Properties

Label 105.4.a.d.1.2
Level 105
Weight 4
Character 105.1
Self dual yes
Analytic conductor 6.195
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.19520055060\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of \(x^{2} - x - 1\)
Character \(\chi\) \(=\) 105.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.236068 q^{2} +3.00000 q^{3} -7.94427 q^{4} -5.00000 q^{5} +0.708204 q^{6} -7.00000 q^{7} -3.76393 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+0.236068 q^{2} +3.00000 q^{3} -7.94427 q^{4} -5.00000 q^{5} +0.708204 q^{6} -7.00000 q^{7} -3.76393 q^{8} +9.00000 q^{9} -1.18034 q^{10} -50.4721 q^{11} -23.8328 q^{12} -80.9706 q^{13} -1.65248 q^{14} -15.0000 q^{15} +62.6656 q^{16} +76.3870 q^{17} +2.12461 q^{18} +4.13777 q^{19} +39.7214 q^{20} -21.0000 q^{21} -11.9149 q^{22} -204.721 q^{23} -11.2918 q^{24} +25.0000 q^{25} -19.1146 q^{26} +27.0000 q^{27} +55.6099 q^{28} -91.1672 q^{29} -3.54102 q^{30} +198.079 q^{31} +44.9048 q^{32} -151.416 q^{33} +18.0325 q^{34} +35.0000 q^{35} -71.4984 q^{36} +155.666 q^{37} +0.976794 q^{38} -242.912 q^{39} +18.8197 q^{40} -156.885 q^{41} -4.95743 q^{42} +354.217 q^{43} +400.964 q^{44} -45.0000 q^{45} -48.3282 q^{46} -175.659 q^{47} +187.997 q^{48} +49.0000 q^{49} +5.90170 q^{50} +229.161 q^{51} +643.252 q^{52} +200.302 q^{53} +6.37384 q^{54} +252.361 q^{55} +26.3475 q^{56} +12.4133 q^{57} -21.5217 q^{58} -312.498 q^{59} +119.164 q^{60} -154.170 q^{61} +46.7601 q^{62} -63.0000 q^{63} -490.724 q^{64} +404.853 q^{65} -35.7446 q^{66} +734.715 q^{67} -606.839 q^{68} -614.164 q^{69} +8.26238 q^{70} -678.577 q^{71} -33.8754 q^{72} -60.8003 q^{73} +36.7477 q^{74} +75.0000 q^{75} -32.8715 q^{76} +353.305 q^{77} -57.3437 q^{78} -1286.26 q^{79} -313.328 q^{80} +81.0000 q^{81} -37.0356 q^{82} +116.170 q^{83} +166.830 q^{84} -381.935 q^{85} +83.6192 q^{86} -273.502 q^{87} +189.974 q^{88} -916.440 q^{89} -10.6231 q^{90} +566.794 q^{91} +1626.36 q^{92} +594.237 q^{93} -41.4676 q^{94} -20.6888 q^{95} +134.714 q^{96} -1416.30 q^{97} +11.5673 q^{98} -454.249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 6q^{3} + 2q^{4} - 10q^{5} - 12q^{6} - 14q^{7} - 12q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 4q^{2} + 6q^{3} + 2q^{4} - 10q^{5} - 12q^{6} - 14q^{7} - 12q^{8} + 18q^{9} + 20q^{10} - 92q^{11} + 6q^{12} + 8q^{13} + 28q^{14} - 30q^{15} + 18q^{16} - 44q^{17} - 36q^{18} - 108q^{19} - 10q^{20} - 42q^{21} + 164q^{22} - 320q^{23} - 36q^{24} + 50q^{25} - 396q^{26} + 54q^{27} - 14q^{28} - 236q^{29} + 60q^{30} - 60q^{31} + 300q^{32} - 276q^{33} + 528q^{34} + 70q^{35} + 18q^{36} + 204q^{37} + 476q^{38} + 24q^{39} + 60q^{40} + 44q^{41} + 84q^{42} + 136q^{43} - 12q^{44} - 90q^{45} + 440q^{46} + 400q^{47} + 54q^{48} + 98q^{49} - 100q^{50} - 132q^{51} + 1528q^{52} + 16q^{53} - 108q^{54} + 460q^{55} + 84q^{56} - 324q^{57} + 592q^{58} - 464q^{59} - 30q^{60} - 684q^{61} + 1140q^{62} - 126q^{63} - 1214q^{64} - 40q^{65} + 492q^{66} + 736q^{67} - 1804q^{68} - 960q^{69} - 140q^{70} - 740q^{71} - 108q^{72} + 424q^{73} - 168q^{74} + 150q^{75} - 1148q^{76} + 644q^{77} - 1188q^{78} - 408q^{79} - 90q^{80} + 162q^{81} - 888q^{82} + 608q^{83} - 42q^{84} + 220q^{85} + 1008q^{86} - 708q^{87} + 532q^{88} - 1332q^{89} + 180q^{90} - 56q^{91} + 480q^{92} - 180q^{93} - 2480q^{94} + 540q^{95} + 900q^{96} - 2448q^{97} - 196q^{98} - 828q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.236068 0.0834626 0.0417313 0.999129i \(-0.486713\pi\)
0.0417313 + 0.999129i \(0.486713\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.94427 −0.993034
\(5\) −5.00000 −0.447214
\(6\) 0.708204 0.0481872
\(7\) −7.00000 −0.377964
\(8\) −3.76393 −0.166344
\(9\) 9.00000 0.333333
\(10\) −1.18034 −0.0373256
\(11\) −50.4721 −1.38345 −0.691724 0.722162i \(-0.743148\pi\)
−0.691724 + 0.722162i \(0.743148\pi\)
\(12\) −23.8328 −0.573328
\(13\) −80.9706 −1.72748 −0.863738 0.503940i \(-0.831883\pi\)
−0.863738 + 0.503940i \(0.831883\pi\)
\(14\) −1.65248 −0.0315459
\(15\) −15.0000 −0.258199
\(16\) 62.6656 0.979150
\(17\) 76.3870 1.08980 0.544899 0.838502i \(-0.316568\pi\)
0.544899 + 0.838502i \(0.316568\pi\)
\(18\) 2.12461 0.0278209
\(19\) 4.13777 0.0499615 0.0249808 0.999688i \(-0.492048\pi\)
0.0249808 + 0.999688i \(0.492048\pi\)
\(20\) 39.7214 0.444098
\(21\) −21.0000 −0.218218
\(22\) −11.9149 −0.115466
\(23\) −204.721 −1.85597 −0.927986 0.372615i \(-0.878461\pi\)
−0.927986 + 0.372615i \(0.878461\pi\)
\(24\) −11.2918 −0.0960387
\(25\) 25.0000 0.200000
\(26\) −19.1146 −0.144180
\(27\) 27.0000 0.192450
\(28\) 55.6099 0.375332
\(29\) −91.1672 −0.583770 −0.291885 0.956453i \(-0.594282\pi\)
−0.291885 + 0.956453i \(0.594282\pi\)
\(30\) −3.54102 −0.0215500
\(31\) 198.079 1.14761 0.573807 0.818991i \(-0.305466\pi\)
0.573807 + 0.818991i \(0.305466\pi\)
\(32\) 44.9048 0.248066
\(33\) −151.416 −0.798734
\(34\) 18.0325 0.0909574
\(35\) 35.0000 0.169031
\(36\) −71.4984 −0.331011
\(37\) 155.666 0.691656 0.345828 0.938298i \(-0.387598\pi\)
0.345828 + 0.938298i \(0.387598\pi\)
\(38\) 0.976794 0.00416992
\(39\) −242.912 −0.997359
\(40\) 18.8197 0.0743912
\(41\) −156.885 −0.597595 −0.298797 0.954317i \(-0.596585\pi\)
−0.298797 + 0.954317i \(0.596585\pi\)
\(42\) −4.95743 −0.0182130
\(43\) 354.217 1.25622 0.628111 0.778124i \(-0.283828\pi\)
0.628111 + 0.778124i \(0.283828\pi\)
\(44\) 400.964 1.37381
\(45\) −45.0000 −0.149071
