# Properties

 Label 3192.2.a.q.1.1 Level $3192$ Weight $2$ Character 3192.1 Self dual yes Analytic conductor $25.488$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3192,2,Mod(1,3192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3192.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3192.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: $$25.4882483252$$ Analytic rank: $$1$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 3192.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.46410 q^{13} +0.732051 q^{15} -4.73205 q^{17} -1.00000 q^{19} -1.00000 q^{21} -4.46410 q^{25} -1.00000 q^{27} +7.66025 q^{29} +2.00000 q^{31} -0.732051 q^{35} -10.0000 q^{37} -1.46410 q^{39} +4.92820 q^{41} -8.92820 q^{43} -0.732051 q^{45} +5.66025 q^{47} +1.00000 q^{49} +4.73205 q^{51} -3.26795 q^{53} +1.00000 q^{57} +8.00000 q^{59} -2.00000 q^{61} +1.00000 q^{63} -1.07180 q^{65} -6.53590 q^{67} +1.26795 q^{71} -10.3923 q^{73} +4.46410 q^{75} +4.00000 q^{79} +1.00000 q^{81} -0.196152 q^{83} +3.46410 q^{85} -7.66025 q^{87} -8.92820 q^{89} +1.46410 q^{91} -2.00000 q^{93} +0.732051 q^{95} +2.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} + 2 q^{35} - 20 q^{37} + 4 q^{39} - 4 q^{41} - 4 q^{43} + 2 q^{45} - 6 q^{47} + 2 q^{49} + 6 q^{51} - 10 q^{53} + 2 q^{57} + 16 q^{59} - 4 q^{61} + 2 q^{63} - 16 q^{65} - 20 q^{67} + 6 q^{71} + 2 q^{75} + 8 q^{79} + 2 q^{81} + 10 q^{83} + 2 q^{87} - 4 q^{89} - 4 q^{91} - 4 q^{93} - 2 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^13 - 2 * q^15 - 6 * q^17 - 2 * q^19 - 2 * q^21 - 2 * q^25 - 2 * q^27 - 2 * q^29 + 4 * q^31 + 2 * q^35 - 20 * q^37 + 4 * q^39 - 4 * q^41 - 4 * q^43 + 2 * q^45 - 6 * q^47 + 2 * q^49 + 6 * q^51 - 10 * q^53 + 2 * q^57 + 16 * q^59 - 4 * q^61 + 2 * q^63 - 16 * q^65 - 20 * q^67 + 6 * q^71 + 2 * q^75 + 8 * q^79 + 2 * q^81 + 10 * q^83 + 2 * q^87 - 4 * q^89 - 4 * q^91 - 4 * q^93 - 2 * q^95 + 4 * 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −0.732051 −0.327383 −0.163692 0.986512i $$-0.552340\pi$$
−0.163692 + 0.986512i $$0.552340\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 1.46410 0.406069 0.203034 0.979172i $$-0.434920\pi$$
0.203034 + 0.979172i $$0.434920\pi$$
$$14$$ 0 0
$$15$$ 0.732051 0.189015
$$16$$ 0 0
$$17$$ −4.73205 −1.14769 −0.573845 0.818964i $$-0.694549\pi$$
−0.573845 + 0.818964i $$0.694549\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −4.46410 −0.892820
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 7.66025 1.42247 0.711237 0.702953i $$-0.248135\pi$$
0.711237 + 0.702953i $$0.248135\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −0.732051 −0.123739
$$36$$ 0 0
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ 0 0
$$39$$ −1.46410 −0.234444
$$40$$ 0 0
$$41$$ 4.92820 0.769656 0.384828 0.922988i $$-0.374261\pi$$
0.384828 + 0.922988i $$0.374261\pi$$
$$42$$ 0 0
$$43$$ −8.92820 −1.36154 −0.680769 0.732498i $$-0.738354\pi$$
−0.680769 + 0.732498i $$0.738354\pi$$
$$44$$ 0 0
$$45$$ −0.732051 −0.109128
$$46$$ 0 0
$$47$$ 5.66025 0.825633 0.412816 0.910814i $$-0.364545\pi$$
0.412816 + 0.910814i $$0.364545\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 4.73205 0.662620
$$52$$ 0 0
$$53$$ −3.26795 −0.448887 −0.224444 0.974487i $$-0.572056\pi$$
−0.224444 + 0.974487i $$0.572056\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ −1.07180 −0.132940
$$66$$ 0 0
$$67$$ −6.53590 −0.798487 −0.399244 0.916845i $$-0.630727\pi$$
−0.399244 + 0.916845i $$0.630727\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 1.26795 0.150478 0.0752389 0.997166i $$-0.476028\pi$$
0.0752389 + 0.997166i $$0.476028\pi$$
$$72$$ 0 0
$$73$$ −10.3923 −1.21633 −0.608164 0.793812i $$-0.708094\pi$$
−0.608164 + 0.793812i $$0.708094\pi$$
$$74$$ 0 0
$$75$$ 4.46410 0.515470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −0.196152 −0.0215305 −0.0107653 0.999942i $$-0.503427\pi$$
−0.0107653 + 0.999942i $$0.503427\pi$$
$$84$$ 0 0
