Properties

Label 2352.2.a.be.1.1
Level $2352$
Weight $2$
Character 2352.1
Self dual yes
Analytic conductor $18.781$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(18.7808145554\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.41421 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.41421 q^{5} +1.00000 q^{9} +2.00000 q^{11} -2.58579 q^{13} -3.41421 q^{15} +2.24264 q^{17} -2.82843 q^{19} +7.65685 q^{23} +6.65685 q^{25} +1.00000 q^{27} -6.82843 q^{29} -1.17157 q^{31} +2.00000 q^{33} -4.00000 q^{37} -2.58579 q^{39} -6.24264 q^{41} -5.65685 q^{43} -3.41421 q^{45} -2.82843 q^{47} +2.24264 q^{51} -2.00000 q^{53} -6.82843 q^{55} -2.82843 q^{57} -1.17157 q^{59} -12.2426 q^{61} +8.82843 q^{65} +5.65685 q^{67} +7.65685 q^{69} -9.31371 q^{71} -13.8995 q^{73} +6.65685 q^{75} -13.6569 q^{79} +1.00000 q^{81} +7.31371 q^{83} -7.65685 q^{85} -6.82843 q^{87} +14.2426 q^{89} -1.17157 q^{93} +9.65685 q^{95} -2.58579 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{5} + 2q^{9} + 4q^{11} - 8q^{13} - 4q^{15} - 4q^{17} + 4q^{23} + 2q^{25} + 2q^{27} - 8q^{29} - 8q^{31} + 4q^{33} - 8q^{37} - 8q^{39} - 4q^{41} - 4q^{45} - 4q^{51} - 4q^{53} - 8q^{55} - 8q^{59} - 16q^{61} + 12q^{65} + 4q^{69} + 4q^{71} - 8q^{73} + 2q^{75} - 16q^{79} + 2q^{81} - 8q^{83} - 4q^{85} - 8q^{87} + 20q^{89} - 8q^{93} + 8q^{95} - 8q^{97} + 4q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −3.41421 −1.52688 −0.763441 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −2.58579 −0.717168 −0.358584 0.933497i \(-0.616740\pi\)
−0.358584 + 0.933497i \(0.616740\pi\)
\(14\) 0 0
\(15\) −3.41421 −0.881546
\(16\) 0 0
\(17\) 2.24264 0.543920 0.271960 0.962309i \(-0.412328\pi\)
0.271960 + 0.962309i \(0.412328\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.65685 1.59656 0.798282 0.602284i \(-0.205742\pi\)
0.798282 + 0.602284i \(0.205742\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.82843 −1.26801 −0.634004 0.773330i \(-0.718590\pi\)
−0.634004 + 0.773330i \(0.718590\pi\)
\(30\) 0 0
\(31\) −1.17157 −0.210421 −0.105210 0.994450i \(-0.533552\pi\)
−0.105210 + 0.994450i \(0.533552\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) −2.58579 −0.414057
\(40\) 0 0
\(41\) −6.24264 −0.974937 −0.487468 0.873141i \(-0.662080\pi\)
−0.487468 + 0.873141i \(0.662080\pi\)
\(42\) 0 0
\(43\) −5.65685 −0.862662 −0.431331 0.902194i \(-0.641956\pi\)
−0.431331 + 0.902194i \(0.641956\pi\)
\(44\) 0 0
\(45\) −3.41421 −0.508961
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.24264 0.314033
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −6.82843 −0.920745
\(56\) 0 0
\(57\) −2.82843 −0.374634
\(58\) 0 0
\(59\) −1.17157 −0.152526 −0.0762629 0.997088i \(-0.524299\pi\)
−0.0762629 + 0.997088i \(0.524299\pi\)
\(60\) 0 0
\(61\) −12.2426 −1.56751 −0.783755 0.621070i \(-0.786698\pi\)
−0.783755 + 0.621070i \(0.786698\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.82843 1.09503
\(66\) 0 0
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) 0 0
\(69\) 7.65685 0.921777
\(70\) 0 0
\(71\) −9.31371 −1.10533 −0.552667 0.833402i \(-0.686390\pi\)
−0.552667 + 0.833402i \(0.686390\pi\)
\(72\) 0 0
\(73\) −13.8995 −1.62681 −0.813406 0.581696i \(-0.802389\pi\)
−0.813406 + 0.581696i \(0.802389\pi\)
\(74\) 0 0
\(75\) 6.65685 0.768667
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −13.6569 −1.53652 −0.768258 0.640140i \(-0.778876\pi\)
−0.768258 + 0.640140i \(0.778876\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.31371 0.802784 0.401392 0.915906i \(-0.368527\pi\)
0.401392 + 0.915906i \(0.368527\pi\)
\(84\) 0 0
\(85\) −7.65685 −0.830502
\(86\) 0 0
\(87\) −6.82843 −0.732084
\(88\) 0 0
\(89\) 14.2426 1.50972 0.754858 0.655888i \(-0.227706\pi\)
0.754858 + 0.655888i \(0.227706\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.17157 −0.121486
\(94\) 0 0
\(95\) 9.65685 0.990772
\(96\) 0 0
\(97\) −2.58579 −0.262547 −0.131273 0.991346i \(-0.541907\pi\)
−0.131273 + 0.991346i \(0.541907\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −2.92893 −0.291440 −0.145720 0.989326i \(-0.546550\pi\)
