 0
$$895$$ −6.24621 −0.208788
$$896$$ 0 0
$$897$$ 26.2462 0.876335
$$898$$ 0 0
$$899$$ −63.5464 −2.11939
$$900$$ 0 0
$$901$$ 2.56155 0.0853377
$$902$$ 0 0
$$903$$ 67.2311 2.23731
$$904$$ 0 0
$$905$$ −4.24621 −0.141149
$$906$$ 0 0
$$907$$ 8.94602 0.297048 0.148524 0.988909i $$-0.452548\pi$$
0.148524 + 0.988909i $$0.452548\pi$$
$$908$$ 0 0
$$909$$ 15.1231 0.501602
$$910$$ 0 0
$$911$$ −2.06913 −0.0685533 −0.0342767 0.999412i $$-0.510913\pi$$
−0.0342767 + 0.999412i $$0.510913\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 5.93087 0.196069
$$916$$ 0 0
$$917$$ 19.6847 0.650045
$$918$$ 0 0
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ −65.6155 −2.16211
$$922$$ 0 0
$$923$$ −7.36932 −0.242564
$$924$$ 0 0
$$925$$ −5.68466 −0.186910
$$926$$ 0 0
$$927$$ 36.4924 1.19857
$$928$$ 0 0
$$929$$ 45.0540 1.47817 0.739086 0.673611i $$-0.235257\pi$$
0.739086 + 0.673611i $$0.235257\pi$$
$$930$$ 0 0
$$931$$ 1.12311 0.0368083
$$932$$ 0 0
$$933$$ 65.1619 2.13331
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 42.0000 1.37208 0.686040 0.727564i $$-0.259347\pi$$
0.686040 + 0.727564i $$0.259347\pi$$
$$938$$ 0 0
$$939$$ −78.1080 −2.54896
$$940$$ 0 0
$$941$$ 4.06913 0.132650 0.0663249 0.997798i $$-0.478873\pi$$
0.0663249 + 0.997798i $$0.478873\pi$$
$$942$$ 0 0
$$943$$ −10.2462 −0.333663
$$944$$ 0 0
$$945$$ −3.68466 −0.119862
$$946$$ 0 0
$$947$$ −39.6847 −1.28958 −0.644789 0.764361i $$-0.723055\pi$$
−0.644789 + 0.764361i $$0.723055\pi$$
$$948$$ 0 0
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ −33.7926 −1.09580
$$952$$ 0 0
$$953$$ 34.1771 1.10710 0.553552 0.832815i $$-0.313272\pi$$
0.553552 + 0.832815i $$0.313272\pi$$
$$954$$ 0 0
$$955$$ 10.2462 0.331560
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 50.2462 1.62253
$$960$$ 0 0
$$961$$ 12.0540 0.388838
$$962$$ 0 0
$$963$$ 56.9848 1.83631
$$964$$ 0 0
$$965$$ −5.19224 −0.167144
$$966$$ 0 0
$$967$$ −11.5464 −0.371307 −0.185654 0.982615i $$-0.559440\pi$$
−0.185654 + 0.982615i $$0.559440\pi$$
$$968$$ 0 0
$$969$$ 3.68466 0.118368
$$970$$ 0 0
$$971$$ 6.24621 0.200450 0.100225 0.994965i $$-0.468044\pi$$
0.100225 + 0.994965i $$0.468044\pi$$
$$972$$ 0 0
$$973$$ −30.7386 −0.985435
$$974$$ 0 0
$$975$$ 5.12311 0.164071
$$976$$ 0 0
$$977$$ −24.8769 −0.795882 −0.397941 0.917411i $$-0.630275\pi$$
−0.397941 + 0.917411i $$0.630275\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −65.8617 −2.10280
$$982$$ 0 0
$$983$$ 18.8769 0.602079 0.301040 0.953612i $$-0.402666\pi$$
0.301040 + 0.953612i $$0.402666\pi$$
$$984$$ 0 0
$$985$$ 5.36932 0.171081
$$986$$ 0 0
$$987$$ 86.1080 2.74085
$$988$$ 0 0
$$989$$ −52.4924 −1.66916
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 0 0
$$993$$ 21.7538 0.690336
$$994$$ 0 0
$$995$$ −0.807764 −0.0256078
$$996$$ 0 0
$$997$$ −13.8617 −0.439006 −0.219503 0.975612i $$-0.570444\pi$$
−0.219503 + 0.975612i $$0.570444\pi$$
$$998$$ 0 0
$$999$$ 8.17708 0.258711
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 4840.2.a.j.1.1 2
4.3 odd 2 9680.2.a.bs.1.2 2
11.10 odd 2 440.2.a.e.1.1 2
33.32 even 2 3960.2.a.w.1.1 2
44.43 even 2 880.2.a.o.1.2 2
55.32 even 4 2200.2.b.i.1849.4 4
55.43 even 4 2200.2.b.i.1849.1 4
55.54 odd 2 2200.2.a.s.1.2 2
88.21 odd 2 3520.2.a.bp.1.2 2
88.43 even 2 3520.2.a.bk.1.1 2
132.131 odd 2 7920.2.a.bu.1.2 2
220.43 odd 4 4400.2.b.t.4049.4 4
220.87 odd 4 4400.2.b.t.4049.1 4
220.219 even 2 4400.2.a.bj.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.e.1.1 2 11.10 odd 2
880.2.a.o.1.2 2 44.43 even 2
2200.2.a.s.1.2 2 55.54 odd 2
2200.2.b.i.1849.1 4 55.43 even 4
2200.2.b.i.1849.4 4 55.32 even 4
3520.2.a.bk.1.1 2 88.43 even 2
3520.2.a.bp.1.2 2 88.21 odd 2
3960.2.a.w.1.1 2 33.32 even 2
4400.2.a.bj.1.1 2 220.219 even 2
4400.2.b.t.4049.1 4 220.87 odd 4
4400.2.b.t.4049.4 4 220.43 odd 4
4840.2.a.j.1.1 2 1.1 even 1 trivial
7920.2.a.bu.1.2 2 132.131 odd 2
9680.2.a.bs.1.2 2 4.3 odd 2