Properties

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

$q$-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 \(-0.473796\pi\)
0.0822304 + 0.996613i \(0.473796\pi\)
\(12\) 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 \(0.187820\pi\)
−0.830911 + 0.556405i \(0.812180\pi\)
\(18\) 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 \(-0.816928\pi\)
0.839117 0.543951i \(-0.183072\pi\)
\(24\) 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 \(-0.632174\pi\)
−0.403407 + 0.915021i \(0.632174\pi\)
\(30\) 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 \(-0.867170\pi\)
−0.914188 + 0.405291i \(0.867170\pi\)
\(42\) 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 \(0.381109\pi\)
−0.364884 + 0.931053i \(0.618891\pi\)
\(48\) 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 \(0.108510\pi\)
−0.942456 + 0.334330i \(0.891490\pi\)
\(54\) 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 \(-0.700541\pi\)
−0.589160 + 0.808016i \(0.700541\pi\)
\(60\) 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 \(-0.801818\pi\)
−0.812360 + 0.583156i \(0.801818\pi\)
\(72\) 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 \(-0.915668\pi\)
0.965109 0.261847i \(-0.0843317\pi\)
\(84\) 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 \(-0.303509\pi\)
0.578830 + 0.815448i \(0.303509\pi\)
\(90\) 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 \(-0.766049\pi\)
−0.741845 + 0.670572i \(0.766049\pi\)
\(102\) 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 \(0.137063\pi\)
−0.908717 + 0.417413i \(0.862937\pi\)
\(108\) 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 \(0.0294561\pi\)
−0.995721 + 0.0924071i \(0.970544\pi\)
\(114\) 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 \(-0.0921285\pi\)
0.958407 + 0.285406i \(0.0921285\pi\)
\(132\) 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 \(-0.919956\pi\)
0.968549 0.248824i \(-0.0800440\pi\)
\(138\) 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 \(-0.509977\pi\)
−0.0313397 + 0.999509i \(0.509977\pi\)
\(150\) 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 \(-0.906630\pi\)
0.957286 0.289141i \(-0.0933697\pi\)
\(168\) 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 \(0.0810282\pi\)
−0.967775 + 0.251817i \(0.918972\pi\)
\(174\) 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 \(-0.423094\pi\)
0.239263 + 0.970955i \(0.423094\pi\)
\(180\) 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 \(-0.396151\pi\)
0.320494 + 0.947250i \(0.396151\pi\)
\(192\) 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 \(-0.773899\pi\)
0.758156 0.652073i \(-0.226101\pi\)
\(198\) 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 \(-0.885887\pi\)
0.936425 0.350867i \(-0.114113\pi\)
\(228\) 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 \(0.000805488\pi\)
−0.999997 + 0.00253051i \(0.999195\pi\)
\(234\) 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 \(-0.207954\pi\)
0.794078 + 0.607816i \(0.207954\pi\)
\(240\) 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 \(-0.480056\pi\)
0.0626165 + 0.998038i \(0.480056\pi\)
\(252\) 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 \(0.0248254\pi\)
−0.996960 + 0.0779122i \(0.975175\pi\)
\(258\) 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 \(0.383781\pi\)
−0.357055 + 0.934084i \(0.616219\pi\)
\(264\) 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 \(-0.341354\pi\)
0.478022 + 0.878348i \(0.341354\pi\)
\(270\) 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 \(-0.544338\pi\)
−0.138841 + 0.990315i \(0.544338\pi\)
\(282\) 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 \(-0.825559\pi\)
0.853556 0.521000i \(-0.174441\pi\)
\(294\) 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 \(-0.813103\pi\)
−0.832521 + 0.553993i \(0.813103\pi\)
\(312\) 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 \(0.237948\pi\)
−0.733366 + 0.679834i \(0.762052\pi\)
\(318\) 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 \(0.294257\pi\)
−0.602286 + 0.798280i \(0.705743\pi\)
\(348\) 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 \(0.00791985\pi\)
−0.999690 + 0.0248784i \(0.992080\pi\)
\(354\) 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 \(-0.280231\pi\)
0.636864 + 0.770976i \(0.280231\pi\)
\(360\) 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 \(0.0903907\pi\)
−0.959950 + 0.280170i \(0.909609\pi\)
\(384\) 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 \(-0.835387\pi\)
−0.869233 + 0.494402i \(0.835387\pi\)
\(390\) 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 \(-0.546064\pi\)
−0.144209 + 0.989547i \(0.546064\pi\)
\(402\) 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 \(-0.561735\pi\)
−0.192731 + 0.981252i \(0.561735\pi\)
\(420\) 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 \(-0.617940\pi\)
−0.362100 + 0.932139i \(0.617940\pi\)
\(432\) 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 \(-0.914163\pi\)
0.963861 0.266407i \(-0.0858366\pi\)
\(444\) 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 \(-0.293259\pi\)
0.604784 + 0.796389i \(0.293259\pi\)
\(450\) 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 \(-0.692161\pi\)
−0.567685 + 0.823246i \(0.692161\pi\)
\(462\) 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 \(0.288144\pi\)
−0.617505 + 0.786567i \(0.711856\pi\)
\(468\) 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 \(-0.571093\pi\)
−0.221492 + 0.975162i \(0.571093\pi\)
