Properties

Label 7448.2.a.bq.1.1
Level $7448$
Weight $2$
Character 7448.1
Self dual yes
Analytic conductor $59.473$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(59.4725794254\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 15 x^{6} - x^{5} + 66 x^{4} + 4 x^{3} - 76 x^{2} + 8 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1064)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.59528\) of defining polynomial
Character \(\chi\) \(=\) 7448.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.59528 q^{3} +1.66621 q^{5} +3.73546 q^{9} +O(q^{10})\) \(q-2.59528 q^{3} +1.66621 q^{5} +3.73546 q^{9} +0.145289 q^{11} -1.47949 q^{13} -4.32429 q^{15} -5.90894 q^{17} +1.00000 q^{19} +6.22006 q^{23} -2.22373 q^{25} -1.90873 q^{27} -3.13688 q^{29} -0.882526 q^{31} -0.377064 q^{33} -1.96240 q^{37} +3.83969 q^{39} -2.00000 q^{41} -2.05954 q^{43} +6.22408 q^{45} -0.174405 q^{47} +15.3353 q^{51} +8.43843 q^{53} +0.242082 q^{55} -2.59528 q^{57} -1.58188 q^{59} -8.55244 q^{61} -2.46515 q^{65} +1.33931 q^{67} -16.1428 q^{69} +7.37135 q^{71} +4.72425 q^{73} +5.77120 q^{75} +11.4986 q^{79} -6.25271 q^{81} +4.37769 q^{83} -9.84555 q^{85} +8.14108 q^{87} -1.39298 q^{89} +2.29040 q^{93} +1.66621 q^{95} +10.6777 q^{97} +0.542720 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{5} + 6 q^{9} + O(q^{10}) \) \( 8 q - q^{5} + 6 q^{9} + 9 q^{11} + 8 q^{15} + 4 q^{17} + 8 q^{19} + 25 q^{23} + 15 q^{25} + 3 q^{27} + 6 q^{29} + 8 q^{33} + 13 q^{37} - 11 q^{39} - 16 q^{41} + 17 q^{43} + 17 q^{45} - 24 q^{47} + 5 q^{51} + 2 q^{53} + 5 q^{55} + 2 q^{59} - 13 q^{61} - 26 q^{65} + 2 q^{67} - 11 q^{69} + 10 q^{71} + 5 q^{73} - 20 q^{75} + 16 q^{79} - 12 q^{81} - 43 q^{83} + 24 q^{85} + 20 q^{87} + 8 q^{89} + 2 q^{93} - q^{95} - 12 q^{97} + 37 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.59528 −1.49838 −0.749192 0.662353i \(-0.769558\pi\)
−0.749192 + 0.662353i \(0.769558\pi\)
\(4\) 0 0
\(5\) 1.66621 0.745153 0.372577 0.928001i \(-0.378474\pi\)
0.372577 + 0.928001i \(0.378474\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 3.73546 1.24515
\(10\) 0 0
\(11\) 0.145289 0.0438062 0.0219031 0.999760i \(-0.493027\pi\)
0.0219031 + 0.999760i \(0.493027\pi\)
\(12\) 0 0
\(13\) −1.47949 −0.410337 −0.205169 0.978727i \(-0.565774\pi\)
−0.205169 + 0.978727i \(0.565774\pi\)
\(14\) 0 0
\(15\) −4.32429 −1.11653
\(16\) 0 0
\(17\) −5.90894 −1.43313 −0.716564 0.697521i \(-0.754286\pi\)
−0.716564 + 0.697521i \(0.754286\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.22006 1.29697 0.648486 0.761227i \(-0.275403\pi\)
0.648486 + 0.761227i \(0.275403\pi\)
\(24\) 0 0
\(25\) −2.22373 −0.444747
\(26\) 0 0
\(27\) −1.90873 −0.367335
\(28\) 0 0
\(29\) −3.13688 −0.582504 −0.291252 0.956646i \(-0.594072\pi\)
−0.291252 + 0.956646i \(0.594072\pi\)
\(30\) 0 0
\(31\) −0.882526 −0.158506 −0.0792532 0.996855i \(-0.525254\pi\)
−0.0792532 + 0.996855i \(0.525254\pi\)
\(32\) 0 0
\(33\) −0.377064 −0.0656384
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.96240 −0.322616 −0.161308 0.986904i \(-0.551571\pi\)
−0.161308 + 0.986904i \(0.551571\pi\)
\(38\) 0 0
\(39\) 3.83969 0.614842
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −2.05954 −0.314076 −0.157038 0.987593i \(-0.550195\pi\)
−0.157038 + 0.987593i \(0.550195\pi\)
\(44\) 0 0
\(45\) 6.22408 0.927831
\(46\) 0 0
\(47\) −0.174405 −0.0254395 −0.0127198 0.999919i \(-0.504049\pi\)
−0.0127198 + 0.999919i \(0.504049\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 15.3353 2.14738
\(52\) 0 0
\(53\) 8.43843 1.15911 0.579553 0.814934i \(-0.303227\pi\)
0.579553 + 0.814934i \(0.303227\pi\)
\(54\) 0 0
\(55\) 0.242082 0.0326423
\(56\) 0 0
\(57\) −2.59528 −0.343753
\(58\) 0 0
\(59\) −1.58188 −0.205943 −0.102971 0.994684i \(-0.532835\pi\)
−0.102971 + 0.994684i \(0.532835\pi\)
\(60\) 0 0
\(61\) −8.55244 −1.09503 −0.547514 0.836797i \(-0.684426\pi\)
−0.547514 + 0.836797i \(0.684426\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.46515 −0.305764
\(66\) 0 0
\(67\) 1.33931 0.163622 0.0818112 0.996648i \(-0.473930\pi\)
0.0818112 + 0.996648i \(0.473930\pi\)
\(68\) 0 0
\(69\) −16.1428 −1.94336
