Properties

Label 2790.2.a.be.1.1
Level $2790$
Weight $2$
Character 2790.1
Self dual yes
Analytic conductor $22.278$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2790.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.2782621639\)
Analytic rank: \(1\)
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 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 2790.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.56155 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.56155 q^{11} +2.00000 q^{13} +1.56155 q^{14} +1.00000 q^{16} -5.12311 q^{17} +4.68466 q^{19} -1.00000 q^{20} -1.56155 q^{22} -5.56155 q^{23} +1.00000 q^{25} -2.00000 q^{26} -1.56155 q^{28} +1.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} +5.12311 q^{34} +1.56155 q^{35} +5.12311 q^{37} -4.68466 q^{38} +1.00000 q^{40} +1.12311 q^{41} -7.80776 q^{43} +1.56155 q^{44} +5.56155 q^{46} +3.12311 q^{47} -4.56155 q^{49} -1.00000 q^{50} +2.00000 q^{52} -11.5616 q^{53} -1.56155 q^{55} +1.56155 q^{56} -1.12311 q^{58} +4.87689 q^{59} +6.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} +9.36932 q^{67} -5.12311 q^{68} -1.56155 q^{70} -4.68466 q^{71} +9.80776 q^{73} -5.12311 q^{74} +4.68466 q^{76} -2.43845 q^{77} -16.6847 q^{79} -1.00000 q^{80} -1.12311 q^{82} +2.24621 q^{83} +5.12311 q^{85} +7.80776 q^{86} -1.56155 q^{88} +1.31534 q^{89} -3.12311 q^{91} -5.56155 q^{92} -3.12311 q^{94} -4.68466 q^{95} -6.00000 q^{97} +4.56155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} + 4 q^{13} - q^{14} + 2 q^{16} - 2 q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 7 q^{23} + 2 q^{25} - 4 q^{26} + q^{28} - 6 q^{29} + 2 q^{31} - 2 q^{32} + 2 q^{34} - q^{35} + 2 q^{37} + 3 q^{38} + 2 q^{40} - 6 q^{41} + 5 q^{43} - q^{44} + 7 q^{46} - 2 q^{47} - 5 q^{49} - 2 q^{50} + 4 q^{52} - 19 q^{53} + q^{55} - q^{56} + 6 q^{58} + 18 q^{59} + 12 q^{61} - 2 q^{62} + 2 q^{64} - 4 q^{65} - 6 q^{67} - 2 q^{68} + q^{70} + 3 q^{71} - q^{73} - 2 q^{74} - 3 q^{76} - 9 q^{77} - 21 q^{79} - 2 q^{80} + 6 q^{82} - 12 q^{83} + 2 q^{85} - 5 q^{86} + q^{88} + 15 q^{89} + 2 q^{91} - 7 q^{92} + 2 q^{94} + 3 q^{95} - 12 q^{97} + 5 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.56155 −0.590211 −0.295106 0.955465i \(-0.595355\pi\)
−0.295106 + 0.955465i \(0.595355\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 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 \(0.410544\pi\)
\(14\) 1.56155 0.417343
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.12311 −1.24254 −0.621268 0.783598i \(-0.713382\pi\)
−0.621268 + 0.783598i \(0.713382\pi\)
\(18\) 0 0
\(19\) 4.68466 1.07473 0.537367 0.843348i \(-0.319419\pi\)
0.537367 + 0.843348i \(0.319419\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.56155 −0.332924
\(23\) −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
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −1.56155 −0.295106
\(29\) 1.12311 0.208555 0.104278 0.994548i \(-0.466747\pi\)
0.104278 + 0.994548i \(0.466747\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 5.12311 0.878605
\(35\) 1.56155 0.263951
\(36\) 0 0
\(37\) 5.12311 0.842233 0.421117 0.907006i \(-0.361638\pi\)
0.421117 + 0.907006i \(0.361638\pi\)
\(38\) −4.68466 −0.759952
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 1.12311 0.175400 0.0876998 0.996147i \(-0.472048\pi\)
0.0876998 + 0.996147i \(0.472048\pi\)
\(42\) 0 0
\(43\) −7.80776 −1.19067 −0.595336 0.803477i \(-0.702981\pi\)
−0.595336 + 0.803477i \(0.702981\pi\)
\(44\) 1.56155 0.235413
\(45\) 0 0
\(46\) 5.56155 0.820006
\(47\) 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
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −11.5616 −1.58810 −0.794051 0.607852i \(-0.792032\pi\)
−0.794051 + 0.607852i \(0.792032\pi\)
\(54\) 0 0
\(55\) −1.56155 −0.210560
\(56\) 1.56155 0.208671
\(57\) 0 0
\(58\) −1.12311 −0.147471
\(59\) 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 \(0.374508\pi\)
