# Properties

 Label 9200.2.a.br.1.2 Level $9200$ Weight $2$ Character 9200.1 Self dual yes Analytic conductor $73.462$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$73.4623698596$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 184) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 9200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.56155 q^{3} -0.561553 q^{9} +O(q^{10})$$ $$q+1.56155 q^{3} -0.561553 q^{9} +3.12311 q^{11} -0.438447 q^{13} -5.12311 q^{17} +3.12311 q^{19} -1.00000 q^{23} -5.56155 q^{27} +3.56155 q^{29} +2.43845 q^{31} +4.87689 q^{33} -8.24621 q^{37} -0.684658 q^{39} -9.80776 q^{41} -8.00000 q^{43} -0.684658 q^{47} -7.00000 q^{49} -8.00000 q^{51} -2.00000 q^{53} +4.87689 q^{57} -10.2462 q^{59} -4.24621 q^{61} +3.12311 q^{67} -1.56155 q^{69} -13.5616 q^{71} +14.6847 q^{73} -3.12311 q^{79} -7.00000 q^{81} +14.2462 q^{83} +5.56155 q^{87} +11.3693 q^{89} +3.80776 q^{93} -11.3693 q^{97} -1.75379 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} + 3q^{9} + O(q^{10})$$ $$2q - q^{3} + 3q^{9} - 2q^{11} - 5q^{13} - 2q^{17} - 2q^{19} - 2q^{23} - 7q^{27} + 3q^{29} + 9q^{31} + 18q^{33} + 11q^{39} + q^{41} - 16q^{43} + 11q^{47} - 14q^{49} - 16q^{51} - 4q^{53} + 18q^{57} - 4q^{59} + 8q^{61} - 2q^{67} + q^{69} - 23q^{71} + 17q^{73} + 2q^{79} - 14q^{81} + 12q^{83} + 7q^{87} - 2q^{89} - 13q^{93} + 2q^{97} - 20q^{99} + O(q^{100})$$

## 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
$$3$$ 1.56155 0.901563 0.450781 0.892634i $$-0.351145\pi$$
0.450781 + 0.892634i $$0.351145\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ −0.561553 −0.187184
$$10$$ 0 0
$$11$$ 3.12311 0.941652 0.470826 0.882226i $$-0.343956\pi$$
0.470826 + 0.882226i $$0.343956\pi$$
$$12$$ 0 0
$$13$$ −0.438447 −0.121603 −0.0608017 0.998150i $$-0.519366\pi$$
−0.0608017 + 0.998150i $$0.519366\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −5.12311 −1.24254 −0.621268 0.783598i $$-0.713382\pi$$
−0.621268 + 0.783598i $$0.713382\pi$$
$$18$$ 0 0
$$19$$ 3.12311 0.716490 0.358245 0.933628i $$-0.383375\pi$$
0.358245 + 0.933628i $$0.383375\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −5.56155 −1.07032
$$28$$ 0 0
$$29$$ 3.56155 0.661364 0.330682 0.943742i $$-0.392721\pi$$
0.330682 + 0.943742i $$0.392721\pi$$
$$30$$ 0 0
$$31$$ 2.43845 0.437958 0.218979 0.975730i $$-0.429727\pi$$
0.218979 + 0.975730i $$0.429727\pi$$
$$32$$ 0 0
$$33$$ 4.87689 0.848958
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.24621 −1.35567 −0.677834 0.735215i $$-0.737081\pi$$
−0.677834 + 0.735215i $$0.737081\pi$$
$$38$$ 0 0
$$39$$ −0.684658 −0.109633
$$40$$ 0 0
$$41$$ −9.80776 −1.53172 −0.765858 0.643010i $$-0.777685\pi$$
−0.765858 + 0.643010i $$0.777685\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −0.684658 −0.0998677 −0.0499338 0.998753i $$-0.515901\pi$$
−0.0499338 + 0.998753i $$0.515901\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ −8.00000 −1.12022
$$52$$ 0 0
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.87689 0.645960
$$58$$ 0 0
$$59$$ −10.2462 −1.33394 −0.666972 0.745083i $$-0.732410\pi$$
−0.666972 + 0.745083i $$0.732410\pi$$
$$60$$ 0 0
$$61$$ −4.24621 −0.543672 −0.271836 0.962344i $$-0.587631\pi$$
−0.271836 + 0.962344i $$0.587631\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 3.12311 0.381548 0.190774 0.981634i $$-0.438900\pi$$
0.190774 + 0.981634i $$0.438900\pi$$
$$68$$ 0 0
$$69$$ −1.56155 −0.187989
$$70$$ 0 0
$$71$$ −13.5616 −1.60946 −0.804730 0.593641i $$-0.797690\pi$$
−0.804730 + 0.593641i $$0.797690\pi$$
$$72$$ 0 0
$$73$$ 14.6847 1.71871 0.859355 0.511380i $$-0.170866\pi$$
0.859355 + 0.511380i $$0.170866\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −3.12311 −0.351377 −0.175688 0.984446i $$-0.556215\pi$$
−0.175688 + 0.984446i $$0.556215\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ 14.2462 1.56372 0.781862 0.623451i $$-0.214270\pi$$