\(46\) −48.3282 −0.154904
\(47\) −175.659 −0.545161 −0.272580 0.962133i \(-0.587877\pi\)
−0.272580 + 0.962133i \(0.587877\pi\)
\(48\) 187.997 0.565313
\(49\) 49.0000 0.142857
\(50\) 5.90170 0.0166925
\(51\) 229.161 0.629195
\(52\) 643.252 1.71544
\(53\) 200.302 0.519124 0.259562 0.965726i \(-0.416422\pi\)
0.259562 + 0.965726i \(0.416422\pi\)
\(54\) 6.37384 0.0160624
\(55\) 252.361 0.618696
\(56\) 26.3475 0.0628721
\(57\) 12.4133 0.0288453
\(58\) −21.5217 −0.0487230
\(59\) −312.498 −0.689556 −0.344778 0.938684i \(-0.612046\pi\)
−0.344778 + 0.938684i \(0.612046\pi\)
\(60\) 119.164 0.256400
\(61\) −154.170 −0.323598 −0.161799 0.986824i \(-0.551730\pi\)
−0.161799 + 0.986824i \(0.551730\pi\)
\(62\) 46.7601 0.0957829
\(63\) −63.0000 −0.125988
\(64\) −490.724 −0.958446
\(65\) 404.853 0.772551
\(66\) −35.7446 −0.0666644
\(67\) 734.715 1.33970 0.669849 0.742498i \(-0.266359\pi\)
0.669849 + 0.742498i \(0.266359\pi\)
\(68\) −606.839 −1.08221
\(69\) −614.164 −1.07155
\(70\) 8.26238 0.0141078
\(71\) −678.577 −1.13426 −0.567129 0.823629i \(-0.691946\pi\)
−0.567129 + 0.823629i \(0.691946\pi\)
\(72\) −33.8754 −0.0554480
\(73\) −60.8003 −0.0974813 −0.0487407 0.998811i \(-0.515521\pi\)
−0.0487407 + 0.998811i \(0.515521\pi\)
\(74\) 36.7477 0.0577274
\(75\) 75.0000 0.115470
\(76\) −32.8715 −0.0496135
\(77\) 353.305 0.522894
\(78\) −57.3437 −0.0832422
\(79\) −1286.26 −1.83184 −0.915919 0.401363i \(-0.868537\pi\)
−0.915919 + 0.401363i \(0.868537\pi\)
\(80\) −313.328 −0.437889
\(81\) 81.0000 0.111111
\(82\) −37.0356 −0.0498768
\(83\) 116.170 0.153631 0.0768153 0.997045i \(-0.475525\pi\)
0.0768153 + 0.997045i \(0.475525\pi\)
\(84\) 166.830 0.216698
\(85\) −381.935 −0.487372
\(86\) 83.6192 0.104848
\(87\) −273.502 −0.337040
\(88\) 189.974 0.230128
\(89\) −916.440 −1.09149 −0.545744 0.837952i \(-0.683753\pi\)
−0.545744 + 0.837952i \(0.683753\pi\)
\(90\) −10.6231 −0.0124419
\(91\) 566.794 0.652925
\(92\) 1626.36 1.84304
\(93\) 594.237 0.662575
\(94\) −41.4676 −0.0455006
\(95\) −20.6888 −0.0223435
\(96\) 134.714 0.143221
\(97\) −1416.30 −1.48251 −0.741256 0.671222i \(-0.765770\pi\)
−0.741256 + 0.671222i \(0.765770\pi\)
\(98\) 11.5673 0.0119232
\(99\) −454.249 −0.461149
\(100\) −198.607 −0.198607
\(101\) −1379.19 −1.35876 −0.679381 0.733785i \(-0.737752\pi\)
−0.679381 + 0.733785i \(0.737752\pi\)
\(102\) 54.0976 0.0525143
\(103\) −1308.43 −1.25169 −0.625844 0.779949i \(-0.715245\pi\)
−0.625844 + 0.779949i \(0.715245\pi\)
\(104\) 304.768 0.287355
\(105\) 105.000 0.0975900
\(106\) 47.2849 0.0433275
\(107\) 1265.65 1.14351 0.571754 0.820425i \(-0.306263\pi\)
0.571754 + 0.820425i \(0.306263\pi\)
\(108\) −214.495 −0.191109
\(109\) 2069.32 1.81839 0.909196 0.416368i \(-0.136697\pi\)
0.909196 + 0.416368i \(0.136697\pi\)
\(110\) 59.5743 0.0516380
\(111\) 466.997 0.399328
\(112\) −438.659 −0.370084
\(113\) −1953.89 −1.62661 −0.813303 0.581840i \(-0.802333\pi\)
−0.813303 + 0.581840i \(0.802333\pi\)
\(114\) 2.93038 0.00240750
\(115\) 1023.61 0.830016
\(116\) 724.257 0.579703
\(117\) −728.735 −0.575826
\(118\) −73.7709 −0.0575522
\(119\) −534.709 −0.411905
\(120\) 56.4590 0.0429498
\(121\) 1216.44 0.913927
\(122\) −36.3947 −0.0270083
\(123\) −470.656 −0.345022
\(124\) −1573.59 −1.13962
\(125\) −125.000 −0.0894427
\(126\) −14.8723 −0.0105153
\(127\) 224.251 0.156685 0.0783426 0.996926i \(-0.475037\pi\)
0.0783426 + 0.996926i \(0.475037\pi\)
\(128\) −475.083 −0.328061
\(129\) 1062.65 0.725280
\(130\) 95.5728 0.0644792
\(131\) −490.898 −0.327404 −0.163702 0.986510i \(-0.552344\pi\)
−0.163702 + 0.986510i \(0.552344\pi\)
\(132\) 1202.89 0.793170
\(133\) −28.9644 −0.0188837
\(134\) 173.443 0.111815
\(135\) −135.000 −0.0860663
\(136\) −287.515 −0.181281
\(137\) 1831.12 1.14192 0.570961 0.820977i \(-0.306571\pi\)
0.570961 + 0.820977i \(0.306571\pi\)
\(138\) −144.984 −0.0894340
\(139\) −3050.84 −1.86165 −0.930823 0.365469i \(-0.880909\pi\)
−0.930823 + 0.365469i \(0.880909\pi\)
\(140\) −278.050 −0.167853
\(141\) −526.978 −0.314749
\(142\) −160.190 −0.0946682
\(143\) 4086.76 2.38987
\(144\) 563.991 0.326383
\(145\) 455.836 0.261070
\(146\) −14.3530 −0.00813605
\(147\) 147.000 0.0824786
\(148\) −1236.65 −0.686838
\(149\) 2246.55 1.23520 0.617599 0.786493i \(-0.288105\pi\)
0.617599 + 0.786493i \(0.288105\pi\)
\(150\) 17.7051 0.00963743
\(151\) −1311.53 −0.706826 −0.353413 0.935467i \(-0.614979\pi\)
−0.353413 + 0.935467i \(0.614979\pi\)
\(152\) −15.5743 −0.00831079
\(153\) 687.483 0.363266
\(154\) 83.4040 0.0436421
\(155\) −990.395 −0.513228
\(156\) 1929.76 0.990412
\(157\) 1790.94 0.910398 0.455199 0.890390i \(-0.349568\pi\)
0.455199 + 0.890390i \(0.349568\pi\)
\(158\) −303.644 −0.152890
\(159\) 600.906 0.299716
\(160\) −224.524 −0.110939
\(161\) 1433.05 0.701491
\(162\) 19.1215 0.00927363
\(163\) −491.108 −0.235991 −0.117996 0.993014i \(-0.537647\pi\)
−0.117996 + 0.993014i \(0.537647\pi\)
\(164\) 1246.34 0.593432
\(165\) 757.082 0.357205
\(166\) 27.4241 0.0128224
\(167\) −826.059 −0.382769 −0.191384 0.981515i \(-0.561298\pi\)
−0.191384 + 0.981515i \(0.561298\pi\)
\(168\) 79.0426 0.0362992
\(169\) 4359.24 1.98418
\(170\) −90.1626 −0.0406774
\(171\) 37.2399 0.0166538
\(172\) −2813.99 −1.24747
\(173\) 2918.00 1.28238 0.641190 0.767382i \(-0.278441\pi\)
0.641190 + 0.767382i \(0.278441\pi\)
\(174\) −64.5650 −0.0281302
\(175\) −175.000 −0.0755929
