Properties

Label 1728.3.b.e.1567.2
Level $1728$
Weight $3$
Character 1728.1567
Analytic conductor $47.085$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.2
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1567
Dual form 1728.3.b.e.1567.3

$q$-expansion

\(f(q)\) \(=\) \(q-3.46410i q^{5} +1.00000i q^{7} +O(q^{10})\) \(q-3.46410i q^{5} +1.00000i q^{7} -17.3205 q^{11} +1.73205i q^{13} +6.00000 q^{17} -1.73205 q^{19} -30.0000i q^{23} +13.0000 q^{25} +20.7846i q^{29} +14.0000i q^{31} +3.46410 q^{35} +19.0526i q^{37} -48.0000 q^{41} -20.7846 q^{43} +66.0000i q^{47} +48.0000 q^{49} +48.4974i q^{53} +60.0000i q^{55} +31.1769 q^{59} +43.3013i q^{61} +6.00000 q^{65} -60.6218 q^{67} -48.0000i q^{71} +49.0000 q^{73} -17.3205i q^{77} +83.0000i q^{79} -13.8564 q^{83} -20.7846i q^{85} +66.0000 q^{89} -1.73205 q^{91} +6.00000i q^{95} +107.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{17} + 52 q^{25} - 192 q^{41} + 192 q^{49} + 24 q^{65} + 196 q^{73} + 264 q^{89} + 428 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 3.46410i − 0.692820i −0.938083 0.346410i \(-0.887401\pi\)
0.938083 0.346410i \(-0.112599\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.142857i 0.997446 + 0.0714286i \(0.0227558\pi\)
−0.997446 + 0.0714286i \(0.977244\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −17.3205 −1.57459 −0.787296 0.616575i \(-0.788520\pi\)
−0.787296 + 0.616575i \(0.788520\pi\)
\(12\) 0 0
\(13\) 1.73205i 0.133235i 0.997779 + 0.0666173i \(0.0212207\pi\)
−0.997779 + 0.0666173i \(0.978779\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 0.352941 0.176471 0.984306i \(-0.443532\pi\)
0.176471 + 0.984306i \(0.443532\pi\)
\(18\) 0 0
\(19\) −1.73205 −0.0911606 −0.0455803 0.998961i \(-0.514514\pi\)
−0.0455803 + 0.998961i \(0.514514\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 30.0000i − 1.30435i −0.758069 0.652174i \(-0.773857\pi\)
0.758069 0.652174i \(-0.226143\pi\)
\(24\) 0 0
\(25\) 13.0000 0.520000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 20.7846i 0.716711i 0.933585 + 0.358355i \(0.116662\pi\)
−0.933585 + 0.358355i \(0.883338\pi\)
\(30\) 0 0
\(31\) 14.0000i 0.451613i 0.974172 + 0.225806i \(0.0725017\pi\)
−0.974172 + 0.225806i \(0.927498\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.46410 0.0989743
\(36\) 0 0
\(37\) 19.0526i 0.514934i 0.966287 + 0.257467i \(0.0828879\pi\)
−0.966287 + 0.257467i \(0.917112\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −48.0000 −1.17073 −0.585366 0.810769i \(-0.699049\pi\)
−0.585366 + 0.810769i \(0.699049\pi\)
\(42\) 0 0
\(43\) −20.7846 −0.483363 −0.241682 0.970356i \(-0.577699\pi\)
−0.241682 + 0.970356i \(0.577699\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 66.0000i 1.40426i 0.712051 + 0.702128i \(0.247766\pi\)
−0.712051 + 0.702128i \(0.752234\pi\)
\(48\) 0 0
\(49\) 48.0000 0.979592
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 48.4974i 0.915046i 0.889198 + 0.457523i \(0.151263\pi\)
−0.889198 + 0.457523i \(0.848737\pi\)
\(54\) 0 0
\(55\) 60.0000i 1.09091i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 31.1769 0.528422 0.264211 0.964465i \(-0.414888\pi\)
0.264211 + 0.964465i \(0.414888\pi\)
\(60\) 0 0
\(61\) 43.3013i 0.709857i 0.934893 + 0.354928i \(0.115495\pi\)
−0.934893 + 0.354928i \(0.884505\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 0.0923077
\(66\) 0 0
\(67\) −60.6218 −0.904803 −0.452401 0.891814i \(-0.649433\pi\)
−0.452401 + 0.891814i \(0.649433\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 48.0000i − 0.676056i −0.941136 0.338028i \(-0.890240\pi\)
0.941136 0.338028i \(-0.109760\pi\)
\(72\) 0 0
\(73\) 49.0000 0.671233 0.335616 0.941999i \(-0.391055\pi\)
0.335616 + 0.941999i \(0.391055\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 17.3205i − 0.224942i
\(78\) 0 0
\(79\) 83.0000i 1.05063i 0.850907 + 0.525316i \(0.176053\pi\)
−0.850907 + 0.525316i \(0.823947\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −13.8564 −0.166945 −0.0834723 0.996510i \(-0.526601\pi\)
−0.0834723 + 0.996510i \(0.526601\pi\)
\(84\) 0 0