\(62\) −1.00000 −0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 9.36932 1.14464 0.572322 0.820029i \(-0.306043\pi\)
0.572322 + 0.820029i \(0.306043\pi\)
\(68\) −5.12311 −0.621268
\(69\) 0 0
\(70\) −1.56155 −0.186641
\(71\) −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 \(0.305408\pi\)
\(74\) −5.12311 −0.595549
\(75\) 0 0
\(76\) 4.68466 0.537367
\(77\) −2.43845 −0.277887
\(78\) 0 0
\(79\) −16.6847 −1.87717 −0.938585 0.345047i \(-0.887863\pi\)
−0.938585 + 0.345047i \(0.887863\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −1.12311 −0.124026
\(83\) 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
\(86\) 7.80776 0.841933
\(87\) 0 0
\(88\) −1.56155 −0.166462
\(89\) 1.31534 0.139426 0.0697130 0.997567i \(-0.477792\pi\)
0.0697130 + 0.997567i \(0.477792\pi\)
\(90\) 0 0
\(91\) −3.12311 −0.327390
\(92\) −5.56155 −0.579832
\(93\) 0 0
\(94\) −3.12311 −0.322124
\(95\) −4.68466 −0.480636
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 4.56155 0.460786
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −3.56155 −0.354388 −0.177194 0.984176i \(-0.556702\pi\)
−0.177194 + 0.984176i \(0.556702\pi\)
\(102\) 0 0
\(103\) −18.2462 −1.79785 −0.898926 0.438100i \(-0.855652\pi\)
−0.898926 + 0.438100i \(0.855652\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 11.5616 1.12296
\(107\) −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 \(0.878151\pi\)
\(110\) 1.56155 0.148888
\(111\) 0 0
\(112\) −1.56155 −0.147553
\(113\) −10.6847 −1.00513 −0.502564 0.864540i \(-0.667610\pi\)
−0.502564 + 0.864540i \(0.667610\pi\)
\(114\) 0 0
\(115\) 5.56155 0.518617
\(116\) 1.12311 0.104278
\(117\) 0 0
\(118\) −4.87689 −0.448955
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) −6.00000 −0.543214
\(123\) 0 0
\(124\) 1.00000 0.0898027
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) −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
\(134\) −9.36932 −0.809386
\(135\) 0 0
\(136\) 5.12311 0.439303
\(137\) −3.75379 −0.320708 −0.160354 0.987060i \(-0.551264\pi\)
−0.160354 + 0.987060i \(0.551264\pi\)
\(138\) 0 0
\(139\) −8.87689 −0.752928 −0.376464 0.926431i \(-0.622860\pi\)
−0.376464 + 0.926431i \(0.622860\pi\)
\(140\) 1.56155 0.131975
\(141\) 0 0
\(142\) 4.68466 0.393128
\(143\) 3.12311 0.261167
\(144\) 0 0
\(145\) −1.12311 −0.0932688
\(146\) −9.80776 −0.811696
\(147\) 0 0
\(148\) 5.12311 0.421117
\(149\) 20.0540 1.64289 0.821443 0.570291i \(-0.193170\pi\)
0.821443 + 0.570291i \(0.193170\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −4.68466 −0.379976
\(153\) 0 0
\(154\) 2.43845 0.196496
\(155\) −1.00000 −0.0803219
\(156\) 0 0
\(157\) −20.0540 −1.60048 −0.800241 0.599679i \(-0.795295\pi\)
−0.800241 + 0.599679i \(0.795295\pi\)
\(158\) 16.6847 1.32736
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 8.68466 0.684447
\(162\) 0 0
\(163\) 9.36932 0.733862 0.366931 0.930248i \(-0.380409\pi\)
0.366931 + 0.930248i \(0.380409\pi\)
\(164\) 1.12311 0.0876998
\(165\) 0 0
\(166\) −2.24621 −0.174340
\(167\) 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
\(170\) −5.12311 −0.392924
\(171\) 0 0
\(172\) −7.80776 −0.595336
\(173\) 8.24621 0.626948 0.313474 0.949597i \(-0.398507\pi\)
0.313474 + 0.949597i \(0.398507\pi\)
\(174\) 0 0
\(175\) −1.56155 −0.118042
\(176\) 1.56155 0.117706
\(177\) 0 0
\(178\) −1.31534 −0.0985890
\(179\) 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 \(0.147917\pi\)
\(182\) 3.12311 0.231500
\(183\) 0 0
\(184\) 5.56155 0.410003
\(185\) −5.12311 −0.376658
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 3.12311 0.227776
\(189\) 0 0
\(190\) 4.68466 0.339861
\(191\) −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 \(0.731806\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) −4.56155 −0.325825
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 3.80776 0.269925 0.134963 0.990851i \(-0.456909\pi\)