\(70\) 0 0
\(71\) 7.37135 0.874819 0.437409 0.899263i \(-0.355896\pi\)
0.437409 + 0.899263i \(0.355896\pi\)
\(72\) 0 0
\(73\) 4.72425 0.552931 0.276466 0.961024i \(-0.410837\pi\)
0.276466 + 0.961024i \(0.410837\pi\)
\(74\) 0 0
\(75\) 5.77120 0.666401
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.4986 1.29369 0.646846 0.762621i \(-0.276088\pi\)
0.646846 + 0.762621i \(0.276088\pi\)
\(80\) 0 0
\(81\) −6.25271 −0.694746
\(82\) 0 0
\(83\) 4.37769 0.480514 0.240257 0.970709i \(-0.422768\pi\)
0.240257 + 0.970709i \(0.422768\pi\)
\(84\) 0 0
\(85\) −9.84555 −1.06790
\(86\) 0 0
\(87\) 8.14108 0.872815
\(88\) 0 0
\(89\) −1.39298 −0.147655 −0.0738277 0.997271i \(-0.523521\pi\)
−0.0738277 + 0.997271i \(0.523521\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.29040 0.237503
\(94\) 0 0
\(95\) 1.66621 0.170950
\(96\) 0 0
\(97\) 10.6777 1.08416 0.542080 0.840327i \(-0.317637\pi\)
0.542080 + 0.840327i \(0.317637\pi\)
\(98\) 0 0
\(99\) 0.542720 0.0545454
\(100\) 0 0
\(101\) −1.20245 −0.119649 −0.0598244 0.998209i \(-0.519054\pi\)
−0.0598244 + 0.998209i \(0.519054\pi\)
\(102\) 0 0
\(103\) −19.5472 −1.92605 −0.963024 0.269417i \(-0.913169\pi\)
−0.963024 + 0.269417i \(0.913169\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.02110 0.775429 0.387714 0.921780i \(-0.373265\pi\)
0.387714 + 0.921780i \(0.373265\pi\)
\(108\) 0 0
\(109\) 20.3842 1.95245 0.976226 0.216755i \(-0.0695472\pi\)
0.976226 + 0.216755i \(0.0695472\pi\)
\(110\) 0 0
\(111\) 5.09297 0.483403
\(112\) 0 0
\(113\) 8.93908 0.840918 0.420459 0.907312i \(-0.361869\pi\)
0.420459 + 0.907312i \(0.361869\pi\)
\(114\) 0 0
\(115\) 10.3639 0.966443
\(116\) 0 0
\(117\) −5.52658 −0.510933
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.9789 −0.998081
\(122\) 0 0
\(123\) 5.19055 0.468016
\(124\) 0 0
\(125\) −12.0363 −1.07656
\(126\) 0 0
\(127\) 1.76063 0.156231 0.0781154 0.996944i \(-0.475110\pi\)
0.0781154 + 0.996944i \(0.475110\pi\)
\(128\) 0 0
\(129\) 5.34506 0.470607
\(130\) 0 0
\(131\) 4.04271 0.353213 0.176607 0.984281i \(-0.443488\pi\)
0.176607 + 0.984281i \(0.443488\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −3.18035 −0.273721
\(136\) 0 0
\(137\) −20.4084 −1.74361 −0.871805 0.489853i \(-0.837051\pi\)
−0.871805 + 0.489853i \(0.837051\pi\)
\(138\) 0 0
\(139\) 11.2435 0.953662 0.476831 0.878995i \(-0.341785\pi\)
0.476831 + 0.878995i \(0.341785\pi\)
\(140\) 0 0
\(141\) 0.452628 0.0381182
\(142\) 0 0
\(143\) −0.214953 −0.0179753
\(144\) 0 0
\(145\) −5.22672 −0.434055
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −5.58697 −0.457702 −0.228851 0.973461i \(-0.573497\pi\)
−0.228851 + 0.973461i \(0.573497\pi\)
\(150\) 0 0
\(151\) −19.3575 −1.57529 −0.787644 0.616131i \(-0.788699\pi\)
−0.787644 + 0.616131i \(0.788699\pi\)
\(152\) 0 0
\(153\) −22.0726 −1.78446
\(154\) 0 0
\(155\) −1.47048 −0.118112
\(156\) 0 0
\(157\) −10.9235 −0.871792 −0.435896 0.899997i \(-0.643568\pi\)
−0.435896 + 0.899997i \(0.643568\pi\)
\(158\) 0 0
\(159\) −21.9000 −1.73679
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.2988 1.27662 0.638310 0.769779i \(-0.279634\pi\)
0.638310 + 0.769779i \(0.279634\pi\)
\(164\) 0 0
\(165\) −0.628269 −0.0489107
\(166\) 0 0
\(167\) 16.7379 1.29522 0.647610 0.761972i \(-0.275769\pi\)
0.647610 + 0.761972i \(0.275769\pi\)
\(168\) 0 0
\(169\) −10.8111 −0.831623
\(170\) 0 0
\(171\) 3.73546 0.285658
\(172\) 0 0
\(173\) −18.2402 −1.38678 −0.693389 0.720564i \(-0.743883\pi\)
−0.693389 + 0.720564i \(0.743883\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.10540 0.308581
\(178\) 0 0
\(179\) −2.03827 −0.152347 −0.0761737 0.997095i \(-0.524270\pi\)
−0.0761737 + 0.997095i \(0.524270\pi\)
\(180\) 0 0
\(181\) −8.70643 −0.647144 −0.323572 0.946204i \(-0.604884\pi\)
−0.323572 + 0.946204i \(0.604884\pi\)
\(182\) 0 0
\(183\) 22.1960 1.64077
\(184\) 0 0
\(185\) −3.26977 −0.240399
\(186\) 0 0
\(187\) −0.858501 −0.0627798
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 14.6580 1.06062 0.530309 0.847805i \(-0.322076\pi\)
0.530309 + 0.847805i \(0.322076\pi\)
\(192\) 0 0
\(193\) 16.7805 1.20789 0.603943 0.797028i \(-0.293596\pi\)
