# Properties

 Label 9200.2.a.j.1.1 Level $9200$ Weight $2$ Character 9200.1 Self dual yes Analytic conductor $73.462$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.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: $$73.4623698596$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4600) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 9200.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{3} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{3} +3.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} -5.00000 q^{13} -4.00000 q^{17} -1.00000 q^{19} -6.00000 q^{21} +1.00000 q^{23} +4.00000 q^{27} +9.00000 q^{29} +2.00000 q^{31} +10.0000 q^{33} -2.00000 q^{37} +10.0000 q^{39} +3.00000 q^{41} +7.00000 q^{43} +12.0000 q^{47} +2.00000 q^{49} +8.00000 q^{51} +12.0000 q^{53} +2.00000 q^{57} +6.00000 q^{59} -10.0000 q^{61} +3.00000 q^{63} +8.00000 q^{67} -2.00000 q^{69} -2.00000 q^{71} +1.00000 q^{73} -15.0000 q^{77} +11.0000 q^{79} -11.0000 q^{81} -9.00000 q^{83} -18.0000 q^{87} -14.0000 q^{89} -15.0000 q^{91} -4.00000 q^{93} -16.0000 q^{97} -5.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ 0 0
$$13$$ −5.00000 −1.38675 −0.693375 0.720577i $$-0.743877\pi$$
−0.693375 + 0.720577i $$0.743877\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ −6.00000 −1.30931
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 0 0
$$33$$ 10.0000 1.74078
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 10.0000 1.60128
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ 7.00000 1.06749 0.533745 0.845645i $$-0.320784\pi$$
0.533745 + 0.845645i $$0.320784\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 8.00000 1.12022
$$52$$ 0 0
$$53$$ 12.0000 1.64833 0.824163 0.566352i $$-0.191646\pi$$
0.824163 + 0.566352i $$0.191646\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ 0 0
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 3.00000 0.377964
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 0 0
$$69$$ −2.00000 −0.240772
$$70$$ 0 0
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ 0 0
$$73$$ 1.00000 0.117041 0.0585206 0.998286i $$-0.481362\pi$$
0.0585206 + 0.998286i $$0.481362\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −15.0000 −1.70941
$$78$$ 0 0
$$79$$ 11.0000 1.23760 0.618798 0.785550i $$-0.287620\pi$$
0.618798 + 0.785550i $$0.287620\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ −9.00000 −0.987878 −0.493939 0.869496i $$-0.664443\pi$$
−0.493939 + 0.869496i $$0.664443\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −18.0000 −1.92980
$$88$$ 0 0
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ −15.0000 −1.57243
$$92$$ 0 0
$$93$$ −4.00000 −0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −16.0000 −1.62455 −0.812277 0.583272i $$-0.801772\pi$$
−0.812277 + 0.583272i $$0.801772\pi$$
$$98$$ 0 0
$$99$$ −5.00000 −0.502519
$$100$$ 0 0
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ 9.00000 0.886796 0.443398 0.896325i $$-0.353773\pi$$
0.443398 + 0.896325i $$0.353773\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −16.0000 −1.54678 −0.773389 0.633932i $$-0.781440\pi$$
−0.773389 + 0.633932i $$0.781440\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ 8.00000 0.752577 0.376288 0.926503i $$-0.377200\pi$$
