Properties

Label 7448.2.a.bn.1.5
Level $7448$
Weight $2$
Character 7448.1
Self dual yes
Analytic conductor $59.473$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.4725794254\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.98211824.1
Defining polynomial: \(x^{6} - 3 x^{5} - 6 x^{4} + 15 x^{3} + 8 x^{2} - 9 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.04143\) of defining polynomial
Character \(\chi\) \(=\) 7448.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.04143 q^{3} +3.17676 q^{5} +1.16742 q^{9} +O(q^{10})\) \(q+2.04143 q^{3} +3.17676 q^{5} +1.16742 q^{9} +2.64156 q^{11} +5.60178 q^{13} +6.48513 q^{15} +7.48513 q^{17} -1.00000 q^{19} +4.87939 q^{23} +5.09182 q^{25} -3.74108 q^{27} +2.60909 q^{29} -3.41086 q^{31} +5.39255 q^{33} -9.05242 q^{37} +11.4356 q^{39} -10.4198 q^{41} +3.77689 q^{43} +3.70862 q^{45} +7.21782 q^{47} +15.2803 q^{51} -9.74609 q^{53} +8.39160 q^{55} -2.04143 q^{57} +3.30969 q^{59} -2.35314 q^{61} +17.7955 q^{65} +13.8674 q^{67} +9.96091 q^{69} +8.57828 q^{71} -14.1861 q^{73} +10.3946 q^{75} -16.8431 q^{79} -11.1394 q^{81} -6.51760 q^{83} +23.7785 q^{85} +5.32626 q^{87} -12.1886 q^{89} -6.96302 q^{93} -3.17676 q^{95} -7.06695 q^{97} +3.08381 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + q^{5} + 3 q^{9} + O(q^{10}) \) \( 6 q + 3 q^{3} + q^{5} + 3 q^{9} + 3 q^{11} + 6 q^{13} - 8 q^{15} - 2 q^{17} - 6 q^{19} + 2 q^{23} + 5 q^{25} + 6 q^{27} - q^{29} + 14 q^{31} + 19 q^{33} - 7 q^{37} + 22 q^{39} - 21 q^{41} + q^{43} - 4 q^{45} + 23 q^{47} + 2 q^{51} - 15 q^{53} + 2 q^{55} - 3 q^{57} + 25 q^{59} + 21 q^{61} + 6 q^{65} + 2 q^{67} - 20 q^{69} + 13 q^{71} - 2 q^{73} + 24 q^{75} - 17 q^{79} - 2 q^{81} + 4 q^{83} + 40 q^{85} + 30 q^{87} - q^{89} + 6 q^{93} - q^{95} - 13 q^{97} + 41 q^{99} + O(q^{100}) \)

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
\(3\) 2.04143 1.17862 0.589309 0.807908i \(-0.299400\pi\)
0.589309 + 0.807908i \(0.299400\pi\)
\(4\) 0 0
\(5\) 3.17676 1.42069 0.710346 0.703853i \(-0.248539\pi\)
0.710346 + 0.703853i \(0.248539\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.16742 0.389140
\(10\) 0 0
\(11\) 2.64156 0.796460 0.398230 0.917286i \(-0.369625\pi\)
0.398230 + 0.917286i \(0.369625\pi\)
\(12\) 0 0
\(13\) 5.60178 1.55365 0.776827 0.629714i \(-0.216828\pi\)
0.776827 + 0.629714i \(0.216828\pi\)
\(14\) 0 0
\(15\) 6.48513 1.67445
\(16\) 0 0
\(17\) 7.48513 1.81541 0.907705 0.419609i \(-0.137833\pi\)
0.907705 + 0.419609i \(0.137833\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.87939 1.01742 0.508711 0.860937i \(-0.330122\pi\)
0.508711 + 0.860937i \(0.330122\pi\)
\(24\) 0 0
\(25\) 5.09182 1.01836
\(26\) 0 0
\(27\) −3.74108 −0.719970
\(28\) 0 0
\(29\) 2.60909 0.484495 0.242248 0.970214i \(-0.422115\pi\)
0.242248 + 0.970214i \(0.422115\pi\)
\(30\) 0 0
\(31\) −3.41086 −0.612608 −0.306304 0.951934i \(-0.599092\pi\)
−0.306304 + 0.951934i \(0.599092\pi\)
\(32\) 0 0
\(33\) 5.39255 0.938722
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.05242 −1.48821 −0.744104 0.668064i \(-0.767123\pi\)
−0.744104 + 0.668064i \(0.767123\pi\)
\(38\) 0 0
\(39\) 11.4356 1.83116
\(40\) 0 0
\(41\) −10.4198 −1.62730 −0.813652 0.581352i \(-0.802524\pi\)
−0.813652 + 0.581352i \(0.802524\pi\)
\(42\) 0 0
\(43\) 3.77689 0.575971 0.287986 0.957635i \(-0.407014\pi\)
0.287986 + 0.957635i \(0.407014\pi\)
\(44\) 0 0
\(45\) 3.70862 0.552848
\(46\) 0 0
\(47\) 7.21782 1.05283 0.526413 0.850229i \(-0.323536\pi\)
0.526413 + 0.850229i \(0.323536\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 15.2803 2.13967
\(52\) 0 0
\(53\) −9.74609 −1.33873 −0.669364 0.742935i \(-0.733433\pi\)
−0.669364 + 0.742935i \(0.733433\pi\)
\(54\) 0 0
\(55\) 8.39160 1.13152
\(56\) 0 0
\(57\) −2.04143 −0.270394
\(58\) 0 0
\(59\) 3.30969 0.430885 0.215443 0.976516i \(-0.430881\pi\)
0.215443 + 0.976516i \(0.430881\pi\)
\(60\) 0 0
\(61\) −2.35314 −0.301289 −0.150644 0.988588i \(-0.548135\pi\)
−0.150644 + 0.988588i \(0.548135\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 17.7955 2.20726
\(66\) 0 0
\(67\) 13.8674 1.69417 0.847086 0.531457i \(-0.178355\pi\)
