Properties

Label 7728.2.a.cb.1.3
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
Defining polynomial: \(x^{4} - x^{3} - 5 x^{2} + 2 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.75660\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.703671 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.703671 q^{5} -1.00000 q^{7} +1.00000 q^{9} -0.606168 q^{11} -1.30984 q^{13} +0.703671 q^{15} -3.67096 q^{17} +0.606168 q^{19} -1.00000 q^{21} +1.00000 q^{23} -4.50485 q^{25} +1.00000 q^{27} +5.63256 q^{29} -1.86597 q^{31} -0.606168 q^{33} -0.703671 q^{35} -2.80118 q^{37} -1.30984 q^{39} -0.329039 q^{41} +2.99431 q^{43} +0.703671 q^{45} +4.19501 q^{47} +1.00000 q^{49} -3.67096 q^{51} -10.0072 q^{53} -0.426543 q^{55} +0.606168 q^{57} +9.73006 q^{59} -13.9386 q^{61} -1.00000 q^{63} -0.921696 q^{65} -8.66527 q^{67} +1.00000 q^{69} +3.86410 q^{71} +5.82757 q^{73} -4.50485 q^{75} +0.606168 q^{77} +1.86597 q^{79} +1.00000 q^{81} -13.5504 q^{83} -2.58315 q^{85} +5.63256 q^{87} -14.9122 q^{89} +1.30984 q^{91} -1.86597 q^{93} +0.426543 q^{95} +5.82757 q^{97} -0.606168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} - 3 q^{15} - 8 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} - q^{25} + 4 q^{27} - 10 q^{29} + 10 q^{31} - 2 q^{33} + 3 q^{35} + q^{39} - 8 q^{41} - 13 q^{43} - 3 q^{45} + 6 q^{47} + 4 q^{49} - 8 q^{51} + 5 q^{53} - 3 q^{55} + 2 q^{57} + q^{59} + 5 q^{61} - 4 q^{63} - 22 q^{65} - 3 q^{67} + 4 q^{69} - 5 q^{71} - 20 q^{73} - q^{75} + 2 q^{77} - 10 q^{79} + 4 q^{81} - 18 q^{83} + 25 q^{85} - 10 q^{87} - 31 q^{89} - q^{91} + 10 q^{93} + 3 q^{95} - 20 q^{97} - 2 q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0.703671 0.314691 0.157346 0.987544i \(-0.449706\pi\)
0.157346 + 0.987544i \(0.449706\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.606168 −0.182766 −0.0913832 0.995816i \(-0.529129\pi\)
−0.0913832 + 0.995816i \(0.529129\pi\)
\(12\) 0 0
\(13\) −1.30984 −0.363284 −0.181642 0.983365i \(-0.558141\pi\)
−0.181642 + 0.983365i \(0.558141\pi\)
\(14\) 0 0
\(15\) 0.703671 0.181687
\(16\) 0 0
\(17\) −3.67096 −0.890339 −0.445169 0.895446i \(-0.646857\pi\)
−0.445169 + 0.895446i \(0.646857\pi\)
\(18\) 0 0
\(19\) 0.606168 0.139064 0.0695322 0.997580i \(-0.477849\pi\)
0.0695322 + 0.997580i \(0.477849\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.50485 −0.900969
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.63256 1.04594 0.522970 0.852351i \(-0.324824\pi\)
0.522970 + 0.852351i \(0.324824\pi\)
\(30\) 0 0
\(31\) −1.86597 −0.335138 −0.167569 0.985860i \(-0.553592\pi\)
−0.167569 + 0.985860i \(0.553592\pi\)
\(32\) 0 0
\(33\) −0.606168 −0.105520
\(34\) 0 0
\(35\) −0.703671 −0.118942
\(36\) 0 0
\(37\) −2.80118 −0.460510 −0.230255 0.973130i \(-0.573956\pi\)
−0.230255 + 0.973130i \(0.573956\pi\)
\(38\) 0 0
\(39\) −1.30984 −0.209742
\(40\) 0 0
\(41\) −0.329039 −0.0513873 −0.0256936 0.999670i \(-0.508179\pi\)
−0.0256936 + 0.999670i \(0.508179\pi\)
\(42\) 0 0
\(43\) 2.99431 0.456628 0.228314 0.973588i \(-0.426679\pi\)
0.228314 + 0.973588i \(0.426679\pi\)
\(44\) 0 0
\(45\) 0.703671 0.104897
\(46\) 0 0
\(47\) 4.19501 0.611905 0.305952 0.952047i \(-0.401025\pi\)
0.305952 + 0.952047i \(0.401025\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.67096 −0.514037
\(52\) 0 0
\(53\) −10.0072 −1.37459 −0.687297 0.726377i \(-0.741203\pi\)
−0.687297 + 0.726377i \(0.741203\pi\)
\(54\) 0 0
\(55\) −0.426543 −0.0575150
\(56\) 0 0
\(57\) 0.606168 0.0802889
\(58\) 0 0
\(59\) 9.73006 1.26675 0.633373 0.773846i \(-0.281670\pi\)
0.633373 + 0.773846i \(0.281670\pi\)
\(60\) 0 0
\(61\) −13.9386 −1.78465 −0.892326 0.451391i \(-0.850928\pi\)
−0.892326 + 0.451391i \(0.850928\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −0.921696 −0.114322
\(66\) 0 0
\(67\) −8.66527 −1.05863 −0.529316 0.848425i \(-0.677551\pi\)
−0.529316 + 0.848425i \(0.677551\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 3.86410 0.458584 0.229292 0.973358i \(-0.426359\pi\)
0.229292 + 0.973358i \(0.426359\pi\)
\(72\) 0 0
\(73\) 5.82757 0.682065 0.341033 0.940051i \(-0.389223\pi\)
0.341033 + 0.940051i \(0.389223\pi\)
\(74\) 0 0
\(75\) −4.50485 −0.520175