0.781862 + 0.623451i $$0.214270\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 5.56155 0.596261
$$88$$ 0 0
$$89$$ 11.3693 1.20515 0.602573 0.798064i $$-0.294142\pi$$
0.602573 + 0.798064i $$0.294142\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 3.80776 0.394847
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −11.3693 −1.15438 −0.577190 0.816610i $$-0.695851\pi$$
−0.577190 + 0.816610i $$0.695851\pi$$
$$98$$ 0 0
$$99$$ −1.75379 −0.176262
$$100$$ 0 0
$$101$$ 12.2462 1.21854 0.609272 0.792961i $$-0.291462\pi$$
0.609272 + 0.792961i $$0.291462\pi$$
$$102$$ 0 0
$$103$$ −14.2462 −1.40372 −0.701860 0.712314i $$-0.747647\pi$$
−0.701860 + 0.712314i $$0.747647\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 11.1231 1.07531 0.537656 0.843165i $$-0.319310\pi$$
0.537656 + 0.843165i $$0.319310\pi$$
$$108$$ 0 0
$$109$$ 13.1231 1.25697 0.628483 0.777824i $$-0.283676\pi$$
0.628483 + 0.777824i $$0.283676\pi$$
$$110$$ 0 0
$$111$$ −12.8769 −1.22222
$$112$$ 0 0
$$113$$ 2.87689 0.270635 0.135318 0.990802i $$-0.456794\pi$$
0.135318 + 0.990802i $$0.456794\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0.246211 0.0227622
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1.24621 −0.113292
$$122$$ 0 0
$$123$$ −15.3153 −1.38094
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0.684658 0.0607536 0.0303768 0.999539i $$-0.490329\pi$$
0.0303768 + 0.999539i $$0.490329\pi$$
$$128$$ 0 0
$$129$$ −12.4924 −1.09990
$$130$$ 0 0
$$131$$ −3.31534 −0.289663 −0.144831 0.989456i $$-0.546264\pi$$
−0.144831 + 0.989456i $$0.546264\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −6.87689 −0.587533 −0.293766 0.955877i $$-0.594909\pi$$
−0.293766 + 0.955877i $$0.594909\pi$$
$$138$$ 0 0
$$139$$ 3.31534 0.281204 0.140602 0.990066i $$-0.455096\pi$$
0.140602 + 0.990066i $$0.455096\pi$$
$$140$$ 0 0
$$141$$ −1.06913 −0.0900370
$$142$$ 0 0
$$143$$ −1.36932 −0.114508
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −10.9309 −0.901563
$$148$$ 0 0
$$149$$ −20.2462 −1.65863 −0.829317 0.558778i $$-0.811270\pi$$
−0.829317 + 0.558778i $$0.811270\pi$$
$$150$$ 0 0
$$151$$ 10.4384 0.849469 0.424734 0.905318i $$-0.360367\pi$$
0.424734 + 0.905318i $$0.360367\pi$$
$$152$$ 0 0
$$153$$ 2.87689 0.232583
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −21.1231 −1.68581 −0.842904 0.538064i $$-0.819156\pi$$
−0.842904 + 0.538064i $$0.819156\pi$$
$$158$$ 0 0
$$159$$ −3.12311 −0.247678
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 18.9309 1.48278 0.741390 0.671074i $$-0.234167\pi$$
0.741390 + 0.671074i $$0.234167\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 0 0
$$169$$ −12.8078 −0.985213
$$170$$ 0 0
$$171$$ −1.75379 −0.134116
$$172$$ 0 0
$$173$$ 10.0000 0.760286 0.380143 0.924928i $$-0.375875\pi$$
0.380143 + 0.924928i $$0.375875\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −16.0000 −1.20263
$$178$$ 0 0
$$179$$ 3.31534 0.247800 0.123900 0.992295i $$-0.460460\pi$$
0.123900 + 0.992295i $$0.460460\pi$$
$$180$$ 0 0
$$181$$ 3.75379 0.279017 0.139508 0.990221i $$-0.455448\pi$$
0.139508 + 0.990221i $$0.455448\pi$$
$$182$$ 0 0
$$183$$ −6.63068 −0.490154
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −16.0000 −1.17004
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −25.3693 −1.83566 −0.917830 0.396974i $$-0.870060\pi$$
−0.917830 + 0.396974i $$0.870060\pi$$
$$192$$ 0 0
$$193$$ 0.438447 0.0315601 0.0157801 0.999875i $$-0.494977\pi$$
0.0157801 + 0.999875i $$0.494977\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 16.9309 1.20627 0.603137 0.797637i $$-0.293917\pi$$
0.603137 + 0.797637i $$0.293917\pi$$
$$198$$ 0 0
$$199$$ −11.1231 −0.788496 −0.394248 0.919004i $$-0.628995\pi$$
−0.394248 + 0.919004i $$0.628995\pi$$
$$200$$ 0 0
$$201$$ 4.87689 0.343990
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0.561553 0.0390306
$$208$$ 0 0
$$209$$ 9.75379 0.674684
