Properties

Label 1575.4.a.m.1.2
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.43845 q^{2} -5.93087 q^{4} +7.00000 q^{7} +20.0388 q^{8} +O(q^{10})\) \(q-1.43845 q^{2} -5.93087 q^{4} +7.00000 q^{7} +20.0388 q^{8} -7.61553 q^{11} -52.3542 q^{13} -10.0691 q^{14} +18.6222 q^{16} -49.7235 q^{17} +140.600 q^{19} +10.9545 q^{22} -23.4470 q^{23} +75.3087 q^{26} -41.5161 q^{28} -157.170 q^{29} +127.892 q^{31} -187.098 q^{32} +71.5246 q^{34} +115.477 q^{37} -202.246 q^{38} +188.617 q^{41} -322.186 q^{43} +45.1667 q^{44} +33.7272 q^{46} -76.6477 q^{47} +49.0000 q^{49} +310.506 q^{52} -424.172 q^{53} +140.272 q^{56} +226.081 q^{58} -107.784 q^{59} +915.511 q^{61} -183.966 q^{62} +120.153 q^{64} +451.723 q^{67} +294.903 q^{68} -907.312 q^{71} -755.956 q^{73} -166.108 q^{74} -833.882 q^{76} -53.3087 q^{77} +22.5834 q^{79} -271.316 q^{82} +1112.25 q^{83} +463.447 q^{86} -152.606 q^{88} -1518.81 q^{89} -366.479 q^{91} +139.061 q^{92} +110.254 q^{94} +549.987 q^{97} -70.4839 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 7 q^{2} + 17 q^{4} + 14 q^{7} - 63 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 7 q^{2} + 17 q^{4} + 14 q^{7} - 63 q^{8} + 26 q^{11} - 14 q^{13} - 49 q^{14} + 297 q^{16} + 16 q^{17} + 174 q^{19} - 176 q^{22} + 184 q^{23} - 138 q^{26} + 119 q^{28} + 32 q^{29} + 330 q^{31} - 1071 q^{32} - 294 q^{34} + 132 q^{37} - 388 q^{38} - 200 q^{41} - 364 q^{43} + 816 q^{44} - 1120 q^{46} + 292 q^{47} + 98 q^{49} + 1190 q^{52} + 34 q^{53} - 441 q^{56} - 826 q^{58} - 364 q^{59} + 792 q^{61} - 1308 q^{62} + 2809 q^{64} + 788 q^{67} + 1802 q^{68} - 454 q^{71} - 778 q^{73} - 258 q^{74} - 68 q^{76} + 182 q^{77} + 408 q^{79} + 1890 q^{82} + 1136 q^{83} + 696 q^{86} - 2944 q^{88} - 36 q^{89} - 98 q^{91} + 4896 q^{92} - 1940 q^{94} + 498 q^{97} - 343 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.43845 −0.508568 −0.254284 0.967130i \(-0.581840\pi\)
−0.254284 + 0.967130i \(0.581840\pi\)
\(3\) 0 0
\(4\) −5.93087 −0.741359
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 20.0388 0.885599
\(9\) 0 0
\(10\) 0 0
\(11\) −7.61553 −0.208743 −0.104371 0.994538i \(-0.533283\pi\)
−0.104371 + 0.994538i \(0.533283\pi\)
\(12\) 0 0
\(13\) −52.3542 −1.11696 −0.558478 0.829519i \(-0.688615\pi\)
−0.558478 + 0.829519i \(0.688615\pi\)
\(14\) −10.0691 −0.192221
\(15\) 0 0
\(16\) 18.6222 0.290971
\(17\) −49.7235 −0.709395 −0.354697 0.934981i \(-0.615416\pi\)
−0.354697 + 0.934981i \(0.615416\pi\)
\(18\) 0 0
\(19\) 140.600 1.69768 0.848840 0.528649i \(-0.177301\pi\)
0.848840 + 0.528649i \(0.177301\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 10.9545 0.106160
\(23\) −23.4470 −0.212566 −0.106283 0.994336i \(-0.533895\pi\)
−0.106283 + 0.994336i \(0.533895\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 75.3087 0.568048
\(27\) 0 0
\(28\) −41.5161 −0.280207
\(29\) −157.170 −1.00641 −0.503204 0.864168i \(-0.667845\pi\)
−0.503204 + 0.864168i \(0.667845\pi\)
\(30\) 0 0
\(31\) 127.892 0.740971 0.370485 0.928838i \(-0.379191\pi\)
0.370485 + 0.928838i \(0.379191\pi\)
\(32\) −187.098 −1.03358
\(33\) 0 0
\(34\) 71.5246 0.360776
\(35\) 0 0
\(36\) 0 0
\(37\) 115.477 0.513090 0.256545 0.966532i \(-0.417416\pi\)
0.256545 + 0.966532i \(0.417416\pi\)
\(38\) −202.246 −0.863386
\(39\) 0 0
\(40\) 0 0
\(41\) 188.617 0.718466 0.359233 0.933248i \(-0.383038\pi\)
0.359233 + 0.933248i \(0.383038\pi\)
\(42\) 0 0
\(43\) −322.186 −1.14262 −0.571312 0.820733i \(-0.693565\pi\)
−0.571312 + 0.820733i \(0.693565\pi\)
\(44\) 45.1667 0.154753
\(45\) 0 0
\(46\) 33.7272 0.108104
\(47\) −76.6477 −0.237877 −0.118938 0.992902i \(-0.537949\pi\)
−0.118938 + 0.992902i \(0.537949\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 310.506 0.828065
\(53\) −424.172 −1.09933 −0.549666 0.835385i \(-0.685245\pi\)
−0.549666 + 0.835385i \(0.685245\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 140.272 0.334725
\(57\) 0 0
\(58\) 226.081 0.511827
\(59\) −107.784 −0.237835 −0.118918 0.992904i \(-0.537942\pi\)
−0.118918 + 0.992904i \(0.537942\pi\)
\(60\) 0 0
\(61\) 915.511 1.92163 0.960813 0.277197i \(-0.0894053\pi\)
0.960813 + 0.277197i \(0.0894053\pi\)
\(62\) −183.966 −0.376834
\(63\) 0 0
\(64\) 120.153 0.234673
\(65\) 0 0
\(66\) 0 0
\(67\) 451.723 0.823684 0.411842 0.911255i \(-0.364886\pi\)
0.411842 + 0.911255i \(0.364886\pi\)
\(68\) 294.903 0.525916
\(69\) 0 0
\(70\) 0 0
