Properties

Label 450.4.c.f.199.2
Level $450$
Weight $4$
Character 450.199
Analytic conductor $26.551$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.5508595026\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 450.199
Dual form 450.4.c.f.199.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} -4.00000 q^{4} +14.0000i q^{7} -8.00000i q^{8} +O(q^{10})\) \(q+2.00000i q^{2} -4.00000 q^{4} +14.0000i q^{7} -8.00000i q^{8} -6.00000 q^{11} -68.0000i q^{13} -28.0000 q^{14} +16.0000 q^{16} -78.0000i q^{17} -44.0000 q^{19} -12.0000i q^{22} +120.000i q^{23} +136.000 q^{26} -56.0000i q^{28} +126.000 q^{29} -244.000 q^{31} +32.0000i q^{32} +156.000 q^{34} -304.000i q^{37} -88.0000i q^{38} +480.000 q^{41} -104.000i q^{43} +24.0000 q^{44} -240.000 q^{46} -600.000i q^{47} +147.000 q^{49} +272.000i q^{52} -258.000i q^{53} +112.000 q^{56} +252.000i q^{58} +534.000 q^{59} +362.000 q^{61} -488.000i q^{62} -64.0000 q^{64} -268.000i q^{67} +312.000i q^{68} +972.000 q^{71} -470.000i q^{73} +608.000 q^{74} +176.000 q^{76} -84.0000i q^{77} -1244.00 q^{79} +960.000i q^{82} +396.000i q^{83} +208.000 q^{86} +48.0000i q^{88} -972.000 q^{89} +952.000 q^{91} -480.000i q^{92} +1200.00 q^{94} -46.0000i q^{97} +294.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} - 12 q^{11} - 56 q^{14} + 32 q^{16} - 88 q^{19} + 272 q^{26} + 252 q^{29} - 488 q^{31} + 312 q^{34} + 960 q^{41} + 48 q^{44} - 480 q^{46} + 294 q^{49} + 224 q^{56} + 1068 q^{59} + 724 q^{61} - 128 q^{64} + 1944 q^{71} + 1216 q^{74} + 352 q^{76} - 2488 q^{79} + 416 q^{86} - 1944 q^{89} + 1904 q^{91} + 2400 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) 0 0
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 14.0000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −6.00000 −0.164461 −0.0822304 0.996613i \(-0.526204\pi\)
−0.0822304 + 0.996613i \(0.526204\pi\)
\(12\) 0 0
\(13\) − 68.0000i − 1.45075i −0.688352 0.725377i \(-0.741665\pi\)
0.688352 0.725377i \(-0.258335\pi\)
\(14\) −28.0000 −0.534522
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 78.0000i − 1.11281i −0.830911 0.556405i \(-0.812180\pi\)
0.830911 0.556405i \(-0.187820\pi\)
\(18\) 0 0
\(19\) −44.0000 −0.531279 −0.265639 0.964072i \(-0.585583\pi\)
−0.265639 + 0.964072i \(0.585583\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 12.0000i − 0.116291i
\(23\) 120.000i 1.08790i 0.839117 + 0.543951i \(0.183072\pi\)
−0.839117 + 0.543951i \(0.816928\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 136.000 1.02584
\(27\) 0 0
\(28\) − 56.0000i − 0.377964i
\(29\) 126.000 0.806814 0.403407 0.915021i \(-0.367826\pi\)
0.403407 + 0.915021i \(0.367826\pi\)
\(30\) 0 0
\(31\) −244.000 −1.41367 −0.706834 0.707380i \(-0.749877\pi\)
−0.706834 + 0.707380i \(0.749877\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 0 0
\(34\) 156.000 0.786876
\(35\) 0 0
\(36\) 0 0
\(37\) − 304.000i − 1.35074i −0.737480 0.675369i \(-0.763984\pi\)
0.737480 0.675369i \(-0.236016\pi\)
\(38\) − 88.0000i − 0.375671i
\(39\) 0 0
\(40\) 0 0
\(41\) 480.000 1.82838 0.914188 0.405291i \(-0.132830\pi\)
0.914188 + 0.405291i \(0.132830\pi\)
\(42\) 0 0
\(43\) − 104.000i − 0.368834i −0.982848 0.184417i \(-0.940960\pi\)
0.982848 0.184417i \(-0.0590396\pi\)
\(44\) 24.0000 0.0822304
\(45\) 0 0
\(46\) −240.000 −0.769262
\(47\) − 600.000i − 1.86211i −0.364884 0.931053i \(-0.618891\pi\)
0.364884 0.931053i \(-0.381109\pi\)
\(48\) 0 0
\(49\) 147.000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 272.000i 0.725377i
\(53\) − 258.000i − 0.668661i −0.942456 0.334330i \(-0.891490\pi\)
0.942456 0.334330i \(-0.108510\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 112.000 0.267261
\(57\) 0 0
\(58\) 252.000i 0.570504i
\(59\) 534.000 1.17832 0.589160 0.808016i \(-0.299459\pi\)
0.589160 + 0.808016i \(0.299459\pi\)
\(60\) 0 0
\(61\) 362.000 0.759825 0.379913 0.925022i \(-0.375954\pi\)
0.379913 + 0.925022i \(0.375954\pi\)
\(62\) − 488.000i − 0.999614i
\(63\) 0 0
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 268.000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 312.000i 0.556405i
\(69\) 0 0
\(70\) 0 0
\(71\) 972.000 1.62472 0.812360 0.583156i \(-0.198182\pi\)
0.812360 + 0.583156i \(0.198182\pi\)
\(72\) 0 0
\(73\) − 470.000i − 0.753553i −0.926304 0.376776i \(-0.877033\pi\)
0.926304 0.376776i \(-0.122967\pi\)
\(74\) 608.000 0.955116
\(75\) 0 0
\(76\) 176.000 0.265639
\(77\) − 84.0000i − 0.124321i
\(78\) 0 0
\(79\) −1244.00 −1.77166 −0.885829 0.464012i \(-0.846409\pi\)
