# Properties

 Label 2070.4.a.bj.1.3 Level $2070$ Weight $4$ Character 2070.1 Self dual yes Analytic conductor $122.134$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 68 x^{2} - 111 x + 342$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 1.3 Root $$8.73081$$ of defining polynomial Character $$\chi$$ $$=$$ 2070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +23.5622 q^{7} +8.00000 q^{8} +O(q^{10})$$ $$q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +23.5622 q^{7} +8.00000 q^{8} +10.0000 q^{10} +32.1352 q^{11} +40.0811 q^{13} +47.1245 q^{14} +16.0000 q^{16} -126.165 q^{17} +0.232742 q^{19} +20.0000 q^{20} +64.2704 q^{22} +23.0000 q^{23} +25.0000 q^{25} +80.1621 q^{26} +94.2490 q^{28} +137.226 q^{29} +112.866 q^{31} +32.0000 q^{32} -252.331 q^{34} +117.811 q^{35} +45.7057 q^{37} +0.465483 q^{38} +40.0000 q^{40} +135.385 q^{41} +543.528 q^{43} +128.541 q^{44} +46.0000 q^{46} -26.4344 q^{47} +212.180 q^{49} +50.0000 q^{50} +160.324 q^{52} -43.6958 q^{53} +160.676 q^{55} +188.498 q^{56} +274.451 q^{58} -202.248 q^{59} +150.279 q^{61} +225.732 q^{62} +64.0000 q^{64} +200.405 q^{65} -420.722 q^{67} -504.661 q^{68} +235.622 q^{70} -667.381 q^{71} +602.960 q^{73} +91.4115 q^{74} +0.930966 q^{76} +757.178 q^{77} -1378.88 q^{79} +80.0000 q^{80} +270.769 q^{82} +485.178 q^{83} -630.826 q^{85} +1087.06 q^{86} +257.082 q^{88} +1127.71 q^{89} +944.400 q^{91} +92.0000 q^{92} -52.8688 q^{94} +1.16371 q^{95} -1486.24 q^{97} +424.359 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8} + O(q^{10})$$ $$4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8} + 40 q^{10} + 39 q^{11} - 20 q^{13} - 2 q^{14} + 64 q^{16} + 23 q^{17} + 53 q^{19} + 80 q^{20} + 78 q^{22} + 92 q^{23} + 100 q^{25} - 40 q^{26} - 4 q^{28} - 161 q^{29} + 388 q^{31} + 128 q^{32} + 46 q^{34} - 5 q^{35} + 466 q^{37} + 106 q^{38} + 160 q^{40} - 484 q^{41} + 894 q^{43} + 156 q^{44} + 184 q^{46} + 265 q^{47} + 1643 q^{49} + 200 q^{50} - 80 q^{52} - 576 q^{53} + 195 q^{55} - 8 q^{56} - 322 q^{58} + 94 q^{59} + 1153 q^{61} + 776 q^{62} + 256 q^{64} - 100 q^{65} - 1472 q^{67} + 92 q^{68} - 10 q^{70} - 200 q^{71} + 1147 q^{73} + 932 q^{74} + 212 q^{76} + 2176 q^{77} - 908 q^{79} + 320 q^{80} - 968 q^{82} + 1048 q^{83} + 115 q^{85} + 1788 q^{86} + 312 q^{88} + 1784 q^{89} + 2329 q^{91} + 368 q^{92} + 530 q^{94} + 265 q^{95} - 2047 q^{97} + 3286 q^{98} + O(q^{100})$$

## 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
$$7$$ 23.5622 1.27224 0.636121 0.771589i $$-0.280538\pi$$
0.636121 + 0.771589i $$0.280538\pi$$
$$8$$ 8.00000 0.353553
$$9$$ 0 0
$$10$$ 10.0000 0.316228
$$11$$ 32.1352 0.880830 0.440415 0.897794i $$-0.354831\pi$$
0.440415 + 0.897794i $$0.354831\pi$$
$$12$$ 0 0
$$13$$ 40.0811 0.855115 0.427557 0.903988i $$-0.359374\pi$$
0.427557 + 0.903988i $$0.359374\pi$$
$$14$$ 47.1245 0.899611
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ −126.165 −1.79997 −0.899987 0.435916i $$-0.856424\pi$$
−0.899987 + 0.435916i $$0.856424\pi$$
$$18$$ 0 0
$$19$$ 0.232742 0.00281024 0.00140512 0.999999i $$-0.499553\pi$$
0.00140512 + 0.999999i $$0.499553\pi$$
$$20$$ 20.0000 0.223607
$$21$$ 0 0
$$22$$ 64.2704 0.622841
$$23$$ 23.0000 0.208514
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 80.1621 0.604657
$$27$$ 0 0
$$28$$ 94.2490 0.636121
$$29$$ 137.226 0.878695 0.439347 0.898317i $$-0.355210\pi$$
0.439347 + 0.898317i $$0.355210\pi$$
$$30$$ 0 0
$$31$$ 112.866 0.653914 0.326957 0.945039i $$-0.393977\pi$$
0.326957 + 0.945039i $$0.393977\pi$$
$$32$$ 32.0000 0.176777
$$33$$ 0 0
$$34$$ −252.331 −1.27277
$$35$$ 117.811 0.568964
$$36$$ 0 0
$$37$$ 45.7057 0.203080 0.101540 0.994831i $$-0.467623\pi$$
0.101540 + 0.994831i $$0.467623\pi$$
$$38$$ 0.465483 0.00198714
$$39$$ 0 0
$$40$$ 40.0000 0.158114
$$41$$ 135.385 0.515696 0.257848 0.966186i $$-0.416987\pi$$
0.257848 + 0.966186i $$0.416987\pi$$
$$42$$ 0 0
$$43$$ 543.528 1.92761 0.963805 0.266608i $$-0.0859030\pi$$
0.963805 + 0.266608i $$0.0859030\pi$$
$$44$$ 128.541 0.440415
$$45$$ 0 0
$$46$$ 46.0000 0.147442
$$47$$ −26.4344 −0.0820394 −0.0410197 0.999158i $$-0.513061\pi$$
−0.0410197 + 0.999158i $$0.513061\pi$$
$$48$$ 0 0
$$49$$ 212.180 0.618599
$$50$$ 50.0000 0.141421
$$51$$ 0 0
$$52$$ 160.324 0.427557
$$53$$ −43.6958 −0.113247 −0.0566235 0.998396i $$-0.518033\pi$$
−0.0566235 + 0.998396i $$0.518033\pi$$
$$54$$ 0 0
$$55$$ 160.676 0.393919
$$56$$ 188.498 0.449805
$$57$$ 0 0
$$58$$ 274.451 0.621331
$$59$$ −202.248 −0.446278 −0.223139 0.974787i $$-0.571630\pi$$
−0.223139 + 0.974787i $$0.571630\pi$$
$$60$$ 0 0
