# Properties

 Label 2070.4.a.e.1.1 Level $2070$ Weight $4$ Character 2070.1 Self dual yes Analytic conductor $122.134$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$122.133953712$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -18.0000 q^{7} -8.00000 q^{8} +O(q^{10})$$ $$q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -18.0000 q^{7} -8.00000 q^{8} -10.0000 q^{10} +32.0000 q^{11} -47.0000 q^{13} +36.0000 q^{14} +16.0000 q^{16} -20.0000 q^{17} +36.0000 q^{19} +20.0000 q^{20} -64.0000 q^{22} +23.0000 q^{23} +25.0000 q^{25} +94.0000 q^{26} -72.0000 q^{28} +27.0000 q^{29} -33.0000 q^{31} -32.0000 q^{32} +40.0000 q^{34} -90.0000 q^{35} +56.0000 q^{37} -72.0000 q^{38} -40.0000 q^{40} +157.000 q^{41} +18.0000 q^{43} +128.000 q^{44} -46.0000 q^{46} -65.0000 q^{47} -19.0000 q^{49} -50.0000 q^{50} -188.000 q^{52} +14.0000 q^{53} +160.000 q^{55} +144.000 q^{56} -54.0000 q^{58} +744.000 q^{59} +552.000 q^{61} +66.0000 q^{62} +64.0000 q^{64} -235.000 q^{65} -156.000 q^{67} -80.0000 q^{68} +180.000 q^{70} -699.000 q^{71} -609.000 q^{73} -112.000 q^{74} +144.000 q^{76} -576.000 q^{77} -644.000 q^{79} +80.0000 q^{80} -314.000 q^{82} -512.000 q^{83} -100.000 q^{85} -36.0000 q^{86} -256.000 q^{88} +102.000 q^{89} +846.000 q^{91} +92.0000 q^{92} +130.000 q^{94} +180.000 q^{95} +578.000 q^{97} +38.0000 q^{98} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −0.707107
$$3$$ 0 0
$$4$$ 4.00000 0.500000
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ −18.0000 −0.971909 −0.485954 0.873984i $$-0.661528\pi$$
−0.485954 + 0.873984i $$0.661528\pi$$
$$8$$ −8.00000 −0.353553
$$9$$ 0 0
$$10$$ −10.0000 −0.316228
$$11$$ 32.0000 0.877124 0.438562 0.898701i $$-0.355488\pi$$
0.438562 + 0.898701i $$0.355488\pi$$
$$12$$ 0 0
$$13$$ −47.0000 −1.00273 −0.501364 0.865237i $$-0.667168\pi$$
−0.501364 + 0.865237i $$0.667168\pi$$
$$14$$ 36.0000 0.687243
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ −20.0000 −0.285336 −0.142668 0.989771i $$-0.545568\pi$$
−0.142668 + 0.989771i $$0.545568\pi$$
$$18$$ 0 0
$$19$$ 36.0000 0.434682 0.217341 0.976096i $$-0.430262\pi$$
0.217341 + 0.976096i $$0.430262\pi$$
$$20$$ 20.0000 0.223607
$$21$$ 0 0
$$22$$ −64.0000 −0.620220
$$23$$ 23.0000 0.208514
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 94.0000 0.709035
$$27$$ 0 0
$$28$$ −72.0000 −0.485954
$$29$$ 27.0000 0.172889 0.0864444 0.996257i $$-0.472450\pi$$
0.0864444 + 0.996257i $$0.472450\pi$$
$$30$$ 0 0
$$31$$ −33.0000 −0.191193 −0.0955964 0.995420i $$-0.530476\pi$$
−0.0955964 + 0.995420i $$0.530476\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ 0 0
$$34$$ 40.0000 0.201763
$$35$$ −90.0000 −0.434651
$$36$$ 0 0
$$37$$ 56.0000 0.248820 0.124410 0.992231i $$-0.460296\pi$$
0.124410 + 0.992231i $$0.460296\pi$$
$$38$$ −72.0000 −0.307367
$$39$$ 0 0
$$40$$ −40.0000 −0.158114
$$41$$ 157.000 0.598031 0.299016 0.954248i $$-0.403342\pi$$
0.299016 + 0.954248i $$0.403342\pi$$
$$42$$ 0 0
$$43$$ 18.0000 0.0638366 0.0319183 0.999490i $$-0.489838\pi$$
0.0319183 + 0.999490i $$0.489838\pi$$
$$44$$ 128.000 0.438562
$$45$$ 0 0
$$46$$ −46.0000 −0.147442
$$47$$ −65.0000 −0.201728 −0.100864 0.994900i $$-0.532161\pi$$
−0.100864 + 0.994900i $$0.532161\pi$$
$$48$$ 0 0
$$49$$ −19.0000 −0.0553936
$$50$$ −50.0000 −0.141421
$$51$$ 0 0
$$52$$ −188.000 −0.501364
$$53$$ 14.0000 0.0362839 0.0181420 0.999835i $$-0.494225\pi$$
0.0181420 + 0.999835i $$0.494225\pi$$
$$54$$ 0 0
$$55$$ 160.000 0.392262
$$56$$ 144.000 0.343622
$$57$$ 0 0
$$58$$ −54.0000 −0.122251
$$59$$ 744.000 1.64170 0.820852 0.571141i $$-0.193499\pi$$
0.820852 + 0.571141i $$0.193499\pi$$
$$60$$ 0 0
$$61$$ 552.000 1.15863 0.579314 0.815104i $$-0.303320\pi$$
0.579314 + 0.815104i $$0.303320\pi$$
$$62$$ 66.0000 0.135194
$$63$$ 0 0
$$64$$ 64.0000 0.125000
$$65$$ −235.000 −0.448433
$$66$$ 0 0
$$67$$ −156.000 −0.284454 −0.142227 0.989834i $$-0.545426\pi$$
−0.142227 + 0.989834i $$0.545426\pi$$
$$68$$ −80.0000 −0.142668
$$69$$ 0 0
$$70$$ 180.000 0.307344
$$71$$ −699.000 −1.16839 −0.584197 0.811612i $$-0.698591\pi$$
−0.584197 + 0.811612i $$0.698591\pi$$
$$72$$ 0 0
$$73$$ −609.000 −0.976412 −0.488206 0.872728i $$-0.662348\pi$$
−0.488206 + 0.872728i $$0.662348\pi$$
$$74$$ −112.000 −0.175942
$$75$$ 0 0
$$76$$ 144.000 0.217341
$$77$$ −576.000 −0.852484
$$78$$ 0 0
$$79$$ −644.000 −0.917160 −0.458580 0.888653i $$-0.651642\pi$$
−0.458580 + 0.888653i $$0.651642\pi$$
$$80$$ 80.0000 0.111803
$$81$$ 0 0
$$82$$ −314.000 −0.422872
