Properties

Label 9025.2.a.bn.1.4
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.0649878242\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.7168.1
Defining polynomial: \( x^{4} - 6x^{2} + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1805)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.10100\) of defining polynomial
Character \(\chi\) \(=\) 9025.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.10100 q^{2} -2.97127 q^{3} +2.41421 q^{4} -6.24264 q^{6} -4.82843 q^{7} +0.870264 q^{8} +5.82843 q^{9} +O(q^{10})\) \(q+2.10100 q^{2} -2.97127 q^{3} +2.41421 q^{4} -6.24264 q^{6} -4.82843 q^{7} +0.870264 q^{8} +5.82843 q^{9} +2.00000 q^{11} -7.17327 q^{12} +1.23074 q^{13} -10.1445 q^{14} -3.00000 q^{16} +3.65685 q^{17} +12.2455 q^{18} +14.3465 q^{21} +4.20201 q^{22} -4.82843 q^{23} -2.58579 q^{24} +2.58579 q^{26} -8.40401 q^{27} -11.6569 q^{28} -2.46148 q^{29} +5.94253 q^{31} -8.04354 q^{32} -5.94253 q^{33} +7.68306 q^{34} +14.0711 q^{36} +7.17327 q^{37} -3.65685 q^{39} +30.1421 q^{42} -0.828427 q^{43} +4.82843 q^{44} -10.1445 q^{46} +6.48528 q^{47} +8.91380 q^{48} +16.3137 q^{49} -10.8655 q^{51} +2.97127 q^{52} +4.71179 q^{53} -17.6569 q^{54} -4.20201 q^{56} -5.17157 q^{58} -11.8851 q^{59} +2.82843 q^{61} +12.4853 q^{62} -28.1421 q^{63} -10.8995 q^{64} -12.4853 q^{66} +5.43275 q^{67} +8.82843 q^{68} +14.3465 q^{69} -10.8655 q^{71} +5.07227 q^{72} +0.343146 q^{73} +15.0711 q^{74} -9.65685 q^{77} -7.68306 q^{78} +11.8851 q^{79} +7.48528 q^{81} -3.17157 q^{83} +34.6356 q^{84} -1.74053 q^{86} +7.31371 q^{87} +1.74053 q^{88} -9.42359 q^{89} -5.94253 q^{91} -11.6569 q^{92} -17.6569 q^{93} +13.6256 q^{94} +23.8995 q^{96} +1.23074 q^{97} +34.2752 q^{98} +11.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 8 q^{6} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 8 q^{6} - 8 q^{7} + 12 q^{9} + 8 q^{11} - 12 q^{16} - 8 q^{17} - 8 q^{23} - 16 q^{24} + 16 q^{26} - 24 q^{28} + 28 q^{36} + 8 q^{39} + 64 q^{42} + 8 q^{43} + 8 q^{44} - 8 q^{47} + 20 q^{49} - 48 q^{54} - 32 q^{58} + 16 q^{62} - 56 q^{63} - 4 q^{64} - 16 q^{66} + 24 q^{68} + 24 q^{73} + 32 q^{74} - 16 q^{77} - 4 q^{81} - 24 q^{83} - 16 q^{87} - 24 q^{92} - 48 q^{93} + 56 q^{96} + 24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.10100 1.48563 0.742817 0.669495i \(-0.233489\pi\)
0.742817 + 0.669495i \(0.233489\pi\)
\(3\) −2.97127 −1.71546 −0.857731 0.514099i \(-0.828126\pi\)
−0.857731 + 0.514099i \(0.828126\pi\)
\(4\) 2.41421 1.20711
\(5\) 0 0
\(6\) −6.24264 −2.54855
\(7\) −4.82843 −1.82497 −0.912487 0.409106i \(-0.865841\pi\)
−0.912487 + 0.409106i \(0.865841\pi\)
\(8\) 0.870264 0.307685
\(9\) 5.82843 1.94281
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −7.17327 −2.07075
\(13\) 1.23074 0.341346 0.170673 0.985328i \(-0.445406\pi\)
0.170673 + 0.985328i \(0.445406\pi\)
\(14\) −10.1445 −2.71124
\(15\) 0 0
\(16\) −3.00000 −0.750000
\(17\) 3.65685 0.886917 0.443459 0.896295i \(-0.353751\pi\)
0.443459 + 0.896295i \(0.353751\pi\)
\(18\) 12.2455 2.88630
\(19\) 0 0
\(20\) 0 0
\(21\) 14.3465 3.13067
\(22\) 4.20201 0.895871
\(23\) −4.82843 −1.00680 −0.503398 0.864054i \(-0.667917\pi\)
−0.503398 + 0.864054i \(0.667917\pi\)
\(24\) −2.58579 −0.527821
\(25\) 0 0
\(26\) 2.58579 0.507114
\(27\) −8.40401 −1.61735
\(28\) −11.6569 −2.20294
\(29\) −2.46148 −0.457085 −0.228543 0.973534i \(-0.573396\pi\)
−0.228543 + 0.973534i \(0.573396\pi\)
\(30\) 0 0
\(31\) 5.94253 1.06731 0.533655 0.845702i \(-0.320818\pi\)
0.533655 + 0.845702i \(0.320818\pi\)
\(32\) −8.04354 −1.42191
\(33\) −5.94253 −1.03446
\(34\) 7.68306 1.31763
\(35\) 0 0
\(36\) 14.0711 2.34518
\(37\) 7.17327 1.17928 0.589639 0.807667i \(-0.299270\pi\)
0.589639 + 0.807667i \(0.299270\pi\)
\(38\) 0 0
\(39\) −3.65685 −0.585565
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 30.1421 4.65103
\(43\) −0.828427 −0.126334 −0.0631670 0.998003i \(-0.520120\pi\)
−0.0631670 + 0.998003i \(0.520120\pi\)
\(44\) 4.82843 0.727913
\(45\) 0 0
\(46\) −10.1445 −1.49573
\(47\) 6.48528 0.945976 0.472988 0.881069i \(-0.343175\pi\)
0.472988 + 0.881069i \(0.343175\pi\)
\(48\) 8.91380 1.28660
\(49\) 16.3137 2.33053
\(50\) 0 0
\(51\) −10.8655 −1.52147
\(52\) 2.97127 0.412041
\(53\) 4.71179 0.647215 0.323607 0.946191i \(-0.395104\pi\)
0.323607 + 0.946191i \(0.395104\pi\)
\(54\) −17.6569 −2.40279
\(55\) 0 0
\(56\) −4.20201 −0.561517
\(57\) 0 0
\(58\) −5.17157 −0.679061
\(59\) −11.8851 −1.54730 −0.773652 0.633611i \(-0.781572\pi\)
−0.773652 + 0.633611i \(0.781572\pi\)
\(60\) 0 0
\(61\) 2.82843 0.362143 0.181071 0.983470i \(-0.442043\pi\)
0.181071 + 0.983470i \(0.442043\pi\)
