Properties

Label 5376.2.c.bn.2689.2
Level $5376$
Weight $2$
Character 5376.2689
Analytic conductor $42.928$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5376.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(42.9275761266\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 672)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2689.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 5376.2689
Dual form 5376.2.c.bn.2689.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +3.46410i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +3.46410i q^{5} +1.00000 q^{7} -1.00000 q^{9} -1.46410i q^{11} -2.00000i q^{13} +3.46410 q^{15} +0.535898 q^{17} -6.92820i q^{19} -1.00000i q^{21} +1.46410 q^{23} -7.00000 q^{25} +1.00000i q^{27} +4.92820i q^{29} -10.9282 q^{31} -1.46410 q^{33} +3.46410i q^{35} -2.00000i q^{37} -2.00000 q^{39} -11.4641 q^{41} -8.00000i q^{43} -3.46410i q^{45} +10.9282 q^{47} +1.00000 q^{49} -0.535898i q^{51} -2.00000i q^{53} +5.07180 q^{55} -6.92820 q^{57} -1.07180i q^{59} +8.92820i q^{61} -1.00000 q^{63} +6.92820 q^{65} +2.92820i q^{67} -1.46410i q^{69} -9.46410 q^{71} -12.9282 q^{73} +7.00000i q^{75} -1.46410i q^{77} -10.9282 q^{79} +1.00000 q^{81} +4.00000i q^{83} +1.85641i q^{85} +4.92820 q^{87} -3.46410 q^{89} -2.00000i q^{91} +10.9282i q^{93} +24.0000 q^{95} -8.92820 q^{97} +1.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} - 4q^{9} + O(q^{10}) \) \( 4q + 4q^{7} - 4q^{9} + 16q^{17} - 8q^{23} - 28q^{25} - 16q^{31} + 8q^{33} - 8q^{39} - 32q^{41} + 16q^{47} + 4q^{49} + 48q^{55} - 4q^{63} - 24q^{71} - 24q^{73} - 16q^{79} + 4q^{81} - 8q^{87} + 96q^{95} - 8q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times\).

\(n\) \(1793\) \(2815\) \(4609\) \(5125\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 3.46410i 1.54919i 0.632456 + 0.774597i \(0.282047\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 1.46410i − 0.441443i −0.975337 0.220722i \(-0.929159\pi\)
0.975337 0.220722i \(-0.0708412\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 3.46410 0.894427
\(16\) 0 0
\(17\) 0.535898 0.129974 0.0649872 0.997886i \(-0.479299\pi\)
0.0649872 + 0.997886i \(0.479299\pi\)
\(18\) 0 0
\(19\) − 6.92820i − 1.58944i −0.606977 0.794719i \(-0.707618\pi\)
0.606977 0.794719i \(-0.292382\pi\)
\(20\) 0 0
\(21\) − 1.00000i − 0.218218i
\(22\) 0 0
\(23\) 1.46410 0.305286 0.152643 0.988281i \(-0.451221\pi\)
0.152643 + 0.988281i \(0.451221\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 4.92820i 0.915144i 0.889172 + 0.457572i \(0.151281\pi\)
−0.889172 + 0.457572i \(0.848719\pi\)
\(30\) 0 0
\(31\) −10.9282 −1.96276 −0.981382 0.192068i \(-0.938481\pi\)
−0.981382 + 0.192068i \(0.938481\pi\)
\(32\) 0 0
\(33\) −1.46410 −0.254867
\(34\) 0 0
\(35\) 3.46410i 0.585540i
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −11.4641 −1.79039 −0.895196 0.445673i \(-0.852964\pi\)
−0.895196 + 0.445673i \(0.852964\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) − 3.46410i − 0.516398i
\(46\) 0 0
\(47\) 10.9282 1.59404 0.797021 0.603951i \(-0.206408\pi\)
0.797021 + 0.603951i \(0.206408\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) − 0.535898i − 0.0750408i
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 5.07180 0.683881
\(56\) 0 0
\(57\) −6.92820 −0.917663
\(58\) 0 0
\(59\) − 1.07180i − 0.139536i −0.997563 0.0697680i \(-0.977774\pi\)
0.997563 0.0697680i \(-0.0222259\pi\)
\(60\) 0 0
\(61\) 8.92820i 1.14314i 0.820554 + 0.571570i \(0.193665\pi\)
−0.820554 + 0.571570i \(0.806335\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 6.92820 0.859338
\(66\) 0 0
\(67\) 2.92820i 0.357737i 0.983873 + 0.178868i \(0.0572437\pi\)
−0.983873 + 0.178868i \(0.942756\pi\)
\(68\) 0 0
\(69\) − 1.46410i − 0.176257i
\(70\) 0 0
\(71\) −9.46410 −1.12318 −0.561591 0.827415i \(-0.689811\pi\)
−0.561591 + 0.827415i \(0.689811\pi\)
\(72\) 0 0
\(73\) −12.9282 −1.51313 −0.756566 0.653917i \(-0.773124\pi\)
−0.756566 + 0.653917i \(0.773124\pi\)
\(74\) 0 0
\(75\) 7.00000i 0.808290i
\(76\) 0 0
\(77\) − 1.46410i − 0.166850i
\(78\) 0 0
\(79\) −10.9282 −1.22952 −0.614759 0.788715i \(-0.710747\pi\)
−0.614759 + 0.788715i \(0.710747\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 1.85641i 0.201356i
