Properties

Label 4235.2.a.v.1.2
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.8166452560\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15952.1
Defining polynomial: \(x^{4} - 6 x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.637875\) of defining polynomial
Character \(\chi\) \(=\) 4235.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.637875 q^{2} -2.56771 q^{3} -1.59312 q^{4} +1.00000 q^{5} +1.63787 q^{6} -1.00000 q^{7} +2.29196 q^{8} +3.59312 q^{9} +O(q^{10})\) \(q-0.637875 q^{2} -2.56771 q^{3} -1.59312 q^{4} +1.00000 q^{5} +1.63787 q^{6} -1.00000 q^{7} +2.29196 q^{8} +3.59312 q^{9} -0.637875 q^{10} +4.09065 q^{12} -2.09065 q^{13} +0.637875 q^{14} -2.56771 q^{15} +1.72425 q^{16} +6.47819 q^{17} -2.29196 q^{18} +1.61246 q^{19} -1.59312 q^{20} +2.56771 q^{21} +0.929832 q^{23} -5.88507 q^{24} +1.00000 q^{25} +1.33358 q^{26} -1.52295 q^{27} +1.59312 q^{28} -2.41116 q^{29} +1.63787 q^{30} -0.749661 q^{31} -5.68377 q^{32} -4.13227 q^{34} -1.00000 q^{35} -5.72425 q^{36} +1.51375 q^{37} -1.02855 q^{38} +5.36819 q^{39} +2.29196 q^{40} +10.4366 q^{41} -1.63787 q^{42} -2.78950 q^{43} +3.59312 q^{45} -0.593116 q^{46} +0.403743 q^{47} -4.42737 q^{48} +1.00000 q^{49} -0.637875 q^{50} -16.6341 q^{51} +3.33065 q^{52} +5.79442 q^{53} +0.971450 q^{54} -2.29196 q^{56} -4.14034 q^{57} +1.53802 q^{58} +8.93097 q^{59} +4.09065 q^{60} -8.13227 q^{61} +0.478189 q^{62} -3.59312 q^{63} +0.177031 q^{64} -2.09065 q^{65} -0.789495 q^{67} -10.3205 q^{68} -2.38754 q^{69} +0.637875 q^{70} -10.9056 q^{71} +8.23527 q^{72} -3.89427 q^{73} -0.965580 q^{74} -2.56771 q^{75} -2.56884 q^{76} -3.42423 q^{78} -5.86887 q^{79} +1.72425 q^{80} -6.86887 q^{81} -6.65722 q^{82} -17.6534 q^{83} -4.09065 q^{84} +6.47819 q^{85} +1.77935 q^{86} +6.19116 q^{87} +13.2099 q^{89} -2.29196 q^{90} +2.09065 q^{91} -1.48133 q^{92} +1.92491 q^{93} -0.257538 q^{94} +1.61246 q^{95} +14.5943 q^{96} -16.6584 q^{97} -0.637875 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} + 6 q^{8} + 4 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} + 6 q^{8} + 4 q^{9} - 8 q^{12} + 16 q^{13} - 2 q^{15} + 12 q^{16} + 2 q^{17} - 6 q^{18} + 6 q^{19} + 4 q^{20} + 2 q^{21} - 2 q^{23} - 10 q^{24} + 4 q^{25} + 2 q^{26} + 10 q^{27} - 4 q^{28} + 12 q^{29} + 4 q^{30} - 6 q^{31} + 12 q^{32} + 8 q^{34} - 4 q^{35} - 28 q^{36} - 6 q^{37} - 10 q^{38} + 12 q^{39} + 6 q^{40} + 18 q^{41} - 4 q^{42} + 6 q^{43} + 4 q^{45} + 8 q^{46} + 4 q^{47} + 2 q^{48} + 4 q^{49} + 4 q^{51} + 30 q^{52} + 34 q^{53} - 2 q^{54} - 6 q^{56} - 28 q^{57} + 32 q^{58} - 10 q^{59} - 8 q^{60} - 8 q^{61} - 22 q^{62} - 4 q^{63} - 16 q^{64} + 16 q^{65} + 14 q^{67} - 44 q^{68} - 10 q^{69} - 12 q^{72} + 2 q^{73} + 48 q^{74} - 2 q^{75} + 38 q^{76} + 14 q^{78} - 8 q^{79} + 12 q^{80} - 12 q^{81} - 34 q^{82} - 10 q^{83} + 8 q^{84} + 2 q^{85} - 24 q^{86} + 32 q^{87} + 10 q^{89} - 6 q^{90} - 16 q^{91} + 10 q^{92} - 26 q^{93} + 54 q^{94} + 6 q^{95} + 8 q^{96} - 34 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.637875 −0.451045 −0.225523 0.974238i \(-0.572409\pi\)
−0.225523 + 0.974238i \(0.572409\pi\)
\(3\) −2.56771 −1.48247 −0.741233 0.671248i \(-0.765759\pi\)
−0.741233 + 0.671248i \(0.765759\pi\)
\(4\) −1.59312 −0.796558
\(5\) 1.00000 0.447214
\(6\) 1.63787 0.668659
\(7\) −1.00000 −0.377964
\(8\) 2.29196 0.810329
\(9\) 3.59312 1.19771
\(10\) −0.637875 −0.201714
\(11\) 0 0
\(12\) 4.09065 1.18087
\(13\) −2.09065 −0.579843 −0.289922 0.957050i \(-0.593629\pi\)
−0.289922 + 0.957050i \(0.593629\pi\)
\(14\) 0.637875 0.170479
\(15\) −2.56771 −0.662979
\(16\) 1.72425 0.431063
\(17\) 6.47819 1.57119 0.785596 0.618740i \(-0.212357\pi\)
0.785596 + 0.618740i \(0.212357\pi\)
\(18\) −2.29196 −0.540220
\(19\) 1.61246 0.369925 0.184962 0.982746i \(-0.440784\pi\)
0.184962 + 0.982746i \(0.440784\pi\)
\(20\) −1.59312 −0.356232
\(21\) 2.56771 0.560319
\(22\) 0 0
\(23\) 0.929832 0.193883 0.0969417 0.995290i \(-0.469094\pi\)
0.0969417 + 0.995290i \(0.469094\pi\)
\(24\) −5.88507 −1.20129
\(25\) 1.00000 0.200000
\(26\) 1.33358 0.261536
\(27\) −1.52295 −0.293091
\(28\) 1.59312 0.301071
\(29\) −2.41116 −0.447742 −0.223871 0.974619i \(-0.571869\pi\)
−0.223871 + 0.974619i \(0.571869\pi\)
\(30\) 1.63787 0.299034
\(31\) −0.749661 −0.134643 −0.0673215 0.997731i \(-0.521445\pi\)
−0.0673215 + 0.997731i \(0.521445\pi\)
\(32\) −5.68377 −1.00476
\(33\) 0 0
\(34\) −4.13227 −0.708679
\(35\) −1.00000 −0.169031
\(36\) −5.72425 −0.954042
\(37\) 1.51375 0.248858 0.124429 0.992228i \(-0.460290\pi\)
0.124429 + 0.992228i \(0.460290\pi\)
\(38\) −1.02855 −0.166853
\(39\) 5.36819 0.859598
\(40\) 2.29196 0.362390
\(41\) 10.4366 1.62992 0.814959 0.579518i \(-0.196759\pi\)
0.814959 + 0.579518i \(0.196759\pi\)
\(42\) −1.63787 −0.252730
\(43\) −2.78950 −0.425394 −0.212697 0.977118i \(-0.568225\pi\)
−0.212697 + 0.977118i \(0.568225\pi\)
\(44\) 0 0
\(45\) 3.59312 0.535630
\(46\) −0.593116 −0.0874502
\(47\) 0.403743 0.0588920 0.0294460 0.999566i \(-0.490626\pi\)
0.0294460 + 0.999566i \(0.490626\pi\)
\(48\) −4.42737 −0.639036
\(49\) 1.00000 0.142857
\(50\) −0.637875 −0.0902091
