Properties

Label 2352.4.a.cm.1.3
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 2 x^{3} - 152 x^{2} - 177 x + 2922\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-6.46638\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} +0.726342 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +0.726342 q^{5} +9.00000 q^{9} +64.4895 q^{11} +71.8475 q^{13} -2.17903 q^{15} +48.9034 q^{17} -34.3968 q^{19} +0.903350 q^{23} -124.472 q^{25} -27.0000 q^{27} +226.686 q^{29} +275.795 q^{31} -193.468 q^{33} +295.209 q^{37} -215.543 q^{39} +186.604 q^{41} +455.317 q^{43} +6.53708 q^{45} -282.334 q^{47} -146.710 q^{51} +356.214 q^{53} +46.8414 q^{55} +103.191 q^{57} -729.370 q^{59} +274.353 q^{61} +52.1859 q^{65} -193.272 q^{67} -2.71005 q^{69} -40.5277 q^{71} -206.472 q^{73} +373.417 q^{75} -937.741 q^{79} +81.0000 q^{81} -911.607 q^{83} +35.5206 q^{85} -680.059 q^{87} -949.975 q^{89} -827.385 q^{93} -24.9839 q^{95} +39.4687 q^{97} +580.405 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{3} + 4q^{5} + 36q^{9} + O(q^{10}) \) \( 4q - 12q^{3} + 4q^{5} + 36q^{9} - 14q^{11} + 22q^{13} - 12q^{15} + 96q^{17} + 26q^{19} - 96q^{23} + 110q^{25} - 108q^{27} - 76q^{29} - 238q^{31} + 42q^{33} + 562q^{37} - 66q^{39} + 428q^{41} + 258q^{43} + 36q^{45} + 80q^{47} - 288q^{51} - 1476q^{55} - 78q^{57} - 262q^{59} - 276q^{61} + 2196q^{65} - 150q^{67} + 288q^{69} + 848q^{71} - 218q^{73} - 330q^{75} - 1762q^{79} + 324q^{81} - 3450q^{83} + 1452q^{85} + 228q^{87} - 344q^{89} + 714q^{93} - 2004q^{95} - 622q^{97} - 126q^{99} + O(q^{100}) \)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 0.726342 0.0649660 0.0324830 0.999472i \(-0.489659\pi\)
0.0324830 + 0.999472i \(0.489659\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 64.4895 1.76766 0.883832 0.467804i \(-0.154955\pi\)
0.883832 + 0.467804i \(0.154955\pi\)
\(12\) 0 0
\(13\) 71.8475 1.53284 0.766420 0.642340i \(-0.222036\pi\)
0.766420 + 0.642340i \(0.222036\pi\)
\(14\) 0 0
\(15\) −2.17903 −0.0375081
\(16\) 0 0
\(17\) 48.9034 0.697694 0.348847 0.937180i \(-0.386573\pi\)
0.348847 + 0.937180i \(0.386573\pi\)
\(18\) 0 0
\(19\) −34.3968 −0.415325 −0.207663 0.978201i \(-0.566586\pi\)
−0.207663 + 0.978201i \(0.566586\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.903350 0.00818963 0.00409482 0.999992i \(-0.498697\pi\)
0.00409482 + 0.999992i \(0.498697\pi\)
\(24\) 0 0
\(25\) −124.472 −0.995779
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 226.686 1.45154 0.725769 0.687939i \(-0.241484\pi\)
0.725769 + 0.687939i \(0.241484\pi\)
\(30\) 0 0
\(31\) 275.795 1.59788 0.798940 0.601411i \(-0.205395\pi\)
0.798940 + 0.601411i \(0.205395\pi\)
\(32\) 0 0
\(33\) −193.468 −1.02056
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 295.209 1.31168 0.655839 0.754901i \(-0.272315\pi\)
0.655839 + 0.754901i \(0.272315\pi\)
\(38\) 0 0
\(39\) −215.543 −0.884985
\(40\) 0 0
\(41\) 186.604 0.710798 0.355399 0.934715i \(-0.384345\pi\)
0.355399 + 0.934715i \(0.384345\pi\)
\(42\) 0 0
\(43\) 455.317 1.61477 0.807386 0.590023i \(-0.200881\pi\)
0.807386 + 0.590023i \(0.200881\pi\)
\(44\) 0 0
\(45\) 6.53708 0.0216553
\(46\) 0 0
\(47\) −282.334 −0.876228 −0.438114 0.898920i \(-0.644353\pi\)
−0.438114 + 0.898920i \(0.644353\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −146.710 −0.402814
\(52\) 0 0
\(53\) 356.214 0.923204 0.461602 0.887087i \(-0.347275\pi\)
0.461602 + 0.887087i \(0.347275\pi\)
\(54\) 0 0
\(55\) 46.8414 0.114838
\(56\) 0 0
\(57\) 103.191 0.239788
\(58\) 0 0
\(59\) −729.370 −1.60942 −0.804711 0.593667i \(-0.797680\pi\)
−0.804711 + 0.593667i \(0.797680\pi\)
\(60\) 0 0
\(61\) 274.353 0.575856 0.287928 0.957652i \(-0.407034\pi\)
0.287928 + 0.957652i \(0.407034\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 52.1859 0.0995825
\(66\) 0 0
\(67\) −193.272 −0.352417 −0.176209 0.984353i \(-0.556383\pi\)
−0.176209 + 0.984353i \(0.556383\pi\)
\(68\) 0 0
\(69\) −2.71005 −0.00472829
\(70\) 0 0
\(71\) −40.5277 −0.0677429 −0.0338715 0.999426i \(-0.510784\pi\)
−0.0338715 + 0.999426i \(0.510784\pi\)
\(72\) 0 0
\(73\) −206.472 −0.331038 −0.165519 0.986207i \(-0.552930\pi\)
−0.165519 + 0.986207i \(0.552930\pi\)