−0.885829 + 0.464012i \(0.846409\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 960.000i 1.29286i
\(83\) 396.000i 0.523695i 0.965109 + 0.261847i \(0.0843317\pi\)
−0.965109 + 0.261847i \(0.915668\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 208.000 0.260805
\(87\) 0 0
\(88\) 48.0000i 0.0581456i
\(89\) −972.000 −1.15766 −0.578830 0.815448i \(-0.696491\pi\)
−0.578830 + 0.815448i \(0.696491\pi\)
\(90\) 0 0
\(91\) 952.000 1.09667
\(92\) − 480.000i − 0.543951i
\(93\) 0 0
\(94\) 1200.00 1.31671
\(95\) 0 0
\(96\) 0 0
\(97\) − 46.0000i − 0.0481504i −0.999710 0.0240752i \(-0.992336\pi\)
0.999710 0.0240752i \(-0.00766412\pi\)
\(98\) 294.000i 0.303046i
\(99\) 0 0
\(100\) 0 0
\(101\) 1506.00 1.48369 0.741845 0.670572i \(-0.233951\pi\)
0.741845 + 0.670572i \(0.233951\pi\)
\(102\) 0 0
\(103\) 1474.00i 1.41007i 0.709171 + 0.705037i \(0.249069\pi\)
−0.709171 + 0.705037i \(0.750931\pi\)
\(104\) −544.000 −0.512919
\(105\) 0 0
\(106\) 516.000 0.472815
\(107\) − 924.000i − 0.834827i −0.908717 0.417413i \(-0.862937\pi\)
0.908717 0.417413i \(-0.137063\pi\)
\(108\) 0 0
\(109\) −698.000 −0.613360 −0.306680 0.951813i \(-0.599218\pi\)
−0.306680 + 0.951813i \(0.599218\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 224.000i 0.188982i
\(113\) − 222.000i − 0.184814i −0.995721 0.0924071i \(-0.970544\pi\)
0.995721 0.0924071i \(-0.0294561\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −504.000 −0.403407
\(117\) 0 0
\(118\) 1068.00i 0.833198i
\(119\) 1092.00 0.841206
\(120\) 0 0
\(121\) −1295.00 −0.972953
\(122\) 724.000i 0.537278i
\(123\) 0 0
\(124\) 976.000 0.706834
\(125\) 0 0
\(126\) 0 0
\(127\) − 1906.00i − 1.33173i −0.746071 0.665867i \(-0.768062\pi\)
0.746071 0.665867i \(-0.231938\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −2874.00 −1.91681 −0.958407 0.285406i \(-0.907872\pi\)
−0.958407 + 0.285406i \(0.907872\pi\)
\(132\) 0 0
\(133\) − 616.000i − 0.401609i
\(134\) 536.000 0.345547
\(135\) 0 0
\(136\) −624.000 −0.393438
\(137\) 798.000i 0.497648i 0.968549 + 0.248824i \(0.0800440\pi\)
−0.968549 + 0.248824i \(0.919956\pi\)
\(138\) 0 0
\(139\) 700.000 0.427146 0.213573 0.976927i \(-0.431490\pi\)
0.213573 + 0.976927i \(0.431490\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1944.00i 1.14885i
\(143\) 408.000i 0.238592i
\(144\) 0 0
\(145\) 0 0
\(146\) 940.000 0.532842
\(147\) 0 0
\(148\) 1216.00i 0.675369i
\(149\) 114.000 0.0626795 0.0313397 0.999509i \(-0.490023\pi\)
0.0313397 + 0.999509i \(0.490023\pi\)
\(150\) 0 0
\(151\) 1064.00 0.573424 0.286712 0.958017i \(-0.407438\pi\)
0.286712 + 0.958017i \(0.407438\pi\)
\(152\) 352.000i 0.187835i
\(153\) 0 0
\(154\) 168.000 0.0879080
\(155\) 0 0
\(156\) 0 0
\(157\) − 1948.00i − 0.990238i −0.868825 0.495119i \(-0.835125\pi\)
0.868825 0.495119i \(-0.164875\pi\)
\(158\) − 2488.00i − 1.25275i
\(159\) 0 0
\(160\) 0 0
\(161\) −1680.00 −0.822376
\(162\) 0 0
\(163\) − 2060.00i − 0.989887i −0.868925 0.494944i \(-0.835189\pi\)
0.868925 0.494944i \(-0.164811\pi\)
\(164\) −1920.00 −0.914188
\(165\) 0 0
\(166\) −792.000 −0.370308
\(167\) 1248.00i 0.578282i 0.957286 + 0.289141i \(0.0933697\pi\)
−0.957286 + 0.289141i \(0.906630\pi\)
\(168\) 0 0
\(169\) −2427.00 −1.10469
\(170\) 0 0
\(171\) 0 0
\(172\) 416.000i 0.184417i
\(173\) − 1146.00i − 0.503634i −0.967775 0.251817i \(-0.918972\pi\)
0.967775 0.251817i \(-0.0810282\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −96.0000 −0.0411152
\(177\) 0 0
\(178\) − 1944.00i − 0.818590i
\(179\) −1146.00 −0.478525 −0.239263 0.970955i \(-0.576906\pi\)
−0.239263 + 0.970955i \(0.576906\pi\)
\(180\) 0 0
\(181\) −118.000 −0.0484579 −0.0242289 0.999706i \(-0.507713\pi\)
−0.0242289 + 0.999706i \(0.507713\pi\)
\(182\) 1904.00i 0.775461i
\(183\) 0 0
\(184\) 960.000 0.384631
\(185\) 0 0
\(186\) 0 0
\(187\) 468.000i 0.183014i
\(188\) 2400.00i 0.931053i
\(189\) 0 0
\(190\) 0 0
\(191\) −1692.00 −0.640989 −0.320494 0.947250i \(-0.603849\pi\)
−0.320494 + 0.947250i \(0.603849\pi\)
\(192\) 0 0
\(193\) − 3350.00i − 1.24942i −0.780856 0.624711i \(-0.785217\pi\)
0.780856 0.624711i \(-0.214783\pi\)
\(194\) 92.0000 0.0340475
\(195\) 0 0
\(196\) −588.000 −0.214286
\(197\) 3606.00i 1.30415i 0.758156 + 0.652073i \(0.226101\pi\)
−0.758156 + 0.652073i \(0.773899\pi\)
\(198\) 0 0
\(199\) −2696.00 −0.960374 −0.480187 0.877166i \(-0.659431\pi\)
−0.480187 + 0.877166i \(0.659431\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 3012.00i 1.04913i
\(203\) 1764.00i 0.609894i
\(204\) 0 0
\(205\) 0 0
