# Properties

 Label 768.4.a.n.1.1 Level $768$ Weight $4$ Character 768.1 Self dual yes Analytic conductor $45.313$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$45.3134668844$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 192) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 768.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -10.3923 q^{5} +3.46410 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -10.3923 q^{5} +3.46410 q^{7} +9.00000 q^{9} +55.4256 q^{13} -31.1769 q^{15} -90.0000 q^{17} -116.000 q^{19} +10.3923 q^{21} +103.923 q^{23} -17.0000 q^{25} +27.0000 q^{27} +259.808 q^{29} -301.377 q^{31} -36.0000 q^{35} -34.6410 q^{37} +166.277 q^{39} +54.0000 q^{41} -20.0000 q^{43} -93.5307 q^{45} +394.908 q^{47} -331.000 q^{49} -270.000 q^{51} -488.438 q^{53} -348.000 q^{57} -324.000 q^{59} -575.041 q^{61} +31.1769 q^{63} -576.000 q^{65} +116.000 q^{67} +311.769 q^{69} -1101.58 q^{71} -1106.00 q^{73} -51.0000 q^{75} -148.956 q^{79} +81.0000 q^{81} +1152.00 q^{83} +935.307 q^{85} +779.423 q^{87} -918.000 q^{89} +192.000 q^{91} -904.131 q^{93} +1205.51 q^{95} +190.000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 18 * q^9 $$2 q + 6 q^{3} + 18 q^{9} - 180 q^{17} - 232 q^{19} - 34 q^{25} + 54 q^{27} - 72 q^{35} + 108 q^{41} - 40 q^{43} - 662 q^{49} - 540 q^{51} - 696 q^{57} - 648 q^{59} - 1152 q^{65} + 232 q^{67} - 2212 q^{73} - 102 q^{75} + 162 q^{81} + 2304 q^{83} - 1836 q^{89} + 384 q^{91} + 380 q^{97}+O(q^{100})$$ 2 * q + 6 * q^3 + 18 * q^9 - 180 * q^17 - 232 * q^19 - 34 * q^25 + 54 * q^27 - 72 * q^35 + 108 * q^41 - 40 * q^43 - 662 * q^49 - 540 * q^51 - 696 * q^57 - 648 * q^59 - 1152 * q^65 + 232 * q^67 - 2212 * q^73 - 102 * q^75 + 162 * q^81 + 2304 * q^83 - 1836 * q^89 + 384 * q^91 + 380 * 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
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −10.3923 −0.929516 −0.464758 0.885438i $$-0.653859\pi$$
−0.464758 + 0.885438i $$0.653859\pi$$
$$6$$ 0 0
$$7$$ 3.46410 0.187044 0.0935220 0.995617i $$-0.470187\pi$$
0.0935220 + 0.995617i $$0.470187\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 55.4256 1.18248 0.591242 0.806494i $$-0.298638\pi$$
0.591242 + 0.806494i $$0.298638\pi$$
$$14$$ 0 0
$$15$$ −31.1769 −0.536656
$$16$$ 0 0
$$17$$ −90.0000 −1.28401 −0.642006 0.766700i $$-0.721898\pi$$
−0.642006 + 0.766700i $$0.721898\pi$$
$$18$$ 0 0
$$19$$ −116.000 −1.40064 −0.700322 0.713827i $$-0.746960\pi$$
−0.700322 + 0.713827i $$0.746960\pi$$
$$20$$ 0 0
$$21$$ 10.3923 0.107990
$$22$$ 0 0
$$23$$ 103.923 0.942150 0.471075 0.882093i $$-0.343866\pi$$
0.471075 + 0.882093i $$0.343866\pi$$
$$24$$ 0 0
$$25$$ −17.0000 −0.136000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 259.808 1.66362 0.831811 0.555058i $$-0.187304\pi$$
0.831811 + 0.555058i $$0.187304\pi$$
$$30$$ 0 0
$$31$$ −301.377 −1.74609 −0.873046 0.487637i $$-0.837859\pi$$
−0.873046 + 0.487637i $$0.837859\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −36.0000 −0.173860
$$36$$ 0 0
$$37$$ −34.6410 −0.153918 −0.0769588 0.997034i $$-0.524521\pi$$
−0.0769588 + 0.997034i $$0.524521\pi$$
$$38$$ 0 0
$$39$$ 166.277 0.682708
$$40$$ 0 0
$$41$$ 54.0000 0.205692 0.102846 0.994697i $$-0.467205\pi$$
0.102846 + 0.994697i $$0.467205\pi$$
$$42$$ 0 0
$$43$$ −20.0000 −0.0709296 −0.0354648 0.999371i $$-0.511291\pi$$
−0.0354648 + 0.999371i $$0.511291\pi$$
$$44$$ 0 0
$$45$$ −93.5307 −0.309839
$$46$$ 0 0
$$47$$ 394.908 1.22560 0.612800 0.790238i $$-0.290043\pi$$
0.612800 + 0.790238i $$0.290043\pi$$
$$48$$ 0 0
$$49$$ −331.000 −0.965015
$$50$$ 0 0
$$51$$ −270.000 −0.741325
$$52$$ 0 0
$$53$$ −488.438 −1.26589 −0.632945 0.774197i $$-0.718154\pi$$
−0.632945 + 0.774197i $$0.718154\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −348.000 −0.808662
$$58$$ 0 0
$$59$$ −324.000 −0.714936 −0.357468 0.933925i $$-0.616360\pi$$
−0.357468 + 0.933925i $$0.616360\pi$$
$$60$$ 0 0
$$61$$ −575.041 −1.20699 −0.603495 0.797366i $$-0.706226\pi$$
−0.603495 + 0.797366i $$0.706226\pi$$
$$62$$ 0 0
$$63$$ 31.1769 0.0623480
$$64$$ 0 0
$$65$$ −576.000 −1.09914
$$66$$ 0 0
$$67$$ 116.000 0.211517 0.105759 0.994392i $$-0.466273\pi$$
0.105759 + 0.994392i $$0.466273\pi$$
$$68$$ 0 0
$$69$$ 311.769 0.543951
$$70$$ 0 0
