Properties

Label 8400.2.a.dh.1.1
Level $8400$
Weight $2$
Character 8400.1
Self dual yes
Analytic conductor $67.074$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.0743376979\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 840)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 8400.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -5.41855 q^{11} +4.34017 q^{13} +1.07838 q^{17} -4.34017 q^{19} +1.00000 q^{21} -6.34017 q^{23} -1.00000 q^{27} -8.83710 q^{29} +4.34017 q^{31} +5.41855 q^{33} -8.68035 q^{37} -4.34017 q^{39} +8.34017 q^{41} -6.15676 q^{43} +6.83710 q^{47} +1.00000 q^{49} -1.07838 q^{51} +6.18342 q^{53} +4.34017 q^{57} +6.83710 q^{59} -4.52359 q^{61} -1.00000 q^{63} +6.34017 q^{69} +14.0989 q^{71} -11.1773 q^{73} +5.41855 q^{77} -0.680346 q^{79} +1.00000 q^{81} +6.83710 q^{83} +8.83710 q^{87} +6.49693 q^{89} -4.34017 q^{91} -4.34017 q^{93} -10.4969 q^{97} -5.41855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{19} + 3 q^{21} - 8 q^{23} - 3 q^{27} + 2 q^{29} + 2 q^{31} + 2 q^{33} - 4 q^{37} - 2 q^{39} + 14 q^{41} - 12 q^{43} - 8 q^{47} + 3 q^{49} + 14 q^{53} + 2 q^{57} - 8 q^{59} + 2 q^{61} - 3 q^{63} + 8 q^{69} + 6 q^{71} + 6 q^{73} + 2 q^{77} + 20 q^{79} + 3 q^{81} - 8 q^{83} - 2 q^{87} + 2 q^{89} - 2 q^{91} - 2 q^{93} - 14 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.41855 −1.63375 −0.816877 0.576812i \(-0.804297\pi\)
−0.816877 + 0.576812i \(0.804297\pi\)
\(12\) 0 0
\(13\) 4.34017 1.20375 0.601874 0.798591i \(-0.294421\pi\)
0.601874 + 0.798591i \(0.294421\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.07838 0.261545 0.130773 0.991412i \(-0.458254\pi\)
0.130773 + 0.991412i \(0.458254\pi\)
\(18\) 0 0
\(19\) −4.34017 −0.995704 −0.497852 0.867262i \(-0.665878\pi\)
−0.497852 + 0.867262i \(0.665878\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −6.34017 −1.32202 −0.661009 0.750378i \(-0.729871\pi\)
−0.661009 + 0.750378i \(0.729871\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −8.83710 −1.64101 −0.820504 0.571640i \(-0.806307\pi\)
−0.820504 + 0.571640i \(0.806307\pi\)
\(30\) 0 0
\(31\) 4.34017 0.779518 0.389759 0.920917i \(-0.372558\pi\)
0.389759 + 0.920917i \(0.372558\pi\)
\(32\) 0 0
\(33\) 5.41855 0.943249
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.68035 −1.42704 −0.713520 0.700635i \(-0.752900\pi\)
−0.713520 + 0.700635i \(0.752900\pi\)
\(38\) 0 0
\(39\) −4.34017 −0.694984
\(40\) 0 0
\(41\) 8.34017 1.30252 0.651258 0.758856i \(-0.274242\pi\)
0.651258 + 0.758856i \(0.274242\pi\)
\(42\) 0 0
\(43\) −6.15676 −0.938896 −0.469448 0.882960i \(-0.655547\pi\)
−0.469448 + 0.882960i \(0.655547\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.83710 0.997294 0.498647 0.866805i \(-0.333830\pi\)
0.498647 + 0.866805i \(0.333830\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.07838 −0.151003
\(52\) 0 0
\(53\) 6.18342 0.849358 0.424679 0.905344i \(-0.360387\pi\)
0.424679 + 0.905344i \(0.360387\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.34017 0.574870
\(58\) 0 0
\(59\) 6.83710 0.890115 0.445057 0.895502i \(-0.353183\pi\)
0.445057 + 0.895502i \(0.353183\pi\)
\(60\) 0 0
\(61\) −4.52359 −0.579186 −0.289593 0.957150i \(-0.593520\pi\)
−0.289593 + 0.957150i \(0.593520\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 6.34017 0.763267
\(70\) 0 0
\(71\) 14.0989 1.67323 0.836616 0.547790i \(-0.184531\pi\)
0.836616 + 0.547790i \(0.184531\pi\)
\(72\) 0 0
\(73\) −11.1773 −1.30820 −0.654101 0.756408i \(-0.726953\pi\)
−0.654101 + 0.756408i \(0.726953\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.41855 0.617501
\(78\) 0 0
\(79\) −0.680346 −0.0765449 −0.0382724 0.999267i \(-0.512185\pi\)
−0.0382724 + 0.999267i \(0.512185\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.83710 0.750469 0.375235 0.926930i \(-0.377562\pi\)
0.375235 + 0.926930i \(0.377562\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.83710 0.947437
\(88\) 0 0
\(89\) 6.49693 0.688673 0.344337 0.938846i \(-0.388104\pi\)