\(74\) 0 0
\(75\) 373.417 0.574914
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −937.741 −1.33549 −0.667747 0.744388i \(-0.732741\pi\)
−0.667747 + 0.744388i \(0.732741\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −911.607 −1.20556 −0.602782 0.797906i \(-0.705941\pi\)
−0.602782 + 0.797906i \(0.705941\pi\)
\(84\) 0 0
\(85\) 35.5206 0.0453264
\(86\) 0 0
\(87\) −680.059 −0.838045
\(88\) 0 0
\(89\) −949.975 −1.13143 −0.565715 0.824601i \(-0.691400\pi\)
−0.565715 + 0.824601i \(0.691400\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −827.385 −0.922536
\(94\) 0 0
\(95\) −24.9839 −0.0269820
\(96\) 0 0
\(97\) 39.4687 0.0413138 0.0206569 0.999787i \(-0.493424\pi\)
0.0206569 + 0.999787i \(0.493424\pi\)
\(98\) 0 0
\(99\) 580.405 0.589221
\(100\) 0 0
\(101\) −316.874 −0.312180 −0.156090 0.987743i \(-0.549889\pi\)
−0.156090 + 0.987743i \(0.549889\pi\)
\(102\) 0 0
\(103\) 322.634 0.308641 0.154321 0.988021i \(-0.450681\pi\)
0.154321 + 0.988021i \(0.450681\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −681.451 −0.615686 −0.307843 0.951437i \(-0.599607\pi\)
−0.307843 + 0.951437i \(0.599607\pi\)
\(108\) 0 0
\(109\) −455.967 −0.400677 −0.200338 0.979727i \(-0.564204\pi\)
−0.200338 + 0.979727i \(0.564204\pi\)
\(110\) 0 0
\(111\) −885.627 −0.757297
\(112\) 0 0
\(113\) −796.025 −0.662688 −0.331344 0.943510i \(-0.607502\pi\)
−0.331344 + 0.943510i \(0.607502\pi\)
\(114\) 0 0
\(115\) 0.656141 0.000532048 0
\(116\) 0 0
\(117\) 646.628 0.510947
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2827.89 2.12464
\(122\) 0 0
\(123\) −559.813 −0.410379
\(124\) 0 0
\(125\) −181.202 −0.129658
\(126\) 0 0
\(127\) −2333.92 −1.63072 −0.815362 0.578952i \(-0.803462\pi\)
−0.815362 + 0.578952i \(0.803462\pi\)
\(128\) 0 0
\(129\) −1365.95 −0.932289
\(130\) 0 0
\(131\) 1886.98 1.25852 0.629259 0.777195i \(-0.283358\pi\)
0.629259 + 0.777195i \(0.283358\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −19.6112 −0.0125027
\(136\) 0 0
\(137\) 2947.26 1.83797 0.918983 0.394297i \(-0.129012\pi\)
0.918983 + 0.394297i \(0.129012\pi\)
\(138\) 0 0
\(139\) −955.433 −0.583013 −0.291506 0.956569i \(-0.594156\pi\)
−0.291506 + 0.956569i \(0.594156\pi\)
\(140\) 0 0
\(141\) 847.003 0.505890
\(142\) 0 0
\(143\) 4633.41 2.70955
\(144\) 0 0
\(145\) 164.652 0.0943006
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2183.74 1.20066 0.600332 0.799751i \(-0.295035\pi\)
0.600332 + 0.799751i \(0.295035\pi\)
\(150\) 0 0
\(151\) −404.719 −0.218116 −0.109058 0.994035i \(-0.534783\pi\)
−0.109058 + 0.994035i \(0.534783\pi\)
\(152\) 0 0
\(153\) 440.130 0.232565
\(154\) 0 0
\(155\) 200.322 0.103808
\(156\) 0 0
\(157\) −929.582 −0.472540 −0.236270 0.971687i \(-0.575925\pi\)
−0.236270 + 0.971687i \(0.575925\pi\)
\(158\) 0 0
\(159\) −1068.64 −0.533012
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1425.68 0.685079 0.342540 0.939503i \(-0.388713\pi\)
0.342540 + 0.939503i \(0.388713\pi\)
\(164\) 0 0
\(165\) −140.524 −0.0663018
\(166\) 0 0
\(167\) −4185.15 −1.93926 −0.969631 0.244572i \(-0.921353\pi\)
−0.969631 + 0.244572i \(0.921353\pi\)
\(168\) 0 0
\(169\) 2965.07 1.34960
\(170\) 0 0
\(171\) −309.572 −0.138442
\(172\) 0 0
\(173\) 2235.55 0.982462 0.491231 0.871029i \(-0.336547\pi\)
0.491231 + 0.871029i \(0.336547\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2188.11 0.929200
\(178\) 0 0
\(179\) 941.479 0.393126 0.196563 0.980491i \(-0.437022\pi\)
0.196563 + 0.980491i \(0.437022\pi\)
\(180\) 0 0
\(181\) −467.540 −0.192000 −0.0960000 0.995381i \(-0.530605\pi\)
−0.0960000 + 0.995381i \(0.530605\pi\)
\(182\) 0 0
\(183\) −823.058 −0.332471
\(184\) 0 0
\(185\) 214.423 0.0852144
\(186\) 0 0
\(187\) 3153.75 1.23329
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −275.403 −0.104332 −0.0521660 0.998638i \(-0.516613\pi\)
−0.0521660 + 0.998638i \(0.516613\pi\)
\(192\) 0 0
\(193\) 1640.30 0.611767 0.305884 0.952069i \(-0.401048\pi\)
0.305884 + 0.952069i \(0.401048\pi\)
\(194\) 0 0
\(195\) −156.558 −0.0574940
\(196\) 0 0
\(197\) 1303.88 0.471560 0.235780 0.971806i \(-0.424236\pi\)
0.235780 + 0.971806i \(0.424236\pi\)
\(198\) 0 0
