# Properties

 Label 1296.4.a.i.1.1 Level $1296$ Weight $4$ Character 1296.1 Self dual yes Analytic conductor $76.466$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1296.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$76.4664753674$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 1296.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-10.3723 q^{5} +5.11684 q^{7} +O(q^{10})$$ $$q-10.3723 q^{5} +5.11684 q^{7} +55.9783 q^{11} +37.5842 q^{13} -23.6495 q^{17} -39.0516 q^{19} +71.0733 q^{23} -17.4158 q^{25} +28.3723 q^{29} -12.8832 q^{31} -53.0733 q^{35} -180.103 q^{37} +215.484 q^{41} -61.2337 q^{43} -61.8776 q^{47} -316.818 q^{49} -492.310 q^{53} -580.622 q^{55} +789.630 q^{59} +521.090 q^{61} -389.834 q^{65} -304.429 q^{67} +270.391 q^{71} -925.464 q^{73} +286.432 q^{77} +1289.03 q^{79} +713.834 q^{83} +245.299 q^{85} +404.804 q^{89} +192.313 q^{91} +405.054 q^{95} +75.0273 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 15 q^{5} - 7 q^{7}+O(q^{10})$$ 2 * q - 15 * q^5 - 7 * q^7 $$2 q - 15 q^{5} - 7 q^{7} + 66 q^{11} - 11 q^{13} - 99 q^{17} + 77 q^{19} + 33 q^{23} - 121 q^{25} + 51 q^{29} - 43 q^{31} + 3 q^{35} - 50 q^{37} - 132 q^{41} - 88 q^{43} + 399 q^{47} - 513 q^{49} - 54 q^{53} - 627 q^{55} + 798 q^{59} + 439 q^{61} - 165 q^{65} - 988 q^{67} + 1368 q^{71} - 455 q^{73} + 165 q^{77} + 803 q^{79} + 813 q^{83} + 594 q^{85} + 396 q^{89} + 781 q^{91} - 132 q^{95} + 736 q^{97}+O(q^{100})$$ 2 * q - 15 * q^5 - 7 * q^7 + 66 * q^11 - 11 * q^13 - 99 * q^17 + 77 * q^19 + 33 * q^23 - 121 * q^25 + 51 * q^29 - 43 * q^31 + 3 * q^35 - 50 * q^37 - 132 * q^41 - 88 * q^43 + 399 * q^47 - 513 * q^49 - 54 * q^53 - 627 * q^55 + 798 * q^59 + 439 * q^61 - 165 * q^65 - 988 * q^67 + 1368 * q^71 - 455 * q^73 + 165 * q^77 + 803 * q^79 + 813 * q^83 + 594 * q^85 + 396 * q^89 + 781 * q^91 - 132 * q^95 + 736 * q^97

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −10.3723 −0.927725 −0.463863 0.885907i $$-0.653537\pi$$
−0.463863 + 0.885907i $$0.653537\pi$$
$$6$$ 0 0
$$7$$ 5.11684 0.276284 0.138142 0.990412i $$-0.455887\pi$$
0.138142 + 0.990412i $$0.455887\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 55.9783 1.53437 0.767185 0.641425i $$-0.221657\pi$$
0.767185 + 0.641425i $$0.221657\pi$$
$$12$$ 0 0
$$13$$ 37.5842 0.801845 0.400923 0.916112i $$-0.368690\pi$$
0.400923 + 0.916112i $$0.368690\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −23.6495 −0.337402 −0.168701 0.985667i $$-0.553957\pi$$
−0.168701 + 0.985667i $$0.553957\pi$$
$$18$$ 0 0
$$19$$ −39.0516 −0.471529 −0.235764 0.971810i $$-0.575759\pi$$
−0.235764 + 0.971810i $$0.575759\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 71.0733 0.644340 0.322170 0.946682i $$-0.395588\pi$$
0.322170 + 0.946682i $$0.395588\pi$$
$$24$$ 0 0
$$25$$ −17.4158 −0.139326
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 28.3723 0.181676 0.0908379 0.995866i $$-0.471045\pi$$
0.0908379 + 0.995866i $$0.471045\pi$$
$$30$$ 0 0
$$31$$ −12.8832 −0.0746414 −0.0373207 0.999303i $$-0.511882\pi$$
−0.0373207 + 0.999303i $$0.511882\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −53.0733 −0.256315
$$36$$ 0 0
$$37$$ −180.103 −0.800237 −0.400119 0.916463i $$-0.631031\pi$$
−0.400119 + 0.916463i $$0.631031\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 215.484 0.820802 0.410401 0.911905i $$-0.365389\pi$$
0.410401 + 0.911905i $$0.365389\pi$$
$$42$$ 0 0
$$43$$ −61.2337 −0.217164 −0.108582 0.994087i $$-0.534631\pi$$
−0.108582 + 0.994087i $$0.534631\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −61.8776 −0.192038 −0.0960189 0.995380i $$-0.530611\pi$$
−0.0960189 + 0.995380i $$0.530611\pi$$
$$48$$ 0 0
$$49$$ −316.818 −0.923667
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −492.310 −1.27592 −0.637962 0.770068i $$-0.720222\pi$$
−0.637962 + 0.770068i $$0.720222\pi$$
$$54$$ 0 0
$$55$$ −580.622 −1.42347
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 789.630 1.74239 0.871196 0.490936i $$-0.163345\pi$$
0.871196 + 0.490936i $$0.163345\pi$$
$$60$$ 0 0
$$61$$ 521.090 1.09375 0.546874 0.837215i $$-0.315818\pi$$
0.546874 + 0.837215i $$0.315818\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −389.834 −0.743892
