Properties

Label 2166.4.a.u.1.1
Level $2166$
Weight $4$
Character 2166.1
Self dual yes
Analytic conductor $127.798$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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 1.1
Root \(6.22121\) of defining polynomial
Character \(\chi\) \(=\) 2166.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.44242 q^{5} -6.00000 q^{6} -9.44242 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.44242 q^{5} -6.00000 q^{6} -9.44242 q^{7} +8.00000 q^{8} +9.00000 q^{9} -16.8848 q^{10} -47.4865 q^{11} -12.0000 q^{12} +67.6545 q^{13} -18.8848 q^{14} +25.3272 q^{15} +16.0000 q^{16} -77.6104 q^{17} +18.0000 q^{18} -33.7697 q^{20} +28.3272 q^{21} -94.9729 q^{22} +170.955 q^{23} -24.0000 q^{24} -53.7256 q^{25} +135.309 q^{26} -27.0000 q^{27} -37.7697 q^{28} +240.123 q^{29} +50.6545 q^{30} +279.104 q^{31} +32.0000 q^{32} +142.459 q^{33} -155.221 q^{34} +79.7168 q^{35} +36.0000 q^{36} -20.0617 q^{37} -202.963 q^{39} -67.5393 q^{40} +71.8213 q^{41} +56.6545 q^{42} +147.318 q^{43} -189.946 q^{44} -75.9817 q^{45} +341.909 q^{46} -207.343 q^{47} -48.0000 q^{48} -253.841 q^{49} -107.451 q^{50} +232.831 q^{51} +270.618 q^{52} -261.523 q^{53} -54.0000 q^{54} +400.901 q^{55} -75.5393 q^{56} +480.245 q^{58} -21.9817 q^{59} +101.309 q^{60} -653.103 q^{61} +558.209 q^{62} -84.9817 q^{63} +64.0000 q^{64} -571.167 q^{65} +284.919 q^{66} -413.795 q^{67} -310.442 q^{68} -512.864 q^{69} +159.434 q^{70} -157.945 q^{71} +72.0000 q^{72} -1102.15 q^{73} -40.1233 q^{74} +161.177 q^{75} +448.387 q^{77} -405.927 q^{78} -417.813 q^{79} -135.079 q^{80} +81.0000 q^{81} +143.643 q^{82} +1431.80 q^{83} +113.309 q^{84} +655.220 q^{85} +294.636 q^{86} -720.368 q^{87} -379.892 q^{88} +1289.39 q^{89} -151.963 q^{90} -638.822 q^{91} +683.819 q^{92} -837.313 q^{93} -414.686 q^{94} -96.0000 q^{96} +138.762 q^{97} -507.682 q^{98} -427.378 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 6q^{2} - 9q^{3} + 12q^{4} + 10q^{5} - 18q^{6} + 7q^{7} + 24q^{8} + 27q^{9} + O(q^{10}) \) \( 3q + 6q^{2} - 9q^{3} + 12q^{4} + 10q^{5} - 18q^{6} + 7q^{7} + 24q^{8} + 27q^{9} + 20q^{10} - 44q^{11} - 36q^{12} - 9q^{13} + 14q^{14} - 30q^{15} + 48q^{16} - 84q^{17} + 54q^{18} + 40q^{20} - 21q^{21} - 88q^{22} - 2q^{23} - 72q^{24} - 83q^{25} - 18q^{26} - 81q^{27} + 28q^{28} + 92q^{29} - 60q^{30} - 109q^{31} + 96q^{32} + 132q^{33} - 168q^{34} + 282q^{35} + 108q^{36} + 245q^{37} + 27q^{39} + 80q^{40} - 688q^{41} - 42q^{42} - 103q^{43} - 176q^{44} + 90q^{45} - 4q^{46} + 322q^{47} - 144q^{48} - 754q^{49} - 166q^{50} + 252q^{51} - 36q^{52} - 1322q^{53} - 162q^{54} - 248q^{55} + 56q^{56} + 184q^{58} + 252q^{59} - 120q^{60} - 435q^{61} - 218q^{62} + 63q^{63} + 192q^{64} - 1582q^{65} + 264q^{66} - 719q^{67} - 336q^{68} + 6q^{69} + 564q^{70} - 62q^{71} + 216q^{72} - 581q^{73} + 490q^{74} + 249q^{75} - 204q^{77} + 54q^{78} - 489q^{79} + 160q^{80} + 243q^{81} - 1376q^{82} + 2496q^{83} - 84q^{84} + 1632q^{85} - 206q^{86} - 276q^{87} - 352q^{88} + 1584q^{89} + 180q^{90} - 1573q^{91} - 8q^{92} + 327q^{93} + 644q^{94} - 288q^{96} + 974q^{97} - 1508q^{98} - 396q^{99} + O(q^{100}) \)

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 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) −8.44242 −0.755113 −0.377556 0.925987i \(-0.623236\pi\)
−0.377556 + 0.925987i \(0.623236\pi\)
\(6\) −6.00000 −0.408248
\(7\) −9.44242 −0.509843 −0.254921 0.966962i \(-0.582050\pi\)
−0.254921 + 0.966962i \(0.582050\pi\)
\(8\) 8.00000 0.353553
\(9\) 9.00000 0.333333
\(10\) −16.8848 −0.533945
\(11\) −47.4865 −1.30161 −0.650805 0.759245i \(-0.725568\pi\)
−0.650805 + 0.759245i \(0.725568\pi\)
\(12\) −12.0000 −0.288675
\(13\) 67.6545 1.44338 0.721692 0.692215i \(-0.243365\pi\)
0.721692 + 0.692215i \(0.243365\pi\)
\(14\) −18.8848 −0.360513
\(15\) 25.3272 0.435964
\(16\) 16.0000 0.250000
\(17\) −77.6104 −1.10725 −0.553626 0.832765i \(-0.686756\pi\)
−0.553626 + 0.832765i \(0.686756\pi\)
\(18\) 18.0000 0.235702
\(19\) 0 0
\(20\) −33.7697 −0.377556
\(21\) 28.3272 0.294358
\(22\) −94.9729 −0.920377
\(23\) 170.955 1.54985 0.774924 0.632054i \(-0.217788\pi\)
0.774924 + 0.632054i \(0.217788\pi\)
\(24\) −24.0000 −0.204124
\(25\) −53.7256 −0.429805
\(26\) 135.309 1.02063
\(27\) −27.0000 −0.192450
\(28\) −37.7697 −0.254921
\(29\) 240.123 1.53757 0.768787 0.639504i \(-0.220860\pi\)
0.768787 + 0.639504i \(0.220860\pi\)
\(30\) 50.6545 0.308273
\(31\) 279.104 1.61705 0.808526 0.588460i \(-0.200265\pi\)
0.808526 + 0.588460i \(0.200265\pi\)
\(32\) 32.0000 0.176777
\(33\) 142.459 0.751485
\(34\) −155.221 −0.782946
\(35\) 79.7168 0.384989
\(36\) 36.0000 0.166667
\(37\) −20.0617 −0.0891383 −0.0445692 0.999006i \(-0.514192\pi\)
−0.0445692 + 0.999006i \(0.514192\pi\)
\(38\) 0 0
\(39\) −202.963 −0.833338
\(40\) −67.5393 −0.266973
\(41\) 71.8213 0.273576 0.136788 0.990600i \(-0.456322\pi\)
0.136788 + 0.990600i \(0.456322\pi\)
\(42\) 56.6545 0.208142
\(43\) 147.318 0.522459 0.261230 0.965277i \(-0.415872\pi\)
0.261230 + 0.965277i \(0.415872\pi\)
\(44\) −189.946 −0.650805
\(45\) −75.9817 −0.251704
\(46\) 341.909 1.09591
\(47\) −207.343 −0.643491 −0.321746 0.946826i \(-0.604270\pi\)
−0.321746 + 0.946826i \(0.604270\pi\)
