# Properties

 Label 4410.2.a.bj.1.1 Level 4410 Weight 2 Character 4410.1 Self dual yes Analytic conductor 35.214 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4410.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.2140272914$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4410.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{13} +1.00000 q^{16} -3.00000 q^{19} +1.00000 q^{20} +1.00000 q^{22} -7.00000 q^{23} +1.00000 q^{25} +1.00000 q^{26} +8.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} +11.0000 q^{37} -3.00000 q^{38} +1.00000 q^{40} +11.0000 q^{41} +8.00000 q^{43} +1.00000 q^{44} -7.00000 q^{46} +5.00000 q^{47} +1.00000 q^{50} +1.00000 q^{52} +11.0000 q^{53} +1.00000 q^{55} +8.00000 q^{58} -4.00000 q^{59} -2.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +6.00000 q^{71} -6.00000 q^{73} +11.0000 q^{74} -3.00000 q^{76} -8.00000 q^{79} +1.00000 q^{80} +11.0000 q^{82} -8.00000 q^{83} +8.00000 q^{86} +1.00000 q^{88} +10.0000 q^{89} -7.00000 q^{92} +5.00000 q^{94} -3.00000 q^{95} -16.0000 q^{97} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −3.00000 −0.688247 −0.344124 0.938924i $$-0.611824\pi$$
−0.344124 + 0.938924i $$0.611824\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ −7.00000 −1.45960 −0.729800 0.683660i $$-0.760387\pi$$
−0.729800 + 0.683660i $$0.760387\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 11.0000 1.80839 0.904194 0.427121i $$-0.140472\pi$$
0.904194 + 0.427121i $$0.140472\pi$$
$$38$$ −3.00000 −0.486664
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 11.0000 1.71791 0.858956 0.512050i $$-0.171114\pi$$
0.858956 + 0.512050i $$0.171114\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ −7.00000 −1.03209
$$47$$ 5.00000 0.729325 0.364662 0.931140i $$-0.381184\pi$$
0.364662 + 0.931140i $$0.381184\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ 11.0000 1.51097 0.755483 0.655168i $$-0.227402\pi$$
0.755483 + 0.655168i $$0.227402\pi$$
$$54$$ 0 0
$$55$$ 1.00000 0.134840
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 8.00000 1.05045
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 0 0
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ 11.0000 1.27872
$$75$$ 0 0
$$76$$ −3.00000 −0.344124
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ 11.0000 1.21475
$$83$$ −8.00000 −0.878114 −0.439057 0.898459i $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ 1.00000 0.106600
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −7.00000 −0.729800
$$93$$ 0 0
$$94$$ 5.00000 0.515711
$$95$$ −3.00000 −0.307794
$$96$$ 0 0
$$97$$ −16.0000 −1.62455 −0.812277 0.583272i $$-0.801772\pi$$
−0.812277 + 0.583272i $$0.801772\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 11.0000 1.06841
$$107$$ 10.0000 0.966736 0.483368 0.875417i $$-0.339413\pi$$
0.483368 + 0.875417i $$0.339413\pi$$
$$108$$ 0 0
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ 1.00000 0.0953463
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ −7.00000 −0.652753
$$116$$ 8.00000 0.742781
$$117$$ 0 0
$$118$$ −4.00000 −0.368230
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ −2.00000 −0.179605
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −17.0000 −1.50851 −0.754253 0.656584i $$-0.772001\pi$$
−0.754253 + 0.656584i $$0.772001\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 1.00000 0.0877058
$$131$$ 5.00000 0.436852 0.218426 0.975854i $$-0.429908\pi$$
0.218426 + 0.975854i $$0.429908\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 6.00000 0.503509
$$143$$ 1.00000 0.0836242
$$144$$ 0 0
$$145$$ 8.00000 0.664364
$$146$$ −6.00000 −0.496564
$$147$$ 0 0
$$148$$ 11.0000 0.904194