\(199\) 1327.36 0.472833 0.236417 0.971652i \(-0.424027\pi\)
0.236417 + 0.971652i \(0.424027\pi\)
\(200\) 0 0
\(201\) 579.817 0.203468
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 135.539 0.0461777
\(206\) 0 0
\(207\) 8.13015 0.00272988
\(208\) 0 0
\(209\) −2218.23 −0.734155
\(210\) 0 0
\(211\) 4753.28 1.55085 0.775426 0.631439i \(-0.217535\pi\)
0.775426 + 0.631439i \(0.217535\pi\)
\(212\) 0 0
\(213\) 121.583 0.0391114
\(214\) 0 0
\(215\) 330.716 0.104905
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 619.417 0.191125
\(220\) 0 0
\(221\) 3513.58 1.06945
\(222\) 0 0
\(223\) −513.149 −0.154094 −0.0770470 0.997027i \(-0.524549\pi\)
−0.0770470 + 0.997027i \(0.524549\pi\)
\(224\) 0 0
\(225\) −1120.25 −0.331926
\(226\) 0 0
\(227\) −3309.34 −0.967616 −0.483808 0.875174i \(-0.660747\pi\)
−0.483808 + 0.875174i \(0.660747\pi\)
\(228\) 0 0
\(229\) −5909.47 −1.70528 −0.852639 0.522501i \(-0.824999\pi\)
−0.852639 + 0.522501i \(0.824999\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −353.572 −0.0994131 −0.0497065 0.998764i \(-0.515829\pi\)
−0.0497065 + 0.998764i \(0.515829\pi\)
\(234\) 0 0
\(235\) −205.071 −0.0569250
\(236\) 0 0
\(237\) 2813.22 0.771048
\(238\) 0 0
\(239\) 1652.55 0.447259 0.223629 0.974674i \(-0.428209\pi\)
0.223629 + 0.974674i \(0.428209\pi\)
\(240\) 0 0
\(241\) −3106.95 −0.830440 −0.415220 0.909721i \(-0.636295\pi\)
−0.415220 + 0.909721i \(0.636295\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2471.33 −0.636627
\(248\) 0 0
\(249\) 2734.82 0.696033
\(250\) 0 0
\(251\) −1771.77 −0.445551 −0.222776 0.974870i \(-0.571512\pi\)
−0.222776 + 0.974870i \(0.571512\pi\)
\(252\) 0 0
\(253\) 58.2566 0.0144765
\(254\) 0 0
\(255\) −106.562 −0.0261692
\(256\) 0 0
\(257\) 1478.13 0.358766 0.179383 0.983779i \(-0.442590\pi\)
0.179383 + 0.983779i \(0.442590\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 2040.18 0.483846
\(262\) 0 0
\(263\) 1431.97 0.335739 0.167869 0.985809i \(-0.446311\pi\)
0.167869 + 0.985809i \(0.446311\pi\)
\(264\) 0 0
\(265\) 258.734 0.0599769
\(266\) 0 0
\(267\) 2849.93 0.653231
\(268\) 0 0
\(269\) 285.398 0.0646879 0.0323439 0.999477i \(-0.489703\pi\)
0.0323439 + 0.999477i \(0.489703\pi\)
\(270\) 0 0
\(271\) 5249.33 1.17666 0.588328 0.808622i \(-0.299786\pi\)
0.588328 + 0.808622i \(0.299786\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8027.16 −1.76020
\(276\) 0 0
\(277\) 7147.87 1.55045 0.775224 0.631687i \(-0.217637\pi\)
0.775224 + 0.631687i \(0.217637\pi\)
\(278\) 0 0
\(279\) 2482.16 0.532627
\(280\) 0 0
\(281\) 4228.36 0.897661 0.448831 0.893617i \(-0.351841\pi\)
0.448831 + 0.893617i \(0.351841\pi\)
\(282\) 0 0
\(283\) −8342.38 −1.75231 −0.876154 0.482031i \(-0.839899\pi\)
−0.876154 + 0.482031i \(0.839899\pi\)
\(284\) 0 0
\(285\) 74.9516 0.0155781
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2521.46 −0.513223
\(290\) 0 0
\(291\) −118.406 −0.0238525
\(292\) 0 0
\(293\) 5038.84 1.00468 0.502342 0.864669i \(-0.332472\pi\)
0.502342 + 0.864669i \(0.332472\pi\)
\(294\) 0 0
\(295\) −529.772 −0.104558
\(296\) 0 0
\(297\) −1741.22 −0.340187
\(298\) 0 0
\(299\) 64.9035 0.0125534
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 950.623 0.180237
\(304\) 0 0
\(305\) 199.274 0.0374111
\(306\) 0 0
\(307\) 4869.67 0.905300 0.452650 0.891688i \(-0.350479\pi\)
0.452650 + 0.891688i \(0.350479\pi\)
\(308\) 0 0
\(309\) −967.901 −0.178194
\(310\) 0 0
\(311\) 1452.84 0.264897 0.132449 0.991190i \(-0.457716\pi\)
0.132449 + 0.991190i \(0.457716\pi\)
\(312\) 0 0
\(313\) 5696.27 1.02867 0.514333 0.857591i \(-0.328040\pi\)
0.514333 + 0.857591i \(0.328040\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3471.47 −0.615070 −0.307535 0.951537i \(-0.599504\pi\)
−0.307535 + 0.951537i \(0.599504\pi\)
\(318\) 0 0
\(319\) 14618.9 2.56583
\(320\) 0 0
\(321\) 2044.35 0.355466
\(322\) 0 0
\(323\) −1682.12 −0.289770
\(324\) 0 0
\(325\) −8943.03 −1.52637
\(326\) 0 0
\(327\) 1367.90 0.231331
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −6254.38 −1.03859 −0.519293 0.854597i \(-0.673805\pi\)
−0.519293 + 0.854597i \(0.673805\pi\)
\(332\) 0 0
\(333\) 2656.88 0.437226
\(334\) 0 0
\(335\) −140.382 −0.0228951