\(48\) −48.0000 −0.144338
\(49\) −253.841 −0.740061
\(50\) −107.451 −0.303918
\(51\) 232.831 0.639273
\(52\) 270.618 0.721692
\(53\) −261.523 −0.677791 −0.338896 0.940824i \(-0.610053\pi\)
−0.338896 + 0.940824i \(0.610053\pi\)
\(54\) −54.0000 −0.136083
\(55\) 400.901 0.982862
\(56\) −75.5393 −0.180257
\(57\) 0 0
\(58\) 480.245 1.08723
\(59\) −21.9817 −0.0485047 −0.0242524 0.999706i \(-0.507721\pi\)
−0.0242524 + 0.999706i \(0.507721\pi\)
\(60\) 101.309 0.217982
\(61\) −653.103 −1.37084 −0.685420 0.728148i \(-0.740382\pi\)
−0.685420 + 0.728148i \(0.740382\pi\)
\(62\) 558.209 1.14343
\(63\) −84.9817 −0.169948
\(64\) 64.0000 0.125000
\(65\) −571.167 −1.08992
\(66\) 284.919 0.531380
\(67\) −413.795 −0.754525 −0.377262 0.926106i \(-0.623135\pi\)
−0.377262 + 0.926106i \(0.623135\pi\)
\(68\) −310.442 −0.553626
\(69\) −512.864 −0.894805
\(70\) 159.434 0.272228
\(71\) −157.945 −0.264009 −0.132005 0.991249i \(-0.542141\pi\)
−0.132005 + 0.991249i \(0.542141\pi\)
\(72\) 72.0000 0.117851
\(73\) −1102.15 −1.76707 −0.883537 0.468362i \(-0.844844\pi\)
−0.883537 + 0.468362i \(0.844844\pi\)
\(74\) −40.1233 −0.0630303
\(75\) 161.177 0.248148
\(76\) 0 0
\(77\) 448.387 0.663616
\(78\) −405.927 −0.589259
\(79\) −417.813 −0.595033 −0.297517 0.954717i \(-0.596158\pi\)
−0.297517 + 0.954717i \(0.596158\pi\)
\(80\) −135.079 −0.188778
\(81\) 81.0000 0.111111
\(82\) 143.643 0.193447
\(83\) 1431.80 1.89350 0.946751 0.321967i \(-0.104344\pi\)
0.946751 + 0.321967i \(0.104344\pi\)
\(84\) 113.309 0.147179
\(85\) 655.220 0.836101
\(86\) 294.636 0.369435
\(87\) −720.368 −0.887719
\(88\) −379.892 −0.460189
\(89\) 1289.39 1.53568 0.767839 0.640642i \(-0.221332\pi\)
0.767839 + 0.640642i \(0.221332\pi\)
\(90\) −151.963 −0.177982
\(91\) −638.822 −0.735898
\(92\) 683.819 0.774924
\(93\) −837.313 −0.933606
\(94\) −414.686 −0.455017
\(95\) 0 0
\(96\) −96.0000 −0.102062
\(97\) 138.762 0.145249 0.0726245 0.997359i \(-0.476863\pi\)
0.0726245 + 0.997359i \(0.476863\pi\)
\(98\) −507.682 −0.523302
\(99\) −427.378 −0.433870
\(100\) −214.902 −0.214902
\(101\) −69.6306 −0.0685990 −0.0342995 0.999412i \(-0.510920\pi\)
−0.0342995 + 0.999412i \(0.510920\pi\)
\(102\) 465.663 0.452034
\(103\) −644.346 −0.616401 −0.308201 0.951321i \(-0.599727\pi\)
−0.308201 + 0.951321i \(0.599727\pi\)
\(104\) 541.236 0.510313
\(105\) −239.150 −0.222273
\(106\) −523.046 −0.479271
\(107\) −189.454 −0.171171 −0.0855853 0.996331i \(-0.527276\pi\)
−0.0855853 + 0.996331i \(0.527276\pi\)
\(108\) −108.000 −0.0962250
\(109\) −239.152 −0.210153 −0.105076 0.994464i \(-0.533509\pi\)
−0.105076 + 0.994464i \(0.533509\pi\)
\(110\) 801.801 0.694988
\(111\) 60.1850 0.0514640
\(112\) −151.079 −0.127461
\(113\) 162.743 0.135483 0.0677413 0.997703i \(-0.478421\pi\)
0.0677413 + 0.997703i \(0.478421\pi\)
\(114\) 0 0
\(115\) −1443.27 −1.17031
\(116\) 960.491 0.768787
\(117\) 608.890 0.481128
\(118\) −43.9635 −0.0342980
\(119\) 732.830 0.564525
\(120\) 202.618 0.154137
\(121\) 923.965 0.694188
\(122\) −1306.21 −0.969331
\(123\) −215.464 −0.157949
\(124\) 1116.42 0.808526
\(125\) 1508.88 1.07966
\(126\) −169.963 −0.120171
\(127\) −97.3783 −0.0680388 −0.0340194 0.999421i \(-0.510831\pi\)
−0.0340194 + 0.999421i \(0.510831\pi\)
\(128\) 128.000 0.0883883
\(129\) −441.953 −0.301642
\(130\) −1142.33 −0.770688
\(131\) 379.398 0.253039 0.126520 0.991964i \(-0.459619\pi\)
0.126520 + 0.991964i \(0.459619\pi\)
\(132\) 569.838 0.375742
\(133\) 0 0
\(134\) −827.591 −0.533530
\(135\) 227.945 0.145321
\(136\) −620.884 −0.391473
\(137\) 2413.68 1.50522 0.752609 0.658468i \(-0.228795\pi\)
0.752609 + 0.658468i \(0.228795\pi\)
\(138\) −1025.73 −0.632723
\(139\) −3109.04 −1.89716 −0.948582 0.316532i \(-0.897481\pi\)
−0.948582 + 0.316532i \(0.897481\pi\)
\(140\) 318.867 0.192494
\(141\) 622.029 0.371520
\(142\) −315.891 −0.186683
\(143\) −3212.67 −1.87872
\(144\) 144.000 0.0833333
\(145\) −2027.22 −1.16104
\(146\) −2204.29 −1.24951
\(147\) 761.522 0.427274
\(148\) −80.2467 −0.0445692
\(149\) −2919.10 −1.60498 −0.802491 0.596665i \(-0.796492\pi\)
−0.802491 + 0.596665i \(0.796492\pi\)
\(150\) 322.354 0.175467
\(151\) −1915.72 −1.03244 −0.516221 0.856455i \(-0.672662\pi\)
−0.516221 + 0.856455i \(0.672662\pi\)
\(152\) 0 0
\(153\) −698.494 −0.369084
\(154\) 896.774 0.469247
\(155\) −2356.32 −1.22106
\(156\) −811.854 −0.416669
\(157\) −2536.69 −1.28949 −0.644744 0.764398i \(-0.723036\pi\)
−0.644744 + 0.764398i \(0.723036\pi\)
\(158\) −835.626 −0.420752
\(159\) 784.569 0.391323
\(160\) −270.157 −0.133486
\(161\) −1614.23 −0.790179
\(162\) 162.000 0.0785674
\(163\) −1731.52 −0.832044 −0.416022 0.909355i \(-0.636576\pi\)
−0.416022 + 0.909355i \(0.636576\pi\)
\(164\) 287.285 0.136788
\(165\) −1202.70 −0.567456
\(166\) 2863.60 1.33891
\(167\) −1430.12 −0.662673 −0.331336 0.943513i \(-0.607499\pi\)
−0.331336 + 0.943513i \(0.607499\pi\)
\(168\) 226.618 0.104071
\(169\) 2380.13 1.08335
\(170\) 1310.44 0.591212
\(171\) 0 0
\(172\) 589.271 0.261230
\(173\) 293.325 0.128908 0.0644539 0.997921i \(-0.479469\pi\)
0.0644539 + 0.997921i \(0.479469\pi\)
\(174\) −1440.74 −0.627712
\(175\) 507.300 0.219133
\(176\) −759.784 −0.325402
\(177\) 65.9452 0.0280042
