# Properties

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

# Learn more

## 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: $$1$$ 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 $$-0.527636\pi$$
−0.0867106 + 0.996234i $$0.527636\pi$$
$$14$$ 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 $$-0.568004\pi$$
−0.212019 + 0.977266i $$0.568004\pi$$
$$30$$ −25.1286 −0.152928
$$31$$ −140.915 −0.816421 −0.408210 0.912888i $$-0.633847\pi$$
−0.408210 + 0.912888i $$0.633847\pi$$
$$32$$ 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 $$-0.612656\pi$$
−0.346578 + 0.938021i $$0.612656\pi$$
$$38$$ 0 0
$$39$$ 24.3859 0.100125
$$40$$ 33.5048 0.132440
$$41$$ −414.563 −1.57912 −0.789559 0.613675i $$-0.789691\pi$$
−0.789559 + 0.613675i $$0.789691\pi$$
$$42$$ −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 $$-0.660104\pi$$
−0.482039 + 0.876150i $$0.660104\pi$$
$$54$$ −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 $$-0.467743\pi$$
0.101164 + 0.994870i $$0.467743\pi$$
$$60$$ −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 $$-0.542304\pi$$
−0.132510 + 0.991182i $$0.542304\pi$$
$$68$$ −424.334 −0.756737
$$69$$ 529.481 0.923797
$$70$$ 26.7042 0.0455967
$$71$$ 887.829 1.48403 0.742014 0.670385i $$-0.233871\pi$$
0.742014 + 0.670385i $$0.233871\pi$$
$$72$$ 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 $$-0.589472\pi$$
−0.277397 + 0.960755i $$0.589472\pi$$
$$80$$ 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 $$-0.581603\pi$$
−0.253565 + 0.967318i $$0.581603\pi$$
$$90$$ 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 $$-0.429480\pi$$
0.219736 + 0.975559i $$0.429480\pi$$
$$98$$ −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 $$-0.591639\pi$$
−0.283931 + 0.958845i $$0.591639\pi$$
$$104$$ −65.0290 −0.0613137
$$105$$ −40.0564 −0.0372295
$$106$$ −743.971 −0.681706
$$107$$ −1778.56 −1.60691 −0.803457 0.595363i $$-0.797008\pi$$
−0.803457 + 0.595363i $$0.797008\pi$$
$$108$$ −108.000 −0.0962250
$$109$$ −1069.99 −0.940243 −0.470121 0.882602i $$-0.655790\pi$$
−0.470121 + 0.882602i $$0.655790\pi$$
$$110$$ 581.311 0.503872
$$111$$ 468.009 0.400194
$$112$$ 51.0097 0.0430354
$$113$$ 583.197 0.485510 0.242755 0.970088i $$-0.421949\pi$$
0.242755 + 0.970088i $$0.421949\pi$$
$$114$$ 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 $$-0.303170\pi$$
0.579700 + 0.814830i $$0.303170\pi$$
$$128$$ 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 $$-0.773059\pi$$
−0.756432 + 0.654073i $$0.773059\pi$$
$$152$$ 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 $$-0.743082\pi$$
−0.691572 + 0.722307i $$0.743082\pi$$
$$168$$ −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 $$-0.496029\pi$$
0.0124754 + 0.999922i $$0.496029\pi$$
$$174$$ 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 $$-0.920468\pi$$
−0.968948 + 0.247266i $$0.920468\pi$$
$$180$$ 150.772 0.0624326
$$181$$ 3775.41 1.55041 0.775204 0.631711i $$-0.217647\pi$$
0.775204 + 0.631711i $$0.217647\pi$$
$$182$$ −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 $$-0.374397\pi$$
0.384433 + 0.923153i $$0.374397\pi$$
$$194$$ 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 $$-0.621928\pi$$
−0.373750 + 0.927529i $$0.621928\pi$$
$$212$$ −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 $$-0.337397\pi$$
0.488902 + 0.872338i $$0.337397\pi$$
$$224$$ 102.019 0.0304306
$$225$$ −967.138 −0.286559
$$226$$ 1166.39 0.343307
$$227$$ −998.044 −0.291817 −0.145909 0.989298i $$-0.546611\pi$$
−0.145909 + 0.989298i $$0.546611\pi$$
$$228$$ 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 $$-0.664241\pi$$
