Properties

Label 245.4.a.p.1.5
Level $245$
Weight $4$
Character 245.1
Self dual yes
Analytic conductor $14.455$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.1163891200.1
Defining polynomial: \( x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.241849\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.65606 q^{2} -0.332888 q^{3} -5.25746 q^{4} -5.00000 q^{5} -0.551283 q^{6} -21.9552 q^{8} -26.8892 q^{9} +O(q^{10})\) \(q+1.65606 q^{2} -0.332888 q^{3} -5.25746 q^{4} -5.00000 q^{5} -0.551283 q^{6} -21.9552 q^{8} -26.8892 q^{9} -8.28031 q^{10} +69.5726 q^{11} +1.75014 q^{12} +68.4326 q^{13} +1.66444 q^{15} +5.70053 q^{16} +104.332 q^{17} -44.5302 q^{18} +71.8929 q^{19} +26.2873 q^{20} +115.217 q^{22} -101.031 q^{23} +7.30861 q^{24} +25.0000 q^{25} +113.329 q^{26} +17.9390 q^{27} -114.661 q^{29} +2.75641 q^{30} -73.6505 q^{31} +185.082 q^{32} -23.1599 q^{33} +172.780 q^{34} +141.369 q^{36} -200.933 q^{37} +119.059 q^{38} -22.7803 q^{39} +109.776 q^{40} +417.308 q^{41} +311.175 q^{43} -365.775 q^{44} +134.446 q^{45} -167.313 q^{46} +149.697 q^{47} -1.89763 q^{48} +41.4016 q^{50} -34.7307 q^{51} -359.781 q^{52} +271.474 q^{53} +29.7082 q^{54} -347.863 q^{55} -23.9323 q^{57} -189.885 q^{58} -518.028 q^{59} -8.75071 q^{60} -219.926 q^{61} -121.970 q^{62} +260.903 q^{64} -342.163 q^{65} -38.3542 q^{66} +80.6950 q^{67} -548.520 q^{68} +33.6319 q^{69} -91.0463 q^{71} +590.357 q^{72} +882.282 q^{73} -332.758 q^{74} -8.32219 q^{75} -377.974 q^{76} -37.7257 q^{78} +599.877 q^{79} -28.5026 q^{80} +720.036 q^{81} +691.087 q^{82} +70.8820 q^{83} -521.659 q^{85} +515.325 q^{86} +38.1691 q^{87} -1527.48 q^{88} +802.592 q^{89} +222.651 q^{90} +531.165 q^{92} +24.5173 q^{93} +247.908 q^{94} -359.465 q^{95} -61.6114 q^{96} +145.648 q^{97} -1870.75 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 16 q^{3} + 14 q^{4} - 30 q^{5} + 24 q^{6} - 66 q^{8} + 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} + 16 q^{3} + 14 q^{4} - 30 q^{5} + 24 q^{6} - 66 q^{8} + 70 q^{9} + 10 q^{10} - 16 q^{11} + 160 q^{12} + 168 q^{13} - 80 q^{15} + 298 q^{16} - 4 q^{17} + 354 q^{18} + 308 q^{19} - 70 q^{20} - 236 q^{22} - 336 q^{23} - 92 q^{24} + 150 q^{25} + 56 q^{26} + 964 q^{27} + 176 q^{29} - 120 q^{30} + 392 q^{31} - 770 q^{32} + 188 q^{33} + 812 q^{34} + 230 q^{36} - 140 q^{37} + 20 q^{38} + 140 q^{39} + 330 q^{40} + 656 q^{41} - 388 q^{43} - 160 q^{44} - 350 q^{45} - 388 q^{46} + 628 q^{47} + 1396 q^{48} - 50 q^{50} + 744 q^{51} + 1520 q^{52} - 676 q^{53} + 2284 q^{54} + 80 q^{55} + 1468 q^{57} - 2012 q^{58} + 996 q^{59} - 800 q^{60} + 740 q^{61} - 364 q^{62} + 1426 q^{64} - 840 q^{65} - 3620 q^{66} + 1768 q^{67} - 2940 q^{68} - 1048 q^{69} - 224 q^{71} + 2858 q^{72} + 2640 q^{73} + 928 q^{74} + 400 q^{75} - 1340 q^{76} + 8 q^{78} + 1636 q^{79} - 1490 q^{80} + 4442 q^{81} - 1756 q^{82} + 140 q^{83} + 20 q^{85} + 1180 q^{86} + 1940 q^{87} - 5652 q^{88} - 1904 q^{89} - 1770 q^{90} - 1952 q^{92} - 1592 q^{93} - 3332 q^{94} - 1540 q^{95} - 6460 q^{96} + 516 q^{97} - 2804 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.65606 0.585506 0.292753 0.956188i \(-0.405429\pi\)
0.292753 + 0.956188i \(0.405429\pi\)
\(3\) −0.332888 −0.0640643 −0.0320321 0.999487i \(-0.510198\pi\)
−0.0320321 + 0.999487i \(0.510198\pi\)
\(4\) −5.25746 −0.657182
\(5\) −5.00000 −0.447214
\(6\) −0.551283 −0.0375100
\(7\) 0 0
\(8\) −21.9552 −0.970291
\(9\) −26.8892 −0.995896
\(10\) −8.28031 −0.261846
\(11\) 69.5726 1.90699 0.953497 0.301402i \(-0.0974546\pi\)
0.953497 + 0.301402i \(0.0974546\pi\)
\(12\) 1.75014 0.0421019
\(13\) 68.4326 1.45998 0.729991 0.683456i \(-0.239524\pi\)
0.729991 + 0.683456i \(0.239524\pi\)
\(14\) 0 0
\(15\) 1.66444 0.0286504
\(16\) 5.70053 0.0890707
\(17\) 104.332 1.48848 0.744240 0.667912i \(-0.232812\pi\)
0.744240 + 0.667912i \(0.232812\pi\)
\(18\) −44.5302 −0.583103
\(19\) 71.8929 0.868072 0.434036 0.900896i \(-0.357089\pi\)
0.434036 + 0.900896i \(0.357089\pi\)
\(20\) 26.2873 0.293901
\(21\) 0 0
\(22\) 115.217 1.11656
\(23\) −101.031 −0.915929 −0.457964 0.888970i \(-0.651421\pi\)
−0.457964 + 0.888970i \(0.651421\pi\)
\(24\) 7.30861 0.0621610
\(25\) 25.0000 0.200000
\(26\) 113.329 0.854829
\(27\) 17.9390 0.127866
\(28\) 0 0
\(29\) −114.661 −0.734205 −0.367102 0.930181i \(-0.619650\pi\)
−0.367102 + 0.930181i \(0.619650\pi\)
\(30\) 2.75641 0.0167750
\(31\) −73.6505 −0.426710 −0.213355 0.976975i \(-0.568439\pi\)
−0.213355 + 0.976975i \(0.568439\pi\)
\(32\) 185.082 1.02244
\(33\) −23.1599 −0.122170
\(34\) 172.780 0.871514
\(35\) 0 0
\(36\) 141.369 0.654485
\(37\) −200.933 −0.892790 −0.446395 0.894836i \(-0.647292\pi\)
−0.446395 + 0.894836i \(0.647292\pi\)
\(38\) 119.059 0.508262
\(39\) −22.7803 −0.0935327
\(40\) 109.776 0.433927
\(41\) 417.308 1.58957 0.794786 0.606889i \(-0.207583\pi\)
0.794786 + 0.606889i \(0.207583\pi\)
\(42\) 0 0
\(43\) 311.175 1.10357 0.551787 0.833985i \(-0.313946\pi\)
0.551787 + 0.833985i \(0.313946\pi\)
\(44\) −365.775 −1.25324
\(45\) 134.446 0.445378
\(46\) −167.313 −0.536282
\(47\) 149.697 0.464586 0.232293 0.972646i \(-0.425377\pi\)