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ −21.1771 −1.45103
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 22.9309 1.54952
$$220$$ 0 0
$$221$$ 2.24621 0.151097
$$222$$ 0 0
$$223$$ −28.4924 −1.90799 −0.953997 0.299817i $$-0.903075\pi$$
−0.953997 + 0.299817i $$0.903075\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −6.24621 −0.414576 −0.207288 0.978280i $$-0.566464\pi$$
−0.207288 + 0.978280i $$0.566464\pi$$
$$228$$ 0 0
$$229$$ 14.8769 0.983093 0.491546 0.870851i $$-0.336432\pi$$
0.491546 + 0.870851i $$0.336432\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 16.0540 1.05173 0.525865 0.850568i $$-0.323742\pi$$
0.525865 + 0.850568i $$0.323742\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −4.87689 −0.316788
$$238$$ 0 0
$$239$$ −26.0540 −1.68529 −0.842646 0.538468i $$-0.819003\pi$$
−0.842646 + 0.538468i $$0.819003\pi$$
$$240$$ 0 0
$$241$$ 6.49242 0.418214 0.209107 0.977893i $$-0.432944\pi$$
0.209107 + 0.977893i $$0.432944\pi$$
$$242$$ 0 0
$$243$$ 5.75379 0.369106
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.36932 −0.0871275
$$248$$ 0 0
$$249$$ 22.2462 1.40980
$$250$$ 0 0
$$251$$ 6.24621 0.394257 0.197129 0.980378i $$-0.436838\pi$$
0.197129 + 0.980378i $$0.436838\pi$$
$$252$$ 0 0
$$253$$ −3.12311 −0.196348
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 5.31534 0.331562 0.165781 0.986163i $$-0.446986\pi$$
0.165781 + 0.986163i $$0.446986\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ −19.1231 −1.17918 −0.589591 0.807702i $$-0.700711\pi$$
−0.589591 + 0.807702i $$0.700711\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 17.7538 1.08651
$$268$$ 0 0
$$269$$ 13.3153 0.811851 0.405925 0.913906i $$-0.366949\pi$$
0.405925 + 0.913906i $$0.366949\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −17.8078 −1.06996 −0.534982 0.844863i $$-0.679682\pi$$
−0.534982 + 0.844863i $$0.679682\pi$$
$$278$$ 0 0
$$279$$ −1.36932 −0.0819789
$$280$$ 0 0
$$281$$ 5.12311 0.305619 0.152809 0.988256i $$-0.451168\pi$$
0.152809 + 0.988256i $$0.451168\pi$$
$$282$$ 0 0
$$283$$ 3.12311 0.185649 0.0928247 0.995682i $$-0.470410\pi$$
0.0928247 + 0.995682i $$0.470410\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 9.24621 0.543895
$$290$$ 0 0
$$291$$ −17.7538 −1.04075
$$292$$ 0 0
$$293$$ 15.3693 0.897885 0.448943 0.893561i $$-0.351801\pi$$
0.448943 + 0.893561i $$0.351801\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −17.3693 −1.00787
$$298$$ 0 0
$$299$$ 0.438447 0.0253561
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 19.1231 1.09859
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −18.2462 −1.04137 −0.520683 0.853750i $$-0.674323\pi$$
−0.520683 + 0.853750i $$0.674323\pi$$
$$308$$ 0 0
$$309$$ −22.2462 −1.26554
$$310$$ 0 0
$$311$$ −5.56155 −0.315367 −0.157683 0.987490i $$-0.550403\pi$$
−0.157683 + 0.987490i $$0.550403\pi$$
$$312$$ 0 0
$$313$$ 15.3693 0.868725 0.434363 0.900738i $$-0.356974\pi$$
0.434363 + 0.900738i $$0.356974\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3.75379 0.210834 0.105417 0.994428i $$-0.466382\pi$$
0.105417 + 0.994428i $$0.466382\pi$$
$$318$$ 0 0
$$319$$ 11.1231 0.622774
$$320$$ 0 0
$$321$$ 17.3693 0.969461
$$322$$ 0 0
$$323$$ −16.0000 −0.890264
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 20.4924 1.13323
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −31.4233 −1.72718 −0.863590 0.504194i $$-0.831789\pi$$
−0.863590 + 0.504194i $$0.831789\pi$$
$$332$$ 0 0
$$333$$ 4.63068 0.253760
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 21.6155 1.17747 0.588736 0.808325i $$-0.299626\pi$$
0.588736 + 0.808325i $$0.299626\pi$$
$$338$$ 0 0
$$339$$ 4.49242 0.243995
$$340$$ 0 0
$$341$$ 7.61553 0.412404
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −16.4924 −0.885360 −0.442680 0.896680i $$-0.645972\pi$$
−0.442680 + 0.896680i $$0.645972\pi$$
$$348$$ 0 0
$$349$$ −34.6847 −1.85663 −0.928314 0.371798i $$-0.878741\pi$$