\(206\) −2948.00 −0.997072
\(207\) 0 0
\(208\) − 1088.00i − 0.362689i
\(209\) 264.000 0.0873745
\(210\) 0 0
\(211\) −4.00000 −0.00130508 −0.000652539 1.00000i \(-0.500208\pi\)
−0.000652539 1.00000i \(0.500208\pi\)
\(212\) 1032.00i 0.334330i
\(213\) 0 0
\(214\) 1848.00 0.590312
\(215\) 0 0
\(216\) 0 0
\(217\) − 3416.00i − 1.06863i
\(218\) − 1396.00i − 0.433711i
\(219\) 0 0
\(220\) 0 0
\(221\) −5304.00 −1.61441
\(222\) 0 0
\(223\) 1162.00i 0.348938i 0.984663 + 0.174469i \(0.0558210\pi\)
−0.984663 + 0.174469i \(0.944179\pi\)
\(224\) −448.000 −0.133631
\(225\) 0 0
\(226\) 444.000 0.130683
\(227\) 2400.00i 0.701734i 0.936425 + 0.350867i \(0.114113\pi\)
−0.936425 + 0.350867i \(0.885887\pi\)
\(228\) 0 0
\(229\) 2314.00 0.667744 0.333872 0.942618i \(-0.391645\pi\)
0.333872 + 0.942618i \(0.391645\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 1008.00i − 0.285252i
\(233\) − 18.0000i − 0.00506103i −0.999997 0.00253051i \(-0.999195\pi\)
0.999997 0.00253051i \(-0.000805488\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2136.00 −0.589160
\(237\) 0 0
\(238\) 2184.00i 0.594822i
\(239\) −5868.00 −1.58816 −0.794078 0.607816i \(-0.792046\pi\)
−0.794078 + 0.607816i \(0.792046\pi\)
\(240\) 0 0
\(241\) −4330.00 −1.15734 −0.578672 0.815560i \(-0.696429\pi\)
−0.578672 + 0.815560i \(0.696429\pi\)
\(242\) − 2590.00i − 0.687981i
\(243\) 0 0
\(244\) −1448.00 −0.379913
\(245\) 0 0
\(246\) 0 0
\(247\) 2992.00i 0.770755i
\(248\) 1952.00i 0.499807i
\(249\) 0 0
\(250\) 0 0
\(251\) −498.000 −0.125233 −0.0626165 0.998038i \(-0.519944\pi\)
−0.0626165 + 0.998038i \(0.519944\pi\)
\(252\) 0 0
\(253\) − 720.000i − 0.178917i
\(254\) 3812.00 0.941678
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 642.000i − 0.155824i −0.996960 0.0779122i \(-0.975175\pi\)
0.996960 0.0779122i \(-0.0248254\pi\)
\(258\) 0 0
\(259\) 4256.00 1.02106
\(260\) 0 0
\(261\) 0 0
\(262\) − 5748.00i − 1.35539i
\(263\) − 7968.00i − 1.86817i −0.357055 0.934084i \(-0.616219\pi\)
0.357055 0.934084i \(-0.383781\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1232.00 0.283980
\(267\) 0 0
\(268\) 1072.00i 0.244339i
\(269\) −4218.00 −0.956045 −0.478022 0.878348i \(-0.658646\pi\)
−0.478022 + 0.878348i \(0.658646\pi\)
\(270\) 0 0
\(271\) 848.000 0.190082 0.0950412 0.995473i \(-0.469702\pi\)
0.0950412 + 0.995473i \(0.469702\pi\)
\(272\) − 1248.00i − 0.278203i
\(273\) 0 0
\(274\) −1596.00 −0.351890
\(275\) 0 0
\(276\) 0 0
\(277\) − 1504.00i − 0.326233i −0.986607 0.163117i \(-0.947845\pi\)
0.986607 0.163117i \(-0.0521547\pi\)
\(278\) 1400.00i 0.302037i
\(279\) 0 0
\(280\) 0 0
\(281\) 1308.00 0.277682 0.138841 0.990315i \(-0.455662\pi\)
0.138841 + 0.990315i \(0.455662\pi\)
\(282\) 0 0
\(283\) 5932.00i 1.24601i 0.782218 + 0.623005i \(0.214088\pi\)
−0.782218 + 0.623005i \(0.785912\pi\)
\(284\) −3888.00 −0.812360
\(285\) 0 0
\(286\) −816.000 −0.168710
\(287\) 6720.00i 1.38212i
\(288\) 0 0
\(289\) −1171.00 −0.238347
\(290\) 0 0
\(291\) 0 0
\(292\) 1880.00i 0.376776i
\(293\) 5226.00i 1.04200i 0.853556 + 0.521000i \(0.174441\pi\)
−0.853556 + 0.521000i \(0.825559\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2432.00 −0.477558
\(297\) 0 0
\(298\) 228.000i 0.0443211i
\(299\) 8160.00 1.57828
\(300\) 0 0
\(301\) 1456.00 0.278812
\(302\) 2128.00i 0.405472i
\(303\) 0 0
\(304\) −704.000 −0.132820
\(305\) 0 0
\(306\) 0 0
\(307\) 4448.00i 0.826908i 0.910525 + 0.413454i \(0.135678\pi\)
−0.910525 + 0.413454i \(0.864322\pi\)
\(308\) 336.000i 0.0621603i
\(309\) 0 0
\(310\) 0 0
\(311\) 9132.00 1.66504 0.832521 0.553993i \(-0.186897\pi\)
0.832521 + 0.553993i \(0.186897\pi\)
\(312\) 0 0
\(313\) 2170.00i 0.391871i 0.980617 + 0.195936i \(0.0627743\pi\)
−0.980617 + 0.195936i \(0.937226\pi\)
\(314\) 3896.00 0.700204
\(315\) 0 0
\(316\) 4976.00 0.885829
\(317\) − 7674.00i − 1.35967i −0.733366 0.679834i \(-0.762052\pi\)
0.733366 0.679834i \(-0.237948\pi\)
\(318\) 0 0
\(319\) −756.000 −0.132689
\(320\) 0 0
\(321\) 0 0
\(322\) − 3360.00i − 0.581508i
\(323\) 3432.00i 0.591212i
\(324\) 0 0
\(325\) 0 0
\(326\) 4120.00 0.699956
\(327\) 0 0
\(328\) − 3840.00i − 0.646428i
\(329\) 8400.00 1.40762
\(330\) 0 0
\(331\) 9596.00 1.59349 0.796743 0.604318i \(-0.206554\pi\)
0.796743 + 0.604318i \(0.206554\pi\)
\(332\) − 1584.00i − 0.261847i
\(333\) 0 0
\(334\) −2496.00 −0.408907
\(335\) 0 0
\(336\) 0 0
\(337\) 12158.0i 1.96525i 0.185608 + 0.982624i \(0.440574\pi\)
−0.185608 + 0.982624i \(0.559426\pi\)
\(338\) − 4854.00i − 0.781133i
\(339\) 0 0
