Properties

Label 1805.2.a.k.1.1
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4129975648\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.17557\) of defining polynomial
Character \(\chi\) \(=\) 1805.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.07768 q^{3} -2.00000 q^{4} -1.00000 q^{5} -0.618034 q^{7} +6.47214 q^{9} +O(q^{10})\) \(q-3.07768 q^{3} -2.00000 q^{4} -1.00000 q^{5} -0.618034 q^{7} +6.47214 q^{9} -5.85410 q^{11} +6.15537 q^{12} +3.07768 q^{13} +3.07768 q^{15} +4.00000 q^{16} -2.85410 q^{17} +2.00000 q^{20} +1.90211 q^{21} +5.47214 q^{23} +1.00000 q^{25} -10.6861 q^{27} +1.23607 q^{28} +3.80423 q^{29} +1.62460 q^{31} +18.0171 q^{33} +0.618034 q^{35} -12.9443 q^{36} +8.33499 q^{37} -9.47214 q^{39} +11.5842 q^{41} -7.38197 q^{43} +11.7082 q^{44} -6.47214 q^{45} +4.70820 q^{47} -12.3107 q^{48} -6.61803 q^{49} +8.78402 q^{51} -6.15537 q^{52} +5.15131 q^{53} +5.85410 q^{55} -4.25325 q^{59} -6.15537 q^{60} -7.23607 q^{61} -4.00000 q^{63} -8.00000 q^{64} -3.07768 q^{65} -7.33094 q^{67} +5.70820 q^{68} -16.8415 q^{69} -0.171513 q^{71} -11.0000 q^{73} -3.07768 q^{75} +3.61803 q^{77} -13.5923 q^{79} -4.00000 q^{80} +13.4721 q^{81} +10.7082 q^{83} -3.80423 q^{84} +2.85410 q^{85} -11.7082 q^{87} -8.22899 q^{89} -1.90211 q^{91} -10.9443 q^{92} -5.00000 q^{93} +8.33499 q^{97} -37.8885 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 4 q^{5} + 2 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} - 4 q^{5} + 2 q^{7} + 8 q^{9} - 10 q^{11} + 16 q^{16} + 2 q^{17} + 8 q^{20} + 4 q^{23} + 4 q^{25} - 4 q^{28} - 2 q^{35} - 16 q^{36} - 20 q^{39} - 34 q^{43} + 20 q^{44} - 8 q^{45} - 8 q^{47} - 22 q^{49} + 10 q^{55} - 20 q^{61} - 16 q^{63} - 32 q^{64} - 4 q^{68} - 44 q^{73} + 10 q^{77} - 16 q^{80} + 36 q^{81} + 16 q^{83} - 2 q^{85} - 20 q^{87} - 8 q^{92} - 20 q^{93} - 80 q^{99}+O(q^{100}) \) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −3.07768 −1.77690 −0.888451 0.458972i \(-0.848218\pi\)
−0.888451 + 0.458972i \(0.848218\pi\)
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.618034 −0.233595 −0.116797 0.993156i \(-0.537263\pi\)
−0.116797 + 0.993156i \(0.537263\pi\)
\(8\) 0 0
\(9\) 6.47214 2.15738
\(10\) 0 0
\(11\) −5.85410 −1.76508 −0.882539 0.470239i \(-0.844168\pi\)
−0.882539 + 0.470239i \(0.844168\pi\)
\(12\) 6.15537 1.77690
\(13\) 3.07768 0.853596 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(14\) 0 0
\(15\) 3.07768 0.794654
\(16\) 4.00000 1.00000
\(17\) −2.85410 −0.692221 −0.346111 0.938194i \(-0.612498\pi\)
−0.346111 + 0.938194i \(0.612498\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 2.00000 0.447214
\(21\) 1.90211 0.415075
\(22\) 0 0
\(23\) 5.47214 1.14102 0.570510 0.821291i \(-0.306746\pi\)
0.570510 + 0.821291i \(0.306746\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −10.6861 −2.05655
\(28\) 1.23607 0.233595
\(29\) 3.80423 0.706427 0.353214 0.935543i \(-0.385089\pi\)
0.353214 + 0.935543i \(0.385089\pi\)
\(30\) 0 0
\(31\) 1.62460 0.291787 0.145893 0.989300i \(-0.453394\pi\)
0.145893 + 0.989300i \(0.453394\pi\)
\(32\) 0 0
\(33\) 18.0171 3.13637
\(34\) 0 0
\(35\) 0.618034 0.104467
\(36\) −12.9443 −2.15738
\(37\) 8.33499 1.37026 0.685132 0.728419i \(-0.259744\pi\)
0.685132 + 0.728419i \(0.259744\pi\)
\(38\) 0 0
\(39\) −9.47214 −1.51676
\(40\) 0 0
\(41\) 11.5842 1.80915 0.904573 0.426318i \(-0.140190\pi\)
0.904573 + 0.426318i \(0.140190\pi\)
\(42\) 0 0
\(43\) −7.38197 −1.12574 −0.562870 0.826546i \(-0.690303\pi\)
−0.562870 + 0.826546i \(0.690303\pi\)
\(44\) 11.7082 1.76508
\(45\) −6.47214 −0.964809
\(46\) 0 0
\(47\) 4.70820 0.686762 0.343381 0.939196i \(-0.388428\pi\)
0.343381 + 0.939196i \(0.388428\pi\)
\(48\) −12.3107 −1.77690
\(49\) −6.61803 −0.945433
\(50\) 0 0
\(51\) 8.78402 1.23001
\(52\) −6.15537 −0.853596
\(53\) 5.15131 0.707587 0.353793 0.935324i \(-0.384892\pi\)
0.353793 + 0.935324i \(0.384892\pi\)
\(54\) 0 0
\(55\) 5.85410 0.789367
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.25325 −0.553727 −0.276863 0.960909i \(-0.589295\pi\)
−0.276863 + 0.960909i \(0.589295\pi\)
\(60\) −6.15537 −0.794654
\(61\) −7.23607 −0.926484 −0.463242 0.886232i \(-0.653314\pi\)
−0.463242 + 0.886232i \(0.653314\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) −8.00000 −1.00000
\(65\) −3.07768 −0.381740
\(66\) 0 0
\(67\) −7.33094 −0.895617 −0.447808 0.894130i \(-0.647795\pi\)
−0.447808 + 0.894130i \(0.647795\pi\)
\(68\) 5.70820 0.692221