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 6.00000 0.488273 0.244137 0.969741i $$-0.421495\pi$$
0.244137 + 0.969741i $$0.421495\pi$$
$$152$$ −3.00000 −0.243332
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −2.00000 −0.160644
$$156$$ 0 0
$$157$$ −7.00000 −0.558661 −0.279330 0.960195i $$-0.590112\pi$$
−0.279330 + 0.960195i $$0.590112\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ 11.0000 0.858956
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ 3.00000 0.232147 0.116073 0.993241i $$-0.462969\pi$$
0.116073 + 0.993241i $$0.462969\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 8.00000 0.609994
$$173$$ 15.0000 1.14043 0.570214 0.821496i $$-0.306860\pi$$
0.570214 + 0.821496i $$0.306860\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ 19.0000 1.42013 0.710063 0.704138i $$-0.248666\pi$$
0.710063 + 0.704138i $$0.248666\pi$$
$$180$$ 0 0
$$181$$ −24.0000 −1.78391 −0.891953 0.452128i $$-0.850665\pi$$
−0.891953 + 0.452128i $$0.850665\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −7.00000 −0.516047
$$185$$ 11.0000 0.808736
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 5.00000 0.364662
$$189$$ 0 0
$$190$$ −3.00000 −0.217643
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ 0 0
$$193$$ 22.0000 1.58359 0.791797 0.610784i $$-0.209146\pi$$
0.791797 + 0.610784i $$0.209146\pi$$
$$194$$ −16.0000 −1.14873
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1.00000 0.0712470 0.0356235 0.999365i $$-0.488658\pi$$
0.0356235 + 0.999365i $$0.488658\pi$$
$$198$$ 0 0
$$199$$ −24.0000 −1.70131 −0.850657 0.525720i $$-0.823796\pi$$
−0.850657 + 0.525720i $$0.823796\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 11.0000 0.768273
$$206$$ −16.0000 −1.11477
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ 5.00000 0.344214 0.172107 0.985078i $$-0.444942\pi$$
0.172107 + 0.985078i $$0.444942\pi$$
$$212$$ 11.0000 0.755483
$$213$$ 0 0
$$214$$ 10.0000 0.683586
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 6.00000 0.406371
$$219$$ 0 0
$$220$$ 1.00000 0.0674200
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 12.0000 0.803579 0.401790 0.915732i $$-0.368388\pi$$
0.401790 + 0.915732i $$0.368388\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ −7.00000 −0.461566
$$231$$ 0 0
$$232$$ 8.00000 0.525226
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ 5.00000 0.326164
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 18.0000 1.16432 0.582162 0.813073i $$-0.302207\pi$$
0.582162 + 0.813073i $$0.302207\pi$$
$$240$$ 0 0
$$241$$ 7.00000 0.450910 0.225455 0.974254i $$-0.427613\pi$$
0.225455 + 0.974254i $$0.427613\pi$$
$$242$$ −10.0000 −0.642824
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3.00000 −0.190885
$$248$$ −2.00000 −0.127000
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ −13.0000 −0.820553 −0.410276 0.911961i $$-0.634568\pi$$
−0.410276 + 0.911961i $$0.634568\pi$$
$$252$$ 0 0
$$253$$ −7.00000 −0.440086
$$254$$ −17.0000 −1.06667
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 20.0000 1.24757 0.623783 0.781598i $$-0.285595\pi$$
0.623783 + 0.781598i $$0.285595\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 1.00000 0.0620174
$$261$$ 0 0
$$262$$ 5.00000 0.308901
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ 11.0000 0.675725
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −20.0000 −1.21942 −0.609711 0.792624i $$-0.708714\pi$$
−0.609711 + 0.792624i $$0.708714\pi$$
$$270$$ 0 0
$$271$$ 32.0000 1.94386 0.971931 0.235267i $$-0.0755965\pi$$
0.971931 + 0.235267i $$0.0755965\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ 20.0000 1.19952
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1.00000 0.0596550 0.0298275 0.999555i $$-0.490504\pi$$