\(62\) 12.4853 1.58563
\(63\) −28.1421 −3.54558
\(64\) −10.8995 −1.36244
\(65\) 0 0
\(66\) −12.4853 −1.53683
\(67\) 5.43275 0.663715 0.331858 0.943329i \(-0.392325\pi\)
0.331858 + 0.943329i \(0.392325\pi\)
\(68\) 8.82843 1.07060
\(69\) 14.3465 1.72712
\(70\) 0 0
\(71\) −10.8655 −1.28950 −0.644748 0.764395i \(-0.723038\pi\)
−0.644748 + 0.764395i \(0.723038\pi\)
\(72\) 5.07227 0.597773
\(73\) 0.343146 0.0401622 0.0200811 0.999798i \(-0.493608\pi\)
0.0200811 + 0.999798i \(0.493608\pi\)
\(74\) 15.0711 1.75198
\(75\) 0 0
\(76\) 0 0
\(77\) −9.65685 −1.10050
\(78\) −7.68306 −0.869935
\(79\) 11.8851 1.33717 0.668587 0.743634i \(-0.266899\pi\)
0.668587 + 0.743634i \(0.266899\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) 0 0
\(83\) −3.17157 −0.348125 −0.174063 0.984735i \(-0.555690\pi\)
−0.174063 + 0.984735i \(0.555690\pi\)
\(84\) 34.6356 3.77906
\(85\) 0 0
\(86\) −1.74053 −0.187686
\(87\) 7.31371 0.784112
\(88\) 1.74053 0.185541
\(89\) −9.42359 −0.998898 −0.499449 0.866343i \(-0.666464\pi\)
−0.499449 + 0.866343i \(0.666464\pi\)
\(90\) 0 0
\(91\) −5.94253 −0.622947
\(92\) −11.6569 −1.21531
\(93\) −17.6569 −1.83093
\(94\) 13.6256 1.40537
\(95\) 0 0
\(96\) 23.8995 2.43923
\(97\) 1.23074 0.124963 0.0624813 0.998046i \(-0.480099\pi\)
0.0624813 + 0.998046i \(0.480099\pi\)
\(98\) 34.2752 3.46231
\(99\) 11.6569 1.17156
\(100\) 0 0
\(101\) −16.4853 −1.64035 −0.820173 0.572115i \(-0.806123\pi\)
−0.820173 + 0.572115i \(0.806123\pi\)
\(102\) −22.8284 −2.26035
\(103\) 8.91380 0.878303 0.439151 0.898413i \(-0.355279\pi\)
0.439151 + 0.898413i \(0.355279\pi\)
\(104\) 1.07107 0.105027
\(105\) 0 0
\(106\) 9.89949 0.961524
\(107\) 6.45232 0.623770 0.311885 0.950120i \(-0.399040\pi\)
0.311885 + 0.950120i \(0.399040\pi\)
\(108\) −20.2891 −1.95232
\(109\) 19.2695 1.84568 0.922842 0.385179i \(-0.125860\pi\)
0.922842 + 0.385179i \(0.125860\pi\)
\(110\) 0 0
\(111\) −21.3137 −2.02301
\(112\) 14.4853 1.36873
\(113\) −9.63475 −0.906361 −0.453181 0.891419i \(-0.649711\pi\)
−0.453181 + 0.891419i \(0.649711\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −5.94253 −0.551750
\(117\) 7.17327 0.663169
\(118\) −24.9706 −2.29873
\(119\) −17.6569 −1.61860
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 5.94253 0.538012
\(123\) 0 0
\(124\) 14.3465 1.28836
\(125\) 0 0
\(126\) −59.1267 −5.26743
\(127\) 12.3949 1.09987 0.549933 0.835209i \(-0.314653\pi\)
0.549933 + 0.835209i \(0.314653\pi\)
\(128\) −6.81280 −0.602172
\(129\) 2.46148 0.216721
\(130\) 0 0
\(131\) −7.31371 −0.639002 −0.319501 0.947586i \(-0.603515\pi\)
−0.319501 + 0.947586i \(0.603515\pi\)
\(132\) −14.3465 −1.24871
\(133\) 0 0
\(134\) 11.4142 0.986038
\(135\) 0 0
\(136\) 3.18243 0.272891
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 30.1421 2.56587
\(139\) 8.34315 0.707656 0.353828 0.935310i \(-0.384880\pi\)
0.353828 + 0.935310i \(0.384880\pi\)
\(140\) 0 0
\(141\) −19.2695 −1.62278
\(142\) −22.8284 −1.91572
\(143\) 2.46148 0.205839
\(144\) −17.4853 −1.45711
\(145\) 0 0
\(146\) 0.720950 0.0596663
\(147\) −48.4724 −3.99793
\(148\) 17.3178 1.42352
\(149\) −9.17157 −0.751365 −0.375682 0.926749i \(-0.622592\pi\)
−0.375682 + 0.926749i \(0.622592\pi\)
\(150\) 0 0
\(151\) −17.8276 −1.45079 −0.725395 0.688333i \(-0.758343\pi\)
−0.725395 + 0.688333i \(0.758343\pi\)
\(152\) 0 0
\(153\) 21.3137 1.72311
\(154\) −20.2891 −1.63494
\(155\) 0 0
\(156\) −8.82843 −0.706840
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 24.9706 1.98655
\(159\) −14.0000 −1.11027
\(160\) 0 0
\(161\) 23.3137 1.83738
\(162\) 15.7266 1.23560
\(163\) −12.8284 −1.00480 −0.502400 0.864635i \(-0.667549\pi\)
−0.502400 + 0.864635i \(0.667549\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −6.66348 −0.517187
\(167\) −22.2408 −1.72104 −0.860521 0.509415i \(-0.829862\pi\)
−0.860521 + 0.509415i \(0.829862\pi\)
\(168\) 12.4853 0.963260
\(169\) −11.4853 −0.883483
\(170\) 0 0
\(171\) 0 0
\(172\) −2.00000 −0.152499
\(173\) −13.1158 −0.997176 −0.498588 0.866839i \(-0.666148\pi\)
−0.498588 + 0.866839i \(0.666148\pi\)
\(174\) 15.3661 1.16490
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 35.3137 2.65434
\(178\) −19.7990 −1.48400
\(179\) −4.92296 −0.367959 −0.183980 0.982930i \(-0.558898\pi\)
−0.183980 + 0.982930i \(0.558898\pi\)
\(180\) 0 0
\(181\) 16.8080 1.24933 0.624665 0.780893i \(-0.285235\pi\)
0.624665 + 0.780893i \(0.285235\pi\)
\(182\) −12.4853 −0.925471
\(183\) −8.40401 −0.621242
\(184\) −4.20201 −0.309776
\(185\) 0 0
\(186\) −37.0971 −2.72009
\(187\) 7.31371 0.534831
\(188\) 15.6569 1.14189
\(189\) 40.5782 2.95163
\(190\) 0 0
