Properties

Label 2790.2.a.be.1.2
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 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 2790.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.56155 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.56155 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.56155 q^{11} +2.00000 q^{13} -2.56155 q^{14} +1.00000 q^{16} +3.12311 q^{17} -7.68466 q^{19} -1.00000 q^{20} +2.56155 q^{22} -1.43845 q^{23} +1.00000 q^{25} -2.00000 q^{26} +2.56155 q^{28} -7.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} -3.12311 q^{34} -2.56155 q^{35} -3.12311 q^{37} +7.68466 q^{38} +1.00000 q^{40} -7.12311 q^{41} +12.8078 q^{43} -2.56155 q^{44} +1.43845 q^{46} -5.12311 q^{47} -0.438447 q^{49} -1.00000 q^{50} +2.00000 q^{52} -7.43845 q^{53} +2.56155 q^{55} -2.56155 q^{56} +7.12311 q^{58} +13.1231 q^{59} +6.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -15.3693 q^{67} +3.12311 q^{68} +2.56155 q^{70} +7.68466 q^{71} -10.8078 q^{73} +3.12311 q^{74} -7.68466 q^{76} -6.56155 q^{77} -4.31534 q^{79} -1.00000 q^{80} +7.12311 q^{82} -14.2462 q^{83} -3.12311 q^{85} -12.8078 q^{86} +2.56155 q^{88} +13.6847 q^{89} +5.12311 q^{91} -1.43845 q^{92} +5.12311 q^{94} +7.68466 q^{95} -6.00000 q^{97} +0.438447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) 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
\(7\) 2.56155 0.968176 0.484088 0.875019i \(-0.339151\pi\)
0.484088 + 0.875019i \(0.339151\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −2.56155 −0.772337 −0.386169 0.922428i \(-0.626202\pi\)
−0.386169 + 0.922428i \(0.626202\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −2.56155 −0.684604
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.12311 0.757464 0.378732 0.925506i \(-0.376360\pi\)
0.378732 + 0.925506i \(0.376360\pi\)
\(18\) 0 0
\(19\) −7.68466 −1.76298 −0.881491 0.472201i \(-0.843460\pi\)
−0.881491 + 0.472201i \(0.843460\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 2.56155 0.546125
\(23\) −1.43845 −0.299937 −0.149968 0.988691i \(-0.547917\pi\)
−0.149968 + 0.988691i \(0.547917\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 2.56155 0.484088
\(29\) −7.12311 −1.32273 −0.661364 0.750065i \(-0.730022\pi\)
−0.661364 + 0.750065i \(0.730022\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.12311 −0.535608
\(35\) −2.56155 −0.432981
\(36\) 0 0
\(37\) −3.12311 −0.513435 −0.256718 0.966486i \(-0.582641\pi\)
−0.256718 + 0.966486i \(0.582641\pi\)
\(38\) 7.68466 1.24662
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −7.12311 −1.11244 −0.556221 0.831034i \(-0.687749\pi\)
−0.556221 + 0.831034i \(0.687749\pi\)
\(42\) 0 0
\(43\) 12.8078 1.95317 0.976583 0.215142i \(-0.0690213\pi\)
0.976583 + 0.215142i \(0.0690213\pi\)
\(44\) −2.56155 −0.386169
\(45\) 0 0
\(46\) 1.43845 0.212087
\(47\) −5.12311 −0.747282 −0.373641 0.927573i \(-0.621891\pi\)
−0.373641 + 0.927573i \(0.621891\pi\)
\(48\) 0 0
\(49\) −0.438447 −0.0626353
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −7.43845 −1.02175 −0.510875 0.859655i \(-0.670678\pi\)
−0.510875 + 0.859655i \(0.670678\pi\)
\(54\) 0 0
\(55\) 2.56155 0.345400
\(56\) −2.56155 −0.342302
\(57\) 0 0
\(58\) 7.12311 0.935310
\(59\) 13.1231 1.70848 0.854241 0.519877i \(-0.174022\pi\)
0.854241 + 0.519877i \(0.174022\pi\)
\(60\) 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
\(67\) −15.3693 −1.87766 −0.938830 0.344380i \(-0.888089\pi\)
−0.938830 + 0.344380i \(0.888089\pi\)
\(68\) 3.12311 0.378732
\(69\) 0 0
\(70\) 2.56155 0.306164
\(71\) 7.68466 0.912001 0.456001 0.889979i \(-0.349281\pi\)
0.456001 + 0.889979i \(0.349281\pi\)
\(72\) 0 0
\(73\) −10.8078 −1.26495 −0.632477 0.774580i \(-0.717962\pi\)
−0.632477 + 0.774580i \(0.717962\pi\)
\(74\) 3.12311 0.363054
\(75\) 0 0
\(76\) −7.68466 −0.881491
\(77\) −6.56155 −0.747758
\(78\) 0 0
\(79\) −4.31534 −0.485514 −0.242757 0.970087i \(-0.578052\pi\)
−0.242757 + 0.970087i \(0.578052\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 7.12311 0.786615
\(83\) −14.2462 −1.56372 −0.781862 0.623451i \(-0.785730\pi\)
−0.781862 + 0.623451i \(0.785730\pi\)
\(84\) 0 0
\(85\) −3.12311 −0.338748
\(86\) −12.8078 −1.38110
\(87\) 0 0
\(88\) 2.56155 0.273062
