# Properties

 Label 768.4.a.g.1.2 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 1.2 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 $$-0.346141\pi$$
0.464758 + 0.885438i $$0.346141\pi$$
$$6$$ 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 $$-0.701362\pi$$
−0.591242 + 0.806494i $$0.701362\pi$$
$$14$$ 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 $$-0.253040\pi$$
0.700322 + 0.713827i $$0.253040\pi$$
$$20$$ 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 $$-0.812696\pi$$
−0.831811 + 0.555058i $$0.812696\pi$$
$$30$$ 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 $$-0.475479\pi$$
0.0769588 + 0.997034i $$0.475479\pi$$
$$38$$ 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 $$-0.488709\pi$$
0.0354648 + 0.999371i $$0.488709\pi$$
$$44$$ 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 $$-0.281846\pi$$
0.632945 + 0.774197i $$0.281846\pi$$
$$54$$ 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 $$-0.383640\pi$$
0.357468 + 0.933925i $$0.383640\pi$$
$$60$$ 0 0
$$61$$ 575.041 1.20699 0.603495 0.797366i $$-0.293774\pi$$
0.603495 + 0.797366i $$0.293774\pi$$
$$62$$ 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 $$-0.533727\pi$$
−0.105759 + 0.994392i $$0.533727\pi$$
$$68$$ 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 $$-0.775653\pi$$
−0.761738 + 0.647886i $$0.775653\pi$$
$$84$$ 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 $$-0.501629\pi$$
−0.00511917 + 0.999987i $$0.501629\pi$$
$$102$$ 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 $$-0.536315\pi$$
−0.113840 + 0.993499i $$0.536315\pi$$
$$108$$ 0 0
$$109$$ 457.261 0.401814 0.200907 0.979610i $$-0.435611\pi$$
0.200907 + 0.979610i $$0.435611\pi$$
$$110$$ 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 $$-0.773005\pi$$
−0.756321 + 0.654201i $$0.773005\pi$$
$$132$$ 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 $$-0.433874\pi$$
0.206250 + 0.978499i $$0.433874\pi$$
$$140$$ 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 $$-0.368027\pi$$
0.402830 + 0.915275i $$0.368027\pi$$
$$150$$ 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 $$-0.671254\pi$$
−0.512430 + 0.858729i $$0.671254\pi$$
$$158$$ 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 $$-0.529717\pi$$
−0.0932224 + 0.995645i $$0.529717\pi$$
$$164$$ 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 $$-0.513815\pi$$
−0.0433877 + 0.999058i $$0.513815\pi$$
$$174$$ 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 $$-0.297638\pi$$
0.593772 + 0.804633i $$0.297638\pi$$
$$180$$ 0 0
$$181$$ −96.9948 −0.0398319 −0.0199159 0.999802i $$-0.506340\pi$$
−0.0199159 + 0.999802i $$0.506340\pi$$
$$182$$ 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 $$-0.555918\pi$$
−0.174769 + 0.984609i $$0.555918\pi$$
$$198$$ 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 $$-0.351102\pi$$
0.450904 + 0.892572i $$0.351102\pi$$
$$212$$ 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 $$-0.812549\pi$$
−0.831555 + 0.555443i $$0.812549\pi$$
$$228$$ 0 0
$$229$$ 5570.28 1.60740 0.803699 0.595036i $$-0.202862\pi$$
0.803699 + 0.595036i $$0.202862\pi$$
$$230$$ 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 $$-0.696710\pi$$
−0.579391 + 0.815050i $$0.696710\pi$$
$$252$$ 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 $$-0.383848\pi$$
0.356859 + 0.934158i $$0.383848\pi$$
$$270$$ 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 $$-0.750321\pi$$
−0.707818 + 0.706394i $$0.750321\pi$$
$$278$$ 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 $$-0.294036\pi$$
0.602840 + 0.797862i $$0.294036\pi$$
$$284$$ 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 $$-0.793434\pi$$
−0.796722 + 0.604346i $$0.793434\pi$$
$$294$$ 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 $$-0.330836\pi$$
0.506779 + 0.862076i $$0.330836\pi$$
$$308$$ 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 $$-0.424568\pi$$
0.234765 + 0.972052i $$0.424568\pi$$
$$318$$ 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 $$-0.388532\pi$$
0.343074 + 0.939308i $$0.388532\pi$$
$$332$$ 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 $$-0.824682\pi$$
−0.852118 + 0.523350i $$0.824682\pi$$
$$348$$ 0 0
$$349$$ −2528.79 −0.387860 −0.193930 0.981015i $$-0.562124\pi$$
−0.193930 + 0.981015i $$0.562124\pi$$
$$350$$ 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 $$-0.387626\pi$$
0.345745 + 0.938328i $$0.387626\pi$$
$$374$$ 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 $$-0.733852\pi$$
−0.670340 + 0.742054i $$0.733852\pi$$
$$380$$ 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 $$-0.276677\pi$$
0.645432 + 0.763818i $$0.276677\pi$$
$$390$$ 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 $$-0.497909\pi$$
0.00656895 + 0.999978i $$0.497909\pi$$
$$398$$ 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 $$-0.708278\pi$$
−0.608625 + 0.793458i $$0.708278\pi$$
$$420$$ 0 0
$$421$$ −900.666 −0.104266 −0.0521328 0.998640i $$-0.516602\pi$$
−0.0521328 + 0.998640i $$0.516602\pi$$
$$422$$ 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 $$-0.530773\pi$$
−0.0965244 + 0.995331i $$0.530773\pi$$
$$444$$ 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 $$-0.470380\pi$$
0.0929188 + 0.995674i $$0.470380\pi$$
$$462$$ 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 $$-0.649658\pi$$
