# Properties

 Label 1620.2.a.h.1.2 Level $1620$ Weight $2$ Character 1620.1 Self dual yes Analytic conductor $12.936$ 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] = [1620,2,Mod(1,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1620.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: $$12.9357651274$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 1620.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} +0.732051 q^{7} +O(q^{10})$$ $$q+1.00000 q^{5} +0.732051 q^{7} +1.73205 q^{11} -1.46410 q^{13} +1.26795 q^{17} +2.46410 q^{19} +3.46410 q^{23} +1.00000 q^{25} +4.26795 q^{29} -7.92820 q^{31} +0.732051 q^{35} +4.19615 q^{37} +0.803848 q^{41} +6.73205 q^{43} +4.73205 q^{47} -6.46410 q^{49} +10.7321 q^{53} +1.73205 q^{55} +4.26795 q^{59} -4.00000 q^{61} -1.46410 q^{65} -14.3923 q^{67} +0.803848 q^{71} +10.1962 q^{73} +1.26795 q^{77} +6.39230 q^{79} -9.12436 q^{83} +1.26795 q^{85} +5.19615 q^{89} -1.07180 q^{91} +2.46410 q^{95} -2.73205 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 - 2 * q^7 $$2 q + 2 q^{5} - 2 q^{7} + 4 q^{13} + 6 q^{17} - 2 q^{19} + 2 q^{25} + 12 q^{29} - 2 q^{31} - 2 q^{35} - 2 q^{37} + 12 q^{41} + 10 q^{43} + 6 q^{47} - 6 q^{49} + 18 q^{53} + 12 q^{59} - 8 q^{61} + 4 q^{65} - 8 q^{67} + 12 q^{71} + 10 q^{73} + 6 q^{77} - 8 q^{79} + 6 q^{83} + 6 q^{85} - 16 q^{91} - 2 q^{95} - 2 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 - 2 * q^7 + 4 * q^13 + 6 * q^17 - 2 * q^19 + 2 * q^25 + 12 * q^29 - 2 * q^31 - 2 * q^35 - 2 * q^37 + 12 * q^41 + 10 * q^43 + 6 * q^47 - 6 * q^49 + 18 * q^53 + 12 * q^59 - 8 * q^61 + 4 * q^65 - 8 * q^67 + 12 * q^71 + 10 * q^73 + 6 * q^77 - 8 * q^79 + 6 * q^83 + 6 * q^85 - 16 * q^91 - 2 * q^95 - 2 * 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0.732051 0.276689 0.138345 0.990384i $$-0.455822\pi$$
0.138345 + 0.990384i $$0.455822\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.73205 0.522233 0.261116 0.965307i $$-0.415909\pi$$
0.261116 + 0.965307i $$0.415909\pi$$
$$12$$ 0 0
$$13$$ −1.46410 −0.406069 −0.203034 0.979172i $$-0.565080\pi$$
−0.203034 + 0.979172i $$0.565080\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.26795 0.307523 0.153761 0.988108i $$-0.450861\pi$$
0.153761 + 0.988108i $$0.450861\pi$$
$$18$$ 0 0
$$19$$ 2.46410 0.565304 0.282652 0.959223i $$-0.408786\pi$$
0.282652 + 0.959223i $$0.408786\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.46410 0.722315 0.361158 0.932505i $$-0.382382\pi$$
0.361158 + 0.932505i $$0.382382\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 4.26795 0.792538 0.396269 0.918134i $$-0.370305\pi$$
0.396269 + 0.918134i $$0.370305\pi$$
$$30$$ 0 0
$$31$$ −7.92820 −1.42395 −0.711974 0.702206i $$-0.752198\pi$$
−0.711974 + 0.702206i $$0.752198\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0.732051 0.123739
$$36$$ 0 0
$$37$$ 4.19615 0.689843 0.344922 0.938631i $$-0.387905\pi$$
0.344922 + 0.938631i $$0.387905\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0.803848 0.125540 0.0627700 0.998028i $$-0.480007\pi$$
0.0627700 + 0.998028i $$0.480007\pi$$
$$42$$ 0 0
$$43$$ 6.73205 1.02663 0.513314 0.858201i $$-0.328418\pi$$
0.513314 + 0.858201i $$0.328418\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.73205 0.690241 0.345120 0.938558i $$-0.387838\pi$$
0.345120 + 0.938558i $$0.387838\pi$$
$$48$$ 0 0
$$49$$ −6.46410 −0.923443
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 10.7321 1.47416 0.737080 0.675805i $$-0.236204\pi$$
0.737080 + 0.675805i $$0.236204\pi$$
$$54$$ 0 0
$$55$$ 1.73205 0.233550
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 4.26795 0.555640 0.277820 0.960633i $$-0.410388\pi$$
0.277820 + 0.960633i $$0.410388\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −1.46410 −0.181599
$$66$$ 0 0
$$67$$ −14.3923 −1.75830 −0.879150 0.476545i $$-0.841889\pi$$
−0.879150 + 0.476545i $$0.841889\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0.803848 0.0953992 0.0476996 0.998862i $$-0.484811\pi$$
0.0476996 + 0.998862i $$0.484811\pi$$
$$72$$ 0 0
$$73$$ 10.1962 1.19337 0.596685 0.802476i $$-0.296484\pi$$