0.134963 + 0.990851i \(0.456909\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 3.56155 0.250590
\(203\) −1.75379 −0.123092
\(204\) 0 0
\(205\) −1.12311 −0.0784411
\(206\) 18.2462 1.27127
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 7.31534 0.506013
\(210\) 0 0
\(211\) −11.3153 −0.778980 −0.389490 0.921031i \(-0.627349\pi\)
−0.389490 + 0.921031i \(0.627349\pi\)
\(212\) −11.5616 −0.794051
\(213\) 0 0
\(214\) 1.56155 0.106746
\(215\) 7.80776 0.532485
\(216\) 0 0
\(217\) −1.56155 −0.106005
\(218\) 19.3693 1.31186
\(219\) 0 0
\(220\) −1.56155 −0.105280
\(221\) −10.2462 −0.689235
\(222\) 0 0
\(223\) −12.8769 −0.862301 −0.431150 0.902280i \(-0.641892\pi\)
−0.431150 + 0.902280i \(0.641892\pi\)
\(224\) 1.56155 0.104336
\(225\) 0 0
\(226\) 10.6847 0.710733
\(227\) 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 \(0.634813\pi\)
\(230\) −5.56155 −0.366718
\(231\) 0 0
\(232\) −1.12311 −0.0737355
\(233\) −20.0540 −1.31378 −0.656890 0.753987i \(-0.728128\pi\)
−0.656890 + 0.753987i \(0.728128\pi\)
\(234\) 0 0
\(235\) −3.12311 −0.203729
\(236\) 4.87689 0.317459
\(237\) 0 0
\(238\) −8.00000 −0.518563
\(239\) −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 \(0.543669\pi\)
\(242\) 8.56155 0.550357
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) 4.56155 0.291427
\(246\) 0 0
\(247\) 9.36932 0.596155
\(248\) −1.00000 −0.0635001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 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
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −10.6847 −0.666491 −0.333245 0.942840i \(-0.608144\pi\)
−0.333245 + 0.942840i \(0.608144\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) 9.36932 0.578838
\(263\) −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
\(266\) 7.31534 0.448532
\(267\) 0 0
\(268\) 9.36932 0.572322
\(269\) −16.2462 −0.990549 −0.495274 0.868737i \(-0.664933\pi\)
−0.495274 + 0.868737i \(0.664933\pi\)
\(270\) 0 0
\(271\) −10.0540 −0.610736 −0.305368 0.952234i \(-0.598779\pi\)
−0.305368 + 0.952234i \(0.598779\pi\)
\(272\) −5.12311 −0.310634
\(273\) 0 0
\(274\) 3.75379 0.226775
\(275\) 1.56155 0.0941652
\(276\) 0 0
\(277\) 19.3693 1.16379 0.581895 0.813264i \(-0.302312\pi\)
0.581895 + 0.813264i \(0.302312\pi\)
\(278\) 8.87689 0.532401
\(279\) 0 0
\(280\) −1.56155 −0.0933206
\(281\) −13.1231 −0.782859 −0.391429 0.920208i \(-0.628019\pi\)
−0.391429 + 0.920208i \(0.628019\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −4.68466 −0.277983
\(285\) 0 0
\(286\) −3.12311 −0.184673
\(287\) −1.75379 −0.103523
\(288\) 0 0
\(289\) 9.24621 0.543895
\(290\) 1.12311 0.0659510
\(291\) 0 0
\(292\) 9.80776 0.573956
\(293\) 6.49242 0.379291 0.189646 0.981853i \(-0.439266\pi\)
0.189646 + 0.981853i \(0.439266\pi\)
\(294\) 0 0
\(295\) −4.87689 −0.283944
\(296\) −5.12311 −0.297774
\(297\) 0 0
\(298\) −20.0540 −1.16170
\(299\) −11.1231 −0.643266
\(300\) 0 0
\(301\) 12.1922 0.702749
\(302\) 0 0
\(303\) 0 0
\(304\) 4.68466 0.268684
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −2.43845 −0.138943
\(309\) 0 0
\(310\) 1.00000 0.0567962
\(311\) 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 \(0.591202\pi\)
\(314\) 20.0540 1.13171
\(315\) 0 0
\(316\) −16.6847 −0.938585
\(317\) −25.1231 −1.41105 −0.705527 0.708683i \(-0.749290\pi\)
−0.705527 + 0.708683i \(0.749290\pi\)
\(318\) 0 0
\(319\) 1.75379 0.0981933
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −8.68466 −0.483977
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) −9.36932 −0.518918
\(327\) 0 0
\(328\) −1.12311 −0.0620131
\(329\) −4.87689 −0.268872
\(330\) 0 0
\(331\) 10.2462 0.563183 0.281591 0.959534i \(-0.409138\pi\)
0.281591 + 0.959534i \(0.409138\pi\)
\(332\) 2.24621 0.123277
\(333\) 0 0
\(334\) −2.43845 −0.133426
\(335\) −9.36932 −0.511900
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 5.12311 0.277839
\(341\) 1.56155 0.0845628
\(342\) 0 0