\(76\) 0 0
\(77\) 0.606168 0.0690792
\(78\) 0 0
\(79\) 1.86597 0.209938 0.104969 0.994476i \(-0.466526\pi\)
0.104969 + 0.994476i \(0.466526\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.5504 −1.48735 −0.743677 0.668539i \(-0.766920\pi\)
−0.743677 + 0.668539i \(0.766920\pi\)
\(84\) 0 0
\(85\) −2.58315 −0.280182
\(86\) 0 0
\(87\) 5.63256 0.603874
\(88\) 0 0
\(89\) −14.9122 −1.58069 −0.790344 0.612663i \(-0.790098\pi\)
−0.790344 + 0.612663i \(0.790098\pi\)
\(90\) 0 0
\(91\) 1.30984 0.137308
\(92\) 0 0
\(93\) −1.86597 −0.193492
\(94\) 0 0
\(95\) 0.426543 0.0437624
\(96\) 0 0
\(97\) 5.82757 0.591700 0.295850 0.955234i \(-0.404397\pi\)
0.295850 + 0.955234i \(0.404397\pi\)
\(98\) 0 0
\(99\) −0.606168 −0.0609221
\(100\) 0 0
\(101\) −2.95378 −0.293912 −0.146956 0.989143i \(-0.546948\pi\)
−0.146956 + 0.989143i \(0.546948\pi\)
\(102\) 0 0
\(103\) −7.36894 −0.726084 −0.363042 0.931773i \(-0.618262\pi\)
−0.363042 + 0.931773i \(0.618262\pi\)
\(104\) 0 0
\(105\) −0.703671 −0.0686713
\(106\) 0 0
\(107\) −19.6398 −1.89865 −0.949323 0.314301i \(-0.898230\pi\)
−0.949323 + 0.314301i \(0.898230\pi\)
\(108\) 0 0
\(109\) 15.6095 1.49512 0.747562 0.664193i \(-0.231224\pi\)
0.747562 + 0.664193i \(0.231224\pi\)
\(110\) 0 0
\(111\) −2.80118 −0.265876
\(112\) 0 0
\(113\) −6.78579 −0.638354 −0.319177 0.947695i \(-0.603406\pi\)
−0.319177 + 0.947695i \(0.603406\pi\)
\(114\) 0 0
\(115\) 0.703671 0.0656177
\(116\) 0 0
\(117\) −1.30984 −0.121095
\(118\) 0 0
\(119\) 3.67096 0.336516
\(120\) 0 0
\(121\) −10.6326 −0.966596
\(122\) 0 0
\(123\) −0.329039 −0.0296685
\(124\) 0 0
\(125\) −6.68829 −0.598219
\(126\) 0 0
\(127\) −9.27144 −0.822707 −0.411354 0.911476i \(-0.634944\pi\)
−0.411354 + 0.911476i \(0.634944\pi\)
\(128\) 0 0
\(129\) 2.99431 0.263634
\(130\) 0 0
\(131\) 2.88330 0.251915 0.125957 0.992036i \(-0.459800\pi\)
0.125957 + 0.992036i \(0.459800\pi\)
\(132\) 0 0
\(133\) −0.606168 −0.0525614
\(134\) 0 0
\(135\) 0.703671 0.0605624
\(136\) 0 0
\(137\) 3.23491 0.276377 0.138189 0.990406i \(-0.455872\pi\)
0.138189 + 0.990406i \(0.455872\pi\)
\(138\) 0 0
\(139\) 7.30539 0.619635 0.309818 0.950796i \(-0.399732\pi\)
0.309818 + 0.950796i \(0.399732\pi\)
\(140\) 0 0
\(141\) 4.19501 0.353283
\(142\) 0 0
\(143\) 0.793982 0.0663961
\(144\) 0 0
\(145\) 3.96347 0.329148
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −20.5633 −1.68461 −0.842306 0.538999i \(-0.818803\pi\)
−0.842306 + 0.538999i \(0.818803\pi\)
\(150\) 0 0
\(151\) −3.17394 −0.258291 −0.129146 0.991626i \(-0.541223\pi\)
−0.129146 + 0.991626i \(0.541223\pi\)
\(152\) 0 0
\(153\) −3.67096 −0.296780
\(154\) 0 0
\(155\) −1.31303 −0.105465
\(156\) 0 0
\(157\) −17.2214 −1.37442 −0.687209 0.726460i \(-0.741164\pi\)
−0.687209 + 0.726460i \(0.741164\pi\)
\(158\) 0 0
\(159\) −10.0072 −0.793622
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 11.6917 0.915762 0.457881 0.889014i \(-0.348609\pi\)
0.457881 + 0.889014i \(0.348609\pi\)
\(164\) 0 0
\(165\) −0.426543 −0.0332063
\(166\) 0 0
\(167\) 12.3548 0.956043 0.478022 0.878348i \(-0.341354\pi\)
0.478022 + 0.878348i \(0.341354\pi\)
\(168\) 0 0
\(169\) −11.2843 −0.868025
\(170\) 0 0
\(171\) 0.606168 0.0463548
\(172\) 0 0
\(173\) −9.37713 −0.712930 −0.356465 0.934309i \(-0.616018\pi\)
−0.356465 + 0.934309i \(0.616018\pi\)
\(174\) 0 0
\(175\) 4.50485 0.340534
\(176\) 0 0
\(177\) 9.73006 0.731356
\(178\) 0 0
\(179\) 4.68197 0.349947 0.174973 0.984573i \(-0.444016\pi\)
0.174973 + 0.984573i \(0.444016\pi\)
\(180\) 0 0
\(181\) 0.930761 0.0691829 0.0345915 0.999402i \(-0.488987\pi\)
0.0345915 + 0.999402i \(0.488987\pi\)
\(182\) 0 0
\(183\) −13.9386 −1.03037
\(184\) 0 0
\(185\) −1.97111 −0.144919
\(186\) 0 0
\(187\) 2.22522 0.162724
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 3.17394 0.229658 0.114829 0.993385i \(-0.463368\pi\)
0.114829 + 0.993385i \(0.463368\pi\)
\(192\) 0 0
\(193\) −16.8789 −1.21497 −0.607483 0.794333i \(-0.707821\pi\)
−0.607483 + 0.794333i \(0.707821\pi\)
\(194\) 0 0
\(195\) −0.921696 −0.0660040
\(196\) 0 0
\(197\) 2.88517 0.205560 0.102780 0.994704i \(-0.467226\pi\)
0.102780 + 0.994704i \(0.467226\pi\)
\(198\) 0 0