−0.493386 + 0.869810i $$0.664241\pi$$
$$242$$ 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 $$-0.409827\pi$$
0.279513 + 0.960142i $$0.409827\pi$$
$$258$$ −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 $$-0.711759\pi$$
−0.617265 + 0.786755i $$0.711759\pi$$
$$270$$ −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 $$-0.383675\pi$$
0.357365 + 0.933965i $$0.383675\pi$$
$$282$$ −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 $$-0.689534\pi$$
−0.560871 + 0.827903i $$0.689534\pi$$
$$294$$ 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 $$-0.325610\pi$$
0.520865 + 0.853639i $$0.325610\pi$$
$$308$$ 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 $$-0.535776\pi$$
−0.112158 + 0.993690i $$0.535776\pi$$
$$318$$ 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 $$-0.638249\pi$$
−0.420797 + 0.907155i $$0.638249\pi$$
$$332$$ 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 $$-0.165179\pi$$
0.868352 + 0.495948i $$0.165179\pi$$
$$338$$ −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 $$-0.386053\pi$$
0.350377 + 0.936609i $$0.386053\pi$$
$$374$$ −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 $$-0.716770\pi$$
−0.629573 + 0.776941i $$0.716770\pi$$
$$380$$ 0 0
$$381$$ −4978.06 −0.669380
$$382$$ −5525.06 −0.740017
$$383$$ 4162.26 0.555304 0.277652 0.960682i $$-0.410444\pi$$
0.277652 + 0.960682i $$0.410444\pi$$
$$384$$ −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 $$-0.701404\pi$$
−0.591347 + 0.806417i $$0.701404\pi$$
$$402$$ 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 $$-0.604916\pi$$
−0.323667 + 0.946171i $$0.604916\pi$$
$$410$$ −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 $$-0.523299\pi$$
−0.0731315 + 0.997322i $$0.523299\pi$$
$$422$$ −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 $$-0.651263\pi$$
−0.457522 + 0.889198i $$0.651263\pi$$
$$432$$ −432.000 −0.0481125
$$433$$ −9242.94 −1.02584 −0.512919 0.858437i $$-0.671436\pi$$
−0.512919 + 0.858437i $$0.671436\pi$$
$$434$$ −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 $$-0.535424\pi$$
−0.111059 + 0.993814i $$0.535424\pi$$
$$440$$ 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 $$-0.744655\pi$$
−0.695134 + 0.718881i $$0.744655\pi$$
$$450$$ −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 $$-0.143748\pi$$
0.899751 + 0.436405i $$0.143748\pi$$
$$488$$ 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 $$-0.357918\pi$$
0.431689 + 0.902023i $$0.357918\pi$$
$$510$$ 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 $$-0.579131\pi$$
−0.246043 + 0.969259i $$0.579131\pi$$
$$522$$ −1191.99 −0.0999466
$$523$$ 6894.09 0.576401 0.288200 0.957570i $$-0.406943\pi$$
0.288200 + 0.957570i $$0.406943\pi$$
$$524$$ 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 $$-0.455114\pi$$
0.140548 + 0.990074i $$0.455114\pi$$
$$548$$ −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 $$-0.433047\pi$$
0.208791 + 0.977960i $$0.433047\pi$$
$$564$$ −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 $$-0.284869\pi$$
0.625564 + 0.780173i $$0.284869\pi$$
$$570$$ 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 $$-0.601368\pi$$
−0.313100 + 0.949720i $$0.601368\pi$$
$$600$$ 2579.03 0.175481
$$601$$ 12700.4 0.862000 0.431000 0.902352i $$-0.358161\pi$$
0.431000 + 0.902352i $$0.358161\pi$$
$$602$$ 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 $$-0.800543\pi$$
−0.810019 + 0.586404i $$0.800543\pi$$
$$608$$ 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 $$-0.717433\pi$$
−0.631190 + 0.775628i $$0.717433\pi$$
$$642$$ 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 $$-0.469630\pi$$