\(178\) 2578.79 1.08589
\(179\) −3843.52 −1.60490 −0.802452 0.596716i \(-0.796472\pi\)
−0.802452 + 0.596716i \(0.796472\pi\)
\(180\) −303.927 −0.125852
\(181\) 1769.28 0.726571 0.363286 0.931678i \(-0.381655\pi\)
0.363286 + 0.931678i \(0.381655\pi\)
\(182\) −1277.64 −0.520359
\(183\) 1959.31 0.791455
\(184\) 1367.64 0.547954
\(185\) 169.369 0.0673095
\(186\) −1674.63 −0.660159
\(187\) 3685.45 1.44121
\(188\) −829.372 −0.321746
\(189\) 254.945 0.0981192
\(190\) 0 0
\(191\) −504.006 −0.190935 −0.0954676 0.995433i \(-0.530435\pi\)
−0.0954676 + 0.995433i \(0.530435\pi\)
\(192\) −192.000 −0.0721688
\(193\) −2167.59 −0.808427 −0.404213 0.914665i \(-0.632455\pi\)
−0.404213 + 0.914665i \(0.632455\pi\)
\(194\) 277.524 0.102707
\(195\) 1713.50 0.629264
\(196\) −1015.36 −0.370030
\(197\) 2111.10 0.763501 0.381750 0.924265i \(-0.375321\pi\)
0.381750 + 0.924265i \(0.375321\pi\)
\(198\) −854.756 −0.306792
\(199\) 2138.59 0.761814 0.380907 0.924613i \(-0.375612\pi\)
0.380907 + 0.924613i \(0.375612\pi\)
\(200\) −429.805 −0.151959
\(201\) 1241.39 0.435625
\(202\) −139.261 −0.0485068
\(203\) −2267.34 −0.783921
\(204\) 931.325 0.319636
\(205\) −606.345 −0.206580
\(206\) −1288.69 −0.435861
\(207\) 1538.59 0.516616
\(208\) 1082.47 0.360846
\(209\) 0 0
\(210\) −478.301 −0.157171
\(211\) −975.953 −0.318424 −0.159212 0.987244i \(-0.550895\pi\)
−0.159212 + 0.987244i \(0.550895\pi\)
\(212\) −1046.09 −0.338896
\(213\) 473.836 0.152426
\(214\) −378.909 −0.121036
\(215\) −1243.72 −0.394516
\(216\) −216.000 −0.0680414
\(217\) −2635.42 −0.824442
\(218\) −478.305 −0.148600
\(219\) 3306.44 1.02022
\(220\) 1603.60 0.491431
\(221\) −5250.70 −1.59819
\(222\) 120.370 0.0363906
\(223\) −3575.45 −1.07368 −0.536839 0.843685i \(-0.680382\pi\)
−0.536839 + 0.843685i \(0.680382\pi\)
\(224\) −302.157 −0.0901283
\(225\) −483.530 −0.143268
\(226\) 325.485 0.0958007
\(227\) −86.5237 −0.0252986 −0.0126493 0.999920i \(-0.504027\pi\)
−0.0126493 + 0.999920i \(0.504027\pi\)
\(228\) 0 0
\(229\) −3928.05 −1.13351 −0.566753 0.823888i \(-0.691801\pi\)
−0.566753 + 0.823888i \(0.691801\pi\)
\(230\) −2886.54 −0.827534
\(231\) −1345.16 −0.383139
\(232\) 1920.98 0.543615
\(233\) 1020.03 0.286799 0.143399 0.989665i \(-0.454197\pi\)
0.143399 + 0.989665i \(0.454197\pi\)
\(234\) 1217.78 0.340209
\(235\) 1750.48 0.485908
\(236\) −87.9270 −0.0242524
\(237\) 1253.44 0.343543
\(238\) 1465.66 0.399179
\(239\) 264.681 0.0716351 0.0358175 0.999358i \(-0.488596\pi\)
0.0358175 + 0.999358i \(0.488596\pi\)
\(240\) 405.236 0.108991
\(241\) −172.387 −0.0460764 −0.0230382 0.999735i \(-0.507334\pi\)
−0.0230382 + 0.999735i \(0.507334\pi\)
\(242\) 1847.93 0.490865
\(243\) −243.000 −0.0641500
\(244\) −2612.41 −0.685420
\(245\) 2143.03 0.558829
\(246\) −430.928 −0.111687
\(247\) 0 0
\(248\) 2232.84 0.571715
\(249\) −4295.41 −1.09321
\(250\) 3017.75 0.763438
\(251\) −4884.87 −1.22841 −0.614203 0.789148i \(-0.710522\pi\)
−0.614203 + 0.789148i \(0.710522\pi\)
\(252\) −339.927 −0.0849738
\(253\) −8118.03 −2.01730
\(254\) −194.757 −0.0481107
\(255\) −1965.66 −0.482723
\(256\) 256.000 0.0625000
\(257\) 4022.64 0.976363 0.488181 0.872742i \(-0.337660\pi\)
0.488181 + 0.872742i \(0.337660\pi\)
\(258\) −883.907 −0.213293
\(259\) 189.431 0.0454465
\(260\) −2284.67 −0.544958
\(261\) 2161.10 0.512525
\(262\) 758.795 0.178926
\(263\) −7142.12 −1.67453 −0.837266 0.546796i \(-0.815847\pi\)
−0.837266 + 0.546796i \(0.815847\pi\)
\(264\) 1139.68 0.265690
\(265\) 2207.89 0.511809
\(266\) 0 0
\(267\) −3868.18 −0.886625
\(268\) −1655.18 −0.377262
\(269\) 5081.26 1.15171 0.575855 0.817552i \(-0.304669\pi\)
0.575855 + 0.817552i \(0.304669\pi\)
\(270\) 455.890 0.102758
\(271\) −2973.80 −0.666590 −0.333295 0.942823i \(-0.608160\pi\)
−0.333295 + 0.942823i \(0.608160\pi\)
\(272\) −1241.77 −0.276813
\(273\) 1916.47 0.424871
\(274\) 4827.37 1.06435
\(275\) 2551.24 0.559438
\(276\) −2051.46 −0.447403
\(277\) 7778.48 1.68723 0.843617 0.536946i \(-0.180422\pi\)
0.843617 + 0.536946i \(0.180422\pi\)
\(278\) −6218.09 −1.34150
\(279\) 2511.94 0.539018
\(280\) 637.734 0.136114
\(281\) 2220.68 0.471440 0.235720 0.971821i \(-0.424255\pi\)
0.235720 + 0.971821i \(0.424255\pi\)
\(282\) 1244.06 0.262704
\(283\) −7616.34 −1.59980 −0.799902 0.600130i \(-0.795115\pi\)
−0.799902 + 0.600130i \(0.795115\pi\)
\(284\) −631.781 −0.132005
\(285\) 0 0
\(286\) −6425.35 −1.32846
\(287\) −678.166 −0.139480
\(288\) 288.000 0.0589256
\(289\) 1110.38 0.226009
\(290\) −4054.43 −0.820981
\(291\) −416.286 −0.0838596
\(292\) −4408.58 −0.883537
\(293\) −9036.61 −1.80179 −0.900894 0.434038i \(-0.857088\pi\)
−0.900894 + 0.434038i \(0.857088\pi\)
\(294\) 1523.04 0.302128
\(295\) 185.579 0.0366265
\(296\) −160.493 −0.0315151
\(297\) 1282.13 0.250495
\(298\) −5838.21 −1.13489
\(299\) 11565.9 2.23703
\(300\) 644.707 0.124074
\(301\) −1391.04 −0.266372
\(302\) −3831.43 −0.730047
\(303\) 208.892 0.0396057
\(304\) 0 0
\(305\) 5513.77 1.03514
\(306\) −1396.99 −0.260982
\(307\) 7797.17 1.44954 0.724769 0.688992i \(-0.241947\pi\)
0.724769 + 0.688992i \(0.241947\pi\)
\(308\) 1793.55 0.331808
\(309\) 1933.04 0.355879
\(310\) −4712.63 −0.863418
\(311\) 2737.49 0.499128 0.249564 0.968358i \(-0.419713\pi\)