\(343\) 18.0540 0.974823
\(344\) 7.80776 0.420966
\(345\) 0 0
\(346\) −8.24621 −0.443319
\(347\) 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 \(0.673475\pi\)
\(350\) 1.56155 0.0834685
\(351\) 0 0
\(352\) −1.56155 −0.0832310
\(353\) −16.2462 −0.864699 −0.432349 0.901706i \(-0.642315\pi\)
−0.432349 + 0.901706i \(0.642315\pi\)
\(354\) 0 0
\(355\) 4.68466 0.248636
\(356\) 1.31534 0.0697130
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 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
\(362\) −24.0540 −1.26425
\(363\) 0 0
\(364\) −3.12311 −0.163695
\(365\) −9.80776 −0.513362
\(366\) 0 0
\(367\) −9.36932 −0.489074 −0.244537 0.969640i \(-0.578636\pi\)
−0.244537 + 0.969640i \(0.578636\pi\)
\(368\) −5.56155 −0.289916
\(369\) 0 0
\(370\) 5.12311 0.266338
\(371\) 18.0540 0.937316
\(372\) 0 0
\(373\) 6.68466 0.346118 0.173059 0.984911i \(-0.444635\pi\)
0.173059 + 0.984911i \(0.444635\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) −3.12311 −0.161062
\(377\) 2.24621 0.115686
\(378\) 0 0
\(379\) 30.0540 1.54377 0.771885 0.635763i \(-0.219314\pi\)
0.771885 + 0.635763i \(0.219314\pi\)
\(380\) −4.68466 −0.240318
\(381\) 0 0
\(382\) 2.24621 0.114926
\(383\) −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
\(386\) 18.4924 0.941240
\(387\) 0 0
\(388\) −6.00000 −0.304604
\(389\) −24.2462 −1.22933 −0.614666 0.788788i \(-0.710709\pi\)
−0.614666 + 0.788788i \(0.710709\pi\)
\(390\) 0 0
\(391\) 28.4924 1.44092
\(392\) 4.56155 0.230393
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 16.6847 0.839496
\(396\) 0 0
\(397\) −28.0540 −1.40799 −0.703994 0.710206i \(-0.748602\pi\)
−0.703994 + 0.710206i \(0.748602\pi\)
\(398\) −3.80776 −0.190866
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 1.31534 0.0656850 0.0328425 0.999461i \(-0.489544\pi\)
0.0328425 + 0.999461i \(0.489544\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) −3.56155 −0.177194
\(405\) 0 0
\(406\) 1.75379 0.0870391
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −12.6307 −0.624547 −0.312274 0.949992i \(-0.601091\pi\)
−0.312274 + 0.949992i \(0.601091\pi\)
\(410\) 1.12311 0.0554662
\(411\) 0 0
\(412\) −18.2462 −0.898926
\(413\) −7.61553 −0.374736
\(414\) 0 0
\(415\) −2.24621 −0.110262
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) −7.31534 −0.357805
\(419\) −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 \(0.351198\pi\)
\(422\) 11.3153 0.550822
\(423\) 0 0
\(424\) 11.5616 0.561479
\(425\) −5.12311 −0.248507
\(426\) 0 0
\(427\) −9.36932 −0.453413
\(428\) −1.56155 −0.0754805
\(429\) 0 0
\(430\) −7.80776 −0.376524
\(431\) 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 \(0.644924\pi\)
\(434\) 1.56155 0.0749569
\(435\) 0 0
\(436\) −19.3693 −0.927622
\(437\) −26.0540 −1.24633
\(438\) 0 0
\(439\) −31.6155 −1.50893 −0.754463 0.656342i \(-0.772103\pi\)
−0.754463 + 0.656342i \(0.772103\pi\)
\(440\) 1.56155 0.0744441
\(441\) 0 0
\(442\) 10.2462 0.487363
\(443\) 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
\(446\) 12.8769 0.609739
\(447\) 0 0
\(448\) −1.56155 −0.0737764
\(449\) 22.4924 1.06148 0.530742 0.847534i \(-0.321913\pi\)
0.530742 + 0.847534i \(0.321913\pi\)
\(450\) 0 0
\(451\) 1.75379 0.0825827
\(452\) −10.6847 −0.502564
\(453\) 0 0
\(454\) −4.68466 −0.219862
\(455\) 3.12311 0.146413
\(456\) 0 0
\(457\) 24.7386 1.15722 0.578612 0.815603i \(-0.303594\pi\)
0.578612 + 0.815603i \(0.303594\pi\)
\(458\) 12.4384 0.581210
\(459\) 0 0
\(460\) 5.56155 0.259309
\(461\) 15.3693 0.715820 0.357910 0.933756i \(-0.383489\pi\)
0.357910 + 0.933756i \(0.383489\pi\)
\(462\) 0 0
\(463\) −18.7386 −0.870858 −0.435429 0.900223i \(-0.643403\pi\)
−0.435429 + 0.900223i \(0.643403\pi\)
\(464\) 1.12311 0.0521389
\(465\) 0 0
\(466\) 20.0540 0.928982
\(467\) 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
\(470\) 3.12311 0.144058
\(471\) 0 0
\(472\) −4.87689 −0.224477
\(473\) −12.1922 −0.560600