$$61$$ 150.279 0.315430 0.157715 0.987485i $$-0.449587\pi$$
0.157715 + 0.987485i $$0.449587\pi$$
$$62$$ 225.732 0.462387
$$63$$ 0 0
$$64$$ 64.0000 0.125000
$$65$$ 200.405 0.382419
$$66$$ 0 0
$$67$$ −420.722 −0.767155 −0.383577 0.923509i $$-0.625308\pi$$
−0.383577 + 0.923509i $$0.625308\pi$$
$$68$$ −504.661 −0.899987
$$69$$ 0 0
$$70$$ 235.622 0.402318
$$71$$ −667.381 −1.11554 −0.557771 0.829995i $$-0.688343\pi$$
−0.557771 + 0.829995i $$0.688343\pi$$
$$72$$ 0 0
$$73$$ 602.960 0.966728 0.483364 0.875420i $$-0.339415\pi$$
0.483364 + 0.875420i $$0.339415\pi$$
$$74$$ 91.4115 0.143600
$$75$$ 0 0
$$76$$ 0.930966 0.00140512
$$77$$ 757.178 1.12063
$$78$$ 0 0
$$79$$ −1378.88 −1.96374 −0.981872 0.189545i $$-0.939299\pi$$
−0.981872 + 0.189545i $$0.939299\pi$$
$$80$$ 80.0000 0.111803
$$81$$ 0 0
$$82$$ 270.769 0.364652
$$83$$ 485.178 0.641629 0.320815 0.947142i $$-0.396043\pi$$
0.320815 + 0.947142i $$0.396043\pi$$
$$84$$ 0 0
$$85$$ −630.826 −0.804973
$$86$$ 1087.06 1.36303
$$87$$ 0 0
$$88$$ 257.082 0.311420
$$89$$ 1127.71 1.34311 0.671556 0.740954i $$-0.265626\pi$$
0.671556 + 0.740954i $$0.265626\pi$$
$$90$$ 0 0
$$91$$ 944.400 1.08791
$$92$$ 92.0000 0.104257
$$93$$ 0 0
$$94$$ −52.8688 −0.0580106
$$95$$ 1.16371 0.00125678
$$96$$ 0 0
$$97$$ −1486.24 −1.55572 −0.777858 0.628441i $$-0.783694\pi$$
−0.777858 + 0.628441i $$0.783694\pi$$
$$98$$ 424.359 0.437416
$$99$$ 0 0
$$100$$ 100.000 0.100000
$$101$$ −1888.86 −1.86087 −0.930437 0.366451i $$-0.880573\pi$$
−0.930437 + 0.366451i $$0.880573\pi$$
$$102$$ 0 0
$$103$$ −1497.17 −1.43224 −0.716118 0.697979i $$-0.754083\pi$$
−0.716118 + 0.697979i $$0.754083\pi$$
$$104$$ 320.649 0.302329
$$105$$ 0 0
$$106$$ −87.3917 −0.0800777
$$107$$ 585.150 0.528678 0.264339 0.964430i $$-0.414846\pi$$
0.264339 + 0.964430i $$0.414846\pi$$
$$108$$ 0 0
$$109$$ −139.166 −0.122290 −0.0611452 0.998129i $$-0.519475\pi$$
−0.0611452 + 0.998129i $$0.519475\pi$$
$$110$$ 321.352 0.278543
$$111$$ 0 0
$$112$$ 376.996 0.318060
$$113$$ 1293.18 1.07656 0.538282 0.842765i $$-0.319073\pi$$
0.538282 + 0.842765i $$0.319073\pi$$
$$114$$ 0 0
$$115$$ 115.000 0.0932505
$$116$$ 548.902 0.439347
$$117$$ 0 0
$$118$$ −404.495 −0.315566
$$119$$ −2972.74 −2.29000
$$120$$ 0 0
$$121$$ −298.328 −0.224138
$$122$$ 300.557 0.223043
$$123$$ 0 0
$$124$$ 451.464 0.326957
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −1298.43 −0.907221 −0.453610 0.891200i $$-0.649864\pi$$
−0.453610 + 0.891200i $$0.649864\pi$$
$$128$$ 128.000 0.0883883
$$129$$ 0 0
$$130$$ 400.811 0.270411
$$131$$ 525.247 0.350313 0.175157 0.984541i $$-0.443957\pi$$
0.175157 + 0.984541i $$0.443957\pi$$
$$132$$ 0 0
$$133$$ 5.48392 0.00357531
$$134$$ −841.444 −0.542460
$$135$$ 0 0
$$136$$ −1009.32 −0.636387
$$137$$ 2428.12 1.51422 0.757109 0.653288i $$-0.226611\pi$$
0.757109 + 0.653288i $$0.226611\pi$$
$$138$$ 0 0
$$139$$ 2456.46 1.49895 0.749476 0.662031i $$-0.230305\pi$$
0.749476 + 0.662031i $$0.230305\pi$$
$$140$$ 471.245 0.284482
$$141$$ 0 0
$$142$$ −1334.76 −0.788808
$$143$$ 1288.01 0.753211
$$144$$ 0 0
$$145$$ 686.128 0.392964
$$146$$ 1205.92 0.683580
$$147$$ 0 0
$$148$$ 182.823 0.101540
$$149$$ 2405.39 1.32253 0.661266 0.750151i $$-0.270019\pi$$
0.661266 + 0.750151i $$0.270019\pi$$
$$150$$ 0 0
$$151$$ −649.276 −0.349916 −0.174958 0.984576i $$-0.555979\pi$$
−0.174958 + 0.984576i $$0.555979\pi$$
$$152$$ 1.86193 0.000993570 0
$$153$$ 0 0
$$154$$ 1514.36 0.792404
$$155$$ 564.330 0.292439
$$156$$ 0 0
$$157$$ 3665.04 1.86307 0.931536 0.363650i $$-0.118470\pi$$
0.931536 + 0.363650i $$0.118470\pi$$
$$158$$ −2757.75 −1.38858
$$159$$ 0 0
$$160$$ 160.000 0.0790569
$$161$$ 541.932 0.265281
$$162$$ 0 0
$$163$$ −1055.27 −0.507088 −0.253544 0.967324i $$-0.581596\pi$$
−0.253544 + 0.967324i $$0.581596\pi$$
$$164$$ 541.539 0.257848
$$165$$ 0 0
$$166$$ 970.356 0.453700
$$167$$ 731.805 0.339095 0.169547 0.985522i $$-0.445769\pi$$
0.169547 + 0.985522i $$0.445769\pi$$
$$168$$ 0 0
$$169$$ −590.507 −0.268779
$$170$$ −1261.65 −0.569202
$$171$$ 0 0
$$172$$ 2174.11 0.963805
$$173$$ 521.773 0.229304 0.114652 0.993406i $$-0.463425\pi$$
0.114652 + 0.993406i $$0.463425\pi$$
$$174$$ 0 0
$$175$$ 589.056 0.254448
$$176$$ 514.163 0.220208
$$177$$ 0 0
$$178$$ 2255.42 0.949723
$$179$$ 1183.07 0.494005 0.247002 0.969015i $$-0.420554\pi$$
0.247002 + 0.969015i $$0.420554\pi$$
$$180$$ 0 0
$$181$$ 1723.92 0.707946 0.353973 0.935256i $$-0.384830\pi$$
0.353973 + 0.935256i $$0.384830\pi$$
$$182$$ 1888.80 0.769270
$$183$$ 0 0
$$184$$ 184.000 0.0737210
$$185$$ 228.529 0.0908204
$$186$$ 0 0
$$187$$ −4054.35 −1.58547
$$188$$ −105.738 −0.0410197