0.847086 + 0.531457i \(0.178355\pi\)
\(68\) 0 0
\(69\) 9.96091 1.19915
\(70\) 0 0
\(71\) 8.57828 1.01805 0.509027 0.860750i \(-0.330005\pi\)
0.509027 + 0.860750i \(0.330005\pi\)
\(72\) 0 0
\(73\) −14.1861 −1.66035 −0.830176 0.557502i \(-0.811760\pi\)
−0.830176 + 0.557502i \(0.811760\pi\)
\(74\) 0 0
\(75\) 10.3946 1.20026
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −16.8431 −1.89500 −0.947500 0.319756i \(-0.896399\pi\)
−0.947500 + 0.319756i \(0.896399\pi\)
\(80\) 0 0
\(81\) −11.1394 −1.23771
\(82\) 0 0
\(83\) −6.51760 −0.715399 −0.357700 0.933837i \(-0.616439\pi\)
−0.357700 + 0.933837i \(0.616439\pi\)
\(84\) 0 0
\(85\) 23.7785 2.57914
\(86\) 0 0
\(87\) 5.32626 0.571035
\(88\) 0 0
\(89\) −12.1886 −1.29199 −0.645997 0.763340i \(-0.723558\pi\)
−0.645997 + 0.763340i \(0.723558\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −6.96302 −0.722031
\(94\) 0 0
\(95\) −3.17676 −0.325929
\(96\) 0 0
\(97\) −7.06695 −0.717540 −0.358770 0.933426i \(-0.616804\pi\)
−0.358770 + 0.933426i \(0.616804\pi\)
\(98\) 0 0
\(99\) 3.08381 0.309935
\(100\) 0 0
\(101\) −4.33427 −0.431276 −0.215638 0.976473i \(-0.569183\pi\)
−0.215638 + 0.976473i \(0.569183\pi\)
\(102\) 0 0
\(103\) −12.9215 −1.27320 −0.636598 0.771196i \(-0.719659\pi\)
−0.636598 + 0.771196i \(0.719659\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.63745 −0.254972 −0.127486 0.991840i \(-0.540691\pi\)
−0.127486 + 0.991840i \(0.540691\pi\)
\(108\) 0 0
\(109\) −8.84895 −0.847576 −0.423788 0.905761i \(-0.639300\pi\)
−0.423788 + 0.905761i \(0.639300\pi\)
\(110\) 0 0
\(111\) −18.4798 −1.75403
\(112\) 0 0
\(113\) 9.47520 0.891352 0.445676 0.895194i \(-0.352963\pi\)
0.445676 + 0.895194i \(0.352963\pi\)
\(114\) 0 0
\(115\) 15.5007 1.44544
\(116\) 0 0
\(117\) 6.53963 0.604589
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −4.02217 −0.365652
\(122\) 0 0
\(123\) −21.2713 −1.91797
\(124\) 0 0
\(125\) 0.291699 0.0260904
\(126\) 0 0
\(127\) −16.3048 −1.44682 −0.723410 0.690419i \(-0.757426\pi\)
−0.723410 + 0.690419i \(0.757426\pi\)
\(128\) 0 0
\(129\) 7.71025 0.678850
\(130\) 0 0
\(131\) 12.8724 1.12467 0.562335 0.826909i \(-0.309903\pi\)
0.562335 + 0.826909i \(0.309903\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −11.8845 −1.02286
\(136\) 0 0
\(137\) −4.67817 −0.399683 −0.199841 0.979828i \(-0.564043\pi\)
−0.199841 + 0.979828i \(0.564043\pi\)
\(138\) 0 0
\(139\) −3.18023 −0.269744 −0.134872 0.990863i \(-0.543062\pi\)
−0.134872 + 0.990863i \(0.543062\pi\)
\(140\) 0 0
\(141\) 14.7346 1.24088
\(142\) 0 0
\(143\) 14.7974 1.23742
\(144\) 0 0
\(145\) 8.28845 0.688318
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.273874 0.0224366 0.0112183 0.999937i \(-0.496429\pi\)
0.0112183 + 0.999937i \(0.496429\pi\)
\(150\) 0 0
\(151\) 22.1226 1.80032 0.900158 0.435564i \(-0.143451\pi\)
0.900158 + 0.435564i \(0.143451\pi\)
\(152\) 0 0
\(153\) 8.73829 0.706449
\(154\) 0 0
\(155\) −10.8355 −0.870327
\(156\) 0 0
\(157\) 1.37859 0.110024 0.0550119 0.998486i \(-0.482480\pi\)
0.0550119 + 0.998486i \(0.482480\pi\)
\(158\) 0 0
\(159\) −19.8959 −1.57785
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.05078 0.160630 0.0803149 0.996770i \(-0.474407\pi\)
0.0803149 + 0.996770i \(0.474407\pi\)
\(164\) 0 0
\(165\) 17.1308 1.33363
\(166\) 0 0
\(167\) 3.51297 0.271842 0.135921 0.990720i \(-0.456601\pi\)
0.135921 + 0.990720i \(0.456601\pi\)
\(168\) 0 0
\(169\) 18.3799 1.41384
\(170\) 0 0
\(171\) −1.16742 −0.0892749
\(172\) 0 0
\(173\) 23.5896 1.79349 0.896744 0.442551i \(-0.145926\pi\)
0.896744 + 0.442551i \(0.145926\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.75649 0.507849
\(178\) 0 0
\(179\) 8.93373 0.667738 0.333869 0.942619i \(-0.391646\pi\)
0.333869 + 0.942619i \(0.391646\pi\)
\(180\) 0 0
\(181\) −25.1244 −1.86748 −0.933741 0.357950i \(-0.883476\pi\)
−0.933741 + 0.357950i \(0.883476\pi\)
\(182\) 0 0
\(183\) −4.80376 −0.355104
\(184\) 0 0
\(185\) −28.7574 −2.11428
\(186\) 0 0
\(187\) 19.7724 1.44590
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.3297 1.32629 0.663147 0.748489i \(-0.269220\pi\)
0.663147 + 0.748489i \(0.269220\pi\)