$$71$$ −1101.58 −1.84132 −0.920662 0.390361i $$-0.872350\pi$$
−0.920662 + 0.390361i $$0.872350\pi$$
$$72$$ 0 0
$$73$$ −1106.00 −1.77325 −0.886627 0.462486i $$-0.846958\pi$$
−0.886627 + 0.462486i $$0.846958\pi$$
$$74$$ 0 0
$$75$$ −51.0000 −0.0785196
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −148.956 −0.212138 −0.106069 0.994359i $$-0.533826\pi$$
−0.106069 + 0.994359i $$0.533826\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 1152.00 1.52348 0.761738 0.647886i $$-0.224347\pi$$
0.761738 + 0.647886i $$0.224347\pi$$
$$84$$ 0 0
$$85$$ 935.307 1.19351
$$86$$ 0 0
$$87$$ 779.423 0.960493
$$88$$ 0 0
$$89$$ −918.000 −1.09335 −0.546673 0.837346i $$-0.684106\pi$$
−0.546673 + 0.837346i $$0.684106\pi$$
$$90$$ 0 0
$$91$$ 192.000 0.221177
$$92$$ 0 0
$$93$$ −904.131 −1.00811
$$94$$ 0 0
$$95$$ 1205.51 1.30192
$$96$$ 0 0
$$97$$ 190.000 0.198882 0.0994411 0.995043i $$-0.468295\pi$$
0.0994411 + 0.995043i $$0.468295\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 10.3923 0.0102383 0.00511917 0.999987i $$-0.498371\pi$$
0.00511917 + 0.999987i $$0.498371\pi$$
$$102$$ 0 0
$$103$$ −793.279 −0.758875 −0.379438 0.925217i $$-0.623882\pi$$
−0.379438 + 0.925217i $$0.623882\pi$$
$$104$$ 0 0
$$105$$ −108.000 −0.100378
$$106$$ 0 0
$$107$$ 252.000 0.227680 0.113840 0.993499i $$-0.463685\pi$$
0.113840 + 0.993499i $$0.463685\pi$$
$$108$$ 0 0
$$109$$ −457.261 −0.401814 −0.200907 0.979610i $$-0.564389\pi$$
−0.200907 + 0.979610i $$0.564389\pi$$
$$110$$ 0 0
$$111$$ −103.923 −0.0888643
$$112$$ 0 0
$$113$$ −2214.00 −1.84315 −0.921573 0.388204i $$-0.873096\pi$$
−0.921573 + 0.388204i $$0.873096\pi$$
$$114$$ 0 0
$$115$$ −1080.00 −0.875744
$$116$$ 0 0
$$117$$ 498.831 0.394162
$$118$$ 0 0
$$119$$ −311.769 −0.240167
$$120$$ 0 0
$$121$$ −1331.00 −1.00000
$$122$$ 0 0
$$123$$ 162.000 0.118756
$$124$$ 0 0
$$125$$ 1475.71 1.05593
$$126$$ 0 0
$$127$$ −696.284 −0.486498 −0.243249 0.969964i $$-0.578213\pi$$
−0.243249 + 0.969964i $$0.578213\pi$$
$$128$$ 0 0
$$129$$ −60.0000 −0.0409512
$$130$$ 0 0
$$131$$ 2268.00 1.51264 0.756321 0.654201i $$-0.226995\pi$$
0.756321 + 0.654201i $$0.226995\pi$$
$$132$$ 0 0
$$133$$ −401.836 −0.261982
$$134$$ 0 0
$$135$$ −280.592 −0.178885
$$136$$ 0 0
$$137$$ −522.000 −0.325529 −0.162764 0.986665i $$-0.552041\pi$$
−0.162764 + 0.986665i $$0.552041\pi$$
$$138$$ 0 0
$$139$$ −676.000 −0.412501 −0.206250 0.978499i $$-0.566126\pi$$
−0.206250 + 0.978499i $$0.566126\pi$$
$$140$$ 0 0
$$141$$ 1184.72 0.707600
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −2700.00 −1.54636
$$146$$ 0 0
$$147$$ −993.000 −0.557151
$$148$$ 0 0
$$149$$ −1465.31 −0.805660 −0.402830 0.915275i $$-0.631973\pi$$
−0.402830 + 0.915275i $$0.631973\pi$$
$$150$$ 0 0
$$151$$ 2386.77 1.28631 0.643153 0.765738i $$-0.277626\pi$$
0.643153 + 0.765738i $$0.277626\pi$$
$$152$$ 0 0
$$153$$ −810.000 −0.428004
$$154$$ 0 0
$$155$$ 3132.00 1.62302
$$156$$ 0 0
$$157$$ 2016.11 1.02486 0.512430 0.858729i $$-0.328746\pi$$
0.512430 + 0.858729i $$0.328746\pi$$
$$158$$ 0 0
$$159$$ −1465.31 −0.730862
$$160$$ 0 0
$$161$$ 360.000 0.176223
$$162$$ 0 0
$$163$$ 388.000 0.186445 0.0932224 0.995645i $$-0.470283\pi$$
0.0932224 + 0.995645i $$0.470283\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 2203.17 1.02088 0.510438 0.859915i $$-0.329483\pi$$
0.510438 + 0.859915i $$0.329483\pi$$
$$168$$ 0 0
$$169$$ 875.000 0.398270
$$170$$ 0 0
$$171$$ −1044.00 −0.466881
$$172$$ 0 0
$$173$$ 197.454 0.0867753 0.0433877 0.999058i $$-0.486185\pi$$
0.0433877 + 0.999058i $$0.486185\pi$$
$$174$$ 0 0
$$175$$ −58.8897 −0.0254380
$$176$$ 0 0
$$177$$ −972.000 −0.412768
$$178$$ 0 0
$$179$$ −2844.00 −1.18754 −0.593772 0.804633i $$-0.702362\pi$$
−0.593772 + 0.804633i $$0.702362\pi$$
$$180$$ 0 0
$$181$$ 96.9948 0.0398319 0.0199159 0.999802i $$-0.493660\pi$$
0.0199159 + 0.999802i $$0.493660\pi$$
$$182$$ 0 0
$$183$$ −1725.12 −0.696856
$$184$$ 0 0
$$185$$ 360.000 0.143069
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 93.5307 0.0359966
$$190$$ 0 0
$$191$$ −3200.83 −1.21259 −0.606293 0.795241i $$-0.707344\pi$$
−0.606293 + 0.795241i $$0.707344\pi$$
$$192$$ 0 0
$$193$$ −1342.00 −0.500514 −0.250257 0.968179i $$-0.580515\pi$$
−0.250257 + 0.968179i $$0.580515\pi$$
$$194$$ 0 0
$$195$$ −1728.00 −0.634588
$$196$$ 0 0
