Properties

Label 100.4.c.b.49.1
Level $100$
Weight $4$
Character 100.49
Analytic conductor $5.900$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 100.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.90019100057\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 100.49
Dual form 100.4.c.b.49.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +26.0000i q^{7} +26.0000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +26.0000i q^{7} +26.0000 q^{9} +45.0000 q^{11} -44.0000i q^{13} +117.000i q^{17} +91.0000 q^{19} +26.0000 q^{21} +18.0000i q^{23} -53.0000i q^{27} -144.000 q^{29} +26.0000 q^{31} -45.0000i q^{33} -214.000i q^{37} -44.0000 q^{39} -459.000 q^{41} +460.000i q^{43} -468.000i q^{47} -333.000 q^{49} +117.000 q^{51} -558.000i q^{53} -91.0000i q^{57} +72.0000 q^{59} -118.000 q^{61} +676.000i q^{63} +251.000i q^{67} +18.0000 q^{69} +108.000 q^{71} -299.000i q^{73} +1170.00i q^{77} +898.000 q^{79} +649.000 q^{81} -927.000i q^{83} +144.000i q^{87} -351.000 q^{89} +1144.00 q^{91} -26.0000i q^{93} +386.000i q^{97} +1170.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 52 q^{9} + 90 q^{11} + 182 q^{19} + 52 q^{21} - 288 q^{29} + 52 q^{31} - 88 q^{39} - 918 q^{41} - 666 q^{49} + 234 q^{51} + 144 q^{59} - 236 q^{61} + 36 q^{69} + 216 q^{71} + 1796 q^{79} + 1298 q^{81} - 702 q^{89} + 2288 q^{91} + 2340 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(1\) \(-1\)

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 0
\(3\) − 1.00000i − 0.192450i −0.995360 0.0962250i \(-0.969323\pi\)
0.995360 0.0962250i \(-0.0306768\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 26.0000i 1.40387i 0.712242 + 0.701934i \(0.247680\pi\)
−0.712242 + 0.701934i \(0.752320\pi\)
\(8\) 0 0
\(9\) 26.0000 0.962963
\(10\) 0 0
\(11\) 45.0000 1.23346 0.616728 0.787177i \(-0.288458\pi\)
0.616728 + 0.787177i \(0.288458\pi\)
\(12\) 0 0
\(13\) − 44.0000i − 0.938723i −0.883006 0.469362i \(-0.844484\pi\)
0.883006 0.469362i \(-0.155516\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 117.000i 1.66922i 0.550845 + 0.834608i \(0.314306\pi\)
−0.550845 + 0.834608i \(0.685694\pi\)
\(18\) 0 0
\(19\) 91.0000 1.09878 0.549390 0.835566i \(-0.314860\pi\)
0.549390 + 0.835566i \(0.314860\pi\)
\(20\) 0 0
\(21\) 26.0000 0.270175
\(22\) 0 0
\(23\) 18.0000i 0.163185i 0.996666 + 0.0815926i \(0.0260006\pi\)
−0.996666 + 0.0815926i \(0.973999\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 53.0000i − 0.377772i
\(28\) 0 0
\(29\) −144.000 −0.922073 −0.461037 0.887381i \(-0.652522\pi\)
−0.461037 + 0.887381i \(0.652522\pi\)
\(30\) 0 0
\(31\) 26.0000 0.150637 0.0753184 0.997160i \(-0.476003\pi\)
0.0753184 + 0.997160i \(0.476003\pi\)
\(32\) 0 0
\(33\) − 45.0000i − 0.237379i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 214.000i − 0.950848i −0.879757 0.475424i \(-0.842295\pi\)
0.879757 0.475424i \(-0.157705\pi\)
\(38\) 0 0
\(39\) −44.0000 −0.180657
\(40\) 0 0
\(41\) −459.000 −1.74838 −0.874192 0.485580i \(-0.838608\pi\)
−0.874192 + 0.485580i \(0.838608\pi\)
\(42\) 0 0
\(43\) 460.000i 1.63138i 0.578489 + 0.815690i \(0.303642\pi\)
−0.578489 + 0.815690i \(0.696358\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 468.000i − 1.45244i −0.687461 0.726221i \(-0.741275\pi\)
0.687461 0.726221i \(-0.258725\pi\)
\(48\) 0 0
\(49\) −333.000 −0.970845
\(50\) 0 0
\(51\) 117.000 0.321241
\(52\) 0 0
\(53\) − 558.000i − 1.44617i −0.690757 0.723087i \(-0.742723\pi\)
0.690757 0.723087i \(-0.257277\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 91.0000i − 0.211460i
\(58\) 0 0
\(59\) 72.0000 0.158875 0.0794373 0.996840i \(-0.474688\pi\)
0.0794373 + 0.996840i \(0.474688\pi\)
\(60\) 0 0
\(61\) −118.000 −0.247678 −0.123839 0.992302i \(-0.539521\pi\)
−0.123839 + 0.992302i \(0.539521\pi\)
\(62\) 0 0
\(63\) 676.000i 1.35187i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 251.000i 0.457680i 0.973464 + 0.228840i \(0.0734932\pi\)
−0.973464 + 0.228840i \(0.926507\pi\)
\(68\) 0 0
\(69\) 18.0000 0.0314050
\(70\) 0 0
\(71\) 108.000 0.180525 0.0902623 0.995918i \(-0.471229\pi\)
0.0902623 + 0.995918i \(0.471229\pi\)
\(72\) 0 0
\(73\) − 299.000i − 0.479388i −0.970849 0.239694i \(-0.922953\pi\)
0.970849 0.239694i \(-0.0770471\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1170.00i 1.73161i
