# Properties

 Label 2200.2.a.s.1.2 Level $2200$ Weight $2$ Character 2200.1 Self dual yes Analytic conductor $17.567$ 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] = [2200,2,Mod(1,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, 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("2200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2200 = 2^{3} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2200.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: $$17.5670884447$$ Analytic rank: $$0$$ 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 1.2 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 2200.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.56155 q^{3} +2.56155 q^{7} +3.56155 q^{9} +O(q^{10})$$ $$q+2.56155 q^{3} +2.56155 q^{7} +3.56155 q^{9} +1.00000 q^{11} -2.00000 q^{13} +0.561553 q^{17} +2.56155 q^{19} +6.56155 q^{21} -5.12311 q^{23} +1.43845 q^{27} +9.68466 q^{29} +6.56155 q^{31} +2.56155 q^{33} +5.68466 q^{37} -5.12311 q^{39} +2.00000 q^{41} -10.2462 q^{43} +13.1231 q^{47} -0.438447 q^{49} +1.43845 q^{51} -4.56155 q^{53} +6.56155 q^{57} +1.12311 q^{59} +2.31534 q^{61} +9.12311 q^{63} -6.24621 q^{67} -13.1231 q^{69} +3.68466 q^{71} +2.00000 q^{73} +2.56155 q^{77} -15.3693 q^{79} -7.00000 q^{81} -5.12311 q^{83} +24.8078 q^{87} -12.5616 q^{89} -5.12311 q^{91} +16.8078 q^{93} -7.12311 q^{97} +3.56155 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + q^7 + 3 * q^9 $$2 q + q^{3} + q^{7} + 3 q^{9} + 2 q^{11} - 4 q^{13} - 3 q^{17} + q^{19} + 9 q^{21} - 2 q^{23} + 7 q^{27} + 7 q^{29} + 9 q^{31} + q^{33} - q^{37} - 2 q^{39} + 4 q^{41} - 4 q^{43} + 18 q^{47} - 5 q^{49} + 7 q^{51} - 5 q^{53} + 9 q^{57} - 6 q^{59} + 17 q^{61} + 10 q^{63} + 4 q^{67} - 18 q^{69} - 5 q^{71} + 4 q^{73} + q^{77} - 6 q^{79} - 14 q^{81} - 2 q^{83} + 29 q^{87} - 21 q^{89} - 2 q^{91} + 13 q^{93} - 6 q^{97} + 3 q^{99}+O(q^{100})$$ 2 * q + q^3 + q^7 + 3 * q^9 + 2 * q^11 - 4 * q^13 - 3 * q^17 + q^19 + 9 * q^21 - 2 * q^23 + 7 * q^27 + 7 * q^29 + 9 * q^31 + q^33 - q^37 - 2 * q^39 + 4 * q^41 - 4 * q^43 + 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^67 - 18 * q^69 - 5 * q^71 + 4 * q^73 + q^77 - 6 * q^79 - 14 * q^81 - 2 * q^83 + 29 * q^87 - 21 * q^89 - 2 * q^91 + 13 * q^93 - 6 * q^97 + 3 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 2.56155 1.47891 0.739457 0.673204i $$-0.235083\pi$$
0.739457 + 0.673204i $$0.235083\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 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
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 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 $$-0.405070\pi$$
0.293830 + 0.955858i $$0.405070\pi$$
$$20$$ 0 0
$$21$$ 6.56155 1.43185
$$22$$ 0 0
$$23$$ −5.12311 −1.06824 −0.534121 0.845408i $$-0.679357\pi$$
−0.534121 + 0.845408i $$0.679357\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.43845 0.276829
$$28$$ 0 0
$$29$$ 9.68466 1.79840 0.899198 0.437542i $$-0.144151\pi$$
0.899198 + 0.437542i $$0.144151\pi$$
$$30$$ 0 0
$$31$$ 6.56155 1.17849 0.589245 0.807955i $$-0.299425\pi$$
0.589245 + 0.807955i $$0.299425\pi$$
$$32$$ 0 0
$$33$$ 2.56155 0.445909
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.68466 0.934552 0.467276 0.884111i $$-0.345235\pi$$
0.467276 + 0.884111i $$0.345235\pi$$
$$38$$ 0 0
$$39$$ −5.12311 −0.820353
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ −10.2462 −1.56253 −0.781266 0.624198i $$-0.785426\pi$$
−0.781266 + 0.624198i $$0.785426\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 13.1231 1.91420 0.957101 0.289755i $$-0.0935738\pi$$
0.957101 + 0.289755i $$0.0935738\pi$$
$$48$$ 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 $$-0.601431\pi$$