$$197$$ 966.484 0.349539 0.174769 0.984609i $$-0.444082\pi$$
0.174769 + 0.984609i $$0.444082\pi$$
$$198$$ 0 0
$$199$$ 1784.01 0.635504 0.317752 0.948174i $$-0.397072\pi$$
0.317752 + 0.948174i $$0.397072\pi$$
$$200$$ 0 0
$$201$$ 348.000 0.122120
$$202$$ 0 0
$$203$$ 900.000 0.311171
$$204$$ 0 0
$$205$$ −561.184 −0.191194
$$206$$ 0 0
$$207$$ 935.307 0.314050
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −2764.00 −0.901809 −0.450904 0.892572i $$-0.648898\pi$$
−0.450904 + 0.892572i $$0.648898\pi$$
$$212$$ 0 0
$$213$$ −3304.75 −1.06309
$$214$$ 0 0
$$215$$ 207.846 0.0659302
$$216$$ 0 0
$$217$$ −1044.00 −0.326596
$$218$$ 0 0
$$219$$ −3318.00 −1.02379
$$220$$ 0 0
$$221$$ −4988.31 −1.51832
$$222$$ 0 0
$$223$$ 4292.02 1.28886 0.644428 0.764665i $$-0.277095\pi$$
0.644428 + 0.764665i $$0.277095\pi$$
$$224$$ 0 0
$$225$$ −153.000 −0.0453333
$$226$$ 0 0
$$227$$ 5688.00 1.66311 0.831555 0.555443i $$-0.187451\pi$$
0.831555 + 0.555443i $$0.187451\pi$$
$$228$$ 0 0
$$229$$ −5570.28 −1.60740 −0.803699 0.595036i $$-0.797138\pi$$
−0.803699 + 0.595036i $$0.797138\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2718.00 −0.764215 −0.382108 0.924118i $$-0.624802\pi$$
−0.382108 + 0.924118i $$0.624802\pi$$
$$234$$ 0 0
$$235$$ −4104.00 −1.13921
$$236$$ 0 0
$$237$$ −446.869 −0.122478
$$238$$ 0 0
$$239$$ −3574.95 −0.967550 −0.483775 0.875192i $$-0.660735\pi$$
−0.483775 + 0.875192i $$0.660735\pi$$
$$240$$ 0 0
$$241$$ 4490.00 1.20011 0.600055 0.799959i $$-0.295146\pi$$
0.600055 + 0.799959i $$0.295146\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 3439.85 0.896996
$$246$$ 0 0
$$247$$ −6429.37 −1.65624
$$248$$ 0 0
$$249$$ 3456.00 0.879579
$$250$$ 0 0
$$251$$ 4608.00 1.15878 0.579391 0.815050i $$-0.303290\pi$$
0.579391 + 0.815050i $$0.303290\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 2805.92 0.689073
$$256$$ 0 0
$$257$$ 4626.00 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ −120.000 −0.0287893
$$260$$ 0 0
$$261$$ 2338.27 0.554541
$$262$$ 0 0
$$263$$ −1995.32 −0.467821 −0.233910 0.972258i $$-0.575152\pi$$
−0.233910 + 0.972258i $$0.575152\pi$$
$$264$$ 0 0
$$265$$ 5076.00 1.17666
$$266$$ 0 0
$$267$$ −2754.00 −0.631244
$$268$$ 0 0
$$269$$ −3148.87 −0.713717 −0.356859 0.934158i $$-0.616152\pi$$
−0.356859 + 0.934158i $$0.616152\pi$$
$$270$$ 0 0
$$271$$ 5345.11 1.19813 0.599063 0.800702i $$-0.295540\pi$$
0.599063 + 0.800702i $$0.295540\pi$$
$$272$$ 0 0
$$273$$ 576.000 0.127696
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 6526.37 1.41564 0.707818 0.706394i $$-0.249679\pi$$
0.707818 + 0.706394i $$0.249679\pi$$
$$278$$ 0 0
$$279$$ −2712.39 −0.582031
$$280$$ 0 0
$$281$$ 1170.00 0.248386 0.124193 0.992258i $$-0.460366\pi$$
0.124193 + 0.992258i $$0.460366\pi$$
$$282$$ 0 0
$$283$$ −5740.00 −1.20568 −0.602840 0.797862i $$-0.705964\pi$$
−0.602840 + 0.797862i $$0.705964\pi$$
$$284$$ 0 0
$$285$$ 3616.52 0.751664
$$286$$ 0 0
$$287$$ 187.061 0.0384735
$$288$$ 0 0
$$289$$ 3187.00 0.648687
$$290$$ 0 0
$$291$$ 570.000 0.114825
$$292$$ 0 0
$$293$$ 7991.68 1.59344 0.796722 0.604346i $$-0.206566\pi$$
0.796722 + 0.604346i $$0.206566\pi$$
$$294$$ 0 0
$$295$$ 3367.11 0.664544
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 5760.00 1.11408
$$300$$ 0 0
$$301$$ −69.2820 −0.0132669
$$302$$ 0 0
$$303$$ 31.1769 0.00591111
$$304$$ 0 0
$$305$$ 5976.00 1.12192
$$306$$ 0 0
$$307$$ −5452.00 −1.01356 −0.506779 0.862076i $$-0.669164\pi$$
−0.506779 + 0.862076i $$0.669164\pi$$
$$308$$ 0 0
$$309$$ −2379.84 −0.438137
$$310$$ 0 0
$$311$$ 2203.17 0.401705 0.200852 0.979622i $$-0.435629\pi$$
0.200852 + 0.979622i $$0.435629\pi$$
$$312$$ 0 0
$$313$$ 1034.00 0.186726 0.0933628 0.995632i $$-0.470238\pi$$
0.0933628 + 0.995632i $$0.470238\pi$$
$$314$$ 0 0
$$315$$ −324.000 −0.0579534
$$316$$ 0 0
$$317$$ −2650.04 −0.469530 −0.234765 0.972052i $$-0.575432\pi$$
−0.234765 + 0.972052i $$0.575432\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 756.000 0.131451
$$322$$ 0 0
$$323$$ 10440.0 1.79844
$$324$$ 0 0
$$325$$ −942.236 −0.160818
$$326$$ 0 0
$$327$$ −1371.78 −0.231987
$$328$$ 0 0
$$329$$ 1368.00 0.229241
$$330$$ 0 0
$$331$$ −4132.00 −0.686149 −0.343074 0.939308i $$-0.611468\pi$$
−0.343074 + 0.939308i $$0.611468\pi$$
$$332$$ 0 0
$$333$$ −311.769 −0.0513058
$$334$$ 0 0
$$335$$ −1205.51 −0.196609
