# Properties

 Label 2166.4.a.t.1.2 Level $2166$ Weight $4$ Character 2166.1 Self dual yes Analytic conductor $127.798$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2166 = 2 \cdot 3 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2166.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$127.798137072$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.14457.1 Defining polynomial: $$x^{3} - x^{2} - 32 x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}\cdot 3$$ Twist minimal: no (minimal twist has level 114) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.0940524$$ of defining polynomial Character $$\chi$$ $$=$$ 2166.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +4.18810 q^{5} -6.00000 q^{6} +3.18810 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +4.18810 q^{5} -6.00000 q^{6} +3.18810 q^{7} -8.00000 q^{8} +9.00000 q^{9} -8.37621 q^{10} +69.4003 q^{11} +12.0000 q^{12} +8.12863 q^{13} -6.37621 q^{14} +12.5643 q^{15} +16.0000 q^{16} -106.084 q^{17} -18.0000 q^{18} +16.7524 q^{20} +9.56431 q^{21} -138.801 q^{22} -176.494 q^{23} -24.0000 q^{24} -107.460 q^{25} -16.2573 q^{26} +27.0000 q^{27} +12.7524 q^{28} +66.2219 q^{29} -25.1286 q^{30} +140.915 q^{31} -32.0000 q^{32} +208.201 q^{33} +212.167 q^{34} +13.3521 q^{35} +36.0000 q^{36} +156.003 q^{37} +24.3859 q^{39} -33.5048 q^{40} +414.563 q^{41} -19.1286 q^{42} +115.850 q^{43} +277.601 q^{44} +37.6929 q^{45} +352.987 q^{46} +620.283 q^{47} +48.0000 q^{48} -332.836 q^{49} +214.920 q^{50} -318.251 q^{51} +32.5145 q^{52} +371.986 q^{53} -54.0000 q^{54} +290.656 q^{55} -25.5048 q^{56} -132.444 q^{58} -91.6929 q^{59} +50.2573 q^{60} +218.621 q^{61} -281.830 q^{62} +28.6929 q^{63} +64.0000 q^{64} +34.0436 q^{65} -416.402 q^{66} +145.342 q^{67} -424.334 q^{68} -529.481 q^{69} -26.7042 q^{70} -887.829 q^{71} -72.0000 q^{72} +199.016 q^{73} -312.006 q^{74} -322.379 q^{75} +221.255 q^{77} -48.7718 q^{78} +389.558 q^{79} +67.0097 q^{80} +81.0000 q^{81} -829.125 q^{82} +380.039 q^{83} +38.2573 q^{84} -444.289 q^{85} -231.701 q^{86} +198.666 q^{87} -555.202 q^{88} +425.799 q^{89} -75.3859 q^{90} +25.9149 q^{91} -705.974 q^{92} +422.744 q^{93} -1240.57 q^{94} -96.0000 q^{96} -419.846 q^{97} +665.672 q^{98} +624.603 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} - 24 q^{8} + 27 q^{9} + O(q^{10})$$ $$3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} - 24 q^{8} + 27 q^{9} - 20 q^{10} - 44 q^{11} + 36 q^{12} + 9 q^{13} - 14 q^{14} + 30 q^{15} + 48 q^{16} - 84 q^{17} - 54 q^{18} + 40 q^{20} + 21 q^{21} + 88 q^{22} - 2 q^{23} - 72 q^{24} - 83 q^{25} - 18 q^{26} + 81 q^{27} + 28 q^{28} - 92 q^{29} - 60 q^{30} + 109 q^{31} - 96 q^{32} - 132 q^{33} + 168 q^{34} + 282 q^{35} + 108 q^{36} - 245 q^{37} + 27 q^{39} - 80 q^{40} + 688 q^{41} - 42 q^{42} - 103 q^{43} - 176 q^{44} + 90 q^{45} + 4 q^{46} + 322 q^{47} + 144 q^{48} - 754 q^{49} + 166 q^{50} - 252 q^{51} + 36 q^{52} + 1322 q^{53} - 162 q^{54} - 248 q^{55} - 56 q^{56} + 184 q^{58} - 252 q^{59} + 120 q^{60} - 435 q^{61} - 218 q^{62} + 63 q^{63} + 192 q^{64} + 1582 q^{65} + 264 q^{66} + 719 q^{67} - 336 q^{68} - 6 q^{69} - 564 q^{70} + 62 q^{71} - 216 q^{72} - 581 q^{73} + 490 q^{74} - 249 q^{75} - 204 q^{77} - 54 q^{78} + 489 q^{79} + 160 q^{80} + 243 q^{81} - 1376 q^{82} + 2496 q^{83} + 84 q^{84} + 1632 q^{85} + 206 q^{86} - 276 q^{87} + 352 q^{88} - 1584 q^{89} - 180 q^{90} + 1573 q^{91} - 8 q^{92} + 327 q^{93} - 644 q^{94} - 288 q^{96} - 974 q^{97} + 1508 q^{98} - 396 q^{99} + O(q^{100})$$

## 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
$$3$$ 3.00000 0.577350
$$4$$ 4.00000 0.500000
$$5$$ 4.18810 0.374595 0.187298 0.982303i $$-0.440027\pi$$
0.187298 + 0.982303i $$0.440027\pi$$
$$6$$ −6.00000 −0.408248
$$7$$ 3.18810 0.172141 0.0860707 0.996289i $$-0.472569\pi$$
0.0860707 + 0.996289i $$0.472569\pi$$
$$8$$ −8.00000 −0.353553
$$9$$ 9.00000 0.333333
$$10$$ −8.37621 −0.264879
$$11$$ 69.4003 1.90227 0.951135 0.308774i $$-0.0999187\pi$$
0.951135 + 0.308774i $$0.0999187\pi$$
$$12$$ 12.0000 0.288675
$$13$$ 8.12863 0.173421 0.0867106 0.996234i $$-0.472364\pi$$
0.0867106 + 0.996234i $$0.472364\pi$$
$$14$$ −6.37621 −0.121722
$$15$$ 12.5643 0.216273
$$16$$ 16.0000 0.250000
$$17$$ −106.084 −1.51347 −0.756737 0.653720i $$-0.773207\pi$$
−0.756737 + 0.653720i $$0.773207\pi$$
$$18$$ −18.0000 −0.235702
$$19$$ 0 0
$$20$$ 16.7524 0.187298
$$21$$ 9.56431 0.0993859
$$22$$ −138.801 −1.34511
$$23$$ −176.494 −1.60006 −0.800031 0.599958i $$-0.795184\pi$$
−0.800031 + 0.599958i $$0.795184\pi$$
$$24$$ −24.0000 −0.204124
$$25$$ −107.460 −0.859678
$$26$$ −16.2573 −0.122627
$$27$$ 27.0000 0.192450
$$28$$ 12.7524 0.0860707
$$29$$ 66.2219 0.424038 0.212019 0.977266i $$-0.431996\pi$$
0.212019 + 0.977266i $$0.431996\pi$$
$$30$$ −25.1286 −0.152928
$$31$$ 140.915 0.816421 0.408210 0.912888i $$-0.366153\pi$$
0.408210 + 0.912888i $$0.366153\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ 208.201 1.09828
$$34$$ 212.167 1.07019
$$35$$ 13.3521 0.0644834
$$36$$ 36.0000 0.166667
$$37$$ 156.003 0.693156 0.346578 0.938021i $$-0.387344\pi$$
0.346578 + 0.938021i $$0.387344\pi$$
$$38$$ 0 0
$$39$$ 24.3859 0.100125
$$40$$ −33.5048 −0.132440
$$41$$ 414.563 1.57912 0.789559 0.613675i $$-0.210309\pi$$