$$83$$ −512.000 −0.677100 −0.338550 0.940948i $$-0.609936\pi$$
−0.338550 + 0.940948i $$0.609936\pi$$
$$84$$ 0 0
$$85$$ −100.000 −0.127606
$$86$$ −36.0000 −0.0451393
$$87$$ 0 0
$$88$$ −256.000 −0.310110
$$89$$ 102.000 0.121483 0.0607415 0.998154i $$-0.480653\pi$$
0.0607415 + 0.998154i $$0.480653\pi$$
$$90$$ 0 0
$$91$$ 846.000 0.974559
$$92$$ 92.0000 0.104257
$$93$$ 0 0
$$94$$ 130.000 0.142643
$$95$$ 180.000 0.194396
$$96$$ 0 0
$$97$$ 578.000 0.605021 0.302510 0.953146i $$-0.402175\pi$$
0.302510 + 0.953146i $$0.402175\pi$$
$$98$$ 38.0000 0.0391692
$$99$$ 0 0
$$100$$ 100.000 0.100000
$$101$$ 6.00000 0.00591111 0.00295556 0.999996i $$-0.499059\pi$$
0.00295556 + 0.999996i $$0.499059\pi$$
$$102$$ 0 0
$$103$$ −160.000 −0.153061 −0.0765304 0.997067i $$-0.524384\pi$$
−0.0765304 + 0.997067i $$0.524384\pi$$
$$104$$ 376.000 0.354518
$$105$$ 0 0
$$106$$ −28.0000 −0.0256566
$$107$$ −380.000 −0.343327 −0.171663 0.985156i $$-0.554914\pi$$
−0.171663 + 0.985156i $$0.554914\pi$$
$$108$$ 0 0
$$109$$ 250.000 0.219685 0.109842 0.993949i $$-0.464965\pi$$
0.109842 + 0.993949i $$0.464965\pi$$
$$110$$ −320.000 −0.277371
$$111$$ 0 0
$$112$$ −288.000 −0.242977
$$113$$ 390.000 0.324674 0.162337 0.986735i $$-0.448097\pi$$
0.162337 + 0.986735i $$0.448097\pi$$
$$114$$ 0 0
$$115$$ 115.000 0.0932505
$$116$$ 108.000 0.0864444
$$117$$ 0 0
$$118$$ −1488.00 −1.16086
$$119$$ 360.000 0.277321
$$120$$ 0 0
$$121$$ −307.000 −0.230654
$$122$$ −1104.00 −0.819274
$$123$$ 0 0
$$124$$ −132.000 −0.0955964
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −769.000 −0.537305 −0.268652 0.963237i $$-0.586578\pi$$
−0.268652 + 0.963237i $$0.586578\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ 0 0
$$130$$ 470.000 0.317090
$$131$$ 213.000 0.142060 0.0710301 0.997474i $$-0.477371\pi$$
0.0710301 + 0.997474i $$0.477371\pi$$
$$132$$ 0 0
$$133$$ −648.000 −0.422472
$$134$$ 312.000 0.201140
$$135$$ 0 0
$$136$$ 160.000 0.100882
$$137$$ −2836.00 −1.76858 −0.884291 0.466936i $$-0.845358\pi$$
−0.884291 + 0.466936i $$0.845358\pi$$
$$138$$ 0 0
$$139$$ −1631.00 −0.995249 −0.497625 0.867393i $$-0.665794\pi$$
−0.497625 + 0.867393i $$0.665794\pi$$
$$140$$ −360.000 −0.217325
$$141$$ 0 0
$$142$$ 1398.00 0.826180
$$143$$ −1504.00 −0.879516
$$144$$ 0 0
$$145$$ 135.000 0.0773182
$$146$$ 1218.00 0.690427
$$147$$ 0 0
$$148$$ 224.000 0.124410
$$149$$ 1966.00 1.08095 0.540473 0.841361i $$-0.318245\pi$$
0.540473 + 0.841361i $$0.318245\pi$$
$$150$$ 0 0
$$151$$ 35.0000 0.0188626 0.00943132 0.999956i $$-0.496998\pi$$
0.00943132 + 0.999956i $$0.496998\pi$$
$$152$$ −288.000 −0.153683
$$153$$ 0 0
$$154$$ 1152.00 0.602797
$$155$$ −165.000 −0.0855040
$$156$$ 0 0
$$157$$ 1702.00 0.865187 0.432594 0.901589i $$-0.357599\pi$$
0.432594 + 0.901589i $$0.357599\pi$$
$$158$$ 1288.00 0.648530
$$159$$ 0 0
$$160$$ −160.000 −0.0790569
$$161$$ −414.000 −0.202657
$$162$$ 0 0
$$163$$ −2045.00 −0.982680 −0.491340 0.870968i $$-0.663493\pi$$
−0.491340 + 0.870968i $$0.663493\pi$$
$$164$$ 628.000 0.299016
$$165$$ 0 0
$$166$$ 1024.00 0.478782
$$167$$ −1016.00 −0.470781 −0.235391 0.971901i $$-0.575637\pi$$
−0.235391 + 0.971901i $$0.575637\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.00546199
$$170$$ 200.000 0.0902312
$$171$$ 0 0
$$172$$ 72.0000 0.0319183
$$173$$ −598.000 −0.262804 −0.131402 0.991329i $$-0.541948\pi$$
−0.131402 + 0.991329i $$0.541948\pi$$
$$174$$ 0 0
$$175$$ −450.000 −0.194382
$$176$$ 512.000 0.219281
$$177$$ 0 0
$$178$$ −204.000 −0.0859014
$$179$$ 4607.00 1.92371 0.961853 0.273567i $$-0.0882036\pi$$
0.961853 + 0.273567i $$0.0882036\pi$$
$$180$$ 0 0
$$181$$ −1212.00 −0.497720 −0.248860 0.968540i $$-0.580056\pi$$
−0.248860 + 0.968540i $$0.580056\pi$$
$$182$$ −1692.00 −0.689117
$$183$$ 0 0
$$184$$ −184.000 −0.0737210
$$185$$ 280.000 0.111276
$$186$$ 0 0
$$187$$ −640.000 −0.250275
$$188$$ −260.000 −0.100864
$$189$$ 0 0
$$190$$ −360.000 −0.137459
$$191$$ 1058.00 0.400807 0.200404 0.979713i $$-0.435775\pi$$
0.200404 + 0.979713i $$0.435775\pi$$
$$192$$ 0 0
$$193$$ 1047.00 0.390491 0.195245 0.980754i $$-0.437450\pi$$
0.195245 + 0.980754i $$0.437450\pi$$
$$194$$ −1156.00 −0.427814
$$195$$ 0 0
$$196$$ −76.0000 −0.0276968
$$197$$ −251.000 −0.0907767 −0.0453883 0.998969i $$-0.514453\pi$$
−0.0453883 + 0.998969i $$0.514453\pi$$
$$198$$ 0 0
$$199$$ −3508.00 −1.24963 −0.624813 0.780775i $$-0.714825\pi$$
−0.624813 + 0.780775i $$0.714825\pi$$
$$200$$ −200.000 −0.0707107
$$201$$ 0 0
$$202$$ −12.0000 −0.00417979
$$203$$ −486.000 −0.168032
$$204$$ 0 0