0.232293 + 0.972646i \(0.425377\pi\)
\(48\) −1.89763 −0.00570625
\(49\) 0 0
\(50\) 41.4016 0.117101
\(51\) −34.7307 −0.0953583
\(52\) −359.781 −0.959475
\(53\) 271.474 0.703582 0.351791 0.936078i \(-0.385573\pi\)
0.351791 + 0.936078i \(0.385573\pi\)
\(54\) 29.7082 0.0748661
\(55\) −347.863 −0.852834
\(56\) 0 0
\(57\) −23.9323 −0.0556124
\(58\) −189.885 −0.429882
\(59\) −518.028 −1.14308 −0.571538 0.820575i \(-0.693653\pi\)
−0.571538 + 0.820575i \(0.693653\pi\)
\(60\) −8.75071 −0.0188285
\(61\) −219.926 −0.461616 −0.230808 0.972999i \(-0.574137\pi\)
−0.230808 + 0.972999i \(0.574137\pi\)
\(62\) −121.970 −0.249842
\(63\) 0 0
\(64\) 260.903 0.509576
\(65\) −342.163 −0.652924
\(66\) −38.3542 −0.0715314
\(67\) 80.6950 0.147141 0.0735706 0.997290i \(-0.476561\pi\)
0.0735706 + 0.997290i \(0.476561\pi\)
\(68\) −548.520 −0.978202
\(69\) 33.6319 0.0586783
\(70\) 0 0
\(71\) −91.0463 −0.152186 −0.0760930 0.997101i \(-0.524245\pi\)
−0.0760930 + 0.997101i \(0.524245\pi\)
\(72\) 590.357 0.966309
\(73\) 882.282 1.41457 0.707283 0.706931i \(-0.249921\pi\)
0.707283 + 0.706931i \(0.249921\pi\)
\(74\) −332.758 −0.522734
\(75\) −8.32219 −0.0128129
\(76\) −377.974 −0.570481
\(77\) 0 0
\(78\) −37.7257 −0.0547640
\(79\) 599.877 0.854322 0.427161 0.904175i \(-0.359514\pi\)
0.427161 + 0.904175i \(0.359514\pi\)
\(80\) −28.5026 −0.0398336
\(81\) 720.036 0.987704
\(82\) 691.087 0.930705
\(83\) 70.8820 0.0937387 0.0468694 0.998901i \(-0.485076\pi\)
0.0468694 + 0.998901i \(0.485076\pi\)
\(84\) 0 0
\(85\) −521.659 −0.665668
\(86\) 515.325 0.646150
\(87\) 38.1691 0.0470363
\(88\) −1527.48 −1.85034
\(89\) 802.592 0.955894 0.477947 0.878389i \(-0.341381\pi\)
0.477947 + 0.878389i \(0.341381\pi\)
\(90\) 222.651 0.260772
\(91\) 0 0
\(92\) 531.165 0.601932
\(93\) 24.5173 0.0273369
\(94\) 247.908 0.272018
\(95\) −359.465 −0.388214
\(96\) −61.6114 −0.0655020
\(97\) 145.648 0.152457 0.0762283 0.997090i \(-0.475712\pi\)
0.0762283 + 0.997090i \(0.475712\pi\)
\(98\) 0 0
\(99\) −1870.75 −1.89917
\(100\) −131.436 −0.131436
\(101\) −619.435 −0.610259 −0.305129 0.952311i \(-0.598700\pi\)
−0.305129 + 0.952311i \(0.598700\pi\)
\(102\) −57.5163 −0.0558329
\(103\) −1822.08 −1.74306 −0.871528 0.490345i \(-0.836871\pi\)
−0.871528 + 0.490345i \(0.836871\pi\)
\(104\) −1502.45 −1.41661
\(105\) 0 0
\(106\) 449.578 0.411952
\(107\) 1089.70 0.984536 0.492268 0.870444i \(-0.336168\pi\)
0.492268 + 0.870444i \(0.336168\pi\)
\(108\) −94.3138 −0.0840310
\(109\) 589.667 0.518164 0.259082 0.965855i \(-0.416580\pi\)
0.259082 + 0.965855i \(0.416580\pi\)
\(110\) −576.083 −0.499340
\(111\) 66.8882 0.0571959
\(112\) 0 0
\(113\) −900.358 −0.749544 −0.374772 0.927117i \(-0.622279\pi\)
−0.374772 + 0.927117i \(0.622279\pi\)
\(114\) −39.6333 −0.0325614
\(115\) 505.154 0.409616
\(116\) 602.823 0.482506
\(117\) −1840.10 −1.45399
\(118\) −857.887 −0.669279
\(119\) 0 0
\(120\) −36.5430 −0.0277992
\(121\) 3509.35 2.63663
\(122\) −364.210 −0.270279
\(123\) −138.917 −0.101835
\(124\) 387.214 0.280426
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1755.75 −1.22676 −0.613378 0.789790i \(-0.710190\pi\)
−0.613378 + 0.789790i \(0.710190\pi\)
\(128\) −1048.58 −0.724082
\(129\) −103.586 −0.0706996
\(130\) −566.643 −0.382291
\(131\) 1809.10 1.20658 0.603289 0.797523i \(-0.293857\pi\)
0.603289 + 0.797523i \(0.293857\pi\)
\(132\) 121.762 0.0802881
\(133\) 0 0
\(134\) 133.636 0.0861521
\(135\) −89.6952 −0.0571832
\(136\) −2290.62 −1.44426
\(137\) −18.5134 −0.0115453 −0.00577265 0.999983i \(-0.501838\pi\)
−0.00577265 + 0.999983i \(0.501838\pi\)
\(138\) 55.6965 0.0343565
\(139\) −625.608 −0.381751 −0.190875 0.981614i \(-0.561133\pi\)
−0.190875 + 0.981614i \(0.561133\pi\)
\(140\) 0 0
\(141\) −49.8323 −0.0297634
\(142\) −150.778 −0.0891059
\(143\) 4761.03 2.78418
\(144\) −153.283 −0.0887052
\(145\) 573.303 0.328346
\(146\) 1461.11 0.828237
\(147\) 0 0
\(148\) 1056.40 0.586725
\(149\) −1028.63 −0.565563 −0.282782 0.959184i \(-0.591257\pi\)
−0.282782 + 0.959184i \(0.591257\pi\)
\(150\) −13.7821 −0.00750201
\(151\) 71.0073 0.0382682 0.0191341 0.999817i \(-0.493909\pi\)
0.0191341 + 0.999817i \(0.493909\pi\)
\(152\) −1578.42 −0.842282
\(153\) −2805.39 −1.48237
\(154\) 0 0
\(155\) 368.252 0.190831
\(156\) 119.767 0.0614680
\(157\) −2061.32 −1.04784 −0.523922 0.851767i \(-0.675532\pi\)
−0.523922 + 0.851767i \(0.675532\pi\)
\(158\) 993.434 0.500211
\(159\) −90.3704 −0.0450745
\(160\) −925.409 −0.457250
\(161\) 0 0
\(162\) 1192.42 0.578307
\(163\) −1963.80 −0.943660 −0.471830 0.881690i \(-0.656406\pi\)
−0.471830 + 0.881690i \(0.656406\pi\)
\(164\) −2193.98 −1.04464
\(165\) 115.799 0.0546362
\(166\) 117.385 0.0548846
\(167\) 2855.04 1.32293 0.661467 0.749974i \(-0.269934\pi\)
0.661467 + 0.749974i \(0.269934\pi\)
\(168\) 0 0
\(169\) 2486.01 1.13155
\(170\) −863.899 −0.389753
\(171\) −1933.14 −0.864509
\(172\) −1635.99 −0.725249
\(173\) −1553.21 −0.682590 −0.341295 0.939956i \(-0.610866\pi\)
−0.341295 + 0.939956i \(0.610866\pi\)
\(174\) 63.2104 0.0275400
\(175\) 0 0
\(176\) 396.601 0.169857
\(177\) 172.445 0.0732303