0.344337 + 0.938846i \(0.388104\pi\)
\(90\) 0 0
\(91\) −4.34017 −0.454974
\(92\) 0 0
\(93\) −4.34017 −0.450055
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.4969 −1.06580 −0.532901 0.846178i \(-0.678898\pi\)
−0.532901 + 0.846178i \(0.678898\pi\)
\(98\) 0 0
\(99\) −5.41855 −0.544585
\(100\) 0 0
\(101\) 18.8638 1.87701 0.938507 0.345259i \(-0.112209\pi\)
0.938507 + 0.345259i \(0.112209\pi\)
\(102\) 0 0
\(103\) −10.1568 −1.00077 −0.500387 0.865802i \(-0.666809\pi\)
−0.500387 + 0.865802i \(0.666809\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.6537 −1.41663 −0.708313 0.705899i \(-0.750543\pi\)
−0.708313 + 0.705899i \(0.750543\pi\)
\(108\) 0 0
\(109\) −12.8371 −1.22957 −0.614786 0.788694i \(-0.710757\pi\)
−0.614786 + 0.788694i \(0.710757\pi\)
\(110\) 0 0
\(111\) 8.68035 0.823902
\(112\) 0 0
\(113\) 1.50307 0.141397 0.0706985 0.997498i \(-0.477477\pi\)
0.0706985 + 0.997498i \(0.477477\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.34017 0.401249
\(118\) 0 0
\(119\) −1.07838 −0.0988547
\(120\) 0 0
\(121\) 18.3607 1.66915
\(122\) 0 0
\(123\) −8.34017 −0.752008
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −19.2039 −1.70407 −0.852037 0.523482i \(-0.824633\pi\)
−0.852037 + 0.523482i \(0.824633\pi\)
\(128\) 0 0
\(129\) 6.15676 0.542072
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 4.34017 0.376341
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.65368 0.739334 0.369667 0.929164i \(-0.379472\pi\)
0.369667 + 0.929164i \(0.379472\pi\)
\(138\) 0 0
\(139\) 6.18342 0.524471 0.262235 0.965004i \(-0.415540\pi\)
0.262235 + 0.965004i \(0.415540\pi\)
\(140\) 0 0
\(141\) −6.83710 −0.575788
\(142\) 0 0
\(143\) −23.5174 −1.96663
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −13.2039 −1.08171 −0.540854 0.841116i \(-0.681899\pi\)
−0.540854 + 0.841116i \(0.681899\pi\)
\(150\) 0 0
\(151\) 18.1568 1.47758 0.738788 0.673938i \(-0.235399\pi\)
0.738788 + 0.673938i \(0.235399\pi\)
\(152\) 0 0
\(153\) 1.07838 0.0871817
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 15.1773 1.21128 0.605639 0.795739i \(-0.292917\pi\)
0.605639 + 0.795739i \(0.292917\pi\)
\(158\) 0 0
\(159\) −6.18342 −0.490377
\(160\) 0 0
\(161\) 6.34017 0.499676
\(162\) 0 0
\(163\) 2.83710 0.222219 0.111109 0.993808i \(-0.464560\pi\)
0.111109 + 0.993808i \(0.464560\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.3607 −1.03388 −0.516941 0.856021i \(-0.672929\pi\)
−0.516941 + 0.856021i \(0.672929\pi\)
\(168\) 0 0
\(169\) 5.83710 0.449008
\(170\) 0 0
\(171\) −4.34017 −0.331901
\(172\) 0 0
\(173\) 2.55479 0.194237 0.0971184 0.995273i \(-0.469037\pi\)
0.0971184 + 0.995273i \(0.469037\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.83710 −0.513908
\(178\) 0 0
\(179\) 11.9421 0.892598 0.446299 0.894884i \(-0.352742\pi\)
0.446299 + 0.894884i \(0.352742\pi\)
\(180\) 0 0
\(181\) 4.15676 0.308969 0.154485 0.987995i \(-0.450628\pi\)
0.154485 + 0.987995i \(0.450628\pi\)
\(182\) 0 0
\(183\) 4.52359 0.334393
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −5.84324 −0.427300
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −6.09890 −0.441301 −0.220650 0.975353i \(-0.570818\pi\)
−0.220650 + 0.975353i \(0.570818\pi\)
\(192\) 0 0
\(193\) 12.6803 0.912751 0.456376 0.889787i \(-0.349147\pi\)
0.456376 + 0.889787i \(0.349147\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.8576 −0.844820 −0.422410 0.906405i \(-0.638816\pi\)
−0.422410 + 0.906405i \(0.638816\pi\)
\(198\) 0 0
\(199\) 5.50307 0.390102 0.195051 0.980793i \(-0.437513\pi\)
0.195051 + 0.980793i \(0.437513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 8.83710 0.620243
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −6.34017 −0.440672
\(208\) 0 0
\(209\) 23.5174 1.62674
\(210\) 0 0
\(211\) 19.1506 1.31838 0.659191 0.751975i \(-0.270899\pi\)
0.659191 + 0.751975i \(0.270899\pi\)
\(212\) 0 0
\(213\) −14.0989 −0.966040
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4.34017 −0.294630
\(218\) 0 0
\(219\) 11.1773 0.755290