\(191\) −13.6569 −0.988175 −0.494088 0.869412i \(-0.664498\pi\)
−0.494088 + 0.869412i \(0.664498\pi\)
\(192\) 32.3853 2.33721
\(193\) 8.19285 0.589734 0.294867 0.955538i \(-0.404725\pi\)
0.294867 + 0.955538i \(0.404725\pi\)
\(194\) 2.58579 0.185649
\(195\) 0 0
\(196\) 39.3848 2.81320
\(197\) 13.3137 0.948562 0.474281 0.880373i \(-0.342708\pi\)
0.474281 + 0.880373i \(0.342708\pi\)
\(198\) 24.4911 1.74051
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −16.1421 −1.13858
\(202\) −34.6356 −2.43695
\(203\) 11.8851 0.834168
\(204\) −26.2316 −1.83658
\(205\) 0 0
\(206\) 18.7279 1.30484
\(207\) −28.1421 −1.95601
\(208\) −3.69222 −0.256009
\(209\) 0 0
\(210\) 0 0
\(211\) 10.8655 0.748011 0.374006 0.927426i \(-0.377984\pi\)
0.374006 + 0.927426i \(0.377984\pi\)
\(212\) 11.3753 0.781257
\(213\) 32.2843 2.21208
\(214\) 13.5563 0.926693
\(215\) 0 0
\(216\) −7.31371 −0.497635
\(217\) −28.6931 −1.94781
\(218\) 40.4853 2.74201
\(219\) −1.01958 −0.0688967
\(220\) 0 0
\(221\) 4.50063 0.302745
\(222\) −44.7802 −3.00545
\(223\) −17.3178 −1.15969 −0.579843 0.814728i \(-0.696886\pi\)
−0.579843 + 0.814728i \(0.696886\pi\)
\(224\) 38.8376 2.59495
\(225\) 0 0
\(226\) −20.2426 −1.34652
\(227\) −22.2408 −1.47617 −0.738086 0.674707i \(-0.764270\pi\)
−0.738086 + 0.674707i \(0.764270\pi\)
\(228\) 0 0
\(229\) 12.4853 0.825051 0.412525 0.910946i \(-0.364647\pi\)
0.412525 + 0.910946i \(0.364647\pi\)
\(230\) 0 0
\(231\) 28.6931 1.88787
\(232\) −2.14214 −0.140638
\(233\) −13.3137 −0.872210 −0.436105 0.899896i \(-0.643642\pi\)
−0.436105 + 0.899896i \(0.643642\pi\)
\(234\) 15.0711 0.985227
\(235\) 0 0
\(236\) −28.6931 −1.86776
\(237\) −35.3137 −2.29387
\(238\) −37.0971 −2.40465
\(239\) −1.65685 −0.107173 −0.0535865 0.998563i \(-0.517065\pi\)
−0.0535865 + 0.998563i \(0.517065\pi\)
\(240\) 0 0
\(241\) 6.96211 0.448469 0.224235 0.974535i \(-0.428012\pi\)
0.224235 + 0.974535i \(0.428012\pi\)
\(242\) −14.7070 −0.945403
\(243\) 2.97127 0.190607
\(244\) 6.82843 0.437145
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 5.17157 0.328395
\(249\) 9.42359 0.597196
\(250\) 0 0
\(251\) −12.9706 −0.818695 −0.409347 0.912379i \(-0.634244\pi\)
−0.409347 + 0.912379i \(0.634244\pi\)
\(252\) −67.9411 −4.27989
\(253\) −9.65685 −0.607121
\(254\) 26.0416 1.63400
\(255\) 0 0
\(256\) 7.48528 0.467830
\(257\) −8.19285 −0.511056 −0.255528 0.966802i \(-0.582249\pi\)
−0.255528 + 0.966802i \(0.582249\pi\)
\(258\) 5.17157 0.321968
\(259\) −34.6356 −2.15215
\(260\) 0 0
\(261\) −14.3465 −0.888029
\(262\) −15.3661 −0.949322
\(263\) −12.1421 −0.748716 −0.374358 0.927284i \(-0.622137\pi\)
−0.374358 + 0.927284i \(0.622137\pi\)
\(264\) −5.17157 −0.318288
\(265\) 0 0
\(266\) 0 0
\(267\) 28.0000 1.71357
\(268\) 13.1158 0.801175
\(269\) −2.46148 −0.150079 −0.0750395 0.997181i \(-0.523908\pi\)
−0.0750395 + 0.997181i \(0.523908\pi\)
\(270\) 0 0
\(271\) −2.97056 −0.180449 −0.0902244 0.995921i \(-0.528758\pi\)
−0.0902244 + 0.995921i \(0.528758\pi\)
\(272\) −10.9706 −0.665188
\(273\) 17.6569 1.06864
\(274\) 4.20201 0.253852
\(275\) 0 0
\(276\) 34.6356 2.08482
\(277\) −5.31371 −0.319270 −0.159635 0.987176i \(-0.551032\pi\)
−0.159635 + 0.987176i \(0.551032\pi\)
\(278\) 17.5290 1.05132
\(279\) 34.6356 2.07358
\(280\) 0 0
\(281\) 23.7701 1.41801 0.709004 0.705205i \(-0.249145\pi\)
0.709004 + 0.705205i \(0.249145\pi\)
\(282\) −40.4853 −2.41086
\(283\) −29.7990 −1.77137 −0.885683 0.464290i \(-0.846309\pi\)
−0.885683 + 0.464290i \(0.846309\pi\)
\(284\) −26.2316 −1.55656
\(285\) 0 0
\(286\) 5.17157 0.305802
\(287\) 0 0
\(288\) −46.8812 −2.76250
\(289\) −3.62742 −0.213377
\(290\) 0 0
\(291\) −3.65685 −0.214369
\(292\) 0.828427 0.0484800
\(293\) −12.0962 −0.706669 −0.353335 0.935497i \(-0.614952\pi\)
−0.353335 + 0.935497i \(0.614952\pi\)
\(294\) −101.841 −5.93947
\(295\) 0 0
\(296\) 6.24264 0.362846
\(297\) −16.8080 −0.975300
\(298\) −19.2695 −1.11625
\(299\) −5.94253 −0.343666
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) −37.4558 −2.15534
\(303\) 48.9822 2.81395
\(304\) 0 0
\(305\) 0 0
\(306\) 44.7802 2.55991
\(307\) 11.3753 0.649221 0.324611 0.945848i \(-0.394767\pi\)
0.324611 + 0.945848i \(0.394767\pi\)
\(308\) −23.3137 −1.32842
\(309\) −26.4853 −1.50670
\(310\) 0 0
\(311\) −31.6569 −1.79510 −0.897548 0.440917i \(-0.854653\pi\)
−0.897548 + 0.440917i \(0.854653\pi\)
\(312\) −3.18243 −0.180170
\(313\) 20.6274 1.16593 0.582965 0.812497i \(-0.301892\pi\)
0.582965 + 0.812497i \(0.301892\pi\)
\(314\) −37.8181 −2.13420
\(315\) 0 0
\(316\) 28.6931 1.61411
\(317\) −15.5773 −0.874907 −0.437454 0.899241i \(-0.644120\pi\)
−0.437454 + 0.899241i \(0.644120\pi\)
\(318\) −29.4140 −1.64946