−0.145720 + 0.989326i \(0.546550\pi\)
\(102\) 0 0
\(103\) 4.48528 0.441948 0.220974 0.975280i \(-0.429076\pi\)
0.220974 + 0.975280i \(0.429076\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.343146 0.0331732 0.0165866 0.999862i \(-0.494720\pi\)
0.0165866 + 0.999862i \(0.494720\pi\)
\(108\) 0 0
\(109\) −5.65685 −0.541828 −0.270914 0.962604i \(-0.587326\pi\)
−0.270914 + 0.962604i \(0.587326\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) −5.31371 −0.499872 −0.249936 0.968262i \(-0.580410\pi\)
−0.249936 + 0.968262i \(0.580410\pi\)
\(114\) 0 0
\(115\) −26.1421 −2.43777
\(116\) 0 0
\(117\) −2.58579 −0.239056
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −6.24264 −0.562880
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) 1.65685 0.147022 0.0735110 0.997294i \(-0.476580\pi\)
0.0735110 + 0.997294i \(0.476580\pi\)
\(128\) 0 0
\(129\) −5.65685 −0.498058
\(130\) 0 0
\(131\) −15.3137 −1.33796 −0.668982 0.743278i \(-0.733270\pi\)
−0.668982 + 0.743278i \(0.733270\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −3.41421 −0.293849
\(136\) 0 0
\(137\) 14.1421 1.20824 0.604122 0.796892i \(-0.293524\pi\)
0.604122 + 0.796892i \(0.293524\pi\)
\(138\) 0 0
\(139\) −17.6569 −1.49763 −0.748817 0.662776i \(-0.769378\pi\)
−0.748817 + 0.662776i \(0.769378\pi\)
\(140\) 0 0
\(141\) −2.82843 −0.238197
\(142\) 0 0
\(143\) −5.17157 −0.432469
\(144\) 0 0
\(145\) 23.3137 1.93610
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 17.3137 1.41839 0.709197 0.705010i \(-0.249058\pi\)
0.709197 + 0.705010i \(0.249058\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 2.24264 0.181307
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −11.7574 −0.938339 −0.469170 0.883108i \(-0.655447\pi\)
−0.469170 + 0.883108i \(0.655447\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 11.3137 0.886158 0.443079 0.896483i \(-0.353886\pi\)
0.443079 + 0.896483i \(0.353886\pi\)
\(164\) 0 0
\(165\) −6.82843 −0.531592
\(166\) 0 0
\(167\) −19.7990 −1.53209 −0.766046 0.642786i \(-0.777779\pi\)
−0.766046 + 0.642786i \(0.777779\pi\)
\(168\) 0 0
\(169\) −6.31371 −0.485670
\(170\) 0 0
\(171\) −2.82843 −0.216295
\(172\) 0 0
\(173\) −21.0711 −1.60200 −0.801002 0.598662i \(-0.795699\pi\)
−0.801002 + 0.598662i \(0.795699\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.17157 −0.0880608
\(178\) 0 0
\(179\) 19.6569 1.46922 0.734611 0.678488i \(-0.237365\pi\)
0.734611 + 0.678488i \(0.237365\pi\)
\(180\) 0 0
\(181\) 2.58579 0.192200 0.0961000 0.995372i \(-0.469363\pi\)
0.0961000 + 0.995372i \(0.469363\pi\)
\(182\) 0 0
\(183\) −12.2426 −0.905002
\(184\) 0 0
\(185\) 13.6569 1.00407
\(186\) 0 0
\(187\) 4.48528 0.327996
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) 5.31371 0.382489 0.191245 0.981542i \(-0.438748\pi\)
0.191245 + 0.981542i \(0.438748\pi\)
\(194\) 0 0
\(195\) 8.82843 0.632217
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 21.6569 1.53521 0.767607 0.640921i \(-0.221447\pi\)
0.767607 + 0.640921i \(0.221447\pi\)
\(200\) 0 0
\(201\) 5.65685 0.399004
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 21.3137 1.48861
\(206\) 0 0
\(207\) 7.65685 0.532188
\(208\) 0 0
\(209\) −5.65685 −0.391293
\(210\) 0 0
\(211\) −12.9706 −0.892930 −0.446465 0.894801i \(-0.647317\pi\)
−0.446465 + 0.894801i \(0.647317\pi\)
\(212\) 0 0
\(213\) −9.31371 −0.638165
\(214\) 0 0
\(215\) 19.3137 1.31718
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −13.8995 −0.939241
\(220\) 0 0
\(221\) −5.79899 −0.390082
\(222\) 0 0
\(223\) 24.9706 1.67215 0.836076 0.548613i \(-0.184844\pi\)
0.836076 + 0.548613i \(0.184844\pi\)
\(224\) 0 0
\(225\) 6.65685 0.443790
\(226\) 0 0
\(227\) 23.7990 1.57959 0.789797 0.613368i \(-0.210186\pi\)
0.789797 + 0.613368i \(0.210186\pi\)
\(228\) 0 0
\(229\) 0.242641 0.0160341 0.00801707 0.999968i \(-0.497448\pi\)
0.00801707 + 0.999968i \(0.497448\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.14214 −0.402385 −0.201192 0.979552i \(-0.564482\pi\)