−0.313289 + 0.949658i $$0.601431\pi$$
$$54$$ 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 $$-0.452644\pi$$
0.148225 + 0.988954i $$0.452644\pi$$
$$62$$ 0 0
$$63$$ 9.12311 1.14940
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −6.24621 −0.763096 −0.381548 0.924349i $$-0.624609\pi$$
−0.381548 + 0.924349i $$0.624609\pi$$
$$68$$ 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
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.56155 0.291916
$$78$$ 0 0
$$79$$ −15.3693 −1.72918 −0.864592 0.502475i $$-0.832423\pi$$
−0.864592 + 0.502475i $$0.832423\pi$$
$$80$$ 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 0
$$86$$ 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
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −7.12311 −0.723242 −0.361621 0.932325i $$-0.617777\pi$$
−0.361621 + 0.932325i $$0.617777\pi$$
$$98$$ 0 0
$$99$$ 3.56155 0.357950
$$100$$ 0 0
$$101$$ −4.24621 −0.422514 −0.211257 0.977431i $$-0.567756\pi$$
−0.211257 + 0.977431i $$0.567756\pi$$
$$102$$ 0 0
$$103$$ −10.2462 −1.00959 −0.504795 0.863239i $$-0.668432\pi$$
−0.504795 + 0.863239i $$0.668432\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 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 $$-0.153729\pi$$
0.885626 + 0.464398i $$0.153729\pi$$
$$110$$ 0 0
$$111$$ 14.5616 1.38212
$$112$$ 0 0
$$113$$ 19.1231 1.79895 0.899475 0.436972i $$-0.143949\pi$$
0.899475 + 0.436972i $$0.143949\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −7.12311 −0.658531
$$118$$ 0 0
$$119$$ 1.43845 0.131862
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 5.12311 0.461935
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 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 $$-0.608975\pi$$
−0.335706 + 0.941967i $$0.608975\pi$$
$$132$$ 0 0
$$133$$ 6.56155 0.568959
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −19.6155 −1.67587 −0.837934 0.545772i $$-0.816237\pi$$
−0.837934 + 0.545772i $$0.816237\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 33.6155 2.83094
$$142$$ 0 0
$$143$$ −2.00000 −0.167248
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −1.12311 −0.0926322
$$148$$ 0 0
$$149$$ 2.31534 0.189680 0.0948401 0.995493i $$-0.469766\pi$$
0.0948401 + 0.995493i $$0.469766\pi$$
$$150$$ 0 0
$$151$$ 13.1231 1.06794 0.533972 0.845502i $$-0.320699\pi$$
0.533972 + 0.845502i $$0.320699\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −14.8078 −1.18179 −0.590894 0.806749i $$-0.701225\pi$$
−0.590894 + 0.806749i $$0.701225\pi$$
$$158$$ 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 $$-0.539899\pi$$
−0.125018 + 0.992155i $$0.539899\pi$$
$$164$$ 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
$$175$$ 0 0
$$176$$ 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
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0.561553 0.0410648
$$188$$ 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
$$195$$ 0 0
$$196$$ 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
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −18.2462 −1.26820
$$208$$ 0 0
$$209$$ 2.56155 0.177186
$$210$$ 0 0
$$211$$ 8.31534 0.572452 0.286226 0.958162i $$-0.407599\pi$$
0.286226 + 0.958162i $$0.407599\pi$$
$$212$$ 0 0
$$213$$ 9.43845 0.646712
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 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 $$-0.644807\pi$$
−0.439394 + 0.898294i $$0.644807\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 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
$$231$$ 6.56155 0.431718
$$232$$ 0 0
$$233$$ −19.9309 −1.30571 −0.652857 0.757481i $$-0.726430\pi$$
−0.652857 + 0.757481i $$0.726430\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −39.3693 −2.55731