\(176\) −3162.87 −1.35460
\(177\) −937.495 −0.398116
\(178\) −216.342 −0.0910984
\(179\) 955.745 0.399082 0.199541 0.979889i \(-0.436055\pi\)
0.199541 + 0.979889i \(0.436055\pi\)
\(180\) 357.492 0.148033
\(181\) 206.080 0.0846289 0.0423145 0.999104i \(-0.486527\pi\)
0.0423145 + 0.999104i \(0.486527\pi\)
\(182\) 133.802 0.0544948
\(183\) −462.511 −0.186829
\(184\) 770.557 0.308730
\(185\) −778.328 −0.309318
\(186\) 140.280 0.0553003
\(187\) −3855.41 −1.50768
\(188\) 1395.49 0.541363
\(189\) −189.000 −0.0727393
\(190\) −4.88397 −0.00186485
\(191\) −2018.63 −0.764727 −0.382364 0.924012i \(-0.624890\pi\)
−0.382364 + 0.924012i \(0.624890\pi\)
\(192\) −1472.17 −0.553359
\(193\) 1031.69 0.384781 0.192390 0.981318i \(-0.438376\pi\)
0.192390 + 0.981318i \(0.438376\pi\)
\(194\) −334.344 −0.123734
\(195\) 1214.56 0.446033
\(196\) −389.269 −0.141862
\(197\) −205.955 −0.0744857 −0.0372429 0.999306i \(-0.511858\pi\)
−0.0372429 + 0.999306i \(0.511858\pi\)
\(198\) −107.234 −0.0384887
\(199\) −1831.38 −0.652376 −0.326188 0.945305i \(-0.605764\pi\)
−0.326188 + 0.945305i \(0.605764\pi\)
\(200\) −94.0983 −0.0332688
\(201\) 2204.15 0.773475
\(202\) −325.584 −0.113406
\(203\) 638.170 0.220644
\(204\) −1820.52 −0.624812
\(205\) 784.427 0.267253
\(206\) −308.879 −0.104469
\(207\) −1842.49 −0.618657
\(208\) −5074.07 −1.69146
\(209\) −208.842 −0.0691191
\(210\) 24.7871 0.00814512
\(211\) 1030.19 0.336119 0.168060 0.985777i \(-0.446250\pi\)
0.168060 + 0.985777i \(0.446250\pi\)
\(212\) −1591.25 −0.515508
\(213\) −2035.73 −0.654864
\(214\) 298.780 0.0954402
\(215\) −1771.08 −0.561800
\(216\) −101.626 −0.0320129
\(217\) −1386.55 −0.433757
\(218\) 488.500 0.151768
\(219\) −182.401 −0.0562809
\(220\) −2004.82 −0.614387
\(221\) −6185.10 −1.88260
\(222\) 110.243 0.0333289
\(223\) 5368.67 1.61217 0.806083 0.591803i \(-0.201584\pi\)
0.806083 + 0.591803i \(0.201584\pi\)
\(224\) −314.334 −0.0937603
\(225\) 225.000 0.0666667
\(226\) −461.251 −0.135761
\(227\) −932.121 −0.272542 −0.136271 0.990672i \(-0.543512\pi\)
−0.136271 + 0.990672i \(0.543512\pi\)
\(228\) −98.6146 −0.0286444
\(229\) 3163.05 0.912752 0.456376 0.889787i \(-0.349147\pi\)
0.456376 + 0.889787i \(0.349147\pi\)
\(230\) 241.641 0.0692753
\(231\) 1059.91 0.301893
\(232\) 343.147 0.0971065
\(233\) 436.562 0.122747 0.0613737 0.998115i \(-0.480452\pi\)
0.0613737 + 0.998115i \(0.480452\pi\)
\(234\) −172.031 −0.0480599
\(235\) 878.297 0.243803
\(236\) 2482.57 0.684753
\(237\) −3858.77 −1.05761
\(238\) −126.228 −0.0343787
\(239\) −1980.82 −0.536103 −0.268051 0.963405i \(-0.586380\pi\)
−0.268051 + 0.963405i \(0.586380\pi\)
\(240\) −939.984 −0.252816
\(241\) 5303.55 1.41756 0.708780 0.705430i \(-0.249246\pi\)
0.708780 + 0.705430i \(0.249246\pi\)
\(242\) 287.162 0.0762787
\(243\) 243.000 0.0641500
\(244\) 1224.77 0.321344
\(245\) −245.000 −0.0638877
\(246\) −111.107 −0.0287964
\(247\) −335.037 −0.0863074
\(248\) −745.556 −0.190899
\(249\) 348.511 0.0886987
\(250\) −29.5085 −0.00746512
\(251\) −2996.04 −0.753420 −0.376710 0.926331i \(-0.622945\pi\)
−0.376710 + 0.926331i \(0.622945\pi\)
\(252\) 500.489 0.125111
\(253\) 10332.7 2.56764
\(254\) 52.9384 0.0130774
\(255\) −1145.80 −0.281385
\(256\) 3813.64 0.931065
\(257\) 968.861 0.235159 0.117580 0.993063i \(-0.462487\pi\)
0.117580 + 0.993063i \(0.462487\pi\)
\(258\) 250.858 0.0605338
\(259\) −1089.66 −0.261421
\(260\) −3216.26 −0.767170
\(261\) −820.505 −0.194590
\(262\) −115.885 −0.0273260
\(263\) −4830.18 −1.13248 −0.566239 0.824241i \(-0.691602\pi\)
−0.566239 + 0.824241i \(0.691602\pi\)
\(264\) 569.921 0.132864
\(265\) −1001.51 −0.232159
\(266\) −6.83756 −0.00157608
\(267\) −2749.32 −0.630171
\(268\) −5836.78 −1.33037
\(269\) −4774.97 −1.08229 −0.541143 0.840930i \(-0.682008\pi\)
−0.541143 + 0.840930i \(0.682008\pi\)
\(270\) −31.8692 −0.00718332
\(271\) 141.909 0.0318094 0.0159047 0.999874i \(-0.494937\pi\)
0.0159047 + 0.999874i \(0.494937\pi\)
\(272\) 4786.84 1.06708
\(273\) 1700.38 0.376966
\(274\) 432.269 0.0953078
\(275\) −1261.80 −0.276689
\(276\) 4879.09 1.06408
\(277\) −3621.13 −0.785460 −0.392730 0.919654i \(-0.628469\pi\)
−0.392730 + 0.919654i \(0.628469\pi\)
\(278\) −720.206 −0.155378
\(279\) 1782.71 0.382538
\(280\) −131.738 −0.0281172
\(281\) 5790.87 1.22937 0.614687 0.788771i \(-0.289282\pi\)
0.614687 + 0.788771i \(0.289282\pi\)
\(282\) −124.403 −0.0262698
\(283\) 526.043 0.110495 0.0552474 0.998473i \(-0.482405\pi\)
0.0552474 + 0.998473i \(0.482405\pi\)
\(284\) 5390.80 1.12636
\(285\) −62.0665 −0.0129000
\(286\) 964.753 0.199465
\(287\) 1098.20 0.225870
\(288\) 404.143 0.0826888
\(289\) 921.972 0.187660
\(290\) 107.608 0.0217896
\(291\) −4248.91 −0.855929
\(292\) 483.014 0.0968023
\(293\) 1914.88 0.381803 0.190901 0.981609i \(-0.438859\pi\)
0.190901 + 0.981609i \(0.438859\pi\)
\(294\) 34.7020 0.00688388
\(295\) 1562.49 0.308379
\(296\) −585.915 −0.115053
\(297\) −1362.75 −0.266245
\(298\) 530.339 0.103093
\(299\) 16576.4 3.20615
\(300\) −595.820 −0.114666
\(301\) −2479.52 −0.474807
\(302\) −309.610 −0.0589936
\(303\) −4137.58 −0.784482
\(304\) 259.296 0.0489199
\(305\) 770.851 0.144717
\(306\) 162.293 0.0303191
\(307\) −6244.17 −1.16083 −0.580413 0.814322i \(-0.697109\pi\)
−0.580413 + 0.814322i \(0.697109\pi\)
\(308\) −2806.75 −0.519251
\(309\) −3925.30 −0.722662
\(310\) −233.800 −0.0428354