0.789559 + 0.613675i $$0.210309\pi$$
$$42$$ −19.1286 −0.0702765
$$43$$ 115.850 0.410861 0.205430 0.978672i $$-0.434141\pi$$
0.205430 + 0.978672i $$0.434141\pi$$
$$44$$ 277.601 0.951135
$$45$$ 37.6929 0.124865
$$46$$ 352.987 1.13142
$$47$$ 620.283 1.92505 0.962527 0.271185i $$-0.0874156\pi$$
0.962527 + 0.271185i $$0.0874156\pi$$
$$48$$ 48.0000 0.144338
$$49$$ −332.836 −0.970367
$$50$$ 214.920 0.607884
$$51$$ −318.251 −0.873804
$$52$$ 32.5145 0.0867106
$$53$$ 371.986 0.964078 0.482039 0.876150i $$-0.339896\pi$$
0.482039 + 0.876150i $$0.339896\pi$$
$$54$$ −54.0000 −0.136083
$$55$$ 290.656 0.712582
$$56$$ −25.5048 −0.0608612
$$57$$ 0 0
$$58$$ −132.444 −0.299840
$$59$$ −91.6929 −0.202329 −0.101164 0.994870i $$-0.532257\pi$$
−0.101164 + 0.994870i $$0.532257\pi$$
$$60$$ 50.2573 0.108136
$$61$$ 218.621 0.458877 0.229438 0.973323i $$-0.426311\pi$$
0.229438 + 0.973323i $$0.426311\pi$$
$$62$$ −281.830 −0.577297
$$63$$ 28.6929 0.0573805
$$64$$ 64.0000 0.125000
$$65$$ 34.0436 0.0649628
$$66$$ −416.402 −0.776599
$$67$$ 145.342 0.265021 0.132510 0.991182i $$-0.457696\pi$$
0.132510 + 0.991182i $$0.457696\pi$$
$$68$$ −424.334 −0.756737
$$69$$ −529.481 −0.923797
$$70$$ −26.7042 −0.0455967
$$71$$ −887.829 −1.48403 −0.742014 0.670385i $$-0.766129\pi$$
−0.742014 + 0.670385i $$0.766129\pi$$
$$72$$ −72.0000 −0.117851
$$73$$ 199.016 0.319083 0.159542 0.987191i $$-0.448998\pi$$
0.159542 + 0.987191i $$0.448998\pi$$
$$74$$ −312.006 −0.490135
$$75$$ −322.379 −0.496335
$$76$$ 0 0
$$77$$ 221.255 0.327460
$$78$$ −48.7718 −0.0707989
$$79$$ 389.558 0.554793 0.277397 0.960755i $$-0.410528\pi$$
0.277397 + 0.960755i $$0.410528\pi$$
$$80$$ 67.0097 0.0936489
$$81$$ 81.0000 0.111111
$$82$$ −829.125 −1.11660
$$83$$ 380.039 0.502586 0.251293 0.967911i $$-0.419144\pi$$
0.251293 + 0.967911i $$0.419144\pi$$
$$84$$ 38.2573 0.0496930
$$85$$ −444.289 −0.566940
$$86$$ −231.701 −0.290523
$$87$$ 198.666 0.244818
$$88$$ −555.202 −0.672554
$$89$$ 425.799 0.507130 0.253565 0.967318i $$-0.418397\pi$$
0.253565 + 0.967318i $$0.418397\pi$$
$$90$$ −75.3859 −0.0882930
$$91$$ 25.9149 0.0298530
$$92$$ −705.974 −0.800031
$$93$$ 422.744 0.471361
$$94$$ −1240.57 −1.36122
$$95$$ 0 0
$$96$$ −96.0000 −0.102062
$$97$$ −419.846 −0.439473 −0.219736 0.975559i $$-0.570520\pi$$
−0.219736 + 0.975559i $$0.570520\pi$$
$$98$$ 665.672 0.686153
$$99$$ 624.603 0.634090
$$100$$ −429.839 −0.429839
$$101$$ −1241.71 −1.22331 −0.611657 0.791123i $$-0.709497\pi$$
−0.611657 + 0.791123i $$0.709497\pi$$
$$102$$ 636.501 0.617873
$$103$$ 593.606 0.567862 0.283931 0.958845i $$-0.408361\pi$$
0.283931 + 0.958845i $$0.408361\pi$$
$$104$$ −65.0290 −0.0613137
$$105$$ 40.0564 0.0372295
$$106$$ −743.971 −0.681706
$$107$$ 1778.56 1.60691 0.803457 0.595363i $$-0.202992\pi$$
0.803457 + 0.595363i $$0.202992\pi$$
$$108$$ 108.000 0.0962250
$$109$$ 1069.99 0.940243 0.470121 0.882602i $$-0.344210\pi$$
0.470121 + 0.882602i $$0.344210\pi$$
$$110$$ −581.311 −0.503872
$$111$$ 468.009 0.400194
$$112$$ 51.0097 0.0430354
$$113$$ −583.197 −0.485510 −0.242755 0.970088i $$-0.578051\pi$$
−0.242755 + 0.970088i $$0.578051\pi$$
$$114$$ 0 0
$$115$$ −739.173 −0.599376
$$116$$ 264.887 0.212019
$$117$$ 73.1577 0.0578071
$$118$$ 183.386 0.143068
$$119$$ −338.206 −0.260532
$$120$$ −100.515 −0.0764640
$$121$$ 3485.40 2.61863
$$122$$ −437.241 −0.324475
$$123$$ 1243.69 0.911704
$$124$$ 563.659 0.408210
$$125$$ −973.566 −0.696627
$$126$$ −57.3859 −0.0405741
$$127$$ −1659.35 −1.15940 −0.579700 0.814830i $$-0.696830\pi$$
−0.579700 + 0.814830i $$0.696830\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ 347.551 0.237211
$$130$$ −68.0871 −0.0459356
$$131$$ 597.546 0.398533 0.199267 0.979945i $$-0.436144\pi$$
0.199267 + 0.979945i $$0.436144\pi$$
$$132$$ 832.804 0.549138
$$133$$ 0 0
$$134$$ −290.685 −0.187398
$$135$$ 113.079 0.0720909
$$136$$ 848.669 0.535094
$$137$$ −1777.31 −1.10836 −0.554182 0.832396i $$-0.686969\pi$$
−0.554182 + 0.832396i $$0.686969\pi$$
$$138$$ 1058.96 0.653223
$$139$$ 163.567 0.0998099 0.0499050 0.998754i $$-0.484108\pi$$
0.0499050 + 0.998754i $$0.484108\pi$$
$$140$$ 53.4085 0.0322417
$$141$$ 1860.85 1.11143
$$142$$ 1775.66 1.04937
$$143$$ 564.129 0.329894
$$144$$ 144.000 0.0833333
$$145$$ 277.344 0.158843
$$146$$ −398.032 −0.225626
$$147$$ −998.508 −0.560242
$$148$$ 624.012 0.346578
$$149$$ −1119.80 −0.615688 −0.307844 0.951437i $$-0.599607\pi$$
−0.307844 + 0.951437i $$0.599607\pi$$
$$150$$ 644.759 0.350962
$$151$$ 2807.15 1.51286 0.756432 0.654073i $$-0.226941\pi$$
0.756432 + 0.654073i $$0.226941\pi$$
$$152$$ 0 0
$$153$$ −954.752 −0.504491
$$154$$ −442.511 −0.231549
$$155$$ 590.166 0.305828
$$156$$ 97.5436 0.0500624
$$157$$ 1003.94 0.510339 0.255170 0.966896i $$-0.417869\pi$$
0.255170 + 0.966896i $$0.417869\pi$$
$$158$$ −779.116 −0.392298
$$159$$ 1115.96 0.556611
$$160$$ −134.019 −0.0662198
$$161$$ −562.680 −0.275437
$$162$$ −162.000 −0.0785674
$$163$$ 1271.99 0.611225 0.305612 0.952156i $$-0.401139\pi$$
0.305612 + 0.952156i $$0.401139\pi$$
$$164$$ 1658.25 0.789559
$$165$$ 871.967 0.411409
$$166$$ −760.077 −0.355382
$$167$$ 2984.98 1.38314 0.691572 0.722307i $$-0.256918\pi$$
0.691572 + 0.722307i $$0.256918\pi$$
$$168$$ −76.5145 −0.0351382
$$169$$ −2130.93 −0.969925