0.376288 + 0.926503i $$0.377200\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −5.00000 −0.462250
$$118$$ 0 0
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ −6.00000 −0.541002
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 10.0000 0.887357 0.443678 0.896186i $$-0.353673\pi$$
0.443678 + 0.896186i $$0.353673\pi$$
$$128$$ 0 0
$$129$$ −14.0000 −1.23263
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 0 0
$$133$$ −3.00000 −0.260133
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ −24.0000 −2.02116
$$142$$ 0 0
$$143$$ 25.0000 2.09061
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −4.00000 −0.329914
$$148$$ 0 0
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ 0 0
$$151$$ 18.0000 1.46482 0.732410 0.680864i $$-0.238396\pi$$
0.732410 + 0.680864i $$0.238396\pi$$
$$152$$ 0 0
$$153$$ −4.00000 −0.323381
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 8.00000 0.638470 0.319235 0.947676i $$-0.396574\pi$$
0.319235 + 0.947676i $$0.396574\pi$$
$$158$$ 0 0
$$159$$ −24.0000 −1.90332
$$160$$ 0 0
$$161$$ 3.00000 0.236433
$$162$$ 0 0
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 14.0000 1.08335 0.541676 0.840587i $$-0.317790\pi$$
0.541676 + 0.840587i $$0.317790\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ 5.00000 0.380143 0.190071 0.981770i $$-0.439128\pi$$
0.190071 + 0.981770i $$0.439128\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −12.0000 −0.901975
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ 0 0
$$183$$ 20.0000 1.47844
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 20.0000 1.46254
$$188$$ 0 0
$$189$$ 12.0000 0.872872
$$190$$ 0 0
$$191$$ 1.00000 0.0723575 0.0361787 0.999345i $$-0.488481\pi$$
0.0361787 + 0.999345i $$0.488481\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −23.0000 −1.63868 −0.819341 0.573306i $$-0.805660\pi$$
−0.819341 + 0.573306i $$0.805660\pi$$
$$198$$ 0 0
$$199$$ −5.00000 −0.354441 −0.177220 0.984171i $$-0.556711\pi$$
−0.177220 + 0.984171i $$0.556711\pi$$
$$200$$ 0 0
$$201$$ −16.0000 −1.12855
$$202$$ 0 0
$$203$$ 27.0000 1.89503
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ 5.00000 0.345857
$$210$$ 0 0
$$211$$ 2.00000 0.137686 0.0688428 0.997628i $$-0.478069\pi$$
0.0688428 + 0.997628i $$0.478069\pi$$
$$212$$ 0 0
$$213$$ 4.00000 0.274075
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 6.00000 0.407307
$$218$$ 0 0
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 20.0000 1.34535
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 0 0
$$229$$ −30.0000 −1.98246 −0.991228 0.132164i $$-0.957808\pi$$
−0.991228 + 0.132164i $$0.957808\pi$$
$$230$$ 0 0
$$231$$ 30.0000 1.97386
$$232$$ 0 0
$$233$$ −25.0000 −1.63780 −0.818902 0.573933i $$-0.805417\pi$$
−0.818902 + 0.573933i $$0.805417\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −22.0000 −1.42905
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −4.00000 −0.257663 −0.128831 0.991667i $$-0.541123\pi$$
−0.128831 + 0.991667i $$0.541123\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.00000 0.318142
$$248$$ 0 0
$$249$$ 18.0000 1.14070
$$250$$ 0 0
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ 0 0
$$253$$ −5.00000 −0.314347
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 9.00000 0.557086
$$262$$ 0 0
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 28.0000 1.71357
$$268$$ 0 0