\(85\) − 20.7846i − 0.244525i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 66.0000 0.741573 0.370787 0.928718i \(-0.379088\pi\)
0.370787 + 0.928718i \(0.379088\pi\)
\(90\) 0 0
\(91\) −1.73205 −0.0190335
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.00000i 0.0631579i
\(96\) 0 0
\(97\) 107.000 1.10309 0.551546 0.834144i \(-0.314038\pi\)
0.551546 + 0.834144i \(0.314038\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 159.349i − 1.57771i −0.614580 0.788855i \(-0.710674\pi\)
0.614580 0.788855i \(-0.289326\pi\)
\(102\) 0 0
\(103\) 95.0000i 0.922330i 0.887314 + 0.461165i \(0.152568\pi\)
−0.887314 + 0.461165i \(0.847432\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 17.3205 0.161874 0.0809370 0.996719i \(-0.474209\pi\)
0.0809370 + 0.996719i \(0.474209\pi\)
\(108\) 0 0
\(109\) 187.061i 1.71616i 0.513516 + 0.858080i \(0.328343\pi\)
−0.513516 + 0.858080i \(0.671657\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 150.000 1.32743 0.663717 0.747984i \(-0.268978\pi\)
0.663717 + 0.747984i \(0.268978\pi\)
\(114\) 0 0
\(115\) −103.923 −0.903679
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.00000i 0.0504202i
\(120\) 0 0
\(121\) 179.000 1.47934
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 131.636i − 1.05309i
\(126\) 0 0
\(127\) 190.000i 1.49606i 0.663663 + 0.748031i \(0.269001\pi\)
−0.663663 + 0.748031i \(0.730999\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −48.4974 −0.370209 −0.185105 0.982719i \(-0.559262\pi\)
−0.185105 + 0.982719i \(0.559262\pi\)
\(132\) 0 0
\(133\) − 1.73205i − 0.0130229i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −222.000 −1.62044 −0.810219 0.586127i \(-0.800652\pi\)
−0.810219 + 0.586127i \(0.800652\pi\)
\(138\) 0 0
\(139\) −164.545 −1.18378 −0.591888 0.806020i \(-0.701617\pi\)
−0.591888 + 0.806020i \(0.701617\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 30.0000i − 0.209790i
\(144\) 0 0
\(145\) 72.0000 0.496552
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 96.9948i 0.650972i 0.945547 + 0.325486i \(0.105528\pi\)
−0.945547 + 0.325486i \(0.894472\pi\)
\(150\) 0 0
\(151\) 109.000i 0.721854i 0.932594 + 0.360927i \(0.117540\pi\)
−0.932594 + 0.360927i \(0.882460\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 48.4974 0.312887
\(156\) 0 0
\(157\) 62.3538i 0.397158i 0.980085 + 0.198579i \(0.0636327\pi\)
−0.980085 + 0.198579i \(0.936367\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 30.0000 0.186335
\(162\) 0 0
\(163\) 84.8705 0.520678 0.260339 0.965517i \(-0.416166\pi\)
0.260339 + 0.965517i \(0.416166\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 234.000i 1.40120i 0.713555 + 0.700599i \(0.247084\pi\)
−0.713555 + 0.700599i \(0.752916\pi\)
\(168\) 0 0
\(169\) 166.000 0.982249
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 180.133i 1.04123i 0.853791 + 0.520616i \(0.174298\pi\)
−0.853791 + 0.520616i \(0.825702\pi\)
\(174\) 0 0
\(175\) 13.0000i 0.0742857i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 193.990 1.08374 0.541871 0.840462i \(-0.317716\pi\)
0.541871 + 0.840462i \(0.317716\pi\)
\(180\) 0 0
\(181\) 164.545i 0.909087i 0.890724 + 0.454544i \(0.150198\pi\)
−0.890724 + 0.454544i \(0.849802\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 66.0000 0.356757
\(186\) 0 0
\(187\) −103.923 −0.555738
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 198.000i − 1.03665i −0.855184 0.518325i \(-0.826556\pi\)
0.855184 0.518325i \(-0.173444\pi\)
\(192\) 0 0
\(193\) 227.000 1.17617 0.588083 0.808801i \(-0.299883\pi\)
0.588083 + 0.808801i \(0.299883\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 232.095i 1.17815i 0.808080 + 0.589073i \(0.200507\pi\)
−0.808080 + 0.589073i \(0.799493\pi\)
\(198\) 0 0
\(199\) 239.000i 1.20101i 0.799623 + 0.600503i \(0.205033\pi\)
−0.799623 + 0.600503i \(0.794967\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −20.7846 −0.102387
\(204\) 0 0
\(205\) 166.277i 0.811107i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 30.0000 0.143541
\(210\) 0 0
\(211\) −292.717 −1.38728 −0.693641 0.720321i \(-0.743995\pi\)
