Properties

Label 7440.2.a.bl.1.2
Level $7440$
Weight $2$
Character 7440.1
Self dual yes
Analytic conductor $59.409$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7440.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.4086991038\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 7440.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.56155 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.56155 q^{7} +1.00000 q^{9} +1.56155 q^{11} +2.00000 q^{13} +1.00000 q^{15} +5.12311 q^{17} -4.68466 q^{19} +1.56155 q^{21} -5.56155 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.12311 q^{29} -1.00000 q^{31} +1.56155 q^{33} +1.56155 q^{35} +5.12311 q^{37} +2.00000 q^{39} -1.12311 q^{41} +7.80776 q^{43} +1.00000 q^{45} +3.12311 q^{47} -4.56155 q^{49} +5.12311 q^{51} +11.5616 q^{53} +1.56155 q^{55} -4.68466 q^{57} +4.87689 q^{59} +6.00000 q^{61} +1.56155 q^{63} +2.00000 q^{65} -9.36932 q^{67} -5.56155 q^{69} -4.68466 q^{71} +9.80776 q^{73} +1.00000 q^{75} +2.43845 q^{77} +16.6847 q^{79} +1.00000 q^{81} +2.24621 q^{83} +5.12311 q^{85} -1.12311 q^{87} -1.31534 q^{89} +3.12311 q^{91} -1.00000 q^{93} -4.68466 q^{95} -6.00000 q^{97} +1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{17} + 3 q^{19} - q^{21} - 7 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} - 2 q^{31} - q^{33} - q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} - 5 q^{43} + 2 q^{45} - 2 q^{47} - 5 q^{49} + 2 q^{51} + 19 q^{53} - q^{55} + 3 q^{57} + 18 q^{59} + 12 q^{61} - q^{63} + 4 q^{65} + 6 q^{67} - 7 q^{69} + 3 q^{71} - q^{73} + 2 q^{75} + 9 q^{77} + 21 q^{79} + 2 q^{81} - 12 q^{83} + 2 q^{85} + 6 q^{87} - 15 q^{89} - 2 q^{91} - 2 q^{93} + 3 q^{95} - 12 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.56155 0.590211 0.295106 0.955465i \(-0.404645\pi\)
0.295106 + 0.955465i \(0.404645\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.56155 0.470826 0.235413 0.971895i \(-0.424356\pi\)
0.235413 + 0.971895i \(0.424356\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 5.12311 1.24254 0.621268 0.783598i \(-0.286618\pi\)
0.621268 + 0.783598i \(0.286618\pi\)
\(18\) 0 0
\(19\) −4.68466 −1.07473 −0.537367 0.843348i \(-0.680581\pi\)
−0.537367 + 0.843348i \(0.680581\pi\)
\(20\) 0 0
\(21\) 1.56155 0.340759
\(22\) 0 0
\(23\) −5.56155 −1.15966 −0.579832 0.814736i \(-0.696882\pi\)
−0.579832 + 0.814736i \(0.696882\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.12311 −0.208555 −0.104278 0.994548i \(-0.533253\pi\)
−0.104278 + 0.994548i \(0.533253\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) 1.56155 0.271831
\(34\) 0 0
\(35\) 1.56155 0.263951
\(36\) 0 0
\(37\) 5.12311 0.842233 0.421117 0.907006i \(-0.361638\pi\)
0.421117 + 0.907006i \(0.361638\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −1.12311 −0.175400 −0.0876998 0.996147i \(-0.527952\pi\)
−0.0876998 + 0.996147i \(0.527952\pi\)
\(42\) 0 0
\(43\) 7.80776 1.19067 0.595336 0.803477i \(-0.297019\pi\)
0.595336 + 0.803477i \(0.297019\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 3.12311 0.455552 0.227776 0.973714i \(-0.426855\pi\)
0.227776 + 0.973714i \(0.426855\pi\)
\(48\) 0 0
\(49\) −4.56155 −0.651650
\(50\) 0 0
\(51\) 5.12311 0.717378
\(52\) 0 0
\(53\) 11.5616 1.58810 0.794051 0.607852i \(-0.207968\pi\)
0.794051 + 0.607852i \(0.207968\pi\)
\(54\) 0 0
\(55\) 1.56155 0.210560
\(56\) 0 0
\(57\) −4.68466 −0.620498
\(58\) 0 0
\(59\) 4.87689 0.634918 0.317459 0.948272i \(-0.397170\pi\)
0.317459 + 0.948272i \(0.397170\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 1.56155 0.196737
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −9.36932 −1.14464 −0.572322 0.820029i \(-0.693957\pi\)
−0.572322 + 0.820029i \(0.693957\pi\)
\(68\) 0 0
\(69\) −5.56155 −0.669532
\(70\) 0 0
\(71\) −4.68466 −0.555967 −0.277983 0.960586i \(-0.589666\pi\)
−0.277983 + 0.960586i \(0.589666\pi\)
\(72\) 0 0
\(73\) 9.80776 1.14791 0.573956 0.818886i \(-0.305408\pi\)