\(199\) 16.3227 1.15709 0.578544 0.815652i \(-0.303621\pi\)
0.578544 + 0.815652i \(0.303621\pi\)
\(200\) 0 0
\(201\) −8.66527 −0.611201
\(202\) 0 0
\(203\) −5.63256 −0.395328
\(204\) 0 0
\(205\) −0.231535 −0.0161711
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −0.367439 −0.0254163
\(210\) 0 0
\(211\) −7.82757 −0.538872 −0.269436 0.963018i \(-0.586837\pi\)
−0.269436 + 0.963018i \(0.586837\pi\)
\(212\) 0 0
\(213\) 3.86410 0.264764
\(214\) 0 0
\(215\) 2.10701 0.143697
\(216\) 0 0
\(217\) 1.86597 0.126670
\(218\) 0 0
\(219\) 5.82757 0.393791
\(220\) 0 0
\(221\) 4.80837 0.323446
\(222\) 0 0
\(223\) −10.7474 −0.719699 −0.359849 0.933010i \(-0.617172\pi\)
−0.359849 + 0.933010i \(0.617172\pi\)
\(224\) 0 0
\(225\) −4.50485 −0.300323
\(226\) 0 0
\(227\) 7.55676 0.501560 0.250780 0.968044i \(-0.419313\pi\)
0.250780 + 0.968044i \(0.419313\pi\)
\(228\) 0 0
\(229\) 9.28495 0.613567 0.306783 0.951779i \(-0.400747\pi\)
0.306783 + 0.951779i \(0.400747\pi\)
\(230\) 0 0
\(231\) 0.606168 0.0398829
\(232\) 0 0
\(233\) −0.629185 −0.0412193 −0.0206096 0.999788i \(-0.506561\pi\)
−0.0206096 + 0.999788i \(0.506561\pi\)
\(234\) 0 0
\(235\) 2.95191 0.192561
\(236\) 0 0
\(237\) 1.86597 0.121208
\(238\) 0 0
\(239\) 10.7474 0.695191 0.347596 0.937645i \(-0.386998\pi\)
0.347596 + 0.937645i \(0.386998\pi\)
\(240\) 0 0
\(241\) −19.5337 −1.25828 −0.629139 0.777292i \(-0.716593\pi\)
−0.629139 + 0.777292i \(0.716593\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0.703671 0.0449559
\(246\) 0 0
\(247\) −0.793982 −0.0505199
\(248\) 0 0
\(249\) −13.5504 −0.858724
\(250\) 0 0
\(251\) 19.1176 1.20669 0.603345 0.797480i \(-0.293834\pi\)
0.603345 + 0.797480i \(0.293834\pi\)
\(252\) 0 0
\(253\) −0.606168 −0.0381094
\(254\) 0 0
\(255\) −2.58315 −0.161763
\(256\) 0 0
\(257\) −27.8848 −1.73941 −0.869703 0.493575i \(-0.835690\pi\)
−0.869703 + 0.493575i \(0.835690\pi\)
\(258\) 0 0
\(259\) 2.80118 0.174057
\(260\) 0 0
\(261\) 5.63256 0.348647
\(262\) 0 0
\(263\) −9.01288 −0.555758 −0.277879 0.960616i \(-0.589631\pi\)
−0.277879 + 0.960616i \(0.589631\pi\)
\(264\) 0 0
\(265\) −7.04178 −0.432573
\(266\) 0 0
\(267\) −14.9122 −0.912611
\(268\) 0 0
\(269\) −1.77291 −0.108096 −0.0540481 0.998538i \(-0.517212\pi\)
−0.0540481 + 0.998538i \(0.517212\pi\)
\(270\) 0 0
\(271\) 18.0880 1.09877 0.549384 0.835570i \(-0.314863\pi\)
0.549384 + 0.835570i \(0.314863\pi\)
\(272\) 0 0
\(273\) 1.30984 0.0792751
\(274\) 0 0
\(275\) 2.73069 0.164667
\(276\) 0 0
\(277\) 12.6788 0.761794 0.380897 0.924617i \(-0.375615\pi\)
0.380897 + 0.924617i \(0.375615\pi\)
\(278\) 0 0
\(279\) −1.86597 −0.111713
\(280\) 0 0
\(281\) −33.3229 −1.98788 −0.993939 0.109933i \(-0.964936\pi\)
−0.993939 + 0.109933i \(0.964936\pi\)
\(282\) 0 0
\(283\) −12.1752 −0.723739 −0.361870 0.932229i \(-0.617861\pi\)
−0.361870 + 0.932229i \(0.617861\pi\)
\(284\) 0 0
\(285\) 0.426543 0.0252662
\(286\) 0 0
\(287\) 0.329039 0.0194226
\(288\) 0 0
\(289\) −3.52405 −0.207297
\(290\) 0 0
\(291\) 5.82757 0.341618
\(292\) 0 0
\(293\) 16.1091 0.941106 0.470553 0.882372i \(-0.344054\pi\)
0.470553 + 0.882372i \(0.344054\pi\)
\(294\) 0 0
\(295\) 6.84677 0.398634
\(296\) 0 0
\(297\) −0.606168 −0.0351734
\(298\) 0 0
\(299\) −1.30984 −0.0757500
\(300\) 0 0
\(301\) −2.99431 −0.172589
\(302\) 0 0
\(303\) −2.95378 −0.169690
\(304\) 0 0
\(305\) −9.80818 −0.561615
\(306\) 0 0
\(307\) −13.1311 −0.749431 −0.374715 0.927140i \(-0.622260\pi\)
−0.374715 + 0.927140i \(0.622260\pi\)
\(308\) 0 0
\(309\) −7.36894 −0.419205
\(310\) 0 0
\(311\) 4.98899 0.282900 0.141450 0.989945i \(-0.454824\pi\)
0.141450 + 0.989945i \(0.454824\pi\)
\(312\) 0 0
\(313\) 22.8945 1.29407 0.647037 0.762459i \(-0.276008\pi\)
0.647037 + 0.762459i \(0.276008\pi\)
\(314\) 0 0
\(315\) −0.703671 −0.0396474
\(316\) 0 0
\(317\) 12.9424 0.726918 0.363459 0.931610i \(-0.381596\pi\)
0.363459 + 0.931610i \(0.381596\pi\)
\(318\) 0 0
\(319\) −3.41428 −0.191163
\(320\) 0 0
\(321\) −19.6398 −1.09618
\(322\) 0 0
\(323\) −2.22522 −0.123814
\(324\) 0 0
\(325\) 5.90062 0.327308
\(326\) 0 0
\(327\) 15.6095 0.863210
\(328\) 0 0
\(329\) −4.19501 −0.231278
\(330\) 0 0
\(331\) −29.1537 −1.60243 −0.801215 0.598376i \(-0.795813\pi\)