$$205$$ 785.000 0.267448
$$206$$ 320.000 0.108230
$$207$$ 0 0
$$208$$ −752.000 −0.250682
$$209$$ 1152.00 0.381270
$$210$$ 0 0
$$211$$ −3296.00 −1.07538 −0.537692 0.843141i $$-0.680704\pi$$
−0.537692 + 0.843141i $$0.680704\pi$$
$$212$$ 56.0000 0.0181420
$$213$$ 0 0
$$214$$ 760.000 0.242769
$$215$$ 90.0000 0.0285486
$$216$$ 0 0
$$217$$ 594.000 0.185822
$$218$$ −500.000 −0.155341
$$219$$ 0 0
$$220$$ 640.000 0.196131
$$221$$ 940.000 0.286114
$$222$$ 0 0
$$223$$ −2720.00 −0.816792 −0.408396 0.912805i $$-0.633912\pi$$
−0.408396 + 0.912805i $$0.633912\pi$$
$$224$$ 576.000 0.171811
$$225$$ 0 0
$$226$$ −780.000 −0.229579
$$227$$ 4134.00 1.20874 0.604368 0.796705i $$-0.293426\pi$$
0.604368 + 0.796705i $$0.293426\pi$$
$$228$$ 0 0
$$229$$ −4510.00 −1.30144 −0.650719 0.759319i $$-0.725532\pi$$
−0.650719 + 0.759319i $$0.725532\pi$$
$$230$$ −230.000 −0.0659380
$$231$$ 0 0
$$232$$ −216.000 −0.0611254
$$233$$ 5003.00 1.40668 0.703342 0.710852i $$-0.251690\pi$$
0.703342 + 0.710852i $$0.251690\pi$$
$$234$$ 0 0
$$235$$ −325.000 −0.0902156
$$236$$ 2976.00 0.820852
$$237$$ 0 0
$$238$$ −720.000 −0.196095
$$239$$ 6309.00 1.70751 0.853756 0.520674i $$-0.174319\pi$$
0.853756 + 0.520674i $$0.174319\pi$$
$$240$$ 0 0
$$241$$ 3038.00 0.812012 0.406006 0.913871i $$-0.366921\pi$$
0.406006 + 0.913871i $$0.366921\pi$$
$$242$$ 614.000 0.163097
$$243$$ 0 0
$$244$$ 2208.00 0.579314
$$245$$ −95.0000 −0.0247728
$$246$$ 0 0
$$247$$ −1692.00 −0.435868
$$248$$ 264.000 0.0675968
$$249$$ 0 0
$$250$$ −250.000 −0.0632456
$$251$$ 1332.00 0.334961 0.167480 0.985875i $$-0.446437\pi$$
0.167480 + 0.985875i $$0.446437\pi$$
$$252$$ 0 0
$$253$$ 736.000 0.182893
$$254$$ 1538.00 0.379932
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ −3301.00 −0.801209 −0.400605 0.916251i $$-0.631200\pi$$
−0.400605 + 0.916251i $$0.631200\pi$$
$$258$$ 0 0
$$259$$ −1008.00 −0.241830
$$260$$ −940.000 −0.224217
$$261$$ 0 0
$$262$$ −426.000 −0.100452
$$263$$ −2072.00 −0.485798 −0.242899 0.970052i $$-0.578098\pi$$
−0.242899 + 0.970052i $$0.578098\pi$$
$$264$$ 0 0
$$265$$ 70.0000 0.0162267
$$266$$ 1296.00 0.298733
$$267$$ 0 0
$$268$$ −624.000 −0.142227
$$269$$ −5721.00 −1.29671 −0.648356 0.761337i $$-0.724543\pi$$
−0.648356 + 0.761337i $$0.724543\pi$$
$$270$$ 0 0
$$271$$ −5900.00 −1.32251 −0.661254 0.750162i $$-0.729975\pi$$
−0.661254 + 0.750162i $$0.729975\pi$$
$$272$$ −320.000 −0.0713340
$$273$$ 0 0
$$274$$ 5672.00 1.25058
$$275$$ 800.000 0.175425
$$276$$ 0 0
$$277$$ 6371.00 1.38194 0.690968 0.722885i $$-0.257185\pi$$
0.690968 + 0.722885i $$0.257185\pi$$
$$278$$ 3262.00 0.703747
$$279$$ 0 0
$$280$$ 720.000 0.153672
$$281$$ −3190.00 −0.677222 −0.338611 0.940926i $$-0.609957\pi$$
−0.338611 + 0.940926i $$0.609957\pi$$
$$282$$ 0 0
$$283$$ −4226.00 −0.887667 −0.443833 0.896109i $$-0.646382\pi$$
−0.443833 + 0.896109i $$0.646382\pi$$
$$284$$ −2796.00 −0.584197
$$285$$ 0 0
$$286$$ 3008.00 0.621912
$$287$$ −2826.00 −0.581232
$$288$$ 0 0
$$289$$ −4513.00 −0.918583
$$290$$ −270.000 −0.0546722
$$291$$ 0 0
$$292$$ −2436.00 −0.488206
$$293$$ −6048.00 −1.20590 −0.602949 0.797780i $$-0.706008\pi$$
−0.602949 + 0.797780i $$0.706008\pi$$
$$294$$ 0 0
$$295$$ 3720.00 0.734192
$$296$$ −448.000 −0.0879712
$$297$$ 0 0
$$298$$ −3932.00 −0.764344
$$299$$ −1081.00 −0.209083
$$300$$ 0 0
$$301$$ −324.000 −0.0620434
$$302$$ −70.0000 −0.0133379
$$303$$ 0 0
$$304$$ 576.000 0.108671
$$305$$ 2760.00 0.518155
$$306$$ 0 0
$$307$$ 8628.00 1.60399 0.801997 0.597328i $$-0.203771\pi$$
0.801997 + 0.597328i $$0.203771\pi$$
$$308$$ −2304.00 −0.426242
$$309$$ 0 0
$$310$$ 330.000 0.0604605
$$311$$ −8247.00 −1.50368 −0.751840 0.659346i $$-0.770833\pi$$
−0.751840 + 0.659346i $$0.770833\pi$$
$$312$$ 0 0
$$313$$ 2620.00 0.473135 0.236567 0.971615i $$-0.423978\pi$$
0.236567 + 0.971615i $$0.423978\pi$$
$$314$$ −3404.00 −0.611780
$$315$$ 0 0
$$316$$ −2576.00 −0.458580
$$317$$ −9906.00 −1.75513 −0.877565 0.479457i $$-0.840834\pi$$
−0.877565 + 0.479457i $$0.840834\pi$$
$$318$$ 0 0
$$319$$ 864.000 0.151645
$$320$$ 320.000 0.0559017
$$321$$ 0 0
$$322$$ 828.000 0.143300
$$323$$ −720.000 −0.124031
$$324$$ 0 0
$$325$$ −1175.00 −0.200545
$$326$$ 4090.00 0.694859
$$327$$ 0 0
$$328$$ −1256.00 −0.211436
$$329$$ 1170.00 0.196061
$$330$$ 0 0
$$331$$ −8115.00 −1.34756 −0.673778 0.738934i $$-0.735329\pi$$
−0.673778 + 0.738934i $$0.735329\pi$$
$$332$$ −2048.00 −0.338550
$$333$$ 0 0
$$334$$ 2032.00 0.332892
$$335$$ −780.000 −0.127212
$$336$$ 0 0
$$337$$ −7586.00 −1.22622 −0.613109 0.789998i $$-0.710082\pi$$