\(340\) 0 0
\(341\) 1464.00 0.232493
\(342\) 0 0
\(343\) 6860.00i 1.07990i
\(344\) −832.000 −0.130402
\(345\) 0 0
\(346\) 2292.00 0.356123
\(347\) − 10320.0i − 1.59656i −0.602286 0.798280i \(-0.705743\pi\)
0.602286 0.798280i \(-0.294257\pi\)
\(348\) 0 0
\(349\) 2158.00 0.330989 0.165494 0.986211i \(-0.447078\pi\)
0.165494 + 0.986211i \(0.447078\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 192.000i − 0.0290728i
\(353\) − 330.000i − 0.0497567i −0.999690 0.0248784i \(-0.992080\pi\)
0.999690 0.0248784i \(-0.00791985\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 3888.00 0.578830
\(357\) 0 0
\(358\) − 2292.00i − 0.338369i
\(359\) −8664.00 −1.27373 −0.636864 0.770976i \(-0.719769\pi\)
−0.636864 + 0.770976i \(0.719769\pi\)
\(360\) 0 0
\(361\) −4923.00 −0.717743
\(362\) − 236.000i − 0.0342649i
\(363\) 0 0
\(364\) −3808.00 −0.548334
\(365\) 0 0
\(366\) 0 0
\(367\) 3782.00i 0.537926i 0.963151 + 0.268963i \(0.0866809\pi\)
−0.963151 + 0.268963i \(0.913319\pi\)
\(368\) 1920.00i 0.271975i
\(369\) 0 0
\(370\) 0 0
\(371\) 3612.00 0.505460
\(372\) 0 0
\(373\) − 11276.0i − 1.56528i −0.622475 0.782640i \(-0.713873\pi\)
0.622475 0.782640i \(-0.286127\pi\)
\(374\) −936.000 −0.129410
\(375\) 0 0
\(376\) −4800.00 −0.658354
\(377\) − 8568.00i − 1.17049i
\(378\) 0 0
\(379\) −980.000 −0.132821 −0.0664106 0.997792i \(-0.521155\pi\)
−0.0664106 + 0.997792i \(0.521155\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 3384.00i − 0.453247i
\(383\) − 4200.00i − 0.560339i −0.959950 0.280170i \(-0.909609\pi\)
0.959950 0.280170i \(-0.0903907\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 6700.00 0.883474
\(387\) 0 0
\(388\) 184.000i 0.0240752i
\(389\) 13338.0 1.73847 0.869233 0.494402i \(-0.164613\pi\)
0.869233 + 0.494402i \(0.164613\pi\)
\(390\) 0 0
\(391\) 9360.00 1.21063
\(392\) − 1176.00i − 0.151523i
\(393\) 0 0
\(394\) −7212.00 −0.922171
\(395\) 0 0
\(396\) 0 0
\(397\) − 7192.00i − 0.909209i −0.890693 0.454605i \(-0.849781\pi\)
0.890693 0.454605i \(-0.150219\pi\)
\(398\) − 5392.00i − 0.679087i
\(399\) 0 0
\(400\) 0 0
\(401\) 2316.00 0.288418 0.144209 0.989547i \(-0.453936\pi\)
0.144209 + 0.989547i \(0.453936\pi\)
\(402\) 0 0
\(403\) 16592.0i 2.05088i
\(404\) −6024.00 −0.741845
\(405\) 0 0
\(406\) −3528.00 −0.431260
\(407\) 1824.00i 0.222143i
\(408\) 0 0
\(409\) 12358.0 1.49404 0.747022 0.664800i \(-0.231483\pi\)
0.747022 + 0.664800i \(0.231483\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 5896.00i − 0.705037i
\(413\) 7476.00i 0.890726i
\(414\) 0 0
\(415\) 0 0
\(416\) 2176.00 0.256460
\(417\) 0 0
\(418\) 528.000i 0.0617831i
\(419\) 3306.00 0.385462 0.192731 0.981252i \(-0.438265\pi\)
0.192731 + 0.981252i \(0.438265\pi\)
\(420\) 0 0
\(421\) −14506.0 −1.67929 −0.839643 0.543139i \(-0.817236\pi\)
−0.839643 + 0.543139i \(0.817236\pi\)
\(422\) − 8.00000i 0 0.000922829i
\(423\) 0 0
\(424\) −2064.00 −0.236407
\(425\) 0 0
\(426\) 0 0
\(427\) 5068.00i 0.574374i
\(428\) 3696.00i 0.417413i
\(429\) 0 0
\(430\) 0 0
\(431\) 6480.00 0.724201 0.362100 0.932139i \(-0.382060\pi\)
0.362100 + 0.932139i \(0.382060\pi\)
\(432\) 0 0
\(433\) − 11894.0i − 1.32007i −0.751236 0.660034i \(-0.770542\pi\)
0.751236 0.660034i \(-0.229458\pi\)
\(434\) 6832.00 0.755637
\(435\) 0 0
\(436\) 2792.00 0.306680
\(437\) − 5280.00i − 0.577979i
\(438\) 0 0
\(439\) 12688.0 1.37942 0.689710 0.724086i \(-0.257738\pi\)
0.689710 + 0.724086i \(0.257738\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 10608.0i − 1.14156i
\(443\) 4968.00i 0.532814i 0.963861 + 0.266407i \(0.0858366\pi\)
−0.963861 + 0.266407i \(0.914163\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −2324.00 −0.246737
\(447\) 0 0
\(448\) − 896.000i − 0.0944911i
\(449\) −11508.0 −1.20957 −0.604784 0.796389i \(-0.706741\pi\)
−0.604784 + 0.796389i \(0.706741\pi\)
\(450\) 0 0
\(451\) −2880.00 −0.300696
\(452\) 888.000i 0.0924071i
\(453\) 0 0
\(454\) −4800.00 −0.496201
\(455\) 0 0
\(456\) 0 0
\(457\) 1082.00i 0.110752i 0.998466 + 0.0553762i \(0.0176358\pi\)
−0.998466 + 0.0553762i \(0.982364\pi\)
\(458\) 4628.00i 0.472166i
\(459\) 0 0
\(460\) 0 0
\(461\) 11238.0 1.13537 0.567685 0.823246i \(-0.307839\pi\)
0.567685 + 0.823246i \(0.307839\pi\)
\(462\) 0 0
\(463\) 2302.00i 0.231065i 0.993304 + 0.115532i \(0.0368574\pi\)
−0.993304 + 0.115532i \(0.963143\pi\)
\(464\) 2016.00 0.201704
\(465\) 0 0
\(466\) 36.0000 0.00357869
\(467\) − 15876.0i − 1.57313i −0.617505 0.786567i \(-0.711856\pi\)
0.617505 0.786567i \(-0.288144\pi\)
\(468\) 0 0