$$170$$ 888.578 0.400887
$$171$$ 0 0
$$172$$ 463.402 0.205430
$$173$$ −56.7747 −0.0249509 −0.0124754 0.999922i $$-0.503971\pi$$
−0.0124754 + 0.999922i $$0.503971\pi$$
$$174$$ −397.331 −0.173113
$$175$$ −342.593 −0.147986
$$176$$ 1110.40 0.475568
$$177$$ −275.079 −0.116815
$$178$$ −851.598 −0.358595
$$179$$ 4640.98 1.93790 0.968948 0.247266i $$-0.0795323\pi$$
0.968948 + 0.247266i $$0.0795323\pi$$
$$180$$ 150.772 0.0624326
$$181$$ −3775.41 −1.55041 −0.775204 0.631711i $$-0.782353\pi$$
−0.775204 + 0.631711i $$0.782353\pi$$
$$182$$ −51.8298 −0.0211093
$$183$$ 655.862 0.264933
$$184$$ 1411.95 0.565708
$$185$$ 653.357 0.259653
$$186$$ −845.489 −0.333302
$$187$$ −7362.23 −2.87904
$$188$$ 2481.13 0.962527
$$189$$ 86.0788 0.0331286
$$190$$ 0 0
$$191$$ −2762.53 −1.04654 −0.523271 0.852166i $$-0.675288\pi$$
−0.523271 + 0.852166i $$0.675288\pi$$
$$192$$ 192.000 0.0721688
$$193$$ −2061.51 −0.768866 −0.384433 0.923153i $$-0.625603\pi$$
−0.384433 + 0.923153i $$0.625603\pi$$
$$194$$ 839.691 0.310754
$$195$$ 102.131 0.0375063
$$196$$ −1331.34 −0.485184
$$197$$ 2094.82 0.757614 0.378807 0.925476i $$-0.376334\pi$$
0.378807 + 0.925476i $$0.376334\pi$$
$$198$$ −1249.21 −0.448370
$$199$$ −1717.36 −0.611762 −0.305881 0.952070i $$-0.598951\pi$$
−0.305881 + 0.952070i $$0.598951\pi$$
$$200$$ 859.678 0.303942
$$201$$ 436.027 0.153010
$$202$$ 2483.42 0.865014
$$203$$ 211.122 0.0729945
$$204$$ −1273.00 −0.436902
$$205$$ 1736.23 0.591530
$$206$$ −1187.21 −0.401539
$$207$$ −1588.44 −0.533354
$$208$$ 130.058 0.0433553
$$209$$ 0 0
$$210$$ −80.1127 −0.0263252
$$211$$ 2291.05 0.747500 0.373750 0.927529i $$-0.378072\pi$$
0.373750 + 0.927529i $$0.378072\pi$$
$$212$$ 1487.94 0.482039
$$213$$ −2663.49 −0.856804
$$214$$ −3557.12 −1.13626
$$215$$ 485.194 0.153907
$$216$$ −216.000 −0.0680414
$$217$$ 449.251 0.140540
$$218$$ −2139.98 −0.664852
$$219$$ 597.048 0.184223
$$220$$ 1162.62 0.356291
$$221$$ −862.314 −0.262468
$$222$$ −936.019 −0.282980
$$223$$ −3256.19 −0.977805 −0.488902 0.872338i $$-0.662603\pi$$
−0.488902 + 0.872338i $$0.662603\pi$$
$$224$$ −102.019 −0.0304306
$$225$$ −967.138 −0.286559
$$226$$ 1166.39 0.343307
$$227$$ 998.044 0.291817 0.145909 0.989298i $$-0.453389\pi$$
0.145909 + 0.989298i $$0.453389\pi$$
$$228$$ 0 0
$$229$$ −1028.59 −0.296816 −0.148408 0.988926i $$-0.547415\pi$$
−0.148408 + 0.988926i $$0.547415\pi$$
$$230$$ 1478.35 0.423823
$$231$$ 663.766 0.189059
$$232$$ −529.775 −0.149920
$$233$$ −125.486 −0.0352827 −0.0176414 0.999844i $$-0.505616\pi$$
−0.0176414 + 0.999844i $$0.505616\pi$$
$$234$$ −146.315 −0.0408758
$$235$$ 2597.81 0.721117
$$236$$ −366.772 −0.101164
$$237$$ 1168.67 0.320310
$$238$$ 676.411 0.184224
$$239$$ 3591.03 0.971901 0.485950 0.873986i $$-0.338474\pi$$
0.485950 + 0.873986i $$0.338474\pi$$
$$240$$ 201.029 0.0540682
$$241$$ 3691.84 0.986773 0.493386 0.869810i $$-0.335759\pi$$
0.493386 + 0.869810i $$0.335759\pi$$
$$242$$ −6970.80 −1.85165
$$243$$ 243.000 0.0641500
$$244$$ 874.482 0.229438
$$245$$ −1393.95 −0.363495
$$246$$ −2487.38 −0.644672
$$247$$ 0 0
$$248$$ −1127.32 −0.288648
$$249$$ 1140.12 0.290168
$$250$$ 1947.13 0.492590
$$251$$ 7294.88 1.83446 0.917228 0.398362i $$-0.130421\pi$$
0.917228 + 0.398362i $$0.130421\pi$$
$$252$$ 114.772 0.0286902
$$253$$ −12248.7 −3.04375
$$254$$ 3318.71 0.819820
$$255$$ −1332.87 −0.327323
$$256$$ 256.000 0.0625000
$$257$$ −2303.20 −0.559026 −0.279513 0.960142i $$-0.590173\pi$$
−0.279513 + 0.960142i $$0.590173\pi$$
$$258$$ −695.102 −0.167733
$$259$$ 497.354 0.119321
$$260$$ 136.174 0.0324814
$$261$$ 595.997 0.141346
$$262$$ −1195.09 −0.281806
$$263$$ 7138.38 1.67365 0.836827 0.547467i $$-0.184408\pi$$
0.836827 + 0.547467i $$0.184408\pi$$
$$264$$ −1665.61 −0.388299
$$265$$ 1557.91 0.361139
$$266$$ 0 0
$$267$$ 1277.40 0.292792
$$268$$ 581.370 0.132510
$$269$$ 5446.66 1.23453 0.617265 0.786755i $$-0.288241\pi$$
0.617265 + 0.786755i $$0.288241\pi$$
$$270$$ −226.158 −0.0509760
$$271$$ −3403.68 −0.762947 −0.381474 0.924380i $$-0.624583\pi$$
−0.381474 + 0.924380i $$0.624583\pi$$
$$272$$ −1697.34 −0.378368
$$273$$ 77.7448 0.0172356
$$274$$ 3554.62 0.783731
$$275$$ −7457.74 −1.63534
$$276$$ −2117.92 −0.461898
$$277$$ −5131.93 −1.11317 −0.556584 0.830791i $$-0.687888\pi$$
−0.556584 + 0.830791i $$0.687888\pi$$
$$278$$ −327.134 −0.0705763
$$279$$ 1268.23 0.272140
$$280$$ −106.817 −0.0227983
$$281$$ −3366.68 −0.714731 −0.357365 0.933965i $$-0.616325\pi$$
−0.357365 + 0.933965i $$0.616325\pi$$
$$282$$ −3721.70 −0.785900
$$283$$ 6685.76 1.40434 0.702168 0.712011i $$-0.252215\pi$$
0.702168 + 0.712011i $$0.252215\pi$$
$$284$$ −3551.32 −0.742014
$$285$$ 0 0
$$286$$ −1128.26 −0.233270
$$287$$ 1321.67 0.271832
$$288$$ −288.000 −0.0589256
$$289$$ 6340.72 1.29060
$$290$$ −554.688 −0.112319
$$291$$ −1259.54 −0.253730
$$292$$ 796.065 0.159542
$$293$$ 5625.93 1.12174 0.560871 0.827903i $$-0.310466\pi$$
0.560871 + 0.827903i $$0.310466\pi$$
$$294$$ 1997.02 0.396151
$$295$$ −384.020 −0.0757915
$$296$$ −1248.02 −0.245067
$$297$$ 1873.81 0.366092
$$298$$ 2239.60 0.435357
$$299$$ −1434.65 −0.277485
$$300$$ −1289.52 −0.248168
$$301$$ 369.343 0.0707262
$$302$$ −5614.29 −1.06976
$$303$$ −3725.13 −0.706281
$$304$$ 0 0
$$305$$ 915.606 0.171893
$$306$$ 1909.50 0.356729