\(78\) 0 0
\(79\) 898.000 1.27890 0.639449 0.768834i \(-0.279163\pi\)
0.639449 + 0.768834i \(0.279163\pi\)
\(80\) 0 0
\(81\) 649.000 0.890261
\(82\) 0 0
\(83\) − 927.000i − 1.22592i −0.790113 0.612961i \(-0.789978\pi\)
0.790113 0.612961i \(-0.210022\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 144.000i 0.177453i
\(88\) 0 0
\(89\) −351.000 −0.418044 −0.209022 0.977911i \(-0.567028\pi\)
−0.209022 + 0.977911i \(0.567028\pi\)
\(90\) 0 0
\(91\) 1144.00 1.31784
\(92\) 0 0
\(93\) − 26.0000i − 0.0289900i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 386.000i 0.404045i 0.979381 + 0.202022i \(0.0647514\pi\)
−0.979381 + 0.202022i \(0.935249\pi\)
\(98\) 0 0
\(99\) 1170.00 1.18777
\(100\) 0 0
\(101\) −954.000 −0.939867 −0.469933 0.882702i \(-0.655722\pi\)
−0.469933 + 0.882702i \(0.655722\pi\)
\(102\) 0 0
\(103\) 772.000i 0.738519i 0.929326 + 0.369259i \(0.120389\pi\)
−0.929326 + 0.369259i \(0.879611\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1197.00i 1.08148i 0.841190 + 0.540740i \(0.181856\pi\)
−0.841190 + 0.540740i \(0.818144\pi\)
\(108\) 0 0
\(109\) 802.000 0.704749 0.352375 0.935859i \(-0.385374\pi\)
0.352375 + 0.935859i \(0.385374\pi\)
\(110\) 0 0
\(111\) −214.000 −0.182991
\(112\) 0 0
\(113\) − 1143.00i − 0.951543i −0.879569 0.475772i \(-0.842169\pi\)
0.879569 0.475772i \(-0.157831\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 1144.00i − 0.903956i
\(118\) 0 0
\(119\) −3042.00 −2.34336
\(120\) 0 0
\(121\) 694.000 0.521412
\(122\) 0 0
\(123\) 459.000i 0.336477i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 2374.00i − 1.65873i −0.558709 0.829364i \(-0.688703\pi\)
0.558709 0.829364i \(-0.311297\pi\)
\(128\) 0 0
\(129\) 460.000 0.313959
\(130\) 0 0
\(131\) −1260.00 −0.840357 −0.420178 0.907442i \(-0.638032\pi\)
−0.420178 + 0.907442i \(0.638032\pi\)
\(132\) 0 0
\(133\) 2366.00i 1.54254i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 891.000i − 0.555644i −0.960633 0.277822i \(-0.910387\pi\)
0.960633 0.277822i \(-0.0896126\pi\)
\(138\) 0 0
\(139\) −389.000 −0.237371 −0.118685 0.992932i \(-0.537868\pi\)
−0.118685 + 0.992932i \(0.537868\pi\)
\(140\) 0 0
\(141\) −468.000 −0.279523
\(142\) 0 0
\(143\) − 1980.00i − 1.15787i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 333.000i 0.186839i
\(148\) 0 0
\(149\) −1296.00 −0.712567 −0.356283 0.934378i \(-0.615956\pi\)
−0.356283 + 0.934378i \(0.615956\pi\)
\(150\) 0 0
\(151\) −2710.00 −1.46051 −0.730254 0.683176i \(-0.760598\pi\)
−0.730254 + 0.683176i \(0.760598\pi\)
\(152\) 0 0
\(153\) 3042.00i 1.60739i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 1846.00i − 0.938388i −0.883095 0.469194i \(-0.844545\pi\)
0.883095 0.469194i \(-0.155455\pi\)
\(158\) 0 0
\(159\) −558.000 −0.278316
\(160\) 0 0
\(161\) −468.000 −0.229090
\(162\) 0 0
\(163\) − 1475.00i − 0.708779i −0.935098 0.354389i \(-0.884689\pi\)
0.935098 0.354389i \(-0.115311\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 1476.00i − 0.683930i −0.939713 0.341965i \(-0.888908\pi\)
0.939713 0.341965i \(-0.111092\pi\)
\(168\) 0 0
\(169\) 261.000 0.118798
\(170\) 0 0
\(171\) 2366.00 1.05809
\(172\) 0 0
\(173\) − 1368.00i − 0.601197i −0.953751 0.300599i \(-0.902814\pi\)
0.953751 0.300599i \(-0.0971864\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 72.0000i − 0.0305754i
\(178\) 0 0
\(179\) 1503.00 0.627595 0.313797 0.949490i \(-0.398399\pi\)
0.313797 + 0.949490i \(0.398399\pi\)
\(180\) 0 0
\(181\) 3770.00 1.54819 0.774094 0.633071i \(-0.218206\pi\)
0.774094 + 0.633071i \(0.218206\pi\)
\(182\) 0 0
\(183\) 118.000i 0.0476656i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5265.00i 2.05890i
\(188\) 0 0
\(189\) 1378.00 0.530343
\(190\) 0 0
\(191\) −4122.00 −1.56156 −0.780779 0.624808i \(-0.785177\pi\)
−0.780779 + 0.624808i \(0.785177\pi\)
\(192\) 0 0
\(193\) 1963.00i 0.732123i 0.930591 + 0.366062i \(0.119294\pi\)
−0.930591 + 0.366062i \(0.880706\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2934.00i 1.06111i 0.847650 + 0.530555i \(0.178017\pi\)
−0.847650 + 0.530555i \(0.821983\pi\)
\(198\) 0 0
\(199\) −1412.00 −0.502985 −0.251493 0.967859i \(-0.580921\pi\)
−0.251493 + 0.967859i \(0.580921\pi\)
\(200\) 0 0