$$238$$ 0 0
$$239$$ 25.6155 1.65693 0.828465 0.560040i $$-0.189214\pi$$
0.828465 + 0.560040i $$0.189214\pi$$
$$240$$ 0 0
$$241$$ −16.2462 −1.04651 −0.523255 0.852176i $$-0.675283\pi$$
−0.523255 + 0.852176i $$0.675283\pi$$
$$242$$ 0 0
$$243$$ −22.2462 −1.42710
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 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
$$253$$ −5.12311 −0.322087
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 28.7386 1.79267 0.896333 0.443381i $$-0.146221\pi$$
0.896333 + 0.443381i $$0.146221\pi$$
$$258$$ 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
$$265$$ 0 0
$$266$$ 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 $$-0.167067\pi$$
0.865396 + 0.501089i $$0.167067\pi$$
$$272$$ 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 $$-0.579104\pi$$
−0.245964 + 0.969279i $$0.579104\pi$$
$$282$$ 0 0
$$283$$ 23.3693 1.38916 0.694581 0.719415i $$-0.255590\pi$$
0.694581 + 0.719415i $$0.255590\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 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
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1.43845 0.0834672
$$298$$ 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
$$305$$ 0 0
$$306$$ 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 $$-0.830642\pi$$
−0.861767 + 0.507305i $$0.830642\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −13.1922 −0.740950 −0.370475 0.928842i $$-0.620805\pi$$
−0.370475 + 0.928842i $$0.620805\pi$$
$$318$$ 0 0
$$319$$ 9.68466 0.542237
$$320$$ 0 0
$$321$$ 40.9848 2.28755
$$322$$ 0 0
$$323$$ 1.43845 0.0800373
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 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
$$335$$ 0 0
$$336$$ 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
$$341$$ 6.56155 0.355328
$$342$$ 0 0
$$343$$ −19.0540 −1.02882
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 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 $$-0.409393\pi$$
0.280823 + 0.959760i $$0.409393\pi$$
$$350$$ 0 0
$$351$$ −2.87689 −0.153557
$$352$$ 0 0
$$353$$ 34.4924 1.83585 0.917923 0.396758i $$-0.129865\pi$$
0.917923 + 0.396758i $$0.129865\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 3.68466 0.195013
$$358$$ 0 0
$$359$$ 26.2462 1.38522 0.692611 0.721311i $$-0.256460\pi$$
0.692611 + 0.721311i $$0.256460\pi$$
$$360$$ 0 0
$$361$$ −12.4384 −0.654655
$$362$$ 0 0
$$363$$ 2.56155 0.134447
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 14.7386 0.769350 0.384675 0.923052i $$-0.374313\pi$$
0.384675 + 0.923052i $$0.374313\pi$$
$$368$$ 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
$$375$$ 0 0
$$376$$ 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 $$-0.204063\pi$$
0.801449 + 0.598064i $$0.204063\pi$$
$$384$$ 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
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 18.0000 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$398$$ 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
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 5.68466 0.281778
$$408$$ 0 0
$$409$$ 7.12311 0.352215 0.176107 0.984371i $$-0.443649\pi$$
0.176107 + 0.984371i $$0.443649\pi$$
$$410$$ 0 0
$$411$$ −50.2462 −2.47846
$$412$$ 0 0
$$413$$ 2.87689 0.141563
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 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 0
$$426$$ 0 0
$$427$$ 5.93087 0.287015
$$428$$ 0 0
$$429$$ −5.12311 −0.247346
$$430$$ 0 0
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ 0 0
$$433$$ −33.3693 −1.60363 −0.801814 0.597574i $$-0.796131\pi$$
−0.801814 + 0.597574i $$0.796131\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −13.1231 −0.627763
$$438$$ 0 0