0.573956 + 0.818886i \(0.305408\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 2.43845 0.277887
\(78\) 0 0
\(79\) 16.6847 1.87717 0.938585 0.345047i \(-0.112137\pi\)
0.938585 + 0.345047i \(0.112137\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.24621 0.246554 0.123277 0.992372i \(-0.460660\pi\)
0.123277 + 0.992372i \(0.460660\pi\)
\(84\) 0 0
\(85\) 5.12311 0.555679
\(86\) 0 0
\(87\) −1.12311 −0.120410
\(88\) 0 0
\(89\) −1.31534 −0.139426 −0.0697130 0.997567i \(-0.522208\pi\)
−0.0697130 + 0.997567i \(0.522208\pi\)
\(90\) 0 0
\(91\) 3.12311 0.327390
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −4.68466 −0.480636
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 1.56155 0.156942
\(100\) 0 0
\(101\) 3.56155 0.354388 0.177194 0.984176i \(-0.443298\pi\)
0.177194 + 0.984176i \(0.443298\pi\)
\(102\) 0 0
\(103\) 18.2462 1.79785 0.898926 0.438100i \(-0.144348\pi\)
0.898926 + 0.438100i \(0.144348\pi\)
\(104\) 0 0
\(105\) 1.56155 0.152392
\(106\) 0 0
\(107\) −1.56155 −0.150961 −0.0754805 0.997147i \(-0.524049\pi\)
−0.0754805 + 0.997147i \(0.524049\pi\)
\(108\) 0 0
\(109\) −19.3693 −1.85524 −0.927622 0.373520i \(-0.878151\pi\)
−0.927622 + 0.373520i \(0.878151\pi\)
\(110\) 0 0
\(111\) 5.12311 0.486264
\(112\) 0 0
\(113\) 10.6847 1.00513 0.502564 0.864540i \(-0.332390\pi\)
0.502564 + 0.864540i \(0.332390\pi\)
\(114\) 0 0
\(115\) −5.56155 −0.518617
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 0 0
\(123\) −1.12311 −0.101267
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 7.80776 0.687435
\(130\) 0 0
\(131\) −9.36932 −0.818601 −0.409301 0.912400i \(-0.634227\pi\)
−0.409301 + 0.912400i \(0.634227\pi\)
\(132\) 0 0
\(133\) −7.31534 −0.634321
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 3.75379 0.320708 0.160354 0.987060i \(-0.448736\pi\)
0.160354 + 0.987060i \(0.448736\pi\)
\(138\) 0 0
\(139\) 8.87689 0.752928 0.376464 0.926431i \(-0.377140\pi\)
0.376464 + 0.926431i \(0.377140\pi\)
\(140\) 0 0
\(141\) 3.12311 0.263013
\(142\) 0 0
\(143\) 3.12311 0.261167
\(144\) 0 0
\(145\) −1.12311 −0.0932688
\(146\) 0 0
\(147\) −4.56155 −0.376231
\(148\) 0 0
\(149\) −20.0540 −1.64289 −0.821443 0.570291i \(-0.806830\pi\)
−0.821443 + 0.570291i \(0.806830\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 5.12311 0.414179
\(154\) 0 0
\(155\) −1.00000 −0.0803219
\(156\) 0 0
\(157\) −20.0540 −1.60048 −0.800241 0.599679i \(-0.795295\pi\)
−0.800241 + 0.599679i \(0.795295\pi\)
\(158\) 0 0
\(159\) 11.5616 0.916891
\(160\) 0 0
\(161\) −8.68466 −0.684447
\(162\) 0 0
\(163\) −9.36932 −0.733862 −0.366931 0.930248i \(-0.619591\pi\)
−0.366931 + 0.930248i \(0.619591\pi\)
\(164\) 0 0
\(165\) 1.56155 0.121567
\(166\) 0 0
\(167\) 2.43845 0.188693 0.0943464 0.995539i \(-0.469924\pi\)
0.0943464 + 0.995539i \(0.469924\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −4.68466 −0.358245
\(172\) 0 0
\(173\) −8.24621 −0.626948 −0.313474 0.949597i \(-0.601493\pi\)
−0.313474 + 0.949597i \(0.601493\pi\)
\(174\) 0 0
\(175\) 1.56155 0.118042
\(176\) 0 0
\(177\) 4.87689 0.366570
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 24.0540 1.78792 0.893959 0.448149i \(-0.147917\pi\)
0.893959 + 0.448149i \(0.147917\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 0 0
\(185\) 5.12311 0.376658
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) 1.56155 0.113586
\(190\) 0 0
\(191\) −2.24621 −0.162530 −0.0812651 0.996693i \(-0.525896\pi\)
−0.0812651 + 0.996693i \(0.525896\pi\)
\(192\) 0 0
\(193\) −18.4924 −1.33111 −0.665557 0.746347i \(-0.731806\pi\)
−0.665557 + 0.746347i \(0.731806\pi\)
\(194\) 0 0
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −3.80776 −0.269925 −0.134963 0.990851i \(-0.543091\pi\)
−0.134963 + 0.990851i \(0.543091\pi\)
\(200\) 0 0
\(201\) −9.36932 −0.660861
\(202\) 0 0
\(203\) −1.75379 −0.123092
\(204\) 0 0
\(205\) −1.12311 −0.0784411
\(206\) 0 0
\(207\) −5.56155 −0.386555
\(208\) 0 0