\(86\) 0 0
\(87\) 4.92820 0.528359
\(88\) 0 0
\(89\) −3.46410 −0.367194 −0.183597 0.983002i \(-0.558774\pi\)
−0.183597 + 0.983002i \(0.558774\pi\)
\(90\) 0 0
\(91\) − 2.00000i − 0.209657i
\(92\) 0 0
\(93\) 10.9282i 1.13320i
\(94\) 0 0
\(95\) 24.0000 2.46235
\(96\) 0 0
\(97\) −8.92820 −0.906522 −0.453261 0.891378i \(-0.649739\pi\)
−0.453261 + 0.891378i \(0.649739\pi\)
\(98\) 0 0
\(99\) 1.46410i 0.147148i
\(100\) 0 0
\(101\) − 15.4641i − 1.53874i −0.638806 0.769368i \(-0.720571\pi\)
0.638806 0.769368i \(-0.279429\pi\)
\(102\) 0 0
\(103\) 10.9282 1.07679 0.538394 0.842693i \(-0.319031\pi\)
0.538394 + 0.842693i \(0.319031\pi\)
\(104\) 0 0
\(105\) 3.46410 0.338062
\(106\) 0 0
\(107\) 12.3923i 1.19801i 0.800746 + 0.599005i \(0.204437\pi\)
−0.800746 + 0.599005i \(0.795563\pi\)
\(108\) 0 0
\(109\) 2.00000i 0.191565i 0.995402 + 0.0957826i \(0.0305354\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −19.8564 −1.86793 −0.933967 0.357360i \(-0.883677\pi\)
−0.933967 + 0.357360i \(0.883677\pi\)
\(114\) 0 0
\(115\) 5.07180i 0.472947i
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 0.535898 0.0491257
\(120\) 0 0
\(121\) 8.85641 0.805128
\(122\) 0 0
\(123\) 11.4641i 1.03368i
\(124\) 0 0
\(125\) − 6.92820i − 0.619677i
\(126\) 0 0
\(127\) 2.92820 0.259836 0.129918 0.991525i \(-0.458529\pi\)
0.129918 + 0.991525i \(0.458529\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) − 12.0000i − 1.04844i −0.851581 0.524222i \(-0.824356\pi\)
0.851581 0.524222i \(-0.175644\pi\)
\(132\) 0 0
\(133\) − 6.92820i − 0.600751i
\(134\) 0 0
\(135\) −3.46410 −0.298142
\(136\) 0 0
\(137\) 8.92820 0.762788 0.381394 0.924413i \(-0.375444\pi\)
0.381394 + 0.924413i \(0.375444\pi\)
\(138\) 0 0
\(139\) − 17.8564i − 1.51456i −0.653090 0.757280i \(-0.726528\pi\)
0.653090 0.757280i \(-0.273472\pi\)
\(140\) 0 0
\(141\) − 10.9282i − 0.920321i
\(142\) 0 0
\(143\) −2.92820 −0.244869
\(144\) 0 0
\(145\) −17.0718 −1.41774
\(146\) 0 0
\(147\) − 1.00000i − 0.0824786i
\(148\) 0 0
\(149\) − 18.0000i − 1.47462i −0.675556 0.737309i \(-0.736096\pi\)
0.675556 0.737309i \(-0.263904\pi\)
\(150\) 0 0
\(151\) −21.8564 −1.77865 −0.889325 0.457277i \(-0.848825\pi\)
−0.889325 + 0.457277i \(0.848825\pi\)
\(152\) 0 0
\(153\) −0.535898 −0.0433248
\(154\) 0 0
\(155\) − 37.8564i − 3.04070i
\(156\) 0 0
\(157\) 0.928203i 0.0740787i 0.999314 + 0.0370393i \(0.0117927\pi\)
−0.999314 + 0.0370393i \(0.988207\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 1.46410 0.115387
\(162\) 0 0
\(163\) − 10.9282i − 0.855963i −0.903788 0.427981i \(-0.859225\pi\)
0.903788 0.427981i \(-0.140775\pi\)
\(164\) 0 0
\(165\) − 5.07180i − 0.394839i
\(166\) 0 0
\(167\) 10.9282 0.845650 0.422825 0.906211i \(-0.361039\pi\)
0.422825 + 0.906211i \(0.361039\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 6.92820i 0.529813i
\(172\) 0 0
\(173\) − 22.3923i − 1.70246i −0.524797 0.851228i \(-0.675859\pi\)
0.524797 0.851228i \(-0.324141\pi\)
\(174\) 0 0
\(175\) −7.00000 −0.529150
\(176\) 0 0
\(177\) −1.07180 −0.0805612
\(178\) 0 0
\(179\) − 9.46410i − 0.707380i −0.935363 0.353690i \(-0.884927\pi\)
0.935363 0.353690i \(-0.115073\pi\)
\(180\) 0 0
\(181\) − 6.00000i − 0.445976i −0.974821 0.222988i \(-0.928419\pi\)
0.974821 0.222988i \(-0.0715812\pi\)
\(182\) 0 0
\(183\) 8.92820 0.659992
\(184\) 0 0
\(185\) 6.92820 0.509372
\(186\) 0 0
\(187\) − 0.784610i − 0.0573763i
\(188\) 0 0
\(189\) 1.00000i 0.0727393i
\(190\) 0 0
\(191\) 9.46410 0.684798 0.342399 0.939555i \(-0.388760\pi\)
0.342399 + 0.939555i \(0.388760\pi\)
\(192\) 0 0
\(193\) 15.8564 1.14137 0.570685 0.821169i \(-0.306678\pi\)
0.570685 + 0.821169i \(0.306678\pi\)
\(194\) 0 0
\(195\) − 6.92820i − 0.496139i
\(196\) 0 0
\(197\) − 7.85641i − 0.559746i −0.960037 0.279873i \(-0.909708\pi\)
0.960037 0.279873i \(-0.0902923\pi\)
\(198\) 0 0
\(199\) −5.85641 −0.415150 −0.207575 0.978219i \(-0.566557\pi\)
−0.207575 + 0.978219i \(0.566557\pi\)
\(200\) 0 0
\(201\) 2.92820 0.206540
\(202\) 0 0
\(203\) 4.92820i 0.345892i
\(204\) 0 0
\(205\) − 39.7128i − 2.77366i
\(206\) 0 0
\(207\) −1.46410 −0.101762
\(208\) 0 0
\(209\) −10.1436 −0.701647
\(210\) 0 0
\(211\) 16.0000i 1.10149i 0.834675 + 0.550743i \(0.185655\pi\)