$$66$$ 0 0
$$67$$ −304.429 −0.555104 −0.277552 0.960711i $$-0.589523\pi$$
−0.277552 + 0.960711i $$0.589523\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 270.391 0.451966 0.225983 0.974131i $$-0.427441\pi$$
0.225983 + 0.974131i $$0.427441\pi$$
$$72$$ 0 0
$$73$$ −925.464 −1.48380 −0.741900 0.670510i $$-0.766075\pi$$
−0.741900 + 0.670510i $$0.766075\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 286.432 0.423921
$$78$$ 0 0
$$79$$ 1289.03 1.83579 0.917897 0.396818i $$-0.129886\pi$$
0.917897 + 0.396818i $$0.129886\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 713.834 0.944018 0.472009 0.881594i $$-0.343529\pi$$
0.472009 + 0.881594i $$0.343529\pi$$
$$84$$ 0 0
$$85$$ 245.299 0.313017
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 404.804 0.482125 0.241063 0.970510i $$-0.422504\pi$$
0.241063 + 0.970510i $$0.422504\pi$$
$$90$$ 0 0
$$91$$ 192.313 0.221537
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 405.054 0.437449
$$96$$ 0 0
$$97$$ 75.0273 0.0785347 0.0392674 0.999229i $$-0.487498\pi$$
0.0392674 + 0.999229i $$0.487498\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1087.88 1.07176 0.535881 0.844294i $$-0.319980\pi$$
0.535881 + 0.844294i $$0.319980\pi$$
$$102$$ 0 0
$$103$$ 1091.82 1.04447 0.522233 0.852803i $$-0.325099\pi$$
0.522233 + 0.852803i $$0.325099\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1029.15 −0.929833 −0.464917 0.885354i $$-0.653916\pi$$
−0.464917 + 0.885354i $$0.653916\pi$$
$$108$$ 0 0
$$109$$ 1776.52 1.56110 0.780548 0.625096i $$-0.214940\pi$$
0.780548 + 0.625096i $$0.214940\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1615.94 1.34526 0.672631 0.739978i $$-0.265164\pi$$
0.672631 + 0.739978i $$0.265164\pi$$
$$114$$ 0 0
$$115$$ −737.193 −0.597770
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −121.011 −0.0932187
$$120$$ 0 0
$$121$$ 1802.56 1.35429
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1477.18 1.05698
$$126$$ 0 0
$$127$$ 1206.10 0.842711 0.421356 0.906895i $$-0.361554\pi$$
0.421356 + 0.906895i $$0.361554\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1027.86 −0.685528 −0.342764 0.939422i $$-0.611363\pi$$
−0.342764 + 0.939422i $$0.611363\pi$$
$$132$$ 0 0
$$133$$ −199.821 −0.130276
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1260.91 0.786326 0.393163 0.919469i $$-0.371381\pi$$
0.393163 + 0.919469i $$0.371381\pi$$
$$138$$ 0 0
$$139$$ −461.832 −0.281813 −0.140907 0.990023i $$-0.545002\pi$$
−0.140907 + 0.990023i $$0.545002\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 2103.90 1.23033
$$144$$ 0 0
$$145$$ −294.285 −0.168545
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1459.32 −0.802365 −0.401182 0.915998i $$-0.631401\pi$$
−0.401182 + 0.915998i $$0.631401\pi$$
$$150$$ 0 0
$$151$$ −1541.32 −0.830666 −0.415333 0.909669i $$-0.636335\pi$$
−0.415333 + 0.909669i $$0.636335\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 133.628 0.0692467
$$156$$ 0 0
$$157$$ −3215.57 −1.63459 −0.817295 0.576220i $$-0.804527\pi$$
−0.817295 + 0.576220i $$0.804527\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 363.671 0.178021
$$162$$ 0 0
$$163$$ −947.587 −0.455342 −0.227671 0.973738i $$-0.573111\pi$$
−0.227671 + 0.973738i $$0.573111\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 685.960 0.317851 0.158926 0.987291i $$-0.449197\pi$$
0.158926 + 0.987291i $$0.449197\pi$$
$$168$$ 0 0
$$169$$ −784.426 −0.357044
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 2212.83 0.972475 0.486237 0.873827i $$-0.338369\pi$$
0.486237 + 0.873827i $$0.338369\pi$$
$$174$$ 0 0
$$175$$ −89.1138 −0.0384936
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 3023.22 1.26238 0.631190 0.775629i $$-0.282567\pi$$
0.631190 + 0.775629i $$0.282567\pi$$
$$180$$ 0 0
$$181$$ 391.445 0.160751 0.0803753 0.996765i $$-0.474388\pi$$
0.0803753 + 0.996765i $$0.474388\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1868.08 0.742400
$$186$$ 0 0
$$187$$ −1323.86 −0.517700
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3485.59 1.32046 0.660231 0.751062i $$-0.270458\pi$$
0.660231 + 0.751062i $$0.270458\pi$$
$$192$$ 0 0
$$193$$ 2215.07 0.826136 0.413068 0.910700i $$-0.364457\pi$$