−0.201192 + 0.979552i \(0.564482\pi\)
\(234\) 0 0
\(235\) 9.65685 0.629944
\(236\) 0 0
\(237\) −13.6569 −0.887108
\(238\) 0 0
\(239\) 15.6569 1.01276 0.506379 0.862311i \(-0.330984\pi\)
0.506379 + 0.862311i \(0.330984\pi\)
\(240\) 0 0
\(241\) 16.2426 1.04628 0.523140 0.852247i \(-0.324760\pi\)
0.523140 + 0.852247i \(0.324760\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 7.31371 0.465360
\(248\) 0 0
\(249\) 7.31371 0.463487
\(250\) 0 0
\(251\) −12.4853 −0.788064 −0.394032 0.919097i \(-0.628920\pi\)
−0.394032 + 0.919097i \(0.628920\pi\)
\(252\) 0 0
\(253\) 15.3137 0.962765
\(254\) 0 0
\(255\) −7.65685 −0.479491
\(256\) 0 0
\(257\) −23.2132 −1.44800 −0.724000 0.689800i \(-0.757698\pi\)
−0.724000 + 0.689800i \(0.757698\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.82843 −0.422669
\(262\) 0 0
\(263\) −5.31371 −0.327657 −0.163829 0.986489i \(-0.552384\pi\)
−0.163829 + 0.986489i \(0.552384\pi\)
\(264\) 0 0
\(265\) 6.82843 0.419467
\(266\) 0 0
\(267\) 14.2426 0.871635
\(268\) 0 0
\(269\) 14.7279 0.897977 0.448989 0.893537i \(-0.351784\pi\)
0.448989 + 0.893537i \(0.351784\pi\)
\(270\) 0 0
\(271\) −10.1421 −0.616091 −0.308045 0.951372i \(-0.599675\pi\)
−0.308045 + 0.951372i \(0.599675\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 13.3137 0.802847
\(276\) 0 0
\(277\) −9.31371 −0.559607 −0.279803 0.960057i \(-0.590269\pi\)
−0.279803 + 0.960057i \(0.590269\pi\)
\(278\) 0 0
\(279\) −1.17157 −0.0701402
\(280\) 0 0
\(281\) 0.485281 0.0289495 0.0144747 0.999895i \(-0.495392\pi\)
0.0144747 + 0.999895i \(0.495392\pi\)
\(282\) 0 0
\(283\) −8.48528 −0.504398 −0.252199 0.967675i \(-0.581154\pi\)
−0.252199 + 0.967675i \(0.581154\pi\)
\(284\) 0 0
\(285\) 9.65685 0.572023
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −11.9706 −0.704151
\(290\) 0 0
\(291\) −2.58579 −0.151581
\(292\) 0 0
\(293\) −16.5858 −0.968952 −0.484476 0.874805i \(-0.660990\pi\)
−0.484476 + 0.874805i \(0.660990\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) −19.7990 −1.14501
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −2.92893 −0.168263
\(304\) 0 0
\(305\) 41.7990 2.39340
\(306\) 0 0
\(307\) 30.1421 1.72030 0.860151 0.510039i \(-0.170369\pi\)
0.860151 + 0.510039i \(0.170369\pi\)
\(308\) 0 0
\(309\) 4.48528 0.255159
\(310\) 0 0
\(311\) 6.14214 0.348289 0.174144 0.984720i \(-0.444284\pi\)
0.174144 + 0.984720i \(0.444284\pi\)
\(312\) 0 0
\(313\) −1.89949 −0.107366 −0.0536829 0.998558i \(-0.517096\pi\)
−0.0536829 + 0.998558i \(0.517096\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0 0
\(319\) −13.6569 −0.764637
\(320\) 0 0
\(321\) 0.343146 0.0191525
\(322\) 0 0
\(323\) −6.34315 −0.352942
\(324\) 0 0
\(325\) −17.2132 −0.954817
\(326\) 0 0
\(327\) −5.65685 −0.312825
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) −19.3137 −1.05522
\(336\) 0 0
\(337\) −29.6569 −1.61551 −0.807756 0.589517i \(-0.799318\pi\)
−0.807756 + 0.589517i \(0.799318\pi\)
\(338\) 0 0
\(339\) −5.31371 −0.288601
\(340\) 0 0
\(341\) −2.34315 −0.126888
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −26.1421 −1.40745
\(346\) 0 0
\(347\) −33.3137 −1.78837 −0.894187 0.447694i \(-0.852245\pi\)
−0.894187 + 0.447694i \(0.852245\pi\)
\(348\) 0 0
\(349\) 9.89949 0.529908 0.264954 0.964261i \(-0.414643\pi\)
0.264954 + 0.964261i \(0.414643\pi\)
\(350\) 0 0
\(351\) −2.58579 −0.138019
\(352\) 0 0
\(353\) 14.7279 0.783888 0.391944 0.919989i \(-0.371803\pi\)
0.391944 + 0.919989i \(0.371803\pi\)
\(354\) 0 0
\(355\) 31.7990 1.68772
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.343146 0.0181105 0.00905527 0.999959i \(-0.497118\pi\)
0.00905527 + 0.999959i \(0.497118\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 47.4558 2.48395
\(366\) 0 0
\(367\) −3.31371 −0.172974 −0.0864871 0.996253i \(-0.527564\pi\)
−0.0864871 + 0.996253i \(0.527564\pi\)
\(368\) 0 0
\(369\) −6.24264 −0.324979
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −10.6863 −0.553315 −0.276658 0.960969i \(-0.589227\pi\)
−0.276658 + 0.960969i \(0.589227\pi\)
