Properties

Label 2736.2.d.a.2015.4
Level $2736$
Weight $2$
Character 2736.2015
Analytic conductor $21.847$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 12 x^{10} + 47 x^{8} - 44 x^{6} + 81 x^{4} + 848 x^{2} + 1600\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2015.4
Root \(0.366740 - 1.25423i\) of defining polynomial
Character \(\chi\) \(=\) 2736.2015
Dual form 2736.2.d.a.2015.9

$q$-expansion

\(f(q)\) \(=\) \(q-1.09425i q^{5} +1.25511i q^{7} +O(q^{10})\) \(q-1.09425i q^{5} +1.25511i q^{7} -2.92198 q^{11} -4.58480 q^{13} -0.680734i q^{17} -1.00000i q^{19} +6.07037 q^{23} +3.80261 q^{25} -1.82773i q^{29} +8.58480i q^{31} +1.37340 q^{35} -0.510210 q^{37} -3.18920i q^{41} +5.38741i q^{43} +10.9937 q^{47} +5.42471 q^{49} +9.03317i q^{53} +3.19739i q^{55} +3.54997 q^{59} +4.29240 q^{61} +5.01693i q^{65} +11.0950i q^{67} +12.1407 q^{71} -5.25511 q^{73} -3.66740i q^{77} -8.26462i q^{79} +6.07037 q^{83} -0.744895 q^{85} +13.9685i q^{89} -5.75441i q^{91} -1.09425 q^{95} -7.67982 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + O(q^{10}) \) \( 12 q + 4 q^{25} + 8 q^{37} - 52 q^{49} + 24 q^{61} - 56 q^{73} - 16 q^{85} + 32 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 1.09425i − 0.489365i −0.969603 0.244682i \(-0.921316\pi\)
0.969603 0.244682i \(-0.0786837\pi\)
\(6\) 0 0
\(7\) 1.25511i 0.474385i 0.971463 + 0.237193i \(0.0762272\pi\)
−0.971463 + 0.237193i \(0.923773\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.92198 −0.881011 −0.440506 0.897750i \(-0.645201\pi\)
−0.440506 + 0.897750i \(0.645201\pi\)
\(12\) 0 0
\(13\) −4.58480 −1.27160 −0.635798 0.771856i \(-0.719329\pi\)
−0.635798 + 0.771856i \(0.719329\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.680734i − 0.165102i −0.996587 0.0825511i \(-0.973693\pi\)
0.996587 0.0825511i \(-0.0263068\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.07037 1.26576 0.632880 0.774250i \(-0.281873\pi\)
0.632880 + 0.774250i \(0.281873\pi\)
\(24\) 0 0
\(25\) 3.80261 0.760522
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 1.82773i − 0.339401i −0.985496 0.169701i \(-0.945720\pi\)
0.985496 0.169701i \(-0.0542801\pi\)
\(30\) 0 0
\(31\) 8.58480i 1.54188i 0.636910 + 0.770938i \(0.280212\pi\)
−0.636910 + 0.770938i \(0.719788\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.37340 0.232147
\(36\) 0 0
\(37\) −0.510210 −0.0838780 −0.0419390 0.999120i \(-0.513354\pi\)
−0.0419390 + 0.999120i \(0.513354\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 3.18920i − 0.498069i −0.968495 0.249035i \(-0.919887\pi\)
0.968495 0.249035i \(-0.0801133\pi\)
\(42\) 0 0
\(43\) 5.38741i 0.821573i 0.911732 + 0.410787i \(0.134746\pi\)
−0.911732 + 0.410787i \(0.865254\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.9937 1.60360 0.801801 0.597591i \(-0.203875\pi\)
0.801801 + 0.597591i \(0.203875\pi\)
\(48\) 0 0
\(49\) 5.42471 0.774959
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.03317i 1.24080i 0.784285 + 0.620401i \(0.213030\pi\)
−0.784285 + 0.620401i \(0.786970\pi\)
\(54\) 0 0
\(55\) 3.19739i 0.431136i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.54997 0.462167 0.231084 0.972934i \(-0.425773\pi\)
0.231084 + 0.972934i \(0.425773\pi\)
\(60\) 0 0
\(61\) 4.29240 0.549586 0.274793 0.961503i \(-0.411391\pi\)
0.274793 + 0.961503i \(0.411391\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.01693i 0.622274i
\(66\) 0 0
\(67\) 11.0950i 1.35547i 0.735306 + 0.677736i \(0.237039\pi\)
−0.735306 + 0.677736i \(0.762961\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.1407 1.44084 0.720421 0.693537i \(-0.243949\pi\)
0.720421 + 0.693537i \(0.243949\pi\)
\(72\) 0 0
\(73\) −5.25511 −0.615064 −0.307532 0.951538i \(-0.599503\pi\)
−0.307532 + 0.951538i \(0.599503\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.66740i − 0.417939i
\(78\) 0 0
\(79\) − 8.26462i − 0.929842i −0.885352 0.464921i \(-0.846083\pi\)
0.885352 0.464921i \(-0.153917\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.07037 0.666310 0.333155 0.942872i \(-0.391887\pi\)