\(51\) −16.6341 −2.32924
\(52\) 3.33065 0.461879
\(53\) 5.79442 0.795925 0.397962 0.917402i \(-0.369717\pi\)
0.397962 + 0.917402i \(0.369717\pi\)
\(54\) 0.971450 0.132198
\(55\) 0 0
\(56\) −2.29196 −0.306276
\(57\) −4.14034 −0.548401
\(58\) 1.53802 0.201952
\(59\) 8.93097 1.16271 0.581357 0.813649i \(-0.302522\pi\)
0.581357 + 0.813649i \(0.302522\pi\)
\(60\) 4.09065 0.528101
\(61\) −8.13227 −1.04123 −0.520615 0.853791i \(-0.674297\pi\)
−0.520615 + 0.853791i \(0.674297\pi\)
\(62\) 0.478189 0.0607301
\(63\) −3.59312 −0.452690
\(64\) 0.177031 0.0221288
\(65\) −2.09065 −0.259314
\(66\) 0 0
\(67\) −0.789495 −0.0964522 −0.0482261 0.998836i \(-0.515357\pi\)
−0.0482261 + 0.998836i \(0.515357\pi\)
\(68\) −10.3205 −1.25155
\(69\) −2.38754 −0.287425
\(70\) 0.637875 0.0762406
\(71\) −10.9056 −1.29425 −0.647126 0.762383i \(-0.724029\pi\)
−0.647126 + 0.762383i \(0.724029\pi\)
\(72\) 8.23527 0.970536
\(73\) −3.89427 −0.455790 −0.227895 0.973686i \(-0.573184\pi\)
−0.227895 + 0.973686i \(0.573184\pi\)
\(74\) −0.965580 −0.112246
\(75\) −2.56771 −0.296493
\(76\) −2.56884 −0.294667
\(77\) 0 0
\(78\) −3.42423 −0.387718
\(79\) −5.86887 −0.660299 −0.330149 0.943929i \(-0.607099\pi\)
−0.330149 + 0.943929i \(0.607099\pi\)
\(80\) 1.72425 0.192777
\(81\) −6.86887 −0.763207
\(82\) −6.65722 −0.735167
\(83\) −17.6534 −1.93772 −0.968858 0.247616i \(-0.920353\pi\)
−0.968858 + 0.247616i \(0.920353\pi\)
\(84\) −4.09065 −0.446327
\(85\) 6.47819 0.702658
\(86\) 1.77935 0.191872
\(87\) 6.19116 0.663762
\(88\) 0 0
\(89\) 13.2099 1.40024 0.700121 0.714024i \(-0.253129\pi\)
0.700121 + 0.714024i \(0.253129\pi\)
\(90\) −2.29196 −0.241594
\(91\) 2.09065 0.219160
\(92\) −1.48133 −0.154439
\(93\) 1.92491 0.199604
\(94\) −0.257538 −0.0265630
\(95\) 1.61246 0.165435
\(96\) 14.5943 1.48952
\(97\) −16.6584 −1.69140 −0.845700 0.533658i \(-0.820817\pi\)
−0.845700 + 0.533658i \(0.820817\pi\)
\(98\) −0.637875 −0.0644351
\(99\) 0 0
\(100\) −1.59312 −0.159312
\(101\) 8.52609 0.848378 0.424189 0.905574i \(-0.360559\pi\)
0.424189 + 0.905574i \(0.360559\pi\)
\(102\) 10.6105 1.05059
\(103\) −20.1975 −1.99012 −0.995060 0.0992731i \(-0.968348\pi\)
−0.995060 + 0.0992731i \(0.968348\pi\)
\(104\) −4.79169 −0.469864
\(105\) 2.56771 0.250582
\(106\) −3.69611 −0.358998
\(107\) −7.32657 −0.708286 −0.354143 0.935191i \(-0.615227\pi\)
−0.354143 + 0.935191i \(0.615227\pi\)
\(108\) 2.42623 0.233464
\(109\) 16.9564 1.62413 0.812063 0.583569i \(-0.198344\pi\)
0.812063 + 0.583569i \(0.198344\pi\)
\(110\) 0 0
\(111\) −3.88686 −0.368924
\(112\) −1.72425 −0.162926
\(113\) −6.55150 −0.616313 −0.308157 0.951336i \(-0.599712\pi\)
−0.308157 + 0.951336i \(0.599712\pi\)
\(114\) 2.64102 0.247354
\(115\) 0.929832 0.0867073
\(116\) 3.84126 0.356652
\(117\) −7.51196 −0.694481
\(118\) −5.69684 −0.524436
\(119\) −6.47819 −0.593855
\(120\) −5.88507 −0.537231
\(121\) 0 0
\(122\) 5.18737 0.469642
\(123\) −26.7981 −2.41630
\(124\) 1.19430 0.107251
\(125\) 1.00000 0.0894427
\(126\) 2.29196 0.204184
\(127\) 10.4944 0.931227 0.465614 0.884988i \(-0.345834\pi\)
0.465614 + 0.884988i \(0.345834\pi\)
\(128\) 11.2546 0.994777
\(129\) 7.16261 0.630632
\(130\) 1.33358 0.116962
\(131\) 15.7193 1.37340 0.686702 0.726939i \(-0.259058\pi\)
0.686702 + 0.726939i \(0.259058\pi\)
\(132\) 0 0
\(133\) −1.61246 −0.139818
\(134\) 0.503599 0.0435043
\(135\) −1.52295 −0.131074
\(136\) 14.8477 1.27318
\(137\) 14.7184 1.25748 0.628738 0.777617i \(-0.283572\pi\)
0.628738 + 0.777617i \(0.283572\pi\)
\(138\) 1.52295 0.129642
\(139\) −2.58391 −0.219165 −0.109582 0.993978i \(-0.534951\pi\)
−0.109582 + 0.993978i \(0.534951\pi\)
\(140\) 1.59312 0.134643
\(141\) −1.03669 −0.0873054
\(142\) 6.95638 0.583766
\(143\) 0 0
\(144\) 6.19543 0.516286
\(145\) −2.41116 −0.200236
\(146\) 2.48406 0.205582
\(147\) −2.56771 −0.211781
\(148\) −2.41157 −0.198230
\(149\) 6.05082 0.495702 0.247851 0.968798i \(-0.420276\pi\)
0.247851 + 0.968798i \(0.420276\pi\)
\(150\) 1.63787 0.133732
\(151\) 12.8252 1.04370 0.521852 0.853036i \(-0.325241\pi\)
0.521852 + 0.853036i \(0.325241\pi\)
\(152\) 3.69570 0.299761
\(153\) 23.2769 1.88182
\(154\) 0 0
\(155\) −0.749661 −0.0602142
\(156\) −8.55214 −0.684720
\(157\) −8.65836 −0.691012 −0.345506 0.938416i \(-0.612293\pi\)
−0.345506 + 0.938416i \(0.612293\pi\)
\(158\) 3.74360 0.297825
\(159\) −14.8784 −1.17993
\(160\) −5.68377 −0.449341
\(161\) −0.929832 −0.0732810
\(162\) 4.38147 0.344241
\(163\) 18.2079 1.42615 0.713075 0.701088i \(-0.247302\pi\)
0.713075 + 0.701088i \(0.247302\pi\)
\(164\) −16.6267 −1.29832
\(165\) 0 0
\(166\) 11.2607 0.873998
\(167\) −5.55536 −0.429887 −0.214943 0.976626i \(-0.568957\pi\)
−0.214943 + 0.976626i \(0.568957\pi\)
\(168\) 5.88507 0.454043
\(169\) −8.62916 −0.663782
\(170\) −4.13227 −0.316931
\(171\) 5.79377 0.443061
\(172\) 4.44399 0.338851
\(173\) 0.618526 0.0470256 0.0235128 0.999724i \(-0.492515\pi\)
0.0235128 + 0.999724i \(0.492515\pi\)
\(174\) −3.94918 −0.299387
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −22.9321 −1.72368
\(178\) −8.42623 −0.631573