\(474\) 0 0
\(475\) 4.68466 0.214947
\(476\) 8.00000 0.366679
\(477\) 0 0
\(478\) 24.0000 1.09773
\(479\) −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
\(482\) 4.24621 0.193410
\(483\) 0 0
\(484\) −8.56155 −0.389161
\(485\) 6.00000 0.272446
\(486\) 0 0
\(487\) 24.0000 1.08754 0.543772 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) −6.00000 −0.271607
\(489\) 0 0
\(490\) −4.56155 −0.206070
\(491\) −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
\(494\) −9.36932 −0.421545
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 7.31534 0.328138
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −16.4924 −0.736093
\(503\) −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
\(506\) 8.68466 0.386080
\(507\) 0 0
\(508\) 0 0
\(509\) 21.6155 0.958091 0.479046 0.877790i \(-0.340983\pi\)
0.479046 + 0.877790i \(0.340983\pi\)
\(510\) 0 0
\(511\) −15.3153 −0.677511
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 10.6847 0.471280
\(515\) 18.2462 0.804024
\(516\) 0 0
\(517\) 4.87689 0.214486
\(518\) 8.00000 0.351500
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) 15.7538 0.690186 0.345093 0.938568i \(-0.387847\pi\)
0.345093 + 0.938568i \(0.387847\pi\)
\(522\) 0 0
\(523\) 39.4233 1.72386 0.861930 0.507027i \(-0.169256\pi\)
0.861930 + 0.507027i \(0.169256\pi\)
\(524\) −9.36932 −0.409301
\(525\) 0 0
\(526\) 12.4924 0.544696
\(527\) −5.12311 −0.223166
\(528\) 0 0
\(529\) 7.93087 0.344820
\(530\) −11.5616 −0.502202
\(531\) 0 0
\(532\) −7.31534 −0.317160
\(533\) 2.24621 0.0972942
\(534\) 0 0
\(535\) 1.56155 0.0675118
\(536\) −9.36932 −0.404693
\(537\) 0 0
\(538\) 16.2462 0.700424
\(539\) −7.12311 −0.306814
\(540\) 0 0
\(541\) 36.2462 1.55835 0.779173 0.626809i \(-0.215639\pi\)
0.779173 + 0.626809i \(0.215639\pi\)
\(542\) 10.0540 0.431855
\(543\) 0 0
\(544\) 5.12311 0.219651
\(545\) 19.3693 0.829690
\(546\) 0 0
\(547\) 35.1231 1.50176 0.750878 0.660441i \(-0.229631\pi\)
0.750878 + 0.660441i \(0.229631\pi\)
\(548\) −3.75379 −0.160354
\(549\) 0 0
\(550\) −1.56155 −0.0665848
\(551\) 5.26137 0.224142
\(552\) 0 0
\(553\) 26.0540 1.10793
\(554\) −19.3693 −0.822923
\(555\) 0 0
\(556\) −8.87689 −0.376464
\(557\) 6.19224 0.262373 0.131187 0.991358i \(-0.458121\pi\)
0.131187 + 0.991358i \(0.458121\pi\)
\(558\) 0 0
\(559\) −15.6155 −0.660466
\(560\) 1.56155 0.0659877
\(561\) 0 0
\(562\) 13.1231 0.553565
\(563\) 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
\(567\) 0 0
\(568\) 4.68466 0.196564
\(569\) −10.1922 −0.427281 −0.213640 0.976912i \(-0.568532\pi\)
−0.213640 + 0.976912i \(0.568532\pi\)
\(570\) 0 0
\(571\) 32.8769 1.37586 0.687928 0.725779i \(-0.258521\pi\)
0.687928 + 0.725779i \(0.258521\pi\)
\(572\) 3.12311 0.130584
\(573\) 0 0
\(574\) 1.75379 0.0732017
\(575\) −5.56155 −0.231933
\(576\) 0 0
\(577\) 6.87689 0.286289 0.143144 0.989702i \(-0.454279\pi\)
0.143144 + 0.989702i \(0.454279\pi\)
\(578\) −9.24621 −0.384592
\(579\) 0 0
\(580\) −1.12311 −0.0466344
\(581\) −3.50758 −0.145519
\(582\) 0 0
\(583\) −18.0540 −0.747719
\(584\) −9.80776 −0.405848
\(585\) 0 0
\(586\) −6.49242 −0.268200
\(587\) −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
\(590\) 4.87689 0.200779
\(591\) 0 0
\(592\) 5.12311 0.210558
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) −8.00000 −0.327968
\(596\) 20.0540 0.821443
\(597\) 0 0
\(598\) 11.1231 0.454858
\(599\) 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 \(0.487012\pi\)
\(602\) −12.1922 −0.496918
\(603\) 0 0
\(604\) 0 0
\(605\) 8.56155 0.348077
\(606\) 0 0
\(607\) −30.0540 −1.21985 −0.609927 0.792458i \(-0.708801\pi\)
−0.609927 + 0.792458i \(0.708801\pi\)
\(608\) −4.68466 −0.189988
\(609\) 0 0
\(610\) 6.00000 0.242933
\(611\) 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
\(615\) 0 0
\(616\) 2.43845 0.0982478
\(617\) 19.5616 0.787518 0.393759 0.919214i \(-0.371174\pi\)
0.393759 + 0.919214i \(0.371174\pi\)