−0.801215 + 0.598376i \(0.795813\pi\)
\(332\) 0 0
\(333\) −2.80118 −0.153503
\(334\) 0 0
\(335\) −6.09750 −0.333142
\(336\) 0 0
\(337\) 30.2043 1.64533 0.822667 0.568523i \(-0.192485\pi\)
0.822667 + 0.568523i \(0.192485\pi\)
\(338\) 0 0
\(339\) −6.78579 −0.368554
\(340\) 0 0
\(341\) 1.13109 0.0612519
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0.703671 0.0378844
\(346\) 0 0
\(347\) −11.9918 −0.643754 −0.321877 0.946781i \(-0.604314\pi\)
−0.321877 + 0.946781i \(0.604314\pi\)
\(348\) 0 0
\(349\) −1.57346 −0.0842252 −0.0421126 0.999113i \(-0.513409\pi\)
−0.0421126 + 0.999113i \(0.513409\pi\)
\(350\) 0 0
\(351\) −1.30984 −0.0699140
\(352\) 0 0
\(353\) −25.8276 −1.37466 −0.687331 0.726344i \(-0.741218\pi\)
−0.687331 + 0.726344i \(0.741218\pi\)
\(354\) 0 0
\(355\) 2.71905 0.144312
\(356\) 0 0
\(357\) 3.67096 0.194288
\(358\) 0 0
\(359\) 18.0433 0.952288 0.476144 0.879367i \(-0.342034\pi\)
0.476144 + 0.879367i \(0.342034\pi\)
\(360\) 0 0
\(361\) −18.6326 −0.980661
\(362\) 0 0
\(363\) −10.6326 −0.558065
\(364\) 0 0
\(365\) 4.10069 0.214640
\(366\) 0 0
\(367\) 22.0477 1.15088 0.575441 0.817843i \(-0.304830\pi\)
0.575441 + 0.817843i \(0.304830\pi\)
\(368\) 0 0
\(369\) −0.329039 −0.0171291
\(370\) 0 0
\(371\) 10.0072 0.519548
\(372\) 0 0
\(373\) 4.09031 0.211788 0.105894 0.994377i \(-0.466230\pi\)
0.105894 + 0.994377i \(0.466230\pi\)
\(374\) 0 0
\(375\) −6.68829 −0.345382
\(376\) 0 0
\(377\) −7.37775 −0.379973
\(378\) 0 0
\(379\) 17.2100 0.884019 0.442010 0.897010i \(-0.354266\pi\)
0.442010 + 0.897010i \(0.354266\pi\)
\(380\) 0 0
\(381\) −9.27144 −0.474990
\(382\) 0 0
\(383\) 27.9458 1.42796 0.713981 0.700165i \(-0.246890\pi\)
0.713981 + 0.700165i \(0.246890\pi\)
\(384\) 0 0
\(385\) 0.426543 0.0217386
\(386\) 0 0
\(387\) 2.99431 0.152209
\(388\) 0 0
\(389\) −28.6897 −1.45463 −0.727313 0.686306i \(-0.759231\pi\)
−0.727313 + 0.686306i \(0.759231\pi\)
\(390\) 0 0
\(391\) −3.67096 −0.185648
\(392\) 0 0
\(393\) 2.88330 0.145443
\(394\) 0 0
\(395\) 1.31303 0.0660656
\(396\) 0 0
\(397\) −18.6995 −0.938500 −0.469250 0.883065i \(-0.655476\pi\)
−0.469250 + 0.883065i \(0.655476\pi\)
\(398\) 0 0
\(399\) −0.606168 −0.0303463
\(400\) 0 0
\(401\) −9.54137 −0.476474 −0.238237 0.971207i \(-0.576569\pi\)
−0.238237 + 0.971207i \(0.576569\pi\)
\(402\) 0 0
\(403\) 2.44412 0.121750
\(404\) 0 0
\(405\) 0.703671 0.0349657
\(406\) 0 0
\(407\) 1.69798 0.0841658
\(408\) 0 0
\(409\) 37.4185 1.85023 0.925114 0.379690i \(-0.123969\pi\)
0.925114 + 0.379690i \(0.123969\pi\)
\(410\) 0 0
\(411\) 3.23491 0.159566
\(412\) 0 0
\(413\) −9.73006 −0.478785
\(414\) 0 0
\(415\) −9.53506 −0.468058
\(416\) 0 0
\(417\) 7.30539 0.357747
\(418\) 0 0
\(419\) −38.4770 −1.87973 −0.939863 0.341553i \(-0.889047\pi\)
−0.939863 + 0.341553i \(0.889047\pi\)
\(420\) 0 0
\(421\) 20.1412 0.981623 0.490812 0.871266i \(-0.336700\pi\)
0.490812 + 0.871266i \(0.336700\pi\)
\(422\) 0 0
\(423\) 4.19501 0.203968
\(424\) 0 0
\(425\) 16.5371 0.802168
\(426\) 0 0
\(427\) 13.9386 0.674535
\(428\) 0 0
\(429\) 0.793982 0.0383338
\(430\) 0 0
\(431\) −3.81005 −0.183524 −0.0917619 0.995781i \(-0.529250\pi\)
−0.0917619 + 0.995781i \(0.529250\pi\)
\(432\) 0 0
\(433\) −17.9874 −0.864418 −0.432209 0.901774i \(-0.642266\pi\)
−0.432209 + 0.901774i \(0.642266\pi\)
\(434\) 0 0
\(435\) 3.96347 0.190034
\(436\) 0 0
\(437\) 0.606168 0.0289969
\(438\) 0 0
\(439\) −25.4713 −1.21568 −0.607840 0.794060i \(-0.707964\pi\)
−0.607840 + 0.794060i \(0.707964\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −12.6851 −0.602687 −0.301344 0.953516i \(-0.597435\pi\)
−0.301344 + 0.953516i \(0.597435\pi\)
\(444\) 0 0
\(445\) −10.4933 −0.497429
\(446\) 0 0
\(447\) −20.5633 −0.972612
\(448\) 0 0
\(449\) 14.5718 0.687684 0.343842 0.939027i \(-0.388271\pi\)
0.343842 + 0.939027i \(0.388271\pi\)
\(450\) 0 0
\(451\) 0.199453 0.00939187
\(452\) 0 0
\(453\) −3.17394 −0.149124
\(454\) 0 0
\(455\) 0.921696 0.0432098
\(456\) 0 0
\(457\) 3.80686 0.178078 0.0890388 0.996028i \(-0.471620\pi\)
0.0890388 + 0.996028i \(0.471620\pi\)
\(458\) 0 0
\(459\) −3.67096 −0.171346
\(460\) 0 0
\(461\) 14.0673 0.655179 0.327590 0.944820i \(-0.393764\pi\)