\(201\) 251.000 0.0880805
\(202\) 0 0
\(203\) − 3744.00i − 1.29447i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 468.000i 0.157141i
\(208\) 0 0
\(209\) 4095.00 1.35530
\(210\) 0 0
\(211\) 3419.00 1.11552 0.557758 0.830004i \(-0.311662\pi\)
0.557758 + 0.830004i \(0.311662\pi\)
\(212\) 0 0
\(213\) − 108.000i − 0.0347420i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 676.000i 0.211474i
\(218\) 0 0
\(219\) −299.000 −0.0922582
\(220\) 0 0
\(221\) 5148.00 1.56693
\(222\) 0 0
\(223\) 100.000i 0.0300291i 0.999887 + 0.0150146i \(0.00477946\pi\)
−0.999887 + 0.0150146i \(0.995221\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4212.00i 1.23154i 0.787925 + 0.615771i \(0.211156\pi\)
−0.787925 + 0.615771i \(0.788844\pi\)
\(228\) 0 0
\(229\) 3484.00 1.00537 0.502684 0.864470i \(-0.332346\pi\)
0.502684 + 0.864470i \(0.332346\pi\)
\(230\) 0 0
\(231\) 1170.00 0.333248
\(232\) 0 0
\(233\) − 918.000i − 0.258112i −0.991637 0.129056i \(-0.958805\pi\)
0.991637 0.129056i \(-0.0411948\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 898.000i − 0.246124i
\(238\) 0 0
\(239\) −3744.00 −1.01330 −0.506651 0.862151i \(-0.669117\pi\)
−0.506651 + 0.862151i \(0.669117\pi\)
\(240\) 0 0
\(241\) −4231.00 −1.13088 −0.565441 0.824789i \(-0.691294\pi\)
−0.565441 + 0.824789i \(0.691294\pi\)
\(242\) 0 0
\(243\) − 2080.00i − 0.549103i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 4004.00i − 1.03145i
\(248\) 0 0
\(249\) −927.000 −0.235929
\(250\) 0 0
\(251\) −2925.00 −0.735555 −0.367778 0.929914i \(-0.619881\pi\)
−0.367778 + 0.929914i \(0.619881\pi\)
\(252\) 0 0
\(253\) 810.000i 0.201282i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 18.0000i − 0.00436891i −0.999998 0.00218445i \(-0.999305\pi\)
0.999998 0.00218445i \(-0.000695334\pi\)
\(258\) 0 0
\(259\) 5564.00 1.33487
\(260\) 0 0
\(261\) −3744.00 −0.887923
\(262\) 0 0
\(263\) 6786.00i 1.59104i 0.605929 + 0.795518i \(0.292801\pi\)
−0.605929 + 0.795518i \(0.707199\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 351.000i 0.0804526i
\(268\) 0 0
\(269\) −7632.00 −1.72986 −0.864928 0.501896i \(-0.832636\pi\)
−0.864928 + 0.501896i \(0.832636\pi\)
\(270\) 0 0
\(271\) 650.000 0.145700 0.0728500 0.997343i \(-0.476791\pi\)
0.0728500 + 0.997343i \(0.476791\pi\)
\(272\) 0 0
\(273\) − 1144.00i − 0.253619i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 3232.00i − 0.701054i −0.936553 0.350527i \(-0.886002\pi\)
0.936553 0.350527i \(-0.113998\pi\)
\(278\) 0 0
\(279\) 676.000 0.145058
\(280\) 0 0
\(281\) 4446.00 0.943865 0.471933 0.881635i \(-0.343557\pi\)
0.471933 + 0.881635i \(0.343557\pi\)
\(282\) 0 0
\(283\) − 2483.00i − 0.521551i −0.965399 0.260776i \(-0.916022\pi\)
0.965399 0.260776i \(-0.0839783\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 11934.0i − 2.45450i
\(288\) 0 0
\(289\) −8776.00 −1.78628
\(290\) 0 0
\(291\) 386.000 0.0777585
\(292\) 0 0
\(293\) 4050.00i 0.807521i 0.914865 + 0.403760i \(0.132297\pi\)
−0.914865 + 0.403760i \(0.867703\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 2385.00i − 0.465965i
\(298\) 0 0
\(299\) 792.000 0.153186
\(300\) 0 0
\(301\) −11960.0 −2.29024
\(302\) 0 0
\(303\) 954.000i 0.180877i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2321.00i 0.431487i 0.976450 + 0.215743i \(0.0692175\pi\)
−0.976450 + 0.215743i \(0.930783\pi\)
\(308\) 0 0
\(309\) 772.000 0.142128
\(310\) 0 0
\(311\) −3258.00 −0.594033 −0.297016 0.954872i \(-0.595992\pi\)
−0.297016 + 0.954872i \(0.595992\pi\)
\(312\) 0 0
\(313\) − 3626.00i − 0.654804i −0.944885 0.327402i \(-0.893827\pi\)
0.944885 0.327402i \(-0.106173\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3852.00i 0.682492i 0.939974 + 0.341246i \(0.110849\pi\)
−0.939974 + 0.341246i \(0.889151\pi\)
\(318\) 0 0
\(319\) −6480.00 −1.13734
\(320\) 0 0
\(321\) 1197.00 0.208131
\(322\) 0 0
\(323\) 10647.0i 1.83410i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 802.000i − 0.135629i
\(328\) 0 0
\(329\) 12168.0 2.03904
\(330\) 0 0
\(331\) 7553.00 1.25423 0.627115 0.778926i \(-0.284235\pi\)
0.627115 + 0.778926i \(0.284235\pi\)
\(332\) 0 0
\(333\) − 5564.00i − 0.915632i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 109.000i − 0.0176190i −0.999961 0.00880951i \(-0.997196\pi\)