\(69\) −16.8415 −2.02748
\(70\) 0 0
\(71\) −0.171513 −0.0203549 −0.0101774 0.999948i \(-0.503240\pi\)
−0.0101774 + 0.999948i \(0.503240\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) −3.07768 −0.355380
\(76\) 0 0
\(77\) 3.61803 0.412313
\(78\) 0 0
\(79\) −13.5923 −1.52925 −0.764627 0.644473i \(-0.777077\pi\)
−0.764627 + 0.644473i \(0.777077\pi\)
\(80\) −4.00000 −0.447214
\(81\) 13.4721 1.49690
\(82\) 0 0
\(83\) 10.7082 1.17538 0.587689 0.809087i \(-0.300038\pi\)
0.587689 + 0.809087i \(0.300038\pi\)
\(84\) −3.80423 −0.415075
\(85\) 2.85410 0.309571
\(86\) 0 0
\(87\) −11.7082 −1.25525
\(88\) 0 0
\(89\) −8.22899 −0.872272 −0.436136 0.899881i \(-0.643653\pi\)
−0.436136 + 0.899881i \(0.643653\pi\)
\(90\) 0 0
\(91\) −1.90211 −0.199396
\(92\) −10.9443 −1.14102
\(93\) −5.00000 −0.518476
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.33499 0.846290 0.423145 0.906062i \(-0.360926\pi\)
0.423145 + 0.906062i \(0.360926\pi\)
\(98\) 0 0
\(99\) −37.8885 −3.80794
\(100\) −2.00000 −0.200000
\(101\) −1.90983 −0.190035 −0.0950176 0.995476i \(-0.530291\pi\)
−0.0950176 + 0.995476i \(0.530291\pi\)
\(102\) 0 0
\(103\) 9.06154 0.892860 0.446430 0.894819i \(-0.352695\pi\)
0.446430 + 0.894819i \(0.352695\pi\)
\(104\) 0 0
\(105\) −1.90211 −0.185627
\(106\) 0 0
\(107\) −2.45714 −0.237541 −0.118770 0.992922i \(-0.537895\pi\)
−0.118770 + 0.992922i \(0.537895\pi\)
\(108\) 21.3723 2.05655
\(109\) −16.2865 −1.55996 −0.779981 0.625804i \(-0.784771\pi\)
−0.779981 + 0.625804i \(0.784771\pi\)
\(110\) 0 0
\(111\) −25.6525 −2.43483
\(112\) −2.47214 −0.233595
\(113\) 2.80017 0.263418 0.131709 0.991288i \(-0.457954\pi\)
0.131709 + 0.991288i \(0.457954\pi\)
\(114\) 0 0
\(115\) −5.47214 −0.510279
\(116\) −7.60845 −0.706427
\(117\) 19.9192 1.84153
\(118\) 0 0
\(119\) 1.76393 0.161699
\(120\) 0 0
\(121\) 23.2705 2.11550
\(122\) 0 0
\(123\) −35.6525 −3.21468
\(124\) −3.24920 −0.291787
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0.620541 0.0550641 0.0275321 0.999621i \(-0.491235\pi\)
0.0275321 + 0.999621i \(0.491235\pi\)
\(128\) 0 0
\(129\) 22.7194 2.00033
\(130\) 0 0
\(131\) 5.23607 0.457477 0.228739 0.973488i \(-0.426540\pi\)
0.228739 + 0.973488i \(0.426540\pi\)
\(132\) −36.0341 −3.13637
\(133\) 0 0
\(134\) 0 0
\(135\) 10.6861 0.919716
\(136\) 0 0
\(137\) 14.5623 1.24414 0.622071 0.782961i \(-0.286292\pi\)
0.622071 + 0.782961i \(0.286292\pi\)
\(138\) 0 0
\(139\) −0.381966 −0.0323979 −0.0161990 0.999869i \(-0.505157\pi\)
−0.0161990 + 0.999869i \(0.505157\pi\)
\(140\) −1.23607 −0.104467
\(141\) −14.4904 −1.22031
\(142\) 0 0
\(143\) −18.0171 −1.50666
\(144\) 25.8885 2.15738
\(145\) −3.80423 −0.315924
\(146\) 0 0
\(147\) 20.3682 1.67994
\(148\) −16.6700 −1.37026
\(149\) −3.76393 −0.308353 −0.154177 0.988043i \(-0.549272\pi\)
−0.154177 + 0.988043i \(0.549272\pi\)
\(150\) 0 0
\(151\) 3.35520 0.273042 0.136521 0.990637i \(-0.456408\pi\)
0.136521 + 0.990637i \(0.456408\pi\)
\(152\) 0 0
\(153\) −18.4721 −1.49338
\(154\) 0 0
\(155\) −1.62460 −0.130491
\(156\) 18.9443 1.51676
\(157\) 15.0344 1.19988 0.599940 0.800045i \(-0.295191\pi\)
0.599940 + 0.800045i \(0.295191\pi\)
\(158\) 0 0
\(159\) −15.8541 −1.25731
\(160\) 0 0
\(161\) −3.38197 −0.266536
\(162\) 0 0
\(163\) −12.0902 −0.946975 −0.473488 0.880800i \(-0.657005\pi\)
−0.473488 + 0.880800i \(0.657005\pi\)
\(164\) −23.1684 −1.80915
\(165\) −18.0171 −1.40263
\(166\) 0 0
\(167\) 2.45714 0.190139 0.0950697 0.995471i \(-0.469693\pi\)
0.0950697 + 0.995471i \(0.469693\pi\)
\(168\) 0 0
\(169\) −3.52786 −0.271374
\(170\) 0 0
\(171\) 0 0
\(172\) 14.7639 1.12574
\(173\) 6.43288 0.489083 0.244541 0.969639i \(-0.421363\pi\)
0.244541 + 0.969639i \(0.421363\pi\)
\(174\) 0 0
\(175\) −0.618034 −0.0467190
\(176\) −23.4164 −1.76508
\(177\) 13.0902 0.983917
\(178\) 0 0
\(179\) −5.42882 −0.405769 −0.202885 0.979203i \(-0.565032\pi\)
−0.202885 + 0.979203i \(0.565032\pi\)
\(180\) 12.9443 0.964809
\(181\) −17.7396 −1.31857 −0.659286 0.751893i \(-0.729141\pi\)
−0.659286 + 0.751893i \(0.729141\pi\)
\(182\) 0 0
\(183\) 22.2703 1.64627
\(184\) 0 0
\(185\) −8.33499 −0.612801
\(186\) 0 0
\(187\) 16.7082 1.22182
\(188\) −9.41641 −0.686762
\(189\) 6.60440 0.480399
\(190\) 0 0
\(191\) 6.09017 0.440669 0.220335 0.975424i \(-0.429285\pi\)
0.220335 + 0.975424i \(0.429285\pi\)
\(192\) 24.6215 1.77690
\(193\) 23.4459 1.68767 0.843836 0.536601i \(-0.180292\pi\)
0.843836 + 0.536601i \(0.180292\pi\)
\(194\) 0 0