0.333155 + 0.942872i \(0.391887\pi\)
\(84\) 0 0
\(85\) −0.744895 −0.0807952
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.9685i 1.48066i 0.672246 + 0.740328i \(0.265330\pi\)
−0.672246 + 0.740328i \(0.734670\pi\)
\(90\) 0 0
\(91\) − 5.75441i − 0.603226i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.09425 −0.112268
\(96\) 0 0
\(97\) −7.67982 −0.779767 −0.389884 0.920864i \(-0.627485\pi\)
−0.389884 + 0.920864i \(0.627485\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 2.96280i − 0.294809i −0.989076 0.147405i \(-0.952908\pi\)
0.989076 0.147405i \(-0.0470919\pi\)
\(102\) 0 0
\(103\) 14.6594i 1.44443i 0.691667 + 0.722217i \(0.256877\pi\)
−0.691667 + 0.722217i \(0.743123\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.84397 −0.564958 −0.282479 0.959274i \(-0.591157\pi\)
−0.282479 + 0.959274i \(0.591157\pi\)
\(108\) 0 0
\(109\) −5.02042 −0.480869 −0.240435 0.970665i \(-0.577290\pi\)
−0.240435 + 0.970665i \(0.577290\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 11.1400i − 1.04797i −0.851728 0.523984i \(-0.824445\pi\)
0.851728 0.523984i \(-0.175555\pi\)
\(114\) 0 0
\(115\) − 6.64252i − 0.619418i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.854393 0.0783221
\(120\) 0 0
\(121\) −2.46201 −0.223819
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 9.63228i − 0.861537i
\(126\) 0 0
\(127\) 0.320184i 0.0284117i 0.999899 + 0.0142059i \(0.00452201\pi\)
−0.999899 + 0.0142059i \(0.995478\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 13.2877 1.16096 0.580478 0.814276i \(-0.302866\pi\)
0.580478 + 0.814276i \(0.302866\pi\)
\(132\) 0 0
\(133\) 1.25511 0.108831
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 21.9074i − 1.87167i −0.352432 0.935837i \(-0.614645\pi\)
0.352432 0.935837i \(-0.385355\pi\)
\(138\) 0 0
\(139\) − 1.25511i − 0.106457i −0.998582 0.0532283i \(-0.983049\pi\)
0.998582 0.0532283i \(-0.0169511\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 13.3967 1.12029
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 17.3040i 1.41760i 0.705411 + 0.708799i \(0.250763\pi\)
−0.705411 + 0.708799i \(0.749237\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.39394 0.754540
\(156\) 0 0
\(157\) −1.67982 −0.134064 −0.0670320 0.997751i \(-0.521353\pi\)
−0.0670320 + 0.997751i \(0.521353\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.61896i 0.600458i
\(162\) 0 0
\(163\) 19.3596i 1.51636i 0.652043 + 0.758182i \(0.273912\pi\)
−0.652043 + 0.758182i \(0.726088\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 25.0847 1.94111 0.970555 0.240881i \(-0.0774364\pi\)
0.970555 + 0.240881i \(0.0774364\pi\)
\(168\) 0 0
\(169\) 8.02042 0.616955
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.59147i 0.729226i 0.931159 + 0.364613i \(0.118799\pi\)
−0.931159 + 0.364613i \(0.881201\pi\)
\(174\) 0 0
\(175\) 4.77268i 0.360780i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −9.39394 −0.702136 −0.351068 0.936350i \(-0.614181\pi\)
−0.351068 + 0.936350i \(0.614181\pi\)
\(180\) 0 0
\(181\) 10.7748 0.800887 0.400443 0.916322i \(-0.368856\pi\)
0.400443 + 0.916322i \(0.368856\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.558299i 0.0410469i
\(186\) 0 0
\(187\) 1.98909i 0.145457i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.89380 0.281746 0.140873 0.990028i \(-0.455009\pi\)
0.140873 + 0.990028i \(0.455009\pi\)
\(192\) 0 0
\(193\) 10.8494 0.780959 0.390479 0.920612i \(-0.372309\pi\)
0.390479 + 0.920612i \(0.372309\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 8.61965i − 0.614125i −0.951689 0.307062i \(-0.900654\pi\)
0.951689 0.307062i \(-0.0993460\pi\)
\(198\) 0 0
\(199\) 10.4078i 0.737792i 0.929471 + 0.368896i \(0.120264\pi\)
−0.929471 + 0.368896i \(0.879736\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.29400 0.161007
\(204\) 0 0
\(205\) −3.48979 −0.243738
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.92198i 0.202118i
\(210\) 0 0
\(211\) 16.1154i 1.10943i 0.832040 + 0.554716i \(0.187173\pi\)