$$439$$ −34.2462 −1.63448 −0.817241 0.576296i $$-0.804498\pi$$
−0.817241 + 0.576296i $$0.804498\pi$$
$$440$$ 0 0
$$441$$ −1.56155 −0.0743597
$$442$$ 0 0
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 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
$$451$$ 2.00000 0.0941763
$$452$$ 0 0
$$453$$ 33.6155 1.57940
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 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 $$-0.341063\pi$$
0.478823 + 0.877911i $$0.341063\pi$$
$$462$$ 0 0
$$463$$ 7.36932 0.342481 0.171241 0.985229i $$-0.445222\pi$$
0.171241 + 0.985229i $$0.445222\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −9.93087 −0.459546 −0.229773 0.973244i $$-0.573798\pi$$
−0.229773 + 0.973244i $$0.573798\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ −37.9309 −1.74776
$$472$$ 0 0
$$473$$ −10.2462 −0.471121
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −16.2462 −0.743863
$$478$$ 0 0
$$479$$ −32.9848 −1.50712 −0.753558 0.657381i $$-0.771664\pi$$
−0.753558 + 0.657381i $$0.771664\pi$$
$$480$$ 0 0
$$481$$ −11.3693 −0.518396
$$482$$ 0 0
$$483$$ −33.6155 −1.52956
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 2.24621 0.101786 0.0508928 0.998704i $$-0.483793\pi$$
0.0508928 + 0.998704i $$0.483793\pi$$
$$488$$ 0 0
$$489$$ −8.17708 −0.369780
$$490$$ 0 0
$$491$$ 7.05398 0.318341 0.159171 0.987251i $$-0.449118\pi$$
0.159171 + 0.987251i $$0.449118\pi$$
$$492$$ 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
$$505$$ 0 0
$$506$$ 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
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 13.1231 0.577154
$$518$$ 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
$$525$$ 0 0
$$526$$ 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
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −16.0000 −0.690451
$$538$$ 0 0
$$539$$ −0.438447 −0.0188853
$$540$$ 0 0
$$541$$ −33.5464 −1.44227 −0.721136 0.692793i $$-0.756380\pi$$
−0.721136 + 0.692793i $$0.756380\pi$$
$$542$$ 0 0
$$543$$ −10.8769 −0.466772
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 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
$$555$$ 0 0
$$556$$ 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
$$561$$ 1.43845 0.0607313
$$562$$ 0 0
$$563$$ 2.87689 0.121247 0.0606233 0.998161i $$-0.480691\pi$$
0.0606233 + 0.998161i $$0.480691\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −17.9309 −0.753026
$$568$$ 0 0
$$569$$ −28.1080 −1.17835 −0.589173 0.808007i $$-0.700546\pi$$
−0.589173 + 0.808007i $$0.700546\pi$$
$$570$$ 0 0
$$571$$ 30.4233 1.27318 0.636588 0.771204i $$-0.280345\pi$$
0.636588 + 0.771204i $$0.280345\pi$$
$$572$$ 0 0
$$573$$ 26.2462 1.09645
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −24.7386 −1.02988 −0.514941 0.857225i $$-0.672186\pi$$
−0.514941 + 0.857225i $$0.672186\pi$$
$$578$$ 0 0
$$579$$ −13.3002 −0.552737
$$580$$ 0 0
$$581$$ −13.1231 −0.544438
$$582$$ 0 0
$$583$$ −4.56155 −0.188920
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −32.3153 −1.33380 −0.666898 0.745149i $$-0.732379\pi$$
−0.666898 + 0.745149i $$0.732379\pi$$
$$588$$ 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
$$595$$ 0 0
$$596$$ 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 $$-0.709580\pi$$
−0.611863 + 0.790964i $$0.709580\pi$$
$$602$$ 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
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.63068 0.266941 0.133471 0.991053i $$-0.457388\pi$$
0.133471 + 0.991053i $$0.457388\pi$$
$$618$$ 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
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 6.56155 0.262043
$$628$$ 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