−0.834675 + 0.550743i \(0.814345\pi\)
\(212\) 0 0
\(213\) 9.46410i 0.648470i
\(214\) 0 0
\(215\) 27.7128 1.89000
\(216\) 0 0
\(217\) −10.9282 −0.741855
\(218\) 0 0
\(219\) 12.9282i 0.873607i
\(220\) 0 0
\(221\) − 1.07180i − 0.0720969i
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) 22.9282i 1.52180i 0.648870 + 0.760899i \(0.275242\pi\)
−0.648870 + 0.760899i \(0.724758\pi\)
\(228\) 0 0
\(229\) − 27.8564i − 1.84080i −0.390974 0.920402i \(-0.627862\pi\)
0.390974 0.920402i \(-0.372138\pi\)
\(230\) 0 0
\(231\) −1.46410 −0.0963308
\(232\) 0 0
\(233\) −12.9282 −0.846955 −0.423477 0.905907i \(-0.639191\pi\)
−0.423477 + 0.905907i \(0.639191\pi\)
\(234\) 0 0
\(235\) 37.8564i 2.46948i
\(236\) 0 0
\(237\) 10.9282i 0.709863i
\(238\) 0 0
\(239\) 14.5359 0.940249 0.470125 0.882600i \(-0.344209\pi\)
0.470125 + 0.882600i \(0.344209\pi\)
\(240\) 0 0
\(241\) −3.07180 −0.197872 −0.0989359 0.995094i \(-0.531544\pi\)
−0.0989359 + 0.995094i \(0.531544\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 3.46410i 0.221313i
\(246\) 0 0
\(247\) −13.8564 −0.881662
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) − 9.07180i − 0.572607i −0.958139 0.286303i \(-0.907573\pi\)
0.958139 0.286303i \(-0.0924265\pi\)
\(252\) 0 0
\(253\) − 2.14359i − 0.134767i
\(254\) 0 0
\(255\) 1.85641 0.116253
\(256\) 0 0
\(257\) 11.4641 0.715111 0.357556 0.933892i \(-0.383610\pi\)
0.357556 + 0.933892i \(0.383610\pi\)
\(258\) 0 0
\(259\) − 2.00000i − 0.124274i
\(260\) 0 0
\(261\) − 4.92820i − 0.305048i
\(262\) 0 0
\(263\) −12.3923 −0.764142 −0.382071 0.924133i \(-0.624789\pi\)
−0.382071 + 0.924133i \(0.624789\pi\)
\(264\) 0 0
\(265\) 6.92820 0.425596
\(266\) 0 0
\(267\) 3.46410i 0.212000i
\(268\) 0 0
\(269\) − 3.46410i − 0.211210i −0.994408 0.105605i \(-0.966322\pi\)
0.994408 0.105605i \(-0.0336779\pi\)
\(270\) 0 0
\(271\) 2.92820 0.177876 0.0889378 0.996037i \(-0.471653\pi\)
0.0889378 + 0.996037i \(0.471653\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) 10.2487i 0.618021i
\(276\) 0 0
\(277\) 0.143594i 0.00862770i 0.999991 + 0.00431385i \(0.00137315\pi\)
−0.999991 + 0.00431385i \(0.998627\pi\)
\(278\) 0 0
\(279\) 10.9282 0.654254
\(280\) 0 0
\(281\) −12.9282 −0.771232 −0.385616 0.922659i \(-0.626011\pi\)
−0.385616 + 0.922659i \(0.626011\pi\)
\(282\) 0 0
\(283\) − 14.9282i − 0.887390i −0.896178 0.443695i \(-0.853667\pi\)
0.896178 0.443695i \(-0.146333\pi\)
\(284\) 0 0
\(285\) − 24.0000i − 1.42164i
\(286\) 0 0
\(287\) −11.4641 −0.676705
\(288\) 0 0
\(289\) −16.7128 −0.983107
\(290\) 0 0
\(291\) 8.92820i 0.523381i
\(292\) 0 0
\(293\) 16.5359i 0.966037i 0.875610 + 0.483019i \(0.160460\pi\)
−0.875610 + 0.483019i \(0.839540\pi\)
\(294\) 0 0
\(295\) 3.71281 0.216168
\(296\) 0 0
\(297\) 1.46410 0.0849558
\(298\) 0 0
\(299\) − 2.92820i − 0.169342i
\(300\) 0 0
\(301\) − 8.00000i − 0.461112i
\(302\) 0 0
\(303\) −15.4641 −0.888389
\(304\) 0 0
\(305\) −30.9282 −1.77094
\(306\) 0 0
\(307\) − 22.9282i − 1.30858i −0.756243 0.654291i \(-0.772967\pi\)
0.756243 0.654291i \(-0.227033\pi\)
\(308\) 0 0
\(309\) − 10.9282i − 0.621684i
\(310\) 0 0
\(311\) −18.9282 −1.07332 −0.536660 0.843799i \(-0.680314\pi\)
−0.536660 + 0.843799i \(0.680314\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) − 3.46410i − 0.195180i
\(316\) 0 0
\(317\) − 0.143594i − 0.00806502i −0.999992 0.00403251i \(-0.998716\pi\)
0.999992 0.00403251i \(-0.00128359\pi\)
\(318\) 0 0
\(319\) 7.21539 0.403984
\(320\) 0 0
\(321\) 12.3923 0.691671
\(322\) 0 0
\(323\) − 3.71281i − 0.206586i
\(324\) 0 0
\(325\) 14.0000i 0.776580i
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) 10.9282 0.602491
\(330\) 0 0
\(331\) 24.0000i 1.31916i 0.751635 + 0.659580i \(0.229266\pi\)
−0.751635 + 0.659580i \(0.770734\pi\)
\(332\) 0 0
\(333\) 2.00000i 0.109599i
\(334\) 0 0
\(335\) −10.1436 −0.554204
\(336\) 0 0
\(337\) 7.85641 0.427966 0.213983 0.976837i \(-0.431356\pi\)
0.213983 + 0.976837i \(0.431356\pi\)
\(338\) 0 0
\(339\) 19.8564i 1.07845i
\(340\) 0 0
\(341\) 16.0000i 0.866449i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 5.07180 0.273056
\(346\) 0 0
\(347\) 23.3205i 1.25191i 0.779859 + 0.625955i \(0.215291\pi\)
−0.779859 + 0.625955i \(0.784709\pi\)
