# Properties

 Label 108.5.c.c.53.1 Level 108 Weight 5 Character 108.53 Analytic conductor 11.164 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 108.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1639560131$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 53.1 Root $$-1.00000i$$ of $$x^{2} + 1$$ Character $$\chi$$ $$=$$ 108.53 Dual form 108.5.c.c.53.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-9.00000i q^{5} +5.00000 q^{7} +O(q^{10})$$ $$q-9.00000i q^{5} +5.00000 q^{7} -117.000i q^{11} -34.0000 q^{13} -450.000i q^{17} -64.0000 q^{19} -612.000i q^{23} +544.000 q^{25} -1062.00i q^{29} -697.000 q^{31} -45.0000i q^{35} -748.000 q^{37} -684.000i q^{41} +2618.00 q^{43} +2646.00i q^{47} -2376.00 q^{49} +1071.00i q^{53} -1053.00 q^{55} +5814.00i q^{59} +6404.00 q^{61} +306.000i q^{65} -5218.00 q^{67} +6570.00i q^{71} -4519.00 q^{73} -585.000i q^{77} +7502.00 q^{79} +5481.00i q^{83} -4050.00 q^{85} -8874.00i q^{89} -170.000 q^{91} +576.000i q^{95} +10571.0 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 10q^{7} + O(q^{10})$$ $$2q + 10q^{7} - 68q^{13} - 128q^{19} + 1088q^{25} - 1394q^{31} - 1496q^{37} + 5236q^{43} - 4752q^{49} - 2106q^{55} + 12808q^{61} - 10436q^{67} - 9038q^{73} + 15004q^{79} - 8100q^{85} - 340q^{91} + 21142q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$55$$ $$\chi(n)$$ $$-1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ − 9.00000i − 0.360000i −0.983667 0.180000i $$-0.942390\pi$$
0.983667 0.180000i $$-0.0576098\pi$$
$$6$$ 0 0
$$7$$ 5.00000 0.102041 0.0510204 0.998698i $$-0.483753\pi$$
0.0510204 + 0.998698i $$0.483753\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 117.000i − 0.966942i −0.875360 0.483471i $$-0.839376\pi$$
0.875360 0.483471i $$-0.160624\pi$$
$$12$$ 0 0
$$13$$ −34.0000 −0.201183 −0.100592 0.994928i $$-0.532074\pi$$
−0.100592 + 0.994928i $$0.532074\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 450.000i − 1.55709i −0.627587 0.778547i $$-0.715957\pi$$
0.627587 0.778547i $$-0.284043\pi$$
$$18$$ 0 0
$$19$$ −64.0000 −0.177285 −0.0886427 0.996063i $$-0.528253\pi$$
−0.0886427 + 0.996063i $$0.528253\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 612.000i − 1.15690i −0.815718 0.578450i $$-0.803658\pi$$
0.815718 0.578450i $$-0.196342\pi$$
$$24$$ 0 0
$$25$$ 544.000 0.870400
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 1062.00i − 1.26278i −0.775464 0.631391i $$-0.782484\pi$$
0.775464 0.631391i $$-0.217516\pi$$
$$30$$ 0 0
$$31$$ −697.000 −0.725286 −0.362643 0.931928i $$-0.618126\pi$$
−0.362643 + 0.931928i $$0.618126\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 45.0000i − 0.0367347i
$$36$$ 0 0
$$37$$ −748.000 −0.546384 −0.273192 0.961959i $$-0.588079\pi$$
−0.273192 + 0.961959i $$0.588079\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 684.000i − 0.406901i −0.979085 0.203450i $$-0.934784\pi$$
0.979085 0.203450i $$-0.0652155\pi$$
$$42$$ 0 0
$$43$$ 2618.00 1.41590 0.707950 0.706262i $$-0.249620\pi$$
0.707950 + 0.706262i $$0.249620\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2646.00i 1.19783i 0.800814 + 0.598914i $$0.204401\pi$$
−0.800814 + 0.598914i $$0.795599\pi$$
$$48$$ 0 0
$$49$$ −2376.00 −0.989588
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1071.00i 0.381274i 0.981661 + 0.190637i $$0.0610554\pi$$
−0.981661 + 0.190637i $$0.938945\pi$$
$$54$$ 0 0
$$55$$ −1053.00 −0.348099
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 5814.00i 1.67021i 0.550091 + 0.835105i $$0.314593\pi$$
−0.550091 + 0.835105i $$0.685407\pi$$
$$60$$ 0 0
$$61$$ 6404.00 1.72104 0.860521 0.509414i $$-0.170138\pi$$
0.860521 + 0.509414i $$0.170138\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 306.000i 0.0724260i
$$66$$ 0 0
$$67$$ −5218.00 −1.16240 −0.581198 0.813762i $$-0.697416\pi$$
−0.581198 + 0.813762i $$0.697416\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6570.00i 1.30331i 0.758514 + 0.651656i $$0.225926\pi$$
−0.758514 + 0.651656i $$0.774074\pi$$
$$72$$ 0 0
$$73$$ −4519.00 −0.848002 −0.424001 0.905662i $$-0.639375\pi$$
−0.424001 + 0.905662i $$0.639375\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 585.000i − 0.0986676i