0.249564 + 0.968358i \(0.419713\pi\)
\(312\) −1623.71 −0.294629
\(313\) −7646.15 −1.38079 −0.690393 0.723435i \(-0.742562\pi\)
−0.690393 + 0.723435i \(0.742562\pi\)
\(314\) −5073.37 −0.911806
\(315\) 717.451 0.128330
\(316\) −1671.25 −0.297517
\(317\) −6420.39 −1.13755 −0.568777 0.822491i \(-0.692583\pi\)
−0.568777 + 0.822491i \(0.692583\pi\)
\(318\) 1569.14 0.276707
\(319\) −11402.6 −2.00132
\(320\) −540.315 −0.0943891
\(321\) 568.363 0.0988253
\(322\) −3228.45 −0.558741
\(323\) 0 0
\(324\) 324.000 0.0555556
\(325\) −3634.78 −0.620373
\(326\) −3463.04 −0.588344
\(327\) 717.457 0.121332
\(328\) 574.570 0.0967236
\(329\) 1957.82 0.328079
\(330\) −2405.40 −0.401252
\(331\) −4594.77 −0.762996 −0.381498 0.924370i \(-0.624592\pi\)
−0.381498 + 0.924370i \(0.624592\pi\)
\(332\) 5727.21 0.946751
\(333\) −180.555 −0.0297128
\(334\) −2860.25 −0.468580
\(335\) 3493.43 0.569751
\(336\) 453.236 0.0735894
\(337\) 6611.27 1.06866 0.534331 0.845276i \(-0.320564\pi\)
0.534331 + 0.845276i \(0.320564\pi\)
\(338\) 4760.26 0.766048
\(339\) −488.228 −0.0782209
\(340\) 2620.88 0.418050
\(341\) −13253.7 −2.10477
\(342\) 0 0
\(343\) 5635.62 0.887157
\(344\) 1178.54 0.184717
\(345\) 4329.81 0.675679
\(346\) 586.649 0.0911516
\(347\) −2498.63 −0.386552 −0.193276 0.981144i \(-0.561911\pi\)
−0.193276 + 0.981144i \(0.561911\pi\)
\(348\) −2881.47 −0.443860
\(349\) 10330.5 1.58447 0.792235 0.610216i \(-0.208917\pi\)
0.792235 + 0.610216i \(0.208917\pi\)
\(350\) 1014.60 0.154950
\(351\) −1826.67 −0.277779
\(352\) −1519.57 −0.230094
\(353\) 2608.83 0.393354 0.196677 0.980468i \(-0.436985\pi\)
0.196677 + 0.980468i \(0.436985\pi\)
\(354\) 131.890 0.0198020
\(355\) 1333.44 0.199357
\(356\) 5157.57 0.767839
\(357\) −2198.49 −0.325928
\(358\) −7687.03 −1.13484
\(359\) 4150.33 0.610156 0.305078 0.952327i \(-0.401317\pi\)
0.305078 + 0.952327i \(0.401317\pi\)
\(360\) −607.854 −0.0889909
\(361\) 0 0
\(362\) 3538.56 0.513764
\(363\) −2771.89 −0.400790
\(364\) −2555.29 −0.367949
\(365\) 9304.77 1.33434
\(366\) 3918.62 0.559643
\(367\) −7370.48 −1.04833 −0.524163 0.851618i \(-0.675622\pi\)
−0.524163 + 0.851618i \(0.675622\pi\)
\(368\) 2735.27 0.387462
\(369\) 646.391 0.0911918
\(370\) 338.738 0.0475950
\(371\) 2469.41 0.345567
\(372\) −3349.25 −0.466803
\(373\) 13166.5 1.82772 0.913858 0.406035i \(-0.133089\pi\)
0.913858 + 0.406035i \(0.133089\pi\)
\(374\) 7370.89 1.01909
\(375\) −4526.63 −0.623344
\(376\) −1658.74 −0.227508
\(377\) 16245.4 2.21931
\(378\) 509.890 0.0693808
\(379\) −14284.2 −1.93596 −0.967978 0.251034i \(-0.919230\pi\)
−0.967978 + 0.251034i \(0.919230\pi\)
\(380\) 0 0
\(381\) 292.135 0.0392822
\(382\) −1008.01 −0.135012
\(383\) −9458.62 −1.26191 −0.630957 0.775818i \(-0.717337\pi\)
−0.630957 + 0.775818i \(0.717337\pi\)
\(384\) −384.000 −0.0510310
\(385\) −3785.47 −0.501105
\(386\) −4335.17 −0.571644
\(387\) 1325.86 0.174153
\(388\) 555.048 0.0726245
\(389\) −7100.96 −0.925534 −0.462767 0.886480i \(-0.653143\pi\)
−0.462767 + 0.886480i \(0.653143\pi\)
\(390\) 3427.00 0.444957
\(391\) −13267.9 −1.71607
\(392\) −2030.73 −0.261651
\(393\) −1138.19 −0.146092
\(394\) 4222.20 0.539877
\(395\) 3527.35 0.449317
\(396\) −1709.51 −0.216935
\(397\) 1916.16 0.242240 0.121120 0.992638i \(-0.461351\pi\)
0.121120 + 0.992638i \(0.461351\pi\)
\(398\) 4277.19 0.538684
\(399\) 0 0
\(400\) −859.610 −0.107451
\(401\) −5568.69 −0.693484 −0.346742 0.937961i \(-0.612712\pi\)
−0.346742 + 0.937961i \(0.612712\pi\)
\(402\) 2482.77 0.308033
\(403\) 18882.7 2.33403
\(404\) −278.522 −0.0342995
\(405\) −683.836 −0.0839014
\(406\) −4534.68 −0.554316
\(407\) 952.658 0.116023
\(408\) 1862.65 0.226017
\(409\) 72.7133 0.00879081 0.00439541 0.999990i \(-0.498601\pi\)
0.00439541 + 0.999990i \(0.498601\pi\)
\(410\) −1212.69 −0.146074
\(411\) −7241.05 −0.869038
\(412\) −2577.38 −0.308201
\(413\) 207.561 0.0247298
\(414\) 3077.18 0.365303
\(415\) −12087.9 −1.42981
\(416\) 2164.94 0.255157
\(417\) 9327.13 1.09533
\(418\) 0 0
\(419\) 340.170 0.0396620 0.0198310 0.999803i \(-0.493687\pi\)
0.0198310 + 0.999803i \(0.493687\pi\)
\(420\) −956.602 −0.111137
\(421\) 1476.99 0.170984 0.0854920 0.996339i \(-0.472754\pi\)
0.0854920 + 0.996339i \(0.472754\pi\)
\(422\) −1951.91 −0.225160
\(423\) −1866.09 −0.214497
\(424\) −2092.18 −0.239635
\(425\) 4169.67 0.475903
\(426\) 947.672 0.107781
\(427\) 6166.87 0.698913
\(428\) −757.817 −0.0855853
\(429\) 9638.02 1.08468
\(430\) −2487.44 −0.278965
\(431\) −12279.0 −1.37230 −0.686149 0.727461i \(-0.740700\pi\)
−0.686149 + 0.727461i \(0.740700\pi\)
\(432\) −432.000 −0.0481125
\(433\) 6211.53 0.689393 0.344697 0.938714i \(-0.387982\pi\)
0.344697 + 0.938714i \(0.387982\pi\)
\(434\) −5270.84 −0.582969
\(435\) 6081.65 0.670328
\(436\) −956.609 −0.105076
\(437\) 0 0
\(438\) 6612.87 0.721405
\(439\) −13469.0 −1.46433 −0.732164 0.681129i \(-0.761489\pi\)
−0.732164 + 0.681129i \(0.761489\pi\)
\(440\) 3207.20 0.347494
\(441\) −2284.57 −0.246687
\(442\) −10501.4 −1.13009
\(443\) 7411.09 0.794834 0.397417 0.917638i \(-0.369907\pi\)
0.397417 + 0.917638i \(0.369907\pi\)
\(444\) 240.740 0.0257320
\(445\) −10885.6 −1.15961
\(446\) −7150.91 −0.759205
\(447\) 8757.31 0.926636