$$336$$ 0 0
$$337$$ −458.000 −0.0740322 −0.0370161 0.999315i $$-0.511785\pi$$
−0.0370161 + 0.999315i $$0.511785\pi$$
$$338$$ 0 0
$$339$$ −6642.00 −1.06414
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −2334.80 −0.367544
$$344$$ 0 0
$$345$$ −3240.00 −0.505611
$$346$$ 0 0
$$347$$ 11016.0 1.70424 0.852118 0.523350i $$-0.175318\pi$$
0.852118 + 0.523350i $$0.175318\pi$$
$$348$$ 0 0
$$349$$ 2528.79 0.387860 0.193930 0.981015i $$-0.437876\pi$$
0.193930 + 0.981015i $$0.437876\pi$$
$$350$$ 0 0
$$351$$ 1496.49 0.227569
$$352$$ 0 0
$$353$$ 5562.00 0.838627 0.419314 0.907841i $$-0.362271\pi$$
0.419314 + 0.907841i $$0.362271\pi$$
$$354$$ 0 0
$$355$$ 11448.0 1.71154
$$356$$ 0 0
$$357$$ −935.307 −0.138660
$$358$$ 0 0
$$359$$ 8875.03 1.30475 0.652376 0.757895i $$-0.273772\pi$$
0.652376 + 0.757895i $$0.273772\pi$$
$$360$$ 0 0
$$361$$ 6597.00 0.961802
$$362$$ 0 0
$$363$$ −3993.00 −0.577350
$$364$$ 0 0
$$365$$ 11493.9 1.64827
$$366$$ 0 0
$$367$$ −12799.9 −1.82056 −0.910282 0.413989i $$-0.864135\pi$$
−0.910282 + 0.413989i $$0.864135\pi$$
$$368$$ 0 0
$$369$$ 486.000 0.0685641
$$370$$ 0 0
$$371$$ −1692.00 −0.236777
$$372$$ 0 0
$$373$$ −4981.38 −0.691491 −0.345745 0.938328i $$-0.612374\pi$$
−0.345745 + 0.938328i $$0.612374\pi$$
$$374$$ 0 0
$$375$$ 4427.12 0.609642
$$376$$ 0 0
$$377$$ 14400.0 1.96721
$$378$$ 0 0
$$379$$ 9892.00 1.34068 0.670340 0.742054i $$-0.266148\pi$$
0.670340 + 0.742054i $$0.266148\pi$$
$$380$$ 0 0
$$381$$ −2088.85 −0.280880
$$382$$ 0 0
$$383$$ 8771.11 1.17019 0.585095 0.810965i $$-0.301057\pi$$
0.585095 + 0.810965i $$0.301057\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −180.000 −0.0236432
$$388$$ 0 0
$$389$$ −9903.87 −1.29086 −0.645432 0.763818i $$-0.723323\pi$$
−0.645432 + 0.763818i $$0.723323\pi$$
$$390$$ 0 0
$$391$$ −9353.07 −1.20973
$$392$$ 0 0
$$393$$ 6804.00 0.873324
$$394$$ 0 0
$$395$$ 1548.00 0.197186
$$396$$ 0 0
$$397$$ −103.923 −0.0131379 −0.00656895 0.999978i $$-0.502091\pi$$
−0.00656895 + 0.999978i $$0.502091\pi$$
$$398$$ 0 0
$$399$$ −1205.51 −0.151255
$$400$$ 0 0
$$401$$ 1062.00 0.132254 0.0661269 0.997811i $$-0.478936\pi$$
0.0661269 + 0.997811i $$0.478936\pi$$
$$402$$ 0 0
$$403$$ −16704.0 −2.06473
$$404$$ 0 0
$$405$$ −841.777 −0.103280
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 8614.00 1.04141 0.520703 0.853738i $$-0.325670\pi$$
0.520703 + 0.853738i $$0.325670\pi$$
$$410$$ 0 0
$$411$$ −1566.00 −0.187944
$$412$$ 0 0
$$413$$ −1122.37 −0.133724
$$414$$ 0 0
$$415$$ −11971.9 −1.41609
$$416$$ 0 0
$$417$$ −2028.00 −0.238157
$$418$$ 0 0
$$419$$ 10440.0 1.21725 0.608625 0.793458i $$-0.291722\pi$$
0.608625 + 0.793458i $$0.291722\pi$$
$$420$$ 0 0
$$421$$ 900.666 0.104266 0.0521328 0.998640i $$-0.483398\pi$$
0.0521328 + 0.998640i $$0.483398\pi$$
$$422$$ 0 0
$$423$$ 3554.17 0.408533
$$424$$ 0 0
$$425$$ 1530.00 0.174626
$$426$$ 0 0
$$427$$ −1992.00 −0.225760
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −394.908 −0.0441346 −0.0220673 0.999756i $$-0.507025\pi$$
−0.0220673 + 0.999756i $$0.507025\pi$$
$$432$$ 0 0
$$433$$ 12958.0 1.43816 0.719078 0.694929i $$-0.244564\pi$$
0.719078 + 0.694929i $$0.244564\pi$$
$$434$$ 0 0
$$435$$ −8100.00 −0.892794
$$436$$ 0 0
$$437$$ −12055.1 −1.31962
$$438$$ 0 0
$$439$$ 11441.9 1.24395 0.621974 0.783038i $$-0.286331\pi$$
0.621974 + 0.783038i $$0.286331\pi$$
$$440$$ 0 0
$$441$$ −2979.00 −0.321672
$$442$$ 0 0
$$443$$ 1800.00 0.193049 0.0965244 0.995331i $$-0.469227\pi$$
0.0965244 + 0.995331i $$0.469227\pi$$
$$444$$ 0 0
$$445$$ 9540.14 1.01628
$$446$$ 0 0
$$447$$ −4395.94 −0.465148
$$448$$ 0 0
$$449$$ −13626.0 −1.43218 −0.716092 0.698006i $$-0.754071\pi$$
−0.716092 + 0.698006i $$0.754071\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 7160.30 0.742649
$$454$$ 0 0
$$455$$ −1995.32 −0.205587
$$456$$ 0 0
$$457$$ −12602.0 −1.28993 −0.644964 0.764213i $$-0.723127\pi$$
−0.644964 + 0.764213i $$0.723127\pi$$
$$458$$ 0 0
$$459$$ −2430.00 −0.247108
$$460$$ 0 0
$$461$$ −1839.44 −0.185838 −0.0929188 0.995674i $$-0.529620\pi$$
−0.0929188 + 0.995674i $$0.529620\pi$$
$$462$$ 0 0
$$463$$ 11012.4 1.10538 0.552688 0.833389i $$-0.313602\pi$$
0.552688 + 0.833389i $$0.313602\pi$$
$$464$$ 0 0
$$465$$ 9396.00 0.937052
$$466$$ 0 0
$$467$$ 9144.00 0.906068 0.453034 0.891493i $$-0.350342\pi$$