$$78$$ 0 0
$$79$$ 7502.00 1.20205 0.601025 0.799230i $$-0.294759\pi$$
0.601025 + 0.799230i $$0.294759\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 5481.00i 0.795616i 0.917469 + 0.397808i $$0.130229\pi$$
−0.917469 + 0.397808i $$0.869771\pi$$
$$84$$ 0 0
$$85$$ −4050.00 −0.560554
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 8874.00i − 1.12031i −0.828387 0.560157i $$-0.810741\pi$$
0.828387 0.560157i $$-0.189259\pi$$
$$90$$ 0 0
$$91$$ −170.000 −0.0205289
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 576.000i 0.0638227i
$$96$$ 0 0
$$97$$ 10571.0 1.12350 0.561749 0.827307i $$-0.310129\pi$$
0.561749 + 0.827307i $$0.310129\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 13113.0i − 1.28546i −0.766092 0.642731i $$-0.777801\pi$$
0.766092 0.642731i $$-0.222199\pi$$
$$102$$ 0 0
$$103$$ −5830.00 −0.549533 −0.274767 0.961511i $$-0.588601\pi$$
−0.274767 + 0.961511i $$0.588601\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 1089.00i − 0.0951175i −0.998868 0.0475587i $$-0.984856\pi$$
0.998868 0.0475587i $$-0.0151441\pi$$
$$108$$ 0 0
$$109$$ −5020.00 −0.422523 −0.211262 0.977430i $$-0.567757\pi$$
−0.211262 + 0.977430i $$0.567757\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 21384.0i − 1.67468i −0.546682 0.837340i $$-0.684109\pi$$
0.546682 0.837340i $$-0.315891\pi$$
$$114$$ 0 0
$$115$$ −5508.00 −0.416484
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 2250.00i − 0.158887i
$$120$$ 0 0
$$121$$ 952.000 0.0650229
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 10521.0i − 0.673344i
$$126$$ 0 0
$$127$$ 9227.00 0.572075 0.286038 0.958218i $$-0.407662\pi$$
0.286038 + 0.958218i $$0.407662\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 4275.00i − 0.249111i −0.992213 0.124556i $$-0.960249\pi$$
0.992213 0.124556i $$-0.0397505\pi$$
$$132$$ 0 0
$$133$$ −320.000 −0.0180903
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 11322.0i − 0.603229i −0.953430 0.301614i $$-0.902474\pi$$
0.953430 0.301614i $$-0.0975255\pi$$
$$138$$ 0 0
$$139$$ 9812.00 0.507841 0.253921 0.967225i $$-0.418280\pi$$
0.253921 + 0.967225i $$0.418280\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 3978.00i 0.194533i
$$144$$ 0 0
$$145$$ −9558.00 −0.454602
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 16479.0i 0.742264i 0.928580 + 0.371132i $$0.121030\pi$$
−0.928580 + 0.371132i $$0.878970\pi$$
$$150$$ 0 0
$$151$$ −25555.0 −1.12078 −0.560392 0.828227i $$-0.689350\pi$$
−0.560392 + 0.828227i $$0.689350\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 6273.00i 0.261103i
$$156$$ 0 0
$$157$$ 21164.0 0.858615 0.429307 0.903158i $$-0.358758\pi$$
0.429307 + 0.903158i $$0.358758\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ − 3060.00i − 0.118051i
$$162$$ 0 0
$$163$$ 33830.0 1.27329 0.636644 0.771158i $$-0.280322\pi$$
0.636644 + 0.771158i $$0.280322\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 27378.0i 0.981677i 0.871250 + 0.490839i $$0.163310\pi$$
−0.871250 + 0.490839i $$0.836690\pi$$
$$168$$ 0 0
$$169$$ −27405.0 −0.959525
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 46197.0i 1.54355i 0.635894 + 0.771777i $$0.280632\pi$$
−0.635894 + 0.771777i $$0.719368\pi$$
$$174$$ 0 0
$$175$$ 2720.00 0.0888163
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 20781.0i 0.648575i 0.945959 + 0.324288i $$0.105125\pi$$
−0.945959 + 0.324288i $$0.894875\pi$$
$$180$$ 0 0
$$181$$ −19504.0 −0.595342 −0.297671 0.954669i $$-0.596210\pi$$
−0.297671 + 0.954669i $$0.596210\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 6732.00i 0.196698i
$$186$$ 0 0
$$187$$ −52650.0 −1.50562
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 7038.00i − 0.192922i −0.995337 0.0964612i $$-0.969248\pi$$
0.995337 0.0964612i $$-0.0307524\pi$$
$$192$$ 0 0
$$193$$ 51527.0 1.38331 0.691656 0.722227i $$-0.256881\pi$$
0.691656 + 0.722227i $$0.256881\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 60435.0i 1.55724i 0.627495 + 0.778621i $$0.284080\pi$$
−0.627495 + 0.778621i $$0.715920\pi$$
$$198$$ 0 0
$$199$$ 45665.0 1.15313 0.576564 0.817052i $$-0.304393\pi$$