−0.693641 + 0.720321i \(0.743995\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 72.0000i 0.334884i
\(216\) 0 0
\(217\) −14.0000 −0.0645161
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 10.3923i 0.0470240i
\(222\) 0 0
\(223\) 130.000i 0.582960i 0.956577 + 0.291480i \(0.0941476\pi\)
−0.956577 + 0.291480i \(0.905852\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −41.5692 −0.183124 −0.0915622 0.995799i \(-0.529186\pi\)
−0.0915622 + 0.995799i \(0.529186\pi\)
\(228\) 0 0
\(229\) − 332.554i − 1.45220i −0.687589 0.726100i \(-0.741331\pi\)
0.687589 0.726100i \(-0.258669\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −276.000 −1.18455 −0.592275 0.805736i \(-0.701770\pi\)
−0.592275 + 0.805736i \(0.701770\pi\)
\(234\) 0 0
\(235\) 228.631 0.972897
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 204.000i − 0.853556i −0.904356 0.426778i \(-0.859648\pi\)
0.904356 0.426778i \(-0.140352\pi\)
\(240\) 0 0
\(241\) 133.000 0.551867 0.275934 0.961177i \(-0.411013\pi\)
0.275934 + 0.961177i \(0.411013\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 166.277i − 0.678681i
\(246\) 0 0
\(247\) − 3.00000i − 0.0121457i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −457.261 −1.82176 −0.910879 0.412673i \(-0.864595\pi\)
−0.910879 + 0.412673i \(0.864595\pi\)
\(252\) 0 0
\(253\) 519.615i 2.05382i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −300.000 −1.16732 −0.583658 0.812000i \(-0.698379\pi\)
−0.583658 + 0.812000i \(0.698379\pi\)
\(258\) 0 0
\(259\) −19.0526 −0.0735620
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 84.0000i 0.319392i 0.987166 + 0.159696i \(0.0510513\pi\)
−0.987166 + 0.159696i \(0.948949\pi\)
\(264\) 0 0
\(265\) 168.000 0.633962
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 93.5307i − 0.347698i −0.984772 0.173849i \(-0.944380\pi\)
0.984772 0.173849i \(-0.0556205\pi\)
\(270\) 0 0
\(271\) 265.000i 0.977860i 0.872323 + 0.488930i \(0.162613\pi\)
−0.872323 + 0.488930i \(0.837387\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −225.167 −0.818788
\(276\) 0 0
\(277\) − 311.769i − 1.12552i −0.826620 0.562760i \(-0.809739\pi\)
0.826620 0.562760i \(-0.190261\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 540.000 1.92171 0.960854 0.277055i \(-0.0893584\pi\)
0.960854 + 0.277055i \(0.0893584\pi\)
\(282\) 0 0
\(283\) −353.338 −1.24855 −0.624273 0.781206i \(-0.714605\pi\)
−0.624273 + 0.781206i \(0.714605\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 48.0000i − 0.167247i
\(288\) 0 0
\(289\) −253.000 −0.875433
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 155.885i − 0.532029i −0.963969 0.266015i \(-0.914293\pi\)
0.963969 0.266015i \(-0.0857069\pi\)
\(294\) 0 0
\(295\) − 108.000i − 0.366102i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 51.9615 0.173784
\(300\) 0 0
\(301\) − 20.7846i − 0.0690519i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 150.000 0.491803
\(306\) 0 0
\(307\) 353.338 1.15094 0.575470 0.817823i \(-0.304819\pi\)
0.575470 + 0.817823i \(0.304819\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 462.000i − 1.48553i −0.669552 0.742765i \(-0.733514\pi\)
0.669552 0.742765i \(-0.266486\pi\)
\(312\) 0 0
\(313\) −35.0000 −0.111821 −0.0559105 0.998436i \(-0.517806\pi\)
−0.0559105 + 0.998436i \(0.517806\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 263.272i 0.830510i 0.909705 + 0.415255i \(0.136308\pi\)
−0.909705 + 0.415255i \(0.863692\pi\)
\(318\) 0 0
\(319\) − 360.000i − 1.12853i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −10.3923 −0.0321743
\(324\) 0 0
\(325\) 22.5167i 0.0692820i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −66.0000 −0.200608
\(330\) 0 0
\(331\) −39.8372 −0.120354 −0.0601770 0.998188i \(-0.519167\pi\)
−0.0601770 + 0.998188i \(0.519167\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 210.000i 0.626866i
\(336\) 0 0
\(337\) 59.0000 0.175074 0.0875371 0.996161i \(-0.472100\pi\)
0.0875371 + 0.996161i \(0.472100\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 242.487i − 0.711106i
\(342\) 0 0