\(209\) −7.31534 −0.506013
\(210\) 0 0
\(211\) 11.3153 0.778980 0.389490 0.921031i \(-0.372651\pi\)
0.389490 + 0.921031i \(0.372651\pi\)
\(212\) 0 0
\(213\) −4.68466 −0.320988
\(214\) 0 0
\(215\) 7.80776 0.532485
\(216\) 0 0
\(217\) −1.56155 −0.106005
\(218\) 0 0
\(219\) 9.80776 0.662747
\(220\) 0 0
\(221\) 10.2462 0.689235
\(222\) 0 0
\(223\) 12.8769 0.862301 0.431150 0.902280i \(-0.358108\pi\)
0.431150 + 0.902280i \(0.358108\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 4.68466 0.310932 0.155466 0.987841i \(-0.450312\pi\)
0.155466 + 0.987841i \(0.450312\pi\)
\(228\) 0 0
\(229\) −12.4384 −0.821956 −0.410978 0.911645i \(-0.634813\pi\)
−0.410978 + 0.911645i \(0.634813\pi\)
\(230\) 0 0
\(231\) 2.43845 0.160438
\(232\) 0 0
\(233\) 20.0540 1.31378 0.656890 0.753987i \(-0.271872\pi\)
0.656890 + 0.753987i \(0.271872\pi\)
\(234\) 0 0
\(235\) 3.12311 0.203729
\(236\) 0 0
\(237\) 16.6847 1.08379
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −4.24621 −0.273523 −0.136761 0.990604i \(-0.543669\pi\)
−0.136761 + 0.990604i \(0.543669\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −4.56155 −0.291427
\(246\) 0 0
\(247\) −9.36932 −0.596155
\(248\) 0 0
\(249\) 2.24621 0.142348
\(250\) 0 0
\(251\) 16.4924 1.04099 0.520496 0.853864i \(-0.325747\pi\)
0.520496 + 0.853864i \(0.325747\pi\)
\(252\) 0 0
\(253\) −8.68466 −0.546000
\(254\) 0 0
\(255\) 5.12311 0.320821
\(256\) 0 0
\(257\) 10.6847 0.666491 0.333245 0.942840i \(-0.391856\pi\)
0.333245 + 0.942840i \(0.391856\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −1.12311 −0.0695185
\(262\) 0 0
\(263\) −12.4924 −0.770316 −0.385158 0.922851i \(-0.625853\pi\)
−0.385158 + 0.922851i \(0.625853\pi\)
\(264\) 0 0
\(265\) 11.5616 0.710221
\(266\) 0 0
\(267\) −1.31534 −0.0804976
\(268\) 0 0
\(269\) 16.2462 0.990549 0.495274 0.868737i \(-0.335067\pi\)
0.495274 + 0.868737i \(0.335067\pi\)
\(270\) 0 0
\(271\) 10.0540 0.610736 0.305368 0.952234i \(-0.401221\pi\)
0.305368 + 0.952234i \(0.401221\pi\)
\(272\) 0 0
\(273\) 3.12311 0.189019
\(274\) 0 0
\(275\) 1.56155 0.0941652
\(276\) 0 0
\(277\) 19.3693 1.16379 0.581895 0.813264i \(-0.302312\pi\)
0.581895 + 0.813264i \(0.302312\pi\)
\(278\) 0 0
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 13.1231 0.782859 0.391429 0.920208i \(-0.371981\pi\)
0.391429 + 0.920208i \(0.371981\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) −4.68466 −0.277495
\(286\) 0 0
\(287\) −1.75379 −0.103523
\(288\) 0 0
\(289\) 9.24621 0.543895
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) 0 0
\(293\) −6.49242 −0.379291 −0.189646 0.981853i \(-0.560734\pi\)
−0.189646 + 0.981853i \(0.560734\pi\)
\(294\) 0 0
\(295\) 4.87689 0.283944
\(296\) 0 0
\(297\) 1.56155 0.0906105
\(298\) 0 0
\(299\) −11.1231 −0.643266
\(300\) 0 0
\(301\) 12.1922 0.702749
\(302\) 0 0
\(303\) 3.56155 0.204606
\(304\) 0 0
\(305\) 6.00000 0.343559
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 18.2462 1.03799
\(310\) 0 0
\(311\) 18.2462 1.03465 0.517324 0.855790i \(-0.326928\pi\)
0.517324 + 0.855790i \(0.326928\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 1.56155 0.0879835
\(316\) 0 0
\(317\) 25.1231 1.41105 0.705527 0.708683i \(-0.250710\pi\)
0.705527 + 0.708683i \(0.250710\pi\)
\(318\) 0 0
\(319\) −1.75379 −0.0981933
\(320\) 0 0
\(321\) −1.56155 −0.0871574
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 0 0
\(327\) −19.3693 −1.07113
\(328\) 0 0
\(329\) 4.87689 0.268872
\(330\) 0 0
\(331\) −10.2462 −0.563183 −0.281591 0.959534i \(-0.590862\pi\)
−0.281591 + 0.959534i \(0.590862\pi\)
\(332\) 0 0
\(333\) 5.12311 0.280744
\(334\) 0 0
\(335\) −9.36932 −0.511900
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 0 0
\(339\) 10.6847 0.580311
\(340\) 0 0
\(341\) −1.56155 −0.0845628
\(342\) 0 0
\(343\) −18.0540 −0.974823
\(344\) 0 0
\(345\) −5.56155 −0.299424
\(346\) 0 0
\(347\) 2.24621 0.120583 0.0602915 0.998181i \(-0.480797\pi\)