0.0298275 + 0.999555i $$0.490504\pi$$
$$282$$ 0 0
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 1.00000 0.0591312
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 8.00000 0.469776
$$291$$ 0 0
$$292$$ −6.00000 −0.351123
$$293$$ −27.0000 −1.57736 −0.788678 0.614806i $$-0.789234\pi$$
−0.788678 + 0.614806i $$0.789234\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 11.0000 0.639362
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −7.00000 −0.404820
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 6.00000 0.345261
$$303$$ 0 0
$$304$$ −3.00000 −0.172062
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −2.00000 −0.113592
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ −12.0000 −0.678280 −0.339140 0.940736i $$-0.610136\pi$$
−0.339140 + 0.940736i $$0.610136\pi$$
$$314$$ −7.00000 −0.395033
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 1.00000 0.0554700
$$326$$ 16.0000 0.886158
$$327$$ 0 0
$$328$$ 11.0000 0.607373
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −13.0000 −0.714545 −0.357272 0.934000i $$-0.616293\pi$$
−0.357272 + 0.934000i $$0.616293\pi$$
$$332$$ −8.00000 −0.439057
$$333$$ 0 0
$$334$$ 3.00000 0.164153
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −12.0000 −0.653682 −0.326841 0.945079i $$-0.605984\pi$$
−0.326841 + 0.945079i $$0.605984\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −2.00000 −0.108306
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ 15.0000 0.806405
$$347$$ −14.0000 −0.751559 −0.375780 0.926709i $$-0.622625\pi$$
−0.375780 + 0.926709i $$0.622625\pi$$
$$348$$ 0 0
$$349$$ 12.0000 0.642345 0.321173 0.947021i $$-0.395923\pi$$
0.321173 + 0.947021i $$0.395923\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1.00000 0.0533002
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 0 0
$$355$$ 6.00000 0.318447
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ 19.0000 1.00418
$$359$$ −4.00000 −0.211112 −0.105556 0.994413i $$-0.533662\pi$$
−0.105556 + 0.994413i $$0.533662\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ −24.0000 −1.26141
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ 25.0000 1.30499 0.652495 0.757793i $$-0.273722\pi$$
0.652495 + 0.757793i $$0.273722\pi$$
$$368$$ −7.00000 −0.364900
$$369$$ 0 0
$$370$$ 11.0000 0.571863
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 5.00000 0.257855
$$377$$ 8.00000 0.412021
$$378$$ 0 0
$$379$$ −19.0000 −0.975964 −0.487982 0.872854i $$-0.662267\pi$$
−0.487982 + 0.872854i $$0.662267\pi$$
$$380$$ −3.00000 −0.153897
$$381$$ 0 0
$$382$$ 6.00000 0.306987
$$383$$ −35.0000 −1.78842 −0.894208 0.447651i $$-0.852261\pi$$
−0.894208 + 0.447651i $$0.852261\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 22.0000 1.11977
$$387$$ 0 0
$$388$$ −16.0000 −0.812277
$$389$$ −10.0000 −0.507020 −0.253510 0.967333i $$-0.581585\pi$$
−0.253510 + 0.967333i $$0.581585\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 1.00000 0.0503793
$$395$$ −8.00000 −0.402524
$$396$$ 0 0
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ −24.0000 −1.20301
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 5.00000 0.249688 0.124844 0.992176i $$-0.460157\pi$$
0.124844 + 0.992176i $$0.460157\pi$$
$$402$$ 0 0
$$403$$ −2.00000 −0.0996271
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 11.0000 0.545250
$$408$$ 0 0
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ 11.0000 0.543251
$$411$$ 0 0
$$412$$ −16.0000 −0.788263
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −8.00000 −0.392705
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ −3.00000 −0.146735
$$419$$ 5.00000 0.244266 0.122133 0.992514i $$-0.461027\pi$$