\(448\) −604.315 −0.0637303
\(449\) −3345.93 −0.351680 −0.175840 0.984419i \(-0.556264\pi\)
−0.175840 + 0.984419i \(0.556264\pi\)
\(450\) −967.061 −0.101306
\(451\) −3410.54 −0.356089
\(452\) 650.970 0.0677413
\(453\) 5747.15 0.596081
\(454\) −173.047 −0.0178888
\(455\) 5393.20 0.555686
\(456\) 0 0
\(457\) −542.238 −0.0555029 −0.0277515 0.999615i \(-0.508835\pi\)
−0.0277515 + 0.999615i \(0.508835\pi\)
\(458\) −7856.10 −0.801510
\(459\) 2095.48 0.213091
\(460\) −5773.08 −0.585155
\(461\) 3823.43 0.386280 0.193140 0.981171i \(-0.438133\pi\)
0.193140 + 0.981171i \(0.438133\pi\)
\(462\) −2690.32 −0.270920
\(463\) −3750.37 −0.376445 −0.188223 0.982126i \(-0.560273\pi\)
−0.188223 + 0.982126i \(0.560273\pi\)
\(464\) 3841.96 0.384394
\(465\) 7068.95 0.704978
\(466\) 2040.05 0.202797
\(467\) −2981.61 −0.295444 −0.147722 0.989029i \(-0.547194\pi\)
−0.147722 + 0.989029i \(0.547194\pi\)
\(468\) 2435.56 0.240564
\(469\) 3907.23 0.384689
\(470\) 3500.95 0.343589
\(471\) 7610.06 0.744486
\(472\) −175.854 −0.0171490
\(473\) −6995.60 −0.680038
\(474\) 2506.88 0.242921
\(475\) 0 0
\(476\) 2931.32 0.282262
\(477\) −2353.71 −0.225930
\(478\) 529.362 0.0506537
\(479\) 15355.8 1.46477 0.732387 0.680889i \(-0.238406\pi\)
0.732387 + 0.680889i \(0.238406\pi\)
\(480\) 810.472 0.0770684
\(481\) −1357.26 −0.128661
\(482\) −344.774 −0.0325810
\(483\) 4842.68 0.456210
\(484\) 3695.86 0.347094
\(485\) −1171.49 −0.109679
\(486\) −486.000 −0.0453609
\(487\) 439.463 0.0408911 0.0204455 0.999791i \(-0.493492\pi\)
0.0204455 + 0.999791i \(0.493492\pi\)
\(488\) −5224.83 −0.484665
\(489\) 5194.56 0.480381
\(490\) 4286.06 0.395152
\(491\) 19765.8 1.81674 0.908370 0.418168i \(-0.137328\pi\)
0.908370 + 0.418168i \(0.137328\pi\)
\(492\) −861.855 −0.0789745
\(493\) −18636.0 −1.70248
\(494\) 0 0
\(495\) 3608.10 0.327621
\(496\) 4465.67 0.404263
\(497\) 1491.38 0.134603
\(498\) −8590.81 −0.773019
\(499\) 8159.72 0.732023 0.366011 0.930610i \(-0.380723\pi\)
0.366011 + 0.930610i \(0.380723\pi\)
\(500\) 6035.50 0.539832
\(501\) 4290.37 0.382594
\(502\) −9769.73 −0.868614
\(503\) −19674.3 −1.74400 −0.872002 0.489503i \(-0.837178\pi\)
−0.872002 + 0.489503i \(0.837178\pi\)
\(504\) −679.854 −0.0600855
\(505\) 587.850 0.0518000
\(506\) −16236.1 −1.42645
\(507\) −7140.39 −0.625475
\(508\) −389.513 −0.0340194
\(509\) −6855.74 −0.597005 −0.298502 0.954409i \(-0.596487\pi\)
−0.298502 + 0.954409i \(0.596487\pi\)
\(510\) −3931.32 −0.341337
\(511\) 10406.9 0.900929
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 8045.28 0.690393
\(515\) 5439.84 0.465452
\(516\) −1767.81 −0.150821
\(517\) 9845.99 0.837574
\(518\) 378.861 0.0321355
\(519\) −879.974 −0.0744250
\(520\) −4569.34 −0.385344
\(521\) −12666.6 −1.06513 −0.532567 0.846388i \(-0.678773\pi\)
−0.532567 + 0.846388i \(0.678773\pi\)
\(522\) 4322.21 0.362410
\(523\) 4717.23 0.394398 0.197199 0.980363i \(-0.436815\pi\)
0.197199 + 0.980363i \(0.436815\pi\)
\(524\) 1517.59 0.126520
\(525\) −1521.90 −0.126516
\(526\) −14284.2 −1.18407
\(527\) −21661.4 −1.79049
\(528\) 2279.35 0.187871
\(529\) 17058.5 1.40203
\(530\) 4415.77 0.361904
\(531\) −197.836 −0.0161682
\(532\) 0 0
\(533\) 4859.03 0.394874
\(534\) −7736.36 −0.626938
\(535\) 1599.45 0.129253
\(536\) −3310.36 −0.266765
\(537\) 11530.5 0.926592
\(538\) 10162.5 0.814382
\(539\) 12054.0 0.963270
\(540\) 911.781 0.0726607
\(541\) 11741.2 0.933072 0.466536 0.884502i \(-0.345502\pi\)
0.466536 + 0.884502i \(0.345502\pi\)
\(542\) −5947.61 −0.471350
\(543\) −5307.83 −0.419486
\(544\) −2483.53 −0.195736
\(545\) 2019.02 0.158689
\(546\) 3832.93 0.300429
\(547\) −22262.1 −1.74015 −0.870073 0.492922i \(-0.835929\pi\)
−0.870073 + 0.492922i \(0.835929\pi\)
\(548\) 9654.73 0.752609
\(549\) −5877.93 −0.456947
\(550\) 5102.48 0.395583
\(551\) 0 0
\(552\) −4102.91 −0.316362
\(553\) 3945.16 0.303373
\(554\) 15557.0 1.19305
\(555\) −508.107 −0.0388611
\(556\) −12436.2 −0.948582
\(557\) −19627.7 −1.49310 −0.746548 0.665332i \(-0.768290\pi\)
−0.746548 + 0.665332i \(0.768290\pi\)
\(558\) 5023.88 0.381143
\(559\) 9966.71 0.754109
\(560\) 1275.47 0.0962471
\(561\) −11056.3 −0.832084
\(562\) 4441.36 0.333358
\(563\) 9810.37 0.734384 0.367192 0.930145i \(-0.380319\pi\)
0.367192 + 0.930145i \(0.380319\pi\)
\(564\) 2488.12 0.185760
\(565\) −1373.94 −0.102305
\(566\) −15232.7 −1.13123
\(567\) −764.836 −0.0566492
\(568\) −1263.56 −0.0933414
\(569\) 2528.37 0.186282 0.0931412 0.995653i \(-0.470309\pi\)
0.0931412 + 0.995653i \(0.470309\pi\)
\(570\) 0 0
\(571\) −21892.0 −1.60447 −0.802233 0.597011i \(-0.796355\pi\)
−0.802233 + 0.597011i \(0.796355\pi\)
\(572\) −12850.7 −0.939361
\(573\) 1512.02 0.110236
\(574\) −1356.33 −0.0986276
\(575\) −9184.64 −0.666132
\(576\) 576.000 0.0416667
\(577\) −4002.61 −0.288788 −0.144394 0.989520i \(-0.546123\pi\)
−0.144394 + 0.989520i \(0.546123\pi\)
\(578\) 2220.76 0.159812
\(579\) 6502.76 0.466745
\(580\) −8108.86 −0.580521
\(581\) −13519.7 −0.965388
\(582\) −832.573 −0.0592977
\(583\) 12418.8 0.882220
\(584\) −8817.16 −0.624755
\(585\) −5140.51 −0.363306
\(586\) −18073.2 −1.27406
\(587\) 24518.5 1.72400 0.861998 0.506912i \(-0.169213\pi\)
0.861998 + 0.506912i \(0.169213\pi\)