\(220\) 0 0
\(221\) 4.68035 0.314834
\(222\) 0 0
\(223\) −12.3135 −0.824574 −0.412287 0.911054i \(-0.635270\pi\)
−0.412287 + 0.911054i \(0.635270\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.2039 1.00912 0.504560 0.863376i \(-0.331655\pi\)
0.504560 + 0.863376i \(0.331655\pi\)
\(228\) 0 0
\(229\) 5.20394 0.343886 0.171943 0.985107i \(-0.444996\pi\)
0.171943 + 0.985107i \(0.444996\pi\)
\(230\) 0 0
\(231\) −5.41855 −0.356514
\(232\) 0 0
\(233\) 11.6598 0.763861 0.381930 0.924191i \(-0.375259\pi\)
0.381930 + 0.924191i \(0.375259\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.680346 0.0441932
\(238\) 0 0
\(239\) −20.6225 −1.33396 −0.666979 0.745077i \(-0.732413\pi\)
−0.666979 + 0.745077i \(0.732413\pi\)
\(240\) 0 0
\(241\) −20.3545 −1.31115 −0.655576 0.755129i \(-0.727574\pi\)
−0.655576 + 0.755129i \(0.727574\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −18.8371 −1.19858
\(248\) 0 0
\(249\) −6.83710 −0.433284
\(250\) 0 0
\(251\) −10.5236 −0.664243 −0.332122 0.943237i \(-0.607764\pi\)
−0.332122 + 0.943237i \(0.607764\pi\)
\(252\) 0 0
\(253\) 34.3545 2.15985
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.8059 1.42259 0.711297 0.702892i \(-0.248108\pi\)
0.711297 + 0.702892i \(0.248108\pi\)
\(258\) 0 0
\(259\) 8.68035 0.539370
\(260\) 0 0
\(261\) −8.83710 −0.547003
\(262\) 0 0
\(263\) 28.0144 1.72744 0.863720 0.503972i \(-0.168128\pi\)
0.863720 + 0.503972i \(0.168128\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.49693 −0.397606
\(268\) 0 0
\(269\) 18.4969 1.12778 0.563889 0.825851i \(-0.309305\pi\)
0.563889 + 0.825851i \(0.309305\pi\)
\(270\) 0 0
\(271\) 29.0205 1.76287 0.881435 0.472304i \(-0.156578\pi\)
0.881435 + 0.472304i \(0.156578\pi\)
\(272\) 0 0
\(273\) 4.34017 0.262679
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −8.68035 −0.521551 −0.260776 0.965399i \(-0.583978\pi\)
−0.260776 + 0.965399i \(0.583978\pi\)
\(278\) 0 0
\(279\) 4.34017 0.259839
\(280\) 0 0
\(281\) 5.63317 0.336046 0.168023 0.985783i \(-0.446262\pi\)
0.168023 + 0.985783i \(0.446262\pi\)
\(282\) 0 0
\(283\) 2.47027 0.146842 0.0734210 0.997301i \(-0.476608\pi\)
0.0734210 + 0.997301i \(0.476608\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.34017 −0.492305
\(288\) 0 0
\(289\) −15.8371 −0.931594
\(290\) 0 0
\(291\) 10.4969 0.615341
\(292\) 0 0
\(293\) −7.60197 −0.444112 −0.222056 0.975034i \(-0.571277\pi\)
−0.222056 + 0.975034i \(0.571277\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.41855 0.314416
\(298\) 0 0
\(299\) −27.5174 −1.59138
\(300\) 0 0
\(301\) 6.15676 0.354869
\(302\) 0 0
\(303\) −18.8638 −1.08369
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −6.15676 −0.351385 −0.175692 0.984445i \(-0.556216\pi\)
−0.175692 + 0.984445i \(0.556216\pi\)
\(308\) 0 0
\(309\) 10.1568 0.577798
\(310\) 0 0
\(311\) 1.52973 0.0867432 0.0433716 0.999059i \(-0.486190\pi\)
0.0433716 + 0.999059i \(0.486190\pi\)
\(312\) 0 0
\(313\) 11.9733 0.676773 0.338387 0.941007i \(-0.390119\pi\)
0.338387 + 0.941007i \(0.390119\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.5441 1.32237 0.661184 0.750223i \(-0.270054\pi\)
0.661184 + 0.750223i \(0.270054\pi\)
\(318\) 0 0
\(319\) 47.8843 2.68101
\(320\) 0 0
\(321\) 14.6537 0.817889
\(322\) 0 0
\(323\) −4.68035 −0.260421
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 12.8371 0.709893
\(328\) 0 0
\(329\) −6.83710 −0.376942
\(330\) 0 0
\(331\) 9.16290 0.503638 0.251819 0.967774i \(-0.418971\pi\)
0.251819 + 0.967774i \(0.418971\pi\)
\(332\) 0 0
\(333\) −8.68035 −0.475680
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.0000 0.871576 0.435788 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) 0 0
\(339\) −1.50307 −0.0816356
\(340\) 0 0
\(341\) −23.5174 −1.27354
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 23.0205 1.23581 0.617903 0.786254i \(-0.287982\pi\)
0.617903 + 0.786254i \(0.287982\pi\)
\(348\) 0 0
\(349\) −3.78992 −0.202870 −0.101435 0.994842i \(-0.532343\pi\)
−0.101435 + 0.994842i \(0.532343\pi\)