0.596685 + 0.802476i $$0.296484\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.26795 0.144496
$$78$$ 0 0
$$79$$ 6.39230 0.719190 0.359595 0.933108i $$-0.382915\pi$$
0.359595 + 0.933108i $$0.382915\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −9.12436 −1.00153 −0.500764 0.865584i $$-0.666948\pi$$
−0.500764 + 0.865584i $$0.666948\pi$$
$$84$$ 0 0
$$85$$ 1.26795 0.137528
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 5.19615 0.550791 0.275396 0.961331i $$-0.411191\pi$$
0.275396 + 0.961331i $$0.411191\pi$$
$$90$$ 0 0
$$91$$ −1.07180 −0.112355
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 2.46410 0.252811
$$96$$ 0 0
$$97$$ −2.73205 −0.277398 −0.138699 0.990335i $$-0.544292\pi$$
−0.138699 + 0.990335i $$0.544292\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 18.1244 1.80344 0.901720 0.432320i $$-0.142305\pi$$
0.901720 + 0.432320i $$0.142305\pi$$
$$102$$ 0 0
$$103$$ −2.39230 −0.235721 −0.117860 0.993030i $$-0.537604\pi$$
−0.117860 + 0.993030i $$0.537604\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 3.46410 0.334887 0.167444 0.985882i $$-0.446449\pi$$
0.167444 + 0.985882i $$0.446449\pi$$
$$108$$ 0 0
$$109$$ 5.92820 0.567819 0.283909 0.958851i $$-0.408368\pi$$
0.283909 + 0.958851i $$0.408368\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 17.6603 1.66134 0.830668 0.556767i $$-0.187959\pi$$
0.830668 + 0.556767i $$0.187959\pi$$
$$114$$ 0 0
$$115$$ 3.46410 0.323029
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0.928203 0.0850883
$$120$$ 0 0
$$121$$ −8.00000 −0.727273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −6.19615 −0.549820 −0.274910 0.961470i $$-0.588648\pi$$
−0.274910 + 0.961470i $$0.588648\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −13.7321 −1.19977 −0.599887 0.800084i $$-0.704788\pi$$
−0.599887 + 0.800084i $$0.704788\pi$$
$$132$$ 0 0
$$133$$ 1.80385 0.156413
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 16.3923 1.40049 0.700245 0.713903i $$-0.253074\pi$$
0.700245 + 0.713903i $$0.253074\pi$$
$$138$$ 0 0
$$139$$ −5.39230 −0.457369 −0.228685 0.973501i $$-0.573442\pi$$
−0.228685 + 0.973501i $$0.573442\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −2.53590 −0.212062
$$144$$ 0 0
$$145$$ 4.26795 0.354434
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 3.39230 0.276062 0.138031 0.990428i $$-0.455923\pi$$
0.138031 + 0.990428i $$0.455923\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −7.92820 −0.636809
$$156$$ 0 0
$$157$$ 6.73205 0.537276 0.268638 0.963241i $$-0.413426\pi$$
0.268638 + 0.963241i $$0.413426\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 2.53590 0.199857
$$162$$ 0 0
$$163$$ 15.2679 1.19588 0.597939 0.801542i $$-0.295986\pi$$
0.597939 + 0.801542i $$0.295986\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −3.12436 −0.241770 −0.120885 0.992667i $$-0.538573\pi$$
−0.120885 + 0.992667i $$0.538573\pi$$
$$168$$ 0 0
$$169$$ −10.8564 −0.835108
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −24.2487 −1.84360 −0.921798 0.387671i $$-0.873280\pi$$
−0.921798 + 0.387671i $$0.873280\pi$$
$$174$$ 0 0
$$175$$ 0.732051 0.0553378
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −12.1244 −0.906217 −0.453108 0.891455i $$-0.649685\pi$$
−0.453108 + 0.891455i $$0.649685\pi$$
$$180$$ 0 0
$$181$$ −9.53590 −0.708798 −0.354399 0.935094i $$-0.615314\pi$$
−0.354399 + 0.935094i $$0.615314\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 4.19615 0.308507
$$186$$ 0 0
$$187$$ 2.19615 0.160599
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 19.0526 1.37859 0.689297 0.724479i $$-0.257919\pi$$
0.689297 + 0.724479i $$0.257919\pi$$
$$192$$ 0 0
$$193$$ 20.5885 1.48199 0.740995 0.671511i $$-0.234354\pi$$
0.740995 + 0.671511i $$0.234354\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −13.8564 −0.987228 −0.493614 0.869681i $$-0.664324\pi$$
−0.493614 + 0.869681i $$0.664324\pi$$
$$198$$ 0 0
$$199$$ −11.8564 −0.840478 −0.420239 0.907413i $$-0.638054\pi$$
−0.420239 + 0.907413i $$0.638054\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 3.12436 0.219287