0.0602915 + 0.998181i \(0.480797\pi\)
\(348\) 0 0
\(349\) −19.3693 −1.03682 −0.518408 0.855133i \(-0.673475\pi\)
−0.518408 + 0.855133i \(0.673475\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 16.2462 0.864699 0.432349 0.901706i \(-0.357685\pi\)
0.432349 + 0.901706i \(0.357685\pi\)
\(354\) 0 0
\(355\) −4.68466 −0.248636
\(356\) 0 0
\(357\) 8.00000 0.423405
\(358\) 0 0
\(359\) 19.3153 1.01942 0.509712 0.860345i \(-0.329752\pi\)
0.509712 + 0.860345i \(0.329752\pi\)
\(360\) 0 0
\(361\) 2.94602 0.155054
\(362\) 0 0
\(363\) −8.56155 −0.449365
\(364\) 0 0
\(365\) 9.80776 0.513362
\(366\) 0 0
\(367\) 9.36932 0.489074 0.244537 0.969640i \(-0.421364\pi\)
0.244537 + 0.969640i \(0.421364\pi\)
\(368\) 0 0
\(369\) −1.12311 −0.0584665
\(370\) 0 0
\(371\) 18.0540 0.937316
\(372\) 0 0
\(373\) 6.68466 0.346118 0.173059 0.984911i \(-0.444635\pi\)
0.173059 + 0.984911i \(0.444635\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −2.24621 −0.115686
\(378\) 0 0
\(379\) −30.0540 −1.54377 −0.771885 0.635763i \(-0.780686\pi\)
−0.771885 + 0.635763i \(0.780686\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −6.24621 −0.319166 −0.159583 0.987184i \(-0.551015\pi\)
−0.159583 + 0.987184i \(0.551015\pi\)
\(384\) 0 0
\(385\) 2.43845 0.124275
\(386\) 0 0
\(387\) 7.80776 0.396891
\(388\) 0 0
\(389\) 24.2462 1.22933 0.614666 0.788788i \(-0.289291\pi\)
0.614666 + 0.788788i \(0.289291\pi\)
\(390\) 0 0
\(391\) −28.4924 −1.44092
\(392\) 0 0
\(393\) −9.36932 −0.472620
\(394\) 0 0
\(395\) 16.6847 0.839496
\(396\) 0 0
\(397\) −28.0540 −1.40799 −0.703994 0.710206i \(-0.748602\pi\)
−0.703994 + 0.710206i \(0.748602\pi\)
\(398\) 0 0
\(399\) −7.31534 −0.366225
\(400\) 0 0
\(401\) −1.31534 −0.0656850 −0.0328425 0.999461i \(-0.510456\pi\)
−0.0328425 + 0.999461i \(0.510456\pi\)
\(402\) 0 0
\(403\) −2.00000 −0.0996271
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −12.6307 −0.624547 −0.312274 0.949992i \(-0.601091\pi\)
−0.312274 + 0.949992i \(0.601091\pi\)
\(410\) 0 0
\(411\) 3.75379 0.185161
\(412\) 0 0
\(413\) 7.61553 0.374736
\(414\) 0 0
\(415\) 2.24621 0.110262
\(416\) 0 0
\(417\) 8.87689 0.434703
\(418\) 0 0
\(419\) −22.2462 −1.08680 −0.543399 0.839474i \(-0.682863\pi\)
−0.543399 + 0.839474i \(0.682863\pi\)
\(420\) 0 0
\(421\) 18.4924 0.901266 0.450633 0.892709i \(-0.351198\pi\)
0.450633 + 0.892709i \(0.351198\pi\)
\(422\) 0 0
\(423\) 3.12311 0.151851
\(424\) 0 0
\(425\) 5.12311 0.248507
\(426\) 0 0
\(427\) 9.36932 0.453413
\(428\) 0 0
\(429\) 3.12311 0.150785
\(430\) 0 0
\(431\) 13.7538 0.662497 0.331248 0.943544i \(-0.392530\pi\)
0.331248 + 0.943544i \(0.392530\pi\)
\(432\) 0 0
\(433\) −18.3002 −0.879451 −0.439725 0.898132i \(-0.644924\pi\)
−0.439725 + 0.898132i \(0.644924\pi\)
\(434\) 0 0
\(435\) −1.12311 −0.0538488
\(436\) 0 0
\(437\) 26.0540 1.24633
\(438\) 0 0
\(439\) 31.6155 1.50893 0.754463 0.656342i \(-0.227897\pi\)
0.754463 + 0.656342i \(0.227897\pi\)
\(440\) 0 0
\(441\) −4.56155 −0.217217
\(442\) 0 0
\(443\) 14.0540 0.667725 0.333862 0.942622i \(-0.391648\pi\)
0.333862 + 0.942622i \(0.391648\pi\)
\(444\) 0 0
\(445\) −1.31534 −0.0623532
\(446\) 0 0
\(447\) −20.0540 −0.948520
\(448\) 0 0
\(449\) −22.4924 −1.06148 −0.530742 0.847534i \(-0.678087\pi\)
−0.530742 + 0.847534i \(0.678087\pi\)
\(450\) 0 0
\(451\) −1.75379 −0.0825827
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.12311 0.146413
\(456\) 0 0
\(457\) 24.7386 1.15722 0.578612 0.815603i \(-0.303594\pi\)
0.578612 + 0.815603i \(0.303594\pi\)
\(458\) 0 0
\(459\) 5.12311 0.239126
\(460\) 0 0
\(461\) −15.3693 −0.715820 −0.357910 0.933756i \(-0.616511\pi\)
−0.357910 + 0.933756i \(0.616511\pi\)
\(462\) 0 0
\(463\) 18.7386 0.870858 0.435429 0.900223i \(-0.356597\pi\)
0.435429 + 0.900223i \(0.356597\pi\)
\(464\) 0 0
\(465\) −1.00000 −0.0463739
\(466\) 0 0
\(467\) 26.2462 1.21453 0.607265 0.794499i \(-0.292267\pi\)
0.607265 + 0.794499i \(0.292267\pi\)
\(468\) 0 0
\(469\) −14.6307 −0.675582