\(89\) 13.6847 1.45057 0.725285 0.688448i \(-0.241708\pi\)
0.725285 + 0.688448i \(0.241708\pi\)
\(90\) 0 0
\(91\) 5.12311 0.537047
\(92\) −1.43845 −0.149968
\(93\) 0 0
\(94\) 5.12311 0.528408
\(95\) 7.68466 0.788429
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0.438447 0.0442899
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0.561553 0.0558766 0.0279383 0.999610i \(-0.491106\pi\)
0.0279383 + 0.999610i \(0.491106\pi\)
\(102\) 0 0
\(103\) −1.75379 −0.172806 −0.0864030 0.996260i \(-0.527537\pi\)
−0.0864030 + 0.996260i \(0.527537\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 7.43845 0.722486
\(107\) 2.56155 0.247635 0.123817 0.992305i \(-0.460486\pi\)
0.123817 + 0.992305i \(0.460486\pi\)
\(108\) 0 0
\(109\) 5.36932 0.514287 0.257144 0.966373i \(-0.417219\pi\)
0.257144 + 0.966373i \(0.417219\pi\)
\(110\) −2.56155 −0.244234
\(111\) 0 0
\(112\) 2.56155 0.242044
\(113\) 1.68466 0.158479 0.0792397 0.996856i \(-0.474751\pi\)
0.0792397 + 0.996856i \(0.474751\pi\)
\(114\) 0 0
\(115\) 1.43845 0.134136
\(116\) −7.12311 −0.661364
\(117\) 0 0
\(118\) −13.1231 −1.20808
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) −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
\(131\) 15.3693 1.34282 0.671412 0.741085i \(-0.265688\pi\)
0.671412 + 0.741085i \(0.265688\pi\)
\(132\) 0 0
\(133\) −19.6847 −1.70688
\(134\) 15.3693 1.32771
\(135\) 0 0
\(136\) −3.12311 −0.267804
\(137\) −20.2462 −1.72975 −0.864875 0.501987i \(-0.832603\pi\)
−0.864875 + 0.501987i \(0.832603\pi\)
\(138\) 0 0
\(139\) −17.1231 −1.45236 −0.726181 0.687503i \(-0.758707\pi\)
−0.726181 + 0.687503i \(0.758707\pi\)
\(140\) −2.56155 −0.216491
\(141\) 0 0
\(142\) −7.68466 −0.644882
\(143\) −5.12311 −0.428416
\(144\) 0 0
\(145\) 7.12311 0.591542
\(146\) 10.8078 0.894457
\(147\) 0 0
\(148\) −3.12311 −0.256718
\(149\) −17.0540 −1.39712 −0.698558 0.715553i \(-0.746175\pi\)
−0.698558 + 0.715553i \(0.746175\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 7.68466 0.623308
\(153\) 0 0
\(154\) 6.56155 0.528745
\(155\) −1.00000 −0.0803219
\(156\) 0 0
\(157\) 17.0540 1.36106 0.680528 0.732722i \(-0.261751\pi\)
0.680528 + 0.732722i \(0.261751\pi\)
\(158\) 4.31534 0.343310
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −3.68466 −0.290392
\(162\) 0 0
\(163\) −15.3693 −1.20382 −0.601909 0.798565i \(-0.705593\pi\)
−0.601909 + 0.798565i \(0.705593\pi\)
\(164\) −7.12311 −0.556221
\(165\) 0 0
\(166\) 14.2462 1.10572
\(167\) 6.56155 0.507748 0.253874 0.967237i \(-0.418295\pi\)
0.253874 + 0.967237i \(0.418295\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 3.12311 0.239531
\(171\) 0 0
\(172\) 12.8078 0.976583
\(173\) −8.24621 −0.626948 −0.313474 0.949597i \(-0.601493\pi\)
−0.313474 + 0.949597i \(0.601493\pi\)
\(174\) 0 0
\(175\) 2.56155 0.193635
\(176\) −2.56155 −0.193084
\(177\) 0 0
\(178\) −13.6847 −1.02571
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −13.0540 −0.970294 −0.485147 0.874433i \(-0.661234\pi\)
−0.485147 + 0.874433i \(0.661234\pi\)
\(182\) −5.12311 −0.379750
\(183\) 0 0
\(184\) 1.43845 0.106044
\(185\) 3.12311 0.229615
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) −5.12311 −0.373641
\(189\) 0 0
\(190\) −7.68466 −0.557504
\(191\) 14.2462 1.03082 0.515410 0.856944i \(-0.327640\pi\)
0.515410 + 0.856944i \(0.327640\pi\)
\(192\) 0 0
\(193\) 14.4924 1.04319 0.521594 0.853194i \(-0.325338\pi\)
0.521594 + 0.853194i \(0.325338\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) −0.438447 −0.0313177
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −16.8078 −1.19147 −0.595735 0.803181i \(-0.703139\pi\)
−0.595735 + 0.803181i \(0.703139\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −0.561553 −0.0395107
\(203\) −18.2462 −1.28063
\(204\) 0 0
\(205\) 7.12311 0.497499
\(206\) 1.75379 0.122192
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 19.6847 1.36162
\(210\) 0 0
\(211\) −23.6847 −1.63052 −0.815260 0.579096i \(-0.803406\pi\)
−0.815260 + 0.579096i \(0.803406\pi\)
\(212\) −7.43845 −0.510875
\(213\) 0 0
\(214\) −2.56155 −0.175104
\(215\) −12.8078 −0.873482
\(216\) 0 0
\(217\) 2.56155 0.173890
\(218\) −5.36932 −0.363656
\(219\) 0 0
\(220\) 2.56155 0.172700
\(221\) 6.24621 0.420166