0.327590 + 0.944820i \(0.393764\pi\)
\(462\) 0 0
\(463\) −23.2966 −1.08268 −0.541342 0.840802i \(-0.682084\pi\)
−0.541342 + 0.840802i \(0.682084\pi\)
\(464\) 0 0
\(465\) −1.31303 −0.0608902
\(466\) 0 0
\(467\) −15.7838 −0.730389 −0.365195 0.930931i \(-0.618998\pi\)
−0.365195 + 0.930931i \(0.618998\pi\)
\(468\) 0 0
\(469\) 8.66527 0.400125
\(470\) 0 0
\(471\) −17.2214 −0.793520
\(472\) 0 0
\(473\) −1.81505 −0.0834563
\(474\) 0 0
\(475\) −2.73069 −0.125293
\(476\) 0 0
\(477\) −10.0072 −0.458198
\(478\) 0 0
\(479\) 23.3608 1.06738 0.533690 0.845680i \(-0.320805\pi\)
0.533690 + 0.845680i \(0.320805\pi\)
\(480\) 0 0
\(481\) 3.66909 0.167296
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 4.10069 0.186203
\(486\) 0 0
\(487\) 19.5558 0.886156 0.443078 0.896483i \(-0.353886\pi\)
0.443078 + 0.896483i \(0.353886\pi\)
\(488\) 0 0
\(489\) 11.6917 0.528715
\(490\) 0 0
\(491\) 36.7045 1.65645 0.828227 0.560393i \(-0.189350\pi\)
0.828227 + 0.560393i \(0.189350\pi\)
\(492\) 0 0
\(493\) −20.6769 −0.931241
\(494\) 0 0
\(495\) −0.426543 −0.0191717
\(496\) 0 0
\(497\) −3.86410 −0.173328
\(498\) 0 0
\(499\) −21.0720 −0.943312 −0.471656 0.881783i \(-0.656344\pi\)
−0.471656 + 0.881783i \(0.656344\pi\)
\(500\) 0 0
\(501\) 12.3548 0.551972
\(502\) 0 0
\(503\) −10.9154 −0.486693 −0.243346 0.969939i \(-0.578245\pi\)
−0.243346 + 0.969939i \(0.578245\pi\)
\(504\) 0 0
\(505\) −2.07849 −0.0924916
\(506\) 0 0
\(507\) −11.2843 −0.501154
\(508\) 0 0
\(509\) −9.53693 −0.422717 −0.211358 0.977409i \(-0.567789\pi\)
−0.211358 + 0.977409i \(0.567789\pi\)
\(510\) 0 0
\(511\) −5.82757 −0.257796
\(512\) 0 0
\(513\) 0.606168 0.0267630
\(514\) 0 0
\(515\) −5.18531 −0.228492
\(516\) 0 0
\(517\) −2.54288 −0.111836
\(518\) 0 0
\(519\) −9.37713 −0.411610
\(520\) 0 0
\(521\) 7.81174 0.342239 0.171119 0.985250i \(-0.445262\pi\)
0.171119 + 0.985250i \(0.445262\pi\)
\(522\) 0 0
\(523\) −0.869786 −0.0380331 −0.0190165 0.999819i \(-0.506054\pi\)
−0.0190165 + 0.999819i \(0.506054\pi\)
\(524\) 0 0
\(525\) 4.50485 0.196608
\(526\) 0 0
\(527\) 6.84990 0.298386
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 9.73006 0.422249
\(532\) 0 0
\(533\) 0.430988 0.0186682
\(534\) 0 0
\(535\) −13.8199 −0.597488
\(536\) 0 0
\(537\) 4.68197 0.202042
\(538\) 0 0
\(539\) −0.606168 −0.0261095
\(540\) 0 0
\(541\) 33.5551 1.44264 0.721322 0.692600i \(-0.243535\pi\)
0.721322 + 0.692600i \(0.243535\pi\)
\(542\) 0 0
\(543\) 0.930761 0.0399428
\(544\) 0 0
\(545\) 10.9840 0.470502
\(546\) 0 0
\(547\) 12.8500 0.549425 0.274712 0.961526i \(-0.411417\pi\)
0.274712 + 0.961526i \(0.411417\pi\)
\(548\) 0 0
\(549\) −13.9386 −0.594884
\(550\) 0 0
\(551\) 3.41428 0.145453
\(552\) 0 0
\(553\) −1.86597 −0.0793490
\(554\) 0 0
\(555\) −1.97111 −0.0836688
\(556\) 0 0
\(557\) −24.6444 −1.04422 −0.522108 0.852879i \(-0.674854\pi\)
−0.522108 + 0.852879i \(0.674854\pi\)
\(558\) 0 0
\(559\) −3.92206 −0.165886
\(560\) 0 0
\(561\) 2.22522 0.0939488
\(562\) 0 0
\(563\) 11.9656 0.504290 0.252145 0.967689i \(-0.418864\pi\)
0.252145 + 0.967689i \(0.418864\pi\)
\(564\) 0 0
\(565\) −4.77497 −0.200884
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 22.0270 0.923421 0.461710 0.887031i \(-0.347236\pi\)
0.461710 + 0.887031i \(0.347236\pi\)
\(570\) 0 0
\(571\) −13.2056 −0.552636 −0.276318 0.961066i \(-0.589114\pi\)
−0.276318 + 0.961066i \(0.589114\pi\)
\(572\) 0 0
\(573\) 3.17394 0.132593
\(574\) 0 0
\(575\) −4.50485 −0.187865
\(576\) 0 0
\(577\) −32.9971 −1.37369 −0.686843 0.726806i \(-0.741004\pi\)
−0.686843 + 0.726806i \(0.741004\pi\)
\(578\) 0 0
\(579\) −16.8789 −0.701461
\(580\) 0 0
\(581\) 13.5504 0.562167
\(582\) 0 0
\(583\) 6.06604 0.251230
\(584\) 0 0
\(585\) −0.921696 −0.0381075
\(586\) 0 0
\(587\) −36.1131 −1.49055 −0.745275 0.666758i \(-0.767682\pi\)
−0.745275 + 0.666758i \(0.767682\pi\)
\(588\) 0 0
\(589\) −1.13109 −0.0466057
\(590\) 0 0
\(591\) 2.88517 0.118680
\(592\) 0 0
\(593\) −37.2461 −1.52951 −0.764757 0.644319i \(-0.777141\pi\)
−0.764757 + 0.644319i \(0.777141\pi\)
\(594\) 0 0
\(595\) 2.58315 0.105899
\(596\) 0 0
\(597\) 16.3227 0.668045
\(598\) 0 0