−0.832040 + 0.554716i \(0.812827\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.89519 0.402049
\(216\) 0 0
\(217\) −10.7748 −0.731443
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.12103i 0.209943i
\(222\) 0 0
\(223\) − 8.26462i − 0.553440i −0.960951 0.276720i \(-0.910753\pi\)
0.960951 0.276720i \(-0.0892474\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.74680 0.182312 0.0911559 0.995837i \(-0.470944\pi\)
0.0911559 + 0.995837i \(0.470944\pi\)
\(228\) 0 0
\(229\) 20.1622 1.33236 0.666179 0.745792i \(-0.267929\pi\)
0.666179 + 0.745792i \(0.267929\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 6.33759i − 0.415189i −0.978215 0.207595i \(-0.933437\pi\)
0.978215 0.207595i \(-0.0665635\pi\)
\(234\) 0 0
\(235\) − 12.0299i − 0.784746i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.40298 0.155436 0.0777178 0.996975i \(-0.475237\pi\)
0.0777178 + 0.996975i \(0.475237\pi\)
\(240\) 0 0
\(241\) −14.0746 −0.906624 −0.453312 0.891352i \(-0.649758\pi\)
−0.453312 + 0.891352i \(0.649758\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 5.93600i − 0.379237i
\(246\) 0 0
\(247\) 4.58480i 0.291724i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.92477 −0.437087 −0.218544 0.975827i \(-0.570131\pi\)
−0.218544 + 0.975827i \(0.570131\pi\)
\(252\) 0 0
\(253\) −17.7375 −1.11515
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.3960i 0.773243i 0.922238 + 0.386621i \(0.126358\pi\)
−0.922238 + 0.386621i \(0.873642\pi\)
\(258\) 0 0
\(259\) − 0.640367i − 0.0397905i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −10.0219 −0.617979 −0.308989 0.951065i \(-0.599991\pi\)
−0.308989 + 0.951065i \(0.599991\pi\)
\(264\) 0 0
\(265\) 9.88457 0.607204
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 6.76304i − 0.412350i −0.978515 0.206175i \(-0.933898\pi\)
0.978515 0.206175i \(-0.0661016\pi\)
\(270\) 0 0
\(271\) 27.9444i 1.69750i 0.528791 + 0.848752i \(0.322645\pi\)
−0.528791 + 0.848752i \(0.677355\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −11.1112 −0.670029
\(276\) 0 0
\(277\) 15.5774 0.935958 0.467979 0.883740i \(-0.344982\pi\)
0.467979 + 0.883740i \(0.344982\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 8.20613i − 0.489537i −0.969582 0.244768i \(-0.921288\pi\)
0.969582 0.244768i \(-0.0787119\pi\)
\(282\) 0 0
\(283\) − 22.5570i − 1.34088i −0.741966 0.670438i \(-0.766106\pi\)
0.741966 0.670438i \(-0.233894\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00278 0.236277
\(288\) 0 0
\(289\) 16.5366 0.972741
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 10.3946i − 0.607261i −0.952790 0.303631i \(-0.901801\pi\)
0.952790 0.303631i \(-0.0981989\pi\)
\(294\) 0 0
\(295\) − 3.88457i − 0.226168i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −27.8315 −1.60954
\(300\) 0 0
\(301\) −6.76177 −0.389742
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 4.69697i − 0.268948i
\(306\) 0 0
\(307\) 5.02042i 0.286531i 0.989684 + 0.143265i \(0.0457602\pi\)
−0.989684 + 0.143265i \(0.954240\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.40298 0.136260 0.0681301 0.997676i \(-0.478297\pi\)
0.0681301 + 0.997676i \(0.478297\pi\)
\(312\) 0 0
\(313\) 5.67982 0.321042 0.160521 0.987032i \(-0.448683\pi\)
0.160521 + 0.987032i \(0.448683\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 27.3652i 1.53698i 0.639860 + 0.768491i \(0.278992\pi\)
−0.639860 + 0.768491i \(0.721008\pi\)
\(318\) 0 0
\(319\) 5.34060i 0.299016i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −0.680734 −0.0378771
\(324\) 0 0
\(325\) −17.4342 −0.967077
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 13.7983i 0.760725i
\(330\) 0 0
\(331\) − 26.0746i − 1.43319i −0.697490 0.716595i \(-0.745700\pi\)
0.697490 0.716595i \(-0.254300\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.1407 0.663320
\(336\) 0 0
\(337\) −31.0394 −1.69083 −0.845413 0.534113i \(-0.820646\pi\)
−0.845413 + 0.534113i \(0.820646\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 25.0847i − 1.35841i