−0.613109 + 0.789998i $$0.710082\pi$$
$$338$$ −24.0000 −0.00386221
$$339$$ 0 0
$$340$$ −400.000 −0.0638031
$$341$$ −1056.00 −0.167700
$$342$$ 0 0
$$343$$ 6516.00 1.02575
$$344$$ −144.000 −0.0225697
$$345$$ 0 0
$$346$$ 1196.00 0.185831
$$347$$ −1356.00 −0.209781 −0.104890 0.994484i $$-0.533449\pi$$
−0.104890 + 0.994484i $$0.533449\pi$$
$$348$$ 0 0
$$349$$ 6649.00 1.01981 0.509904 0.860231i $$-0.329681\pi$$
0.509904 + 0.860231i $$0.329681\pi$$
$$350$$ 900.000 0.137449
$$351$$ 0 0
$$352$$ −1024.00 −0.155055
$$353$$ −10691.0 −1.61197 −0.805984 0.591938i $$-0.798363\pi$$
−0.805984 + 0.591938i $$0.798363\pi$$
$$354$$ 0 0
$$355$$ −3495.00 −0.522522
$$356$$ 408.000 0.0607415
$$357$$ 0 0
$$358$$ −9214.00 −1.36027
$$359$$ 6420.00 0.943829 0.471915 0.881644i $$-0.343563\pi$$
0.471915 + 0.881644i $$0.343563\pi$$
$$360$$ 0 0
$$361$$ −5563.00 −0.811051
$$362$$ 2424.00 0.351941
$$363$$ 0 0
$$364$$ 3384.00 0.487280
$$365$$ −3045.00 −0.436665
$$366$$ 0 0
$$367$$ −524.000 −0.0745302 −0.0372651 0.999305i $$-0.511865\pi$$
−0.0372651 + 0.999305i $$0.511865\pi$$
$$368$$ 368.000 0.0521286
$$369$$ 0 0
$$370$$ −560.000 −0.0786838
$$371$$ −252.000 −0.0352647
$$372$$ 0 0
$$373$$ 5566.00 0.772645 0.386322 0.922364i $$-0.373745\pi$$
0.386322 + 0.922364i $$0.373745\pi$$
$$374$$ 1280.00 0.176971
$$375$$ 0 0
$$376$$ 520.000 0.0713217
$$377$$ −1269.00 −0.173360
$$378$$ 0 0
$$379$$ 2240.00 0.303591 0.151796 0.988412i $$-0.451494\pi$$
0.151796 + 0.988412i $$0.451494\pi$$
$$380$$ 720.000 0.0971979
$$381$$ 0 0
$$382$$ −2116.00 −0.283414
$$383$$ −8778.00 −1.17111 −0.585555 0.810633i $$-0.699123\pi$$
−0.585555 + 0.810633i $$0.699123\pi$$
$$384$$ 0 0
$$385$$ −2880.00 −0.381243
$$386$$ −2094.00 −0.276119
$$387$$ 0 0
$$388$$ 2312.00 0.302510
$$389$$ −4056.00 −0.528656 −0.264328 0.964433i $$-0.585150\pi$$
−0.264328 + 0.964433i $$0.585150\pi$$
$$390$$ 0 0
$$391$$ −460.000 −0.0594967
$$392$$ 152.000 0.0195846
$$393$$ 0 0
$$394$$ 502.000 0.0641888
$$395$$ −3220.00 −0.410167
$$396$$ 0 0
$$397$$ −9151.00 −1.15687 −0.578433 0.815730i $$-0.696335\pi$$
−0.578433 + 0.815730i $$0.696335\pi$$
$$398$$ 7016.00 0.883619
$$399$$ 0 0
$$400$$ 400.000 0.0500000
$$401$$ −15930.0 −1.98381 −0.991903 0.126997i $$-0.959466\pi$$
−0.991903 + 0.126997i $$0.959466\pi$$
$$402$$ 0 0
$$403$$ 1551.00 0.191714
$$404$$ 24.0000 0.00295556
$$405$$ 0 0
$$406$$ 972.000 0.118817
$$407$$ 1792.00 0.218246
$$408$$ 0 0
$$409$$ −5891.00 −0.712203 −0.356102 0.934447i $$-0.615894\pi$$
−0.356102 + 0.934447i $$0.615894\pi$$
$$410$$ −1570.00 −0.189114
$$411$$ 0 0
$$412$$ −640.000 −0.0765304
$$413$$ −13392.0 −1.59559
$$414$$ 0 0
$$415$$ −2560.00 −0.302808
$$416$$ 1504.00 0.177259
$$417$$ 0 0
$$418$$ −2304.00 −0.269599
$$419$$ −15282.0 −1.78180 −0.890900 0.454199i $$-0.849926\pi$$
−0.890900 + 0.454199i $$0.849926\pi$$
$$420$$ 0 0
$$421$$ −10934.0 −1.26577 −0.632887 0.774244i $$-0.718130\pi$$
−0.632887 + 0.774244i $$0.718130\pi$$
$$422$$ 6592.00 0.760411
$$423$$ 0 0
$$424$$ −112.000 −0.0128283
$$425$$ −500.000 −0.0570672
$$426$$ 0 0
$$427$$ −9936.00 −1.12608
$$428$$ −1520.00 −0.171663
$$429$$ 0 0
$$430$$ −180.000 −0.0201869
$$431$$ 2794.00 0.312256 0.156128 0.987737i $$-0.450099\pi$$
0.156128 + 0.987737i $$0.450099\pi$$
$$432$$ 0 0
$$433$$ −15062.0 −1.67167 −0.835835 0.548980i $$-0.815016\pi$$
−0.835835 + 0.548980i $$0.815016\pi$$
$$434$$ −1188.00 −0.131396
$$435$$ 0 0
$$436$$ 1000.00 0.109842
$$437$$ 828.000 0.0906376
$$438$$ 0 0
$$439$$ 261.000 0.0283755 0.0141878 0.999899i $$-0.495484\pi$$
0.0141878 + 0.999899i $$0.495484\pi$$
$$440$$ −1280.00 −0.138685
$$441$$ 0 0
$$442$$ −1880.00 −0.202313
$$443$$ 7083.00 0.759647 0.379823 0.925059i $$-0.375985\pi$$
0.379823 + 0.925059i $$0.375985\pi$$
$$444$$ 0 0
$$445$$ 510.000 0.0543288
$$446$$ 5440.00 0.577559
$$447$$ 0 0
$$448$$ −1152.00 −0.121489
$$449$$ 10370.0 1.08996 0.544978 0.838450i $$-0.316538\pi$$
0.544978 + 0.838450i $$0.316538\pi$$
$$450$$ 0 0
$$451$$ 5024.00 0.524547
$$452$$ 1560.00 0.162337
$$453$$ 0 0
$$454$$ −8268.00 −0.854706
$$455$$ 4230.00 0.435836
$$456$$ 0 0
$$457$$ −10496.0 −1.07436 −0.537180 0.843468i $$-0.680510\pi$$
−0.537180 + 0.843468i $$0.680510\pi$$
$$458$$ 9020.00 0.920255
$$459$$ 0 0
$$460$$ 460.000 0.0466252
$$461$$ −18021.0 −1.82065 −0.910327 0.413889i $$-0.864170\pi$$
−0.910327 + 0.413889i $$0.864170\pi$$
$$462$$ 0 0
$$463$$ −17188.0 −1.72526 −0.862629 0.505838i $$-0.831183\pi$$
−0.862629 + 0.505838i $$0.831183\pi$$
$$464$$ 432.000 0.0432222