−0.453034 + 0.891493i $$0.649658\pi$$
$$468$$ 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 $$-0.494207\pi$$
0.0181988 + 0.999834i $$0.494207\pi$$
$$492$$ 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 $$-0.311635\pi$$
0.557827 + 0.829957i $$0.311635\pi$$
$$500$$ 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 $$-0.630518\pi$$
−0.398640 + 0.917107i $$0.630518\pi$$
$$510$$ 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 $$-0.222071\pi$$
0.766350 + 0.642423i $$0.222071\pi$$
$$524$$ 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 $$-0.266267\pi$$
0.670062 + 0.742305i $$0.266267\pi$$
$$542$$ 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 $$-0.520965\pi$$
−0.0658159 + 0.997832i $$0.520965\pi$$
$$548$$ 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 $$-0.527589\pi$$
−0.0865652 + 0.996246i $$0.527589\pi$$
$$558$$ 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 $$-0.403303\pi$$
0.299132 + 0.954212i $$0.403303\pi$$
$$564$$ 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 $$-0.336347\pi$$
0.491777 + 0.870721i $$0.336347\pi$$
$$572$$ 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 $$-0.271878\pi$$
0.656875 + 0.754000i $$0.271878\pi$$
$$588$$ 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 $$-0.688103\pi$$
−0.557144 + 0.830416i $$0.688103\pi$$
$$614$$ 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 $$-0.646168\pi$$
−0.443231 + 0.896407i $$0.646168\pi$$
$$620$$ 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 $$-0.662033\pi$$
−0.487340 + 0.873212i $$0.662033\pi$$
$$644$$ 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 $$-0.235326\pi$$
0.738941 + 0.673770i $$0.235326\pi$$
$$654$$ 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 $$-0.185326\pi$$
0.835245 + 0.549878i $$0.185326\pi$$
$$660$$ 0 0
$$661$$ 25863.0 1.52187 0.760933 0.648830i $$-0.224742\pi$$
0.760933 + 0.648830i $$0.224742\pi$$
$$662$$ 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 $$-0.454683\pi$$
0.141887 + 0.989883i $$0.454683\pi$$
$$678$$ 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 $$-0.572528\pi$$
−0.225886 + 0.974154i $$0.572528\pi$$
$$684$$ 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 $$-0.322296\pi$$
0.529722 + 0.848171i $$0.322296\pi$$
$$692$$ 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 $$-0.553453\pi$$
−0.167140 + 0.985933i $$0.553453\pi$$
$$702$$ 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 $$-0.369769\pi$$
0.397814 + 0.917466i $$0.369769\pi$$
$$710$$ 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 $$-0.601034\pi$$
−0.312106 + 0.950047i $$0.601034\pi$$
$$734$$ 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 $$-0.589756\pi$$
−0.278256 + 0.960507i $$0.589756\pi$$
$$740$$ 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 $$-0.113240\pi$$
0.937385 + 0.348296i $$0.113240\pi$$
$$758$$ 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 $$-0.536172\pi$$
−0.113393 + 0.993550i $$0.536172\pi$$
$$774$$ 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 $$-0.747611\pi$$
−0.701781 + 0.712393i $$0.747611\pi$$
$$788$$ 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 $$-0.449136\pi$$
0.159116 + 0.987260i $$0.449136\pi$$
$$798$$ 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 $$-0.575488\pi$$
−0.234935 + 0.972011i $$0.575488\pi$$
$$812$$ 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 $$-0.732171\pi$$
−0.666411 + 0.745584i $$0.732171\pi$$
$$822$$ 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 $$-0.567725\pi$$
−0.211163 + 0.977451i $$0.567725\pi$$
$$828$$ 0 0
$$829$$ −9796.48 −0.410429 −0.205215 0.978717i $$-0.565789\pi$$
−0.205215 + 0.978717i $$0.565789\pi$$
$$830$$ 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 $$-0.804316\pi$$
−0.816911 + 0.576763i $$0.804316\pi$$
$$854$$ 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 $$-0.324889\pi$$
0.522796 + 0.852458i $$0.324889\pi$$
$$860$$ 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 $$-0.484922\pi$$
0.0473500 + 0.998878i $$0.484922\pi$$
$$878$$ 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 $$-0.613022\pi$$
−0.347655 + 0.937622i $$0.613022\pi$$
$$884$$ 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 $$-0.403022\pi$$
0.299975 + 0.953947i $$0.403022\pi$$
$$908$$ 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 $$-0.485269\pi$$
0.0462627 + 0.998929i $$0.485269\pi$$
$$942$$ 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 $$-0.625304\pi$$
−0.383565 + 0.923514i $$0.625304\pi$$
$$948$$ 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 $$-0.203810\pi$$
0.801925 + 0.597425i $$0.203810\pi$$
$$972$$ 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 $$-0.728845\pi$$
−0.658585 + 0.752506i $$0.728845\pi$$
$$998$$ 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.g.1.2 2
3.2 odd 2 2304.4.a.bg.1.1 2
4.3 odd 2 768.4.a.n.1.2 2
8.3 odd 2 inner 768.4.a.g.1.1 2
8.5 even 2 768.4.a.n.1.1 2
12.11 even 2 2304.4.a.be.1.1 2
16.3 odd 4 192.4.d.a.97.3 yes 4
16.5 even 4 192.4.d.a.97.4 yes 4
16.11 odd 4 192.4.d.a.97.2 yes 4
16.13 even 4 192.4.d.a.97.1 4
24.5 odd 2 2304.4.a.be.1.2 2
24.11 even 2 2304.4.a.bg.1.2 2
48.5 odd 4 576.4.d.g.289.1 4
48.11 even 4 576.4.d.g.289.2 4
48.29 odd 4 576.4.d.g.289.3 4
48.35 even 4 576.4.d.g.289.4 4

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