\(342\) 0 0
\(343\) 15.5943i 0.842014i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −13.2877 −0.713323 −0.356662 0.934234i \(-0.616085\pi\)
−0.356662 + 0.934234i \(0.616085\pi\)
\(348\) 0 0
\(349\) 1.21426 0.0649981 0.0324991 0.999472i \(-0.489653\pi\)
0.0324991 + 0.999472i \(0.489653\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 19.3751i 1.03123i 0.856820 + 0.515615i \(0.172437\pi\)
−0.856820 + 0.515615i \(0.827563\pi\)
\(354\) 0 0
\(355\) − 13.2850i − 0.705097i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −36.0784 −1.90415 −0.952073 0.305872i \(-0.901052\pi\)
−0.952073 + 0.305872i \(0.901052\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.75041i 0.300990i
\(366\) 0 0
\(367\) 5.75441i 0.300378i 0.988657 + 0.150189i \(0.0479882\pi\)
−0.988657 + 0.150189i \(0.952012\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −11.3376 −0.588618
\(372\) 0 0
\(373\) 2.58480 0.133836 0.0669180 0.997758i \(-0.478683\pi\)
0.0669180 + 0.997758i \(0.478683\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.37979i 0.431581i
\(378\) 0 0
\(379\) − 7.53063i − 0.386822i −0.981118 0.193411i \(-0.938045\pi\)
0.981118 0.193411i \(-0.0619552\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 26.5755 1.35794 0.678972 0.734164i \(-0.262426\pi\)
0.678972 + 0.734164i \(0.262426\pi\)
\(384\) 0 0
\(385\) −4.01306 −0.204524
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 7.61743i − 0.386219i −0.981177 0.193110i \(-0.938143\pi\)
0.981177 0.193110i \(-0.0618573\pi\)
\(390\) 0 0
\(391\) − 4.13231i − 0.208980i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.04358 −0.455032
\(396\) 0 0
\(397\) 11.7245 0.588435 0.294217 0.955739i \(-0.404941\pi\)
0.294217 + 0.955739i \(0.404941\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 4.73778i − 0.236594i −0.992978 0.118297i \(-0.962257\pi\)
0.992978 0.118297i \(-0.0377434\pi\)
\(402\) 0 0
\(403\) − 39.3596i − 1.96064i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.49083 0.0738975
\(408\) 0 0
\(409\) −25.7544 −1.27347 −0.636737 0.771081i \(-0.719716\pi\)
−0.636737 + 0.771081i \(0.719716\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.45559i 0.219245i
\(414\) 0 0
\(415\) − 6.64252i − 0.326068i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −10.0732 −0.492106 −0.246053 0.969256i \(-0.579134\pi\)
−0.246053 + 0.969256i \(0.579134\pi\)
\(420\) 0 0
\(421\) −2.77483 −0.135237 −0.0676185 0.997711i \(-0.521540\pi\)
−0.0676185 + 0.997711i \(0.521540\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 2.58857i − 0.125564i
\(426\) 0 0
\(427\) 5.38741i 0.260715i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 21.5347 1.03729 0.518645 0.854990i \(-0.326437\pi\)
0.518645 + 0.854990i \(0.326437\pi\)
\(432\) 0 0
\(433\) 24.3055 1.16805 0.584023 0.811737i \(-0.301478\pi\)
0.584023 + 0.811737i \(0.301478\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 6.07037i − 0.290385i
\(438\) 0 0
\(439\) − 6.39478i − 0.305206i −0.988288 0.152603i \(-0.951234\pi\)
0.988288 0.152603i \(-0.0487656\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −23.5873 −1.12067 −0.560333 0.828267i \(-0.689327\pi\)
−0.560333 + 0.828267i \(0.689327\pi\)
\(444\) 0 0
\(445\) 15.2850 0.724580
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 2.57314i − 0.121434i −0.998155 0.0607171i \(-0.980661\pi\)
0.998155 0.0607171i \(-0.0193387\pi\)
\(450\) 0 0
\(451\) 9.31879i 0.438805i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.29678 −0.295197
\(456\) 0 0
\(457\) −7.06508 −0.330490 −0.165245 0.986253i \(-0.552842\pi\)
−0.165245 + 0.986253i \(0.552842\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 23.0425i − 1.07319i −0.843839 0.536597i \(-0.819710\pi\)
0.843839 0.536597i \(-0.180290\pi\)
\(462\) 0 0
\(463\) 27.5774i 1.28163i 0.767694 + 0.640816i \(0.221404\pi\)
−0.767694 + 0.640816i \(0.778596\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.5128 0.532747 0.266373 0.963870i \(-0.414175\pi\)
0.266373 + 0.963870i \(0.414175\pi\)