\(343\) 97.0000i 0.282799i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 630.466 1.81691 0.908453 0.417987i \(-0.137264\pi\)
0.908453 + 0.417987i \(0.137264\pi\)
\(348\) 0 0
\(349\) 185.329i 0.531030i 0.964107 + 0.265515i \(0.0855420\pi\)
−0.964107 + 0.265515i \(0.914458\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.0000 0.0339943 0.0169972 0.999856i \(-0.494589\pi\)
0.0169972 + 0.999856i \(0.494589\pi\)
\(354\) 0 0
\(355\) −166.277 −0.468386
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 318.000i 0.885794i 0.896573 + 0.442897i \(0.146049\pi\)
−0.896573 + 0.442897i \(0.853951\pi\)
\(360\) 0 0
\(361\) −358.000 −0.991690
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 169.741i − 0.465044i
\(366\) 0 0
\(367\) − 109.000i − 0.297003i −0.988912 0.148501i \(-0.952555\pi\)
0.988912 0.148501i \(-0.0474449\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −48.4974 −0.130721
\(372\) 0 0
\(373\) − 729.193i − 1.95494i −0.211070 0.977471i \(-0.567695\pi\)
0.211070 0.977471i \(-0.432305\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −36.0000 −0.0954907
\(378\) 0 0
\(379\) 438.209 1.15622 0.578112 0.815957i \(-0.303790\pi\)
0.578112 + 0.815957i \(0.303790\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 516.000i 1.34726i 0.739069 + 0.673629i \(0.235266\pi\)
−0.739069 + 0.673629i \(0.764734\pi\)
\(384\) 0 0
\(385\) −60.0000 −0.155844
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 543.864i 1.39811i 0.715069 + 0.699054i \(0.246395\pi\)
−0.715069 + 0.699054i \(0.753605\pi\)
\(390\) 0 0
\(391\) − 180.000i − 0.460358i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 287.520 0.727900
\(396\) 0 0
\(397\) 685.892i 1.72769i 0.503759 + 0.863844i \(0.331950\pi\)
−0.503759 + 0.863844i \(0.668050\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −252.000 −0.628429 −0.314214 0.949352i \(-0.601741\pi\)
−0.314214 + 0.949352i \(0.601741\pi\)
\(402\) 0 0
\(403\) −24.2487 −0.0601705
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 330.000i − 0.810811i
\(408\) 0 0
\(409\) −11.0000 −0.0268949 −0.0134474 0.999910i \(-0.504281\pi\)
−0.0134474 + 0.999910i \(0.504281\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 31.1769i 0.0754889i
\(414\) 0 0
\(415\) 48.0000i 0.115663i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 135.100 0.322434 0.161217 0.986919i \(-0.448458\pi\)
0.161217 + 0.986919i \(0.448458\pi\)
\(420\) 0 0
\(421\) − 601.022i − 1.42760i −0.700347 0.713802i \(-0.746971\pi\)
0.700347 0.713802i \(-0.253029\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 78.0000 0.183529
\(426\) 0 0
\(427\) −43.3013 −0.101408
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 486.000i 1.12761i 0.825908 + 0.563805i \(0.190663\pi\)
−0.825908 + 0.563805i \(0.809337\pi\)
\(432\) 0 0
\(433\) −494.000 −1.14088 −0.570439 0.821340i \(-0.693227\pi\)
−0.570439 + 0.821340i \(0.693227\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 51.9615i 0.118905i
\(438\) 0 0
\(439\) − 610.000i − 1.38952i −0.719241 0.694761i \(-0.755510\pi\)
0.719241 0.694761i \(-0.244490\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 145.492 0.328425 0.164212 0.986425i \(-0.447492\pi\)
0.164212 + 0.986425i \(0.447492\pi\)
\(444\) 0 0
\(445\) − 228.631i − 0.513777i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −366.000 −0.815145 −0.407572 0.913173i \(-0.633625\pi\)
−0.407572 + 0.913173i \(0.633625\pi\)
\(450\) 0 0
\(451\) 831.384 1.84342
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6.00000i 0.0131868i
\(456\) 0 0
\(457\) −398.000 −0.870897 −0.435449 0.900214i \(-0.643410\pi\)
−0.435449 + 0.900214i \(0.643410\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 710.141i − 1.54044i −0.637781 0.770218i \(-0.720147\pi\)
0.637781 0.770218i \(-0.279853\pi\)
\(462\) 0 0
\(463\) − 875.000i − 1.88985i −0.327289 0.944924i \(-0.606135\pi\)
0.327289 0.944924i \(-0.393865\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 349.874 0.749195 0.374598 0.927187i \(-0.377781\pi\)
0.374598 + 0.927187i \(0.377781\pi\)
\(468\) 0 0