\(71\) −907.312 −1.51659 −0.758297 0.651909i \(-0.773968\pi\)
−0.758297 + 0.651909i \(0.773968\pi\)
\(72\) 0 0
\(73\) −755.956 −1.21203 −0.606014 0.795454i \(-0.707232\pi\)
−0.606014 + 0.795454i \(0.707232\pi\)
\(74\) −166.108 −0.260941
\(75\) 0 0
\(76\) −833.882 −1.25859
\(77\) −53.3087 −0.0788973
\(78\) 0 0
\(79\) 22.5834 0.0321624 0.0160812 0.999871i \(-0.494881\pi\)
0.0160812 + 0.999871i \(0.494881\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −271.316 −0.365389
\(83\) 1112.25 1.47091 0.735454 0.677575i \(-0.236969\pi\)
0.735454 + 0.677575i \(0.236969\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 463.447 0.581102
\(87\) 0 0
\(88\) −152.606 −0.184862
\(89\) −1518.81 −1.80892 −0.904458 0.426562i \(-0.859725\pi\)
−0.904458 + 0.426562i \(0.859725\pi\)
\(90\) 0 0
\(91\) −366.479 −0.422170
\(92\) 139.061 0.157588
\(93\) 0 0
\(94\) 110.254 0.120977
\(95\) 0 0
\(96\) 0 0
\(97\) 549.987 0.575698 0.287849 0.957676i \(-0.407060\pi\)
0.287849 + 0.957676i \(0.407060\pi\)
\(98\) −70.4839 −0.0726526
\(99\) 0 0
\(100\) 0 0
\(101\) 533.299 0.525398 0.262699 0.964878i \(-0.415387\pi\)
0.262699 + 0.964878i \(0.415387\pi\)
\(102\) 0 0
\(103\) −1357.79 −1.29890 −0.649451 0.760404i \(-0.725001\pi\)
−0.649451 + 0.760404i \(0.725001\pi\)
\(104\) −1049.12 −0.989176
\(105\) 0 0
\(106\) 610.149 0.559084
\(107\) 913.693 0.825515 0.412757 0.910841i \(-0.364566\pi\)
0.412757 + 0.910841i \(0.364566\pi\)
\(108\) 0 0
\(109\) −160.489 −0.141028 −0.0705139 0.997511i \(-0.522464\pi\)
−0.0705139 + 0.997511i \(0.522464\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 130.355 0.109977
\(113\) −1788.48 −1.48891 −0.744453 0.667675i \(-0.767290\pi\)
−0.744453 + 0.667675i \(0.767290\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 932.157 0.746109
\(117\) 0 0
\(118\) 155.042 0.120955
\(119\) −348.064 −0.268126
\(120\) 0 0
\(121\) −1273.00 −0.956427
\(122\) −1316.91 −0.977277
\(123\) 0 0
\(124\) −758.511 −0.549325
\(125\) 0 0
\(126\) 0 0
\(127\) 271.015 0.189360 0.0946799 0.995508i \(-0.469817\pi\)
0.0946799 + 0.995508i \(0.469817\pi\)
\(128\) 1323.95 0.914231
\(129\) 0 0
\(130\) 0 0
\(131\) 763.151 0.508984 0.254492 0.967075i \(-0.418092\pi\)
0.254492 + 0.967075i \(0.418092\pi\)
\(132\) 0 0
\(133\) 984.203 0.641663
\(134\) −649.780 −0.418899
\(135\) 0 0
\(136\) −996.400 −0.628240
\(137\) −240.934 −0.150251 −0.0751254 0.997174i \(-0.523936\pi\)
−0.0751254 + 0.997174i \(0.523936\pi\)
\(138\) 0 0
\(139\) −103.150 −0.0629427 −0.0314714 0.999505i \(-0.510019\pi\)
−0.0314714 + 0.999505i \(0.510019\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1305.12 0.771291
\(143\) 398.705 0.233156
\(144\) 0 0
\(145\) 0 0
\(146\) 1087.40 0.616398
\(147\) 0 0
\(148\) −684.881 −0.380384
\(149\) 3256.71 1.79061 0.895303 0.445458i \(-0.146959\pi\)
0.895303 + 0.445458i \(0.146959\pi\)
\(150\) 0 0
\(151\) 1471.04 0.792793 0.396396 0.918080i \(-0.370261\pi\)
0.396396 + 0.918080i \(0.370261\pi\)
\(152\) 2817.47 1.50346
\(153\) 0 0
\(154\) 76.6817 0.0401246
\(155\) 0 0
\(156\) 0 0
\(157\) 1394.22 0.708730 0.354365 0.935107i \(-0.384697\pi\)
0.354365 + 0.935107i \(0.384697\pi\)
\(158\) −32.4850 −0.0163567
\(159\) 0 0
\(160\) 0 0
\(161\) −164.129 −0.0803426
\(162\) 0 0
\(163\) 3674.27 1.76559 0.882794 0.469760i \(-0.155659\pi\)
0.882794 + 0.469760i \(0.155659\pi\)
\(164\) −1118.67 −0.532641
\(165\) 0 0
\(166\) −1599.91 −0.748056
\(167\) 4041.09 1.87251 0.936254 0.351325i \(-0.114269\pi\)
0.936254 + 0.351325i \(0.114269\pi\)
\(168\) 0 0
\(169\) 543.958 0.247591
\(170\) 0 0
\(171\) 0 0
\(172\) 1910.84 0.847094
\(173\) 59.4582 0.0261302 0.0130651 0.999915i \(-0.495841\pi\)
0.0130651 + 0.999915i \(0.495841\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −141.818 −0.0607381
\(177\) 0 0
\(178\) 2184.73 0.919957
\(179\) 2973.32 1.24155 0.620773 0.783991i \(-0.286819\pi\)
0.620773 + 0.783991i \(0.286819\pi\)
\(180\) 0 0
\(181\) −676.220 −0.277696 −0.138848 0.990314i \(-0.544340\pi\)
−0.138848 + 0.990314i \(0.544340\pi\)
\(182\) 527.161 0.214702
\(183\) 0 0
\(184\) −469.849 −0.188249
\(185\) 0 0
\(186\) 0 0
\(187\) 378.671 0.148081
\(188\) 454.588 0.176352
\(189\) 0 0
\(190\) 0 0
\(191\) −16.5589 −0.00627308 −0.00313654 0.999995i \(-0.500998\pi\)
−0.00313654 + 0.999995i \(0.500998\pi\)
\(192\) 0 0
\(193\) −2694.68 −1.00501 −0.502506 0.864574i \(-0.667589\pi\)
−0.502506 + 0.864574i \(0.667589\pi\)