\(222\) 0 0
\(223\) −21.1231 −1.41451 −0.707254 0.706960i \(-0.750066\pi\)
−0.707254 + 0.706960i \(0.750066\pi\)
\(224\) −2.56155 −0.171151
\(225\) 0 0
\(226\) −1.68466 −0.112062
\(227\) −7.68466 −0.510049 −0.255024 0.966935i \(-0.582083\pi\)
−0.255024 + 0.966935i \(0.582083\pi\)
\(228\) 0 0
\(229\) −16.5616 −1.09442 −0.547209 0.836996i \(-0.684310\pi\)
−0.547209 + 0.836996i \(0.684310\pi\)
\(230\) −1.43845 −0.0948484
\(231\) 0 0
\(232\) 7.12311 0.467655
\(233\) 17.0540 1.11724 0.558622 0.829423i \(-0.311330\pi\)
0.558622 + 0.829423i \(0.311330\pi\)
\(234\) 0 0
\(235\) 5.12311 0.334195
\(236\) 13.1231 0.854241
\(237\) 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
\(241\) 12.2462 0.788848 0.394424 0.918929i \(-0.370944\pi\)
0.394424 + 0.918929i \(0.370944\pi\)
\(242\) 4.43845 0.285314
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) 0.438447 0.0280114
\(246\) 0 0
\(247\) −15.3693 −0.977926
\(248\) −1.00000 −0.0635001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −16.4924 −1.04099 −0.520496 0.853864i \(-0.674253\pi\)
−0.520496 + 0.853864i \(0.674253\pi\)
\(252\) 0 0
\(253\) 3.68466 0.231652
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 1.68466 0.105086 0.0525431 0.998619i \(-0.483267\pi\)
0.0525431 + 0.998619i \(0.483267\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) −15.3693 −0.949520
\(263\) 20.4924 1.26362 0.631808 0.775125i \(-0.282313\pi\)
0.631808 + 0.775125i \(0.282313\pi\)
\(264\) 0 0
\(265\) 7.43845 0.456940
\(266\) 19.6847 1.20694
\(267\) 0 0
\(268\) −15.3693 −0.938830
\(269\) 0.246211 0.0150118 0.00750588 0.999972i \(-0.497611\pi\)
0.00750588 + 0.999972i \(0.497611\pi\)
\(270\) 0 0
\(271\) 27.0540 1.64341 0.821706 0.569912i \(-0.193023\pi\)
0.821706 + 0.569912i \(0.193023\pi\)
\(272\) 3.12311 0.189366
\(273\) 0 0
\(274\) 20.2462 1.22312
\(275\) −2.56155 −0.154467
\(276\) 0 0
\(277\) −5.36932 −0.322611 −0.161305 0.986905i \(-0.551570\pi\)
−0.161305 + 0.986905i \(0.551570\pi\)
\(278\) 17.1231 1.02698
\(279\) 0 0
\(280\) 2.56155 0.153082
\(281\) −4.87689 −0.290931 −0.145466 0.989363i \(-0.546468\pi\)
−0.145466 + 0.989363i \(0.546468\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 7.68466 0.456001
\(285\) 0 0
\(286\) 5.12311 0.302936
\(287\) −18.2462 −1.07704
\(288\) 0 0
\(289\) −7.24621 −0.426248
\(290\) −7.12311 −0.418283
\(291\) 0 0
\(292\) −10.8078 −0.632477
\(293\) −26.4924 −1.54770 −0.773852 0.633367i \(-0.781673\pi\)
−0.773852 + 0.633367i \(0.781673\pi\)
\(294\) 0 0
\(295\) −13.1231 −0.764057
\(296\) 3.12311 0.181527
\(297\) 0 0
\(298\) 17.0540 0.987910
\(299\) −2.87689 −0.166375
\(300\) 0 0
\(301\) 32.8078 1.89101
\(302\) 0 0
\(303\) 0 0
\(304\) −7.68466 −0.440745
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −6.56155 −0.373879
\(309\) 0 0
\(310\) 1.00000 0.0567962
\(311\) 1.75379 0.0994482 0.0497241 0.998763i \(-0.484166\pi\)
0.0497241 + 0.998763i \(0.484166\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −17.0540 −0.962412
\(315\) 0 0
\(316\) −4.31534 −0.242757
\(317\) −16.8769 −0.947901 −0.473950 0.880552i \(-0.657172\pi\)
−0.473950 + 0.880552i \(0.657172\pi\)
\(318\) 0 0
\(319\) 18.2462 1.02159
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 3.68466 0.205338
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 15.3693 0.851228
\(327\) 0 0
\(328\) 7.12311 0.393308
\(329\) −13.1231 −0.723500
\(330\) 0 0
\(331\) −6.24621 −0.343323 −0.171661 0.985156i \(-0.554914\pi\)
−0.171661 + 0.985156i \(0.554914\pi\)
\(332\) −14.2462 −0.781862
\(333\) 0 0
\(334\) −6.56155 −0.359032
\(335\) 15.3693 0.839715
\(336\) 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
\(340\) −3.12311 −0.169374
\(341\) −2.56155 −0.138716
\(342\) 0 0
\(343\) −19.0540 −1.02882
\(344\) −12.8078 −0.690548
\(345\) 0 0
\(346\) 8.24621 0.443319
\(347\) −14.2462 −0.764777 −0.382388 0.924002i \(-0.624898\pi\)
−0.382388 + 0.924002i \(0.624898\pi\)
\(348\) 0 0
\(349\) 5.36932 0.287413 0.143706 0.989620i \(-0.454098\pi\)
0.143706 + 0.989620i \(0.454098\pi\)
\(350\) −2.56155 −0.136921
\(351\) 0 0
\(352\) 2.56155 0.136531
\(353\) 0.246211 0.0131045 0.00655225 0.999979i \(-0.497914\pi\)
0.00655225 + 0.999979i \(0.497914\pi\)
\(354\) 0 0