0.603943 + 0.797028i \(0.293596\pi\)
\(194\) 0 0
\(195\) 6.39774 0.458152
\(196\) 0 0
\(197\) 24.0559 1.71391 0.856955 0.515391i \(-0.172353\pi\)
0.856955 + 0.515391i \(0.172353\pi\)
\(198\) 0 0
\(199\) 6.26427 0.444062 0.222031 0.975040i \(-0.428731\pi\)
0.222031 + 0.975040i \(0.428731\pi\)
\(200\) 0 0
\(201\) −3.47587 −0.245169
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −3.33243 −0.232747
\(206\) 0 0
\(207\) 23.2348 1.61493
\(208\) 0 0
\(209\) 0.145289 0.0100498
\(210\) 0 0
\(211\) −4.39876 −0.302823 −0.151412 0.988471i \(-0.548382\pi\)
−0.151412 + 0.988471i \(0.548382\pi\)
\(212\) 0 0
\(213\) −19.1307 −1.31081
\(214\) 0 0
\(215\) −3.43163 −0.234035
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −12.2607 −0.828504
\(220\) 0 0
\(221\) 8.74222 0.588066
\(222\) 0 0
\(223\) 19.7309 1.32128 0.660639 0.750703i \(-0.270285\pi\)
0.660639 + 0.750703i \(0.270285\pi\)
\(224\) 0 0
\(225\) −8.30667 −0.553778
\(226\) 0 0
\(227\) −8.14720 −0.540748 −0.270374 0.962755i \(-0.587147\pi\)
−0.270374 + 0.962755i \(0.587147\pi\)
\(228\) 0 0
\(229\) −16.9770 −1.12187 −0.560937 0.827859i \(-0.689559\pi\)
−0.560937 + 0.827859i \(0.689559\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.19191 −0.209109 −0.104555 0.994519i \(-0.533342\pi\)
−0.104555 + 0.994519i \(0.533342\pi\)
\(234\) 0 0
\(235\) −0.290595 −0.0189563
\(236\) 0 0
\(237\) −29.8420 −1.93845
\(238\) 0 0
\(239\) −19.9212 −1.28859 −0.644297 0.764775i \(-0.722850\pi\)
−0.644297 + 0.764775i \(0.722850\pi\)
\(240\) 0 0
\(241\) −8.29145 −0.534099 −0.267050 0.963683i \(-0.586049\pi\)
−0.267050 + 0.963683i \(0.586049\pi\)
\(242\) 0 0
\(243\) 21.9537 1.40833
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.47949 −0.0941378
\(248\) 0 0
\(249\) −11.3613 −0.719995
\(250\) 0 0
\(251\) −10.2504 −0.647002 −0.323501 0.946228i \(-0.604860\pi\)
−0.323501 + 0.946228i \(0.604860\pi\)
\(252\) 0 0
\(253\) 0.903703 0.0568153
\(254\) 0 0
\(255\) 25.5519 1.60012
\(256\) 0 0
\(257\) −9.14284 −0.570315 −0.285157 0.958481i \(-0.592046\pi\)
−0.285157 + 0.958481i \(0.592046\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −11.7177 −0.725308
\(262\) 0 0
\(263\) 2.67921 0.165207 0.0826036 0.996582i \(-0.473676\pi\)
0.0826036 + 0.996582i \(0.473676\pi\)
\(264\) 0 0
\(265\) 14.0602 0.863712
\(266\) 0 0
\(267\) 3.61516 0.221244
\(268\) 0 0
\(269\) 25.2063 1.53685 0.768426 0.639938i \(-0.221040\pi\)
0.768426 + 0.639938i \(0.221040\pi\)
\(270\) 0 0
\(271\) −12.6703 −0.769663 −0.384831 0.922987i \(-0.625740\pi\)
−0.384831 + 0.922987i \(0.625740\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.323083 −0.0194826
\(276\) 0 0
\(277\) 22.0486 1.32477 0.662386 0.749163i \(-0.269544\pi\)
0.662386 + 0.749163i \(0.269544\pi\)
\(278\) 0 0
\(279\) −3.29664 −0.197365
\(280\) 0 0
\(281\) 10.9385 0.652539 0.326270 0.945277i \(-0.394208\pi\)
0.326270 + 0.945277i \(0.394208\pi\)
\(282\) 0 0
\(283\) 20.6217 1.22583 0.612916 0.790148i \(-0.289996\pi\)
0.612916 + 0.790148i \(0.289996\pi\)
\(284\) 0 0
\(285\) −4.32429 −0.256149
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.9156 1.05386
\(290\) 0 0
\(291\) −27.7117 −1.62449
\(292\) 0 0
\(293\) 23.8496 1.39331 0.696654 0.717407i \(-0.254671\pi\)
0.696654 + 0.717407i \(0.254671\pi\)
\(294\) 0 0
\(295\) −2.63574 −0.153459
\(296\) 0 0
\(297\) −0.277316 −0.0160915
\(298\) 0 0
\(299\) −9.20252 −0.532196
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 3.12070 0.179280
\(304\) 0 0
\(305\) −14.2502 −0.815964
\(306\) 0 0
\(307\) −14.6943 −0.838647 −0.419323 0.907837i \(-0.637733\pi\)
−0.419323 + 0.907837i \(0.637733\pi\)
\(308\) 0 0
\(309\) 50.7305 2.88596
\(310\) 0 0
\(311\) 5.91507 0.335413 0.167706 0.985837i \(-0.446364\pi\)
0.167706 + 0.985837i \(0.446364\pi\)
\(312\) 0 0
\(313\) 0.910652 0.0514731 0.0257365 0.999669i \(-0.491807\pi\)
0.0257365 + 0.999669i \(0.491807\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −25.4108 −1.42721 −0.713607 0.700546i \(-0.752940\pi\)
−0.713607 + 0.700546i \(0.752940\pi\)
\(318\) 0 0
\(319\) −0.455753 −0.0255173
\(320\) 0 0
\(321\) −20.8170 −1.16189
\(322\) 0 0
\(323\) −5.90894 −0.328782