\(469\) 3752.00 0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) − 4272.00i − 0.416599i
\(473\) 624.000i 0.0606587i
\(474\) 0 0
\(475\) 0 0
\(476\) −4368.00 −0.420603
\(477\) 0 0
\(478\) − 11736.0i − 1.12300i
\(479\) 4644.00 0.442985 0.221492 0.975162i \(-0.428907\pi\)
0.221492 + 0.975162i \(0.428907\pi\)
\(480\) 0 0
\(481\) −20672.0 −1.95959
\(482\) − 8660.00i − 0.818366i
\(483\) 0 0
\(484\) 5180.00 0.486476
\(485\) 0 0
\(486\) 0 0
\(487\) 2426.00i 0.225734i 0.993610 + 0.112867i \(0.0360034\pi\)
−0.993610 + 0.112867i \(0.963997\pi\)
\(488\) − 2896.00i − 0.268639i
\(489\) 0 0
\(490\) 0 0
\(491\) −234.000 −0.0215077 −0.0107538 0.999942i \(-0.503423\pi\)
−0.0107538 + 0.999942i \(0.503423\pi\)
\(492\) 0 0
\(493\) − 9828.00i − 0.897831i
\(494\) −5984.00 −0.545006
\(495\) 0 0
\(496\) −3904.00 −0.353417
\(497\) 13608.0i 1.22817i
\(498\) 0 0
\(499\) −14204.0 −1.27427 −0.637133 0.770754i \(-0.719880\pi\)
−0.637133 + 0.770754i \(0.719880\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 996.000i − 0.0885531i
\(503\) 4920.00i 0.436127i 0.975935 + 0.218064i \(0.0699740\pi\)
−0.975935 + 0.218064i \(0.930026\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1440.00 0.126513
\(507\) 0 0
\(508\) 7624.00i 0.665867i
\(509\) −4458.00 −0.388207 −0.194104 0.980981i \(-0.562180\pi\)
−0.194104 + 0.980981i \(0.562180\pi\)
\(510\) 0 0
\(511\) 6580.00 0.569632
\(512\) 512.000i 0.0441942i
\(513\) 0 0
\(514\) 1284.00 0.110184
\(515\) 0 0
\(516\) 0 0
\(517\) 3600.00i 0.306243i
\(518\) 8512.00i 0.722000i
\(519\) 0 0
\(520\) 0 0
\(521\) −4212.00 −0.354186 −0.177093 0.984194i \(-0.556669\pi\)
−0.177093 + 0.984194i \(0.556669\pi\)
\(522\) 0 0
\(523\) 11212.0i 0.937412i 0.883354 + 0.468706i \(0.155280\pi\)
−0.883354 + 0.468706i \(0.844720\pi\)
\(524\) 11496.0 0.958407
\(525\) 0 0
\(526\) 15936.0 1.32099
\(527\) 19032.0i 1.57314i
\(528\) 0 0
\(529\) −2233.00 −0.183529
\(530\) 0 0
\(531\) 0 0
\(532\) 2464.00i 0.200804i
\(533\) − 32640.0i − 2.65252i
\(534\) 0 0
\(535\) 0 0
\(536\) −2144.00 −0.172774
\(537\) 0 0
\(538\) − 8436.00i − 0.676026i
\(539\) −882.000 −0.0704832
\(540\) 0 0
\(541\) 14018.0 1.11401 0.557006 0.830508i \(-0.311950\pi\)
0.557006 + 0.830508i \(0.311950\pi\)
\(542\) 1696.00i 0.134409i
\(543\) 0 0
\(544\) 2496.00 0.196719
\(545\) 0 0
\(546\) 0 0
\(547\) 18200.0i 1.42262i 0.702876 + 0.711312i \(0.251899\pi\)
−0.702876 + 0.711312i \(0.748101\pi\)
\(548\) − 3192.00i − 0.248824i
\(549\) 0 0
\(550\) 0 0
\(551\) −5544.00 −0.428643
\(552\) 0 0
\(553\) − 17416.0i − 1.33925i
\(554\) 3008.00 0.230682
\(555\) 0 0
\(556\) −2800.00 −0.213573
\(557\) 11826.0i 0.899612i 0.893126 + 0.449806i \(0.148507\pi\)
−0.893126 + 0.449806i \(0.851493\pi\)
\(558\) 0 0
\(559\) −7072.00 −0.535087
\(560\) 0 0
\(561\) 0 0
\(562\) 2616.00i 0.196351i
\(563\) − 2952.00i − 0.220980i −0.993877 0.110490i \(-0.964758\pi\)
0.993877 0.110490i \(-0.0352421\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −11864.0 −0.881062
\(567\) 0 0
\(568\) − 7776.00i − 0.574426i
\(569\) 3084.00 0.227220 0.113610 0.993525i \(-0.463759\pi\)
0.113610 + 0.993525i \(0.463759\pi\)
\(570\) 0 0
\(571\) −4756.00 −0.348568 −0.174284 0.984695i \(-0.555761\pi\)
−0.174284 + 0.984695i \(0.555761\pi\)
\(572\) − 1632.00i − 0.119296i
\(573\) 0 0
\(574\) −13440.0 −0.977308
\(575\) 0 0
\(576\) 0 0
\(577\) − 11014.0i − 0.794660i −0.917676 0.397330i \(-0.869937\pi\)
0.917676 0.397330i \(-0.130063\pi\)
\(578\) − 2342.00i − 0.168537i
\(579\) 0 0
\(580\) 0 0
\(581\) −5544.00 −0.395876
\(582\) 0 0
\(583\) 1548.00i 0.109968i
\(584\) −3760.00 −0.266421
\(585\) 0 0
\(586\) −10452.0 −0.736806
\(587\) 852.000i 0.0599077i 0.999551 + 0.0299538i \(0.00953603\pi\)
−0.999551 + 0.0299538i \(0.990464\pi\)
\(588\) 0 0
\(589\) 10736.0 0.751051
\(590\) 0 0
\(591\) 0 0
\(592\) − 4864.00i − 0.337684i
\(593\) 15546.0i 1.07656i 0.842767 + 0.538278i \(0.180925\pi\)
−0.842767 + 0.538278i \(0.819075\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −456.000 −0.0313397
\(597\) 0 0
\(598\) 16320.0i 1.11601i
\(599\) −8616.00 −0.587713 −0.293857 0.955850i \(-0.594939\pi\)
−0.293857 + 0.955850i \(0.594939\pi\)
\(600\) 0 0
\(601\) 17510.0 1.18843 0.594216 0.804305i \(-0.297462\pi\)
0.594216 + 0.804305i \(0.297462\pi\)
\(602\) 2912.00i 0.197150i
\(603\) 0 0
\(604\) −4256.00 −0.286712
\(605\) 0 0
\(606\) 0 0
\(607\) − 13894.0i − 0.929061i −0.885557 0.464531i \(-0.846223\pi\)
0.885557 0.464531i \(-0.153777\pi\)
\(608\) − 1408.00i − 0.0939177i
\(609\) 0 0
\(610\) 0 0