$$465$$ 0 0
$$466$$ −10006.0 −0.994676
$$467$$ 15246.0 1.51071 0.755354 0.655317i $$-0.227465\pi$$
0.755354 + 0.655317i $$0.227465\pi$$
$$468$$ 0 0
$$469$$ 2808.00 0.276464
$$470$$ 650.000 0.0637921
$$471$$ 0 0
$$472$$ −5952.00 −0.580430
$$473$$ 576.000 0.0559926
$$474$$ 0 0
$$475$$ 900.000 0.0869365
$$476$$ 1440.00 0.138660
$$477$$ 0 0
$$478$$ −12618.0 −1.20739
$$479$$ 8556.00 0.816145 0.408073 0.912949i $$-0.366201\pi$$
0.408073 + 0.912949i $$0.366201\pi$$
$$480$$ 0 0
$$481$$ −2632.00 −0.249499
$$482$$ −6076.00 −0.574179
$$483$$ 0 0
$$484$$ −1228.00 −0.115327
$$485$$ 2890.00 0.270573
$$486$$ 0 0
$$487$$ −1805.00 −0.167951 −0.0839757 0.996468i $$-0.526762\pi$$
−0.0839757 + 0.996468i $$0.526762\pi$$
$$488$$ −4416.00 −0.409637
$$489$$ 0 0
$$490$$ 190.000 0.0175170
$$491$$ −5245.00 −0.482085 −0.241042 0.970515i $$-0.577489\pi$$
−0.241042 + 0.970515i $$0.577489\pi$$
$$492$$ 0 0
$$493$$ −540.000 −0.0493314
$$494$$ 3384.00 0.308205
$$495$$ 0 0
$$496$$ −528.000 −0.0477982
$$497$$ 12582.0 1.13557
$$498$$ 0 0
$$499$$ 9027.00 0.809828 0.404914 0.914355i $$-0.367302\pi$$
0.404914 + 0.914355i $$0.367302\pi$$
$$500$$ 500.000 0.0447214
$$501$$ 0 0
$$502$$ −2664.00 −0.236853
$$503$$ −3522.00 −0.312203 −0.156102 0.987741i $$-0.549893\pi$$
−0.156102 + 0.987741i $$0.549893\pi$$
$$504$$ 0 0
$$505$$ 30.0000 0.00264353
$$506$$ −1472.00 −0.129325
$$507$$ 0 0
$$508$$ −3076.00 −0.268652
$$509$$ −3949.00 −0.343883 −0.171941 0.985107i $$-0.555004\pi$$
−0.171941 + 0.985107i $$0.555004\pi$$
$$510$$ 0 0
$$511$$ 10962.0 0.948983
$$512$$ −512.000 −0.0441942
$$513$$ 0 0
$$514$$ 6602.00 0.566540
$$515$$ −800.000 −0.0684509
$$516$$ 0 0
$$517$$ −2080.00 −0.176941
$$518$$ 2016.00 0.171000
$$519$$ 0 0
$$520$$ 1880.00 0.158545
$$521$$ −3236.00 −0.272115 −0.136057 0.990701i $$-0.543443\pi$$
−0.136057 + 0.990701i $$0.543443\pi$$
$$522$$ 0 0
$$523$$ 12394.0 1.03624 0.518118 0.855309i $$-0.326633\pi$$
0.518118 + 0.855309i $$0.326633\pi$$
$$524$$ 852.000 0.0710301
$$525$$ 0 0
$$526$$ 4144.00 0.343511
$$527$$ 660.000 0.0545542
$$528$$ 0 0
$$529$$ 529.000 0.0434783
$$530$$ −140.000 −0.0114740
$$531$$ 0 0
$$532$$ −2592.00 −0.211236
$$533$$ −7379.00 −0.599662
$$534$$ 0 0
$$535$$ −1900.00 −0.153540
$$536$$ 1248.00 0.100570
$$537$$ 0 0
$$538$$ 11442.0 0.916914
$$539$$ −608.000 −0.0485870
$$540$$ 0 0
$$541$$ −7159.00 −0.568927 −0.284463 0.958687i $$-0.591815\pi$$
−0.284463 + 0.958687i $$0.591815\pi$$
$$542$$ 11800.0 0.935154
$$543$$ 0 0
$$544$$ 640.000 0.0504408
$$545$$ 1250.00 0.0982461
$$546$$ 0 0
$$547$$ −19761.0 −1.54464 −0.772321 0.635232i $$-0.780904\pi$$
−0.772321 + 0.635232i $$0.780904\pi$$
$$548$$ −11344.0 −0.884291
$$549$$ 0 0
$$550$$ −1600.00 −0.124044
$$551$$ 972.000 0.0751517
$$552$$ 0 0
$$553$$ 11592.0 0.891396
$$554$$ −12742.0 −0.977176
$$555$$ 0 0
$$556$$ −6524.00 −0.497625
$$557$$ −18010.0 −1.37003 −0.685016 0.728528i $$-0.740205\pi$$
−0.685016 + 0.728528i $$0.740205\pi$$
$$558$$ 0 0
$$559$$ −846.000 −0.0640107
$$560$$ −1440.00 −0.108663
$$561$$ 0 0
$$562$$ 6380.00 0.478868
$$563$$ 2648.00 0.198224 0.0991118 0.995076i $$-0.468400\pi$$
0.0991118 + 0.995076i $$0.468400\pi$$
$$564$$ 0 0
$$565$$ 1950.00 0.145198
$$566$$ 8452.00 0.627675
$$567$$ 0 0
$$568$$ 5592.00 0.413090
$$569$$ 1566.00 0.115378 0.0576890 0.998335i $$-0.481627\pi$$
0.0576890 + 0.998335i $$0.481627\pi$$
$$570$$ 0 0
$$571$$ 2864.00 0.209903 0.104952 0.994477i $$-0.466531\pi$$
0.104952 + 0.994477i $$0.466531\pi$$
$$572$$ −6016.00 −0.439758
$$573$$ 0 0
$$574$$ 5652.00 0.410993
$$575$$ 575.000 0.0417029
$$576$$ 0 0
$$577$$ −929.000 −0.0670273 −0.0335137 0.999438i $$-0.510670\pi$$
−0.0335137 + 0.999438i $$0.510670\pi$$
$$578$$ 9026.00 0.649537
$$579$$ 0 0
$$580$$ 540.000 0.0386591
$$581$$ 9216.00 0.658079
$$582$$ 0 0
$$583$$ 448.000 0.0318255
$$584$$ 4872.00 0.345214
$$585$$ 0 0
$$586$$ 12096.0 0.852698
$$587$$ 19499.0 1.37106 0.685528 0.728046i $$-0.259571\pi$$
0.685528 + 0.728046i $$0.259571\pi$$
$$588$$ 0 0
$$589$$ −1188.00 −0.0831081
$$590$$ −7440.00 −0.519152
$$591$$ 0 0
$$592$$ 896.000 0.0622050
$$593$$ −6570.00 −0.454971 −0.227485 0.973782i $$-0.573050\pi$$
−0.227485 + 0.973782i $$0.573050\pi$$
$$594$$ 0 0
$$595$$ 1800.00 0.124022
$$596$$ 7864.00 0.540473
$$597$$ 0 0
$$598$$ 2162.00 0.147844
$$599$$ −1880.00 −0.128238 −0.0641191 0.997942i $$-0.520424\pi$$
−0.0641191 + 0.997942i $$0.520424\pi$$
$$600$$ 0 0
$$601$$ 3701.00 0.251193 0.125596 0.992081i $$-0.459916\pi$$
0.125596 + 0.992081i $$0.459916\pi$$