$$269$$ 5.00000 0.304855 0.152428 0.988315i $$-0.451291\pi$$
0.152428 + 0.988315i $$0.451291\pi$$
$$270$$ 0 0
$$271$$ −12.0000 −0.728948 −0.364474 0.931214i $$-0.618751\pi$$
−0.364474 + 0.931214i $$0.618751\pi$$
$$272$$ 0 0
$$273$$ 30.0000 1.81568
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 13.0000 0.781094 0.390547 0.920583i $$-0.372286\pi$$
0.390547 + 0.920583i $$0.372286\pi$$
$$278$$ 0 0
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ −8.00000 −0.477240 −0.238620 0.971113i $$-0.576695\pi$$
−0.238620 + 0.971113i $$0.576695\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 9.00000 0.531253
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 32.0000 1.87587
$$292$$ 0 0
$$293$$ 28.0000 1.63578 0.817889 0.575376i $$-0.195144\pi$$
0.817889 + 0.575376i $$0.195144\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −20.0000 −1.16052
$$298$$ 0 0
$$299$$ −5.00000 −0.289157
$$300$$ 0 0
$$301$$ 21.0000 1.21042
$$302$$ 0 0
$$303$$ 28.0000 1.60856
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 0 0
$$309$$ −18.0000 −1.02398
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 17.0000 0.954815 0.477408 0.878682i $$-0.341577\pi$$
0.477408 + 0.878682i $$0.341577\pi$$
$$318$$ 0 0
$$319$$ −45.0000 −2.51952
$$320$$ 0 0
$$321$$ 32.0000 1.78607
$$322$$ 0 0
$$323$$ 4.00000 0.222566
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 20.0000 1.10600
$$328$$ 0 0
$$329$$ 36.0000 1.98474
$$330$$ 0 0
$$331$$ −26.0000 −1.42909 −0.714545 0.699590i $$-0.753366\pi$$
−0.714545 + 0.699590i $$0.753366\pi$$
$$332$$ 0 0
$$333$$ −2.00000 −0.109599
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 32.0000 1.74315 0.871576 0.490261i $$-0.163099\pi$$
0.871576 + 0.490261i $$0.163099\pi$$
$$338$$ 0 0
$$339$$ −16.0000 −0.869001
$$340$$ 0 0
$$341$$ −10.0000 −0.541530
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −32.0000 −1.71785 −0.858925 0.512101i $$-0.828867\pi$$
−0.858925 + 0.512101i $$0.828867\pi$$
$$348$$ 0 0
$$349$$ −7.00000 −0.374701 −0.187351 0.982293i $$-0.559990\pi$$
−0.187351 + 0.982293i $$0.559990\pi$$
$$350$$ 0 0
$$351$$ −20.0000 −1.06752
$$352$$ 0 0
$$353$$ 21.0000 1.11772 0.558859 0.829263i $$-0.311239\pi$$
0.558859 + 0.829263i $$0.311239\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 24.0000 1.27021
$$358$$ 0 0
$$359$$ 3.00000 0.158334 0.0791670 0.996861i $$-0.474774\pi$$
0.0791670 + 0.996861i $$0.474774\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ −28.0000 −1.46962
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 15.0000 0.782994 0.391497 0.920179i $$-0.371957\pi$$
0.391497 + 0.920179i $$0.371957\pi$$
$$368$$ 0 0
$$369$$ 3.00000 0.156174
$$370$$ 0 0
$$371$$ 36.0000 1.86903
$$372$$ 0 0
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −45.0000 −2.31762
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ −20.0000 −1.02463
$$382$$ 0 0
$$383$$ 23.0000 1.17525 0.587623 0.809135i $$-0.300064\pi$$
0.587623 + 0.809135i $$0.300064\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 7.00000 0.355830
$$388$$ 0 0
$$389$$ −4.00000 −0.202808 −0.101404 0.994845i $$-0.532333\pi$$
−0.101404 + 0.994845i $$0.532333\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ 0 0
$$393$$ 12.0000 0.605320
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −6.00000 −0.301131 −0.150566 0.988600i $$-0.548110\pi$$