\(194\) −791.127 −0.292781
\(195\) 0 0
\(196\) −290.613 −0.105908
\(197\) 1027.38 0.371561 0.185781 0.982591i \(-0.440519\pi\)
0.185781 + 0.982591i \(0.440519\pi\)
\(198\) 0 0
\(199\) 2823.77 1.00589 0.502944 0.864319i \(-0.332250\pi\)
0.502944 + 0.864319i \(0.332250\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −767.123 −0.267201
\(203\) −1100.19 −0.380386
\(204\) 0 0
\(205\) 0 0
\(206\) 1953.11 0.660579
\(207\) 0 0
\(208\) −974.948 −0.325002
\(209\) −1070.75 −0.354378
\(210\) 0 0
\(211\) 5151.16 1.68067 0.840333 0.542071i \(-0.182360\pi\)
0.840333 + 0.542071i \(0.182360\pi\)
\(212\) 2515.71 0.814999
\(213\) 0 0
\(214\) −1314.30 −0.419830
\(215\) 0 0
\(216\) 0 0
\(217\) 895.244 0.280061
\(218\) 230.855 0.0717222
\(219\) 0 0
\(220\) 0 0
\(221\) 2603.23 0.792363
\(222\) 0 0
\(223\) 114.496 0.0343822 0.0171911 0.999852i \(-0.494528\pi\)
0.0171911 + 0.999852i \(0.494528\pi\)
\(224\) −1309.68 −0.390656
\(225\) 0 0
\(226\) 2572.64 0.757209
\(227\) 4744.75 1.38731 0.693657 0.720306i \(-0.255999\pi\)
0.693657 + 0.720306i \(0.255999\pi\)
\(228\) 0 0
\(229\) −5384.47 −1.55378 −0.776891 0.629635i \(-0.783204\pi\)
−0.776891 + 0.629635i \(0.783204\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3149.51 −0.891274
\(233\) −1608.91 −0.452373 −0.226187 0.974084i \(-0.572626\pi\)
−0.226187 + 0.974084i \(0.572626\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 639.253 0.176321
\(237\) 0 0
\(238\) 500.672 0.136360
\(239\) 3113.11 0.842554 0.421277 0.906932i \(-0.361582\pi\)
0.421277 + 0.906932i \(0.361582\pi\)
\(240\) 0 0
\(241\) 7136.38 1.90745 0.953724 0.300685i \(-0.0972151\pi\)
0.953724 + 0.300685i \(0.0972151\pi\)
\(242\) 1831.15 0.486408
\(243\) 0 0
\(244\) −5429.78 −1.42461
\(245\) 0 0
\(246\) 0 0
\(247\) −7361.01 −1.89624
\(248\) 2562.81 0.656203
\(249\) 0 0
\(250\) 0 0
\(251\) 225.504 0.0567079 0.0283539 0.999598i \(-0.490973\pi\)
0.0283539 + 0.999598i \(0.490973\pi\)
\(252\) 0 0
\(253\) 178.561 0.0443717
\(254\) −389.841 −0.0963024
\(255\) 0 0
\(256\) −2865.65 −0.699621
\(257\) −4423.05 −1.07355 −0.536775 0.843725i \(-0.680358\pi\)
−0.536775 + 0.843725i \(0.680358\pi\)
\(258\) 0 0
\(259\) 808.341 0.193930
\(260\) 0 0
\(261\) 0 0
\(262\) −1097.75 −0.258853
\(263\) 6540.56 1.53349 0.766746 0.641950i \(-0.221875\pi\)
0.766746 + 0.641950i \(0.221875\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1415.72 −0.326329
\(267\) 0 0
\(268\) −2679.11 −0.610645
\(269\) −2262.97 −0.512920 −0.256460 0.966555i \(-0.582556\pi\)
−0.256460 + 0.966555i \(0.582556\pi\)
\(270\) 0 0
\(271\) 1615.68 0.362160 0.181080 0.983468i \(-0.442041\pi\)
0.181080 + 0.983468i \(0.442041\pi\)
\(272\) −925.959 −0.206414
\(273\) 0 0
\(274\) 346.571 0.0764127
\(275\) 0 0
\(276\) 0 0
\(277\) −4691.55 −1.01765 −0.508823 0.860871i \(-0.669919\pi\)
−0.508823 + 0.860871i \(0.669919\pi\)
\(278\) 148.375 0.0320107
\(279\) 0 0
\(280\) 0 0
\(281\) 8119.00 1.72363 0.861813 0.507226i \(-0.169329\pi\)
0.861813 + 0.507226i \(0.169329\pi\)
\(282\) 0 0
\(283\) 3633.75 0.763265 0.381632 0.924314i \(-0.375362\pi\)
0.381632 + 0.924314i \(0.375362\pi\)
\(284\) 5381.15 1.12434
\(285\) 0 0
\(286\) −573.515 −0.118576
\(287\) 1320.32 0.271554
\(288\) 0 0
\(289\) −2440.58 −0.496759
\(290\) 0 0
\(291\) 0 0
\(292\) 4483.48 0.898547
\(293\) −7981.99 −1.59151 −0.795756 0.605618i \(-0.792926\pi\)
−0.795756 + 0.605618i \(0.792926\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2314.03 0.454392
\(297\) 0 0
\(298\) −4684.61 −0.910645
\(299\) 1227.55 0.237427
\(300\) 0 0
\(301\) −2255.30 −0.431871
\(302\) −2116.02 −0.403189
\(303\) 0 0
\(304\) 2618.28 0.493977
\(305\) 0 0
\(306\) 0 0
\(307\) 7118.15 1.32330 0.661652 0.749811i \(-0.269856\pi\)
0.661652 + 0.749811i \(0.269856\pi\)
\(308\) 316.167 0.0584912
\(309\) 0 0
\(310\) 0 0
\(311\) 9155.92 1.66940 0.834702 0.550703i \(-0.185640\pi\)
0.834702 + 0.550703i \(0.185640\pi\)
\(312\) 0 0
\(313\) 6163.44 1.11303 0.556515 0.830838i \(-0.312138\pi\)
0.556515 + 0.830838i \(0.312138\pi\)
\(314\) −2005.51 −0.360438
\(315\) 0 0
\(316\) −133.939 −0.0238438
\(317\) 8658.37 1.53408 0.767038 0.641601i \(-0.221730\pi\)
0.767038 + 0.641601i \(0.221730\pi\)
\(318\) 0 0
\(319\) 1196.94 0.210080
\(320\) 0 0
\(321\) 0 0
\(322\) 236.090 0.0408597
\(323\) −6991.14 −1.20433
\(324\) 0 0
\(325\) 0 0
\(326\) −5285.24 −0.897922
\(327\) 0 0
\(328\) 3779.67 0.636272
\(329\) −536.534 −0.0899090