$$189$$ 0 0
$$190$$ 2.32742 0.000888676 0
$$191$$ 2798.12 1.06002 0.530012 0.847990i $$-0.322187\pi$$
0.530012 + 0.847990i $$0.322187\pi$$
$$192$$ 0 0
$$193$$ −497.686 −0.185618 −0.0928089 0.995684i $$-0.529585\pi$$
−0.0928089 + 0.995684i $$0.529585\pi$$
$$194$$ −2972.47 −1.10006
$$195$$ 0 0
$$196$$ 848.718 0.309300
$$197$$ −296.489 −0.107228 −0.0536141 0.998562i $$-0.517074\pi$$
−0.0536141 + 0.998562i $$0.517074\pi$$
$$198$$ 0 0
$$199$$ 2347.43 0.836207 0.418103 0.908399i $$-0.362695\pi$$
0.418103 + 0.908399i $$0.362695\pi$$
$$200$$ 200.000 0.0707107
$$201$$ 0 0
$$202$$ −3777.71 −1.31584
$$203$$ 3233.34 1.11791
$$204$$ 0 0
$$205$$ 676.923 0.230626
$$206$$ −2994.34 −1.01274
$$207$$ 0 0
$$208$$ 641.297 0.213779
$$209$$ 7.47920 0.00247535
$$210$$ 0 0
$$211$$ 2838.61 0.926153 0.463076 0.886318i $$-0.346746\pi$$
0.463076 + 0.886318i $$0.346746\pi$$
$$212$$ −174.783 −0.0566235
$$213$$ 0 0
$$214$$ 1170.30 0.373832
$$215$$ 2717.64 0.862053
$$216$$ 0 0
$$217$$ 2659.38 0.831937
$$218$$ −278.331 −0.0864723
$$219$$ 0 0
$$220$$ 642.704 0.196960
$$221$$ −5056.84 −1.53918
$$222$$ 0 0
$$223$$ −2124.68 −0.638024 −0.319012 0.947751i $$-0.603351\pi$$
−0.319012 + 0.947751i $$0.603351\pi$$
$$224$$ 753.992 0.224903
$$225$$ 0 0
$$226$$ 2586.35 0.761246
$$227$$ −1551.97 −0.453780 −0.226890 0.973920i $$-0.572856\pi$$
−0.226890 + 0.973920i $$0.572856\pi$$
$$228$$ 0 0
$$229$$ −158.531 −0.0457469 −0.0228735 0.999738i $$-0.507281\pi$$
−0.0228735 + 0.999738i $$0.507281\pi$$
$$230$$ 230.000 0.0659380
$$231$$ 0 0
$$232$$ 1097.80 0.310666
$$233$$ 728.999 0.204971 0.102486 0.994734i $$-0.467320\pi$$
0.102486 + 0.994734i $$0.467320\pi$$
$$234$$ 0 0
$$235$$ −132.172 −0.0366891
$$236$$ −808.991 −0.223139
$$237$$ 0 0
$$238$$ −5945.47 −1.61928
$$239$$ 6201.60 1.67844 0.839222 0.543789i $$-0.183011\pi$$
0.839222 + 0.543789i $$0.183011\pi$$
$$240$$ 0 0
$$241$$ −7223.04 −1.93061 −0.965305 0.261126i $$-0.915906\pi$$
−0.965305 + 0.261126i $$0.915906\pi$$
$$242$$ −596.656 −0.158490
$$243$$ 0 0
$$244$$ 601.115 0.157715
$$245$$ 1060.90 0.276646
$$246$$ 0 0
$$247$$ 9.32853 0.00240308
$$248$$ 902.928 0.231193
$$249$$ 0 0
$$250$$ 250.000 0.0632456
$$251$$ −5042.78 −1.26812 −0.634059 0.773285i $$-0.718612\pi$$
−0.634059 + 0.773285i $$0.718612\pi$$
$$252$$ 0 0
$$253$$ 739.110 0.183666
$$254$$ −2596.86 −0.641502
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ −2981.15 −0.723577 −0.361788 0.932260i $$-0.617834\pi$$
−0.361788 + 0.932260i $$0.617834\pi$$
$$258$$ 0 0
$$259$$ 1076.93 0.258367
$$260$$ 801.621 0.191209
$$261$$ 0 0
$$262$$ 1050.49 0.247709
$$263$$ −7242.86 −1.69815 −0.849076 0.528271i $$-0.822841\pi$$
−0.849076 + 0.528271i $$0.822841\pi$$
$$264$$ 0 0
$$265$$ −218.479 −0.0506456
$$266$$ 10.9678 0.00252812
$$267$$ 0 0
$$268$$ −1682.89 −0.383577
$$269$$ 2567.58 0.581962 0.290981 0.956729i $$-0.406018\pi$$
0.290981 + 0.956729i $$0.406018\pi$$
$$270$$ 0 0
$$271$$ 8066.27 1.80808 0.904042 0.427443i $$-0.140586\pi$$
0.904042 + 0.427443i $$0.140586\pi$$
$$272$$ −2018.64 −0.449994
$$273$$ 0 0
$$274$$ 4856.23 1.07071
$$275$$ 803.380 0.176166
$$276$$ 0 0
$$277$$ 8991.54 1.95036 0.975179 0.221418i $$-0.0710684\pi$$
0.975179 + 0.221418i $$0.0710684\pi$$
$$278$$ 4912.92 1.05992
$$279$$ 0 0
$$280$$ 942.490 0.201159
$$281$$ −968.130 −0.205529 −0.102765 0.994706i $$-0.532769\pi$$
−0.102765 + 0.994706i $$0.532769\pi$$
$$282$$ 0 0
$$283$$ −2252.65 −0.473167 −0.236583 0.971611i $$-0.576028\pi$$
−0.236583 + 0.971611i $$0.576028\pi$$
$$284$$ −2669.52 −0.557771
$$285$$ 0 0
$$286$$ 2576.03 0.532600
$$287$$ 3189.97 0.656090
$$288$$ 0 0
$$289$$ 11004.7 2.23991
$$290$$ 1372.26 0.277868
$$291$$ 0 0
$$292$$ 2411.84 0.483364
$$293$$ −2734.35 −0.545196 −0.272598 0.962128i $$-0.587883\pi$$
−0.272598 + 0.962128i $$0.587883\pi$$
$$294$$ 0 0
$$295$$ −1011.24 −0.199582
$$296$$ 365.646 0.0717998
$$297$$ 0 0
$$298$$ 4810.78 0.935172
$$299$$ 921.865 0.178304
$$300$$ 0 0
$$301$$ 12806.7 2.45239
$$302$$ −1298.55 −0.247428
$$303$$ 0 0
$$304$$ 3.72387 0.000702560 0
$$305$$ 751.394 0.141065
$$306$$ 0 0
$$307$$ −7977.09 −1.48299 −0.741493 0.670961i $$-0.765882\pi$$
−0.741493 + 0.670961i $$0.765882\pi$$
$$308$$ 3028.71 0.560314
$$309$$ 0 0
$$310$$ 1128.66 0.206786
$$311$$ −7729.43 −1.40931 −0.704655 0.709550i $$-0.748898\pi$$
−0.704655 + 0.709550i $$0.748898\pi$$
$$312$$ 0 0
$$313$$ 5506.25 0.994350 0.497175 0.867650i $$-0.334371\pi$$
0.497175 + 0.867650i $$0.334371\pi$$
$$314$$ 7330.08 1.31739
$$315$$ 0 0
$$316$$ −5515.51 −0.981872
$$317$$ −5231.37 −0.926887 −0.463443 0.886126i $$-0.653386\pi$$