\(336\) 0 0
\(337\) 8006.96 1.29426 0.647132 0.762378i \(-0.275968\pi\)
0.647132 + 0.762378i \(0.275968\pi\)
\(338\) 0 0
\(339\) 2388.08 0.382603
\(340\) 0 0
\(341\) 17785.9 2.82451
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1.96842 −0.000307178 0
\(346\) 0 0
\(347\) −7635.59 −1.18127 −0.590634 0.806940i \(-0.701122\pi\)
−0.590634 + 0.806940i \(0.701122\pi\)
\(348\) 0 0
\(349\) −10358.0 −1.58869 −0.794345 0.607468i \(-0.792185\pi\)
−0.794345 + 0.607468i \(0.792185\pi\)
\(350\) 0 0
\(351\) −1939.88 −0.294995
\(352\) 0 0
\(353\) 3384.23 0.510268 0.255134 0.966906i \(-0.417880\pi\)
0.255134 + 0.966906i \(0.417880\pi\)
\(354\) 0 0
\(355\) −29.4369 −0.00440099
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8194.61 1.20472 0.602361 0.798224i \(-0.294227\pi\)
0.602361 + 0.798224i \(0.294227\pi\)
\(360\) 0 0
\(361\) −5675.86 −0.827505
\(362\) 0 0
\(363\) −8483.67 −1.22666
\(364\) 0 0
\(365\) −149.970 −0.0215062
\(366\) 0 0
\(367\) −805.510 −0.114570 −0.0572851 0.998358i \(-0.518244\pi\)
−0.0572851 + 0.998358i \(0.518244\pi\)
\(368\) 0 0
\(369\) 1679.44 0.236933
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 7905.71 1.09743 0.548716 0.836009i \(-0.315117\pi\)
0.548716 + 0.836009i \(0.315117\pi\)
\(374\) 0 0
\(375\) 543.607 0.0748580
\(376\) 0 0
\(377\) 16286.8 2.22497
\(378\) 0 0
\(379\) −3324.24 −0.450540 −0.225270 0.974296i \(-0.572326\pi\)
−0.225270 + 0.974296i \(0.572326\pi\)
\(380\) 0 0
\(381\) 7001.76 0.941499
\(382\) 0 0
\(383\) 8470.76 1.13012 0.565060 0.825050i \(-0.308853\pi\)
0.565060 + 0.825050i \(0.308853\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4097.85 0.538258
\(388\) 0 0
\(389\) −7908.40 −1.03078 −0.515388 0.856957i \(-0.672352\pi\)
−0.515388 + 0.856957i \(0.672352\pi\)
\(390\) 0 0
\(391\) 44.1769 0.00571386
\(392\) 0 0
\(393\) −5660.93 −0.726606
\(394\) 0 0
\(395\) −681.120 −0.0867617
\(396\) 0 0
\(397\) −9781.58 −1.23658 −0.618292 0.785949i \(-0.712175\pi\)
−0.618292 + 0.785949i \(0.712175\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −795.653 −0.0990849 −0.0495424 0.998772i \(-0.515776\pi\)
−0.0495424 + 0.998772i \(0.515776\pi\)
\(402\) 0 0
\(403\) 19815.2 2.44929
\(404\) 0 0
\(405\) 58.8337 0.00721844
\(406\) 0 0
\(407\) 19037.9 2.31860
\(408\) 0 0
\(409\) 8584.72 1.03787 0.518933 0.854815i \(-0.326329\pi\)
0.518933 + 0.854815i \(0.326329\pi\)
\(410\) 0 0
\(411\) −8841.78 −1.06115
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −662.138 −0.0783207
\(416\) 0 0
\(417\) 2866.30 0.336603
\(418\) 0 0
\(419\) 7447.09 0.868292 0.434146 0.900843i \(-0.357050\pi\)
0.434146 + 0.900843i \(0.357050\pi\)
\(420\) 0 0
\(421\) −4446.76 −0.514779 −0.257390 0.966308i \(-0.582862\pi\)
−0.257390 + 0.966308i \(0.582862\pi\)
\(422\) 0 0
\(423\) −2541.01 −0.292076
\(424\) 0 0
\(425\) −6087.12 −0.694750
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −13900.2 −1.56436
\(430\) 0 0
\(431\) 8963.70 1.00178 0.500889 0.865512i \(-0.333006\pi\)
0.500889 + 0.865512i \(0.333006\pi\)
\(432\) 0 0
\(433\) −16173.6 −1.79504 −0.897520 0.440973i \(-0.854633\pi\)
−0.897520 + 0.440973i \(0.854633\pi\)
\(434\) 0 0
\(435\) −493.955 −0.0544445
\(436\) 0 0
\(437\) −31.0724 −0.00340136
\(438\) 0 0
\(439\) 6298.43 0.684755 0.342378 0.939562i \(-0.388768\pi\)
0.342378 + 0.939562i \(0.388768\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 16784.7 1.80015 0.900074 0.435738i \(-0.143512\pi\)
0.900074 + 0.435738i \(0.143512\pi\)
\(444\) 0 0
\(445\) −690.007 −0.0735044
\(446\) 0 0
\(447\) −6551.22 −0.693203
\(448\) 0 0
\(449\) 2733.88 0.287349 0.143674 0.989625i \(-0.454108\pi\)
0.143674 + 0.989625i \(0.454108\pi\)
\(450\) 0 0
\(451\) 12034.0 1.25645
\(452\) 0 0
\(453\) 1214.16 0.125930
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9229.20 0.944691 0.472345 0.881414i \(-0.343408\pi\)
0.472345 + 0.881414i \(0.343408\pi\)
\(458\) 0 0
\(459\) −1320.39 −0.134271
\(460\) 0 0
\(461\) −19726.7 −1.99298 −0.996491 0.0837048i \(-0.973325\pi\)
−0.996491 + 0.0837048i \(0.973325\pi\)
\(462\) 0 0
\(463\) −368.924 −0.0370310 −0.0185155 0.999829i \(-0.505894\pi\)
−0.0185155 + 0.999829i \(0.505894\pi\)
\(464\) 0 0