\(324\) 0 0
\(325\) 3.28999 0.182496
\(326\) 0 0
\(327\) −52.9026 −2.92552
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9.79525 0.538396 0.269198 0.963085i \(-0.413241\pi\)
0.269198 + 0.963085i \(0.413241\pi\)
\(332\) 0 0
\(333\) −7.33046 −0.401707
\(334\) 0 0
\(335\) 2.23157 0.121924
\(336\) 0 0
\(337\) 23.8358 1.29842 0.649210 0.760609i \(-0.275100\pi\)
0.649210 + 0.760609i \(0.275100\pi\)
\(338\) 0 0
\(339\) −23.1994 −1.26002
\(340\) 0 0
\(341\) −0.128221 −0.00694356
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −26.8973 −1.44810
\(346\) 0 0
\(347\) 26.6393 1.43007 0.715035 0.699089i \(-0.246411\pi\)
0.715035 + 0.699089i \(0.246411\pi\)
\(348\) 0 0
\(349\) −6.31857 −0.338225 −0.169113 0.985597i \(-0.554090\pi\)
−0.169113 + 0.985597i \(0.554090\pi\)
\(350\) 0 0
\(351\) 2.82395 0.150731
\(352\) 0 0
\(353\) 3.12471 0.166311 0.0831557 0.996537i \(-0.473500\pi\)
0.0831557 + 0.996537i \(0.473500\pi\)
\(354\) 0 0
\(355\) 12.2822 0.651874
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.1244 −0.798234 −0.399117 0.916900i \(-0.630683\pi\)
−0.399117 + 0.916900i \(0.630683\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 28.4933 1.49551
\(364\) 0 0
\(365\) 7.87161 0.412019
\(366\) 0 0
\(367\) 4.45173 0.232378 0.116189 0.993227i \(-0.462932\pi\)
0.116189 + 0.993227i \(0.462932\pi\)
\(368\) 0 0
\(369\) −7.47092 −0.388921
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 11.6612 0.603792 0.301896 0.953341i \(-0.402380\pi\)
0.301896 + 0.953341i \(0.402380\pi\)
\(374\) 0 0
\(375\) 31.2375 1.61310
\(376\) 0 0
\(377\) 4.64099 0.239023
\(378\) 0 0
\(379\) 36.0926 1.85395 0.926976 0.375122i \(-0.122399\pi\)
0.926976 + 0.375122i \(0.122399\pi\)
\(380\) 0 0
\(381\) −4.56933 −0.234094
\(382\) 0 0
\(383\) 14.5992 0.745984 0.372992 0.927834i \(-0.378332\pi\)
0.372992 + 0.927834i \(0.378332\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −7.69332 −0.391073
\(388\) 0 0
\(389\) 4.53987 0.230181 0.115090 0.993355i \(-0.463284\pi\)
0.115090 + 0.993355i \(0.463284\pi\)
\(390\) 0 0
\(391\) −36.7539 −1.85873
\(392\) 0 0
\(393\) −10.4920 −0.529249
\(394\) 0 0
\(395\) 19.1591 0.963998
\(396\) 0 0
\(397\) 21.8844 1.09835 0.549175 0.835708i \(-0.314942\pi\)
0.549175 + 0.835708i \(0.314942\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8.07346 0.403169 0.201585 0.979471i \(-0.435391\pi\)
0.201585 + 0.979471i \(0.435391\pi\)
\(402\) 0 0
\(403\) 1.30569 0.0650411
\(404\) 0 0
\(405\) −10.4184 −0.517692
\(406\) 0 0
\(407\) −0.285114 −0.0141326
\(408\) 0 0
\(409\) 37.5265 1.85557 0.927784 0.373118i \(-0.121711\pi\)
0.927784 + 0.373118i \(0.121711\pi\)
\(410\) 0 0
\(411\) 52.9655 2.61260
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7.29417 0.358057
\(416\) 0 0
\(417\) −29.1800 −1.42895
\(418\) 0 0
\(419\) −24.9638 −1.21956 −0.609782 0.792569i \(-0.708743\pi\)
−0.609782 + 0.792569i \(0.708743\pi\)
\(420\) 0 0
\(421\) 17.4710 0.851485 0.425743 0.904844i \(-0.360013\pi\)
0.425743 + 0.904844i \(0.360013\pi\)
\(422\) 0 0
\(423\) −0.651482 −0.0316761
\(424\) 0 0
\(425\) 13.1399 0.637379
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0.557863 0.0269339
\(430\) 0 0
\(431\) 25.5759 1.23195 0.615973 0.787767i \(-0.288763\pi\)
0.615973 + 0.787767i \(0.288763\pi\)
\(432\) 0 0
\(433\) −21.5603 −1.03612 −0.518060 0.855344i \(-0.673346\pi\)
−0.518060 + 0.855344i \(0.673346\pi\)
\(434\) 0 0
\(435\) 13.5648 0.650381
\(436\) 0 0
\(437\) 6.22006 0.297546
\(438\) 0 0
\(439\) 25.8430 1.23342 0.616710 0.787190i \(-0.288465\pi\)
0.616710 + 0.787190i \(0.288465\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 29.0570 1.38054 0.690270 0.723552i \(-0.257492\pi\)
0.690270 + 0.723552i \(0.257492\pi\)
\(444\) 0 0
\(445\) −2.32100 −0.110026
\(446\) 0 0
\(447\) 14.4997 0.685814
\(448\) 0 0
\(449\) 29.6266 1.39817 0.699083 0.715040i \(-0.253592\pi\)
0.699083 + 0.715040i \(0.253592\pi\)
\(450\) 0 0
\(451\) −0.290577 −0.0136827
\(452\) 0 0
\(453\) 50.2380 2.36039
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4.03231 −0.188624 −0.0943119 0.995543i \(-0.530065\pi\)
−0.0943119 + 0.995543i \(0.530065\pi\)