\(611\) −40800.0 −2.70146
\(612\) 0 0
\(613\) 6496.00i 0.428011i 0.976832 + 0.214006i \(0.0686511\pi\)
−0.976832 + 0.214006i \(0.931349\pi\)
\(614\) −8896.00 −0.584712
\(615\) 0 0
\(616\) −672.000 −0.0439540
\(617\) 570.000i 0.0371918i 0.999827 + 0.0185959i \(0.00591960\pi\)
−0.999827 + 0.0185959i \(0.994080\pi\)
\(618\) 0 0
\(619\) 2140.00 0.138956 0.0694781 0.997583i \(-0.477867\pi\)
0.0694781 + 0.997583i \(0.477867\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 18264.0i 1.17736i
\(623\) − 13608.0i − 0.875109i
\(624\) 0 0
\(625\) 0 0
\(626\) −4340.00 −0.277095
\(627\) 0 0
\(628\) 7792.00i 0.495119i
\(629\) −23712.0 −1.50312
\(630\) 0 0
\(631\) 14660.0 0.924890 0.462445 0.886648i \(-0.346972\pi\)
0.462445 + 0.886648i \(0.346972\pi\)
\(632\) 9952.00i 0.626375i
\(633\) 0 0
\(634\) 15348.0 0.961431
\(635\) 0 0
\(636\) 0 0
\(637\) − 9996.00i − 0.621752i
\(638\) − 1512.00i − 0.0938255i
\(639\) 0 0
\(640\) 0 0
\(641\) 456.000 0.0280982 0.0140491 0.999901i \(-0.495528\pi\)
0.0140491 + 0.999901i \(0.495528\pi\)
\(642\) 0 0
\(643\) 23452.0i 1.43835i 0.694831 + 0.719173i \(0.255479\pi\)
−0.694831 + 0.719173i \(0.744521\pi\)
\(644\) 6720.00 0.411188
\(645\) 0 0
\(646\) −6864.00 −0.418050
\(647\) − 7224.00i − 0.438956i −0.975617 0.219478i \(-0.929565\pi\)
0.975617 0.219478i \(-0.0704355\pi\)
\(648\) 0 0
\(649\) −3204.00 −0.193787
\(650\) 0 0
\(651\) 0 0
\(652\) 8240.00i 0.494944i
\(653\) 19146.0i 1.14738i 0.819072 + 0.573691i \(0.194489\pi\)
−0.819072 + 0.573691i \(0.805511\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 7680.00 0.457094
\(657\) 0 0
\(658\) 16800.0i 0.995338i
\(659\) 27810.0 1.64389 0.821945 0.569567i \(-0.192889\pi\)
0.821945 + 0.569567i \(0.192889\pi\)
\(660\) 0 0
\(661\) −30598.0 −1.80049 −0.900245 0.435383i \(-0.856613\pi\)
−0.900245 + 0.435383i \(0.856613\pi\)
\(662\) 19192.0i 1.12676i
\(663\) 0 0
\(664\) 3168.00 0.185154
\(665\) 0 0
\(666\) 0 0
\(667\) 15120.0i 0.877734i
\(668\) − 4992.00i − 0.289141i
\(669\) 0 0
\(670\) 0 0
\(671\) −2172.00 −0.124961
\(672\) 0 0
\(673\) 3778.00i 0.216391i 0.994130 + 0.108196i \(0.0345073\pi\)
−0.994130 + 0.108196i \(0.965493\pi\)
\(674\) −24316.0 −1.38964
\(675\) 0 0
\(676\) 9708.00 0.552344
\(677\) 27198.0i 1.54402i 0.635608 + 0.772012i \(0.280749\pi\)
−0.635608 + 0.772012i \(0.719251\pi\)
\(678\) 0 0
\(679\) 644.000 0.0363983
\(680\) 0 0
\(681\) 0 0
\(682\) 2928.00i 0.164397i
\(683\) 32316.0i 1.81045i 0.424933 + 0.905225i \(0.360298\pi\)
−0.424933 + 0.905225i \(0.639702\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −13720.0 −0.763604
\(687\) 0 0
\(688\) − 1664.00i − 0.0922084i
\(689\) −17544.0 −0.970063
\(690\) 0 0
\(691\) 29324.0 1.61438 0.807191 0.590291i \(-0.200987\pi\)
0.807191 + 0.590291i \(0.200987\pi\)
\(692\) 4584.00i 0.251817i
\(693\) 0 0
\(694\) 20640.0 1.12894
\(695\) 0 0
\(696\) 0 0
\(697\) − 37440.0i − 2.03464i
\(698\) 4316.00i 0.234044i
\(699\) 0 0
\(700\) 0 0
\(701\) −22782.0 −1.22748 −0.613741 0.789508i \(-0.710336\pi\)
−0.613741 + 0.789508i \(0.710336\pi\)
\(702\) 0 0
\(703\) 13376.0i 0.717618i
\(704\) 384.000 0.0205576
\(705\) 0 0
\(706\) 660.000 0.0351833
\(707\) 21084.0i 1.12156i
\(708\) 0 0
\(709\) −26054.0 −1.38008 −0.690041 0.723770i \(-0.742408\pi\)
−0.690041 + 0.723770i \(0.742408\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 7776.00i 0.409295i
\(713\) − 29280.0i − 1.53793i
\(714\) 0 0
\(715\) 0 0
\(716\) 4584.00 0.239263
\(717\) 0 0
\(718\) − 17328.0i − 0.900662i
\(719\) −5976.00 −0.309968 −0.154984 0.987917i \(-0.549533\pi\)
−0.154984 + 0.987917i \(0.549533\pi\)
\(720\) 0 0
\(721\) −20636.0 −1.06592
\(722\) − 9846.00i − 0.507521i
\(723\) 0 0
\(724\) 472.000 0.0242289
\(725\) 0 0
\(726\) 0 0
\(727\) − 5110.00i − 0.260687i −0.991469 0.130343i \(-0.958392\pi\)
0.991469 0.130343i \(-0.0416080\pi\)
\(728\) − 7616.00i − 0.387730i
\(729\) 0 0
\(730\) 0 0
\(731\) −8112.00 −0.410442
\(732\) 0 0
\(733\) − 17336.0i − 0.873560i −0.899568 0.436780i \(-0.856119\pi\)
0.899568 0.436780i \(-0.143881\pi\)
\(734\) −7564.00 −0.380371
\(735\) 0 0
\(736\) −3840.00 −0.192316
\(737\) 1608.00i 0.0803683i
\(738\) 0 0
\(739\) 13660.0 0.679961 0.339981 0.940432i \(-0.389580\pi\)
0.339981 + 0.940432i \(0.389580\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 7224.00i 0.357414i
\(743\) 1320.00i 0.0651765i 0.999469 + 0.0325882i \(0.0103750\pi\)
−0.999469 + 0.0325882i \(0.989625\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 22552.0 1.10682
\(747\) 0 0