\(350\) 0 0
\(351\) −4.34017 −0.231661
\(352\) 0 0
\(353\) 28.5958 1.52200 0.761001 0.648751i \(-0.224708\pi\)
0.761001 + 0.648751i \(0.224708\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.07838 0.0570738
\(358\) 0 0
\(359\) 11.2618 0.594375 0.297187 0.954819i \(-0.403951\pi\)
0.297187 + 0.954819i \(0.403951\pi\)
\(360\) 0 0
\(361\) −0.162899 −0.00857361
\(362\) 0 0
\(363\) −18.3607 −0.963686
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 25.3607 1.32382 0.661909 0.749584i \(-0.269747\pi\)
0.661909 + 0.749584i \(0.269747\pi\)
\(368\) 0 0
\(369\) 8.34017 0.434172
\(370\) 0 0
\(371\) −6.18342 −0.321027
\(372\) 0 0
\(373\) −21.3074 −1.10325 −0.551627 0.834091i \(-0.685993\pi\)
−0.551627 + 0.834091i \(0.685993\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −38.3545 −1.97536
\(378\) 0 0
\(379\) 1.84324 0.0946811 0.0473406 0.998879i \(-0.484925\pi\)
0.0473406 + 0.998879i \(0.484925\pi\)
\(380\) 0 0
\(381\) 19.2039 0.983847
\(382\) 0 0
\(383\) −4.99386 −0.255174 −0.127587 0.991827i \(-0.540723\pi\)
−0.127587 + 0.991827i \(0.540723\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −6.15676 −0.312965
\(388\) 0 0
\(389\) 16.8371 0.853675 0.426837 0.904328i \(-0.359628\pi\)
0.426837 + 0.904328i \(0.359628\pi\)
\(390\) 0 0
\(391\) −6.83710 −0.345767
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −36.8515 −1.84952 −0.924761 0.380548i \(-0.875736\pi\)
−0.924761 + 0.380548i \(0.875736\pi\)
\(398\) 0 0
\(399\) −4.34017 −0.217280
\(400\) 0 0
\(401\) −24.3545 −1.21621 −0.608104 0.793857i \(-0.708070\pi\)
−0.608104 + 0.793857i \(0.708070\pi\)
\(402\) 0 0
\(403\) 18.8371 0.938343
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 47.0349 2.33143
\(408\) 0 0
\(409\) −28.0410 −1.38654 −0.693270 0.720678i \(-0.743831\pi\)
−0.693270 + 0.720678i \(0.743831\pi\)
\(410\) 0 0
\(411\) −8.65368 −0.426855
\(412\) 0 0
\(413\) −6.83710 −0.336432
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −6.18342 −0.302803
\(418\) 0 0
\(419\) −0.482553 −0.0235742 −0.0117871 0.999931i \(-0.503752\pi\)
−0.0117871 + 0.999931i \(0.503752\pi\)
\(420\) 0 0
\(421\) 21.1506 1.03082 0.515409 0.856944i \(-0.327640\pi\)
0.515409 + 0.856944i \(0.327640\pi\)
\(422\) 0 0
\(423\) 6.83710 0.332431
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4.52359 0.218912
\(428\) 0 0
\(429\) 23.5174 1.13543
\(430\) 0 0
\(431\) 28.7382 1.38427 0.692135 0.721768i \(-0.256670\pi\)
0.692135 + 0.721768i \(0.256670\pi\)
\(432\) 0 0
\(433\) 9.02052 0.433498 0.216749 0.976227i \(-0.430455\pi\)
0.216749 + 0.976227i \(0.430455\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 27.5174 1.31634
\(438\) 0 0
\(439\) 17.8166 0.850339 0.425170 0.905114i \(-0.360214\pi\)
0.425170 + 0.905114i \(0.360214\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −11.7009 −0.555925 −0.277962 0.960592i \(-0.589659\pi\)
−0.277962 + 0.960592i \(0.589659\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 13.2039 0.624525
\(448\) 0 0
\(449\) 10.7337 0.506553 0.253277 0.967394i \(-0.418492\pi\)
0.253277 + 0.967394i \(0.418492\pi\)
\(450\) 0 0
\(451\) −45.1917 −2.12799
\(452\) 0 0
\(453\) −18.1568 −0.853079
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 20.9939 0.982051 0.491026 0.871145i \(-0.336622\pi\)
0.491026 + 0.871145i \(0.336622\pi\)
\(458\) 0 0
\(459\) −1.07838 −0.0503344
\(460\) 0 0
\(461\) −15.3751 −0.716088 −0.358044 0.933705i \(-0.616556\pi\)
−0.358044 + 0.933705i \(0.616556\pi\)
\(462\) 0 0
\(463\) 29.1917 1.35665 0.678326 0.734762i \(-0.262706\pi\)
0.678326 + 0.734762i \(0.262706\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −15.2039 −0.703554 −0.351777 0.936084i \(-0.614423\pi\)
−0.351777 + 0.936084i \(0.614423\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −15.1773 −0.699332
\(472\) 0 0
\(473\) 33.3607 1.53393
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.18342 0.283119
\(478\) 0 0
\(479\) 18.7838 0.858253 0.429126 0.903244i \(-0.358822\pi\)
0.429126 + 0.903244i \(0.358822\pi\)