$$635$$ 0 0
$$636$$ 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 $$-0.654671\pi$$
−0.467016 + 0.884249i $$0.654671\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 13.1231 0.515923 0.257961 0.966155i $$-0.416949\pi$$
0.257961 + 0.966155i $$0.416949\pi$$
$$648$$ 0 0
$$649$$ 1.12311 0.0440858
$$650$$ 0 0
$$651$$ 43.0540 1.68742
$$652$$ 0 0
$$653$$ 10.8078 0.422940 0.211470 0.977384i $$-0.432175\pi$$
0.211470 + 0.977384i $$0.432175\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 7.12311 0.277899
$$658$$ 0 0
$$659$$ 19.1922 0.747623 0.373812 0.927505i $$-0.378051\pi$$
0.373812 + 0.927505i $$0.378051\pi$$
$$660$$ 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
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −49.6155 −1.92112
$$668$$ 0 0
$$669$$ −33.6155 −1.29965
$$670$$ 0 0
$$671$$ 2.31534 0.0893828
$$672$$ 0 0
$$673$$ 35.7926 1.37970 0.689852 0.723951i $$-0.257676\pi$$
0.689852 + 0.723951i $$0.257676\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 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 $$-0.334762\pi$$
0.496109 + 0.868260i $$0.334762\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 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
$$693$$ 9.12311 0.346558
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 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 $$-0.275591\pi$$
0.648036 + 0.761609i $$0.275591\pi$$
$$702$$ 0 0
$$703$$ 14.5616 0.549199
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 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
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −17.6155 −0.653324 −0.326662 0.945141i $$-0.605924\pi$$
−0.326662 + 0.945141i $$0.605924\pi$$
$$728$$ 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
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −6.24621 −0.230082
$$738$$ 0 0
$$739$$ −42.7386 −1.57217 −0.786083 0.618121i $$-0.787894\pi$$
−0.786083 + 0.618121i $$0.787894\pi$$
$$740$$ 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
$$745$$ 0 0
$$746$$ 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
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 26.0000 0.944986 0.472493 0.881334i $$-0.343354\pi$$
0.472493 + 0.881334i $$0.343354\pi$$
$$758$$ 0 0
$$759$$ −13.1231 −0.476339
$$760$$ 0 0
$$761$$ −36.7386 −1.33177 −0.665887 0.746052i $$-0.731947\pi$$
−0.665887 + 0.746052i $$0.731947\pi$$
$$762$$ 0 0
$$763$$ 47.3693 1.71488
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −2.24621 −0.0811060
$$768$$ 0 0
$$769$$ 24.7386 0.892098 0.446049 0.895009i $$-0.352831\pi$$
0.446049 + 0.895009i $$0.352831\pi$$
$$770$$ 0 0
$$771$$ 73.6155 2.65120
$$772$$ 0 0
$$773$$ 37.6847 1.35542 0.677711 0.735328i $$-0.262972\pi$$
0.677711 + 0.735328i $$0.262972\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 37.3002 1.33814
$$778$$ 0 0
$$779$$ 5.12311 0.183554
$$780$$ 0 0
$$781$$ 3.68466 0.131847
$$782$$ 0 0
$$783$$ 13.9309 0.497849
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 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
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 23.7538 0.841402 0.420701 0.907199i $$-0.361784\pi$$
0.420701 + 0.907199i $$0.361784\pi$$
$$798$$ 0 0
$$799$$ 7.36932 0.260708
$$800$$ 0 0
$$801$$ −44.7386 −1.58076
$$802$$ 0 0
$$803$$ 2.00000 0.0705785
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −50.2462 −1.76875
$$808$$ 0 0
$$809$$ 6.49242 0.228261 0.114131 0.993466i $$-0.463592\pi$$
0.114131 + 0.993466i $$0.463592\pi$$
$$810$$ 0 0
$$811$$ 27.1922 0.954849 0.477424 0.878673i $$-0.341570\pi$$
0.477424 + 0.878673i $$0.341570\pi$$
$$812$$ 0 0
$$813$$ 72.9848 2.55969