\(348\) 0 0
\(349\) − 10.7846i − 0.577287i −0.957437 0.288643i \(-0.906796\pi\)
0.957437 0.288643i \(-0.0932042\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 8.53590 0.454320 0.227160 0.973857i \(-0.427056\pi\)
0.227160 + 0.973857i \(0.427056\pi\)
\(354\) 0 0
\(355\) − 32.7846i − 1.74003i
\(356\) 0 0
\(357\) − 0.535898i − 0.0283628i
\(358\) 0 0
\(359\) −7.32051 −0.386362 −0.193181 0.981163i \(-0.561880\pi\)
−0.193181 + 0.981163i \(0.561880\pi\)
\(360\) 0 0
\(361\) −29.0000 −1.52632
\(362\) 0 0
\(363\) − 8.85641i − 0.464841i
\(364\) 0 0
\(365\) − 44.7846i − 2.34413i
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 11.4641 0.596797
\(370\) 0 0
\(371\) − 2.00000i − 0.103835i
\(372\) 0 0
\(373\) 30.0000i 1.55334i 0.629907 + 0.776671i \(0.283093\pi\)
−0.629907 + 0.776671i \(0.716907\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) 0 0
\(377\) 9.85641 0.507631
\(378\) 0 0
\(379\) − 8.00000i − 0.410932i −0.978664 0.205466i \(-0.934129\pi\)
0.978664 0.205466i \(-0.0658711\pi\)
\(380\) 0 0
\(381\) − 2.92820i − 0.150016i
\(382\) 0 0
\(383\) 35.7128 1.82484 0.912420 0.409256i \(-0.134212\pi\)
0.912420 + 0.409256i \(0.134212\pi\)
\(384\) 0 0
\(385\) 5.07180 0.258483
\(386\) 0 0
\(387\) 8.00000i 0.406663i
\(388\) 0 0
\(389\) 14.7846i 0.749609i 0.927104 + 0.374805i \(0.122290\pi\)
−0.927104 + 0.374805i \(0.877710\pi\)
\(390\) 0 0
\(391\) 0.784610 0.0396794
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) − 37.8564i − 1.90476i
\(396\) 0 0
\(397\) − 10.7846i − 0.541264i −0.962683 0.270632i \(-0.912767\pi\)
0.962683 0.270632i \(-0.0872327\pi\)
\(398\) 0 0
\(399\) −6.92820 −0.346844
\(400\) 0 0
\(401\) 4.92820 0.246103 0.123051 0.992400i \(-0.460732\pi\)
0.123051 + 0.992400i \(0.460732\pi\)
\(402\) 0 0
\(403\) 21.8564i 1.08875i
\(404\) 0 0
\(405\) 3.46410i 0.172133i
\(406\) 0 0
\(407\) −2.92820 −0.145146
\(408\) 0 0
\(409\) −4.92820 −0.243684 −0.121842 0.992550i \(-0.538880\pi\)
−0.121842 + 0.992550i \(0.538880\pi\)
\(410\) 0 0
\(411\) − 8.92820i − 0.440396i
\(412\) 0 0
\(413\) − 1.07180i − 0.0527397i
\(414\) 0 0
\(415\) −13.8564 −0.680184
\(416\) 0 0
\(417\) −17.8564 −0.874432
\(418\) 0 0
\(419\) − 17.0718i − 0.834012i −0.908904 0.417006i \(-0.863079\pi\)
0.908904 0.417006i \(-0.136921\pi\)
\(420\) 0 0
\(421\) − 4.14359i − 0.201946i −0.994889 0.100973i \(-0.967804\pi\)
0.994889 0.100973i \(-0.0321956\pi\)
\(422\) 0 0
\(423\) −10.9282 −0.531347
\(424\) 0 0
\(425\) −3.75129 −0.181964
\(426\) 0 0
\(427\) 8.92820i 0.432066i
\(428\) 0 0
\(429\) 2.92820i 0.141375i
\(430\) 0 0
\(431\) −10.2487 −0.493663 −0.246832 0.969058i \(-0.579389\pi\)
−0.246832 + 0.969058i \(0.579389\pi\)
\(432\) 0 0
\(433\) −27.8564 −1.33869 −0.669347 0.742950i \(-0.733426\pi\)
−0.669347 + 0.742950i \(0.733426\pi\)
\(434\) 0 0
\(435\) 17.0718i 0.818530i
\(436\) 0 0
\(437\) − 10.1436i − 0.485234i
\(438\) 0 0
\(439\) −13.8564 −0.661330 −0.330665 0.943748i \(-0.607273\pi\)
−0.330665 + 0.943748i \(0.607273\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) − 28.3923i − 1.34896i −0.738294 0.674480i \(-0.764368\pi\)
0.738294 0.674480i \(-0.235632\pi\)
\(444\) 0 0
\(445\) − 12.0000i − 0.568855i
\(446\) 0 0
\(447\) −18.0000 −0.851371
\(448\) 0 0
\(449\) 7.85641 0.370767 0.185383 0.982666i \(-0.440647\pi\)
0.185383 + 0.982666i \(0.440647\pi\)
\(450\) 0 0
\(451\) 16.7846i 0.790356i
\(452\) 0 0
\(453\) 21.8564i 1.02690i
\(454\) 0 0
\(455\) 6.92820 0.324799
\(456\) 0 0
\(457\) −37.7128 −1.76413 −0.882065 0.471127i \(-0.843847\pi\)
−0.882065 + 0.471127i \(0.843847\pi\)
\(458\) 0 0
\(459\) 0.535898i 0.0250136i
\(460\) 0 0
\(461\) 27.1769i 1.26576i 0.774252 + 0.632878i \(0.218126\pi\)
−0.774252 + 0.632878i \(0.781874\pi\)
\(462\) 0 0
\(463\) 32.7846 1.52363 0.761815 0.647795i \(-0.224309\pi\)
0.761815 + 0.647795i \(0.224309\pi\)
\(464\) 0 0
\(465\) −37.8564 −1.75555
\(466\) 0 0
\(467\) − 20.7846i − 0.961797i −0.876776 0.480899i \(-0.840311\pi\)
0.876776 0.480899i \(-0.159689\pi\)
\(468\) 0 0
\(469\) 2.92820i 0.135212i
\(470\) 0 0
\(471\) 0.928203 0.0427693
\(472\) 0 0
\(473\) −11.7128 −0.538556
\(474\) 0 0
\(475\) 48.4974i 2.22521i
\(476\) 0 0
\(477\) 2.00000i 0.0915737i
\(478\) 0 0
\(479\) −5.07180 −0.231736 −0.115868 0.993265i \(-0.536965\pi\)