$$307$$ −5603.54 −1.04173 −0.520865 0.853639i $$-0.674390\pi$$
−0.520865 + 0.853639i $$0.674390\pi$$
$$308$$ 885.022 0.163730
$$309$$ 1780.82 0.327855
$$310$$ −1180.33 −0.216253
$$311$$ 6668.20 1.21582 0.607908 0.794008i $$-0.292009\pi$$
0.607908 + 0.794008i $$0.292009\pi$$
$$312$$ −195.087 −0.0353995
$$313$$ 4491.40 0.811083 0.405541 0.914077i $$-0.367083\pi$$
0.405541 + 0.914077i $$0.367083\pi$$
$$314$$ −2007.88 −0.360865
$$315$$ 120.169 0.0214945
$$316$$ 1558.23 0.277397
$$317$$ 1266.05 0.224316 0.112158 0.993690i $$-0.464224\pi$$
0.112158 + 0.993690i $$0.464224\pi$$
$$318$$ −2231.91 −0.393583
$$319$$ 4595.82 0.806635
$$320$$ 268.039 0.0468244
$$321$$ 5335.68 0.927752
$$322$$ 1125.36 0.194764
$$323$$ 0 0
$$324$$ 324.000 0.0555556
$$325$$ −873.501 −0.149086
$$326$$ −2543.97 −0.432201
$$327$$ 3209.97 0.542849
$$328$$ −3316.50 −0.558302
$$329$$ 1977.53 0.331382
$$330$$ −1743.93 −0.290910
$$331$$ 5068.09 0.841594 0.420797 0.907155i $$-0.361751\pi$$
0.420797 + 0.907155i $$0.361751\pi$$
$$332$$ 1520.15 0.251293
$$333$$ 1404.03 0.231052
$$334$$ −5969.97 −0.978031
$$335$$ 608.709 0.0992757
$$336$$ 153.029 0.0248465
$$337$$ −10744.1 −1.73670 −0.868352 0.495948i $$-0.834821\pi$$
−0.868352 + 0.495948i $$0.834821\pi$$
$$338$$ 4261.85 0.685841
$$339$$ −1749.59 −0.280309
$$340$$ −1777.16 −0.283470
$$341$$ 9779.53 1.55305
$$342$$ 0 0
$$343$$ −2154.64 −0.339182
$$344$$ −926.803 −0.145261
$$345$$ −2217.52 −0.346050
$$346$$ 113.549 0.0176429
$$347$$ −4851.34 −0.750528 −0.375264 0.926918i $$-0.622448\pi$$
−0.375264 + 0.926918i $$0.622448\pi$$
$$348$$ 794.662 0.122409
$$349$$ −7611.35 −1.16741 −0.583705 0.811966i $$-0.698398\pi$$
−0.583705 + 0.811966i $$0.698398\pi$$
$$350$$ 685.186 0.104642
$$351$$ 219.473 0.0333749
$$352$$ −2220.81 −0.336277
$$353$$ 5567.64 0.839477 0.419739 0.907645i $$-0.362122\pi$$
0.419739 + 0.907645i $$0.362122\pi$$
$$354$$ 550.158 0.0826004
$$355$$ −3718.32 −0.555910
$$356$$ 1703.20 0.253565
$$357$$ −1014.62 −0.150418
$$358$$ −9281.96 −1.37030
$$359$$ 4341.31 0.638233 0.319116 0.947716i $$-0.396614\pi$$
0.319116 + 0.947716i $$0.396614\pi$$
$$360$$ −301.544 −0.0441465
$$361$$ 0 0
$$362$$ 7550.82 1.09630
$$363$$ 10456.2 1.51187
$$364$$ 103.660 0.0149265
$$365$$ 833.500 0.119527
$$366$$ −1311.72 −0.187336
$$367$$ −8964.48 −1.27505 −0.637524 0.770431i $$-0.720041\pi$$
−0.637524 + 0.770431i $$0.720041\pi$$
$$368$$ −2823.90 −0.400016
$$369$$ 3731.06 0.526372
$$370$$ −1306.71 −0.183602
$$371$$ 1185.93 0.165958
$$372$$ 1690.98 0.235680
$$373$$ −5048.11 −0.700755 −0.350377 0.936609i $$-0.613947\pi$$
−0.350377 + 0.936609i $$0.613947\pi$$
$$374$$ 14724.5 2.03579
$$375$$ −2920.70 −0.402198
$$376$$ −4962.26 −0.680609
$$377$$ 538.293 0.0735371
$$378$$ −172.158 −0.0234255
$$379$$ 9290.41 1.25915 0.629573 0.776941i $$-0.283230\pi$$
0.629573 + 0.776941i $$0.283230\pi$$
$$380$$ 0 0
$$381$$ −4978.06 −0.669380
$$382$$ 5525.06 0.740017
$$383$$ −4162.26 −0.555304 −0.277652 0.960682i $$-0.589556\pi$$
−0.277652 + 0.960682i $$0.589556\pi$$
$$384$$ −384.000 −0.0510310
$$385$$ 926.641 0.122665
$$386$$ 4123.03 0.543670
$$387$$ 1042.65 0.136954
$$388$$ −1679.38 −0.219736
$$389$$ 4185.23 0.545500 0.272750 0.962085i $$-0.412067\pi$$
0.272750 + 0.962085i $$0.412067\pi$$
$$390$$ −204.261 −0.0265210
$$391$$ 18723.1 2.42165
$$392$$ 2662.69 0.343077
$$393$$ 1792.64 0.230093
$$394$$ −4189.64 −0.535714
$$395$$ 1631.51 0.207823
$$396$$ 2498.41 0.317045
$$397$$ −4529.76 −0.572650 −0.286325 0.958133i $$-0.592434\pi$$
−0.286325 + 0.958133i $$0.592434\pi$$
$$398$$ 3434.72 0.432581
$$399$$ 0 0
$$400$$ −1719.36 −0.214920
$$401$$ 9497.05 1.18269 0.591347 0.806417i $$-0.298596\pi$$
0.591347 + 0.806417i $$0.298596\pi$$
$$402$$ −872.055 −0.108194
$$403$$ 1145.44 0.141585
$$404$$ −4966.84 −0.611657
$$405$$ 339.236 0.0416217
$$406$$ −422.245 −0.0516149
$$407$$ 10826.7 1.31857
$$408$$ 2546.01 0.308936
$$409$$ 5354.44 0.647335 0.323667 0.946171i $$-0.395084\pi$$
0.323667 + 0.946171i $$0.395084\pi$$
$$410$$ −3472.46 −0.418275
$$411$$ −5331.93 −0.639914
$$412$$ 2374.43 0.283931
$$413$$ −292.327 −0.0348292
$$414$$ 3176.88 0.377138
$$415$$ 1591.64 0.188267
$$416$$ −260.116 −0.0306568
$$417$$ 490.701 0.0576253
$$418$$ 0 0
$$419$$ 14506.1 1.69134 0.845668 0.533709i $$-0.179202\pi$$
0.845668 + 0.533709i $$0.179202\pi$$
$$420$$ 160.225 0.0186148
$$421$$ 1263.45 0.146263 0.0731315 0.997322i $$-0.476701\pi$$
0.0731315 + 0.997322i $$0.476701\pi$$
$$422$$ −4582.10 −0.528562
$$423$$ 5582.55 0.641685
$$424$$ −2975.88 −0.340853
$$425$$ 11399.7 1.30110
$$426$$ 5326.98 0.605852
$$427$$ 696.985 0.0789918
$$428$$ 7114.24 0.803457
$$429$$ 1692.39 0.190464
$$430$$ −970.387 −0.108828
$$431$$ 8187.62 0.915043 0.457522 0.889198i $$-0.348737\pi$$
0.457522 + 0.889198i $$0.348737\pi$$
$$432$$ 432.000 0.0481125
$$433$$ 9242.94 1.02584 0.512919 0.858437i $$-0.328564\pi$$
0.512919 + 0.858437i $$0.328564\pi$$
$$434$$ −898.502 −0.0993767
$$435$$ 832.032 0.0917078
$$436$$ 4279.96 0.470121
$$437$$ 0 0
$$438$$ −1194.10 −0.130265
$$439$$ 2043.05 0.222117 0.111059 0.993814i $$-0.464576\pi$$
0.111059 + 0.993814i $$0.464576\pi$$
$$440$$ −2325.25 −0.251936
$$441$$ −2995.52 −0.323456
$$442$$ 1724.63 0.185593
$$443$$ −9669.13 −1.03701 −0.518504 0.855075i $$-0.673511\pi$$