$$85$$ 3.46410 0.375735
$$86$$ 0 0
$$87$$ −7.66025 −0.821265
$$88$$ 0 0
$$89$$ −8.92820 −0.946388 −0.473194 0.880958i $$-0.656899\pi$$
−0.473194 + 0.880958i $$0.656899\pi$$
$$90$$ 0 0
$$91$$ 1.46410 0.153480
$$92$$ 0 0
$$93$$ −2.00000 −0.207390
$$94$$ 0 0
$$95$$ 0.732051 0.0751068
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 4.73205 0.470857 0.235428 0.971892i $$-0.424351\pi$$
0.235428 + 0.971892i $$0.424351\pi$$
$$102$$ 0 0
$$103$$ 10.9282 1.07679 0.538394 0.842693i $$-0.319031\pi$$
0.538394 + 0.842693i $$0.319031\pi$$
$$104$$ 0 0
$$105$$ 0.732051 0.0714408
$$106$$ 0 0
$$107$$ −1.26795 −0.122577 −0.0612886 0.998120i $$-0.519521\pi$$
−0.0612886 + 0.998120i $$0.519521\pi$$
$$108$$ 0 0
$$109$$ −18.3923 −1.76166 −0.880832 0.473430i $$-0.843016\pi$$
−0.880832 + 0.473430i $$0.843016\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 0 0
$$113$$ −10.1962 −0.959173 −0.479587 0.877495i $$-0.659213\pi$$
−0.479587 + 0.877495i $$0.659213\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1.46410 0.135356
$$118$$ 0 0
$$119$$ −4.73205 −0.433786
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ −4.92820 −0.444361
$$124$$ 0 0
$$125$$ 6.92820 0.619677
$$126$$ 0 0
$$127$$ −0.392305 −0.0348114 −0.0174057 0.999849i $$-0.505541\pi$$
−0.0174057 + 0.999849i $$0.505541\pi$$
$$128$$ 0 0
$$129$$ 8.92820 0.786084
$$130$$ 0 0
$$131$$ −16.5885 −1.44934 −0.724670 0.689096i $$-0.758008\pi$$
−0.724670 + 0.689096i $$0.758008\pi$$
$$132$$ 0 0
$$133$$ −1.00000 −0.0867110
$$134$$ 0 0
$$135$$ 0.732051 0.0630049
$$136$$ 0 0
$$137$$ −0.928203 −0.0793018 −0.0396509 0.999214i $$-0.512625\pi$$
−0.0396509 + 0.999214i $$0.512625\pi$$
$$138$$ 0 0
$$139$$ 19.3205 1.63874 0.819372 0.573262i $$-0.194322\pi$$
0.819372 + 0.573262i $$0.194322\pi$$
$$140$$ 0 0
$$141$$ −5.66025 −0.476679
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −5.60770 −0.465694
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ −6.39230 −0.523678 −0.261839 0.965112i $$-0.584329\pi$$
−0.261839 + 0.965112i $$0.584329\pi$$
$$150$$ 0 0
$$151$$ −9.46410 −0.770178 −0.385089 0.922880i $$-0.625829\pi$$
−0.385089 + 0.922880i $$0.625829\pi$$
$$152$$ 0 0
$$153$$ −4.73205 −0.382564
$$154$$ 0 0
$$155$$ −1.46410 −0.117599
$$156$$ 0 0
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ 0 0
$$159$$ 3.26795 0.259165
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −20.9282 −1.63922 −0.819612 0.572919i $$-0.805811\pi$$
−0.819612 + 0.572919i $$0.805811\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −23.3205 −1.80460 −0.902298 0.431114i $$-0.858121\pi$$
−0.902298 + 0.431114i $$0.858121\pi$$
$$168$$ 0 0
$$169$$ −10.8564 −0.835108
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ −3.07180 −0.233544 −0.116772 0.993159i $$-0.537255\pi$$
−0.116772 + 0.993159i $$0.537255\pi$$
$$174$$ 0 0
$$175$$ −4.46410 −0.337454
$$176$$ 0 0
$$177$$ −8.00000 −0.601317
$$178$$ 0 0
$$179$$ −3.80385 −0.284313 −0.142156 0.989844i $$-0.545404\pi$$
−0.142156 + 0.989844i $$0.545404\pi$$
$$180$$ 0 0
$$181$$ −8.92820 −0.663628 −0.331814 0.943345i $$-0.607661\pi$$
−0.331814 + 0.943345i $$0.607661\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ 7.32051 0.538214
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −1.07180 −0.0775525 −0.0387762 0.999248i $$-0.512346\pi$$
−0.0387762 + 0.999248i $$0.512346\pi$$
$$192$$ 0 0
$$193$$ 3.85641 0.277590 0.138795 0.990321i $$-0.455677\pi$$
0.138795 + 0.990321i $$0.455677\pi$$
$$194$$ 0 0
$$195$$ 1.07180 0.0767530
$$196$$ 0 0
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ 8.39230 0.594915 0.297457 0.954735i $$-0.403861\pi$$
0.297457 + 0.954735i $$0.403861\pi$$
$$200$$ 0 0
$$201$$ 6.53590 0.461007
$$202$$ 0 0
$$203$$ 7.66025 0.537644
$$204$$ 0 0
$$205$$ −3.60770 −0.251972
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ −1.26795 −0.0868784
$$214$$ 0 0
$$215$$ 6.53590 0.445745
$$216$$ 0 0
$$217$$ 2.00000 0.135769
$$218$$ 0 0