\(618\) 0 0
\(619\) −13.3693 −0.537358 −0.268679 0.963230i \(-0.586587\pi\)
−0.268679 + 0.963230i \(0.586587\pi\)
\(620\) −1.00000 −0.0401610
\(621\) 0 0
\(622\) −18.2462 −0.731606
\(623\) −2.05398 −0.0822908
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) −20.0540 −0.800241
\(629\) −26.2462 −1.04650
\(630\) 0 0
\(631\) −24.3002 −0.967375 −0.483688 0.875241i \(-0.660703\pi\)
−0.483688 + 0.875241i \(0.660703\pi\)
\(632\) 16.6847 0.663680
\(633\) 0 0
\(634\) 25.1231 0.997766
\(635\) 0 0
\(636\) 0 0
\(637\) −9.12311 −0.361471
\(638\) −1.75379 −0.0694332
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 38.4924 1.52036 0.760180 0.649713i \(-0.225111\pi\)
0.760180 + 0.649713i \(0.225111\pi\)
\(642\) 0 0
\(643\) 30.0540 1.18521 0.592607 0.805492i \(-0.298099\pi\)
0.592607 + 0.805492i \(0.298099\pi\)
\(644\) 8.68466 0.342223
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 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
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 9.36932 0.366931
\(653\) −23.7538 −0.929558 −0.464779 0.885427i \(-0.653866\pi\)
−0.464779 + 0.885427i \(0.653866\pi\)
\(654\) 0 0
\(655\) 9.36932 0.366090
\(656\) 1.12311 0.0438499
\(657\) 0 0
\(658\) 4.87689 0.190121
\(659\) −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 \(0.540298\pi\)
\(662\) −10.2462 −0.398230
\(663\) 0 0
\(664\) −2.24621 −0.0871699
\(665\) 7.31534 0.283677
\(666\) 0 0
\(667\) −6.24621 −0.241854
\(668\) 2.43845 0.0943464
\(669\) 0 0
\(670\) 9.36932 0.361968
\(671\) 9.36932 0.361698
\(672\) 0 0
\(673\) 28.2462 1.08881 0.544406 0.838822i \(-0.316755\pi\)
0.544406 + 0.838822i \(0.316755\pi\)
\(674\) 10.0000 0.385186
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 29.4233 1.13083 0.565414 0.824807i \(-0.308716\pi\)
0.565414 + 0.824807i \(0.308716\pi\)
\(678\) 0 0
\(679\) 9.36932 0.359561
\(680\) −5.12311 −0.196462
\(681\) 0 0
\(682\) −1.56155 −0.0597949
\(683\) −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
\(686\) −18.0540 −0.689304
\(687\) 0 0
\(688\) −7.80776 −0.297668
\(689\) −23.1231 −0.880920
\(690\) 0 0
\(691\) −22.0540 −0.838973 −0.419486 0.907762i \(-0.637790\pi\)
−0.419486 + 0.907762i \(0.637790\pi\)
\(692\) 8.24621 0.313474
\(693\) 0 0
\(694\) −2.24621 −0.0852650
\(695\) 8.87689 0.336720
\(696\) 0 0
\(697\) −5.75379 −0.217940
\(698\) 19.3693 0.733139
\(699\) 0 0
\(700\) −1.56155 −0.0590211
\(701\) −25.8078 −0.974746 −0.487373 0.873194i \(-0.662045\pi\)
−0.487373 + 0.873194i \(0.662045\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) 1.56155 0.0588532
\(705\) 0 0
\(706\) 16.2462 0.611434
\(707\) 5.56155 0.209164
\(708\) 0 0
\(709\) −20.0540 −0.753143 −0.376571 0.926388i \(-0.622897\pi\)
−0.376571 + 0.926388i \(0.622897\pi\)
\(710\) −4.68466 −0.175812
\(711\) 0 0
\(712\) −1.31534 −0.0492945
\(713\) −5.56155 −0.208282
\(714\) 0 0
\(715\) −3.12311 −0.116798
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −19.3153 −0.720842
\(719\) 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
\(722\) −2.94602 −0.109640
\(723\) 0 0
\(724\) 24.0540 0.893959
\(725\) 1.12311 0.0417111
\(726\) 0 0
\(727\) −14.0540 −0.521233 −0.260617 0.965442i \(-0.583926\pi\)
−0.260617 + 0.965442i \(0.583926\pi\)
\(728\) 3.12311 0.115750
\(729\) 0 0
\(730\) 9.80776 0.363002
\(731\) 40.0000 1.47945
\(732\) 0 0
\(733\) 26.4924 0.978520 0.489260 0.872138i \(-0.337267\pi\)
0.489260 + 0.872138i \(0.337267\pi\)
\(734\) 9.36932 0.345828
\(735\) 0 0
\(736\) 5.56155 0.205002
\(737\) 14.6307 0.538928
\(738\) 0 0
\(739\) −12.9848 −0.477655 −0.238828 0.971062i \(-0.576763\pi\)
−0.238828 + 0.971062i \(0.576763\pi\)
\(740\) −5.12311 −0.188329
\(741\) 0 0
\(742\) −18.0540 −0.662782
\(743\) −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
\(746\) −6.68466 −0.244743
\(747\) 0 0
\(748\) −8.00000 −0.292509
\(749\) 2.43845 0.0890989
\(750\) 0 0