\(311\) −9658.78 −1.76109 −0.880545 0.473962i \(-0.842823\pi\)
−0.880545 + 0.473962i \(0.842823\pi\)
\(312\) 914.303 0.165905
\(313\) 2198.34 0.396988 0.198494 0.980102i \(-0.436395\pi\)
0.198494 + 0.980102i \(0.436395\pi\)
\(314\) 422.783 0.0759842
\(315\) 315.000 0.0563436
\(316\) 10218.4 1.81908
\(317\) 3030.78 0.536990 0.268495 0.963281i \(-0.413474\pi\)
0.268495 + 0.963281i \(0.413474\pi\)
\(318\) 141.855 0.0250151
\(319\) 4601.40 0.807615
\(320\) 2453.62 0.428630
\(321\) 3796.96 0.660204
\(322\) 338.297 0.0585483
\(323\) 316.072 0.0544480
\(324\) −643.486 −0.110337
\(325\) −2024.26 −0.345495
\(326\) −115.935 −0.0196965
\(327\) 6207.96 1.04985
\(328\) 590.506 0.0994062
\(329\) 1229.62 0.206051
\(330\) 178.723 0.0298132
\(331\) 4753.74 0.789393 0.394696 0.918812i \(-0.370850\pi\)
0.394696 + 0.918812i \(0.370850\pi\)
\(332\) −922.888 −0.152560
\(333\) 1400.99 0.230552
\(334\) −195.006 −0.0319469
\(335\) −3673.58 −0.599131
\(336\) −1315.98 −0.213668
\(337\) 8824.40 1.42640 0.713199 0.700962i \(-0.247246\pi\)
0.713199 + 0.700962i \(0.247246\pi\)
\(338\) 1029.08 0.165605
\(339\) −5861.67 −0.939122
\(340\) 3034.20 0.483977
\(341\) −9997.47 −1.58766
\(342\) 8.79115 0.00138997
\(343\) −343.000 −0.0539949
\(344\) −1333.25 −0.208965
\(345\) 3070.82 0.479210
\(346\) 688.847 0.107031
\(347\) −3413.97 −0.528160 −0.264080 0.964501i \(-0.585068\pi\)
−0.264080 + 0.964501i \(0.585068\pi\)
\(348\) 2172.77 0.334692
\(349\) −5676.32 −0.870621 −0.435310 0.900280i \(-0.643361\pi\)
−0.435310 + 0.900280i \(0.643361\pi\)
\(350\) −41.3119 −0.00630918
\(351\) −2186.21 −0.332453
\(352\) −2266.44 −0.343187
\(353\) 6225.80 0.938713 0.469357 0.883009i \(-0.344486\pi\)
0.469357 + 0.883009i \(0.344486\pi\)
\(354\) −221.313 −0.0332278
\(355\) 3392.89 0.507256
\(356\) 7280.45 1.08388
\(357\) −1604.13 −0.237813
\(358\) 225.621 0.0333084
\(359\) −4907.73 −0.721505 −0.360752 0.932662i \(-0.617480\pi\)
−0.360752 + 0.932662i \(0.617480\pi\)
\(360\) 169.377 0.0247971
\(361\) −6841.88 −0.997504
\(362\) 48.6490 0.00706335
\(363\) 3649.31 0.527656
\(364\) −4502.77 −0.648377
\(365\) 304.001 0.0435950
\(366\) −109.184 −0.0155933
\(367\) −3906.48 −0.555631 −0.277816 0.960634i \(-0.589610\pi\)
−0.277816 + 0.960634i \(0.589610\pi\)
\(368\) −12829.0 −1.81728
\(369\) −1411.97 −0.199198
\(370\) −183.738 −0.0258165
\(371\) −1402.11 −0.196210
\(372\) −4720.78 −0.657960
\(373\) −2102.52 −0.291862 −0.145931 0.989295i \(-0.546618\pi\)
−0.145931 + 0.989295i \(0.546618\pi\)
\(374\) −910.140 −0.125835
\(375\) −375.000 −0.0516398
\(376\) 661.170 0.0906842
\(377\) 7381.86 1.00845
\(378\) −44.6168 −0.00607101
\(379\) −6612.76 −0.896239 −0.448120 0.893974i \(-0.647906\pi\)
−0.448120 + 0.893974i \(0.647906\pi\)
\(380\) 164.358 0.0221878
\(381\) 672.752 0.0904623
\(382\) −476.534 −0.0638262
\(383\) 2.37457 0.000316801 0 0.000158401 1.00000i \(-0.499950\pi\)
0.000158401 1.00000i \(0.499950\pi\)
\(384\) −1425.25 −0.189406
\(385\) −1766.52 −0.233845
\(386\) 243.549 0.0321148
\(387\) 3187.95 0.418741
\(388\) 11251.5 1.47218
\(389\) 7716.98 1.00583 0.502913 0.864337i \(-0.332261\pi\)
0.502913 + 0.864337i \(0.332261\pi\)
\(390\) 286.718 0.0372271
\(391\) −15638.0 −2.02263
\(392\) −184.433 −0.0237634
\(393\) −1472.69 −0.189027
\(394\) −48.6194 −0.00621678
\(395\) 6431.28 0.819223
\(396\) 3608.68 0.457937
\(397\) 6403.95 0.809584 0.404792 0.914409i \(-0.367344\pi\)
0.404792 + 0.914409i \(0.367344\pi\)
\(398\) −432.329 −0.0544490
\(399\) −86.8931 −0.0109025
\(400\) 1566.64 0.195830
\(401\) −10969.9 −1.36611 −0.683054 0.730368i \(-0.739349\pi\)
−0.683054 + 0.730368i \(0.739349\pi\)
\(402\) 520.328 0.0645562
\(403\) −16038.6 −1.98248
\(404\) 10956.7 1.34930
\(405\) −405.000 −0.0496904
\(406\) 150.652 0.0184155
\(407\) −7856.78 −0.956870
\(408\) −862.546 −0.104663
\(409\) −400.353 −0.0484015 −0.0242007 0.999707i \(-0.507704\pi\)
−0.0242007 + 0.999707i \(0.507704\pi\)
\(410\) 185.178 0.0223056
\(411\) 5493.37 0.659289
\(412\) 10394.6 1.24297
\(413\) 2187.49 0.260628
\(414\) −434.953 −0.0516348
\(415\) −580.851 −0.0687057
\(416\) −3635.97 −0.428529
\(417\) −9152.52 −1.07482
\(418\) −49.3009 −0.00576887
\(419\) −15815.4 −1.84399 −0.921995 0.387201i \(-0.873442\pi\)
−0.921995 + 0.387201i \(0.873442\pi\)
\(420\) −834.149 −0.0969102
\(421\) 1936.53 0.224182 0.112091 0.993698i \(-0.464245\pi\)
0.112091 + 0.993698i \(0.464245\pi\)
\(422\) 243.195 0.0280534
\(423\) −1580.93 −0.181720
\(424\) −753.923 −0.0863531
\(425\) 1909.67 0.217960
\(426\) −480.571 −0.0546567
\(427\) 1079.19 0.122309
\(428\) −10054.7 −1.13554
\(429\) 12260.3 1.37979
\(430\) −418.096 −0.0468893
\(431\) 2030.91 0.226973 0.113487 0.993540i \(-0.463798\pi\)
0.113487 + 0.993540i \(0.463798\pi\)
\(432\) 1691.97 0.188438
\(433\) −10784.1 −1.19689 −0.598443 0.801165i \(-0.704214\pi\)
−0.598443 + 0.801165i \(0.704214\pi\)
\(434\) −327.321 −0.0362025
\(435\) 1367.51 0.150729
\(436\) −16439.2 −1.80573
\(437\) −847.089 −0.0927272
\(438\) −43.0590 −0.00469735
\(439\) −6304.19 −0.685382 −0.342691 0.939448i \(-0.611338\pi\)
−0.342691 + 0.939448i \(0.611338\pi\)
\(440\) −949.868 −0.102916
\(441\) 441.000 0.0476190
\(442\) −1460.10 −0.157127
\(443\) 15494.8 1.66181 0.830905 0.556414i \(-0.187823\pi\)
0.830905 + 0.556414i \(0.187823\pi\)
\(444\) −3709.95 −0.396546
\(445\) 4582.20 0.488128
\(446\) 1267.37 0.134556
\(447\) 6739.65 0.713142