$$204$$ 0 0
$$205$$ 0.803848 0.0561432
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 4.26795 0.295220
$$210$$ 0 0
$$211$$ −6.07180 −0.418000 −0.209000 0.977916i $$-0.567021\pi$$
−0.209000 + 0.977916i $$0.567021\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 6.73205 0.459122
$$216$$ 0 0
$$217$$ −5.80385 −0.393991
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1.85641 −0.124875
$$222$$ 0 0
$$223$$ 21.8564 1.46361 0.731807 0.681512i $$-0.238677\pi$$
0.731807 + 0.681512i $$0.238677\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −15.8038 −1.04894 −0.524469 0.851429i $$-0.675736\pi$$
−0.524469 + 0.851429i $$0.675736\pi$$
$$228$$ 0 0
$$229$$ 9.85641 0.651330 0.325665 0.945485i $$-0.394412\pi$$
0.325665 + 0.945485i $$0.394412\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.58846 0.431624 0.215812 0.976435i $$-0.430760\pi$$
0.215812 + 0.976435i $$0.430760\pi$$
$$234$$ 0 0
$$235$$ 4.73205 0.308685
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −15.4641 −1.00029 −0.500145 0.865942i $$-0.666720\pi$$
−0.500145 + 0.865942i $$0.666720\pi$$
$$240$$ 0 0
$$241$$ −24.3205 −1.56662 −0.783311 0.621630i $$-0.786471\pi$$
−0.783311 + 0.621630i $$0.786471\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −6.46410 −0.412976
$$246$$ 0 0
$$247$$ −3.60770 −0.229552
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −27.4641 −1.73352 −0.866759 0.498727i $$-0.833801\pi$$
−0.866759 + 0.498727i $$0.833801\pi$$
$$252$$ 0 0
$$253$$ 6.00000 0.377217
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −8.53590 −0.532455 −0.266227 0.963910i $$-0.585777\pi$$
−0.266227 + 0.963910i $$0.585777\pi$$
$$258$$ 0 0
$$259$$ 3.07180 0.190872
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −24.2487 −1.49524 −0.747620 0.664127i $$-0.768803\pi$$
−0.747620 + 0.664127i $$0.768803\pi$$
$$264$$ 0 0
$$265$$ 10.7321 0.659265
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 19.7321 1.20308 0.601542 0.798841i $$-0.294553\pi$$
0.601542 + 0.798841i $$0.294553\pi$$
$$270$$ 0 0
$$271$$ −24.7846 −1.50556 −0.752779 0.658273i $$-0.771287\pi$$
−0.752779 + 0.658273i $$0.771287\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.73205 0.104447
$$276$$ 0 0
$$277$$ −19.1244 −1.14907 −0.574536 0.818480i $$-0.694817\pi$$
−0.574536 + 0.818480i $$0.694817\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 5.32051 0.317395 0.158697 0.987327i $$-0.449271\pi$$
0.158697 + 0.987327i $$0.449271\pi$$
$$282$$ 0 0
$$283$$ 23.4641 1.39480 0.697398 0.716684i $$-0.254341\pi$$
0.697398 + 0.716684i $$0.254341\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0.588457 0.0347355
$$288$$ 0 0
$$289$$ −15.3923 −0.905430
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 6.58846 0.384902 0.192451 0.981307i $$-0.438356\pi$$
0.192451 + 0.981307i $$0.438356\pi$$
$$294$$ 0 0
$$295$$ 4.26795 0.248490
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −5.07180 −0.293310
$$300$$ 0 0
$$301$$ 4.92820 0.284057
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −4.00000 −0.229039
$$306$$ 0 0
$$307$$ −12.1962 −0.696071 −0.348036 0.937481i $$-0.613151\pi$$
−0.348036 + 0.937481i $$0.613151\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −28.5167 −1.61703 −0.808516 0.588475i $$-0.799729\pi$$
−0.808516 + 0.588475i $$0.799729\pi$$
$$312$$ 0 0
$$313$$ 1.07180 0.0605815 0.0302908 0.999541i $$-0.490357\pi$$
0.0302908 + 0.999541i $$0.490357\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 21.1244 1.18646 0.593231 0.805032i $$-0.297852\pi$$
0.593231 + 0.805032i $$0.297852\pi$$
$$318$$ 0 0
$$319$$ 7.39230 0.413890
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 3.12436 0.173844
$$324$$ 0 0
$$325$$ −1.46410 −0.0812137
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 3.46410 0.190982
$$330$$ 0 0
$$331$$ −8.60770 −0.473122 −0.236561 0.971617i $$-0.576020\pi$$
−0.236561 + 0.971617i $$0.576020\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −14.3923 −0.786336
$$336$$ 0 0
$$337$$ 14.2487 0.776177 0.388088 0.921622i $$-0.373136\pi$$