\(374\) 0 0
\(375\) −5.65685 −0.292119
\(376\) 0 0
\(377\) 17.6569 0.909374
\(378\) 0 0
\(379\) −8.68629 −0.446185 −0.223092 0.974797i \(-0.571615\pi\)
−0.223092 + 0.974797i \(0.571615\pi\)
\(380\) 0 0
\(381\) 1.65685 0.0848832
\(382\) 0 0
\(383\) −18.3431 −0.937291 −0.468645 0.883386i \(-0.655258\pi\)
−0.468645 + 0.883386i \(0.655258\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −5.65685 −0.287554
\(388\) 0 0
\(389\) −18.1421 −0.919843 −0.459921 0.887960i \(-0.652122\pi\)
−0.459921 + 0.887960i \(0.652122\pi\)
\(390\) 0 0
\(391\) 17.1716 0.868404
\(392\) 0 0
\(393\) −15.3137 −0.772474
\(394\) 0 0
\(395\) 46.6274 2.34608
\(396\) 0 0
\(397\) 2.38478 0.119688 0.0598442 0.998208i \(-0.480940\pi\)
0.0598442 + 0.998208i \(0.480940\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.14214 −0.306724 −0.153362 0.988170i \(-0.549010\pi\)
−0.153362 + 0.988170i \(0.549010\pi\)
\(402\) 0 0
\(403\) 3.02944 0.150907
\(404\) 0 0
\(405\) −3.41421 −0.169654
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 21.4142 1.05886 0.529432 0.848352i \(-0.322405\pi\)
0.529432 + 0.848352i \(0.322405\pi\)
\(410\) 0 0
\(411\) 14.1421 0.697580
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −24.9706 −1.22576
\(416\) 0 0
\(417\) −17.6569 −0.864660
\(418\) 0 0
\(419\) 33.1716 1.62054 0.810269 0.586059i \(-0.199321\pi\)
0.810269 + 0.586059i \(0.199321\pi\)
\(420\) 0 0
\(421\) 16.6274 0.810371 0.405185 0.914235i \(-0.367207\pi\)
0.405185 + 0.914235i \(0.367207\pi\)
\(422\) 0 0
\(423\) −2.82843 −0.137523
\(424\) 0 0
\(425\) 14.9289 0.724160
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −5.17157 −0.249686
\(430\) 0 0
\(431\) 26.9706 1.29913 0.649563 0.760308i \(-0.274952\pi\)
0.649563 + 0.760308i \(0.274952\pi\)
\(432\) 0 0
\(433\) 20.2426 0.972799 0.486400 0.873736i \(-0.338310\pi\)
0.486400 + 0.873736i \(0.338310\pi\)
\(434\) 0 0
\(435\) 23.3137 1.11781
\(436\) 0 0
\(437\) −21.6569 −1.03599
\(438\) 0 0
\(439\) 12.6863 0.605484 0.302742 0.953073i \(-0.402098\pi\)
0.302742 + 0.953073i \(0.402098\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −34.9706 −1.66150 −0.830751 0.556645i \(-0.812089\pi\)
−0.830751 + 0.556645i \(0.812089\pi\)
\(444\) 0 0
\(445\) −48.6274 −2.30516
\(446\) 0 0
\(447\) 17.3137 0.818910
\(448\) 0 0
\(449\) −5.31371 −0.250769 −0.125385 0.992108i \(-0.540017\pi\)
−0.125385 + 0.992108i \(0.540017\pi\)
\(450\) 0 0
\(451\) −12.4853 −0.587909
\(452\) 0 0
\(453\) −12.0000 −0.563809
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 0 0
\(459\) 2.24264 0.104678
\(460\) 0 0
\(461\) 16.5858 0.772477 0.386239 0.922399i \(-0.373774\pi\)
0.386239 + 0.922399i \(0.373774\pi\)
\(462\) 0 0
\(463\) 26.6274 1.23748 0.618741 0.785595i \(-0.287643\pi\)
0.618741 + 0.785595i \(0.287643\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) 0.201010 0.00930164 0.00465082 0.999989i \(-0.498520\pi\)
0.00465082 + 0.999989i \(0.498520\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −11.7574 −0.541751
\(472\) 0 0
\(473\) −11.3137 −0.520205
\(474\) 0 0
\(475\) −18.8284 −0.863907
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) 1.85786 0.0848880 0.0424440 0.999099i \(-0.486486\pi\)
0.0424440 + 0.999099i \(0.486486\pi\)
\(480\) 0 0
\(481\) 10.3431 0.471607
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.82843 0.400878
\(486\) 0 0
\(487\) −26.6274 −1.20660 −0.603302 0.797513i \(-0.706149\pi\)
−0.603302 + 0.797513i \(0.706149\pi\)
\(488\) 0 0
\(489\) 11.3137 0.511624
\(490\) 0 0
\(491\) −5.02944 −0.226975 −0.113488 0.993539i \(-0.536202\pi\)
−0.113488 + 0.993539i \(0.536202\pi\)
\(492\) 0 0
\(493\) −15.3137 −0.689695
\(494\) 0 0
\(495\) −6.82843 −0.306915
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −3.31371 −0.148342 −0.0741710 0.997246i \(-0.523631\pi\)
−0.0741710 + 0.997246i \(0.523631\pi\)
\(500\) 0 0
\(501\) −19.7990 −0.884554
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) −6.31371 −0.280402
\(508\) 0 0
\(509\) −5.55635 −0.246281 −0.123140 0.992389i \(-0.539297\pi\)