0.999961 0.00880951i \(-0.00280419\pi\)
\(338\) 0 0
\(339\) −1143.00 −0.183125
\(340\) 0 0
\(341\) 1170.00 0.185804
\(342\) 0 0
\(343\) 260.000i 0.0409291i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2835.00i − 0.438590i −0.975659 0.219295i \(-0.929624\pi\)
0.975659 0.219295i \(-0.0703757\pi\)
\(348\) 0 0
\(349\) −2990.00 −0.458599 −0.229299 0.973356i \(-0.573644\pi\)
−0.229299 + 0.973356i \(0.573644\pi\)
\(350\) 0 0
\(351\) −2332.00 −0.354624
\(352\) 0 0
\(353\) − 9126.00i − 1.37600i −0.725711 0.688000i \(-0.758489\pi\)
0.725711 0.688000i \(-0.241511\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3042.00i 0.450980i
\(358\) 0 0
\(359\) 9594.00 1.41045 0.705226 0.708983i \(-0.250846\pi\)
0.705226 + 0.708983i \(0.250846\pi\)
\(360\) 0 0
\(361\) 1422.00 0.207319
\(362\) 0 0
\(363\) − 694.000i − 0.100346i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 9764.00i 1.38876i 0.719606 + 0.694382i \(0.244322\pi\)
−0.719606 + 0.694382i \(0.755678\pi\)
\(368\) 0 0
\(369\) −11934.0 −1.68363
\(370\) 0 0
\(371\) 14508.0 2.03024
\(372\) 0 0
\(373\) − 6722.00i − 0.933115i −0.884491 0.466558i \(-0.845494\pi\)
0.884491 0.466558i \(-0.154506\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6336.00i 0.865572i
\(378\) 0 0
\(379\) 13537.0 1.83469 0.917347 0.398089i \(-0.130326\pi\)
0.917347 + 0.398089i \(0.130326\pi\)
\(380\) 0 0
\(381\) −2374.00 −0.319222
\(382\) 0 0
\(383\) 8658.00i 1.15510i 0.816355 + 0.577550i \(0.195991\pi\)
−0.816355 + 0.577550i \(0.804009\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 11960.0i 1.57096i
\(388\) 0 0
\(389\) 8874.00 1.15663 0.578316 0.815813i \(-0.303710\pi\)
0.578316 + 0.815813i \(0.303710\pi\)
\(390\) 0 0
\(391\) −2106.00 −0.272391
\(392\) 0 0
\(393\) 1260.00i 0.161727i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5876.00i 0.742841i 0.928465 + 0.371421i \(0.121129\pi\)
−0.928465 + 0.371421i \(0.878871\pi\)
\(398\) 0 0
\(399\) 2366.00 0.296863
\(400\) 0 0
\(401\) −1755.00 −0.218555 −0.109277 0.994011i \(-0.534854\pi\)
−0.109277 + 0.994011i \(0.534854\pi\)
\(402\) 0 0
\(403\) − 1144.00i − 0.141406i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 9630.00i − 1.17283i
\(408\) 0 0
\(409\) −4589.00 −0.554796 −0.277398 0.960755i \(-0.589472\pi\)
−0.277398 + 0.960755i \(0.589472\pi\)
\(410\) 0 0
\(411\) −891.000 −0.106934
\(412\) 0 0
\(413\) 1872.00i 0.223039i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 389.000i 0.0456820i
\(418\) 0 0
\(419\) −5409.00 −0.630661 −0.315330 0.948982i \(-0.602115\pi\)
−0.315330 + 0.948982i \(0.602115\pi\)
\(420\) 0 0
\(421\) 12116.0 1.40261 0.701304 0.712863i \(-0.252602\pi\)
0.701304 + 0.712863i \(0.252602\pi\)
\(422\) 0 0
\(423\) − 12168.0i − 1.39865i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 3068.00i − 0.347707i
\(428\) 0 0
\(429\) −1980.00 −0.222833
\(430\) 0 0
\(431\) 9126.00 1.01992 0.509958 0.860199i \(-0.329661\pi\)
0.509958 + 0.860199i \(0.329661\pi\)
\(432\) 0 0
\(433\) − 629.000i − 0.0698102i −0.999391 0.0349051i \(-0.988887\pi\)
0.999391 0.0349051i \(-0.0111129\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1638.00i 0.179305i
\(438\) 0 0
\(439\) −4472.00 −0.486189 −0.243094 0.970003i \(-0.578162\pi\)
−0.243094 + 0.970003i \(0.578162\pi\)
\(440\) 0 0
\(441\) −8658.00 −0.934888
\(442\) 0 0
\(443\) 3393.00i 0.363897i 0.983308 + 0.181948i \(0.0582404\pi\)
−0.983308 + 0.181948i \(0.941760\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1296.00i 0.137134i
\(448\) 0 0
\(449\) 5031.00 0.528792 0.264396 0.964414i \(-0.414827\pi\)
0.264396 + 0.964414i \(0.414827\pi\)
\(450\) 0 0
\(451\) −20655.0 −2.15655
\(452\) 0 0
\(453\) 2710.00i 0.281075i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 6487.00i − 0.664002i −0.943279 0.332001i \(-0.892276\pi\)
0.943279 0.332001i \(-0.107724\pi\)
\(458\) 0 0
\(459\) 6201.00 0.630584
\(460\) 0 0
\(461\) 2700.00 0.272780 0.136390 0.990655i \(-0.456450\pi\)
0.136390 + 0.990655i \(0.456450\pi\)
\(462\) 0 0
\(463\) 2932.00i 0.294302i 0.989114 + 0.147151i \(0.0470102\pi\)
−0.989114 + 0.147151i \(0.952990\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 15660.0i − 1.55173i −0.630898 0.775866i \(-0.717314\pi\)
0.630898 0.775866i \(-0.282686\pi\)
\(468\) 0 0
\(469\) −6526.00 −0.642522