−0.115868 + 0.993265i \(0.536965\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) − 1.46410i − 0.0666189i
\(484\) 0 0
\(485\) − 30.9282i − 1.40438i
\(486\) 0 0
\(487\) 27.7128 1.25579 0.627894 0.778299i \(-0.283917\pi\)
0.627894 + 0.778299i \(0.283917\pi\)
\(488\) 0 0
\(489\) −10.9282 −0.494190
\(490\) 0 0
\(491\) − 6.53590i − 0.294961i −0.989065 0.147480i \(-0.952884\pi\)
0.989065 0.147480i \(-0.0471164\pi\)
\(492\) 0 0
\(493\) 2.64102i 0.118945i
\(494\) 0 0
\(495\) −5.07180 −0.227960
\(496\) 0 0
\(497\) −9.46410 −0.424523
\(498\) 0 0
\(499\) 21.8564i 0.978427i 0.872164 + 0.489214i \(0.162716\pi\)
−0.872164 + 0.489214i \(0.837284\pi\)
\(500\) 0 0
\(501\) − 10.9282i − 0.488236i
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 53.5692 2.38380
\(506\) 0 0
\(507\) − 9.00000i − 0.399704i
\(508\) 0 0
\(509\) 40.2487i 1.78399i 0.452043 + 0.891996i \(0.350696\pi\)
−0.452043 + 0.891996i \(0.649304\pi\)
\(510\) 0 0
\(511\) −12.9282 −0.571910
\(512\) 0 0
\(513\) 6.92820 0.305888
\(514\) 0 0
\(515\) 37.8564i 1.66815i
\(516\) 0 0
\(517\) − 16.0000i − 0.703679i
\(518\) 0 0
\(519\) −22.3923 −0.982913
\(520\) 0 0
\(521\) −30.3923 −1.33151 −0.665756 0.746170i \(-0.731891\pi\)
−0.665756 + 0.746170i \(0.731891\pi\)
\(522\) 0 0
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) 0 0
\(525\) 7.00000i 0.305505i
\(526\) 0 0
\(527\) −5.85641 −0.255109
\(528\) 0 0
\(529\) −20.8564 −0.906800
\(530\) 0 0
\(531\) 1.07180i 0.0465120i
\(532\) 0 0
\(533\) 22.9282i 0.993131i
\(534\) 0 0
\(535\) −42.9282 −1.85595
\(536\) 0 0
\(537\) −9.46410 −0.408406
\(538\) 0 0
\(539\) − 1.46410i − 0.0630633i
\(540\) 0 0
\(541\) 7.85641i 0.337773i 0.985635 + 0.168887i \(0.0540172\pi\)
−0.985635 + 0.168887i \(0.945983\pi\)
\(542\) 0 0
\(543\) −6.00000 −0.257485
\(544\) 0 0
\(545\) −6.92820 −0.296772
\(546\) 0 0
\(547\) − 10.9282i − 0.467256i −0.972326 0.233628i \(-0.924940\pi\)
0.972326 0.233628i \(-0.0750598\pi\)
\(548\) 0 0
\(549\) − 8.92820i − 0.381046i
\(550\) 0 0
\(551\) 34.1436 1.45457
\(552\) 0 0
\(553\) −10.9282 −0.464714
\(554\) 0 0
\(555\) − 6.92820i − 0.294086i
\(556\) 0 0
\(557\) 4.14359i 0.175570i 0.996139 + 0.0877848i \(0.0279788\pi\)
−0.996139 + 0.0877848i \(0.972021\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) −0.784610 −0.0331262
\(562\) 0 0
\(563\) − 33.0718i − 1.39381i −0.717163 0.696905i \(-0.754560\pi\)
0.717163 0.696905i \(-0.245440\pi\)
\(564\) 0 0
\(565\) − 68.7846i − 2.89379i
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 24.9282 1.04504 0.522522 0.852626i \(-0.324991\pi\)
0.522522 + 0.852626i \(0.324991\pi\)
\(570\) 0 0
\(571\) 16.7846i 0.702414i 0.936298 + 0.351207i \(0.114229\pi\)
−0.936298 + 0.351207i \(0.885771\pi\)
\(572\) 0 0
\(573\) − 9.46410i − 0.395369i
\(574\) 0 0
\(575\) −10.2487 −0.427401
\(576\) 0 0
\(577\) 23.8564 0.993155 0.496578 0.867992i \(-0.334590\pi\)
0.496578 + 0.867992i \(0.334590\pi\)
\(578\) 0 0
\(579\) − 15.8564i − 0.658970i
\(580\) 0 0
\(581\) 4.00000i 0.165948i
\(582\) 0 0
\(583\) −2.92820 −0.121274
\(584\) 0 0
\(585\) −6.92820 −0.286446
\(586\) 0 0
\(587\) − 20.7846i − 0.857873i −0.903335 0.428936i \(-0.858888\pi\)
0.903335 0.428936i \(-0.141112\pi\)
\(588\) 0 0
\(589\) 75.7128i 3.11969i
\(590\) 0 0
\(591\) −7.85641 −0.323169
\(592\) 0 0
\(593\) 8.53590 0.350527 0.175264 0.984522i \(-0.443922\pi\)
0.175264 + 0.984522i \(0.443922\pi\)
\(594\) 0 0
\(595\) 1.85641i 0.0761052i
\(596\) 0 0
\(597\) 5.85641i 0.239687i
\(598\) 0 0
\(599\) 28.3923 1.16008 0.580039 0.814589i \(-0.303037\pi\)
0.580039 + 0.814589i \(0.303037\pi\)
\(600\) 0 0
\(601\) −43.5692 −1.77723 −0.888613 0.458658i \(-0.848330\pi\)
−0.888613 + 0.458658i \(0.848330\pi\)
\(602\) 0 0
\(603\) − 2.92820i − 0.119246i
\(604\) 0 0
\(605\) 30.6795i 1.24730i
\(606\) 0 0
\(607\) 13.8564 0.562414 0.281207 0.959647i \(-0.409265\pi\)
0.281207 + 0.959647i \(0.409265\pi\)
\(608\) 0 0
\(609\) 4.92820 0.199701
\(610\) 0 0
\(611\) − 21.8564i − 0.884216i
\(612\) 0 0
\(613\) 27.8564i 1.12511i 0.826760 + 0.562555i \(0.190181\pi\)
−0.826760 + 0.562555i \(0.809819\pi\)
\(614\) 0 0
\(615\) −39.7128 −1.60138
\(616\) 0 0
\(617\) −18.7846 −0.756240 −0.378120 0.925757i \(-0.623429\pi\)