\(179\) 6.27083 0.468704 0.234352 0.972152i \(-0.424703\pi\)
0.234352 + 0.972152i \(0.424703\pi\)
\(180\) −5.72425 −0.426660
\(181\) −18.6584 −1.38687 −0.693433 0.720522i \(-0.743902\pi\)
−0.693433 + 0.720522i \(0.743902\pi\)
\(182\) −1.33358 −0.0988512
\(183\) 20.8813 1.54359
\(184\) 2.13113 0.157109
\(185\) 1.51375 0.111293
\(186\) −1.22785 −0.0900303
\(187\) 0 0
\(188\) −0.643210 −0.0469109
\(189\) 1.52295 0.110778
\(190\) −1.02855 −0.0746189
\(191\) −1.96758 −0.142369 −0.0711847 0.997463i \(-0.522678\pi\)
−0.0711847 + 0.997463i \(0.522678\pi\)
\(192\) −0.454563 −0.0328052
\(193\) −20.1246 −1.44860 −0.724301 0.689484i \(-0.757837\pi\)
−0.724301 + 0.689484i \(0.757837\pi\)
\(194\) 10.6259 0.762898
\(195\) 5.36819 0.384424
\(196\) −1.59312 −0.113794
\(197\) 20.7981 1.48180 0.740900 0.671615i \(-0.234399\pi\)
0.740900 + 0.671615i \(0.234399\pi\)
\(198\) 0 0
\(199\) 0.189373 0.0134243 0.00671214 0.999977i \(-0.497863\pi\)
0.00671214 + 0.999977i \(0.497863\pi\)
\(200\) 2.29196 0.162066
\(201\) 2.02719 0.142987
\(202\) −5.43857 −0.382657
\(203\) 2.41116 0.169230
\(204\) 26.5000 1.85537
\(205\) 10.4366 0.728922
\(206\) 12.8835 0.897635
\(207\) 3.34099 0.232215
\(208\) −3.60481 −0.249949
\(209\) 0 0
\(210\) −1.63787 −0.113024
\(211\) 10.8547 0.747271 0.373636 0.927576i \(-0.378111\pi\)
0.373636 + 0.927576i \(0.378111\pi\)
\(212\) −9.23118 −0.634000
\(213\) 28.0023 1.91868
\(214\) 4.67343 0.319469
\(215\) −2.78950 −0.190242
\(216\) −3.49053 −0.237501
\(217\) 0.749661 0.0508903
\(218\) −10.8160 −0.732555
\(219\) 9.99935 0.675694
\(220\) 0 0
\(221\) −13.5437 −0.911045
\(222\) 2.47933 0.166402
\(223\) −12.5106 −0.837772 −0.418886 0.908039i \(-0.637579\pi\)
−0.418886 + 0.908039i \(0.637579\pi\)
\(224\) 5.68377 0.379763
\(225\) 3.59312 0.239541
\(226\) 4.17903 0.277985
\(227\) 24.3154 1.61387 0.806934 0.590642i \(-0.201125\pi\)
0.806934 + 0.590642i \(0.201125\pi\)
\(228\) 6.59604 0.436833
\(229\) −19.8619 −1.31251 −0.656257 0.754538i \(-0.727861\pi\)
−0.656257 + 0.754538i \(0.727861\pi\)
\(230\) −0.593116 −0.0391089
\(231\) 0 0
\(232\) −5.52628 −0.362818
\(233\) −2.32751 −0.152481 −0.0762403 0.997089i \(-0.524292\pi\)
−0.0762403 + 0.997089i \(0.524292\pi\)
\(234\) 4.79169 0.313243
\(235\) 0.403743 0.0263373
\(236\) −14.2281 −0.926168
\(237\) 15.0695 0.978871
\(238\) 4.13227 0.267855
\(239\) 4.32164 0.279544 0.139772 0.990184i \(-0.455363\pi\)
0.139772 + 0.990184i \(0.455363\pi\)
\(240\) −4.42737 −0.285786
\(241\) 29.5621 1.90426 0.952132 0.305686i \(-0.0988858\pi\)
0.952132 + 0.305686i \(0.0988858\pi\)
\(242\) 0 0
\(243\) 22.2061 1.42452
\(244\) 12.9557 0.829401
\(245\) 1.00000 0.0638877
\(246\) 17.0938 1.08986
\(247\) −3.37111 −0.214498
\(248\) −1.71819 −0.109105
\(249\) 45.3288 2.87260
\(250\) −0.637875 −0.0403427
\(251\) 7.48247 0.472289 0.236145 0.971718i \(-0.424116\pi\)
0.236145 + 0.971718i \(0.424116\pi\)
\(252\) 5.72425 0.360594
\(253\) 0 0
\(254\) −6.69411 −0.420026
\(255\) −16.6341 −1.04167
\(256\) −7.53310 −0.470818
\(257\) −22.6669 −1.41392 −0.706962 0.707252i \(-0.749935\pi\)
−0.706962 + 0.707252i \(0.749935\pi\)
\(258\) −4.56884 −0.284444
\(259\) −1.51375 −0.0940596
\(260\) 3.33065 0.206558
\(261\) −8.66358 −0.536262
\(262\) −10.0270 −0.619468
\(263\) −5.45706 −0.336497 −0.168248 0.985745i \(-0.553811\pi\)
−0.168248 + 0.985745i \(0.553811\pi\)
\(264\) 0 0
\(265\) 5.79442 0.355948
\(266\) 1.02855 0.0630645
\(267\) −33.9190 −2.07581
\(268\) 1.25776 0.0768298
\(269\) 17.2121 1.04944 0.524721 0.851274i \(-0.324170\pi\)
0.524721 + 0.851274i \(0.324170\pi\)
\(270\) 0.971450 0.0591205
\(271\) −12.2829 −0.746136 −0.373068 0.927804i \(-0.621694\pi\)
−0.373068 + 0.927804i \(0.621694\pi\)
\(272\) 11.1700 0.677282
\(273\) −5.36819 −0.324897
\(274\) −9.38848 −0.567179
\(275\) 0 0
\(276\) 3.80362 0.228951
\(277\) 18.1895 1.09290 0.546449 0.837492i \(-0.315979\pi\)
0.546449 + 0.837492i \(0.315979\pi\)
\(278\) 1.64821 0.0988533
\(279\) −2.69362 −0.161263
\(280\) −2.29196 −0.136971
\(281\) 8.03870 0.479548 0.239774 0.970829i \(-0.422927\pi\)
0.239774 + 0.970829i \(0.422927\pi\)
\(282\) 0.661281 0.0393787
\(283\) −13.3675 −0.794618 −0.397309 0.917685i \(-0.630056\pi\)
−0.397309 + 0.917685i \(0.630056\pi\)
\(284\) 17.3738 1.03095
\(285\) −4.14034 −0.245252
\(286\) 0 0
\(287\) −10.4366 −0.616051
\(288\) −20.4224 −1.20340
\(289\) 24.9669 1.46864
\(290\) 1.53802 0.0903156
\(291\) 42.7738 2.50744
\(292\) 6.20403 0.363063
\(293\) 29.6222 1.73054 0.865272 0.501302i \(-0.167145\pi\)
0.865272 + 0.501302i \(0.167145\pi\)
\(294\) 1.63787 0.0955228
\(295\) 8.93097 0.519981
\(296\) 3.46944 0.201657
\(297\) 0 0
\(298\) −3.85966 −0.223584
\(299\) −1.94396 −0.112422
\(300\) 4.09065 0.236174
\(301\) 2.78950 0.160784
\(302\) −8.18090 −0.470758
\(303\) −21.8925 −1.25769
\(304\) 2.78029 0.159461
\(305\) −8.13227 −0.465653
\(306\) −14.8477 −0.848788
\(307\) 33.1613 1.89262 0.946308 0.323266i \(-0.104781\pi\)
0.946308 + 0.323266i \(0.104781\pi\)
\(308\) 0 0
\(309\) 51.8613 2.95029
\(310\) 0.478189 0.0271593
\(311\) 18.2303 1.03375 0.516874 0.856061i \(-0.327096\pi\)