\(355\) −7.68466 −0.407859
\(356\) 13.6847 0.725285
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 31.6847 1.67225 0.836126 0.548537i \(-0.184815\pi\)
0.836126 + 0.548537i \(0.184815\pi\)
\(360\) 0 0
\(361\) 40.0540 2.10810
\(362\) 13.0540 0.686102
\(363\) 0 0
\(364\) 5.12311 0.268524
\(365\) 10.8078 0.565704
\(366\) 0 0
\(367\) 15.3693 0.802272 0.401136 0.916019i \(-0.368616\pi\)
0.401136 + 0.916019i \(0.368616\pi\)
\(368\) −1.43845 −0.0749842
\(369\) 0 0
\(370\) −3.12311 −0.162363
\(371\) −19.0540 −0.989233
\(372\) 0 0
\(373\) −5.68466 −0.294340 −0.147170 0.989111i \(-0.547017\pi\)
−0.147170 + 0.989111i \(0.547017\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 5.12311 0.264204
\(377\) −14.2462 −0.733717
\(378\) 0 0
\(379\) −7.05398 −0.362338 −0.181169 0.983452i \(-0.557988\pi\)
−0.181169 + 0.983452i \(0.557988\pi\)
\(380\) 7.68466 0.394215
\(381\) 0 0
\(382\) −14.2462 −0.728900
\(383\) 10.2462 0.523557 0.261778 0.965128i \(-0.415691\pi\)
0.261778 + 0.965128i \(0.415691\pi\)
\(384\) 0 0
\(385\) 6.56155 0.334408
\(386\) −14.4924 −0.737645
\(387\) 0 0
\(388\) −6.00000 −0.304604
\(389\) −7.75379 −0.393133 −0.196566 0.980491i \(-0.562979\pi\)
−0.196566 + 0.980491i \(0.562979\pi\)
\(390\) 0 0
\(391\) −4.49242 −0.227192
\(392\) 0.438447 0.0221449
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 4.31534 0.217128
\(396\) 0 0
\(397\) 9.05398 0.454406 0.227203 0.973847i \(-0.427042\pi\)
0.227203 + 0.973847i \(0.427042\pi\)
\(398\) 16.8078 0.842497
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 13.6847 0.683379 0.341690 0.939813i \(-0.389001\pi\)
0.341690 + 0.939813i \(0.389001\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) 0.561553 0.0279383
\(405\) 0 0
\(406\) 18.2462 0.905544
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −37.3693 −1.84779 −0.923897 0.382642i \(-0.875014\pi\)
−0.923897 + 0.382642i \(0.875014\pi\)
\(410\) −7.12311 −0.351785
\(411\) 0 0
\(412\) −1.75379 −0.0864030
\(413\) 33.6155 1.65411
\(414\) 0 0
\(415\) 14.2462 0.699319
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) −19.6847 −0.962808
\(419\) −5.75379 −0.281091 −0.140545 0.990074i \(-0.544886\pi\)
−0.140545 + 0.990074i \(0.544886\pi\)
\(420\) 0 0
\(421\) −14.4924 −0.706317 −0.353159 0.935563i \(-0.614892\pi\)
−0.353159 + 0.935563i \(0.614892\pi\)
\(422\) 23.6847 1.15295
\(423\) 0 0
\(424\) 7.43845 0.361243
\(425\) 3.12311 0.151493
\(426\) 0 0
\(427\) 15.3693 0.743773
\(428\) 2.56155 0.123817
\(429\) 0 0
\(430\) 12.8078 0.617645
\(431\) 30.2462 1.45691 0.728454 0.685094i \(-0.240239\pi\)
0.728454 + 0.685094i \(0.240239\pi\)
\(432\) 0 0
\(433\) 35.3002 1.69642 0.848209 0.529661i \(-0.177681\pi\)
0.848209 + 0.529661i \(0.177681\pi\)
\(434\) −2.56155 −0.122958
\(435\) 0 0
\(436\) 5.36932 0.257144
\(437\) 11.0540 0.528783
\(438\) 0 0
\(439\) 9.61553 0.458924 0.229462 0.973318i \(-0.426303\pi\)
0.229462 + 0.973318i \(0.426303\pi\)
\(440\) −2.56155 −0.122117
\(441\) 0 0
\(442\) −6.24621 −0.297102
\(443\) −23.0540 −1.09533 −0.547664 0.836699i \(-0.684483\pi\)
−0.547664 + 0.836699i \(0.684483\pi\)
\(444\) 0 0
\(445\) −13.6847 −0.648715
\(446\) 21.1231 1.00021
\(447\) 0 0
\(448\) 2.56155 0.121022
\(449\) −10.4924 −0.495168 −0.247584 0.968866i \(-0.579637\pi\)
−0.247584 + 0.968866i \(0.579637\pi\)
\(450\) 0 0
\(451\) 18.2462 0.859181
\(452\) 1.68466 0.0792397
\(453\) 0 0
\(454\) 7.68466 0.360659
\(455\) −5.12311 −0.240175
\(456\) 0 0
\(457\) −24.7386 −1.15722 −0.578612 0.815603i \(-0.696406\pi\)
−0.578612 + 0.815603i \(0.696406\pi\)
\(458\) 16.5616 0.773871
\(459\) 0 0
\(460\) 1.43845 0.0670679
\(461\) −9.36932 −0.436373 −0.218186 0.975907i \(-0.570014\pi\)
−0.218186 + 0.975907i \(0.570014\pi\)
\(462\) 0 0
\(463\) 30.7386 1.42855 0.714273 0.699867i \(-0.246758\pi\)
0.714273 + 0.699867i \(0.246758\pi\)
\(464\) −7.12311 −0.330682
\(465\) 0 0
\(466\) −17.0540 −0.790010
\(467\) 9.75379 0.451352 0.225676 0.974202i \(-0.427541\pi\)
0.225676 + 0.974202i \(0.427541\pi\)
\(468\) 0 0
\(469\) −39.3693 −1.81791
\(470\) −5.12311 −0.236311
\(471\) 0 0
\(472\) −13.1231 −0.604040
\(473\) −32.8078 −1.50850
\(474\) 0 0
\(475\) −7.68466 −0.352596
\(476\) 8.00000 0.366679
\(477\) 0 0
\(478\) 24.0000 1.09773