\(468\) 0 0
\(469\) −13.9254 −0.643016
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 15.7419i − 0.723815i
\(474\) 0 0
\(475\) − 3.80261i − 0.174476i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.52040 −0.115160 −0.0575800 0.998341i \(-0.518338\pi\)
−0.0575800 + 0.998341i \(0.518338\pi\)
\(480\) 0 0
\(481\) 2.33921 0.106659
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.40366i 0.381590i
\(486\) 0 0
\(487\) − 10.4546i − 0.473745i −0.971541 0.236873i \(-0.923878\pi\)
0.971541 0.236873i \(-0.0761224\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −41.0018 −1.85038 −0.925192 0.379499i \(-0.876096\pi\)
−0.925192 + 0.379499i \(0.876096\pi\)
\(492\) 0 0
\(493\) −1.24420 −0.0560359
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15.2379i 0.683514i
\(498\) 0 0
\(499\) − 33.0841i − 1.48105i −0.672030 0.740524i \(-0.734577\pi\)
0.672030 0.740524i \(-0.265423\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −14.6611 −0.653708 −0.326854 0.945075i \(-0.605989\pi\)
−0.326854 + 0.945075i \(0.605989\pi\)
\(504\) 0 0
\(505\) −3.24205 −0.144269
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.92351i 0.129582i 0.997899 + 0.0647911i \(0.0206381\pi\)
−0.997899 + 0.0647911i \(0.979362\pi\)
\(510\) 0 0
\(511\) − 6.59571i − 0.291777i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.0411 0.706854
\(516\) 0 0
\(517\) −32.1236 −1.41279
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 3.91075i − 0.171333i −0.996324 0.0856664i \(-0.972698\pi\)
0.996324 0.0856664i \(-0.0273019\pi\)
\(522\) 0 0
\(523\) − 11.8290i − 0.517246i −0.965978 0.258623i \(-0.916731\pi\)
0.965978 0.258623i \(-0.0832688\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.84397 0.254567
\(528\) 0 0
\(529\) 13.8494 0.602149
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 14.6219i 0.633343i
\(534\) 0 0
\(535\) 6.39478i 0.276470i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −15.8509 −0.682748
\(540\) 0 0
\(541\) 44.9948 1.93448 0.967239 0.253869i \(-0.0817033\pi\)
0.967239 + 0.253869i \(0.0817033\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.49361i 0.235320i
\(546\) 0 0
\(547\) − 11.4152i − 0.488079i −0.969765 0.244039i \(-0.921527\pi\)
0.969765 0.244039i \(-0.0784726\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.82773 −0.0778640
\(552\) 0 0
\(553\) 10.3730 0.441103
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 39.6234i 1.67890i 0.543440 + 0.839448i \(0.317122\pi\)
−0.543440 + 0.839448i \(0.682878\pi\)
\(558\) 0 0
\(559\) − 24.7002i − 1.04471i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −27.8315 −1.17296 −0.586478 0.809965i \(-0.699486\pi\)
−0.586478 + 0.809965i \(0.699486\pi\)
\(564\) 0 0
\(565\) −12.1900 −0.512838
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 34.6523i − 1.45270i −0.687326 0.726349i \(-0.741216\pi\)
0.687326 0.726349i \(-0.258784\pi\)
\(570\) 0 0
\(571\) − 27.9444i − 1.16944i −0.811236 0.584719i \(-0.801205\pi\)
0.811236 0.584719i \(-0.198795\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 23.0833 0.962639
\(576\) 0 0
\(577\) 16.2924 0.678262 0.339131 0.940739i \(-0.389867\pi\)
0.339131 + 0.940739i \(0.389867\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.61896i 0.316088i
\(582\) 0 0
\(583\) − 26.3948i − 1.09316i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.01915 0.248437 0.124218 0.992255i \(-0.460358\pi\)
0.124218 + 0.992255i \(0.460358\pi\)
\(588\) 0 0
\(589\) 8.58480 0.353731
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 7.52388i − 0.308969i −0.987995 0.154484i \(-0.950628\pi\)
0.987995 0.154484i \(-0.0493716\pi\)
\(594\) 0 0
\(595\) − 0.934921i − 0.0383280i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.1407 −0.496057 −0.248029 0.968753i \(-0.579783\pi\)
−0.248029 + 0.968753i \(0.579783\pi\)
\(600\) 0 0
\(601\) 28.6256 1.16766 0.583832 0.811874i \(-0.301553\pi\)
0.583832 + 0.811874i \(0.301553\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.69406i 0.109529i
\(606\) 0 0