\(458\) 0 0
\(459\) 11.2785 0.526438
\(460\) 0 0
\(461\) 17.6181 0.820555 0.410277 0.911961i \(-0.365432\pi\)
0.410277 + 0.911961i \(0.365432\pi\)
\(462\) 0 0
\(463\) −26.2595 −1.22038 −0.610192 0.792253i \(-0.708908\pi\)
−0.610192 + 0.792253i \(0.708908\pi\)
\(464\) 0 0
\(465\) 3.81630 0.176976
\(466\) 0 0
\(467\) 34.9518 1.61738 0.808688 0.588238i \(-0.200178\pi\)
0.808688 + 0.588238i \(0.200178\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 28.3496 1.30628
\(472\) 0 0
\(473\) −0.299227 −0.0137585
\(474\) 0 0
\(475\) −2.22373 −0.102032
\(476\) 0 0
\(477\) 31.5214 1.44327
\(478\) 0 0
\(479\) −27.0479 −1.23585 −0.617925 0.786237i \(-0.712026\pi\)
−0.617925 + 0.786237i \(0.712026\pi\)
\(480\) 0 0
\(481\) 2.90335 0.132381
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 17.7914 0.807865
\(486\) 0 0
\(487\) 19.2167 0.870790 0.435395 0.900239i \(-0.356609\pi\)
0.435395 + 0.900239i \(0.356609\pi\)
\(488\) 0 0
\(489\) −42.2999 −1.91287
\(490\) 0 0
\(491\) 31.6972 1.43048 0.715238 0.698881i \(-0.246318\pi\)
0.715238 + 0.698881i \(0.246318\pi\)
\(492\) 0 0
\(493\) 18.5356 0.834803
\(494\) 0 0
\(495\) 0.904287 0.0406447
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 21.5590 0.965112 0.482556 0.875865i \(-0.339708\pi\)
0.482556 + 0.875865i \(0.339708\pi\)
\(500\) 0 0
\(501\) −43.4396 −1.94074
\(502\) 0 0
\(503\) −7.27173 −0.324230 −0.162115 0.986772i \(-0.551832\pi\)
−0.162115 + 0.986772i \(0.551832\pi\)
\(504\) 0 0
\(505\) −2.00355 −0.0891566
\(506\) 0 0
\(507\) 28.0578 1.24609
\(508\) 0 0
\(509\) 40.6805 1.80313 0.901567 0.432640i \(-0.142418\pi\)
0.901567 + 0.432640i \(0.142418\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1.90873 −0.0842723
\(514\) 0 0
\(515\) −32.5699 −1.43520
\(516\) 0 0
\(517\) −0.0253390 −0.00111441
\(518\) 0 0
\(519\) 47.3384 2.07792
\(520\) 0 0
\(521\) −2.03869 −0.0893167 −0.0446583 0.999002i \(-0.514220\pi\)
−0.0446583 + 0.999002i \(0.514220\pi\)
\(522\) 0 0
\(523\) −37.5930 −1.64383 −0.821914 0.569611i \(-0.807094\pi\)
−0.821914 + 0.569611i \(0.807094\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.21479 0.227160
\(528\) 0 0
\(529\) 15.6891 0.682135
\(530\) 0 0
\(531\) −5.90904 −0.256430
\(532\) 0 0
\(533\) 2.95898 0.128168
\(534\) 0 0
\(535\) 13.3649 0.577813
\(536\) 0 0
\(537\) 5.28987 0.228275
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 32.3190 1.38950 0.694751 0.719250i \(-0.255514\pi\)
0.694751 + 0.719250i \(0.255514\pi\)
\(542\) 0 0
\(543\) 22.5956 0.969670
\(544\) 0 0
\(545\) 33.9644 1.45488
\(546\) 0 0
\(547\) 5.52462 0.236216 0.118108 0.993001i \(-0.462317\pi\)
0.118108 + 0.993001i \(0.462317\pi\)
\(548\) 0 0
\(549\) −31.9473 −1.36348
\(550\) 0 0
\(551\) −3.13688 −0.133636
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 8.48597 0.360209
\(556\) 0 0
\(557\) −31.7060 −1.34342 −0.671712 0.740812i \(-0.734441\pi\)
−0.671712 + 0.740812i \(0.734441\pi\)
\(558\) 0 0
\(559\) 3.04707 0.128877
\(560\) 0 0
\(561\) 2.22805 0.0940683
\(562\) 0 0
\(563\) 34.7554 1.46477 0.732383 0.680893i \(-0.238408\pi\)
0.732383 + 0.680893i \(0.238408\pi\)
\(564\) 0 0
\(565\) 14.8944 0.626613
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −35.8986 −1.50495 −0.752473 0.658623i \(-0.771139\pi\)
−0.752473 + 0.658623i \(0.771139\pi\)
\(570\) 0 0
\(571\) −43.0882 −1.80318 −0.901592 0.432588i \(-0.857600\pi\)
−0.901592 + 0.432588i \(0.857600\pi\)
\(572\) 0 0
\(573\) −38.0416 −1.58921
\(574\) 0 0
\(575\) −13.8317 −0.576824
\(576\) 0 0
\(577\) 46.6165 1.94067 0.970335 0.241766i \(-0.0777265\pi\)
0.970335 + 0.241766i \(0.0777265\pi\)
\(578\) 0 0
\(579\) −43.5500 −1.80988
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1.22601 0.0507760
\(584\) 0 0
\(585\) −9.20847 −0.380723
\(586\) 0 0
\(587\) 46.4370 1.91666 0.958331 0.285661i \(-0.0922132\pi\)
0.958331 + 0.285661i \(0.0922132\pi\)
\(588\) 0 0
\(589\) −0.882526 −0.0363639
\(590\) 0 0
\(591\) −62.4317 −2.56810
\(592\) 0 0
\(593\) 40.3003 1.65494 0.827468 0.561513i \(-0.189781\pi\)
0.827468 + 0.561513i \(0.189781\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −16.2575 −0.665376
\(598\) 0 0