0.516874 + 0.856061i \(0.327096\pi\)
\(312\) 12.3037 0.696557
\(313\) −1.66851 −0.0943096 −0.0471548 0.998888i \(-0.515015\pi\)
−0.0471548 + 0.998888i \(0.515015\pi\)
\(314\) 5.52295 0.311678
\(315\) −3.59312 −0.202449
\(316\) 9.34978 0.525966
\(317\) 25.9699 1.45861 0.729306 0.684187i \(-0.239843\pi\)
0.729306 + 0.684187i \(0.239843\pi\)
\(318\) 9.49053 0.532203
\(319\) 0 0
\(320\) 0.177031 0.00989631
\(321\) 18.8125 1.05001
\(322\) 0.593116 0.0330531
\(323\) 10.4459 0.581223
\(324\) 10.9429 0.607939
\(325\) −2.09065 −0.115969
\(326\) −11.6143 −0.643258
\(327\) −43.5390 −2.40771
\(328\) 23.9202 1.32077
\(329\) −0.403743 −0.0222591
\(330\) 0 0
\(331\) −19.1053 −1.05012 −0.525060 0.851065i \(-0.675957\pi\)
−0.525060 + 0.851065i \(0.675957\pi\)
\(332\) 28.1240 1.54350
\(333\) 5.43907 0.298059
\(334\) 3.54363 0.193899
\(335\) −0.789495 −0.0431347
\(336\) 4.42737 0.241533
\(337\) −13.2515 −0.721854 −0.360927 0.932594i \(-0.617540\pi\)
−0.360927 + 0.932594i \(0.617540\pi\)
\(338\) 5.50432 0.299396
\(339\) 16.8223 0.913663
\(340\) −10.3205 −0.559708
\(341\) 0 0
\(342\) −3.69570 −0.199841
\(343\) −1.00000 −0.0539949
\(344\) −6.39340 −0.344709
\(345\) −2.38754 −0.128541
\(346\) −0.394542 −0.0212107
\(347\) −23.2839 −1.24994 −0.624972 0.780647i \(-0.714890\pi\)
−0.624972 + 0.780647i \(0.714890\pi\)
\(348\) −9.86323 −0.528725
\(349\) 25.6464 1.37282 0.686411 0.727214i \(-0.259185\pi\)
0.686411 + 0.727214i \(0.259185\pi\)
\(350\) 0.637875 0.0340958
\(351\) 3.18396 0.169947
\(352\) 0 0
\(353\) −9.17638 −0.488410 −0.244205 0.969724i \(-0.578527\pi\)
−0.244205 + 0.969724i \(0.578527\pi\)
\(354\) 14.6278 0.777459
\(355\) −10.9056 −0.578807
\(356\) −21.0448 −1.11537
\(357\) 16.6341 0.880369
\(358\) −4.00000 −0.211407
\(359\) 29.8203 1.57386 0.786928 0.617045i \(-0.211670\pi\)
0.786928 + 0.617045i \(0.211670\pi\)
\(360\) 8.23527 0.434037
\(361\) −16.4000 −0.863156
\(362\) 11.9017 0.625539
\(363\) 0 0
\(364\) −3.33065 −0.174574
\(365\) −3.89427 −0.203836
\(366\) −13.3196 −0.696229
\(367\) −6.51394 −0.340025 −0.170012 0.985442i \(-0.554381\pi\)
−0.170012 + 0.985442i \(0.554381\pi\)
\(368\) 1.60326 0.0835759
\(369\) 37.4998 1.95216
\(370\) −0.965580 −0.0501981
\(371\) −5.79442 −0.300831
\(372\) −3.06660 −0.158996
\(373\) 19.0702 0.987416 0.493708 0.869628i \(-0.335641\pi\)
0.493708 + 0.869628i \(0.335641\pi\)
\(374\) 0 0
\(375\) −2.56771 −0.132596
\(376\) 0.925362 0.0477219
\(377\) 5.04091 0.259620
\(378\) −0.971450 −0.0499660
\(379\) 14.1305 0.725834 0.362917 0.931821i \(-0.381781\pi\)
0.362917 + 0.931821i \(0.381781\pi\)
\(380\) −2.56884 −0.131779
\(381\) −26.9465 −1.38051
\(382\) 1.25507 0.0642150
\(383\) −0.380117 −0.0194231 −0.00971153 0.999953i \(-0.503091\pi\)
−0.00971153 + 0.999953i \(0.503091\pi\)
\(384\) −28.8986 −1.47472
\(385\) 0 0
\(386\) 12.8370 0.653385
\(387\) −10.0230 −0.509497
\(388\) 26.5387 1.34730
\(389\) 17.1827 0.871196 0.435598 0.900141i \(-0.356537\pi\)
0.435598 + 0.900141i \(0.356537\pi\)
\(390\) −3.42423 −0.173393
\(391\) 6.02363 0.304628
\(392\) 2.29196 0.115761
\(393\) −40.3626 −2.03602
\(394\) −13.2665 −0.668359
\(395\) −5.86887 −0.295295
\(396\) 0 0
\(397\) 10.2049 0.512171 0.256086 0.966654i \(-0.417567\pi\)
0.256086 + 0.966654i \(0.417567\pi\)
\(398\) −0.120796 −0.00605496
\(399\) 4.14034 0.207276
\(400\) 1.72425 0.0862125
\(401\) 16.6636 0.832140 0.416070 0.909333i \(-0.363407\pi\)
0.416070 + 0.909333i \(0.363407\pi\)
\(402\) −1.29309 −0.0644937
\(403\) 1.56728 0.0780718
\(404\) −13.5830 −0.675782
\(405\) −6.86887 −0.341317
\(406\) −1.53802 −0.0763306
\(407\) 0 0
\(408\) −38.1246 −1.88745
\(409\) 25.1844 1.24529 0.622645 0.782504i \(-0.286058\pi\)
0.622645 + 0.782504i \(0.286058\pi\)
\(410\) −6.65722 −0.328777
\(411\) −37.7925 −1.86417
\(412\) 32.1770 1.58525
\(413\) −8.93097 −0.439464
\(414\) −2.13113 −0.104740
\(415\) −17.6534 −0.866573
\(416\) 11.8828 0.582602
\(417\) 6.63473 0.324904
\(418\) 0 0
\(419\) −36.1916 −1.76808 −0.884039 0.467413i \(-0.845186\pi\)
−0.884039 + 0.467413i \(0.845186\pi\)
\(420\) −4.09065 −0.199603
\(421\) −15.6327 −0.761893 −0.380946 0.924597i \(-0.624402\pi\)
−0.380946 + 0.924597i \(0.624402\pi\)
\(422\) −6.92396 −0.337053
\(423\) 1.45070 0.0705353
\(424\) 13.2806 0.644961
\(425\) 6.47819 0.314238
\(426\) −17.8619 −0.865414
\(427\) 8.13227 0.393548
\(428\) 11.6721 0.564191
\(429\) 0 0
\(430\) 1.77935 0.0858078
\(431\) −20.6525 −0.994795 −0.497398 0.867523i \(-0.665711\pi\)
−0.497398 + 0.867523i \(0.665711\pi\)
\(432\) −2.62594 −0.126341
\(433\) −21.0158 −1.00995 −0.504976 0.863133i \(-0.668499\pi\)
−0.504976 + 0.863133i \(0.668499\pi\)
\(434\) −0.478189 −0.0229538
\(435\) 6.19116 0.296843
\(436\) −27.0135 −1.29371
\(437\) 1.49932 0.0717223
\(438\) −6.37833 −0.304769
\(439\) 7.55642 0.360648 0.180324 0.983607i \(-0.442285\pi\)
0.180324 + 0.983607i \(0.442285\pi\)
\(440\) 0 0
\(441\) 3.59312 0.171101
\(442\) 8.63915 0.410923
\(443\) 28.8876 1.37249 0.686245 0.727371i \(-0.259258\pi\)
0.686245 + 0.727371i \(0.259258\pi\)
\(444\) 6.19221 0.293869
\(445\) 13.2099 0.626207
\(446\) 7.98020 0.377873