−0.123140 + 0.992389i \(0.539297\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2.82843 −0.124878
\(514\) 0 0
\(515\) −15.3137 −0.674803
\(516\) 0 0
\(517\) −5.65685 −0.248788
\(518\) 0 0
\(519\) −21.0711 −0.924917
\(520\) 0 0
\(521\) 35.4142 1.55152 0.775762 0.631025i \(-0.217366\pi\)
0.775762 + 0.631025i \(0.217366\pi\)
\(522\) 0 0
\(523\) −25.6569 −1.12190 −0.560948 0.827851i \(-0.689563\pi\)
−0.560948 + 0.827851i \(0.689563\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.62742 −0.114452
\(528\) 0 0
\(529\) 35.6274 1.54902
\(530\) 0 0
\(531\) −1.17157 −0.0508419
\(532\) 0 0
\(533\) 16.1421 0.699194
\(534\) 0 0
\(535\) −1.17157 −0.0506515
\(536\) 0 0
\(537\) 19.6569 0.848256
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 17.3137 0.744374 0.372187 0.928158i \(-0.378608\pi\)
0.372187 + 0.928158i \(0.378608\pi\)
\(542\) 0 0
\(543\) 2.58579 0.110967
\(544\) 0 0
\(545\) 19.3137 0.827308
\(546\) 0 0
\(547\) 36.9706 1.58075 0.790374 0.612625i \(-0.209886\pi\)
0.790374 + 0.612625i \(0.209886\pi\)
\(548\) 0 0
\(549\) −12.2426 −0.522503
\(550\) 0 0
\(551\) 19.3137 0.822792
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 13.6569 0.579701
\(556\) 0 0
\(557\) 26.0000 1.10166 0.550828 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) 0 0
\(559\) 14.6274 0.618674
\(560\) 0 0
\(561\) 4.48528 0.189369
\(562\) 0 0
\(563\) −1.17157 −0.0493759 −0.0246880 0.999695i \(-0.507859\pi\)
−0.0246880 + 0.999695i \(0.507859\pi\)
\(564\) 0 0
\(565\) 18.1421 0.763245
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −16.4853 −0.691099 −0.345549 0.938401i \(-0.612307\pi\)
−0.345549 + 0.938401i \(0.612307\pi\)
\(570\) 0 0
\(571\) −22.3431 −0.935032 −0.467516 0.883985i \(-0.654851\pi\)
−0.467516 + 0.883985i \(0.654851\pi\)
\(572\) 0 0
\(573\) 18.0000 0.751961
\(574\) 0 0
\(575\) 50.9706 2.12562
\(576\) 0 0
\(577\) 33.8995 1.41125 0.705627 0.708583i \(-0.250665\pi\)
0.705627 + 0.708583i \(0.250665\pi\)
\(578\) 0 0
\(579\) 5.31371 0.220830
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) 0 0
\(585\) 8.82843 0.365011
\(586\) 0 0
\(587\) −22.8284 −0.942230 −0.471115 0.882072i \(-0.656148\pi\)
−0.471115 + 0.882072i \(0.656148\pi\)
\(588\) 0 0
\(589\) 3.31371 0.136539
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 0 0
\(593\) 6.92893 0.284537 0.142269 0.989828i \(-0.454560\pi\)
0.142269 + 0.989828i \(0.454560\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 21.6569 0.886356
\(598\) 0 0
\(599\) 2.00000 0.0817178 0.0408589 0.999165i \(-0.486991\pi\)
0.0408589 + 0.999165i \(0.486991\pi\)
\(600\) 0 0
\(601\) 15.0711 0.614762 0.307381 0.951587i \(-0.400547\pi\)
0.307381 + 0.951587i \(0.400547\pi\)
\(602\) 0 0
\(603\) 5.65685 0.230365
\(604\) 0 0
\(605\) 23.8995 0.971653
\(606\) 0 0
\(607\) −18.3431 −0.744525 −0.372263 0.928127i \(-0.621418\pi\)
−0.372263 + 0.928127i \(0.621418\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.31371 0.295881
\(612\) 0 0
\(613\) 4.68629 0.189278 0.0946388 0.995512i \(-0.469830\pi\)
0.0946388 + 0.995512i \(0.469830\pi\)
\(614\) 0 0
\(615\) 21.3137 0.859452
\(616\) 0 0
\(617\) −24.4853 −0.985740 −0.492870 0.870103i \(-0.664052\pi\)
−0.492870 + 0.870103i \(0.664052\pi\)
\(618\) 0 0
\(619\) 28.9706 1.16443 0.582213 0.813037i \(-0.302187\pi\)
0.582213 + 0.813037i \(0.302187\pi\)
\(620\) 0 0
\(621\) 7.65685 0.307259
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −13.9706 −0.558823
\(626\) 0 0
\(627\) −5.65685 −0.225913
\(628\) 0 0
\(629\) −8.97056 −0.357680
\(630\) 0 0
\(631\) −23.3137 −0.928104 −0.464052 0.885808i \(-0.653605\pi\)
−0.464052 + 0.885808i \(0.653605\pi\)
\(632\) 0 0
\(633\) −12.9706 −0.515534
\(634\) 0 0
\(635\) −5.65685 −0.224485
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −9.31371 −0.368445
\(640\) 0 0
\(641\) −10.8284 −0.427697 −0.213849 0.976867i \(-0.568600\pi\)
−0.213849 + 0.976867i \(0.568600\pi\)
\(642\) 0 0
\(643\) −34.4264 −1.35764 −0.678822 0.734302i \(-0.737509\pi\)
−0.678822 + 0.734302i \(0.737509\pi\)