\(480\) 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 \(-0.496577\pi\)
0.0107538 + 0.999942i \(0.496577\pi\)
\(492\) 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 \(-0.930026\pi\)
0.975935 0.218064i \(-0.0699740\pi\)
\(504\) 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 \(-0.437820\pi\)
0.194104 + 0.980981i \(0.437820\pi\)
\(510\) 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 \(-0.443331\pi\)
0.177093 + 0.984194i \(0.443331\pi\)
\(522\) 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 \(-0.851493\pi\)
0.893126 0.449806i \(-0.148507\pi\)
\(558\) 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 \(0.0352421\pi\)
−0.993877 + 0.110490i \(0.964758\pi\)
\(564\) 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 \(-0.536241\pi\)
−0.113610 + 0.993525i \(0.536241\pi\)
\(570\) 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 \(-0.990464\pi\)
0.999551 0.0299538i \(-0.00953603\pi\)
\(588\) 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 \(-0.819075\pi\)
0.842767 0.538278i \(-0.180925\pi\)
\(594\) 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 \(-0.405061\pi\)
0.293857 + 0.955850i \(0.405061\pi\)
\(600\) 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 \(-0.994080\pi\)
0.999827 0.0185959i \(-0.00591960\pi\)
\(618\) 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 \(-0.504472\pi\)
−0.0140491 + 0.999901i \(0.504472\pi\)
\(642\) 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 \(0.0704355\pi\)
−0.975617 + 0.219478i \(0.929565\pi\)
\(648\) 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 \(-0.805511\pi\)
0.819072 0.573691i \(-0.194489\pi\)
\(654\) 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 \(-0.807111\pi\)
−0.821945 + 0.569567i \(0.807111\pi\)
\(660\) 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 \(-0.719251\pi\)
0.635608 0.772012i \(-0.280749\pi\)
\(678\) 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 \(-0.639702\pi\)
0.424933 0.905225i \(-0.360298\pi\)
\(684\) 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 \(-0.289664\pi\)
0.613741 + 0.789508i \(0.289664\pi\)
\(702\) 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 \(-0.450467\pi\)
0.154984 + 0.987917i \(0.450467\pi\)
\(720\) 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 \(-0.989625\pi\)
0.999469 0.0325882i \(-0.0103750\pi\)
\(744\) 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 \(-0.398871\pi\)
0.312388 + 0.949955i \(0.398871\pi\)
\(762\) 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 \(0.0899233\pi\)
−0.960361 + 0.278760i \(0.910077\pi\)
\(774\) 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 \(0.163293\pi\)
−0.871275 + 0.490794i \(0.836707\pi\)
\(798\) 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 \(-0.653824\pi\)
−0.464661 + 0.885489i \(0.653824\pi\)
\(810\) 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 \(-0.799391\pi\)
−0.807892 + 0.589331i \(0.799391\pi\)
\(822\) 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 \(-0.894824\pi\)
0.945906 0.324440i \(-0.105176\pi\)
\(828\) 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 \(-0.691844\pi\)
−0.566866 + 0.823810i \(0.691844\pi\)
\(840\) 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 \(-0.921129\pi\)
0.969459 0.245254i \(-0.0788713\pi\)
\(858\) 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 \(-0.973150\pi\)
0.996444 0.0842529i \(-0.0268504\pi\)
\(864\) 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 \(-0.777140\pi\)
−0.764756 + 0.644320i \(0.777140\pi\)
\(882\) 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 \(-0.965083\pi\)
0.993990 0.109474i \(-0.0349168\pi\)
\(888\) 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 \(-0.697118\pi\)
−0.580437 + 0.814305i \(0.697118\pi\)
\(912\) 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 \(-0.507892\pi\)
−0.0247921 + 0.999693i \(0.507892\pi\)
\(930\) 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 \(-0.437303\pi\)
0.195698 + 0.980664i \(0.437303\pi\)
\(942\) 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 \(0.165568\pi\)
−0.867746 + 0.497008i \(0.834432\pi\)
\(948\) 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 \(-0.710726\pi\)
0.614707 0.788755i \(-0.289274\pi\)
\(954\) 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 \(-0.597492\pi\)
−0.301515 + 0.953461i \(0.597492\pi\)
\(972\) 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 \(0.139960\pi\)
−0.904881 + 0.425665i \(0.860040\pi\)
\(978\) 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 \(-0.923797\pi\)
0.971480 0.237120i \(-0.0762035\pi\)
\(984\) 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.g.199.1 2
3.2 odd 2 450.4.c.f.199.2 2
5.2 odd 4 450.4.a.m.1.1 1
5.3 odd 4 90.4.a.b.1.1 1
5.4 even 2 inner 450.4.c.g.199.2 2
15.2 even 4 450.4.a.c.1.1 1
15.8 even 4 90.4.a.e.1.1 yes 1
15.14 odd 2 450.4.c.f.199.1 2
20.3 even 4 720.4.a.e.1.1 1
45.13 odd 12 810.4.e.u.541.1 2
45.23 even 12 810.4.e.a.541.1 2
45.38 even 12 810.4.e.a.271.1 2
45.43 odd 12 810.4.e.u.271.1 2
60.23 odd 4 720.4.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.4.a.b.1.1 1 5.3 odd 4
90.4.a.e.1.1 yes 1 15.8 even 4
450.4.a.c.1.1 1 15.2 even 4
450.4.a.m.1.1 1 5.2 odd 4
450.4.c.f.199.1 2 15.14 odd 2
450.4.c.f.199.2 2 3.2 odd 2
450.4.c.g.199.1 2 1.1 even 1 trivial
450.4.c.g.199.2 2 5.4 even 2 inner
720.4.a.e.1.1 1 20.3 even 4
720.4.a.t.1.1 1 60.23 odd 4
810.4.e.a.271.1 2 45.38 even 12
810.4.e.a.541.1 2 45.23 even 12
810.4.e.u.271.1 2 45.43 odd 12
810.4.e.u.541.1 2 45.13 odd 12