−0.378120 + 0.925757i \(0.623429\pi\)
\(618\) 0 0
\(619\) − 4.00000i − 0.160774i −0.996764 0.0803868i \(-0.974384\pi\)
0.996764 0.0803868i \(-0.0256155\pi\)
\(620\) 0 0
\(621\) 1.46410i 0.0587524i
\(622\) 0 0
\(623\) −3.46410 −0.138786
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 10.1436i 0.405096i
\(628\) 0 0
\(629\) − 1.07180i − 0.0427353i
\(630\) 0 0
\(631\) −26.9282 −1.07199 −0.535997 0.844220i \(-0.680064\pi\)
−0.535997 + 0.844220i \(0.680064\pi\)
\(632\) 0 0
\(633\) 16.0000 0.635943
\(634\) 0 0
\(635\) 10.1436i 0.402536i
\(636\) 0 0
\(637\) − 2.00000i − 0.0792429i
\(638\) 0 0
\(639\) 9.46410 0.374394
\(640\) 0 0
\(641\) −6.78461 −0.267976 −0.133988 0.990983i \(-0.542778\pi\)
−0.133988 + 0.990983i \(0.542778\pi\)
\(642\) 0 0
\(643\) 12.7846i 0.504176i 0.967704 + 0.252088i \(0.0811172\pi\)
−0.967704 + 0.252088i \(0.918883\pi\)
\(644\) 0 0
\(645\) − 27.7128i − 1.09119i
\(646\) 0 0
\(647\) 5.07180 0.199393 0.0996965 0.995018i \(-0.468213\pi\)
0.0996965 + 0.995018i \(0.468213\pi\)
\(648\) 0 0
\(649\) −1.56922 −0.0615972
\(650\) 0 0
\(651\) 10.9282i 0.428310i
\(652\) 0 0
\(653\) 20.9282i 0.818984i 0.912314 + 0.409492i \(0.134294\pi\)
−0.912314 + 0.409492i \(0.865706\pi\)
\(654\) 0 0
\(655\) 41.5692 1.62424
\(656\) 0 0
\(657\) 12.9282 0.504377
\(658\) 0 0
\(659\) − 14.5359i − 0.566238i −0.959085 0.283119i \(-0.908631\pi\)
0.959085 0.283119i \(-0.0913692\pi\)
\(660\) 0 0
\(661\) 26.7846i 1.04180i 0.853618 + 0.520900i \(0.174404\pi\)
−0.853618 + 0.520900i \(0.825596\pi\)
\(662\) 0 0
\(663\) −1.07180 −0.0416251
\(664\) 0 0
\(665\) 24.0000 0.930680
\(666\) 0 0
\(667\) 7.21539i 0.279381i
\(668\) 0 0
\(669\) 24.0000i 0.927894i
\(670\) 0 0
\(671\) 13.0718 0.504631
\(672\) 0 0
\(673\) 31.8564 1.22797 0.613987 0.789316i \(-0.289565\pi\)
0.613987 + 0.789316i \(0.289565\pi\)
\(674\) 0 0
\(675\) − 7.00000i − 0.269430i
\(676\) 0 0
\(677\) − 1.60770i − 0.0617887i −0.999523 0.0308944i \(-0.990164\pi\)
0.999523 0.0308944i \(-0.00983555\pi\)
\(678\) 0 0
\(679\) −8.92820 −0.342633
\(680\) 0 0
\(681\) 22.9282 0.878611
\(682\) 0 0
\(683\) − 18.2487i − 0.698268i −0.937073 0.349134i \(-0.886476\pi\)
0.937073 0.349134i \(-0.113524\pi\)
\(684\) 0 0
\(685\) 30.9282i 1.18171i
\(686\) 0 0
\(687\) −27.8564 −1.06279
\(688\) 0 0
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) 41.8564i 1.59229i 0.605104 + 0.796146i \(0.293131\pi\)
−0.605104 + 0.796146i \(0.706869\pi\)
\(692\) 0 0
\(693\) 1.46410i 0.0556166i
\(694\) 0 0
\(695\) 61.8564 2.34635
\(696\) 0 0
\(697\) −6.14359 −0.232705
\(698\) 0 0
\(699\) 12.9282i 0.488990i
\(700\) 0 0
\(701\) − 28.6410i − 1.08176i −0.841101 0.540878i \(-0.818092\pi\)
0.841101 0.540878i \(-0.181908\pi\)
\(702\) 0 0
\(703\) −13.8564 −0.522604
\(704\) 0 0
\(705\) 37.8564 1.42575
\(706\) 0 0
\(707\) − 15.4641i − 0.581587i
\(708\) 0 0
\(709\) 17.7128i 0.665219i 0.943065 + 0.332609i \(0.107929\pi\)
−0.943065 + 0.332609i \(0.892071\pi\)
\(710\) 0 0
\(711\) 10.9282 0.409840
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) − 10.1436i − 0.379349i
\(716\) 0 0
\(717\) − 14.5359i − 0.542853i
\(718\) 0 0
\(719\) −13.8564 −0.516757 −0.258378 0.966044i \(-0.583188\pi\)
−0.258378 + 0.966044i \(0.583188\pi\)
\(720\) 0 0
\(721\) 10.9282 0.406988
\(722\) 0 0
\(723\) 3.07180i 0.114241i
\(724\) 0 0
\(725\) − 34.4974i − 1.28120i
\(726\) 0 0
\(727\) −21.0718 −0.781510 −0.390755 0.920495i \(-0.627786\pi\)
−0.390755 + 0.920495i \(0.627786\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 4.28719i − 0.158567i
\(732\) 0 0
\(733\) − 5.71281i − 0.211008i −0.994419 0.105504i \(-0.966354\pi\)
0.994419 0.105504i \(-0.0336455\pi\)
\(734\) 0 0
\(735\) 3.46410 0.127775
\(736\) 0 0
\(737\) 4.28719 0.157921
\(738\) 0 0
\(739\) 37.0718i 1.36371i 0.731488 + 0.681854i \(0.238826\pi\)
−0.731488 + 0.681854i \(0.761174\pi\)
\(740\) 0 0
\(741\) 13.8564i 0.509028i
\(742\) 0 0
\(743\) −11.6077 −0.425845 −0.212923 0.977069i \(-0.568298\pi\)
−0.212923 + 0.977069i \(0.568298\pi\)
\(744\) 0 0
\(745\) 62.3538 2.28447
\(746\) 0 0
\(747\) − 4.00000i − 0.146352i
\(748\) 0 0
\(749\) 12.3923i 0.452805i
\(750\) 0 0
\(751\) 27.7128 1.01125 0.505627 0.862752i \(-0.331261\pi\)
0.505627 + 0.862752i \(0.331261\pi\)