0.122133 + 0.992514i $$0.461027\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 5.00000 0.243396
$$423$$ 0 0
$$424$$ 11.0000 0.534207
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 10.0000 0.483368
$$429$$ 0 0
$$430$$ 8.00000 0.385794
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −32.0000 −1.53782 −0.768911 0.639356i $$-0.779201\pi$$
−0.768911 + 0.639356i $$0.779201\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 6.00000 0.287348
$$437$$ 21.0000 1.00457
$$438$$ 0 0
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ 1.00000 0.0476731
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −16.0000 −0.760183 −0.380091 0.924949i $$-0.624107\pi$$
−0.380091 + 0.924949i $$0.624107\pi$$
$$444$$ 0 0
$$445$$ 10.0000 0.474045
$$446$$ 12.0000 0.568216
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −11.0000 −0.519122 −0.259561 0.965727i $$-0.583578\pi$$
−0.259561 + 0.965727i $$0.583578\pi$$
$$450$$ 0 0
$$451$$ 11.0000 0.517970
$$452$$ −6.00000 −0.282216
$$453$$ 0 0
$$454$$ −8.00000 −0.375459
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −18.0000 −0.842004 −0.421002 0.907060i $$-0.638322\pi$$
−0.421002 + 0.907060i $$0.638322\pi$$
$$458$$ 14.0000 0.654177
$$459$$ 0 0
$$460$$ −7.00000 −0.326377
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 0 0
$$463$$ −13.0000 −0.604161 −0.302081 0.953282i $$-0.597681\pi$$
−0.302081 + 0.953282i $$0.597681\pi$$
$$464$$ 8.00000 0.371391
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 5.00000 0.230633
$$471$$ 0 0
$$472$$ −4.00000 −0.184115
$$473$$ 8.00000 0.367840
$$474$$ 0 0
$$475$$ −3.00000 −0.137649
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 18.0000 0.823301
$$479$$ −22.0000 −1.00521 −0.502603 0.864517i $$-0.667624\pi$$
−0.502603 + 0.864517i $$0.667624\pi$$
$$480$$ 0 0
$$481$$ 11.0000 0.501557
$$482$$ 7.00000 0.318841
$$483$$ 0 0
$$484$$ −10.0000 −0.454545
$$485$$ −16.0000 −0.726523
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −3.00000 −0.134976
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ −13.0000 −0.580218
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −7.00000 −0.311188
$$507$$ 0 0
$$508$$ −17.0000 −0.754253
$$509$$ 34.0000 1.50702 0.753512 0.657434i $$-0.228358\pi$$
0.753512 + 0.657434i $$0.228358\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 20.0000 0.882162
$$515$$ −16.0000 −0.705044
$$516$$ 0 0
$$517$$ 5.00000 0.219900
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 1.00000 0.0438529
$$521$$ 33.0000 1.44576 0.722878 0.690976i $$-0.242819\pi$$
0.722878 + 0.690976i $$0.242819\pi$$
$$522$$ 0 0
$$523$$ 2.00000 0.0874539 0.0437269 0.999044i $$-0.486077\pi$$
0.0437269 + 0.999044i $$0.486077\pi$$
$$524$$ 5.00000 0.218426
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 26.0000 1.13043
$$530$$ 11.0000 0.477809
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 11.0000 0.476463
$$534$$ 0 0
$$535$$ 10.0000 0.432338
$$536$$ 0 0
$$537$$ 0 0
$$538$$ −20.0000 −0.862261
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 32.0000 1.37452
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 6.00000 0.257012
$$546$$ 0 0
$$547$$ 16.0000 0.684111 0.342055 0.939680i $$-0.388877\pi$$
0.342055 + 0.939680i $$0.388877\pi$$
$$548$$ 18.0000 0.768922
$$549$$ 0 0
$$550$$ 1.00000 0.0426401
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ 20.0000 0.848189
$$557$$ 33.0000 1.39825 0.699127 0.714997i $$-0.253572\pi$$
0.699127 + 0.714997i $$0.253572\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 1.00000 0.0421825
$$563$$ 38.0000 1.60151 0.800755 0.598993i $$-0.204432\pi$$
0.800755 + 0.598993i $$0.204432\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ −14.0000 −0.588464