0.413068 + 0.910700i $$0.364457\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3975.11 1.43764 0.718820 0.695196i $$-0.244682\pi$$
0.718820 + 0.695196i $$0.244682\pi$$
$$198$$ 0 0
$$199$$ 1555.34 0.554046 0.277023 0.960863i $$-0.410652\pi$$
0.277023 + 0.960863i $$0.410652\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 145.177 0.0501941
$$204$$ 0 0
$$205$$ −2235.06 −0.761479
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −2186.04 −0.723500
$$210$$ 0 0
$$211$$ −1747.73 −0.570231 −0.285115 0.958493i $$-0.592032\pi$$
−0.285115 + 0.958493i $$0.592032\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 635.133 0.201468
$$216$$ 0 0
$$217$$ −65.9211 −0.0206222
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −888.847 −0.270544
$$222$$ 0 0
$$223$$ 2541.94 0.763323 0.381662 0.924302i $$-0.375352\pi$$
0.381662 + 0.924302i $$0.375352\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −2993.26 −0.875197 −0.437598 0.899171i $$-0.644171\pi$$
−0.437598 + 0.899171i $$0.644171\pi$$
$$228$$ 0 0
$$229$$ −4305.31 −1.24237 −0.621185 0.783664i $$-0.713348\pi$$
−0.621185 + 0.783664i $$0.713348\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 5581.34 1.56930 0.784648 0.619942i $$-0.212844\pi$$
0.784648 + 0.619942i $$0.212844\pi$$
$$234$$ 0 0
$$235$$ 641.812 0.178158
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1409.63 0.381512 0.190756 0.981638i $$-0.438906\pi$$
0.190756 + 0.981638i $$0.438906\pi$$
$$240$$ 0 0
$$241$$ 626.572 0.167473 0.0837366 0.996488i $$-0.473315\pi$$
0.0837366 + 0.996488i $$0.473315\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 3286.12 0.856909
$$246$$ 0 0
$$247$$ −1467.72 −0.378093
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 1705.53 0.428892 0.214446 0.976736i $$-0.431205\pi$$
0.214446 + 0.976736i $$0.431205\pi$$
$$252$$ 0 0
$$253$$ 3978.56 0.988656
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 3597.38 0.873146 0.436573 0.899669i $$-0.356192\pi$$
0.436573 + 0.899669i $$0.356192\pi$$
$$258$$ 0 0
$$259$$ −921.560 −0.221092
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 4137.50 0.970074 0.485037 0.874494i $$-0.338806\pi$$
0.485037 + 0.874494i $$0.338806\pi$$
$$264$$ 0 0
$$265$$ 5106.37 1.18371
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −6090.99 −1.38057 −0.690287 0.723536i $$-0.742516\pi$$
−0.690287 + 0.723536i $$0.742516\pi$$
$$270$$ 0 0
$$271$$ 3196.62 0.716534 0.358267 0.933619i $$-0.383368\pi$$
0.358267 + 0.933619i $$0.383368\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −974.905 −0.213778
$$276$$ 0 0
$$277$$ −3119.36 −0.676622 −0.338311 0.941034i $$-0.609856\pi$$
−0.338311 + 0.941034i $$0.609856\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −4948.33 −1.05051 −0.525254 0.850946i $$-0.676030\pi$$
−0.525254 + 0.850946i $$0.676030\pi$$
$$282$$ 0 0
$$283$$ 4544.93 0.954658 0.477329 0.878725i $$-0.341605\pi$$
0.477329 + 0.878725i $$0.341605\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1102.60 0.226774
$$288$$ 0 0
$$289$$ −4353.70 −0.886160
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −6860.11 −1.36782 −0.683911 0.729566i $$-0.739722\pi$$
−0.683911 + 0.729566i $$0.739722\pi$$
$$294$$ 0 0
$$295$$ −8190.27 −1.61646
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 2671.24 0.516661
$$300$$ 0 0
$$301$$ −313.323 −0.0599988
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −5404.89 −1.01470
$$306$$ 0 0
$$307$$ −6332.25 −1.17720 −0.588600 0.808424i $$-0.700321\pi$$
−0.588600 + 0.808424i $$0.700321\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 7077.67 1.29048 0.645238 0.763982i $$-0.276758\pi$$
0.645238 + 0.763982i $$0.276758\pi$$
$$312$$ 0 0
$$313$$ 1381.30 0.249443 0.124721 0.992192i $$-0.460196\pi$$
0.124721 + 0.992192i $$0.460196\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8174.93 1.44842 0.724211 0.689578i $$-0.242204\pi$$
0.724211 + 0.689578i $$0.242204\pi$$
$$318$$ 0 0
$$319$$ 1588.23 0.278758
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 923.549 0.159095
$$324$$ 0 0
$$325$$ −654.559 −0.111718
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −316.618 −0.0530569
$$330$$ 0 0
$$331$$ 9661.28 1.60433 0.802163 0.597105i $$-0.203682\pi$$