\(330\) 0 0
\(331\) −128.477 −0.0213346 −0.0106673 0.999943i \(-0.503396\pi\)
−0.0106673 + 0.999943i \(0.503396\pi\)
\(332\) −6596.61 −1.09047
\(333\) 0 0
\(334\) −5812.89 −0.952297
\(335\) 0 0
\(336\) 0 0
\(337\) −7784.57 −1.25832 −0.629158 0.777277i \(-0.716600\pi\)
−0.629158 + 0.777277i \(0.716600\pi\)
\(338\) −782.455 −0.125917
\(339\) 0 0
\(340\) 0 0
\(341\) −973.965 −0.154672
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) −6456.22 −1.01191
\(345\) 0 0
\(346\) −85.5274 −0.0132890
\(347\) 3740.26 0.578638 0.289319 0.957233i \(-0.406571\pi\)
0.289319 + 0.957233i \(0.406571\pi\)
\(348\) 0 0
\(349\) 5676.86 0.870703 0.435351 0.900261i \(-0.356624\pi\)
0.435351 + 0.900261i \(0.356624\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1424.85 0.215752
\(353\) 909.564 0.137142 0.0685711 0.997646i \(-0.478156\pi\)
0.0685711 + 0.997646i \(0.478156\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 9007.87 1.34106
\(357\) 0 0
\(358\) −4276.97 −0.631410
\(359\) −2678.57 −0.393788 −0.196894 0.980425i \(-0.563085\pi\)
−0.196894 + 0.980425i \(0.563085\pi\)
\(360\) 0 0
\(361\) 12909.5 1.88212
\(362\) 972.706 0.141227
\(363\) 0 0
\(364\) 2173.54 0.312979
\(365\) 0 0
\(366\) 0 0
\(367\) 716.898 0.101967 0.0509833 0.998700i \(-0.483764\pi\)
0.0509833 + 0.998700i \(0.483764\pi\)
\(368\) −436.633 −0.0618508
\(369\) 0 0
\(370\) 0 0
\(371\) −2969.21 −0.415508
\(372\) 0 0
\(373\) 2006.15 0.278484 0.139242 0.990258i \(-0.455533\pi\)
0.139242 + 0.990258i \(0.455533\pi\)
\(374\) −544.698 −0.0753092
\(375\) 0 0
\(376\) −1535.93 −0.210664
\(377\) 8228.53 1.12411
\(378\) 0 0
\(379\) 7277.53 0.986336 0.493168 0.869934i \(-0.335839\pi\)
0.493168 + 0.869934i \(0.335839\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 23.8191 0.00319029
\(383\) 5953.94 0.794339 0.397170 0.917745i \(-0.369992\pi\)
0.397170 + 0.917745i \(0.369992\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3876.16 0.511117
\(387\) 0 0
\(388\) −3261.90 −0.426799
\(389\) −1867.98 −0.243472 −0.121736 0.992563i \(-0.538846\pi\)
−0.121736 + 0.992563i \(0.538846\pi\)
\(390\) 0 0
\(391\) 1165.86 0.150794
\(392\) 981.902 0.126514
\(393\) 0 0
\(394\) −1477.83 −0.188964
\(395\) 0 0
\(396\) 0 0
\(397\) 2160.07 0.273075 0.136538 0.990635i \(-0.456403\pi\)
0.136538 + 0.990635i \(0.456403\pi\)
\(398\) −4061.85 −0.511563
\(399\) 0 0
\(400\) 0 0
\(401\) −1954.81 −0.243438 −0.121719 0.992565i \(-0.538841\pi\)
−0.121719 + 0.992565i \(0.538841\pi\)
\(402\) 0 0
\(403\) −6695.68 −0.827632
\(404\) −3162.93 −0.389509
\(405\) 0 0
\(406\) 1582.57 0.193452
\(407\) −879.420 −0.107104
\(408\) 0 0
\(409\) 14895.1 1.80077 0.900384 0.435096i \(-0.143286\pi\)
0.900384 + 0.435096i \(0.143286\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8052.86 0.962952
\(413\) −754.489 −0.0898934
\(414\) 0 0
\(415\) 0 0
\(416\) 9795.34 1.15446
\(417\) 0 0
\(418\) 1540.21 0.180225
\(419\) −12608.9 −1.47013 −0.735067 0.677994i \(-0.762849\pi\)
−0.735067 + 0.677994i \(0.762849\pi\)
\(420\) 0 0
\(421\) −7862.86 −0.910243 −0.455122 0.890429i \(-0.650404\pi\)
−0.455122 + 0.890429i \(0.650404\pi\)
\(422\) −7409.67 −0.854732
\(423\) 0 0
\(424\) −8499.91 −0.973567
\(425\) 0 0
\(426\) 0 0
\(427\) 6408.58 0.726307
\(428\) −5419.00 −0.612002
\(429\) 0 0
\(430\) 0 0
\(431\) −14291.0 −1.59715 −0.798575 0.601896i \(-0.794412\pi\)
−0.798575 + 0.601896i \(0.794412\pi\)
\(432\) 0 0
\(433\) 13759.0 1.52705 0.763527 0.645776i \(-0.223466\pi\)
0.763527 + 0.645776i \(0.223466\pi\)
\(434\) −1287.76 −0.142430
\(435\) 0 0
\(436\) 951.838 0.104552
\(437\) −3296.65 −0.360870
\(438\) 0 0
\(439\) −6093.13 −0.662436 −0.331218 0.943554i \(-0.607459\pi\)
−0.331218 + 0.943554i \(0.607459\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3744.61 −0.402970
\(443\) 13449.5 1.44244 0.721222 0.692704i \(-0.243581\pi\)
0.721222 + 0.692704i \(0.243581\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −164.697 −0.0174857
\(447\) 0 0
\(448\) 841.068 0.0886981
\(449\) −6893.83 −0.724588 −0.362294 0.932064i \(-0.618006\pi\)
−0.362294 + 0.932064i \(0.618006\pi\)
\(450\) 0 0
\(451\) −1436.42 −0.149974
\(452\) 10607.3 1.10381
\(453\) 0 0
\(454\) −6825.07 −0.705543
\(455\) 0 0
\(456\) 0 0
\(457\) 11820.1 1.20990 0.604948 0.796265i \(-0.293194\pi\)
0.604948 + 0.796265i \(0.293194\pi\)
\(458\) 7745.28 0.790203
\(459\) 0 0
\(460\) 0 0