\(644\) 0 0
\(645\) 19.3137 0.760477
\(646\) 0 0
\(647\) 26.8284 1.05473 0.527367 0.849638i \(-0.323179\pi\)
0.527367 + 0.849638i \(0.323179\pi\)
\(648\) 0 0
\(649\) −2.34315 −0.0919765
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 36.4853 1.42778 0.713890 0.700258i \(-0.246932\pi\)
0.713890 + 0.700258i \(0.246932\pi\)
\(654\) 0 0
\(655\) 52.2843 2.04292
\(656\) 0 0
\(657\) −13.8995 −0.542271
\(658\) 0 0
\(659\) −9.31371 −0.362811 −0.181405 0.983408i \(-0.558065\pi\)
−0.181405 + 0.983408i \(0.558065\pi\)
\(660\) 0 0
\(661\) 23.5563 0.916236 0.458118 0.888891i \(-0.348524\pi\)
0.458118 + 0.888891i \(0.348524\pi\)
\(662\) 0 0
\(663\) −5.79899 −0.225214
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −52.2843 −2.02446
\(668\) 0 0
\(669\) 24.9706 0.965418
\(670\) 0 0
\(671\) −24.4853 −0.945244
\(672\) 0 0
\(673\) 23.3137 0.898677 0.449339 0.893361i \(-0.351660\pi\)
0.449339 + 0.893361i \(0.351660\pi\)
\(674\) 0 0
\(675\) 6.65685 0.256222
\(676\) 0 0
\(677\) −31.4142 −1.20735 −0.603673 0.797232i \(-0.706297\pi\)
−0.603673 + 0.797232i \(0.706297\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 23.7990 0.911979
\(682\) 0 0
\(683\) 19.6569 0.752149 0.376074 0.926590i \(-0.377274\pi\)
0.376074 + 0.926590i \(0.377274\pi\)
\(684\) 0 0
\(685\) −48.2843 −1.84485
\(686\) 0 0
\(687\) 0.242641 0.00925732
\(688\) 0 0
\(689\) 5.17157 0.197021
\(690\) 0 0
\(691\) −0.686292 −0.0261078 −0.0130539 0.999915i \(-0.504155\pi\)
−0.0130539 + 0.999915i \(0.504155\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 60.2843 2.28671
\(696\) 0 0
\(697\) −14.0000 −0.530288
\(698\) 0 0
\(699\) −6.14214 −0.232317
\(700\) 0 0
\(701\) −17.1716 −0.648561 −0.324281 0.945961i \(-0.605122\pi\)
−0.324281 + 0.945961i \(0.605122\pi\)
\(702\) 0 0
\(703\) 11.3137 0.426705
\(704\) 0 0
\(705\) 9.65685 0.363698
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −36.2843 −1.36268 −0.681342 0.731965i \(-0.738603\pi\)
−0.681342 + 0.731965i \(0.738603\pi\)
\(710\) 0 0
\(711\) −13.6569 −0.512172
\(712\) 0 0
\(713\) −8.97056 −0.335950
\(714\) 0 0
\(715\) 17.6569 0.660329
\(716\) 0 0
\(717\) 15.6569 0.584716
\(718\) 0 0
\(719\) 41.9411 1.56414 0.782070 0.623191i \(-0.214164\pi\)
0.782070 + 0.623191i \(0.214164\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 16.2426 0.604070
\(724\) 0 0
\(725\) −45.4558 −1.68819
\(726\) 0 0
\(727\) 12.4853 0.463053 0.231527 0.972829i \(-0.425628\pi\)
0.231527 + 0.972829i \(0.425628\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.6863 −0.469219
\(732\) 0 0
\(733\) −49.6985 −1.83566 −0.917828 0.396979i \(-0.870059\pi\)
−0.917828 + 0.396979i \(0.870059\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11.3137 0.416746
\(738\) 0 0
\(739\) −4.68629 −0.172388 −0.0861940 0.996278i \(-0.527470\pi\)
−0.0861940 + 0.996278i \(0.527470\pi\)
\(740\) 0 0
\(741\) 7.31371 0.268676
\(742\) 0 0
\(743\) 50.9706 1.86993 0.934964 0.354742i \(-0.115431\pi\)
0.934964 + 0.354742i \(0.115431\pi\)
\(744\) 0 0
\(745\) −59.1127 −2.16572
\(746\) 0 0
\(747\) 7.31371 0.267595
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −13.6569 −0.498346 −0.249173 0.968459i \(-0.580159\pi\)
−0.249173 + 0.968459i \(0.580159\pi\)
\(752\) 0 0
\(753\) −12.4853 −0.454989
\(754\) 0 0
\(755\) 40.9706 1.49107
\(756\) 0 0
\(757\) 26.3431 0.957458 0.478729 0.877963i \(-0.341098\pi\)
0.478729 + 0.877963i \(0.341098\pi\)
\(758\) 0 0
\(759\) 15.3137 0.555852
\(760\) 0 0
\(761\) 18.5269 0.671600 0.335800 0.941933i \(-0.390993\pi\)
0.335800 + 0.941933i \(0.390993\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −7.65685 −0.276834
\(766\) 0 0
\(767\) 3.02944 0.109387
\(768\) 0 0
\(769\) −29.6985 −1.07095 −0.535477 0.844550i \(-0.679868\pi\)
−0.535477 + 0.844550i \(0.679868\pi\)
\(770\) 0 0
\(771\) −23.2132 −0.836003
\(772\) 0 0
\(773\) 9.55635 0.343718 0.171859 0.985122i \(-0.445023\pi\)
0.171859 + 0.985122i \(0.445023\pi\)
\(774\) 0 0
\(775\) −7.79899 −0.280148
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 17.6569 0.632622