\(480\) 0 0
\(481\) −37.6742 −1.71780
\(482\) 0 0
\(483\) −6.34017 −0.288488
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −16.5113 −0.748199 −0.374099 0.927389i \(-0.622048\pi\)
−0.374099 + 0.927389i \(0.622048\pi\)
\(488\) 0 0
\(489\) −2.83710 −0.128298
\(490\) 0 0
\(491\) −39.4063 −1.77838 −0.889190 0.457538i \(-0.848731\pi\)
−0.889190 + 0.457538i \(0.848731\pi\)
\(492\) 0 0
\(493\) −9.52973 −0.429198
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −14.0989 −0.632422
\(498\) 0 0
\(499\) 31.5174 1.41091 0.705457 0.708752i \(-0.250742\pi\)
0.705457 + 0.708752i \(0.250742\pi\)
\(500\) 0 0
\(501\) 13.3607 0.596912
\(502\) 0 0
\(503\) 20.3668 0.908112 0.454056 0.890973i \(-0.349977\pi\)
0.454056 + 0.890973i \(0.349977\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −5.83710 −0.259235
\(508\) 0 0
\(509\) 11.1773 0.495424 0.247712 0.968834i \(-0.420321\pi\)
0.247712 + 0.968834i \(0.420321\pi\)
\(510\) 0 0
\(511\) 11.1773 0.494454
\(512\) 0 0
\(513\) 4.34017 0.191623
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −37.0472 −1.62933
\(518\) 0 0
\(519\) −2.55479 −0.112143
\(520\) 0 0
\(521\) 32.6537 1.43058 0.715292 0.698826i \(-0.246294\pi\)
0.715292 + 0.698826i \(0.246294\pi\)
\(522\) 0 0
\(523\) −15.6865 −0.685922 −0.342961 0.939350i \(-0.611430\pi\)
−0.342961 + 0.939350i \(0.611430\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.68035 0.203879
\(528\) 0 0
\(529\) 17.1978 0.747730
\(530\) 0 0
\(531\) 6.83710 0.296705
\(532\) 0 0
\(533\) 36.1978 1.56790
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −11.9421 −0.515341
\(538\) 0 0
\(539\) −5.41855 −0.233394
\(540\) 0 0
\(541\) 30.1978 1.29830 0.649152 0.760658i \(-0.275124\pi\)
0.649152 + 0.760658i \(0.275124\pi\)
\(542\) 0 0
\(543\) −4.15676 −0.178383
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −9.36069 −0.400234 −0.200117 0.979772i \(-0.564132\pi\)
−0.200117 + 0.979772i \(0.564132\pi\)
\(548\) 0 0
\(549\) −4.52359 −0.193062
\(550\) 0 0
\(551\) 38.3545 1.63396
\(552\) 0 0
\(553\) 0.680346 0.0289313
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7.49079 −0.317395 −0.158697 0.987327i \(-0.550729\pi\)
−0.158697 + 0.987327i \(0.550729\pi\)
\(558\) 0 0
\(559\) −26.7214 −1.13019
\(560\) 0 0
\(561\) 5.84324 0.246702
\(562\) 0 0
\(563\) −31.7152 −1.33664 −0.668319 0.743875i \(-0.732986\pi\)
−0.668319 + 0.743875i \(0.732986\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 30.6803 1.28619 0.643094 0.765788i \(-0.277651\pi\)
0.643094 + 0.765788i \(0.277651\pi\)
\(570\) 0 0
\(571\) 10.6393 0.445241 0.222621 0.974905i \(-0.428539\pi\)
0.222621 + 0.974905i \(0.428539\pi\)
\(572\) 0 0
\(573\) 6.09890 0.254785
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 30.0677 1.25173 0.625867 0.779930i \(-0.284745\pi\)
0.625867 + 0.779930i \(0.284745\pi\)
\(578\) 0 0
\(579\) −12.6803 −0.526977
\(580\) 0 0
\(581\) −6.83710 −0.283651
\(582\) 0 0
\(583\) −33.5052 −1.38764
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −10.0410 −0.414438 −0.207219 0.978295i \(-0.566441\pi\)
−0.207219 + 0.978295i \(0.566441\pi\)
\(588\) 0 0
\(589\) −18.8371 −0.776169
\(590\) 0 0
\(591\) 11.8576 0.487757
\(592\) 0 0
\(593\) −24.2823 −0.997155 −0.498578 0.866845i \(-0.666144\pi\)
−0.498578 + 0.866845i \(0.666144\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5.50307 −0.225226
\(598\) 0 0
\(599\) 9.90110 0.404548 0.202274 0.979329i \(-0.435167\pi\)
0.202274 + 0.979329i \(0.435167\pi\)
\(600\) 0 0
\(601\) 17.6865 0.721447 0.360723 0.932673i \(-0.382530\pi\)
0.360723 + 0.932673i \(0.382530\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −28.3135 −1.14921 −0.574605 0.818431i \(-0.694844\pi\)
−0.574605 + 0.818431i \(0.694844\pi\)
\(608\) 0 0
\(609\) −8.83710 −0.358097
\(610\) 0 0
\(611\) 29.6742 1.20049
\(612\) 0 0
\(613\) −23.6865 −0.956688 −0.478344 0.878172i \(-0.658763\pi\)
−0.478344 + 0.878172i \(0.658763\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.7009 −0.551576 −0.275788 0.961218i \(-0.588939\pi\)