\(751\) 36.8769 1.34566 0.672828 0.739798i \(-0.265079\pi\)
0.672828 + 0.739798i \(0.265079\pi\)
\(752\) 3.12311 0.113888
\(753\) 0 0
\(754\) −2.24621 −0.0818022
\(755\) 0 0
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) −30.0540 −1.09161
\(759\) 0 0
\(760\) 4.68466 0.169930
\(761\) −41.4233 −1.50159 −0.750797 0.660533i \(-0.770330\pi\)
−0.750797 + 0.660533i \(0.770330\pi\)
\(762\) 0 0
\(763\) 30.2462 1.09499
\(764\) −2.24621 −0.0812651
\(765\) 0 0
\(766\) 6.24621 0.225685
\(767\) 9.75379 0.352189
\(768\) 0 0
\(769\) −16.4384 −0.592786 −0.296393 0.955066i \(-0.595784\pi\)
−0.296393 + 0.955066i \(0.595784\pi\)
\(770\) −2.43845 −0.0878755
\(771\) 0 0
\(772\) −18.4924 −0.665557
\(773\) 15.5616 0.559710 0.279855 0.960042i \(-0.409714\pi\)
0.279855 + 0.960042i \(0.409714\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 24.2462 0.869269
\(779\) 5.26137 0.188508
\(780\) 0 0
\(781\) −7.31534 −0.261764
\(782\) −28.4924 −1.01889
\(783\) 0 0
\(784\) −4.56155 −0.162913
\(785\) 20.0540 0.715757
\(786\) 0 0
\(787\) 10.9309 0.389643 0.194822 0.980839i \(-0.437587\pi\)
0.194822 + 0.980839i \(0.437587\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) −16.6847 −0.593614
\(791\) 16.6847 0.593238
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 28.0540 0.995598
\(795\) 0 0
\(796\) 3.80776 0.134963
\(797\) −38.9848 −1.38091 −0.690457 0.723373i \(-0.742591\pi\)
−0.690457 + 0.723373i \(0.742591\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −1.31534 −0.0464463
\(803\) 15.3153 0.540467
\(804\) 0 0
\(805\) −8.68466 −0.306094
\(806\) −2.00000 −0.0704470
\(807\) 0 0
\(808\) 3.56155 0.125295
\(809\) 25.3153 0.890040 0.445020 0.895521i \(-0.353197\pi\)
0.445020 + 0.895521i \(0.353197\pi\)
\(810\) 0 0
\(811\) 53.6695 1.88459 0.942296 0.334782i \(-0.108663\pi\)
0.942296 + 0.334782i \(0.108663\pi\)
\(812\) −1.75379 −0.0615459
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) −9.36932 −0.328193
\(816\) 0 0
\(817\) −36.5767 −1.27966
\(818\) 12.6307 0.441621
\(819\) 0 0
\(820\) −1.12311 −0.0392205
\(821\) 21.6155 0.754387 0.377194 0.926134i \(-0.376889\pi\)
0.377194 + 0.926134i \(0.376889\pi\)
\(822\) 0 0
\(823\) 15.6155 0.544323 0.272162 0.962252i \(-0.412261\pi\)
0.272162 + 0.962252i \(0.412261\pi\)
\(824\) 18.2462 0.635637
\(825\) 0 0
\(826\) 7.61553 0.264978
\(827\) −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 \(0.725182\pi\)
\(830\) 2.24621 0.0779671
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) 23.3693 0.809699
\(834\) 0 0
\(835\) −2.43845 −0.0843859
\(836\) 7.31534 0.253006
\(837\) 0 0
\(838\) 22.2462 0.768483
\(839\) −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
\(842\) −18.4924 −0.637291
\(843\) 0 0
\(844\) −11.3153 −0.389490
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 13.3693 0.459375
\(848\) −11.5616 −0.397025
\(849\) 0 0
\(850\) 5.12311 0.175721
\(851\) −28.4924 −0.976708
\(852\) 0 0
\(853\) −38.7926 −1.32823 −0.664117 0.747629i \(-0.731192\pi\)
−0.664117 + 0.747629i \(0.731192\pi\)
\(854\) 9.36932 0.320611
\(855\) 0 0
\(856\) 1.56155 0.0533728
\(857\) 53.2311 1.81834 0.909169 0.416427i \(-0.136718\pi\)
0.909169 + 0.416427i \(0.136718\pi\)
\(858\) 0 0
\(859\) −22.7386 −0.775832 −0.387916 0.921695i \(-0.626805\pi\)
−0.387916 + 0.921695i \(0.626805\pi\)
\(860\) 7.80776 0.266243
\(861\) 0 0
\(862\) −13.7538 −0.468456
\(863\) 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
\(866\) 18.3002 0.621866
\(867\) 0 0
\(868\) −1.56155 −0.0530026
\(869\) −26.0540 −0.883821
\(870\) 0 0
\(871\) 18.7386 0.634934
\(872\) 19.3693 0.655928
\(873\) 0 0
\(874\) 26.0540 0.881289
\(875\) 1.56155 0.0527901
\(876\) 0 0
\(877\) −50.0000 −1.68838 −0.844190 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) 31.6155 1.06697
\(879\) 0 0
\(880\) −1.56155 −0.0526399
\(881\) −42.4924 −1.43161 −0.715803 0.698302i \(-0.753939\pi\)
−0.715803 + 0.698302i \(0.753939\pi\)
\(882\) 0 0