\(469\) − 60.6218i − 0.129258i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 360.000 0.761099
\(474\) 0 0
\(475\) −22.5167 −0.0474035
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 228.000i − 0.475992i −0.971266 0.237996i \(-0.923510\pi\)
0.971266 0.237996i \(-0.0764905\pi\)
\(480\) 0 0
\(481\) −33.0000 −0.0686071
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 370.659i − 0.764245i
\(486\) 0 0
\(487\) − 251.000i − 0.515400i −0.966225 0.257700i \(-0.917035\pi\)
0.966225 0.257700i \(-0.0829647\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −703.213 −1.43220 −0.716102 0.697995i \(-0.754076\pi\)
−0.716102 + 0.697995i \(0.754076\pi\)
\(492\) 0 0
\(493\) 124.708i 0.252957i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 48.0000 0.0965795
\(498\) 0 0
\(499\) −374.123 −0.749745 −0.374873 0.927076i \(-0.622314\pi\)
−0.374873 + 0.927076i \(0.622314\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 762.000i − 1.51491i −0.652887 0.757455i \(-0.726442\pi\)
0.652887 0.757455i \(-0.273558\pi\)
\(504\) 0 0
\(505\) −552.000 −1.09307
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 204.382i 0.401536i 0.979639 + 0.200768i \(0.0643438\pi\)
−0.979639 + 0.200768i \(0.935656\pi\)
\(510\) 0 0
\(511\) 49.0000i 0.0958904i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 329.090 0.639009
\(516\) 0 0
\(517\) − 1143.15i − 2.21113i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 726.000 1.39347 0.696737 0.717327i \(-0.254634\pi\)
0.696737 + 0.717327i \(0.254634\pi\)
\(522\) 0 0
\(523\) 833.116 1.59296 0.796478 0.604667i \(-0.206694\pi\)
0.796478 + 0.604667i \(0.206694\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 84.0000i 0.159393i
\(528\) 0 0
\(529\) −371.000 −0.701323
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 83.1384i − 0.155982i
\(534\) 0 0
\(535\) − 60.0000i − 0.112150i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −831.384 −1.54246
\(540\) 0 0
\(541\) 646.055i 1.19419i 0.802172 + 0.597093i \(0.203678\pi\)
−0.802172 + 0.597093i \(0.796322\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 648.000 1.18899
\(546\) 0 0
\(547\) 313.501 0.573128 0.286564 0.958061i \(-0.407487\pi\)
0.286564 + 0.958061i \(0.407487\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 36.0000i − 0.0653358i
\(552\) 0 0
\(553\) −83.0000 −0.150090
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 682.428i 1.22518i 0.790399 + 0.612592i \(0.209873\pi\)
−0.790399 + 0.612592i \(0.790127\pi\)
\(558\) 0 0
\(559\) − 36.0000i − 0.0644007i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −651.251 −1.15675 −0.578376 0.815770i \(-0.696313\pi\)
−0.578376 + 0.815770i \(0.696313\pi\)
\(564\) 0 0
\(565\) − 519.615i − 0.919673i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −846.000 −1.48682 −0.743409 0.668837i \(-0.766793\pi\)
−0.743409 + 0.668837i \(0.766793\pi\)
\(570\) 0 0
\(571\) −375.855 −0.658240 −0.329120 0.944288i \(-0.606752\pi\)
−0.329120 + 0.944288i \(0.606752\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 390.000i − 0.678261i
\(576\) 0 0
\(577\) −791.000 −1.37088 −0.685442 0.728127i \(-0.740391\pi\)
−0.685442 + 0.728127i \(0.740391\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 13.8564i − 0.0238492i
\(582\) 0 0
\(583\) − 840.000i − 1.44082i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −273.664 −0.466208 −0.233104 0.972452i \(-0.574888\pi\)
−0.233104 + 0.972452i \(0.574888\pi\)
\(588\) 0 0
\(589\) − 24.2487i − 0.0411693i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −768.000 −1.29511 −0.647555 0.762019i \(-0.724208\pi\)
−0.647555 + 0.762019i \(0.724208\pi\)
\(594\) 0 0
\(595\) 20.7846 0.0349321
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 564.000i 0.941569i 0.882248 + 0.470785i \(0.156029\pi\)
−0.882248 + 0.470785i \(0.843971\pi\)
\(600\) 0 0
\(601\) −178.000 −0.296173 −0.148087 0.988974i \(-0.547311\pi\)
−0.148087 + 0.988974i \(0.547311\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 620.074i − 1.02492i
\(606\) 0 0
\(607\) − 1103.00i − 1.81713i −0.417740 0.908567i \(-0.637178\pi\)