−0.518504 + 0.855075i $$0.673511\pi$$
$$444$$ 1872.04 0.200097
$$445$$ 1783.29 0.189969
$$446$$ 6512.38 0.691412
$$447$$ −3359.40 −0.355467
$$448$$ 204.039 0.0215177
$$449$$ 13227.2 1.39027 0.695134 0.718881i $$-0.255345\pi$$
0.695134 + 0.718881i $$0.255345\pi$$
$$450$$ 1934.28 0.202628
$$451$$ 28770.8 3.00391
$$452$$ −2332.79 −0.242755
$$453$$ 8421.44 0.873452
$$454$$ −1996.09 −0.206346
$$455$$ 108.534 0.0111828
$$456$$ 0 0
$$457$$ −5380.98 −0.550791 −0.275396 0.961331i $$-0.588809\pi$$
−0.275396 + 0.961331i $$0.588809\pi$$
$$458$$ 2057.17 0.209880
$$459$$ −2864.26 −0.291268
$$460$$ −2956.69 −0.299688
$$461$$ −2844.67 −0.287396 −0.143698 0.989622i $$-0.545899\pi$$
−0.143698 + 0.989622i $$0.545899\pi$$
$$462$$ −1327.53 −0.133685
$$463$$ 6625.39 0.665028 0.332514 0.943098i $$-0.392103\pi$$
0.332514 + 0.943098i $$0.392103\pi$$
$$464$$ 1059.55 0.106009
$$465$$ 1770.50 0.176570
$$466$$ 250.972 0.0249487
$$467$$ −18635.1 −1.84653 −0.923264 0.384167i $$-0.874489\pi$$
−0.923264 + 0.384167i $$0.874489\pi$$
$$468$$ 292.631 0.0289035
$$469$$ 463.367 0.0456211
$$470$$ −5195.62 −0.509906
$$471$$ 3011.83 0.294645
$$472$$ 733.544 0.0715341
$$473$$ 8040.05 0.781569
$$474$$ −2337.35 −0.226493
$$475$$ 0 0
$$476$$ −1352.82 −0.130266
$$477$$ 3347.87 0.321359
$$478$$ −7182.05 −0.687237
$$479$$ 1556.62 0.148484 0.0742420 0.997240i $$-0.476346\pi$$
0.0742420 + 0.997240i $$0.476346\pi$$
$$480$$ −402.058 −0.0382320
$$481$$ 1268.09 0.120208
$$482$$ −7383.68 −0.697754
$$483$$ −1688.04 −0.159024
$$484$$ 13941.6 1.30932
$$485$$ −1758.36 −0.164625
$$486$$ −486.000 −0.0453609
$$487$$ −19339.5 −1.79950 −0.899751 0.436405i $$-0.856252\pi$$
−0.899751 + 0.436405i $$0.856252\pi$$
$$488$$ −1748.96 −0.162238
$$489$$ 3815.96 0.352891
$$490$$ 2787.90 0.257030
$$491$$ −6838.25 −0.628525 −0.314262 0.949336i $$-0.601757\pi$$
−0.314262 + 0.949336i $$0.601757\pi$$
$$492$$ 4974.75 0.455852
$$493$$ −7025.05 −0.641770
$$494$$ 0 0
$$495$$ 2615.90 0.237527
$$496$$ 2254.64 0.204105
$$497$$ −2830.49 −0.255463
$$498$$ −2280.23 −0.205180
$$499$$ −18115.0 −1.62513 −0.812563 0.582874i $$-0.801928\pi$$
−0.812563 + 0.582874i $$0.801928\pi$$
$$500$$ −3894.26 −0.348314
$$501$$ 8954.95 0.798559
$$502$$ −14589.8 −1.29716
$$503$$ −5831.85 −0.516957 −0.258478 0.966017i $$-0.583221\pi$$
−0.258478 + 0.966017i $$0.583221\pi$$
$$504$$ −229.544 −0.0202871
$$505$$ −5200.41 −0.458248
$$506$$ 24497.4 2.15226
$$507$$ −6392.78 −0.559987
$$508$$ −6637.41 −0.579700
$$509$$ −9914.65 −0.863377 −0.431689 0.902023i $$-0.642082\pi$$
−0.431689 + 0.902023i $$0.642082\pi$$
$$510$$ 2665.73 0.231452
$$511$$ 634.484 0.0549275
$$512$$ −512.000 −0.0441942
$$513$$ 0 0
$$514$$ 4606.40 0.395291
$$515$$ 2486.09 0.212718
$$516$$ 1390.20 0.118605
$$517$$ 43047.8 3.66197
$$518$$ −994.709 −0.0843726
$$519$$ −170.324 −0.0144054
$$520$$ −272.348 −0.0229678
$$521$$ 5851.91 0.492086 0.246043 0.969259i $$-0.420869\pi$$
0.246043 + 0.969259i $$0.420869\pi$$
$$522$$ −1191.99 −0.0999466
$$523$$ −6894.09 −0.576401 −0.288200 0.957570i $$-0.593057\pi$$
−0.288200 + 0.957570i $$0.593057\pi$$
$$524$$ 2390.19 0.199267
$$525$$ −1027.78 −0.0854399
$$526$$ −14276.8 −1.18345
$$527$$ −14948.7 −1.23563
$$528$$ 3331.21 0.274569
$$529$$ 18983.0 1.56020
$$530$$ −3115.83 −0.255364
$$531$$ −825.236 −0.0674430
$$532$$ 0 0
$$533$$ 3369.83 0.273853
$$534$$ −2554.79 −0.207035
$$535$$ 7448.79 0.601943
$$536$$ −1162.74 −0.0936991
$$537$$ 13922.9 1.11884
$$538$$ −10893.3 −0.872945
$$539$$ −23098.9 −1.84590
$$540$$ 452.315 0.0360455
$$541$$ −5624.99 −0.447019 −0.223509 0.974702i $$-0.571751\pi$$
−0.223509 + 0.974702i $$0.571751\pi$$
$$542$$ 6807.36 0.539485
$$543$$ −11326.2 −0.895129
$$544$$ 3394.67 0.267547
$$545$$ 4481.23 0.352211
$$546$$ −155.490 −0.0121874
$$547$$ −3596.14 −0.281096 −0.140548 0.990074i $$-0.544886\pi$$
−0.140548 + 0.990074i $$0.544886\pi$$
$$548$$ −7109.24 −0.554182
$$549$$ 1967.59 0.152959
$$550$$ 14915.5 1.15636
$$551$$ 0 0
$$552$$ 4235.85 0.326611
$$553$$ 1241.95 0.0955029
$$554$$ 10263.9 0.787129
$$555$$ 1960.07 0.149911
$$556$$ 654.269 0.0499050
$$557$$ 18381.5 1.39829 0.699145 0.714980i $$-0.253564\pi$$
0.699145 + 0.714980i $$0.253564\pi$$
$$558$$ −2536.47 −0.192432
$$559$$ 941.705 0.0712520
$$560$$ 213.634 0.0161209
$$561$$ −22086.7 −1.66221
$$562$$ 6733.36 0.505391
$$563$$ −5578.33 −0.417582 −0.208791 0.977960i $$-0.566953\pi$$
−0.208791 + 0.977960i $$0.566953\pi$$
$$564$$ 7443.39 0.555715
$$565$$ −2442.49 −0.181870
$$566$$ −13371.5 −0.993016
$$567$$ 258.236 0.0191268
$$568$$ 7102.63 0.524683
$$569$$ −16981.3 −1.25113 −0.625564 0.780173i $$-0.715131\pi$$
−0.625564 + 0.780173i $$0.715131\pi$$
$$570$$ 0 0
$$571$$ −19520.5 −1.43066 −0.715332 0.698785i $$-0.753724\pi$$
−0.715332 + 0.698785i $$0.753724\pi$$
$$572$$ 2256.52 0.164947
$$573$$ −8287.59 −0.604221
$$574$$ −2643.34 −0.192214
$$575$$ 18966.0 1.37554
$$576$$ 576.000 0.0416667
$$577$$ 6371.35 0.459693 0.229846 0.973227i $$-0.426178\pi$$
0.229846 + 0.973227i $$0.426178\pi$$
$$578$$ −12681.4 −0.912593
$$579$$ −6184.54 −0.443905
$$580$$ 1109.38 0.0794213
$$581$$ 1211.60 0.0865159
$$582$$ 2519.07 0.179414
$$583$$ 25815.9 1.83394
$$584$$ −1592.13 −0.112813
$$585$$ 306.392 0.0216543
$$586$$ −11251.9 −0.793191