\(883\) −31.4233 −1.05748 −0.528739 0.848784i \(-0.677335\pi\)
−0.528739 + 0.848784i \(0.677335\pi\)
\(884\) −10.2462 −0.344617
\(885\) 0 0
\(886\) −14.0540 −0.472153
\(887\) 57.3693 1.92627 0.963137 0.269013i \(-0.0866974\pi\)
0.963137 + 0.269013i \(0.0866974\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1.31534 0.0440903
\(891\) 0 0
\(892\) −12.8769 −0.431150
\(893\) 14.6307 0.489597
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 1.56155 0.0521678
\(897\) 0 0
\(898\) −22.4924 −0.750582
\(899\) 1.12311 0.0374577
\(900\) 0 0
\(901\) 59.2311 1.97327
\(902\) −1.75379 −0.0583948
\(903\) 0 0
\(904\) 10.6847 0.355366
\(905\) −24.0540 −0.799581
\(906\) 0 0
\(907\) 24.0000 0.796907 0.398453 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) 4.68466 0.155466
\(909\) 0 0
\(910\) −3.12311 −0.103530
\(911\) 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
\(914\) −24.7386 −0.818281
\(915\) 0 0
\(916\) −12.4384 −0.410978
\(917\) 14.6307 0.483148
\(918\) 0 0
\(919\) 39.6155 1.30680 0.653398 0.757015i \(-0.273343\pi\)
0.653398 + 0.757015i \(0.273343\pi\)
\(920\) −5.56155 −0.183359
\(921\) 0 0
\(922\) −15.3693 −0.506161
\(923\) −9.36932 −0.308395
\(924\) 0 0
\(925\) 5.12311 0.168447
\(926\) 18.7386 0.615790
\(927\) 0 0
\(928\) −1.12311 −0.0368677
\(929\) −11.5616 −0.379322 −0.189661 0.981850i \(-0.560739\pi\)
−0.189661 + 0.981850i \(0.560739\pi\)
\(930\) 0 0
\(931\) −21.3693 −0.700351
\(932\) −20.0540 −0.656890
\(933\) 0 0
\(934\) −26.2462 −0.858802
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) 16.6307 0.543301 0.271650 0.962396i \(-0.412431\pi\)
0.271650 + 0.962396i \(0.412431\pi\)
\(938\) 14.6307 0.477709
\(939\) 0 0
\(940\) −3.12311 −0.101864
\(941\) −27.3693 −0.892214 −0.446107 0.894980i \(-0.647190\pi\)
−0.446107 + 0.894980i \(0.647190\pi\)
\(942\) 0 0
\(943\) −6.24621 −0.203405
\(944\) 4.87689 0.158729
\(945\) 0 0
\(946\) 12.1922 0.396404
\(947\) 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
\(950\) −4.68466 −0.151990
\(951\) 0 0
\(952\) −8.00000 −0.259281
\(953\) 58.1080 1.88230 0.941151 0.337988i \(-0.109746\pi\)
0.941151 + 0.337988i \(0.109746\pi\)
\(954\) 0 0
\(955\) 2.24621 0.0726857
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 6.43845 0.208017
\(959\) 5.86174 0.189285
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −10.2462 −0.330351
\(963\) 0 0
\(964\) −4.24621 −0.136761
\(965\) 18.4924 0.595292
\(966\) 0 0
\(967\) −25.7538 −0.828186 −0.414093 0.910235i \(-0.635901\pi\)
−0.414093 + 0.910235i \(0.635901\pi\)
\(968\) 8.56155 0.275179
\(969\) 0 0
\(970\) −6.00000 −0.192648
\(971\) 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
\(974\) −24.0000 −0.769010
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 22.9848 0.735350 0.367675 0.929954i \(-0.380154\pi\)
0.367675 + 0.929954i \(0.380154\pi\)
\(978\) 0 0
\(979\) 2.05398 0.0656453
\(980\) 4.56155 0.145713
\(981\) 0 0
\(982\) 2.93087 0.0935278
\(983\) 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
\(986\) 5.75379 0.183238
\(987\) 0 0
\(988\) 9.36932 0.298078
\(989\) 43.4233 1.38078
\(990\) 0 0
\(991\) 23.3153 0.740636 0.370318 0.928905i \(-0.379249\pi\)
0.370318 + 0.928905i \(0.379249\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 0 0
\(994\) −7.31534 −0.232029
\(995\) −3.80776 −0.120714
\(996\) 0 0
\(997\) −62.4924 −1.97915 −0.989577 0.144002i \(-0.954003\pi\)
−0.989577 + 0.144002i \(0.954003\pi\)
\(998\) −20.0000 −0.633089
\(999\) 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 2790.2.a.be.1.1 2
3.2 odd 2 930.2.a.p.1.1 2
12.11 even 2 7440.2.a.bl.1.2 2
15.2 even 4 4650.2.d.bd.3349.3 4
15.8 even 4 4650.2.d.bd.3349.2 4
15.14 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 3.2 odd 2
2790.2.a.be.1.1 2 1.1 even 1 trivial
4650.2.a.ce.1.2 2 15.14 odd 2
4650.2.d.bd.3349.2 4 15.8 even 4
4650.2.d.bd.3349.3 4 15.2 even 4
7440.2.a.bl.1.2 2 12.11 even 2