\(461\) −8443.38 −0.853031 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(462\) 0 0
\(463\) 1269.74 0.127451 0.0637257 0.997967i \(-0.479702\pi\)
0.0637257 + 0.997967i \(0.479702\pi\)
\(464\) −2926.86 −0.292836
\(465\) 0 0
\(466\) 2314.33 0.230063
\(467\) −16481.8 −1.63316 −0.816579 0.577233i \(-0.804132\pi\)
−0.816579 + 0.577233i \(0.804132\pi\)
\(468\) 0 0
\(469\) 3162.06 0.311323
\(470\) 0 0
\(471\) 0 0
\(472\) −2159.87 −0.210627
\(473\) 2453.61 0.238514
\(474\) 0 0
\(475\) 0 0
\(476\) 2064.32 0.198778
\(477\) 0 0
\(478\) −4478.05 −0.428496
\(479\) 1400.21 0.133564 0.0667822 0.997768i \(-0.478727\pi\)
0.0667822 + 0.997768i \(0.478727\pi\)
\(480\) 0 0
\(481\) −6045.72 −0.573100
\(482\) −10265.3 −0.970066
\(483\) 0 0
\(484\) 7550.02 0.709055
\(485\) 0 0
\(486\) 0 0
\(487\) 14165.9 1.31811 0.659055 0.752094i \(-0.270956\pi\)
0.659055 + 0.752094i \(0.270956\pi\)
\(488\) 18345.8 1.70179
\(489\) 0 0
\(490\) 0 0
\(491\) −4739.28 −0.435603 −0.217801 0.975993i \(-0.569888\pi\)
−0.217801 + 0.975993i \(0.569888\pi\)
\(492\) 0 0
\(493\) 7815.06 0.713940
\(494\) 10588.4 0.964364
\(495\) 0 0
\(496\) 2381.63 0.215601
\(497\) −6351.19 −0.573219
\(498\) 0 0
\(499\) −11370.0 −1.02003 −0.510013 0.860167i \(-0.670360\pi\)
−0.510013 + 0.860167i \(0.670360\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −324.375 −0.0288398
\(503\) 9212.48 0.816629 0.408314 0.912841i \(-0.366117\pi\)
0.408314 + 0.912841i \(0.366117\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −256.851 −0.0225660
\(507\) 0 0
\(508\) −1607.36 −0.140384
\(509\) 15938.3 1.38792 0.693960 0.720014i \(-0.255864\pi\)
0.693960 + 0.720014i \(0.255864\pi\)
\(510\) 0 0
\(511\) −5291.69 −0.458103
\(512\) −6469.49 −0.558426
\(513\) 0 0
\(514\) 6362.33 0.545973
\(515\) 0 0
\(516\) 0 0
\(517\) 583.713 0.0496550
\(518\) −1162.76 −0.0986265
\(519\) 0 0
\(520\) 0 0
\(521\) −6442.99 −0.541790 −0.270895 0.962609i \(-0.587320\pi\)
−0.270895 + 0.962609i \(0.587320\pi\)
\(522\) 0 0
\(523\) −986.655 −0.0824922 −0.0412461 0.999149i \(-0.513133\pi\)
−0.0412461 + 0.999149i \(0.513133\pi\)
\(524\) −4526.15 −0.377339
\(525\) 0 0
\(526\) −9408.26 −0.779885
\(527\) −6359.24 −0.525641
\(528\) 0 0
\(529\) −11617.2 −0.954815
\(530\) 0 0
\(531\) 0 0
\(532\) −5837.18 −0.475703
\(533\) −9874.91 −0.802495
\(534\) 0 0
\(535\) 0 0
\(536\) 9052.01 0.729454
\(537\) 0 0
\(538\) 3255.16 0.260855
\(539\) −373.161 −0.0298204
\(540\) 0 0
\(541\) −12681.1 −1.00777 −0.503885 0.863771i \(-0.668097\pi\)
−0.503885 + 0.863771i \(0.668097\pi\)
\(542\) −2324.06 −0.184183
\(543\) 0 0
\(544\) 9303.14 0.733215
\(545\) 0 0
\(546\) 0 0
\(547\) 14826.4 1.15892 0.579462 0.814999i \(-0.303263\pi\)
0.579462 + 0.814999i \(0.303263\pi\)
\(548\) 1428.95 0.111390
\(549\) 0 0
\(550\) 0 0
\(551\) −22098.2 −1.70856
\(552\) 0 0
\(553\) 158.083 0.0121562
\(554\) 6748.55 0.517542
\(555\) 0 0
\(556\) 611.767 0.0466632
\(557\) 1926.46 0.146547 0.0732737 0.997312i \(-0.476655\pi\)
0.0732737 + 0.997312i \(0.476655\pi\)
\(558\) 0 0
\(559\) 16867.8 1.27626
\(560\) 0 0
\(561\) 0 0
\(562\) −11678.8 −0.876581
\(563\) −18624.8 −1.39422 −0.697108 0.716966i \(-0.745530\pi\)
−0.697108 + 0.716966i \(0.745530\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −5226.96 −0.388172
\(567\) 0 0
\(568\) −18181.5 −1.34309
\(569\) 20093.9 1.48045 0.740227 0.672357i \(-0.234718\pi\)
0.740227 + 0.672357i \(0.234718\pi\)
\(570\) 0 0
\(571\) 4535.25 0.332389 0.166195 0.986093i \(-0.446852\pi\)
0.166195 + 0.986093i \(0.446852\pi\)
\(572\) −2364.66 −0.172852
\(573\) 0 0
\(574\) −1899.21 −0.138104
\(575\) 0 0
\(576\) 0 0
\(577\) −10034.6 −0.723994 −0.361997 0.932179i \(-0.617905\pi\)
−0.361997 + 0.932179i \(0.617905\pi\)
\(578\) 3510.64 0.252636
\(579\) 0 0
\(580\) 0 0
\(581\) 7785.75 0.555951
\(582\) 0 0
\(583\) 3230.30 0.229477
\(584\) −15148.5 −1.07337
\(585\) 0 0
\(586\) 11481.7 0.809391
\(587\) −11192.6 −0.786999 −0.393499 0.919325i \(-0.628736\pi\)
−0.393499 + 0.919325i \(0.628736\pi\)
\(588\) 0 0
\(589\) 17981.7 1.25793
\(590\) 0 0
\(591\) 0 0
\(592\) 2150.44 0.149295
\(593\) 20317.6 1.40699 0.703493 0.710703i \(-0.251623\pi\)
0.703493 + 0.710703i \(0.251623\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −19315.1 −1.32748
\(597\) 0 0
\(598\) −1765.76 −0.120748
\(599\) 26376.5 1.79919 0.899594 0.436727i \(-0.143862\pi\)
0.899594 + 0.436727i \(0.143862\pi\)
\(600\) 0 0
