Properties

Label 21.6.g.a.17.1
Level $21$
Weight $6$
Character 21.17
Analytic conductor $3.368$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.36806021607\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 17.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.17
Dual form 21.6.g.a.5.1

$q$-expansion

\(f(q)\) \(=\) \(q+(13.5000 + 7.79423i) q^{3} +(-16.0000 + 27.7128i) q^{4} +(105.500 + 75.3442i) q^{7} +(121.500 + 210.444i) q^{9} +O(q^{10})\) \(q+(13.5000 + 7.79423i) q^{3} +(-16.0000 + 27.7128i) q^{4} +(105.500 + 75.3442i) q^{7} +(121.500 + 210.444i) q^{9} +(-432.000 + 249.415i) q^{12} -1141.42i q^{13} +(-512.000 - 886.810i) q^{16} +(-139.500 + 80.5404i) q^{19} +(837.000 + 1839.44i) q^{21} +(1562.50 - 2706.33i) q^{25} +3788.00i q^{27} +(-3776.00 + 1718.19i) q^{28} +(8962.50 + 5174.50i) q^{31} -7776.00 q^{36} +(-3330.50 - 5768.60i) q^{37} +(8896.50 - 15409.2i) q^{39} -22475.0 q^{43} -15962.6i q^{48} +(5453.50 + 15897.6i) q^{49} +(31632.0 + 18262.7i) q^{52} -2511.00 q^{57} +(-37614.0 + 21716.5i) q^{61} +(-3037.50 + 31356.2i) q^{63} +32768.0 q^{64} +(18969.5 - 32856.1i) q^{67} +(-40513.5 - 23390.5i) q^{73} +(42187.5 - 24357.0i) q^{75} -5154.58i q^{76} +(-45428.5 - 78684.5i) q^{79} +(-29524.5 + 51137.9i) q^{81} +(-64368.0 - 6235.38i) q^{84} +(85999.5 - 120420. i) q^{91} +(80662.5 + 139712. i) q^{93} +127631. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 27 q^{3} - 32 q^{4} + 211 q^{7} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 27 q^{3} - 32 q^{4} + 211 q^{7} + 243 q^{9} - 864 q^{12} - 1024 q^{16} - 279 q^{19} + 1674 q^{21} + 3125 q^{25} - 7552 q^{28} + 17925 q^{31} - 15552 q^{36} - 6661 q^{37} + 17793 q^{39} - 44950 q^{43} + 10907 q^{49} + 63264 q^{52} - 5022 q^{57} - 75228 q^{61} - 6075 q^{63} + 65536 q^{64} + 37939 q^{67} - 81027 q^{73} + 84375 q^{75} - 90857 q^{79} - 59049 q^{81} - 128736 q^{84} + 171999 q^{91} + 161325 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(3\) 13.5000 + 7.79423i 0.866025 + 0.500000i
\(4\) −16.0000 + 27.7128i −0.500000 + 0.866025i
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 105.500 + 75.3442i 0.813781 + 0.581172i
\(8\) 0 0
\(9\) 121.500 + 210.444i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) −432.000 + 249.415i −0.866025 + 0.500000i
\(13\) 1141.42i 1.87322i −0.350380 0.936608i \(-0.613948\pi\)
0.350380 0.936608i \(-0.386052\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −512.000 886.810i −0.500000 0.866025i
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −139.500 + 80.5404i −0.0886523 + 0.0511835i −0.543671 0.839299i \(-0.682966\pi\)
0.455018 + 0.890482i \(0.349633\pi\)
\(20\) 0 0
\(21\) 837.000 + 1839.44i 0.414169 + 0.910200i
\(22\) 0 0
\(23\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) 1562.50 2706.33i 0.500000 0.866025i
\(26\) 0 0
\(27\) 3788.00i 1.00000i
\(28\) −3776.00 + 1718.19i −0.910200 + 0.414169i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 8962.50 + 5174.50i 1.67504 + 0.967084i 0.964748 + 0.263176i \(0.0847701\pi\)
0.710291 + 0.703908i \(0.248563\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −7776.00 −1.00000
\(37\) −3330.50 5768.60i −0.399949 0.692733i 0.593770 0.804635i \(-0.297639\pi\)
−0.993719 + 0.111902i \(0.964306\pi\)
\(38\) 0 0