\(752\) 0 0
\(753\) −9.07180 −0.330595
\(754\) 0 0
\(755\) − 75.7128i − 2.75547i
\(756\) 0 0
\(757\) − 26.0000i − 0.944986i −0.881334 0.472493i \(-0.843354\pi\)
0.881334 0.472493i \(-0.156646\pi\)
\(758\) 0 0
\(759\) −2.14359 −0.0778075
\(760\) 0 0
\(761\) −27.4641 −0.995573 −0.497786 0.867300i \(-0.665854\pi\)
−0.497786 + 0.867300i \(0.665854\pi\)
\(762\) 0 0
\(763\) 2.00000i 0.0724049i
\(764\) 0 0
\(765\) − 1.85641i − 0.0671185i
\(766\) 0 0
\(767\) −2.14359 −0.0774007
\(768\) 0 0
\(769\) 4.14359 0.149422 0.0747109 0.997205i \(-0.476197\pi\)
0.0747109 + 0.997205i \(0.476197\pi\)
\(770\) 0 0
\(771\) − 11.4641i − 0.412870i
\(772\) 0 0
\(773\) − 5.32051i − 0.191365i −0.995412 0.0956827i \(-0.969497\pi\)
0.995412 0.0956827i \(-0.0305034\pi\)
\(774\) 0 0
\(775\) 76.4974 2.74787
\(776\) 0 0
\(777\) −2.00000 −0.0717496
\(778\) 0 0
\(779\) 79.4256i 2.84572i
\(780\) 0 0
\(781\) 13.8564i 0.495821i
\(782\) 0 0
\(783\) −4.92820 −0.176120
\(784\) 0 0
\(785\) −3.21539 −0.114762
\(786\) 0 0
\(787\) − 23.7128i − 0.845270i −0.906300 0.422635i \(-0.861105\pi\)
0.906300 0.422635i \(-0.138895\pi\)
\(788\) 0 0
\(789\) 12.3923i 0.441178i
\(790\) 0 0
\(791\) −19.8564 −0.706013
\(792\) 0 0
\(793\) 17.8564 0.634100
\(794\) 0 0
\(795\) − 6.92820i − 0.245718i
\(796\) 0 0
\(797\) − 6.39230i − 0.226427i −0.993571 0.113214i \(-0.963886\pi\)
0.993571 0.113214i \(-0.0361144\pi\)
\(798\) 0 0
\(799\) 5.85641 0.207185
\(800\) 0 0
\(801\) 3.46410 0.122398
\(802\) 0 0
\(803\) 18.9282i 0.667962i
\(804\) 0 0
\(805\) 5.07180i 0.178757i
\(806\) 0 0
\(807\) −3.46410 −0.121942
\(808\) 0 0
\(809\) 38.0000 1.33601 0.668004 0.744157i \(-0.267149\pi\)
0.668004 + 0.744157i \(0.267149\pi\)
\(810\) 0 0
\(811\) − 37.5692i − 1.31923i −0.751602 0.659617i \(-0.770719\pi\)
0.751602 0.659617i \(-0.229281\pi\)
\(812\) 0 0
\(813\) − 2.92820i − 0.102697i
\(814\) 0 0
\(815\) 37.8564 1.32605
\(816\) 0 0
\(817\) −55.4256 −1.93910
\(818\) 0 0
\(819\) 2.00000i 0.0698857i
\(820\) 0 0
\(821\) − 12.1436i − 0.423814i −0.977290 0.211907i \(-0.932033\pi\)
0.977290 0.211907i \(-0.0679674\pi\)
\(822\) 0 0
\(823\) 21.0718 0.734517 0.367258 0.930119i \(-0.380296\pi\)
0.367258 + 0.930119i \(0.380296\pi\)
\(824\) 0 0
\(825\) 10.2487 0.356814
\(826\) 0 0
\(827\) − 37.1769i − 1.29277i −0.763012 0.646384i \(-0.776280\pi\)
0.763012 0.646384i \(-0.223720\pi\)
\(828\) 0 0
\(829\) − 21.7128i − 0.754117i −0.926189 0.377059i \(-0.876936\pi\)
0.926189 0.377059i \(-0.123064\pi\)
\(830\) 0 0
\(831\) 0.143594 0.00498120
\(832\) 0 0
\(833\) 0.535898 0.0185678
\(834\) 0 0
\(835\) 37.8564i 1.31007i
\(836\) 0 0
\(837\) − 10.9282i − 0.377734i
\(838\) 0 0
\(839\) 10.9282 0.377283 0.188642 0.982046i \(-0.439592\pi\)
0.188642 + 0.982046i \(0.439592\pi\)
\(840\) 0 0
\(841\) 4.71281 0.162511
\(842\) 0 0
\(843\) 12.9282i 0.445271i
\(844\) 0 0
\(845\) 31.1769i 1.07252i
\(846\) 0 0
\(847\) 8.85641 0.304310
\(848\) 0 0
\(849\) −14.9282 −0.512335
\(850\) 0 0
\(851\) − 2.92820i − 0.100378i
\(852\) 0 0
\(853\) 23.0718i 0.789963i 0.918689 + 0.394982i \(0.129249\pi\)
−0.918689 + 0.394982i \(0.870751\pi\)
\(854\) 0 0
\(855\) −24.0000 −0.820783
\(856\) 0 0
\(857\) −55.1769 −1.88481 −0.942404 0.334477i \(-0.891440\pi\)
−0.942404 + 0.334477i \(0.891440\pi\)
\(858\) 0 0
\(859\) − 38.9282i − 1.32821i −0.747638 0.664107i \(-0.768812\pi\)
0.747638 0.664107i \(-0.231188\pi\)
\(860\) 0 0
\(861\) 11.4641i 0.390696i
\(862\) 0 0
\(863\) −9.46410 −0.322162 −0.161081 0.986941i \(-0.551498\pi\)
−0.161081 + 0.986941i \(0.551498\pi\)
\(864\) 0 0
\(865\) 77.5692 2.63743
\(866\) 0 0
\(867\) 16.7128i 0.567597i
\(868\) 0 0
\(869\) 16.0000i 0.542763i
\(870\) 0 0
\(871\) 5.85641 0.198437
\(872\) 0 0
\(873\) 8.92820 0.302174
\(874\) 0 0
\(875\) − 6.92820i − 0.234216i
\(876\) 0 0
\(877\) − 8.14359i − 0.274990i −0.990502 0.137495i \(-0.956095\pi\)
0.990502 0.137495i \(-0.0439050\pi\)
\(878\) 0 0
\(879\) 16.5359 0.557742
\(880\) 0 0
\(881\) −13.3205 −0.448779 −0.224390 0.974500i \(-0.572039\pi\)
−0.224390 + 0.974500i \(0.572039\pi\)
\(882\) 0 0
\(883\) 33.5692i 1.12969i 0.825196 + 0.564847i \(0.191065\pi\)
−0.825196 + 0.564847i \(0.808935\pi\)
\(884\) 0 0
\(885\) − 3.71281i − 0.124805i
\(886\) 0 0