0.417740 0.908567i \(-0.362822\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −114.315 −0.187096
\(612\) 0 0
\(613\) − 226.899i − 0.370145i −0.982725 0.185072i \(-0.940748\pi\)
0.982725 0.185072i \(-0.0592519\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 162.000 0.262561 0.131280 0.991345i \(-0.458091\pi\)
0.131280 + 0.991345i \(0.458091\pi\)
\(618\) 0 0
\(619\) 351.606 0.568023 0.284012 0.958821i \(-0.408335\pi\)
0.284012 + 0.958821i \(0.408335\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 66.0000i 0.105939i
\(624\) 0 0
\(625\) −131.000 −0.209600
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 114.315i 0.181741i
\(630\) 0 0
\(631\) 85.0000i 0.134707i 0.997729 + 0.0673534i \(0.0214555\pi\)
−0.997729 + 0.0673534i \(0.978545\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 658.179 1.03650
\(636\) 0 0
\(637\) 83.1384i 0.130516i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −720.000 −1.12324 −0.561622 0.827394i \(-0.689823\pi\)
−0.561622 + 0.827394i \(0.689823\pi\)
\(642\) 0 0
\(643\) 394.908 0.614164 0.307082 0.951683i \(-0.400647\pi\)
0.307082 + 0.951683i \(0.400647\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 312.000i 0.482226i 0.970497 + 0.241113i \(0.0775124\pi\)
−0.970497 + 0.241113i \(0.922488\pi\)
\(648\) 0 0
\(649\) −540.000 −0.832049
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 720.533i 1.10342i 0.834036 + 0.551710i \(0.186024\pi\)
−0.834036 + 0.551710i \(0.813976\pi\)
\(654\) 0 0
\(655\) 168.000i 0.256489i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −200.918 −0.304883 −0.152441 0.988312i \(-0.548714\pi\)
−0.152441 + 0.988312i \(0.548714\pi\)
\(660\) 0 0
\(661\) − 933.575i − 1.41237i −0.708028 0.706184i \(-0.750415\pi\)
0.708028 0.706184i \(-0.249585\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.00000 −0.00902256
\(666\) 0 0
\(667\) 623.538 0.934840
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 750.000i − 1.11773i
\(672\) 0 0
\(673\) −323.000 −0.479941 −0.239970 0.970780i \(-0.577138\pi\)
−0.239970 + 0.970780i \(0.577138\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 737.854i 1.08989i 0.838473 + 0.544944i \(0.183449\pi\)
−0.838473 + 0.544944i \(0.816551\pi\)
\(678\) 0 0
\(679\) 107.000i 0.157585i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1060.02 1.55200 0.775999 0.630734i \(-0.217246\pi\)
0.775999 + 0.630734i \(0.217246\pi\)
\(684\) 0 0
\(685\) 769.031i 1.12267i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −84.0000 −0.121916
\(690\) 0 0
\(691\) −561.184 −0.812134 −0.406067 0.913843i \(-0.633100\pi\)
−0.406067 + 0.913843i \(0.633100\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 570.000i 0.820144i
\(696\) 0 0
\(697\) −288.000 −0.413199
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 557.720i 0.795607i 0.917471 + 0.397803i \(0.130227\pi\)
−0.917471 + 0.397803i \(0.869773\pi\)
\(702\) 0 0
\(703\) − 33.0000i − 0.0469417i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 159.349 0.225387
\(708\) 0 0
\(709\) − 912.791i − 1.28743i −0.765264 0.643717i \(-0.777391\pi\)
0.765264 0.643717i \(-0.222609\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 420.000 0.589060
\(714\) 0 0
\(715\) −103.923 −0.145347
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 432.000i − 0.600834i −0.953808 0.300417i \(-0.902874\pi\)
0.953808 0.300417i \(-0.0971259\pi\)
\(720\) 0 0
\(721\) −95.0000 −0.131761
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 270.200i 0.372690i
\(726\) 0 0
\(727\) − 758.000i − 1.04264i −0.853361 0.521320i \(-0.825440\pi\)
0.853361 0.521320i \(-0.174560\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −124.708 −0.170599
\(732\) 0 0
\(733\) 270.200i 0.368622i 0.982868 + 0.184311i \(0.0590054\pi\)
−0.982868 + 0.184311i \(0.940995\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1050.00 1.42469
\(738\) 0 0
\(739\) 852.169 1.15314 0.576569 0.817048i \(-0.304391\pi\)
0.576569 + 0.817048i \(0.304391\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 150.000i 0.201884i 0.994892 + 0.100942i \(0.0321857\pi\)
−0.994892 + 0.100942i \(0.967814\pi\)