−0.463443 + 0.886126i $$0.653386\pi$$
$$318$$ 0 0
$$319$$ 4409.77 0.773981
$$320$$ 320.000 0.0559017
$$321$$ 0 0
$$322$$ 1083.86 0.187582
$$323$$ −29.3639 −0.00505836
$$324$$ 0 0
$$325$$ 1002.03 0.171023
$$326$$ −2110.55 −0.358566
$$327$$ 0 0
$$328$$ 1083.08 0.182326
$$329$$ −622.854 −0.104374
$$330$$ 0 0
$$331$$ −3551.61 −0.589771 −0.294885 0.955533i $$-0.595281\pi$$
−0.294885 + 0.955533i $$0.595281\pi$$
$$332$$ 1940.71 0.320815
$$333$$ 0 0
$$334$$ 1463.61 0.239776
$$335$$ −2103.61 −0.343082
$$336$$ 0 0
$$337$$ −7002.99 −1.13198 −0.565990 0.824412i $$-0.691506\pi$$
−0.565990 + 0.824412i $$0.691506\pi$$
$$338$$ −1181.01 −0.190055
$$339$$ 0 0
$$340$$ −2523.31 −0.402487
$$341$$ 3626.97 0.575987
$$342$$ 0 0
$$343$$ −3082.42 −0.485234
$$344$$ 4348.22 0.681513
$$345$$ 0 0
$$346$$ 1043.55 0.162143
$$347$$ 10268.2 1.58854 0.794272 0.607562i $$-0.207852\pi$$
0.794272 + 0.607562i $$0.207852\pi$$
$$348$$ 0 0
$$349$$ 4515.58 0.692588 0.346294 0.938126i $$-0.387440\pi$$
0.346294 + 0.938126i $$0.387440\pi$$
$$350$$ 1178.11 0.179922
$$351$$ 0 0
$$352$$ 1028.33 0.155710
$$353$$ −7838.63 −1.18189 −0.590947 0.806711i $$-0.701246\pi$$
−0.590947 + 0.806711i $$0.701246\pi$$
$$354$$ 0 0
$$355$$ −3336.90 −0.498886
$$356$$ 4510.84 0.671556
$$357$$ 0 0
$$358$$ 2366.14 0.349314
$$359$$ 12464.8 1.83250 0.916250 0.400606i $$-0.131201\pi$$
0.916250 + 0.400606i $$0.131201\pi$$
$$360$$ 0 0
$$361$$ −6858.95 −0.999992
$$362$$ 3447.85 0.500594
$$363$$ 0 0
$$364$$ 3777.60 0.543956
$$365$$ 3014.80 0.432334
$$366$$ 0 0
$$367$$ 3936.14 0.559850 0.279925 0.960022i $$-0.409690\pi$$
0.279925 + 0.960022i $$0.409690\pi$$
$$368$$ 368.000 0.0521286
$$369$$ 0 0
$$370$$ 457.057 0.0642197
$$371$$ −1029.57 −0.144077
$$372$$ 0 0
$$373$$ 1920.72 0.266625 0.133313 0.991074i $$-0.457439\pi$$
0.133313 + 0.991074i $$0.457439\pi$$
$$374$$ −8108.69 −1.12110
$$375$$ 0 0
$$376$$ −211.475 −0.0290053
$$377$$ 5500.15 0.751385
$$378$$ 0 0
$$379$$ −13074.5 −1.77201 −0.886004 0.463678i $$-0.846530\pi$$
−0.886004 + 0.463678i $$0.846530\pi$$
$$380$$ 4.65483 0.000628389 0
$$381$$ 0 0
$$382$$ 5596.23 0.749550
$$383$$ −8838.22 −1.17914 −0.589571 0.807716i $$-0.700703\pi$$
−0.589571 + 0.807716i $$0.700703\pi$$
$$384$$ 0 0
$$385$$ 3785.89 0.501160
$$386$$ −995.372 −0.131252
$$387$$ 0 0
$$388$$ −5944.94 −0.777858
$$389$$ 12687.3 1.65365 0.826827 0.562456i $$-0.190144\pi$$
0.826827 + 0.562456i $$0.190144\pi$$
$$390$$ 0 0
$$391$$ −2901.80 −0.375321
$$392$$ 1697.44 0.218708
$$393$$ 0 0
$$394$$ −592.977 −0.0758218
$$395$$ −6894.38 −0.878213
$$396$$ 0 0
$$397$$ 8060.49 1.01900 0.509501 0.860470i $$-0.329830\pi$$
0.509501 + 0.860470i $$0.329830\pi$$
$$398$$ 4694.87 0.591287
$$399$$ 0 0
$$400$$ 400.000 0.0500000
$$401$$ −12325.1 −1.53488 −0.767440 0.641121i $$-0.778470\pi$$
−0.767440 + 0.641121i $$0.778470\pi$$
$$402$$ 0 0
$$403$$ 4523.79 0.559171
$$404$$ −7555.43 −0.930437
$$405$$ 0 0
$$406$$ 6466.69 0.790483
$$407$$ 1468.76 0.178879
$$408$$ 0 0
$$409$$ −10806.7 −1.30650 −0.653250 0.757143i $$-0.726595\pi$$
−0.653250 + 0.757143i $$0.726595\pi$$
$$410$$ 1353.85 0.163077
$$411$$ 0 0
$$412$$ −5988.67 −0.716118
$$413$$ −4765.41 −0.567774
$$414$$ 0 0
$$415$$ 2425.89 0.286945
$$416$$ 1282.59 0.151164
$$417$$ 0 0
$$418$$ 14.9584 0.00175033
$$419$$ 1823.74 0.212639 0.106319 0.994332i $$-0.466093\pi$$
0.106319 + 0.994332i $$0.466093\pi$$
$$420$$ 0 0
$$421$$ 5995.43 0.694060 0.347030 0.937854i $$-0.387190\pi$$
0.347030 + 0.937854i $$0.387190\pi$$
$$422$$ 5677.23 0.654889
$$423$$ 0 0
$$424$$ −349.567 −0.0400388
$$425$$ −3154.13 −0.359995
$$426$$ 0 0
$$427$$ 3540.91 0.401303
$$428$$ 2340.60 0.264339
$$429$$ 0 0
$$430$$ 5435.28 0.609564
$$431$$ −3887.80 −0.434499 −0.217249 0.976116i $$-0.569708\pi$$
−0.217249 + 0.976116i $$0.569708\pi$$
$$432$$ 0 0
$$433$$ −4422.93 −0.490884 −0.245442 0.969411i $$-0.578933\pi$$
−0.245442 + 0.969411i $$0.578933\pi$$
$$434$$ 5318.75 0.588268
$$435$$ 0 0
$$436$$ −556.662 −0.0611452
$$437$$ 5.35306 0.000585976 0
$$438$$ 0 0
$$439$$ −1748.08 −0.190048 −0.0950242 0.995475i $$-0.530293\pi$$
−0.0950242 + 0.995475i $$0.530293\pi$$
$$440$$ 1285.41 0.139271
$$441$$ 0 0
$$442$$ −10113.7 −1.08837
$$443$$ −3371.31 −0.361571 −0.180785 0.983523i $$-0.557864\pi$$
−0.180785 + 0.983523i $$0.557864\pi$$
$$444$$ 0 0
$$445$$ 5638.55 0.600658
$$446$$ −4249.37 −0.451151
$$447$$ 0 0
$$448$$ 1507.98 0.159030
$$449$$ −13314.7 −1.39947 −0.699734 0.714404i $$-0.746698\pi$$
−0.699734 + 0.714404i $$0.746698\pi$$
$$450$$ 0 0
$$451$$ 4350.61 0.454240
$$452$$ 5172.70 0.538282
$$453$$ 0 0
$$454$$ −3103.94 −0.320871
$$455$$ 4722.00 0.486529