$$219$$ 10.3923 0.702247
$$220$$ 0 0
$$221$$ −6.92820 −0.466041
$$222$$ 0 0
$$223$$ −7.07180 −0.473563 −0.236781 0.971563i $$-0.576092\pi$$
−0.236781 + 0.971563i $$0.576092\pi$$
$$224$$ 0 0
$$225$$ −4.46410 −0.297607
$$226$$ 0 0
$$227$$ −4.39230 −0.291528 −0.145764 0.989319i $$-0.546564\pi$$
−0.145764 + 0.989319i $$0.546564\pi$$
$$228$$ 0 0
$$229$$ −20.2487 −1.33807 −0.669036 0.743230i $$-0.733293\pi$$
−0.669036 + 0.743230i $$0.733293\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 18.3923 1.20492 0.602460 0.798149i $$-0.294187\pi$$
0.602460 + 0.798149i $$0.294187\pi$$
$$234$$ 0 0
$$235$$ −4.14359 −0.270298
$$236$$ 0 0
$$237$$ −4.00000 −0.259828
$$238$$ 0 0
$$239$$ 16.3923 1.06033 0.530165 0.847894i $$-0.322130\pi$$
0.530165 + 0.847894i $$0.322130\pi$$
$$240$$ 0 0
$$241$$ 0.928203 0.0597908 0.0298954 0.999553i $$-0.490483\pi$$
0.0298954 + 0.999553i $$0.490483\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −0.732051 −0.0467690
$$246$$ 0 0
$$247$$ −1.46410 −0.0931586
$$248$$ 0 0
$$249$$ 0.196152 0.0124307
$$250$$ 0 0
$$251$$ 17.6603 1.11471 0.557353 0.830276i $$-0.311817\pi$$
0.557353 + 0.830276i $$0.311817\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −3.46410 −0.216930
$$256$$ 0 0
$$257$$ 4.53590 0.282942 0.141471 0.989942i $$-0.454817\pi$$
0.141471 + 0.989942i $$0.454817\pi$$
$$258$$ 0 0
$$259$$ −10.0000 −0.621370
$$260$$ 0 0
$$261$$ 7.66025 0.474158
$$262$$ 0 0
$$263$$ 15.3205 0.944703 0.472351 0.881410i $$-0.343405\pi$$
0.472351 + 0.881410i $$0.343405\pi$$
$$264$$ 0 0
$$265$$ 2.39230 0.146958
$$266$$ 0 0
$$267$$ 8.92820 0.546397
$$268$$ 0 0
$$269$$ −31.1769 −1.90089 −0.950445 0.310893i $$-0.899372\pi$$
−0.950445 + 0.310893i $$0.899372\pi$$
$$270$$ 0 0
$$271$$ 12.7846 0.776610 0.388305 0.921531i $$-0.373061\pi$$
0.388305 + 0.921531i $$0.373061\pi$$
$$272$$ 0 0
$$273$$ −1.46410 −0.0886115
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −8.39230 −0.504245 −0.252122 0.967695i $$-0.581129\pi$$
−0.252122 + 0.967695i $$0.581129\pi$$
$$278$$ 0 0
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ 6.19615 0.369631 0.184816 0.982773i $$-0.440831\pi$$
0.184816 + 0.982773i $$0.440831\pi$$
$$282$$ 0 0
$$283$$ 0.392305 0.0233201 0.0116601 0.999932i $$-0.496288\pi$$
0.0116601 + 0.999932i $$0.496288\pi$$
$$284$$ 0 0
$$285$$ −0.732051 −0.0433629
$$286$$ 0 0
$$287$$ 4.92820 0.290903
$$288$$ 0 0
$$289$$ 5.39230 0.317194
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 0 0
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ −5.85641 −0.340973
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −8.92820 −0.514613
$$302$$ 0 0
$$303$$ −4.73205 −0.271849
$$304$$ 0 0
$$305$$ 1.46410 0.0838342
$$306$$ 0 0
$$307$$ −22.7846 −1.30039 −0.650193 0.759769i $$-0.725312\pi$$
−0.650193 + 0.759769i $$0.725312\pi$$
$$308$$ 0 0
$$309$$ −10.9282 −0.621684
$$310$$ 0 0
$$311$$ −12.5885 −0.713826 −0.356913 0.934138i $$-0.616171\pi$$
−0.356913 + 0.934138i $$0.616171\pi$$
$$312$$ 0 0
$$313$$ −2.00000 −0.113047 −0.0565233 0.998401i $$-0.518002\pi$$
−0.0565233 + 0.998401i $$0.518002\pi$$
$$314$$ 0 0
$$315$$ −0.732051 −0.0412464
$$316$$ 0 0
$$317$$ 1.80385 0.101314 0.0506571 0.998716i $$-0.483868\pi$$
0.0506571 + 0.998716i $$0.483868\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 1.26795 0.0707700
$$322$$ 0 0
$$323$$ 4.73205 0.263298
$$324$$ 0 0
$$325$$ −6.53590 −0.362546
$$326$$ 0 0
$$327$$ 18.3923 1.01710
$$328$$ 0 0
$$329$$ 5.66025 0.312060
$$330$$ 0 0
$$331$$ −26.2487 −1.44276 −0.721380 0.692540i $$-0.756492\pi$$
−0.721380 + 0.692540i $$0.756492\pi$$
$$332$$ 0 0
$$333$$ −10.0000 −0.547997
$$334$$ 0 0
$$335$$ 4.78461 0.261411
$$336$$ 0 0
$$337$$ 9.60770 0.523365 0.261682 0.965154i $$-0.415723\pi$$
0.261682 + 0.965154i $$0.415723\pi$$
$$338$$ 0 0
$$339$$ 10.1962 0.553779
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 4.39230 0.235791 0.117896 0.993026i $$-0.462385\pi$$
0.117896 + 0.993026i $$0.462385\pi$$