\(192\) 0 0
\(193\) −0.833709 −0.0600117 −0.0300059 0.999550i \(-0.509553\pi\)
−0.0300059 + 0.999550i \(0.509553\pi\)
\(194\) 0 0
\(195\) 36.3283 2.60152
\(196\) 0 0
\(197\) 4.66174 0.332135 0.166068 0.986114i \(-0.446893\pi\)
0.166068 + 0.986114i \(0.446893\pi\)
\(198\) 0 0
\(199\) −14.3836 −1.01963 −0.509814 0.860285i \(-0.670286\pi\)
−0.509814 + 0.860285i \(0.670286\pi\)
\(200\) 0 0
\(201\) 28.3092 1.99678
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −33.1013 −2.31190
\(206\) 0 0
\(207\) 5.69630 0.395920
\(208\) 0 0
\(209\) −2.64156 −0.182720
\(210\) 0 0
\(211\) −21.9757 −1.51287 −0.756436 0.654067i \(-0.773061\pi\)
−0.756436 + 0.654067i \(0.773061\pi\)
\(212\) 0 0
\(213\) 17.5119 1.19990
\(214\) 0 0
\(215\) 11.9983 0.818277
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −28.9598 −1.95692
\(220\) 0 0
\(221\) 41.9300 2.82052
\(222\) 0 0
\(223\) 2.44849 0.163963 0.0819814 0.996634i \(-0.473875\pi\)
0.0819814 + 0.996634i \(0.473875\pi\)
\(224\) 0 0
\(225\) 5.94430 0.396287
\(226\) 0 0
\(227\) 8.70936 0.578061 0.289030 0.957320i \(-0.406667\pi\)
0.289030 + 0.957320i \(0.406667\pi\)
\(228\) 0 0
\(229\) 18.2921 1.20878 0.604388 0.796690i \(-0.293418\pi\)
0.604388 + 0.796690i \(0.293418\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.96121 −0.325020 −0.162510 0.986707i \(-0.551959\pi\)
−0.162510 + 0.986707i \(0.551959\pi\)
\(234\) 0 0
\(235\) 22.9293 1.49574
\(236\) 0 0
\(237\) −34.3840 −2.23348
\(238\) 0 0
\(239\) −4.42911 −0.286495 −0.143248 0.989687i \(-0.545755\pi\)
−0.143248 + 0.989687i \(0.545755\pi\)
\(240\) 0 0
\(241\) 17.8698 1.15109 0.575547 0.817769i \(-0.304789\pi\)
0.575547 + 0.817769i \(0.304789\pi\)
\(242\) 0 0
\(243\) −11.5170 −0.738817
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −5.60178 −0.356433
\(248\) 0 0
\(249\) −13.3052 −0.843183
\(250\) 0 0
\(251\) −11.4381 −0.721968 −0.360984 0.932572i \(-0.617559\pi\)
−0.360984 + 0.932572i \(0.617559\pi\)
\(252\) 0 0
\(253\) 12.8892 0.810336
\(254\) 0 0
\(255\) 48.5420 3.03982
\(256\) 0 0
\(257\) −2.18402 −0.136235 −0.0681176 0.997677i \(-0.521699\pi\)
−0.0681176 + 0.997677i \(0.521699\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 3.04590 0.188537
\(262\) 0 0
\(263\) 19.3509 1.19322 0.596612 0.802530i \(-0.296513\pi\)
0.596612 + 0.802530i \(0.296513\pi\)
\(264\) 0 0
\(265\) −30.9610 −1.90192
\(266\) 0 0
\(267\) −24.8822 −1.52277
\(268\) 0 0
\(269\) −11.5091 −0.701725 −0.350862 0.936427i \(-0.614112\pi\)
−0.350862 + 0.936427i \(0.614112\pi\)
\(270\) 0 0
\(271\) 7.84158 0.476342 0.238171 0.971223i \(-0.423452\pi\)
0.238171 + 0.971223i \(0.423452\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 13.4503 0.811086
\(276\) 0 0
\(277\) −12.3647 −0.742925 −0.371463 0.928448i \(-0.621144\pi\)
−0.371463 + 0.928448i \(0.621144\pi\)
\(278\) 0 0
\(279\) −3.98191 −0.238391
\(280\) 0 0
\(281\) 7.02949 0.419344 0.209672 0.977772i \(-0.432760\pi\)
0.209672 + 0.977772i \(0.432760\pi\)
\(282\) 0 0
\(283\) 31.3214 1.86186 0.930931 0.365195i \(-0.118998\pi\)
0.930931 + 0.365195i \(0.118998\pi\)
\(284\) 0 0
\(285\) −6.48513 −0.384146
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 39.0271 2.29571
\(290\) 0 0
\(291\) −14.4266 −0.845705
\(292\) 0 0
\(293\) −11.0475 −0.645402 −0.322701 0.946501i \(-0.604591\pi\)
−0.322701 + 0.946501i \(0.604591\pi\)
\(294\) 0 0
\(295\) 10.5141 0.612155
\(296\) 0 0
\(297\) −9.88227 −0.573427
\(298\) 0 0
\(299\) 27.3333 1.58072
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −8.84809 −0.508310
\(304\) 0 0
\(305\) −7.47536 −0.428038
\(306\) 0 0
\(307\) −8.24429 −0.470527 −0.235263 0.971932i \(-0.575595\pi\)
−0.235263 + 0.971932i \(0.575595\pi\)
\(308\) 0 0
\(309\) −26.3783 −1.50061
\(310\) 0 0
\(311\) 14.6155 0.828766 0.414383 0.910103i \(-0.363997\pi\)
0.414383 + 0.910103i \(0.363997\pi\)
\(312\) 0 0
\(313\) −0.0179410 −0.00101408 −0.000507041 1.00000i \(-0.500161\pi\)
−0.000507041 1.00000i \(0.500161\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −21.4897 −1.20698 −0.603491 0.797370i \(-0.706224\pi\)
−0.603491 + 0.797370i \(0.706224\pi\)
\(318\) 0 0
\(319\) 6.89205 0.385881