$$587$$ 10759.0 0.756513 0.378257 0.925701i $$-0.376524\pi$$
0.378257 + 0.925701i $$0.376524\pi$$
$$588$$ −3994.03 −0.280121
$$589$$ 0 0
$$590$$ 768.039 0.0535927
$$591$$ 6284.47 0.437408
$$592$$ 2496.05 0.173289
$$593$$ 3611.76 0.250113 0.125057 0.992150i $$-0.460089\pi$$
0.125057 + 0.992150i $$0.460089\pi$$
$$594$$ −3747.62 −0.258866
$$595$$ −1416.44 −0.0975939
$$596$$ −4479.19 −0.307844
$$597$$ −5152.09 −0.353201
$$598$$ 2869.30 0.196211
$$599$$ 9180.23 0.626201 0.313100 0.949720i $$-0.398632\pi$$
0.313100 + 0.949720i $$0.398632\pi$$
$$600$$ 2579.03 0.175481
$$601$$ −12700.4 −0.862000 −0.431000 0.902352i $$-0.641839\pi$$
−0.431000 + 0.902352i $$0.641839\pi$$
$$602$$ −738.686 −0.0500110
$$603$$ 1308.08 0.0883403
$$604$$ 11228.6 0.756432
$$605$$ 14597.2 0.980928
$$606$$ 7450.26 0.499416
$$607$$ 24227.5 1.62004 0.810019 0.586404i $$-0.199457\pi$$
0.810019 + 0.586404i $$0.199457\pi$$
$$608$$ 0 0
$$609$$ 633.367 0.0421434
$$610$$ −1831.21 −0.121547
$$611$$ 5042.05 0.333845
$$612$$ −3819.01 −0.252246
$$613$$ −16213.5 −1.06828 −0.534141 0.845396i $$-0.679365\pi$$
−0.534141 + 0.845396i $$0.679365\pi$$
$$614$$ 11207.1 0.736614
$$615$$ 5208.70 0.341520
$$616$$ −1770.04 −0.115774
$$617$$ −21515.3 −1.40385 −0.701924 0.712252i $$-0.747675\pi$$
−0.701924 + 0.712252i $$0.747675\pi$$
$$618$$ −3561.64 −0.231829
$$619$$ −15878.5 −1.03103 −0.515516 0.856880i $$-0.672400\pi$$
−0.515516 + 0.856880i $$0.672400\pi$$
$$620$$ 2360.66 0.152914
$$621$$ −4765.33 −0.307932
$$622$$ −13336.4 −0.859712
$$623$$ 1357.49 0.0872982
$$624$$ 390.174 0.0250312
$$625$$ 9355.08 0.598725
$$626$$ −8982.80 −0.573522
$$627$$ 0 0
$$628$$ 4015.77 0.255170
$$629$$ −16549.4 −1.04907
$$630$$ −240.338 −0.0151989
$$631$$ 5559.75 0.350761 0.175380 0.984501i $$-0.443884\pi$$
0.175380 + 0.984501i $$0.443884\pi$$
$$632$$ −3116.46 −0.196149
$$633$$ 6873.16 0.431569
$$634$$ −2532.09 −0.158616
$$635$$ −6949.55 −0.434306
$$636$$ 4463.83 0.278305
$$637$$ −2705.50 −0.168282
$$638$$ −9191.64 −0.570377
$$639$$ −7990.46 −0.494676
$$640$$ −536.077 −0.0331099
$$641$$ 20486.9 1.26238 0.631190 0.775628i $$-0.282567\pi$$
0.631190 + 0.775628i $$0.282567\pi$$
$$642$$ −10671.4 −0.656020
$$643$$ −4102.30 −0.251600 −0.125800 0.992056i $$-0.540150\pi$$
−0.125800 + 0.992056i $$0.540150\pi$$
$$644$$ −2250.72 −0.137719
$$645$$ 1455.58 0.0888580
$$646$$ 0 0
$$647$$ 22860.4 1.38908 0.694539 0.719455i $$-0.255608\pi$$
0.694539 + 0.719455i $$0.255608\pi$$
$$648$$ −648.000 −0.0392837
$$649$$ −6363.52 −0.384884
$$650$$ 1747.00 0.105420
$$651$$ 1347.75 0.0811408
$$652$$ 5087.94 0.305612
$$653$$ 26388.5 1.58141 0.790706 0.612197i $$-0.209714\pi$$
0.790706 + 0.612197i $$0.209714\pi$$
$$654$$ −6419.94 −0.383853
$$655$$ 2502.59 0.149289
$$656$$ 6633.00 0.394779
$$657$$ 1791.15 0.106361
$$658$$ −3955.05 −0.234322
$$659$$ −3223.29 −0.190534 −0.0952668 0.995452i $$-0.530370\pi$$
−0.0952668 + 0.995452i $$0.530370\pi$$
$$660$$ 3487.87 0.205705
$$661$$ 27480.2 1.61703 0.808513 0.588478i $$-0.200273\pi$$
0.808513 + 0.588478i $$0.200273\pi$$
$$662$$ −10136.2 −0.595097
$$663$$ −2586.94 −0.151536
$$664$$ −3040.31 −0.177691
$$665$$ 0 0
$$666$$ −2808.06 −0.163378
$$667$$ −11687.7 −0.678487
$$668$$ 11939.9 0.691572
$$669$$ −9768.56 −0.564536
$$670$$ −1217.42 −0.0701985
$$671$$ 15172.3 0.872908
$$672$$ −306.058 −0.0175691
$$673$$ 2165.00 0.124004 0.0620020 0.998076i $$-0.480251\pi$$
0.0620020 + 0.998076i $$0.480251\pi$$
$$674$$ 21488.2 1.22804
$$675$$ −2901.41 −0.165445
$$676$$ −8523.70 −0.484963
$$677$$ −11999.0 −0.681179 −0.340589 0.940212i $$-0.610627\pi$$
−0.340589 + 0.940212i $$0.610627\pi$$
$$678$$ 3499.18 0.198209
$$679$$ −1338.51 −0.0756515
$$680$$ 3554.31 0.200444
$$681$$ 2994.13 0.168481
$$682$$ −19559.1 −1.09817
$$683$$ −5234.35 −0.293246 −0.146623 0.989192i $$-0.546840\pi$$
−0.146623 + 0.989192i $$0.546840\pi$$
$$684$$ 0 0
$$685$$ −7443.56 −0.415188
$$686$$ 4309.27 0.239838
$$687$$ −3085.76 −0.171367
$$688$$ 1853.61 0.102715
$$689$$ 3023.73 0.167192
$$690$$ 4435.04 0.244694
$$691$$ −5160.63 −0.284109 −0.142055 0.989859i $$-0.545371\pi$$
−0.142055 + 0.989859i $$0.545371\pi$$
$$692$$ −227.099 −0.0124754
$$693$$ 1991.30 0.109153
$$694$$ 9702.67 0.530704
$$695$$ 685.036 0.0373884
$$696$$ −1589.32 −0.0865563
$$697$$ −43978.3 −2.38995
$$698$$ 15222.7 0.825484
$$699$$ −376.459 −0.0203705
$$700$$ −1370.37 −0.0739931
$$701$$ −3315.03 −0.178612 −0.0893059 0.996004i $$-0.528465\pi$$
−0.0893059 + 0.996004i $$0.528465\pi$$
$$702$$ −438.946 −0.0235996
$$703$$ 0 0
$$704$$ 4441.62 0.237784
$$705$$ 7793.43 0.416337
$$706$$ −11135.3 −0.593600
$$707$$ −3958.70 −0.210583
$$708$$ −1100.32 −0.0584073
$$709$$ −18333.2 −0.971111 −0.485555 0.874206i $$-0.661383\pi$$
−0.485555 + 0.874206i $$0.661383\pi$$
$$710$$ 7436.64 0.393088
$$711$$ 3506.02 0.184931
$$712$$ −3406.39 −0.179298
$$713$$ −24870.6 −1.30632
$$714$$ 2029.23 0.106362
$$715$$ 2362.63 0.123577
$$716$$ 18563.9 0.968948
$$717$$ 10773.1 0.561127
$$718$$ −8682.62 −0.451299
$$719$$ 9747.00 0.505566 0.252783 0.967523i $$-0.418654\pi$$
0.252783 + 0.967523i $$0.418654\pi$$
$$720$$ 603.087 0.0312163
$$721$$ 1892.48 0.0977526
$$722$$ 0 0
$$723$$ 11075.5 0.569713
$$724$$ −15101.6 −0.775204
$$725$$ −7116.19 −0.364536
$$726$$ −20912.4 −1.06905
