Properties

Label 224.3.g.b.15.5
Level 224
Weight 3
Character 224.15
Analytic conductor 6.104
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.292213762624.3
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 15.5
Root \(-1.05468 + 1.69931i\) of \(x^{8} - x^{7} - 2 x^{6} - 2 x^{5} + 24 x^{4} - 8 x^{3} - 32 x^{2} - 64 x + 256\)
Character \(\chi\) \(=\) 224.15
Dual form 224.3.g.b.15.6

$q$-expansion

\(f(q)\) \(=\) \(q+3.44128 q^{3} -4.88287i q^{5} +2.64575i q^{7} +2.84239 q^{9} +O(q^{10})\) \(q+3.44128 q^{3} -4.88287i q^{5} +2.64575i q^{7} +2.84239 q^{9} +21.4776 q^{11} -13.0760i q^{13} -16.8033i q^{15} -0.234889 q^{17} -4.55872 q^{19} +9.10476i q^{21} -10.9523i q^{23} +1.15761 q^{25} -21.1900 q^{27} +34.6435i q^{29} +34.1079i q^{31} +73.9103 q^{33} +12.9189 q^{35} -54.2370i q^{37} -44.9982i q^{39} -37.8300 q^{41} +4.84714 q^{43} -13.8790i q^{45} +72.3368i q^{47} -7.00000 q^{49} -0.808319 q^{51} +21.6707i q^{53} -104.872i q^{55} -15.6878 q^{57} -34.9007 q^{59} +63.6012i q^{61} +7.52026i q^{63} -63.8485 q^{65} -18.4344 q^{67} -37.6899i q^{69} +47.5244i q^{71} +55.9103 q^{73} +3.98365 q^{75} +56.8243i q^{77} +95.0135i q^{79} -98.5023 q^{81} -71.5156 q^{83} +1.14693i q^{85} +119.218i q^{87} -159.756 q^{89} +34.5959 q^{91} +117.375i q^{93} +22.2596i q^{95} -90.4794 q^{97} +61.0477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} + 48q^{9} + O(q^{10}) \) \( 8q + 8q^{3} + 48q^{9} + 32q^{11} - 80q^{17} - 56q^{19} - 16q^{25} + 32q^{27} + 32q^{33} - 56q^{35} + 128q^{41} - 56q^{49} + 368q^{51} + 56q^{57} - 104q^{59} - 72q^{65} - 304q^{67} - 112q^{73} - 72q^{75} + 48q^{81} - 72q^{83} - 512q^{89} + 56q^{91} + 64q^{97} - 256q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 3.44128 1.14709 0.573546 0.819173i \(-0.305567\pi\)
0.573546 + 0.819173i \(0.305567\pi\)
\(4\) 0 0
\(5\) − 4.88287i − 0.976573i −0.872683 0.488287i \(-0.837622\pi\)
0.872683 0.488287i \(-0.162378\pi\)
\(6\) 0 0
\(7\) 2.64575i 0.377964i
\(8\) 0 0
\(9\) 2.84239 0.315821
\(10\) 0 0
\(11\) 21.4776 1.95251 0.976253 0.216632i \(-0.0695071\pi\)
0.976253 + 0.216632i \(0.0695071\pi\)
\(12\) 0 0
\(13\) − 13.0760i − 1.00585i −0.864331 0.502924i \(-0.832258\pi\)
0.864331 0.502924i \(-0.167742\pi\)
\(14\) 0 0
\(15\) − 16.8033i − 1.12022i
\(16\) 0 0
\(17\) −0.234889 −0.0138170 −0.00690851 0.999976i \(-0.502199\pi\)
−0.00690851 + 0.999976i \(0.502199\pi\)
\(18\) 0 0