0.576564 + 0.817052i $$0.304393\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 5310.00i − 0.128855i
$$204$$ 0 0
$$205$$ −6156.00 −0.146484
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 7488.00i 0.171425i
$$210$$ 0 0
$$211$$ 13124.0 0.294782 0.147391 0.989078i $$-0.452912\pi$$
0.147391 + 0.989078i $$0.452912\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ − 23562.0i − 0.509724i
$$216$$ 0 0
$$217$$ −3485.00 −0.0740088
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 15300.0i 0.313261i
$$222$$ 0 0
$$223$$ −78850.0 −1.58559 −0.792797 0.609486i $$-0.791376\pi$$
−0.792797 + 0.609486i $$0.791376\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 22302.0i − 0.432805i −0.976304 0.216402i $$-0.930568\pi$$
0.976304 0.216402i $$-0.0694323\pi$$
$$228$$ 0 0
$$229$$ 8870.00 0.169142 0.0845712 0.996417i $$-0.473048\pi$$
0.0845712 + 0.996417i $$0.473048\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 26010.0i − 0.479103i −0.970884 0.239551i $$-0.923000\pi$$
0.970884 0.239551i $$-0.0770003\pi$$
$$234$$ 0 0
$$235$$ 23814.0 0.431218
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 64152.0i − 1.12309i −0.827446 0.561545i $$-0.810207\pi$$
0.827446 0.561545i $$-0.189793\pi$$
$$240$$ 0 0
$$241$$ 33422.0 0.575438 0.287719 0.957715i $$-0.407103\pi$$
0.287719 + 0.957715i $$0.407103\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 21384.0i 0.356252i
$$246$$ 0 0
$$247$$ 2176.00 0.0356669
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 110466.i − 1.75340i −0.481037 0.876700i $$-0.659740\pi$$
0.481037 0.876700i $$-0.340260\pi$$
$$252$$ 0 0
$$253$$ −71604.0 −1.11866
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 15174.0i − 0.229739i −0.993381 0.114869i $$-0.963355\pi$$
0.993381 0.114869i $$-0.0366449\pi$$
$$258$$ 0 0
$$259$$ −3740.00 −0.0557535
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 69948.0i − 1.01126i −0.862750 0.505631i $$-0.831260\pi$$
0.862750 0.505631i $$-0.168740\pi$$
$$264$$ 0 0
$$265$$ 9639.00 0.137259
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 50346.0i 0.695762i 0.937539 + 0.347881i $$0.113099\pi$$
−0.937539 + 0.347881i $$0.886901\pi$$
$$270$$ 0 0
$$271$$ 108323. 1.47497 0.737483 0.675366i $$-0.236014\pi$$
0.737483 + 0.675366i $$0.236014\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 63648.0i − 0.841626i
$$276$$ 0 0
$$277$$ 30716.0 0.400318 0.200159 0.979763i $$-0.435854\pi$$
0.200159 + 0.979763i $$0.435854\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 129258.i − 1.63699i −0.574517 0.818493i $$-0.694810\pi$$
0.574517 0.818493i $$-0.305190\pi$$
$$282$$ 0 0
$$283$$ −63976.0 −0.798811 −0.399406 0.916774i $$-0.630783\pi$$
−0.399406 + 0.916774i $$0.630783\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 3420.00i − 0.0415205i
$$288$$ 0 0
$$289$$ −118979. −1.42454
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 108630.i 1.26536i 0.774413 + 0.632681i $$0.218045\pi$$
−0.774413 + 0.632681i $$0.781955\pi$$
$$294$$ 0 0
$$295$$ 52326.0 0.601275
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 20808.0i 0.232749i
$$300$$ 0 0
$$301$$ 13090.0 0.144480
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ − 57636.0i − 0.619575i
$$306$$ 0 0
$$307$$ 6410.00 0.0680113 0.0340057 0.999422i $$-0.489174\pi$$
0.0340057 + 0.999422i $$0.489174\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ − 70992.0i − 0.733987i −0.930223 0.366994i $$-0.880387\pi$$
0.930223 0.366994i $$-0.119613\pi$$
$$312$$ 0 0
$$313$$ 160961. 1.64298 0.821489 0.570224i $$-0.193143\pi$$
0.821489 + 0.570224i $$0.193143\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 194733.i 1.93785i 0.247347 + 0.968927i $$0.420441\pi$$
−0.247347 + 0.968927i $$0.579559\pi$$
$$318$$ 0 0
$$319$$ −124254. −1.22104
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 28800.0i 0.276050i
$$324$$ 0 0
$$325$$ −18496.0 −0.175110
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 13230.0i 0.122227i
$$330$$ 0 0
$$331$$ −33286.0 −0.303812 −0.151906 0.988395i $$-0.548541\pi$$
−0.151906 + 0.988395i $$0.548541\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 46962.0i 0.418463i