\(448\) 3435.07 0.362259
\(449\) −242.018 −0.0254377 −0.0127189 0.999919i \(-0.504049\pi\)
−0.0127189 + 0.999919i \(0.504049\pi\)
\(450\) 53.1153 0.00556418
\(451\) 7918.34 0.826741
\(452\) 15522.2 1.61528
\(453\) −3934.59 −0.408086
\(454\) −220.044 −0.0227471
\(455\) −2833.97 −0.291997
\(456\) −46.7228 −0.00479824
\(457\) −11670.4 −1.19457 −0.597283 0.802030i \(-0.703753\pi\)
−0.597283 + 0.802030i \(0.703753\pi\)
\(458\) 746.695 0.0761807
\(459\) 2062.45 0.209732
\(460\) −8131.81 −0.824234
\(461\) 12128.1 1.22530 0.612649 0.790355i \(-0.290104\pi\)
0.612649 + 0.790355i \(0.290104\pi\)
\(462\) 250.212 0.0251968
\(463\) 5161.64 0.518103 0.259051 0.965864i \(-0.416590\pi\)
0.259051 + 0.965864i \(0.416590\pi\)
\(464\) −5713.05 −0.571598
\(465\) −2971.18 −0.296313
\(466\) 103.058 0.0102448
\(467\) 11680.2 1.15738 0.578691 0.815547i \(-0.303564\pi\)
0.578691 + 0.815547i \(0.303564\pi\)
\(468\) 5789.27 0.571814
\(469\) −5143.01 −0.506358
\(470\) 207.338 0.0203485
\(471\) 5372.82 0.525619
\(472\) 1176.22 0.114703
\(473\) −17878.1 −1.73792
\(474\) −910.932 −0.0882711
\(475\) 103.444 0.00999230
\(476\) 4247.87 0.409036
\(477\) 1802.72 0.173041
\(478\) −467.608 −0.0447445
\(479\) −18458.7 −1.76075 −0.880373 0.474282i \(-0.842708\pi\)
−0.880373 + 0.474282i \(0.842708\pi\)
\(480\) −673.572 −0.0640505
\(481\) −12604.3 −1.19482
\(482\) 1252.00 0.118313
\(483\) 4299.15 0.405006
\(484\) −9663.70 −0.907560
\(485\) 7081.51 0.663000
\(486\) 57.3645 0.00535413
\(487\) 8630.11 0.803014 0.401507 0.915856i \(-0.368487\pi\)
0.401507 + 0.915856i \(0.368487\pi\)
\(488\) 580.286 0.0538286
\(489\) −1473.33 −0.136250
\(490\) −57.8367 −0.00533223
\(491\) 17801.6 1.63621 0.818103 0.575072i \(-0.195026\pi\)
0.818103 + 0.575072i \(0.195026\pi\)
\(492\) 3739.02 0.342618
\(493\) −6963.99 −0.636191
\(494\) −79.0916 −0.00720344
\(495\) 2271.25 0.206232
\(496\) 12412.7 1.12369
\(497\) 4750.04 0.428709
\(498\) 82.2723 0.00740303
\(499\) 200.167 0.0179574 0.00897868 0.999960i \(-0.497142\pi\)
0.00897868 + 0.999960i \(0.497142\pi\)
\(500\) 993.034 0.0888197
\(501\) −2478.18 −0.220992
\(502\) −707.269 −0.0628824
\(503\) 16400.4 1.45379 0.726896 0.686747i \(-0.240962\pi\)
0.726896 + 0.686747i \(0.240962\pi\)
\(504\) 237.128 0.0209574
\(505\) 6895.97 0.607657
\(506\) 2439.23 0.214302
\(507\) 13077.7 1.14556
\(508\) −1781.51 −0.155594
\(509\) −15006.6 −1.30679 −0.653394 0.757018i \(-0.726656\pi\)
−0.653394 + 0.757018i \(0.726656\pi\)
\(510\) −270.488 −0.0234851
\(511\) 425.602 0.0368445
\(512\) 4700.94 0.405770
\(513\) 111.720 0.00961510
\(514\) 228.717 0.0196270
\(515\) 6542.17 0.559772
\(516\) −8441.98 −0.720228
\(517\) 8865.91 0.754201
\(518\) −257.234 −0.0218189
\(519\) 8754.01 0.740382
\(520\) −1523.84 −0.128509
\(521\) 7113.18 0.598146 0.299073 0.954230i \(-0.403323\pi\)
0.299073 + 0.954230i \(0.403323\pi\)
\(522\) −193.695 −0.0162410
\(523\) −8888.46 −0.743146 −0.371573 0.928404i \(-0.621181\pi\)
−0.371573 + 0.928404i \(0.621181\pi\)
\(524\) 3899.83 0.325123
\(525\) −525.000 −0.0436436
\(526\) −1140.25 −0.0945196
\(527\) 15130.7 1.25067
\(528\) −9488.60 −0.782081
\(529\) 29743.8 2.44463
\(530\) −236.424 −0.0193766
\(531\) −2812.49 −0.229852
\(532\) 230.101 0.0187521
\(533\) 12703.1 1.03233
\(534\) −649.026 −0.0525957
\(535\) −6328.27 −0.511392
\(536\) −2765.42 −0.222850
\(537\) 2867.23 0.230410
\(538\) −1127.22 −0.0903305
\(539\) −2473.13 −0.197635
\(540\) 1072.48 0.0854668
\(541\) −653.827 −0.0519597 −0.0259799 0.999662i \(-0.508271\pi\)
−0.0259799 + 0.999662i \(0.508271\pi\)
\(542\) 33.5001 0.00265489
\(543\) 618.241 0.0488605
\(544\) 3430.14 0.270342
\(545\) −10346.6 −0.813210
\(546\) 401.406 0.0314626
\(547\) 1138.52 0.0889940 0.0444970 0.999010i \(-0.485831\pi\)
0.0444970 + 0.999010i \(0.485831\pi\)
\(548\) −14546.9 −1.13397
\(549\) −1387.53 −0.107866
\(550\) −297.871 −0.0230932
\(551\) −377.229 −0.0291660
\(552\) 2311.67 0.178245
\(553\) 9003.80 0.692370
\(554\) −854.832 −0.0655566
\(555\) −2334.98 −0.178585
\(556\) 24236.7 1.84868
\(557\) −19804.8 −1.50657 −0.753283 0.657696i \(-0.771531\pi\)
−0.753283 + 0.657696i \(0.771531\pi\)
\(558\) 420.841 0.0319276
\(559\) −28681.1 −2.17009
\(560\) 2193.30 0.165507
\(561\) −11566.2 −0.870458
\(562\) 1367.04 0.102607
\(563\) 10276.3 0.769265 0.384632 0.923070i \(-0.374328\pi\)
0.384632 + 0.923070i \(0.374328\pi\)
\(564\) 4186.46 0.312556
\(565\) 9769.45 0.727440
\(566\) 124.182 0.00922219
\(567\) −567.000 −0.0419961
\(568\) 2554.12 0.188677
\(569\) 4139.03 0.304951 0.152475 0.988307i \(-0.451276\pi\)
0.152475 + 0.988307i \(0.451276\pi\)
\(570\) −14.6519 −0.00107667
\(571\) −4486.81 −0.328839 −0.164420 0.986390i \(-0.552575\pi\)
−0.164420 + 0.986390i \(0.552575\pi\)
\(572\) −32466.3 −2.37323
\(573\) −6055.89 −0.441516
\(574\) 259.249 0.0188517
\(575\) −5118.03 −0.371194
\(576\) −4416.52 −0.319482
\(577\) −1104.77 −0.0797093 −0.0398547 0.999205i \(-0.512689\pi\)
−0.0398547 + 0.999205i \(0.512689\pi\)
\(578\) 217.648 0.0156626
\(579\) 3095.07 0.222153
\(580\) −3621.28 −0.259251
\(581\) −813.192 −0.0580669
\(582\) −1003.03 −0.0714381
\(583\) −10109.7 −0.718181
\(584\) 228.848 0.0162154
\(585\) 3643.68 0.257517
\(586\) 452.041 0.0318663
\(587\) −10413.2 −0.732199 −0.366100 0.930576i \(-0.619307\pi\)
−0.366100 + 0.930576i \(0.619307\pi\)
\(588\) −1167.81 −0.0819041
\(589\) 819.605 0.0573365
\(590\) 368.854 0.0257381