\(195\) 9.47214 0.678314
\(196\) 13.2361 0.945433
\(197\) −8.65248 −0.616463 −0.308232 0.951311i \(-0.599737\pi\)
−0.308232 + 0.951311i \(0.599737\pi\)
\(198\) 0 0
\(199\) −7.56231 −0.536078 −0.268039 0.963408i \(-0.586376\pi\)
−0.268039 + 0.963408i \(0.586376\pi\)
\(200\) 0 0
\(201\) 22.5623 1.59142
\(202\) 0 0
\(203\) −2.35114 −0.165018
\(204\) −17.5680 −1.23001
\(205\) −11.5842 −0.809075
\(206\) 0 0
\(207\) 35.4164 2.46161
\(208\) 12.3107 0.853596
\(209\) 0 0
\(210\) 0 0
\(211\) −8.78402 −0.604717 −0.302359 0.953194i \(-0.597774\pi\)
−0.302359 + 0.953194i \(0.597774\pi\)
\(212\) −10.3026 −0.707587
\(213\) 0.527864 0.0361686
\(214\) 0 0
\(215\) 7.38197 0.503446
\(216\) 0 0
\(217\) −1.00406 −0.0681598
\(218\) 0 0
\(219\) 33.8545 2.28768
\(220\) −11.7082 −0.789367
\(221\) −8.78402 −0.590877
\(222\) 0 0
\(223\) −12.5882 −0.842971 −0.421486 0.906835i \(-0.638491\pi\)
−0.421486 + 0.906835i \(0.638491\pi\)
\(224\) 0 0
\(225\) 6.47214 0.431476
\(226\) 0 0
\(227\) 9.61657 0.638274 0.319137 0.947709i \(-0.396607\pi\)
0.319137 + 0.947709i \(0.396607\pi\)
\(228\) 0 0
\(229\) 1.85410 0.122523 0.0612613 0.998122i \(-0.480488\pi\)
0.0612613 + 0.998122i \(0.480488\pi\)
\(230\) 0 0
\(231\) −11.1352 −0.732640
\(232\) 0 0
\(233\) −4.94427 −0.323910 −0.161955 0.986798i \(-0.551780\pi\)
−0.161955 + 0.986798i \(0.551780\pi\)
\(234\) 0 0
\(235\) −4.70820 −0.307129
\(236\) 8.50651 0.553727
\(237\) 41.8328 2.71733
\(238\) 0 0
\(239\) −13.9443 −0.901980 −0.450990 0.892529i \(-0.648929\pi\)
−0.450990 + 0.892529i \(0.648929\pi\)
\(240\) 12.3107 0.794654
\(241\) −3.18368 −0.205079 −0.102540 0.994729i \(-0.532697\pi\)
−0.102540 + 0.994729i \(0.532697\pi\)
\(242\) 0 0
\(243\) −9.40456 −0.603303
\(244\) 14.4721 0.926484
\(245\) 6.61803 0.422811
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −32.9565 −2.08853
\(250\) 0 0
\(251\) 12.8541 0.811344 0.405672 0.914019i \(-0.367038\pi\)
0.405672 + 0.914019i \(0.367038\pi\)
\(252\) 8.00000 0.503953
\(253\) −32.0344 −2.01399
\(254\) 0 0
\(255\) −8.78402 −0.550077
\(256\) 16.0000 1.00000
\(257\) −5.36331 −0.334554 −0.167277 0.985910i \(-0.553497\pi\)
−0.167277 + 0.985910i \(0.553497\pi\)
\(258\) 0 0
\(259\) −5.15131 −0.320087
\(260\) 6.15537 0.381740
\(261\) 24.6215 1.52403
\(262\) 0 0
\(263\) 22.0902 1.36214 0.681069 0.732219i \(-0.261515\pi\)
0.681069 + 0.732219i \(0.261515\pi\)
\(264\) 0 0
\(265\) −5.15131 −0.316442
\(266\) 0 0
\(267\) 25.3262 1.54994
\(268\) 14.6619 0.895617
\(269\) −22.3763 −1.36431 −0.682154 0.731208i \(-0.738957\pi\)
−0.682154 + 0.731208i \(0.738957\pi\)
\(270\) 0 0
\(271\) 8.41641 0.511260 0.255630 0.966775i \(-0.417717\pi\)
0.255630 + 0.966775i \(0.417717\pi\)
\(272\) −11.4164 −0.692221
\(273\) 5.85410 0.354306
\(274\) 0 0
\(275\) −5.85410 −0.353016
\(276\) 33.6830 2.02748
\(277\) −19.3607 −1.16327 −0.581635 0.813450i \(-0.697587\pi\)
−0.581635 + 0.813450i \(0.697587\pi\)
\(278\) 0 0
\(279\) 10.5146 0.629494
\(280\) 0 0
\(281\) 15.6659 0.934551 0.467276 0.884112i \(-0.345236\pi\)
0.467276 + 0.884112i \(0.345236\pi\)
\(282\) 0 0
\(283\) −24.0000 −1.42665 −0.713326 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) 0.343027 0.0203549
\(285\) 0 0
\(286\) 0 0
\(287\) −7.15942 −0.422607
\(288\) 0 0
\(289\) −8.85410 −0.520830
\(290\) 0 0
\(291\) −25.6525 −1.50377
\(292\) 22.0000 1.28745
\(293\) 22.5478 1.31726 0.658629 0.752467i \(-0.271136\pi\)
0.658629 + 0.752467i \(0.271136\pi\)
\(294\) 0 0
\(295\) 4.25325 0.247634
\(296\) 0 0
\(297\) 62.5577 3.62997
\(298\) 0 0
\(299\) 16.8415 0.973969
\(300\) 6.15537 0.355380
\(301\) 4.56231 0.262967
\(302\) 0 0
\(303\) 5.87785 0.337674
\(304\) 0 0
\(305\) 7.23607 0.414336
\(306\) 0 0
\(307\) −12.2047 −0.696561 −0.348280 0.937390i \(-0.613234\pi\)
−0.348280 + 0.937390i \(0.613234\pi\)
\(308\) −7.23607 −0.412313
\(309\) −27.8885 −1.58652
\(310\) 0 0
\(311\) −27.0344 −1.53298 −0.766491 0.642255i \(-0.777999\pi\)
−0.766491 + 0.642255i \(0.777999\pi\)
\(312\) 0 0
\(313\) 13.6738 0.772887 0.386443 0.922313i \(-0.373703\pi\)
0.386443 + 0.922313i \(0.373703\pi\)
\(314\) 0 0
\(315\) 4.00000 0.225374
\(316\) 27.1846 1.52925
\(317\) −34.4095 −1.93263 −0.966316 0.257357i \(-0.917148\pi\)
−0.966316 + 0.257357i \(0.917148\pi\)
\(318\) 0 0
\(319\) −22.2703 −1.24690
\(320\) 8.00000 0.447214
\(321\) 7.56231 0.422087
\(322\) 0 0
\(323\) 0 0
\(324\) −26.9443 −1.49690
\(325\) 3.07768 0.170719
\(326\) 0 0
\(327\) 50.1246 2.77190
\(328\) 0 0