\(39\) 8896.50 15409.2i 0.936608 1.62225i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −22475.0 −1.85365 −0.926827 0.375489i \(-0.877475\pi\)
−0.926827 + 0.375489i \(0.877475\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 15962.6i 1.00000i
\(49\) 5453.50 + 15897.6i 0.324478 + 0.945893i
\(50\) 0 0
\(51\) 0 0
\(52\) 31632.0 + 18262.7i 1.62225 + 0.936608i
\(53\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2511.00 −0.102367
\(58\) 0 0
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) −37614.0 + 21716.5i −1.29427 + 0.747247i −0.979408 0.201890i \(-0.935292\pi\)
−0.314862 + 0.949137i \(0.601958\pi\)
\(62\) 0 0
\(63\) −3037.50 + 31356.2i −0.0964195 + 0.995341i
\(64\) 32768.0 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 18969.5 32856.1i 0.516260 0.894189i −0.483561 0.875310i \(-0.660657\pi\)
0.999822 0.0188789i \(-0.00600969\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −40513.5 23390.5i −0.889800 0.513727i −0.0159232 0.999873i \(-0.505069\pi\)
−0.873877 + 0.486147i \(0.838402\pi\)
\(74\) 0 0
\(75\) 42187.5 24357.0i 0.866025 0.500000i
\(76\) 5154.58i 0.102367i
\(77\) 0 0
\(78\) 0 0
\(79\) −45428.5 78684.5i −0.818956 1.41847i −0.906452 0.422309i \(-0.861220\pi\)
0.0874958 0.996165i \(-0.472114\pi\)
\(80\) 0 0
\(81\) −29524.5 + 51137.9i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −64368.0 6235.38i −0.995341 0.0964195i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 85999.5 120420.i 1.08866 1.52439i
\(92\) 0 0
\(93\) 80662.5 + 139712.i 0.967084 + 1.67504i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 127631.i 1.37730i 0.725095 + 0.688649i \(0.241796\pi\)
−0.725095 + 0.688649i \(0.758204\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 50000.0 + 86602.5i 0.500000 + 0.866025i
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) 35161.5 20300.5i 0.326569 0.188544i −0.327748 0.944765i \(-0.606290\pi\)
0.654317 + 0.756221i \(0.272956\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) −104976. 60607.9i −0.866025 0.500000i
\(109\) −123922. + 214638.i −0.999034 + 1.73038i −0.461467 + 0.887157i \(0.652677\pi\)
−0.537567 + 0.843221i \(0.680656\pi\)
\(110\) 0 0
\(111\) 103835.i 0.799899i
\(112\) 12800.0 132135.i 0.0964195 0.995341i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 240206. 138683.i 1.62225 0.936608i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −80525.5 139474.i −0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) −286800. + 165584.i −1.67504 + 0.967084i
\(125\) 0 0
\(126\) 0 0
\(127\) 347111. 1.90967 0.954837 0.297131i \(-0.0960299\pi\)
0.954837 + 0.297131i \(0.0960299\pi\)
\(128\) 0 0
\(129\) −303412. 175175.i −1.60531 0.926827i
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) −20785.5 2013.51i −0.101890 0.00987017i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) 29024.0i 0.127415i 0.997969 + 0.0637074i \(0.0202924\pi\)
−0.997969 + 0.0637074i \(0.979708\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 124416. 215495.i 0.500000 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) −50287.5 + 257124.i −0.191941 + 0.981407i
\(148\) 213152. 0.799899
\(149\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) −204362. + 353965.i −0.729387 + 1.26333i 0.227756 + 0.973718i \(0.426861\pi\)
−0.957143 + 0.289616i \(0.906472\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 284688. + 493094.i 0.936608 + 1.62225i