\(319\) −4.92296 −0.275633
\(320\) 0 0
\(321\) −19.1716 −1.07005
\(322\) 48.9822 2.72967
\(323\) 0 0
\(324\) 18.0711 1.00395
\(325\) 0 0
\(326\) −26.9526 −1.49276
\(327\) −57.2548 −3.16620
\(328\) 0 0
\(329\) −31.3137 −1.72638
\(330\) 0 0
\(331\) −29.7127 −1.63316 −0.816578 0.577235i \(-0.804132\pi\)
−0.816578 + 0.577235i \(0.804132\pi\)
\(332\) −7.65685 −0.420224
\(333\) 41.8089 2.29111
\(334\) −46.7279 −2.55684
\(335\) 0 0
\(336\) −43.0396 −2.34800
\(337\) 30.9434 1.68559 0.842797 0.538231i \(-0.180907\pi\)
0.842797 + 0.538231i \(0.180907\pi\)
\(338\) −24.1306 −1.31253
\(339\) 28.6274 1.55483
\(340\) 0 0
\(341\) 11.8851 0.643612
\(342\) 0 0
\(343\) −44.9706 −2.42818
\(344\) −0.720950 −0.0388710
\(345\) 0 0
\(346\) −27.5563 −1.48144
\(347\) −24.8284 −1.33286 −0.666430 0.745568i \(-0.732178\pi\)
−0.666430 + 0.745568i \(0.732178\pi\)
\(348\) 17.6569 0.946507
\(349\) −6.68629 −0.357909 −0.178954 0.983857i \(-0.557271\pi\)
−0.178954 + 0.983857i \(0.557271\pi\)
\(350\) 0 0
\(351\) −10.3431 −0.552076
\(352\) −16.0871 −0.857444
\(353\) 3.65685 0.194635 0.0973174 0.995253i \(-0.468974\pi\)
0.0973174 + 0.995253i \(0.468974\pi\)
\(354\) 74.1942 3.94338
\(355\) 0 0
\(356\) −22.7506 −1.20578
\(357\) 52.4632 2.77665
\(358\) −10.3431 −0.546652
\(359\) −25.3137 −1.33601 −0.668003 0.744158i \(-0.732851\pi\)
−0.668003 + 0.744158i \(0.732851\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 35.3137 1.85605
\(363\) 20.7989 1.09166
\(364\) −14.3465 −0.751963
\(365\) 0 0
\(366\) −17.6569 −0.922939
\(367\) −8.82843 −0.460840 −0.230420 0.973091i \(-0.574010\pi\)
−0.230420 + 0.973091i \(0.574010\pi\)
\(368\) 14.4853 0.755097
\(369\) 0 0
\(370\) 0 0
\(371\) −22.7506 −1.18115
\(372\) −42.6274 −2.21013
\(373\) −23.9813 −1.24170 −0.620852 0.783928i \(-0.713213\pi\)
−0.620852 + 0.783928i \(0.713213\pi\)
\(374\) 15.3661 0.794563
\(375\) 0 0
\(376\) 5.64391 0.291062
\(377\) −3.02944 −0.156024
\(378\) 85.2548 4.38504
\(379\) 4.92296 0.252875 0.126438 0.991975i \(-0.459646\pi\)
0.126438 + 0.991975i \(0.459646\pi\)
\(380\) 0 0
\(381\) −36.8284 −1.88678
\(382\) −28.6931 −1.46807
\(383\) 12.8172 0.654927 0.327464 0.944864i \(-0.393806\pi\)
0.327464 + 0.944864i \(0.393806\pi\)
\(384\) 20.2426 1.03300
\(385\) 0 0
\(386\) 17.2132 0.876129
\(387\) −4.82843 −0.245443
\(388\) 2.97127 0.150843
\(389\) 24.6274 1.24866 0.624330 0.781161i \(-0.285372\pi\)
0.624330 + 0.781161i \(0.285372\pi\)
\(390\) 0 0
\(391\) −17.6569 −0.892946
\(392\) 14.1972 0.717069
\(393\) 21.7310 1.09618
\(394\) 27.9721 1.40922
\(395\) 0 0
\(396\) 28.1421 1.41420
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −8.40401 −0.421255
\(399\) 0 0
\(400\) 0 0
\(401\) 16.8080 0.839353 0.419676 0.907674i \(-0.362144\pi\)
0.419676 + 0.907674i \(0.362144\pi\)
\(402\) −33.9147 −1.69151
\(403\) 7.31371 0.364322
\(404\) −39.7990 −1.98007
\(405\) 0 0
\(406\) 24.9706 1.23927
\(407\) 14.3465 0.711132
\(408\) −9.45584 −0.468134
\(409\) 38.1167 1.88475 0.942374 0.334560i \(-0.108588\pi\)
0.942374 + 0.334560i \(0.108588\pi\)
\(410\) 0 0
\(411\) −5.94253 −0.293124
\(412\) 21.5198 1.06021
\(413\) 57.3862 2.82379
\(414\) −59.1267 −2.90592
\(415\) 0 0
\(416\) −9.89949 −0.485363
\(417\) −24.7897 −1.21396
\(418\) 0 0
\(419\) 8.97056 0.438241 0.219120 0.975698i \(-0.429681\pi\)
0.219120 + 0.975698i \(0.429681\pi\)
\(420\) 0 0
\(421\) 16.8080 0.819173 0.409586 0.912271i \(-0.365673\pi\)
0.409586 + 0.912271i \(0.365673\pi\)
\(422\) 22.8284 1.11127
\(423\) 37.7990 1.83785
\(424\) 4.10051 0.199138
\(425\) 0 0
\(426\) 67.8294 3.28634
\(427\) −13.6569 −0.660901
\(428\) 15.5773 0.752956
\(429\) −7.31371 −0.353109
\(430\) 0 0
\(431\) 5.94253 0.286242 0.143121 0.989705i \(-0.454286\pi\)
0.143121 + 0.989705i \(0.454286\pi\)
\(432\) 25.2120 1.21301
\(433\) −33.4049 −1.60534 −0.802668 0.596426i \(-0.796587\pi\)
−0.802668 + 0.596426i \(0.796587\pi\)
\(434\) −60.2843 −2.89374
\(435\) 0 0
\(436\) 46.5207 2.22794
\(437\) 0 0
\(438\) −2.14214 −0.102355
\(439\) −28.6931 −1.36945 −0.684723 0.728803i \(-0.740077\pi\)
−0.684723 + 0.728803i \(0.740077\pi\)
\(440\) 0 0
\(441\) 95.0833 4.52777
\(442\) 9.45584 0.449769
\(443\) 8.14214 0.386845 0.193422 0.981116i \(-0.438041\pi\)
0.193422 + 0.981116i \(0.438041\pi\)
\(444\) −51.4558 −2.44199
\(445\) 0 0
\(446\) −36.3848 −1.72287
\(447\) 27.2512 1.28894
\(448\) 52.6274 2.48641
\(449\) 2.46148 0.116164 0.0580822 0.998312i \(-0.481501\pi\)
0.0580822 + 0.998312i \(0.481501\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −23.2603 −1.09407
\(453\) 52.9706 2.48877
\(454\) −46.7279 −2.19305
\(455\) 0 0
\(456\) 0 0