$$348$$ 0 0
$$349$$ −32.9282 −1.76261 −0.881303 0.472551i $$-0.843333\pi$$
−0.881303 + 0.472551i $$0.843333\pi$$
$$350$$ 0 0
$$351$$ −1.46410 −0.0781480
$$352$$ 0 0
$$353$$ 5.80385 0.308908 0.154454 0.988000i $$-0.450638\pi$$
0.154454 + 0.988000i $$0.450638\pi$$
$$354$$ 0 0
$$355$$ −0.928203 −0.0492639
$$356$$ 0 0
$$357$$ 4.73205 0.250447
$$358$$ 0 0
$$359$$ 17.0718 0.901015 0.450507 0.892773i $$-0.351243\pi$$
0.450507 + 0.892773i $$0.351243\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ 7.60770 0.398205
$$366$$ 0 0
$$367$$ 25.4641 1.32922 0.664608 0.747193i $$-0.268599\pi$$
0.664608 + 0.747193i $$0.268599\pi$$
$$368$$ 0 0
$$369$$ 4.92820 0.256552
$$370$$ 0 0
$$371$$ −3.26795 −0.169663
$$372$$ 0 0
$$373$$ −12.9282 −0.669397 −0.334698 0.942325i $$-0.608634\pi$$
−0.334698 + 0.942325i $$0.608634\pi$$
$$374$$ 0 0
$$375$$ −6.92820 −0.357771
$$376$$ 0 0
$$377$$ 11.2154 0.577622
$$378$$ 0 0
$$379$$ 19.3205 0.992428 0.496214 0.868200i $$-0.334723\pi$$
0.496214 + 0.868200i $$0.334723\pi$$
$$380$$ 0 0
$$381$$ 0.392305 0.0200984
$$382$$ 0 0
$$383$$ 23.3205 1.19162 0.595811 0.803125i $$-0.296831\pi$$
0.595811 + 0.803125i $$0.296831\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −8.92820 −0.453846
$$388$$ 0 0
$$389$$ −27.4641 −1.39249 −0.696243 0.717807i $$-0.745146\pi$$
−0.696243 + 0.717807i $$0.745146\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 16.5885 0.836777
$$394$$ 0 0
$$395$$ −2.92820 −0.147334
$$396$$ 0 0
$$397$$ 33.7128 1.69200 0.845999 0.533185i $$-0.179005\pi$$
0.845999 + 0.533185i $$0.179005\pi$$
$$398$$ 0 0
$$399$$ 1.00000 0.0500626
$$400$$ 0 0
$$401$$ −10.1962 −0.509172 −0.254586 0.967050i $$-0.581939\pi$$
−0.254586 + 0.967050i $$0.581939\pi$$
$$402$$ 0 0
$$403$$ 2.92820 0.145864
$$404$$ 0 0
$$405$$ −0.732051 −0.0363759
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −7.60770 −0.376176 −0.188088 0.982152i $$-0.560229\pi$$
−0.188088 + 0.982152i $$0.560229\pi$$
$$410$$ 0 0
$$411$$ 0.928203 0.0457849
$$412$$ 0 0
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ 0.143594 0.00704873
$$416$$ 0 0
$$417$$ −19.3205 −0.946129
$$418$$ 0 0
$$419$$ 17.2679 0.843595 0.421797 0.906690i $$-0.361399\pi$$
0.421797 + 0.906690i $$0.361399\pi$$
$$420$$ 0 0
$$421$$ 16.2487 0.791914 0.395957 0.918269i $$-0.370413\pi$$
0.395957 + 0.918269i $$0.370413\pi$$
$$422$$ 0 0
$$423$$ 5.66025 0.275211
$$424$$ 0 0
$$425$$ 21.1244 1.02468
$$426$$ 0 0
$$427$$ −2.00000 −0.0967868
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −38.0526 −1.83293 −0.916464 0.400118i $$-0.868969\pi$$
−0.916464 + 0.400118i $$0.868969\pi$$
$$432$$ 0 0
$$433$$ 11.8564 0.569783 0.284891 0.958560i $$-0.408043\pi$$
0.284891 + 0.958560i $$0.408043\pi$$
$$434$$ 0 0
$$435$$ 5.60770 0.268868
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 8.14359 0.388673 0.194336 0.980935i $$-0.437745\pi$$
0.194336 + 0.980935i $$0.437745\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −41.4641 −1.97002 −0.985009 0.172500i $$-0.944815\pi$$
−0.985009 + 0.172500i $$0.944815\pi$$
$$444$$ 0 0
$$445$$ 6.53590 0.309831
$$446$$ 0 0
$$447$$ 6.39230 0.302346
$$448$$ 0 0
$$449$$ 8.73205 0.412091 0.206045 0.978542i $$-0.433941\pi$$
0.206045 + 0.978542i $$0.433941\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 9.46410 0.444662
$$454$$ 0 0
$$455$$ −1.07180 −0.0502466
$$456$$ 0 0
$$457$$ 0.392305 0.0183512 0.00917562 0.999958i $$-0.497079\pi$$
0.00917562 + 0.999958i $$0.497079\pi$$
$$458$$ 0 0
$$459$$ 4.73205 0.220873
$$460$$ 0 0
$$461$$ 31.6603 1.47457 0.737283 0.675585i $$-0.236109\pi$$
0.737283 + 0.675585i $$0.236109\pi$$
$$462$$ 0 0
$$463$$ −5.85641 −0.272170 −0.136085 0.990697i $$-0.543452\pi$$
−0.136085 + 0.990697i $$0.543452\pi$$
$$464$$ 0 0
$$465$$ 1.46410 0.0678961
$$466$$ 0 0
$$467$$ 38.0526 1.76086 0.880431 0.474174i $$-0.157253\pi$$
0.880431 + 0.474174i $$0.157253\pi$$
$$468$$ 0 0
$$469$$ −6.53590 −0.301800
$$470$$ 0 0