\(479\) −10.5616 −0.482570 −0.241285 0.970454i \(-0.577569\pi\)
−0.241285 + 0.970454i \(0.577569\pi\)
\(480\) 0 0
\(481\) −6.24621 −0.284803
\(482\) −12.2462 −0.557800
\(483\) 0 0
\(484\) −4.43845 −0.201748
\(485\) 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
\(490\) −0.438447 −0.0198070
\(491\) 25.9309 1.17024 0.585122 0.810945i \(-0.301047\pi\)
0.585122 + 0.810945i \(0.301047\pi\)
\(492\) 0 0
\(493\) −22.2462 −1.00192
\(494\) 15.3693 0.691498
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 19.6847 0.882978
\(498\) 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
\(503\) −26.2462 −1.17026 −0.585130 0.810939i \(-0.698957\pi\)
−0.585130 + 0.810939i \(0.698957\pi\)
\(504\) 0 0
\(505\) −0.561553 −0.0249888
\(506\) −3.68466 −0.163803
\(507\) 0 0
\(508\) 0 0
\(509\) −19.6155 −0.869443 −0.434721 0.900565i \(-0.643153\pi\)
−0.434721 + 0.900565i \(0.643153\pi\)
\(510\) 0 0
\(511\) −27.6847 −1.22470
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −1.68466 −0.0743071
\(515\) 1.75379 0.0772812
\(516\) 0 0
\(517\) 13.1231 0.577154
\(518\) 8.00000 0.351500
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) 32.2462 1.41273 0.706366 0.707847i \(-0.250333\pi\)
0.706366 + 0.707847i \(0.250333\pi\)
\(522\) 0 0
\(523\) −22.4233 −0.980502 −0.490251 0.871581i \(-0.663095\pi\)
−0.490251 + 0.871581i \(0.663095\pi\)
\(524\) 15.3693 0.671412
\(525\) 0 0
\(526\) −20.4924 −0.893512
\(527\) 3.12311 0.136045
\(528\) 0 0
\(529\) −20.9309 −0.910038
\(530\) −7.43845 −0.323105
\(531\) 0 0
\(532\) −19.6847 −0.853438
\(533\) −14.2462 −0.617072
\(534\) 0 0
\(535\) −2.56155 −0.110746
\(536\) 15.3693 0.663853
\(537\) 0 0
\(538\) −0.246211 −0.0106149
\(539\) 1.12311 0.0483756
\(540\) 0 0
\(541\) 19.7538 0.849282 0.424641 0.905362i \(-0.360400\pi\)
0.424641 + 0.905362i \(0.360400\pi\)
\(542\) −27.0540 −1.16207
\(543\) 0 0
\(544\) −3.12311 −0.133902
\(545\) −5.36932 −0.229996
\(546\) 0 0
\(547\) 26.8769 1.14917 0.574587 0.818444i \(-0.305163\pi\)
0.574587 + 0.818444i \(0.305163\pi\)
\(548\) −20.2462 −0.864875
\(549\) 0 0
\(550\) 2.56155 0.109225
\(551\) 54.7386 2.33194
\(552\) 0 0
\(553\) −11.0540 −0.470063
\(554\) 5.36932 0.228120
\(555\) 0 0
\(556\) −17.1231 −0.726181
\(557\) 26.8078 1.13588 0.567941 0.823069i \(-0.307740\pi\)
0.567941 + 0.823069i \(0.307740\pi\)
\(558\) 0 0
\(559\) 25.6155 1.08342
\(560\) −2.56155 −0.108245
\(561\) 0 0
\(562\) 4.87689 0.205719
\(563\) −16.4924 −0.695073 −0.347536 0.937667i \(-0.612982\pi\)
−0.347536 + 0.937667i \(0.612982\pi\)
\(564\) 0 0
\(565\) −1.68466 −0.0708741
\(566\) 0 0
\(567\) 0 0
\(568\) −7.68466 −0.322441
\(569\) −30.8078 −1.29153 −0.645764 0.763537i \(-0.723461\pi\)
−0.645764 + 0.763537i \(0.723461\pi\)
\(570\) 0 0
\(571\) 41.1231 1.72095 0.860474 0.509494i \(-0.170167\pi\)
0.860474 + 0.509494i \(0.170167\pi\)
\(572\) −5.12311 −0.214208
\(573\) 0 0
\(574\) 18.2462 0.761582
\(575\) −1.43845 −0.0599874
\(576\) 0 0
\(577\) 15.1231 0.629583 0.314792 0.949161i \(-0.398065\pi\)
0.314792 + 0.949161i \(0.398065\pi\)
\(578\) 7.24621 0.301403
\(579\) 0 0
\(580\) 7.12311 0.295771
\(581\) −36.4924 −1.51396
\(582\) 0 0
\(583\) 19.0540 0.789135
\(584\) 10.8078 0.447228
\(585\) 0 0
\(586\) 26.4924 1.09439
\(587\) 0.492423 0.0203245 0.0101622 0.999948i \(-0.496765\pi\)
0.0101622 + 0.999948i \(0.496765\pi\)
\(588\) 0 0
\(589\) −7.68466 −0.316641
\(590\) 13.1231 0.540270
\(591\) 0 0
\(592\) −3.12311 −0.128359
\(593\) 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
\(596\) −17.0540 −0.698558
\(597\) 0 0
\(598\) 2.87689 0.117645
\(599\) −38.4233 −1.56993 −0.784967 0.619538i \(-0.787320\pi\)
−0.784967 + 0.619538i \(0.787320\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) −32.8078 −1.33714
\(603\) 0 0
\(604\) 0 0
\(605\) 4.43845 0.180449
\(606\) 0 0
\(607\) 7.05398 0.286312 0.143156 0.989700i \(-0.454275\pi\)
0.143156 + 0.989700i \(0.454275\pi\)
\(608\) 7.68466 0.311654
\(609\) 0 0
\(610\) 6.00000 0.242933
\(611\) −10.2462 −0.414517
\(612\) 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
\(616\) 6.56155 0.264372
\(617\) 15.4384 0.621528 0.310764 0.950487i \(-0.399415\pi\)
0.310764 + 0.950487i \(0.399415\pi\)
\(618\) 0 0
\(619\) 11.3693 0.456971 0.228486 0.973547i \(-0.426623\pi\)