−0.275788 + 0.961218i \(0.588939\pi\)
\(618\) 0 0
\(619\) 2.49693 0.100360 0.0501800 0.998740i \(-0.484020\pi\)
0.0501800 + 0.998740i \(0.484020\pi\)
\(620\) 0 0
\(621\) 6.34017 0.254422
\(622\) 0 0
\(623\) −6.49693 −0.260294
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −23.5174 −0.939196
\(628\) 0 0
\(629\) −9.36069 −0.373235
\(630\) 0 0
\(631\) −8.68035 −0.345559 −0.172780 0.984961i \(-0.555275\pi\)
−0.172780 + 0.984961i \(0.555275\pi\)
\(632\) 0 0
\(633\) −19.1506 −0.761169
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4.34017 0.171964
\(638\) 0 0
\(639\) 14.0989 0.557744
\(640\) 0 0
\(641\) 3.30737 0.130633 0.0653166 0.997865i \(-0.479194\pi\)
0.0653166 + 0.997865i \(0.479194\pi\)
\(642\) 0 0
\(643\) 6.15676 0.242799 0.121399 0.992604i \(-0.461262\pi\)
0.121399 + 0.992604i \(0.461262\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.95282 0.273344 0.136672 0.990616i \(-0.456359\pi\)
0.136672 + 0.990616i \(0.456359\pi\)
\(648\) 0 0
\(649\) −37.0472 −1.45423
\(650\) 0 0
\(651\) 4.34017 0.170105
\(652\) 0 0
\(653\) −38.7480 −1.51633 −0.758164 0.652064i \(-0.773903\pi\)
−0.758164 + 0.652064i \(0.773903\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −11.1773 −0.436067
\(658\) 0 0
\(659\) −9.22076 −0.359190 −0.179595 0.983741i \(-0.557479\pi\)
−0.179595 + 0.983741i \(0.557479\pi\)
\(660\) 0 0
\(661\) 25.8843 1.00678 0.503391 0.864059i \(-0.332086\pi\)
0.503391 + 0.864059i \(0.332086\pi\)
\(662\) 0 0
\(663\) −4.68035 −0.181770
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 56.0288 2.16944
\(668\) 0 0
\(669\) 12.3135 0.476068
\(670\) 0 0
\(671\) 24.5113 0.946248
\(672\) 0 0
\(673\) −40.0821 −1.54505 −0.772525 0.634984i \(-0.781007\pi\)
−0.772525 + 0.634984i \(0.781007\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 33.5897 1.29096 0.645478 0.763779i \(-0.276658\pi\)
0.645478 + 0.763779i \(0.276658\pi\)
\(678\) 0 0
\(679\) 10.4969 0.402835
\(680\) 0 0
\(681\) −15.2039 −0.582616
\(682\) 0 0
\(683\) 18.7070 0.715804 0.357902 0.933759i \(-0.383492\pi\)
0.357902 + 0.933759i \(0.383492\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −5.20394 −0.198543
\(688\) 0 0
\(689\) 26.8371 1.02241
\(690\) 0 0
\(691\) 19.1773 0.729538 0.364769 0.931098i \(-0.381148\pi\)
0.364769 + 0.931098i \(0.381148\pi\)
\(692\) 0 0
\(693\) 5.41855 0.205834
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 8.99386 0.340667
\(698\) 0 0
\(699\) −11.6598 −0.441015
\(700\) 0 0
\(701\) 21.4641 0.810689 0.405344 0.914164i \(-0.367152\pi\)
0.405344 + 0.914164i \(0.367152\pi\)
\(702\) 0 0
\(703\) 37.6742 1.42091
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −18.8638 −0.709445
\(708\) 0 0
\(709\) 2.62702 0.0986599 0.0493299 0.998783i \(-0.484291\pi\)
0.0493299 + 0.998783i \(0.484291\pi\)
\(710\) 0 0
\(711\) −0.680346 −0.0255150
\(712\) 0 0
\(713\) −27.5174 −1.03054
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 20.6225 0.770161
\(718\) 0 0
\(719\) 4.36683 0.162855 0.0814277 0.996679i \(-0.474052\pi\)
0.0814277 + 0.996679i \(0.474052\pi\)
\(720\) 0 0
\(721\) 10.1568 0.378257
\(722\) 0 0
\(723\) 20.3545 0.756994
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 28.1445 1.04382 0.521910 0.853000i \(-0.325220\pi\)
0.521910 + 0.853000i \(0.325220\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.63931 −0.245564
\(732\) 0 0
\(733\) 26.7480 0.987962 0.493981 0.869473i \(-0.335541\pi\)
0.493981 + 0.869473i \(0.335541\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −19.0349 −0.700210 −0.350105 0.936710i \(-0.613854\pi\)
−0.350105 + 0.936710i \(0.613854\pi\)
\(740\) 0 0
\(741\) 18.8371 0.691998
\(742\) 0 0
\(743\) 33.6598 1.23486 0.617430 0.786626i \(-0.288174\pi\)
0.617430 + 0.786626i \(0.288174\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6.83710 0.250156
\(748\) 0 0
\(749\) 14.6537 0.535434
\(750\) 0 0
\(751\) 49.8720 1.81985 0.909927 0.414767i \(-0.136137\pi\)
0.909927 + 0.414767i \(0.136137\pi\)
\(752\) 0 0
\(753\) 10.5236 0.383501