\(780\) 0 0
\(781\) −18.6274 −0.666541
\(782\) 0 0
\(783\) −6.82843 −0.244028
\(784\) 0 0
\(785\) 40.1421 1.43273
\(786\) 0 0
\(787\) 24.6863 0.879971 0.439986 0.898005i \(-0.354984\pi\)
0.439986 + 0.898005i \(0.354984\pi\)
\(788\) 0 0
\(789\) −5.31371 −0.189173
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 31.6569 1.12417
\(794\) 0 0
\(795\) 6.82843 0.242179
\(796\) 0 0
\(797\) 8.38478 0.297004 0.148502 0.988912i \(-0.452555\pi\)
0.148502 + 0.988912i \(0.452555\pi\)
\(798\) 0 0
\(799\) −6.34315 −0.224404
\(800\) 0 0
\(801\) 14.2426 0.503239
\(802\) 0 0
\(803\) −27.7990 −0.981005
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14.7279 0.518447
\(808\) 0 0
\(809\) 19.9411 0.701093 0.350546 0.936545i \(-0.385996\pi\)
0.350546 + 0.936545i \(0.385996\pi\)
\(810\) 0 0
\(811\) −17.6569 −0.620016 −0.310008 0.950734i \(-0.600332\pi\)
−0.310008 + 0.950734i \(0.600332\pi\)
\(812\) 0 0
\(813\) −10.1421 −0.355700
\(814\) 0 0
\(815\) −38.6274 −1.35306
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −10.6863 −0.372954 −0.186477 0.982459i \(-0.559707\pi\)
−0.186477 + 0.982459i \(0.559707\pi\)
\(822\) 0 0
\(823\) −8.97056 −0.312694 −0.156347 0.987702i \(-0.549972\pi\)
−0.156347 + 0.987702i \(0.549972\pi\)
\(824\) 0 0
\(825\) 13.3137 0.463524
\(826\) 0 0
\(827\) −47.6569 −1.65719 −0.828596 0.559848i \(-0.810860\pi\)
−0.828596 + 0.559848i \(0.810860\pi\)
\(828\) 0 0
\(829\) 0.727922 0.0252818 0.0126409 0.999920i \(-0.495976\pi\)
0.0126409 + 0.999920i \(0.495976\pi\)
\(830\) 0 0
\(831\) −9.31371 −0.323089
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 67.5980 2.33932
\(836\) 0 0
\(837\) −1.17157 −0.0404955
\(838\) 0 0
\(839\) 50.8284 1.75479 0.877396 0.479767i \(-0.159279\pi\)
0.877396 + 0.479767i \(0.159279\pi\)
\(840\) 0 0
\(841\) 17.6274 0.607842
\(842\) 0 0
\(843\) 0.485281 0.0167140
\(844\) 0 0
\(845\) 21.5563 0.741561
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −8.48528 −0.291214
\(850\) 0 0
\(851\) −30.6274 −1.04989
\(852\) 0 0
\(853\) 49.4975 1.69476 0.847381 0.530986i \(-0.178178\pi\)
0.847381 + 0.530986i \(0.178178\pi\)
\(854\) 0 0
\(855\) 9.65685 0.330257
\(856\) 0 0
\(857\) −15.4142 −0.526540 −0.263270 0.964722i \(-0.584801\pi\)
−0.263270 + 0.964722i \(0.584801\pi\)
\(858\) 0 0
\(859\) 57.4558 1.96037 0.980184 0.198089i \(-0.0634735\pi\)
0.980184 + 0.198089i \(0.0634735\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −17.3137 −0.589365 −0.294683 0.955595i \(-0.595214\pi\)
−0.294683 + 0.955595i \(0.595214\pi\)
\(864\) 0 0
\(865\) 71.9411 2.44607
\(866\) 0 0
\(867\) −11.9706 −0.406542
\(868\) 0 0
\(869\) −27.3137 −0.926554
\(870\) 0 0
\(871\) −14.6274 −0.495631
\(872\) 0 0
\(873\) −2.58579 −0.0875156
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −11.3137 −0.382037 −0.191018 0.981586i \(-0.561179\pi\)
−0.191018 + 0.981586i \(0.561179\pi\)
\(878\) 0 0
\(879\) −16.5858 −0.559425
\(880\) 0 0
\(881\) −21.7574 −0.733024 −0.366512 0.930413i \(-0.619448\pi\)
−0.366512 + 0.930413i \(0.619448\pi\)
\(882\) 0 0
\(883\) 4.68629 0.157706 0.0788531 0.996886i \(-0.474874\pi\)
0.0788531 + 0.996886i \(0.474874\pi\)
\(884\) 0 0
\(885\) 4.00000 0.134459
\(886\) 0 0
\(887\) −2.82843 −0.0949693 −0.0474846 0.998872i \(-0.515121\pi\)
−0.0474846 + 0.998872i \(0.515121\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) −67.1127 −2.24333
\(896\) 0 0
\(897\) −19.7990 −0.661069
\(898\) 0 0
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) −4.48528 −0.149426
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.82843 −0.293467
\(906\) 0 0
\(907\) 16.0000 0.531271 0.265636 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) 0 0
\(909\) −2.92893 −0.0971465
\(910\) 0 0
\(911\) 1.02944 0.0341068 0.0170534 0.999855i \(-0.494571\pi\)
0.0170534 + 0.999855i \(0.494571\pi\)
\(912\) 0 0
\(913\) 14.6274 0.484097
\(914\) 0 0
\(915\) 41.7990 1.38183
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 8.28427 0.273273 0.136636 0.990621i \(-0.456371\pi\)
0.136636 + 0.990621i \(0.456371\pi\)