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 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 $$-0.511111\pi$$
−0.0349002 + 0.999391i $$0.511111\pi$$
$$822$$ 0 0
$$823$$ 52.4924 1.82977 0.914885 0.403714i $$-0.132281\pi$$
0.914885 + 0.403714i $$0.132281\pi$$
$$824$$ 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
$$835$$ 0 0
$$836$$ 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
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.56155 0.0880160
$$848$$ 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
$$855$$ 0 0
$$856$$ 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 $$-0.552329\pi$$
−0.163658 + 0.986517i $$0.552329\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −42.7386 −1.45148
$$868$$ 0 0
$$869$$ −15.3693 −0.521368
$$870$$ 0 0
$$871$$ 12.4924 0.423290
$$872$$ 0 0
$$873$$ −25.3693 −0.858621
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 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 $$-0.532563\pi$$
−0.102121 + 0.994772i $$0.532563\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 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
$$891$$ −7.00000 −0.234509
$$892$$ 0 0
$$893$$ 33.6155 1.12490
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 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
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −8.94602 −0.297048 −0.148524 0.988909i $$-0.547452\pi$$
−0.148524 + 0.988909i $$0.547452\pi$$
$$908$$ 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
$$913$$ −5.12311 −0.169550
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −19.6847 −0.650045
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 65.6155 2.16211
$$922$$ 0 0
$$923$$ −7.36932 −0.242564
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 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 $$-0.521127\pi$$
−0.0663249 + 0.997798i $$0.521127\pi$$
$$942$$ 0 0
$$943$$ −10.2462 −0.333663
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 39.6847 1.28958 0.644789 0.764361i $$-0.276945\pi$$
0.644789 + 0.764361i $$0.276945\pi$$
$$948$$ 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
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 24.8078 0.801921
$$958$$ 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
$$965$$ 0 0
$$966$$ 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
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 24.8769 0.795882 0.397941 0.917411i $$-0.369725\pi$$
0.397941 + 0.917411i $$0.369725\pi$$
$$978$$ 0 0
$$979$$ −12.5616 −0.401469
$$980$$ 0 0
$$981$$ 65.8617 2.10280
$$982$$ 0 0
$$983$$ −18.8769 −0.602079 −0.301040 0.953612i $$-0.597334\pi$$
−0.301040 + 0.953612i $$0.597334\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 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 0
$$996$$ 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 2200.2.a.s.1.2 2
4.3 odd 2 4400.2.a.bj.1.1 2
5.2 odd 4 2200.2.b.i.1849.1 4
5.3 odd 4 2200.2.b.i.1849.4 4
5.4 even 2 440.2.a.e.1.1 2
15.14 odd 2 3960.2.a.w.1.1 2
20.3 even 4 4400.2.b.t.4049.1 4
20.7 even 4 4400.2.b.t.4049.4 4
20.19 odd 2 880.2.a.o.1.2 2
40.19 odd 2 3520.2.a.bk.1.1 2
40.29 even 2 3520.2.a.bp.1.2 2
55.54 odd 2 4840.2.a.j.1.1 2
60.59 even 2 7920.2.a.bu.1.2 2
220.219 even 2 9680.2.a.bs.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.e.1.1 2 5.4 even 2
880.2.a.o.1.2 2 20.19 odd 2
2200.2.a.s.1.2 2 1.1 even 1 trivial
2200.2.b.i.1849.1 4 5.2 odd 4
2200.2.b.i.1849.4 4 5.3 odd 4
3520.2.a.bk.1.1 2 40.19 odd 2
3520.2.a.bp.1.2 2 40.29 even 2
3960.2.a.w.1.1 2 15.14 odd 2
4400.2.a.bj.1.1 2 4.3 odd 2
4400.2.b.t.4049.1 4 20.3 even 4
4400.2.b.t.4049.4 4 20.7 even 4
4840.2.a.j.1.1 2 55.54 odd 2
7920.2.a.bu.1.2 2 60.59 even 2
9680.2.a.bs.1.2 2 220.219 even 2