\(607\) 30.7748i 1.24911i 0.780980 + 0.624556i \(0.214720\pi\)
−0.780980 + 0.624556i \(0.785280\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −50.4042 −2.03913
\(612\) 0 0
\(613\) 8.14322 0.328901 0.164451 0.986385i \(-0.447415\pi\)
0.164451 + 0.986385i \(0.447415\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 6.93822i − 0.279322i −0.990199 0.139661i \(-0.955399\pi\)
0.990199 0.139661i \(-0.0446013\pi\)
\(618\) 0 0
\(619\) − 42.1900i − 1.69576i −0.530188 0.847880i \(-0.677879\pi\)
0.530188 0.847880i \(-0.322121\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −17.5319 −0.702401
\(624\) 0 0
\(625\) 8.47291 0.338917
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.347317i 0.0138485i
\(630\) 0 0
\(631\) 0.614738i 0.0244723i 0.999925 + 0.0122362i \(0.00389499\pi\)
−0.999925 + 0.0122362i \(0.996105\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0.350362 0.0139037
\(636\) 0 0
\(637\) −24.8712 −0.985434
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 41.0784i 1.62250i 0.584700 + 0.811250i \(0.301212\pi\)
−0.584700 + 0.811250i \(0.698788\pi\)
\(642\) 0 0
\(643\) 17.5366i 0.691576i 0.938313 + 0.345788i \(0.112388\pi\)
−0.938313 + 0.345788i \(0.887612\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −30.3006 −1.19124 −0.595621 0.803266i \(-0.703094\pi\)
−0.595621 + 0.803266i \(0.703094\pi\)
\(648\) 0 0
\(649\) −10.3730 −0.407175
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 36.6895i − 1.43577i −0.696162 0.717885i \(-0.745111\pi\)
0.696162 0.717885i \(-0.254889\pi\)
\(654\) 0 0
\(655\) − 14.5401i − 0.568130i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −40.7754 −1.58838 −0.794192 0.607667i \(-0.792105\pi\)
−0.794192 + 0.607667i \(0.792105\pi\)
\(660\) 0 0
\(661\) 20.7748 0.808047 0.404024 0.914749i \(-0.367611\pi\)
0.404024 + 0.914749i \(0.367611\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 1.37340i − 0.0532582i
\(666\) 0 0
\(667\) − 11.0950i − 0.429601i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −12.5423 −0.484191
\(672\) 0 0
\(673\) 0.394777 0.0152175 0.00760876 0.999971i \(-0.497578\pi\)
0.00760876 + 0.999971i \(0.497578\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 20.5579i − 0.790103i −0.918659 0.395051i \(-0.870727\pi\)
0.918659 0.395051i \(-0.129273\pi\)
\(678\) 0 0
\(679\) − 9.63898i − 0.369910i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −22.4403 −0.858654 −0.429327 0.903149i \(-0.641249\pi\)
−0.429327 + 0.903149i \(0.641249\pi\)
\(684\) 0 0
\(685\) −23.9722 −0.915931
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 41.4153i − 1.57780i
\(690\) 0 0
\(691\) − 23.1974i − 0.882470i −0.897392 0.441235i \(-0.854540\pi\)
0.897392 0.441235i \(-0.145460\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.37340 −0.0520961
\(696\) 0 0
\(697\) −2.17100 −0.0822324
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 45.3892i − 1.71433i −0.515045 0.857163i \(-0.672225\pi\)
0.515045 0.857163i \(-0.327775\pi\)
\(702\) 0 0
\(703\) 0.510210i 0.0192429i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.71862 0.139853
\(708\) 0 0
\(709\) −40.8494 −1.53413 −0.767066 0.641568i \(-0.778284\pi\)
−0.767066 + 0.641568i \(0.778284\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 52.1130i 1.95165i
\(714\) 0 0
\(715\) − 14.6594i − 0.548230i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 30.3006 1.13002 0.565012 0.825083i \(-0.308872\pi\)
0.565012 + 0.825083i \(0.308872\pi\)
\(720\) 0 0
\(721\) −18.3991 −0.685218
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 6.95015i − 0.258122i
\(726\) 0 0
\(727\) − 17.5366i − 0.650397i −0.945646 0.325198i \(-0.894569\pi\)
0.945646 0.325198i \(-0.105431\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.66740 0.135644
\(732\) 0 0
\(733\) 6.51021 0.240460 0.120230 0.992746i \(-0.461637\pi\)
0.120230 + 0.992746i \(0.461637\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 32.4195i − 1.19419i
\(738\) 0 0
\(739\) 20.9349i 0.770104i 0.922895 + 0.385052i \(0.125816\pi\)