0.802163 + 0.597105i $$0.203682\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 3157.63 0.514984
$$336$$ 0 0
$$337$$ −4956.02 −0.801103 −0.400552 0.916274i $$-0.631181\pi$$
−0.400552 + 0.916274i $$0.631181\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −721.177 −0.114528
$$342$$ 0 0
$$343$$ −3376.19 −0.531478
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −1015.60 −0.157120 −0.0785598 0.996909i $$-0.525032\pi$$
−0.0785598 + 0.996909i $$0.525032\pi$$
$$348$$ 0 0
$$349$$ 12158.6 1.86485 0.932426 0.361360i $$-0.117687\pi$$
0.932426 + 0.361360i $$0.117687\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −4236.08 −0.638708 −0.319354 0.947635i $$-0.603466\pi$$
−0.319354 + 0.947635i $$0.603466\pi$$
$$354$$ 0 0
$$355$$ −2804.58 −0.419300
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −517.939 −0.0761443 −0.0380721 0.999275i $$-0.512122\pi$$
−0.0380721 + 0.999275i $$0.512122\pi$$
$$360$$ 0 0
$$361$$ −5333.97 −0.777660
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 9599.18 1.37656
$$366$$ 0 0
$$367$$ 4616.29 0.656590 0.328295 0.944575i $$-0.393526\pi$$
0.328295 + 0.944575i $$0.393526\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2519.07 −0.352517
$$372$$ 0 0
$$373$$ 4765.42 0.661512 0.330756 0.943716i $$-0.392696\pi$$
0.330756 + 0.943716i $$0.392696\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1066.35 0.145676
$$378$$ 0 0
$$379$$ 2000.33 0.271108 0.135554 0.990770i $$-0.456719\pi$$
0.135554 + 0.990770i $$0.456719\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 990.294 0.132119 0.0660596 0.997816i $$-0.478957\pi$$
0.0660596 + 0.997816i $$0.478957\pi$$
$$384$$ 0 0
$$385$$ −2970.95 −0.393283
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −404.411 −0.0527106 −0.0263553 0.999653i $$-0.508390\pi$$
−0.0263553 + 0.999653i $$0.508390\pi$$
$$390$$ 0 0
$$391$$ −1680.85 −0.217402
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −13370.2 −1.70311
$$396$$ 0 0
$$397$$ 2919.61 0.369096 0.184548 0.982824i $$-0.440918\pi$$
0.184548 + 0.982824i $$0.440918\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 10186.2 1.26852 0.634258 0.773121i $$-0.281306\pi$$
0.634258 + 0.773121i $$0.281306\pi$$
$$402$$ 0 0
$$403$$ −484.203 −0.0598508
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −10081.9 −1.22786
$$408$$ 0 0
$$409$$ 6914.24 0.835910 0.417955 0.908468i $$-0.362747\pi$$
0.417955 + 0.908468i $$0.362747\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 4040.41 0.481394
$$414$$ 0 0
$$415$$ −7404.09 −0.875789
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 5120.31 0.597002 0.298501 0.954409i $$-0.403513\pi$$
0.298501 + 0.954409i $$0.403513\pi$$
$$420$$ 0 0
$$421$$ 1866.49 0.216074 0.108037 0.994147i $$-0.465543\pi$$
0.108037 + 0.994147i $$0.465543\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 411.874 0.0470090
$$426$$ 0 0
$$427$$ 2666.33 0.302185
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −4090.64 −0.457168 −0.228584 0.973524i $$-0.573410\pi$$
−0.228584 + 0.973524i $$0.573410\pi$$
$$432$$ 0 0
$$433$$ 633.052 0.0702599 0.0351299 0.999383i $$-0.488815\pi$$
0.0351299 + 0.999383i $$0.488815\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −2775.53 −0.303825
$$438$$ 0 0
$$439$$ 11306.5 1.22923 0.614614 0.788828i $$-0.289312\pi$$
0.614614 + 0.788828i $$0.289312\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −8281.30 −0.888163 −0.444082 0.895986i $$-0.646470\pi$$
−0.444082 + 0.895986i $$0.646470\pi$$
$$444$$ 0 0
$$445$$ −4198.74 −0.447280
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −6888.40 −0.724017 −0.362008 0.932175i $$-0.617909\pi$$
−0.362008 + 0.932175i $$0.617909\pi$$
$$450$$ 0 0
$$451$$ 12062.4 1.25941
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −1994.72 −0.205525
$$456$$ 0 0
$$457$$ 4283.60 0.438465 0.219233 0.975673i $$-0.429645\pi$$
0.219233 + 0.975673i $$0.429645\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −13778.3 −1.39202 −0.696009 0.718033i $$-0.745042\pi$$
−0.696009 + 0.718033i $$0.745042\pi$$
$$462$$ 0 0
$$463$$ −5734.53 −0.575608 −0.287804 0.957689i $$-0.592925\pi$$
−0.287804 + 0.957689i $$0.592925\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −8950.97 −0.886941 −0.443470 0.896289i $$-0.646253\pi$$