$$567$$ 0 0
$$568$$ 6.00000 0.251754
$$569$$ 9.00000 0.377300 0.188650 0.982044i $$-0.439589\pi$$
0.188650 + 0.982044i $$0.439589\pi$$
$$570$$ 0 0
$$571$$ −32.0000 −1.33916 −0.669579 0.742741i $$-0.733526\pi$$
−0.669579 + 0.742741i $$0.733526\pi$$
$$572$$ 1.00000 0.0418121
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −7.00000 −0.291920
$$576$$ 0 0
$$577$$ −14.0000 −0.582828 −0.291414 0.956597i $$-0.594126\pi$$
−0.291414 + 0.956597i $$0.594126\pi$$
$$578$$ −17.0000 −0.707107
$$579$$ 0 0
$$580$$ 8.00000 0.332182
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 11.0000 0.455573
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ −27.0000 −1.11536
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ 0 0
$$589$$ 6.00000 0.247226
$$590$$ −4.00000 −0.164677
$$591$$ 0 0
$$592$$ 11.0000 0.452097
$$593$$ −16.0000 −0.657041 −0.328521 0.944497i $$-0.606550\pi$$
−0.328521 + 0.944497i $$0.606550\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ −7.00000 −0.286251
$$599$$ −2.00000 −0.0817178 −0.0408589 0.999165i $$-0.513009\pi$$
−0.0408589 + 0.999165i $$0.513009\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 6.00000 0.244137
$$605$$ −10.0000 −0.406558
$$606$$ 0 0
$$607$$ −37.0000 −1.50178 −0.750892 0.660425i $$-0.770376\pi$$
−0.750892 + 0.660425i $$0.770376\pi$$
$$608$$ −3.00000 −0.121666
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 5.00000 0.202278
$$612$$ 0 0
$$613$$ −41.0000 −1.65597 −0.827987 0.560747i $$-0.810514\pi$$
−0.827987 + 0.560747i $$0.810514\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −28.0000 −1.12724 −0.563619 0.826035i $$-0.690591\pi$$
−0.563619 + 0.826035i $$0.690591\pi$$
$$618$$ 0 0
$$619$$ 29.0000 1.16561 0.582804 0.812613i $$-0.301955\pi$$
0.582804 + 0.812613i $$0.301955\pi$$
$$620$$ −2.00000 −0.0803219
$$621$$ 0 0
$$622$$ −12.0000 −0.481156
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −12.0000 −0.479616
$$627$$ 0 0
$$628$$ −7.00000 −0.279330
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −26.0000 −1.03504 −0.517522 0.855670i $$-0.673145\pi$$
−0.517522 + 0.855670i $$0.673145\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 0 0
$$634$$ 18.0000 0.714871
$$635$$ −17.0000 −0.674624
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 8.00000 0.316723
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ 3.00000 0.118493 0.0592464 0.998243i $$-0.481130\pi$$
0.0592464 + 0.998243i $$0.481130\pi$$
$$642$$ 0 0
$$643$$ −10.0000 −0.394362 −0.197181 0.980367i $$-0.563179\pi$$
−0.197181 + 0.980367i $$0.563179\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 17.0000 0.668339 0.334169 0.942513i $$-0.391544\pi$$
0.334169 + 0.942513i $$0.391544\pi$$
$$648$$ 0 0
$$649$$ −4.00000 −0.157014
$$650$$ 1.00000 0.0392232
$$651$$ 0 0
$$652$$ 16.0000 0.626608
$$653$$ −7.00000 −0.273931 −0.136966 0.990576i $$-0.543735\pi$$
−0.136966 + 0.990576i $$0.543735\pi$$
$$654$$ 0 0
$$655$$ 5.00000 0.195366
$$656$$ 11.0000 0.429478
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ −24.0000 −0.933492 −0.466746 0.884391i $$-0.654574\pi$$
−0.466746 + 0.884391i $$0.654574\pi$$
$$662$$ −13.0000 −0.505259
$$663$$ 0 0
$$664$$ −8.00000 −0.310460
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −56.0000 −2.16833
$$668$$ 3.00000 0.116073
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 28.0000 1.07932 0.539660 0.841883i $$-0.318553\pi$$
0.539660 + 0.841883i $$0.318553\pi$$
$$674$$ −12.0000 −0.462223
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ 13.0000 0.499631 0.249815 0.968294i $$-0.419630\pi$$
0.249815 + 0.968294i $$0.419630\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −2.00000 −0.0765840