\(178\) 1329.14 0.559682
\(179\) 269.841 0.112675 0.0563376 0.998412i \(-0.482058\pi\)
0.0563376 + 0.998412i \(0.482058\pi\)
\(180\) −706.844 −0.292695
\(181\) 2229.61 0.915613 0.457806 0.889052i \(-0.348635\pi\)
0.457806 + 0.889052i \(0.348635\pi\)
\(182\) 0 0
\(183\) 73.2105 0.0295731
\(184\) 2218.15 0.888717
\(185\) 1004.67 0.399268
\(186\) 40.6022 0.0160059
\(187\) 7258.63 2.83852
\(188\) −787.026 −0.305318
\(189\) 0 0
\(190\) −595.296 −0.227302
\(191\) −465.920 −0.176507 −0.0882533 0.996098i \(-0.528129\pi\)
−0.0882533 + 0.996098i \(0.528129\pi\)
\(192\) −86.8513 −0.0326456
\(193\) −4414.46 −1.64642 −0.823212 0.567734i \(-0.807820\pi\)
−0.823212 + 0.567734i \(0.807820\pi\)
\(194\) 241.202 0.0892643
\(195\) 113.902 0.0418291
\(196\) 0 0
\(197\) −289.812 −0.104814 −0.0524068 0.998626i \(-0.516689\pi\)
−0.0524068 + 0.998626i \(0.516689\pi\)
\(198\) −3098.08 −1.11197
\(199\) 4817.73 1.71618 0.858091 0.513498i \(-0.171651\pi\)
0.858091 + 0.513498i \(0.171651\pi\)
\(200\) −548.879 −0.194058
\(201\) −26.8624 −0.00942649
\(202\) −1025.82 −0.357310
\(203\) 0 0
\(204\) 182.595 0.0626678
\(205\) −2086.54 −0.710879
\(206\) −3017.48 −1.02057
\(207\) 2716.63 0.912170
\(208\) 390.102 0.130042
\(209\) 5001.78 1.65541
\(210\) 0 0
\(211\) 2022.01 0.659719 0.329859 0.944030i \(-0.392999\pi\)
0.329859 + 0.944030i \(0.392999\pi\)
\(212\) −1427.26 −0.462382
\(213\) 30.3082 0.00974968
\(214\) 1804.61 0.576452
\(215\) −1555.87 −0.493533
\(216\) −393.855 −0.124067
\(217\) 0 0
\(218\) 976.525 0.303388
\(219\) −293.701 −0.0906231
\(220\) 1828.88 0.560467
\(221\) 7139.69 2.17315
\(222\) 110.771 0.0334886
\(223\) 4343.86 1.30442 0.652211 0.758037i \(-0.273842\pi\)
0.652211 + 0.758037i \(0.273842\pi\)
\(224\) 0 0
\(225\) −672.230 −0.199179
\(226\) −1491.05 −0.438863
\(227\) −2647.59 −0.774127 −0.387064 0.922053i \(-0.626511\pi\)
−0.387064 + 0.922053i \(0.626511\pi\)
\(228\) 125.823 0.0365475
\(229\) −1445.09 −0.417006 −0.208503 0.978022i \(-0.566859\pi\)
−0.208503 + 0.978022i \(0.566859\pi\)
\(230\) 836.566 0.239833
\(231\) 0 0
\(232\) 2517.39 0.712392
\(233\) −6245.91 −1.75615 −0.878076 0.478522i \(-0.841173\pi\)
−0.878076 + 0.478522i \(0.841173\pi\)
\(234\) −3047.31 −0.851321
\(235\) −748.485 −0.207769
\(236\) 2723.51 0.751209
\(237\) −199.692 −0.0547315
\(238\) 0 0
\(239\) 1340.24 0.362731 0.181366 0.983416i \(-0.441948\pi\)
0.181366 + 0.983416i \(0.441948\pi\)
\(240\) 9.48817 0.00255191
\(241\) 3369.92 0.900729 0.450364 0.892845i \(-0.351294\pi\)
0.450364 + 0.892845i \(0.351294\pi\)
\(242\) 5811.70 1.54376
\(243\) −724.045 −0.191142
\(244\) 1156.25 0.303366
\(245\) 0 0
\(246\) −230.054 −0.0596249
\(247\) 4919.82 1.26737
\(248\) 1617.01 0.414033
\(249\) −23.5958 −0.00600530
\(250\) −207.008 −0.0523693
\(251\) −3592.64 −0.903449 −0.451724 0.892158i \(-0.649191\pi\)
−0.451724 + 0.892158i \(0.649191\pi\)
\(252\) 0 0
\(253\) −7028.97 −1.74667
\(254\) −2907.64 −0.718273
\(255\) 173.654 0.0426455
\(256\) −3823.74 −0.933531
\(257\) −2.84763 −0.000691167 0 −0.000345584 1.00000i \(-0.500110\pi\)
−0.000345584 1.00000i \(0.500110\pi\)
\(258\) −171.545 −0.0413951
\(259\) 0 0
\(260\) 1798.91 0.429090
\(261\) 3083.13 0.731191
\(262\) 2995.98 0.706459
\(263\) −2817.07 −0.660488 −0.330244 0.943896i \(-0.607131\pi\)
−0.330244 + 0.943896i \(0.607131\pi\)
\(264\) 508.479 0.118541
\(265\) −1357.37 −0.314652
\(266\) 0 0
\(267\) −267.173 −0.0612386
\(268\) −424.250 −0.0966986
\(269\) −1447.43 −0.328072 −0.164036 0.986454i \(-0.552451\pi\)
−0.164036 + 0.986454i \(0.552451\pi\)
\(270\) −148.541 −0.0334811
\(271\) 8054.50 1.80545 0.902724 0.430221i \(-0.141564\pi\)
0.902724 + 0.430221i \(0.141564\pi\)
\(272\) 594.746 0.132580
\(273\) 0 0
\(274\) −30.6593 −0.00675985
\(275\) 1739.32 0.381399
\(276\) −176.818 −0.0385623
\(277\) −571.300 −0.123921 −0.0619604 0.998079i \(-0.519735\pi\)
−0.0619604 + 0.998079i \(0.519735\pi\)
\(278\) −1036.05 −0.223518
\(279\) 1980.40 0.424959
\(280\) 0 0
\(281\) −1784.48 −0.378837 −0.189418 0.981896i \(-0.560660\pi\)
−0.189418 + 0.981896i \(0.560660\pi\)
\(282\) −82.5254 −0.0174267
\(283\) 3321.05 0.697582 0.348791 0.937201i \(-0.386592\pi\)
0.348791 + 0.937201i \(0.386592\pi\)
\(284\) 478.672 0.100014
\(285\) 119.661 0.0248706
\(286\) 7884.57 1.63015
\(287\) 0 0
\(288\) −4976.70 −1.01825
\(289\) 5972.11 1.21557
\(290\) 949.425 0.192249
\(291\) −48.4843 −0.00976701
\(292\) −4638.56 −0.929627
\(293\) −5049.54 −1.00682 −0.503408 0.864049i \(-0.667921\pi\)
−0.503408 + 0.864049i \(0.667921\pi\)
\(294\) 0 0
\(295\) 2590.14 0.511199
\(296\) 4411.52 0.866266
\(297\) 1248.07 0.243839
\(298\) −1703.48 −0.331141
\(299\) −6913.79 −1.33724
\(300\) 43.7536 0.00842038
\(301\) 0 0
\(302\) 117.593 0.0224063
\(303\) 206.202 0.0390958
\(304\) 409.828 0.0773198
\(305\) 1099.63 0.206441
\(306\) −4645.91 −0.867938
\(307\) −1535.73 −0.285500 −0.142750 0.989759i \(-0.545595\pi\)
−0.142750 + 0.989759i \(0.545595\pi\)
\(308\) 0 0
\(309\) 606.548 0.111668
\(310\) 609.849 0.111733
\(311\) −9283.05 −1.69258 −0.846291 0.532720i \(-0.821170\pi\)