−0.443470 + 0.896289i $$0.646253\pi$$
$$468$$ 0 0
$$469$$ −1557.72 −0.153366
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −3427.75 −0.333210
$$474$$ 0 0
$$475$$ 680.114 0.0656964
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 9681.01 0.923459 0.461729 0.887021i $$-0.347229\pi$$
0.461729 + 0.887021i $$0.347229\pi$$
$$480$$ 0 0
$$481$$ −6769.04 −0.641666
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −778.204 −0.0728586
$$486$$ 0 0
$$487$$ −8704.66 −0.809950 −0.404975 0.914328i $$-0.632720\pi$$
−0.404975 + 0.914328i $$0.632720\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −15595.7 −1.43345 −0.716725 0.697356i $$-0.754360\pi$$
−0.716725 + 0.697356i $$0.754360\pi$$
$$492$$ 0 0
$$493$$ −670.989 −0.0612979
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1383.55 0.124871
$$498$$ 0 0
$$499$$ 9696.28 0.869870 0.434935 0.900462i $$-0.356771\pi$$
0.434935 + 0.900462i $$0.356771\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 20949.7 1.85706 0.928532 0.371253i $$-0.121072\pi$$
0.928532 + 0.371253i $$0.121072\pi$$
$$504$$ 0 0
$$505$$ −11283.8 −0.994300
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −11274.7 −0.981816 −0.490908 0.871211i $$-0.663335\pi$$
−0.490908 + 0.871211i $$0.663335\pi$$
$$510$$ 0 0
$$511$$ −4735.46 −0.409950
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −11324.6 −0.968977
$$516$$ 0 0
$$517$$ −3463.80 −0.294657
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −8675.49 −0.729520 −0.364760 0.931102i $$-0.618849\pi$$
−0.364760 + 0.931102i $$0.618849\pi$$
$$522$$ 0 0
$$523$$ 4226.14 0.353339 0.176670 0.984270i $$-0.443468\pi$$
0.176670 + 0.984270i $$0.443468\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 304.680 0.0251842
$$528$$ 0 0
$$529$$ −7115.58 −0.584826
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 8098.78 0.658156
$$534$$ 0 0
$$535$$ 10674.7 0.862630
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −17734.9 −1.41725
$$540$$ 0 0
$$541$$ 13357.8 1.06154 0.530771 0.847515i $$-0.321902\pi$$
0.530771 + 0.847515i $$0.321902\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −18426.5 −1.44827
$$546$$ 0 0
$$547$$ −21671.1 −1.69395 −0.846974 0.531634i $$-0.821578\pi$$
−0.846974 + 0.531634i $$0.821578\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −1107.98 −0.0856654
$$552$$ 0 0
$$553$$ 6595.79 0.507200
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −7477.63 −0.568828 −0.284414 0.958702i $$-0.591799\pi$$
−0.284414 + 0.958702i $$0.591799\pi$$
$$558$$ 0 0
$$559$$ −2301.42 −0.174132
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −23304.7 −1.74454 −0.872269 0.489026i $$-0.837352\pi$$
−0.872269 + 0.489026i $$0.837352\pi$$
$$564$$ 0 0
$$565$$ −16761.0 −1.24803
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 14649.1 1.07930 0.539650 0.841890i $$-0.318557\pi$$
0.539650 + 0.841890i $$0.318557\pi$$
$$570$$ 0 0
$$571$$ −23164.0 −1.69769 −0.848846 0.528640i $$-0.822702\pi$$
−0.848846 + 0.528640i $$0.822702\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1237.80 −0.0897735
$$576$$ 0 0
$$577$$ 7865.97 0.567529 0.283765 0.958894i $$-0.408417\pi$$
0.283765 + 0.958894i $$0.408417\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 3652.58 0.260817
$$582$$ 0 0
$$583$$ −27558.6 −1.95774
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −956.182 −0.0672332 −0.0336166 0.999435i $$-0.510703\pi$$
−0.0336166 + 0.999435i $$0.510703\pi$$
$$588$$ 0 0
$$589$$ 503.108 0.0351956
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 16966.0 1.17489 0.587444 0.809265i $$-0.300134\pi$$
0.587444 + 0.809265i $$0.300134\pi$$
$$594$$ 0 0
$$595$$ 1255.16 0.0864813
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −6191.41 −0.422327 −0.211164 0.977451i $$-0.567725\pi$$
−0.211164 + 0.977451i $$0.567725\pi$$
$$600$$ 0 0
$$601$$ −2718.54 −0.184512 −0.0922559 0.995735i $$-0.529408\pi$$
−0.0922559 + 0.995735i $$0.529408\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −18696.7 −1.25641
$$606$$ 0 0
$$607$$ −16825.0 −1.12505 −0.562524 0.826781i $$-0.690170\pi$$
−0.562524 + 0.826781i $$0.690170\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −2325.62 −0.153985