\(19\) −4.55872 −0.239933 −0.119966 0.992778i \(-0.538279\pi\)
−0.119966 + 0.992778i \(0.538279\pi\)
\(20\) 0 0
\(21\) 9.10476i 0.433560i
\(22\) 0 0
\(23\) − 10.9523i − 0.476187i −0.971242 0.238094i \(-0.923478\pi\)
0.971242 0.238094i \(-0.0765225\pi\)
\(24\) 0 0
\(25\) 1.15761 0.0463043
\(26\) 0 0
\(27\) −21.1900 −0.784816
\(28\) 0 0
\(29\) 34.6435i 1.19460i 0.802016 + 0.597302i \(0.203761\pi\)
−0.802016 + 0.597302i \(0.796239\pi\)
\(30\) 0 0
\(31\) 34.1079i 1.10025i 0.835081 + 0.550127i \(0.185421\pi\)
−0.835081 + 0.550127i \(0.814579\pi\)
\(32\) 0 0
\(33\) 73.9103 2.23971
\(34\) 0 0
\(35\) 12.9189 0.369110
\(36\) 0 0
\(37\) − 54.2370i − 1.46586i −0.680302 0.732932i \(-0.738151\pi\)
0.680302 0.732932i \(-0.261849\pi\)
\(38\) 0 0
\(39\) − 44.9982i − 1.15380i
\(40\) 0 0
\(41\) −37.8300 −0.922682 −0.461341 0.887223i \(-0.652632\pi\)
−0.461341 + 0.887223i \(0.652632\pi\)
\(42\) 0 0
\(43\) 4.84714 0.112724 0.0563621 0.998410i \(-0.482050\pi\)
0.0563621 + 0.998410i \(0.482050\pi\)
\(44\) 0 0
\(45\) − 13.8790i − 0.308423i
\(46\) 0 0
\(47\) 72.3368i 1.53908i 0.638598 + 0.769541i \(0.279515\pi\)
−0.638598 + 0.769541i \(0.720485\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) −0.808319 −0.0158494
\(52\) 0 0
\(53\) 21.6707i 0.408881i 0.978879 + 0.204440i \(0.0655374\pi\)
−0.978879 + 0.204440i \(0.934463\pi\)
\(54\) 0 0
\(55\) − 104.872i − 1.90677i
\(56\) 0 0
\(57\) −15.6878 −0.275225
\(58\) 0 0
\(59\) −34.9007 −0.591537 −0.295768 0.955260i \(-0.595576\pi\)
−0.295768 + 0.955260i \(0.595576\pi\)
\(60\) 0 0
\(61\) 63.6012i 1.04264i 0.853360 + 0.521321i \(0.174561\pi\)
−0.853360 + 0.521321i \(0.825439\pi\)
\(62\) 0 0
\(63\) 7.52026i 0.119369i
\(64\) 0 0
\(65\) −63.8485 −0.982284
\(66\) 0 0
\(67\) −18.4344 −0.275140 −0.137570 0.990492i \(-0.543929\pi\)
−0.137570 + 0.990492i \(0.543929\pi\)
\(68\) 0 0
\(69\) − 37.6899i − 0.546231i
\(70\) 0 0
\(71\) 47.5244i 0.669358i 0.942332 + 0.334679i \(0.108628\pi\)
−0.942332 + 0.334679i \(0.891372\pi\)
\(72\) 0 0
\(73\) 55.9103 0.765894 0.382947 0.923770i \(-0.374909\pi\)
0.382947 + 0.923770i \(0.374909\pi\)
\(74\) 0 0
\(75\) 3.98365 0.0531153
\(76\) 0 0
\(77\) 56.8243i 0.737978i
\(78\) 0 0
\(79\) 95.0135i 1.20270i 0.798985 + 0.601351i \(0.205371\pi\)
−0.798985 + 0.601351i \(0.794629\pi\)
\(80\) 0 0
\(81\) −98.5023 −1.21608
\(82\) 0 0
\(83\) −71.5156 −0.861634 −0.430817 0.902439i \(-0.641775\pi\)
−0.430817 + 0.902439i \(0.641775\pi\)
\(84\) 0 0