\(329\) −2.90983 −0.160424
\(330\) 0 0
\(331\) 23.5519 1.29453 0.647265 0.762265i \(-0.275913\pi\)
0.647265 + 0.762265i \(0.275913\pi\)
\(332\) −21.4164 −1.17538
\(333\) 53.9452 2.95618
\(334\) 0 0
\(335\) 7.33094 0.400532
\(336\) 7.60845 0.415075
\(337\) 0.171513 0.00934293 0.00467147 0.999989i \(-0.498513\pi\)
0.00467147 + 0.999989i \(0.498513\pi\)
\(338\) 0 0
\(339\) −8.61803 −0.468067
\(340\) −5.70820 −0.309571
\(341\) −9.51057 −0.515026
\(342\) 0 0
\(343\) 8.41641 0.454443
\(344\) 0 0
\(345\) 16.8415 0.906716
\(346\) 0 0
\(347\) −24.7082 −1.32641 −0.663203 0.748440i \(-0.730803\pi\)
−0.663203 + 0.748440i \(0.730803\pi\)
\(348\) 23.4164 1.25525
\(349\) 10.1246 0.541958 0.270979 0.962585i \(-0.412653\pi\)
0.270979 + 0.962585i \(0.412653\pi\)
\(350\) 0 0
\(351\) −32.8885 −1.75546
\(352\) 0 0
\(353\) 15.3820 0.818699 0.409350 0.912378i \(-0.365756\pi\)
0.409350 + 0.912378i \(0.365756\pi\)
\(354\) 0 0
\(355\) 0.171513 0.00910299
\(356\) 16.4580 0.872272
\(357\) −5.42882 −0.287324
\(358\) 0 0
\(359\) −6.43769 −0.339768 −0.169884 0.985464i \(-0.554339\pi\)
−0.169884 + 0.985464i \(0.554339\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) −71.6193 −3.75904
\(364\) 3.80423 0.199396
\(365\) 11.0000 0.575766
\(366\) 0 0
\(367\) −13.6525 −0.712653 −0.356327 0.934361i \(-0.615971\pi\)
−0.356327 + 0.934361i \(0.615971\pi\)
\(368\) 21.8885 1.14102
\(369\) 74.9745 3.90301
\(370\) 0 0
\(371\) −3.18368 −0.165289
\(372\) 10.0000 0.518476
\(373\) −34.9241 −1.80830 −0.904150 0.427214i \(-0.859495\pi\)
−0.904150 + 0.427214i \(0.859495\pi\)
\(374\) 0 0
\(375\) 3.07768 0.158931
\(376\) 0 0
\(377\) 11.7082 0.603003
\(378\) 0 0
\(379\) 30.7768 1.58090 0.790450 0.612527i \(-0.209847\pi\)
0.790450 + 0.612527i \(0.209847\pi\)
\(380\) 0 0
\(381\) −1.90983 −0.0978436
\(382\) 0 0
\(383\) −21.2008 −1.08331 −0.541654 0.840601i \(-0.682202\pi\)
−0.541654 + 0.840601i \(0.682202\pi\)
\(384\) 0 0
\(385\) −3.61803 −0.184392
\(386\) 0 0
\(387\) −47.7771 −2.42865
\(388\) −16.6700 −0.846290
\(389\) 7.43769 0.377106 0.188553 0.982063i \(-0.439620\pi\)
0.188553 + 0.982063i \(0.439620\pi\)
\(390\) 0 0
\(391\) −15.6180 −0.789838
\(392\) 0 0
\(393\) −16.1150 −0.812892
\(394\) 0 0
\(395\) 13.5923 0.683903
\(396\) 75.7771 3.80794
\(397\) −16.4721 −0.826713 −0.413356 0.910569i \(-0.635644\pi\)
−0.413356 + 0.910569i \(0.635644\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 8.16348 0.407665 0.203832 0.979006i \(-0.434660\pi\)
0.203832 + 0.979006i \(0.434660\pi\)
\(402\) 0 0
\(403\) 5.00000 0.249068
\(404\) 3.81966 0.190035
\(405\) −13.4721 −0.669436
\(406\) 0 0
\(407\) −48.7939 −2.41862
\(408\) 0 0
\(409\) −4.25325 −0.210310 −0.105155 0.994456i \(-0.533534\pi\)
−0.105155 + 0.994456i \(0.533534\pi\)
\(410\) 0 0
\(411\) −44.8182 −2.21072
\(412\) −18.1231 −0.892860
\(413\) 2.62866 0.129348
\(414\) 0 0
\(415\) −10.7082 −0.525645
\(416\) 0 0
\(417\) 1.17557 0.0575679
\(418\) 0 0
\(419\) −23.2918 −1.13788 −0.568939 0.822379i \(-0.692646\pi\)
−0.568939 + 0.822379i \(0.692646\pi\)
\(420\) 3.80423 0.185627
\(421\) 16.8415 0.820805 0.410402 0.911905i \(-0.365388\pi\)
0.410402 + 0.911905i \(0.365388\pi\)
\(422\) 0 0
\(423\) 30.4721 1.48161
\(424\) 0 0
\(425\) −2.85410 −0.138444
\(426\) 0 0
\(427\) 4.47214 0.216422
\(428\) 4.91428 0.237541
\(429\) 55.4508 2.67719
\(430\) 0 0
\(431\) −5.77185 −0.278020 −0.139010 0.990291i \(-0.544392\pi\)
−0.139010 + 0.990291i \(0.544392\pi\)
\(432\) −42.7445 −2.05655
\(433\) −6.04937 −0.290714 −0.145357 0.989379i \(-0.546433\pi\)
−0.145357 + 0.989379i \(0.546433\pi\)
\(434\) 0 0
\(435\) 11.7082 0.561365
\(436\) 32.5729 1.55996
\(437\) 0 0
\(438\) 0 0
\(439\) −16.4985 −0.787429 −0.393715 0.919233i \(-0.628810\pi\)
−0.393715 + 0.919233i \(0.628810\pi\)
\(440\) 0 0
\(441\) −42.8328 −2.03966
\(442\) 0 0
\(443\) 17.9443 0.852558 0.426279 0.904592i \(-0.359824\pi\)
0.426279 + 0.904592i \(0.359824\pi\)
\(444\) 51.3050 2.43483
\(445\) 8.22899 0.390092
\(446\) 0 0
\(447\) 11.5842 0.547913
\(448\) 4.94427 0.233595
\(449\) −0.514540 −0.0242827 −0.0121413 0.999926i \(-0.503865\pi\)
−0.0121413 + 0.999926i \(0.503865\pi\)
\(450\) 0 0
\(451\) −67.8150 −3.19329
\(452\) −5.60034 −0.263418
\(453\) −10.3262 −0.485169
\(454\) 0 0
\(455\) 1.90211 0.0891724
\(456\) 0 0
\(457\) −12.6525 −0.591858 −0.295929 0.955210i \(-0.595629\pi\)
−0.295929 + 0.955210i \(0.595629\pi\)
\(458\) 0 0
\(459\) 30.4993 1.42359
\(460\) 10.9443 0.510279