\(470\) 0 0
\(471\) −1846.00 −0.180593
\(472\) 0 0
\(473\) 20700.0i 2.01223i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 14508.0i − 1.39261i
\(478\) 0 0
\(479\) −10134.0 −0.966669 −0.483334 0.875436i \(-0.660574\pi\)
−0.483334 + 0.875436i \(0.660574\pi\)
\(480\) 0 0
\(481\) −9416.00 −0.892583
\(482\) 0 0
\(483\) 468.000i 0.0440885i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1898.00i 0.176605i 0.996094 + 0.0883025i \(0.0281442\pi\)
−0.996094 + 0.0883025i \(0.971856\pi\)
\(488\) 0 0
\(489\) −1475.00 −0.136405
\(490\) 0 0
\(491\) −6300.00 −0.579053 −0.289526 0.957170i \(-0.593498\pi\)
−0.289526 + 0.957170i \(0.593498\pi\)
\(492\) 0 0
\(493\) − 16848.0i − 1.53914i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2808.00i 0.253433i
\(498\) 0 0
\(499\) −18044.0 −1.61876 −0.809379 0.587286i \(-0.800196\pi\)
−0.809379 + 0.587286i \(0.800196\pi\)
\(500\) 0 0
\(501\) −1476.00 −0.131622
\(502\) 0 0
\(503\) 6876.00i 0.609514i 0.952430 + 0.304757i \(0.0985753\pi\)
−0.952430 + 0.304757i \(0.901425\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 261.000i − 0.0228628i
\(508\) 0 0
\(509\) 4806.00 0.418511 0.209256 0.977861i \(-0.432896\pi\)
0.209256 + 0.977861i \(0.432896\pi\)
\(510\) 0 0
\(511\) 7774.00 0.672997
\(512\) 0 0
\(513\) − 4823.00i − 0.415089i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 21060.0i − 1.79152i
\(518\) 0 0
\(519\) −1368.00 −0.115700
\(520\) 0 0
\(521\) 7749.00 0.651612 0.325806 0.945437i \(-0.394364\pi\)
0.325806 + 0.945437i \(0.394364\pi\)
\(522\) 0 0
\(523\) − 8153.00i − 0.681655i −0.940126 0.340828i \(-0.889293\pi\)
0.940126 0.340828i \(-0.110707\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3042.00i 0.251445i
\(528\) 0 0
\(529\) 11843.0 0.973371
\(530\) 0 0
\(531\) 1872.00 0.152990
\(532\) 0 0
\(533\) 20196.0i 1.64125i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 1503.00i − 0.120781i
\(538\) 0 0
\(539\) −14985.0 −1.19749
\(540\) 0 0
\(541\) −10576.0 −0.840476 −0.420238 0.907414i \(-0.638053\pi\)
−0.420238 + 0.907414i \(0.638053\pi\)
\(542\) 0 0
\(543\) − 3770.00i − 0.297949i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 7553.00i 0.590389i 0.955437 + 0.295195i \(0.0953845\pi\)
−0.955437 + 0.295195i \(0.904615\pi\)
\(548\) 0 0
\(549\) −3068.00 −0.238505
\(550\) 0 0
\(551\) −13104.0 −1.01316
\(552\) 0 0
\(553\) 23348.0i 1.79540i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13500.0i 1.02695i 0.858103 + 0.513477i \(0.171643\pi\)
−0.858103 + 0.513477i \(0.828357\pi\)
\(558\) 0 0
\(559\) 20240.0 1.53141
\(560\) 0 0
\(561\) 5265.00 0.396236
\(562\) 0 0
\(563\) − 23184.0i − 1.73550i −0.496997 0.867752i \(-0.665564\pi\)
0.496997 0.867752i \(-0.334436\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 16874.0i 1.24981i
\(568\) 0 0
\(569\) 8055.00 0.593468 0.296734 0.954960i \(-0.404103\pi\)
0.296734 + 0.954960i \(0.404103\pi\)
\(570\) 0 0
\(571\) 3068.00 0.224854 0.112427 0.993660i \(-0.464138\pi\)
0.112427 + 0.993660i \(0.464138\pi\)
\(572\) 0 0
\(573\) 4122.00i 0.300522i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 12419.0i 0.896031i 0.894026 + 0.448015i \(0.147869\pi\)
−0.894026 + 0.448015i \(0.852131\pi\)
\(578\) 0 0
\(579\) 1963.00 0.140897
\(580\) 0 0
\(581\) 24102.0 1.72103
\(582\) 0 0
\(583\) − 25110.0i − 1.78379i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 12393.0i − 0.871403i −0.900091 0.435702i \(-0.856500\pi\)
0.900091 0.435702i \(-0.143500\pi\)
\(588\) 0 0
\(589\) 2366.00 0.165517
\(590\) 0 0
\(591\) 2934.00 0.204211
\(592\) 0 0
\(593\) − 23751.0i − 1.64475i −0.568946 0.822375i \(-0.692649\pi\)
0.568946 0.822375i \(-0.307351\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1412.00i 0.0967995i
\(598\) 0 0
\(599\) 11610.0 0.791939 0.395970 0.918264i \(-0.370409\pi\)
0.395970 + 0.918264i \(0.370409\pi\)
\(600\) 0 0
\(601\) 26675.0 1.81048 0.905238 0.424905i \(-0.139693\pi\)
0.905238 + 0.424905i \(0.139693\pi\)
\(602\) 0 0
\(603\) 6526.00i 0.440728i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 17264.0i 1.15441i 0.816601 + 0.577203i \(0.195856\pi\)
−0.816601 + 0.577203i \(0.804144\pi\)
\(608\) 0 0
\(609\) −3744.00 −0.249121
\(610\) 0 0
\(611\) −20592.0 −1.36344
\(612\) 0 0