−0.150566 + 0.988600i $$0.548110\pi$$
$$398$$ 0 0
$$399$$ 6.00000 0.300376
$$400$$ 0 0
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 0 0
$$403$$ −10.0000 −0.498135
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 10.0000 0.495682
$$408$$ 0 0
$$409$$ −35.0000 −1.73064 −0.865319 0.501221i $$-0.832884\pi$$
−0.865319 + 0.501221i $$0.832884\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 0 0
$$413$$ 18.0000 0.885722
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 24.0000 1.17529
$$418$$ 0 0
$$419$$ −3.00000 −0.146560 −0.0732798 0.997311i $$-0.523347\pi$$
−0.0732798 + 0.997311i $$0.523347\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 0 0
$$423$$ 12.0000 0.583460
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −30.0000 −1.45180
$$428$$ 0 0
$$429$$ −50.0000 −2.41402
$$430$$ 0 0
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ 0 0
$$433$$ 6.00000 0.288342 0.144171 0.989553i $$-0.453949\pi$$
0.144171 + 0.989553i $$0.453949\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −1.00000 −0.0478365
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ −14.0000 −0.665160 −0.332580 0.943075i $$-0.607919\pi$$
−0.332580 + 0.943075i $$0.607919\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −28.0000 −1.32435
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ −15.0000 −0.706322
$$452$$ 0 0
$$453$$ −36.0000 −1.69143
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −28.0000 −1.30978 −0.654892 0.755722i $$-0.727286\pi$$
−0.654892 + 0.755722i $$0.727286\pi$$
$$458$$ 0 0
$$459$$ −16.0000 −0.746816
$$460$$ 0 0
$$461$$ 41.0000 1.90956 0.954780 0.297313i $$-0.0960904\pi$$
0.954780 + 0.297313i $$0.0960904\pi$$
$$462$$ 0 0
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 21.0000 0.971764 0.485882 0.874024i $$-0.338498\pi$$
0.485882 + 0.874024i $$0.338498\pi$$
$$468$$ 0 0
$$469$$ 24.0000 1.10822
$$470$$ 0 0
$$471$$ −16.0000 −0.737241
$$472$$ 0 0
$$473$$ −35.0000 −1.60930
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 12.0000 0.549442
$$478$$ 0 0
$$479$$ 17.0000 0.776750 0.388375 0.921501i $$-0.373037\pi$$
0.388375 + 0.921501i $$0.373037\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ 0 0
$$483$$ −6.00000 −0.273009
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 0 0
$$489$$ 32.0000 1.44709
$$490$$ 0 0
$$491$$ 2.00000 0.0902587 0.0451294 0.998981i $$-0.485630\pi$$
0.0451294 + 0.998981i $$0.485630\pi$$
$$492$$ 0 0
$$493$$ −36.0000 −1.62136
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −6.00000 −0.269137
$$498$$ 0 0
$$499$$ −24.0000 −1.07439 −0.537194 0.843459i $$-0.680516\pi$$
−0.537194 + 0.843459i $$0.680516\pi$$
$$500$$ 0 0
$$501$$ −28.0000 −1.25095
$$502$$ 0 0
$$503$$ 5.00000 0.222939 0.111469 0.993768i $$-0.464444\pi$$
0.111469 + 0.993768i $$0.464444\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −24.0000 −1.06588
$$508$$ 0 0
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ 3.00000 0.132712
$$512$$ 0 0
$$513$$ −4.00000 −0.176604
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −60.0000 −2.63880
$$518$$ 0 0
$$519$$ −10.0000 −0.438951
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ −7.00000 −0.306089 −0.153044 0.988219i $$-0.548908\pi$$
−0.153044 + 0.988219i $$0.548908\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ 0 0