0.453034 + 0.891493i $$0.350342\pi$$
$$468$$ 0 0
$$469$$ 401.836 0.0395630
$$470$$ 0 0
$$471$$ 6048.32 0.591703
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 1972.00 0.190488
$$476$$ 0 0
$$477$$ −4395.94 −0.421963
$$478$$ 0 0
$$479$$ −6173.03 −0.588837 −0.294418 0.955677i $$-0.595126\pi$$
−0.294418 + 0.955677i $$0.595126\pi$$
$$480$$ 0 0
$$481$$ −1920.00 −0.182005
$$482$$ 0 0
$$483$$ 1080.00 0.101743
$$484$$ 0 0
$$485$$ −1974.54 −0.184864
$$486$$ 0 0
$$487$$ 3204.29 0.298153 0.149076 0.988826i $$-0.452370\pi$$
0.149076 + 0.988826i $$0.452370\pi$$
$$488$$ 0 0
$$489$$ 1164.00 0.107644
$$490$$ 0 0
$$491$$ −396.000 −0.0363976 −0.0181988 0.999834i $$-0.505793\pi$$
−0.0181988 + 0.999834i $$0.505793\pi$$
$$492$$ 0 0
$$493$$ −23382.7 −2.13611
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −3816.00 −0.344408
$$498$$ 0 0
$$499$$ −12436.0 −1.11565 −0.557827 0.829957i $$-0.688365\pi$$
−0.557827 + 0.829957i $$0.688365\pi$$
$$500$$ 0 0
$$501$$ 6609.51 0.589403
$$502$$ 0 0
$$503$$ −16482.2 −1.46104 −0.730522 0.682890i $$-0.760723\pi$$
−0.730522 + 0.682890i $$0.760723\pi$$
$$504$$ 0 0
$$505$$ −108.000 −0.00951671
$$506$$ 0 0
$$507$$ 2625.00 0.229942
$$508$$ 0 0
$$509$$ 9155.62 0.797280 0.398640 0.917107i $$-0.369482\pi$$
0.398640 + 0.917107i $$0.369482\pi$$
$$510$$ 0 0
$$511$$ −3831.30 −0.331676
$$512$$ 0 0
$$513$$ −3132.00 −0.269554
$$514$$ 0 0
$$515$$ 8244.00 0.705386
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 592.361 0.0500998
$$520$$ 0 0
$$521$$ −7650.00 −0.643287 −0.321644 0.946861i $$-0.604235\pi$$
−0.321644 + 0.946861i $$0.604235\pi$$
$$522$$ 0 0
$$523$$ −18332.0 −1.53270 −0.766350 0.642423i $$-0.777929\pi$$
−0.766350 + 0.642423i $$0.777929\pi$$
$$524$$ 0 0
$$525$$ −176.669 −0.0146866
$$526$$ 0 0
$$527$$ 27123.9 2.24200
$$528$$ 0 0
$$529$$ −1367.00 −0.112353
$$530$$ 0 0
$$531$$ −2916.00 −0.238312
$$532$$ 0 0
$$533$$ 2992.98 0.243228
$$534$$ 0 0
$$535$$ −2618.86 −0.211632
$$536$$ 0 0
$$537$$ −8532.00 −0.685629
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −16863.2 −1.34012 −0.670062 0.742305i $$-0.733733\pi$$
−0.670062 + 0.742305i $$0.733733\pi$$
$$542$$ 0 0
$$543$$ 290.985 0.0229969
$$544$$ 0 0
$$545$$ 4752.00 0.373492
$$546$$ 0 0
$$547$$ 1684.00 0.131632 0.0658159 0.997832i $$-0.479035\pi$$
0.0658159 + 0.997832i $$0.479035\pi$$
$$548$$ 0 0
$$549$$ −5175.37 −0.402330
$$550$$ 0 0
$$551$$ −30137.7 −2.33014
$$552$$ 0 0
$$553$$ −516.000 −0.0396791
$$554$$ 0 0
$$555$$ 1080.00 0.0826008
$$556$$ 0 0
$$557$$ 2275.91 0.173130 0.0865652 0.996246i $$-0.472411\pi$$
0.0865652 + 0.996246i $$0.472411\pi$$
$$558$$ 0 0
$$559$$ −1108.51 −0.0838731
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −7992.00 −0.598264 −0.299132 0.954212i $$-0.596697\pi$$
−0.299132 + 0.954212i $$0.596697\pi$$
$$564$$ 0 0
$$565$$ 23008.6 1.71323
$$566$$ 0 0
$$567$$ 280.592 0.0207827
$$568$$ 0 0
$$569$$ 5526.00 0.407139 0.203569 0.979061i $$-0.434746\pi$$
0.203569 + 0.979061i $$0.434746\pi$$
$$570$$ 0 0
$$571$$ −13420.0 −0.983554 −0.491777 0.870721i $$-0.663653\pi$$
−0.491777 + 0.870721i $$0.663653\pi$$
$$572$$ 0 0
$$573$$ −9602.49 −0.700087
$$574$$ 0 0
$$575$$ −1766.69 −0.128132
$$576$$ 0 0
$$577$$ −10178.0 −0.734343 −0.367171 0.930153i $$-0.619674\pi$$
−0.367171 + 0.930153i $$0.619674\pi$$
$$578$$ 0 0
$$579$$ −4026.00 −0.288972
$$580$$ 0 0
$$581$$ 3990.65 0.284957
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −5184.00 −0.366380
$$586$$ 0 0
$$587$$ −18684.0 −1.31375 −0.656875 0.754000i $$-0.728122\pi$$
−0.656875 + 0.754000i $$0.728122\pi$$
$$588$$ 0 0
$$589$$ 34959.7 2.44565
$$590$$ 0 0
$$591$$ 2899.45 0.201806
$$592$$ 0 0
$$593$$ −5094.00 −0.352758 −0.176379 0.984322i $$-0.556438\pi$$
−0.176379 + 0.984322i $$0.556438\pi$$
$$594$$ 0 0
$$595$$ 3240.00 0.223239
$$596$$ 0 0
$$597$$ 5352.04 0.366908
$$598$$ 0 0
$$599$$ 19433.6 1.32560 0.662801 0.748795i $$-0.269367\pi$$
0.662801 + 0.748795i $$0.269367\pi$$
$$600$$ 0 0
$$601$$ −27722.0 −1.88154 −0.940769 0.339049i $$-0.889895\pi$$
−0.940769 + 0.339049i $$0.889895\pi$$
$$602$$ 0 0
$$603$$ 1044.00 0.0705057
$$604$$ 0 0
$$605$$ 13832.2 0.929516
$$606$$ 0 0
$$607$$ 26684.0 1.78430 0.892149 0.451741i $$-0.149197\pi$$
0.892149 + 0.451741i $$0.149197\pi$$
$$608$$ 0 0