\(457\) 17.3137 0.809901 0.404951 0.914339i \(-0.367289\pi\)
0.404951 + 0.914339i \(0.367289\pi\)
\(458\) 26.2316 1.22572
\(459\) −30.7322 −1.43446
\(460\) 0 0
\(461\) −33.3137 −1.55157 −0.775787 0.630995i \(-0.782647\pi\)
−0.775787 + 0.630995i \(0.782647\pi\)
\(462\) 60.2843 2.80468
\(463\) 27.1716 1.26277 0.631385 0.775469i \(-0.282487\pi\)
0.631385 + 0.775469i \(0.282487\pi\)
\(464\) 7.38443 0.342814
\(465\) 0 0
\(466\) −27.9721 −1.29578
\(467\) −21.5147 −0.995582 −0.497791 0.867297i \(-0.665855\pi\)
−0.497791 + 0.867297i \(0.665855\pi\)
\(468\) 17.3178 0.800516
\(469\) −26.2316 −1.21126
\(470\) 0 0
\(471\) 53.4828 2.46436
\(472\) −10.3431 −0.476082
\(473\) −1.65685 −0.0761822
\(474\) −74.1942 −3.40785
\(475\) 0 0
\(476\) −42.6274 −1.95382
\(477\) 27.4624 1.25741
\(478\) −3.48106 −0.159220
\(479\) 10.0000 0.456912 0.228456 0.973554i \(-0.426632\pi\)
0.228456 + 0.973554i \(0.426632\pi\)
\(480\) 0 0
\(481\) 8.82843 0.402542
\(482\) 14.6274 0.666261
\(483\) −69.2713 −3.15195
\(484\) −16.8995 −0.768159
\(485\) 0 0
\(486\) 6.24264 0.283172
\(487\) 15.8759 0.719406 0.359703 0.933067i \(-0.382878\pi\)
0.359703 + 0.933067i \(0.382878\pi\)
\(488\) 2.46148 0.111426
\(489\) 38.1167 1.72370
\(490\) 0 0
\(491\) −24.2843 −1.09593 −0.547967 0.836500i \(-0.684598\pi\)
−0.547967 + 0.836500i \(0.684598\pi\)
\(492\) 0 0
\(493\) −9.00127 −0.405397
\(494\) 0 0
\(495\) 0 0
\(496\) −17.8276 −0.800483
\(497\) 52.4632 2.35330
\(498\) 19.7990 0.887214
\(499\) 34.9706 1.56550 0.782749 0.622338i \(-0.213817\pi\)
0.782749 + 0.622338i \(0.213817\pi\)
\(500\) 0 0
\(501\) 66.0833 2.95238
\(502\) −27.2512 −1.21628
\(503\) 14.4853 0.645867 0.322933 0.946422i \(-0.395331\pi\)
0.322933 + 0.946422i \(0.395331\pi\)
\(504\) −24.4911 −1.09092
\(505\) 0 0
\(506\) −20.2891 −0.901960
\(507\) 34.1258 1.51558
\(508\) 29.9238 1.32766
\(509\) −4.50063 −0.199487 −0.0997435 0.995013i \(-0.531802\pi\)
−0.0997435 + 0.995013i \(0.531802\pi\)
\(510\) 0 0
\(511\) −1.65685 −0.0732949
\(512\) 29.3522 1.29720
\(513\) 0 0
\(514\) −17.2132 −0.759242
\(515\) 0 0
\(516\) 5.94253 0.261605
\(517\) 12.9706 0.570445
\(518\) −72.7696 −3.19731
\(519\) 38.9706 1.71062
\(520\) 0 0
\(521\) 4.92296 0.215679 0.107839 0.994168i \(-0.465607\pi\)
0.107839 + 0.994168i \(0.465607\pi\)
\(522\) −30.1421 −1.31929
\(523\) 7.89422 0.345190 0.172595 0.984993i \(-0.444785\pi\)
0.172595 + 0.984993i \(0.444785\pi\)
\(524\) −17.6569 −0.771343
\(525\) 0 0
\(526\) −25.5107 −1.11232
\(527\) 21.7310 0.946616
\(528\) 17.8276 0.775847
\(529\) 0.313708 0.0136395
\(530\) 0 0
\(531\) −69.2713 −3.00612
\(532\) 0 0
\(533\) 0 0
\(534\) 58.8281 2.54574
\(535\) 0 0
\(536\) 4.72792 0.204215
\(537\) 14.6274 0.631220
\(538\) −5.17157 −0.222962
\(539\) 32.6274 1.40536
\(540\) 0 0
\(541\) −6.14214 −0.264071 −0.132036 0.991245i \(-0.542151\pi\)
−0.132036 + 0.991245i \(0.542151\pi\)
\(542\) −6.24116 −0.268081
\(543\) −49.9411 −2.14318
\(544\) −29.4140 −1.26112
\(545\) 0 0
\(546\) 37.0971 1.58761
\(547\) 13.8368 0.591617 0.295809 0.955247i \(-0.404411\pi\)
0.295809 + 0.955247i \(0.404411\pi\)
\(548\) 4.82843 0.206260
\(549\) 16.4853 0.703575
\(550\) 0 0
\(551\) 0 0
\(552\) 12.4853 0.531409
\(553\) −57.3862 −2.44031
\(554\) −11.1641 −0.474318
\(555\) 0 0
\(556\) 20.1421 0.854217
\(557\) −16.6274 −0.704526 −0.352263 0.935901i \(-0.614588\pi\)
−0.352263 + 0.935901i \(0.614588\pi\)
\(558\) 72.7696 3.08058
\(559\) −1.01958 −0.0431235
\(560\) 0 0
\(561\) −21.7310 −0.917483
\(562\) 49.9411 2.10664
\(563\) 9.93338 0.418642 0.209321 0.977847i \(-0.432875\pi\)
0.209321 + 0.977847i \(0.432875\pi\)
\(564\) −46.5207 −1.95887
\(565\) 0 0
\(566\) −62.6078 −2.63160
\(567\) −36.1421 −1.51783
\(568\) −9.45584 −0.396758
\(569\) −9.42359 −0.395057 −0.197529 0.980297i \(-0.563292\pi\)
−0.197529 + 0.980297i \(0.563292\pi\)
\(570\) 0 0
\(571\) −9.31371 −0.389767 −0.194883 0.980826i \(-0.562433\pi\)
−0.194883 + 0.980826i \(0.562433\pi\)
\(572\) 5.94253 0.248470
\(573\) 40.5782 1.69518
\(574\) 0 0
\(575\) 0 0
\(576\) −63.5269 −2.64695
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −7.62121 −0.317001
\(579\) −24.3431 −1.01167
\(580\) 0 0
\(581\) 15.3137 0.635320
\(582\) −7.68306 −0.318473
\(583\) 9.42359 0.390285
\(584\) 0.298627 0.0123573
\(585\) 0 0
\(586\) −25.4142 −1.04985
\(587\) −1.51472 −0.0625191 −0.0312596 0.999511i \(-0.509952\pi\)
−0.0312596 + 0.999511i \(0.509952\pi\)
\(588\) −117.023 −4.82593
\(589\) 0 0
\(590\) 0 0
\(591\) −39.5586 −1.62722
\(592\) −21.5198 −0.884459
\(593\) −13.3137 −0.546728 −0.273364 0.961911i \(-0.588136\pi\)
−0.273364 + 0.961911i \(0.588136\pi\)
\(594\) −35.3137 −1.44894
\(595\) 0 0