\(85\) 1.14693i 0.0134933i
\(86\) 0 0
\(87\) 119.218i 1.37032i
\(88\) 0 0
\(89\) −159.756 −1.79501 −0.897504 0.441006i \(-0.854622\pi\)
−0.897504 + 0.441006i \(0.854622\pi\)
\(90\) 0 0
\(91\) 34.5959 0.380175
\(92\) 0 0
\(93\) 117.375i 1.26209i
\(94\) 0 0
\(95\) 22.2596i 0.234312i
\(96\) 0 0
\(97\) −90.4794 −0.932777 −0.466389 0.884580i \(-0.654445\pi\)
−0.466389 + 0.884580i \(0.654445\pi\)
\(98\) 0 0
\(99\) 61.0477 0.616643
\(100\) 0 0
\(101\) − 181.147i − 1.79353i −0.442503 0.896767i \(-0.645909\pi\)
0.442503 0.896767i \(-0.354091\pi\)
\(102\) 0 0
\(103\) − 39.3003i − 0.381556i −0.981633 0.190778i \(-0.938899\pi\)
0.981633 0.190778i \(-0.0611010\pi\)
\(104\) 0 0
\(105\) 44.4574 0.423403
\(106\) 0 0
\(107\) −38.4498 −0.359344 −0.179672 0.983727i \(-0.557504\pi\)
−0.179672 + 0.983727i \(0.557504\pi\)
\(108\) 0 0
\(109\) 27.8786i 0.255767i 0.991789 + 0.127883i \(0.0408183\pi\)
−0.991789 + 0.127883i \(0.959182\pi\)
\(110\) 0 0
\(111\) − 186.644i − 1.68148i
\(112\) 0 0
\(113\) 82.4419 0.729574 0.364787 0.931091i \(-0.381142\pi\)
0.364787 + 0.931091i \(0.381142\pi\)
\(114\) 0 0
\(115\) −53.4786 −0.465032
\(116\) 0 0
\(117\) − 37.1672i − 0.317668i
\(118\) 0 0
\(119\) − 0.621458i − 0.00522234i
\(120\) 0 0
\(121\) 340.286 2.81228
\(122\) 0 0
\(123\) −130.183 −1.05840
\(124\) 0 0
\(125\) − 127.724i − 1.02179i
\(126\) 0 0
\(127\) − 25.1408i − 0.197959i −0.995089 0.0989796i \(-0.968442\pi\)
0.995089 0.0989796i \(-0.0315579\pi\)
\(128\) 0 0
\(129\) 16.6804 0.129305
\(130\) 0 0
\(131\) 126.398 0.964872 0.482436 0.875931i \(-0.339752\pi\)
0.482436 + 0.875931i \(0.339752\pi\)
\(132\) 0 0
\(133\) − 12.0612i − 0.0906861i
\(134\) 0 0
\(135\) 103.468i 0.766431i
\(136\) 0 0
\(137\) 34.9456 0.255078 0.127539 0.991834i \(-0.459292\pi\)
0.127539 + 0.991834i \(0.459292\pi\)
\(138\) 0 0
\(139\) 119.148 0.857177 0.428589 0.903500i \(-0.359011\pi\)
0.428589 + 0.903500i \(0.359011\pi\)
\(140\) 0 0
\(141\) 248.931i 1.76547i
\(142\) 0 0
\(143\) − 280.841i − 1.96392i
\(144\) 0 0
\(145\) 169.160 1.16662
\(146\) 0 0
\(147\) −24.0889 −0.163870
\(148\) 0 0
\(149\) 121.932i 0.818334i 0.912460 + 0.409167i \(0.134181\pi\)
−0.912460 + 0.409167i \(0.865819\pi\)
\(150\) 0 0
\(151\) − 220.404i − 1.45963i −0.683645 0.729815i \(-0.739606\pi\)
0.683645 0.729815i \(-0.260394\pi\)
\(152\) 0 0
\(153\) −0.667647 −0.00436371
\(154\) 0 0
\(155\) 166.544 1.07448
\(156\) 0 0