\(470\) 0 0
\(471\) −20.0540 −0.924038
\(472\) 0 0
\(473\) 12.1922 0.560600
\(474\) 0 0
\(475\) −4.68466 −0.214947
\(476\) 0 0
\(477\) 11.5616 0.529367
\(478\) 0 0
\(479\) −6.43845 −0.294180 −0.147090 0.989123i \(-0.546991\pi\)
−0.147090 + 0.989123i \(0.546991\pi\)
\(480\) 0 0
\(481\) 10.2462 0.467187
\(482\) 0 0
\(483\) −8.68466 −0.395166
\(484\) 0 0
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 0 0
\(489\) −9.36932 −0.423695
\(490\) 0 0
\(491\) −2.93087 −0.132268 −0.0661341 0.997811i \(-0.521067\pi\)
−0.0661341 + 0.997811i \(0.521067\pi\)
\(492\) 0 0
\(493\) −5.75379 −0.259138
\(494\) 0 0
\(495\) 1.56155 0.0701866
\(496\) 0 0
\(497\) −7.31534 −0.328138
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 2.43845 0.108942
\(502\) 0 0
\(503\) −9.75379 −0.434900 −0.217450 0.976071i \(-0.569774\pi\)
−0.217450 + 0.976071i \(0.569774\pi\)
\(504\) 0 0
\(505\) 3.56155 0.158487
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −21.6155 −0.958091 −0.479046 0.877790i \(-0.659017\pi\)
−0.479046 + 0.877790i \(0.659017\pi\)
\(510\) 0 0
\(511\) 15.3153 0.677511
\(512\) 0 0
\(513\) −4.68466 −0.206833
\(514\) 0 0
\(515\) 18.2462 0.804024
\(516\) 0 0
\(517\) 4.87689 0.214486
\(518\) 0 0
\(519\) −8.24621 −0.361968
\(520\) 0 0
\(521\) −15.7538 −0.690186 −0.345093 0.938568i \(-0.612153\pi\)
−0.345093 + 0.938568i \(0.612153\pi\)
\(522\) 0 0
\(523\) −39.4233 −1.72386 −0.861930 0.507027i \(-0.830744\pi\)
−0.861930 + 0.507027i \(0.830744\pi\)
\(524\) 0 0
\(525\) 1.56155 0.0681518
\(526\) 0 0
\(527\) −5.12311 −0.223166
\(528\) 0 0
\(529\) 7.93087 0.344820
\(530\) 0 0
\(531\) 4.87689 0.211639
\(532\) 0 0
\(533\) −2.24621 −0.0972942
\(534\) 0 0
\(535\) −1.56155 −0.0675118
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) −7.12311 −0.306814
\(540\) 0 0
\(541\) 36.2462 1.55835 0.779173 0.626809i \(-0.215639\pi\)
0.779173 + 0.626809i \(0.215639\pi\)
\(542\) 0 0
\(543\) 24.0540 1.03225
\(544\) 0 0
\(545\) −19.3693 −0.829690
\(546\) 0 0
\(547\) −35.1231 −1.50176 −0.750878 0.660441i \(-0.770369\pi\)
−0.750878 + 0.660441i \(0.770369\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 5.26137 0.224142
\(552\) 0 0
\(553\) 26.0540 1.10793
\(554\) 0 0
\(555\) 5.12311 0.217464
\(556\) 0 0
\(557\) −6.19224 −0.262373 −0.131187 0.991358i \(-0.541879\pi\)
−0.131187 + 0.991358i \(0.541879\pi\)
\(558\) 0 0
\(559\) 15.6155 0.660466
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 16.4924 0.695073 0.347536 0.937667i \(-0.387018\pi\)
0.347536 + 0.937667i \(0.387018\pi\)
\(564\) 0 0
\(565\) 10.6847 0.449507
\(566\) 0 0
\(567\) 1.56155 0.0655791
\(568\) 0 0
\(569\) 10.1922 0.427281 0.213640 0.976912i \(-0.431468\pi\)
0.213640 + 0.976912i \(0.431468\pi\)
\(570\) 0 0
\(571\) −32.8769 −1.37586 −0.687928 0.725779i \(-0.741479\pi\)
−0.687928 + 0.725779i \(0.741479\pi\)
\(572\) 0 0
\(573\) −2.24621 −0.0938368
\(574\) 0 0
\(575\) −5.56155 −0.231933
\(576\) 0 0
\(577\) 6.87689 0.286289 0.143144 0.989702i \(-0.454279\pi\)
0.143144 + 0.989702i \(0.454279\pi\)
\(578\) 0 0
\(579\) −18.4924 −0.768519
\(580\) 0 0
\(581\) 3.50758 0.145519
\(582\) 0 0
\(583\) 18.0540 0.747719
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) −32.4924 −1.34111 −0.670553 0.741862i \(-0.733943\pi\)
−0.670553 + 0.741862i \(0.733943\pi\)
\(588\) 0 0
\(589\) 4.68466 0.193028
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 8.00000 0.327968
\(596\) 0 0
\(597\) −3.80776 −0.155841
\(598\) 0 0
\(599\) 23.4233 0.957050 0.478525 0.878074i \(-0.341172\pi\)
0.478525 + 0.878074i \(0.341172\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) −9.36932 −0.381548
\(604\) 0 0
\(605\) −8.56155 −0.348077
\(606\) 0 0
\(607\) 30.0540 1.21985 0.609927 0.792458i \(-0.291199\pi\)
0.609927 + 0.792458i \(0.291199\pi\)
\(608\) 0 0
\(609\) −1.75379 −0.0710671
\(610\) 0 0
\(611\) 6.24621 0.252695
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) −1.12311 −0.0452880
\(616\) 0 0