\(157\) 526518. + 303985.i 1.70476 + 0.984246i 0.940785 + 0.339004i \(0.110090\pi\)
0.763978 + 0.645242i \(0.223243\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −339124. 587380.i −0.999746 1.73161i −0.519405 0.854528i \(-0.673846\pi\)
−0.480341 0.877082i \(-0.659487\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −931550. −2.50893
\(170\) 0 0
\(171\) −33898.5 19571.3i −0.0886523 0.0511835i
\(172\) 359600. 622845.i 0.926827 1.60531i
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 368750. 167792.i 0.910200 0.414169i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) 0 0
\(181\) 222265.i 0.504284i −0.967690 0.252142i \(-0.918865\pi\)
0.967690 0.252142i \(-0.0811351\pi\)
\(182\) 0 0
\(183\) −677052. −1.49449
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −285404. + 399633.i −0.581172 + 0.813781i
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 442368. + 255401.i 0.866025 + 0.500000i
\(193\) 328188. 568437.i 0.634204 1.09847i −0.352480 0.935820i \(-0.614661\pi\)
0.986683 0.162653i \(-0.0520053\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −527824. 103230.i −0.981407 0.191941i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 558627. + 322523.i 0.999975 + 0.577336i 0.908241 0.418448i \(-0.137426\pi\)
0.0917343 + 0.995784i \(0.470759\pi\)
\(200\) 0 0
\(201\) 512176. 295705.i 0.894189 0.516260i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.01222e6 + 584408.i −1.62225 + 0.936608i
\(209\) 0 0
\(210\) 0 0
\(211\) 288976. 0.446844 0.223422 0.974722i \(-0.428277\pi\)
0.223422 + 0.974722i \(0.428277\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 555675. + 1.22118e6i 0.801072 + 1.76048i
\(218\) 0 0
\(219\) −364622. 631543.i −0.513727 0.889800i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.45377e6i 1.95764i 0.204718 + 0.978821i \(0.434372\pi\)
−0.204718 + 0.978821i \(0.565628\pi\)
\(224\) 0 0
\(225\) 759375. 1.00000
\(226\) 0 0
\(227\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(228\) 40176.0 69586.9i 0.0511835 0.0886523i
\(229\) 540064. 311806.i 0.680545 0.392913i −0.119515 0.992832i \(-0.538134\pi\)
0.800060 + 0.599919i \(0.204801\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.41632e6i 1.63791i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.08506e6 626462.i −1.20341 0.694788i −0.242096 0.970252i \(-0.577835\pi\)
−0.961311 + 0.275465i \(0.911168\pi\)
\(242\) 0 0
\(243\) −797161. + 460241.i −0.866025 + 0.500000i
\(244\) 1.38985e6i 1.49449i
\(245\) 0 0
\(246\) 0 0
\(247\) 91930.5 + 159228.i 0.0958776 + 0.166065i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −820368. 585877.i −0.813781 0.581172i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −524288. + 908093.i −0.500000 + 0.866025i
\(257\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) 83262.5 859521.i 0.0771259 0.796172i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 607024. + 1.05140e6i 0.516260 + 0.894189i
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) 760575. 439118.i 0.629099 0.363210i −0.151304 0.988487i \(-0.548347\pi\)
0.780403 + 0.625277i \(0.215014\pi\)
\(272\) 0 0
\(273\) 2.09957e6 955370.i 1.70500 0.775827i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −369194. + 639464.i −0.289105 + 0.500745i −0.973596 0.228276i \(-0.926691\pi\)
0.684491 + 0.729021i \(0.260024\pi\)