$$456$$ 0 0
$$457$$ 3767.20 0.385607 0.192803 0.981237i $$-0.438242\pi$$
0.192803 + 0.981237i $$0.438242\pi$$
$$458$$ −317.063 −0.0323480
$$459$$ 0 0
$$460$$ 460.000 0.0466252
$$461$$ −9674.06 −0.977366 −0.488683 0.872461i $$-0.662523\pi$$
−0.488683 + 0.872461i $$0.662523\pi$$
$$462$$ 0 0
$$463$$ 2977.73 0.298892 0.149446 0.988770i $$-0.452251\pi$$
0.149446 + 0.988770i $$0.452251\pi$$
$$464$$ 2195.61 0.219674
$$465$$ 0 0
$$466$$ 1458.00 0.144937
$$467$$ 10701.0 1.06035 0.530176 0.847888i $$-0.322126\pi$$
0.530176 + 0.847888i $$0.322126\pi$$
$$468$$ 0 0
$$469$$ −9913.16 −0.976006
$$470$$ −264.344 −0.0259431
$$471$$ 0 0
$$472$$ −1617.98 −0.157783
$$473$$ 17466.4 1.69790
$$474$$ 0 0
$$475$$ 5.81854 0.000562048 0
$$476$$ −11890.9 −1.14500
$$477$$ 0 0
$$478$$ 12403.2 1.18684
$$479$$ −2337.15 −0.222937 −0.111469 0.993768i $$-0.535555\pi$$
−0.111469 + 0.993768i $$0.535555\pi$$
$$480$$ 0 0
$$481$$ 1831.94 0.173657
$$482$$ −14446.1 −1.36515
$$483$$ 0 0
$$484$$ −1193.31 −0.112069
$$485$$ −7431.18 −0.695737
$$486$$ 0 0
$$487$$ −7183.85 −0.668442 −0.334221 0.942495i $$-0.608473\pi$$
−0.334221 + 0.942495i $$0.608473\pi$$
$$488$$ 1202.23 0.111521
$$489$$ 0 0
$$490$$ 2121.80 0.195618
$$491$$ 12084.2 1.11070 0.555350 0.831617i $$-0.312584\pi$$
0.555350 + 0.831617i $$0.312584\pi$$
$$492$$ 0 0
$$493$$ −17313.1 −1.58163
$$494$$ 18.6571 0.00169923
$$495$$ 0 0
$$496$$ 1805.86 0.163478
$$497$$ −15725.0 −1.41924
$$498$$ 0 0
$$499$$ 7145.78 0.641060 0.320530 0.947238i $$-0.396139\pi$$
0.320530 + 0.947238i $$0.396139\pi$$
$$500$$ 500.000 0.0447214
$$501$$ 0 0
$$502$$ −10085.6 −0.896695
$$503$$ −20436.2 −1.81154 −0.905770 0.423771i $$-0.860706\pi$$
−0.905770 + 0.423771i $$0.860706\pi$$
$$504$$ 0 0
$$505$$ −9444.29 −0.832208
$$506$$ 1478.22 0.129871
$$507$$ 0 0
$$508$$ −5193.72 −0.453610
$$509$$ −19721.7 −1.71738 −0.858690 0.512495i $$-0.828721\pi$$
−0.858690 + 0.512495i $$0.828721\pi$$
$$510$$ 0 0
$$511$$ 14207.1 1.22991
$$512$$ 512.000 0.0441942
$$513$$ 0 0
$$514$$ −5962.30 −0.511646
$$515$$ −7485.84 −0.640516
$$516$$ 0 0
$$517$$ −849.475 −0.0722628
$$518$$ 2153.86 0.182693
$$519$$ 0 0
$$520$$ 1603.24 0.135205
$$521$$ −3483.23 −0.292904 −0.146452 0.989218i $$-0.546785\pi$$
−0.146452 + 0.989218i $$0.546785\pi$$
$$522$$ 0 0
$$523$$ 15689.0 1.31172 0.655862 0.754881i $$-0.272305\pi$$
0.655862 + 0.754881i $$0.272305\pi$$
$$524$$ 2100.99 0.175157
$$525$$ 0 0
$$526$$ −14485.7 −1.20077
$$527$$ −14239.8 −1.17703
$$528$$ 0 0
$$529$$ 529.000 0.0434783
$$530$$ −436.958 −0.0358118
$$531$$ 0 0
$$532$$ 21.9357 0.00178765
$$533$$ 5426.36 0.440979
$$534$$ 0 0
$$535$$ 2925.75 0.236432
$$536$$ −3365.78 −0.271230
$$537$$ 0 0
$$538$$ 5135.15 0.411510
$$539$$ 6818.43 0.544881
$$540$$ 0 0
$$541$$ 7620.73 0.605621 0.302810 0.953051i $$-0.402075\pi$$
0.302810 + 0.953051i $$0.402075\pi$$
$$542$$ 16132.5 1.27851
$$543$$ 0 0
$$544$$ −4037.29 −0.318194
$$545$$ −695.828 −0.0546899
$$546$$ 0 0
$$547$$ 5925.62 0.463183 0.231592 0.972813i $$-0.425607\pi$$
0.231592 + 0.972813i $$0.425607\pi$$
$$548$$ 9712.47 0.757109
$$549$$ 0 0
$$550$$ 1606.76 0.124568
$$551$$ 31.9381 0.00246934
$$552$$ 0 0
$$553$$ −32489.4 −2.49836
$$554$$ 17983.1 1.37911
$$555$$ 0 0
$$556$$ 9825.85 0.749476
$$557$$ 5456.16 0.415054 0.207527 0.978229i $$-0.433459\pi$$
0.207527 + 0.978229i $$0.433459\pi$$
$$558$$ 0 0
$$559$$ 21785.2 1.64833
$$560$$ 1884.98 0.142241
$$561$$ 0 0
$$562$$ −1936.26 −0.145331
$$563$$ 9194.29 0.688265 0.344132 0.938921i $$-0.388173\pi$$
0.344132 + 0.938921i $$0.388173\pi$$
$$564$$ 0 0
$$565$$ 6465.88 0.481454
$$566$$ −4505.30 −0.334580
$$567$$ 0 0
$$568$$ −5339.05 −0.394404
$$569$$ −338.831 −0.0249640 −0.0124820 0.999922i $$-0.503973\pi$$
−0.0124820 + 0.999922i $$0.503973\pi$$
$$570$$ 0 0
$$571$$ −1725.34 −0.126451 −0.0632254 0.997999i $$-0.520139\pi$$
−0.0632254 + 0.997999i $$0.520139\pi$$
$$572$$ 5152.06 0.376605
$$573$$ 0 0
$$574$$ 6379.93 0.463926
$$575$$ 575.000 0.0417029
$$576$$ 0 0
$$577$$ 23300.0 1.68109 0.840547 0.541738i $$-0.182234\pi$$
0.840547 + 0.541738i $$0.182234\pi$$
$$578$$ 22009.3 1.58385
$$579$$ 0 0
$$580$$ 2744.51 0.196482
$$581$$ 11431.9 0.816307
$$582$$ 0 0
$$583$$ −1404.18 −0.0997513
$$584$$ 4823.68 0.341790
$$585$$ 0 0
$$586$$ −5468.70 −0.385512
$$587$$ −15653.3 −1.10065 −0.550325 0.834950i $$-0.685496\pi$$
−0.550325 + 0.834950i $$0.685496\pi$$
$$588$$ 0 0
$$589$$ 26.2686 0.00183766
$$590$$ −2022.48 −0.141126
$$591$$ 0 0
$$592$$ 731.292 0.0507701
$$593$$ 2658.08 0.184072 0.0920358 0.995756i $$-0.470663\pi$$
0.0920358 + 0.995756i $$0.470663\pi$$
$$594$$ 0 0