\(461\) −6.41641 −0.298842 −0.149421 0.988774i \(-0.547741\pi\)
−0.149421 + 0.988774i \(0.547741\pi\)
\(462\) 0 0
\(463\) −17.2918 −0.803618 −0.401809 0.915724i \(-0.631618\pi\)
−0.401809 + 0.915724i \(0.631618\pi\)
\(464\) 15.2169 0.706427
\(465\) 5.00000 0.231869
\(466\) 0 0
\(467\) 10.9443 0.506441 0.253220 0.967409i \(-0.418510\pi\)
0.253220 + 0.967409i \(0.418510\pi\)
\(468\) −39.8384 −1.84153
\(469\) 4.53077 0.209211
\(470\) 0 0
\(471\) −46.2713 −2.13207
\(472\) 0 0
\(473\) 43.2148 1.98702
\(474\) 0 0
\(475\) 0 0
\(476\) −3.52786 −0.161699
\(477\) 33.3400 1.52653
\(478\) 0 0
\(479\) 25.1459 1.14895 0.574473 0.818524i \(-0.305207\pi\)
0.574473 + 0.818524i \(0.305207\pi\)
\(480\) 0 0
\(481\) 25.6525 1.16965
\(482\) 0 0
\(483\) 10.4086 0.473609
\(484\) −46.5410 −2.11550
\(485\) −8.33499 −0.378473
\(486\) 0 0
\(487\) 19.9192 0.902624 0.451312 0.892366i \(-0.350956\pi\)
0.451312 + 0.892366i \(0.350956\pi\)
\(488\) 0 0
\(489\) 37.2097 1.68268
\(490\) 0 0
\(491\) −30.6180 −1.38177 −0.690886 0.722963i \(-0.742779\pi\)
−0.690886 + 0.722963i \(0.742779\pi\)
\(492\) 71.3050 3.21468
\(493\) −10.8576 −0.489004
\(494\) 0 0
\(495\) 37.8885 1.70296
\(496\) 6.49839 0.291787
\(497\) 0.106001 0.00475480
\(498\) 0 0
\(499\) 16.1246 0.721837 0.360918 0.932597i \(-0.382463\pi\)
0.360918 + 0.932597i \(0.382463\pi\)
\(500\) 2.00000 0.0894427
\(501\) −7.56231 −0.337859
\(502\) 0 0
\(503\) 25.1459 1.12120 0.560600 0.828087i \(-0.310571\pi\)
0.560600 + 0.828087i \(0.310571\pi\)
\(504\) 0 0
\(505\) 1.90983 0.0849863
\(506\) 0 0
\(507\) 10.8576 0.482205
\(508\) −1.24108 −0.0550641
\(509\) −22.2048 −0.984211 −0.492106 0.870536i \(-0.663773\pi\)
−0.492106 + 0.870536i \(0.663773\pi\)
\(510\) 0 0
\(511\) 6.79837 0.300742
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −9.06154 −0.399299
\(516\) −45.4387 −2.00033
\(517\) −27.5623 −1.21219
\(518\) 0 0
\(519\) −19.7984 −0.869052
\(520\) 0 0
\(521\) 41.7405 1.82868 0.914342 0.404943i \(-0.132709\pi\)
0.914342 + 0.404943i \(0.132709\pi\)
\(522\) 0 0
\(523\) 23.1684 1.01308 0.506541 0.862216i \(-0.330924\pi\)
0.506541 + 0.862216i \(0.330924\pi\)
\(524\) −10.4721 −0.457477
\(525\) 1.90211 0.0830150
\(526\) 0 0
\(527\) −4.63677 −0.201981
\(528\) 72.0683 3.13637
\(529\) 6.94427 0.301925
\(530\) 0 0
\(531\) −27.5276 −1.19460
\(532\) 0 0
\(533\) 35.6525 1.54428
\(534\) 0 0
\(535\) 2.45714 0.106232
\(536\) 0 0
\(537\) 16.7082 0.721012
\(538\) 0 0
\(539\) 38.7426 1.66876
\(540\) −21.3723 −0.919716
\(541\) −28.9443 −1.24441 −0.622206 0.782854i \(-0.713763\pi\)
−0.622206 + 0.782854i \(0.713763\pi\)
\(542\) 0 0
\(543\) 54.5967 2.34297
\(544\) 0 0
\(545\) 16.2865 0.697636
\(546\) 0 0
\(547\) −26.9726 −1.15327 −0.576633 0.817003i \(-0.695634\pi\)
−0.576633 + 0.817003i \(0.695634\pi\)
\(548\) −29.1246 −1.24414
\(549\) −46.8328 −1.99878
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.40051 0.357226
\(554\) 0 0
\(555\) 25.6525 1.08889
\(556\) 0.763932 0.0323979
\(557\) −34.3607 −1.45591 −0.727954 0.685626i \(-0.759529\pi\)
−0.727954 + 0.685626i \(0.759529\pi\)
\(558\) 0 0
\(559\) −22.7194 −0.960926
\(560\) 2.47214 0.104467
\(561\) −51.4226 −2.17106
\(562\) 0 0
\(563\) −11.5842 −0.488215 −0.244108 0.969748i \(-0.578495\pi\)
−0.244108 + 0.969748i \(0.578495\pi\)
\(564\) 28.9807 1.22031
\(565\) −2.80017 −0.117804
\(566\) 0 0
\(567\) −8.32624 −0.349669
\(568\) 0 0
\(569\) −36.5892 −1.53390 −0.766949 0.641708i \(-0.778226\pi\)
−0.766949 + 0.641708i \(0.778226\pi\)
\(570\) 0 0
\(571\) −38.5410 −1.61289 −0.806446 0.591308i \(-0.798612\pi\)
−0.806446 + 0.591308i \(0.798612\pi\)
\(572\) 36.0341 1.50666
\(573\) −18.7436 −0.783026
\(574\) 0 0
\(575\) 5.47214 0.228204
\(576\) −51.7771 −2.15738
\(577\) −12.1246 −0.504754 −0.252377 0.967629i \(-0.581212\pi\)
−0.252377 + 0.967629i \(0.581212\pi\)
\(578\) 0 0
\(579\) −72.1591 −2.99883
\(580\) 7.60845 0.315924
\(581\) −6.61803 −0.274562
\(582\) 0 0
\(583\) −30.1563 −1.24895
\(584\) 0 0
\(585\) −19.9192 −0.823557
\(586\) 0 0
\(587\) −19.8885 −0.820888 −0.410444 0.911886i \(-0.634626\pi\)
−0.410444 + 0.911886i \(0.634626\pi\)
\(588\) −40.7364 −1.67994
\(589\) 0 0
\(590\) 0 0
\(591\) 26.6296 1.09539
\(592\) 33.3400 1.37026
\(593\) −24.2148 −0.994382 −0.497191 0.867641i \(-0.665635\pi\)
−0.497191 + 0.867641i \(0.665635\pi\)
\(594\) 0 0
\(595\) −1.76393 −0.0723142
\(596\) 7.52786 0.308353
\(597\) 23.2744 0.952557
\(598\) 0 0