$$533$$ −15.0000 −0.649722
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 24.0000 1.03568
$$538$$ 0 0
$$539$$ −10.0000 −0.430730
$$540$$ 0 0
$$541$$ −23.0000 −0.988847 −0.494424 0.869221i $$-0.664621\pi$$
−0.494424 + 0.869221i $$0.664621\pi$$
$$542$$ 0 0
$$543$$ 40.0000 1.71656
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −44.0000 −1.88130 −0.940652 0.339372i $$-0.889785\pi$$
−0.940652 + 0.339372i $$0.889785\pi$$
$$548$$ 0 0
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ −9.00000 −0.383413
$$552$$ 0 0
$$553$$ 33.0000 1.40330
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 42.0000 1.77960 0.889799 0.456354i $$-0.150845\pi$$
0.889799 + 0.456354i $$0.150845\pi$$
$$558$$ 0 0
$$559$$ −35.0000 −1.48034
$$560$$ 0 0
$$561$$ −40.0000 −1.68880
$$562$$ 0 0
$$563$$ 39.0000 1.64365 0.821827 0.569737i $$-0.192955\pi$$
0.821827 + 0.569737i $$0.192955\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −33.0000 −1.38587
$$568$$ 0 0
$$569$$ −4.00000 −0.167689 −0.0838444 0.996479i $$-0.526720\pi$$
−0.0838444 + 0.996479i $$0.526720\pi$$
$$570$$ 0 0
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ 0 0
$$573$$ −2.00000 −0.0835512
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −23.0000 −0.957503 −0.478751 0.877951i $$-0.658910\pi$$
−0.478751 + 0.877951i $$0.658910\pi$$
$$578$$ 0 0
$$579$$ −4.00000 −0.166234
$$580$$ 0 0
$$581$$ −27.0000 −1.12015
$$582$$ 0 0
$$583$$ −60.0000 −2.48495
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −30.0000 −1.23823 −0.619116 0.785299i $$-0.712509\pi$$
−0.619116 + 0.785299i $$0.712509\pi$$
$$588$$ 0 0
$$589$$ −2.00000 −0.0824086
$$590$$ 0 0
$$591$$ 46.0000 1.89219
$$592$$ 0 0
$$593$$ −17.0000 −0.698106 −0.349053 0.937103i $$-0.613497\pi$$
−0.349053 + 0.937103i $$0.613497\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 10.0000 0.409273
$$598$$ 0 0
$$599$$ 14.0000 0.572024 0.286012 0.958226i $$-0.407670\pi$$
0.286012 + 0.958226i $$0.407670\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 0 0
$$603$$ 8.00000 0.325785
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −40.0000 −1.62355 −0.811775 0.583970i $$-0.801498\pi$$
−0.811775 + 0.583970i $$0.801498\pi$$
$$608$$ 0 0
$$609$$ −54.0000 −2.18819
$$610$$ 0 0
$$611$$ −60.0000 −2.42734
$$612$$ 0 0
$$613$$ 18.0000 0.727013 0.363507 0.931592i $$-0.381579\pi$$
0.363507 + 0.931592i $$0.381579\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −32.0000 −1.28827 −0.644136 0.764911i $$-0.722783\pi$$
−0.644136 + 0.764911i $$0.722783\pi$$
$$618$$ 0 0
$$619$$ 24.0000 0.964641 0.482321 0.875995i $$-0.339794\pi$$
0.482321 + 0.875995i $$0.339794\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 0 0
$$623$$ −42.0000 −1.68269
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −10.0000 −0.399362
$$628$$ 0 0
$$629$$ 8.00000 0.318981
$$630$$ 0 0
$$631$$ 25.0000 0.995234 0.497617 0.867397i $$-0.334208\pi$$
0.497617 + 0.867397i $$0.334208\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −10.0000 −0.396214
$$638$$ 0 0
$$639$$ −2.00000 −0.0791188
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ −29.0000 −1.14365 −0.571824 0.820376i $$-0.693764\pi$$
−0.571824 + 0.820376i $$0.693764\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 32.0000 1.25805 0.629025 0.777385i $$-0.283454\pi$$
0.629025 + 0.777385i $$0.283454\pi$$