\(748\) − 1872.00i − 0.0915068i
\(749\) 12936.0 0.631070
\(750\) 0 0
\(751\) 15860.0 0.770625 0.385313 0.922786i \(-0.374094\pi\)
0.385313 + 0.922786i \(0.374094\pi\)
\(752\) − 9600.00i − 0.465527i
\(753\) 0 0
\(754\) 17136.0 0.827661
\(755\) 0 0
\(756\) 0 0
\(757\) 22160.0i 1.06396i 0.846756 + 0.531981i \(0.178552\pi\)
−0.846756 + 0.531981i \(0.821448\pi\)
\(758\) − 1960.00i − 0.0939187i
\(759\) 0 0
\(760\) 0 0
\(761\) −13116.0 −0.624776 −0.312388 0.949955i \(-0.601129\pi\)
−0.312388 + 0.949955i \(0.601129\pi\)
\(762\) 0 0
\(763\) − 9772.00i − 0.463657i
\(764\) 6768.00 0.320494
\(765\) 0 0
\(766\) 8400.00 0.396220
\(767\) − 36312.0i − 1.70945i
\(768\) 0 0
\(769\) −32846.0 −1.54026 −0.770128 0.637889i \(-0.779808\pi\)
−0.770128 + 0.637889i \(0.779808\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 13400.0i 0.624711i
\(773\) − 11982.0i − 0.557520i −0.960361 0.278760i \(-0.910077\pi\)
0.960361 0.278760i \(-0.0899233\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −368.000 −0.0170238
\(777\) 0 0
\(778\) 26676.0i 1.22928i
\(779\) −21120.0 −0.971377
\(780\) 0 0
\(781\) −5832.00 −0.267203
\(782\) 18720.0i 0.856043i
\(783\) 0 0
\(784\) 2352.00 0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) − 21076.0i − 0.954610i −0.878738 0.477305i \(-0.841614\pi\)
0.878738 0.477305i \(-0.158386\pi\)
\(788\) − 14424.0i − 0.652073i
\(789\) 0 0
\(790\) 0 0
\(791\) 3108.00 0.139706
\(792\) 0 0
\(793\) − 24616.0i − 1.10232i
\(794\) 14384.0 0.642908
\(795\) 0 0
\(796\) 10784.0 0.480187
\(797\) − 22086.0i − 0.981589i −0.871275 0.490794i \(-0.836707\pi\)
0.871275 0.490794i \(-0.163293\pi\)
\(798\) 0 0
\(799\) −46800.0 −2.07217
\(800\) 0 0
\(801\) 0 0
\(802\) 4632.00i 0.203942i
\(803\) 2820.00i 0.123930i
\(804\) 0 0
\(805\) 0 0
\(806\) −33184.0 −1.45019
\(807\) 0 0
\(808\) − 12048.0i − 0.524563i
\(809\) 21384.0 0.929322 0.464661 0.885489i \(-0.346176\pi\)
0.464661 + 0.885489i \(0.346176\pi\)
\(810\) 0 0
\(811\) 5228.00 0.226362 0.113181 0.993574i \(-0.463896\pi\)
0.113181 + 0.993574i \(0.463896\pi\)
\(812\) − 7056.00i − 0.304947i
\(813\) 0 0
\(814\) −3648.00 −0.157079
\(815\) 0 0
\(816\) 0 0
\(817\) 4576.00i 0.195953i
\(818\) 24716.0i 1.05645i
\(819\) 0 0
\(820\) 0 0
\(821\) 38010.0 1.61578 0.807892 0.589331i \(-0.200609\pi\)
0.807892 + 0.589331i \(0.200609\pi\)
\(822\) 0 0
\(823\) − 38642.0i − 1.63667i −0.574745 0.818333i \(-0.694899\pi\)
0.574745 0.818333i \(-0.305101\pi\)
\(824\) 11792.0 0.498536
\(825\) 0 0
\(826\) −14952.0 −0.629839
\(827\) 15432.0i 0.648879i 0.945906 + 0.324440i \(0.105176\pi\)
−0.945906 + 0.324440i \(0.894824\pi\)
\(828\) 0 0
\(829\) 3886.00 0.162806 0.0814031 0.996681i \(-0.474060\pi\)
0.0814031 + 0.996681i \(0.474060\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 4352.00i 0.181344i
\(833\) − 11466.0i − 0.476919i
\(834\) 0 0
\(835\) 0 0
\(836\) −1056.00 −0.0436872
\(837\) 0 0
\(838\) 6612.00i 0.272563i
\(839\) 27552.0 1.13373 0.566866 0.823810i \(-0.308156\pi\)
0.566866 + 0.823810i \(0.308156\pi\)
\(840\) 0 0
\(841\) −8513.00 −0.349051
\(842\) − 29012.0i − 1.18743i
\(843\) 0 0
\(844\) 16.0000 0.000652539 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 18130.0i − 0.735483i
\(848\) − 4128.00i − 0.167165i
\(849\) 0 0
\(850\) 0 0
\(851\) 36480.0 1.46947
\(852\) 0 0
\(853\) − 15104.0i − 0.606273i −0.952947 0.303137i \(-0.901966\pi\)
0.952947 0.303137i \(-0.0980339\pi\)
\(854\) −10136.0 −0.406144
\(855\) 0 0
\(856\) −7392.00 −0.295156
\(857\) 12306.0i 0.490508i 0.969459 + 0.245254i \(0.0788713\pi\)
−0.969459 + 0.245254i \(0.921129\pi\)
\(858\) 0 0
\(859\) 47500.0 1.88670 0.943352 0.331793i \(-0.107654\pi\)
0.943352 + 0.331793i \(0.107654\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 12960.0i 0.512087i
\(863\) 4272.00i 0.168506i 0.996444 + 0.0842529i \(0.0268504\pi\)
−0.996444 + 0.0842529i \(0.973150\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 23788.0 0.933429
\(867\) 0 0
\(868\) 13664.0i 0.534316i
\(869\) 7464.00 0.291368
\(870\) 0 0
\(871\) −18224.0 −0.708951
\(872\) 5584.00i 0.216856i
\(873\) 0 0
\(874\) 10560.0 0.408693
\(875\) 0 0
\(876\) 0 0
\(877\) − 27796.0i − 1.07024i −0.844775 0.535122i \(-0.820266\pi\)
0.844775 0.535122i \(-0.179734\pi\)
\(878\) 25376.0i 0.975397i
\(879\) 0 0
\(880\) 0 0
\(881\) 39996.0 1.52951 0.764756 0.644320i \(-0.222860\pi\)
0.764756 + 0.644320i \(0.222860\pi\)
\(882\) 0 0
\(883\) 3772.00i 0.143758i 0.997413 + 0.0718788i \(0.0228995\pi\)
−0.997413 + 0.0718788i \(0.977101\pi\)
\(884\) 21216.0 0.807207