\(599\) −27.2501 −1.11341 −0.556705 0.830710i \(-0.687935\pi\)
−0.556705 + 0.830710i \(0.687935\pi\)
\(600\) 0 0
\(601\) 36.0341 1.46986 0.734932 0.678141i \(-0.237214\pi\)
0.734932 + 0.678141i \(0.237214\pi\)
\(602\) 0 0
\(603\) −47.4468 −1.93218
\(604\) −6.71040 −0.273042
\(605\) −23.2705 −0.946081
\(606\) 0 0
\(607\) −42.5325 −1.72634 −0.863171 0.504911i \(-0.831525\pi\)
−0.863171 + 0.504911i \(0.831525\pi\)
\(608\) 0 0
\(609\) 7.23607 0.293220
\(610\) 0 0
\(611\) 14.4904 0.586217
\(612\) 36.9443 1.49338
\(613\) −23.2705 −0.939887 −0.469944 0.882696i \(-0.655726\pi\)
−0.469944 + 0.882696i \(0.655726\pi\)
\(614\) 0 0
\(615\) 35.6525 1.43765
\(616\) 0 0
\(617\) 0.888544 0.0357714 0.0178857 0.999840i \(-0.494306\pi\)
0.0178857 + 0.999840i \(0.494306\pi\)
\(618\) 0 0
\(619\) −20.9098 −0.840437 −0.420219 0.907423i \(-0.638047\pi\)
−0.420219 + 0.907423i \(0.638047\pi\)
\(620\) 3.24920 0.130491
\(621\) −58.4760 −2.34656
\(622\) 0 0
\(623\) 5.08580 0.203758
\(624\) −37.8885 −1.51676
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) −30.0689 −1.19988
\(629\) −23.7889 −0.948527
\(630\) 0 0
\(631\) −24.4721 −0.974220 −0.487110 0.873341i \(-0.661949\pi\)
−0.487110 + 0.873341i \(0.661949\pi\)
\(632\) 0 0
\(633\) 27.0344 1.07452
\(634\) 0 0
\(635\) −0.620541 −0.0246254
\(636\) 31.7082 1.25731
\(637\) −20.3682 −0.807018
\(638\) 0 0
\(639\) −1.11006 −0.0439132
\(640\) 0 0
\(641\) −19.3642 −0.764838 −0.382419 0.923989i \(-0.624909\pi\)
−0.382419 + 0.923989i \(0.624909\pi\)
\(642\) 0 0
\(643\) 14.9443 0.589345 0.294672 0.955598i \(-0.404790\pi\)
0.294672 + 0.955598i \(0.404790\pi\)
\(644\) 6.76393 0.266536
\(645\) −22.7194 −0.894574
\(646\) 0 0
\(647\) 45.3607 1.78331 0.891656 0.452713i \(-0.149544\pi\)
0.891656 + 0.452713i \(0.149544\pi\)
\(648\) 0 0
\(649\) 24.8990 0.977371
\(650\) 0 0
\(651\) 3.09017 0.121113
\(652\) 24.1803 0.946975
\(653\) −9.32624 −0.364964 −0.182482 0.983209i \(-0.558413\pi\)
−0.182482 + 0.983209i \(0.558413\pi\)
\(654\) 0 0
\(655\) −5.23607 −0.204590
\(656\) 46.3368 1.80915
\(657\) −71.1935 −2.77752
\(658\) 0 0
\(659\) 29.6013 1.15310 0.576551 0.817061i \(-0.304398\pi\)
0.576551 + 0.817061i \(0.304398\pi\)
\(660\) 36.0341 1.40263
\(661\) 20.0907 0.781438 0.390719 0.920510i \(-0.372226\pi\)
0.390719 + 0.920510i \(0.372226\pi\)
\(662\) 0 0
\(663\) 27.0344 1.04993
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 20.8172 0.806047
\(668\) −4.91428 −0.190139
\(669\) 38.7426 1.49788
\(670\) 0 0
\(671\) 42.3607 1.63532
\(672\) 0 0
\(673\) −0.661030 −0.0254808 −0.0127404 0.999919i \(-0.504056\pi\)
−0.0127404 + 0.999919i \(0.504056\pi\)
\(674\) 0 0
\(675\) −10.6861 −0.411310
\(676\) 7.05573 0.271374
\(677\) 4.04125 0.155318 0.0776590 0.996980i \(-0.475255\pi\)
0.0776590 + 0.996980i \(0.475255\pi\)
\(678\) 0 0
\(679\) −5.15131 −0.197689
\(680\) 0 0
\(681\) −29.5967 −1.13415
\(682\) 0 0
\(683\) 26.8011 1.02552 0.512758 0.858533i \(-0.328624\pi\)
0.512758 + 0.858533i \(0.328624\pi\)
\(684\) 0 0
\(685\) −14.5623 −0.556397
\(686\) 0 0
\(687\) −5.70634 −0.217710
\(688\) −29.5279 −1.12574
\(689\) 15.8541 0.603993
\(690\) 0 0
\(691\) −28.9443 −1.10109 −0.550546 0.834805i \(-0.685580\pi\)
−0.550546 + 0.834805i \(0.685580\pi\)
\(692\) −12.8658 −0.489083
\(693\) 23.4164 0.889516
\(694\) 0 0
\(695\) 0.381966 0.0144888
\(696\) 0 0
\(697\) −33.0625 −1.25233
\(698\) 0 0
\(699\) 15.2169 0.575556
\(700\) 1.23607 0.0467190
\(701\) 25.0000 0.944237 0.472118 0.881535i \(-0.343489\pi\)
0.472118 + 0.881535i \(0.343489\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 46.8328 1.76508
\(705\) 14.4904 0.545739
\(706\) 0 0
\(707\) 1.18034 0.0443913
\(708\) −26.1803 −0.983917
\(709\) −25.5279 −0.958719 −0.479360 0.877619i \(-0.659131\pi\)
−0.479360 + 0.877619i \(0.659131\pi\)
\(710\) 0 0
\(711\) −87.9713 −3.29918
\(712\) 0 0
\(713\) 8.89002 0.332934
\(714\) 0 0
\(715\) 18.0171 0.673800
\(716\) 10.8576 0.405769
\(717\) 42.9161 1.60273
\(718\) 0 0
\(719\) 8.23607 0.307154 0.153577 0.988137i \(-0.450921\pi\)
0.153577 + 0.988137i \(0.450921\pi\)
\(720\) −25.8885 −0.964809
\(721\) −5.60034 −0.208567
\(722\) 0 0
\(723\) 9.79837 0.364405
\(724\) 35.4791 1.31857
\(725\) 3.80423 0.141285
\(726\) 0 0
\(727\) 21.0689 0.781402 0.390701 0.920518i \(-0.372233\pi\)
0.390701 + 0.920518i \(0.372233\pi\)
\(728\) 0 0
\(729\) −11.4721 −0.424894
\(730\) 0 0
\(731\) 21.0689 0.779261
\(732\) −44.5407 −1.64627
\(733\) −25.7984 −0.952885 −0.476442 0.879206i \(-0.658074\pi\)