\(596\) −22.1421 −0.906977
\(597\) 11.8851 0.486423
\(598\) −12.4853 −0.510561
\(599\) 9.84591 0.402293 0.201147 0.979561i \(-0.435533\pi\)
0.201147 + 0.979561i \(0.435533\pi\)
\(600\) 0 0
\(601\) 4.92296 0.200812 0.100406 0.994947i \(-0.467986\pi\)
0.100406 + 0.994947i \(0.467986\pi\)
\(602\) 8.40401 0.342522
\(603\) 31.6644 1.28947
\(604\) −43.0396 −1.75126
\(605\) 0 0
\(606\) 102.912 4.18050
\(607\) −21.8184 −0.885583 −0.442792 0.896625i \(-0.646012\pi\)
−0.442792 + 0.896625i \(0.646012\pi\)
\(608\) 0 0
\(609\) −35.3137 −1.43098
\(610\) 0 0
\(611\) 7.98169 0.322905
\(612\) 51.4558 2.07998
\(613\) −42.2843 −1.70785 −0.853923 0.520400i \(-0.825783\pi\)
−0.853923 + 0.520400i \(0.825783\pi\)
\(614\) 23.8995 0.964505
\(615\) 0 0
\(616\) −8.40401 −0.338607
\(617\) 26.2843 1.05816 0.529082 0.848570i \(-0.322536\pi\)
0.529082 + 0.848570i \(0.322536\pi\)
\(618\) −55.6457 −2.23840
\(619\) 14.9706 0.601718 0.300859 0.953669i \(-0.402727\pi\)
0.300859 + 0.953669i \(0.402727\pi\)
\(620\) 0 0
\(621\) 40.5782 1.62835
\(622\) −66.5111 −2.66685
\(623\) 45.5011 1.82296
\(624\) 10.9706 0.439174
\(625\) 0 0
\(626\) 43.3383 1.73215
\(627\) 0 0
\(628\) −43.4558 −1.73408
\(629\) 26.2316 1.04592
\(630\) 0 0
\(631\) −38.2843 −1.52407 −0.762036 0.647534i \(-0.775800\pi\)
−0.762036 + 0.647534i \(0.775800\pi\)
\(632\) 10.3431 0.411428
\(633\) −32.2843 −1.28318
\(634\) −32.7279 −1.29979
\(635\) 0 0
\(636\) −33.7990 −1.34022
\(637\) 20.0779 0.795516
\(638\) −10.3431 −0.409489
\(639\) −63.3287 −2.50525
\(640\) 0 0
\(641\) −35.6552 −1.40830 −0.704148 0.710053i \(-0.748671\pi\)
−0.704148 + 0.710053i \(0.748671\pi\)
\(642\) −40.2795 −1.58971
\(643\) −26.4853 −1.04448 −0.522239 0.852799i \(-0.674903\pi\)
−0.522239 + 0.852799i \(0.674903\pi\)
\(644\) 56.2843 2.21791
\(645\) 0 0
\(646\) 0 0
\(647\) 32.8284 1.29062 0.645309 0.763921i \(-0.276728\pi\)
0.645309 + 0.763921i \(0.276728\pi\)
\(648\) 6.51417 0.255901
\(649\) −23.7701 −0.933059
\(650\) 0 0
\(651\) 85.2548 3.34140
\(652\) −30.9706 −1.21290
\(653\) −16.3431 −0.639557 −0.319778 0.947492i \(-0.603608\pi\)
−0.319778 + 0.947492i \(0.603608\pi\)
\(654\) −120.293 −4.70381
\(655\) 0 0
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) −65.7902 −2.56477
\(659\) 33.6160 1.30950 0.654748 0.755848i \(-0.272775\pi\)
0.654748 + 0.755848i \(0.272775\pi\)
\(660\) 0 0
\(661\) −28.6931 −1.11603 −0.558016 0.829830i \(-0.688437\pi\)
−0.558016 + 0.829830i \(0.688437\pi\)
\(662\) −62.4264 −2.42627
\(663\) −13.3726 −0.519348
\(664\) −2.76011 −0.107113
\(665\) 0 0
\(666\) 87.8406 3.40375
\(667\) 11.8851 0.460192
\(668\) −53.6940 −2.07748
\(669\) 51.4558 1.98940
\(670\) 0 0
\(671\) 5.65685 0.218380
\(672\) −115.397 −4.45153
\(673\) 7.59560 0.292789 0.146394 0.989226i \(-0.453233\pi\)
0.146394 + 0.989226i \(0.453233\pi\)
\(674\) 65.0122 2.50418
\(675\) 0 0
\(676\) −27.7279 −1.06646
\(677\) 21.5198 0.827074 0.413537 0.910487i \(-0.364293\pi\)
0.413537 + 0.910487i \(0.364293\pi\)
\(678\) 60.1463 2.30990
\(679\) −5.94253 −0.228054
\(680\) 0 0
\(681\) 66.0833 2.53232
\(682\) 24.9706 0.956172
\(683\) −13.8368 −0.529449 −0.264724 0.964324i \(-0.585281\pi\)
−0.264724 + 0.964324i \(0.585281\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −94.4833 −3.60739
\(687\) −37.0971 −1.41534
\(688\) 2.48528 0.0947505
\(689\) 5.79899 0.220924
\(690\) 0 0
\(691\) −28.6274 −1.08904 −0.544519 0.838748i \(-0.683288\pi\)
−0.544519 + 0.838748i \(0.683288\pi\)
\(692\) −31.6644 −1.20370
\(693\) −56.2843 −2.13806
\(694\) −52.1646 −1.98014
\(695\) 0 0
\(696\) 6.36486 0.241259
\(697\) 0 0
\(698\) −14.0479 −0.531722
\(699\) 39.5586 1.49624
\(700\) 0 0
\(701\) −7.51472 −0.283827 −0.141914 0.989879i \(-0.545325\pi\)
−0.141914 + 0.989879i \(0.545325\pi\)
\(702\) −21.7310 −0.820183
\(703\) 0 0
\(704\) −21.7990 −0.821580
\(705\) 0 0
\(706\) 7.68306 0.289156
\(707\) 79.5980 2.99359
\(708\) 85.2548 3.20407
\(709\) 17.3137 0.650230 0.325115 0.945674i \(-0.394597\pi\)
0.325115 + 0.945674i \(0.394597\pi\)
\(710\) 0 0
\(711\) 69.2713 2.59787
\(712\) −8.20101 −0.307346
\(713\) −28.6931 −1.07456
\(714\) 110.225 4.12508
\(715\) 0 0
\(716\) −11.8851 −0.444166
\(717\) 4.92296 0.183851
\(718\) −53.1842 −1.98482
\(719\) −35.9411 −1.34038 −0.670189 0.742191i \(-0.733787\pi\)
−0.670189 + 0.742191i \(0.733787\pi\)
\(720\) 0 0
\(721\) −43.0396 −1.60288
\(722\) 0 0
\(723\) −20.6863 −0.769331
\(724\) 40.5782 1.50808
\(725\) 0 0
\(726\) 43.6985 1.62180
\(727\) 10.4853 0.388878 0.194439 0.980915i \(-0.437711\pi\)
0.194439 + 0.980915i \(0.437711\pi\)
\(728\) −5.17157 −0.191671
\(729\) −31.2843 −1.15868
\(730\) 0 0