−0.846291 + 0.532720i \(0.821170\pi\)
\(312\) 500.147 0.0907539
\(313\) 6025.43 1.08811 0.544054 0.839050i \(-0.316889\pi\)
0.544054 + 0.839050i \(0.316889\pi\)
\(314\) −3413.68 −0.613519
\(315\) 0 0
\(316\) −3153.83 −0.561445
\(317\) −6977.58 −1.23628 −0.618139 0.786069i \(-0.712113\pi\)
−0.618139 + 0.786069i \(0.712113\pi\)
\(318\) −149.659 −0.0263914
\(319\) −7977.24 −1.40012
\(320\) −1304.51 −0.227889
\(321\) −362.748 −0.0630736
\(322\) 0 0
\(323\) 7500.71 1.29211
\(324\) −3785.56 −0.649102
\(325\) 1710.81 0.291997
\(326\) −3252.17 −0.552519
\(327\) −196.293 −0.0331958
\(328\) −9162.06 −1.54235
\(329\) 0 0
\(330\) 191.771 0.0319898
\(331\) 984.878 0.163546 0.0817731 0.996651i \(-0.473942\pi\)
0.0817731 + 0.996651i \(0.473942\pi\)
\(332\) −372.659 −0.0616034
\(333\) 5402.93 0.889125
\(334\) 4728.13 0.774586
\(335\) −403.475 −0.0658035
\(336\) 0 0
\(337\) 51.9653 0.00839979 0.00419990 0.999991i \(-0.498663\pi\)
0.00419990 + 0.999991i \(0.498663\pi\)
\(338\) 4116.99 0.662529
\(339\) 299.718 0.0480190
\(340\) 2742.60 0.437465
\(341\) −5124.06 −0.813734
\(342\) −3201.40 −0.506176
\(343\) 0 0
\(344\) −6831.89 −1.07079
\(345\) −168.159 −0.0262417
\(346\) −2572.21 −0.399661
\(347\) −11300.5 −1.74825 −0.874123 0.485704i \(-0.838563\pi\)
−0.874123 + 0.485704i \(0.838563\pi\)
\(348\) −200.672 −0.0309114
\(349\) −2016.91 −0.309349 −0.154674 0.987966i \(-0.549433\pi\)
−0.154674 + 0.987966i \(0.549433\pi\)
\(350\) 0 0
\(351\) 1227.61 0.186682
\(352\) 12876.6 1.94979
\(353\) −7589.41 −1.14432 −0.572158 0.820143i \(-0.693894\pi\)
−0.572158 + 0.820143i \(0.693894\pi\)
\(354\) 285.580 0.0428768
\(355\) 455.232 0.0680597
\(356\) −4219.59 −0.628196
\(357\) 0 0
\(358\) 446.873 0.0659720
\(359\) 8734.24 1.28405 0.642027 0.766682i \(-0.278094\pi\)
0.642027 + 0.766682i \(0.278094\pi\)
\(360\) −2951.78 −0.432146
\(361\) −1690.41 −0.246451
\(362\) 3692.38 0.536097
\(363\) −1168.22 −0.168914
\(364\) 0 0
\(365\) −4411.41 −0.632613
\(366\) 121.241 0.0173152
\(367\) −1890.54 −0.268898 −0.134449 0.990921i \(-0.542926\pi\)
−0.134449 + 0.990921i \(0.542926\pi\)
\(368\) −575.928 −0.0815825
\(369\) −11221.1 −1.58305
\(370\) 1663.79 0.233774
\(371\) 0 0
\(372\) −128.899 −0.0179653
\(373\) 2713.88 0.376728 0.188364 0.982099i \(-0.439682\pi\)
0.188364 + 0.982099i \(0.439682\pi\)
\(374\) 12020.7 1.66197
\(375\) 41.6110 0.00573008
\(376\) −3286.63 −0.450784
\(377\) −7846.52 −1.07193
\(378\) 0 0
\(379\) 8941.19 1.21182 0.605908 0.795535i \(-0.292810\pi\)
0.605908 + 0.795535i \(0.292810\pi\)
\(380\) 1889.87 0.255127
\(381\) 584.469 0.0785912
\(382\) −771.592 −0.103346
\(383\) −9293.88 −1.23994 −0.619968 0.784627i \(-0.712854\pi\)
−0.619968 + 0.784627i \(0.712854\pi\)
\(384\) 349.060 0.0463878
\(385\) 0 0
\(386\) −7310.62 −0.963992
\(387\) −8367.23 −1.09904
\(388\) −765.737 −0.100192
\(389\) 10454.8 1.36267 0.681333 0.731974i \(-0.261401\pi\)
0.681333 + 0.731974i \(0.261401\pi\)
\(390\) 188.628 0.0244912
\(391\) −10540.7 −1.36334
\(392\) 0 0
\(393\) −602.226 −0.0772985
\(394\) −479.947 −0.0613690
\(395\) −2999.39 −0.382065
\(396\) 9835.40 1.24810
\(397\) −3626.03 −0.458401 −0.229200 0.973379i \(-0.573611\pi\)
−0.229200 + 0.973379i \(0.573611\pi\)
\(398\) 7978.47 1.00484
\(399\) 0 0
\(400\) 142.513 0.0178141
\(401\) −8422.52 −1.04888 −0.524440 0.851448i \(-0.675725\pi\)
−0.524440 + 0.851448i \(0.675725\pi\)
\(402\) −44.4857 −0.00551927
\(403\) −5040.09 −0.622989
\(404\) 3256.65 0.401051
\(405\) −3600.18 −0.441715
\(406\) 0 0
\(407\) −13979.5 −1.70254
\(408\) 762.519 0.0925253
\(409\) −14580.7 −1.76276 −0.881379 0.472409i \(-0.843384\pi\)
−0.881379 + 0.472409i \(0.843384\pi\)
\(410\) −3455.44 −0.416224
\(411\) 6.16288 0.000739641 0
\(412\) 9579.51 1.14551
\(413\) 0 0
\(414\) 4498.92 0.534081
\(415\) −354.410 −0.0419212
\(416\) 12665.6 1.49275
\(417\) 208.257 0.0244566
\(418\) 8283.26 0.969252
\(419\) −2537.53 −0.295863 −0.147931 0.988998i \(-0.547261\pi\)
−0.147931 + 0.988998i \(0.547261\pi\)
\(420\) 0 0
\(421\) 9649.52 1.11708 0.558538 0.829479i \(-0.311363\pi\)
0.558538 + 0.829479i \(0.311363\pi\)
\(422\) 3348.57 0.386269
\(423\) −4025.23 −0.462680
\(424\) −5960.27 −0.682680
\(425\) 2608.29 0.297696
\(426\) 50.1922 0.00570850
\(427\) 0 0
\(428\) −5729.06 −0.647020
\(429\) −1584.89 −0.178366
\(430\) −2576.62 −0.288967
\(431\) −7262.56 −0.811660 −0.405830 0.913949i \(-0.633017\pi\)
−0.405830 + 0.913949i \(0.633017\pi\)
\(432\) 102.262 0.0113891
\(433\) 11345.0 1.25914 0.629570 0.776944i \(-0.283231\pi\)
0.629570 + 0.776944i \(0.283231\pi\)
\(434\) 0 0
\(435\) −190.845 −0.0210353
\(436\) −3100.15 −0.340528
\(437\) −7263.40 −0.795092
\(438\) −486.387 −0.0530604
\(439\) 11705.9 1.27265 0.636323 0.771423i \(-0.280455\pi\)
0.636323 + 0.771423i \(0.280455\pi\)
\(440\) 7637.40 0.827497
\(441\) 0 0
\(442\) 11823.8 1.27240
\(443\) 15078.0 1.61710 0.808551 0.588426i \(-0.200252\pi\)
0.808551 + 0.588426i \(0.200252\pi\)
\(444\) −351.662 −0.0375881
\(445\) −4012.96 −0.427489
\(446\) 7193.70 0.763747
\(447\) 342.419 0.0362324
\(448\) 0 0
\(449\) 1075.45 0.113037 0.0565185 0.998402i \(-0.482000\pi\)