$$595$$ −14863.7 −1.02412
$$596$$ 9621.57 0.661266
$$597$$ 0 0
$$598$$ 1843.73 0.126080
$$599$$ −19417.6 −1.32451 −0.662256 0.749278i $$-0.730401\pi$$
−0.662256 + 0.749278i $$0.730401\pi$$
$$600$$ 0 0
$$601$$ −18469.0 −1.25352 −0.626760 0.779213i $$-0.715619\pi$$
−0.626760 + 0.779213i $$0.715619\pi$$
$$602$$ 25613.5 1.73410
$$603$$ 0 0
$$604$$ −2597.10 −0.174958
$$605$$ −1491.64 −0.100238
$$606$$ 0 0
$$607$$ 3968.56 0.265369 0.132684 0.991158i $$-0.457640\pi$$
0.132684 + 0.991158i $$0.457640\pi$$
$$608$$ 7.44773 0.000496785 0
$$609$$ 0 0
$$610$$ 1502.79 0.0997477
$$611$$ −1059.52 −0.0701531
$$612$$ 0 0
$$613$$ −11478.7 −0.756311 −0.378156 0.925742i $$-0.623442\pi$$
−0.378156 + 0.925742i $$0.623442\pi$$
$$614$$ −15954.2 −1.04863
$$615$$ 0 0
$$616$$ 6057.42 0.396202
$$617$$ 15691.1 1.02382 0.511911 0.859039i $$-0.328938\pi$$
0.511911 + 0.859039i $$0.328938\pi$$
$$618$$ 0 0
$$619$$ −18249.4 −1.18499 −0.592494 0.805575i $$-0.701856\pi$$
−0.592494 + 0.805575i $$0.701856\pi$$
$$620$$ 2257.32 0.146220
$$621$$ 0 0
$$622$$ −15458.9 −0.996533
$$623$$ 26571.4 1.70876
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 11012.5 0.703111
$$627$$ 0 0
$$628$$ 14660.2 0.931536
$$629$$ −5766.48 −0.365540
$$630$$ 0 0
$$631$$ −23528.8 −1.48442 −0.742208 0.670169i $$-0.766222\pi$$
−0.742208 + 0.670169i $$0.766222\pi$$
$$632$$ −11031.0 −0.694288
$$633$$ 0 0
$$634$$ −10462.7 −0.655408
$$635$$ −6492.15 −0.405722
$$636$$ 0 0
$$637$$ 8504.38 0.528973
$$638$$ 8819.55 0.547287
$$639$$ 0 0
$$640$$ 640.000 0.0395285
$$641$$ −3748.79 −0.230996 −0.115498 0.993308i $$-0.536846\pi$$
−0.115498 + 0.993308i $$0.536846\pi$$
$$642$$ 0 0
$$643$$ 15924.3 0.976660 0.488330 0.872659i $$-0.337606\pi$$
0.488330 + 0.872659i $$0.337606\pi$$
$$644$$ 2167.73 0.132640
$$645$$ 0 0
$$646$$ −58.7278 −0.00357680
$$647$$ 2854.11 0.173426 0.0867129 0.996233i $$-0.472364\pi$$
0.0867129 + 0.996233i $$0.472364\pi$$
$$648$$ 0 0
$$649$$ −6499.27 −0.393095
$$650$$ 2004.05 0.120931
$$651$$ 0 0
$$652$$ −4221.09 −0.253544
$$653$$ −7925.97 −0.474988 −0.237494 0.971389i $$-0.576326\pi$$
−0.237494 + 0.971389i $$0.576326\pi$$
$$654$$ 0 0
$$655$$ 2626.24 0.156665
$$656$$ 2166.15 0.128924
$$657$$ 0 0
$$658$$ −1245.71 −0.0738035
$$659$$ 5799.43 0.342813 0.171406 0.985200i $$-0.445169\pi$$
0.171406 + 0.985200i $$0.445169\pi$$
$$660$$ 0 0
$$661$$ 22324.8 1.31367 0.656833 0.754036i $$-0.271896\pi$$
0.656833 + 0.754036i $$0.271896\pi$$
$$662$$ −7103.22 −0.417031
$$663$$ 0 0
$$664$$ 3881.42 0.226850
$$665$$ 27.4196 0.00159893
$$666$$ 0 0
$$667$$ 3156.19 0.183221
$$668$$ 2927.22 0.169547
$$669$$ 0 0
$$670$$ −4207.22 −0.242596
$$671$$ 4829.24 0.277840
$$672$$ 0 0
$$673$$ −23253.5 −1.33188 −0.665940 0.746005i $$-0.731969\pi$$
−0.665940 + 0.746005i $$0.731969\pi$$
$$674$$ −14006.0 −0.800430
$$675$$ 0 0
$$676$$ −2362.03 −0.134389
$$677$$ 19003.7 1.07884 0.539419 0.842038i $$-0.318644\pi$$
0.539419 + 0.842038i $$0.318644\pi$$
$$678$$ 0 0
$$679$$ −35019.1 −1.97925
$$680$$ −5046.61 −0.284601
$$681$$ 0 0
$$682$$ 7253.94 0.407284
$$683$$ 11222.5 0.628722 0.314361 0.949304i $$-0.398210\pi$$
0.314361 + 0.949304i $$0.398210\pi$$
$$684$$ 0 0
$$685$$ 12140.6 0.677179
$$686$$ −6164.85 −0.343112
$$687$$ 0 0
$$688$$ 8696.45 0.481902
$$689$$ −1751.38 −0.0968391
$$690$$ 0 0
$$691$$ 4297.68 0.236601 0.118301 0.992978i $$-0.462255\pi$$
0.118301 + 0.992978i $$0.462255\pi$$
$$692$$ 2087.09 0.114652
$$693$$ 0 0
$$694$$ 20536.4 1.12327
$$695$$ 12282.3 0.670352
$$696$$ 0 0
$$697$$ −17080.8 −0.928240
$$698$$ 9031.15 0.489734
$$699$$ 0 0
$$700$$ 2356.22 0.127224
$$701$$ −25769.6 −1.38845 −0.694225 0.719758i $$-0.744253\pi$$
−0.694225 + 0.719758i $$0.744253\pi$$
$$702$$ 0 0
$$703$$ 10.6376 0.000570705 0
$$704$$ 2056.65 0.110104
$$705$$ 0 0
$$706$$ −15677.3 −0.835725
$$707$$ −44505.7 −2.36748
$$708$$ 0 0
$$709$$ −4900.01 −0.259554 −0.129777 0.991543i $$-0.541426\pi$$
−0.129777 + 0.991543i $$0.541426\pi$$
$$710$$ −6673.81 −0.352766
$$711$$ 0 0
$$712$$ 9021.67 0.474862
$$713$$ 2595.92 0.136350
$$714$$ 0 0
$$715$$ 6440.07 0.336846
$$716$$ 4732.28 0.247002
$$717$$ 0 0
$$718$$ 24929.6 1.29577
$$719$$ −7631.85 −0.395855 −0.197928 0.980217i $$-0.563421\pi$$
−0.197928 + 0.980217i $$0.563421\pi$$
$$720$$ 0 0
$$721$$ −35276.6 −1.82215
$$722$$ −13717.9 −0.707101
$$723$$ 0 0
$$724$$ 6895.70 0.353973
$$725$$ 3430.64 0.175739
$$726$$ 0 0
$$727$$ −36985.6 −1.88682 −0.943410 0.331628i $$-0.892402\pi$$
−0.943410 + 0.331628i $$0.892402\pi$$
$$728$$ 7555.20 0.384635
$$729$$ 0 0
$$730$$ 6029.60 0.305706
$$731$$ −68574.3 −3.46965
$$732$$ 0 0