\(591\) −617.865 −0.0430044
\(592\) 9754.89 0.677235
\(593\) −3235.16 −0.224034 −0.112017 0.993706i \(-0.535731\pi\)
−0.112017 + 0.993706i \(0.535731\pi\)
\(594\) −321.701 −0.0222215
\(595\) 2673.54 0.184209
\(596\) −17847.2 −1.22659
\(597\) −5494.13 −0.376649
\(598\) 3913.16 0.267594
\(599\) 569.048 0.0388158 0.0194079 0.999812i \(-0.493822\pi\)
0.0194079 + 0.999812i \(0.493822\pi\)
\(600\) −282.295 −0.0192077
\(601\) 3760.89 0.255258 0.127629 0.991822i \(-0.459263\pi\)
0.127629 + 0.991822i \(0.459263\pi\)
\(602\) −585.335 −0.0396287
\(603\) 6612.44 0.446566
\(604\) 10419.1 0.701902
\(605\) −6082.18 −0.408720
\(606\) −976.751 −0.0654749
\(607\) −2224.05 −0.148717 −0.0743585 0.997232i \(-0.523691\pi\)
−0.0743585 + 0.997232i \(0.523691\pi\)
\(608\) 185.806 0.0123938
\(609\) 1914.51 0.127389
\(610\) 181.973 0.0120785
\(611\) 14223.2 0.941753
\(612\) −5461.55 −0.360735
\(613\) 5914.50 0.389697 0.194849 0.980833i \(-0.437578\pi\)
0.194849 + 0.980833i \(0.437578\pi\)
\(614\) −1474.05 −0.0968856
\(615\) 2353.28 0.154298
\(616\) −1329.82 −0.0869802
\(617\) −18591.2 −1.21306 −0.606528 0.795062i \(-0.707438\pi\)
−0.606528 + 0.795062i \(0.707438\pi\)
\(618\) −926.638 −0.0603153
\(619\) 5125.97 0.332844 0.166422 0.986055i \(-0.446779\pi\)
0.166422 + 0.986055i \(0.446779\pi\)
\(620\) 7867.96 0.509653
\(621\) −5527.48 −0.357182
\(622\) −2280.13 −0.146985
\(623\) 6415.08 0.412544
\(624\) −15222.2 −0.976565
\(625\) 625.000 0.0400000
\(626\) 518.957 0.0331337
\(627\) −626.526 −0.0399060
\(628\) −14227.7 −0.904056
\(629\) 11890.8 0.753765
\(630\) 74.3614 0.00470259
\(631\) −10649.0 −0.671839 −0.335919 0.941891i \(-0.609047\pi\)
−0.335919 + 0.941891i \(0.609047\pi\)
\(632\) 4841.38 0.304715
\(633\) 3090.57 0.194058
\(634\) 715.471 0.0448186
\(635\) −1121.25 −0.0700718
\(636\) −4773.76 −0.297629
\(637\) −3967.56 −0.246782
\(638\) 1086.24 0.0674056
\(639\) −6107.20 −0.378086
\(640\) 2375.41 0.146713
\(641\) 24025.7 1.48044 0.740218 0.672367i \(-0.234722\pi\)
0.740218 + 0.672367i \(0.234722\pi\)
\(642\) 896.341 0.0551024
\(643\) 14929.3 0.915638 0.457819 0.889045i \(-0.348631\pi\)
0.457819 + 0.889045i \(0.348631\pi\)
\(644\) −11384.5 −0.696605
\(645\) −5313.25 −0.324355
\(646\) 74.6144 0.00454437
\(647\) 14479.1 0.879801 0.439901 0.898046i \(-0.355014\pi\)
0.439901 + 0.898046i \(0.355014\pi\)
\(648\) −304.878 −0.0184827
\(649\) 15772.5 0.953965
\(650\) −477.864 −0.0288360
\(651\) −4159.66 −0.250430
\(652\) 3901.50 0.234347
\(653\) 898.168 0.0538255 0.0269127 0.999638i \(-0.491432\pi\)
0.0269127 + 0.999638i \(0.491432\pi\)
\(654\) 1465.50 0.0876232
\(655\) 2454.49 0.146420
\(656\) −9831.33 −0.585135
\(657\) −547.203 −0.0324938
\(658\) 290.273 0.0171976
\(659\) −30198.0 −1.78505 −0.892526 0.450997i \(-0.851069\pi\)
−0.892526 + 0.450997i \(0.851069\pi\)
\(660\) −6014.47 −0.354716
\(661\) −19337.8 −1.13790 −0.568952 0.822371i \(-0.692651\pi\)
−0.568952 + 0.822371i \(0.692651\pi\)
\(662\) 1122.20 0.0658848
\(663\) −18555.3 −1.08692
\(664\) −437.257 −0.0255555
\(665\) 144.822 0.00844504
\(666\) 330.729 0.0192425
\(667\) 18663.9 1.08346
\(668\) 6562.44 0.380102
\(669\) 16106.0 0.930784
\(670\) −867.214 −0.0500051
\(671\) 7781.30 0.447681
\(672\) −943.001 −0.0541325
\(673\) 10132.2 0.580336 0.290168 0.956976i \(-0.406289\pi\)
0.290168 + 0.956976i \(0.406289\pi\)
\(674\) 2083.16 0.119051
\(675\) 675.000 0.0384900
\(676\) −34631.0 −1.97035
\(677\) −33177.3 −1.88347 −0.941733 0.336361i \(-0.890804\pi\)
−0.941733 + 0.336361i \(0.890804\pi\)
\(678\) −1383.75 −0.0783816
\(679\) 9914.11 0.560337
\(680\) 1437.58 0.0810714
\(681\) −2796.36 −0.157352
\(682\) −2360.08 −0.132511
\(683\) −11423.6 −0.639987 −0.319994 0.947420i \(-0.603681\pi\)
−0.319994 + 0.947420i \(0.603681\pi\)
\(684\) −295.844 −0.0165378
\(685\) −9155.61 −0.510683
\(686\) −80.9713 −0.00450656
\(687\) 9489.15 0.526978
\(688\) 22197.2 1.23003
\(689\) −16218.6 −0.896775
\(690\) 724.922 0.0399961
\(691\) −19737.4 −1.08661 −0.543304 0.839536i \(-0.682827\pi\)
−0.543304 + 0.839536i \(0.682827\pi\)
\(692\) −23181.4 −1.27345
\(693\) 3179.74 0.174298
\(694\) −805.929 −0.0440816
\(695\) 15254.2 0.832554
\(696\) 1029.44 0.0560645
\(697\) −11984.0 −0.651258
\(698\) −1340.00 −0.0726643
\(699\) 1309.69 0.0708682
\(700\) 1390.25 0.0750663
\(701\) 11307.8 0.609258 0.304629 0.952471i \(-0.401468\pi\)
0.304629 + 0.952471i \(0.401468\pi\)
\(702\) −516.093 −0.0277474
\(703\) 644.108 0.0345562
\(704\) 24767.9 1.32596
\(705\) 2634.89 0.140760
\(706\) 1469.71 0.0783475
\(707\) 9654.36 0.513564
\(708\) 7447.72 0.395342
\(709\) 30859.2 1.63461 0.817307 0.576202i \(-0.195466\pi\)
0.817307 + 0.576202i \(0.195466\pi\)
\(710\) 800.952 0.0423369
\(711\) −11576.3 −0.610613
\(712\) 3449.42 0.181562
\(713\) −40551.0 −2.12994
\(714\) −378.683 −0.0198485
\(715\) −20433.8 −1.06878
\(716\) −7592.69 −0.396302
\(717\) −5942.46 −0.309519
\(718\) −1158.56 −0.0602187
\(719\) 33152.4 1.71958 0.859789 0.510650i \(-0.170595\pi\)
0.859789 + 0.510650i \(0.170595\pi\)
\(720\) −2819.95 −0.145963
\(721\) 9159.03 0.473093
\(722\) −1615.15 −0.0832543
\(723\) 15910.7 0.818428
\(724\) −1637.16 −0.0840394
\(725\) −2279.18 −0.116754
\(726\) 861.485 0.0440395
\(727\) −16743.0 −0.854146 −0.427073 0.904217i \(-0.640455\pi\)
−0.427073 + 0.904217i \(0.640455\pi\)
\(728\) −2133.37 −0.108610
\(729\) 729.000 0.0370370