$$612$$ 0 0
$$613$$ −20175.1 −1.32930 −0.664652 0.747153i $$-0.731420\pi$$
−0.664652 + 0.747153i $$0.731420\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −11310.6 −0.738004 −0.369002 0.929429i $$-0.620301\pi$$
−0.369002 + 0.929429i $$0.620301\pi$$
$$618$$ 0 0
$$619$$ 17059.9 1.10775 0.553873 0.832601i $$-0.313149\pi$$
0.553873 + 0.832601i $$0.313149\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 2071.32 0.133203
$$624$$ 0 0
$$625$$ −13144.7 −0.841262
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 4259.34 0.270002
$$630$$ 0 0
$$631$$ 13186.3 0.831916 0.415958 0.909384i $$-0.363446\pi$$
0.415958 + 0.909384i $$0.363446\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −12510.0 −0.781805
$$636$$ 0 0
$$637$$ −11907.4 −0.740638
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −16362.0 −1.00820 −0.504102 0.863644i $$-0.668177\pi$$
−0.504102 + 0.863644i $$0.668177\pi$$
$$642$$ 0 0
$$643$$ 28044.9 1.72004 0.860019 0.510262i $$-0.170452\pi$$
0.860019 + 0.510262i $$0.170452\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 21247.7 1.29109 0.645543 0.763724i $$-0.276631\pi$$
0.645543 + 0.763724i $$0.276631\pi$$
$$648$$ 0 0
$$649$$ 44202.1 2.67347
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −1259.86 −0.0755007 −0.0377504 0.999287i $$-0.512019\pi$$
−0.0377504 + 0.999287i $$0.512019\pi$$
$$654$$ 0 0
$$655$$ 10661.2 0.635981
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −12046.7 −0.712098 −0.356049 0.934467i $$-0.615876\pi$$
−0.356049 + 0.934467i $$0.615876\pi$$
$$660$$ 0 0
$$661$$ 13108.1 0.771324 0.385662 0.922640i $$-0.373973\pi$$
0.385662 + 0.922640i $$0.373973\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2072.60 0.120860
$$666$$ 0 0
$$667$$ 2016.51 0.117061
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 29169.7 1.67822
$$672$$ 0 0
$$673$$ 2743.65 0.157147 0.0785734 0.996908i $$-0.474963\pi$$
0.0785734 + 0.996908i $$0.474963\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −25004.0 −1.41947 −0.709735 0.704468i $$-0.751186\pi$$
−0.709735 + 0.704468i $$0.751186\pi$$
$$678$$ 0 0
$$679$$ 383.903 0.0216979
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −4846.23 −0.271502 −0.135751 0.990743i $$-0.543345\pi$$
−0.135751 + 0.990743i $$0.543345\pi$$
$$684$$ 0 0
$$685$$ −13078.5 −0.729494
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −18503.1 −1.02309
$$690$$ 0 0
$$691$$ −3484.58 −0.191837 −0.0959187 0.995389i $$-0.530579\pi$$
−0.0959187 + 0.995389i $$0.530579\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 4790.25 0.261445
$$696$$ 0 0
$$697$$ −5096.07 −0.276940
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −15701.4 −0.845981 −0.422991 0.906134i $$-0.639020\pi$$
−0.422991 + 0.906134i $$0.639020\pi$$
$$702$$ 0 0
$$703$$ 7033.32 0.377335
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 5566.50 0.296110
$$708$$ 0 0
$$709$$ 15643.4 0.828634 0.414317 0.910133i $$-0.364020\pi$$
0.414317 + 0.910133i $$0.364020\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −915.649 −0.0480944
$$714$$ 0 0
$$715$$ −21822.2 −1.14141
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −6964.13 −0.361222 −0.180611 0.983555i $$-0.557807\pi$$
−0.180611 + 0.983555i $$0.557807\pi$$
$$720$$ 0 0
$$721$$ 5586.66 0.288569
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −494.125 −0.0253122
$$726$$ 0 0
$$727$$ 14207.2 0.724782 0.362391 0.932026i $$-0.381961\pi$$
0.362391 + 0.932026i $$0.381961\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 1448.14 0.0732716
$$732$$ 0 0
$$733$$ 26530.5 1.33687 0.668437 0.743769i $$-0.266964\pi$$
0.668437 + 0.743769i $$0.266964\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −17041.4 −0.851735
$$738$$ 0 0
$$739$$ 5683.47 0.282909 0.141455 0.989945i $$-0.454822\pi$$
0.141455 + 0.989945i $$0.454822\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −15568.6 −0.768715 −0.384358 0.923184i $$-0.625577\pi$$
−0.384358 + 0.923184i $$0.625577\pi$$
$$744$$ 0 0
$$745$$ 15136.5 0.744374
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −5266.02 −0.256898
$$750$$ 0 0
$$751$$ 8261.64 0.401427 0.200713 0.979650i $$-0.435674\pi$$