\(599\) 1.20727 0.0493279 0.0246639 0.999696i \(-0.492148\pi\)
0.0246639 + 0.999696i \(0.492148\pi\)
\(600\) 0 0
\(601\) 31.6967 1.29293 0.646467 0.762942i \(-0.276246\pi\)
0.646467 + 0.762942i \(0.276246\pi\)
\(602\) 0 0
\(603\) −8.66527 −0.352877
\(604\) 0 0
\(605\) −7.48183 −0.304180
\(606\) 0 0
\(607\) 6.98399 0.283471 0.141736 0.989905i \(-0.454732\pi\)
0.141736 + 0.989905i \(0.454732\pi\)
\(608\) 0 0
\(609\) −5.63256 −0.228243
\(610\) 0 0
\(611\) −5.49478 −0.222295
\(612\) 0 0
\(613\) 34.1241 1.37826 0.689129 0.724638i \(-0.257993\pi\)
0.689129 + 0.724638i \(0.257993\pi\)
\(614\) 0 0
\(615\) −0.231535 −0.00933641
\(616\) 0 0
\(617\) 7.85646 0.316289 0.158145 0.987416i \(-0.449449\pi\)
0.158145 + 0.987416i \(0.449449\pi\)
\(618\) 0 0
\(619\) 23.1163 0.929121 0.464561 0.885541i \(-0.346212\pi\)
0.464561 + 0.885541i \(0.346212\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 14.9122 0.597444
\(624\) 0 0
\(625\) 17.8179 0.712715
\(626\) 0 0
\(627\) −0.367439 −0.0146741
\(628\) 0 0
\(629\) 10.2830 0.410010
\(630\) 0 0
\(631\) −10.2793 −0.409211 −0.204605 0.978845i \(-0.565591\pi\)
−0.204605 + 0.978845i \(0.565591\pi\)
\(632\) 0 0
\(633\) −7.82757 −0.311118
\(634\) 0 0
\(635\) −6.52405 −0.258899
\(636\) 0 0
\(637\) −1.30984 −0.0518977
\(638\) 0 0
\(639\) 3.86410 0.152861
\(640\) 0 0
\(641\) −25.5923 −1.01083 −0.505417 0.862875i \(-0.668661\pi\)
−0.505417 + 0.862875i \(0.668661\pi\)
\(642\) 0 0
\(643\) 10.6494 0.419973 0.209987 0.977704i \(-0.432658\pi\)
0.209987 + 0.977704i \(0.432658\pi\)
\(644\) 0 0
\(645\) 2.10701 0.0829635
\(646\) 0 0
\(647\) −9.58910 −0.376986 −0.188493 0.982074i \(-0.560360\pi\)
−0.188493 + 0.982074i \(0.560360\pi\)
\(648\) 0 0
\(649\) −5.89805 −0.231519
\(650\) 0 0
\(651\) 1.86597 0.0731331
\(652\) 0 0
\(653\) 0.851214 0.0333106 0.0166553 0.999861i \(-0.494698\pi\)
0.0166553 + 0.999861i \(0.494698\pi\)
\(654\) 0 0
\(655\) 2.02889 0.0792754
\(656\) 0 0
\(657\) 5.82757 0.227355
\(658\) 0 0
\(659\) −11.5158 −0.448591 −0.224296 0.974521i \(-0.572008\pi\)
−0.224296 + 0.974521i \(0.572008\pi\)
\(660\) 0 0
\(661\) 22.5601 0.877487 0.438744 0.898612i \(-0.355424\pi\)
0.438744 + 0.898612i \(0.355424\pi\)
\(662\) 0 0
\(663\) 4.80837 0.186742
\(664\) 0 0
\(665\) −0.426543 −0.0165406
\(666\) 0 0
\(667\) 5.63256 0.218094
\(668\) 0 0
\(669\) −10.7474 −0.415518
\(670\) 0 0
\(671\) 8.44912 0.326175
\(672\) 0 0
\(673\) 43.3135 1.66961 0.834806 0.550545i \(-0.185580\pi\)
0.834806 + 0.550545i \(0.185580\pi\)
\(674\) 0 0
\(675\) −4.50485 −0.173392
\(676\) 0 0
\(677\) 3.09688 0.119023 0.0595113 0.998228i \(-0.481046\pi\)
0.0595113 + 0.998228i \(0.481046\pi\)
\(678\) 0 0
\(679\) −5.82757 −0.223642
\(680\) 0 0
\(681\) 7.55676 0.289576
\(682\) 0 0
\(683\) −8.17243 −0.312709 −0.156355 0.987701i \(-0.549974\pi\)
−0.156355 + 0.987701i \(0.549974\pi\)
\(684\) 0 0
\(685\) 2.27631 0.0869735
\(686\) 0 0
\(687\) 9.28495 0.354243
\(688\) 0 0
\(689\) 13.1078 0.499368
\(690\) 0 0
\(691\) 28.8958 1.09925 0.549624 0.835412i \(-0.314771\pi\)
0.549624 + 0.835412i \(0.314771\pi\)
\(692\) 0 0
\(693\) 0.606168 0.0230264
\(694\) 0 0
\(695\) 5.14060 0.194994
\(696\) 0 0
\(697\) 1.20789 0.0457521
\(698\) 0 0
\(699\) −0.629185 −0.0237980
\(700\) 0 0
\(701\) 12.7451 0.481375 0.240687 0.970603i \(-0.422627\pi\)
0.240687 + 0.970603i \(0.422627\pi\)
\(702\) 0 0
\(703\) −1.69798 −0.0640406
\(704\) 0 0
\(705\) 2.95191 0.111175
\(706\) 0 0
\(707\) 2.95378 0.111088
\(708\) 0 0
\(709\) 37.4384 1.40603 0.703014 0.711176i \(-0.251837\pi\)
0.703014 + 0.711176i \(0.251837\pi\)
\(710\) 0 0
\(711\) 1.86597 0.0699793
\(712\) 0 0
\(713\) −1.86597 −0.0698811
\(714\) 0 0
\(715\) 0.558703 0.0208943
\(716\) 0 0
\(717\) 10.7474 0.401369
\(718\) 0 0
\(719\) −39.2379 −1.46333 −0.731664 0.681666i \(-0.761256\pi\)
−0.731664 + 0.681666i \(0.761256\pi\)
\(720\) 0 0
\(721\) 7.36894 0.274434
\(722\) 0 0
\(723\) −19.5337 −0.726468
\(724\) 0 0
\(725\) −25.3738 −0.942360
\(726\) 0 0
\(727\) 2.97448 0.110317 0.0551587 0.998478i \(-0.482434\pi\)
0.0551587 + 0.998478i \(0.482434\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −10.9920 −0.406554
\(732\) 0 0
\(733\) 15.2227 0.562262 0.281131 0.959669i \(-0.409291\pi\)