0.388088 + 0.921622i $$0.373136\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −13.7321 −0.743632
$$342$$ 0 0
$$343$$ −9.85641 −0.532196
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −32.1962 −1.72838 −0.864190 0.503166i $$-0.832169\pi$$
−0.864190 + 0.503166i $$0.832169\pi$$
$$348$$ 0 0
$$349$$ −25.0000 −1.33822 −0.669110 0.743164i $$-0.733324\pi$$
−0.669110 + 0.743164i $$0.733324\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −22.9808 −1.22314 −0.611571 0.791189i $$-0.709462\pi$$
−0.611571 + 0.791189i $$0.709462\pi$$
$$354$$ 0 0
$$355$$ 0.803848 0.0426638
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −26.6603 −1.40707 −0.703537 0.710658i $$-0.748397\pi$$
−0.703537 + 0.710658i $$0.748397\pi$$
$$360$$ 0 0
$$361$$ −12.9282 −0.680432
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 10.1962 0.533691
$$366$$ 0 0
$$367$$ −5.60770 −0.292719 −0.146360 0.989231i $$-0.546756\pi$$
−0.146360 + 0.989231i $$0.546756\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 7.85641 0.407884
$$372$$ 0 0
$$373$$ 24.0526 1.24539 0.622697 0.782463i $$-0.286037\pi$$
0.622697 + 0.782463i $$0.286037\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6.24871 −0.321825
$$378$$ 0 0
$$379$$ 11.4641 0.588871 0.294436 0.955671i $$-0.404868\pi$$
0.294436 + 0.955671i $$0.404868\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 30.2487 1.54564 0.772818 0.634627i $$-0.218846\pi$$
0.772818 + 0.634627i $$0.218846\pi$$
$$384$$ 0 0
$$385$$ 1.26795 0.0646207
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −20.5359 −1.04121 −0.520606 0.853797i $$-0.674294\pi$$
−0.520606 + 0.853797i $$0.674294\pi$$
$$390$$ 0 0
$$391$$ 4.39230 0.222128
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 6.39230 0.321632
$$396$$ 0 0
$$397$$ 2.67949 0.134480 0.0672399 0.997737i $$-0.478581\pi$$
0.0672399 + 0.997737i $$0.478581\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −26.7846 −1.33756 −0.668780 0.743461i $$-0.733183\pi$$
−0.668780 + 0.743461i $$0.733183\pi$$
$$402$$ 0 0
$$403$$ 11.6077 0.578220
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 7.26795 0.360259
$$408$$ 0 0
$$409$$ 9.85641 0.487368 0.243684 0.969855i $$-0.421644\pi$$
0.243684 + 0.969855i $$0.421644\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 3.12436 0.153739
$$414$$ 0 0
$$415$$ −9.12436 −0.447897
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −14.5359 −0.710125 −0.355063 0.934843i $$-0.615540\pi$$
−0.355063 + 0.934843i $$0.615540\pi$$
$$420$$ 0 0
$$421$$ −27.7846 −1.35414 −0.677070 0.735919i $$-0.736750\pi$$
−0.677070 + 0.735919i $$0.736750\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 1.26795 0.0615046
$$426$$ 0 0
$$427$$ −2.92820 −0.141706
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 2.41154 0.116160 0.0580800 0.998312i $$-0.481502\pi$$
0.0580800 + 0.998312i $$0.481502\pi$$
$$432$$ 0 0
$$433$$ 4.53590 0.217981 0.108991 0.994043i $$-0.465238\pi$$
0.108991 + 0.994043i $$0.465238\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 8.53590 0.408327
$$438$$ 0 0
$$439$$ 27.3923 1.30736 0.653682 0.756770i $$-0.273224\pi$$
0.653682 + 0.756770i $$0.273224\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −40.7321 −1.93524 −0.967619 0.252415i $$-0.918775\pi$$
−0.967619 + 0.252415i $$0.918775\pi$$
$$444$$ 0 0
$$445$$ 5.19615 0.246321
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0.124356 0.00586871 0.00293435 0.999996i $$-0.499066\pi$$
0.00293435 + 0.999996i $$0.499066\pi$$
$$450$$ 0 0
$$451$$ 1.39230 0.0655611
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −1.07180 −0.0502466
$$456$$ 0 0
$$457$$ 11.8038 0.552161 0.276080 0.961135i $$-0.410964\pi$$
0.276080 + 0.961135i $$0.410964\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 21.5885 1.00547 0.502737 0.864439i $$-0.332326\pi$$
0.502737 + 0.864439i $$0.332326\pi$$
$$462$$ 0 0
$$463$$ 18.3923 0.854763 0.427381 0.904071i $$-0.359436\pi$$
0.427381 + 0.904071i $$0.359436\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −15.8038 −0.731315 −0.365657 0.930750i $$-0.619156\pi$$