$$336$$ 0 0
$$337$$ −127690. −1.12434 −0.562169 0.827022i $$-0.690033\pi$$
−0.562169 + 0.827022i $$0.690033\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 81549.0i 0.701310i
$$342$$ 0 0
$$343$$ −23885.0 −0.203019
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 138807.i 1.15280i 0.817169 + 0.576398i $$0.195542\pi$$
−0.817169 + 0.576398i $$0.804458\pi$$
$$348$$ 0 0
$$349$$ 203792. 1.67316 0.836578 0.547848i $$-0.184553\pi$$
0.836578 + 0.547848i $$0.184553\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 32328.0i − 0.259436i −0.991551 0.129718i $$-0.958593\pi$$
0.991551 0.129718i $$-0.0414071\pi$$
$$354$$ 0 0
$$355$$ 59130.0 0.469193
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 185922.i 1.44259i 0.692630 + 0.721293i $$0.256452\pi$$
−0.692630 + 0.721293i $$0.743548\pi$$
$$360$$ 0 0
$$361$$ −126225. −0.968570
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 40671.0i 0.305281i
$$366$$ 0 0
$$367$$ 151079. 1.12169 0.560844 0.827922i $$-0.310477\pi$$
0.560844 + 0.827922i $$0.310477\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 5355.00i 0.0389056i
$$372$$ 0 0
$$373$$ −6478.00 −0.0465611 −0.0232806 0.999729i $$-0.507411\pi$$
−0.0232806 + 0.999729i $$0.507411\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 36108.0i 0.254051i
$$378$$ 0 0
$$379$$ 99008.0 0.689274 0.344637 0.938736i $$-0.388002\pi$$
0.344637 + 0.938736i $$0.388002\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 91062.0i 0.620783i 0.950609 + 0.310391i $$0.100460\pi$$
−0.950609 + 0.310391i $$0.899540\pi$$
$$384$$ 0 0
$$385$$ −5265.00 −0.0355203
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 95319.0i 0.629913i 0.949106 + 0.314956i $$0.101990\pi$$
−0.949106 + 0.314956i $$0.898010\pi$$
$$390$$ 0 0
$$391$$ −275400. −1.80140
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 67518.0i − 0.432738i
$$396$$ 0 0
$$397$$ −163438. −1.03698 −0.518492 0.855083i $$-0.673506\pi$$
−0.518492 + 0.855083i $$0.673506\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ − 284616.i − 1.76999i −0.465601 0.884994i $$-0.654162\pi$$
0.465601 0.884994i $$-0.345838\pi$$
$$402$$ 0 0
$$403$$ 23698.0 0.145916
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 87516.0i 0.528322i
$$408$$ 0 0
$$409$$ 107525. 0.642781 0.321390 0.946947i $$-0.395850\pi$$
0.321390 + 0.946947i $$0.395850\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 29070.0i 0.170430i
$$414$$ 0 0
$$415$$ 49329.0 0.286422
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 150282.i 0.856010i 0.903776 + 0.428005i $$0.140783\pi$$
−0.903776 + 0.428005i $$0.859217\pi$$
$$420$$ 0 0
$$421$$ 134420. 0.758402 0.379201 0.925314i $$-0.376199\pi$$
0.379201 + 0.925314i $$0.376199\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ − 244800.i − 1.35529i
$$426$$ 0 0
$$427$$ 32020.0 0.175617
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 237726.i − 1.27974i −0.768483 0.639871i $$-0.778988\pi$$
0.768483 0.639871i $$-0.221012\pi$$
$$432$$ 0 0
$$433$$ 112187. 0.598366 0.299183 0.954196i $$-0.403286\pi$$
0.299183 + 0.954196i $$0.403286\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 39168.0i 0.205101i
$$438$$ 0 0
$$439$$ −204643. −1.06186 −0.530931 0.847415i $$-0.678158\pi$$
−0.530931 + 0.847415i $$0.678158\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 50490.0i − 0.257275i −0.991692 0.128638i $$-0.958940\pi$$
0.991692 0.128638i $$-0.0410604\pi$$
$$444$$ 0 0
$$445$$ −79866.0 −0.403313
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 363528.i − 1.80321i −0.432565 0.901603i $$-0.642391\pi$$
0.432565 0.901603i $$-0.357609\pi$$
$$450$$ 0 0
$$451$$ −80028.0 −0.393449
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 1530.00i 0.00739041i
$$456$$ 0 0
$$457$$ 6677.00 0.0319705 0.0159852 0.999872i $$-0.494912\pi$$
0.0159852 + 0.999872i $$0.494912\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 229347.i − 1.07917i −0.841930 0.539587i $$-0.818581\pi$$
0.841930 0.539587i $$-0.181419\pi$$
$$462$$ 0 0
$$463$$ 238799. 1.11396 0.556981 0.830525i $$-0.311960\pi$$