\(887\) 26.9282 0.904161 0.452080 0.891977i \(-0.350682\pi\)
0.452080 + 0.891977i \(0.350682\pi\)
\(888\) 0 0
\(889\) 2.92820 0.0982088
\(890\) 0 0
\(891\) − 1.46410i − 0.0490492i
\(892\) 0 0
\(893\) − 75.7128i − 2.53363i
\(894\) 0 0
\(895\) 32.7846 1.09587
\(896\) 0 0
\(897\) −2.92820 −0.0977699
\(898\) 0 0
\(899\) − 53.8564i − 1.79621i
\(900\) 0 0
\(901\) − 1.07180i − 0.0357067i
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) 0 0
\(905\) 20.7846 0.690904
\(906\) 0 0
\(907\) − 43.7128i − 1.45146i −0.687980 0.725730i \(-0.741502\pi\)
0.687980 0.725730i \(-0.258498\pi\)
\(908\) 0 0
\(909\) 15.4641i 0.512912i
\(910\) 0 0
\(911\) 29.1769 0.966674 0.483337 0.875434i \(-0.339425\pi\)
0.483337 + 0.875434i \(0.339425\pi\)
\(912\) 0 0
\(913\) 5.85641 0.193819
\(914\) 0 0
\(915\) 30.9282i 1.02245i
\(916\) 0 0
\(917\) − 12.0000i − 0.396275i
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) −22.9282 −0.755510
\(922\) 0 0
\(923\) 18.9282i 0.623029i
\(924\) 0 0
\(925\) 14.0000i 0.460317i
\(926\) 0 0
\(927\) −10.9282 −0.358929
\(928\) 0 0
\(929\) 13.6077 0.446454 0.223227 0.974766i \(-0.428341\pi\)
0.223227 + 0.974766i \(0.428341\pi\)
\(930\) 0 0
\(931\) − 6.92820i − 0.227063i
\(932\) 0 0
\(933\) 18.9282i 0.619682i
\(934\) 0 0
\(935\) 2.71797 0.0888870
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) − 14.0000i − 0.456873i
\(940\) 0 0
\(941\) 36.5359i 1.19104i 0.803342 + 0.595518i \(0.203053\pi\)
−0.803342 + 0.595518i \(0.796947\pi\)
\(942\) 0 0
\(943\) −16.7846 −0.546582
\(944\) 0 0
\(945\) −3.46410 −0.112687
\(946\) 0 0
\(947\) − 26.2487i − 0.852969i −0.904495 0.426484i \(-0.859752\pi\)
0.904495 0.426484i \(-0.140248\pi\)
\(948\) 0 0
\(949\) 25.8564i 0.839334i
\(950\) 0 0
\(951\) −0.143594 −0.00465634
\(952\) 0 0
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) 0 0
\(955\) 32.7846i 1.06089i
\(956\) 0 0
\(957\) − 7.21539i − 0.233240i
\(958\) 0 0
\(959\) 8.92820 0.288307
\(960\) 0 0
\(961\) 88.4256 2.85244
\(962\) 0 0
\(963\) − 12.3923i − 0.399336i
\(964\) 0 0
\(965\) 54.9282i 1.76820i
\(966\) 0 0
\(967\) −34.9282 −1.12322 −0.561608 0.827404i \(-0.689817\pi\)
−0.561608 + 0.827404i \(0.689817\pi\)
\(968\) 0 0
\(969\) −3.71281 −0.119273
\(970\) 0 0
\(971\) 1.85641i 0.0595749i 0.999556 + 0.0297875i \(0.00948305\pi\)
−0.999556 + 0.0297875i \(0.990517\pi\)
\(972\) 0 0
\(973\) − 17.8564i − 0.572450i
\(974\) 0 0
\(975\) 14.0000 0.448359
\(976\) 0 0
\(977\) 38.4974 1.23164 0.615821 0.787886i \(-0.288825\pi\)
0.615821 + 0.787886i \(0.288825\pi\)
\(978\) 0 0
\(979\) 5.07180i 0.162095i
\(980\) 0 0
\(981\) − 2.00000i − 0.0638551i
\(982\) 0 0
\(983\) 3.71281 0.118420 0.0592102 0.998246i \(-0.481142\pi\)
0.0592102 + 0.998246i \(0.481142\pi\)
\(984\) 0 0
\(985\) 27.2154 0.867154
\(986\) 0 0
\(987\) − 10.9282i − 0.347849i
\(988\) 0 0
\(989\) − 11.7128i − 0.372446i
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) 24.0000 0.761617
\(994\) 0 0
\(995\) − 20.2872i − 0.643147i
\(996\) 0 0
\(997\) 31.0718i 0.984054i 0.870580 + 0.492027i \(0.163744\pi\)
−0.870580 + 0.492027i \(0.836256\pi\)
\(998\) 0 0
\(999\) 2.00000 0.0632772
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 5376.2.c.bn.2689.2 4
4.3 odd 2 5376.2.c.bh.2689.4 4
8.3 odd 2 5376.2.c.bh.2689.1 4
8.5 even 2 inner 5376.2.c.bn.2689.3 4
16.3 odd 4 672.2.a.i.1.2 2
16.5 even 4 1344.2.a.u.1.1 2
16.11 odd 4 1344.2.a.v.1.1 2
16.13 even 4 672.2.a.j.1.2 yes 2
48.5 odd 4 4032.2.a.br.1.2 2
48.11 even 4 4032.2.a.bs.1.2 2
48.29 odd 4 2016.2.a.s.1.1 2
48.35 even 4 2016.2.a.t.1.1 2
112.13 odd 4 4704.2.a.bm.1.1 2
112.27 even 4 9408.2.a.do.1.2 2
112.69 odd 4 9408.2.a.dx.1.2 2
112.83 even 4 4704.2.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.a.i.1.2 2 16.3 odd 4
672.2.a.j.1.2 yes 2 16.13 even 4
1344.2.a.u.1.1 2 16.5 even 4
1344.2.a.v.1.1 2 16.11 odd 4
2016.2.a.s.1.1 2 48.29 odd 4
2016.2.a.t.1.1 2 48.35 even 4
4032.2.a.br.1.2 2 48.5 odd 4
4032.2.a.bs.1.2 2 48.11 even 4
4704.2.a.bm.1.1 2 112.13 odd 4
4704.2.a.bn.1.1 2 112.83 even 4
5376.2.c.bh.2689.1 4 8.3 odd 2
5376.2.c.bh.2689.4 4 4.3 odd 2
5376.2.c.bn.2689.2 4 1.1 even 1 trivial
5376.2.c.bn.2689.3 4 8.5 even 2 inner
9408.2.a.do.1.2 2 112.27 even 4
9408.2.a.dx.1.2 2 112.69 odd 4