\(320\) 0 0
\(321\) −5.38416 −0.300515
\(322\) 0 0
\(323\) −7.48513 −0.416484
\(324\) 0 0
\(325\) 28.5233 1.58219
\(326\) 0 0
\(327\) −18.0645 −0.998968
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −15.1332 −0.831797 −0.415898 0.909411i \(-0.636533\pi\)
−0.415898 + 0.909411i \(0.636533\pi\)
\(332\) 0 0
\(333\) −10.5680 −0.579122
\(334\) 0 0
\(335\) 44.0534 2.40689
\(336\) 0 0
\(337\) −25.4252 −1.38500 −0.692500 0.721418i \(-0.743491\pi\)
−0.692500 + 0.721418i \(0.743491\pi\)
\(338\) 0 0
\(339\) 19.3429 1.05056
\(340\) 0 0
\(341\) −9.00998 −0.487918
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 31.6435 1.70363
\(346\) 0 0
\(347\) −7.49458 −0.402330 −0.201165 0.979557i \(-0.564473\pi\)
−0.201165 + 0.979557i \(0.564473\pi\)
\(348\) 0 0
\(349\) 6.60122 0.353355 0.176678 0.984269i \(-0.443465\pi\)
0.176678 + 0.984269i \(0.443465\pi\)
\(350\) 0 0
\(351\) −20.9567 −1.11858
\(352\) 0 0
\(353\) 22.8798 1.21777 0.608885 0.793259i \(-0.291617\pi\)
0.608885 + 0.793259i \(0.291617\pi\)
\(354\) 0 0
\(355\) 27.2512 1.44634
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 19.6109 1.03502 0.517511 0.855676i \(-0.326858\pi\)
0.517511 + 0.855676i \(0.326858\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −8.21097 −0.430964
\(364\) 0 0
\(365\) −45.0657 −2.35885
\(366\) 0 0
\(367\) −0.544256 −0.0284099 −0.0142050 0.999899i \(-0.504522\pi\)
−0.0142050 + 0.999899i \(0.504522\pi\)
\(368\) 0 0
\(369\) −12.1643 −0.633249
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −13.2016 −0.683553 −0.341776 0.939781i \(-0.611029\pi\)
−0.341776 + 0.939781i \(0.611029\pi\)
\(374\) 0 0
\(375\) 0.595483 0.0307506
\(376\) 0 0
\(377\) 14.6155 0.752738
\(378\) 0 0
\(379\) −20.8032 −1.06859 −0.534293 0.845299i \(-0.679422\pi\)
−0.534293 + 0.845299i \(0.679422\pi\)
\(380\) 0 0
\(381\) −33.2851 −1.70525
\(382\) 0 0
\(383\) 11.9617 0.611215 0.305607 0.952158i \(-0.401141\pi\)
0.305607 + 0.952158i \(0.401141\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.40923 0.224134
\(388\) 0 0
\(389\) 18.7881 0.952597 0.476298 0.879284i \(-0.341978\pi\)
0.476298 + 0.879284i \(0.341978\pi\)
\(390\) 0 0
\(391\) 36.5228 1.84704
\(392\) 0 0
\(393\) 26.2781 1.32556
\(394\) 0 0
\(395\) −53.5066 −2.69221
\(396\) 0 0
\(397\) 25.9270 1.30124 0.650618 0.759405i \(-0.274510\pi\)
0.650618 + 0.759405i \(0.274510\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.15411 0.0576333 0.0288166 0.999585i \(-0.490826\pi\)
0.0288166 + 0.999585i \(0.490826\pi\)
\(402\) 0 0
\(403\) −19.1069 −0.951781
\(404\) 0 0
\(405\) −35.3872 −1.75840
\(406\) 0 0
\(407\) −23.9125 −1.18530
\(408\) 0 0
\(409\) 21.6754 1.07178 0.535890 0.844288i \(-0.319976\pi\)
0.535890 + 0.844288i \(0.319976\pi\)
\(410\) 0 0
\(411\) −9.55013 −0.471073
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −20.7049 −1.01636
\(416\) 0 0
\(417\) −6.49221 −0.317925
\(418\) 0 0
\(419\) −19.9806 −0.976117 −0.488058 0.872811i \(-0.662295\pi\)
−0.488058 + 0.872811i \(0.662295\pi\)
\(420\) 0 0
\(421\) −30.0972 −1.46685 −0.733425 0.679771i \(-0.762079\pi\)
−0.733425 + 0.679771i \(0.762079\pi\)
\(422\) 0 0
\(423\) 8.42623 0.409697
\(424\) 0 0
\(425\) 38.1129 1.84875
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 30.2079 1.45845
\(430\) 0 0
\(431\) −31.7226 −1.52802 −0.764012 0.645202i \(-0.776773\pi\)
−0.764012 + 0.645202i \(0.776773\pi\)
\(432\) 0 0
\(433\) 4.35424 0.209252 0.104626 0.994512i \(-0.466636\pi\)
0.104626 + 0.994512i \(0.466636\pi\)
\(434\) 0 0
\(435\) 16.9203 0.811264
\(436\) 0 0
\(437\) −4.87939 −0.233413
\(438\) 0 0
\(439\) −19.4310 −0.927390 −0.463695 0.885995i \(-0.653477\pi\)
−0.463695 + 0.885995i \(0.653477\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.70100 0.318374 0.159187 0.987248i \(-0.449113\pi\)
0.159187 + 0.987248i \(0.449113\pi\)
\(444\) 0 0
\(445\) −38.7204 −1.83552
\(446\) 0 0
\(447\) 0.559093 0.0264442
\(448\) 0 0
\(449\) 8.79763 0.415186 0.207593 0.978215i \(-0.433437\pi\)
0.207593 + 0.978215i \(0.433437\pi\)
\(450\) 0 0
\(451\) −27.5246 −1.29608
\(452\) 0 0
\(453\) 45.1617 2.12188