\(599\) −19.7924 −0.808695 −0.404348 0.914605i \(-0.632501\pi\)
−0.404348 + 0.914605i \(0.632501\pi\)
\(600\) 0 0
\(601\) 39.2064 1.59926 0.799631 0.600492i \(-0.205029\pi\)
0.799631 + 0.600492i \(0.205029\pi\)
\(602\) 0 0
\(603\) 5.00293 0.203735
\(604\) 0 0
\(605\) −18.2932 −0.743723
\(606\) 0 0
\(607\) −17.2293 −0.699316 −0.349658 0.936877i \(-0.613702\pi\)
−0.349658 + 0.936877i \(0.613702\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.258030 0.0104388
\(612\) 0 0
\(613\) 24.1131 0.973917 0.486959 0.873425i \(-0.338106\pi\)
0.486959 + 0.873425i \(0.338106\pi\)
\(614\) 0 0
\(615\) 8.64857 0.348744
\(616\) 0 0
\(617\) −44.6161 −1.79618 −0.898089 0.439814i \(-0.855044\pi\)
−0.898089 + 0.439814i \(0.855044\pi\)
\(618\) 0 0
\(619\) −20.5549 −0.826172 −0.413086 0.910692i \(-0.635549\pi\)
−0.413086 + 0.910692i \(0.635549\pi\)
\(620\) 0 0
\(621\) −11.8724 −0.476423
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −8.93635 −0.357454
\(626\) 0 0
\(627\) −0.377064 −0.0150585
\(628\) 0 0
\(629\) 11.5957 0.462350
\(630\) 0 0
\(631\) 3.35794 0.133678 0.0668388 0.997764i \(-0.478709\pi\)
0.0668388 + 0.997764i \(0.478709\pi\)
\(632\) 0 0
\(633\) 11.4160 0.453745
\(634\) 0 0
\(635\) 2.93359 0.116416
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 27.5354 1.08928
\(640\) 0 0
\(641\) −4.65861 −0.184004 −0.0920020 0.995759i \(-0.529327\pi\)
−0.0920020 + 0.995759i \(0.529327\pi\)
\(642\) 0 0
\(643\) 9.50546 0.374859 0.187429 0.982278i \(-0.439984\pi\)
0.187429 + 0.982278i \(0.439984\pi\)
\(644\) 0 0
\(645\) 8.90602 0.350674
\(646\) 0 0
\(647\) 13.0308 0.512294 0.256147 0.966638i \(-0.417547\pi\)
0.256147 + 0.966638i \(0.417547\pi\)
\(648\) 0 0
\(649\) −0.229828 −0.00902156
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.02455 0.274892 0.137446 0.990509i \(-0.456111\pi\)
0.137446 + 0.990509i \(0.456111\pi\)
\(654\) 0 0
\(655\) 6.73602 0.263198
\(656\) 0 0
\(657\) 17.6472 0.688485
\(658\) 0 0
\(659\) −7.92031 −0.308532 −0.154266 0.988029i \(-0.549301\pi\)
−0.154266 + 0.988029i \(0.549301\pi\)
\(660\) 0 0
\(661\) 10.7536 0.418265 0.209133 0.977887i \(-0.432936\pi\)
0.209133 + 0.977887i \(0.432936\pi\)
\(662\) 0 0
\(663\) −22.6885 −0.881148
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −19.5116 −0.755492
\(668\) 0 0
\(669\) −51.2071 −1.97978
\(670\) 0 0
\(671\) −1.24257 −0.0479690
\(672\) 0 0
\(673\) −32.6941 −1.26027 −0.630133 0.776487i \(-0.717000\pi\)
−0.630133 + 0.776487i \(0.717000\pi\)
\(674\) 0 0
\(675\) 4.24450 0.163371
\(676\) 0 0
\(677\) 36.5994 1.40663 0.703314 0.710879i \(-0.251703\pi\)
0.703314 + 0.710879i \(0.251703\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 21.1442 0.810249
\(682\) 0 0
\(683\) 2.97440 0.113812 0.0569062 0.998380i \(-0.481876\pi\)
0.0569062 + 0.998380i \(0.481876\pi\)
\(684\) 0 0
\(685\) −34.0048 −1.29926
\(686\) 0 0
\(687\) 44.0601 1.68100
\(688\) 0 0
\(689\) −12.4846 −0.475625
\(690\) 0 0
\(691\) −1.82312 −0.0693546 −0.0346773 0.999399i \(-0.511040\pi\)
−0.0346773 + 0.999399i \(0.511040\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18.7341 0.710625
\(696\) 0 0
\(697\) 11.8179 0.447634
\(698\) 0 0
\(699\) 8.28390 0.313326
\(700\) 0 0
\(701\) −49.7699 −1.87978 −0.939891 0.341474i \(-0.889074\pi\)
−0.939891 + 0.341474i \(0.889074\pi\)
\(702\) 0 0
\(703\) −1.96240 −0.0740133
\(704\) 0 0
\(705\) 0.754175 0.0284039
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −14.9525 −0.561552 −0.280776 0.959773i \(-0.590592\pi\)
−0.280776 + 0.959773i \(0.590592\pi\)
\(710\) 0 0
\(711\) 42.9525 1.61084
\(712\) 0 0
\(713\) −5.48936 −0.205578
\(714\) 0 0
\(715\) −0.358158 −0.0133943
\(716\) 0 0
\(717\) 51.7010 1.93081
\(718\) 0 0
\(719\) 14.2696 0.532166 0.266083 0.963950i \(-0.414270\pi\)
0.266083 + 0.963950i \(0.414270\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 21.5186 0.800286
\(724\) 0 0
\(725\) 6.97559 0.259067
\(726\) 0 0
\(727\) −2.40032 −0.0890231 −0.0445115 0.999009i \(-0.514173\pi\)
−0.0445115 + 0.999009i \(0.514173\pi\)
\(728\) 0 0
\(729\) −38.2178 −1.41547
\(730\) 0 0
\(731\) 12.1697 0.450111
\(732\) 0 0