−0.928314 + 0.371798i $$0.878741\pi$$
$$350$$ 0 0
$$351$$ 2.43845 0.130155
$$352$$ 0 0
$$353$$ −15.5616 −0.828258 −0.414129 0.910218i $$-0.635914\pi$$
−0.414129 + 0.910218i $$0.635914\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 23.6155 1.24638 0.623190 0.782071i $$-0.285836\pi$$
0.623190 + 0.782071i $$0.285836\pi$$
$$360$$ 0 0
$$361$$ −9.24621 −0.486643
$$362$$ 0 0
$$363$$ −1.94602 −0.102140
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −14.2462 −0.743646 −0.371823 0.928304i $$-0.621267\pi$$
−0.371823 + 0.928304i $$0.621267\pi$$
$$368$$ 0 0
$$369$$ 5.50758 0.286713
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10.4924 0.543277 0.271639 0.962399i $$-0.412434\pi$$
0.271639 + 0.962399i $$0.412434\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −1.56155 −0.0804241
$$378$$ 0 0
$$379$$ 12.4924 0.641693 0.320846 0.947131i $$-0.396033\pi$$
0.320846 + 0.947131i $$0.396033\pi$$
$$380$$ 0 0
$$381$$ 1.06913 0.0547732
$$382$$ 0 0
$$383$$ 9.75379 0.498395 0.249198 0.968453i $$-0.419833\pi$$
0.249198 + 0.968453i $$0.419833\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 4.49242 0.228363
$$388$$ 0 0
$$389$$ −23.3693 −1.18487 −0.592436 0.805618i $$-0.701834\pi$$
−0.592436 + 0.805618i $$0.701834\pi$$
$$390$$ 0 0
$$391$$ 5.12311 0.259087
$$392$$ 0 0
$$393$$ −5.17708 −0.261149
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −16.4384 −0.825022 −0.412511 0.910953i $$-0.635348\pi$$
−0.412511 + 0.910953i $$0.635348\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −28.2462 −1.41055 −0.705274 0.708935i $$-0.749176\pi$$
−0.705274 + 0.708935i $$0.749176\pi$$
$$402$$ 0 0
$$403$$ −1.06913 −0.0532572
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −25.7538 −1.27657
$$408$$ 0 0
$$409$$ 26.3002 1.30046 0.650230 0.759737i $$-0.274672\pi$$
0.650230 + 0.759737i $$0.274672\pi$$
$$410$$ 0 0
$$411$$ −10.7386 −0.529698
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 5.17708 0.253523
$$418$$ 0 0
$$419$$ −20.4924 −1.00112 −0.500560 0.865702i $$-0.666873\pi$$
−0.500560 + 0.865702i $$0.666873\pi$$
$$420$$ 0 0
$$421$$ −18.8769 −0.920004 −0.460002 0.887918i $$-0.652151\pi$$
−0.460002 + 0.887918i $$0.652151\pi$$
$$422$$ 0 0
$$423$$ 0.384472 0.0186937
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −2.13826 −0.103236
$$430$$ 0 0
$$431$$ −4.49242 −0.216392 −0.108196 0.994130i $$-0.534507\pi$$
−0.108196 + 0.994130i $$0.534507\pi$$
$$432$$ 0 0
$$433$$ 24.7386 1.18886 0.594431 0.804146i $$-0.297377\pi$$
0.594431 + 0.804146i $$0.297377\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −3.12311 −0.149398
$$438$$ 0 0
$$439$$ 26.0540 1.24349 0.621744 0.783220i $$-0.286424\pi$$
0.621744 + 0.783220i $$0.286424\pi$$
$$440$$ 0 0
$$441$$ 3.93087 0.187184
$$442$$ 0 0
$$443$$ −34.9309 −1.65962 −0.829808 0.558049i $$-0.811550\pi$$
−0.829808 + 0.558049i $$0.811550\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −31.6155 −1.49536
$$448$$ 0 0
$$449$$ 8.24621 0.389163 0.194581 0.980886i $$-0.437665\pi$$
0.194581 + 0.980886i $$0.437665\pi$$
$$450$$ 0 0
$$451$$ −30.6307 −1.44234
$$452$$ 0 0
$$453$$ 16.3002 0.765850
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −19.3693 −0.906058 −0.453029 0.891496i $$-0.649657\pi$$
−0.453029 + 0.891496i $$0.649657\pi$$
$$458$$ 0 0
$$459$$ 28.4924 1.32991
$$460$$ 0 0
$$461$$ 9.80776 0.456793 0.228397 0.973568i $$-0.426652\pi$$
0.228397 + 0.973568i $$0.426652\pi$$
$$462$$ 0 0
$$463$$ −20.4924 −0.952364 −0.476182 0.879347i $$-0.657980\pi$$
−0.476182 + 0.879347i $$0.657980\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 9.36932 0.433560 0.216780 0.976220i $$-0.430445\pi$$
0.216780 + 0.976220i $$0.430445\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −32.9848 −1.51986
$$472$$ 0 0
$$473$$ −24.9848 −1.14880
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 1.12311 0.0514235
$$478$$ 0 0