$$471$$ 14.0000 0.645086
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 4.46410 0.204827
$$476$$ 0 0
$$477$$ −3.26795 −0.149629
$$478$$ 0 0
$$479$$ 10.3397 0.472435 0.236218 0.971700i $$-0.424092\pi$$
0.236218 + 0.971700i $$0.424092\pi$$
$$480$$ 0 0
$$481$$ −14.6410 −0.667573
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −1.46410 −0.0664814
$$486$$ 0 0
$$487$$ −1.85641 −0.0841218 −0.0420609 0.999115i $$-0.513392\pi$$
−0.0420609 + 0.999115i $$0.513392\pi$$
$$488$$ 0 0
$$489$$ 20.9282 0.946406
$$490$$ 0 0
$$491$$ −15.3205 −0.691405 −0.345702 0.938344i $$-0.612359\pi$$
−0.345702 + 0.938344i $$0.612359\pi$$
$$492$$ 0 0
$$493$$ −36.2487 −1.63256
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1.26795 0.0568753
$$498$$ 0 0
$$499$$ 17.8564 0.799363 0.399681 0.916654i $$-0.369121\pi$$
0.399681 + 0.916654i $$0.369121\pi$$
$$500$$ 0 0
$$501$$ 23.3205 1.04188
$$502$$ 0 0
$$503$$ 1.26795 0.0565351 0.0282675 0.999600i $$-0.491001\pi$$
0.0282675 + 0.999600i $$0.491001\pi$$
$$504$$ 0 0
$$505$$ −3.46410 −0.154150
$$506$$ 0 0
$$507$$ 10.8564 0.482150
$$508$$ 0 0
$$509$$ 40.2487 1.78399 0.891996 0.452043i $$-0.149304\pi$$
0.891996 + 0.452043i $$0.149304\pi$$
$$510$$ 0 0
$$511$$ −10.3923 −0.459728
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 3.07180 0.134837
$$520$$ 0 0
$$521$$ −12.2487 −0.536626 −0.268313 0.963332i $$-0.586466\pi$$
−0.268313 + 0.963332i $$0.586466\pi$$
$$522$$ 0 0
$$523$$ −22.0000 −0.961993 −0.480996 0.876723i $$-0.659725\pi$$
−0.480996 + 0.876723i $$0.659725\pi$$
$$524$$ 0 0
$$525$$ 4.46410 0.194829
$$526$$ 0 0
$$527$$ −9.46410 −0.412263
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 8.00000 0.347170
$$532$$ 0 0
$$533$$ 7.21539 0.312533
$$534$$ 0 0
$$535$$ 0.928203 0.0401297
$$536$$ 0 0
$$537$$ 3.80385 0.164148
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ 0 0
$$543$$ 8.92820 0.383146
$$544$$ 0 0
$$545$$ 13.4641 0.576739
$$546$$ 0 0
$$547$$ 43.7128 1.86902 0.934512 0.355930i $$-0.115836\pi$$
0.934512 + 0.355930i $$0.115836\pi$$
$$548$$ 0 0
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −7.66025 −0.326338
$$552$$ 0 0
$$553$$ 4.00000 0.170097
$$554$$ 0 0
$$555$$ −7.32051 −0.310738
$$556$$ 0 0
$$557$$ 27.4641 1.16369 0.581846 0.813299i $$-0.302331\pi$$
0.581846 + 0.813299i $$0.302331\pi$$
$$558$$ 0 0
$$559$$ −13.0718 −0.552878
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −18.5359 −0.781195 −0.390597 0.920562i $$-0.627732\pi$$
−0.390597 + 0.920562i $$0.627732\pi$$
$$564$$ 0 0
$$565$$ 7.46410 0.314017
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −2.58846 −0.108514 −0.0542569 0.998527i $$-0.517279\pi$$
−0.0542569 + 0.998527i $$0.517279\pi$$
$$570$$ 0 0
$$571$$ 1.85641 0.0776882 0.0388441 0.999245i $$-0.487632\pi$$
0.0388441 + 0.999245i $$0.487632\pi$$
$$572$$ 0 0
$$573$$ 1.07180 0.0447750
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −10.0000 −0.416305 −0.208153 0.978096i $$-0.566745\pi$$
−0.208153 + 0.978096i $$0.566745\pi$$
$$578$$ 0 0
$$579$$ −3.85641 −0.160267
$$580$$ 0 0
$$581$$ −0.196152 −0.00813777
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −1.07180 −0.0443133
$$586$$ 0 0
$$587$$ −39.9090 −1.64722 −0.823610 0.567157i $$-0.808043\pi$$
−0.823610 + 0.567157i $$0.808043\pi$$
$$588$$ 0 0
$$589$$ −2.00000 −0.0824086
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 0 0
$$593$$ 38.1962 1.56853 0.784264 0.620427i $$-0.213041\pi$$
0.784264 + 0.620427i $$0.213041\pi$$
$$594$$ 0 0
$$595$$ 3.46410 0.142014
$$596$$ 0 0
$$597$$ −8.39230 −0.343474
$$598$$ 0 0
$$599$$ −10.7321 −0.438500 −0.219250 0.975669i $$-0.570361\pi$$
−0.219250 + 0.975669i $$0.570361\pi$$
$$600$$ 0 0
$$601$$ 34.0000 1.38689 0.693444 0.720510i $$-0.256092\pi$$
0.693444 + 0.720510i $$0.256092\pi$$
$$602$$ 0 0
$$603$$ −6.53590 −0.266162
$$604$$ 0 0
$$605$$ 8.05256 0.327383
$$606$$ 0 0
$$607$$ −26.9282 −1.09298 −0.546491 0.837465i $$-0.684037\pi$$