\(278\) 0 0
\(279\) 2.51481e6i 1.93417i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −1.16534e6 672807.i −0.864939 0.499373i 0.000724409 1.00000i \(-0.499769\pi\)
−0.865663 + 0.500627i \(0.833103\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 709928. + 1.22963e6i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) −994788. + 1.72302e6i −0.688649 + 1.19278i
\(292\) 1.29643e6 748495.i 0.889800 0.513727i
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 1.55885e6i 1.00000i
\(301\) −2.37111e6 1.69336e6i −1.50847 1.07729i
\(302\) 0 0
\(303\) 0 0
\(304\) 142848. + 82473.3i 0.0886523 + 0.0511835i
\(305\) 0 0
\(306\) 0 0
\(307\) 808260.i 0.489446i −0.969593 0.244723i \(-0.921303\pi\)
0.969593 0.244723i \(-0.0786971\pi\)
\(308\) 0 0
\(309\) 632907. 0.377089
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 2.01744e6 1.16477e6i 1.16396 0.672014i 0.211712 0.977332i \(-0.432096\pi\)
0.952250 + 0.305318i \(0.0987629\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 2.90742e6 1.63791
\(317\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −944784. 1.63641e6i −0.500000 0.866025i
\(325\) −3.08906e6 1.78347e6i −1.62225 0.936608i
\(326\) 0 0
\(327\) −3.34588e6 + 1.93175e6i −1.73038 + 0.999034i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −273076. 472981.i −0.136998 0.237287i 0.789361 0.613929i \(-0.210412\pi\)
−0.926359 + 0.376642i \(0.877079\pi\)
\(332\) 0 0
\(333\) 809312. 1.40177e6i 0.399949 0.692733i
\(334\) 0 0
\(335\) 0 0
\(336\) 1.20269e6 1.68405e6i 0.581172 0.813781i
\(337\) −2.63172e6 −1.26231 −0.631155 0.775657i \(-0.717419\pi\)
−0.631155 + 0.775657i \(0.717419\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −622450. + 2.08809e6i −0.285673 + 0.958327i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0 0
\(349\) 1.55867e6i 0.685001i −0.939518 0.342501i \(-0.888726\pi\)
0.939518 0.342501i \(-0.111274\pi\)
\(350\) 0 0
\(351\) 4.32370e6 1.87322
\(352\) 0 0
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0 0
\(361\) −1.22508e6 + 2.12189e6i −0.494761 + 0.856950i
\(362\) 0 0
\(363\) 2.51054e6i 1.00000i
\(364\) 1.96118e6 + 4.31001e6i 0.775827 + 1.70500i
\(365\) 0 0
\(366\) 0 0
\(367\) 3.86895e6 + 2.23374e6i 1.49944 + 0.865700i 1.00000 0.000650122i \(-0.000206940\pi\)
0.499437 + 0.866350i \(0.333540\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −5.16240e6 −1.93417
\(373\) 2.65864e6 + 4.60489e6i 0.989434 + 1.71375i 0.620276 + 0.784384i \(0.287021\pi\)
0.369158 + 0.929367i \(0.379646\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −69893.0 −0.0249940 −0.0124970 0.999922i \(-0.503978\pi\)
−0.0124970 + 0.999922i \(0.503978\pi\)
\(380\) 0 0
\(381\) 4.68600e6 + 2.70546e6i 1.65383 + 0.954837i
\(382\) 0 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.73071e6 4.72973e6i −0.926827 1.60531i
\(388\) −3.53702e6 2.04210e6i −1.19278 0.688649i
\(389\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −3.28777e6 + 1.89820e6i −1.04695 + 0.604456i −0.921794 0.387681i \(-0.873276\pi\)
−0.125156 + 0.992137i \(0.539943\pi\)
\(398\) 0 0
\(399\) −264910. 189189.i −0.0833042 0.0594928i
\(400\) −3.20000e6 −1.00000
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) 5.90629e6 1.02300e7i 1.81156 3.13771i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 3.77979e6 + 2.18226e6i 1.11727 + 0.645057i 0.940703 0.339231i \(-0.110167\pi\)