$$479$$ 6.24621 0.285397 0.142698 0.989766i $$-0.454422\pi$$
0.142698 + 0.989766i $$0.454422\pi$$
$$480$$ 0 0
$$481$$ 3.61553 0.164854
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 24.6847 1.11857 0.559284 0.828976i $$-0.311076\pi$$
0.559284 + 0.828976i $$0.311076\pi$$
$$488$$ 0 0
$$489$$ 29.5616 1.33682
$$490$$ 0 0
$$491$$ 7.80776 0.352359 0.176180 0.984358i $$-0.443626\pi$$
0.176180 + 0.984358i $$0.443626\pi$$
$$492$$ 0 0
$$493$$ −18.2462 −0.821768
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −14.0540 −0.629142 −0.314571 0.949234i $$-0.601861\pi$$
−0.314571 + 0.949234i $$0.601861\pi$$
$$500$$ 0 0
$$501$$ −24.9848 −1.11624
$$502$$ 0 0
$$503$$ −17.3693 −0.774460 −0.387230 0.921983i $$-0.626568\pi$$
−0.387230 + 0.921983i $$0.626568\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −20.0000 −0.888231
$$508$$ 0 0
$$509$$ 38.3002 1.69763 0.848813 0.528693i $$-0.177318\pi$$
0.848813 + 0.528693i $$0.177318\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −17.3693 −0.766874
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −2.13826 −0.0940406
$$518$$ 0 0
$$519$$ 15.6155 0.685446
$$520$$ 0 0
$$521$$ −21.6155 −0.946993 −0.473497 0.880796i $$-0.657008\pi$$
−0.473497 + 0.880796i $$0.657008\pi$$
$$522$$ 0 0
$$523$$ 22.2462 0.972759 0.486379 0.873748i $$-0.338317\pi$$
0.486379 + 0.873748i $$0.338317\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −12.4924 −0.544178
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 5.75379 0.249693
$$532$$ 0 0
$$533$$ 4.30019 0.186262
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 5.17708 0.223408
$$538$$ 0 0
$$539$$ −21.8617 −0.941652
$$540$$ 0 0
$$541$$ 7.06913 0.303926 0.151963 0.988386i $$-0.451441\pi$$
0.151963 + 0.988386i $$0.451441\pi$$
$$542$$ 0 0
$$543$$ 5.86174 0.251551
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −26.5464 −1.13504 −0.567521 0.823359i $$-0.692097\pi$$
−0.567521 + 0.823359i $$0.692097\pi$$
$$548$$ 0 0
$$549$$ 2.38447 0.101767
$$550$$ 0 0
$$551$$ 11.1231 0.473860
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 9.12311 0.386558 0.193279 0.981144i $$-0.438088\pi$$
0.193279 + 0.981144i $$0.438088\pi$$
$$558$$ 0 0
$$559$$ 3.50758 0.148355
$$560$$ 0 0
$$561$$ −24.9848 −1.05486
$$562$$ 0 0
$$563$$ 22.2462 0.937566 0.468783 0.883313i $$-0.344693\pi$$
0.468783 + 0.883313i $$0.344693\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 8.24621 0.345699 0.172850 0.984948i $$-0.444703\pi$$
0.172850 + 0.984948i $$0.444703\pi$$
$$570$$ 0 0
$$571$$ 44.4924 1.86195 0.930975 0.365083i $$-0.118959\pi$$
0.930975 + 0.365083i $$0.118959\pi$$
$$572$$ 0 0
$$573$$ −39.6155 −1.65496
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 12.9309 0.538319 0.269160 0.963096i $$-0.413254\pi$$
0.269160 + 0.963096i $$0.413254\pi$$
$$578$$ 0 0
$$579$$ 0.684658 0.0284534
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −6.24621 −0.258692
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 14.0540 0.580070 0.290035 0.957016i $$-0.406333\pi$$
0.290035 + 0.957016i $$0.406333\pi$$
$$588$$ 0 0
$$589$$ 7.61553 0.313792
$$590$$ 0 0
$$591$$ 26.4384 1.08753
$$592$$ 0 0
$$593$$ −4.73863 −0.194592 −0.0972962 0.995255i $$-0.531019\pi$$
−0.0972962 + 0.995255i $$0.531019\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −17.3693 −0.710879
$$598$$ 0 0
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ 0 0
$$601$$ −18.1922 −0.742077 −0.371038 0.928618i $$-0.620998\pi$$
−0.371038 + 0.928618i $$0.620998\pi$$
$$602$$ 0 0
$$603$$ −1.75379 −0.0714198
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 34.7386 1.41000 0.704999 0.709208i $$-0.250948\pi$$
0.704999 + 0.709208i $$0.250948\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0.300187 0.0121442
$$612$$ 0 0
$$613$$ −46.1080 −1.86228 −0.931141 0.364659i $$-0.881186\pi$$
−0.931141 + 0.364659i $$0.881186\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 30.9848 1.24740 0.623701 0.781663i $$-0.285628\pi$$