0.228486 + 0.973547i \(0.426623\pi\)
\(620\) −1.00000 −0.0401610
\(621\) 0 0
\(622\) −1.75379 −0.0703205
\(623\) 35.0540 1.40441
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 17.0540 0.680528
\(629\) −9.75379 −0.388909
\(630\) 0 0
\(631\) 29.3002 1.16642 0.583211 0.812321i \(-0.301796\pi\)
0.583211 + 0.812321i \(0.301796\pi\)
\(632\) 4.31534 0.171655
\(633\) 0 0
\(634\) 16.8769 0.670267
\(635\) 0 0
\(636\) 0 0
\(637\) −0.876894 −0.0347438
\(638\) −18.2462 −0.722374
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 5.50758 0.217536 0.108768 0.994067i \(-0.465309\pi\)
0.108768 + 0.994067i \(0.465309\pi\)
\(642\) 0 0
\(643\) −7.05398 −0.278182 −0.139091 0.990280i \(-0.544418\pi\)
−0.139091 + 0.990280i \(0.544418\pi\)
\(644\) −3.68466 −0.145196
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 45.9309 1.80573 0.902864 0.429925i \(-0.141460\pi\)
0.902864 + 0.429925i \(0.141460\pi\)
\(648\) 0 0
\(649\) −33.6155 −1.31952
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −15.3693 −0.601909
\(653\) −40.2462 −1.57496 −0.787478 0.616343i \(-0.788614\pi\)
−0.787478 + 0.616343i \(0.788614\pi\)
\(654\) 0 0
\(655\) −15.3693 −0.600529
\(656\) −7.12311 −0.278111
\(657\) 0 0
\(658\) 13.1231 0.511592
\(659\) 30.7386 1.19741 0.598704 0.800971i \(-0.295683\pi\)
0.598704 + 0.800971i \(0.295683\pi\)
\(660\) 0 0
\(661\) 26.4924 1.03044 0.515218 0.857059i \(-0.327711\pi\)
0.515218 + 0.857059i \(0.327711\pi\)
\(662\) 6.24621 0.242766
\(663\) 0 0
\(664\) 14.2462 0.552860
\(665\) 19.6847 0.763338
\(666\) 0 0
\(667\) 10.2462 0.396735
\(668\) 6.56155 0.253874
\(669\) 0 0
\(670\) −15.3693 −0.593769
\(671\) −15.3693 −0.593326
\(672\) 0 0
\(673\) 11.7538 0.453075 0.226538 0.974002i \(-0.427259\pi\)
0.226538 + 0.974002i \(0.427259\pi\)
\(674\) 10.0000 0.385186
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −32.4233 −1.24613 −0.623064 0.782171i \(-0.714112\pi\)
−0.623064 + 0.782171i \(0.714112\pi\)
\(678\) 0 0
\(679\) −15.3693 −0.589820
\(680\) 3.12311 0.119766
\(681\) 0 0
\(682\) 2.56155 0.0980869
\(683\) −31.6847 −1.21238 −0.606190 0.795320i \(-0.707303\pi\)
−0.606190 + 0.795320i \(0.707303\pi\)
\(684\) 0 0
\(685\) 20.2462 0.773568
\(686\) 19.0540 0.727484
\(687\) 0 0
\(688\) 12.8078 0.488291
\(689\) −14.8769 −0.566765
\(690\) 0 0
\(691\) 15.0540 0.572680 0.286340 0.958128i \(-0.407561\pi\)
0.286340 + 0.958128i \(0.407561\pi\)
\(692\) −8.24621 −0.313474
\(693\) 0 0
\(694\) 14.2462 0.540779
\(695\) 17.1231 0.649516
\(696\) 0 0
\(697\) −22.2462 −0.842635
\(698\) −5.36932 −0.203232
\(699\) 0 0
\(700\) 2.56155 0.0968176
\(701\) −5.19224 −0.196108 −0.0980540 0.995181i \(-0.531262\pi\)
−0.0980540 + 0.995181i \(0.531262\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) −2.56155 −0.0965422
\(705\) 0 0
\(706\) −0.246211 −0.00926628
\(707\) 1.43845 0.0540984
\(708\) 0 0
\(709\) 17.0540 0.640475 0.320238 0.947337i \(-0.396237\pi\)
0.320238 + 0.947337i \(0.396237\pi\)
\(710\) 7.68466 0.288400
\(711\) 0 0
\(712\) −13.6847 −0.512854
\(713\) −1.43845 −0.0538703
\(714\) 0 0
\(715\) 5.12311 0.191593
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −31.6847 −1.18246
\(719\) 34.8769 1.30069 0.650344 0.759640i \(-0.274625\pi\)
0.650344 + 0.759640i \(0.274625\pi\)
\(720\) 0 0
\(721\) −4.49242 −0.167307
\(722\) −40.0540 −1.49065
\(723\) 0 0
\(724\) −13.0540 −0.485147
\(725\) −7.12311 −0.264546
\(726\) 0 0
\(727\) 23.0540 0.855025 0.427512 0.904010i \(-0.359390\pi\)
0.427512 + 0.904010i \(0.359390\pi\)
\(728\) −5.12311 −0.189875
\(729\) 0 0
\(730\) −10.8078 −0.400013
\(731\) 40.0000 1.47945
\(732\) 0 0
\(733\) −6.49242 −0.239803 −0.119902 0.992786i \(-0.538258\pi\)
−0.119902 + 0.992786i \(0.538258\pi\)
\(734\) −15.3693 −0.567292
\(735\) 0 0
\(736\) 1.43845 0.0530219
\(737\) 39.3693 1.45019
\(738\) 0 0
\(739\) 52.9848 1.94908 0.974540 0.224216i \(-0.0719821\pi\)
0.974540 + 0.224216i \(0.0719821\pi\)
\(740\) 3.12311 0.114808
\(741\) 0 0
\(742\) 19.0540 0.699493
\(743\) 50.4233 1.84985 0.924926 0.380148i \(-0.124127\pi\)
0.924926 + 0.380148i \(0.124127\pi\)
\(744\) 0 0
\(745\) 17.0540 0.624809
\(746\) 5.68466 0.208130
\(747\) 0 0
\(748\) −8.00000 −0.292509
\(749\) 6.56155 0.239754