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −32.8664 −1.53743 −0.768713 0.639594i \(-0.779102\pi\)
−0.768713 + 0.639594i \(0.779102\pi\)
\(458\) 0 0
\(459\) −28.0024 −1.30704
\(460\) 0 0
\(461\) 20.5086 0.955180 0.477590 0.878583i \(-0.341510\pi\)
0.477590 + 0.878583i \(0.341510\pi\)
\(462\) 0 0
\(463\) 21.6865 1.00786 0.503928 0.863745i \(-0.331888\pi\)
0.503928 + 0.863745i \(0.331888\pi\)
\(464\) 0 0
\(465\) −22.1198 −1.02578
\(466\) 0 0
\(467\) 12.7524 0.590110 0.295055 0.955480i \(-0.404662\pi\)
0.295055 + 0.955480i \(0.404662\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2.81430 0.129676
\(472\) 0 0
\(473\) 9.97689 0.458738
\(474\) 0 0
\(475\) −5.09182 −0.233629
\(476\) 0 0
\(477\) −11.3778 −0.520953
\(478\) 0 0
\(479\) 16.3071 0.745088 0.372544 0.928014i \(-0.378486\pi\)
0.372544 + 0.928014i \(0.378486\pi\)
\(480\) 0 0
\(481\) −50.7096 −2.31216
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −22.4500 −1.01940
\(486\) 0 0
\(487\) 4.75145 0.215309 0.107654 0.994188i \(-0.465666\pi\)
0.107654 + 0.994188i \(0.465666\pi\)
\(488\) 0 0
\(489\) 4.18652 0.189321
\(490\) 0 0
\(491\) −13.8998 −0.627290 −0.313645 0.949540i \(-0.601550\pi\)
−0.313645 + 0.949540i \(0.601550\pi\)
\(492\) 0 0
\(493\) 19.5293 0.879558
\(494\) 0 0
\(495\) 9.79653 0.440321
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 18.6230 0.833681 0.416841 0.908980i \(-0.363137\pi\)
0.416841 + 0.908980i \(0.363137\pi\)
\(500\) 0 0
\(501\) 7.17148 0.320398
\(502\) 0 0
\(503\) 14.0137 0.624840 0.312420 0.949944i \(-0.398860\pi\)
0.312420 + 0.949944i \(0.398860\pi\)
\(504\) 0 0
\(505\) −13.7690 −0.612710
\(506\) 0 0
\(507\) 37.5213 1.66638
\(508\) 0 0
\(509\) 12.6112 0.558980 0.279490 0.960149i \(-0.409835\pi\)
0.279490 + 0.960149i \(0.409835\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.74108 0.165172
\(514\) 0 0
\(515\) −41.0486 −1.80882
\(516\) 0 0
\(517\) 19.0663 0.838534
\(518\) 0 0
\(519\) 48.1565 2.11384
\(520\) 0 0
\(521\) 4.03072 0.176589 0.0882945 0.996094i \(-0.471858\pi\)
0.0882945 + 0.996094i \(0.471858\pi\)
\(522\) 0 0
\(523\) −44.0062 −1.92426 −0.962129 0.272596i \(-0.912118\pi\)
−0.962129 + 0.272596i \(0.912118\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −25.5307 −1.11214
\(528\) 0 0
\(529\) 0.808435 0.0351493
\(530\) 0 0
\(531\) 3.86380 0.167675
\(532\) 0 0
\(533\) −58.3696 −2.52827
\(534\) 0 0
\(535\) −8.37856 −0.362237
\(536\) 0 0
\(537\) 18.2375 0.787008
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 32.6699 1.40459 0.702295 0.711886i \(-0.252159\pi\)
0.702295 + 0.711886i \(0.252159\pi\)
\(542\) 0 0
\(543\) −51.2896 −2.20105
\(544\) 0 0
\(545\) −28.1110 −1.20414
\(546\) 0 0
\(547\) −17.1209 −0.732038 −0.366019 0.930607i \(-0.619280\pi\)
−0.366019 + 0.930607i \(0.619280\pi\)
\(548\) 0 0
\(549\) −2.74710 −0.117244
\(550\) 0 0
\(551\) −2.60909 −0.111151
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −58.7061 −2.49193
\(556\) 0 0
\(557\) −14.2650 −0.604427 −0.302213 0.953240i \(-0.597726\pi\)
−0.302213 + 0.953240i \(0.597726\pi\)
\(558\) 0 0
\(559\) 21.1573 0.894860
\(560\) 0 0
\(561\) 40.3639 1.70416
\(562\) 0 0
\(563\) 29.2283 1.23182 0.615912 0.787815i \(-0.288788\pi\)
0.615912 + 0.787815i \(0.288788\pi\)
\(564\) 0 0
\(565\) 30.1005 1.26634
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 44.4290 1.86256 0.931280 0.364305i \(-0.118693\pi\)
0.931280 + 0.364305i \(0.118693\pi\)
\(570\) 0 0
\(571\) 1.46048 0.0611192 0.0305596 0.999533i \(-0.490271\pi\)
0.0305596 + 0.999533i \(0.490271\pi\)
\(572\) 0 0
\(573\) 37.4188 1.56319
\(574\) 0 0
\(575\) 24.8450 1.03611
\(576\) 0 0
\(577\) −10.6800 −0.444616 −0.222308 0.974977i \(-0.571359\pi\)
−0.222308 + 0.974977i \(0.571359\pi\)
\(578\) 0 0
\(579\) −1.70196 −0.0707309
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −25.7449 −1.06624
\(584\) 0 0
\(585\) 20.7749 0.858935
\(586\) 0 0
\(587\) −18.5240 −0.764569 −0.382285 0.924045i \(-0.624863\pi\)
−0.382285 + 0.924045i \(0.624863\pi\)
\(588\) 0 0
\(589\) 3.41086 0.140542
\(590\) 0 0
\(591\) 9.51660 0.391461
\(592\) 0 0
\(593\) −37.3332 −1.53309 −0.766546 0.642189i \(-0.778026\pi\)