\(588\) 3046.09 0.213637
\(589\) 0 0
\(590\) 371.158 0.0258989
\(591\) −6333.30 −0.440807
\(592\) −320.987 −0.0222846
\(593\) −17698.7 −1.22563 −0.612815 0.790226i \(-0.709963\pi\)
−0.612815 + 0.790226i \(0.709963\pi\)
\(594\) 2564.27 0.177127
\(595\) −6186.86 −0.426280
\(596\) −11676.4 −0.802491
\(597\) −6415.78 −0.439833
\(598\) 23131.7 1.58182
\(599\) 3608.01 0.246109 0.123054 0.992400i \(-0.460731\pi\)
0.123054 + 0.992400i \(0.460731\pi\)
\(600\) 1289.41 0.0877336
\(601\) 23195.0 1.57429 0.787143 0.616771i \(-0.211559\pi\)
0.787143 + 0.616771i \(0.211559\pi\)
\(602\) −2782.07 −0.188353
\(603\) −3724.16 −0.251508
\(604\) −7662.86 −0.516221
\(605\) −7800.50 −0.524190
\(606\) 417.783 0.0280054
\(607\) −3485.49 −0.233067 −0.116534 0.993187i \(-0.537178\pi\)
−0.116534 + 0.993187i \(0.537178\pi\)
\(608\) 0 0
\(609\) 6802.02 0.452597
\(610\) 11027.5 0.731954
\(611\) −14027.7 −0.928804
\(612\) −2793.98 −0.184542
\(613\) −10953.2 −0.721692 −0.360846 0.932625i \(-0.617512\pi\)
−0.360846 + 0.932625i \(0.617512\pi\)
\(614\) 15594.3 1.02498
\(615\) 1819.03 0.119269
\(616\) 3587.10 0.234624
\(617\) 7009.52 0.457362 0.228681 0.973501i \(-0.426559\pi\)
0.228681 + 0.973501i \(0.426559\pi\)
\(618\) 3866.08 0.251645
\(619\) 24225.1 1.57301 0.786503 0.617587i \(-0.211890\pi\)
0.786503 + 0.617587i \(0.211890\pi\)
\(620\) −9425.26 −0.610529
\(621\) −4615.78 −0.298268
\(622\) 5474.98 0.352937
\(623\) −12175.0 −0.782954
\(624\) −3247.42 −0.208334
\(625\) −6022.86 −0.385463
\(626\) −15292.3 −0.976363
\(627\) 0 0
\(628\) −10146.7 −0.644744
\(629\) 1556.99 0.0986986
\(630\) 1434.90 0.0907427
\(631\) 14904.5 0.940314 0.470157 0.882583i \(-0.344197\pi\)
0.470157 + 0.882583i \(0.344197\pi\)
\(632\) −3342.50 −0.210376
\(633\) 2927.86 0.183842
\(634\) −12840.8 −0.804373
\(635\) 822.108 0.0513769
\(636\) 3138.28 0.195662
\(637\) −17173.5 −1.06819
\(638\) −22805.2 −1.41515
\(639\) −1421.51 −0.0880031
\(640\) −1080.63 −0.0667432
\(641\) −16749.1 −1.03206 −0.516028 0.856572i \(-0.672590\pi\)
−0.516028 + 0.856572i \(0.672590\pi\)
\(642\) 1136.73 0.0698801
\(643\) −7109.61 −0.436043 −0.218022 0.975944i \(-0.569960\pi\)
−0.218022 + 0.975944i \(0.569960\pi\)
\(644\) −6456.90 −0.395089
\(645\) 3731.15 0.227774
\(646\) 0 0
\(647\) 6637.82 0.403338 0.201669 0.979454i \(-0.435363\pi\)
0.201669 + 0.979454i \(0.435363\pi\)
\(648\) 648.000 0.0392837
\(649\) 1043.84 0.0631342
\(650\) −7269.56 −0.438670
\(651\) 7906.26 0.475992
\(652\) −6926.08 −0.416022
\(653\) −18762.0 −1.12437 −0.562184 0.827012i \(-0.690039\pi\)
−0.562184 + 0.827012i \(0.690039\pi\)
\(654\) 1434.91 0.0857944
\(655\) −3203.03 −0.191073
\(656\) 1149.14 0.0683939
\(657\) −9919.31 −0.589024
\(658\) 3915.64 0.231987
\(659\) 6156.57 0.363924 0.181962 0.983306i \(-0.441755\pi\)
0.181962 + 0.983306i \(0.441755\pi\)
\(660\) −4810.81 −0.283728
\(661\) 26165.2 1.53965 0.769824 0.638257i \(-0.220344\pi\)
0.769824 + 0.638257i \(0.220344\pi\)
\(662\) −9189.55 −0.539520
\(663\) 15752.1 0.922715
\(664\) 11454.4 0.669454
\(665\) 0 0
\(666\) −361.110 −0.0210101
\(667\) 41050.1 2.38301
\(668\) −5720.50 −0.331336
\(669\) 10726.4 0.619888
\(670\) 6986.87 0.402875
\(671\) 31013.6 1.78430
\(672\) 906.472 0.0520356
\(673\) 31028.8 1.77723 0.888613 0.458657i \(-0.151669\pi\)
0.888613 + 0.458657i \(0.151669\pi\)
\(674\) 13222.5 0.755658
\(675\) 1450.59 0.0827160
\(676\) 9520.52 0.541677
\(677\) −4341.55 −0.246469 −0.123234 0.992378i \(-0.539327\pi\)
−0.123234 + 0.992378i \(0.539327\pi\)
\(678\) −976.455 −0.0553105
\(679\) −1310.25 −0.0740542
\(680\) 5241.76 0.295606
\(681\) 259.571 0.0146062
\(682\) −26507.4 −1.48830
\(683\) 16457.2 0.921986 0.460993 0.887404i \(-0.347493\pi\)
0.460993 + 0.887404i \(0.347493\pi\)
\(684\) 0 0
\(685\) −20377.3 −1.13661
\(686\) 11271.2 0.627315
\(687\) 11784.2 0.654430
\(688\) 2357.08 0.130615
\(689\) −17693.2 −0.978313
\(690\) 8659.62 0.477777
\(691\) 2522.21 0.138856 0.0694280 0.997587i \(-0.477883\pi\)
0.0694280 + 0.997587i \(0.477883\pi\)
\(692\) 1173.30 0.0644539
\(693\) 4035.48 0.221205
\(694\) −4997.26 −0.273333
\(695\) 26247.8 1.43257
\(696\) −5762.95 −0.313856
\(697\) −5574.08 −0.302917
\(698\) 20661.1 1.12039
\(699\) −3060.08 −0.165583
\(700\) 2029.20 0.109566
\(701\) 4019.82 0.216586 0.108293 0.994119i \(-0.465462\pi\)
0.108293 + 0.994119i \(0.465462\pi\)
\(702\) −3653.34 −0.196420
\(703\) 0 0
\(704\) −3039.13 −0.162701
\(705\) −5251.43 −0.280539
\(706\) 5217.65 0.278143
\(707\) 657.481 0.0349747
\(708\) 263.781 0.0140021
\(709\) −10052.9 −0.532503 −0.266252 0.963904i \(-0.585785\pi\)
−0.266252 + 0.963904i \(0.585785\pi\)
\(710\) 2666.88 0.140966
\(711\) −3760.32 −0.198344
\(712\) 10315.1 0.542944
\(713\) 47714.2 2.50619
\(714\) −4396.98 −0.230466
\(715\) 27122.7 1.41865
\(716\) −15374.1 −0.802452
\(717\) −794.043 −0.0413585
\(718\) 8300.66 0.431445
\(719\) 10430.4 0.541011 0.270505 0.962718i \(-0.412809\pi\)
0.270505 + 0.962718i \(0.412809\pi\)
\(720\) −1215.71 −0.0629261
\(721\) 6084.18 0.314268
\(722\) 0 0
\(723\) 517.161 0.0266022
\(724\) 7077.11 0.363286
\(725\) −12900.7 −0.660857
\(726\) −5543.79 −0.283401
\(727\) −21693.5 −1.10669 −0.553346 0.832951i \(-0.686649\pi\)
−0.553346 + 0.832951i \(0.686649\pi\)