0.0565185 + 0.998402i \(0.482000\pi\)
\(450\) −1113.25 −0.116621
\(451\) 29033.2 3.03131
\(452\) 4733.59 0.492587
\(453\) −23.6375 −0.00245162
\(454\) −4384.58 −0.453257
\(455\) 0 0
\(456\) 525.437 0.0539602
\(457\) −10736.9 −1.09902 −0.549511 0.835487i \(-0.685186\pi\)
−0.549511 + 0.835487i \(0.685186\pi\)
\(458\) −2393.16 −0.244159
\(459\) 1871.61 0.190325
\(460\) −2655.82 −0.269192
\(461\) 452.568 0.0457228 0.0228614 0.999739i \(-0.492722\pi\)
0.0228614 + 0.999739i \(0.492722\pi\)
\(462\) 0 0
\(463\) 7118.15 0.714489 0.357244 0.934011i \(-0.383716\pi\)
0.357244 + 0.934011i \(0.383716\pi\)
\(464\) −653.626 −0.0653962
\(465\) −122.587 −0.0122254
\(466\) −10343.6 −1.02824
\(467\) 973.800 0.0964927 0.0482463 0.998835i \(-0.484637\pi\)
0.0482463 + 0.998835i \(0.484637\pi\)
\(468\) 9674.23 0.955537
\(469\) 0 0
\(470\) −1239.54 −0.121650
\(471\) 686.188 0.0671293
\(472\) 11373.4 1.10912
\(473\) 21649.2 2.10451
\(474\) −330.702 −0.0320457
\(475\) 1797.32 0.173614
\(476\) 0 0
\(477\) −7299.72 −0.700695
\(478\) 2219.52 0.212381
\(479\) 9714.00 0.926606 0.463303 0.886200i \(-0.346664\pi\)
0.463303 + 0.886200i \(0.346664\pi\)
\(480\) 308.057 0.0292934
\(481\) −13750.4 −1.30346
\(482\) 5580.80 0.527383
\(483\) 0 0
\(484\) −18450.3 −1.73274
\(485\) −728.239 −0.0681806
\(486\) −1199.06 −0.111915
\(487\) 923.389 0.0859194 0.0429597 0.999077i \(-0.486321\pi\)
0.0429597 + 0.999077i \(0.486321\pi\)
\(488\) 4828.50 0.447902
\(489\) 653.724 0.0604549
\(490\) 0 0
\(491\) 1289.11 0.118486 0.0592430 0.998244i \(-0.481131\pi\)
0.0592430 + 0.998244i \(0.481131\pi\)
\(492\) 730.348 0.0669240
\(493\) −11962.7 −1.09285
\(494\) 8147.52 0.742053
\(495\) 9353.76 0.849334
\(496\) −419.846 −0.0380074
\(497\) 0 0
\(498\) −39.0760 −0.00351614
\(499\) −19338.3 −1.73487 −0.867436 0.497549i \(-0.834234\pi\)
−0.867436 + 0.497549i \(0.834234\pi\)
\(500\) 657.182 0.0587802
\(501\) −950.409 −0.0847528
\(502\) −5949.64 −0.528975
\(503\) −1772.84 −0.157151 −0.0785757 0.996908i \(-0.525037\pi\)
−0.0785757 + 0.996908i \(0.525037\pi\)
\(504\) 0 0
\(505\) 3097.18 0.272916
\(506\) −11640.4 −1.02269
\(507\) −827.563 −0.0724919
\(508\) 9230.80 0.806202
\(509\) 3151.75 0.274458 0.137229 0.990539i \(-0.456180\pi\)
0.137229 + 0.990539i \(0.456180\pi\)
\(510\) 287.581 0.0249692
\(511\) 0 0
\(512\) 2056.31 0.177494
\(513\) 1289.69 0.110997
\(514\) −4.71585 −0.000404683 0
\(515\) 9110.40 0.779519
\(516\) 544.600 0.0464626
\(517\) 10414.8 0.885964
\(518\) 0 0
\(519\) 517.043 0.0437296
\(520\) 7512.24 0.633526
\(521\) −11029.4 −0.927458 −0.463729 0.885977i \(-0.653489\pi\)
−0.463729 + 0.885977i \(0.653489\pi\)
\(522\) 5105.85 0.428117
\(523\) 14448.4 1.20800 0.604002 0.796983i \(-0.293572\pi\)
0.604002 + 0.796983i \(0.293572\pi\)
\(524\) −9511.26 −0.792941
\(525\) 0 0
\(526\) −4665.25 −0.386720
\(527\) −7684.08 −0.635149
\(528\) −132.023 −0.0108818
\(529\) −1959.79 −0.161074
\(530\) −2247.89 −0.184231
\(531\) 13929.4 1.13838
\(532\) 0 0
\(533\) 28557.4 2.32075
\(534\) −442.455 −0.0358556
\(535\) −5448.51 −0.440298
\(536\) −1771.67 −0.142770
\(537\) −89.8266 −0.00721845
\(538\) −2397.04 −0.192089
\(539\) 0 0
\(540\) 471.569 0.0375798
\(541\) −22274.8 −1.77018 −0.885091 0.465418i \(-0.845904\pi\)
−0.885091 + 0.465418i \(0.845904\pi\)
\(542\) 13338.8 1.05710
\(543\) −742.211 −0.0586580
\(544\) 19309.9 1.52188
\(545\) −2948.33 −0.231730
\(546\) 0 0
\(547\) 18642.6 1.45722 0.728609 0.684930i \(-0.240167\pi\)
0.728609 + 0.684930i \(0.240167\pi\)
\(548\) 97.3334 0.00758737
\(549\) 5913.62 0.459721
\(550\) 2880.42 0.223311
\(551\) −8243.28 −0.637343
\(552\) −738.394 −0.0569350
\(553\) 0 0
\(554\) −946.108 −0.0725565
\(555\) −334.441 −0.0255788
\(556\) 3289.11 0.250880
\(557\) −21431.8 −1.63033 −0.815165 0.579229i \(-0.803354\pi\)
−0.815165 + 0.579229i \(0.803354\pi\)
\(558\) 3279.67 0.248816
\(559\) 21294.5 1.61120
\(560\) 0 0
\(561\) −2416.31 −0.181848
\(562\) −2955.21 −0.221811
\(563\) −6154.85 −0.460739 −0.230370 0.973103i \(-0.573993\pi\)
−0.230370 + 0.973103i \(0.573993\pi\)
\(564\) 261.991 0.0195600
\(565\) 4501.79 0.335206
\(566\) 5499.86 0.408439
\(567\) 0 0
\(568\) 1998.94 0.147665
\(569\) −8389.99 −0.618149 −0.309074 0.951038i \(-0.600019\pi\)
−0.309074 + 0.951038i \(0.600019\pi\)
\(570\) 198.167 0.0145619
\(571\) −600.502 −0.0440109 −0.0220055 0.999758i \(-0.507005\pi\)
−0.0220055 + 0.999758i \(0.507005\pi\)
\(572\) −25030.9 −1.82971
\(573\) 155.099 0.0113078
\(574\) 0 0
\(575\) −2525.77 −0.183186
\(576\) −7015.46 −0.507484
\(577\) 4062.35 0.293098 0.146549 0.989203i \(-0.453183\pi\)
0.146549 + 0.989203i \(0.453183\pi\)
\(578\) 9890.18 0.711725
\(579\) 1469.52 0.105477
\(580\) −3014.12 −0.215783
\(581\) 0 0
\(582\) −80.2931 −0.00571865
\(583\) 18887.2 1.34173
\(584\) −19370.6 −1.37254
\(585\) 9200.48 0.650244
\(586\) −8362.35 −0.589498
\(587\) −8387.50 −0.589760 −0.294880 0.955534i \(-0.595280\pi\)
−0.294880 + 0.955534i \(0.595280\pi\)
\(588\) 0 0
\(589\) −5294.95 −0.370415
\(590\) 4289.43 0.299310
\(591\) 96.4749 0.00671480
\(592\) −1145.43 −0.0795214