0.200713 + 0.979650i $$0.435674\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 15987.0 0.770630
$$756$$ 0 0
$$757$$ −13381.5 −0.642481 −0.321240 0.946998i $$-0.604100\pi$$
−0.321240 + 0.946998i $$0.604100\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −5449.84 −0.259601 −0.129801 0.991540i $$-0.541434\pi$$
−0.129801 + 0.991540i $$0.541434\pi$$
$$762$$ 0 0
$$763$$ 9090.15 0.431305
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 29677.6 1.39713
$$768$$ 0 0
$$769$$ −19364.0 −0.908039 −0.454020 0.890992i $$-0.650010\pi$$
−0.454020 + 0.890992i $$0.650010\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 1865.54 0.0868033 0.0434017 0.999058i $$-0.486180\pi$$
0.0434017 + 0.999058i $$0.486180\pi$$
$$774$$ 0 0
$$775$$ 224.370 0.0103995
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −8414.98 −0.387032
$$780$$ 0 0
$$781$$ 15136.0 0.693483
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 33352.8 1.51645
$$786$$ 0 0
$$787$$ 19207.3 0.869970 0.434985 0.900438i $$-0.356754\pi$$
0.434985 + 0.900438i $$0.356754\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 8268.50 0.371674
$$792$$ 0 0
$$793$$ 19584.7 0.877017
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −186.074 −0.00826988 −0.00413494 0.999991i $$-0.501316\pi$$
−0.00413494 + 0.999991i $$0.501316\pi$$
$$798$$ 0 0
$$799$$ 1463.37 0.0647940
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −51805.9 −2.27670
$$804$$ 0 0
$$805$$ −3772.10 −0.165154
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −5903.09 −0.256541 −0.128270 0.991739i $$-0.540943\pi$$
−0.128270 + 0.991739i $$0.540943\pi$$
$$810$$ 0 0
$$811$$ −23111.0 −1.00066 −0.500331 0.865834i $$-0.666788\pi$$
−0.500331 + 0.865834i $$0.666788\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 9828.64 0.422432
$$816$$ 0 0
$$817$$ 2391.27 0.102399
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 9644.29 0.409973 0.204987 0.978765i $$-0.434285\pi$$
0.204987 + 0.978765i $$0.434285\pi$$
$$822$$ 0 0
$$823$$ 33573.4 1.42199 0.710994 0.703198i $$-0.248245\pi$$
0.710994 + 0.703198i $$0.248245\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −25916.1 −1.08971 −0.544855 0.838530i $$-0.683415\pi$$
−0.544855 + 0.838530i $$0.683415\pi$$
$$828$$ 0 0
$$829$$ −28650.6 −1.20033 −0.600166 0.799876i $$-0.704899\pi$$
−0.600166 + 0.799876i $$0.704899\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 7492.58 0.311647
$$834$$ 0 0
$$835$$ −7114.97 −0.294878
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 712.960 0.0293374 0.0146687 0.999892i $$-0.495331\pi$$
0.0146687 + 0.999892i $$0.495331\pi$$
$$840$$ 0 0
$$841$$ −23584.0 −0.966994
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 8136.29 0.331239
$$846$$ 0 0
$$847$$ 9223.44 0.374169
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −12800.5 −0.515625
$$852$$ 0 0
$$853$$ −30367.2 −1.21894 −0.609469 0.792810i $$-0.708617\pi$$
−0.609469 + 0.792810i $$0.708617\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −9080.70 −0.361950 −0.180975 0.983488i $$-0.557925\pi$$
−0.180975 + 0.983488i $$0.557925\pi$$
$$858$$ 0 0
$$859$$ −26160.2 −1.03909 −0.519543 0.854444i $$-0.673898\pi$$
−0.519543 + 0.854444i $$0.673898\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −40102.0 −1.58180 −0.790898 0.611949i $$-0.790386\pi$$
−0.790898 + 0.611949i $$0.790386\pi$$
$$864$$ 0 0
$$865$$ −22952.1 −0.902189
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 72157.9 2.81679
$$870$$ 0 0
$$871$$ −11441.7 −0.445108
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 7558.48 0.292027
$$876$$ 0 0
$$877$$ 25252.2 0.972299 0.486149 0.873876i $$-0.338401\pi$$
0.486149 + 0.873876i $$0.338401\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 2049.26 0.0783670 0.0391835 0.999232i $$-0.487524\pi$$
0.0391835 + 0.999232i $$0.487524\pi$$
$$882$$ 0 0
$$883$$ −39413.4 −1.50211 −0.751057 0.660237i $$-0.770456\pi$$
−0.751057 + 0.660237i $$0.770456\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 36968.5 1.39941 0.699707 0.714430i $$-0.253314\pi$$
0.699707 + 0.714430i $$0.253314\pi$$
$$888$$ 0 0
$$889$$ 6171.44 0.232827
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 2416.42 0.0905514
$$894$$ 0 0
$$895$$ −31357.7 −1.17114