\(465\) −600.965 −0.0599335
\(466\) 0 0
\(467\) 13309.6 1.31883 0.659414 0.751780i \(-0.270804\pi\)
0.659414 + 0.751780i \(0.270804\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2788.75 0.272821
\(472\) 0 0
\(473\) 29363.2 2.85438
\(474\) 0 0
\(475\) 4281.46 0.413572
\(476\) 0 0
\(477\) 3205.93 0.307735
\(478\) 0 0
\(479\) −11562.8 −1.10296 −0.551479 0.834189i \(-0.685936\pi\)
−0.551479 + 0.834189i \(0.685936\pi\)
\(480\) 0 0
\(481\) 21210.0 2.01059
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 28.6677 0.00268399
\(486\) 0 0
\(487\) −2628.13 −0.244542 −0.122271 0.992497i \(-0.539018\pi\)
−0.122271 + 0.992497i \(0.539018\pi\)
\(488\) 0 0
\(489\) −4277.04 −0.395531
\(490\) 0 0
\(491\) 12319.9 1.13236 0.566181 0.824281i \(-0.308420\pi\)
0.566181 + 0.824281i \(0.308420\pi\)
\(492\) 0 0
\(493\) 11085.7 1.01273
\(494\) 0 0
\(495\) 421.573 0.0382794
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 3305.81 0.296570 0.148285 0.988945i \(-0.452625\pi\)
0.148285 + 0.988945i \(0.452625\pi\)
\(500\) 0 0
\(501\) 12555.5 1.11963
\(502\) 0 0
\(503\) 3072.72 0.272377 0.136189 0.990683i \(-0.456515\pi\)
0.136189 + 0.990683i \(0.456515\pi\)
\(504\) 0 0
\(505\) −230.159 −0.0202811
\(506\) 0 0
\(507\) −8895.20 −0.779190
\(508\) 0 0
\(509\) 13569.8 1.18167 0.590836 0.806791i \(-0.298798\pi\)
0.590836 + 0.806791i \(0.298798\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 928.715 0.0799294
\(514\) 0 0
\(515\) 234.342 0.0200512
\(516\) 0 0
\(517\) −18207.6 −1.54888
\(518\) 0 0
\(519\) −6706.66 −0.567225
\(520\) 0 0
\(521\) −10733.4 −0.902566 −0.451283 0.892381i \(-0.649033\pi\)
−0.451283 + 0.892381i \(0.649033\pi\)
\(522\) 0 0
\(523\) −8348.15 −0.697971 −0.348986 0.937128i \(-0.613474\pi\)
−0.348986 + 0.937128i \(0.613474\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 13487.3 1.11483
\(528\) 0 0
\(529\) −12166.2 −0.999933
\(530\) 0 0
\(531\) −6564.33 −0.536474
\(532\) 0 0
\(533\) 13407.1 1.08954
\(534\) 0 0
\(535\) −494.967 −0.0399987
\(536\) 0 0
\(537\) −2824.44 −0.226971
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −23107.3 −1.83634 −0.918170 0.396186i \(-0.870334\pi\)
−0.918170 + 0.396186i \(0.870334\pi\)
\(542\) 0 0
\(543\) 1402.62 0.110851
\(544\) 0 0
\(545\) −331.188 −0.0260304
\(546\) 0 0
\(547\) −13935.2 −1.08926 −0.544630 0.838676i \(-0.683330\pi\)
−0.544630 + 0.838676i \(0.683330\pi\)
\(548\) 0 0
\(549\) 2469.17 0.191952
\(550\) 0 0
\(551\) −7797.29 −0.602860
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −643.268 −0.0491986
\(556\) 0 0
\(557\) −15047.5 −1.14467 −0.572337 0.820018i \(-0.693963\pi\)
−0.572337 + 0.820018i \(0.693963\pi\)
\(558\) 0 0
\(559\) 32713.4 2.47519
\(560\) 0 0
\(561\) −9461.25 −0.712040
\(562\) 0 0
\(563\) −15442.2 −1.15597 −0.577983 0.816049i \(-0.696160\pi\)
−0.577983 + 0.816049i \(0.696160\pi\)
\(564\) 0 0
\(565\) −578.187 −0.0430522
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13362.1 −0.984481 −0.492240 0.870459i \(-0.663822\pi\)
−0.492240 + 0.870459i \(0.663822\pi\)
\(570\) 0 0
\(571\) 22185.3 1.62596 0.812981 0.582290i \(-0.197843\pi\)
0.812981 + 0.582290i \(0.197843\pi\)
\(572\) 0 0
\(573\) 826.208 0.0602362
\(574\) 0 0
\(575\) −112.442 −0.00815507
\(576\) 0 0
\(577\) −18951.4 −1.36734 −0.683671 0.729791i \(-0.739617\pi\)
−0.683671 + 0.729791i \(0.739617\pi\)
\(578\) 0 0
\(579\) −4920.89 −0.353204
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 22972.1 1.63191
\(584\) 0 0
\(585\) 469.673 0.0331942
\(586\) 0 0
\(587\) −19579.5 −1.37672 −0.688359 0.725371i \(-0.741668\pi\)
−0.688359 + 0.725371i \(0.741668\pi\)
\(588\) 0 0
\(589\) −9486.48 −0.663640
\(590\) 0 0
\(591\) −3911.63 −0.272255
\(592\) 0 0
\(593\) 5613.01 0.388699 0.194350 0.980932i \(-0.437740\pi\)
0.194350 + 0.980932i \(0.437740\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3982.07 −0.272990
\(598\) 0 0
\(599\) −21925.5 −1.49558 −0.747790 0.663936i \(-0.768885\pi\)
−0.747790 + 0.663936i \(0.768885\pi\)
\(600\) 0 0
\(601\) −2067.51 −0.140326 −0.0701628 0.997536i \(-0.522352\pi\)
−0.0701628 + 0.997536i \(0.522352\pi\)
\(602\) 0 0
\(603\) −1739.45 −0.117472
\(604\) 0 0
\(605\) 2054.02 0.138029