\(593\) −15342.6 −1.06247 −0.531236 0.847224i \(-0.678272\pi\)
−0.531236 + 0.847224i \(0.678272\pi\)
\(594\) 2066.88 0.142769
\(595\) 0 0
\(596\) 5408.00 0.371678
\(597\) −1603.76 −0.109946
\(598\) −11449.7 −0.782963
\(599\) −19708.5 −1.34435 −0.672175 0.740392i \(-0.734640\pi\)
−0.672175 + 0.740392i \(0.734640\pi\)
\(600\) 182.715 0.0124322
\(601\) −19002.5 −1.28973 −0.644866 0.764295i \(-0.723087\pi\)
−0.644866 + 0.764295i \(0.723087\pi\)
\(602\) 0 0
\(603\) −2169.82 −0.146537
\(604\) −373.318 −0.0251492
\(605\) −17546.8 −1.17914
\(606\) 341.484 0.0228908
\(607\) 29453.1 1.96946 0.984732 0.174076i \(-0.0556939\pi\)
0.984732 + 0.174076i \(0.0556939\pi\)
\(608\) 13306.1 0.887553
\(609\) 0 0
\(610\) 1821.05 0.120873
\(611\) 10244.2 0.678288
\(612\) 14749.2 0.974188
\(613\) 9987.27 0.658045 0.329023 0.944322i \(-0.393281\pi\)
0.329023 + 0.944322i \(0.393281\pi\)
\(614\) −2543.26 −0.167162
\(615\) 694.583 0.0455419
\(616\) 0 0
\(617\) −21076.1 −1.37519 −0.687593 0.726096i \(-0.741333\pi\)
−0.687593 + 0.726096i \(0.741333\pi\)
\(618\) 1004.48 0.0653821
\(619\) −314.668 −0.0204323 −0.0102161 0.999948i \(-0.503252\pi\)
−0.0102161 + 0.999948i \(0.503252\pi\)
\(620\) −1936.07 −0.125410
\(621\) −1812.39 −0.117116
\(622\) −15373.3 −0.991018
\(623\) 0 0
\(624\) −129.860 −0.00833103
\(625\) 625.000 0.0400000
\(626\) 9978.49 0.637094
\(627\) −1665.03 −0.106052
\(628\) 10837.3 0.688624
\(629\) −20963.7 −1.32890
\(630\) 0 0
\(631\) 3314.96 0.209138 0.104569 0.994518i \(-0.466654\pi\)
0.104569 + 0.994518i \(0.466654\pi\)
\(632\) −13170.4 −0.828941
\(633\) −673.101 −0.0422644
\(634\) −11555.3 −0.723849
\(635\) 8778.77 0.548622
\(636\) 475.119 0.0296221
\(637\) 0 0
\(638\) −13210.8 −0.819782
\(639\) 2448.16 0.151561
\(640\) 5242.92 0.323820
\(641\) 3005.12 0.185172 0.0925858 0.995705i \(-0.470487\pi\)
0.0925858 + 0.995705i \(0.470487\pi\)
\(642\) −600.733 −0.0369300
\(643\) 21225.7 1.30180 0.650902 0.759162i \(-0.274391\pi\)
0.650902 + 0.759162i \(0.274391\pi\)
\(644\) 0 0
\(645\) 517.931 0.0316178
\(646\) 12421.6 0.756537
\(647\) −2740.35 −0.166514 −0.0832568 0.996528i \(-0.526532\pi\)
−0.0832568 + 0.996528i \(0.526532\pi\)
\(648\) −15808.5 −0.958360
\(649\) −36040.6 −2.17984
\(650\) 2833.21 0.170966
\(651\) 0 0
\(652\) 10324.6 0.620157
\(653\) 22790.7 1.36580 0.682900 0.730511i \(-0.260718\pi\)
0.682900 + 0.730511i \(0.260718\pi\)
\(654\) −325.073 −0.0194363
\(655\) −9045.49 −0.539598
\(656\) 2378.87 0.141584
\(657\) −23723.8 −1.40876
\(658\) 0 0
\(659\) 19405.1 1.14706 0.573532 0.819183i \(-0.305573\pi\)
0.573532 + 0.819183i \(0.305573\pi\)
\(660\) −608.810 −0.0359059
\(661\) 15637.3 0.920150 0.460075 0.887880i \(-0.347823\pi\)
0.460075 + 0.887880i \(0.347823\pi\)
\(662\) 1631.02 0.0957574
\(663\) −2376.71 −0.139222
\(664\) −1556.23 −0.0909538
\(665\) 0 0
\(666\) 8947.59 0.520589
\(667\) 11584.2 0.672479
\(668\) −15010.3 −0.869409
\(669\) −1446.02 −0.0835668
\(670\) −668.179 −0.0385284
\(671\) −15300.8 −0.880299
\(672\) 0 0
\(673\) −2579.54 −0.147747 −0.0738735 0.997268i \(-0.523536\pi\)
−0.0738735 + 0.997268i \(0.523536\pi\)
\(674\) 86.0578 0.00491813
\(675\) 448.476 0.0255731
\(676\) −13070.1 −0.743634
\(677\) 8159.56 0.463216 0.231608 0.972809i \(-0.425601\pi\)
0.231608 + 0.972809i \(0.425601\pi\)
\(678\) 496.351 0.0281154
\(679\) 0 0
\(680\) 11453.1 0.645892
\(681\) 881.351 0.0495939
\(682\) −8485.76 −0.476446
\(683\) −20529.3 −1.15012 −0.575059 0.818112i \(-0.695021\pi\)
−0.575059 + 0.818112i \(0.695021\pi\)
\(684\) 10163.4 0.568140
\(685\) 92.5670 0.00516321
\(686\) 0 0
\(687\) 481.053 0.0267151
\(688\) 1773.86 0.0982961
\(689\) 18577.7 1.02722
\(690\) −278.482 −0.0153647
\(691\) −917.295 −0.0505001 −0.0252500 0.999681i \(-0.508038\pi\)
−0.0252500 + 0.999681i \(0.508038\pi\)
\(692\) 8165.92 0.448586
\(693\) 0 0
\(694\) −18714.3 −1.02361
\(695\) 3128.04 0.170724
\(696\) −838.009 −0.0456389
\(697\) 43538.4 2.36605
\(698\) −3340.13 −0.181126
\(699\) 2079.19 0.112507
\(700\) 0 0
\(701\) −10491.3 −0.565266 −0.282633 0.959228i \(-0.591208\pi\)
−0.282633 + 0.959228i \(0.591208\pi\)
\(702\) 2033.01 0.109303
\(703\) −14445.7 −0.775006
\(704\) 18151.7 0.971758
\(705\) 249.162 0.0133106
\(706\) −12568.5 −0.670005
\(707\) 0 0
\(708\) −906.623 −0.0481257
\(709\) 23837.6 1.26268 0.631339 0.775507i \(-0.282506\pi\)
0.631339 + 0.775507i \(0.282506\pi\)
\(710\) 753.892 0.0398494
\(711\) −16130.2 −0.850816
\(712\) −17621.0 −0.927495
\(713\) 7440.96 0.390836
\(714\) 0 0
\(715\) −23805.2 −1.24512
\(716\) −1418.68 −0.0740481
\(717\) −446.148 −0.0232381
\(718\) 14464.4 0.751822
\(719\) −3926.19 −0.203647 −0.101824 0.994802i \(-0.532468\pi\)
−0.101824 + 0.994802i \(0.532468\pi\)
\(720\) 766.413 0.0396702
\(721\) 0 0
\(722\) −2799.42 −0.144299
\(723\) −1121.80 −0.0577045
\(724\) −11722.1 −0.601724
\(725\) −2866.51 −0.146841
\(726\) −1934.64 −0.0989000
\(727\) −21071.2 −1.07495 −0.537474 0.843281i \(-0.680621\pi\)
−0.537474 + 0.843281i \(0.680621\pi\)
\(728\) 0 0
\(729\) −19200.0 −0.975459
\(730\) −7305.57 −0.370399
\(731\) 32465.4 1.64265
\(732\) −384.901 −0.0194349