0.0952668 + 0.995452i $$0.469630\pi$$
$$660$$ −3487.87 −0.205705
$$661$$ −27480.2 −1.61703 −0.808513 0.588478i $$-0.799727\pi$$
−0.808513 + 0.588478i $$0.799727\pi$$
$$662$$ −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 $$-0.519749\pi$$
−0.0620020 + 0.998076i $$0.519749\pi$$
$$674$$ 21488.2 1.22804
$$675$$ 2901.41 0.165445
$$676$$ −8523.70 −0.484963
$$677$$ 11999.0 0.681179 0.340589 0.940212i $$-0.389373\pi$$
0.340589 + 0.940212i $$0.389373\pi$$
$$678$$ −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 $$-0.453160\pi$$
0.146623 + 0.989192i $$0.453160\pi$$
$$684$$ 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 $$-0.319666\pi$$
0.536713 + 0.843765i $$0.319666\pi$$
$$744$$ 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 $$-0.875795\pi$$
−0.924833 + 0.380374i $$0.875795\pi$$
$$752$$ 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 $$-0.377931\pi$$
0.374159 + 0.927364i $$0.377931\pi$$
$$774$$ 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 $$-0.261667\pi$$
0.680720 + 0.732544i $$0.261667\pi$$
$$788$$ 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 $$-0.646662\pi$$
−0.444621 + 0.895719i $$0.646662\pi$$
$$798$$ 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 $$-0.423599\pi$$
0.237723 + 0.971333i $$0.423599\pi$$
$$812$$ −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 $$-0.115228\pi$$
0.935191 + 0.354145i $$0.115228\pi$$
$$828$$ −6353.77 −0.266677
$$829$$ 23302.5 0.976271 0.488136 0.872768i $$-0.337677\pi$$
0.488136 + 0.872768i $$0.337677\pi$$
$$830$$ 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 $$-0.692892\pi$$
−0.569575 + 0.821939i $$0.692892\pi$$
$$840$$ −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 $$-0.430178\pi$$
0.217597 + 0.976039i $$0.430178\pi$$
$$858$$ 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 $$-0.648283\pi$$
−0.449176 + 0.893443i $$0.648283\pi$$
$$864$$ −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 $$-0.495061\pi$$
0.0155152 + 0.999880i $$0.495061\pi$$
$$878$$ −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 $$-0.318209\pi$$
0.540568 + 0.841300i $$0.318209\pi$$
$$888$$ 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 $$-0.769348\pi$$
−0.748755 + 0.662847i $$0.769348\pi$$
$$908$$ −3992.18 −0.145909
$$909$$ −11175.4 −0.407772
$$910$$ −217.069 −0.00790743
$$911$$ 30047.9 1.09279 0.546395 0.837527i $$-0.316000\pi$$
0.546395 + 0.837527i $$0.316000\pi$$
$$912$$ 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 $$-0.535423\pi$$
−0.111054 + 0.993814i $$0.535423\pi$$
$$942$$ −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 $$-0.725378\pi$$
−0.650351 + 0.759633i $$0.725378\pi$$
$$954$$ −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 $$-0.488555\pi$$
0.0359478 + 0.999354i $$0.488555\pi$$
$$972$$ −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 $$-0.427946\pi$$
0.224436 + 0.974489i $$0.427946\pi$$
$$978$$ −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 $$-0.599121\pi$$
−0.306389 + 0.951906i $$0.599121\pi$$
$$984$$ 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 $$-0.725897\pi$$
−0.651589 + 0.758572i $$0.725897\pi$$
$$992$$ −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.u.1.2 3
19.7 even 3 114.4.e.d.49.2 yes 6
19.11 even 3 114.4.e.d.7.2 6
19.18 odd 2 2166.4.a.t.1.2 3
57.11 odd 6 342.4.g.h.235.2 6
57.26 odd 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 19.11 even 3
114.4.e.d.49.2 yes 6 19.7 even 3
342.4.g.h.163.2 6 57.26 odd 6
342.4.g.h.235.2 6 57.11 odd 6
2166.4.a.t.1.2 3 19.18 odd 2
2166.4.a.u.1.2 3 1.1 even 1 trivial