0.623701 + 0.781663i $$0.285628\pi$$
$$618$$ 0 0
$$619$$ 4.49242 0.180566 0.0902829 0.995916i $$-0.471223\pi$$
0.0902829 + 0.995916i $$0.471223\pi$$
$$620$$ 0 0
$$621$$ 5.56155 0.223177
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 15.2311 0.608270
$$628$$ 0 0
$$629$$ 42.2462 1.68447
$$630$$ 0 0
$$631$$ −42.7386 −1.70140 −0.850699 0.525653i $$-0.823821\pi$$
−0.850699 + 0.525653i $$0.823821\pi$$
$$632$$ 0 0
$$633$$ 6.24621 0.248265
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 3.06913 0.121603
$$638$$ 0 0
$$639$$ 7.61553 0.301266
$$640$$ 0 0
$$641$$ 11.3693 0.449061 0.224531 0.974467i $$-0.427915\pi$$
0.224531 + 0.974467i $$0.427915\pi$$
$$642$$ 0 0
$$643$$ 1.36932 0.0540006 0.0270003 0.999635i $$-0.491404\pi$$
0.0270003 + 0.999635i $$0.491404\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.3002 0.955339 0.477669 0.878540i $$-0.341482\pi$$
0.477669 + 0.878540i $$0.341482\pi$$
$$648$$ 0 0
$$649$$ −32.0000 −1.25611
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 12.0540 0.471709 0.235854 0.971788i $$-0.424211\pi$$
0.235854 + 0.971788i $$0.424211\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −8.24621 −0.321715
$$658$$ 0 0
$$659$$ 14.2462 0.554954 0.277477 0.960732i $$-0.410502\pi$$
0.277477 + 0.960732i $$0.410502\pi$$
$$660$$ 0 0
$$661$$ 16.2462 0.631904 0.315952 0.948775i $$-0.397676\pi$$
0.315952 + 0.948775i $$0.397676\pi$$
$$662$$ 0 0
$$663$$ 3.50758 0.136223
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −3.56155 −0.137904
$$668$$ 0 0
$$669$$ −44.4924 −1.72018
$$670$$ 0 0
$$671$$ −13.2614 −0.511949
$$672$$ 0 0
$$673$$ 40.0540 1.54397 0.771984 0.635642i $$-0.219265\pi$$
0.771984 + 0.635642i $$0.219265\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −26.9848 −1.03711 −0.518556 0.855044i $$-0.673530\pi$$
−0.518556 + 0.855044i $$0.673530\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −9.75379 −0.373766
$$682$$ 0 0
$$683$$ 28.6847 1.09759 0.548794 0.835958i $$-0.315087\pi$$
0.548794 + 0.835958i $$0.315087\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 23.2311 0.886320
$$688$$ 0 0
$$689$$ 0.876894 0.0334070
$$690$$ 0 0
$$691$$ 44.9848 1.71130 0.855652 0.517551i $$-0.173156\pi$$
0.855652 + 0.517551i $$0.173156\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 50.2462 1.90321
$$698$$ 0 0
$$699$$ 25.0691 0.948202
$$700$$ 0 0
$$701$$ −27.8617 −1.05232 −0.526162 0.850385i $$-0.676369\pi$$
−0.526162 + 0.850385i $$0.676369\pi$$
$$702$$ 0 0
$$703$$ −25.7538 −0.971323
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −12.6307 −0.474355 −0.237178 0.971466i $$-0.576222\pi$$
−0.237178 + 0.971466i $$0.576222\pi$$
$$710$$ 0 0
$$711$$ 1.75379 0.0657722
$$712$$ 0 0
$$713$$ −2.43845 −0.0913206
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −40.6847 −1.51940
$$718$$ 0 0
$$719$$ 28.4924 1.06259 0.531294 0.847187i $$-0.321706\pi$$
0.531294 + 0.847187i $$0.321706\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 10.1383 0.377046
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −14.6307 −0.542622 −0.271311 0.962492i $$-0.587457\pi$$
−0.271311 + 0.962492i $$0.587457\pi$$
$$728$$ 0 0
$$729$$ 29.9848 1.11055
$$730$$ 0 0
$$731$$ 40.9848 1.51588
$$732$$ 0 0
$$733$$ −24.6307 −0.909755 −0.454878 0.890554i $$-0.650317\pi$$
−0.454878 + 0.890554i $$0.650317\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 9.75379 0.359285
$$738$$ 0 0
$$739$$ −12.6847 −0.466613 −0.233306 0.972403i $$-0.574954\pi$$
−0.233306 + 0.972403i $$0.574954\pi$$
$$740$$ 0 0
$$741$$ −2.13826 −0.0785510
$$742$$ 0 0
$$743$$ 17.7538 0.651323 0.325662 0.945486i $$-0.394413\pi$$
0.325662 + 0.945486i $$0.394413\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −8.00000 −0.292705
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 24.9848 0.911710 0.455855 0.890054i $$-0.349334\pi$$