\(617\) −19.5616 −0.787518 −0.393759 0.919214i \(-0.628826\pi\)
−0.393759 + 0.919214i \(0.628826\pi\)
\(618\) 0 0
\(619\) 13.3693 0.537358 0.268679 0.963230i \(-0.413413\pi\)
0.268679 + 0.963230i \(0.413413\pi\)
\(620\) 0 0
\(621\) −5.56155 −0.223177
\(622\) 0 0
\(623\) −2.05398 −0.0822908
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −7.31534 −0.292147
\(628\) 0 0
\(629\) 26.2462 1.04650
\(630\) 0 0
\(631\) 24.3002 0.967375 0.483688 0.875241i \(-0.339297\pi\)
0.483688 + 0.875241i \(0.339297\pi\)
\(632\) 0 0
\(633\) 11.3153 0.449744
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −9.12311 −0.361471
\(638\) 0 0
\(639\) −4.68466 −0.185322
\(640\) 0 0
\(641\) −38.4924 −1.52036 −0.760180 0.649713i \(-0.774889\pi\)
−0.760180 + 0.649713i \(0.774889\pi\)
\(642\) 0 0
\(643\) −30.0540 −1.18521 −0.592607 0.805492i \(-0.701901\pi\)
−0.592607 + 0.805492i \(0.701901\pi\)
\(644\) 0 0
\(645\) 7.80776 0.307430
\(646\) 0 0
\(647\) 17.0691 0.671057 0.335528 0.942030i \(-0.391085\pi\)
0.335528 + 0.942030i \(0.391085\pi\)
\(648\) 0 0
\(649\) 7.61553 0.298936
\(650\) 0 0
\(651\) −1.56155 −0.0612021
\(652\) 0 0
\(653\) 23.7538 0.929558 0.464779 0.885427i \(-0.346134\pi\)
0.464779 + 0.885427i \(0.346134\pi\)
\(654\) 0 0
\(655\) −9.36932 −0.366090
\(656\) 0 0
\(657\) 9.80776 0.382637
\(658\) 0 0
\(659\) −18.7386 −0.729954 −0.364977 0.931017i \(-0.618923\pi\)
−0.364977 + 0.931017i \(0.618923\pi\)
\(660\) 0 0
\(661\) −6.49242 −0.252526 −0.126263 0.991997i \(-0.540298\pi\)
−0.126263 + 0.991997i \(0.540298\pi\)
\(662\) 0 0
\(663\) 10.2462 0.397930
\(664\) 0 0
\(665\) −7.31534 −0.283677
\(666\) 0 0
\(667\) 6.24621 0.241854
\(668\) 0 0
\(669\) 12.8769 0.497849
\(670\) 0 0
\(671\) 9.36932 0.361698
\(672\) 0 0
\(673\) 28.2462 1.08881 0.544406 0.838822i \(-0.316755\pi\)
0.544406 + 0.838822i \(0.316755\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −29.4233 −1.13083 −0.565414 0.824807i \(-0.691284\pi\)
−0.565414 + 0.824807i \(0.691284\pi\)
\(678\) 0 0
\(679\) −9.36932 −0.359561
\(680\) 0 0
\(681\) 4.68466 0.179517
\(682\) 0 0
\(683\) −19.3153 −0.739081 −0.369541 0.929215i \(-0.620485\pi\)
−0.369541 + 0.929215i \(0.620485\pi\)
\(684\) 0 0
\(685\) 3.75379 0.143425
\(686\) 0 0
\(687\) −12.4384 −0.474556
\(688\) 0 0
\(689\) 23.1231 0.880920
\(690\) 0 0
\(691\) 22.0540 0.838973 0.419486 0.907762i \(-0.362210\pi\)
0.419486 + 0.907762i \(0.362210\pi\)
\(692\) 0 0
\(693\) 2.43845 0.0926289
\(694\) 0 0
\(695\) 8.87689 0.336720
\(696\) 0 0
\(697\) −5.75379 −0.217940
\(698\) 0 0
\(699\) 20.0540 0.758511
\(700\) 0 0
\(701\) 25.8078 0.974746 0.487373 0.873194i \(-0.337955\pi\)
0.487373 + 0.873194i \(0.337955\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) 3.12311 0.117623
\(706\) 0 0
\(707\) 5.56155 0.209164
\(708\) 0 0
\(709\) −20.0540 −0.753143 −0.376571 0.926388i \(-0.622897\pi\)
−0.376571 + 0.926388i \(0.622897\pi\)
\(710\) 0 0
\(711\) 16.6847 0.625724
\(712\) 0 0
\(713\) 5.56155 0.208282
\(714\) 0 0
\(715\) 3.12311 0.116798
\(716\) 0 0
\(717\) −24.0000 −0.896296
\(718\) 0 0
\(719\) 43.1231 1.60822 0.804110 0.594480i \(-0.202642\pi\)
0.804110 + 0.594480i \(0.202642\pi\)
\(720\) 0 0
\(721\) 28.4924 1.06111
\(722\) 0 0
\(723\) −4.24621 −0.157918
\(724\) 0 0
\(725\) −1.12311 −0.0417111
\(726\) 0 0
\(727\) 14.0540 0.521233 0.260617 0.965442i \(-0.416074\pi\)
0.260617 + 0.965442i \(0.416074\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 40.0000 1.47945
\(732\) 0 0
\(733\) 26.4924 0.978520 0.489260 0.872138i \(-0.337267\pi\)
0.489260 + 0.872138i \(0.337267\pi\)
\(734\) 0 0
\(735\) −4.56155 −0.168255
\(736\) 0 0
\(737\) −14.6307 −0.538928
\(738\) 0 0
\(739\) 12.9848 0.477655 0.238828 0.971062i \(-0.423237\pi\)
0.238828 + 0.971062i \(0.423237\pi\)
\(740\) 0 0
\(741\) −9.36932 −0.344190
\(742\) 0 0
\(743\) −11.4233 −0.419080 −0.209540 0.977800i \(-0.567197\pi\)
−0.209540 + 0.977800i \(0.567197\pi\)
\(744\) 0 0
\(745\) −20.0540 −0.734721