−0.766546 + 0.642189i \(0.778026\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −29.3631 −1.20175
\(598\) 0 0
\(599\) 37.5271 1.53331 0.766657 0.642057i \(-0.221919\pi\)
0.766657 + 0.642057i \(0.221919\pi\)
\(600\) 0 0
\(601\) 38.2640 1.56082 0.780411 0.625267i \(-0.215010\pi\)
0.780411 + 0.625267i \(0.215010\pi\)
\(602\) 0 0
\(603\) 16.1891 0.659270
\(604\) 0 0
\(605\) −12.7775 −0.519479
\(606\) 0 0
\(607\) 26.1112 1.05982 0.529910 0.848054i \(-0.322226\pi\)
0.529910 + 0.848054i \(0.322226\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 40.4326 1.63573
\(612\) 0 0
\(613\) −27.5292 −1.11190 −0.555948 0.831217i \(-0.687645\pi\)
−0.555948 + 0.831217i \(0.687645\pi\)
\(614\) 0 0
\(615\) −67.5739 −2.72484
\(616\) 0 0
\(617\) −19.0678 −0.767641 −0.383821 0.923408i \(-0.625392\pi\)
−0.383821 + 0.923408i \(0.625392\pi\)
\(618\) 0 0
\(619\) 41.8369 1.68157 0.840783 0.541373i \(-0.182095\pi\)
0.840783 + 0.541373i \(0.182095\pi\)
\(620\) 0 0
\(621\) −18.2542 −0.732514
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −24.5325 −0.981298
\(626\) 0 0
\(627\) −5.39255 −0.215358
\(628\) 0 0
\(629\) −67.7585 −2.70171
\(630\) 0 0
\(631\) −23.9463 −0.953286 −0.476643 0.879097i \(-0.658146\pi\)
−0.476643 + 0.879097i \(0.658146\pi\)
\(632\) 0 0
\(633\) −44.8619 −1.78310
\(634\) 0 0
\(635\) −51.7966 −2.05548
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 10.0145 0.396166
\(640\) 0 0
\(641\) 18.2101 0.719258 0.359629 0.933095i \(-0.382903\pi\)
0.359629 + 0.933095i \(0.382903\pi\)
\(642\) 0 0
\(643\) −29.5698 −1.16612 −0.583060 0.812429i \(-0.698145\pi\)
−0.583060 + 0.812429i \(0.698145\pi\)
\(644\) 0 0
\(645\) 24.4936 0.964436
\(646\) 0 0
\(647\) −22.2490 −0.874698 −0.437349 0.899292i \(-0.644083\pi\)
−0.437349 + 0.899292i \(0.644083\pi\)
\(648\) 0 0
\(649\) 8.74275 0.343183
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 32.4918 1.27150 0.635751 0.771894i \(-0.280691\pi\)
0.635751 + 0.771894i \(0.280691\pi\)
\(654\) 0 0
\(655\) 40.8927 1.59781
\(656\) 0 0
\(657\) −16.5611 −0.646110
\(658\) 0 0
\(659\) 20.1618 0.785391 0.392695 0.919669i \(-0.371543\pi\)
0.392695 + 0.919669i \(0.371543\pi\)
\(660\) 0 0
\(661\) 29.5870 1.15080 0.575400 0.817872i \(-0.304846\pi\)
0.575400 + 0.817872i \(0.304846\pi\)
\(662\) 0 0
\(663\) 85.5971 3.32431
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.7307 0.492937
\(668\) 0 0
\(669\) 4.99840 0.193249
\(670\) 0 0
\(671\) −6.21595 −0.239964
\(672\) 0 0
\(673\) 9.97995 0.384699 0.192349 0.981326i \(-0.438389\pi\)
0.192349 + 0.981326i \(0.438389\pi\)
\(674\) 0 0
\(675\) −19.0489 −0.733192
\(676\) 0 0
\(677\) −3.68285 −0.141543 −0.0707717 0.997493i \(-0.522546\pi\)
−0.0707717 + 0.997493i \(0.522546\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 17.7795 0.681313
\(682\) 0 0
\(683\) 33.7957 1.29316 0.646579 0.762847i \(-0.276199\pi\)
0.646579 + 0.762847i \(0.276199\pi\)
\(684\) 0 0
\(685\) −14.8614 −0.567826
\(686\) 0 0
\(687\) 37.3419 1.42468
\(688\) 0 0
\(689\) −54.5954 −2.07992
\(690\) 0 0
\(691\) −11.1423 −0.423873 −0.211936 0.977283i \(-0.567977\pi\)
−0.211936 + 0.977283i \(0.567977\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −10.1029 −0.383223
\(696\) 0 0
\(697\) −77.9937 −2.95422
\(698\) 0 0
\(699\) −10.1279 −0.383074
\(700\) 0 0
\(701\) −36.7249 −1.38708 −0.693539 0.720419i \(-0.743950\pi\)
−0.693539 + 0.720419i \(0.743950\pi\)
\(702\) 0 0
\(703\) 9.05242 0.341418
\(704\) 0 0
\(705\) 46.8085 1.76291
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 15.4053 0.578560 0.289280 0.957245i \(-0.406584\pi\)
0.289280 + 0.957245i \(0.406584\pi\)
\(710\) 0 0
\(711\) −19.6630 −0.737421
\(712\) 0 0
\(713\) −16.6429 −0.623282
\(714\) 0 0
\(715\) 47.0079 1.75800
\(716\) 0 0
\(717\) −9.04170 −0.337668
\(718\) 0 0
\(719\) 32.0902 1.19676 0.598380 0.801212i \(-0.295811\pi\)
0.598380 + 0.801212i \(0.295811\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 36.4798 1.35670
\(724\) 0 0
\(725\) 13.2850 0.493393
\(726\) 0 0
\(727\) −41.0731 −1.52332 −0.761658 0.647979i \(-0.775614\pi\)
−0.761658 + 0.647979i \(0.775614\pi\)
\(728\) 0 0