\(733\) −12.4073 −0.458273 −0.229137 0.973394i \(-0.573590\pi\)
−0.229137 + 0.973394i \(0.573590\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.194586 0.00716767
\(738\) 0 0
\(739\) −0.0480938 −0.00176916 −0.000884580 1.00000i \(-0.500282\pi\)
−0.000884580 1.00000i \(0.500282\pi\)
\(740\) 0 0
\(741\) 3.83969 0.141055
\(742\) 0 0
\(743\) −28.5507 −1.04742 −0.523712 0.851895i \(-0.675453\pi\)
−0.523712 + 0.851895i \(0.675453\pi\)
\(744\) 0 0
\(745\) −9.30908 −0.341058
\(746\) 0 0
\(747\) 16.3527 0.598314
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −10.8556 −0.396128 −0.198064 0.980189i \(-0.563465\pi\)
−0.198064 + 0.980189i \(0.563465\pi\)
\(752\) 0 0
\(753\) 26.6027 0.969457
\(754\) 0 0
\(755\) −32.2537 −1.17383
\(756\) 0 0
\(757\) −20.1699 −0.733089 −0.366545 0.930400i \(-0.619459\pi\)
−0.366545 + 0.930400i \(0.619459\pi\)
\(758\) 0 0
\(759\) −2.34536 −0.0851312
\(760\) 0 0
\(761\) −29.9226 −1.08469 −0.542347 0.840155i \(-0.682464\pi\)
−0.542347 + 0.840155i \(0.682464\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −36.7777 −1.32970
\(766\) 0 0
\(767\) 2.34037 0.0845059
\(768\) 0 0
\(769\) −15.7701 −0.568685 −0.284343 0.958723i \(-0.591775\pi\)
−0.284343 + 0.958723i \(0.591775\pi\)
\(770\) 0 0
\(771\) 23.7282 0.854550
\(772\) 0 0
\(773\) 35.2277 1.26705 0.633526 0.773721i \(-0.281607\pi\)
0.633526 + 0.773721i \(0.281607\pi\)
\(774\) 0 0
\(775\) 1.96250 0.0704952
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) 1.07097 0.0383224
\(782\) 0 0
\(783\) 5.98745 0.213974
\(784\) 0 0
\(785\) −18.2009 −0.649618
\(786\) 0 0
\(787\) 45.9116 1.63657 0.818286 0.574811i \(-0.194924\pi\)
0.818286 + 0.574811i \(0.194924\pi\)
\(788\) 0 0
\(789\) −6.95329 −0.247544
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 12.6533 0.449331
\(794\) 0 0
\(795\) −36.4902 −1.29417
\(796\) 0 0
\(797\) −2.14407 −0.0759468 −0.0379734 0.999279i \(-0.512090\pi\)
−0.0379734 + 0.999279i \(0.512090\pi\)
\(798\) 0 0
\(799\) 1.03055 0.0364581
\(800\) 0 0
\(801\) −5.20342 −0.183854
\(802\) 0 0
\(803\) 0.686379 0.0242218
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −65.4172 −2.30279
\(808\) 0 0
\(809\) −23.3363 −0.820461 −0.410230 0.911982i \(-0.634552\pi\)
−0.410230 + 0.911982i \(0.634552\pi\)
\(810\) 0 0
\(811\) 41.5527 1.45911 0.729556 0.683921i \(-0.239727\pi\)
0.729556 + 0.683921i \(0.239727\pi\)
\(812\) 0 0
\(813\) 32.8828 1.15325
\(814\) 0 0
\(815\) 27.1573 0.951278
\(816\) 0 0
\(817\) −2.05954 −0.0720540
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 31.1695 1.08782 0.543912 0.839142i \(-0.316943\pi\)
0.543912 + 0.839142i \(0.316943\pi\)
\(822\) 0 0
\(823\) −17.2573 −0.601551 −0.300775 0.953695i \(-0.597245\pi\)
−0.300775 + 0.953695i \(0.597245\pi\)
\(824\) 0 0
\(825\) 0.838490 0.0291925
\(826\) 0 0
\(827\) −35.1385 −1.22188 −0.610942 0.791675i \(-0.709209\pi\)
−0.610942 + 0.791675i \(0.709209\pi\)
\(828\) 0 0
\(829\) −29.2471 −1.01579 −0.507897 0.861418i \(-0.669577\pi\)
−0.507897 + 0.861418i \(0.669577\pi\)
\(830\) 0 0
\(831\) −57.2222 −1.98502
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 27.8890 0.965138
\(836\) 0 0
\(837\) 1.68450 0.0582249
\(838\) 0 0
\(839\) −16.6637 −0.575294 −0.287647 0.957737i \(-0.592873\pi\)
−0.287647 + 0.957737i \(0.592873\pi\)
\(840\) 0 0
\(841\) −19.1600 −0.660689
\(842\) 0 0
\(843\) −28.3886 −0.977754
\(844\) 0 0
\(845\) −18.0136 −0.619687
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −53.5190 −1.83677
\(850\) 0 0
\(851\) −12.2062 −0.418424
\(852\) 0 0
\(853\) 20.0759 0.687386 0.343693 0.939082i \(-0.388322\pi\)
0.343693 + 0.939082i \(0.388322\pi\)
\(854\) 0 0
\(855\) 6.22408 0.212859
\(856\) 0 0
\(857\) 21.4823 0.733823 0.366911 0.930256i \(-0.380415\pi\)
0.366911 + 0.930256i \(0.380415\pi\)
\(858\) 0 0
\(859\) 13.9460 0.475830 0.237915 0.971286i \(-0.423536\pi\)
0.237915 + 0.971286i \(0.423536\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −10.0298 −0.341419 −0.170709 0.985321i \(-0.554606\pi\)
−0.170709 + 0.985321i \(0.554606\pi\)
\(864\) 0 0
\(865\) −30.3921 −1.03336
\(866\) 0 0
\(867\) −46.4958 −1.57908
\(868\) 0 0
\(869\) 1.67061 0.0566717
\(870\) 0 0