\(613\) 26026.0i 1.71481i 0.514640 + 0.857406i \(0.327926\pi\)
−0.514640 + 0.857406i \(0.672074\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 5022.00i 0.327679i 0.986487 + 0.163840i \(0.0523880\pi\)
−0.986487 + 0.163840i \(0.947612\pi\)
\(618\) 0 0
\(619\) −7820.00 −0.507774 −0.253887 0.967234i \(-0.581709\pi\)
−0.253887 + 0.967234i \(0.581709\pi\)
\(620\) 0 0
\(621\) 954.000 0.0616469
\(622\) 0 0
\(623\) − 9126.00i − 0.586879i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 4095.00i − 0.260827i
\(628\) 0 0
\(629\) 25038.0 1.58717
\(630\) 0 0
\(631\) 15002.0 0.946466 0.473233 0.880937i \(-0.343087\pi\)
0.473233 + 0.880937i \(0.343087\pi\)
\(632\) 0 0
\(633\) − 3419.00i − 0.214681i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 14652.0i 0.911355i
\(638\) 0 0
\(639\) 2808.00 0.173838
\(640\) 0 0
\(641\) −918.000 −0.0565660 −0.0282830 0.999600i \(-0.509004\pi\)
−0.0282830 + 0.999600i \(0.509004\pi\)
\(642\) 0 0
\(643\) 23452.0i 1.43835i 0.694831 + 0.719173i \(0.255479\pi\)
−0.694831 + 0.719173i \(0.744521\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20556.0i 1.24906i 0.781002 + 0.624528i \(0.214709\pi\)
−0.781002 + 0.624528i \(0.785291\pi\)
\(648\) 0 0
\(649\) 3240.00 0.195965
\(650\) 0 0
\(651\) 676.000 0.0406982
\(652\) 0 0
\(653\) − 30654.0i − 1.83703i −0.395381 0.918517i \(-0.629387\pi\)
0.395381 0.918517i \(-0.370613\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 7774.00i − 0.461633i
\(658\) 0 0
\(659\) −8919.00 −0.527215 −0.263608 0.964630i \(-0.584912\pi\)
−0.263608 + 0.964630i \(0.584912\pi\)
\(660\) 0 0
\(661\) −22912.0 −1.34822 −0.674110 0.738631i \(-0.735473\pi\)
−0.674110 + 0.738631i \(0.735473\pi\)
\(662\) 0 0
\(663\) − 5148.00i − 0.301556i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 2592.00i − 0.150469i
\(668\) 0 0
\(669\) 100.000 0.00577911
\(670\) 0 0
\(671\) −5310.00 −0.305500
\(672\) 0 0
\(673\) 1222.00i 0.0699920i 0.999387 + 0.0349960i \(0.0111419\pi\)
−0.999387 + 0.0349960i \(0.988858\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 144.000i 0.00817484i 0.999992 + 0.00408742i \(0.00130107\pi\)
−0.999992 + 0.00408742i \(0.998699\pi\)
\(678\) 0 0
\(679\) −10036.0 −0.567226
\(680\) 0 0
\(681\) 4212.00 0.237011
\(682\) 0 0
\(683\) 12519.0i 0.701356i 0.936496 + 0.350678i \(0.114049\pi\)
−0.936496 + 0.350678i \(0.885951\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 3484.00i − 0.193483i
\(688\) 0 0
\(689\) −24552.0 −1.35756
\(690\) 0 0
\(691\) 11873.0 0.653647 0.326824 0.945085i \(-0.394022\pi\)
0.326824 + 0.945085i \(0.394022\pi\)
\(692\) 0 0
\(693\) 30420.0i 1.66748i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 53703.0i − 2.91843i
\(698\) 0 0
\(699\) −918.000 −0.0496737
\(700\) 0 0
\(701\) −3060.00 −0.164871 −0.0824355 0.996596i \(-0.526270\pi\)
−0.0824355 + 0.996596i \(0.526270\pi\)
\(702\) 0 0
\(703\) − 19474.0i − 1.04477i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 24804.0i − 1.31945i
\(708\) 0 0
\(709\) −4004.00 −0.212092 −0.106046 0.994361i \(-0.533819\pi\)
−0.106046 + 0.994361i \(0.533819\pi\)
\(710\) 0 0
\(711\) 23348.0 1.23153
\(712\) 0 0
\(713\) 468.000i 0.0245817i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3744.00i 0.195010i
\(718\) 0 0
\(719\) −10314.0 −0.534975 −0.267488 0.963561i \(-0.586193\pi\)
−0.267488 + 0.963561i \(0.586193\pi\)
\(720\) 0 0
\(721\) −20072.0 −1.03678
\(722\) 0 0
\(723\) 4231.00i 0.217638i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 9872.00i 0.503621i 0.967777 + 0.251810i \(0.0810259\pi\)
−0.967777 + 0.251810i \(0.918974\pi\)
\(728\) 0 0
\(729\) 15443.0 0.784586
\(730\) 0 0
\(731\) −53820.0 −2.72313
\(732\) 0 0
\(733\) − 7436.00i − 0.374700i −0.982293 0.187350i \(-0.940010\pi\)
0.982293 0.187350i \(-0.0599898\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11295.0i 0.564527i
\(738\) 0 0
\(739\) 16900.0 0.841240 0.420620 0.907237i \(-0.361813\pi\)
0.420620 + 0.907237i \(0.361813\pi\)
\(740\) 0 0
\(741\) −4004.00 −0.198503
\(742\) 0 0
\(743\) 23058.0i 1.13851i 0.822160 + 0.569257i \(0.192769\pi\)
−0.822160 + 0.569257i \(0.807231\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 24102.0i − 1.18052i
\(748\) 0 0
\(749\) −31122.0 −1.51826
\(750\) 0 0
\(751\) −8224.00 −0.399598 −0.199799 0.979837i \(-0.564029\pi\)