\(744\) 0 0
\(745\) 336.000 0.451007
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 17.3205i 0.0231248i
\(750\) 0 0
\(751\) − 457.000i − 0.608522i −0.952589 0.304261i \(-0.901591\pi\)
0.952589 0.304261i \(-0.0984095\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 377.587 0.500115
\(756\) 0 0
\(757\) − 621.806i − 0.821409i −0.911769 0.410704i \(-0.865283\pi\)
0.911769 0.410704i \(-0.134717\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1458.00 −1.91590 −0.957950 0.286935i \(-0.907364\pi\)
−0.957950 + 0.286935i \(0.907364\pi\)
\(762\) 0 0
\(763\) −187.061 −0.245166
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 54.0000i 0.0704042i
\(768\) 0 0
\(769\) 47.0000 0.0611183 0.0305592 0.999533i \(-0.490271\pi\)
0.0305592 + 0.999533i \(0.490271\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 706.677i − 0.914200i −0.889415 0.457100i \(-0.848888\pi\)
0.889415 0.457100i \(-0.151112\pi\)
\(774\) 0 0
\(775\) 182.000i 0.234839i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 83.1384 0.106725
\(780\) 0 0
\(781\) 831.384i 1.06451i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 216.000 0.275159
\(786\) 0 0
\(787\) −708.409 −0.900138 −0.450069 0.892994i \(-0.648601\pi\)
−0.450069 + 0.892994i \(0.648601\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 150.000i 0.189633i
\(792\) 0 0
\(793\) −75.0000 −0.0945776
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1129.30i − 1.41693i −0.705743 0.708467i \(-0.749387\pi\)
0.705743 0.708467i \(-0.250613\pi\)
\(798\) 0 0
\(799\) 396.000i 0.495620i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −848.705 −1.05692
\(804\) 0 0
\(805\) − 103.923i − 0.129097i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 288.000 0.355995 0.177998 0.984031i \(-0.443038\pi\)
0.177998 + 0.984031i \(0.443038\pi\)
\(810\) 0 0
\(811\) −187.061 −0.230655 −0.115328 0.993328i \(-0.536792\pi\)
−0.115328 + 0.993328i \(0.536792\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 294.000i − 0.360736i
\(816\) 0 0
\(817\) 36.0000 0.0440636
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 114.315i 0.139239i 0.997574 + 0.0696196i \(0.0221785\pi\)
−0.997574 + 0.0696196i \(0.977821\pi\)
\(822\) 0 0
\(823\) 601.000i 0.730255i 0.930957 + 0.365128i \(0.118975\pi\)
−0.930957 + 0.365128i \(0.881025\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −820.992 −0.992735 −0.496368 0.868112i \(-0.665333\pi\)
−0.496368 + 0.868112i \(0.665333\pi\)
\(828\) 0 0
\(829\) 642.591i 0.775140i 0.921840 + 0.387570i \(0.126685\pi\)
−0.921840 + 0.387570i \(0.873315\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 288.000 0.345738
\(834\) 0 0
\(835\) 810.600 0.970778
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 972.000i 1.15852i 0.815142 + 0.579261i \(0.196659\pi\)
−0.815142 + 0.579261i \(0.803341\pi\)
\(840\) 0 0
\(841\) 409.000 0.486326
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 575.041i − 0.680522i
\(846\) 0 0
\(847\) 179.000i 0.211334i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 571.577 0.671653
\(852\) 0 0
\(853\) − 334.286i − 0.391894i −0.980614 0.195947i \(-0.937222\pi\)
0.980614 0.195947i \(-0.0627781\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.0000 0.0280047 0.0140023 0.999902i \(-0.495543\pi\)
0.0140023 + 0.999902i \(0.495543\pi\)
\(858\) 0 0
\(859\) −105.655 −0.122998 −0.0614989 0.998107i \(-0.519588\pi\)
−0.0614989 + 0.998107i \(0.519588\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 150.000i − 0.173812i −0.996217 0.0869061i \(-0.972302\pi\)
0.996217 0.0869061i \(-0.0276980\pi\)
\(864\) 0 0
\(865\) 624.000 0.721387
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 1437.60i − 1.65432i
\(870\) 0 0
\(871\) − 105.000i − 0.120551i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 131.636 0.150441
\(876\) 0 0
\(877\) 39.8372i 0.0454244i 0.999742 + 0.0227122i \(0.00723013\pi\)
−0.999742 + 0.0227122i \(0.992770\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1014.00 1.15096 0.575482 0.817814i \(-0.304814\pi\)
0.575482 + 0.817814i \(0.304814\pi\)