\(601\) 9266.13 0.628907 0.314454 0.949273i \(-0.398179\pi\)
0.314454 + 0.949273i \(0.398179\pi\)
\(602\) 3244.13 0.219636
\(603\) 0 0
\(604\) −8724.56 −0.587744
\(605\) 0 0
\(606\) 0 0
\(607\) −11338.0 −0.758149 −0.379075 0.925366i \(-0.623758\pi\)
−0.379075 + 0.925366i \(0.623758\pi\)
\(608\) −26306.0 −1.75469
\(609\) 0 0
\(610\) 0 0
\(611\) 4012.83 0.265698
\(612\) 0 0
\(613\) −25712.5 −1.69416 −0.847078 0.531469i \(-0.821640\pi\)
−0.847078 + 0.531469i \(0.821640\pi\)
\(614\) −10239.1 −0.672990
\(615\) 0 0
\(616\) −1068.24 −0.0698714
\(617\) 663.465 0.0432903 0.0216451 0.999766i \(-0.493110\pi\)
0.0216451 + 0.999766i \(0.493110\pi\)
\(618\) 0 0
\(619\) −12768.7 −0.829108 −0.414554 0.910025i \(-0.636062\pi\)
−0.414554 + 0.910025i \(0.636062\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −13170.3 −0.849005
\(623\) −10631.7 −0.683706
\(624\) 0 0
\(625\) 0 0
\(626\) −8865.78 −0.566051
\(627\) 0 0
\(628\) −8268.92 −0.525423
\(629\) −5741.93 −0.363984
\(630\) 0 0
\(631\) −14937.6 −0.942400 −0.471200 0.882026i \(-0.656179\pi\)
−0.471200 + 0.882026i \(0.656179\pi\)
\(632\) 452.544 0.0284829
\(633\) 0 0
\(634\) −12454.6 −0.780182
\(635\) 0 0
\(636\) 0 0
\(637\) −2565.35 −0.159565
\(638\) −1721.73 −0.106840
\(639\) 0 0
\(640\) 0 0
\(641\) −10903.0 −0.671829 −0.335914 0.941893i \(-0.609045\pi\)
−0.335914 + 0.941893i \(0.609045\pi\)
\(642\) 0 0
\(643\) 7623.47 0.467559 0.233779 0.972290i \(-0.424891\pi\)
0.233779 + 0.972290i \(0.424891\pi\)
\(644\) 973.426 0.0595627
\(645\) 0 0
\(646\) 10056.4 0.612482
\(647\) 5384.84 0.327202 0.163601 0.986527i \(-0.447689\pi\)
0.163601 + 0.986527i \(0.447689\pi\)
\(648\) 0 0
\(649\) 820.833 0.0496464
\(650\) 0 0
\(651\) 0 0
\(652\) −21791.6 −1.30893
\(653\) 297.318 0.0178177 0.00890883 0.999960i \(-0.497164\pi\)
0.00890883 + 0.999960i \(0.497164\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3512.47 0.209053
\(657\) 0 0
\(658\) 771.776 0.0457248
\(659\) −10324.7 −0.610309 −0.305155 0.952303i \(-0.598708\pi\)
−0.305155 + 0.952303i \(0.598708\pi\)
\(660\) 0 0
\(661\) 4272.98 0.251437 0.125718 0.992066i \(-0.459876\pi\)
0.125718 + 0.992066i \(0.459876\pi\)
\(662\) 184.808 0.0108501
\(663\) 0 0
\(664\) 22288.2 1.30263
\(665\) 0 0
\(666\) 0 0
\(667\) 3685.17 0.213928
\(668\) −23967.2 −1.38820
\(669\) 0 0
\(670\) 0 0
\(671\) −6972.10 −0.401125
\(672\) 0 0
\(673\) 23033.4 1.31927 0.659637 0.751584i \(-0.270710\pi\)
0.659637 + 0.751584i \(0.270710\pi\)
\(674\) 11197.7 0.639940
\(675\) 0 0
\(676\) −3226.15 −0.183554
\(677\) 5113.50 0.290292 0.145146 0.989410i \(-0.453635\pi\)
0.145146 + 0.989410i \(0.453635\pi\)
\(678\) 0 0
\(679\) 3849.91 0.217593
\(680\) 0 0
\(681\) 0 0
\(682\) 1401.00 0.0786613
\(683\) 1341.02 0.0751286 0.0375643 0.999294i \(-0.488040\pi\)
0.0375643 + 0.999294i \(0.488040\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −493.387 −0.0274601
\(687\) 0 0
\(688\) −5999.80 −0.332471
\(689\) 22207.2 1.22790
\(690\) 0 0
\(691\) −16809.3 −0.925404 −0.462702 0.886514i \(-0.653120\pi\)
−0.462702 + 0.886514i \(0.653120\pi\)
\(692\) −352.639 −0.0193718
\(693\) 0 0
\(694\) −5380.16 −0.294277
\(695\) 0 0
\(696\) 0 0
\(697\) −9378.71 −0.509676
\(698\) −8165.86 −0.442812
\(699\) 0 0
\(700\) 0 0
\(701\) −13467.0 −0.725592 −0.362796 0.931869i \(-0.618178\pi\)
−0.362796 + 0.931869i \(0.618178\pi\)
\(702\) 0 0
\(703\) 16236.1 0.871064
\(704\) −915.025 −0.0489863
\(705\) 0 0
\(706\) −1308.36 −0.0697461
\(707\) 3733.09 0.198582
\(708\) 0 0
\(709\) 35514.3 1.88119 0.940597 0.339525i \(-0.110266\pi\)
0.940597 + 0.339525i \(0.110266\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −30435.2 −1.60198
\(713\) −2998.68 −0.157506
\(714\) 0 0
\(715\) 0 0
\(716\) −17634.4 −0.920431
\(717\) 0 0
\(718\) 3852.99 0.200268
\(719\) −4993.61 −0.259013 −0.129506 0.991579i \(-0.541339\pi\)
−0.129506 + 0.991579i \(0.541339\pi\)
\(720\) 0 0
\(721\) −9504.51 −0.490938
\(722\) −18569.6 −0.957186
\(723\) 0 0
\(724\) 4010.57 0.205872
\(725\) 0 0
\(726\) 0 0
\(727\) 4223.35 0.215454 0.107727 0.994181i \(-0.465643\pi\)
0.107727 + 0.994181i \(0.465643\pi\)
\(728\) −7343.81 −0.373873
\(729\) 0 0
\(730\) 0 0
\(731\) 16020.2 0.810572
\(732\) 0 0
\(733\) −19030.9 −0.958968 −0.479484 0.877551i \(-0.659176\pi\)
−0.479484 + 0.877551i \(0.659176\pi\)
\(734\) −1031.22 −0.0518570
\(735\) 0 0
\(736\) 4386.87 0.219704
\(737\) −3440.11 −0.171938
\(738\) 0 0