−0.365657 + 0.930750i $$0.619156\pi$$
$$468$$ 0 0
$$469$$ −10.5359 −0.486503
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 11.6603 0.536139
$$474$$ 0 0
$$475$$ 2.46410 0.113061
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 20.6603 0.943991 0.471996 0.881601i $$-0.343534\pi$$
0.471996 + 0.881601i $$0.343534\pi$$
$$480$$ 0 0
$$481$$ −6.14359 −0.280124
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −2.73205 −0.124056
$$486$$ 0 0
$$487$$ −37.4641 −1.69766 −0.848830 0.528666i $$-0.822693\pi$$
−0.848830 + 0.528666i $$0.822693\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 7.73205 0.348943 0.174471 0.984662i $$-0.444178\pi$$
0.174471 + 0.984662i $$0.444178\pi$$
$$492$$ 0 0
$$493$$ 5.41154 0.243724
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0.588457 0.0263959
$$498$$ 0 0
$$499$$ 6.60770 0.295801 0.147901 0.989002i $$-0.452748\pi$$
0.147901 + 0.989002i $$0.452748\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0.679492 0.0302970 0.0151485 0.999885i $$-0.495178\pi$$
0.0151485 + 0.999885i $$0.495178\pi$$
$$504$$ 0 0
$$505$$ 18.1244 0.806523
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −14.7846 −0.655316 −0.327658 0.944796i $$-0.606259\pi$$
−0.327658 + 0.944796i $$0.606259\pi$$
$$510$$ 0 0
$$511$$ 7.46410 0.330192
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −2.39230 −0.105418
$$516$$ 0 0
$$517$$ 8.19615 0.360466
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −10.3923 −0.455295 −0.227648 0.973744i $$-0.573103\pi$$
−0.227648 + 0.973744i $$0.573103\pi$$
$$522$$ 0 0
$$523$$ 24.3923 1.06660 0.533301 0.845926i $$-0.320952\pi$$
0.533301 + 0.845926i $$0.320952\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −10.0526 −0.437896
$$528$$ 0 0
$$529$$ −11.0000 −0.478261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −1.17691 −0.0509778
$$534$$ 0 0
$$535$$ 3.46410 0.149766
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −11.1962 −0.482252
$$540$$ 0 0
$$541$$ −4.46410 −0.191927 −0.0959634 0.995385i $$-0.530593\pi$$
−0.0959634 + 0.995385i $$0.530593\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 5.92820 0.253936
$$546$$ 0 0
$$547$$ −36.7846 −1.57280 −0.786398 0.617720i $$-0.788057\pi$$
−0.786398 + 0.617720i $$0.788057\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 10.5167 0.448025
$$552$$ 0 0
$$553$$ 4.67949 0.198992
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −16.3923 −0.694564 −0.347282 0.937761i $$-0.612895\pi$$
−0.347282 + 0.937761i $$0.612895\pi$$
$$558$$ 0 0
$$559$$ −9.85641 −0.416882
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −19.2679 −0.812047 −0.406024 0.913863i $$-0.633085\pi$$
−0.406024 + 0.913863i $$0.633085\pi$$
$$564$$ 0 0
$$565$$ 17.6603 0.742972
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 4.94744 0.207408 0.103704 0.994608i $$-0.466931\pi$$
0.103704 + 0.994608i $$0.466931\pi$$
$$570$$ 0 0
$$571$$ −26.8564 −1.12391 −0.561953 0.827169i $$-0.689950\pi$$
−0.561953 + 0.827169i $$0.689950\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 3.46410 0.144463
$$576$$ 0 0
$$577$$ 22.1962 0.924038 0.462019 0.886870i $$-0.347125\pi$$
0.462019 + 0.886870i $$0.347125\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −6.67949 −0.277112
$$582$$ 0 0
$$583$$ 18.5885 0.769855
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −14.1962 −0.585938 −0.292969 0.956122i $$-0.594643\pi$$
−0.292969 + 0.956122i $$0.594643\pi$$
$$588$$ 0 0
$$589$$ −19.5359 −0.804963
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 36.9282 1.51646 0.758230 0.651987i $$-0.226065\pi$$
0.758230 + 0.651987i $$0.226065\pi$$
$$594$$ 0 0
$$595$$ 0.928203 0.0380526
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 45.5885 1.86269 0.931347 0.364133i $$-0.118635\pi$$
0.931347 + 0.364133i $$0.118635\pi$$
$$600$$ 0 0
$$601$$ 10.3205 0.420982 0.210491 0.977596i $$-0.432494\pi$$
0.210491 + 0.977596i $$0.432494\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −8.00000 −0.325246
$$606$$ 0 0
$$607$$ −28.5885 −1.16037 −0.580185 0.814485i $$-0.697020\pi$$