0.455855 + 0.890054i $$0.349334\pi$$
$$752$$ 0 0
$$753$$ 9.75379 0.355448
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 34.4924 1.25365 0.626824 0.779161i $$-0.284354\pi$$
0.626824 + 0.779161i $$0.284354\pi$$
$$758$$ 0 0
$$759$$ −4.87689 −0.177020
$$760$$ 0 0
$$761$$ −30.6847 −1.11232 −0.556159 0.831076i $$-0.687725\pi$$
−0.556159 + 0.831076i $$0.687725\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 4.49242 0.162212
$$768$$ 0 0
$$769$$ −6.00000 −0.216366 −0.108183 0.994131i $$-0.534503\pi$$
−0.108183 + 0.994131i $$0.534503\pi$$
$$770$$ 0 0
$$771$$ 8.30019 0.298924
$$772$$ 0 0
$$773$$ −27.3693 −0.984406 −0.492203 0.870480i $$-0.663808\pi$$
−0.492203 + 0.870480i $$0.663808\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −30.6307 −1.09746
$$780$$ 0 0
$$781$$ −42.3542 −1.51555
$$782$$ 0 0
$$783$$ −19.8078 −0.707872
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −9.75379 −0.347685 −0.173843 0.984773i $$-0.555618\pi$$
−0.173843 + 0.984773i $$0.555618\pi$$
$$788$$ 0 0
$$789$$ −29.8617 −1.06311
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 1.86174 0.0661123
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 4.24621 0.150409 0.0752043 0.997168i $$-0.476039\pi$$
0.0752043 + 0.997168i $$0.476039\pi$$
$$798$$ 0 0
$$799$$ 3.50758 0.124089
$$800$$ 0 0
$$801$$ −6.38447 −0.225584
$$802$$ 0 0
$$803$$ 45.8617 1.61843
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 20.7926 0.731935
$$808$$ 0 0
$$809$$ 26.0000 0.914111 0.457056 0.889438i $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ 0 0
$$811$$ −5.06913 −0.178001 −0.0890006 0.996032i $$-0.528367\pi$$
−0.0890006 + 0.996032i $$0.528367\pi$$
$$812$$ 0 0
$$813$$ −37.4773 −1.31439
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −24.9848 −0.874109
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 24.7386 0.863384 0.431692 0.902021i $$-0.357917\pi$$
0.431692 + 0.902021i $$0.357917\pi$$
$$822$$ 0 0
$$823$$ 53.5616 1.86704 0.933519 0.358527i $$-0.116721\pi$$
0.933519 + 0.358527i $$0.116721\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −28.4924 −0.990779 −0.495389 0.868671i $$-0.664975\pi$$
−0.495389 + 0.868671i $$0.664975\pi$$
$$828$$ 0 0
$$829$$ −3.75379 −0.130374 −0.0651872 0.997873i $$-0.520764\pi$$
−0.0651872 + 0.997873i $$0.520764\pi$$
$$830$$ 0 0
$$831$$ −27.8078 −0.964641
$$832$$ 0 0
$$833$$ 35.8617 1.24254
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −13.5616 −0.468756
$$838$$ 0 0
$$839$$ −33.7538 −1.16531 −0.582655 0.812720i $$-0.697986\pi$$
−0.582655 + 0.812720i $$0.697986\pi$$
$$840$$ 0 0
$$841$$ −16.3153 −0.562598
$$842$$ 0 0
$$843$$ 8.00000 0.275535
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 4.87689 0.167375
$$850$$ 0 0
$$851$$ 8.24621 0.282676
$$852$$ 0 0
$$853$$ −30.9848 −1.06090 −0.530450 0.847716i $$-0.677977\pi$$
−0.530450 + 0.847716i $$0.677977\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −12.4384 −0.424889 −0.212445 0.977173i $$-0.568143\pi$$
−0.212445 + 0.977173i $$0.568143\pi$$
$$858$$ 0 0
$$859$$ −54.0540 −1.84430 −0.922149 0.386835i $$-0.873568\pi$$
−0.922149 + 0.386835i $$0.873568\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 37.1771 1.26552 0.632761 0.774347i $$-0.281921\pi$$
0.632761 + 0.774347i $$0.281921\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 14.4384 0.490355
$$868$$ 0 0
$$869$$ −9.75379 −0.330875
$$870$$ 0 0
$$871$$ −1.36932 −0.0463975
$$872$$ 0 0
$$873$$ 6.38447 0.216082
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 34.9848 1.18135 0.590677 0.806908i $$-0.298861\pi$$
0.590677 + 0.806908i $$0.298861\pi$$
$$878$$ 0 0
$$879$$ 24.0000 0.809500
$$880$$ 0 0
$$881$$ 13.1231 0.442129 0.221064 0.975259i $$-0.429047\pi$$
0.221064 + 0.975259i $$0.429047\pi$$
$$882$$ 0 0
$$883$$ −28.9848 −0.975418 −0.487709 0.873006i $$-0.662167\pi$$