\(871\) −1.98149 −0.0671403
\(872\) 0 0
\(873\) 39.8863 1.34995
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 14.2658 0.481722 0.240861 0.970560i \(-0.422570\pi\)
0.240861 + 0.970560i \(0.422570\pi\)
\(878\) 0 0
\(879\) −61.8963 −2.08771
\(880\) 0 0
\(881\) 33.3253 1.12276 0.561379 0.827559i \(-0.310271\pi\)
0.561379 + 0.827559i \(0.310271\pi\)
\(882\) 0 0
\(883\) 40.1473 1.35107 0.675533 0.737330i \(-0.263914\pi\)
0.675533 + 0.737330i \(0.263914\pi\)
\(884\) 0 0
\(885\) 6.84048 0.229940
\(886\) 0 0
\(887\) −17.5383 −0.588879 −0.294440 0.955670i \(-0.595133\pi\)
−0.294440 + 0.955670i \(0.595133\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.908448 −0.0304341
\(892\) 0 0
\(893\) −0.174405 −0.00583623
\(894\) 0 0
\(895\) −3.39619 −0.113522
\(896\) 0 0
\(897\) 23.8831 0.797433
\(898\) 0 0
\(899\) 2.76838 0.0923307
\(900\) 0 0
\(901\) −49.8621 −1.66115
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −14.5068 −0.482221
\(906\) 0 0
\(907\) 30.7972 1.02260 0.511301 0.859401i \(-0.329164\pi\)
0.511301 + 0.859401i \(0.329164\pi\)
\(908\) 0 0
\(909\) −4.49172 −0.148981
\(910\) 0 0
\(911\) −26.5837 −0.880758 −0.440379 0.897812i \(-0.645156\pi\)
−0.440379 + 0.897812i \(0.645156\pi\)
\(912\) 0 0
\(913\) 0.636029 0.0210495
\(914\) 0 0
\(915\) 36.9832 1.22263
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −12.1469 −0.400690 −0.200345 0.979725i \(-0.564206\pi\)
−0.200345 + 0.979725i \(0.564206\pi\)
\(920\) 0 0
\(921\) 38.1357 1.25661
\(922\) 0 0
\(923\) −10.9059 −0.358971
\(924\) 0 0
\(925\) 4.36385 0.143482
\(926\) 0 0
\(927\) −73.0180 −2.39823
\(928\) 0 0
\(929\) −26.8183 −0.879881 −0.439940 0.898027i \(-0.645000\pi\)
−0.439940 + 0.898027i \(0.645000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −15.3512 −0.502577
\(934\) 0 0
\(935\) −1.43045 −0.0467806
\(936\) 0 0
\(937\) 42.6816 1.39435 0.697174 0.716902i \(-0.254441\pi\)
0.697174 + 0.716902i \(0.254441\pi\)
\(938\) 0 0
\(939\) −2.36339 −0.0771264
\(940\) 0 0
\(941\) 26.4552 0.862416 0.431208 0.902253i \(-0.358088\pi\)
0.431208 + 0.902253i \(0.358088\pi\)
\(942\) 0 0
\(943\) −12.4401 −0.405106
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −7.25108 −0.235628 −0.117814 0.993036i \(-0.537589\pi\)
−0.117814 + 0.993036i \(0.537589\pi\)
\(948\) 0 0
\(949\) −6.98949 −0.226888
\(950\) 0 0
\(951\) 65.9482 2.13852
\(952\) 0 0
\(953\) −19.1832 −0.621404 −0.310702 0.950507i \(-0.600564\pi\)
−0.310702 + 0.950507i \(0.600564\pi\)
\(954\) 0 0
\(955\) 24.4234 0.790322
\(956\) 0 0
\(957\) 1.18281 0.0382347
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −30.2211 −0.974876
\(962\) 0 0
\(963\) 29.9625 0.965528
\(964\) 0 0
\(965\) 27.9599 0.900060
\(966\) 0 0
\(967\) −25.6363 −0.824407 −0.412203 0.911092i \(-0.635241\pi\)
−0.412203 + 0.911092i \(0.635241\pi\)
\(968\) 0 0
\(969\) 15.3353 0.492642
\(970\) 0 0
\(971\) 36.2893 1.16458 0.582289 0.812982i \(-0.302157\pi\)
0.582289 + 0.812982i \(0.302157\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −8.53844 −0.273449
\(976\) 0 0
\(977\) 2.29103 0.0732967 0.0366483 0.999328i \(-0.488332\pi\)
0.0366483 + 0.999328i \(0.488332\pi\)
\(978\) 0 0
\(979\) −0.202384 −0.00646822
\(980\) 0 0
\(981\) 76.1444 2.43110
\(982\) 0 0
\(983\) −10.6705 −0.340336 −0.170168 0.985415i \(-0.554431\pi\)
−0.170168 + 0.985415i \(0.554431\pi\)
\(984\) 0 0
\(985\) 40.0822 1.27713
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12.8104 −0.407348
\(990\) 0 0
\(991\) 18.7242 0.594792 0.297396 0.954754i \(-0.403882\pi\)
0.297396 + 0.954754i \(0.403882\pi\)
\(992\) 0 0
\(993\) −25.4214 −0.806723
\(994\) 0 0
\(995\) 10.4376 0.330895
\(996\) 0 0
\(997\) −14.4493 −0.457614 −0.228807 0.973472i \(-0.573482\pi\)
−0.228807 + 0.973472i \(0.573482\pi\)
\(998\) 0 0
\(999\) 3.74568 0.118508
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7448.2.a.bq.1.1 8
7.2 even 3 1064.2.q.n.305.8 16
7.4 even 3 1064.2.q.n.457.8 yes 16
7.6 odd 2 7448.2.a.br.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1064.2.q.n.305.8 16 7.2 even 3
1064.2.q.n.457.8 yes 16 7.4 even 3
7448.2.a.bq.1.1 8 1.1 even 1 trivial
7448.2.a.br.1.8 8 7.6 odd 2