$$602$$ 648.000 0.0438713
$$603$$ 0 0
$$604$$ 140.000 0.00943132
$$605$$ −1535.00 −0.103151
$$606$$ 0 0
$$607$$ −3080.00 −0.205953 −0.102976 0.994684i $$-0.532837\pi$$
−0.102976 + 0.994684i $$0.532837\pi$$
$$608$$ −1152.00 −0.0768417
$$609$$ 0 0
$$610$$ −5520.00 −0.366391
$$611$$ 3055.00 0.202278
$$612$$ 0 0
$$613$$ 24004.0 1.58159 0.790793 0.612083i $$-0.209668\pi$$
0.790793 + 0.612083i $$0.209668\pi$$
$$614$$ −17256.0 −1.13419
$$615$$ 0 0
$$616$$ 4608.00 0.301399
$$617$$ −780.000 −0.0508940 −0.0254470 0.999676i $$-0.508101\pi$$
−0.0254470 + 0.999676i $$0.508101\pi$$
$$618$$ 0 0
$$619$$ 21892.0 1.42151 0.710754 0.703440i $$-0.248354\pi$$
0.710754 + 0.703440i $$0.248354\pi$$
$$620$$ −660.000 −0.0427520
$$621$$ 0 0
$$622$$ 16494.0 1.06326
$$623$$ −1836.00 −0.118070
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ −5240.00 −0.334557
$$627$$ 0 0
$$628$$ 6808.00 0.432594
$$629$$ −1120.00 −0.0709973
$$630$$ 0 0
$$631$$ 8050.00 0.507869 0.253935 0.967221i $$-0.418275\pi$$
0.253935 + 0.967221i $$0.418275\pi$$
$$632$$ 5152.00 0.324265
$$633$$ 0 0
$$634$$ 19812.0 1.24106
$$635$$ −3845.00 −0.240290
$$636$$ 0 0
$$637$$ 893.000 0.0555447
$$638$$ −1728.00 −0.107229
$$639$$ 0 0
$$640$$ −640.000 −0.0395285
$$641$$ 25890.0 1.59531 0.797655 0.603114i $$-0.206074\pi$$
0.797655 + 0.603114i $$0.206074\pi$$
$$642$$ 0 0
$$643$$ 4774.00 0.292797 0.146398 0.989226i $$-0.453232\pi$$
0.146398 + 0.989226i $$0.453232\pi$$
$$644$$ −1656.00 −0.101328
$$645$$ 0 0
$$646$$ 1440.00 0.0877029
$$647$$ −3349.00 −0.203497 −0.101749 0.994810i $$-0.532444\pi$$
−0.101749 + 0.994810i $$0.532444\pi$$
$$648$$ 0 0
$$649$$ 23808.0 1.43998
$$650$$ 2350.00 0.141807
$$651$$ 0 0
$$652$$ −8180.00 −0.491340
$$653$$ −24813.0 −1.48699 −0.743497 0.668739i $$-0.766834\pi$$
−0.743497 + 0.668739i $$0.766834\pi$$
$$654$$ 0 0
$$655$$ 1065.00 0.0635313
$$656$$ 2512.00 0.149508
$$657$$ 0 0
$$658$$ −2340.00 −0.138636
$$659$$ −18180.0 −1.07465 −0.537323 0.843376i $$-0.680565\pi$$
−0.537323 + 0.843376i $$0.680565\pi$$
$$660$$ 0 0
$$661$$ −29250.0 −1.72117 −0.860585 0.509307i $$-0.829902\pi$$
−0.860585 + 0.509307i $$0.829902\pi$$
$$662$$ 16230.0 0.952865
$$663$$ 0 0
$$664$$ 4096.00 0.239391
$$665$$ −3240.00 −0.188935
$$666$$ 0 0
$$667$$ 621.000 0.0360498
$$668$$ −4064.00 −0.235391
$$669$$ 0 0
$$670$$ 1560.00 0.0899523
$$671$$ 17664.0 1.01626
$$672$$ 0 0
$$673$$ −23027.0 −1.31891 −0.659454 0.751745i $$-0.729213\pi$$
−0.659454 + 0.751745i $$0.729213\pi$$
$$674$$ 15172.0 0.867068
$$675$$ 0 0
$$676$$ 48.0000 0.00273100
$$677$$ −20106.0 −1.14141 −0.570706 0.821154i $$-0.693331\pi$$
−0.570706 + 0.821154i $$0.693331\pi$$
$$678$$ 0 0
$$679$$ −10404.0 −0.588025
$$680$$ 800.000 0.0451156
$$681$$ 0 0
$$682$$ 2112.00 0.118582
$$683$$ 18745.0 1.05016 0.525079 0.851054i $$-0.324036\pi$$
0.525079 + 0.851054i $$0.324036\pi$$
$$684$$ 0 0
$$685$$ −14180.0 −0.790934
$$686$$ −13032.0 −0.725312
$$687$$ 0 0
$$688$$ 288.000 0.0159592
$$689$$ −658.000 −0.0363829
$$690$$ 0 0
$$691$$ 24424.0 1.34462 0.672310 0.740270i $$-0.265302\pi$$
0.672310 + 0.740270i $$0.265302\pi$$
$$692$$ −2392.00 −0.131402
$$693$$ 0 0
$$694$$ 2712.00 0.148337
$$695$$ −8155.00 −0.445089
$$696$$ 0 0
$$697$$ −3140.00 −0.170640
$$698$$ −13298.0 −0.721113
$$699$$ 0 0
$$700$$ −1800.00 −0.0971909
$$701$$ 27278.0 1.46972 0.734862 0.678217i $$-0.237247\pi$$
0.734862 + 0.678217i $$0.237247\pi$$
$$702$$ 0 0
$$703$$ 2016.00 0.108158
$$704$$ 2048.00 0.109640
$$705$$ 0 0
$$706$$ 21382.0 1.13983
$$707$$ −108.000 −0.00574506
$$708$$ 0 0
$$709$$ 12214.0 0.646977 0.323488 0.946232i $$-0.395144\pi$$
0.323488 + 0.946232i $$0.395144\pi$$
$$710$$ 6990.00 0.369479
$$711$$ 0 0
$$712$$ −816.000 −0.0429507
$$713$$ −759.000 −0.0398664
$$714$$ 0 0
$$715$$ −7520.00 −0.393332
$$716$$ 18428.0 0.961853
$$717$$ 0 0
$$718$$ −12840.0 −0.667388
$$719$$ −12932.0 −0.670768 −0.335384 0.942082i $$-0.608866\pi$$
−0.335384 + 0.942082i $$0.608866\pi$$
$$720$$ 0 0
$$721$$ 2880.00 0.148761
$$722$$ 11126.0 0.573500
$$723$$ 0 0
$$724$$ −4848.00 −0.248860
$$725$$ 675.000 0.0345778
$$726$$ 0 0
$$727$$ 10046.0 0.512497 0.256249 0.966611i $$-0.417513\pi$$
0.256249 + 0.966611i $$0.417513\pi$$
$$728$$ −6768.00 −0.344559
$$729$$ 0 0
$$730$$ 6090.00 0.308769
$$731$$ −360.000 −0.0182149
$$732$$ 0 0
$$733$$ −5924.00 −0.298510 −0.149255 0.988799i $$-0.547688\pi$$
−0.149255 + 0.988799i $$0.547688\pi$$
$$734$$ 1048.00 0.0527008
$$735$$ 0 0
$$736$$ −736.000 −0.0368605
$$737$$ −4992.00 −0.249502
$$738$$ 0 0