−0.922895 + 0.385052i \(0.874184\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −19.6935 −0.722484 −0.361242 0.932472i \(-0.617647\pi\)
−0.361242 + 0.932472i \(0.617647\pi\)
\(744\) 0 0
\(745\) 18.9349 0.693722
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 7.33479i − 0.268008i
\(750\) 0 0
\(751\) 34.3392i 1.25306i 0.779399 + 0.626528i \(0.215525\pi\)
−0.779399 + 0.626528i \(0.784475\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −16.2178 −0.589446 −0.294723 0.955583i \(-0.595227\pi\)
−0.294723 + 0.955583i \(0.595227\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 48.6038i 1.76189i 0.473222 + 0.880943i \(0.343091\pi\)
−0.473222 + 0.880943i \(0.656909\pi\)
\(762\) 0 0
\(763\) − 6.30115i − 0.228117i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −16.2759 −0.587690
\(768\) 0 0
\(769\) −37.9912 −1.37000 −0.685000 0.728543i \(-0.740198\pi\)
−0.685000 + 0.728543i \(0.740198\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6.36494i 0.228931i 0.993427 + 0.114466i \(0.0365156\pi\)
−0.993427 + 0.114466i \(0.963484\pi\)
\(774\) 0 0
\(775\) 32.6447i 1.17263i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.18920 −0.114265
\(780\) 0 0
\(781\) −35.4751 −1.26940
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.83814i 0.0656061i
\(786\) 0 0
\(787\) 28.9050i 1.03035i 0.857085 + 0.515176i \(0.172273\pi\)
−0.857085 + 0.515176i \(0.827727\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 13.9819 0.497140
\(792\) 0 0
\(793\) −19.6798 −0.698851
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 33.7705i 1.19621i 0.801417 + 0.598106i \(0.204080\pi\)
−0.801417 + 0.598106i \(0.795920\pi\)
\(798\) 0 0
\(799\) − 7.48382i − 0.264758i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 15.3553 0.541878
\(804\) 0 0
\(805\) 8.33706 0.293843
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 39.5841i 1.39170i 0.718186 + 0.695851i \(0.244973\pi\)
−0.718186 + 0.695851i \(0.755027\pi\)
\(810\) 0 0
\(811\) 19.0394i 0.668565i 0.942473 + 0.334283i \(0.108494\pi\)
−0.942473 + 0.334283i \(0.891506\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 21.1843 0.742055
\(816\) 0 0
\(817\) 5.38741 0.188482
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 24.8413i 0.866968i 0.901161 + 0.433484i \(0.142716\pi\)
−0.901161 + 0.433484i \(0.857284\pi\)
\(822\) 0 0
\(823\) 1.25511i 0.0437502i 0.999761 + 0.0218751i \(0.00696362\pi\)
−0.999761 + 0.0218751i \(0.993036\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 32.4195 1.12733 0.563667 0.826002i \(-0.309390\pi\)
0.563667 + 0.826002i \(0.309390\pi\)
\(828\) 0 0
\(829\) 8.90499 0.309283 0.154641 0.987971i \(-0.450578\pi\)
0.154641 + 0.987971i \(0.450578\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 3.69279i − 0.127947i
\(834\) 0 0
\(835\) − 27.4489i − 0.949910i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −31.0311 −1.07131 −0.535656 0.844436i \(-0.679936\pi\)
−0.535656 + 0.844436i \(0.679936\pi\)
\(840\) 0 0
\(841\) 25.6594 0.884807
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 8.77636i − 0.301916i
\(846\) 0 0
\(847\) − 3.09008i − 0.106176i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −3.09717 −0.106169
\(852\) 0 0
\(853\) 45.5088 1.55819 0.779096 0.626904i \(-0.215678\pi\)
0.779096 + 0.626904i \(0.215678\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 30.4046i − 1.03860i −0.854592 0.519301i \(-0.826192\pi\)
0.854592 0.519301i \(-0.173808\pi\)
\(858\) 0 0
\(859\) 52.1236i 1.77843i 0.457487 + 0.889216i \(0.348750\pi\)
−0.457487 + 0.889216i \(0.651250\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −11.6879 −0.397862 −0.198931 0.980013i \(-0.563747\pi\)
−0.198931 + 0.980013i \(0.563747\pi\)
\(864\) 0 0
\(865\) 10.4955 0.356857
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 24.1491i 0.819202i
\(870\) 0 0
\(871\) − 50.8685i − 1.72361i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12.0895 0.408700
\(876\) 0 0
\(877\) 46.7939 1.58012 0.790058 0.613032i \(-0.210050\pi\)