\(739\) −27772.5 −1.38245 −0.691224 0.722641i \(-0.742928\pi\)
−0.691224 + 0.722641i \(0.742928\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4271.05 0.211314
\(743\) 26880.7 1.32726 0.663631 0.748060i \(-0.269015\pi\)
0.663631 + 0.748060i \(0.269015\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −2885.74 −0.141628
\(747\) 0 0
\(748\) −2245.85 −0.109781
\(749\) 6395.85 0.312015
\(750\) 0 0
\(751\) −35166.6 −1.70872 −0.854360 0.519681i \(-0.826051\pi\)
−0.854360 + 0.519681i \(0.826051\pi\)
\(752\) −1427.35 −0.0692154
\(753\) 0 0
\(754\) −11836.3 −0.571688
\(755\) 0 0
\(756\) 0 0
\(757\) −14589.2 −0.700467 −0.350233 0.936662i \(-0.613898\pi\)
−0.350233 + 0.936662i \(0.613898\pi\)
\(758\) −10468.3 −0.501619
\(759\) 0 0
\(760\) 0 0
\(761\) 782.826 0.0372897 0.0186448 0.999826i \(-0.494065\pi\)
0.0186448 + 0.999826i \(0.494065\pi\)
\(762\) 0 0
\(763\) −1123.42 −0.0533035
\(764\) 98.2085 0.00465060
\(765\) 0 0
\(766\) −8564.42 −0.403975
\(767\) 5642.95 0.265652
\(768\) 0 0
\(769\) −16548.4 −0.776008 −0.388004 0.921658i \(-0.626835\pi\)
−0.388004 + 0.921658i \(0.626835\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 15981.8 0.745075
\(773\) 5744.94 0.267310 0.133655 0.991028i \(-0.457329\pi\)
0.133655 + 0.991028i \(0.457329\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 11021.1 0.509837
\(777\) 0 0
\(778\) 2687.00 0.123822
\(779\) 26519.7 1.21973
\(780\) 0 0
\(781\) 6909.66 0.316578
\(782\) −1677.03 −0.0766888
\(783\) 0 0
\(784\) 912.486 0.0415673
\(785\) 0 0
\(786\) 0 0
\(787\) 34744.3 1.57370 0.786848 0.617146i \(-0.211711\pi\)
0.786848 + 0.617146i \(0.211711\pi\)
\(788\) −6093.24 −0.275460
\(789\) 0 0
\(790\) 0 0
\(791\) −12519.4 −0.562753
\(792\) 0 0
\(793\) −47930.8 −2.14637
\(794\) −3107.15 −0.138877
\(795\) 0 0
\(796\) −16747.4 −0.745724
\(797\) 21748.7 0.966600 0.483300 0.875455i \(-0.339438\pi\)
0.483300 + 0.875455i \(0.339438\pi\)
\(798\) 0 0
\(799\) 3811.19 0.168749
\(800\) 0 0
\(801\) 0 0
\(802\) 2811.89 0.123805
\(803\) 5757.01 0.253002
\(804\) 0 0
\(805\) 0 0
\(806\) 9631.38 0.420907
\(807\) 0 0
\(808\) 10686.7 0.465292
\(809\) 42350.6 1.84050 0.920252 0.391325i \(-0.127983\pi\)
0.920252 + 0.391325i \(0.127983\pi\)
\(810\) 0 0
\(811\) −18910.7 −0.818796 −0.409398 0.912356i \(-0.634261\pi\)
−0.409398 + 0.912356i \(0.634261\pi\)
\(812\) 6525.10 0.282003
\(813\) 0 0
\(814\) 1265.00 0.0544696
\(815\) 0 0
\(816\) 0 0
\(817\) −45299.4 −1.93981
\(818\) −21425.8 −0.915813
\(819\) 0 0
\(820\) 0 0
\(821\) −6593.42 −0.280283 −0.140141 0.990132i \(-0.544756\pi\)
−0.140141 + 0.990132i \(0.544756\pi\)
\(822\) 0 0
\(823\) −26762.4 −1.13351 −0.566755 0.823886i \(-0.691801\pi\)
−0.566755 + 0.823886i \(0.691801\pi\)
\(824\) −27208.5 −1.15031
\(825\) 0 0
\(826\) 1085.29 0.0457169
\(827\) −24016.7 −1.00985 −0.504924 0.863164i \(-0.668479\pi\)
−0.504924 + 0.863164i \(0.668479\pi\)
\(828\) 0 0
\(829\) −28422.3 −1.19077 −0.595383 0.803442i \(-0.703000\pi\)
−0.595383 + 0.803442i \(0.703000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −6290.49 −0.262120
\(833\) −2436.45 −0.101342
\(834\) 0 0
\(835\) 0 0
\(836\) 6350.46 0.262721
\(837\) 0 0
\(838\) 18137.3 0.747663
\(839\) −6637.09 −0.273108 −0.136554 0.990633i \(-0.543603\pi\)
−0.136554 + 0.990633i \(0.543603\pi\)
\(840\) 0 0
\(841\) 313.546 0.0128560
\(842\) 11310.3 0.462921
\(843\) 0 0
\(844\) −30550.9 −1.24598
\(845\) 0 0
\(846\) 0 0
\(847\) −8911.03 −0.361495
\(848\) −7899.01 −0.319874
\(849\) 0 0
\(850\) 0 0
\(851\) −2707.59 −0.109066
\(852\) 0 0
\(853\) 1406.88 0.0564720 0.0282360 0.999601i \(-0.491011\pi\)
0.0282360 + 0.999601i \(0.491011\pi\)
\(854\) −9218.40 −0.369376
\(855\) 0 0
\(856\) 18309.3 0.731075
\(857\) 27943.4 1.11380 0.556901 0.830579i \(-0.311990\pi\)
0.556901 + 0.830579i \(0.311990\pi\)
\(858\) 0 0
\(859\) 1936.63 0.0769233 0.0384616 0.999260i \(-0.487754\pi\)
0.0384616 + 0.999260i \(0.487754\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 20556.8 0.812259
\(863\) 7947.70 0.313491 0.156746 0.987639i \(-0.449900\pi\)
0.156746 + 0.987639i \(0.449900\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −19791.6 −0.776611
\(867\) 0 0
\(868\) −5309.58 −0.207625
\(869\) −171.984 −0.00671365
\(870\) 0 0
\(871\) −23649.6 −0.920019
\(872\) −3216.00 −0.124894
\(873\) 0 0
\(874\) 4742.06 0.183527
\(875\) 0 0
\(876\) 0 0
\(877\) −38655.0 −1.48835 −0.744177 0.667983i \(-0.767158\pi\)