$$739$$ 829.000 0.0412656 0.0206328 0.999787i $$-0.493432\pi$$
0.0206328 + 0.999787i $$0.493432\pi$$
$$740$$ 1120.00 0.0556379
$$741$$ 0 0
$$742$$ 504.000 0.0249359
$$743$$ −7072.00 −0.349188 −0.174594 0.984641i $$-0.555861\pi$$
−0.174594 + 0.984641i $$0.555861\pi$$
$$744$$ 0 0
$$745$$ 9830.00 0.483414
$$746$$ −11132.0 −0.546342
$$747$$ 0 0
$$748$$ −2560.00 −0.125138
$$749$$ 6840.00 0.333682
$$750$$ 0 0
$$751$$ 16234.0 0.788798 0.394399 0.918939i $$-0.370953\pi$$
0.394399 + 0.918939i $$0.370953\pi$$
$$752$$ −1040.00 −0.0504320
$$753$$ 0 0
$$754$$ 2538.00 0.122584
$$755$$ 175.000 0.00843563
$$756$$ 0 0
$$757$$ 9128.00 0.438260 0.219130 0.975696i $$-0.429678\pi$$
0.219130 + 0.975696i $$0.429678\pi$$
$$758$$ −4480.00 −0.214671
$$759$$ 0 0
$$760$$ −1440.00 −0.0687293
$$761$$ −165.000 −0.00785972 −0.00392986 0.999992i $$-0.501251\pi$$
−0.00392986 + 0.999992i $$0.501251\pi$$
$$762$$ 0 0
$$763$$ −4500.00 −0.213514
$$764$$ 4232.00 0.200404
$$765$$ 0 0
$$766$$ 17556.0 0.828099
$$767$$ −34968.0 −1.64618
$$768$$ 0 0
$$769$$ −20834.0 −0.976974 −0.488487 0.872571i $$-0.662451\pi$$
−0.488487 + 0.872571i $$0.662451\pi$$
$$770$$ 5760.00 0.269579
$$771$$ 0 0
$$772$$ 4188.00 0.195245
$$773$$ 31782.0 1.47881 0.739404 0.673262i $$-0.235107\pi$$
0.739404 + 0.673262i $$0.235107\pi$$
$$774$$ 0 0
$$775$$ −825.000 −0.0382385
$$776$$ −4624.00 −0.213907
$$777$$ 0 0
$$778$$ 8112.00 0.373817
$$779$$ 5652.00 0.259954
$$780$$ 0 0
$$781$$ −22368.0 −1.02483
$$782$$ 920.000 0.0420705
$$783$$ 0 0
$$784$$ −304.000 −0.0138484
$$785$$ 8510.00 0.386923
$$786$$ 0 0
$$787$$ 33104.0 1.49940 0.749701 0.661776i $$-0.230197\pi$$
0.749701 + 0.661776i $$0.230197\pi$$
$$788$$ −1004.00 −0.0453883
$$789$$ 0 0
$$790$$ 6440.00 0.290032
$$791$$ −7020.00 −0.315553
$$792$$ 0 0
$$793$$ −25944.0 −1.16179
$$794$$ 18302.0 0.818027
$$795$$ 0 0
$$796$$ −14032.0 −0.624813
$$797$$ −4736.00 −0.210486 −0.105243 0.994447i $$-0.533562\pi$$
−0.105243 + 0.994447i $$0.533562\pi$$
$$798$$ 0 0
$$799$$ 1300.00 0.0575603
$$800$$ −800.000 −0.0353553
$$801$$ 0 0
$$802$$ 31860.0 1.40276
$$803$$ −19488.0 −0.856434
$$804$$ 0 0
$$805$$ −2070.00 −0.0906309
$$806$$ −3102.00 −0.135562
$$807$$ 0 0
$$808$$ −48.0000 −0.00208989
$$809$$ 7470.00 0.324637 0.162318 0.986738i $$-0.448103\pi$$
0.162318 + 0.986738i $$0.448103\pi$$
$$810$$ 0 0
$$811$$ 19919.0 0.862455 0.431227 0.902243i $$-0.358081\pi$$
0.431227 + 0.902243i $$0.358081\pi$$
$$812$$ −1944.00 −0.0840160
$$813$$ 0 0
$$814$$ −3584.00 −0.154323
$$815$$ −10225.0 −0.439468
$$816$$ 0 0
$$817$$ 648.000 0.0277487
$$818$$ 11782.0 0.503604
$$819$$ 0 0
$$820$$ 3140.00 0.133724
$$821$$ 22694.0 0.964709 0.482354 0.875976i $$-0.339782\pi$$
0.482354 + 0.875976i $$0.339782\pi$$
$$822$$ 0 0
$$823$$ −31907.0 −1.35141 −0.675704 0.737173i $$-0.736160\pi$$
−0.675704 + 0.737173i $$0.736160\pi$$
$$824$$ 1280.00 0.0541152
$$825$$ 0 0
$$826$$ 26784.0 1.12825
$$827$$ 15236.0 0.640638 0.320319 0.947310i $$-0.396210\pi$$
0.320319 + 0.947310i $$0.396210\pi$$
$$828$$ 0 0
$$829$$ 27286.0 1.14316 0.571581 0.820545i $$-0.306330\pi$$
0.571581 + 0.820545i $$0.306330\pi$$
$$830$$ 5120.00 0.214118
$$831$$ 0 0
$$832$$ −3008.00 −0.125341
$$833$$ 380.000 0.0158058
$$834$$ 0 0
$$835$$ −5080.00 −0.210540
$$836$$ 4608.00 0.190635
$$837$$ 0 0
$$838$$ 30564.0 1.25992
$$839$$ −23054.0 −0.948644 −0.474322 0.880351i $$-0.657307\pi$$
−0.474322 + 0.880351i $$0.657307\pi$$
$$840$$ 0 0
$$841$$ −23660.0 −0.970109
$$842$$ 21868.0 0.895037
$$843$$ 0 0
$$844$$ −13184.0 −0.537692
$$845$$ 60.0000 0.00244268
$$846$$ 0 0
$$847$$ 5526.00 0.224174
$$848$$ 224.000 0.00907098
$$849$$ 0 0
$$850$$ 1000.00 0.0403526
$$851$$ 1288.00 0.0518826
$$852$$ 0 0
$$853$$ −34506.0 −1.38507 −0.692534 0.721385i $$-0.743506\pi$$
−0.692534 + 0.721385i $$0.743506\pi$$
$$854$$ 19872.0 0.796260
$$855$$ 0 0
$$856$$ 3040.00 0.121384
$$857$$ 22263.0 0.887386 0.443693 0.896179i $$-0.353668\pi$$
0.443693 + 0.896179i $$0.353668\pi$$
$$858$$ 0 0
$$859$$ 12851.0 0.510443 0.255221 0.966883i $$-0.417852\pi$$
0.255221 + 0.966883i $$0.417852\pi$$
$$860$$ 360.000 0.0142743
$$861$$ 0 0
$$862$$ −5588.00 −0.220798
$$863$$ −15723.0 −0.620182 −0.310091 0.950707i $$-0.600360\pi$$
−0.310091 + 0.950707i $$0.600360\pi$$
$$864$$ 0 0
$$865$$ −2990.00 −0.117530
$$866$$ 30124.0 1.18205
$$867$$ 0 0
$$868$$ 2376.00 0.0929109
$$869$$ −20608.0 −0.804463
$$870$$ 0 0
$$871$$ 7332.00 0.285230
$$872$$ −2000.00 −0.0776704
$$873$$ 0 0
$$874$$ −1656.00 −0.0640904
$$875$$ −2250.00 −0.0869302