$$648$$ 0 0
$$649$$ −30.0000 −1.17760
$$650$$ 0 0
$$651$$ −12.0000 −0.470317
$$652$$ 0 0
$$653$$ −29.0000 −1.13486 −0.567429 0.823422i $$-0.692062\pi$$
−0.567429 + 0.823422i $$0.692062\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 1.00000 0.0390137
$$658$$ 0 0
$$659$$ 45.0000 1.75295 0.876476 0.481446i $$-0.159888\pi$$
0.876476 + 0.481446i $$0.159888\pi$$
$$660$$ 0 0
$$661$$ 8.00000 0.311164 0.155582 0.987823i $$-0.450275\pi$$
0.155582 + 0.987823i $$0.450275\pi$$
$$662$$ 0 0
$$663$$ −40.0000 −1.55347
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 9.00000 0.348481
$$668$$ 0 0
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ 50.0000 1.93023
$$672$$ 0 0
$$673$$ 15.0000 0.578208 0.289104 0.957298i $$-0.406643\pi$$
0.289104 + 0.957298i $$0.406643\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 0 0
$$679$$ −48.0000 −1.84207
$$680$$ 0 0
$$681$$ 40.0000 1.53280
$$682$$ 0 0
$$683$$ 34.0000 1.30097 0.650487 0.759517i $$-0.274565\pi$$
0.650487 + 0.759517i $$0.274565\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 60.0000 2.28914
$$688$$ 0 0
$$689$$ −60.0000 −2.28582
$$690$$ 0 0
$$691$$ 16.0000 0.608669 0.304334 0.952565i $$-0.401566\pi$$
0.304334 + 0.952565i $$0.401566\pi$$
$$692$$ 0 0
$$693$$ −15.0000 −0.569803
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ 0 0
$$699$$ 50.0000 1.89117
$$700$$ 0 0
$$701$$ 8.00000 0.302156 0.151078 0.988522i $$-0.451726\pi$$
0.151078 + 0.988522i $$0.451726\pi$$
$$702$$ 0 0
$$703$$ 2.00000 0.0754314
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −42.0000 −1.57957
$$708$$ 0 0
$$709$$ −6.00000 −0.225335 −0.112667 0.993633i $$-0.535939\pi$$
−0.112667 + 0.993633i $$0.535939\pi$$
$$710$$ 0 0
$$711$$ 11.0000 0.412532
$$712$$ 0 0
$$713$$ 2.00000 0.0749006
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 24.0000 0.896296
$$718$$ 0 0
$$719$$ 34.0000 1.26799 0.633993 0.773339i $$-0.281415\pi$$
0.633993 + 0.773339i $$0.281415\pi$$
$$720$$ 0 0
$$721$$ 27.0000 1.00553
$$722$$ 0 0
$$723$$ 8.00000 0.297523
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −16.0000 −0.593407 −0.296704 0.954970i $$-0.595887\pi$$
−0.296704 + 0.954970i $$0.595887\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −28.0000 −1.03562
$$732$$ 0 0
$$733$$ −26.0000 −0.960332 −0.480166 0.877178i $$-0.659424\pi$$
−0.480166 + 0.877178i $$0.659424\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −40.0000 −1.47342
$$738$$ 0 0
$$739$$ 26.0000 0.956425 0.478213 0.878244i $$-0.341285\pi$$
0.478213 + 0.878244i $$0.341285\pi$$
$$740$$ 0 0
$$741$$ −10.0000 −0.367359
$$742$$ 0 0
$$743$$ −21.0000 −0.770415 −0.385208 0.922830i $$-0.625870\pi$$
−0.385208 + 0.922830i $$0.625870\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −9.00000 −0.329293
$$748$$ 0 0
$$749$$ −48.0000 −1.75388
$$750$$ 0 0
$$751$$ −19.0000 −0.693320 −0.346660 0.937991i $$-0.612684\pi$$
−0.346660 + 0.937991i $$0.612684\pi$$
$$752$$ 0 0
$$753$$ 40.0000 1.45768
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 0 0
$$759$$ 10.0000 0.362977
$$760$$ 0 0
$$761$$ 1.00000 0.0362500 0.0181250 0.999836i $$-0.494230\pi$$
0.0181250 + 0.999836i $$0.494230\pi$$
$$762$$ 0 0
$$763$$ −30.0000 −1.08607
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −30.0000 −1.08324
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 4.00000 0.144056