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 0 0
$$685$$ 18.0000 0.687745
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 8.00000 0.304997
$$689$$ 11.0000 0.419067
$$690$$ 0 0
$$691$$ −12.0000 −0.456502 −0.228251 0.973602i $$-0.573301\pi$$
−0.228251 + 0.973602i $$0.573301\pi$$
$$692$$ 15.0000 0.570214
$$693$$ 0 0
$$694$$ −14.0000 −0.531433
$$695$$ 20.0000 0.758643
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 12.0000 0.454207
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ −33.0000 −1.24462
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ −24.0000 −0.903252
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 44.0000 1.65245 0.826227 0.563337i $$-0.190483\pi$$
0.826227 + 0.563337i $$0.190483\pi$$
$$710$$ 6.00000 0.225176
$$711$$ 0 0
$$712$$ 10.0000 0.374766
$$713$$ 14.0000 0.524304
$$714$$ 0 0
$$715$$ 1.00000 0.0373979
$$716$$ 19.0000 0.710063
$$717$$ 0 0
$$718$$ −4.00000 −0.149279
$$719$$ −2.00000 −0.0745874 −0.0372937 0.999304i $$-0.511874\pi$$
−0.0372937 + 0.999304i $$0.511874\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −10.0000 −0.372161
$$723$$ 0 0
$$724$$ −24.0000 −0.891953
$$725$$ 8.00000 0.297113
$$726$$ 0 0
$$727$$ 11.0000 0.407967 0.203984 0.978974i $$-0.434611\pi$$
0.203984 + 0.978974i $$0.434611\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −6.00000 −0.222070
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 11.0000 0.406294 0.203147 0.979148i $$-0.434883\pi$$
0.203147 + 0.979148i $$0.434883\pi$$
$$734$$ 25.0000 0.922767
$$735$$ 0 0
$$736$$ −7.00000 −0.258023
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −19.0000 −0.698926 −0.349463 0.936950i $$-0.613636\pi$$
−0.349463 + 0.936950i $$0.613636\pi$$
$$740$$ 11.0000 0.404368
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −49.0000 −1.79764 −0.898818 0.438322i $$-0.855573\pi$$
−0.898818 + 0.438322i $$0.855573\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 6.00000 0.219676
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 26.0000 0.948753 0.474377 0.880322i $$-0.342673\pi$$
0.474377 + 0.880322i $$0.342673\pi$$
$$752$$ 5.00000 0.182331
$$753$$ 0 0
$$754$$ 8.00000 0.291343
$$755$$ 6.00000 0.218362
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ −19.0000 −0.690111
$$759$$ 0 0
$$760$$ −3.00000 −0.108821
$$761$$ −27.0000 −0.978749 −0.489375 0.872074i $$-0.662775\pi$$
−0.489375 + 0.872074i $$0.662775\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 6.00000 0.217072
$$765$$ 0 0
$$766$$ −35.0000 −1.26460
$$767$$ −4.00000 −0.144432
$$768$$ 0 0
$$769$$ −5.00000 −0.180305 −0.0901523 0.995928i $$-0.528735\pi$$
−0.0901523 + 0.995928i $$0.528735\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 22.0000 0.791797
$$773$$ −33.0000 −1.18693 −0.593464 0.804861i $$-0.702240\pi$$
−0.593464 + 0.804861i $$0.702240\pi$$
$$774$$ 0 0
$$775$$ −2.00000 −0.0718421
$$776$$ −16.0000 −0.574367
$$777$$ 0 0
$$778$$ −10.0000 −0.358517
$$779$$ −33.0000 −1.18235
$$780$$ 0 0
$$781$$ 6.00000 0.214697
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −7.00000 −0.249841
$$786$$ 0 0
$$787$$ 22.0000 0.784215 0.392108 0.919919i $$-0.371746\pi$$
0.392108 + 0.919919i $$0.371746\pi$$
$$788$$ 1.00000 0.0356235
$$789$$ 0 0
$$790$$ −8.00000 −0.284627
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ −24.0000 −0.850657
$$797$$ 18.0000 0.637593 0.318796 0.947823i $$-0.396721\pi$$
0.318796 + 0.947823i $$0.396721\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ 5.00000 0.176556
$$803$$ −6.00000 −0.211735
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −2.00000 −0.0704470
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 39.0000 1.37117 0.685583 0.727994i $$-0.259547\pi$$