\(157\) 6.77014i 0.0431219i 0.999768 + 0.0215610i \(0.00686360\pi\)
−0.999768 + 0.0215610i \(0.993136\pi\)
\(158\) 0 0
\(159\) 74.5748i 0.469024i
\(160\) 0 0
\(161\) 28.9771 0.179982
\(162\) 0 0
\(163\) 207.243 1.27143 0.635715 0.771924i \(-0.280706\pi\)
0.635715 + 0.771924i \(0.280706\pi\)
\(164\) 0 0
\(165\) − 360.894i − 2.18724i
\(166\) 0 0
\(167\) 165.529i 0.991193i 0.868553 + 0.495596i \(0.165050\pi\)
−0.868553 + 0.495596i \(0.834950\pi\)
\(168\) 0 0
\(169\) −1.98237 −0.0117300
\(170\) 0 0
\(171\) −12.9577 −0.0757759
\(172\) 0 0
\(173\) 88.8530i 0.513601i 0.966464 + 0.256800i \(0.0826683\pi\)
−0.966464 + 0.256800i \(0.917332\pi\)
\(174\) 0 0
\(175\) 3.06274i 0.0175014i
\(176\) 0 0
\(177\) −120.103 −0.678548
\(178\) 0 0
\(179\) −80.3791 −0.449045 −0.224523 0.974469i \(-0.572082\pi\)
−0.224523 + 0.974469i \(0.572082\pi\)
\(180\) 0 0
\(181\) − 276.353i − 1.52681i −0.645919 0.763406i \(-0.723526\pi\)
0.645919 0.763406i \(-0.276474\pi\)
\(182\) 0 0
\(183\) 218.869i 1.19601i
\(184\) 0 0
\(185\) −264.832 −1.43152
\(186\) 0 0
\(187\) −5.04485 −0.0269778
\(188\) 0 0
\(189\) − 56.0636i − 0.296633i
\(190\) 0 0
\(191\) 203.015i 1.06290i 0.847088 + 0.531452i \(0.178353\pi\)
−0.847088 + 0.531452i \(0.821647\pi\)
\(192\) 0 0
\(193\) 87.3328 0.452502 0.226251 0.974069i \(-0.427353\pi\)
0.226251 + 0.974069i \(0.427353\pi\)
\(194\) 0 0
\(195\) −219.720 −1.12677
\(196\) 0 0
\(197\) − 21.6639i − 0.109969i −0.998487 0.0549845i \(-0.982489\pi\)
0.998487 0.0549845i \(-0.0175109\pi\)
\(198\) 0 0
\(199\) − 181.933i − 0.914235i −0.889406 0.457118i \(-0.848882\pi\)
0.889406 0.457118i \(-0.151118\pi\)
\(200\) 0 0
\(201\) −63.4378 −0.315611
\(202\) 0 0
\(203\) −91.6582 −0.451518
\(204\) 0 0
\(205\) 184.719i 0.901067i
\(206\) 0 0
\(207\) − 31.1307i − 0.150390i
\(208\) 0 0
\(209\) −97.9103 −0.468470
\(210\) 0 0
\(211\) 21.4204 0.101519 0.0507594 0.998711i \(-0.483836\pi\)
0.0507594 + 0.998711i \(0.483836\pi\)
\(212\) 0 0
\(213\) 163.545i 0.767815i
\(214\) 0 0
\(215\) − 23.6680i − 0.110084i
\(216\) 0 0
\(217\) −90.2410 −0.415857
\(218\) 0 0
\(219\) 192.403 0.878552
\(220\) 0 0
\(221\) 3.07142i 0.0138978i
\(222\) 0 0
\(223\) − 195.958i − 0.878735i −0.898307 0.439367i \(-0.855203\pi\)
0.898307 0.439367i \(-0.144797\pi\)
\(224\) 0 0
\(225\) 3.29038 0.0146239
\(226\) 0 0
\(227\) 27.2652 0.120111 0.0600554 0.998195i \(-0.480872\pi\)
0.0600554 + 0.998195i \(0.480872\pi\)
\(228\) 0 0