−0.546491 + 0.837465i $$0.684037\pi$$
$$608$$ 0 0
$$609$$ −7.66025 −0.310409
$$610$$ 0 0
$$611$$ 8.28719 0.335264
$$612$$ 0 0
$$613$$ 30.2487 1.22173 0.610867 0.791733i $$-0.290821\pi$$
0.610867 + 0.791733i $$0.290821\pi$$
$$614$$ 0 0
$$615$$ 3.60770 0.145476
$$616$$ 0 0
$$617$$ 44.6410 1.79718 0.898590 0.438790i $$-0.144593\pi$$
0.898590 + 0.438790i $$0.144593\pi$$
$$618$$ 0 0
$$619$$ 49.1769 1.97659 0.988294 0.152564i $$-0.0487531\pi$$
0.988294 + 0.152564i $$0.0487531\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −8.92820 −0.357701
$$624$$ 0 0
$$625$$ 17.2487 0.689948
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 47.3205 1.88679
$$630$$ 0 0
$$631$$ 27.0718 1.07771 0.538856 0.842398i $$-0.318857\pi$$
0.538856 + 0.842398i $$0.318857\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ 0 0
$$635$$ 0.287187 0.0113967
$$636$$ 0 0
$$637$$ 1.46410 0.0580098
$$638$$ 0 0
$$639$$ 1.26795 0.0501593
$$640$$ 0 0
$$641$$ −23.6603 −0.934524 −0.467262 0.884119i $$-0.654759\pi$$
−0.467262 + 0.884119i $$0.654759\pi$$
$$642$$ 0 0
$$643$$ −0.392305 −0.0154710 −0.00773550 0.999970i $$-0.502462\pi$$
−0.00773550 + 0.999970i $$0.502462\pi$$
$$644$$ 0 0
$$645$$ −6.53590 −0.257351
$$646$$ 0 0
$$647$$ 13.6603 0.537040 0.268520 0.963274i $$-0.413465\pi$$
0.268520 + 0.963274i $$0.413465\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −2.00000 −0.0783862
$$652$$ 0 0
$$653$$ −1.21539 −0.0475619 −0.0237809 0.999717i $$-0.507570\pi$$
−0.0237809 + 0.999717i $$0.507570\pi$$
$$654$$ 0 0
$$655$$ 12.1436 0.474489
$$656$$ 0 0
$$657$$ −10.3923 −0.405442
$$658$$ 0 0
$$659$$ −6.05256 −0.235774 −0.117887 0.993027i $$-0.537612\pi$$
−0.117887 + 0.993027i $$0.537612\pi$$
$$660$$ 0 0
$$661$$ −13.4641 −0.523693 −0.261846 0.965110i $$-0.584331\pi$$
−0.261846 + 0.965110i $$0.584331\pi$$
$$662$$ 0 0
$$663$$ 6.92820 0.269069
$$664$$ 0 0
$$665$$ 0.732051 0.0283877
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 7.07180 0.273411
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −20.9282 −0.806723 −0.403361 0.915041i $$-0.632158\pi$$
−0.403361 + 0.915041i $$0.632158\pi$$
$$674$$ 0 0
$$675$$ 4.46410 0.171823
$$676$$ 0 0
$$677$$ −5.32051 −0.204484 −0.102242 0.994760i $$-0.532602\pi$$
−0.102242 + 0.994760i $$0.532602\pi$$
$$678$$ 0 0
$$679$$ 2.00000 0.0767530
$$680$$ 0 0
$$681$$ 4.39230 0.168313
$$682$$ 0 0
$$683$$ 1.26795 0.0485167 0.0242584 0.999706i $$-0.492278\pi$$
0.0242584 + 0.999706i $$0.492278\pi$$
$$684$$ 0 0
$$685$$ 0.679492 0.0259621
$$686$$ 0 0
$$687$$ 20.2487 0.772537
$$688$$ 0 0
$$689$$ −4.78461 −0.182279
$$690$$ 0 0
$$691$$ 32.7846 1.24719 0.623593 0.781749i $$-0.285672\pi$$
0.623593 + 0.781749i $$0.285672\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −14.1436 −0.536497
$$696$$ 0 0
$$697$$ −23.3205 −0.883327
$$698$$ 0 0
$$699$$ −18.3923 −0.695661
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ 0 0
$$703$$ 10.0000 0.377157
$$704$$ 0 0
$$705$$ 4.14359 0.156057
$$706$$ 0 0
$$707$$ 4.73205 0.177967
$$708$$ 0 0
$$709$$ −20.3923 −0.765849 −0.382925 0.923780i $$-0.625083\pi$$
−0.382925 + 0.923780i $$0.625083\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −16.3923 −0.612182
$$718$$ 0 0
$$719$$ −4.98076 −0.185751 −0.0928755 0.995678i $$-0.529606\pi$$
−0.0928755 + 0.995678i $$0.529606\pi$$
$$720$$ 0 0
$$721$$ 10.9282 0.406988
$$722$$ 0 0
$$723$$ −0.928203 −0.0345202
$$724$$ 0 0
$$725$$ −34.1962 −1.27001
$$726$$ 0 0
$$727$$ 5.07180 0.188103 0.0940513 0.995567i $$-0.470018\pi$$
0.0940513 + 0.995567i $$0.470018\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 42.2487 1.56263
$$732$$ 0 0
$$733$$ −44.2487 −1.63436 −0.817182 0.576380i $$-0.804465\pi$$
−0.817182 + 0.576380i $$0.804465\pi$$
$$734$$ 0 0
$$735$$ 0.732051 0.0270021
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −47.8564 −1.76043 −0.880213 0.474578i $$-0.842601\pi$$
−0.880213 + 0.474578i $$0.842601\pi$$
$$740$$ 0 0
$$741$$ 1.46410 0.0537851
$$742$$ 0 0