\(885\) 0 0
\(886\) −9936.00 −0.376757
\(887\) 5784.00i 0.218949i 0.993990 + 0.109474i \(0.0349168\pi\)
−0.993990 + 0.109474i \(0.965083\pi\)
\(888\) 0 0
\(889\) 26684.0 1.00670
\(890\) 0 0
\(891\) 0 0
\(892\) − 4648.00i − 0.174469i
\(893\) 26400.0i 0.989297i
\(894\) 0 0
\(895\) 0 0
\(896\) 1792.00 0.0668153
\(897\) 0 0
\(898\) − 23016.0i − 0.855294i
\(899\) −30744.0 −1.14057
\(900\) 0 0
\(901\) −20124.0 −0.744093
\(902\) − 5760.00i − 0.212624i
\(903\) 0 0
\(904\) −1776.00 −0.0653417
\(905\) 0 0
\(906\) 0 0
\(907\) − 8440.00i − 0.308981i −0.987994 0.154490i \(-0.950626\pi\)
0.987994 0.154490i \(-0.0493736\pi\)
\(908\) − 9600.00i − 0.350867i
\(909\) 0 0
\(910\) 0 0
\(911\) 31920.0 1.16087 0.580437 0.814305i \(-0.302882\pi\)
0.580437 + 0.814305i \(0.302882\pi\)
\(912\) 0 0
\(913\) − 2376.00i − 0.0861272i
\(914\) −2164.00 −0.0783137
\(915\) 0 0
\(916\) −9256.00 −0.333872
\(917\) − 40236.0i − 1.44897i
\(918\) 0 0
\(919\) −34652.0 −1.24381 −0.621906 0.783092i \(-0.713642\pi\)
−0.621906 + 0.783092i \(0.713642\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 22476.0i 0.802828i
\(923\) − 66096.0i − 2.35707i
\(924\) 0 0
\(925\) 0 0
\(926\) −4604.00 −0.163388
\(927\) 0 0
\(928\) 4032.00i 0.142626i
\(929\) 1404.00 0.0495842 0.0247921 0.999693i \(-0.492108\pi\)
0.0247921 + 0.999693i \(0.492108\pi\)
\(930\) 0 0
\(931\) −6468.00 −0.227691
\(932\) 72.0000i 0.00253051i
\(933\) 0 0
\(934\) 31752.0 1.11237
\(935\) 0 0
\(936\) 0 0
\(937\) − 7654.00i − 0.266857i −0.991058 0.133429i \(-0.957401\pi\)
0.991058 0.133429i \(-0.0425987\pi\)
\(938\) 7504.00i 0.261209i
\(939\) 0 0
\(940\) 0 0
\(941\) −11298.0 −0.391397 −0.195698 0.980664i \(-0.562697\pi\)
−0.195698 + 0.980664i \(0.562697\pi\)
\(942\) 0 0
\(943\) 57600.0i 1.98909i
\(944\) 8544.00 0.294580
\(945\) 0 0
\(946\) −1248.00 −0.0428922
\(947\) − 28968.0i − 0.994016i −0.867746 0.497008i \(-0.834432\pi\)
0.867746 0.497008i \(-0.165568\pi\)
\(948\) 0 0
\(949\) −31960.0 −1.09322
\(950\) 0 0
\(951\) 0 0
\(952\) − 8736.00i − 0.297411i
\(953\) 46410.0i 1.57751i 0.614707 + 0.788755i \(0.289274\pi\)
−0.614707 + 0.788755i \(0.710726\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 23472.0 0.794078
\(957\) 0 0
\(958\) 9288.00i 0.313238i
\(959\) −11172.0 −0.376186
\(960\) 0 0
\(961\) 29745.0 0.998456
\(962\) − 41344.0i − 1.38564i
\(963\) 0 0
\(964\) 17320.0 0.578672
\(965\) 0 0
\(966\) 0 0
\(967\) − 41506.0i − 1.38029i −0.723670 0.690146i \(-0.757546\pi\)
0.723670 0.690146i \(-0.242454\pi\)
\(968\) 10360.0i 0.343991i
\(969\) 0 0
\(970\) 0 0
\(971\) 18246.0 0.603030 0.301515 0.953461i \(-0.402508\pi\)
0.301515 + 0.953461i \(0.402508\pi\)
\(972\) 0 0
\(973\) 9800.00i 0.322892i
\(974\) −4852.00 −0.159618
\(975\) 0 0
\(976\) 5792.00 0.189956
\(977\) − 25998.0i − 0.851330i −0.904881 0.425665i \(-0.860040\pi\)
0.904881 0.425665i \(-0.139960\pi\)
\(978\) 0 0
\(979\) 5832.00 0.190390
\(980\) 0 0
\(981\) 0 0
\(982\) − 468.000i − 0.0152082i
\(983\) 14616.0i 0.474240i 0.971480 + 0.237120i \(0.0762035\pi\)
−0.971480 + 0.237120i \(0.923797\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 19656.0 0.634863
\(987\) 0 0
\(988\) − 11968.0i − 0.385377i
\(989\) 12480.0 0.401255
\(990\) 0 0
\(991\) −2968.00 −0.0951379 −0.0475689 0.998868i \(-0.515147\pi\)
−0.0475689 + 0.998868i \(0.515147\pi\)
\(992\) − 7808.00i − 0.249903i
\(993\) 0 0
\(994\) −27216.0 −0.868450
\(995\) 0 0
\(996\) 0 0
\(997\) − 9052.00i − 0.287542i −0.989611 0.143771i \(-0.954077\pi\)
0.989611 0.143771i \(-0.0459229\pi\)
\(998\) − 28408.0i − 0.901042i
\(999\) 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 450.4.c.f.199.2 2
3.2 odd 2 450.4.c.g.199.1 2
5.2 odd 4 450.4.a.c.1.1 1
5.3 odd 4 90.4.a.e.1.1 yes 1
5.4 even 2 inner 450.4.c.f.199.1 2
15.2 even 4 450.4.a.m.1.1 1
15.8 even 4 90.4.a.b.1.1 1
15.14 odd 2 450.4.c.g.199.2 2
20.3 even 4 720.4.a.t.1.1 1
45.13 odd 12 810.4.e.a.541.1 2
45.23 even 12 810.4.e.u.541.1 2
45.38 even 12 810.4.e.u.271.1 2
45.43 odd 12 810.4.e.a.271.1 2
60.23 odd 4 720.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.4.a.b.1.1 1 15.8 even 4
90.4.a.e.1.1 yes 1 5.3 odd 4
450.4.a.c.1.1 1 5.2 odd 4
450.4.a.m.1.1 1 15.2 even 4
450.4.c.f.199.1 2 5.4 even 2 inner
450.4.c.f.199.2 2 1.1 even 1 trivial
450.4.c.g.199.1 2 3.2 odd 2
450.4.c.g.199.2 2 15.14 odd 2
720.4.a.e.1.1 1 60.23 odd 4
720.4.a.t.1.1 1 20.3 even 4
810.4.e.a.271.1 2 45.43 odd 12
810.4.e.a.541.1 2 45.13 odd 12
810.4.e.u.271.1 2 45.38 even 12
810.4.e.u.541.1 2 45.23 even 12