$$727$$ 32723.6 1.66939 0.834697 0.550709i $$-0.185643\pi$$
0.834697 + 0.550709i $$0.185643\pi$$
$$728$$ −207.319 −0.0105546
$$729$$ 729.000 0.0370370
$$730$$ −1667.00 −0.0845185
$$731$$ −12289.8 −0.621827
$$732$$ 2623.45 0.132466
$$733$$ 36938.2 1.86131 0.930657 0.365894i $$-0.119237\pi$$
0.930657 + 0.365894i $$0.119237\pi$$
$$734$$ 17929.0 0.901594
$$735$$ −4181.86 −0.209864
$$736$$ 5647.79 0.282854
$$737$$ 10086.8 0.504142
$$738$$ −7462.13 −0.372202
$$739$$ −9987.19 −0.497138 −0.248569 0.968614i $$-0.579960\pi$$
−0.248569 + 0.968614i $$0.579960\pi$$
$$740$$ 2613.43 0.129826
$$741$$ 0 0
$$742$$ −2371.86 −0.117350
$$743$$ −21739.8 −1.07343 −0.536713 0.843765i $$-0.680334\pi$$
−0.536713 + 0.843765i $$0.680334\pi$$
$$744$$ −3381.96 −0.166651
$$745$$ −4689.83 −0.230634
$$746$$ 10096.2 0.495508
$$747$$ 3420.35 0.167529
$$748$$ −29448.9 −1.43952
$$749$$ 5670.23 0.276617
$$750$$ 5841.40 0.284397
$$751$$ 38067.4 1.84967 0.924833 0.380374i $$-0.124205\pi$$
0.924833 + 0.380374i $$0.124205\pi$$
$$752$$ 9924.53 0.481264
$$753$$ 21884.6 1.05912
$$754$$ −1076.59 −0.0519986
$$755$$ 11756.6 0.566712
$$756$$ 344.315 0.0165643
$$757$$ −11365.2 −0.545675 −0.272838 0.962060i $$-0.587962\pi$$
−0.272838 + 0.962060i $$0.587962\pi$$
$$758$$ −18580.8 −0.890351
$$759$$ −36746.1 −1.75731
$$760$$ 0 0
$$761$$ 24549.3 1.16940 0.584699 0.811250i $$-0.301213\pi$$
0.584699 + 0.811250i $$0.301213\pi$$
$$762$$ 9956.12 0.473323
$$763$$ 3411.24 0.161855
$$764$$ −11050.1 −0.523271
$$765$$ −3998.60 −0.188980
$$766$$ 8324.51 0.392659
$$767$$ −745.338 −0.0350881
$$768$$ 768.000 0.0360844
$$769$$ 14222.5 0.666941 0.333471 0.942760i $$-0.391780\pi$$
0.333471 + 0.942760i $$0.391780\pi$$
$$770$$ −1853.28 −0.0867372
$$771$$ −6909.60 −0.322754
$$772$$ −8246.06 −0.384433
$$773$$ −16082.6 −0.748319 −0.374159 0.927364i $$-0.622069\pi$$
−0.374159 + 0.927364i $$0.622069\pi$$
$$774$$ −2085.31 −0.0968409
$$775$$ −15142.7 −0.701859
$$776$$ 3358.77 0.155377
$$777$$ 1492.06 0.0688899
$$778$$ −8370.46 −0.385727
$$779$$ 0 0
$$780$$ 408.523 0.0187531
$$781$$ −61615.6 −2.82302
$$782$$ −37446.1 −1.71237
$$783$$ 1787.99 0.0816061
$$784$$ −5325.38 −0.242592
$$785$$ 4204.61 0.191171
$$786$$ −3585.28 −0.162701
$$787$$ −30058.0 −1.36144 −0.680720 0.732544i $$-0.738333\pi$$
−0.680720 + 0.732544i $$0.738333\pi$$
$$788$$ 8379.29 0.378807
$$789$$ 21415.1 0.966285
$$790$$ −3263.02 −0.146953
$$791$$ −1859.29 −0.0835764
$$792$$ −4996.82 −0.224185
$$793$$ 1777.09 0.0795790
$$794$$ 9059.52 0.404925
$$795$$ 4673.74 0.208504
$$796$$ −6869.45 −0.305881
$$797$$ 20008.2 0.889243 0.444621 0.895719i $$-0.353338\pi$$
0.444621 + 0.895719i $$0.353338\pi$$
$$798$$ 0 0
$$799$$ −65801.8 −2.91352
$$800$$ 3438.71 0.151971
$$801$$ 3832.19 0.169043
$$802$$ −18994.1 −0.836291
$$803$$ 13811.8 0.606983
$$804$$ 1744.11 0.0765050
$$805$$ −2356.56 −0.103178
$$806$$ −2290.89 −0.100116
$$807$$ 16340.0 0.712757
$$808$$ 9933.68 0.432507
$$809$$ 35302.3 1.53419 0.767097 0.641532i $$-0.221701\pi$$
0.767097 + 0.641532i $$0.221701\pi$$
$$810$$ −678.473 −0.0294310
$$811$$ −10980.8 −0.475446 −0.237723 0.971333i $$-0.576401\pi$$
−0.237723 + 0.971333i $$0.576401\pi$$
$$812$$ 844.489 0.0364972
$$813$$ −10211.0 −0.440488
$$814$$ −21653.3 −0.932369
$$815$$ 5327.21 0.228962
$$816$$ −5092.01 −0.218451
$$817$$ 0 0
$$818$$ −10708.9 −0.457735
$$819$$ 233.234 0.00995100
$$820$$ 6944.93 0.295765
$$821$$ −5626.68 −0.239187 −0.119594 0.992823i $$-0.538159\pi$$
−0.119594 + 0.992823i $$0.538159\pi$$
$$822$$ 10663.9 0.452487
$$823$$ 28766.3 1.21838 0.609191 0.793023i $$-0.291494\pi$$
0.609191 + 0.793023i $$0.291494\pi$$
$$824$$ −4748.85 −0.200769
$$825$$ −22373.2 −0.944164
$$826$$ 584.653 0.0246280
$$827$$ −44482.4 −1.87038 −0.935191 0.354145i $$-0.884772\pi$$
−0.935191 + 0.354145i $$0.884772\pi$$
$$828$$ −6353.77 −0.266677
$$829$$ −23302.5 −0.976271 −0.488136 0.872768i $$-0.662323\pi$$
−0.488136 + 0.872768i $$0.662323\pi$$
$$830$$ −3183.28 −0.133125
$$831$$ −15395.8 −0.642688
$$832$$ 520.232 0.0216777
$$833$$ 35308.4 1.46862
$$834$$ −981.403 −0.0407472
$$835$$ 12501.4 0.518120
$$836$$ 0 0
$$837$$ 3804.70 0.157120
$$838$$ −29012.2 −1.19596
$$839$$ 27683.7 1.13915 0.569575 0.821939i $$-0.307108\pi$$
0.569575 + 0.821939i $$0.307108\pi$$
$$840$$ −320.451 −0.0131626
$$841$$ −20003.7 −0.820192
$$842$$ −2526.90 −0.103424
$$843$$ −10100.0 −0.412650
$$844$$ 9164.21 0.373750
$$845$$ −8924.54 −0.363330
$$846$$ −11165.1 −0.453740
$$847$$ 11111.8 0.450776
$$848$$ 5951.77 0.241020
$$849$$ 20057.3 0.810794
$$850$$ −22799.4 −0.920017
$$851$$ −27533.5 −1.10909
$$852$$ −10654.0 −0.428402
$$853$$ −30609.5 −1.22866 −0.614331 0.789049i $$-0.710574\pi$$
−0.614331 + 0.789049i $$0.710574\pi$$
$$854$$ −1393.97 −0.0558556
$$855$$ 0 0
$$856$$ −14228.5 −0.568130
$$857$$ −10918.3 −0.435194 −0.217597 0.976039i $$-0.569822\pi$$
−0.217597 + 0.976039i $$0.569822\pi$$
$$858$$ −3384.78 −0.134679
$$859$$ 14639.5 0.581482 0.290741 0.956802i $$-0.406098\pi$$
0.290741 + 0.956802i $$0.406098\pi$$
$$860$$ 1940.77 0.0769533
$$861$$ 3965.01 0.156942
$$862$$ −16375.2 −0.647033
$$863$$ 22775.2 0.898353 0.449176 0.893443i $$-0.351717\pi$$
0.449176 + 0.893443i $$0.351717\pi$$
$$864$$ −864.000 −0.0340207
$$865$$ −237.778 −0.00934648
$$866$$ −18485.9 −0.725377