\(730\) 71.7650 0.00363855
\(731\) 27057.5 1.36903
\(732\) 3674.31 0.185528
\(733\) −8827.55 −0.444820 −0.222410 0.974953i \(-0.571392\pi\)
−0.222410 + 0.974953i \(0.571392\pi\)
\(734\) −922.195 −0.0463744
\(735\) −735.000 −0.0368856
\(736\) −9192.97 −0.460404
\(737\) −37082.6 −1.85340
\(738\) −333.321 −0.0166256
\(739\) 36154.0 1.79966 0.899829 0.436243i \(-0.143691\pi\)
0.899829 + 0.436243i \(0.143691\pi\)
\(740\) 6183.25 0.307163
\(741\) −1005.11 −0.0498296
\(742\) −330.994 −0.0163762
\(743\) −1820.69 −0.0898987 −0.0449494 0.998989i \(-0.514313\pi\)
−0.0449494 + 0.998989i \(0.514313\pi\)
\(744\) −2236.67 −0.110215
\(745\) −11232.8 −0.552398
\(746\) −496.338 −0.0243596
\(747\) 1045.53 0.0512102
\(748\) 30628.5 1.49718
\(749\) −8859.57 −0.432205
\(750\) −88.5255 −0.00430999
\(751\) 27764.4 1.34905 0.674526 0.738251i \(-0.264348\pi\)
0.674526 + 0.738251i \(0.264348\pi\)
\(752\) −11007.8 −0.533795
\(753\) −8988.12 −0.434987
\(754\) 1742.62 0.0841678
\(755\) 6557.65 0.316102
\(756\) 1501.47 0.0722326
\(757\) −13518.3 −0.649050 −0.324525 0.945877i \(-0.605205\pi\)
−0.324525 + 0.945877i \(0.605205\pi\)
\(758\) −1561.06 −0.0748025
\(759\) 30998.2 1.48243
\(760\) 77.8714 0.00371670
\(761\) 30695.2 1.46216 0.731078 0.682294i \(-0.239017\pi\)
0.731078 + 0.682294i \(0.239017\pi\)
\(762\) 158.815 0.00755022
\(763\) −14485.2 −0.687288
\(764\) 16036.5 0.759400
\(765\) −3437.41 −0.162457
\(766\) 0.560560 2.64410e−5 0
\(767\) 25303.2 1.19119
\(768\) 11440.9 0.537551
\(769\) 4536.39 0.212726 0.106363 0.994327i \(-0.466079\pi\)
0.106363 + 0.994327i \(0.466079\pi\)
\(770\) −417.020 −0.0195173
\(771\) 2906.58 0.135769
\(772\) −8196.03 −0.382100
\(773\) 31238.9 1.45354 0.726768 0.686883i \(-0.241022\pi\)
0.726768 + 0.686883i \(0.241022\pi\)
\(774\) 752.573 0.0349492
\(775\) 4951.97 0.229523
\(776\) 5330.86 0.246607
\(777\) −3268.98 −0.150932
\(778\) 1821.73 0.0839490
\(779\) −649.155 −0.0298567
\(780\) −9648.78 −0.442926
\(781\) 34249.2 1.56919
\(782\) −3691.64 −0.168814
\(783\) −2461.51 −0.112347
\(784\) 3070.62 0.139879
\(785\) −8954.70 −0.407143
\(786\) −347.656 −0.0157767
\(787\) 39597.2 1.79350 0.896752 0.442534i \(-0.145921\pi\)
0.896752 + 0.442534i \(0.145921\pi\)
\(788\) 1636.16 0.0739669
\(789\) −14490.5 −0.653836
\(790\) 1518.22 0.0683745
\(791\) 13677.2 0.614799
\(792\) 1709.76 0.0767093
\(793\) 12483.3 0.559008
\(794\) 1511.77 0.0675700
\(795\) −3004.53 −0.134037
\(796\) 14549.0 0.647832
\(797\) −16567.0 −0.736302 −0.368151 0.929766i \(-0.620009\pi\)
−0.368151 + 0.929766i \(0.620009\pi\)
\(798\) −20.5127 −0.000909951 0
\(799\) −13418.1 −0.594115
\(800\) 1122.62 0.0496133
\(801\) −8247.96 −0.363829
\(802\) −2589.63 −0.114019
\(803\) 3068.72 0.134860
\(804\) −17510.3 −0.768087
\(805\) −7165.25 −0.313717
\(806\) −3786.19 −0.165463
\(807\) −14324.9 −0.624859
\(808\) 5191.20 0.226022
\(809\) −12141.6 −0.527657 −0.263828 0.964570i \(-0.584985\pi\)
−0.263828 + 0.964570i \(0.584985\pi\)
\(810\) −95.6075 −0.00414729
\(811\) 30295.6 1.31174 0.655870 0.754873i \(-0.272302\pi\)
0.655870 + 0.754873i \(0.272302\pi\)
\(812\) −5069.80 −0.219107
\(813\) 425.726 0.0183651
\(814\) −1854.73 −0.0798629
\(815\) 2455.54 0.105538
\(816\) 14360.5 0.616077
\(817\) 1465.67 0.0627628
\(818\) −94.5106 −0.00403971
\(819\) 5101.15 0.217642
\(820\) −6231.70 −0.265391
\(821\) −14914.8 −0.634018 −0.317009 0.948423i \(-0.602678\pi\)
−0.317009 + 0.948423i \(0.602678\pi\)
\(822\) 1296.81 0.0550260
\(823\) −31077.6 −1.31628 −0.658138 0.752897i \(-0.728656\pi\)
−0.658138 + 0.752897i \(0.728656\pi\)
\(824\) 4924.85 0.208210
\(825\) −3785.41 −0.159747
\(826\) 516.396 0.0217527
\(827\) 15527.8 0.652908 0.326454 0.945213i \(-0.394146\pi\)
0.326454 + 0.945213i \(0.394146\pi\)
\(828\) 14637.3 0.614348
\(829\) −40221.5 −1.68510 −0.842551 0.538617i \(-0.818947\pi\)
−0.842551 + 0.538617i \(0.818947\pi\)
\(830\) −137.120 −0.00573436
\(831\) −10863.4 −0.453486
\(832\) 39734.2 1.65569
\(833\) 3742.96 0.155685
\(834\) −2160.62 −0.0897075
\(835\) 4130.29 0.171179
\(836\) 1659.10 0.0686377
\(837\) 5348.13 0.220858
\(838\) −3733.51 −0.153904
\(839\) −21153.7 −0.870448 −0.435224 0.900322i \(-0.643331\pi\)
−0.435224 + 0.900322i \(0.643331\pi\)
\(840\) −395.213 −0.0162335
\(841\) −16077.5 −0.659213
\(842\) 457.152 0.0187108
\(843\) 17372.6 0.709780
\(844\) −8184.10 −0.333778
\(845\) −21796.2 −0.887351
\(846\) −373.208 −0.0151669
\(847\) −8515.06 −0.345432
\(848\) 12552.0 0.508301
\(849\) 1578.13 0.0637942
\(850\) 450.813 0.0181915
\(851\) −31868.1 −1.28369
\(852\) 16172.4 0.650302
\(853\) 636.075 0.0255320 0.0127660 0.999919i \(-0.495936\pi\)
0.0127660 + 0.999919i \(0.495936\pi\)
\(854\) 254.763 0.0102082
\(855\) −186.200 −0.00744782
\(856\) −4763.83 −0.190215
\(857\) −3941.46 −0.157104 −0.0785518 0.996910i \(-0.525030\pi\)
−0.0785518 + 0.996910i \(0.525030\pi\)
\(858\) 2894.26 0.115161
\(859\) −21781.6 −0.865165 −0.432583 0.901594i \(-0.642398\pi\)
−0.432583 + 0.901594i \(0.642398\pi\)
\(860\) 14070.0 0.557886
\(861\) 3294.59 0.130406
\(862\) 479.432 0.0189438
\(863\) −44697.9 −1.76308 −0.881538 0.472112i \(-0.843492\pi\)
−0.881538 + 0.472112i \(0.843492\pi\)
\(864\) 1212.43 0.0477404
\(865\) −14590.0 −0.573498
\(866\) −2545.79 −0.0998953
\(867\) 2765.92 0.108345
\(868\) 11015.2 0.430736
\(869\) 64920.1 2.53425
\(870\) 322.825 0.0125802
\(871\) −59490.3 −2.31430