\(728\) −5110.58 −0.260179
\(729\) 729.000 0.0370370
\(730\) 18609.5 0.943520
\(731\) −11433.4 −0.578495
\(732\) 7837.24 0.395728
\(733\) −3382.86 −0.170462 −0.0852310 0.996361i \(-0.527163\pi\)
−0.0852310 + 0.996361i \(0.527163\pi\)
\(734\) −14741.0 −0.741279
\(735\) −6429.09 −0.322640
\(736\) 5470.55 0.273977
\(737\) 19649.7 0.982097
\(738\) 1292.78 0.0644824
\(739\) 8017.89 0.399111 0.199555 0.979887i \(-0.436050\pi\)
0.199555 + 0.979887i \(0.436050\pi\)
\(740\) 677.476 0.0336547
\(741\) 0 0
\(742\) 4938.82 0.244353
\(743\) 9357.91 0.462057 0.231029 0.972947i \(-0.425791\pi\)
0.231029 + 0.972947i \(0.425791\pi\)
\(744\) −6698.51 −0.330080
\(745\) 24644.3 1.21194
\(746\) 26333.1 1.29239
\(747\) 12886.2 0.631167
\(748\) 14741.8 0.720606
\(749\) 1788.91 0.0872700
\(750\) −9053.26 −0.440771
\(751\) −12967.3 −0.630070 −0.315035 0.949080i \(-0.602016\pi\)
−0.315035 + 0.949080i \(0.602016\pi\)
\(752\) −3317.49 −0.160873
\(753\) 14654.6 0.709221
\(754\) 32490.8 1.56929
\(755\) 16173.3 0.779610
\(756\) 1019.78 0.0490596
\(757\) 27781.4 1.33386 0.666931 0.745119i \(-0.267608\pi\)
0.666931 + 0.745119i \(0.267608\pi\)
\(758\) −28568.3 −1.36893
\(759\) 24354.1 1.16469
\(760\) 0 0
\(761\) 26066.1 1.24165 0.620825 0.783949i \(-0.286798\pi\)
0.620825 + 0.783949i \(0.286798\pi\)
\(762\) 584.270 0.0277767
\(763\) 2258.18 0.107145
\(764\) −2016.03 −0.0954676
\(765\) 5896.98 0.278700
\(766\) −18917.2 −0.892307
\(767\) −1487.16 −0.0700109
\(768\) −768.000 −0.0360844
\(769\) −36835.4 −1.72733 −0.863665 0.504066i \(-0.831837\pi\)
−0.863665 + 0.504066i \(0.831837\pi\)
\(770\) −7570.94 −0.354335
\(771\) −12067.9 −0.563703
\(772\) −8670.35 −0.404213
\(773\) 11208.4 0.521524 0.260762 0.965403i \(-0.416026\pi\)
0.260762 + 0.965403i \(0.416026\pi\)
\(774\) 2651.72 0.123145
\(775\) −14995.1 −0.695017
\(776\) 1110.10 0.0513533
\(777\) −568.292 −0.0262385
\(778\) −14201.9 −0.654452
\(779\) 0 0
\(780\) 6854.01 0.314632
\(781\) 7500.26 0.343637
\(782\) −26535.7 −1.21345
\(783\) −6483.31 −0.295906
\(784\) −4061.45 −0.185015
\(785\) 21415.8 0.973709
\(786\) −2276.39 −0.103303
\(787\) −25466.9 −1.15349 −0.576745 0.816924i \(-0.695677\pi\)
−0.576745 + 0.816924i \(0.695677\pi\)
\(788\) 8444.40 0.381750
\(789\) 21426.4 0.966791
\(790\) 7054.70 0.317715
\(791\) −1536.68 −0.0690748
\(792\) −3419.03 −0.153396
\(793\) −44185.4 −1.97865
\(794\) 3832.32 0.171290
\(795\) −6623.66 −0.295493
\(796\) 8554.38 0.380907
\(797\) 31745.4 1.41089 0.705446 0.708764i \(-0.250747\pi\)
0.705446 + 0.708764i \(0.250747\pi\)
\(798\) 0 0
\(799\) 16092.0 0.712507
\(800\) −1719.22 −0.0759795
\(801\) 11604.5 0.511893
\(802\) −11137.4 −0.490367
\(803\) 52337.0 2.30004
\(804\) 4965.55 0.217813
\(805\) 13628.0 0.596674
\(806\) 37765.3 1.65041
\(807\) −15243.8 −0.664940
\(808\) −557.045 −0.0242534
\(809\) −14106.2 −0.613040 −0.306520 0.951864i \(-0.599165\pi\)
−0.306520 + 0.951864i \(0.599165\pi\)
\(810\) −1367.67 −0.0593273
\(811\) 42675.4 1.84776 0.923882 0.382678i \(-0.124998\pi\)
0.923882 + 0.382678i \(0.124998\pi\)
\(812\) −9069.35 −0.391961
\(813\) 8921.41 0.384856
\(814\) 1905.32 0.0820409
\(815\) 14618.2 0.628287
\(816\) 3725.30 0.159818
\(817\) 0 0
\(818\) 145.427 0.00621604
\(819\) −5749.40 −0.245299
\(820\) −2425.38 −0.103290
\(821\) −19439.3 −0.826354 −0.413177 0.910651i \(-0.635581\pi\)
−0.413177 + 0.910651i \(0.635581\pi\)
\(822\) −14482.1 −0.614503
\(823\) −25023.4 −1.05986 −0.529928 0.848042i \(-0.677781\pi\)
−0.529928 + 0.848042i \(0.677781\pi\)
\(824\) −5154.77 −0.217931
\(825\) −7653.72 −0.322992
\(826\) 415.122 0.0174866
\(827\) −33545.7 −1.41052 −0.705260 0.708949i \(-0.749170\pi\)
−0.705260 + 0.708949i \(0.749170\pi\)
\(828\) 6154.37 0.258308
\(829\) 5555.52 0.232752 0.116376 0.993205i \(-0.462872\pi\)
0.116376 + 0.993205i \(0.462872\pi\)
\(830\) −24175.7 −1.01103
\(831\) −23335.4 −0.974124
\(832\) 4329.89 0.180423
\(833\) 19700.7 0.819434
\(834\) 18654.3 0.774514
\(835\) 12073.7 0.500392
\(836\) 0 0
\(837\) −7535.82 −0.311202
\(838\) 680.340 0.0280453
\(839\) 40982.4 1.68638 0.843188 0.537618i \(-0.180676\pi\)
0.843188 + 0.537618i \(0.180676\pi\)
\(840\) −1913.20 −0.0785855
\(841\) 33269.9 1.36414
\(842\) 2953.99 0.120904
\(843\) −6662.04 −0.272186
\(844\) −3903.81 −0.159212
\(845\) −20094.1 −0.818055
\(846\) −3732.17 −0.151672
\(847\) −8724.46 −0.353927
\(848\) −4184.37 −0.169448
\(849\) 22849.0 0.923648
\(850\) 8339.34 0.336514
\(851\) −3429.64 −0.138151
\(852\) 1895.34 0.0762129
\(853\) −31609.3 −1.26879 −0.634397 0.773007i \(-0.718752\pi\)
−0.634397 + 0.773007i \(0.718752\pi\)
\(854\) 12333.7 0.494206
\(855\) 0 0
\(856\) −1515.63 −0.0605179
\(857\) 20963.7 0.835596 0.417798 0.908540i \(-0.362802\pi\)
0.417798 + 0.908540i \(0.362802\pi\)
\(858\) 19276.0 0.766985
\(859\) 3896.64 0.154775 0.0773875 0.997001i \(-0.475342\pi\)
0.0773875 + 0.997001i \(0.475342\pi\)
\(860\) −4974.87 −0.197258
\(861\) 2034.50 0.0805291
\(862\) −24558.1 −0.970361
\(863\) 2630.83 0.103771 0.0518856 0.998653i \(-0.483477\pi\)
0.0518856 + 0.998653i \(0.483477\pi\)
\(864\) −864.000 −0.0340207
\(865\) −2476.37 −0.0973400
\(866\) 12423.1 0.487475
\(867\) −3331.14 −0.130486