−0.744177 + 0.667983i \(0.767158\pi\)
\(878\) 8764.65 0.336893
\(879\) 0 0
\(880\) 0 0
\(881\) 18879.4 0.721978 0.360989 0.932570i \(-0.382439\pi\)
0.360989 + 0.932570i \(0.382439\pi\)
\(882\) 0 0
\(883\) 35098.1 1.33765 0.668825 0.743420i \(-0.266798\pi\)
0.668825 + 0.743420i \(0.266798\pi\)
\(884\) −15439.4 −0.587425
\(885\) 0 0
\(886\) −19346.3 −0.733581
\(887\) −48816.4 −1.84791 −0.923954 0.382504i \(-0.875062\pi\)
−0.923954 + 0.382504i \(0.875062\pi\)
\(888\) 0 0
\(889\) 1897.11 0.0715713
\(890\) 0 0
\(891\) 0 0
\(892\) −679.062 −0.0254895
\(893\) −10776.7 −0.403839
\(894\) 0 0
\(895\) 0 0
\(896\) 9267.63 0.345547
\(897\) 0 0
\(898\) 9916.41 0.368502
\(899\) −20100.8 −0.745718
\(900\) 0 0
\(901\) 21091.3 0.779860
\(902\) 2066.22 0.0762721
\(903\) 0 0
\(904\) −35839.1 −1.31857
\(905\) 0 0
\(906\) 0 0
\(907\) 6010.83 0.220051 0.110026 0.993929i \(-0.464907\pi\)
0.110026 + 0.993929i \(0.464907\pi\)
\(908\) −28140.5 −1.02850
\(909\) 0 0
\(910\) 0 0
\(911\) −25780.1 −0.937576 −0.468788 0.883311i \(-0.655309\pi\)
−0.468788 + 0.883311i \(0.655309\pi\)
\(912\) 0 0
\(913\) −8470.37 −0.307041
\(914\) −17002.6 −0.615314
\(915\) 0 0
\(916\) 31934.6 1.15191
\(917\) 5342.06 0.192378
\(918\) 0 0
\(919\) 26731.8 0.959522 0.479761 0.877399i \(-0.340723\pi\)
0.479761 + 0.877399i \(0.340723\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 12145.4 0.433824
\(923\) 47501.6 1.69397
\(924\) 0 0
\(925\) 0 0
\(926\) −1826.46 −0.0648177
\(927\) 0 0
\(928\) 29406.2 1.04020
\(929\) 30464.6 1.07590 0.537949 0.842977i \(-0.319199\pi\)
0.537949 + 0.842977i \(0.319199\pi\)
\(930\) 0 0
\(931\) 6889.42 0.242526
\(932\) 9542.22 0.335371
\(933\) 0 0
\(934\) 23708.1 0.830572
\(935\) 0 0
\(936\) 0 0
\(937\) 28533.4 0.994819 0.497409 0.867516i \(-0.334285\pi\)
0.497409 + 0.867516i \(0.334285\pi\)
\(938\) −4548.46 −0.158329
\(939\) 0 0
\(940\) 0 0
\(941\) −34455.8 −1.19365 −0.596827 0.802370i \(-0.703572\pi\)
−0.596827 + 0.802370i \(0.703572\pi\)
\(942\) 0 0
\(943\) −4422.50 −0.152722
\(944\) −2007.17 −0.0692033
\(945\) 0 0
\(946\) −3529.39 −0.121301
\(947\) 2477.68 0.0850198 0.0425099 0.999096i \(-0.486465\pi\)
0.0425099 + 0.999096i \(0.486465\pi\)
\(948\) 0 0
\(949\) 39577.5 1.35378
\(950\) 0 0
\(951\) 0 0
\(952\) −6974.80 −0.237452
\(953\) 41690.6 1.41709 0.708547 0.705663i \(-0.249351\pi\)
0.708547 + 0.705663i \(0.249351\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −18463.5 −0.624635
\(957\) 0 0
\(958\) −2014.13 −0.0679266
\(959\) −1686.54 −0.0567895
\(960\) 0 0
\(961\) −13434.6 −0.450962
\(962\) 8696.44 0.291460
\(963\) 0 0
\(964\) −42325.0 −1.41410
\(965\) 0 0
\(966\) 0 0
\(967\) 20641.3 0.686430 0.343215 0.939257i \(-0.388484\pi\)
0.343215 + 0.939257i \(0.388484\pi\)
\(968\) −25509.5 −0.847010
\(969\) 0 0
\(970\) 0 0
\(971\) −6626.17 −0.218995 −0.109497 0.993987i \(-0.534924\pi\)
−0.109497 + 0.993987i \(0.534924\pi\)
\(972\) 0 0
\(973\) −722.048 −0.0237901
\(974\) −20377.0 −0.670349
\(975\) 0 0
\(976\) 17048.8 0.559138
\(977\) 41961.0 1.37405 0.687027 0.726632i \(-0.258915\pi\)
0.687027 + 0.726632i \(0.258915\pi\)
\(978\) 0 0
\(979\) 11566.5 0.377598
\(980\) 0 0
\(981\) 0 0
\(982\) 6817.21 0.221533
\(983\) −16781.7 −0.544510 −0.272255 0.962225i \(-0.587769\pi\)
−0.272255 + 0.962225i \(0.587769\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −11241.6 −0.363087
\(987\) 0 0
\(988\) 43657.2 1.40579
\(989\) 7554.27 0.242884
\(990\) 0 0
\(991\) 50319.8 1.61298 0.806489 0.591250i \(-0.201365\pi\)
0.806489 + 0.591250i \(0.201365\pi\)
\(992\) −23928.3 −0.765851
\(993\) 0 0
\(994\) 9135.85 0.291521
\(995\) 0 0
\(996\) 0 0
\(997\) −12949.5 −0.411348 −0.205674 0.978621i \(-0.565939\pi\)
−0.205674 + 0.978621i \(0.565939\pi\)
\(998\) 16355.2 0.518753
\(999\) 0 0
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 1575.4.a.m.1.2 2
3.2 odd 2 525.4.a.p.1.1 2
5.4 even 2 315.4.a.m.1.1 2
15.2 even 4 525.4.d.i.274.3 4
15.8 even 4 525.4.d.i.274.2 4
15.14 odd 2 105.4.a.c.1.2 2
35.34 odd 2 2205.4.a.bh.1.1 2
60.59 even 2 1680.4.a.bk.1.1 2
105.104 even 2 735.4.a.k.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.c.1.2 2 15.14 odd 2
315.4.a.m.1.1 2 5.4 even 2
525.4.a.p.1.1 2 3.2 odd 2
525.4.d.i.274.2 4 15.8 even 4
525.4.d.i.274.3 4 15.2 even 4
735.4.a.k.1.2 2 105.104 even 2
1575.4.a.m.1.2 2 1.1 even 1 trivial
1680.4.a.bk.1.1 2 60.59 even 2
2205.4.a.bh.1.1 2 35.34 odd 2