0.176569 + 0.984288i \(0.443500\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.29923e6i 0.377089i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −226220. + 391824.i −0.0637074 + 0.110344i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −4.87580e6 −1.34073 −0.670364 0.742033i \(-0.733862\pi\)
−0.670364 + 0.742033i \(0.733862\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −5.60449e6 542911.i −1.48753 0.144098i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 3.35923e6 1.93945e6i 0.866025 0.500000i
\(433\) 2.18049e6i 0.558901i 0.960160 + 0.279450i \(0.0901522\pi\)
−0.960160 + 0.279450i \(0.909848\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.96549e6 6.86843e6i −0.999034 1.73038i
\(437\) 0 0
\(438\) 0 0
\(439\) 6.07908e6 3.50976e6i 1.50548 0.869192i 0.505505 0.862824i \(-0.331306\pi\)
0.999980 0.00636830i \(-0.00202711\pi\)
\(440\) 0 0
\(441\) −2.68296e6 + 3.07922e6i −0.656929 + 0.753953i
\(442\) 0 0
\(443\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(444\) 2.87755e6 + 1.66136e6i 0.692733 + 0.399949i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 3.45702e6 + 2.46888e6i 0.813781 + 0.581172i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −5.51777e6 + 3.18569e6i −1.26333 + 0.729387i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 592880. + 1.02690e6i 0.132793 + 0.230005i 0.924752 0.380569i \(-0.124272\pi\)
−0.791959 + 0.610574i \(0.790939\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 2.92217e6 0.633510 0.316755 0.948507i \(-0.397407\pi\)
0.316755 + 0.948507i \(0.397407\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) 8.87569e6i 1.87322i
\(469\) 4.47680e6 2.03708e6i 0.939801 0.427638i
\(470\) 0 0
\(471\) 4.73866e6 + 8.20760e6i 0.984246 + 1.70476i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 503377.i 0.102367i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) −6.58440e6 + 3.80150e6i −1.29764 + 0.749191i
\(482\) 0 0
\(483\) 0 0
\(484\) 5.15363e6 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −682437. + 1.18202e6i −0.130389 + 0.225840i −0.923827 0.382811i \(-0.874956\pi\)
0.793438 + 0.608652i \(0.208289\pi\)
\(488\) 0 0
\(489\) 1.05728e7i 1.99949i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.05974e7i 1.93417i
\(497\) 0 0
\(498\) 0 0
\(499\) −1.08685e6 1.88247e6i −0.195397 0.338437i 0.751634 0.659581i \(-0.229266\pi\)
−0.947030 + 0.321144i \(0.895933\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.25759e7 7.26071e6i −2.17280 1.25447i
\(508\) −5.55378e6 + 9.61942e6i −0.954837 + 1.65383i
\(509\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) −2.51184e6 5.52015e6i −0.425539 0.935188i
\(512\) 0 0
\(513\) −305086. 528425.i −0.0511835 0.0886523i
\(514\) 0 0
\(515\) 0 0
\(516\) 9.70920e6 5.60561e6i 1.60531 0.926827i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(522\) 0 0
\(523\) −3.77071e6 + 2.17702e6i −0.602794 + 0.348023i −0.770140 0.637875i \(-0.779814\pi\)
0.167346 + 0.985898i \(0.446480\pi\)
\(524\) 0 0
\(525\) 6.28594e6 + 608924.i 0.995341 + 0.0964195i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.21817e6 + 5.57404e6i −0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 388368. 543809.i 0.0594928 0.0833042i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.51118e6 + 2.61743e6i 0.221984 + 0.384488i 0.955410 0.295281i \(-0.0954134\pi\)