\(733\) −22840.7 −1.15094 −0.575470 0.817823i \(-0.695181\pi\)
−0.575470 + 0.817823i \(0.695181\pi\)
\(734\) −3130.86 −0.157441
\(735\) 0 0
\(736\) −18699.0 −0.936485
\(737\) 5614.16 0.280597
\(738\) −18582.8 −0.926885
\(739\) 10837.3 0.539456 0.269728 0.962937i \(-0.413066\pi\)
0.269728 + 0.962937i \(0.413066\pi\)
\(740\) −5281.99 −0.262392
\(741\) −1637.75 −0.0811931
\(742\) 0 0
\(743\) −21631.9 −1.06810 −0.534050 0.845453i \(-0.679331\pi\)
−0.534050 + 0.845453i \(0.679331\pi\)
\(744\) −538.282 −0.0265247
\(745\) 5143.17 0.252928
\(746\) 4494.36 0.220577
\(747\) −1905.96 −0.0933540
\(748\) −38161.9 −1.86543
\(749\) 0 0
\(750\) 68.9103 0.00335500
\(751\) −16179.2 −0.786133 −0.393067 0.919510i \(-0.628586\pi\)
−0.393067 + 0.919510i \(0.628586\pi\)
\(752\) 853.352 0.0413811
\(753\) 1195.95 0.0578788
\(754\) −12994.3 −0.627620
\(755\) −355.037 −0.0171140
\(756\) 0 0
\(757\) −40930.9 −1.96520 −0.982601 0.185727i \(-0.940536\pi\)
−0.982601 + 0.185727i \(0.940536\pi\)
\(758\) 14807.2 0.709526
\(759\) 2339.86 0.111899
\(760\) 7892.11 0.376680
\(761\) 3183.97 0.151667 0.0758337 0.997120i \(-0.475838\pi\)
0.0758337 + 0.997120i \(0.475838\pi\)
\(762\) 967.917 0.0460157
\(763\) 0 0
\(764\) 2449.55 0.115997
\(765\) 14027.0 0.662936
\(766\) −15391.2 −0.725990
\(767\) −35450.0 −1.66887
\(768\) 1272.88 0.0598059
\(769\) 33595.8 1.57542 0.787708 0.616048i \(-0.211267\pi\)
0.787708 + 0.616048i \(0.211267\pi\)
\(770\) 0 0
\(771\) 0.947940 4.42791e−5 0
\(772\) 23208.8 1.08200
\(773\) 34386.0 1.59997 0.799986 0.600019i \(-0.204840\pi\)
0.799986 + 0.600019i \(0.204840\pi\)
\(774\) −13856.7 −0.643498
\(775\) −1841.26 −0.0853420
\(776\) −3197.72 −0.147927
\(777\) 0 0
\(778\) 17313.7 0.797849
\(779\) 30001.5 1.37986
\(780\) −598.834 −0.0274893
\(781\) −6334.33 −0.290218
\(782\) −17456.1 −0.798245
\(783\) −2056.90 −0.0938795
\(784\) 0 0
\(785\) 10306.6 0.468610
\(786\) −997.324 −0.0452588
\(787\) −8212.43 −0.371972 −0.185986 0.982552i \(-0.559548\pi\)
−0.185986 + 0.982552i \(0.559548\pi\)
\(788\) 1523.68 0.0688816
\(789\) 937.769 0.0423136
\(790\) −4967.17 −0.223701
\(791\) 0 0
\(792\) 41072.7 1.84274
\(793\) −15050.1 −0.673951
\(794\) −6004.92 −0.268396
\(795\) 451.852 0.0201579
\(796\) −25329.0 −1.12784
\(797\) 36798.3 1.63546 0.817732 0.575600i \(-0.195231\pi\)
0.817732 + 0.575600i \(0.195231\pi\)
\(798\) 0 0
\(799\) 15618.2 0.691528
\(800\) 4627.05 0.204488
\(801\) −21581.0 −0.951971
\(802\) −13948.2 −0.614125
\(803\) 61382.7 2.69757
\(804\) 141.228 0.00619492
\(805\) 0 0
\(806\) −8346.70 −0.364764
\(807\) 481.832 0.0210177
\(808\) 13599.8 0.592128
\(809\) 10186.2 0.442678 0.221339 0.975197i \(-0.428957\pi\)
0.221339 + 0.975197i \(0.428957\pi\)
\(810\) −5962.12 −0.258627
\(811\) 21196.9 0.917786 0.458893 0.888492i \(-0.348246\pi\)
0.458893 + 0.888492i \(0.348246\pi\)
\(812\) 0 0
\(813\) −2681.24 −0.115665
\(814\) −23150.8 −0.996851
\(815\) 9818.99 0.422018
\(816\) −197.983 −0.00849364
\(817\) 22371.3 0.957982
\(818\) −24146.5 −1.03211
\(819\) 0 0
\(820\) 10969.9 0.467177
\(821\) 7563.94 0.321539 0.160769 0.986992i \(-0.448602\pi\)
0.160769 + 0.986992i \(0.448602\pi\)
\(822\) 10.2061 0.000433065 0
\(823\) 8399.80 0.355770 0.177885 0.984051i \(-0.443075\pi\)
0.177885 + 0.984051i \(0.443075\pi\)
\(824\) 40004.1 1.69127
\(825\) −578.997 −0.0244340
\(826\) 0 0
\(827\) 5479.54 0.230402 0.115201 0.993342i \(-0.463249\pi\)
0.115201 + 0.993342i \(0.463249\pi\)
\(828\) −14282.6 −0.599462
\(829\) 9252.56 0.387641 0.193821 0.981037i \(-0.437912\pi\)
0.193821 + 0.981037i \(0.437912\pi\)
\(830\) −586.925 −0.0245452
\(831\) 190.179 0.00793890
\(832\) 17854.2 0.743972
\(833\) 0 0
\(834\) 344.887 0.0143195
\(835\) −14275.2 −0.591634
\(836\) −26296.6 −1.08790
\(837\) −1321.22 −0.0545615
\(838\) −4202.31 −0.173230
\(839\) −34517.6 −1.42036 −0.710178 0.704022i \(-0.751386\pi\)
−0.710178 + 0.704022i \(0.751386\pi\)
\(840\) 0 0
\(841\) −11241.9 −0.460943
\(842\) 15980.2 0.654055
\(843\) 594.031 0.0242699
\(844\) −10630.6 −0.433555
\(845\) −12430.1 −0.506044
\(846\) −6666.04 −0.270902
\(847\) 0 0
\(848\) 1547.55 0.0626686
\(849\) −1105.54 −0.0446901
\(850\) 4319.50 0.174303
\(851\) 20300.4 0.817732
\(852\) −159.344 −0.00640732
\(853\) −1498.57 −0.0601525 −0.0300763 0.999548i \(-0.509575\pi\)
−0.0300763 + 0.999548i \(0.509575\pi\)
\(854\) 0 0
\(855\) 9665.71 0.386620
\(856\) −23924.6 −0.955287
\(857\) −9357.02 −0.372963 −0.186482 0.982458i \(-0.559709\pi\)
−0.186482 + 0.982458i \(0.559709\pi\)
\(858\) −2624.67 −0.104435
\(859\) −29960.9 −1.19005 −0.595025 0.803707i \(-0.702858\pi\)
−0.595025 + 0.803707i \(0.702858\pi\)
\(860\) 8179.94 0.324341
\(861\) 0 0
\(862\) −12027.3 −0.475232
\(863\) −33941.0 −1.33878 −0.669389 0.742912i \(-0.733444\pi\)
−0.669389 + 0.742912i \(0.733444\pi\)
\(864\) 3320.19 0.130735
\(865\) 7766.03 0.305264
\(866\) 18788.1 0.737235
\(867\) −1988.04 −0.0778747
\(868\) 0 0
\(869\) 41735.0 1.62919
\(870\) −316.052 −0.0123163
\(871\) 5522.16 0.214824
\(872\) −12946.2 −0.502769
\(873\) −3916.35 −0.151831
\(874\) −12028.6 −0.465532