$$733$$ −22227.3 −1.12003 −0.560015 0.828482i $$-0.689205\pi$$
−0.560015 + 0.828482i $$0.689205\pi$$
$$734$$ 7872.29 0.395874
$$735$$ 0 0
$$736$$ 736.000 0.0368605
$$737$$ −13520.0 −0.675733
$$738$$ 0 0
$$739$$ −17899.1 −0.890974 −0.445487 0.895288i $$-0.646970\pi$$
−0.445487 + 0.895288i $$0.646970\pi$$
$$740$$ 914.115 0.0454102
$$741$$ 0 0
$$742$$ −2059.14 −0.101878
$$743$$ 29843.4 1.47355 0.736776 0.676137i $$-0.236347\pi$$
0.736776 + 0.676137i $$0.236347\pi$$
$$744$$ 0 0
$$745$$ 12027.0 0.591455
$$746$$ 3841.44 0.188532
$$747$$ 0 0
$$748$$ −16217.4 −0.792736
$$749$$ 13787.4 0.672606
$$750$$ 0 0
$$751$$ −9399.65 −0.456722 −0.228361 0.973577i $$-0.573337\pi$$
−0.228361 + 0.973577i $$0.573337\pi$$
$$752$$ −422.950 −0.0205098
$$753$$ 0 0
$$754$$ 11000.3 0.531309
$$755$$ −3246.38 −0.156487
$$756$$ 0 0
$$757$$ 29871.6 1.43422 0.717109 0.696961i $$-0.245465\pi$$
0.717109 + 0.696961i $$0.245465\pi$$
$$758$$ −26149.0 −1.25300
$$759$$ 0 0
$$760$$ 9.30966 0.000444338 0
$$761$$ 273.955 0.0130497 0.00652487 0.999979i $$-0.497923\pi$$
0.00652487 + 0.999979i $$0.497923\pi$$
$$762$$ 0 0
$$763$$ −3279.05 −0.155583
$$764$$ 11192.5 0.530012
$$765$$ 0 0
$$766$$ −17676.4 −0.833780
$$767$$ −8106.31 −0.381619
$$768$$ 0 0
$$769$$ −13273.8 −0.622450 −0.311225 0.950336i $$-0.600739\pi$$
−0.311225 + 0.950336i $$0.600739\pi$$
$$770$$ 7571.78 0.354374
$$771$$ 0 0
$$772$$ −1990.74 −0.0928089
$$773$$ −34161.1 −1.58951 −0.794753 0.606933i $$-0.792400\pi$$
−0.794753 + 0.606933i $$0.792400\pi$$
$$774$$ 0 0
$$775$$ 2821.65 0.130783
$$776$$ −11889.9 −0.550028
$$777$$ 0 0
$$778$$ 25374.6 1.16931
$$779$$ 31.5096 0.00144923
$$780$$ 0 0
$$781$$ −21446.4 −0.982603
$$782$$ −5803.60 −0.265392
$$783$$ 0 0
$$784$$ 3394.87 0.154650
$$785$$ 18325.2 0.833191
$$786$$ 0 0
$$787$$ −38489.6 −1.74334 −0.871669 0.490095i $$-0.836962\pi$$
−0.871669 + 0.490095i $$0.836962\pi$$
$$788$$ −1185.95 −0.0536141
$$789$$ 0 0
$$790$$ −13788.8 −0.620990
$$791$$ 30470.1 1.36965
$$792$$ 0 0
$$793$$ 6023.33 0.269729
$$794$$ 16121.0 0.720544
$$795$$ 0 0
$$796$$ 9389.73 0.418103
$$797$$ −17176.8 −0.763405 −0.381702 0.924285i $$-0.624662\pi$$
−0.381702 + 0.924285i $$0.624662\pi$$
$$798$$ 0 0
$$799$$ 3335.10 0.147669
$$800$$ 800.000 0.0353553
$$801$$ 0 0
$$802$$ −24650.2 −1.08532
$$803$$ 19376.2 0.851523
$$804$$ 0 0
$$805$$ 2709.66 0.118637
$$806$$ 9047.58 0.395394
$$807$$ 0 0
$$808$$ −15110.9 −0.657918
$$809$$ −8226.61 −0.357518 −0.178759 0.983893i $$-0.557208\pi$$
−0.178759 + 0.983893i $$0.557208\pi$$
$$810$$ 0 0
$$811$$ 27867.9 1.20662 0.603312 0.797505i $$-0.293847\pi$$
0.603312 + 0.797505i $$0.293847\pi$$
$$812$$ 12933.4 0.558956
$$813$$ 0 0
$$814$$ 2937.53 0.126487
$$815$$ −5276.37 −0.226777
$$816$$ 0 0
$$817$$ 126.502 0.00541705
$$818$$ −21613.5 −0.923835
$$819$$ 0 0
$$820$$ 2707.69 0.115313
$$821$$ 17637.5 0.749761 0.374880 0.927073i $$-0.377684\pi$$
0.374880 + 0.927073i $$0.377684\pi$$
$$822$$ 0 0
$$823$$ 28048.6 1.18799 0.593993 0.804470i $$-0.297551\pi$$
0.593993 + 0.804470i $$0.297551\pi$$
$$824$$ −11977.3 −0.506372
$$825$$ 0 0
$$826$$ −9530.82 −0.401477
$$827$$ 32592.9 1.37046 0.685228 0.728329i $$-0.259703\pi$$
0.685228 + 0.728329i $$0.259703\pi$$
$$828$$ 0 0
$$829$$ 18815.9 0.788303 0.394152 0.919045i $$-0.371038\pi$$
0.394152 + 0.919045i $$0.371038\pi$$
$$830$$ 4851.78 0.202901
$$831$$ 0 0
$$832$$ 2565.19 0.106889
$$833$$ −26769.7 −1.11346
$$834$$ 0 0
$$835$$ 3659.03 0.151648
$$836$$ 29.9168 0.00123767
$$837$$ 0 0
$$838$$ 3647.48 0.150358
$$839$$ −7612.72 −0.313254 −0.156627 0.987658i $$-0.550062\pi$$
−0.156627 + 0.987658i $$0.550062\pi$$
$$840$$ 0 0
$$841$$ −5558.14 −0.227896
$$842$$ 11990.9 0.490775
$$843$$ 0 0
$$844$$ 11354.5 0.463076
$$845$$ −2952.54 −0.120202
$$846$$ 0 0
$$847$$ −7029.28 −0.285158
$$848$$ −699.134 −0.0283117
$$849$$ 0 0
$$850$$ −6308.26 −0.254555
$$851$$ 1051.23 0.0423452
$$852$$ 0 0
$$853$$ −31421.4 −1.26125 −0.630627 0.776086i $$-0.717202\pi$$
−0.630627 + 0.776086i $$0.717202\pi$$
$$854$$ 7081.81 0.283764
$$855$$ 0 0
$$856$$ 4681.20 0.186916
$$857$$ −20909.5 −0.833435 −0.416718 0.909036i $$-0.636820\pi$$
−0.416718 + 0.909036i $$0.636820\pi$$
$$858$$ 0 0
$$859$$ 23304.5 0.925655 0.462828 0.886448i $$-0.346835\pi$$
0.462828 + 0.886448i $$0.346835\pi$$
$$860$$ 10870.6 0.431027
$$861$$ 0 0
$$862$$ −7775.61 −0.307237
$$863$$ 19146.3 0.755211 0.377606 0.925966i $$-0.376747\pi$$
0.377606 + 0.925966i $$0.376747\pi$$
$$864$$ 0 0
$$865$$ 2608.86 0.102548
$$866$$ −8845.87 −0.347107
$$867$$ 0 0
$$868$$ 10637.5 0.415968
$$869$$ −44310.5 −1.72972
$$870$$ 0 0
$$871$$ −16863.0 −0.656005