−0.733426 + 0.679769i \(0.762080\pi\)
\(542\) 0 0
\(543\) 1.73239e6 3.00058e6i 0.252142 0.436723i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.27982e7 1.82886 0.914430 0.404744i \(-0.132639\pi\)
0.914430 + 0.404744i \(0.132639\pi\)
\(548\) 0 0
\(549\) −9.14020e6 5.27710e6i −1.29427 0.747247i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1.13571e6 1.17240e7i 0.157927 1.63028i
\(554\) 0 0
\(555\) 0 0
\(556\) −804336. 464384.i −0.110344 0.0637074i
\(557\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) 2.56534e7i 3.47229i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −6.96778e6 + 3.17055e6i −0.910200 + 0.414169i
\(568\) 0 0
\(569\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(570\) 0 0
\(571\) −2.94390e6 + 5.09898e6i −0.377862 + 0.654475i −0.990751 0.135693i \(-0.956674\pi\)
0.612889 + 0.790169i \(0.290007\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 3.98131e6 + 6.89583e6i 0.500000 + 0.866025i
\(577\) 8.17725e6 + 4.72114e6i 1.02251 + 0.590347i 0.914830 0.403839i \(-0.132324\pi\)
0.107680 + 0.994186i \(0.465658\pi\)
\(578\) 0 0
\(579\) 8.86106e6 5.11594e6i 1.09847 0.634204i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −6.32102e6 5.50759e6i −0.753953 0.656929i
\(589\) −1.66702e6 −0.197995
\(590\) 0 0
\(591\) 0 0
\(592\) −3.41043e6 + 5.90704e6i −0.399949 + 0.692733i
\(593\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.02764e6 + 8.70813e6i 0.577336 + 0.999975i
\(598\) 0 0
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) 3.11249e6i 0.351498i −0.984435 0.175749i \(-0.943765\pi\)
0.984435 0.175749i \(-0.0562346\pi\)
\(602\) 0 0
\(603\) 9.21918e6 1.03252
\(604\) −6.53958e6 1.13269e7i −0.729387 1.26333i
\(605\) 0 0
\(606\) 0 0
\(607\) 7.92463e6 4.57529e6i 0.872986 0.504019i 0.00464665 0.999989i \(-0.498521\pi\)
0.868339 + 0.495970i \(0.165188\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 7.91172e6 1.37035e7i 0.850394 1.47292i −0.0304599 0.999536i \(-0.509697\pi\)
0.880853 0.473389i \(-0.156969\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −1.26698e7 7.31491e6i −1.32905 0.767330i −0.343901 0.939006i \(-0.611748\pi\)
−0.985153 + 0.171676i \(0.945082\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −1.82200e7 −1.87322
\(625\) −4.88281e6 8.45728e6i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) −1.68486e7 + 9.72753e6i −1.70476 + 0.984246i
\(629\) 0 0
\(630\) 0 0
\(631\) −1.62439e7 −1.62411 −0.812057 0.583579i \(-0.801652\pi\)
−0.812057 + 0.583579i \(0.801652\pi\)
\(632\) 0 0
\(633\) 3.90118e6 + 2.25235e6i 0.386978 + 0.223422i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.81459e7 6.22474e6i 1.77186 0.607817i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) 1.61402e7i 1.53951i −0.638342 0.769753i \(-0.720380\pi\)
0.638342 0.769753i \(-0.279620\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −2.01656e6 + 2.08170e7i −0.186492 + 1.92516i
\(652\) 2.17039e7 1.99949
\(653\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.13678e7i 1.02745i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 1.94473e7 + 1.12279e7i 1.73123 + 0.999527i 0.880993 + 0.473129i \(0.156876\pi\)