0.556981 + 0.830525i $$0.311960\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 263133.i 1.20654i 0.797537 + 0.603270i $$0.206136\pi$$
−0.797537 + 0.603270i $$0.793864\pi$$
$$468$$ 0 0
$$469$$ −26090.0 −0.118612
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 306306.i − 1.36909i
$$474$$ 0 0
$$475$$ −34816.0 −0.154309
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 9342.00i − 0.0407163i −0.999793 0.0203582i $$-0.993519\pi$$
0.999793 0.0203582i $$-0.00648066\pi$$
$$480$$ 0 0
$$481$$ 25432.0 0.109923
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 95139.0i − 0.404460i
$$486$$ 0 0
$$487$$ 331262. 1.39673 0.698367 0.715740i $$-0.253910\pi$$
0.698367 + 0.715740i $$0.253910\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ − 396297.i − 1.64383i −0.569608 0.821917i $$-0.692905\pi$$
0.569608 0.821917i $$-0.307095\pi$$
$$492$$ 0 0
$$493$$ −477900. −1.96627
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 32850.0i 0.132991i
$$498$$ 0 0
$$499$$ 45050.0 0.180923 0.0904615 0.995900i $$-0.471166\pi$$
0.0904615 + 0.995900i $$0.471166\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 233172.i 0.921596i 0.887505 + 0.460798i $$0.152437\pi$$
−0.887505 + 0.460798i $$0.847563\pi$$
$$504$$ 0 0
$$505$$ −118017. −0.462766
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 163449.i 0.630880i 0.948946 + 0.315440i $$0.102152\pi$$
−0.948946 + 0.315440i $$0.897848\pi$$
$$510$$ 0 0
$$511$$ −22595.0 −0.0865308
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 52470.0i 0.197832i
$$516$$ 0 0
$$517$$ 309582. 1.15823
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 384606.i − 1.41690i −0.705759 0.708452i $$-0.749394\pi$$
0.705759 0.708452i $$-0.250606\pi$$
$$522$$ 0 0
$$523$$ −214642. −0.784714 −0.392357 0.919813i $$-0.628340\pi$$
−0.392357 + 0.919813i $$0.628340\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 313650.i 1.12934i
$$528$$ 0 0
$$529$$ −94703.0 −0.338417
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 23256.0i 0.0818617i
$$534$$ 0 0
$$535$$ −9801.00 −0.0342423
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 277992.i 0.956874i
$$540$$ 0 0
$$541$$ 545156. 1.86263 0.931314 0.364217i $$-0.118663\pi$$
0.931314 + 0.364217i $$0.118663\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 45180.0i 0.152108i
$$546$$ 0 0
$$547$$ −491422. −1.64240 −0.821202 0.570638i $$-0.806696\pi$$
−0.821202 + 0.570638i $$0.806696\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 67968.0i 0.223873i
$$552$$ 0 0
$$553$$ 37510.0 0.122658
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 445077.i 1.43458i 0.696775 + 0.717290i $$0.254618\pi$$
−0.696775 + 0.717290i $$0.745382\pi$$
$$558$$ 0 0
$$559$$ −89012.0 −0.284856
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 153765.i 0.485111i 0.970138 + 0.242555i $$0.0779856\pi$$
−0.970138 + 0.242555i $$0.922014\pi$$
$$564$$ 0 0
$$565$$ −192456. −0.602885
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 269712.i 0.833059i 0.909122 + 0.416529i $$0.136754\pi$$
−0.909122 + 0.416529i $$0.863246\pi$$
$$570$$ 0 0
$$571$$ 589718. 1.80872 0.904362 0.426767i $$-0.140347\pi$$
0.904362 + 0.426767i $$0.140347\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 332928.i − 1.00697i
$$576$$ 0 0
$$577$$ 184094. 0.552953 0.276476 0.961021i $$-0.410833\pi$$
0.276476 + 0.961021i $$0.410833\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 27405.0i 0.0811853i
$$582$$ 0 0
$$583$$ 125307. 0.368670
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 411543.i 1.19437i 0.802103 + 0.597185i $$0.203714\pi$$
−0.802103 + 0.597185i $$0.796286\pi$$
$$588$$ 0 0
$$589$$ 44608.0 0.128583
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 339966.i 0.966777i 0.875406 + 0.483388i $$0.160594\pi$$
−0.875406 + 0.483388i $$0.839406\pi$$
$$594$$ 0 0
$$595$$ −20250.0 −0.0571994
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 299574.i 0.834931i 0.908693 + 0.417465i $$0.137082\pi$$
−0.908693 + 0.417465i $$0.862918\pi$$
$$600$$ 0 0
$$601$$ −516115. −1.42889 −0.714443 0.699694i $$-0.753320\pi$$