\(872\) −7788.78 −0.302478
\(873\) −12746.7 −0.494171
\(874\) −199.971 −0.00773926
\(875\) 875.000 0.0338062
\(876\) 1449.04 0.0558888
\(877\) −20171.7 −0.776682 −0.388341 0.921516i \(-0.626952\pi\)
−0.388341 + 0.921516i \(0.626952\pi\)
\(878\) −1488.22 −0.0572038
\(879\) 5744.63 0.220434
\(880\) 15814.3 0.605797
\(881\) 11577.6 0.442744 0.221372 0.975189i \(-0.428946\pi\)
0.221372 + 0.975189i \(0.428946\pi\)
\(882\) 104.106 0.00397441
\(883\) −35388.3 −1.34871 −0.674355 0.738407i \(-0.735578\pi\)
−0.674355 + 0.738407i \(0.735578\pi\)
\(884\) 49136.1 1.86949
\(885\) 4687.48 0.178043
\(886\) 3657.83 0.138699
\(887\) −41705.0 −1.57871 −0.789356 0.613936i \(-0.789585\pi\)
−0.789356 + 0.613936i \(0.789585\pi\)
\(888\) −1757.74 −0.0664257
\(889\) −1569.75 −0.0592215
\(890\) 1081.71 0.0407405
\(891\) −4088.24 −0.153716
\(892\) −42650.2 −1.60093
\(893\) −726.838 −0.0272371
\(894\) 1591.02 0.0595207
\(895\) −4778.72 −0.178475
\(896\) 3325.58 0.123995
\(897\) 49729.2 1.85107
\(898\) −57.1328 −0.00212310
\(899\) −18058.3 −0.669942
\(900\) −1787.46 −0.0662023
\(901\) 15300.5 0.565740
\(902\) 1869.27 0.0690020
\(903\) −7438.55 −0.274130
\(904\) 7354.31 0.270576
\(905\) −1030.40 −0.0378472
\(906\) −928.830 −0.0340600
\(907\) −28645.1 −1.04867 −0.524336 0.851511i \(-0.675687\pi\)
−0.524336 + 0.851511i \(0.675687\pi\)
\(908\) 7405.02 0.270643
\(909\) −12412.8 −0.452921
\(910\) −669.010 −0.0243708
\(911\) 13337.8 0.485074 0.242537 0.970142i \(-0.422020\pi\)
0.242537 + 0.970142i \(0.422020\pi\)
\(912\) 777.887 0.0282439
\(913\) −5863.36 −0.212540
\(914\) −2755.00 −0.0997016
\(915\) 2312.55 0.0835527
\(916\) −25128.1 −0.906394
\(917\) 3436.29 0.123747
\(918\) 486.878 0.0175048
\(919\) −28911.0 −1.03774 −0.518871 0.854853i \(-0.673647\pi\)
−0.518871 + 0.854853i \(0.673647\pi\)
\(920\) −3852.79 −0.138068
\(921\) −18732.5 −0.670203
\(922\) 2863.06 0.102267
\(923\) 54944.8 1.95940
\(924\) −8420.25 −0.299790
\(925\) 3891.64 0.138331
\(926\) 1218.50 0.0432422
\(927\) −11775.9 −0.417229
\(928\) −4093.84 −0.144814
\(929\) 7093.88 0.250530 0.125265 0.992123i \(-0.460022\pi\)
0.125265 + 0.992123i \(0.460022\pi\)
\(930\) −701.401 −0.0247310
\(931\) 202.751 0.00713736
\(932\) −3468.17 −0.121892
\(933\) −28976.3 −1.01677
\(934\) 2757.33 0.0965981
\(935\) 19277.1 0.674254
\(936\) 2742.91 0.0957851
\(937\) −19271.1 −0.671888 −0.335944 0.941882i \(-0.609055\pi\)
−0.335944 + 0.941882i \(0.609055\pi\)
\(938\) −1214.10 −0.0422620
\(939\) 6595.01 0.229201
\(940\) −6977.43 −0.242105
\(941\) −18115.2 −0.627563 −0.313782 0.949495i \(-0.601596\pi\)
−0.313782 + 0.949495i \(0.601596\pi\)
\(942\) 1268.35 0.0438695
\(943\) 32117.8 1.10912
\(944\) −19582.9 −0.675180
\(945\) 945.000 0.0325300
\(946\) −4220.44 −0.145051
\(947\) −2475.55 −0.0849468 −0.0424734 0.999098i \(-0.513524\pi\)
−0.0424734 + 0.999098i \(0.513524\pi\)
\(948\) 30655.1 1.05024
\(949\) 4923.04 0.168397
\(950\) 24.4199 0.000833984 0
\(951\) 9092.35 0.310031
\(952\) 2012.61 0.0685179
\(953\) 12866.6 0.437344 0.218672 0.975798i \(-0.429827\pi\)
0.218672 + 0.975798i \(0.429827\pi\)
\(954\) 425.564 0.0144425
\(955\) 10093.2 0.341997
\(956\) 15736.2 0.532368
\(957\) 13804.2 0.466277
\(958\) −4357.50 −0.146957
\(959\) −12817.9 −0.431606
\(960\) 7360.87 0.247470
\(961\) 9444.26 0.317017
\(962\) −2975.48 −0.0997228
\(963\) 11390.9 0.381169
\(964\) −42132.9 −1.40768
\(965\) −5158.45 −0.172079
\(966\) 1014.89 0.0338029
\(967\) −2142.86 −0.0712614 −0.0356307 0.999365i \(-0.511344\pi\)
−0.0356307 + 0.999365i \(0.511344\pi\)
\(968\) −4578.58 −0.152026
\(969\) 948.215 0.0314356
\(970\) 1671.72 0.0553357
\(971\) 2879.06 0.0951529 0.0475765 0.998868i \(-0.484850\pi\)
0.0475765 + 0.998868i \(0.484850\pi\)
\(972\) −1930.46 −0.0637032
\(973\) 21355.9 0.703636
\(974\) 2037.29 0.0670217
\(975\) −6072.79 −0.199472
\(976\) −9661.18 −0.316851
\(977\) 48741.1 1.59607 0.798037 0.602608i \(-0.205872\pi\)
0.798037 + 0.602608i \(0.205872\pi\)
\(978\) −347.805 −0.0113718
\(979\) 46254.7 1.51002
\(980\) 1946.35 0.0634426
\(981\) 18623.9 0.606131
\(982\) 4202.40 0.136562
\(983\) −45756.8 −1.48466 −0.742328 0.670037i \(-0.766278\pi\)
−0.742328 + 0.670037i \(0.766278\pi\)
\(984\) 1771.52 0.0573922
\(985\) 1029.78 0.0333110
\(986\) −1643.97 −0.0530982
\(987\) 3688.85 0.118964
\(988\) 2661.63 0.0857062
\(989\) −72515.7 −2.33151
\(990\) 536.168 0.0172127
\(991\) −51552.1 −1.65248 −0.826240 0.563319i \(-0.809524\pi\)
−0.826240 + 0.563319i \(0.809524\pi\)
\(992\) 8894.70 0.284684
\(993\) 14261.2 0.455756
\(994\) 1121.33 0.0357812
\(995\) 9156.88 0.291751
\(996\) −2768.67 −0.0880808
\(997\) 25565.3 0.812097 0.406048 0.913852i \(-0.366906\pi\)
0.406048 + 0.913852i \(0.366906\pi\)
\(998\) 47.2531 0.00149877
\(999\) 4202.97 0.133109
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 105.4.a.d.1.2 2
3.2 odd 2 315.4.a.l.1.1 2
4.3 odd 2 1680.4.a.bd.1.2 2
5.2 odd 4 525.4.d.k.274.3 4
5.3 odd 4 525.4.d.k.274.2 4
5.4 even 2 525.4.a.o.1.1 2
7.6 odd 2 735.4.a.m.1.2 2
15.14 odd 2 1575.4.a.n.1.2 2
21.20 even 2 2205.4.a.be.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.d.1.2 2 1.1 even 1 trivial
315.4.a.l.1.1 2 3.2 odd 2
525.4.a.o.1.1 2 5.4 even 2
525.4.d.k.274.2 4 5.3 odd 4
525.4.d.k.274.3 4 5.2 odd 4
735.4.a.m.1.2 2 7.6 odd 2
1575.4.a.n.1.2 2 15.14 odd 2
1680.4.a.bd.1.2 2 4.3 odd 2
2205.4.a.be.1.1 2 21.20 even 2