−0.199799 + 0.979837i \(0.564029\pi\)
\(752\) 0 0
\(753\) 2925.00i 0.141558i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 7696.00i − 0.369506i −0.982785 0.184753i \(-0.940852\pi\)
0.982785 0.184753i \(-0.0591485\pi\)
\(758\) 0 0
\(759\) 810.000 0.0387367
\(760\) 0 0
\(761\) −6363.00 −0.303099 −0.151550 0.988450i \(-0.548426\pi\)
−0.151550 + 0.988450i \(0.548426\pi\)
\(762\) 0 0
\(763\) 20852.0i 0.989375i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 3168.00i − 0.149139i
\(768\) 0 0
\(769\) −8333.00 −0.390762 −0.195381 0.980727i \(-0.562594\pi\)
−0.195381 + 0.980727i \(0.562594\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.000840797 0
\(772\) 0 0
\(773\) − 32760.0i − 1.52431i −0.647392 0.762157i \(-0.724140\pi\)
0.647392 0.762157i \(-0.275860\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 5564.00i − 0.256895i
\(778\) 0 0
\(779\) −41769.0 −1.92109
\(780\) 0 0
\(781\) 4860.00 0.222669
\(782\) 0 0
\(783\) 7632.00i 0.348334i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 43732.0i − 1.98078i −0.138286 0.990392i \(-0.544159\pi\)
0.138286 0.990392i \(-0.455841\pi\)
\(788\) 0 0
\(789\) 6786.00 0.306195
\(790\) 0 0
\(791\) 29718.0 1.33584
\(792\) 0 0
\(793\) 5192.00i 0.232501i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 16866.0i 0.749591i 0.927107 + 0.374796i \(0.122287\pi\)
−0.927107 + 0.374796i \(0.877713\pi\)
\(798\) 0 0
\(799\) 54756.0 2.42444
\(800\) 0 0
\(801\) −9126.00 −0.402561
\(802\) 0 0
\(803\) − 13455.0i − 0.591303i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 7632.00i 0.332911i
\(808\) 0 0
\(809\) −16146.0 −0.701685 −0.350842 0.936434i \(-0.614105\pi\)
−0.350842 + 0.936434i \(0.614105\pi\)
\(810\) 0 0
\(811\) 32444.0 1.40476 0.702382 0.711801i \(-0.252120\pi\)
0.702382 + 0.711801i \(0.252120\pi\)
\(812\) 0 0
\(813\) − 650.000i − 0.0280400i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 41860.0i 1.79253i
\(818\) 0 0
\(819\) 29744.0 1.26903
\(820\) 0 0
\(821\) −2574.00 −0.109419 −0.0547096 0.998502i \(-0.517423\pi\)
−0.0547096 + 0.998502i \(0.517423\pi\)
\(822\) 0 0
\(823\) 27604.0i 1.16916i 0.811338 + 0.584578i \(0.198740\pi\)
−0.811338 + 0.584578i \(0.801260\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11655.0i 0.490065i 0.969515 + 0.245033i \(0.0787987\pi\)
−0.969515 + 0.245033i \(0.921201\pi\)
\(828\) 0 0
\(829\) −33428.0 −1.40049 −0.700243 0.713905i \(-0.746925\pi\)
−0.700243 + 0.713905i \(0.746925\pi\)
\(830\) 0 0
\(831\) −3232.00 −0.134918
\(832\) 0 0
\(833\) − 38961.0i − 1.62055i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 1378.00i − 0.0569064i
\(838\) 0 0
\(839\) 17712.0 0.728827 0.364414 0.931237i \(-0.381269\pi\)
0.364414 + 0.931237i \(0.381269\pi\)
\(840\) 0 0
\(841\) −3653.00 −0.149781
\(842\) 0 0
\(843\) − 4446.00i − 0.181647i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 18044.0i 0.731994i
\(848\) 0 0
\(849\) −2483.00 −0.100373
\(850\) 0 0
\(851\) 3852.00 0.155164
\(852\) 0 0
\(853\) 10270.0i 0.412237i 0.978527 + 0.206118i \(0.0660832\pi\)
−0.978527 + 0.206118i \(0.933917\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 38709.0i 1.54291i 0.636284 + 0.771455i \(0.280471\pi\)
−0.636284 + 0.771455i \(0.719529\pi\)
\(858\) 0 0
\(859\) −15509.0 −0.616019 −0.308009 0.951383i \(-0.599663\pi\)
−0.308009 + 0.951383i \(0.599663\pi\)
\(860\) 0 0
\(861\) −11934.0 −0.472369
\(862\) 0 0
\(863\) − 15912.0i − 0.627637i −0.949483 0.313819i \(-0.898392\pi\)
0.949483 0.313819i \(-0.101608\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 8776.00i 0.343770i
\(868\) 0 0
\(869\) 40410.0 1.57746
\(870\) 0 0
\(871\) 11044.0 0.429635
\(872\) 0 0
\(873\) 10036.0i 0.389080i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 10972.0i − 0.422461i −0.977436 0.211230i \(-0.932253\pi\)
0.977436 0.211230i \(-0.0677470\pi\)
\(878\) 0 0
\(879\) 4050.00 0.155407
\(880\) 0 0
\(881\) −18738.0 −0.716571 −0.358286 0.933612i \(-0.616639\pi\)
−0.358286 + 0.933612i \(0.616639\pi\)
\(882\) 0 0
\(883\) 21367.0i 0.814334i 0.913354 + 0.407167i \(0.133483\pi\)
−0.913354 + 0.407167i \(0.866517\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 20124.0i 0.761779i 0.924621 + 0.380889i \(0.124382\pi\)
−0.924621 + 0.380889i \(0.875618\pi\)