−0.714443 + 0.699694i $$0.753320\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 8568.00i − 0.0234082i
$$606$$ 0 0
$$607$$ −486574. −1.32060 −0.660300 0.751002i $$-0.729571\pi$$
−0.660300 + 0.751002i $$0.729571\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 89964.0i − 0.240983i
$$612$$ 0 0
$$613$$ −189550. −0.504432 −0.252216 0.967671i $$-0.581159\pi$$
−0.252216 + 0.967671i $$0.581159\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 148248.i 0.389420i 0.980861 + 0.194710i $$0.0623766\pi$$
−0.980861 + 0.194710i $$0.937623\pi$$
$$618$$ 0 0
$$619$$ 11390.0 0.0297264 0.0148632 0.999890i $$-0.495269\pi$$
0.0148632 + 0.999890i $$0.495269\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 44370.0i − 0.114318i
$$624$$ 0 0
$$625$$ 245311. 0.627996
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 336600.i 0.850771i
$$630$$ 0 0
$$631$$ −6715.00 −0.0168650 −0.00843252 0.999964i $$-0.502684\pi$$
−0.00843252 + 0.999964i $$0.502684\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 83043.0i − 0.205947i
$$636$$ 0 0
$$637$$ 80784.0 0.199089
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 441108.i − 1.07357i −0.843720 0.536783i $$-0.819639\pi$$
0.843720 0.536783i $$-0.180361\pi$$
$$642$$ 0 0
$$643$$ 547448. 1.32410 0.662050 0.749459i $$-0.269687\pi$$
0.662050 + 0.749459i $$0.269687\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 42228.0i − 0.100877i −0.998727 0.0504385i $$-0.983938\pi$$
0.998727 0.0504385i $$-0.0160619\pi$$
$$648$$ 0 0
$$649$$ 680238. 1.61500
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 258993.i 0.607382i 0.952771 + 0.303691i $$0.0982190\pi$$
−0.952771 + 0.303691i $$0.901781\pi$$
$$654$$ 0 0
$$655$$ −38475.0 −0.0896801
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 313083.i − 0.720923i −0.932774 0.360461i $$-0.882619\pi$$
0.932774 0.360461i $$-0.117381\pi$$
$$660$$ 0 0
$$661$$ −686320. −1.57081 −0.785405 0.618982i $$-0.787545\pi$$
−0.785405 + 0.618982i $$0.787545\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2880.00i 0.00651252i
$$666$$ 0 0
$$667$$ −649944. −1.46091
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ − 749268.i − 1.66415i
$$672$$ 0 0
$$673$$ 214727. 0.474085 0.237043 0.971499i $$-0.423822\pi$$
0.237043 + 0.971499i $$0.423822\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 343638.i 0.749763i 0.927073 + 0.374881i $$0.122317\pi$$
−0.927073 + 0.374881i $$0.877683\pi$$
$$678$$ 0 0
$$679$$ 52855.0 0.114643
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 626238.i − 1.34245i −0.741254 0.671225i $$-0.765769\pi$$
0.741254 0.671225i $$-0.234231\pi$$
$$684$$ 0 0
$$685$$ −101898. −0.217162
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ − 36414.0i − 0.0767061i
$$690$$ 0 0
$$691$$ 684476. 1.43351 0.716757 0.697323i $$-0.245626\pi$$
0.716757 + 0.697323i $$0.245626\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 88308.0i − 0.182823i
$$696$$ 0 0
$$697$$ −307800. −0.633582
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 809523.i − 1.64738i −0.567042 0.823689i $$-0.691912\pi$$
0.567042 0.823689i $$-0.308088\pi$$
$$702$$ 0 0
$$703$$ 47872.0 0.0968659
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 65565.0i − 0.131170i
$$708$$ 0 0
$$709$$ −319648. −0.635886 −0.317943 0.948110i $$-0.602992\pi$$
−0.317943 + 0.948110i $$0.602992\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 426564.i 0.839083i
$$714$$ 0 0
$$715$$ 35802.0 0.0700318
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ − 55836.0i − 0.108008i −0.998541 0.0540041i $$-0.982802\pi$$
0.998541 0.0540041i $$-0.0171984\pi$$
$$720$$ 0 0
$$721$$ −29150.0 −0.0560748
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 577728.i − 1.09913i
$$726$$ 0 0
$$727$$ 863873. 1.63449 0.817243 0.576294i $$-0.195502\pi$$
0.817243 + 0.576294i $$0.195502\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 1.17810e6i − 2.20469i
$$732$$ 0 0
$$733$$ 207608. 0.386399 0.193200 0.981159i $$-0.438114\pi$$
0.193200 + 0.981159i $$0.438114\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 610506.i 1.12397i
$$738$$ 0 0
$$739$$ −803590. −1.47145 −0.735725 0.677280i $$-0.763159\pi$$