\(447\) −15.5367 −0.734862
\(448\) −0.177031 −0.00836391
\(449\) 30.5594 1.44219 0.721094 0.692837i \(-0.243640\pi\)
0.721094 + 0.692837i \(0.243640\pi\)
\(450\) −2.29196 −0.108044
\(451\) 0 0
\(452\) 10.4373 0.490929
\(453\) −32.9315 −1.54725
\(454\) −15.5102 −0.727927
\(455\) 2.09065 0.0980114
\(456\) −9.48947 −0.444385
\(457\) −14.2177 −0.665076 −0.332538 0.943090i \(-0.607905\pi\)
−0.332538 + 0.943090i \(0.607905\pi\)
\(458\) 12.6694 0.592003
\(459\) −9.86594 −0.460503
\(460\) −1.48133 −0.0690674
\(461\) 23.9831 1.11701 0.558503 0.829502i \(-0.311376\pi\)
0.558503 + 0.829502i \(0.311376\pi\)
\(462\) 0 0
\(463\) 33.3680 1.55074 0.775370 0.631507i \(-0.217563\pi\)
0.775370 + 0.631507i \(0.217563\pi\)
\(464\) −4.15745 −0.193005
\(465\) 1.92491 0.0892655
\(466\) 1.48466 0.0687756
\(467\) −35.1440 −1.62627 −0.813136 0.582073i \(-0.802242\pi\)
−0.813136 + 0.582073i \(0.802242\pi\)
\(468\) 11.9674 0.553195
\(469\) 0.789495 0.0364555
\(470\) −0.257538 −0.0118793
\(471\) 22.2321 1.02440
\(472\) 20.4694 0.942180
\(473\) 0 0
\(474\) −9.61246 −0.441515
\(475\) 1.61246 0.0739850
\(476\) 10.3205 0.473040
\(477\) 20.8200 0.953283
\(478\) −2.75667 −0.126087
\(479\) 12.4977 0.571036 0.285518 0.958373i \(-0.407834\pi\)
0.285518 + 0.958373i \(0.407834\pi\)
\(480\) 14.5943 0.666133
\(481\) −3.16472 −0.144299
\(482\) −18.8569 −0.858910
\(483\) 2.38754 0.108637
\(484\) 0 0
\(485\) −16.6584 −0.756417
\(486\) −14.1647 −0.642523
\(487\) 22.6577 1.02672 0.513360 0.858174i \(-0.328401\pi\)
0.513360 + 0.858174i \(0.328401\pi\)
\(488\) −18.6388 −0.843740
\(489\) −46.7524 −2.11422
\(490\) −0.637875 −0.0288162
\(491\) −8.70104 −0.392672 −0.196336 0.980537i \(-0.562904\pi\)
−0.196336 + 0.980537i \(0.562904\pi\)
\(492\) 42.6924 1.92472
\(493\) −15.6200 −0.703488
\(494\) 2.15034 0.0967485
\(495\) 0 0
\(496\) −1.29260 −0.0580396
\(497\) 10.9056 0.489181
\(498\) −28.9141 −1.29567
\(499\) 15.3351 0.686494 0.343247 0.939245i \(-0.388473\pi\)
0.343247 + 0.939245i \(0.388473\pi\)
\(500\) −1.59312 −0.0712463
\(501\) 14.2645 0.637293
\(502\) −4.77288 −0.213024
\(503\) −15.4820 −0.690307 −0.345154 0.938546i \(-0.612173\pi\)
−0.345154 + 0.938546i \(0.612173\pi\)
\(504\) −8.23527 −0.366828
\(505\) 8.52609 0.379406
\(506\) 0 0
\(507\) 22.1572 0.984034
\(508\) −16.7188 −0.741777
\(509\) 21.5587 0.955572 0.477786 0.878476i \(-0.341439\pi\)
0.477786 + 0.878476i \(0.341439\pi\)
\(510\) 10.6105 0.469839
\(511\) 3.89427 0.172273
\(512\) −17.7041 −0.782417
\(513\) −2.45570 −0.108422
\(514\) 14.4586 0.637744
\(515\) −20.1975 −0.890009
\(516\) −11.4109 −0.502335
\(517\) 0 0
\(518\) 0.965580 0.0424252
\(519\) −1.58819 −0.0697139
\(520\) −4.79169 −0.210130
\(521\) 1.43971 0.0630749 0.0315375 0.999503i \(-0.489960\pi\)
0.0315375 + 0.999503i \(0.489960\pi\)
\(522\) 5.52628 0.241879
\(523\) −45.0273 −1.96890 −0.984452 0.175652i \(-0.943797\pi\)
−0.984452 + 0.175652i \(0.943797\pi\)
\(524\) −25.0427 −1.09400
\(525\) 2.56771 0.112064
\(526\) 3.48092 0.151775
\(527\) −4.85644 −0.211550
\(528\) 0 0
\(529\) −22.1354 −0.962409
\(530\) −3.69611 −0.160549
\(531\) 32.0900 1.39259
\(532\) 2.56884 0.111373
\(533\) −21.8193 −0.945097
\(534\) 21.6361 0.936285
\(535\) −7.32657 −0.316755
\(536\) −1.80949 −0.0781581
\(537\) −16.1016 −0.694837
\(538\) −10.9792 −0.473346
\(539\) 0 0
\(540\) 2.42623 0.104408
\(541\) 19.9577 0.858050 0.429025 0.903293i \(-0.358857\pi\)
0.429025 + 0.903293i \(0.358857\pi\)
\(542\) 7.83498 0.336541
\(543\) 47.9092 2.05598
\(544\) −36.8205 −1.57867
\(545\) 16.9564 0.726331
\(546\) 3.42423 0.146543
\(547\) −12.7180 −0.543781 −0.271891 0.962328i \(-0.587649\pi\)
−0.271891 + 0.962328i \(0.587649\pi\)
\(548\) −23.4481 −1.00165
\(549\) −29.2202 −1.24709
\(550\) 0 0
\(551\) −3.88791 −0.165631
\(552\) −5.47213 −0.232909
\(553\) 5.86887 0.249570
\(554\) −11.6026 −0.492947
\(555\) −3.88686 −0.164988
\(556\) 4.11648 0.174577
\(557\) 0.445281 0.0188672 0.00943359 0.999956i \(-0.496997\pi\)
0.00943359 + 0.999956i \(0.496997\pi\)
\(558\) 1.71819 0.0727368
\(559\) 5.83187 0.246662
\(560\) −1.72425 −0.0728629
\(561\) 0 0
\(562\) −5.12768 −0.216298
\(563\) 13.2186 0.557100 0.278550 0.960422i \(-0.410146\pi\)
0.278550 + 0.960422i \(0.410146\pi\)
\(564\) 1.65157 0.0695438
\(565\) −6.55150 −0.275624
\(566\) 8.52681 0.358409
\(567\) 6.86887 0.288465
\(568\) −24.9951 −1.04877
\(569\) 32.9020 1.37932 0.689662 0.724132i \(-0.257759\pi\)
0.689662 + 0.724132i \(0.257759\pi\)
\(570\) 2.64102 0.110620
\(571\) 8.04097 0.336504 0.168252 0.985744i \(-0.446188\pi\)
0.168252 + 0.985744i \(0.446188\pi\)
\(572\) 0 0
\(573\) 5.05218 0.211058
\(574\) 6.65722 0.277867
\(575\) 0.929832 0.0387767
\(576\) 0.636092 0.0265038
\(577\) 36.6548 1.52596 0.762980 0.646423i \(-0.223736\pi\)
0.762980 + 0.646423i \(0.223736\pi\)
\(578\) −15.9258 −0.662425
\(579\) 51.6741 2.14750
\(580\) 3.84126 0.159500
\(581\) 17.6534 0.732388
\(582\) −27.2843 −1.13097
\(583\) 0 0
\(584\) −8.92551 −0.369340
\(585\) −7.51196 −0.310581
\(586\) −18.8952 −0.780554
\(587\) −8.45979 −0.349173 −0.174586 0.984642i \(-0.555859\pi\)