\(731\) −3.02944 −0.112048
\(732\) −20.2891 −0.749906
\(733\) 24.3431 0.899135 0.449567 0.893246i \(-0.351578\pi\)
0.449567 + 0.893246i \(0.351578\pi\)
\(734\) −18.5486 −0.684640
\(735\) 0 0
\(736\) 38.8376 1.43157
\(737\) 10.8655 0.400235
\(738\) 0 0
\(739\) 46.6274 1.71522 0.857609 0.514303i \(-0.171949\pi\)
0.857609 + 0.514303i \(0.171949\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −47.7990 −1.75476
\(743\) 5.43275 0.199308 0.0996540 0.995022i \(-0.468226\pi\)
0.0996540 + 0.995022i \(0.468226\pi\)
\(744\) −15.3661 −0.563349
\(745\) 0 0
\(746\) −50.3848 −1.84472
\(747\) −18.4853 −0.676341
\(748\) 17.6569 0.645599
\(749\) −31.1546 −1.13836
\(750\) 0 0
\(751\) 22.7506 0.830180 0.415090 0.909780i \(-0.363750\pi\)
0.415090 + 0.909780i \(0.363750\pi\)
\(752\) −19.4558 −0.709482
\(753\) 38.5390 1.40444
\(754\) −6.36486 −0.231794
\(755\) 0 0
\(756\) 97.9643 3.56293
\(757\) 29.3137 1.06542 0.532712 0.846296i \(-0.321173\pi\)
0.532712 + 0.846296i \(0.321173\pi\)
\(758\) 10.3431 0.375680
\(759\) 28.6931 1.04149
\(760\) 0 0
\(761\) −14.8284 −0.537530 −0.268765 0.963206i \(-0.586616\pi\)
−0.268765 + 0.963206i \(0.586616\pi\)
\(762\) −77.3766 −2.80306
\(763\) −93.0414 −3.36832
\(764\) −32.9706 −1.19283
\(765\) 0 0
\(766\) 26.9289 0.972982
\(767\) −14.6274 −0.528165
\(768\) −22.2408 −0.802545
\(769\) −14.1421 −0.509978 −0.254989 0.966944i \(-0.582072\pi\)
−0.254989 + 0.966944i \(0.582072\pi\)
\(770\) 0 0
\(771\) 24.3431 0.876697
\(772\) 19.7793 0.711872
\(773\) −22.5394 −0.810686 −0.405343 0.914165i \(-0.632848\pi\)
−0.405343 + 0.914165i \(0.632848\pi\)
\(774\) −10.1445 −0.364638
\(775\) 0 0
\(776\) 1.07107 0.0384491
\(777\) 102.912 3.69194
\(778\) 51.7423 1.85505
\(779\) 0 0
\(780\) 0 0
\(781\) −21.7310 −0.777596
\(782\) −37.0971 −1.32659
\(783\) 20.6863 0.739268
\(784\) −48.9411 −1.74790
\(785\) 0 0
\(786\) 45.6569 1.62853
\(787\) 23.6827 0.844196 0.422098 0.906550i \(-0.361294\pi\)
0.422098 + 0.906550i \(0.361294\pi\)
\(788\) 32.1421 1.14502
\(789\) 36.0775 1.28439
\(790\) 0 0
\(791\) 46.5207 1.65409
\(792\) 10.1445 0.360471
\(793\) 3.48106 0.123616
\(794\) 29.4140 1.04387
\(795\) 0 0
\(796\) −9.65685 −0.342278
\(797\) −7.17327 −0.254090 −0.127045 0.991897i \(-0.540549\pi\)
−0.127045 + 0.991897i \(0.540549\pi\)
\(798\) 0 0
\(799\) 23.7157 0.839002
\(800\) 0 0
\(801\) −54.9247 −1.94067
\(802\) 35.3137 1.24697
\(803\) 0.686292 0.0242187
\(804\) −38.9706 −1.37439
\(805\) 0 0
\(806\) 15.3661 0.541249
\(807\) 7.31371 0.257455
\(808\) −14.3465 −0.504710
\(809\) 9.31371 0.327453 0.163726 0.986506i \(-0.447649\pi\)
0.163726 + 0.986506i \(0.447649\pi\)
\(810\) 0 0
\(811\) 44.4815 1.56196 0.780979 0.624557i \(-0.214721\pi\)
0.780979 + 0.624557i \(0.214721\pi\)
\(812\) 28.6931 1.00693
\(813\) 8.82633 0.309553
\(814\) 30.1421 1.05648
\(815\) 0 0
\(816\) 32.5965 1.14110
\(817\) 0 0
\(818\) 80.0833 2.80005
\(819\) −34.6356 −1.21027
\(820\) 0 0
\(821\) 25.3137 0.883455 0.441727 0.897149i \(-0.354366\pi\)
0.441727 + 0.897149i \(0.354366\pi\)
\(822\) −12.4853 −0.435474
\(823\) 54.0833 1.88522 0.942612 0.333890i \(-0.108361\pi\)
0.942612 + 0.333890i \(0.108361\pi\)
\(824\) 7.75736 0.270240
\(825\) 0 0
\(826\) 120.569 4.19512
\(827\) −1.95169 −0.0678669 −0.0339334 0.999424i \(-0.510803\pi\)
−0.0339334 + 0.999424i \(0.510803\pi\)
\(828\) −67.9411 −2.36112
\(829\) −12.3074 −0.427453 −0.213727 0.976893i \(-0.568560\pi\)
−0.213727 + 0.976893i \(0.568560\pi\)
\(830\) 0 0
\(831\) 15.7884 0.547695
\(832\) −13.4144 −0.465062
\(833\) 59.6569 2.06699
\(834\) −52.0833 −1.80350
\(835\) 0 0
\(836\) 0 0
\(837\) −49.9411 −1.72622
\(838\) 18.8472 0.651065
\(839\) −18.8472 −0.650677 −0.325338 0.945598i \(-0.605478\pi\)
−0.325338 + 0.945598i \(0.605478\pi\)
\(840\) 0 0
\(841\) −22.9411 −0.791073
\(842\) 35.3137 1.21699
\(843\) −70.6274 −2.43254
\(844\) 26.2316 0.902929
\(845\) 0 0
\(846\) 79.4158 2.73037
\(847\) 33.7990 1.16135
\(848\) −14.1354 −0.485411
\(849\) 88.5408 3.03871
\(850\) 0 0
\(851\) −34.6356 −1.18729
\(852\) 77.9411 2.67022
\(853\) −47.6569 −1.63174 −0.815870 0.578236i \(-0.803741\pi\)
−0.815870 + 0.578236i \(0.803741\pi\)
\(854\) −28.6931 −0.981857
\(855\) 0 0
\(856\) 5.61522 0.191924
\(857\) −36.8859 −1.26000 −0.630000 0.776595i \(-0.716945\pi\)
−0.630000 + 0.776595i \(0.716945\pi\)
\(858\) −15.3661 −0.524591
\(859\) 12.2843 0.419134 0.209567 0.977794i \(-0.432795\pi\)
0.209567 + 0.977794i \(0.432795\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 12.4853 0.425250
\(863\) 24.2799 0.826498 0.413249 0.910618i \(-0.364394\pi\)
0.413249 + 0.910618i \(0.364394\pi\)
\(864\) 67.5980 2.29973