0.685583 + 0.727994i $$0.259547\pi$$
$$810$$ 0 0
$$811$$ 9.00000 0.316033 0.158016 0.987436i $$-0.449490\pi$$
0.158016 + 0.987436i $$0.449490\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 11.0000 0.385550
$$815$$ 16.0000 0.560456
$$816$$ 0 0
$$817$$ −24.0000 −0.839654
$$818$$ 2.00000 0.0699284
$$819$$ 0 0
$$820$$ 11.0000 0.384137
$$821$$ −42.0000 −1.46581 −0.732905 0.680331i $$-0.761836\pi$$
−0.732905 + 0.680331i $$0.761836\pi$$
$$822$$ 0 0
$$823$$ −24.0000 −0.836587 −0.418294 0.908312i $$-0.637372\pi$$
−0.418294 + 0.908312i $$0.637372\pi$$
$$824$$ −16.0000 −0.557386
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 10.0000 0.347734 0.173867 0.984769i $$-0.444374\pi$$
0.173867 + 0.984769i $$0.444374\pi$$
$$828$$ 0 0
$$829$$ −8.00000 −0.277851 −0.138926 0.990303i $$-0.544365\pi$$
−0.138926 + 0.990303i $$0.544365\pi$$
$$830$$ −8.00000 −0.277684
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 3.00000 0.103819
$$836$$ −3.00000 −0.103757
$$837$$ 0 0
$$838$$ 5.00000 0.172722
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ 22.0000 0.758170
$$843$$ 0 0
$$844$$ 5.00000 0.172107
$$845$$ −12.0000 −0.412813
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 11.0000 0.377742
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −77.0000 −2.63953
$$852$$ 0 0
$$853$$ −41.0000 −1.40381 −0.701907 0.712269i $$-0.747668\pi$$
−0.701907 + 0.712269i $$0.747668\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 10.0000 0.341793
$$857$$ 14.0000 0.478231 0.239115 0.970991i $$-0.423143\pi$$
0.239115 + 0.970991i $$0.423143\pi$$
$$858$$ 0 0
$$859$$ 12.0000 0.409435 0.204717 0.978821i $$-0.434372\pi$$
0.204717 + 0.978821i $$0.434372\pi$$
$$860$$ 8.00000 0.272798
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 53.0000 1.80414 0.902070 0.431589i $$-0.142047\pi$$
0.902070 + 0.431589i $$0.142047\pi$$
$$864$$ 0 0
$$865$$ 15.0000 0.510015
$$866$$ −32.0000 −1.08740
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −8.00000 −0.271381
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 6.00000 0.203186
$$873$$ 0 0
$$874$$ 21.0000 0.710336
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 17.0000 0.574049 0.287025 0.957923i $$-0.407334\pi$$
0.287025 + 0.957923i $$0.407334\pi$$
$$878$$ 32.0000 1.07995
$$879$$ 0 0
$$880$$ 1.00000 0.0337100
$$881$$ 53.0000 1.78562 0.892808 0.450438i $$-0.148732\pi$$
0.892808 + 0.450438i $$0.148732\pi$$
$$882$$ 0 0
$$883$$ −42.0000 −1.41341 −0.706706 0.707507i $$-0.749820\pi$$
−0.706706 + 0.707507i $$0.749820\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −16.0000 −0.537531
$$887$$ −44.0000 −1.47738 −0.738688 0.674048i $$-0.764554\pi$$
−0.738688 + 0.674048i $$0.764554\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 10.0000 0.335201
$$891$$ 0 0
$$892$$ 12.0000 0.401790
$$893$$ −15.0000 −0.501956
$$894$$ 0 0
$$895$$ 19.0000 0.635100
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −11.0000 −0.367075
$$899$$ −16.0000 −0.533630
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 11.0000 0.366260
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ −24.0000 −0.797787
$$906$$ 0 0
$$907$$ −10.0000 −0.332045 −0.166022 0.986122i $$-0.553092\pi$$
−0.166022 + 0.986122i $$0.553092\pi$$
$$908$$ −8.00000 −0.265489
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 2.00000 0.0662630 0.0331315 0.999451i $$-0.489452\pi$$
0.0331315 + 0.999451i $$0.489452\pi$$
$$912$$ 0 0
$$913$$ −8.00000 −0.264761
$$914$$ −18.0000 −0.595387
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −36.0000 −1.18753 −0.593765 0.804638i $$-0.702359\pi$$
−0.593765 + 0.804638i $$0.702359\pi$$
$$920$$ −7.00000 −0.230783
$$921$$ 0 0