$$609$$ 2700.00 0.179654
$$610$$ 0 0
$$611$$ 21888.0 1.44925
$$612$$ 0 0
$$613$$ 16911.7 1.11429 0.557144 0.830416i $$-0.311897\pi$$
0.557144 + 0.830416i $$0.311897\pi$$
$$614$$ 0 0
$$615$$ −1683.55 −0.110386
$$616$$ 0 0
$$617$$ −17694.0 −1.15451 −0.577256 0.816563i $$-0.695876\pi$$
−0.577256 + 0.816563i $$0.695876\pi$$
$$618$$ 0 0
$$619$$ 13652.0 0.886462 0.443231 0.896407i $$-0.353832\pi$$
0.443231 + 0.896407i $$0.353832\pi$$
$$620$$ 0 0
$$621$$ 2805.92 0.181317
$$622$$ 0 0
$$623$$ −3180.05 −0.204504
$$624$$ 0 0
$$625$$ −13211.0 −0.845504
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 3117.69 0.197632
$$630$$ 0 0
$$631$$ 9162.55 0.578059 0.289030 0.957320i $$-0.406667\pi$$
0.289030 + 0.957320i $$0.406667\pi$$
$$632$$ 0 0
$$633$$ −8292.00 −0.520659
$$634$$ 0 0
$$635$$ 7236.00 0.452208
$$636$$ 0 0
$$637$$ −18345.9 −1.14112
$$638$$ 0 0
$$639$$ −9914.26 −0.613775
$$640$$ 0 0
$$641$$ −5202.00 −0.320541 −0.160270 0.987073i $$-0.551237\pi$$
−0.160270 + 0.987073i $$0.551237\pi$$
$$642$$ 0 0
$$643$$ 15892.0 0.974680 0.487340 0.873212i $$-0.337967\pi$$
0.487340 + 0.873212i $$0.337967\pi$$
$$644$$ 0 0
$$645$$ 623.538 0.0380648
$$646$$ 0 0
$$647$$ −478.046 −0.0290478 −0.0145239 0.999895i $$-0.504623\pi$$
−0.0145239 + 0.999895i $$0.504623\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −3132.00 −0.188560
$$652$$ 0 0
$$653$$ −24660.9 −1.47788 −0.738941 0.673770i $$-0.764674\pi$$
−0.738941 + 0.673770i $$0.764674\pi$$
$$654$$ 0 0
$$655$$ −23569.7 −1.40602
$$656$$ 0 0
$$657$$ −9954.00 −0.591085
$$658$$ 0 0
$$659$$ −28260.0 −1.67049 −0.835245 0.549878i $$-0.814674\pi$$
−0.835245 + 0.549878i $$0.814674\pi$$
$$660$$ 0 0
$$661$$ −25863.0 −1.52187 −0.760933 0.648830i $$-0.775258\pi$$
−0.760933 + 0.648830i $$0.775258\pi$$
$$662$$ 0 0
$$663$$ −14964.9 −0.876605
$$664$$ 0 0
$$665$$ 4176.00 0.243516
$$666$$ 0 0
$$667$$ 27000.0 1.56738
$$668$$ 0 0
$$669$$ 12876.1 0.744122
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 190.000 0.0108826 0.00544128 0.999985i $$-0.498268\pi$$
0.00544128 + 0.999985i $$0.498268\pi$$
$$674$$ 0 0
$$675$$ −459.000 −0.0261732
$$676$$ 0 0
$$677$$ −4998.70 −0.283775 −0.141887 0.989883i $$-0.545317\pi$$
−0.141887 + 0.989883i $$0.545317\pi$$
$$678$$ 0 0
$$679$$ 658.179 0.0371997
$$680$$ 0 0
$$681$$ 17064.0 0.960197
$$682$$ 0 0
$$683$$ 8064.00 0.451772 0.225886 0.974154i $$-0.427472\pi$$
0.225886 + 0.974154i $$0.427472\pi$$
$$684$$ 0 0
$$685$$ 5424.78 0.302584
$$686$$ 0 0
$$687$$ −16710.8 −0.928032
$$688$$ 0 0
$$689$$ −27072.0 −1.49690
$$690$$ 0 0
$$691$$ −19244.0 −1.05944 −0.529722 0.848171i $$-0.677704\pi$$
−0.529722 + 0.848171i $$0.677704\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 7025.20 0.383426
$$696$$ 0 0
$$697$$ −4860.00 −0.264111
$$698$$ 0 0
$$699$$ −8154.00 −0.441220
$$700$$ 0 0
$$701$$ 6204.21 0.334279 0.167140 0.985933i $$-0.446547\pi$$
0.167140 + 0.985933i $$0.446547\pi$$
$$702$$ 0 0
$$703$$ 4018.36 0.215584
$$704$$ 0 0
$$705$$ −12312.0 −0.657726
$$706$$ 0 0
$$707$$ 36.0000 0.00191502
$$708$$ 0 0
$$709$$ −15020.3 −0.795629 −0.397814 0.917466i $$-0.630231\pi$$
−0.397814 + 0.917466i $$0.630231\pi$$
$$710$$ 0 0
$$711$$ −1340.61 −0.0707127
$$712$$ 0 0
$$713$$ −31320.0 −1.64508
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −10724.9 −0.558615
$$718$$ 0 0
$$719$$ −30740.4 −1.59447 −0.797236 0.603668i $$-0.793705\pi$$
−0.797236 + 0.603668i $$0.793705\pi$$
$$720$$ 0 0
$$721$$ −2748.00 −0.141943
$$722$$ 0 0
$$723$$ 13470.0 0.692883
$$724$$ 0 0
$$725$$ −4416.73 −0.226253
$$726$$ 0 0
$$727$$ 12127.8 0.618701 0.309351 0.950948i $$-0.399888\pi$$
0.309351 + 0.950948i $$0.399888\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 1800.00 0.0910744
$$732$$ 0 0
$$733$$ 12387.6 0.624212 0.312106 0.950047i $$-0.398966\pi$$
0.312106 + 0.950047i $$0.398966\pi$$
$$734$$ 0 0
$$735$$ 10319.6 0.517881
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 11180.0 0.556513 0.278256 0.960507i $$-0.410244\pi$$
0.278256 + 0.960507i $$0.410244\pi$$
$$740$$ 0 0
$$741$$ −19288.1 −0.956230
$$742$$ 0 0
$$743$$ 35500.1 1.75286 0.876429 0.481532i $$-0.159919\pi$$
0.876429 + 0.481532i $$0.159919\pi$$
$$744$$ 0 0
$$745$$ 15228.0 0.748873
$$746$$ 0 0
$$747$$ 10368.0 0.507825
$$748$$ 0 0
$$749$$ 872.954 0.0425862
$$750$$ 0 0