\(865\) 0 0
\(866\) −70.1838 −2.38494
\(867\) 10.7780 0.366041
\(868\) −69.2713 −2.35122
\(869\) 23.7701 0.806347
\(870\) 0 0
\(871\) 6.68629 0.226556
\(872\) 16.7696 0.567889
\(873\) 7.17327 0.242779
\(874\) 0 0
\(875\) 0 0
\(876\) −2.46148 −0.0831656
\(877\) −23.9813 −0.809791 −0.404895 0.914363i \(-0.632692\pi\)
−0.404895 + 0.914363i \(0.632692\pi\)
\(878\) −60.2843 −2.03450
\(879\) 35.9411 1.21226
\(880\) 0 0
\(881\) −12.4853 −0.420640 −0.210320 0.977633i \(-0.567451\pi\)
−0.210320 + 0.977633i \(0.567451\pi\)
\(882\) 199.770 6.72661
\(883\) −32.1421 −1.08167 −0.540834 0.841129i \(-0.681891\pi\)
−0.540834 + 0.841129i \(0.681891\pi\)
\(884\) 10.8655 0.365446
\(885\) 0 0
\(886\) 17.1067 0.574709
\(887\) 31.6644 1.06319 0.531593 0.847000i \(-0.321594\pi\)
0.531593 + 0.847000i \(0.321594\pi\)
\(888\) −18.5486 −0.622449
\(889\) −59.8477 −2.00723
\(890\) 0 0
\(891\) 14.9706 0.501533
\(892\) −41.8089 −1.39987
\(893\) 0 0
\(894\) 57.2548 1.91489
\(895\) 0 0
\(896\) 32.8951 1.09895
\(897\) 17.6569 0.589545
\(898\) 5.17157 0.172578
\(899\) −14.6274 −0.487852
\(900\) 0 0
\(901\) 17.2303 0.574026
\(902\) 0 0
\(903\) −11.8851 −0.395510
\(904\) −8.38478 −0.278874
\(905\) 0 0
\(906\) 111.291 3.69741
\(907\) 33.1063 1.09928 0.549638 0.835403i \(-0.314766\pi\)
0.549638 + 0.835403i \(0.314766\pi\)
\(908\) −53.6940 −1.78190
\(909\) −96.0833 −3.18688
\(910\) 0 0
\(911\) 56.3666 1.86751 0.933754 0.357914i \(-0.116512\pi\)
0.933754 + 0.357914i \(0.116512\pi\)
\(912\) 0 0
\(913\) −6.34315 −0.209927
\(914\) 36.3762 1.20322
\(915\) 0 0
\(916\) 30.1421 0.995924
\(917\) 35.3137 1.16616
\(918\) −64.5685 −2.13108
\(919\) −56.2843 −1.85665 −0.928323 0.371774i \(-0.878750\pi\)
−0.928323 + 0.371774i \(0.878750\pi\)
\(920\) 0 0
\(921\) −33.7990 −1.11371
\(922\) −69.9922 −2.30507
\(923\) −13.3726 −0.440164
\(924\) 69.2713 2.27886
\(925\) 0 0
\(926\) 57.0876 1.87601
\(927\) 51.9534 1.70637
\(928\) 19.7990 0.649934
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −32.1421 −1.05285
\(933\) 94.0610 3.07942
\(934\) −45.2025 −1.47907
\(935\) 0 0
\(936\) 6.24264 0.204047
\(937\) 22.9706 0.750416 0.375208 0.926941i \(-0.377571\pi\)
0.375208 + 0.926941i \(0.377571\pi\)
\(938\) −55.1127 −1.79949
\(939\) −61.2896 −2.00011
\(940\) 0 0
\(941\) −28.6931 −0.935368 −0.467684 0.883896i \(-0.654911\pi\)
−0.467684 + 0.883896i \(0.654911\pi\)
\(942\) 112.368 3.66113
\(943\) 0 0
\(944\) 35.6552 1.16048
\(945\) 0 0
\(946\) −3.48106 −0.113179
\(947\) −24.1421 −0.784514 −0.392257 0.919856i \(-0.628306\pi\)
−0.392257 + 0.919856i \(0.628306\pi\)
\(948\) −85.2548 −2.76895
\(949\) 0.422323 0.0137092
\(950\) 0 0
\(951\) 46.2843 1.50087
\(952\) −15.3661 −0.498019
\(953\) 27.4624 0.889593 0.444796 0.895632i \(-0.353276\pi\)
0.444796 + 0.895632i \(0.353276\pi\)
\(954\) 57.6985 1.86806
\(955\) 0 0
\(956\) −4.00000 −0.129369
\(957\) 14.6274 0.472837
\(958\) 21.0100 0.678803
\(959\) −9.65685 −0.311836
\(960\) 0 0
\(961\) 4.31371 0.139152
\(962\) 18.5486 0.598029
\(963\) 37.6069 1.21187
\(964\) 16.8080 0.541350
\(965\) 0 0
\(966\) −145.539 −4.68264
\(967\) −38.4853 −1.23760 −0.618802 0.785547i \(-0.712382\pi\)
−0.618802 + 0.785547i \(0.712382\pi\)
\(968\) −6.09185 −0.195799
\(969\) 0 0
\(970\) 0 0
\(971\) 46.5207 1.49292 0.746460 0.665430i \(-0.231752\pi\)
0.746460 + 0.665430i \(0.231752\pi\)
\(972\) 7.17327 0.230083
\(973\) −40.2843 −1.29145
\(974\) 33.3553 1.06877
\(975\) 0 0
\(976\) −8.48528 −0.271607
\(977\) −21.0975 −0.674969 −0.337484 0.941331i \(-0.609576\pi\)
−0.337484 + 0.941331i \(0.609576\pi\)
\(978\) 80.0833 2.56078
\(979\) −18.8472 −0.602358
\(980\) 0 0
\(981\) 112.311 3.58581
\(982\) −51.0213 −1.62816
\(983\) −56.4541 −1.80061 −0.900303 0.435265i \(-0.856655\pi\)
−0.900303 + 0.435265i \(0.856655\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −18.9117 −0.602271
\(987\) 93.0414 2.96154
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 12.9046 0.409930 0.204965 0.978769i \(-0.434292\pi\)
0.204965 + 0.978769i \(0.434292\pi\)
\(992\) −47.7990 −1.51762
\(993\) 88.2843 2.80162
\(994\) 110.225 3.49614
\(995\) 0 0
\(996\) 22.7506 0.720879
\(997\) −3.65685 −0.115814 −0.0579069 0.998322i \(-0.518443\pi\)
−0.0579069 + 0.998322i \(0.518443\pi\)
\(998\) 73.4733 2.32576
\(999\) −60.2843 −1.90731
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 9025.2.a.bn.1.4 4
5.4 even 2 1805.2.a.m.1.1 4
19.18 odd 2 inner 9025.2.a.bn.1.1 4
95.94 odd 2 1805.2.a.m.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.m.1.1 4 5.4 even 2
1805.2.a.m.1.4 yes 4 95.94 odd 2
9025.2.a.bn.1.1 4 19.18 odd 2 inner
9025.2.a.bn.1.4 4 1.1 even 1 trivial