0.281131 + 0.959669i \(0.409291\pi\)
\(734\) 0 0
\(735\) 0.703671 0.0259553
\(736\) 0 0
\(737\) 5.25261 0.193482
\(738\) 0 0
\(739\) −24.8412 −0.913797 −0.456898 0.889519i \(-0.651040\pi\)
−0.456898 + 0.889519i \(0.651040\pi\)
\(740\) 0 0
\(741\) −0.793982 −0.0291677
\(742\) 0 0
\(743\) −37.7445 −1.38471 −0.692355 0.721557i \(-0.743427\pi\)
−0.692355 + 0.721557i \(0.743427\pi\)
\(744\) 0 0
\(745\) −14.4698 −0.530133
\(746\) 0 0
\(747\) −13.5504 −0.495785
\(748\) 0 0
\(749\) 19.6398 0.717621
\(750\) 0 0
\(751\) −0.171115 −0.00624406 −0.00312203 0.999995i \(-0.500994\pi\)
−0.00312203 + 0.999995i \(0.500994\pi\)
\(752\) 0 0
\(753\) 19.1176 0.696683
\(754\) 0 0
\(755\) −2.23341 −0.0812820
\(756\) 0 0
\(757\) 32.2997 1.17395 0.586976 0.809604i \(-0.300318\pi\)
0.586976 + 0.809604i \(0.300318\pi\)
\(758\) 0 0
\(759\) −0.606168 −0.0220025
\(760\) 0 0
\(761\) −49.5783 −1.79721 −0.898607 0.438755i \(-0.855420\pi\)
−0.898607 + 0.438755i \(0.855420\pi\)
\(762\) 0 0
\(763\) −15.6095 −0.565103
\(764\) 0 0
\(765\) −2.58315 −0.0933940
\(766\) 0 0
\(767\) −12.7448 −0.460189
\(768\) 0 0
\(769\) 4.48884 0.161872 0.0809358 0.996719i \(-0.474209\pi\)
0.0809358 + 0.996719i \(0.474209\pi\)
\(770\) 0 0
\(771\) −27.8848 −1.00425
\(772\) 0 0
\(773\) −14.1123 −0.507585 −0.253793 0.967259i \(-0.581678\pi\)
−0.253793 + 0.967259i \(0.581678\pi\)
\(774\) 0 0
\(775\) 8.40590 0.301949
\(776\) 0 0
\(777\) 2.80118 0.100492
\(778\) 0 0
\(779\) −0.199453 −0.00714614
\(780\) 0 0
\(781\) −2.34229 −0.0838137
\(782\) 0 0
\(783\) 5.63256 0.201291
\(784\) 0 0
\(785\) −12.1182 −0.432517
\(786\) 0 0
\(787\) 46.4230 1.65480 0.827400 0.561613i \(-0.189819\pi\)
0.827400 + 0.561613i \(0.189819\pi\)
\(788\) 0 0
\(789\) −9.01288 −0.320867
\(790\) 0 0
\(791\) 6.78579 0.241275
\(792\) 0 0
\(793\) 18.2573 0.648336
\(794\) 0 0
\(795\) −7.04178 −0.249746
\(796\) 0 0
\(797\) −2.19269 −0.0776692 −0.0388346 0.999246i \(-0.512365\pi\)
−0.0388346 + 0.999246i \(0.512365\pi\)
\(798\) 0 0
\(799\) −15.3997 −0.544803
\(800\) 0 0
\(801\) −14.9122 −0.526896
\(802\) 0 0
\(803\) −3.53248 −0.124659
\(804\) 0 0
\(805\) −0.703671 −0.0248012
\(806\) 0 0
\(807\) −1.77291 −0.0624094
\(808\) 0 0
\(809\) −3.23003 −0.113562 −0.0567809 0.998387i \(-0.518084\pi\)
−0.0567809 + 0.998387i \(0.518084\pi\)
\(810\) 0 0
\(811\) 2.50966 0.0881261 0.0440631 0.999029i \(-0.485970\pi\)
0.0440631 + 0.999029i \(0.485970\pi\)
\(812\) 0 0
\(813\) 18.0880 0.634374
\(814\) 0 0
\(815\) 8.22709 0.288182
\(816\) 0 0
\(817\) 1.81505 0.0635007
\(818\) 0 0
\(819\) 1.30984 0.0457695
\(820\) 0 0
\(821\) −35.5129 −1.23941 −0.619705 0.784835i \(-0.712748\pi\)
−0.619705 + 0.784835i \(0.712748\pi\)
\(822\) 0 0
\(823\) −37.7942 −1.31742 −0.658712 0.752395i \(-0.728898\pi\)
−0.658712 + 0.752395i \(0.728898\pi\)
\(824\) 0 0
\(825\) 2.73069 0.0950705
\(826\) 0 0
\(827\) −20.6589 −0.718380 −0.359190 0.933264i \(-0.616947\pi\)
−0.359190 + 0.933264i \(0.616947\pi\)
\(828\) 0 0
\(829\) 13.5376 0.470181 0.235091 0.971973i \(-0.424461\pi\)
0.235091 + 0.971973i \(0.424461\pi\)
\(830\) 0 0
\(831\) 12.6788 0.439822
\(832\) 0 0
\(833\) −3.67096 −0.127191
\(834\) 0 0
\(835\) 8.69372 0.300859
\(836\) 0 0
\(837\) −1.86597 −0.0644973
\(838\) 0 0
\(839\) −33.4538 −1.15495 −0.577477 0.816407i \(-0.695963\pi\)
−0.577477 + 0.816407i \(0.695963\pi\)
\(840\) 0 0
\(841\) 2.72574 0.0939910
\(842\) 0 0
\(843\) −33.3229 −1.14770
\(844\) 0 0
\(845\) −7.94045 −0.273160
\(846\) 0 0
\(847\) 10.6326 0.365339
\(848\) 0 0
\(849\) −12.1752 −0.417851
\(850\) 0 0
\(851\) −2.80118 −0.0960230
\(852\) 0 0
\(853\) 8.48884 0.290652 0.145326 0.989384i \(-0.453577\pi\)
0.145326 + 0.989384i \(0.453577\pi\)
\(854\) 0 0
\(855\) 0.426543 0.0145875
\(856\) 0 0
\(857\) −20.4736 −0.699364 −0.349682 0.936869i \(-0.613710\pi\)
−0.349682 + 0.936869i \(0.613710\pi\)
\(858\) 0 0
\(859\) −3.26512 −0.111405 −0.0557023 0.998447i \(-0.517740\pi\)
−0.0557023 + 0.998447i \(0.517740\pi\)
\(860\) 0 0
\(861\) 0.329039 0.0112136
\(862\) 0 0
\(863\) 5.49478 0.187045 0.0935223 0.995617i \(-0.470187\pi\)
0.0935223 + 0.995617i \(0.470187\pi\)
\(864\) 0 0
\(865\) −6.59842 −0.224353