$$743$$ −5.26795 −0.193262 −0.0966312 0.995320i $$-0.530807\pi$$
−0.0966312 + 0.995320i $$0.530807\pi$$
$$744$$ 0 0
$$745$$ 4.67949 0.171443
$$746$$ 0 0
$$747$$ −0.196152 −0.00717684
$$748$$ 0 0
$$749$$ −1.26795 −0.0463299
$$750$$ 0 0
$$751$$ 13.1769 0.480832 0.240416 0.970670i $$-0.422716\pi$$
0.240416 + 0.970670i $$0.422716\pi$$
$$752$$ 0 0
$$753$$ −17.6603 −0.643575
$$754$$ 0 0
$$755$$ 6.92820 0.252143
$$756$$ 0 0
$$757$$ 18.7846 0.682738 0.341369 0.939929i $$-0.389109\pi$$
0.341369 + 0.939929i $$0.389109\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 16.3397 0.592315 0.296158 0.955139i $$-0.404295\pi$$
0.296158 + 0.955139i $$0.404295\pi$$
$$762$$ 0 0
$$763$$ −18.3923 −0.665846
$$764$$ 0 0
$$765$$ 3.46410 0.125245
$$766$$ 0 0
$$767$$ 11.7128 0.422925
$$768$$ 0 0
$$769$$ 27.8564 1.00453 0.502264 0.864714i $$-0.332501\pi$$
0.502264 + 0.864714i $$0.332501\pi$$
$$770$$ 0 0
$$771$$ −4.53590 −0.163356
$$772$$ 0 0
$$773$$ −14.0000 −0.503545 −0.251773 0.967786i $$-0.581013\pi$$
−0.251773 + 0.967786i $$0.581013\pi$$
$$774$$ 0 0
$$775$$ −8.92820 −0.320711
$$776$$ 0 0
$$777$$ 10.0000 0.358748
$$778$$ 0 0
$$779$$ −4.92820 −0.176571
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −7.66025 −0.273755
$$784$$ 0 0
$$785$$ 10.2487 0.365792
$$786$$ 0 0
$$787$$ 0.143594 0.00511856 0.00255928 0.999997i $$-0.499185\pi$$
0.00255928 + 0.999997i $$0.499185\pi$$
$$788$$ 0 0
$$789$$ −15.3205 −0.545425
$$790$$ 0 0
$$791$$ −10.1962 −0.362533
$$792$$ 0 0
$$793$$ −2.92820 −0.103984
$$794$$ 0 0
$$795$$ −2.39230 −0.0848463
$$796$$ 0 0
$$797$$ 30.7846 1.09045 0.545223 0.838291i $$-0.316445\pi$$
0.545223 + 0.838291i $$0.316445\pi$$
$$798$$ 0 0
$$799$$ −26.7846 −0.947571
$$800$$ 0 0
$$801$$ −8.92820 −0.315463
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 31.1769 1.09748
$$808$$ 0 0
$$809$$ 30.7846 1.08233 0.541165 0.840917i $$-0.317984\pi$$
0.541165 + 0.840917i $$0.317984\pi$$
$$810$$ 0 0
$$811$$ 4.00000 0.140459 0.0702295 0.997531i $$-0.477627\pi$$
0.0702295 + 0.997531i $$0.477627\pi$$
$$812$$ 0 0
$$813$$ −12.7846 −0.448376
$$814$$ 0 0
$$815$$ 15.3205 0.536654
$$816$$ 0 0
$$817$$ 8.92820 0.312358
$$818$$ 0 0
$$819$$ 1.46410 0.0511599
$$820$$ 0 0
$$821$$ 28.5359 0.995910 0.497955 0.867203i $$-0.334085\pi$$
0.497955 + 0.867203i $$0.334085\pi$$
$$822$$ 0 0
$$823$$ −48.6410 −1.69552 −0.847760 0.530381i $$-0.822049\pi$$
−0.847760 + 0.530381i $$0.822049\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 11.9090 0.414115 0.207058 0.978329i $$-0.433611\pi$$
0.207058 + 0.978329i $$0.433611\pi$$
$$828$$ 0 0
$$829$$ 17.7128 0.615191 0.307596 0.951517i $$-0.400476\pi$$
0.307596 + 0.951517i $$0.400476\pi$$
$$830$$ 0 0
$$831$$ 8.39230 0.291126
$$832$$ 0 0
$$833$$ −4.73205 −0.163956
$$834$$ 0 0
$$835$$ 17.0718 0.590794
$$836$$ 0 0
$$837$$ −2.00000 −0.0691301
$$838$$ 0 0
$$839$$ 12.7846 0.441374 0.220687 0.975345i $$-0.429170\pi$$
0.220687 + 0.975345i $$0.429170\pi$$
$$840$$ 0 0
$$841$$ 29.6795 1.02343
$$842$$ 0 0
$$843$$ −6.19615 −0.213407
$$844$$ 0 0
$$845$$ 7.94744 0.273400
$$846$$ 0 0
$$847$$ −11.0000 −0.377964
$$848$$ 0 0
$$849$$ −0.392305 −0.0134639
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −43.8564 −1.50161 −0.750807 0.660521i $$-0.770335\pi$$
−0.750807 + 0.660521i $$0.770335\pi$$
$$854$$ 0 0
$$855$$ 0.732051 0.0250356
$$856$$ 0 0
$$857$$ 3.07180 0.104931 0.0524653 0.998623i $$-0.483292\pi$$
0.0524653 + 0.998623i $$0.483292\pi$$
$$858$$ 0 0
$$859$$ 18.1436 0.619051 0.309526 0.950891i $$-0.399830\pi$$
0.309526 + 0.950891i $$0.399830\pi$$
$$860$$ 0 0
$$861$$ −4.92820 −0.167953
$$862$$ 0 0
$$863$$ 8.98076 0.305709 0.152854 0.988249i $$-0.451153\pi$$
0.152854 + 0.988249i $$0.451153\pi$$
$$864$$ 0 0
$$865$$ 2.24871 0.0764585
$$866$$ 0 0
$$867$$ −5.39230 −0.183132
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −9.56922 −0.324241
$$872$$ 0 0
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ 6.92820 0.234216