0.850238 + 0.526398i \(0.176458\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −1.13310e7 + 1.96259e7i −0.978821 + 1.69537i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.73115e7 −1.47332 −0.736661 0.676262i \(-0.763599\pi\)
−0.736661 + 0.676262i \(0.763599\pi\)
\(674\) 0 0
\(675\) 1.02516e7 + 5.91874e6i 0.866025 + 0.500000i
\(676\) 1.49048e7 2.58159e7i 1.25447 2.17280i
\(677\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(678\) 0 0
\(679\) −9.61628e6 + 1.34651e7i −0.800447 + 1.12082i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) 1.08475e6 626282.i 0.0886523 0.0511835i
\(685\) 0 0
\(686\) 0 0
\(687\) 9.72116e6 0.785826
\(688\) 1.15072e7 + 1.99311e7i 0.926827 + 1.60531i
\(689\) 0 0
\(690\) 0 0
\(691\) −2.08726e7 + 1.20508e7i −1.66296 + 0.960112i −0.691673 + 0.722211i \(0.743126\pi\)
−0.971289 + 0.237901i \(0.923541\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.25000e6 + 1.29038e7i −0.0964195 + 0.995341i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 929210. + 536479.i 0.0709129 + 0.0409416i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.13507e7 1.96600e7i −0.848021 1.46882i −0.882971 0.469427i \(-0.844461\pi\)
0.0349502 0.999389i \(-0.488873\pi\)
\(710\) 0 0
\(711\) 1.10391e7 1.91203e7i 0.818956 1.41847i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) 5.23906e6 + 507513.i 0.375332 + 0.0363587i
\(722\) 0 0
\(723\) −9.76558e6 1.69145e7i −0.694788 1.20341i
\(724\) 6.15960e6 + 3.55625e6i 0.436723 + 0.252142i
\(725\) 0 0
\(726\) 0 0
\(727\) 2.37945e7i 1.66971i 0.550474 + 0.834853i \(0.314447\pi\)
−0.550474 + 0.834853i \(0.685553\pi\)
\(728\) 0 0
\(729\) −1.43489e7 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 1.08328e7 1.87630e7i 0.747247 1.29427i
\(733\) 2.00852e7 1.15962e7i 1.38075 0.797177i 0.388503 0.921448i \(-0.372992\pi\)
0.992248 + 0.124270i \(0.0396590\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −6.40852e6 + 1.10999e7i −0.431665 + 0.747666i −0.997017 0.0771842i \(-0.975407\pi\)
0.565352 + 0.824850i \(0.308740\pi\)
\(740\) 0 0
\(741\) 2.86611e6i 0.191755i
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.36610e7 2.36615e7i −0.883857 1.53088i −0.847019 0.531563i \(-0.821605\pi\)
−0.0368381 0.999321i \(-0.511729\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −6.50851e6 1.43035e7i −0.414169 0.910200i
\(757\) −1.98526e7 −1.25915 −0.629575 0.776940i \(-0.716771\pi\)
−0.629575 + 0.776940i \(0.716771\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) −2.92455e7 + 1.33076e7i −1.81864 + 0.827537i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.41558e7 + 8.17284e6i −0.866025 + 0.500000i
\(769\) 2.12277e7i 1.29446i −0.762296 0.647228i \(-0.775928\pi\)
0.762296 0.647228i \(-0.224072\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.05020e7 + 1.81900e7i 0.634204 + 1.09847i
\(773\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(774\) 0 0
\(775\) 2.80078e7 1.61703e7i 1.67504 0.967084i
\(776\) 0 0
\(777\) 7.82334e6 1.09546e7i 0.464879 0.650942i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.13060e7 1.29758e7i 0.656929 0.753953i
\(785\) 0 0
\(786\) 0 0
\(787\) 9.14931e6 + 5.28235e6i 0.526564 + 0.304012i 0.739616 0.673029i \(-0.235007\pi\)
−0.213052 + 0.977041i \(0.568340\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.47876e7 + 4.29334e7i 1.39975 + 2.42445i
\(794\) 0 0
\(795\) 0 0
\(796\) −1.78761e7 + 1.03208e7i −0.999975 + 0.577336i