\(746\) 0 0
\(747\) 2.24621 0.0821846
\(748\) 0 0
\(749\) −2.43845 −0.0890989
\(750\) 0 0
\(751\) −36.8769 −1.34566 −0.672828 0.739798i \(-0.734921\pi\)
−0.672828 + 0.739798i \(0.734921\pi\)
\(752\) 0 0
\(753\) 16.4924 0.601017
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 0 0
\(759\) −8.68466 −0.315233
\(760\) 0 0
\(761\) 41.4233 1.50159 0.750797 0.660533i \(-0.229670\pi\)
0.750797 + 0.660533i \(0.229670\pi\)
\(762\) 0 0
\(763\) −30.2462 −1.09499
\(764\) 0 0
\(765\) 5.12311 0.185226
\(766\) 0 0
\(767\) 9.75379 0.352189
\(768\) 0 0
\(769\) −16.4384 −0.592786 −0.296393 0.955066i \(-0.595784\pi\)
−0.296393 + 0.955066i \(0.595784\pi\)
\(770\) 0 0
\(771\) 10.6847 0.384799
\(772\) 0 0
\(773\) −15.5616 −0.559710 −0.279855 0.960042i \(-0.590286\pi\)
−0.279855 + 0.960042i \(0.590286\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 0 0
\(777\) 8.00000 0.286998
\(778\) 0 0
\(779\) 5.26137 0.188508
\(780\) 0 0
\(781\) −7.31534 −0.261764
\(782\) 0 0
\(783\) −1.12311 −0.0401365
\(784\) 0 0
\(785\) −20.0540 −0.715757
\(786\) 0 0
\(787\) −10.9309 −0.389643 −0.194822 0.980839i \(-0.562413\pi\)
−0.194822 + 0.980839i \(0.562413\pi\)
\(788\) 0 0
\(789\) −12.4924 −0.444742
\(790\) 0 0
\(791\) 16.6847 0.593238
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 0 0
\(795\) 11.5616 0.410046
\(796\) 0 0
\(797\) 38.9848 1.38091 0.690457 0.723373i \(-0.257409\pi\)
0.690457 + 0.723373i \(0.257409\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −1.31534 −0.0464753
\(802\) 0 0
\(803\) 15.3153 0.540467
\(804\) 0 0
\(805\) −8.68466 −0.306094
\(806\) 0 0
\(807\) 16.2462 0.571894
\(808\) 0 0
\(809\) −25.3153 −0.890040 −0.445020 0.895521i \(-0.646803\pi\)
−0.445020 + 0.895521i \(0.646803\pi\)
\(810\) 0 0
\(811\) −53.6695 −1.88459 −0.942296 0.334782i \(-0.891337\pi\)
−0.942296 + 0.334782i \(0.891337\pi\)
\(812\) 0 0
\(813\) 10.0540 0.352608
\(814\) 0 0
\(815\) −9.36932 −0.328193
\(816\) 0 0
\(817\) −36.5767 −1.27966
\(818\) 0 0
\(819\) 3.12311 0.109130
\(820\) 0 0
\(821\) −21.6155 −0.754387 −0.377194 0.926134i \(-0.623111\pi\)
−0.377194 + 0.926134i \(0.623111\pi\)
\(822\) 0 0
\(823\) −15.6155 −0.544323 −0.272162 0.962252i \(-0.587739\pi\)
−0.272162 + 0.962252i \(0.587739\pi\)
\(824\) 0 0
\(825\) 1.56155 0.0543663
\(826\) 0 0
\(827\) −18.2462 −0.634483 −0.317241 0.948345i \(-0.602757\pi\)
−0.317241 + 0.948345i \(0.602757\pi\)
\(828\) 0 0
\(829\) −37.4233 −1.29976 −0.649882 0.760035i \(-0.725182\pi\)
−0.649882 + 0.760035i \(0.725182\pi\)
\(830\) 0 0
\(831\) 19.3693 0.671914
\(832\) 0 0
\(833\) −23.3693 −0.809699
\(834\) 0 0
\(835\) 2.43845 0.0843859
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) −34.9309 −1.20595 −0.602974 0.797761i \(-0.706018\pi\)
−0.602974 + 0.797761i \(0.706018\pi\)
\(840\) 0 0
\(841\) −27.7386 −0.956505
\(842\) 0 0
\(843\) 13.1231 0.451984
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −13.3693 −0.459375
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −28.4924 −0.976708
\(852\) 0 0
\(853\) −38.7926 −1.32823 −0.664117 0.747629i \(-0.731192\pi\)
−0.664117 + 0.747629i \(0.731192\pi\)
\(854\) 0 0
\(855\) −4.68466 −0.160212
\(856\) 0 0
\(857\) −53.2311 −1.81834 −0.909169 0.416427i \(-0.863282\pi\)
−0.909169 + 0.416427i \(0.863282\pi\)
\(858\) 0 0
\(859\) 22.7386 0.775832 0.387916 0.921695i \(-0.373195\pi\)
0.387916 + 0.921695i \(0.373195\pi\)
\(860\) 0 0
\(861\) −1.75379 −0.0597690
\(862\) 0 0
\(863\) 30.5464 1.03981 0.519906 0.854224i \(-0.325967\pi\)
0.519906 + 0.854224i \(0.325967\pi\)
\(864\) 0 0
\(865\) −8.24621 −0.280380
\(866\) 0 0
\(867\) 9.24621 0.314018
\(868\) 0 0
\(869\) 26.0540 0.883821
\(870\) 0 0
\(871\) −18.7386 −0.634934
\(872\) 0 0
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 1.56155 0.0527901
\(876\) 0 0
\(877\) −50.0000 −1.68838 −0.844190 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) 0 0
\(879\) −6.49242 −0.218984
\(880\) 0 0