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −365.525 −0.0135605
$$900$$ 0 0
$$901$$ 11642.9 0.430499
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −4060.17 −0.149132
$$906$$ 0 0
$$907$$ −2710.62 −0.0992334 −0.0496167 0.998768i $$-0.515800\pi$$
−0.0496167 + 0.998768i $$0.515800\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 22996.6 0.836345 0.418172 0.908368i $$-0.362671\pi$$
0.418172 + 0.908368i $$0.362671\pi$$
$$912$$ 0 0
$$913$$ 39959.2 1.44847
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −5259.38 −0.189400
$$918$$ 0 0
$$919$$ 39103.8 1.40361 0.701804 0.712370i $$-0.252378\pi$$
0.701804 + 0.712370i $$0.252378\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 10162.5 0.362407
$$924$$ 0 0
$$925$$ 3136.64 0.111494
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 35954.6 1.26979 0.634894 0.772600i $$-0.281044\pi$$
0.634894 + 0.772600i $$0.281044\pi$$
$$930$$ 0 0
$$931$$ 12372.2 0.435536
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 13731.4 0.480283
$$936$$ 0 0
$$937$$ −7263.94 −0.253258 −0.126629 0.991950i $$-0.540416\pi$$
−0.126629 + 0.991950i $$0.540416\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −7478.91 −0.259092 −0.129546 0.991573i $$-0.541352\pi$$
−0.129546 + 0.991573i $$0.541352\pi$$
$$942$$ 0 0
$$943$$ 15315.1 0.528875
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 13491.4 0.462947 0.231473 0.972841i $$-0.425645\pi$$
0.231473 + 0.972841i $$0.425645\pi$$
$$948$$ 0 0
$$949$$ −34782.9 −1.18978
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 13981.6 0.475246 0.237623 0.971357i $$-0.423632\pi$$
0.237623 + 0.971357i $$0.423632\pi$$
$$954$$ 0 0
$$955$$ −36153.5 −1.22503
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 6451.87 0.217249
$$960$$ 0 0
$$961$$ −29625.0 −0.994429
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −22975.3 −0.766427
$$966$$ 0 0
$$967$$ 9081.47 0.302007 0.151003 0.988533i $$-0.451750\pi$$
0.151003 + 0.988533i $$0.451750\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −9709.13 −0.320887 −0.160443 0.987045i $$-0.551292\pi$$
−0.160443 + 0.987045i $$0.551292\pi$$
$$972$$ 0 0
$$973$$ −2363.12 −0.0778604
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −10854.9 −0.355455 −0.177727 0.984080i $$-0.556875\pi$$
−0.177727 + 0.984080i $$0.556875\pi$$
$$978$$ 0 0
$$979$$ 22660.2 0.739759
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −7510.10 −0.243678 −0.121839 0.992550i $$-0.538879\pi$$
−0.121839 + 0.992550i $$0.538879\pi$$
$$984$$ 0 0
$$985$$ −41231.0 −1.33373
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −4352.08 −0.139927
$$990$$ 0 0
$$991$$ −46125.6 −1.47854 −0.739268 0.673412i $$-0.764828\pi$$
−0.739268 + 0.673412i $$0.764828\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −16132.4 −0.514003
$$996$$ 0 0
$$997$$ 45350.1 1.44057 0.720287 0.693677i $$-0.244010\pi$$
0.720287 + 0.693677i $$0.244010\pi$$
$$998$$ 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 1296.4.a.i.1.1 2
3.2 odd 2 1296.4.a.u.1.2 2
4.3 odd 2 81.4.a.a.1.2 2
9.2 odd 6 144.4.i.c.49.2 4
9.4 even 3 432.4.i.c.289.2 4
9.5 odd 6 144.4.i.c.97.2 4
9.7 even 3 432.4.i.c.145.2 4
12.11 even 2 81.4.a.d.1.1 2
20.19 odd 2 2025.4.a.n.1.1 2
36.7 odd 6 27.4.c.a.10.1 4
36.11 even 6 9.4.c.a.4.2 4
36.23 even 6 9.4.c.a.7.2 yes 4
36.31 odd 6 27.4.c.a.19.1 4
60.59 even 2 2025.4.a.g.1.2 2
180.23 odd 12 225.4.k.b.124.2 8
180.47 odd 12 225.4.k.b.49.2 8
180.59 even 6 225.4.e.b.151.1 4
180.83 odd 12 225.4.k.b.49.3 8
180.119 even 6 225.4.e.b.76.1 4
180.167 odd 12 225.4.k.b.124.3 8

By twisted newform
Twist Min Dim Char Parity Ord Type
9.4.c.a.4.2 4 36.11 even 6
9.4.c.a.7.2 yes 4 36.23 even 6
27.4.c.a.10.1 4 36.7 odd 6
27.4.c.a.19.1 4 36.31 odd 6
81.4.a.a.1.2 2 4.3 odd 2
81.4.a.d.1.1 2 12.11 even 2
144.4.i.c.49.2 4 9.2 odd 6
144.4.i.c.97.2 4 9.5 odd 6
225.4.e.b.76.1 4 180.119 even 6
225.4.e.b.151.1 4 180.59 even 6
225.4.k.b.49.2 8 180.47 odd 12
225.4.k.b.49.3 8 180.83 odd 12
225.4.k.b.124.2 8 180.23 odd 12
225.4.k.b.124.3 8 180.167 odd 12
432.4.i.c.145.2 4 9.7 even 3
432.4.i.c.289.2 4 9.4 even 3
1296.4.a.i.1.1 2 1.1 even 1 trivial
1296.4.a.u.1.2 2 3.2 odd 2
2025.4.a.g.1.2 2 60.59 even 2
2025.4.a.n.1.1 2 20.19 odd 2