−0.487709 + 0.873006i $$0.662167\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 7.31534 0.245625 0.122813 0.992430i $$-0.460809\pi$$
0.122813 + 0.992430i $$0.460809\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −21.8617 −0.732396
$$892$$ 0 0
$$893$$ −2.13826 −0.0715542
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0.684658 0.0228601
$$898$$ 0 0
$$899$$ 8.68466 0.289650
$$900$$ 0 0
$$901$$ 10.2462 0.341351
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 51.1231 1.69751 0.848757 0.528782i $$-0.177351\pi$$
0.848757 + 0.528782i $$0.177351\pi$$
$$908$$ 0 0
$$909$$ −6.87689 −0.228092
$$910$$ 0 0
$$911$$ 28.8769 0.956734 0.478367 0.878160i $$-0.341229\pi$$
0.478367 + 0.878160i $$0.341229\pi$$
$$912$$ 0 0
$$913$$ 44.4924 1.47248
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −54.2462 −1.78942 −0.894709 0.446650i $$-0.852617\pi$$
−0.894709 + 0.446650i $$0.852617\pi$$
$$920$$ 0 0
$$921$$ −28.4924 −0.938857
$$922$$ 0 0
$$923$$ 5.94602 0.195716
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ 14.1922 0.465632 0.232816 0.972521i $$-0.425206\pi$$
0.232816 + 0.972521i $$0.425206\pi$$
$$930$$ 0 0
$$931$$ −21.8617 −0.716490
$$932$$ 0 0
$$933$$ −8.68466 −0.284323
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −32.2462 −1.05344 −0.526719 0.850040i $$-0.676578\pi$$
−0.526719 + 0.850040i $$0.676578\pi$$
$$938$$ 0 0
$$939$$ 24.0000 0.783210
$$940$$ 0 0
$$941$$ 30.8769 1.00656 0.503279 0.864124i $$-0.332127\pi$$
0.503279 + 0.864124i $$0.332127\pi$$
$$942$$ 0 0
$$943$$ 9.80776 0.319385
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1.56155 0.0507436 0.0253718 0.999678i $$-0.491923\pi$$
0.0253718 + 0.999678i $$0.491923\pi$$
$$948$$ 0 0
$$949$$ −6.43845 −0.209001
$$950$$ 0 0
$$951$$ 5.86174 0.190080
$$952$$ 0 0
$$953$$ −27.7538 −0.899033 −0.449517 0.893272i $$-0.648404\pi$$
−0.449517 + 0.893272i $$0.648404\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 17.3693 0.561470
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −25.0540 −0.808193
$$962$$ 0 0
$$963$$ −6.24621 −0.201281
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −2.05398 −0.0660514 −0.0330257 0.999455i $$-0.510514\pi$$
−0.0330257 + 0.999455i $$0.510514\pi$$
$$968$$ 0 0
$$969$$ −24.9848 −0.802629
$$970$$ 0 0
$$971$$ −6.63068 −0.212789 −0.106394 0.994324i $$-0.533931\pi$$
−0.106394 + 0.994324i $$0.533931\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −25.2311 −0.807213 −0.403607 0.914933i $$-0.632244\pi$$
−0.403607 + 0.914933i $$0.632244\pi$$
$$978$$ 0 0
$$979$$ 35.5076 1.13483
$$980$$ 0 0
$$981$$ −7.36932 −0.235284
$$982$$ 0 0
$$983$$ −52.1080 −1.66199 −0.830993 0.556283i $$-0.812227\pi$$
−0.830993 + 0.556283i $$0.812227\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ 56.9848 1.81018 0.905092 0.425217i $$-0.139802\pi$$
0.905092 + 0.425217i $$0.139802\pi$$
$$992$$ 0 0
$$993$$ −49.0691 −1.55716
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −14.0000 −0.443384 −0.221692 0.975117i $$-0.571158\pi$$
−0.221692 + 0.975117i $$0.571158\pi$$
$$998$$ 0 0
$$999$$ 45.8617 1.45100
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.br.1.2 2
4.3 odd 2 4600.2.a.s.1.1 2
5.4 even 2 368.2.a.i.1.1 2
15.14 odd 2 3312.2.a.t.1.1 2
20.3 even 4 4600.2.e.o.4049.2 4
20.7 even 4 4600.2.e.o.4049.3 4
20.19 odd 2 184.2.a.e.1.2 2
40.19 odd 2 1472.2.a.u.1.1 2
40.29 even 2 1472.2.a.p.1.2 2
60.59 even 2 1656.2.a.j.1.2 2
115.114 odd 2 8464.2.a.bd.1.1 2
140.139 even 2 9016.2.a.w.1.1 2
460.459 even 2 4232.2.a.o.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.a.e.1.2 2 20.19 odd 2
368.2.a.i.1.1 2 5.4 even 2
1472.2.a.p.1.2 2 40.29 even 2
1472.2.a.u.1.1 2 40.19 odd 2
1656.2.a.j.1.2 2 60.59 even 2
3312.2.a.t.1.1 2 15.14 odd 2
4232.2.a.o.1.2 2 460.459 even 2
4600.2.a.s.1.1 2 4.3 odd 2
4600.2.e.o.4049.2 4 20.3 even 4
4600.2.e.o.4049.3 4 20.7 even 4
8464.2.a.bd.1.1 2 115.114 odd 2
9016.2.a.w.1.1 2 140.139 even 2
9200.2.a.br.1.2 2 1.1 even 1 trivial