\(606\) 0 0
\(607\) 10167.2 0.679859 0.339930 0.940451i \(-0.389597\pi\)
0.339930 + 0.940451i \(0.389597\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −20285.0 −1.34312
\(612\) 0 0
\(613\) −18186.1 −1.19825 −0.599127 0.800654i \(-0.704485\pi\)
−0.599127 + 0.800654i \(0.704485\pi\)
\(614\) 0 0
\(615\) −406.616 −0.0266607
\(616\) 0 0
\(617\) 6584.41 0.429625 0.214812 0.976655i \(-0.431086\pi\)
0.214812 + 0.976655i \(0.431086\pi\)
\(618\) 0 0
\(619\) 7779.72 0.505159 0.252579 0.967576i \(-0.418721\pi\)
0.252579 + 0.967576i \(0.418721\pi\)
\(620\) 0 0
\(621\) −24.3905 −0.00157610
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15427.4 0.987356
\(626\) 0 0
\(627\) 6654.70 0.423865
\(628\) 0 0
\(629\) 14436.7 0.915150
\(630\) 0 0
\(631\) −3787.78 −0.238969 −0.119484 0.992836i \(-0.538124\pi\)
−0.119484 + 0.992836i \(0.538124\pi\)
\(632\) 0 0
\(633\) −14259.9 −0.895384
\(634\) 0 0
\(635\) −1695.22 −0.105942
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −364.749 −0.0225810
\(640\) 0 0
\(641\) −19930.9 −1.22812 −0.614058 0.789261i \(-0.710464\pi\)
−0.614058 + 0.789261i \(0.710464\pi\)
\(642\) 0 0
\(643\) −3185.18 −0.195352 −0.0976759 0.995218i \(-0.531141\pi\)
−0.0976759 + 0.995218i \(0.531141\pi\)
\(644\) 0 0
\(645\) −992.148 −0.0605671
\(646\) 0 0
\(647\) 16943.0 1.02952 0.514759 0.857335i \(-0.327881\pi\)
0.514759 + 0.857335i \(0.327881\pi\)
\(648\) 0 0
\(649\) −47036.7 −2.84492
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −20078.8 −1.20328 −0.601640 0.798767i \(-0.705486\pi\)
−0.601640 + 0.798767i \(0.705486\pi\)
\(654\) 0 0
\(655\) 1370.59 0.0817609
\(656\) 0 0
\(657\) −1858.25 −0.110346
\(658\) 0 0
\(659\) 9442.46 0.558158 0.279079 0.960268i \(-0.409971\pi\)
0.279079 + 0.960268i \(0.409971\pi\)
\(660\) 0 0
\(661\) 761.990 0.0448381 0.0224190 0.999749i \(-0.492863\pi\)
0.0224190 + 0.999749i \(0.492863\pi\)
\(662\) 0 0
\(663\) −10540.8 −0.617449
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 204.777 0.0118876
\(668\) 0 0
\(669\) 1539.45 0.0889663
\(670\) 0 0
\(671\) 17692.8 1.01792
\(672\) 0 0
\(673\) −16111.6 −0.922818 −0.461409 0.887188i \(-0.652656\pi\)
−0.461409 + 0.887188i \(0.652656\pi\)
\(674\) 0 0
\(675\) 3360.76 0.191638
\(676\) 0 0
\(677\) 31241.5 1.77357 0.886786 0.462180i \(-0.152933\pi\)
0.886786 + 0.462180i \(0.152933\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 9928.03 0.558653
\(682\) 0 0
\(683\) 30113.5 1.68706 0.843530 0.537082i \(-0.180473\pi\)
0.843530 + 0.537082i \(0.180473\pi\)
\(684\) 0 0
\(685\) 2140.72 0.119405
\(686\) 0 0
\(687\) 17728.4 0.984542
\(688\) 0 0
\(689\) 25593.1 1.41512
\(690\) 0 0
\(691\) 9750.85 0.536816 0.268408 0.963305i \(-0.413502\pi\)
0.268408 + 0.963305i \(0.413502\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −693.971 −0.0378760
\(696\) 0 0
\(697\) 9125.58 0.495920
\(698\) 0 0
\(699\) 1060.71 0.0573962
\(700\) 0 0
\(701\) 11851.6 0.638559 0.319280 0.947661i \(-0.396559\pi\)
0.319280 + 0.947661i \(0.396559\pi\)
\(702\) 0 0
\(703\) −10154.3 −0.544773
\(704\) 0 0
\(705\) 615.214 0.0328657
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1322.39 0.0700472 0.0350236 0.999386i \(-0.488849\pi\)
0.0350236 + 0.999386i \(0.488849\pi\)
\(710\) 0 0
\(711\) −8439.67 −0.445165
\(712\) 0 0
\(713\) 249.140 0.0130861
\(714\) 0 0
\(715\) 3365.44 0.176028
\(716\) 0 0
\(717\) −4957.66 −0.258225
\(718\) 0 0
\(719\) −7301.56 −0.378723 −0.189362 0.981907i \(-0.560642\pi\)
−0.189362 + 0.981907i \(0.560642\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 9320.84 0.479455
\(724\) 0 0
\(725\) −28216.2 −1.44541
\(726\) 0 0
\(727\) 6088.72 0.310616 0.155308 0.987866i \(-0.450363\pi\)
0.155308 + 0.987866i \(0.450363\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 22266.5 1.12662
\(732\) 0 0
\(733\) −26578.5 −1.33929 −0.669644 0.742683i \(-0.733553\pi\)
−0.669644 + 0.742683i \(0.733553\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12464.0 −0.622955
\(738\) 0 0
\(739\) −25219.8 −1.25538 −0.627689 0.778464i \(-0.715999\pi\)
−0.627689 + 0.778464i \(0.715999\pi\)
\(740\) 0 0
\(741\) 7413.98 0.367557
\(742\) 0 0
\(743\) −2634.28 −0.130071 −0.0650353 0.997883i \(-0.520716\pi\)