\(868\) −10541.7 −0.412221
\(869\) 19840.5 0.774501
\(870\) 12163.3 0.473993
\(871\) −27995.1 −1.08907
\(872\) −1913.22 −0.0743002
\(873\) 1248.86 0.0484164
\(874\) 0 0
\(875\) −14247.4 −0.550459
\(876\) 13225.7 0.510110
\(877\) −36577.3 −1.40836 −0.704178 0.710023i \(-0.748684\pi\)
−0.704178 + 0.710023i \(0.748684\pi\)
\(878\) −26938.0 −1.03544
\(879\) 27109.8 1.04026
\(880\) 6414.41 0.245716
\(881\) 6389.06 0.244328 0.122164 0.992510i \(-0.461017\pi\)
0.122164 + 0.992510i \(0.461017\pi\)
\(882\) −4569.13 −0.174434
\(883\) −17737.8 −0.676020 −0.338010 0.941142i \(-0.609754\pi\)
−0.338010 + 0.941142i \(0.609754\pi\)
\(884\) −21002.8 −0.799095
\(885\) −556.737 −0.0211463
\(886\) 14822.2 0.562033
\(887\) 29621.0 1.12128 0.560641 0.828059i \(-0.310555\pi\)
0.560641 + 0.828059i \(0.310555\pi\)
\(888\) 481.480 0.0181953
\(889\) 919.486 0.0346891
\(890\) −21771.2 −0.819968
\(891\) −3846.40 −0.144623
\(892\) −14301.8 −0.536839
\(893\) 0 0
\(894\) 17514.6 0.655231
\(895\) 32448.6 1.21188
\(896\) −1208.63 −0.0450641
\(897\) −34697.6 −1.29155
\(898\) −6691.87 −0.248675
\(899\) 67019.3 2.48634
\(900\) −1934.12 −0.0716341
\(901\) 20296.9 0.750486
\(902\) −6821.08 −0.251793
\(903\) 4173.11 0.153790
\(904\) 1301.94 0.0479003
\(905\) −14937.0 −0.548643
\(906\) 11494.3 0.421493
\(907\) 14352.3 0.525423 0.262712 0.964874i \(-0.415383\pi\)
0.262712 + 0.964874i \(0.415383\pi\)
\(908\) −346.095 −0.0126493
\(909\) −626.675 −0.0228663
\(910\) 10786.4 0.392929
\(911\) 50.5532 0.00183853 0.000919266 1.00000i \(-0.499707\pi\)
0.000919266 1.00000i \(0.499707\pi\)
\(912\) 0 0
\(913\) −67991.2 −2.46460
\(914\) −1084.48 −0.0392465
\(915\) −16541.3 −0.597638
\(916\) −15712.2 −0.566753
\(917\) −3582.43 −0.129010
\(918\) 4190.96 0.150678
\(919\) −38362.9 −1.37701 −0.688507 0.725230i \(-0.741734\pi\)
−0.688507 + 0.725230i \(0.741734\pi\)
\(920\) −11546.2 −0.413767
\(921\) −23391.5 −0.836891
\(922\) 7646.86 0.273141
\(923\) −10685.7 −0.381066
\(924\) −5380.64 −0.191569
\(925\) 1077.83 0.0383121
\(926\) −7500.73 −0.266187
\(927\) −5799.12 −0.205467
\(928\) 7683.93 0.271807
\(929\) −38087.0 −1.34510 −0.672548 0.740053i \(-0.734800\pi\)
−0.672548 + 0.740053i \(0.734800\pi\)
\(930\) 14137.9 0.498494
\(931\) 0 0
\(932\) 4080.10 0.143399
\(933\) −8212.47 −0.288172
\(934\) −5963.22 −0.208910
\(935\) −31114.1 −1.08828
\(936\) 4871.12 0.170104
\(937\) 44240.1 1.54243 0.771217 0.636573i \(-0.219649\pi\)
0.771217 + 0.636573i \(0.219649\pi\)
\(938\) 7814.46 0.272016
\(939\) 22938.5 0.797197
\(940\) 7001.90 0.242954
\(941\) 24209.1 0.838675 0.419338 0.907830i \(-0.362262\pi\)
0.419338 + 0.907830i \(0.362262\pi\)
\(942\) 15220.1 0.526431
\(943\) 12278.2 0.424001
\(944\) −351.708 −0.0121262
\(945\) −2152.35 −0.0740911
\(946\) −13991.2 −0.480860
\(947\) −20524.7 −0.704292 −0.352146 0.935945i \(-0.614548\pi\)
−0.352146 + 0.935945i \(0.614548\pi\)
\(948\) 5013.76 0.171771
\(949\) −74565.1 −2.55056
\(950\) 0 0
\(951\) 19261.2 0.656768
\(952\) 5862.64 0.199590
\(953\) −4911.36 −0.166941 −0.0834703 0.996510i \(-0.526600\pi\)
−0.0834703 + 0.996510i \(0.526600\pi\)
\(954\) −4707.41 −0.159757
\(955\) 4255.03 0.144178
\(956\) 1058.72 0.0358175
\(957\) 34207.7 1.15546
\(958\) 30711.7 1.03575
\(959\) −22791.0 −0.767424
\(960\) 1620.94 0.0544956
\(961\) 48108.3 1.61486
\(962\) −2714.52 −0.0909769
\(963\) −1705.09 −0.0570568
\(964\) −689.548 −0.0230382
\(965\) 18299.7 0.610453
\(966\) 9685.35 0.322589
\(967\) −40394.7 −1.34334 −0.671668 0.740852i \(-0.734422\pi\)
−0.671668 + 0.740852i \(0.734422\pi\)
\(968\) 7391.72 0.245433
\(969\) 0 0
\(970\) −2342.98 −0.0775550
\(971\) −26791.9 −0.885471 −0.442735 0.896652i \(-0.645992\pi\)
−0.442735 + 0.896652i \(0.645992\pi\)
\(972\) −972.000 −0.0320750
\(973\) 29356.9 0.967255
\(974\) 878.926 0.0289144
\(975\) 10904.3 0.358173
\(976\) −10449.7 −0.342710
\(977\) −8122.31 −0.265973 −0.132987 0.991118i \(-0.542457\pi\)
−0.132987 + 0.991118i \(0.542457\pi\)
\(978\) 10389.1 0.339680
\(979\) −61228.7 −1.99885
\(980\) 8572.12 0.279415
\(981\) −2152.37 −0.0700509
\(982\) 39531.6 1.28463
\(983\) −32918.1 −1.06808 −0.534041 0.845459i \(-0.679327\pi\)
−0.534041 + 0.845459i \(0.679327\pi\)
\(984\) −1723.71 −0.0558434
\(985\) −17822.8 −0.576529
\(986\) −37272.1 −1.20384
\(987\) −5873.46 −0.189417
\(988\) 0 0
\(989\) 25184.7 0.809733
\(990\) 7216.21 0.231663
\(991\) −31770.1 −1.01838 −0.509189 0.860655i \(-0.670054\pi\)
−0.509189 + 0.860655i \(0.670054\pi\)
\(992\) 8931.34 0.285857
\(993\) 13784.3 0.440516
\(994\) 2982.77 0.0951788
\(995\) −18054.9 −0.575255
\(996\) −17181.6 −0.546607
\(997\) 15506.5 0.492574 0.246287 0.969197i \(-0.420789\pi\)
0.246287 + 0.969197i \(0.420789\pi\)
\(998\) 16319.4 0.517618
\(999\) 541.665 0.0171547
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2166.4.a.u.1.1 3
19.7 even 3 114.4.e.d.49.3 yes 6
19.11 even 3 114.4.e.d.7.3 6
19.18 odd 2 2166.4.a.t.1.1 3
57.11 odd 6 342.4.g.h.235.1 6
57.26 odd 6 342.4.g.h.163.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.4.e.d.7.3 6 19.11 even 3
114.4.e.d.49.3 yes 6 19.7 even 3
342.4.g.h.163.1 6 57.26 odd 6
342.4.g.h.235.1 6 57.11 odd 6
2166.4.a.t.1.1 3 19.18 odd 2
2166.4.a.u.1.1 3 1.1 even 1 trivial