$$751$$ 37970.0 1.84493 0.922467 0.386076i $$-0.126170\pi$$
0.922467 + 0.386076i $$0.126170\pi$$
$$752$$ 0 0
$$753$$ 13824.0 0.669023
$$754$$ 0 0
$$755$$ −24804.0 −1.19564
$$756$$ 0 0
$$757$$ −39047.4 −1.87477 −0.937385 0.348296i $$-0.886760\pi$$
−0.937385 + 0.348296i $$0.886760\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 12222.0 0.582191 0.291095 0.956694i $$-0.405980\pi$$
0.291095 + 0.956694i $$0.405980\pi$$
$$762$$ 0 0
$$763$$ −1584.00 −0.0751568
$$764$$ 0 0
$$765$$ 8417.77 0.397837
$$766$$ 0 0
$$767$$ −17957.9 −0.845401
$$768$$ 0 0
$$769$$ −34030.0 −1.59578 −0.797889 0.602804i $$-0.794050\pi$$
−0.797889 + 0.602804i $$0.794050\pi$$
$$770$$ 0 0
$$771$$ 13878.0 0.648254
$$772$$ 0 0
$$773$$ 4873.99 0.226786 0.113393 0.993550i $$-0.463828\pi$$
0.113393 + 0.993550i $$0.463828\pi$$
$$774$$ 0 0
$$775$$ 5123.41 0.237469
$$776$$ 0 0
$$777$$ −360.000 −0.0166215
$$778$$ 0 0
$$779$$ −6264.00 −0.288102
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 7014.81 0.320164
$$784$$ 0 0
$$785$$ −20952.0 −0.952623
$$786$$ 0 0
$$787$$ 30988.0 1.40356 0.701781 0.712393i $$-0.252389\pi$$
0.701781 + 0.712393i $$0.252389\pi$$
$$788$$ 0 0
$$789$$ −5985.97 −0.270096
$$790$$ 0 0
$$791$$ −7669.52 −0.344749
$$792$$ 0 0
$$793$$ −31872.0 −1.42725
$$794$$ 0 0
$$795$$ 15228.0 0.679348
$$796$$ 0 0
$$797$$ −7160.30 −0.318232 −0.159116 0.987260i $$-0.550864\pi$$
−0.159116 + 0.987260i $$0.550864\pi$$
$$798$$ 0 0
$$799$$ −35541.7 −1.57369
$$800$$ 0 0
$$801$$ −8262.00 −0.364449
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −3741.23 −0.163803
$$806$$ 0 0
$$807$$ −9446.61 −0.412065
$$808$$ 0 0
$$809$$ −37530.0 −1.63101 −0.815503 0.578752i $$-0.803540\pi$$
−0.815503 + 0.578752i $$0.803540\pi$$
$$810$$ 0 0
$$811$$ 10852.0 0.469871 0.234935 0.972011i $$-0.424512\pi$$
0.234935 + 0.972011i $$0.424512\pi$$
$$812$$ 0 0
$$813$$ 16035.3 0.691739
$$814$$ 0 0
$$815$$ −4032.21 −0.173303
$$816$$ 0 0
$$817$$ 2320.00 0.0993470
$$818$$ 0 0
$$819$$ 1728.00 0.0737255
$$820$$ 0 0
$$821$$ 31353.6 1.33282 0.666411 0.745584i $$-0.267829\pi$$
0.666411 + 0.745584i $$0.267829\pi$$
$$822$$ 0 0
$$823$$ −32947.1 −1.39546 −0.697729 0.716361i $$-0.745806\pi$$
−0.697729 + 0.716361i $$0.745806\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 10044.0 0.422327 0.211163 0.977451i $$-0.432275\pi$$
0.211163 + 0.977451i $$0.432275\pi$$
$$828$$ 0 0
$$829$$ 9796.48 0.410429 0.205215 0.978717i $$-0.434211\pi$$
0.205215 + 0.978717i $$0.434211\pi$$
$$830$$ 0 0
$$831$$ 19579.1 0.817318
$$832$$ 0 0
$$833$$ 29790.0 1.23909
$$834$$ 0 0
$$835$$ −22896.0 −0.948921
$$836$$ 0 0
$$837$$ −8137.17 −0.336036
$$838$$ 0 0
$$839$$ 21054.8 0.866380 0.433190 0.901303i $$-0.357388\pi$$
0.433190 + 0.901303i $$0.357388\pi$$
$$840$$ 0 0
$$841$$ 43111.0 1.76764
$$842$$ 0 0
$$843$$ 3510.00 0.143405
$$844$$ 0 0
$$845$$ −9093.27 −0.370199
$$846$$ 0 0
$$847$$ −4610.72 −0.187044
$$848$$ 0 0
$$849$$ −17220.0 −0.696100
$$850$$ 0 0
$$851$$ −3600.00 −0.145013
$$852$$ 0 0
$$853$$ 40703.2 1.63382 0.816911 0.576763i $$-0.195684\pi$$
0.816911 + 0.576763i $$0.195684\pi$$
$$854$$ 0 0
$$855$$ 10849.6 0.433973
$$856$$ 0 0
$$857$$ 18342.0 0.731098 0.365549 0.930792i $$-0.380881\pi$$
0.365549 + 0.930792i $$0.380881\pi$$
$$858$$ 0 0
$$859$$ −26324.0 −1.04559 −0.522796 0.852458i $$-0.675111\pi$$
−0.522796 + 0.852458i $$0.675111\pi$$
$$860$$ 0 0
$$861$$ 561.184 0.0222127
$$862$$ 0 0
$$863$$ 12761.8 0.503378 0.251689 0.967808i $$-0.419014\pi$$
0.251689 + 0.967808i $$0.419014\pi$$
$$864$$ 0 0
$$865$$ −2052.00 −0.0806591
$$866$$ 0 0
$$867$$ 9561.00 0.374520
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 6429.37 0.250116
$$872$$ 0 0
$$873$$ 1710.00 0.0662941
$$874$$ 0 0
$$875$$ 5112.00 0.197505
$$876$$ 0 0
$$877$$ −2459.51 −0.0946999 −0.0473500 0.998878i $$-0.515078\pi$$
−0.0473500 + 0.998878i $$0.515078\pi$$
$$878$$ 0 0
$$879$$ 23975.0 0.919975
$$880$$ 0 0
$$881$$ 37314.0 1.42695 0.713474 0.700682i $$-0.247121\pi$$
0.713474 + 0.700682i $$0.247121\pi$$
$$882$$ 0 0
$$883$$ 18244.0 0.695311 0.347655 0.937622i $$-0.386978\pi$$
0.347655 + 0.937622i $$0.386978\pi$$
$$884$$ 0 0
$$885$$ 10101.3 0.383675
$$886$$ 0 0
$$887$$ −17957.9 −0.679783 −0.339891 0.940465i $$-0.610390\pi$$
−0.339891 + 0.940465i $$0.610390\pi$$
$$888$$ 0 0