\(750\) 0 0
\(751\) 45.1231 1.64657 0.823283 0.567631i \(-0.192140\pi\)
0.823283 + 0.567631i \(0.192140\pi\)
\(752\) −5.12311 −0.186820
\(753\) 0 0
\(754\) 14.2462 0.518816
\(755\) 0 0
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 7.05398 0.256212
\(759\) 0 0
\(760\) −7.68466 −0.278752
\(761\) 20.4233 0.740344 0.370172 0.928963i \(-0.379299\pi\)
0.370172 + 0.928963i \(0.379299\pi\)
\(762\) 0 0
\(763\) 13.7538 0.497921
\(764\) 14.2462 0.515410
\(765\) 0 0
\(766\) −10.2462 −0.370211
\(767\) 26.2462 0.947696
\(768\) 0 0
\(769\) −20.5616 −0.741469 −0.370734 0.928739i \(-0.620894\pi\)
−0.370734 + 0.928739i \(0.620894\pi\)
\(770\) −6.56155 −0.236462
\(771\) 0 0
\(772\) 14.4924 0.521594
\(773\) 11.4384 0.411412 0.205706 0.978614i \(-0.434051\pi\)
0.205706 + 0.978614i \(0.434051\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 7.75379 0.277987
\(779\) 54.7386 1.96122
\(780\) 0 0
\(781\) −19.6847 −0.704372
\(782\) 4.49242 0.160649
\(783\) 0 0
\(784\) −0.438447 −0.0156588
\(785\) −17.0540 −0.608682
\(786\) 0 0
\(787\) −17.9309 −0.639166 −0.319583 0.947558i \(-0.603543\pi\)
−0.319583 + 0.947558i \(0.603543\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) −4.31534 −0.153533
\(791\) 4.31534 0.153436
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) −9.05398 −0.321314
\(795\) 0 0
\(796\) −16.8078 −0.595735
\(797\) 26.9848 0.955852 0.477926 0.878400i \(-0.341389\pi\)
0.477926 + 0.878400i \(0.341389\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −13.6847 −0.483222
\(803\) 27.6847 0.976970
\(804\) 0 0
\(805\) 3.68466 0.129867
\(806\) −2.00000 −0.0704470
\(807\) 0 0
\(808\) −0.561553 −0.0197554
\(809\) 37.6847 1.32492 0.662461 0.749096i \(-0.269512\pi\)
0.662461 + 0.749096i \(0.269512\pi\)
\(810\) 0 0
\(811\) −24.6695 −0.866263 −0.433132 0.901331i \(-0.642592\pi\)
−0.433132 + 0.901331i \(0.642592\pi\)
\(812\) −18.2462 −0.640316
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) 15.3693 0.538364
\(816\) 0 0
\(817\) −98.4233 −3.44340
\(818\) 37.3693 1.30659
\(819\) 0 0
\(820\) 7.12311 0.248750
\(821\) −19.6155 −0.684587 −0.342293 0.939593i \(-0.611204\pi\)
−0.342293 + 0.939593i \(0.611204\pi\)
\(822\) 0 0
\(823\) −25.6155 −0.892901 −0.446451 0.894808i \(-0.647312\pi\)
−0.446451 + 0.894808i \(0.647312\pi\)
\(824\) 1.75379 0.0610961
\(825\) 0 0
\(826\) −33.6155 −1.16963
\(827\) −1.75379 −0.0609852 −0.0304926 0.999535i \(-0.509708\pi\)
−0.0304926 + 0.999535i \(0.509708\pi\)
\(828\) 0 0
\(829\) 24.4233 0.848256 0.424128 0.905602i \(-0.360581\pi\)
0.424128 + 0.905602i \(0.360581\pi\)
\(830\) −14.2462 −0.494493
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) −1.36932 −0.0474440
\(834\) 0 0
\(835\) −6.56155 −0.227072
\(836\) 19.6847 0.680808
\(837\) 0 0
\(838\) 5.75379 0.198761
\(839\) −6.06913 −0.209530 −0.104765 0.994497i \(-0.533409\pi\)
−0.104765 + 0.994497i \(0.533409\pi\)
\(840\) 0 0
\(841\) 21.7386 0.749608
\(842\) 14.4924 0.499442
\(843\) 0 0
\(844\) −23.6847 −0.815260
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −11.3693 −0.390654
\(848\) −7.43845 −0.255437
\(849\) 0 0
\(850\) −3.12311 −0.107122
\(851\) 4.49242 0.153998
\(852\) 0 0
\(853\) 47.7926 1.63639 0.818194 0.574942i \(-0.194976\pi\)
0.818194 + 0.574942i \(0.194976\pi\)
\(854\) −15.3693 −0.525927
\(855\) 0 0
\(856\) −2.56155 −0.0875521
\(857\) −29.2311 −0.998514 −0.499257 0.866454i \(-0.666394\pi\)
−0.499257 + 0.866454i \(0.666394\pi\)
\(858\) 0 0
\(859\) 26.7386 0.912310 0.456155 0.889900i \(-0.349226\pi\)
0.456155 + 0.889900i \(0.349226\pi\)
\(860\) −12.8078 −0.436741
\(861\) 0 0
\(862\) −30.2462 −1.03019
\(863\) −39.5464 −1.34618 −0.673088 0.739563i \(-0.735032\pi\)
−0.673088 + 0.739563i \(0.735032\pi\)
\(864\) 0 0
\(865\) 8.24621 0.280380
\(866\) −35.3002 −1.19955
\(867\) 0 0
\(868\) 2.56155 0.0869448
\(869\) 11.0540 0.374980
\(870\) 0 0
\(871\) −30.7386 −1.04154
\(872\) −5.36932 −0.181828
\(873\) 0 0
\(874\) −11.0540 −0.373906
\(875\) −2.56155 −0.0865963
\(876\) 0 0
\(877\) −50.0000 −1.68838 −0.844190 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) −9.61553 −0.324508
\(879\) 0 0
\(880\) 2.56155 0.0863499
\(881\) −9.50758 −0.320318 −0.160159 0.987091i \(-0.551201\pi\)
−0.160159 + 0.987091i \(0.551201\pi\)