$$772$$ 0 0
$$773$$ −18.0000 −0.647415 −0.323708 0.946157i $$-0.604929\pi$$
−0.323708 + 0.946157i $$0.604929\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 12.0000 0.430498
$$778$$ 0 0
$$779$$ −3.00000 −0.107486
$$780$$ 0 0
$$781$$ 10.0000 0.357828
$$782$$ 0 0
$$783$$ 36.0000 1.28654
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 25.0000 0.891154 0.445577 0.895244i $$-0.352999\pi$$
0.445577 + 0.895244i $$0.352999\pi$$
$$788$$ 0 0
$$789$$ −32.0000 −1.13923
$$790$$ 0 0
$$791$$ 24.0000 0.853342
$$792$$ 0 0
$$793$$ 50.0000 1.77555
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 18.0000 0.637593 0.318796 0.947823i $$-0.396721\pi$$
0.318796 + 0.947823i $$0.396721\pi$$
$$798$$ 0 0
$$799$$ −48.0000 −1.69812
$$800$$ 0 0
$$801$$ −14.0000 −0.494666
$$802$$ 0 0
$$803$$ −5.00000 −0.176446
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −10.0000 −0.352017
$$808$$ 0 0
$$809$$ −23.0000 −0.808637 −0.404318 0.914618i $$-0.632491\pi$$
−0.404318 + 0.914618i $$0.632491\pi$$
$$810$$ 0 0
$$811$$ −40.0000 −1.40459 −0.702295 0.711886i $$-0.747841\pi$$
−0.702295 + 0.711886i $$0.747841\pi$$
$$812$$ 0 0
$$813$$ 24.0000 0.841717
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −7.00000 −0.244899
$$818$$ 0 0
$$819$$ −15.0000 −0.524142
$$820$$ 0 0
$$821$$ 51.0000 1.77991 0.889956 0.456046i $$-0.150735\pi$$
0.889956 + 0.456046i $$0.150735\pi$$
$$822$$ 0 0
$$823$$ −12.0000 −0.418294 −0.209147 0.977884i $$-0.567069\pi$$
−0.209147 + 0.977884i $$0.567069\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −7.00000 −0.243414 −0.121707 0.992566i $$-0.538837\pi$$
−0.121707 + 0.992566i $$0.538837\pi$$
$$828$$ 0 0
$$829$$ −31.0000 −1.07667 −0.538337 0.842729i $$-0.680947\pi$$
−0.538337 + 0.842729i $$0.680947\pi$$
$$830$$ 0 0
$$831$$ −26.0000 −0.901930
$$832$$ 0 0
$$833$$ −8.00000 −0.277184
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 8.00000 0.276520
$$838$$ 0 0
$$839$$ −15.0000 −0.517858 −0.258929 0.965896i $$-0.583369\pi$$
−0.258929 + 0.965896i $$0.583369\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ 16.0000 0.551069
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 42.0000 1.44314
$$848$$ 0 0
$$849$$ −8.00000 −0.274559
$$850$$ 0 0
$$851$$ −2.00000 −0.0685591
$$852$$ 0 0
$$853$$ 19.0000 0.650548 0.325274 0.945620i $$-0.394544\pi$$
0.325274 + 0.945620i $$0.394544\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −38.0000 −1.29806 −0.649028 0.760765i $$-0.724824\pi$$
−0.649028 + 0.760765i $$0.724824\pi$$
$$858$$ 0 0
$$859$$ 2.00000 0.0682391 0.0341196 0.999418i $$-0.489137\pi$$
0.0341196 + 0.999418i $$0.489137\pi$$
$$860$$ 0 0
$$861$$ −18.0000 −0.613438
$$862$$ 0 0
$$863$$ −36.0000 −1.22545 −0.612727 0.790295i $$-0.709928\pi$$
−0.612727 + 0.790295i $$0.709928\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 2.00000 0.0679236
$$868$$ 0 0
$$869$$ −55.0000 −1.86575
$$870$$ 0 0
$$871$$ −40.0000 −1.35535
$$872$$ 0 0
$$873$$ −16.0000 −0.541518
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 0 0
$$879$$ −56.0000 −1.88883
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 0 0
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ 0 0
$$889$$ 30.0000 1.00617
$$890$$ 0 0
$$891$$ 55.0000 1.84257
$$892$$ 0 0
$$893$$ −12.0000 −0.401565