$$922$$ 12.0000 0.395199
$$923$$ 6.00000 0.197492
$$924$$ 0 0
$$925$$ 11.0000 0.361678
$$926$$ −13.0000 −0.427207
$$927$$ 0 0
$$928$$ 8.00000 0.262613
$$929$$ 21.0000 0.688988 0.344494 0.938789i $$-0.388051\pi$$
0.344494 + 0.938789i $$0.388051\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −18.0000 −0.589610
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 12.0000 0.392023 0.196011 0.980602i $$-0.437201\pi$$
0.196011 + 0.980602i $$0.437201\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 5.00000 0.163082
$$941$$ −54.0000 −1.76035 −0.880175 0.474650i $$-0.842575\pi$$
−0.880175 + 0.474650i $$0.842575\pi$$
$$942$$ 0 0
$$943$$ −77.0000 −2.50746
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ −22.0000 −0.714904 −0.357452 0.933932i $$-0.616354\pi$$
−0.357452 + 0.933932i $$0.616354\pi$$
$$948$$ 0 0
$$949$$ −6.00000 −0.194768
$$950$$ −3.00000 −0.0973329
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 20.0000 0.647864 0.323932 0.946080i $$-0.394995\pi$$
0.323932 + 0.946080i $$0.394995\pi$$
$$954$$ 0 0
$$955$$ 6.00000 0.194155
$$956$$ 18.0000 0.582162
$$957$$ 0 0
$$958$$ −22.0000 −0.710788
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 11.0000 0.354654
$$963$$ 0 0
$$964$$ 7.00000 0.225455
$$965$$ 22.0000 0.708205
$$966$$ 0 0
$$967$$ −20.0000 −0.643157 −0.321578 0.946883i $$-0.604213\pi$$
−0.321578 + 0.946883i $$0.604213\pi$$
$$968$$ −10.0000 −0.321412
$$969$$ 0 0
$$970$$ −16.0000 −0.513729
$$971$$ 45.0000 1.44412 0.722059 0.691831i $$-0.243196\pi$$
0.722059 + 0.691831i $$0.243196\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −54.0000 −1.72761 −0.863807 0.503824i $$-0.831926\pi$$
−0.863807 + 0.503824i $$0.831926\pi$$
$$978$$ 0 0
$$979$$ 10.0000 0.319601
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −20.0000 −0.638226
$$983$$ −49.0000 −1.56286 −0.781429 0.623995i $$-0.785509\pi$$
−0.781429 + 0.623995i $$0.785509\pi$$
$$984$$ 0 0
$$985$$ 1.00000 0.0318626
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −3.00000 −0.0954427
$$989$$ −56.0000 −1.78070
$$990$$ 0 0
$$991$$ 22.0000 0.698853 0.349427 0.936964i $$-0.386376\pi$$
0.349427 + 0.936964i $$0.386376\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −24.0000 −0.760851
$$996$$ 0 0
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ −32.0000 −1.01294
$$999$$ 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 4410.2.a.bj.1.1 1
3.2 odd 2 1470.2.a.a.1.1 1
7.2 even 3 630.2.k.c.361.1 2
7.4 even 3 630.2.k.c.541.1 2
7.6 odd 2 4410.2.a.ba.1.1 1
15.14 odd 2 7350.2.a.cp.1.1 1
21.2 odd 6 210.2.i.d.151.1 yes 2
21.5 even 6 1470.2.i.m.361.1 2
21.11 odd 6 210.2.i.d.121.1 2
21.17 even 6 1470.2.i.m.961.1 2
21.20 even 2 1470.2.a.h.1.1 1
84.11 even 6 1680.2.bg.g.961.1 2
84.23 even 6 1680.2.bg.g.1201.1 2
105.2 even 12 1050.2.o.i.949.1 4
105.23 even 12 1050.2.o.i.949.2 4
105.32 even 12 1050.2.o.i.499.2 4
105.44 odd 6 1050.2.i.b.151.1 2
105.53 even 12 1050.2.o.i.499.1 4
105.74 odd 6 1050.2.i.b.751.1 2
105.104 even 2 7350.2.a.bu.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.i.d.121.1 2 21.11 odd 6
210.2.i.d.151.1 yes 2 21.2 odd 6
630.2.k.c.361.1 2 7.2 even 3
630.2.k.c.541.1 2 7.4 even 3
1050.2.i.b.151.1 2 105.44 odd 6
1050.2.i.b.751.1 2 105.74 odd 6
1050.2.o.i.499.1 4 105.53 even 12
1050.2.o.i.499.2 4 105.32 even 12
1050.2.o.i.949.1 4 105.2 even 12
1050.2.o.i.949.2 4 105.23 even 12
1470.2.a.a.1.1 1 3.2 odd 2
1470.2.a.h.1.1 1 21.20 even 2
1470.2.i.m.361.1 2 21.5 even 6
1470.2.i.m.961.1 2 21.17 even 6
1680.2.bg.g.961.1 2 84.11 even 6
1680.2.bg.g.1201.1 2 84.23 even 6
4410.2.a.ba.1.1 1 7.6 odd 2
4410.2.a.bj.1.1 1 1.1 even 1 trivial
7350.2.a.bu.1.1 1 105.104 even 2
7350.2.a.cp.1.1 1 15.14 odd 2