$$876$$ 0 0
$$877$$ −886.000 −0.0341141 −0.0170571 0.999855i $$-0.505430\pi$$
−0.0170571 + 0.999855i $$0.505430\pi$$
$$878$$ −522.000 −0.0200645
$$879$$ 0 0
$$880$$ 2560.00 0.0980654
$$881$$ 37120.0 1.41953 0.709764 0.704439i $$-0.248801\pi$$
0.709764 + 0.704439i $$0.248801\pi$$
$$882$$ 0 0
$$883$$ −7524.00 −0.286753 −0.143376 0.989668i $$-0.545796\pi$$
−0.143376 + 0.989668i $$0.545796\pi$$
$$884$$ 3760.00 0.143057
$$885$$ 0 0
$$886$$ −14166.0 −0.537151
$$887$$ −9221.00 −0.349054 −0.174527 0.984652i $$-0.555840\pi$$
−0.174527 + 0.984652i $$0.555840\pi$$
$$888$$ 0 0
$$889$$ 13842.0 0.522211
$$890$$ −1020.00 −0.0384163
$$891$$ 0 0
$$892$$ −10880.0 −0.408396
$$893$$ −2340.00 −0.0876877
$$894$$ 0 0
$$895$$ 23035.0 0.860307
$$896$$ 2304.00 0.0859054
$$897$$ 0 0
$$898$$ −20740.0 −0.770716
$$899$$ −891.000 −0.0330551
$$900$$ 0 0
$$901$$ −280.000 −0.0103531
$$902$$ −10048.0 −0.370911
$$903$$ 0 0
$$904$$ −3120.00 −0.114789
$$905$$ −6060.00 −0.222587
$$906$$ 0 0
$$907$$ 29116.0 1.06591 0.532955 0.846143i $$-0.321081\pi$$
0.532955 + 0.846143i $$0.321081\pi$$
$$908$$ 16536.0 0.604368
$$909$$ 0 0
$$910$$ −8460.00 −0.308183
$$911$$ −11440.0 −0.416053 −0.208026 0.978123i $$-0.566704\pi$$
−0.208026 + 0.978123i $$0.566704\pi$$
$$912$$ 0 0
$$913$$ −16384.0 −0.593901
$$914$$ 20992.0 0.759687
$$915$$ 0 0
$$916$$ −18040.0 −0.650719
$$917$$ −3834.00 −0.138070
$$918$$ 0 0
$$919$$ 2958.00 0.106176 0.0530878 0.998590i $$-0.483094\pi$$
0.0530878 + 0.998590i $$0.483094\pi$$
$$920$$ −920.000 −0.0329690
$$921$$ 0 0
$$922$$ 36042.0 1.28740
$$923$$ 32853.0 1.17158
$$924$$ 0 0
$$925$$ 1400.00 0.0497640
$$926$$ 34376.0 1.21994
$$927$$ 0 0
$$928$$ −864.000 −0.0305627
$$929$$ −20907.0 −0.738360 −0.369180 0.929358i $$-0.620361\pi$$
−0.369180 + 0.929358i $$0.620361\pi$$
$$930$$ 0 0
$$931$$ −684.000 −0.0240786
$$932$$ 20012.0 0.703342
$$933$$ 0 0
$$934$$ −30492.0 −1.06823
$$935$$ −3200.00 −0.111926
$$936$$ 0 0
$$937$$ 9748.00 0.339865 0.169932 0.985456i $$-0.445645\pi$$
0.169932 + 0.985456i $$0.445645\pi$$
$$938$$ −5616.00 −0.195489
$$939$$ 0 0
$$940$$ −1300.00 −0.0451078
$$941$$ −19624.0 −0.679834 −0.339917 0.940455i $$-0.610399\pi$$
−0.339917 + 0.940455i $$0.610399\pi$$
$$942$$ 0 0
$$943$$ 3611.00 0.124698
$$944$$ 11904.0 0.410426
$$945$$ 0 0
$$946$$ −1152.00 −0.0395928
$$947$$ 41859.0 1.43636 0.718181 0.695856i $$-0.244975\pi$$
0.718181 + 0.695856i $$0.244975\pi$$
$$948$$ 0 0
$$949$$ 28623.0 0.979075
$$950$$ −1800.00 −0.0614734
$$951$$ 0 0
$$952$$ −2880.00 −0.0980476
$$953$$ −29226.0 −0.993413 −0.496707 0.867918i $$-0.665458\pi$$
−0.496707 + 0.867918i $$0.665458\pi$$
$$954$$ 0 0
$$955$$ 5290.00 0.179246
$$956$$ 25236.0 0.853756
$$957$$ 0 0
$$958$$ −17112.0 −0.577102
$$959$$ 51048.0 1.71890
$$960$$ 0 0
$$961$$ −28702.0 −0.963445
$$962$$ 5264.00 0.176422
$$963$$ 0 0
$$964$$ 12152.0 0.406006
$$965$$ 5235.00 0.174633
$$966$$ 0 0
$$967$$ −29849.0 −0.992636 −0.496318 0.868141i $$-0.665315\pi$$
−0.496318 + 0.868141i $$0.665315\pi$$
$$968$$ 2456.00 0.0815484
$$969$$ 0 0
$$970$$ −5780.00 −0.191324
$$971$$ 9390.00 0.310339 0.155170 0.987888i $$-0.450408\pi$$
0.155170 + 0.987888i $$0.450408\pi$$
$$972$$ 0 0
$$973$$ 29358.0 0.967291
$$974$$ 3610.00 0.118760
$$975$$ 0 0
$$976$$ 8832.00 0.289657
$$977$$ −33536.0 −1.09817 −0.549085 0.835767i $$-0.685024\pi$$
−0.549085 + 0.835767i $$0.685024\pi$$
$$978$$ 0 0
$$979$$ 3264.00 0.106556
$$980$$ −380.000 −0.0123864
$$981$$ 0 0
$$982$$ 10490.0 0.340885
$$983$$ −28994.0 −0.940758 −0.470379 0.882465i $$-0.655883\pi$$
−0.470379 + 0.882465i $$0.655883\pi$$
$$984$$ 0 0
$$985$$ −1255.00 −0.0405966
$$986$$ 1080.00 0.0348826
$$987$$ 0 0
$$988$$ −6768.00 −0.217934
$$989$$ 414.000 0.0133109
$$990$$ 0 0
$$991$$ 11272.0 0.361319 0.180659 0.983546i $$-0.442177\pi$$
0.180659 + 0.983546i $$0.442177\pi$$
$$992$$ 1056.00 0.0337984
$$993$$ 0 0
$$994$$ −25164.0 −0.802971
$$995$$ −17540.0 −0.558850
$$996$$ 0 0
$$997$$ 61186.0 1.94361 0.971805 0.235784i $$-0.0757658\pi$$
0.971805 + 0.235784i $$0.0757658\pi$$
$$998$$ −18054.0 −0.572635
$$999$$ 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 2070.4.a.e.1.1 1
3.2 odd 2 230.4.a.e.1.1 1
12.11 even 2 1840.4.a.d.1.1 1
15.2 even 4 1150.4.b.f.599.2 2
15.8 even 4 1150.4.b.f.599.1 2
15.14 odd 2 1150.4.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.e.1.1 1 3.2 odd 2
1150.4.a.b.1.1 1 15.14 odd 2
1150.4.b.f.599.1 2 15.8 even 4
1150.4.b.f.599.2 2 15.2 even 4
1840.4.a.d.1.1 1 12.11 even 2
2070.4.a.e.1.1 1 1.1 even 1 trivial