−0.174586 + 0.984642i \(0.555859\pi\)
\(588\) 4.09065 0.168696
\(589\) −1.20880 −0.0498078
\(590\) −5.69684 −0.234535
\(591\) −53.4033 −2.19672
\(592\) 2.61008 0.107274
\(593\) −23.3601 −0.959285 −0.479643 0.877464i \(-0.659234\pi\)
−0.479643 + 0.877464i \(0.659234\pi\)
\(594\) 0 0
\(595\) −6.47819 −0.265580
\(596\) −9.63966 −0.394856
\(597\) −0.486254 −0.0199010
\(598\) 1.24000 0.0507074
\(599\) −5.52401 −0.225705 −0.112852 0.993612i \(-0.535999\pi\)
−0.112852 + 0.993612i \(0.535999\pi\)
\(600\) −5.88507 −0.240257
\(601\) −33.2675 −1.35701 −0.678504 0.734597i \(-0.737371\pi\)
−0.678504 + 0.734597i \(0.737371\pi\)
\(602\) −1.77935 −0.0725208
\(603\) −2.83675 −0.115521
\(604\) −20.4321 −0.831370
\(605\) 0 0
\(606\) 13.9647 0.567276
\(607\) −15.0511 −0.610906 −0.305453 0.952207i \(-0.598808\pi\)
−0.305453 + 0.952207i \(0.598808\pi\)
\(608\) −9.16488 −0.371685
\(609\) −6.19116 −0.250878
\(610\) 5.18737 0.210030
\(611\) −0.844088 −0.0341481
\(612\) −37.0828 −1.49898
\(613\) 30.7859 1.24343 0.621716 0.783243i \(-0.286436\pi\)
0.621716 + 0.783243i \(0.286436\pi\)
\(614\) −21.1528 −0.853656
\(615\) −26.7981 −1.08060
\(616\) 0 0
\(617\) −3.31442 −0.133433 −0.0667167 0.997772i \(-0.521252\pi\)
−0.0667167 + 0.997772i \(0.521252\pi\)
\(618\) −33.0810 −1.33071
\(619\) 13.8215 0.555531 0.277766 0.960649i \(-0.410406\pi\)
0.277766 + 0.960649i \(0.410406\pi\)
\(620\) 1.19430 0.0479641
\(621\) −1.41609 −0.0568256
\(622\) −11.6287 −0.466267
\(623\) −13.2099 −0.529242
\(624\) 9.25610 0.370541
\(625\) 1.00000 0.0400000
\(626\) 1.06430 0.0425379
\(627\) 0 0
\(628\) 13.7938 0.550431
\(629\) 9.80634 0.391004
\(630\) 2.29196 0.0913138
\(631\) 29.6806 1.18157 0.590784 0.806830i \(-0.298819\pi\)
0.590784 + 0.806830i \(0.298819\pi\)
\(632\) −13.4512 −0.535060
\(633\) −27.8718 −1.10780
\(634\) −16.5655 −0.657901
\(635\) 10.4944 0.416457
\(636\) 23.7030 0.939884
\(637\) −2.09065 −0.0828347
\(638\) 0 0
\(639\) −39.1849 −1.55013
\(640\) 11.2546 0.444878
\(641\) −45.7358 −1.80646 −0.903228 0.429162i \(-0.858809\pi\)
−0.903228 + 0.429162i \(0.858809\pi\)
\(642\) −12.0000 −0.473602
\(643\) −11.4658 −0.452166 −0.226083 0.974108i \(-0.572592\pi\)
−0.226083 + 0.974108i \(0.572592\pi\)
\(644\) 1.48133 0.0583726
\(645\) 7.16261 0.282027
\(646\) −6.66314 −0.262158
\(647\) −31.5824 −1.24163 −0.620816 0.783956i \(-0.713199\pi\)
−0.620816 + 0.783956i \(0.713199\pi\)
\(648\) −15.7431 −0.618449
\(649\) 0 0
\(650\) 1.33358 0.0523071
\(651\) −1.92491 −0.0754431
\(652\) −29.0072 −1.13601
\(653\) 3.60724 0.141162 0.0705811 0.997506i \(-0.477515\pi\)
0.0705811 + 0.997506i \(0.477515\pi\)
\(654\) 27.7724 1.08599
\(655\) 15.7193 0.614205
\(656\) 17.9953 0.702597
\(657\) −13.9926 −0.545903
\(658\) 0.257538 0.0100399
\(659\) −19.0672 −0.742753 −0.371377 0.928482i \(-0.621114\pi\)
−0.371377 + 0.928482i \(0.621114\pi\)
\(660\) 0 0
\(661\) 0.139279 0.00541732 0.00270866 0.999996i \(-0.499138\pi\)
0.00270866 + 0.999996i \(0.499138\pi\)
\(662\) 12.1868 0.473652
\(663\) 34.7761 1.35059
\(664\) −40.4609 −1.57019
\(665\) −1.61246 −0.0625287
\(666\) −3.46944 −0.134438
\(667\) −2.24198 −0.0868096
\(668\) 8.85034 0.342430
\(669\) 32.1236 1.24197
\(670\) 0.503599 0.0194557
\(671\) 0 0
\(672\) −14.5943 −0.562986
\(673\) 30.8673 1.18985 0.594923 0.803783i \(-0.297183\pi\)
0.594923 + 0.803783i \(0.297183\pi\)
\(674\) 8.45278 0.325589
\(675\) −1.52295 −0.0586183
\(676\) 13.7473 0.528741
\(677\) 20.0624 0.771061 0.385530 0.922695i \(-0.374018\pi\)
0.385530 + 0.922695i \(0.374018\pi\)
\(678\) −10.7305 −0.412104
\(679\) 16.6584 0.639289
\(680\) 14.8477 0.569385
\(681\) −62.4347 −2.39250
\(682\) 0 0
\(683\) −39.1013 −1.49617 −0.748085 0.663603i \(-0.769026\pi\)
−0.748085 + 0.663603i \(0.769026\pi\)
\(684\) −9.23015 −0.352924
\(685\) 14.7184 0.562360
\(686\) 0.637875 0.0243542
\(687\) 50.9996 1.94576
\(688\) −4.80979 −0.183372
\(689\) −12.1141 −0.461511
\(690\) 1.52295 0.0579776
\(691\) −47.0562 −1.79010 −0.895052 0.445962i \(-0.852862\pi\)
−0.895052 + 0.445962i \(0.852862\pi\)
\(692\) −0.985383 −0.0374587
\(693\) 0 0
\(694\) 14.8522 0.563782
\(695\) −2.58391 −0.0980135
\(696\) 14.1899 0.537865
\(697\) 67.6101 2.56091
\(698\) −16.3592 −0.619205
\(699\) 5.97637 0.226047
\(700\) 1.59312 0.0602141
\(701\) 12.7715 0.482373 0.241187 0.970479i \(-0.422463\pi\)
0.241187 + 0.970479i \(0.422463\pi\)
\(702\) −2.03097 −0.0766539
\(703\) 2.44086 0.0920589
\(704\) 0 0
\(705\) −1.03669 −0.0390442
\(706\) 5.85338 0.220295
\(707\) −8.52609 −0.320657
\(708\) 36.5335 1.37301
\(709\) −14.2137 −0.533808 −0.266904 0.963723i \(-0.586001\pi\)
−0.266904 + 0.963723i \(0.586001\pi\)
\(710\) 6.95638 0.261068
\(711\) −21.0875 −0.790844
\(712\) 30.2764 1.13466
\(713\) −0.697058 −0.0261050
\(714\) −10.6105 −0.397087
\(715\) 0 0
\(716\) −9.99015 −0.373350
\(717\) −11.0967 −0.414414
\(718\) −19.0216 −0.709881
\(719\) −6.32971 −0.236058 −0.118029 0.993010i \(-0.537658\pi\)
−0.118029 + 0.993010i \(0.537658\pi\)
\(720\) 6.19543 0.230890
\(721\) 20.1975 0.752195
\(722\) 10.4611 0.389322
\(723\) −75.9069 −2.82301