0.790058 + 0.613032i \(0.210050\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 43.2550i − 1.45730i −0.684888 0.728648i \(-0.740149\pi\)
0.684888 0.728648i \(-0.259851\pi\)
\(882\) 0 0
\(883\) 8.78574i 0.295664i 0.989013 + 0.147832i \(0.0472294\pi\)
−0.989013 + 0.147832i \(0.952771\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −7.68516 −0.258042 −0.129021 0.991642i \(-0.541184\pi\)
−0.129021 + 0.991642i \(0.541184\pi\)
\(888\) 0 0
\(889\) −0.401864 −0.0134781
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 10.9937i − 0.367892i
\(894\) 0 0
\(895\) 10.2793i 0.343601i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 15.6907 0.523315
\(900\) 0 0
\(901\) 6.14919 0.204859
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 11.7904i − 0.391926i
\(906\) 0 0
\(907\) 21.2295i 0.704913i 0.935828 + 0.352457i \(0.114654\pi\)
−0.935828 + 0.352457i \(0.885346\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 6.74958 0.223624 0.111812 0.993729i \(-0.464335\pi\)
0.111812 + 0.993729i \(0.464335\pi\)
\(912\) 0 0
\(913\) −17.7375 −0.587027
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.6775i 0.550740i
\(918\) 0 0
\(919\) − 2.83039i − 0.0933661i −0.998910 0.0466830i \(-0.985135\pi\)
0.998910 0.0466830i \(-0.0148651\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −55.6629 −1.83217
\(924\) 0 0
\(925\) −1.94013 −0.0637911
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 39.7323i − 1.30358i −0.758401 0.651788i \(-0.774019\pi\)
0.758401 0.651788i \(-0.225981\pi\)
\(930\) 0 0
\(931\) − 5.42471i − 0.177788i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.17657 0.0711815
\(936\) 0 0
\(937\) −15.7245 −0.513696 −0.256848 0.966452i \(-0.582684\pi\)
−0.256848 + 0.966452i \(0.582684\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 44.1756i 1.44008i 0.693931 + 0.720041i \(0.255877\pi\)
−0.693931 + 0.720041i \(0.744123\pi\)
\(942\) 0 0
\(943\) − 19.3596i − 0.630436i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13.1703 0.427978 0.213989 0.976836i \(-0.431354\pi\)
0.213989 + 0.976836i \(0.431354\pi\)
\(948\) 0 0
\(949\) 24.0936 0.782112
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 9.88407i 0.320177i 0.987103 + 0.160088i \(0.0511779\pi\)
−0.987103 + 0.160088i \(0.948822\pi\)
\(954\) 0 0
\(955\) − 4.26080i − 0.137876i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 27.4961 0.887895
\(960\) 0 0
\(961\) −42.6988 −1.37738
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 11.8720i − 0.382173i
\(966\) 0 0
\(967\) 31.5088i 1.01326i 0.862165 + 0.506628i \(0.169108\pi\)
−0.862165 + 0.506628i \(0.830892\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −35.5166 −1.13978 −0.569891 0.821720i \(-0.693015\pi\)
−0.569891 + 0.821720i \(0.693015\pi\)
\(972\) 0 0
\(973\) 1.57529 0.0505014
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 14.4790i − 0.463226i −0.972808 0.231613i \(-0.925600\pi\)
0.972808 0.231613i \(-0.0744002\pi\)
\(978\) 0 0
\(979\) − 40.8157i − 1.30447i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 15.6907 0.500456 0.250228 0.968187i \(-0.419494\pi\)
0.250228 + 0.968187i \(0.419494\pi\)
\(984\) 0 0
\(985\) −9.43207 −0.300531
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 32.7036i 1.03991i
\(990\) 0 0
\(991\) − 11.0950i − 0.352445i −0.984350 0.176222i \(-0.943612\pi\)
0.984350 0.176222i \(-0.0563878\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 11.3888 0.361049
\(996\) 0 0
\(997\) 24.8772 0.787869 0.393934 0.919139i \(-0.371114\pi\)
0.393934 + 0.919139i \(0.371114\pi\)
\(998\) 0 0
\(999\) 0 0
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 2736.2.d.a.2015.4 yes 12
3.2 odd 2 inner 2736.2.d.a.2015.10 yes 12
4.3 odd 2 inner 2736.2.d.a.2015.3 12
12.11 even 2 inner 2736.2.d.a.2015.9 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2736.2.d.a.2015.3 12 4.3 odd 2 inner
2736.2.d.a.2015.4 yes 12 1.1 even 1 trivial
2736.2.d.a.2015.9 yes 12 12.11 even 2 inner
2736.2.d.a.2015.10 yes 12 3.2 odd 2 inner