−0.735725 + 0.677280i $$0.763159\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 619650.i − 1.12245i −0.827662 0.561227i $$-0.810329\pi$$
0.827662 0.561227i $$-0.189671\pi$$
$$744$$ 0 0
$$745$$ 148311. 0.267215
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 5445.00i − 0.00970587i
$$750$$ 0 0
$$751$$ −1.06598e6 −1.89004 −0.945020 0.327011i $$-0.893959\pi$$
−0.945020 + 0.327011i $$0.893959\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 229995.i 0.403482i
$$756$$ 0 0
$$757$$ 750410. 1.30950 0.654752 0.755844i $$-0.272773\pi$$
0.654752 + 0.755844i $$0.272773\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 413208.i 0.713509i 0.934198 + 0.356754i $$0.116117\pi$$
−0.934198 + 0.356754i $$0.883883\pi$$
$$762$$ 0 0
$$763$$ −25100.0 −0.0431146
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 197676.i − 0.336019i
$$768$$ 0 0
$$769$$ −733381. −1.24016 −0.620079 0.784539i $$-0.712899\pi$$
−0.620079 + 0.784539i $$0.712899\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 231822.i − 0.387968i −0.981005 0.193984i $$-0.937859\pi$$
0.981005 0.193984i $$-0.0621410\pi$$
$$774$$ 0 0
$$775$$ −379168. −0.631289
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 43776.0i 0.0721375i
$$780$$ 0 0
$$781$$ 768690. 1.26023
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ − 190476.i − 0.309101i
$$786$$ 0 0
$$787$$ 502724. 0.811671 0.405836 0.913946i $$-0.366981\pi$$
0.405836 + 0.913946i $$0.366981\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ − 106920.i − 0.170886i
$$792$$ 0 0
$$793$$ −217736. −0.346245
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 596241.i 0.938653i 0.883025 + 0.469327i $$0.155503\pi$$
−0.883025 + 0.469327i $$0.844497\pi$$
$$798$$ 0 0
$$799$$ 1.19070e6 1.86513
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 528723.i 0.819968i
$$804$$ 0 0
$$805$$ −27540.0 −0.0424984
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 376038.i 0.574559i 0.957847 + 0.287280i $$0.0927509\pi$$
−0.957847 + 0.287280i $$0.907249\pi$$
$$810$$ 0 0
$$811$$ −331072. −0.503362 −0.251681 0.967810i $$-0.580983\pi$$
−0.251681 + 0.967810i $$0.580983\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 304470.i − 0.458384i
$$816$$ 0 0
$$817$$ −167552. −0.251018
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 274626.i 0.407432i 0.979030 + 0.203716i $$0.0653019\pi$$
−0.979030 + 0.203716i $$0.934698\pi$$
$$822$$ 0 0
$$823$$ −541195. −0.799013 −0.399507 0.916730i $$-0.630819\pi$$
−0.399507 + 0.916730i $$0.630819\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 358362.i − 0.523975i −0.965071 0.261988i $$-0.915622\pi$$
0.965071 0.261988i $$-0.0843780\pi$$
$$828$$ 0 0
$$829$$ 39626.0 0.0576595 0.0288298 0.999584i $$-0.490822\pi$$
0.0288298 + 0.999584i $$0.490822\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 1.06920e6i 1.54088i
$$834$$ 0 0
$$835$$ 246402. 0.353404
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 1.05543e6i 1.49936i 0.661801 + 0.749679i $$0.269792\pi$$
−0.661801 + 0.749679i $$0.730208\pi$$
$$840$$ 0 0
$$841$$ −420563. −0.594619
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 246645.i 0.345429i
$$846$$ 0 0
$$847$$ 4760.00 0.00663499
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 457776.i 0.632112i
$$852$$ 0 0
$$853$$ −33034.0 −0.0454008 −0.0227004 0.999742i $$-0.507226\pi$$
−0.0227004 + 0.999742i $$0.507226\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 456246.i 0.621209i 0.950539 + 0.310604i $$0.100531\pi$$
−0.950539 + 0.310604i $$0.899469\pi$$
$$858$$ 0 0
$$859$$ 343604. 0.465663 0.232832 0.972517i $$-0.425201\pi$$
0.232832 + 0.972517i $$0.425201\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 74556.0i 0.100106i 0.998747 + 0.0500531i $$0.0159391\pi$$
−0.998747 + 0.0500531i $$0.984061\pi$$
$$864$$ 0 0
$$865$$ 415773. 0.555679
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 877734.i − 1.16231i
$$870$$ 0 0
$$871$$ 177412. 0.233855
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 52605.0i − 0.0687086i
$$876$$ 0 0
$$877$$ −696094. −0.905042 −0.452521 0.891754i $$-0.649475\pi$$