−0.476442 + 0.879206i \(0.658074\pi\)
\(734\) 0 0
\(735\) −20.3682 −0.751293
\(736\) 0 0
\(737\) 42.9161 1.58083
\(738\) 0 0
\(739\) −38.4164 −1.41317 −0.706585 0.707628i \(-0.749765\pi\)
−0.706585 + 0.707628i \(0.749765\pi\)
\(740\) 16.6700 0.612801
\(741\) 0 0
\(742\) 0 0
\(743\) −38.4508 −1.41062 −0.705312 0.708897i \(-0.749193\pi\)
−0.705312 + 0.708897i \(0.749193\pi\)
\(744\) 0 0
\(745\) 3.76393 0.137900
\(746\) 0 0
\(747\) 69.3050 2.53574
\(748\) −33.4164 −1.22182
\(749\) 1.51860 0.0554883
\(750\) 0 0
\(751\) 6.53888 0.238607 0.119304 0.992858i \(-0.461934\pi\)
0.119304 + 0.992858i \(0.461934\pi\)
\(752\) 18.8328 0.686762
\(753\) −39.5609 −1.44168
\(754\) 0 0
\(755\) −3.35520 −0.122108
\(756\) −13.2088 −0.480399
\(757\) 52.7771 1.91822 0.959108 0.283041i \(-0.0913431\pi\)
0.959108 + 0.283041i \(0.0913431\pi\)
\(758\) 0 0
\(759\) 98.5919 3.57866
\(760\) 0 0
\(761\) −39.0689 −1.41625 −0.708123 0.706089i \(-0.750458\pi\)
−0.708123 + 0.706089i \(0.750458\pi\)
\(762\) 0 0
\(763\) 10.0656 0.364399
\(764\) −12.1803 −0.440669
\(765\) 18.4721 0.667861
\(766\) 0 0
\(767\) −13.0902 −0.472659
\(768\) −49.2429 −1.77690
\(769\) 18.3607 0.662103 0.331052 0.943613i \(-0.392597\pi\)
0.331052 + 0.943613i \(0.392597\pi\)
\(770\) 0 0
\(771\) 16.5066 0.594470
\(772\) −46.8918 −1.68767
\(773\) 21.3723 0.768707 0.384354 0.923186i \(-0.374424\pi\)
0.384354 + 0.923186i \(0.374424\pi\)
\(774\) 0 0
\(775\) 1.62460 0.0583573
\(776\) 0 0
\(777\) 15.8541 0.568763
\(778\) 0 0
\(779\) 0 0
\(780\) −18.9443 −0.678314
\(781\) 1.00406 0.0359280
\(782\) 0 0
\(783\) −40.6525 −1.45280
\(784\) −26.4721 −0.945433
\(785\) −15.0344 −0.536602
\(786\) 0 0
\(787\) 22.0583 0.786294 0.393147 0.919476i \(-0.371386\pi\)
0.393147 + 0.919476i \(0.371386\pi\)
\(788\) 17.3050 0.616463
\(789\) −67.9866 −2.42039
\(790\) 0 0
\(791\) −1.73060 −0.0615330
\(792\) 0 0
\(793\) −22.2703 −0.790843
\(794\) 0 0
\(795\) 15.8541 0.562287
\(796\) 15.1246 0.536078
\(797\) −12.4822 −0.442144 −0.221072 0.975258i \(-0.570956\pi\)
−0.221072 + 0.975258i \(0.570956\pi\)
\(798\) 0 0
\(799\) −13.4377 −0.475391
\(800\) 0 0
\(801\) −53.2592 −1.88182
\(802\) 0 0
\(803\) 64.3951 2.27245
\(804\) −45.1246 −1.59142
\(805\) 3.38197 0.119199
\(806\) 0 0
\(807\) 68.8673 2.42424
\(808\) 0 0
\(809\) 42.2361 1.48494 0.742471 0.669879i \(-0.233654\pi\)
0.742471 + 0.669879i \(0.233654\pi\)
\(810\) 0 0
\(811\) 31.9524 1.12200 0.561000 0.827816i \(-0.310417\pi\)
0.561000 + 0.827816i \(0.310417\pi\)
\(812\) 4.70228 0.165018
\(813\) −25.9030 −0.908459
\(814\) 0 0
\(815\) 12.0902 0.423500
\(816\) 35.1361 1.23001
\(817\) 0 0
\(818\) 0 0
\(819\) −12.3107 −0.430172
\(820\) 23.1684 0.809075
\(821\) 17.5967 0.614131 0.307065 0.951688i \(-0.400653\pi\)
0.307065 + 0.951688i \(0.400653\pi\)
\(822\) 0 0
\(823\) 18.8885 0.658413 0.329207 0.944258i \(-0.393219\pi\)
0.329207 + 0.944258i \(0.393219\pi\)
\(824\) 0 0
\(825\) 18.0171 0.627274
\(826\) 0 0
\(827\) 25.0705 0.871787 0.435893 0.899998i \(-0.356433\pi\)
0.435893 + 0.899998i \(0.356433\pi\)
\(828\) −70.8328 −2.46161
\(829\) 32.0584 1.11343 0.556717 0.830702i \(-0.312061\pi\)
0.556717 + 0.830702i \(0.312061\pi\)
\(830\) 0 0
\(831\) 59.5860 2.06702
\(832\) −24.6215 −0.853596
\(833\) 18.8885 0.654449
\(834\) 0 0
\(835\) −2.45714 −0.0850329
\(836\) 0 0
\(837\) −17.3607 −0.600073
\(838\) 0 0
\(839\) 24.7930 0.855949 0.427974 0.903791i \(-0.359227\pi\)
0.427974 + 0.903791i \(0.359227\pi\)
\(840\) 0 0
\(841\) −14.5279 −0.500961
\(842\) 0 0
\(843\) −48.2148 −1.66061
\(844\) 17.5680 0.604717
\(845\) 3.52786 0.121362
\(846\) 0 0
\(847\) −14.3820 −0.494170
\(848\) 20.6052 0.707587
\(849\) 73.8644 2.53502
\(850\) 0 0
\(851\) 45.6102 1.56350
\(852\) −1.05573 −0.0361686
\(853\) −30.2705 −1.03644 −0.518221 0.855247i \(-0.673406\pi\)
−0.518221 + 0.855247i \(0.673406\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −47.1693 −1.61127 −0.805636 0.592410i \(-0.798176\pi\)
−0.805636 + 0.592410i \(0.798176\pi\)
\(858\) 0 0
\(859\) 6.83282 0.233133 0.116566 0.993183i \(-0.462811\pi\)
0.116566 + 0.993183i \(0.462811\pi\)
\(860\) −14.7639 −0.503446
\(861\) 22.0344 0.750932
\(862\) 0 0
\(863\) −9.02105 −0.307080 −0.153540 0.988142i \(-0.549067\pi\)
−0.153540 + 0.988142i \(0.549067\pi\)
\(864\) 0 0
\(865\) −6.43288 −0.218725
\(866\) 0 0
\(867\) 27.2501 0.925463
\(868\) 2.00811 0.0681598
\(869\) 79.5707 2.69925
\(870\) 0 0