\(729\) 9.90703 0.366927
\(730\) 0 0
\(731\) 28.2705 1.04562
\(732\) 0 0
\(733\) 24.7871 0.915534 0.457767 0.889072i \(-0.348649\pi\)
0.457767 + 0.889072i \(0.348649\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 36.6315 1.34934
\(738\) 0 0
\(739\) −38.0986 −1.40148 −0.700740 0.713417i \(-0.747147\pi\)
−0.700740 + 0.713417i \(0.747147\pi\)
\(740\) 0 0
\(741\) −11.4356 −0.420098
\(742\) 0 0
\(743\) −33.2083 −1.21830 −0.609148 0.793057i \(-0.708488\pi\)
−0.609148 + 0.793057i \(0.708488\pi\)
\(744\) 0 0
\(745\) 0.870031 0.0318755
\(746\) 0 0
\(747\) −7.60878 −0.278391
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 37.8448 1.38098 0.690488 0.723344i \(-0.257396\pi\)
0.690488 + 0.723344i \(0.257396\pi\)
\(752\) 0 0
\(753\) −23.3501 −0.850925
\(754\) 0 0
\(755\) 70.2784 2.55769
\(756\) 0 0
\(757\) 19.5426 0.710289 0.355145 0.934811i \(-0.384432\pi\)
0.355145 + 0.934811i \(0.384432\pi\)
\(758\) 0 0
\(759\) 26.3123 0.955077
\(760\) 0 0
\(761\) −29.9515 −1.08574 −0.542870 0.839817i \(-0.682662\pi\)
−0.542870 + 0.839817i \(0.682662\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 27.7595 1.00365
\(766\) 0 0
\(767\) 18.5402 0.669447
\(768\) 0 0
\(769\) −15.2561 −0.550150 −0.275075 0.961423i \(-0.588703\pi\)
−0.275075 + 0.961423i \(0.588703\pi\)
\(770\) 0 0
\(771\) −4.45851 −0.160569
\(772\) 0 0
\(773\) 15.2561 0.548724 0.274362 0.961626i \(-0.411533\pi\)
0.274362 + 0.961626i \(0.411533\pi\)
\(774\) 0 0
\(775\) −17.3675 −0.623858
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 10.4198 0.373329
\(780\) 0 0
\(781\) 22.6600 0.810839
\(782\) 0 0
\(783\) −9.76079 −0.348822
\(784\) 0 0
\(785\) 4.37947 0.156310
\(786\) 0 0
\(787\) −1.08333 −0.0386166 −0.0193083 0.999814i \(-0.506146\pi\)
−0.0193083 + 0.999814i \(0.506146\pi\)
\(788\) 0 0
\(789\) 39.5033 1.40636
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −13.1818 −0.468098
\(794\) 0 0
\(795\) −63.2046 −2.24164
\(796\) 0 0
\(797\) −17.9472 −0.635722 −0.317861 0.948137i \(-0.602965\pi\)
−0.317861 + 0.948137i \(0.602965\pi\)
\(798\) 0 0
\(799\) 54.0263 1.91131
\(800\) 0 0
\(801\) −14.2293 −0.502767
\(802\) 0 0
\(803\) −37.4733 −1.32240
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −23.4951 −0.827066
\(808\) 0 0
\(809\) 26.3254 0.925551 0.462775 0.886476i \(-0.346854\pi\)
0.462775 + 0.886476i \(0.346854\pi\)
\(810\) 0 0
\(811\) 52.2631 1.83521 0.917603 0.397498i \(-0.130122\pi\)
0.917603 + 0.397498i \(0.130122\pi\)
\(812\) 0 0
\(813\) 16.0080 0.561425
\(814\) 0 0
\(815\) 6.51485 0.228205
\(816\) 0 0
\(817\) −3.77689 −0.132137
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −0.737155 −0.0257269 −0.0128634 0.999917i \(-0.504095\pi\)
−0.0128634 + 0.999917i \(0.504095\pi\)
\(822\) 0 0
\(823\) 38.7222 1.34977 0.674885 0.737922i \(-0.264193\pi\)
0.674885 + 0.737922i \(0.264193\pi\)
\(824\) 0 0
\(825\) 27.4579 0.955961
\(826\) 0 0
\(827\) −49.5257 −1.72218 −0.861088 0.508456i \(-0.830216\pi\)
−0.861088 + 0.508456i \(0.830216\pi\)
\(828\) 0 0
\(829\) −34.9798 −1.21490 −0.607450 0.794358i \(-0.707808\pi\)
−0.607450 + 0.794358i \(0.707808\pi\)
\(830\) 0 0
\(831\) −25.2417 −0.875625
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 11.1599 0.386204
\(836\) 0 0
\(837\) 12.7603 0.441060
\(838\) 0 0
\(839\) 26.6963 0.921660 0.460830 0.887488i \(-0.347552\pi\)
0.460830 + 0.887488i \(0.347552\pi\)
\(840\) 0 0
\(841\) −22.1927 −0.765264
\(842\) 0 0
\(843\) 14.3502 0.494246
\(844\) 0 0
\(845\) 58.3887 2.00863
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 63.9403 2.19442
\(850\) 0 0
\(851\) −44.1703 −1.51414
\(852\) 0 0
\(853\) −12.4509 −0.426312 −0.213156 0.977018i \(-0.568374\pi\)
−0.213156 + 0.977018i \(0.568374\pi\)
\(854\) 0 0
\(855\) −3.70862 −0.126832
\(856\) 0 0
\(857\) 21.5641 0.736615 0.368307 0.929704i \(-0.379937\pi\)
0.368307 + 0.929704i \(0.379937\pi\)
\(858\) 0 0
\(859\) 25.1796 0.859117 0.429558 0.903039i \(-0.358669\pi\)
0.429558 + 0.903039i \(0.358669\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.39527 −0.183657 −0.0918286 0.995775i \(-0.529271\pi\)
−0.0918286 + 0.995775i \(0.529271\pi\)
\(864\) 0 0