\(888\) 0 0
\(889\) 61724.0 2.32864
\(890\) 0 0
\(891\) 29205.0 1.09810
\(892\) 0 0
\(893\) − 42588.0i − 1.59592i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 792.000i − 0.0294806i
\(898\) 0 0
\(899\) −3744.00 −0.138898
\(900\) 0 0
\(901\) 65286.0 2.41398
\(902\) 0 0
\(903\) 11960.0i 0.440757i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 23132.0i 0.846842i 0.905933 + 0.423421i \(0.139171\pi\)
−0.905933 + 0.423421i \(0.860829\pi\)
\(908\) 0 0
\(909\) −24804.0 −0.905057
\(910\) 0 0
\(911\) 31212.0 1.13513 0.567563 0.823330i \(-0.307886\pi\)
0.567563 + 0.823330i \(0.307886\pi\)
\(912\) 0 0
\(913\) − 41715.0i − 1.51212i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 32760.0i − 1.17975i
\(918\) 0 0
\(919\) 6994.00 0.251045 0.125523 0.992091i \(-0.459939\pi\)
0.125523 + 0.992091i \(0.459939\pi\)
\(920\) 0 0
\(921\) 2321.00 0.0830397
\(922\) 0 0
\(923\) − 4752.00i − 0.169463i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 20072.0i 0.711166i
\(928\) 0 0
\(929\) −19422.0 −0.685915 −0.342958 0.939351i \(-0.611429\pi\)
−0.342958 + 0.939351i \(0.611429\pi\)
\(930\) 0 0
\(931\) −30303.0 −1.06675
\(932\) 0 0
\(933\) 3258.00i 0.114322i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 11699.0i 0.407887i 0.978983 + 0.203943i \(0.0653758\pi\)
−0.978983 + 0.203943i \(0.934624\pi\)
\(938\) 0 0
\(939\) −3626.00 −0.126017
\(940\) 0 0
\(941\) −42948.0 −1.48785 −0.743924 0.668264i \(-0.767038\pi\)
−0.743924 + 0.668264i \(0.767038\pi\)
\(942\) 0 0
\(943\) − 8262.00i − 0.285310i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3816.00i − 0.130943i −0.997854 0.0654717i \(-0.979145\pi\)
0.997854 0.0654717i \(-0.0208552\pi\)
\(948\) 0 0
\(949\) −13156.0 −0.450012
\(950\) 0 0
\(951\) 3852.00 0.131346
\(952\) 0 0
\(953\) 43407.0i 1.47544i 0.675109 + 0.737718i \(0.264097\pi\)
−0.675109 + 0.737718i \(0.735903\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 6480.00i 0.218881i
\(958\) 0 0
\(959\) 23166.0 0.780051
\(960\) 0 0
\(961\) −29115.0 −0.977309
\(962\) 0 0
\(963\) 31122.0i 1.04143i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 43216.0i − 1.43716i −0.695445 0.718580i \(-0.744793\pi\)
0.695445 0.718580i \(-0.255207\pi\)
\(968\) 0 0
\(969\) 10647.0 0.352973
\(970\) 0 0
\(971\) −47619.0 −1.57381 −0.786903 0.617076i \(-0.788317\pi\)
−0.786903 + 0.617076i \(0.788317\pi\)
\(972\) 0 0
\(973\) − 10114.0i − 0.333237i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 4671.00i − 0.152957i −0.997071 0.0764783i \(-0.975632\pi\)
0.997071 0.0764783i \(-0.0243676\pi\)
\(978\) 0 0
\(979\) −15795.0 −0.515639
\(980\) 0 0
\(981\) 20852.0 0.678647
\(982\) 0 0
\(983\) − 9054.00i − 0.293772i −0.989153 0.146886i \(-0.953075\pi\)
0.989153 0.146886i \(-0.0469250\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 12168.0i − 0.392413i
\(988\) 0 0
\(989\) −8280.00 −0.266217
\(990\) 0 0
\(991\) 8126.00 0.260475 0.130238 0.991483i \(-0.458426\pi\)
0.130238 + 0.991483i \(0.458426\pi\)
\(992\) 0 0
\(993\) − 7553.00i − 0.241377i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 38468.0i 1.22196i 0.791646 + 0.610980i \(0.209224\pi\)
−0.791646 + 0.610980i \(0.790776\pi\)
\(998\) 0 0
\(999\) −11342.0 −0.359204
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 100.4.c.b.49.1 2
3.2 odd 2 900.4.d.a.649.2 2
4.3 odd 2 400.4.c.l.49.2 2
5.2 odd 4 100.4.a.b.1.1 1
5.3 odd 4 100.4.a.c.1.1 yes 1
5.4 even 2 inner 100.4.c.b.49.2 2
15.2 even 4 900.4.a.c.1.1 1
15.8 even 4 900.4.a.p.1.1 1
15.14 odd 2 900.4.d.a.649.1 2
20.3 even 4 400.4.a.i.1.1 1
20.7 even 4 400.4.a.l.1.1 1
20.19 odd 2 400.4.c.l.49.1 2
40.3 even 4 1600.4.a.bd.1.1 1
40.13 odd 4 1600.4.a.x.1.1 1
40.27 even 4 1600.4.a.y.1.1 1
40.37 odd 4 1600.4.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
100.4.a.b.1.1 1 5.2 odd 4
100.4.a.c.1.1 yes 1 5.3 odd 4
100.4.c.b.49.1 2 1.1 even 1 trivial
100.4.c.b.49.2 2 5.4 even 2 inner
400.4.a.i.1.1 1 20.3 even 4
400.4.a.l.1.1 1 20.7 even 4
400.4.c.l.49.1 2 20.19 odd 2
400.4.c.l.49.2 2 4.3 odd 2
900.4.a.c.1.1 1 15.2 even 4
900.4.a.p.1.1 1 15.8 even 4
900.4.d.a.649.1 2 15.14 odd 2
900.4.d.a.649.2 2 3.2 odd 2
1600.4.a.x.1.1 1 40.13 odd 4
1600.4.a.y.1.1 1 40.27 even 4
1600.4.a.bc.1.1 1 40.37 odd 4
1600.4.a.bd.1.1 1 40.3 even 4