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 10.0000 0.333890
$$898$$ 0 0
$$899$$ 18.0000 0.600334
$$900$$ 0 0
$$901$$ −48.0000 −1.59911
$$902$$ 0 0
$$903$$ −42.0000 −1.39767
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −15.0000 −0.498067 −0.249033 0.968495i $$-0.580113\pi$$
−0.249033 + 0.968495i $$0.580113\pi$$
$$908$$ 0 0
$$909$$ −14.0000 −0.464351
$$910$$ 0 0
$$911$$ −5.00000 −0.165657 −0.0828287 0.996564i $$-0.526395\pi$$
−0.0828287 + 0.996564i $$0.526395\pi$$
$$912$$ 0 0
$$913$$ 45.0000 1.48928
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −18.0000 −0.594412
$$918$$ 0 0
$$919$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$920$$ 0 0
$$921$$ 32.0000 1.05444
$$922$$ 0 0
$$923$$ 10.0000 0.329154
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 9.00000 0.295599
$$928$$ 0 0
$$929$$ −13.0000 −0.426516 −0.213258 0.976996i $$-0.568408\pi$$
−0.213258 + 0.976996i $$0.568408\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 0 0
$$933$$ 36.0000 1.17859
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −44.0000 −1.43742 −0.718709 0.695311i $$-0.755266\pi$$
−0.718709 + 0.695311i $$0.755266\pi$$
$$938$$ 0 0
$$939$$ 20.0000 0.652675
$$940$$ 0 0
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ 0 0
$$943$$ 3.00000 0.0976934
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −18.0000 −0.584921 −0.292461 0.956278i $$-0.594474\pi$$
−0.292461 + 0.956278i $$0.594474\pi$$
$$948$$ 0 0
$$949$$ −5.00000 −0.162307
$$950$$ 0 0
$$951$$ −34.0000 −1.10253
$$952$$ 0 0
$$953$$ −2.00000 −0.0647864 −0.0323932 0.999475i $$-0.510313\pi$$
−0.0323932 + 0.999475i $$0.510313\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 90.0000 2.90929
$$958$$ 0 0
$$959$$ 18.0000 0.581250
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 0 0
$$963$$ −16.0000 −0.515593
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −12.0000 −0.385894 −0.192947 0.981209i $$-0.561805\pi$$
−0.192947 + 0.981209i $$0.561805\pi$$
$$968$$ 0 0
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ 41.0000 1.31575 0.657876 0.753126i $$-0.271455\pi$$
0.657876 + 0.753126i $$0.271455\pi$$
$$972$$ 0 0
$$973$$ −36.0000 −1.15411
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −20.0000 −0.639857 −0.319928 0.947442i $$-0.603659\pi$$
−0.319928 + 0.947442i $$0.603659\pi$$
$$978$$ 0 0
$$979$$ 70.0000 2.23721
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 0 0
$$983$$ 1.00000 0.0318950 0.0159475 0.999873i $$-0.494924\pi$$
0.0159475 + 0.999873i $$0.494924\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −72.0000 −2.29179
$$988$$ 0 0
$$989$$ 7.00000 0.222587
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ 0 0
$$993$$ 52.0000 1.65017
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 5.00000 0.158352 0.0791758 0.996861i $$-0.474771\pi$$
0.0791758 + 0.996861i $$0.474771\pi$$
$$998$$ 0 0
$$999$$ −8.00000 −0.253109
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 9200.2.a.j.1.1 1
4.3 odd 2 4600.2.a.m.1.1 yes 1
5.4 even 2 9200.2.a.bc.1.1 1
20.3 even 4 4600.2.e.d.4049.2 2
20.7 even 4 4600.2.e.d.4049.1 2
20.19 odd 2 4600.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.d.1.1 1 20.19 odd 2
4600.2.a.m.1.1 yes 1 4.3 odd 2
4600.2.e.d.4049.1 2 20.7 even 4
4600.2.e.d.4049.2 2 20.3 even 4
9200.2.a.j.1.1 1 1.1 even 1 trivial
9200.2.a.bc.1.1 1 5.4 even 2