\(724\) 29.7249 1.10472
\(725\) −2.41116 −0.0895483
\(726\) 0 0
\(727\) 27.5329 1.02114 0.510569 0.859837i \(-0.329435\pi\)
0.510569 + 0.859837i \(0.329435\pi\)
\(728\) 4.79169 0.177592
\(729\) −36.4121 −1.34860
\(730\) 2.48406 0.0919391
\(731\) −18.0709 −0.668376
\(732\) −33.2663 −1.22956
\(733\) −23.4494 −0.866124 −0.433062 0.901364i \(-0.642567\pi\)
−0.433062 + 0.901364i \(0.642567\pi\)
\(734\) 4.15508 0.153367
\(735\) −2.56771 −0.0947113
\(736\) −5.28495 −0.194806
\(737\) 0 0
\(738\) −23.9202 −0.880514
\(739\) 5.55014 0.204165 0.102083 0.994776i \(-0.467449\pi\)
0.102083 + 0.994776i \(0.467449\pi\)
\(740\) −2.41157 −0.0886512
\(741\) 8.65601 0.317987
\(742\) 3.69611 0.135689
\(743\) −46.9974 −1.72417 −0.862083 0.506767i \(-0.830840\pi\)
−0.862083 + 0.506767i \(0.830840\pi\)
\(744\) 4.41181 0.161745
\(745\) 6.05082 0.221685
\(746\) −12.1644 −0.445369
\(747\) −63.4308 −2.32081
\(748\) 0 0
\(749\) 7.32657 0.267707
\(750\) 1.63787 0.0598067
\(751\) 47.1787 1.72157 0.860787 0.508966i \(-0.169972\pi\)
0.860787 + 0.508966i \(0.169972\pi\)
\(752\) 0.696155 0.0253862
\(753\) −19.2128 −0.700153
\(754\) −3.21547 −0.117100
\(755\) 12.8252 0.466758
\(756\) −2.42623 −0.0882412
\(757\) 48.2852 1.75496 0.877478 0.479617i \(-0.159224\pi\)
0.877478 + 0.479617i \(0.159224\pi\)
\(758\) −9.01348 −0.327384
\(759\) 0 0
\(760\) 3.69570 0.134057
\(761\) 42.2057 1.52995 0.764977 0.644058i \(-0.222750\pi\)
0.764977 + 0.644058i \(0.222750\pi\)
\(762\) 17.1885 0.622674
\(763\) −16.9564 −0.613862
\(764\) 3.13459 0.113405
\(765\) 23.2769 0.841578
\(766\) 0.242467 0.00876068
\(767\) −18.6716 −0.674191
\(768\) 19.3428 0.697972
\(769\) −24.6277 −0.888099 −0.444049 0.896002i \(-0.646458\pi\)
−0.444049 + 0.896002i \(0.646458\pi\)
\(770\) 0 0
\(771\) 58.2020 2.09609
\(772\) 32.0609 1.15390
\(773\) −44.9737 −1.61759 −0.808796 0.588089i \(-0.799880\pi\)
−0.808796 + 0.588089i \(0.799880\pi\)
\(774\) 6.39340 0.229806
\(775\) −0.749661 −0.0269286
\(776\) −38.1803 −1.37059
\(777\) 3.88686 0.139440
\(778\) −10.9604 −0.392949
\(779\) 16.8286 0.602947
\(780\) −8.55214 −0.306216
\(781\) 0 0
\(782\) −3.84232 −0.137401
\(783\) 3.67207 0.131229
\(784\) 1.72425 0.0615804
\(785\) −8.65836 −0.309030
\(786\) 25.7463 0.918340
\(787\) 1.51416 0.0539739 0.0269870 0.999636i \(-0.491409\pi\)
0.0269870 + 0.999636i \(0.491409\pi\)
\(788\) −33.1337 −1.18034
\(789\) 14.0121 0.498845
\(790\) 3.74360 0.133191
\(791\) 6.55150 0.232944
\(792\) 0 0
\(793\) 17.0018 0.603751
\(794\) −6.50947 −0.231012
\(795\) −14.8784 −0.527681
\(796\) −0.301693 −0.0106932
\(797\) 31.3462 1.11034 0.555169 0.831737i \(-0.312653\pi\)
0.555169 + 0.831737i \(0.312653\pi\)
\(798\) −2.64102 −0.0934909
\(799\) 2.61553 0.0925306
\(800\) −5.68377 −0.200952
\(801\) 47.4646 1.67708
\(802\) −10.6293 −0.375333
\(803\) 0 0
\(804\) −3.22955 −0.113898
\(805\) −0.929832 −0.0327723
\(806\) −0.999729 −0.0352139
\(807\) −44.1957 −1.55576
\(808\) 19.5414 0.687465
\(809\) −9.70948 −0.341367 −0.170684 0.985326i \(-0.554598\pi\)
−0.170684 + 0.985326i \(0.554598\pi\)
\(810\) 4.38147 0.153949
\(811\) 23.3079 0.818452 0.409226 0.912433i \(-0.365799\pi\)
0.409226 + 0.912433i \(0.365799\pi\)
\(812\) −3.84126 −0.134802
\(813\) 31.5390 1.10612
\(814\) 0 0
\(815\) 18.2079 0.637793
\(816\) −28.6813 −1.00405
\(817\) −4.49796 −0.157364
\(818\) −16.0645 −0.561683
\(819\) 7.51196 0.262489
\(820\) −16.6267 −0.580628
\(821\) 3.74985 0.130871 0.0654354 0.997857i \(-0.479156\pi\)
0.0654354 + 0.997857i \(0.479156\pi\)
\(822\) 24.1069 0.840823
\(823\) −31.1092 −1.08440 −0.542200 0.840249i \(-0.682409\pi\)
−0.542200 + 0.840249i \(0.682409\pi\)
\(824\) −46.2918 −1.61265
\(825\) 0 0
\(826\) 5.69684 0.198218
\(827\) −15.3613 −0.534164 −0.267082 0.963674i \(-0.586060\pi\)
−0.267082 + 0.963674i \(0.586060\pi\)
\(828\) −5.32259 −0.184973
\(829\) −5.22493 −0.181469 −0.0907347 0.995875i \(-0.528922\pi\)
−0.0907347 + 0.995875i \(0.528922\pi\)
\(830\) 11.2607 0.390864
\(831\) −46.7052 −1.62018
\(832\) −0.370110 −0.0128313
\(833\) 6.47819 0.224456
\(834\) −4.23213 −0.146547
\(835\) −5.55536 −0.192251
\(836\) 0 0
\(837\) 1.14169 0.0394627
\(838\) 23.0857 0.797483
\(839\) −52.4526 −1.81087 −0.905433 0.424489i \(-0.860454\pi\)
−0.905433 + 0.424489i \(0.860454\pi\)
\(840\) 5.88507 0.203054
\(841\) −23.1863 −0.799528
\(842\) 9.97172 0.343648
\(843\) −20.6410 −0.710914
\(844\) −17.2929 −0.595245
\(845\) −8.62916 −0.296852
\(846\) −0.925362 −0.0318146
\(847\) 0 0
\(848\) 9.99103 0.343093
\(849\) 34.3239 1.17799
\(850\) −4.13227 −0.141736
\(851\) 1.40753 0.0482495
\(852\) −44.6109 −1.52834
\(853\) 35.1937 1.20501 0.602504 0.798116i \(-0.294170\pi\)
0.602504 + 0.798116i \(0.294170\pi\)
\(854\) −5.18737 −0.177508
\(855\) 5.79377 0.198143
\(856\) −16.7922 −0.573945
\(857\) −54.5011 −1.86172 −0.930862 0.365372i \(-0.880942\pi\)
−0.930862 + 0.365372i \(0.880942\pi\)
\(858\) 0 0
\(859\) 11.5923 0.395523 0.197762 0.980250i \(-0.436633\pi\)
0.197762 + 0.980250i \(0.436633\pi\)
\(860\) 4.44399 0.151539
\(861\) 26.7981 0.913275
\(862\) 13.1737 0.448698