$$872$$ −1113.32 −0.0432362
$$873$$ 0 0
$$874$$ 10.7061 0.000414347 0
$$875$$ 2945.28 0.113793
$$876$$ 0 0
$$877$$ 31389.0 1.20859 0.604293 0.796762i $$-0.293456\pi$$
0.604293 + 0.796762i $$0.293456\pi$$
$$878$$ −3496.16 −0.134384
$$879$$ 0 0
$$880$$ 2570.82 0.0984798
$$881$$ −44496.3 −1.70161 −0.850806 0.525480i $$-0.823886\pi$$
−0.850806 + 0.525480i $$0.823886\pi$$
$$882$$ 0 0
$$883$$ 8906.65 0.339448 0.169724 0.985492i $$-0.445712\pi$$
0.169724 + 0.985492i $$0.445712\pi$$
$$884$$ −20227.4 −0.769592
$$885$$ 0 0
$$886$$ −6742.62 −0.255669
$$887$$ 36584.7 1.38489 0.692444 0.721472i $$-0.256534\pi$$
0.692444 + 0.721472i $$0.256534\pi$$
$$888$$ 0 0
$$889$$ −30593.9 −1.15420
$$890$$ 11277.1 0.424729
$$891$$ 0 0
$$892$$ −8498.74 −0.319012
$$893$$ −6.15238 −0.000230551 0
$$894$$ 0 0
$$895$$ 5915.35 0.220926
$$896$$ 3015.97 0.112451
$$897$$ 0 0
$$898$$ −26629.5 −0.989573
$$899$$ 15488.1 0.574591
$$900$$ 0 0
$$901$$ 5512.90 0.203842
$$902$$ 8701.23 0.321197
$$903$$ 0 0
$$904$$ 10345.4 0.380623
$$905$$ 8619.62 0.316603
$$906$$ 0 0
$$907$$ −21346.1 −0.781463 −0.390731 0.920505i $$-0.627778\pi$$
−0.390731 + 0.920505i $$0.627778\pi$$
$$908$$ −6207.89 −0.226890
$$909$$ 0 0
$$910$$ 9444.00 0.344028
$$911$$ 10208.0 0.371246 0.185623 0.982621i $$-0.440570\pi$$
0.185623 + 0.982621i $$0.440570\pi$$
$$912$$ 0 0
$$913$$ 15591.3 0.565166
$$914$$ 7534.40 0.272665
$$915$$ 0 0
$$916$$ −634.125 −0.0228735
$$917$$ 12376.0 0.445683
$$918$$ 0 0
$$919$$ 1758.00 0.0631025 0.0315513 0.999502i $$-0.489955\pi$$
0.0315513 + 0.999502i $$0.489955\pi$$
$$920$$ 920.000 0.0329690
$$921$$ 0 0
$$922$$ −19348.1 −0.691102
$$923$$ −26749.3 −0.953917
$$924$$ 0 0
$$925$$ 1142.64 0.0406161
$$926$$ 5955.46 0.211348
$$927$$ 0 0
$$928$$ 4391.22 0.155333
$$929$$ 845.008 0.0298426 0.0149213 0.999889i $$-0.495250\pi$$
0.0149213 + 0.999889i $$0.495250\pi$$
$$930$$ 0 0
$$931$$ 49.3830 0.00173841
$$932$$ 2915.99 0.102486
$$933$$ 0 0
$$934$$ 21402.0 0.749782
$$935$$ −20271.7 −0.709045
$$936$$ 0 0
$$937$$ 3851.61 0.134287 0.0671433 0.997743i $$-0.478612\pi$$
0.0671433 + 0.997743i $$0.478612\pi$$
$$938$$ −19826.3 −0.690141
$$939$$ 0 0
$$940$$ −528.688 −0.0183446
$$941$$ 4379.36 0.151714 0.0758571 0.997119i $$-0.475831\pi$$
0.0758571 + 0.997119i $$0.475831\pi$$
$$942$$ 0 0
$$943$$ 3113.85 0.107530
$$944$$ −3235.96 −0.111570
$$945$$ 0 0
$$946$$ 34932.8 1.20059
$$947$$ −29746.9 −1.02074 −0.510371 0.859954i $$-0.670492\pi$$
−0.510371 + 0.859954i $$0.670492\pi$$
$$948$$ 0 0
$$949$$ 24167.3 0.826663
$$950$$ 11.6371 0.000397428 0
$$951$$ 0 0
$$952$$ −23781.9 −0.809638
$$953$$ −4306.61 −0.146385 −0.0731924 0.997318i $$-0.523319\pi$$
−0.0731924 + 0.997318i $$0.523319\pi$$
$$954$$ 0 0
$$955$$ 13990.6 0.474057
$$956$$ 24806.4 0.839222
$$957$$ 0 0
$$958$$ −4674.29 −0.157640
$$959$$ 57211.9 1.92645
$$960$$ 0 0
$$961$$ −17052.3 −0.572397
$$962$$ 3663.87 0.122794
$$963$$ 0 0
$$964$$ −28892.2 −0.965305
$$965$$ −2488.43 −0.0830108
$$966$$ 0 0
$$967$$ −12233.4 −0.406824 −0.203412 0.979093i $$-0.565203\pi$$
−0.203412 + 0.979093i $$0.565203\pi$$
$$968$$ −2386.63 −0.0792449
$$969$$ 0 0
$$970$$ −14862.4 −0.491960
$$971$$ −48207.7 −1.59326 −0.796631 0.604465i $$-0.793387\pi$$
−0.796631 + 0.604465i $$0.793387\pi$$
$$972$$ 0 0
$$973$$ 57879.8 1.90703
$$974$$ −14367.7 −0.472660
$$975$$ 0 0
$$976$$ 2404.46 0.0788575
$$977$$ −28662.9 −0.938595 −0.469298 0.883040i $$-0.655493\pi$$
−0.469298 + 0.883040i $$0.655493\pi$$
$$978$$ 0 0
$$979$$ 36239.2 1.18305
$$980$$ 4243.59 0.138323
$$981$$ 0 0
$$982$$ 24168.4 0.785383
$$983$$ 20944.1 0.679567 0.339783 0.940504i $$-0.389646\pi$$
0.339783 + 0.940504i $$0.389646\pi$$
$$984$$ 0 0
$$985$$ −1482.44 −0.0479539
$$986$$ −34626.2 −1.11838
$$987$$ 0 0
$$988$$ 37.3141 0.00120154
$$989$$ 12501.1 0.401934
$$990$$ 0 0
$$991$$ −36440.2 −1.16808 −0.584038 0.811727i $$-0.698528\pi$$
−0.584038 + 0.811727i $$0.698528\pi$$
$$992$$ 3611.71 0.115597
$$993$$ 0 0
$$994$$ −31450.0 −1.00355
$$995$$ 11737.2 0.373963
$$996$$ 0 0
$$997$$ −11945.0 −0.379439 −0.189720 0.981838i $$-0.560758\pi$$
−0.189720 + 0.981838i $$0.560758\pi$$
$$998$$ 14291.6 0.453298
$$999$$ 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.bj.1.3 4
3.2 odd 2 230.4.a.h.1.4 4
12.11 even 2 1840.4.a.m.1.1 4
15.2 even 4 1150.4.b.n.599.1 8
15.8 even 4 1150.4.b.n.599.8 8
15.14 odd 2 1150.4.a.p.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.4 4 3.2 odd 2
1150.4.a.p.1.1 4 15.14 odd 2
1150.4.b.n.599.1 8 15.2 even 4
1150.4.b.n.599.8 8 15.8 even 4
1840.4.a.m.1.1 4 12.11 even 2
2070.4.a.bj.1.3 4 1.1 even 1 trivial