\(881\) 42.4924 1.43161 0.715803 0.698302i \(-0.246061\pi\)
0.715803 + 0.698302i \(0.246061\pi\)
\(882\) 0 0
\(883\) 31.4233 1.05748 0.528739 0.848784i \(-0.322665\pi\)
0.528739 + 0.848784i \(0.322665\pi\)
\(884\) 0 0
\(885\) 4.87689 0.163935
\(886\) 0 0
\(887\) 57.3693 1.92627 0.963137 0.269013i \(-0.0866974\pi\)
0.963137 + 0.269013i \(0.0866974\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.56155 0.0523140
\(892\) 0 0
\(893\) −14.6307 −0.489597
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) −11.1231 −0.371390
\(898\) 0 0
\(899\) 1.12311 0.0374577
\(900\) 0 0
\(901\) 59.2311 1.97327
\(902\) 0 0
\(903\) 12.1922 0.405732
\(904\) 0 0
\(905\) 24.0540 0.799581
\(906\) 0 0
\(907\) −24.0000 −0.796907 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) 0 0
\(909\) 3.56155 0.118129
\(910\) 0 0
\(911\) 44.4924 1.47410 0.737050 0.675838i \(-0.236218\pi\)
0.737050 + 0.675838i \(0.236218\pi\)
\(912\) 0 0
\(913\) 3.50758 0.116084
\(914\) 0 0
\(915\) 6.00000 0.198354
\(916\) 0 0
\(917\) −14.6307 −0.483148
\(918\) 0 0
\(919\) −39.6155 −1.30680 −0.653398 0.757015i \(-0.726657\pi\)
−0.653398 + 0.757015i \(0.726657\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −9.36932 −0.308395
\(924\) 0 0
\(925\) 5.12311 0.168447
\(926\) 0 0
\(927\) 18.2462 0.599284
\(928\) 0 0
\(929\) 11.5616 0.379322 0.189661 0.981850i \(-0.439261\pi\)
0.189661 + 0.981850i \(0.439261\pi\)
\(930\) 0 0
\(931\) 21.3693 0.700351
\(932\) 0 0
\(933\) 18.2462 0.597354
\(934\) 0 0
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) 16.6307 0.543301 0.271650 0.962396i \(-0.412431\pi\)
0.271650 + 0.962396i \(0.412431\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 27.3693 0.892214 0.446107 0.894980i \(-0.352810\pi\)
0.446107 + 0.894980i \(0.352810\pi\)
\(942\) 0 0
\(943\) 6.24621 0.203405
\(944\) 0 0
\(945\) 1.56155 0.0507973
\(946\) 0 0
\(947\) 0.876894 0.0284952 0.0142476 0.999898i \(-0.495465\pi\)
0.0142476 + 0.999898i \(0.495465\pi\)
\(948\) 0 0
\(949\) 19.6155 0.636747
\(950\) 0 0
\(951\) 25.1231 0.814673
\(952\) 0 0
\(953\) −58.1080 −1.88230 −0.941151 0.337988i \(-0.890254\pi\)
−0.941151 + 0.337988i \(0.890254\pi\)
\(954\) 0 0
\(955\) −2.24621 −0.0726857
\(956\) 0 0
\(957\) −1.75379 −0.0566919
\(958\) 0 0
\(959\) 5.86174 0.189285
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −1.56155 −0.0503203
\(964\) 0 0
\(965\) −18.4924 −0.595292
\(966\) 0 0
\(967\) 25.7538 0.828186 0.414093 0.910235i \(-0.364099\pi\)
0.414093 + 0.910235i \(0.364099\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 28.8769 0.926704 0.463352 0.886174i \(-0.346647\pi\)
0.463352 + 0.886174i \(0.346647\pi\)
\(972\) 0 0
\(973\) 13.8617 0.444387
\(974\) 0 0
\(975\) 2.00000 0.0640513
\(976\) 0 0
\(977\) −22.9848 −0.735350 −0.367675 0.929954i \(-0.619846\pi\)
−0.367675 + 0.929954i \(0.619846\pi\)
\(978\) 0 0
\(979\) −2.05398 −0.0656453
\(980\) 0 0
\(981\) −19.3693 −0.618415
\(982\) 0 0
\(983\) 24.9848 0.796893 0.398446 0.917192i \(-0.369549\pi\)
0.398446 + 0.917192i \(0.369549\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 4.87689 0.155233
\(988\) 0 0
\(989\) −43.4233 −1.38078
\(990\) 0 0
\(991\) −23.3153 −0.740636 −0.370318 0.928905i \(-0.620751\pi\)
−0.370318 + 0.928905i \(0.620751\pi\)
\(992\) 0 0
\(993\) −10.2462 −0.325154
\(994\) 0 0
\(995\) −3.80776 −0.120714
\(996\) 0 0
\(997\) −62.4924 −1.97915 −0.989577 0.144002i \(-0.954003\pi\)
−0.989577 + 0.144002i \(0.954003\pi\)
\(998\) 0 0
\(999\) 5.12311 0.162088
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 7440.2.a.bl.1.2 2
4.3 odd 2 930.2.a.p.1.1 2
12.11 even 2 2790.2.a.be.1.1 2
20.3 even 4 4650.2.d.bd.3349.2 4
20.7 even 4 4650.2.d.bd.3349.3 4
20.19 odd 2 4650.2.a.ce.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.p.1.1 2 4.3 odd 2
2790.2.a.be.1.1 2 12.11 even 2
4650.2.a.ce.1.2 2 20.19 odd 2
4650.2.d.bd.3349.2 4 20.3 even 4
4650.2.d.bd.3349.3 4 20.7 even 4
7440.2.a.bl.1.2 2 1.1 even 1 trivial