−0.0650353 + 0.997883i \(0.520716\pi\)
\(744\) 0 0
\(745\) 1586.14 0.0780023
\(746\) 0 0
\(747\) −8204.46 −0.401855
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −5767.29 −0.280228 −0.140114 0.990135i \(-0.544747\pi\)
−0.140114 + 0.990135i \(0.544747\pi\)
\(752\) 0 0
\(753\) 5315.32 0.257239
\(754\) 0 0
\(755\) −293.965 −0.0141702
\(756\) 0 0
\(757\) 33378.7 1.60260 0.801302 0.598260i \(-0.204141\pi\)
0.801302 + 0.598260i \(0.204141\pi\)
\(758\) 0 0
\(759\) −174.770 −0.00835802
\(760\) 0 0
\(761\) 3626.09 0.172727 0.0863637 0.996264i \(-0.472475\pi\)
0.0863637 + 0.996264i \(0.472475\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 319.685 0.0151088
\(766\) 0 0
\(767\) −52403.4 −2.46698
\(768\) 0 0
\(769\) 3426.15 0.160663 0.0803316 0.996768i \(-0.474402\pi\)
0.0803316 + 0.996768i \(0.474402\pi\)
\(770\) 0 0
\(771\) −4434.38 −0.207134
\(772\) 0 0
\(773\) 13108.2 0.609920 0.304960 0.952365i \(-0.401357\pi\)
0.304960 + 0.952365i \(0.401357\pi\)
\(774\) 0 0
\(775\) −34328.9 −1.59114
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6418.60 −0.295212
\(780\) 0 0
\(781\) −2613.61 −0.119747
\(782\) 0 0
\(783\) −6120.53 −0.279348
\(784\) 0 0
\(785\) −675.194 −0.0306990
\(786\) 0 0
\(787\) −39319.7 −1.78094 −0.890468 0.455045i \(-0.849623\pi\)
−0.890468 + 0.455045i \(0.849623\pi\)
\(788\) 0 0
\(789\) −4295.92 −0.193839
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 19711.5 0.882696
\(794\) 0 0
\(795\) −776.201 −0.0346277
\(796\) 0 0
\(797\) −14597.8 −0.648785 −0.324393 0.945923i \(-0.605160\pi\)
−0.324393 + 0.945923i \(0.605160\pi\)
\(798\) 0 0
\(799\) −13807.1 −0.611339
\(800\) 0 0
\(801\) −8549.78 −0.377143
\(802\) 0 0
\(803\) −13315.3 −0.585164
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −856.194 −0.0373475
\(808\) 0 0
\(809\) 26827.4 1.16589 0.582943 0.812513i \(-0.301901\pi\)
0.582943 + 0.812513i \(0.301901\pi\)
\(810\) 0 0
\(811\) 5177.09 0.224158 0.112079 0.993699i \(-0.464249\pi\)
0.112079 + 0.993699i \(0.464249\pi\)
\(812\) 0 0
\(813\) −15748.0 −0.679343
\(814\) 0 0
\(815\) 1035.53 0.0445068
\(816\) 0 0
\(817\) −15661.5 −0.670656
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 20356.8 0.865355 0.432678 0.901549i \(-0.357569\pi\)
0.432678 + 0.901549i \(0.357569\pi\)
\(822\) 0 0
\(823\) −57.5738 −0.00243851 −0.00121926 0.999999i \(-0.500388\pi\)
−0.00121926 + 0.999999i \(0.500388\pi\)
\(824\) 0 0
\(825\) 24081.5 1.01625
\(826\) 0 0
\(827\) 12296.6 0.517044 0.258522 0.966005i \(-0.416765\pi\)
0.258522 + 0.966005i \(0.416765\pi\)
\(828\) 0 0
\(829\) 22871.7 0.958224 0.479112 0.877754i \(-0.340959\pi\)
0.479112 + 0.877754i \(0.340959\pi\)
\(830\) 0 0
\(831\) −21443.6 −0.895151
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −3039.85 −0.125986
\(836\) 0 0
\(837\) −7446.47 −0.307512
\(838\) 0 0
\(839\) −37571.3 −1.54601 −0.773007 0.634398i \(-0.781248\pi\)
−0.773007 + 0.634398i \(0.781248\pi\)
\(840\) 0 0
\(841\) 26997.6 1.10696
\(842\) 0 0
\(843\) −12685.1 −0.518265
\(844\) 0 0
\(845\) 2153.65 0.0876780
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 25027.1 1.01170
\(850\) 0 0
\(851\) 266.677 0.0107422
\(852\) 0 0
\(853\) −38354.5 −1.53955 −0.769774 0.638317i \(-0.779631\pi\)
−0.769774 + 0.638317i \(0.779631\pi\)
\(854\) 0 0
\(855\) −224.855 −0.00899401
\(856\) 0 0
\(857\) 36545.7 1.45668 0.728341 0.685215i \(-0.240292\pi\)
0.728341 + 0.685215i \(0.240292\pi\)
\(858\) 0 0
\(859\) 18564.3 0.737374 0.368687 0.929554i \(-0.379807\pi\)
0.368687 + 0.929554i \(0.379807\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −37457.7 −1.47749 −0.738746 0.673984i \(-0.764582\pi\)
−0.738746 + 0.673984i \(0.764582\pi\)
\(864\) 0 0
\(865\) 1623.78 0.0638266
\(866\) 0 0
\(867\) 7564.39 0.296309
\(868\) 0 0
\(869\) −60474.4 −2.36071
\(870\) 0 0
\(871\) −13886.1 −0.540199
\(872\) 0 0
\(873\) 355.218 0.0137713
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −27128.8 −1.04455 −0.522277 0.852776i \(-0.674917\pi\)
−0.522277 + 0.852776i \(0.674917\pi\)
\(878\) 0 0
\(879\) −15116.5 −0.580055
\(880\) 0 0
\(881\) 23647.8 0.904329 0.452165 0.891935i \(-0.350652\pi\)