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 7.63317 0.277432 0.138716 0.990332i \(-0.455702\pi\)
0.138716 + 0.990332i \(0.455702\pi\)
\(758\) 0 0
\(759\) −34.3545 −1.24699
\(760\) 0 0
\(761\) 19.5974 0.710406 0.355203 0.934789i \(-0.384412\pi\)
0.355203 + 0.934789i \(0.384412\pi\)
\(762\) 0 0
\(763\) 12.8371 0.464734
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 29.6742 1.07147
\(768\) 0 0
\(769\) −32.4079 −1.16866 −0.584329 0.811517i \(-0.698642\pi\)
−0.584329 + 0.811517i \(0.698642\pi\)
\(770\) 0 0
\(771\) −22.8059 −0.821335
\(772\) 0 0
\(773\) 12.0845 0.434650 0.217325 0.976099i \(-0.430267\pi\)
0.217325 + 0.976099i \(0.430267\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −8.68035 −0.311406
\(778\) 0 0
\(779\) −36.1978 −1.29692
\(780\) 0 0
\(781\) −76.3956 −2.73365
\(782\) 0 0
\(783\) 8.83710 0.315812
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 25.0472 0.892836 0.446418 0.894825i \(-0.352700\pi\)
0.446418 + 0.894825i \(0.352700\pi\)
\(788\) 0 0
\(789\) −28.0144 −0.997338
\(790\) 0 0
\(791\) −1.50307 −0.0534431
\(792\) 0 0
\(793\) −19.6332 −0.697194
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.2762 0.753641 0.376820 0.926286i \(-0.377017\pi\)
0.376820 + 0.926286i \(0.377017\pi\)
\(798\) 0 0
\(799\) 7.37298 0.260837
\(800\) 0 0
\(801\) 6.49693 0.229558
\(802\) 0 0
\(803\) 60.5646 2.13728
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −18.4969 −0.651123
\(808\) 0 0
\(809\) −49.6619 −1.74602 −0.873010 0.487702i \(-0.837835\pi\)
−0.873010 + 0.487702i \(0.837835\pi\)
\(810\) 0 0
\(811\) −25.8166 −0.906543 −0.453271 0.891373i \(-0.649743\pi\)
−0.453271 + 0.891373i \(0.649743\pi\)
\(812\) 0 0
\(813\) −29.0205 −1.01779
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 26.7214 0.934863
\(818\) 0 0
\(819\) −4.34017 −0.151658
\(820\) 0 0
\(821\) 24.8904 0.868682 0.434341 0.900749i \(-0.356981\pi\)
0.434341 + 0.900749i \(0.356981\pi\)
\(822\) 0 0
\(823\) 18.4703 0.643833 0.321917 0.946768i \(-0.395673\pi\)
0.321917 + 0.946768i \(0.395673\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −21.6886 −0.754186 −0.377093 0.926175i \(-0.623076\pi\)
−0.377093 + 0.926175i \(0.623076\pi\)
\(828\) 0 0
\(829\) 47.5052 1.64992 0.824961 0.565189i \(-0.191197\pi\)
0.824961 + 0.565189i \(0.191197\pi\)
\(830\) 0 0
\(831\) 8.68035 0.301118
\(832\) 0 0
\(833\) 1.07838 0.0373636
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.34017 −0.150018
\(838\) 0 0
\(839\) −11.4641 −0.395785 −0.197893 0.980224i \(-0.563410\pi\)
−0.197893 + 0.980224i \(0.563410\pi\)
\(840\) 0 0
\(841\) 49.0944 1.69291
\(842\) 0 0
\(843\) −5.63317 −0.194017
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −18.3607 −0.630881
\(848\) 0 0
\(849\) −2.47027 −0.0847793
\(850\) 0 0
\(851\) 55.0349 1.88657
\(852\) 0 0
\(853\) −39.8043 −1.36287 −0.681437 0.731877i \(-0.738644\pi\)
−0.681437 + 0.731877i \(0.738644\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −16.9627 −0.579433 −0.289717 0.957112i \(-0.593561\pi\)
−0.289717 + 0.957112i \(0.593561\pi\)
\(858\) 0 0
\(859\) 41.0493 1.40058 0.700292 0.713857i \(-0.253053\pi\)
0.700292 + 0.713857i \(0.253053\pi\)
\(860\) 0 0
\(861\) 8.34017 0.284232
\(862\) 0 0
\(863\) 50.0554 1.70391 0.851953 0.523618i \(-0.175418\pi\)
0.851953 + 0.523618i \(0.175418\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 15.8371 0.537856
\(868\) 0 0
\(869\) 3.68649 0.125056
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −10.4969 −0.355267
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6.32580 −0.213607 −0.106803 0.994280i \(-0.534062\pi\)
−0.106803 + 0.994280i \(0.534062\pi\)
\(878\) 0 0
\(879\) 7.60197 0.256408
\(880\) 0 0
\(881\) 20.5380 0.691942 0.345971 0.938245i \(-0.387550\pi\)
0.345971 + 0.938245i \(0.387550\pi\)
\(882\) 0 0
\(883\) −5.30737 −0.178607 −0.0893036 0.996004i \(-0.528464\pi\)
−0.0893036 + 0.996004i \(0.528464\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.15061 −0.105787 −0.0528936 0.998600i \(-0.516844\pi\)