\(920\) 0 0
\(921\) 30.1421 0.993217
\(922\) 0 0
\(923\) 24.0833 0.792710
\(924\) 0 0
\(925\) −26.6274 −0.875504
\(926\) 0 0
\(927\) 4.48528 0.147316
\(928\) 0 0
\(929\) 39.2132 1.28654 0.643272 0.765638i \(-0.277577\pi\)
0.643272 + 0.765638i \(0.277577\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 6.14214 0.201084
\(934\) 0 0
\(935\) −15.3137 −0.500812
\(936\) 0 0
\(937\) −30.5858 −0.999194 −0.499597 0.866258i \(-0.666519\pi\)
−0.499597 + 0.866258i \(0.666519\pi\)
\(938\) 0 0
\(939\) −1.89949 −0.0619877
\(940\) 0 0
\(941\) −35.2132 −1.14792 −0.573959 0.818884i \(-0.694593\pi\)
−0.573959 + 0.818884i \(0.694593\pi\)
\(942\) 0 0
\(943\) −47.7990 −1.55655
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −30.6863 −0.997170 −0.498585 0.866841i \(-0.666147\pi\)
−0.498585 + 0.866841i \(0.666147\pi\)
\(948\) 0 0
\(949\) 35.9411 1.16670
\(950\) 0 0
\(951\) 10.0000 0.324272
\(952\) 0 0
\(953\) 2.00000 0.0647864 0.0323932 0.999475i \(-0.489687\pi\)
0.0323932 + 0.999475i \(0.489687\pi\)
\(954\) 0 0
\(955\) −61.4558 −1.98866
\(956\) 0 0
\(957\) −13.6569 −0.441463
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29.6274 −0.955723
\(962\) 0 0
\(963\) 0.343146 0.0110577
\(964\) 0 0
\(965\) −18.1421 −0.584016
\(966\) 0 0
\(967\) −33.6569 −1.08233 −0.541166 0.840916i \(-0.682017\pi\)
−0.541166 + 0.840916i \(0.682017\pi\)
\(968\) 0 0
\(969\) −6.34315 −0.203771
\(970\) 0 0
\(971\) −50.6274 −1.62471 −0.812356 0.583162i \(-0.801815\pi\)
−0.812356 + 0.583162i \(0.801815\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −17.2132 −0.551264
\(976\) 0 0
\(977\) −21.1716 −0.677339 −0.338669 0.940905i \(-0.609977\pi\)
−0.338669 + 0.940905i \(0.609977\pi\)
\(978\) 0 0
\(979\) 28.4853 0.910394
\(980\) 0 0
\(981\) −5.65685 −0.180609
\(982\) 0 0
\(983\) −53.2548 −1.69857 −0.849283 0.527938i \(-0.822965\pi\)
−0.849283 + 0.527938i \(0.822965\pi\)
\(984\) 0 0
\(985\) −6.82843 −0.217572
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −43.3137 −1.37730
\(990\) 0 0
\(991\) 12.9706 0.412024 0.206012 0.978550i \(-0.433951\pi\)
0.206012 + 0.978550i \(0.433951\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) −73.9411 −2.34409
\(996\) 0 0
\(997\) −26.3848 −0.835614 −0.417807 0.908536i \(-0.637201\pi\)
−0.417807 + 0.908536i \(0.637201\pi\)
\(998\) 0 0
\(999\) −4.00000 −0.126554
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 2352.2.a.be.1.1 2
3.2 odd 2 7056.2.a.cv.1.2 2
4.3 odd 2 147.2.a.d.1.2 2
7.2 even 3 2352.2.q.bb.1537.2 4
7.3 odd 6 2352.2.q.bd.961.1 4
7.4 even 3 2352.2.q.bb.961.2 4
7.5 odd 6 2352.2.q.bd.1537.1 4
7.6 odd 2 2352.2.a.bc.1.2 2
8.3 odd 2 9408.2.a.ef.1.2 2
8.5 even 2 9408.2.a.dq.1.2 2
12.11 even 2 441.2.a.j.1.1 2
20.19 odd 2 3675.2.a.bf.1.1 2
21.20 even 2 7056.2.a.cf.1.1 2
28.3 even 6 147.2.e.d.79.1 4
28.11 odd 6 147.2.e.e.79.1 4
28.19 even 6 147.2.e.d.67.1 4
28.23 odd 6 147.2.e.e.67.1 4
28.27 even 2 147.2.a.e.1.2 yes 2
56.13 odd 2 9408.2.a.dt.1.1 2
56.27 even 2 9408.2.a.di.1.1 2
84.11 even 6 441.2.e.f.226.2 4
84.23 even 6 441.2.e.f.361.2 4
84.47 odd 6 441.2.e.g.361.2 4
84.59 odd 6 441.2.e.g.226.2 4
84.83 odd 2 441.2.a.i.1.1 2
140.139 even 2 3675.2.a.bd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.a.d.1.2 2 4.3 odd 2
147.2.a.e.1.2 yes 2 28.27 even 2
147.2.e.d.67.1 4 28.19 even 6
147.2.e.d.79.1 4 28.3 even 6
147.2.e.e.67.1 4 28.23 odd 6
147.2.e.e.79.1 4 28.11 odd 6
441.2.a.i.1.1 2 84.83 odd 2
441.2.a.j.1.1 2 12.11 even 2
441.2.e.f.226.2 4 84.11 even 6
441.2.e.f.361.2 4 84.23 even 6
441.2.e.g.226.2 4 84.59 odd 6
441.2.e.g.361.2 4 84.47 odd 6
2352.2.a.bc.1.2 2 7.6 odd 2
2352.2.a.be.1.1 2 1.1 even 1 trivial
2352.2.q.bb.961.2 4 7.4 even 3
2352.2.q.bb.1537.2 4 7.2 even 3
2352.2.q.bd.961.1 4 7.3 odd 6
2352.2.q.bd.1537.1 4 7.5 odd 6
3675.2.a.bd.1.1 2 140.139 even 2
3675.2.a.bf.1.1 2 20.19 odd 2
7056.2.a.cf.1.1 2 21.20 even 2
7056.2.a.cv.1.2 2 3.2 odd 2
9408.2.a.di.1.1 2 56.27 even 2
9408.2.a.dq.1.2 2 8.5 even 2
9408.2.a.dt.1.1 2 56.13 odd 2
9408.2.a.ef.1.2 2 8.3 odd 2