\(866\) 0 0
\(867\) −3.52405 −0.119683
\(868\) 0 0
\(869\) −1.13109 −0.0383696
\(870\) 0 0
\(871\) 11.3501 0.384584
\(872\) 0 0
\(873\) 5.82757 0.197233
\(874\) 0 0
\(875\) 6.68829 0.226105
\(876\) 0 0
\(877\) 44.8417 1.51420 0.757098 0.653301i \(-0.226616\pi\)
0.757098 + 0.653301i \(0.226616\pi\)
\(878\) 0 0
\(879\) 16.1091 0.543348
\(880\) 0 0
\(881\) 52.8202 1.77956 0.889779 0.456393i \(-0.150859\pi\)
0.889779 + 0.456393i \(0.150859\pi\)
\(882\) 0 0
\(883\) 28.4157 0.956265 0.478133 0.878288i \(-0.341314\pi\)
0.478133 + 0.878288i \(0.341314\pi\)
\(884\) 0 0
\(885\) 6.84677 0.230152
\(886\) 0 0
\(887\) 12.6864 0.425968 0.212984 0.977056i \(-0.431682\pi\)
0.212984 + 0.977056i \(0.431682\pi\)
\(888\) 0 0
\(889\) 9.27144 0.310954
\(890\) 0 0
\(891\) −0.606168 −0.0203074
\(892\) 0 0
\(893\) 2.54288 0.0850942
\(894\) 0 0
\(895\) 3.29457 0.110125
\(896\) 0 0
\(897\) −1.30984 −0.0437343
\(898\) 0 0
\(899\) −10.5102 −0.350534
\(900\) 0 0
\(901\) 36.7360 1.22385
\(902\) 0 0
\(903\) −2.99431 −0.0996444
\(904\) 0 0
\(905\) 0.654950 0.0217713
\(906\) 0 0
\(907\) −9.67622 −0.321294 −0.160647 0.987012i \(-0.551358\pi\)
−0.160647 + 0.987012i \(0.551358\pi\)
\(908\) 0 0
\(909\) −2.95378 −0.0979707
\(910\) 0 0
\(911\) 7.64075 0.253149 0.126575 0.991957i \(-0.459602\pi\)
0.126575 + 0.991957i \(0.459602\pi\)
\(912\) 0 0
\(913\) 8.21384 0.271838
\(914\) 0 0
\(915\) −9.80818 −0.324248
\(916\) 0 0
\(917\) −2.88330 −0.0952148
\(918\) 0 0
\(919\) −53.8533 −1.77646 −0.888229 0.459401i \(-0.848064\pi\)
−0.888229 + 0.459401i \(0.848064\pi\)
\(920\) 0 0
\(921\) −13.1311 −0.432684
\(922\) 0 0
\(923\) −5.06134 −0.166596
\(924\) 0 0
\(925\) 12.6189 0.414906
\(926\) 0 0
\(927\) −7.36894 −0.242028
\(928\) 0 0
\(929\) 29.0290 0.952409 0.476205 0.879335i \(-0.342012\pi\)
0.476205 + 0.879335i \(0.342012\pi\)
\(930\) 0 0
\(931\) 0.606168 0.0198663
\(932\) 0 0
\(933\) 4.98899 0.163332
\(934\) 0 0
\(935\) 1.56582 0.0512079
\(936\) 0 0
\(937\) −2.96685 −0.0969227 −0.0484613 0.998825i \(-0.515432\pi\)
−0.0484613 + 0.998825i \(0.515432\pi\)
\(938\) 0 0
\(939\) 22.8945 0.747134
\(940\) 0 0
\(941\) −17.5895 −0.573403 −0.286701 0.958020i \(-0.592559\pi\)
−0.286701 + 0.958020i \(0.592559\pi\)
\(942\) 0 0
\(943\) −0.329039 −0.0107150
\(944\) 0 0
\(945\) −0.703671 −0.0228904
\(946\) 0 0
\(947\) 27.8041 0.903512 0.451756 0.892142i \(-0.350798\pi\)
0.451756 + 0.892142i \(0.350798\pi\)
\(948\) 0 0
\(949\) −7.63318 −0.247783
\(950\) 0 0
\(951\) 12.9424 0.419686
\(952\) 0 0
\(953\) −42.3634 −1.37229 −0.686143 0.727467i \(-0.740698\pi\)
−0.686143 + 0.727467i \(0.740698\pi\)
\(954\) 0 0
\(955\) 2.23341 0.0722714
\(956\) 0 0
\(957\) −3.41428 −0.110368
\(958\) 0 0
\(959\) −3.23491 −0.104461
\(960\) 0 0
\(961\) −27.5182 −0.887683
\(962\) 0 0
\(963\) −19.6398 −0.632882
\(964\) 0 0
\(965\) −11.8772 −0.382339
\(966\) 0 0
\(967\) −5.41747 −0.174214 −0.0871070 0.996199i \(-0.527762\pi\)
−0.0871070 + 0.996199i \(0.527762\pi\)
\(968\) 0 0
\(969\) −2.22522 −0.0714843
\(970\) 0 0
\(971\) 10.0872 0.323713 0.161857 0.986814i \(-0.448252\pi\)
0.161857 + 0.986814i \(0.448252\pi\)
\(972\) 0 0
\(973\) −7.30539 −0.234200
\(974\) 0 0
\(975\) 5.90062 0.188971
\(976\) 0 0
\(977\) −3.58039 −0.114547 −0.0572734 0.998359i \(-0.518241\pi\)
−0.0572734 + 0.998359i \(0.518241\pi\)
\(978\) 0 0
\(979\) 9.03929 0.288897
\(980\) 0 0
\(981\) 15.6095 0.498374
\(982\) 0 0
\(983\) 30.1294 0.960979 0.480489 0.877001i \(-0.340459\pi\)
0.480489 + 0.877001i \(0.340459\pi\)
\(984\) 0 0
\(985\) 2.03021 0.0646879
\(986\) 0 0
\(987\) −4.19501 −0.133529
\(988\) 0 0
\(989\) 2.99431 0.0952135
\(990\) 0 0
\(991\) 43.6374 1.38619 0.693094 0.720847i \(-0.256247\pi\)
0.693094 + 0.720847i \(0.256247\pi\)
\(992\) 0 0
\(993\) −29.1537 −0.925164
\(994\) 0 0
\(995\) 11.4858 0.364125
\(996\) 0 0
\(997\) 11.3569 0.359675 0.179838 0.983696i \(-0.442443\pi\)
0.179838 + 0.983696i \(0.442443\pi\)
\(998\) 0 0
\(999\) −2.80118 −0.0886253
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 7728.2.a.cb.1.3 4
4.3 odd 2 3864.2.a.r.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.r.1.3 4 4.3 odd 2
7728.2.a.cb.1.3 4 1.1 even 1 trivial