\(863\) 26.6514 0.907225 0.453613 0.891199i \(-0.350135\pi\)
0.453613 + 0.891199i \(0.350135\pi\)
\(864\) 8.65609 0.294486
\(865\) 0.618526 0.0210305
\(866\) 13.4054 0.455535
\(867\) −64.1078 −2.17721
\(868\) −1.19430 −0.0405371
\(869\) 0 0
\(870\) −3.94918 −0.133890
\(871\) 1.65056 0.0559272
\(872\) 38.8633 1.31608
\(873\) −59.8554 −2.02580
\(874\) −0.956379 −0.0323500
\(875\) −1.00000 −0.0338062
\(876\) −15.9301 −0.538229
\(877\) 12.0449 0.406729 0.203365 0.979103i \(-0.434812\pi\)
0.203365 + 0.979103i \(0.434812\pi\)
\(878\) −4.82005 −0.162669
\(879\) −76.0610 −2.56547
\(880\) 0 0
\(881\) −6.11314 −0.205957 −0.102979 0.994684i \(-0.532837\pi\)
−0.102979 + 0.994684i \(0.532837\pi\)
\(882\) −2.29196 −0.0771742
\(883\) 11.5321 0.388088 0.194044 0.980993i \(-0.437840\pi\)
0.194044 + 0.980993i \(0.437840\pi\)
\(884\) 21.5766 0.725700
\(885\) −22.9321 −0.770854
\(886\) −18.4266 −0.619055
\(887\) 19.6325 0.659196 0.329598 0.944121i \(-0.393087\pi\)
0.329598 + 0.944121i \(0.393087\pi\)
\(888\) −8.90851 −0.298950
\(889\) −10.4944 −0.351971
\(890\) −8.42623 −0.282448
\(891\) 0 0
\(892\) 19.9308 0.667334
\(893\) 0.651022 0.0217856
\(894\) 9.91048 0.331456
\(895\) 6.27083 0.209611
\(896\) −11.2546 −0.375990
\(897\) 4.99151 0.166662
\(898\) −19.4931 −0.650492
\(899\) 1.80755 0.0602853
\(900\) −5.72425 −0.190808
\(901\) 37.5373 1.25055
\(902\) 0 0
\(903\) −7.16261 −0.238357
\(904\) −15.0158 −0.499417
\(905\) −18.6584 −0.620225
\(906\) 21.0061 0.697882
\(907\) −30.3586 −1.00804 −0.504021 0.863691i \(-0.668147\pi\)
−0.504021 + 0.863691i \(0.668147\pi\)
\(908\) −38.7372 −1.28554
\(909\) 30.6352 1.01611
\(910\) −1.33358 −0.0442076
\(911\) 14.7391 0.488328 0.244164 0.969734i \(-0.421487\pi\)
0.244164 + 0.969734i \(0.421487\pi\)
\(912\) −7.13898 −0.236395
\(913\) 0 0
\(914\) 9.06911 0.299980
\(915\) 20.8813 0.690314
\(916\) 31.6424 1.04549
\(917\) −15.7193 −0.519098
\(918\) 6.29324 0.207708
\(919\) −4.78135 −0.157722 −0.0788611 0.996886i \(-0.525128\pi\)
−0.0788611 + 0.996886i \(0.525128\pi\)
\(920\) 2.13113 0.0702614
\(921\) −85.1485 −2.80574
\(922\) −15.2982 −0.503821
\(923\) 22.7998 0.750463
\(924\) 0 0
\(925\) 1.51375 0.0497717
\(926\) −21.2846 −0.699454
\(927\) −72.5720 −2.38358
\(928\) 13.7045 0.449872
\(929\) 1.68714 0.0553534 0.0276767 0.999617i \(-0.491189\pi\)
0.0276767 + 0.999617i \(0.491189\pi\)
\(930\) −1.22785 −0.0402628
\(931\) 1.61246 0.0528464
\(932\) 3.70800 0.121460
\(933\) −46.8102 −1.53250
\(934\) 22.4175 0.733523
\(935\) 0 0
\(936\) −17.2171 −0.562758
\(937\) 22.1923 0.724991 0.362495 0.931986i \(-0.381925\pi\)
0.362495 + 0.931986i \(0.381925\pi\)
\(938\) −0.503599 −0.0164431
\(939\) 4.28424 0.139811
\(940\) −0.643210 −0.0209792
\(941\) 15.0418 0.490350 0.245175 0.969479i \(-0.421155\pi\)
0.245175 + 0.969479i \(0.421155\pi\)
\(942\) −14.1813 −0.462052
\(943\) 9.70426 0.316014
\(944\) 15.3992 0.501202
\(945\) 1.52295 0.0495415
\(946\) 0 0
\(947\) 10.7171 0.348259 0.174129 0.984723i \(-0.444289\pi\)
0.174129 + 0.984723i \(0.444289\pi\)
\(948\) −24.0075 −0.779727
\(949\) 8.14158 0.264287
\(950\) −1.02855 −0.0333706
\(951\) −66.6830 −2.16234
\(952\) −14.8477 −0.481218
\(953\) −31.2924 −1.01366 −0.506831 0.862045i \(-0.669183\pi\)
−0.506831 + 0.862045i \(0.669183\pi\)
\(954\) −13.2806 −0.429974
\(955\) −1.96758 −0.0636695
\(956\) −6.88488 −0.222673
\(957\) 0 0
\(958\) −7.97198 −0.257563
\(959\) −14.7184 −0.475281
\(960\) −0.454563 −0.0146709
\(961\) −30.4380 −0.981871
\(962\) 2.01869 0.0650853
\(963\) −26.3252 −0.848318
\(964\) −47.0959 −1.51686
\(965\) −20.1246 −0.647834
\(966\) −1.52295 −0.0490000
\(967\) 30.8263 0.991307 0.495654 0.868520i \(-0.334929\pi\)
0.495654 + 0.868520i \(0.334929\pi\)
\(968\) 0 0
\(969\) −26.8219 −0.861643
\(970\) 10.6259 0.341179
\(971\) −27.1359 −0.870833 −0.435416 0.900229i \(-0.643399\pi\)
−0.435416 + 0.900229i \(0.643399\pi\)
\(972\) −35.3769 −1.13471
\(973\) 2.58391 0.0828365
\(974\) −14.4528 −0.463097
\(975\) 5.36819 0.171920
\(976\) −14.0221 −0.448836
\(977\) 42.5686 1.36189 0.680945 0.732334i \(-0.261569\pi\)
0.680945 + 0.732334i \(0.261569\pi\)
\(978\) 29.8222 0.953608
\(979\) 0 0
\(980\) −1.59312 −0.0508902
\(981\) 60.9262 1.94523
\(982\) 5.55017 0.177113
\(983\) 42.1199 1.34342 0.671708 0.740816i \(-0.265561\pi\)
0.671708 + 0.740816i \(0.265561\pi\)
\(984\) −61.4200 −1.95800
\(985\) 20.7981 0.662681
\(986\) 9.96358 0.317305
\(987\) 1.03669 0.0329983
\(988\) 5.37056 0.170860
\(989\) −2.59376 −0.0824768
\(990\) 0 0
\(991\) −36.5439 −1.16086 −0.580428 0.814312i \(-0.697115\pi\)
−0.580428 + 0.814312i \(0.697115\pi\)
\(992\) 4.26090 0.135284
\(993\) 49.0567 1.55677
\(994\) −6.95638 −0.220643
\(995\) 0.189373 0.00600352
\(996\) −72.2141 −2.28819
\(997\) −15.0094 −0.475352 −0.237676 0.971344i \(-0.576386\pi\)
−0.237676 + 0.971344i \(0.576386\pi\)
\(998\) −9.78189 −0.309640
\(999\) −2.30536 −0.0729383
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 4235.2.a.v.1.2 yes 4
11.10 odd 2 4235.2.a.u.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4235.2.a.u.1.3 4 11.10 odd 2
4235.2.a.v.1.2 yes 4 1.1 even 1 trivial