−0.580185 + 0.814485i $$0.697020\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6.92820 −0.280285
$$612$$ 0 0
$$613$$ 3.60770 0.145713 0.0728567 0.997342i $$-0.476788\pi$$
0.0728567 + 0.997342i $$0.476788\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.0000 0.483102 0.241551 0.970388i $$-0.422344\pi$$
0.241551 + 0.970388i $$0.422344\pi$$
$$618$$ 0 0
$$619$$ −10.0000 −0.401934 −0.200967 0.979598i $$-0.564408\pi$$
−0.200967 + 0.979598i $$0.564408\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 3.80385 0.152398
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 5.32051 0.212143
$$630$$ 0 0
$$631$$ −13.9282 −0.554473 −0.277237 0.960802i $$-0.589419\pi$$
−0.277237 + 0.960802i $$0.589419\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −6.19615 −0.245887
$$636$$ 0 0
$$637$$ 9.46410 0.374981
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 23.1962 0.916193 0.458096 0.888902i $$-0.348531\pi$$
0.458096 + 0.888902i $$0.348531\pi$$
$$642$$ 0 0
$$643$$ −16.5885 −0.654185 −0.327092 0.944992i $$-0.606069\pi$$
−0.327092 + 0.944992i $$0.606069\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 48.2487 1.89685 0.948426 0.316998i $$-0.102675\pi$$
0.948426 + 0.316998i $$0.102675\pi$$
$$648$$ 0 0
$$649$$ 7.39230 0.290173
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 9.46410 0.370359 0.185179 0.982705i $$-0.440713\pi$$
0.185179 + 0.982705i $$0.440713\pi$$
$$654$$ 0 0
$$655$$ −13.7321 −0.536556
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 9.46410 0.368669 0.184335 0.982864i $$-0.440987\pi$$
0.184335 + 0.982864i $$0.440987\pi$$
$$660$$ 0 0
$$661$$ −5.39230 −0.209736 −0.104868 0.994486i $$-0.533442\pi$$
−0.104868 + 0.994486i $$0.533442\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 1.80385 0.0699502
$$666$$ 0 0
$$667$$ 14.7846 0.572462
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −6.92820 −0.267460
$$672$$ 0 0
$$673$$ −17.6077 −0.678727 −0.339363 0.940655i $$-0.610212\pi$$
−0.339363 + 0.940655i $$0.610212\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 28.6410 1.10076 0.550382 0.834913i $$-0.314482\pi$$
0.550382 + 0.834913i $$0.314482\pi$$
$$678$$ 0 0
$$679$$ −2.00000 −0.0767530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −14.5359 −0.556201 −0.278100 0.960552i $$-0.589705\pi$$
−0.278100 + 0.960552i $$0.589705\pi$$
$$684$$ 0 0
$$685$$ 16.3923 0.626318
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −15.7128 −0.598610
$$690$$ 0 0
$$691$$ −10.0000 −0.380418 −0.190209 0.981744i $$-0.560917\pi$$
−0.190209 + 0.981744i $$0.560917\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −5.39230 −0.204542
$$696$$ 0 0
$$697$$ 1.01924 0.0386064
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 42.1244 1.59101 0.795507 0.605944i $$-0.207204\pi$$
0.795507 + 0.605944i $$0.207204\pi$$
$$702$$ 0 0
$$703$$ 10.3397 0.389971
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 13.2679 0.498993
$$708$$ 0 0
$$709$$ 17.4641 0.655878 0.327939 0.944699i $$-0.393646\pi$$
0.327939 + 0.944699i $$0.393646\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −27.4641 −1.02854
$$714$$ 0 0
$$715$$ −2.53590 −0.0948372
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 6.80385 0.253741 0.126870 0.991919i $$-0.459507\pi$$
0.126870 + 0.991919i $$0.459507\pi$$
$$720$$ 0 0
$$721$$ −1.75129 −0.0652214
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 4.26795 0.158508
$$726$$ 0 0
$$727$$ 33.1769 1.23046 0.615232 0.788346i $$-0.289062\pi$$
0.615232 + 0.788346i $$0.289062\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 8.53590 0.315712
$$732$$ 0 0
$$733$$ −39.5692 −1.46152 −0.730761 0.682633i $$-0.760835\pi$$
−0.730761 + 0.682633i $$0.760835\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −24.9282 −0.918242
$$738$$ 0 0
$$739$$ 36.1769 1.33079 0.665395 0.746492i $$-0.268263\pi$$
0.665395 + 0.746492i $$0.268263\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 36.5885 1.34230 0.671150 0.741321i $$-0.265801\pi$$
0.671150 + 0.741321i $$0.265801\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 2.53590 0.0926597