\(871\) −22.5623 −0.764495
\(872\) 0 0
\(873\) 53.9452 1.82577
\(874\) 0 0
\(875\) 0.618034 0.0208934
\(876\) −67.7090 −2.28768
\(877\) −26.4581 −0.893426 −0.446713 0.894677i \(-0.647405\pi\)
−0.446713 + 0.894677i \(0.647405\pi\)
\(878\) 0 0
\(879\) −69.3951 −2.34064
\(880\) 23.4164 0.789367
\(881\) 29.5967 0.997140 0.498570 0.866850i \(-0.333859\pi\)
0.498570 + 0.866850i \(0.333859\pi\)
\(882\) 0 0
\(883\) −32.0902 −1.07992 −0.539960 0.841691i \(-0.681561\pi\)
−0.539960 + 0.841691i \(0.681561\pi\)
\(884\) 17.5680 0.590877
\(885\) −13.0902 −0.440021
\(886\) 0 0
\(887\) 58.7940 1.97411 0.987055 0.160385i \(-0.0512736\pi\)
0.987055 + 0.160385i \(0.0512736\pi\)
\(888\) 0 0
\(889\) −0.383516 −0.0128627
\(890\) 0 0
\(891\) −78.8673 −2.64215
\(892\) 25.1765 0.842971
\(893\) 0 0
\(894\) 0 0
\(895\) 5.42882 0.181466
\(896\) 0 0
\(897\) −51.8328 −1.73065
\(898\) 0 0
\(899\) 6.18034 0.206126
\(900\) −12.9443 −0.431476
\(901\) −14.7024 −0.489807
\(902\) 0 0
\(903\) −14.0413 −0.467266
\(904\) 0 0
\(905\) 17.7396 0.589683
\(906\) 0 0
\(907\) 36.5237 1.21275 0.606374 0.795179i \(-0.292623\pi\)
0.606374 + 0.795179i \(0.292623\pi\)
\(908\) −19.2331 −0.638274
\(909\) −12.3607 −0.409978
\(910\) 0 0
\(911\) −41.0139 −1.35885 −0.679426 0.733744i \(-0.737771\pi\)
−0.679426 + 0.733744i \(0.737771\pi\)
\(912\) 0 0
\(913\) −62.6869 −2.07463
\(914\) 0 0
\(915\) −22.2703 −0.736234
\(916\) −3.70820 −0.122523
\(917\) −3.23607 −0.106864
\(918\) 0 0
\(919\) 9.59675 0.316567 0.158284 0.987394i \(-0.449404\pi\)
0.158284 + 0.987394i \(0.449404\pi\)
\(920\) 0 0
\(921\) 37.5623 1.23772
\(922\) 0 0
\(923\) −0.527864 −0.0173749
\(924\) 22.2703 0.732640
\(925\) 8.33499 0.274053
\(926\) 0 0
\(927\) 58.6475 1.92624
\(928\) 0 0
\(929\) −23.5410 −0.772356 −0.386178 0.922424i \(-0.626205\pi\)
−0.386178 + 0.922424i \(0.626205\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 9.88854 0.323910
\(933\) 83.2035 2.72396
\(934\) 0 0
\(935\) −16.7082 −0.546417
\(936\) 0 0
\(937\) −19.4377 −0.635002 −0.317501 0.948258i \(-0.602844\pi\)
−0.317501 + 0.948258i \(0.602844\pi\)
\(938\) 0 0
\(939\) −42.0835 −1.37334
\(940\) 9.41641 0.307129
\(941\) −4.49028 −0.146379 −0.0731895 0.997318i \(-0.523318\pi\)
−0.0731895 + 0.997318i \(0.523318\pi\)
\(942\) 0 0
\(943\) 63.3903 2.06427
\(944\) −17.0130 −0.553727
\(945\) −6.60440 −0.214841
\(946\) 0 0
\(947\) −38.1033 −1.23819 −0.619096 0.785315i \(-0.712501\pi\)
−0.619096 + 0.785315i \(0.712501\pi\)
\(948\) −83.6656 −2.71733
\(949\) −33.8545 −1.09896
\(950\) 0 0
\(951\) 105.902 3.43410
\(952\) 0 0
\(953\) −23.3804 −0.757365 −0.378682 0.925527i \(-0.623623\pi\)
−0.378682 + 0.925527i \(0.623623\pi\)
\(954\) 0 0
\(955\) −6.09017 −0.197073
\(956\) 27.8885 0.901980
\(957\) 68.5410 2.21562
\(958\) 0 0
\(959\) −9.00000 −0.290625
\(960\) −24.6215 −0.794654
\(961\) −28.3607 −0.914861
\(962\) 0 0
\(963\) −15.9030 −0.512466
\(964\) 6.36737 0.205079
\(965\) −23.4459 −0.754750
\(966\) 0 0
\(967\) −42.0689 −1.35284 −0.676422 0.736514i \(-0.736470\pi\)
−0.676422 + 0.736514i \(0.736470\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −21.8618 −0.701578 −0.350789 0.936454i \(-0.614087\pi\)
−0.350789 + 0.936454i \(0.614087\pi\)
\(972\) 18.8091 0.603303
\(973\) 0.236068 0.00756799
\(974\) 0 0
\(975\) −9.47214 −0.303351
\(976\) −28.9443 −0.926484
\(977\) −32.8505 −1.05098 −0.525490 0.850800i \(-0.676118\pi\)
−0.525490 + 0.850800i \(0.676118\pi\)
\(978\) 0 0
\(979\) 48.1734 1.53963
\(980\) −13.2361 −0.422811
\(981\) −105.408 −3.36543
\(982\) 0 0
\(983\) −5.81234 −0.185385 −0.0926924 0.995695i \(-0.529547\pi\)
−0.0926924 + 0.995695i \(0.529547\pi\)
\(984\) 0 0
\(985\) 8.65248 0.275691
\(986\) 0 0
\(987\) 8.95554 0.285058
\(988\) 0 0
\(989\) −40.3951 −1.28449
\(990\) 0 0
\(991\) 30.6708 0.974291 0.487146 0.873321i \(-0.338038\pi\)
0.487146 + 0.873321i \(0.338038\pi\)
\(992\) 0 0
\(993\) −72.4853 −2.30025
\(994\) 0 0
\(995\) 7.56231 0.239741
\(996\) 65.9129 2.08853
\(997\) 9.23607 0.292509 0.146255 0.989247i \(-0.453278\pi\)
0.146255 + 0.989247i \(0.453278\pi\)
\(998\) 0 0
\(999\) −89.0689 −2.81801
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 1805.2.a.k.1.1 4
5.4 even 2 9025.2.a.bi.1.4 4
19.18 odd 2 inner 1805.2.a.k.1.4 yes 4
95.94 odd 2 9025.2.a.bi.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.k.1.1 4 1.1 even 1 trivial
1805.2.a.k.1.4 yes 4 19.18 odd 2 inner
9025.2.a.bi.1.1 4 95.94 odd 2
9025.2.a.bi.1.4 4 5.4 even 2