0.452165 + 0.891935i \(0.350652\pi\)
\(882\) 0 0
\(883\) 4488.47 0.171063 0.0855316 0.996335i \(-0.472741\pi\)
0.0855316 + 0.996335i \(0.472741\pi\)
\(884\) 0 0
\(885\) 1589.32 0.0603664
\(886\) 0 0
\(887\) 41026.8 1.55304 0.776520 0.630093i \(-0.216983\pi\)
0.776520 + 0.630093i \(0.216983\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 5223.65 0.196407
\(892\) 0 0
\(893\) 9711.41 0.363919
\(894\) 0 0
\(895\) 683.836 0.0255398
\(896\) 0 0
\(897\) −194.710 −0.00724771
\(898\) 0 0
\(899\) 62519.0 2.31938
\(900\) 0 0
\(901\) 17420.1 0.644114
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −339.594 −0.0124735
\(906\) 0 0
\(907\) 736.305 0.0269555 0.0134777 0.999909i \(-0.495710\pi\)
0.0134777 + 0.999909i \(0.495710\pi\)
\(908\) 0 0
\(909\) −2851.87 −0.104060
\(910\) 0 0
\(911\) −1287.54 −0.0468256 −0.0234128 0.999726i \(-0.507453\pi\)
−0.0234128 + 0.999726i \(0.507453\pi\)
\(912\) 0 0
\(913\) −58789.0 −2.13103
\(914\) 0 0
\(915\) −597.821 −0.0215993
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −34280.0 −1.23046 −0.615230 0.788348i \(-0.710937\pi\)
−0.615230 + 0.788348i \(0.710937\pi\)
\(920\) 0 0
\(921\) −14609.0 −0.522675
\(922\) 0 0
\(923\) −2911.81 −0.103839
\(924\) 0 0
\(925\) −36745.4 −1.30614
\(926\) 0 0
\(927\) 2903.70 0.102880
\(928\) 0 0
\(929\) −31475.8 −1.11161 −0.555806 0.831312i \(-0.687590\pi\)
−0.555806 + 0.831312i \(0.687590\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −4358.52 −0.152939
\(934\) 0 0
\(935\) 2290.70 0.0801219
\(936\) 0 0
\(937\) 52560.6 1.83253 0.916264 0.400575i \(-0.131190\pi\)
0.916264 + 0.400575i \(0.131190\pi\)
\(938\) 0 0
\(939\) −17088.8 −0.593900
\(940\) 0 0
\(941\) 15976.7 0.553482 0.276741 0.960945i \(-0.410746\pi\)
0.276741 + 0.960945i \(0.410746\pi\)
\(942\) 0 0
\(943\) 168.569 0.00582118
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 8504.43 0.291824 0.145912 0.989298i \(-0.453388\pi\)
0.145912 + 0.989298i \(0.453388\pi\)
\(948\) 0 0
\(949\) −14834.5 −0.507428
\(950\) 0 0
\(951\) 10414.4 0.355111
\(952\) 0 0
\(953\) 29417.5 0.999923 0.499961 0.866048i \(-0.333348\pi\)
0.499961 + 0.866048i \(0.333348\pi\)
\(954\) 0 0
\(955\) −200.036 −0.00677804
\(956\) 0 0
\(957\) −43856.6 −1.48138
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 46272.0 1.55322
\(962\) 0 0
\(963\) −6133.06 −0.205229
\(964\) 0 0
\(965\) 1191.42 0.0397441
\(966\) 0 0
\(967\) −3461.97 −0.115129 −0.0575643 0.998342i \(-0.518333\pi\)
−0.0575643 + 0.998342i \(0.518333\pi\)
\(968\) 0 0
\(969\) 5046.36 0.167299
\(970\) 0 0
\(971\) 43558.8 1.43962 0.719809 0.694172i \(-0.244229\pi\)
0.719809 + 0.694172i \(0.244229\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 26829.1 0.881250
\(976\) 0 0
\(977\) −5504.95 −0.180265 −0.0901325 0.995930i \(-0.528729\pi\)
−0.0901325 + 0.995930i \(0.528729\pi\)
\(978\) 0 0
\(979\) −61263.4 −1.99999
\(980\) 0 0
\(981\) −4103.71 −0.133559
\(982\) 0 0
\(983\) 30789.7 0.999023 0.499511 0.866307i \(-0.333513\pi\)
0.499511 + 0.866307i \(0.333513\pi\)
\(984\) 0 0
\(985\) 947.060 0.0306354
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 411.311 0.0132244
\(990\) 0 0
\(991\) 21629.1 0.693312 0.346656 0.937992i \(-0.387317\pi\)
0.346656 + 0.937992i \(0.387317\pi\)
\(992\) 0 0
\(993\) 18763.1 0.599627
\(994\) 0 0
\(995\) 964.114 0.0307181
\(996\) 0 0
\(997\) −38195.8 −1.21331 −0.606656 0.794964i \(-0.707490\pi\)
−0.606656 + 0.794964i \(0.707490\pi\)
\(998\) 0 0
\(999\) −7970.64 −0.252432
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 2352.4.a.cm.1.3 4
4.3 odd 2 1176.4.a.bd.1.3 4
7.2 even 3 336.4.q.m.193.2 8
7.4 even 3 336.4.q.m.289.2 8
7.6 odd 2 2352.4.a.cp.1.2 4
28.11 odd 6 168.4.q.f.121.2 yes 8
28.23 odd 6 168.4.q.f.25.2 8
28.27 even 2 1176.4.a.ba.1.2 4
84.11 even 6 504.4.s.j.289.3 8
84.23 even 6 504.4.s.j.361.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.f.25.2 8 28.23 odd 6
168.4.q.f.121.2 yes 8 28.11 odd 6
336.4.q.m.193.2 8 7.2 even 3
336.4.q.m.289.2 8 7.4 even 3
504.4.s.j.289.3 8 84.11 even 6
504.4.s.j.361.3 8 84.23 even 6
1176.4.a.ba.1.2 4 28.27 even 2
1176.4.a.bd.1.3 4 4.3 odd 2
2352.4.a.cm.1.3 4 1.1 even 1 trivial
2352.4.a.cp.1.2 4 7.6 odd 2