$$750$$ 0 0
$$751$$ −0.784610 −0.0286308 −0.0143154 0.999898i $$-0.504557\pi$$
−0.0143154 + 0.999898i $$0.504557\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 3.39230 0.123459
$$756$$ 0 0
$$757$$ 0.392305 0.0142586 0.00712928 0.999975i $$-0.497731\pi$$
0.00712928 + 0.999975i $$0.497731\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 5.87564 0.212992 0.106496 0.994313i $$-0.466037\pi$$
0.106496 + 0.994313i $$0.466037\pi$$
$$762$$ 0 0
$$763$$ 4.33975 0.157109
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −6.24871 −0.225628
$$768$$ 0 0
$$769$$ −13.2487 −0.477761 −0.238880 0.971049i $$-0.576780\pi$$
−0.238880 + 0.971049i $$0.576780\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −0.339746 −0.0122198 −0.00610991 0.999981i $$-0.501945\pi$$
−0.00610991 + 0.999981i $$0.501945\pi$$
$$774$$ 0 0
$$775$$ −7.92820 −0.284789
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 1.98076 0.0709682
$$780$$ 0 0
$$781$$ 1.39230 0.0498206
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 6.73205 0.240277
$$786$$ 0 0
$$787$$ −25.8038 −0.919808 −0.459904 0.887969i $$-0.652116\pi$$
−0.459904 + 0.887969i $$0.652116\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 12.9282 0.459674
$$792$$ 0 0
$$793$$ 5.85641 0.207967
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 11.3205 0.400993 0.200496 0.979694i $$-0.435744\pi$$
0.200496 + 0.979694i $$0.435744\pi$$
$$798$$ 0 0
$$799$$ 6.00000 0.212265
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 17.6603 0.623217
$$804$$ 0 0
$$805$$ 2.53590 0.0893787
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 39.5885 1.39186 0.695928 0.718112i $$-0.254993\pi$$
0.695928 + 0.718112i $$0.254993\pi$$
$$810$$ 0 0
$$811$$ 23.2487 0.816373 0.408186 0.912899i $$-0.366161\pi$$
0.408186 + 0.912899i $$0.366161\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 15.2679 0.534813
$$816$$ 0 0
$$817$$ 16.5885 0.580357
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −20.6603 −0.721048 −0.360524 0.932750i $$-0.617402\pi$$
−0.360524 + 0.932750i $$0.617402\pi$$
$$822$$ 0 0
$$823$$ −3.07180 −0.107076 −0.0535381 0.998566i $$-0.517050\pi$$
−0.0535381 + 0.998566i $$0.517050\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 38.5359 1.34002 0.670012 0.742350i $$-0.266289\pi$$
0.670012 + 0.742350i $$0.266289\pi$$
$$828$$ 0 0
$$829$$ −10.2154 −0.354795 −0.177398 0.984139i $$-0.556768\pi$$
−0.177398 + 0.984139i $$0.556768\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −8.19615 −0.283980
$$834$$ 0 0
$$835$$ −3.12436 −0.108123
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 39.5885 1.36675 0.683373 0.730070i $$-0.260512\pi$$
0.683373 + 0.730070i $$0.260512\pi$$
$$840$$ 0 0
$$841$$ −10.7846 −0.371883
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −10.8564 −0.373472
$$846$$ 0 0
$$847$$ −5.85641 −0.201229
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 14.5359 0.498284
$$852$$ 0 0
$$853$$ 10.1962 0.349110 0.174555 0.984647i $$-0.444151\pi$$
0.174555 + 0.984647i $$0.444151\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 42.9282 1.46640 0.733200 0.680013i $$-0.238026\pi$$
0.733200 + 0.680013i $$0.238026\pi$$
$$858$$ 0 0
$$859$$ −33.7846 −1.15272 −0.576358 0.817197i $$-0.695527\pi$$
−0.576358 + 0.817197i $$0.695527\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 4.48334 0.152615 0.0763073 0.997084i $$-0.475687\pi$$
0.0763073 + 0.997084i $$0.475687\pi$$
$$864$$ 0 0
$$865$$ −24.2487 −0.824481
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 11.0718 0.375585
$$870$$ 0 0
$$871$$ 21.0718 0.713991
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0.732051 0.0247478
$$876$$ 0 0
$$877$$ −34.2487 −1.15650 −0.578248 0.815861i $$-0.696264\pi$$
−0.578248 + 0.815861i $$0.696264\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −17.1962 −0.579353 −0.289677 0.957125i $$-0.593548\pi$$
−0.289677 + 0.957125i $$0.593548\pi$$
$$882$$ 0 0
$$883$$ 32.8372 1.10506 0.552529 0.833493i $$-0.313663\pi$$
0.552529 + 0.833493i $$0.313663\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −16.7321 −0.561807 −0.280904 0.959736i $$-0.590634\pi$$