−0.0528936 + 0.998600i \(0.516844\pi\)
\(888\) 0 0
\(889\) 19.2039 0.644079
\(890\) 0 0
\(891\) −5.41855 −0.181528
\(892\) 0 0
\(893\) −29.6742 −0.993009
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 27.5174 0.918781
\(898\) 0 0
\(899\) −38.3545 −1.27920
\(900\) 0 0
\(901\) 6.66806 0.222145
\(902\) 0 0
\(903\) −6.15676 −0.204884
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 11.9467 0.396683 0.198341 0.980133i \(-0.436445\pi\)
0.198341 + 0.980133i \(0.436445\pi\)
\(908\) 0 0
\(909\) 18.8638 0.625672
\(910\) 0 0
\(911\) −46.0989 −1.52732 −0.763662 0.645616i \(-0.776601\pi\)
−0.763662 + 0.645616i \(0.776601\pi\)
\(912\) 0 0
\(913\) −37.0472 −1.22608
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) −7.25726 −0.239395 −0.119697 0.992810i \(-0.538192\pi\)
−0.119697 + 0.992810i \(0.538192\pi\)
\(920\) 0 0
\(921\) 6.15676 0.202872
\(922\) 0 0
\(923\) 61.1917 2.01415
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −10.1568 −0.333592
\(928\) 0 0
\(929\) −44.9048 −1.47328 −0.736639 0.676286i \(-0.763588\pi\)
−0.736639 + 0.676286i \(0.763588\pi\)
\(930\) 0 0
\(931\) −4.34017 −0.142243
\(932\) 0 0
\(933\) −1.52973 −0.0500812
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 43.6886 1.42724 0.713622 0.700531i \(-0.247054\pi\)
0.713622 + 0.700531i \(0.247054\pi\)
\(938\) 0 0
\(939\) −11.9733 −0.390735
\(940\) 0 0
\(941\) 14.6660 0.478097 0.239048 0.971008i \(-0.423165\pi\)
0.239048 + 0.971008i \(0.423165\pi\)
\(942\) 0 0
\(943\) −52.8781 −1.72195
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.96719 0.0964209 0.0482104 0.998837i \(-0.484648\pi\)
0.0482104 + 0.998837i \(0.484648\pi\)
\(948\) 0 0
\(949\) −48.5113 −1.57474
\(950\) 0 0
\(951\) −23.5441 −0.763470
\(952\) 0 0
\(953\) 45.5851 1.47665 0.738324 0.674446i \(-0.235618\pi\)
0.738324 + 0.674446i \(0.235618\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −47.8843 −1.54788
\(958\) 0 0
\(959\) −8.65368 −0.279442
\(960\) 0 0
\(961\) −12.1629 −0.392352
\(962\) 0 0
\(963\) −14.6537 −0.472208
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 32.3668 1.04085 0.520424 0.853908i \(-0.325774\pi\)
0.520424 + 0.853908i \(0.325774\pi\)
\(968\) 0 0
\(969\) 4.68035 0.150354
\(970\) 0 0
\(971\) 11.3197 0.363265 0.181632 0.983366i \(-0.441862\pi\)
0.181632 + 0.983366i \(0.441862\pi\)
\(972\) 0 0
\(973\) −6.18342 −0.198231
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −39.6886 −1.26975 −0.634875 0.772615i \(-0.718948\pi\)
−0.634875 + 0.772615i \(0.718948\pi\)
\(978\) 0 0
\(979\) −35.2039 −1.12512
\(980\) 0 0
\(981\) −12.8371 −0.409857
\(982\) 0 0
\(983\) 43.5174 1.38799 0.693996 0.719979i \(-0.255849\pi\)
0.693996 + 0.719979i \(0.255849\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.83710 0.217627
\(988\) 0 0
\(989\) 39.0349 1.24124
\(990\) 0 0
\(991\) 3.80221 0.120781 0.0603905 0.998175i \(-0.480765\pi\)
0.0603905 + 0.998175i \(0.480765\pi\)
\(992\) 0 0
\(993\) −9.16290 −0.290776
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −23.8043 −0.753890 −0.376945 0.926236i \(-0.623025\pi\)
−0.376945 + 0.926236i \(0.623025\pi\)
\(998\) 0 0
\(999\) 8.68035 0.274634
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 8400.2.a.dh.1.1 3
4.3 odd 2 4200.2.a.bq.1.3 3
5.2 odd 4 1680.2.t.i.1009.4 6
5.3 odd 4 1680.2.t.i.1009.1 6
5.4 even 2 8400.2.a.dk.1.1 3
15.2 even 4 5040.2.t.ba.1009.5 6
15.8 even 4 5040.2.t.ba.1009.6 6
20.3 even 4 840.2.t.e.169.4 yes 6
20.7 even 4 840.2.t.e.169.1 6
20.19 odd 2 4200.2.a.bo.1.3 3
60.23 odd 4 2520.2.t.j.1009.6 6
60.47 odd 4 2520.2.t.j.1009.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.e.169.1 6 20.7 even 4
840.2.t.e.169.4 yes 6 20.3 even 4
1680.2.t.i.1009.1 6 5.3 odd 4
1680.2.t.i.1009.4 6 5.2 odd 4
2520.2.t.j.1009.5 6 60.47 odd 4
2520.2.t.j.1009.6 6 60.23 odd 4
4200.2.a.bo.1.3 3 20.19 odd 2
4200.2.a.bq.1.3 3 4.3 odd 2
5040.2.t.ba.1009.5 6 15.2 even 4
5040.2.t.ba.1009.6 6 15.8 even 4
8400.2.a.dh.1.1 3 1.1 even 1 trivial
8400.2.a.dk.1.1 3 5.4 even 2