\(882\) 0 0
\(883\) 30.4233 1.02383 0.511913 0.859038i \(-0.328937\pi\)
0.511913 + 0.859038i \(0.328937\pi\)
\(884\) 6.24621 0.210083
\(885\) 0 0
\(886\) 23.0540 0.774513
\(887\) 32.6307 1.09563 0.547816 0.836599i \(-0.315460\pi\)
0.547816 + 0.836599i \(0.315460\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 13.6847 0.458711
\(891\) 0 0
\(892\) −21.1231 −0.707254
\(893\) 39.3693 1.31744
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) −2.56155 −0.0855755
\(897\) 0 0
\(898\) 10.4924 0.350137
\(899\) −7.12311 −0.237569
\(900\) 0 0
\(901\) −23.2311 −0.773939
\(902\) −18.2462 −0.607532
\(903\) 0 0
\(904\) −1.68466 −0.0560309
\(905\) 13.0540 0.433929
\(906\) 0 0
\(907\) 24.0000 0.796907 0.398453 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) −7.68466 −0.255024
\(909\) 0 0
\(910\) 5.12311 0.169829
\(911\) 11.5076 0.381263 0.190632 0.981662i \(-0.438946\pi\)
0.190632 + 0.981662i \(0.438946\pi\)
\(912\) 0 0
\(913\) 36.4924 1.20772
\(914\) 24.7386 0.818281
\(915\) 0 0
\(916\) −16.5616 −0.547209
\(917\) 39.3693 1.30009
\(918\) 0 0
\(919\) −1.61553 −0.0532914 −0.0266457 0.999645i \(-0.508483\pi\)
−0.0266457 + 0.999645i \(0.508483\pi\)
\(920\) −1.43845 −0.0474242
\(921\) 0 0
\(922\) 9.36932 0.308562
\(923\) 15.3693 0.505887
\(924\) 0 0
\(925\) −3.12311 −0.102687
\(926\) −30.7386 −1.01013
\(927\) 0 0
\(928\) 7.12311 0.233827
\(929\) −7.43845 −0.244048 −0.122024 0.992527i \(-0.538938\pi\)
−0.122024 + 0.992527i \(0.538938\pi\)
\(930\) 0 0
\(931\) 3.36932 0.110425
\(932\) 17.0540 0.558622
\(933\) 0 0
\(934\) −9.75379 −0.319154
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) 41.3693 1.35148 0.675738 0.737142i \(-0.263825\pi\)
0.675738 + 0.737142i \(0.263825\pi\)
\(938\) 39.3693 1.28545
\(939\) 0 0
\(940\) 5.12311 0.167097
\(941\) −2.63068 −0.0857578 −0.0428789 0.999080i \(-0.513653\pi\)
−0.0428789 + 0.999080i \(0.513653\pi\)
\(942\) 0 0
\(943\) 10.2462 0.333663
\(944\) 13.1231 0.427121
\(945\) 0 0
\(946\) 32.8078 1.06667
\(947\) 9.12311 0.296461 0.148231 0.988953i \(-0.452642\pi\)
0.148231 + 0.988953i \(0.452642\pi\)
\(948\) 0 0
\(949\) −21.6155 −0.701670
\(950\) 7.68466 0.249323
\(951\) 0 0
\(952\) −8.00000 −0.259281
\(953\) −16.1080 −0.521788 −0.260894 0.965367i \(-0.584017\pi\)
−0.260894 + 0.965367i \(0.584017\pi\)
\(954\) 0 0
\(955\) −14.2462 −0.460997
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 10.5616 0.341228
\(959\) −51.8617 −1.67470
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 6.24621 0.201386
\(963\) 0 0
\(964\) 12.2462 0.394424
\(965\) −14.4924 −0.466528
\(966\) 0 0
\(967\) −42.2462 −1.35855 −0.679273 0.733885i \(-0.737705\pi\)
−0.679273 + 0.733885i \(0.737705\pi\)
\(968\) 4.43845 0.142657
\(969\) 0 0
\(970\) −6.00000 −0.192648
\(971\) 37.1231 1.19134 0.595669 0.803230i \(-0.296887\pi\)
0.595669 + 0.803230i \(0.296887\pi\)
\(972\) 0 0
\(973\) −43.8617 −1.40614
\(974\) −24.0000 −0.769010
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) −42.9848 −1.37521 −0.687604 0.726086i \(-0.741337\pi\)
−0.687604 + 0.726086i \(0.741337\pi\)
\(978\) 0 0
\(979\) −35.0540 −1.12033
\(980\) 0.438447 0.0140057
\(981\) 0 0
\(982\) −25.9309 −0.827487
\(983\) −40.9848 −1.30721 −0.653607 0.756834i \(-0.726745\pi\)
−0.653607 + 0.756834i \(0.726745\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 22.2462 0.708464
\(987\) 0 0
\(988\) −15.3693 −0.488963
\(989\) −18.4233 −0.585827
\(990\) 0 0
\(991\) 35.6847 1.13356 0.566780 0.823869i \(-0.308189\pi\)
0.566780 + 0.823869i \(0.308189\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 0 0
\(994\) −19.6847 −0.624359
\(995\) 16.8078 0.532842
\(996\) 0 0
\(997\) −29.5076 −0.934514 −0.467257 0.884121i \(-0.654758\pi\)
−0.467257 + 0.884121i \(0.654758\pi\)
\(998\) −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.1.2 2
3.2 odd 2 930.2.a.p.1.2 2
12.11 even 2 7440.2.a.bl.1.1 2
15.2 even 4 4650.2.d.bd.3349.4 4
15.8 even 4 4650.2.d.bd.3349.1 4
15.14 odd 2 4650.2.a.ce.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.p.1.2 2 3.2 odd 2
2790.2.a.be.1.2 2 1.1 even 1 trivial
4650.2.a.ce.1.1 2 15.14 odd 2
4650.2.d.bd.3349.1 4 15.8 even 4
4650.2.d.bd.3349.4 4 15.2 even 4
7440.2.a.bl.1.1 2 12.11 even 2