−0.452521 + 0.891754i $$0.649475\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 672426.i 0.866349i 0.901310 + 0.433174i $$0.142607\pi$$
−0.901310 + 0.433174i $$0.857393\pi$$
$$882$$ 0 0
$$883$$ −1.44813e6 −1.85731 −0.928657 0.370938i $$-0.879036\pi$$
−0.928657 + 0.370938i $$0.879036\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 1.50964e6i 1.91879i 0.282071 + 0.959393i $$0.408978\pi$$
−0.282071 + 0.959393i $$0.591022\pi$$
$$888$$ 0 0
$$889$$ 46135.0 0.0583750
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 169344.i − 0.212357i
$$894$$ 0 0
$$895$$ 187029. 0.233487
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 740214.i 0.915879i
$$900$$ 0 0
$$901$$ 481950. 0.593680
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 175536.i 0.214323i
$$906$$ 0 0
$$907$$ 374828. 0.455635 0.227818 0.973704i $$-0.426841\pi$$
0.227818 + 0.973704i $$0.426841\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 1.21149e6i − 1.45977i −0.683572 0.729883i $$-0.739575\pi$$
0.683572 0.729883i $$-0.260425\pi$$
$$912$$ 0 0
$$913$$ 641277. 0.769315
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 21375.0i − 0.0254195i
$$918$$ 0 0
$$919$$ 656777. 0.777655 0.388827 0.921311i $$-0.372880\pi$$
0.388827 + 0.921311i $$0.372880\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ − 223380.i − 0.262205i
$$924$$ 0 0
$$925$$ −406912. −0.475573
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 578988.i 0.670870i 0.942063 + 0.335435i $$0.108883\pi$$
−0.942063 + 0.335435i $$0.891117\pi$$
$$930$$ 0 0
$$931$$ 152064. 0.175439
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 473850.i 0.542023i
$$936$$ 0 0
$$937$$ 195173. 0.222301 0.111150 0.993804i $$-0.464547\pi$$
0.111150 + 0.993804i $$0.464547\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 1.31932e6i − 1.48995i −0.667095 0.744973i $$-0.732462\pi$$
0.667095 0.744973i $$-0.267538\pi$$
$$942$$ 0 0
$$943$$ −418608. −0.470743
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 972621.i 1.08454i 0.840206 + 0.542268i $$0.182434\pi$$
−0.840206 + 0.542268i $$0.817566\pi$$
$$948$$ 0 0
$$949$$ 153646. 0.170604
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 534384.i − 0.588393i −0.955745 0.294197i $$-0.904948\pi$$
0.955745 0.294197i $$-0.0950520\pi$$
$$954$$ 0 0
$$955$$ −63342.0 −0.0694520
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ − 56610.0i − 0.0615540i
$$960$$ 0 0
$$961$$ −437712. −0.473960
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 463743.i − 0.497992i
$$966$$ 0 0
$$967$$ −783619. −0.838015 −0.419008 0.907983i $$-0.637622\pi$$
−0.419008 + 0.907983i $$0.637622\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 276777.i − 0.293556i −0.989169 0.146778i $$-0.953110\pi$$
0.989169 0.146778i $$-0.0468904\pi$$
$$972$$ 0 0
$$973$$ 49060.0 0.0518205
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 648234.i − 0.679114i −0.940585 0.339557i $$-0.889723\pi$$
0.940585 0.339557i $$-0.110277\pi$$
$$978$$ 0 0
$$979$$ −1.03826e6 −1.08328
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 290862.i − 0.301009i −0.988609 0.150505i $$-0.951910\pi$$
0.988609 0.150505i $$-0.0480899\pi$$
$$984$$ 0 0
$$985$$ 543915. 0.560607
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 1.60222e6i − 1.63806i
$$990$$ 0 0
$$991$$ 881243. 0.897322 0.448661 0.893702i $$-0.351901\pi$$
0.448661 + 0.893702i $$0.351901\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 410985.i − 0.415126i
$$996$$ 0 0
$$997$$ −690166. −0.694326 −0.347163 0.937805i $$-0.612855\pi$$
−0.347163 + 0.937805i $$0.612855\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 108.5.c.c.53.1 2
3.2 odd 2 inner 108.5.c.c.53.2 yes 2
4.3 odd 2 432.5.e.d.161.1 2
9.2 odd 6 324.5.g.d.53.1 4
9.4 even 3 324.5.g.d.269.1 4
9.5 odd 6 324.5.g.d.269.2 4
9.7 even 3 324.5.g.d.53.2 4
12.11 even 2 432.5.e.d.161.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
108.5.c.c.53.1 2 1.1 even 1 trivial
108.5.c.c.53.2 yes 2 3.2 odd 2 inner
324.5.g.d.53.1 4 9.2 odd 6
324.5.g.d.53.2 4 9.7 even 3
324.5.g.d.269.1 4 9.4 even 3
324.5.g.d.269.2 4 9.5 odd 6
432.5.e.d.161.1 2 4.3 odd 2
432.5.e.d.161.2 2 12.11 even 2