−0.280904 + 0.959736i $$0.590634\pi$$
$$888$$ 0 0
$$889$$ −4.53590 −0.152129
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 11.6603 0.390196
$$894$$ 0 0
$$895$$ −12.1244 −0.405273
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −33.8372 −1.12853
$$900$$ 0 0
$$901$$ 13.6077 0.453338
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −9.53590 −0.316984
$$906$$ 0 0
$$907$$ −42.7846 −1.42064 −0.710320 0.703879i $$-0.751450\pi$$
−0.710320 + 0.703879i $$0.751450\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 17.1962 0.569734 0.284867 0.958567i $$-0.408051\pi$$
0.284867 + 0.958567i $$0.408051\pi$$
$$912$$ 0 0
$$913$$ −15.8038 −0.523031
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −10.0526 −0.331965
$$918$$ 0 0
$$919$$ 30.6077 1.00965 0.504827 0.863220i $$-0.331556\pi$$
0.504827 + 0.863220i $$0.331556\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −1.17691 −0.0387386
$$924$$ 0 0
$$925$$ 4.19615 0.137969
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −46.5167 −1.52616 −0.763081 0.646303i $$-0.776314\pi$$
−0.763081 + 0.646303i $$0.776314\pi$$
$$930$$ 0 0
$$931$$ −15.9282 −0.522026
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 2.19615 0.0718219
$$936$$ 0 0
$$937$$ 32.9282 1.07572 0.537859 0.843035i $$-0.319233\pi$$
0.537859 + 0.843035i $$0.319233\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 15.4641 0.504115 0.252058 0.967712i $$-0.418893\pi$$
0.252058 + 0.967712i $$0.418893\pi$$
$$942$$ 0 0
$$943$$ 2.78461 0.0906794
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −16.6410 −0.540760 −0.270380 0.962754i $$-0.587149\pi$$
−0.270380 + 0.962754i $$0.587149\pi$$
$$948$$ 0 0
$$949$$ −14.9282 −0.484590
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 9.46410 0.306572 0.153286 0.988182i $$-0.451014\pi$$
0.153286 + 0.988182i $$0.451014\pi$$
$$954$$ 0 0
$$955$$ 19.0526 0.616526
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ 31.8564 1.02763
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 20.5885 0.662766
$$966$$ 0 0
$$967$$ 38.5885 1.24092 0.620461 0.784238i $$-0.286946\pi$$
0.620461 + 0.784238i $$0.286946\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −25.7321 −0.825781 −0.412890 0.910781i $$-0.635481\pi$$
−0.412890 + 0.910781i $$0.635481\pi$$
$$972$$ 0 0
$$973$$ −3.94744 −0.126549
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 16.3923 0.524436 0.262218 0.965009i $$-0.415546\pi$$
0.262218 + 0.965009i $$0.415546\pi$$
$$978$$ 0 0
$$979$$ 9.00000 0.287641
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 49.2679 1.57140 0.785702 0.618605i $$-0.212302\pi$$
0.785702 + 0.618605i $$0.212302\pi$$
$$984$$ 0 0
$$985$$ −13.8564 −0.441502
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 23.3205 0.741549
$$990$$ 0 0
$$991$$ 14.2154 0.451567 0.225783 0.974178i $$-0.427506\pi$$
0.225783 + 0.974178i $$0.427506\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −11.8564 −0.375873
$$996$$ 0 0
$$997$$ 41.8038 1.32394 0.661971 0.749530i $$-0.269720\pi$$
0.661971 + 0.749530i $$0.269720\pi$$
$$998$$ 0 0
$$999$$ 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 1620.2.a.h.1.2 yes 2
3.2 odd 2 1620.2.a.g.1.2 2
4.3 odd 2 6480.2.a.bp.1.1 2
5.2 odd 4 8100.2.d.l.649.3 4
5.3 odd 4 8100.2.d.l.649.2 4
5.4 even 2 8100.2.a.s.1.1 2
9.2 odd 6 1620.2.i.n.1081.1 4
9.4 even 3 1620.2.i.m.541.1 4
9.5 odd 6 1620.2.i.n.541.1 4
9.7 even 3 1620.2.i.m.1081.1 4
12.11 even 2 6480.2.a.bh.1.1 2
15.2 even 4 8100.2.d.m.649.3 4
15.8 even 4 8100.2.d.m.649.2 4
15.14 odd 2 8100.2.a.t.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1620.2.a.g.1.2 2 3.2 odd 2
1620.2.a.h.1.2 yes 2 1.1 even 1 trivial
1620.2.i.m.541.1 4 9.4 even 3
1620.2.i.m.1081.1 4 9.7 even 3
1620.2.i.n.541.1 4 9.5 odd 6
1620.2.i.n.1081.1 4 9.2 odd 6
6480.2.a.bh.1.1 2 12.11 even 2
6480.2.a.bp.1.1 2 4.3 odd 2
8100.2.a.s.1.1 2 5.4 even 2
8100.2.a.t.1.1 2 15.14 odd 2
8100.2.d.l.649.2 4 5.3 odd 4
8100.2.d.l.649.3 4 5.2 odd 4
8100.2.d.m.649.2 4 15.8 even 4
8100.2.d.m.649.3 4 15.2 even 4