$$876$$ 0 0
$$877$$ 1.71281 0.0578376 0.0289188 0.999582i $$-0.490794\pi$$
0.0289188 + 0.999582i $$0.490794\pi$$
$$878$$ 0 0
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ 0.732051 0.0246634 0.0123317 0.999924i $$-0.496075\pi$$
0.0123317 + 0.999924i $$0.496075\pi$$
$$882$$ 0 0
$$883$$ 22.9282 0.771595 0.385798 0.922583i $$-0.373926\pi$$
0.385798 + 0.922583i $$0.373926\pi$$
$$884$$ 0 0
$$885$$ 5.85641 0.196861
$$886$$ 0 0
$$887$$ −6.24871 −0.209811 −0.104906 0.994482i $$-0.533454\pi$$
−0.104906 + 0.994482i $$0.533454\pi$$
$$888$$ 0 0
$$889$$ −0.392305 −0.0131575
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −5.66025 −0.189413
$$894$$ 0 0
$$895$$ 2.78461 0.0930792
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 15.3205 0.510968
$$900$$ 0 0
$$901$$ 15.4641 0.515184
$$902$$ 0 0
$$903$$ 8.92820 0.297112
$$904$$ 0 0
$$905$$ 6.53590 0.217261
$$906$$ 0 0
$$907$$ −54.6410 −1.81433 −0.907163 0.420780i $$-0.861756\pi$$
−0.907163 + 0.420780i $$0.861756\pi$$
$$908$$ 0 0
$$909$$ 4.73205 0.156952
$$910$$ 0 0
$$911$$ −9.26795 −0.307061 −0.153530 0.988144i $$-0.549064\pi$$
−0.153530 + 0.988144i $$0.549064\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −1.46410 −0.0484017
$$916$$ 0 0
$$917$$ −16.5885 −0.547799
$$918$$ 0 0
$$919$$ −2.14359 −0.0707106 −0.0353553 0.999375i $$-0.511256\pi$$
−0.0353553 + 0.999375i $$0.511256\pi$$
$$920$$ 0 0
$$921$$ 22.7846 0.750778
$$922$$ 0 0
$$923$$ 1.85641 0.0611044
$$924$$ 0 0
$$925$$ 44.6410 1.46779
$$926$$ 0 0
$$927$$ 10.9282 0.358929
$$928$$ 0 0
$$929$$ 46.9808 1.54139 0.770694 0.637205i $$-0.219910\pi$$
0.770694 + 0.637205i $$0.219910\pi$$
$$930$$ 0 0
$$931$$ −1.00000 −0.0327737
$$932$$ 0 0
$$933$$ 12.5885 0.412128
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −20.6410 −0.674313 −0.337156 0.941449i $$-0.609465\pi$$
−0.337156 + 0.941449i $$0.609465\pi$$
$$938$$ 0 0
$$939$$ 2.00000 0.0652675
$$940$$ 0 0
$$941$$ −44.5359 −1.45183 −0.725914 0.687785i $$-0.758583\pi$$
−0.725914 + 0.687785i $$0.758583\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0.732051 0.0238136
$$946$$ 0 0
$$947$$ 22.9282 0.745066 0.372533 0.928019i $$-0.378489\pi$$
0.372533 + 0.928019i $$0.378489\pi$$
$$948$$ 0 0
$$949$$ −15.2154 −0.493912
$$950$$ 0 0
$$951$$ −1.80385 −0.0584938
$$952$$ 0 0
$$953$$ 29.8038 0.965441 0.482721 0.875774i $$-0.339649\pi$$
0.482721 + 0.875774i $$0.339649\pi$$
$$954$$ 0 0
$$955$$ 0.784610 0.0253894
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −0.928203 −0.0299732
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 0 0
$$963$$ −1.26795 −0.0408591
$$964$$ 0 0
$$965$$ −2.82309 −0.0908783
$$966$$ 0 0
$$967$$ 32.6410 1.04966 0.524832 0.851206i $$-0.324128\pi$$
0.524832 + 0.851206i $$0.324128\pi$$
$$968$$ 0 0
$$969$$ −4.73205 −0.152015
$$970$$ 0 0
$$971$$ 59.7128 1.91628 0.958138 0.286308i $$-0.0924280\pi$$
0.958138 + 0.286308i $$0.0924280\pi$$
$$972$$ 0 0
$$973$$ 19.3205 0.619387
$$974$$ 0 0
$$975$$ 6.53590 0.209316
$$976$$ 0 0
$$977$$ 58.3013 1.86522 0.932611 0.360882i $$-0.117524\pi$$
0.932611 + 0.360882i $$0.117524\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −18.3923 −0.587221
$$982$$ 0 0
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ 0 0
$$985$$ 13.1769 0.419851
$$986$$ 0 0
$$987$$ −5.66025 −0.180168
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 28.7846 0.914373 0.457187 0.889371i $$-0.348857\pi$$
0.457187 + 0.889371i $$0.348857\pi$$
$$992$$ 0 0
$$993$$ 26.2487 0.832978
$$994$$ 0 0
$$995$$ −6.14359 −0.194765
$$996$$ 0 0
$$997$$ 7.85641 0.248815 0.124407 0.992231i $$-0.460297\pi$$
0.124407 + 0.992231i $$0.460297\pi$$
$$998$$ 0 0
$$999$$ 10.0000 0.316386
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 3192.2.a.q.1.1 2
3.2 odd 2 9576.2.a.bj.1.2 2
4.3 odd 2 6384.2.a.bs.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.q.1.1 2 1.1 even 1 trivial
6384.2.a.bs.1.1 2 4.3 odd 2
9576.2.a.bj.1.2 2 3.2 odd 2