\(882\) 0 0
\(883\) −476.314 −0.539427 −0.269713 0.962941i \(-0.586929\pi\)
−0.269713 + 0.962941i \(0.586929\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 768.000i − 0.865840i −0.901432 0.432920i \(-0.857483\pi\)
0.901432 0.432920i \(-0.142517\pi\)
\(888\) 0 0
\(889\) −190.000 −0.213723
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 114.315i − 0.128013i
\(894\) 0 0
\(895\) − 672.000i − 0.750838i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −290.985 −0.323676
\(900\) 0 0
\(901\) 290.985i 0.322957i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 570.000 0.629834
\(906\) 0 0
\(907\) 209.578 0.231067 0.115534 0.993304i \(-0.463142\pi\)
0.115534 + 0.993304i \(0.463142\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 600.000i 0.658617i 0.944222 + 0.329308i \(0.106816\pi\)
−0.944222 + 0.329308i \(0.893184\pi\)
\(912\) 0 0
\(913\) 240.000 0.262870
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 48.4974i − 0.0528870i
\(918\) 0 0
\(919\) − 422.000i − 0.459195i −0.973286 0.229597i \(-0.926259\pi\)
0.973286 0.229597i \(-0.0737409\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 83.1384 0.0900741
\(924\) 0 0
\(925\) 247.683i 0.267766i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1116.00 1.20129 0.600646 0.799515i \(-0.294910\pi\)
0.600646 + 0.799515i \(0.294910\pi\)
\(930\) 0 0
\(931\) −83.1384 −0.0893001
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 360.000i 0.385027i
\(936\) 0 0
\(937\) −275.000 −0.293490 −0.146745 0.989174i \(-0.546880\pi\)
−0.146745 + 0.989174i \(0.546880\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 329.090i 0.349723i 0.984593 + 0.174862i \(0.0559478\pi\)
−0.984593 + 0.174862i \(0.944052\pi\)
\(942\) 0 0
\(943\) 1440.00i 1.52704i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 162.813 0.171925 0.0859624 0.996298i \(-0.472604\pi\)
0.0859624 + 0.996298i \(0.472604\pi\)
\(948\) 0 0
\(949\) 84.8705i 0.0894315i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1506.00 −1.58027 −0.790136 0.612931i \(-0.789990\pi\)
−0.790136 + 0.612931i \(0.789990\pi\)
\(954\) 0 0
\(955\) −685.892 −0.718212
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 222.000i − 0.231491i
\(960\) 0 0
\(961\) 765.000 0.796046
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 786.351i − 0.814872i
\(966\) 0 0
\(967\) 1307.00i 1.35160i 0.737084 + 0.675801i \(0.236202\pi\)
−0.737084 + 0.675801i \(0.763798\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 627.002 0.645729 0.322864 0.946445i \(-0.395354\pi\)
0.322864 + 0.946445i \(0.395354\pi\)
\(972\) 0 0
\(973\) − 164.545i − 0.169111i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −840.000 −0.859775 −0.429887 0.902883i \(-0.641447\pi\)
−0.429887 + 0.902883i \(0.641447\pi\)
\(978\) 0 0
\(979\) −1143.15 −1.16767
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1134.00i − 1.15361i −0.816881 0.576806i \(-0.804299\pi\)
0.816881 0.576806i \(-0.195701\pi\)
\(984\) 0 0
\(985\) 804.000 0.816244
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 623.538i 0.630473i
\(990\) 0 0
\(991\) 1355.00i 1.36731i 0.729807 + 0.683653i \(0.239610\pi\)
−0.729807 + 0.683653i \(0.760390\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 827.920 0.832081
\(996\) 0 0
\(997\) 394.908i 0.396096i 0.980192 + 0.198048i \(0.0634602\pi\)
−0.980192 + 0.198048i \(0.936540\pi\)
\(998\) 0 0
\(999\) 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 1728.3.b.e.1567.2 yes 4
3.2 odd 2 1728.3.b.b.1567.4 yes 4
4.3 odd 2 inner 1728.3.b.e.1567.1 yes 4
8.3 odd 2 inner 1728.3.b.e.1567.3 yes 4
8.5 even 2 inner 1728.3.b.e.1567.4 yes 4
12.11 even 2 1728.3.b.b.1567.3 yes 4
24.5 odd 2 1728.3.b.b.1567.2 yes 4
24.11 even 2 1728.3.b.b.1567.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.3.b.b.1567.1 4 24.11 even 2
1728.3.b.b.1567.2 yes 4 24.5 odd 2
1728.3.b.b.1567.3 yes 4 12.11 even 2
1728.3.b.b.1567.4 yes 4 3.2 odd 2
1728.3.b.e.1567.1 yes 4 4.3 odd 2 inner
1728.3.b.e.1567.2 yes 4 1.1 even 1 trivial
1728.3.b.e.1567.3 yes 4 8.3 odd 2 inner
1728.3.b.e.1567.4 yes 4 8.5 even 2 inner