\(865\) 74.9387 2.54799
\(866\) 0 0
\(867\) 79.6710 2.70577
\(868\) 0 0
\(869\) −44.4921 −1.50929
\(870\) 0 0
\(871\) 77.6821 2.63216
\(872\) 0 0
\(873\) −8.25010 −0.279224
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 6.02445 0.203431 0.101716 0.994814i \(-0.467567\pi\)
0.101716 + 0.994814i \(0.467567\pi\)
\(878\) 0 0
\(879\) −22.5527 −0.760682
\(880\) 0 0
\(881\) −36.2622 −1.22170 −0.610852 0.791745i \(-0.709173\pi\)
−0.610852 + 0.791745i \(0.709173\pi\)
\(882\) 0 0
\(883\) 14.3386 0.482533 0.241267 0.970459i \(-0.422437\pi\)
0.241267 + 0.970459i \(0.422437\pi\)
\(884\) 0 0
\(885\) 21.4638 0.721497
\(886\) 0 0
\(887\) 20.0310 0.672575 0.336287 0.941759i \(-0.390829\pi\)
0.336287 + 0.941759i \(0.390829\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −29.4253 −0.985786
\(892\) 0 0
\(893\) −7.21782 −0.241535
\(894\) 0 0
\(895\) 28.3803 0.948650
\(896\) 0 0
\(897\) 55.7988 1.86307
\(898\) 0 0
\(899\) −8.89923 −0.296806
\(900\) 0 0
\(901\) −72.9507 −2.43034
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −79.8143 −2.65312
\(906\) 0 0
\(907\) −15.2833 −0.507475 −0.253737 0.967273i \(-0.581660\pi\)
−0.253737 + 0.967273i \(0.581660\pi\)
\(908\) 0 0
\(909\) −5.05992 −0.167827
\(910\) 0 0
\(911\) 41.1998 1.36501 0.682506 0.730880i \(-0.260890\pi\)
0.682506 + 0.730880i \(0.260890\pi\)
\(912\) 0 0
\(913\) −17.2166 −0.569787
\(914\) 0 0
\(915\) −15.2604 −0.504493
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −27.0599 −0.892625 −0.446312 0.894877i \(-0.647263\pi\)
−0.446312 + 0.894877i \(0.647263\pi\)
\(920\) 0 0
\(921\) −16.8301 −0.554571
\(922\) 0 0
\(923\) 48.0536 1.58170
\(924\) 0 0
\(925\) −46.0933 −1.51554
\(926\) 0 0
\(927\) −15.0849 −0.495452
\(928\) 0 0
\(929\) −52.3083 −1.71618 −0.858090 0.513500i \(-0.828349\pi\)
−0.858090 + 0.513500i \(0.828349\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 29.8364 0.976799
\(934\) 0 0
\(935\) 62.8122 2.05418
\(936\) 0 0
\(937\) −17.8565 −0.583345 −0.291673 0.956518i \(-0.594212\pi\)
−0.291673 + 0.956518i \(0.594212\pi\)
\(938\) 0 0
\(939\) −0.0366251 −0.00119522
\(940\) 0 0
\(941\) 20.6849 0.674310 0.337155 0.941449i \(-0.390535\pi\)
0.337155 + 0.941449i \(0.390535\pi\)
\(942\) 0 0
\(943\) −50.8424 −1.65566
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −32.7059 −1.06280 −0.531399 0.847122i \(-0.678334\pi\)
−0.531399 + 0.847122i \(0.678334\pi\)
\(948\) 0 0
\(949\) −79.4671 −2.57961
\(950\) 0 0
\(951\) −43.8696 −1.42257
\(952\) 0 0
\(953\) 51.9928 1.68421 0.842107 0.539311i \(-0.181315\pi\)
0.842107 + 0.539311i \(0.181315\pi\)
\(954\) 0 0
\(955\) 58.2293 1.88425
\(956\) 0 0
\(957\) 14.0696 0.454806
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −19.3660 −0.624711
\(962\) 0 0
\(963\) −3.07902 −0.0992199
\(964\) 0 0
\(965\) −2.64850 −0.0852581
\(966\) 0 0
\(967\) 46.7892 1.50464 0.752320 0.658798i \(-0.228935\pi\)
0.752320 + 0.658798i \(0.228935\pi\)
\(968\) 0 0
\(969\) −15.2803 −0.490875
\(970\) 0 0
\(971\) −15.2980 −0.490936 −0.245468 0.969405i \(-0.578942\pi\)
−0.245468 + 0.969405i \(0.578942\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 58.2281 1.86479
\(976\) 0 0
\(977\) 44.6983 1.43002 0.715012 0.699112i \(-0.246421\pi\)
0.715012 + 0.699112i \(0.246421\pi\)
\(978\) 0 0
\(979\) −32.1970 −1.02902
\(980\) 0 0
\(981\) −10.3305 −0.329826
\(982\) 0 0
\(983\) −23.2120 −0.740347 −0.370174 0.928963i \(-0.620702\pi\)
−0.370174 + 0.928963i \(0.620702\pi\)
\(984\) 0 0
\(985\) 14.8092 0.471862
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 18.4289 0.586006
\(990\) 0 0
\(991\) −46.9794 −1.49235 −0.746174 0.665751i \(-0.768111\pi\)
−0.746174 + 0.665751i \(0.768111\pi\)
\(992\) 0 0
\(993\) −30.8933 −0.980370
\(994\) 0 0
\(995\) −45.6933 −1.44858
\(996\) 0 0
\(997\) −32.9126 −1.04235 −0.521176 0.853449i \(-0.674506\pi\)
−0.521176 + 0.853449i \(0.674506\pi\)
\(998\) 0 0
\(999\) 33.8658 1.07147
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 7448.2.a.bn.1.5 yes 6
7.6 odd 2 7448.2.a.bm.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7448.2.a.bm.1.2 6 7.6 odd 2
7448.2.a.bn.1.5 yes 6 1.1 even 1 trivial