\(875\) 0 0
\(876\) 1544.12 0.0595559
\(877\) −39436.6 −1.51845 −0.759224 0.650830i \(-0.774421\pi\)
−0.759224 + 0.650830i \(0.774421\pi\)
\(878\) 19385.7 0.745142
\(879\) 1680.93 0.0645010
\(880\) −1983.00 −0.0759625
\(881\) 32411.4 1.23946 0.619732 0.784813i \(-0.287241\pi\)
0.619732 + 0.784813i \(0.287241\pi\)
\(882\) 0 0
\(883\) −17121.4 −0.652526 −0.326263 0.945279i \(-0.605790\pi\)
−0.326263 + 0.945279i \(0.605790\pi\)
\(884\) −37536.6 −1.42816
\(885\) −862.226 −0.0327496
\(886\) 24970.1 0.946824
\(887\) −687.797 −0.0260360 −0.0130180 0.999915i \(-0.504144\pi\)
−0.0130180 + 0.999915i \(0.504144\pi\)
\(888\) −1468.54 −0.0554967
\(889\) 0 0
\(890\) −6645.71 −0.250297
\(891\) 50094.8 1.88355
\(892\) −22837.6 −0.857243
\(893\) 10762.2 0.403294
\(894\) 567.068 0.0212143
\(895\) −1349.20 −0.0503898
\(896\) 0 0
\(897\) 2301.52 0.0856693
\(898\) 1781.01 0.0661839
\(899\) 8444.81 0.313293
\(900\) 3534.22 0.130897
\(901\) 28323.4 1.04727
\(902\) 48080.8 1.77485
\(903\) 0 0
\(904\) 19767.5 0.727276
\(905\) −11148.1 −0.409474
\(906\) −39.1451 −0.00143544
\(907\) 19104.2 0.699388 0.349694 0.936864i \(-0.386286\pi\)
0.349694 + 0.936864i \(0.386286\pi\)
\(908\) 13919.6 0.508743
\(909\) 16656.1 0.607754
\(910\) 0 0
\(911\) 23135.3 0.841390 0.420695 0.907202i \(-0.361786\pi\)
0.420695 + 0.907202i \(0.361786\pi\)
\(912\) −136.427 −0.00495344
\(913\) 4931.45 0.178759
\(914\) −17781.0 −0.643484
\(915\) −366.052 −0.0132255
\(916\) 7597.50 0.274049
\(917\) 0 0
\(918\) 3099.50 0.111437
\(919\) 47373.9 1.70046 0.850230 0.526412i \(-0.176463\pi\)
0.850230 + 0.526412i \(0.176463\pi\)
\(920\) −11090.7 −0.397447
\(921\) 511.224 0.0182903
\(922\) 749.481 0.0267710
\(923\) −6230.53 −0.222189
\(924\) 0 0
\(925\) −5023.33 −0.178558
\(926\) 11788.1 0.418338
\(927\) 48994.2 1.73590
\(928\) −21221.6 −0.750682
\(929\) −7617.72 −0.269031 −0.134515 0.990912i \(-0.542948\pi\)
−0.134515 + 0.990912i \(0.542948\pi\)
\(930\) −203.011 −0.00715806
\(931\) 0 0
\(932\) 32837.6 1.15411
\(933\) 3090.21 0.108434
\(934\) 1612.67 0.0564971
\(935\) −36293.2 −1.26943
\(936\) 40399.6 1.41079
\(937\) −52091.2 −1.81616 −0.908082 0.418792i \(-0.862454\pi\)
−0.908082 + 0.418792i \(0.862454\pi\)
\(938\) 0 0
\(939\) −2005.79 −0.0697088
\(940\) 3935.13 0.136542
\(941\) −56765.1 −1.96651 −0.983257 0.182225i \(-0.941670\pi\)
−0.983257 + 0.182225i \(0.941670\pi\)
\(942\) 1136.37 0.0393046
\(943\) −42160.9 −1.45594
\(944\) −2953.03 −0.101815
\(945\) 0 0
\(946\) 35852.5 1.23220
\(947\) −47629.2 −1.63436 −0.817181 0.576381i \(-0.804465\pi\)
−0.817181 + 0.576381i \(0.804465\pi\)
\(948\) 1049.87 0.0359686
\(949\) 60376.8 2.06524
\(950\) 2976.48 0.101652
\(951\) 2322.75 0.0792012
\(952\) 0 0
\(953\) 12893.5 0.438259 0.219129 0.975696i \(-0.429678\pi\)
0.219129 + 0.975696i \(0.429678\pi\)
\(954\) −12088.8 −0.410261
\(955\) 2329.60 0.0789362
\(956\) −7046.24 −0.238380
\(957\) 2655.52 0.0896979
\(958\) 16087.0 0.542534
\(959\) 0 0
\(960\) 434.257 0.0145996
\(961\) −24366.6 −0.817918
\(962\) −22771.5 −0.763183
\(963\) −29301.2 −0.980496
\(964\) −17717.2 −0.591943
\(965\) 22072.3 0.736303
\(966\) 0 0
\(967\) −28420.0 −0.945114 −0.472557 0.881300i \(-0.656669\pi\)
−0.472557 + 0.881300i \(0.656669\pi\)
\(968\) −77048.4 −2.55830
\(969\) −2496.89 −0.0827779
\(970\) −1206.01 −0.0399202
\(971\) 29273.8 0.967497 0.483749 0.875207i \(-0.339275\pi\)
0.483749 + 0.875207i \(0.339275\pi\)
\(972\) 3806.64 0.125615
\(973\) 0 0
\(974\) 1529.19 0.0503063
\(975\) −569.509 −0.0187065
\(976\) −1253.69 −0.0411165
\(977\) 12055.0 0.394753 0.197377 0.980328i \(-0.436758\pi\)
0.197377 + 0.980328i \(0.436758\pi\)
\(978\) 1082.61 0.0353967
\(979\) 55838.4 1.82288
\(980\) 0 0
\(981\) −15855.7 −0.516037
\(982\) 2134.84 0.0693743
\(983\) 10605.1 0.344099 0.172049 0.985088i \(-0.444961\pi\)
0.172049 + 0.985088i \(0.444961\pi\)
\(984\) 3049.94 0.0988094
\(985\) 1449.06 0.0468740
\(986\) −19811.0 −0.639870
\(987\) 0 0
\(988\) −25865.7 −0.832893
\(989\) −31438.2 −1.01080
\(990\) 15490.4 0.497290
\(991\) 45229.1 1.44980 0.724898 0.688856i \(-0.241887\pi\)
0.724898 + 0.688856i \(0.241887\pi\)
\(992\) −13631.4 −0.436287
\(993\) −327.854 −0.0104775
\(994\) 0 0
\(995\) −24088.7 −0.767500
\(996\) 124.054 0.00394658
\(997\) −49676.8 −1.57802 −0.789008 0.614383i \(-0.789405\pi\)
−0.789008 + 0.614383i \(0.789405\pi\)
\(998\) −32025.4 −1.01578
\(999\) −3604.55 −0.114157
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 245.4.a.p.1.5 yes 6
3.2 odd 2 2205.4.a.ca.1.2 6
5.4 even 2 1225.4.a.bi.1.2 6
7.2 even 3 245.4.e.p.116.2 12
7.3 odd 6 245.4.e.q.226.2 12
7.4 even 3 245.4.e.p.226.2 12
7.5 odd 6 245.4.e.q.116.2 12
7.6 odd 2 245.4.a.o.1.5 6
21.20 even 2 2205.4.a.bz.1.2 6
35.34 odd 2 1225.4.a.bj.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.o.1.5 6 7.6 odd 2
245.4.a.p.1.5 yes 6 1.1 even 1 trivial
245.4.e.p.116.2 12 7.2 even 3
245.4.e.p.226.2 12 7.4 even 3
245.4.e.q.116.2 12 7.5 odd 6
245.4.e.q.226.2 12 7.3 odd 6
1225.4.a.bi.1.2 6 5.4 even 2
1225.4.a.bj.1.2 6 35.34 odd 2
2205.4.a.bz.1.2 6 21.20 even 2
2205.4.a.ca.1.2 6 3.2 odd 2