$$867$$ 19022.2 0.745129
$$868$$ 1797.00 0.0702700
$$869$$ 27035.4 1.05537
$$870$$ −1664.06 −0.0648472
$$871$$ 1181.43 0.0459603
$$872$$ −8559.92 −0.332426
$$873$$ −3778.61 −0.146491
$$874$$ 0 0
$$875$$ −3103.83 −0.119918
$$876$$ 2388.19 0.0921114
$$877$$ −805.908 −0.0310303 −0.0155152 0.999880i $$-0.504939\pi$$
−0.0155152 + 0.999880i $$0.504939\pi$$
$$878$$ −4086.10 −0.157061
$$879$$ 16877.8 0.647638
$$880$$ 4650.49 0.178146
$$881$$ −901.690 −0.0344821 −0.0172410 0.999851i $$-0.505488\pi$$
−0.0172410 + 0.999851i $$0.505488\pi$$
$$882$$ 5991.05 0.228718
$$883$$ −25406.3 −0.968279 −0.484140 0.874991i $$-0.660867\pi$$
−0.484140 + 0.874991i $$0.660867\pi$$
$$884$$ −3449.26 −0.131234
$$885$$ −1152.06 −0.0437582
$$886$$ 19338.3 0.733275
$$887$$ −28560.5 −1.08114 −0.540568 0.841300i $$-0.681791\pi$$
−0.540568 + 0.841300i $$0.681791\pi$$
$$888$$ −3744.07 −0.141490
$$889$$ −5290.19 −0.199581
$$890$$ −3566.58 −0.134328
$$891$$ 5621.42 0.211363
$$892$$ −13024.8 −0.488902
$$893$$ 0 0
$$894$$ 6718.79 0.251353
$$895$$ 19436.9 0.725927
$$896$$ −408.077 −0.0152153
$$897$$ −4303.95 −0.160206
$$898$$ −26454.4 −0.983067
$$899$$ 9331.64 0.346193
$$900$$ −3868.55 −0.143280
$$901$$ −39461.6 −1.45911
$$902$$ −57541.6 −2.12408
$$903$$ 1108.03 0.0408338
$$904$$ 4665.58 0.171654
$$905$$ −15811.8 −0.580776
$$906$$ −16842.9 −0.617624
$$907$$ 40905.4 1.49751 0.748755 0.662847i $$-0.230652\pi$$
0.748755 + 0.662847i $$0.230652\pi$$
$$908$$ 3992.18 0.145909
$$909$$ −11175.4 −0.407772
$$910$$ −217.069 −0.00790743
$$911$$ −30047.9 −1.09279 −0.546395 0.837527i $$-0.684000\pi$$
−0.546395 + 0.837527i $$0.684000\pi$$
$$912$$ 0 0
$$913$$ 26374.8 0.956055
$$914$$ 10762.0 0.389468
$$915$$ 2746.82 0.0992426
$$916$$ −4114.34 −0.148408
$$917$$ 1905.04 0.0686041
$$918$$ 5728.51 0.205958
$$919$$ −49600.1 −1.78036 −0.890182 0.455605i $$-0.849423\pi$$
−0.890182 + 0.455605i $$0.849423\pi$$
$$920$$ 5913.39 0.211912
$$921$$ −16810.6 −0.601443
$$922$$ 5689.33 0.203219
$$923$$ −7216.84 −0.257362
$$924$$ 2655.07 0.0945295
$$925$$ −16764.1 −0.595891
$$926$$ −13250.8 −0.470246
$$927$$ 5342.46 0.189287
$$928$$ −2119.10 −0.0749600
$$929$$ 27556.8 0.973205 0.486603 0.873623i $$-0.338236\pi$$
0.486603 + 0.873623i $$0.338236\pi$$
$$930$$ −3541.00 −0.124854
$$931$$ 0 0
$$932$$ −501.945 −0.0176414
$$933$$ 20004.6 0.701952
$$934$$ 37270.1 1.30569
$$935$$ −30833.8 −1.07847
$$936$$ −585.261 −0.0204379
$$937$$ −26880.8 −0.937202 −0.468601 0.883410i $$-0.655242\pi$$
−0.468601 + 0.883410i $$0.655242\pi$$
$$938$$ −926.734 −0.0322590
$$939$$ 13474.2 0.468279
$$940$$ 10391.2 0.360558
$$941$$ 6411.34 0.222108 0.111054 0.993814i $$-0.464577\pi$$
0.111054 + 0.993814i $$0.464577\pi$$
$$942$$ −6023.65 −0.208345
$$943$$ −73167.6 −2.52669
$$944$$ −1467.09 −0.0505822
$$945$$ 360.507 0.0124098
$$946$$ −16080.1 −0.552653
$$947$$ −1989.83 −0.0682796 −0.0341398 0.999417i $$-0.510869\pi$$
−0.0341398 + 0.999417i $$0.510869\pi$$
$$948$$ 4674.69 0.160155
$$949$$ 1617.73 0.0553358
$$950$$ 0 0
$$951$$ 3798.14 0.129509
$$952$$ 2705.64 0.0921118
$$953$$ 38266.4 1.30070 0.650351 0.759633i $$-0.274622\pi$$
0.650351 + 0.759633i $$0.274622\pi$$
$$954$$ −6695.74 −0.227235
$$955$$ −11569.8 −0.392030
$$956$$ 14364.1 0.485950
$$957$$ 13787.5 0.465711
$$958$$ −3113.24 −0.104994
$$959$$ −5666.25 −0.190795
$$960$$ 804.116 0.0270341
$$961$$ −9934.01 −0.333457
$$962$$ −2536.18 −0.0849998
$$963$$ 16007.0 0.535638
$$964$$ 14767.4 0.493386
$$965$$ −8633.84 −0.288014
$$966$$ 3376.08 0.112447
$$967$$ 220.116 0.00732002 0.00366001 0.999993i $$-0.498835\pi$$
0.00366001 + 0.999993i $$0.498835\pi$$
$$968$$ −27883.2 −0.925827
$$969$$ 0 0
$$970$$ 3516.72 0.116407
$$971$$ −2175.36 −0.0718955 −0.0359478 0.999354i $$-0.511445\pi$$
−0.0359478 + 0.999354i $$0.511445\pi$$
$$972$$ 972.000 0.0320750
$$973$$ 521.469 0.0171814
$$974$$ 38679.0 1.27244
$$975$$ −2620.50 −0.0860751
$$976$$ 3497.93 0.114719
$$977$$ −13707.7 −0.448872 −0.224436 0.974489i $$-0.572054\pi$$
−0.224436 + 0.974489i $$0.572054\pi$$
$$978$$ −7631.91 −0.249531
$$979$$ 29550.6 0.964699
$$980$$ −5575.81 −0.181748
$$981$$ 9629.91 0.313414
$$982$$ 13676.5 0.444434
$$983$$ 18885.7 0.612779 0.306389 0.951906i $$-0.400879\pi$$
0.306389 + 0.951906i $$0.400879\pi$$
$$984$$ −9949.50 −0.322336
$$985$$ 8773.33 0.283799
$$986$$ 14050.1 0.453800
$$987$$ 5932.58 0.191323
$$988$$ 0 0
$$989$$ −20446.8 −0.657403
$$990$$ −5231.80 −0.167957
$$991$$ 40655.0 1.30318 0.651589 0.758572i $$-0.274103\pi$$
0.651589 + 0.758572i $$0.274103\pi$$
$$992$$ −4509.27 −0.144324
$$993$$ 15204.3 0.485894
$$994$$ 5660.99 0.180639
$$995$$ −7192.49 −0.229163
$$996$$ 4560.46 0.145084
$$997$$ −16433.3 −0.522014 −0.261007 0.965337i $$-0.584055\pi$$
−0.261007 + 0.965337i $$0.584055\pi$$
$$998$$ 36229.9 1.14914
$$999$$ 4212.08 0.133398
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 2166.4.a.t.1.2 3
19.8 odd 6 114.4.e.d.7.2 6
19.12 odd 6 114.4.e.d.49.2 yes 6
19.18 odd 2 2166.4.a.u.1.2 3
57.8 even 6 342.4.g.h.235.2 6
57.50 even 6 342.4.g.h.163.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
114.4.e.d.7.2 6 19.8 odd 6
114.4.e.d.49.2 yes 6 19.12 odd 6
342.4.g.h.163.2 6 57.50 even 6
342.4.g.h.235.2 6 57.8 even 6
2166.4.a.t.1.2 3 1.1 even 1 trivial
2166.4.a.u.1.2 3 19.18 odd 2