$$889$$ −2412.00 −0.0909965
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −45809.3 −1.71663
$$894$$ 0 0
$$895$$ 29555.7 1.10384
$$896$$ 0 0
$$897$$ 17280.0 0.643213
$$898$$ 0 0
$$899$$ −78300.0 −2.90484
$$900$$ 0 0
$$901$$ 43959.4 1.62542
$$902$$ 0 0
$$903$$ −207.846 −0.00765967
$$904$$ 0 0
$$905$$ −1008.00 −0.0370244
$$906$$ 0 0
$$907$$ −16388.0 −0.599950 −0.299975 0.953947i $$-0.596978\pi$$
−0.299975 + 0.953947i $$0.596978\pi$$
$$908$$ 0 0
$$909$$ 93.5307 0.00341278
$$910$$ 0 0
$$911$$ −25107.8 −0.913127 −0.456564 0.889691i $$-0.650920\pi$$
−0.456564 + 0.889691i $$0.650920\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 17928.0 0.647739
$$916$$ 0 0
$$917$$ 7856.58 0.282930
$$918$$ 0 0
$$919$$ −27155.1 −0.974716 −0.487358 0.873202i $$-0.662039\pi$$
−0.487358 + 0.873202i $$0.662039\pi$$
$$920$$ 0 0
$$921$$ −16356.0 −0.585178
$$922$$ 0 0
$$923$$ −61056.0 −2.17734
$$924$$ 0 0
$$925$$ 588.897 0.0209328
$$926$$ 0 0
$$927$$ −7139.51 −0.252958
$$928$$ 0 0
$$929$$ 48006.0 1.69540 0.847700 0.530477i $$-0.177987\pi$$
0.847700 + 0.530477i $$0.177987\pi$$
$$930$$ 0 0
$$931$$ 38396.0 1.35164
$$932$$ 0 0
$$933$$ 6609.51 0.231924
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −7894.00 −0.275225 −0.137612 0.990486i $$-0.543943\pi$$
−0.137612 + 0.990486i $$0.543943\pi$$
$$938$$ 0 0
$$939$$ 3102.00 0.107806
$$940$$ 0 0
$$941$$ −2670.82 −0.0925253 −0.0462627 0.998929i $$-0.514731\pi$$
−0.0462627 + 0.998929i $$0.514731\pi$$
$$942$$ 0 0
$$943$$ 5611.84 0.193793
$$944$$ 0 0
$$945$$ −972.000 −0.0334594
$$946$$ 0 0
$$947$$ 22356.0 0.767130 0.383565 0.923514i $$-0.374696\pi$$
0.383565 + 0.923514i $$0.374696\pi$$
$$948$$ 0 0
$$949$$ −61300.7 −2.09685
$$950$$ 0 0
$$951$$ −7950.11 −0.271083
$$952$$ 0 0
$$953$$ 14958.0 0.508434 0.254217 0.967147i $$-0.418182\pi$$
0.254217 + 0.967147i $$0.418182\pi$$
$$954$$ 0 0
$$955$$ 33264.0 1.12712
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −1808.26 −0.0608882
$$960$$ 0 0
$$961$$ 61037.0 2.04884
$$962$$ 0 0
$$963$$ 2268.00 0.0758933
$$964$$ 0 0
$$965$$ 13946.5 0.465236
$$966$$ 0 0
$$967$$ −17476.4 −0.581182 −0.290591 0.956847i $$-0.593852\pi$$
−0.290591 + 0.956847i $$0.593852\pi$$
$$968$$ 0 0
$$969$$ 31320.0 1.03833
$$970$$ 0 0
$$971$$ −48528.0 −1.60385 −0.801925 0.597425i $$-0.796190\pi$$
−0.801925 + 0.597425i $$0.796190\pi$$
$$972$$ 0 0
$$973$$ −2341.73 −0.0771557
$$974$$ 0 0
$$975$$ −2826.71 −0.0928483
$$976$$ 0 0
$$977$$ −39978.0 −1.30912 −0.654560 0.756010i $$-0.727146\pi$$
−0.654560 + 0.756010i $$0.727146\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −4115.35 −0.133938
$$982$$ 0 0
$$983$$ −17957.9 −0.582674 −0.291337 0.956621i $$-0.594100\pi$$
−0.291337 + 0.956621i $$0.594100\pi$$
$$984$$ 0 0
$$985$$ −10044.0 −0.324902
$$986$$ 0 0
$$987$$ 4104.00 0.132352
$$988$$ 0 0
$$989$$ −2078.46 −0.0668263
$$990$$ 0 0
$$991$$ 1666.23 0.0534103 0.0267052 0.999643i $$-0.491498\pi$$
0.0267052 + 0.999643i $$0.491498\pi$$
$$992$$ 0 0
$$993$$ −12396.0 −0.396148
$$994$$ 0 0
$$995$$ −18540.0 −0.590711
$$996$$ 0 0
$$997$$ 41465.3 1.31717 0.658585 0.752506i $$-0.271155\pi$$
0.658585 + 0.752506i $$0.271155\pi$$
$$998$$ 0 0
$$999$$ −935.307 −0.0296214
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 768.4.a.n.1.1 2
3.2 odd 2 2304.4.a.be.1.2 2
4.3 odd 2 768.4.a.g.1.1 2
8.3 odd 2 inner 768.4.a.n.1.2 2
8.5 even 2 768.4.a.g.1.2 2
12.11 even 2 2304.4.a.bg.1.2 2
16.3 odd 4 192.4.d.a.97.2 yes 4
16.5 even 4 192.4.d.a.97.1 4
16.11 odd 4 192.4.d.a.97.3 yes 4
16.13 even 4 192.4.d.a.97.4 yes 4
24.5 odd 2 2304.4.a.bg.1.1 2
24.11 even 2 2304.4.a.be.1.1 2
48.5 odd 4 576.4.d.g.289.3 4
48.11 even 4 576.4.d.g.289.4 4
48.29 odd 4 576.4.d.g.289.1 4
48.35 even 4 576.4.d.g.289.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
192.4.d.a.97.1 4 16.5 even 4
192.4.d.a.97.2 yes 4 16.3 odd 4
192.4.d.a.97.3 yes 4 16.11 odd 4
192.4.d.a.97.4 yes 4 16.13 even 4
576.4.d.g.289.1 4 48.29 odd 4
576.4.d.g.289.2 4 48.35 even 4
576.4.d.g.289.3 4 48.5 odd 4
576.4.d.g.289.4 4 48.11 even 4
768.4.a.g.1.1 2 4.3 odd 2
768.4.a.g.1.2 2 8.5 even 2
768.4.a.n.1.1 2 1.1 even 1 trivial
768.4.a.n.1.2 2 8.3 odd 2 inner
2304.4.a.be.1.1 2 24.11 even 2
2304.4.a.be.1.2 2 3.2 odd 2
2304.4.a.bg.1.1 2 24.5 odd 2
2304.4.a.bg.1.2 2 12.11 even 2