\(229\) − 176.347i − 0.770076i −0.922901 0.385038i \(-0.874188\pi\)
0.922901 0.385038i \(-0.125812\pi\)
\(230\) 0 0
\(231\) 195.548i 0.846529i
\(232\) 0 0
\(233\) 71.8366 0.308312 0.154156 0.988047i \(-0.450734\pi\)
0.154156 + 0.988047i \(0.450734\pi\)
\(234\) 0 0
\(235\) 353.211 1.50303
\(236\) 0 0
\(237\) 326.968i 1.37961i
\(238\) 0 0
\(239\) 71.0926i 0.297459i 0.988878 + 0.148729i \(0.0475183\pi\)
−0.988878 + 0.148729i \(0.952482\pi\)
\(240\) 0 0
\(241\) 56.1113 0.232827 0.116413 0.993201i \(-0.462860\pi\)
0.116413 + 0.993201i \(0.462860\pi\)
\(242\) 0 0
\(243\) −148.264 −0.610138
\(244\) 0 0
\(245\) 34.1801i 0.139510i
\(246\) 0 0
\(247\) 59.6100i 0.241336i
\(248\) 0 0
\(249\) −246.105 −0.988374
\(250\) 0 0
\(251\) −368.953 −1.46993 −0.734966 0.678104i \(-0.762802\pi\)
−0.734966 + 0.678104i \(0.762802\pi\)
\(252\) 0 0
\(253\) − 235.229i − 0.929759i
\(254\) 0 0
\(255\) 3.94691i 0.0154781i
\(256\) 0 0
\(257\) 23.7428 0.0923845 0.0461923 0.998933i \(-0.485291\pi\)
0.0461923 + 0.998933i \(0.485291\pi\)
\(258\) 0 0
\(259\) 143.498 0.554044
\(260\) 0 0
\(261\) 98.4705i 0.377282i
\(262\) 0 0
\(263\) 73.9707i 0.281257i 0.990062 + 0.140629i \(0.0449124\pi\)
−0.990062 + 0.140629i \(0.955088\pi\)
\(264\) 0 0
\(265\) 105.815 0.399302
\(266\) 0 0
\(267\) −549.764 −2.05904
\(268\) 0 0
\(269\) 335.593i 1.24756i 0.781601 + 0.623779i \(0.214403\pi\)
−0.781601 + 0.623779i \(0.785597\pi\)
\(270\) 0 0
\(271\) − 187.276i − 0.691054i −0.938409 0.345527i \(-0.887700\pi\)
0.938409 0.345527i \(-0.112300\pi\)
\(272\) 0 0
\(273\) 119.054 0.436096
\(274\) 0 0
\(275\) 24.8626 0.0904095
\(276\) 0 0
\(277\) 132.592i 0.478670i 0.970937 + 0.239335i \(0.0769294\pi\)
−0.970937 + 0.239335i \(0.923071\pi\)
\(278\) 0 0
\(279\) 96.9480i 0.347484i
\(280\) 0 0
\(281\) 331.520 1.17979 0.589894 0.807481i \(-0.299170\pi\)
0.589894 + 0.807481i \(0.299170\pi\)
\(282\) 0 0
\(283\) 66.7158 0.235745 0.117873 0.993029i \(-0.462393\pi\)
0.117873 + 0.993029i \(0.462393\pi\)
\(284\) 0 0
\(285\) 76.6016i 0.268778i
\(286\) 0 0
\(287\) − 100.089i − 0.348741i
\(288\) 0 0
\(289\) −288.945 −0.999809
\(290\) 0 0
\(291\) −311.365 −1.06998
\(292\) 0 0
\(293\) − 289.215i − 0.987082i −0.869723 0.493541i \(-0.835702\pi\)
0.869723 0.493541i \(-0.164298\pi\)
\(294\) 0 0
\(295\) 170.415i 0.577679i
\(296\) 0 0
\(297\) −455.111 −1.53236
\(298\) 0 0
\(299\) −143.213 −0.478972
\(300\) 0 0
\(301\) 12.8243i 0.0426058i
\(302\) 0 0
\(303\) − 623.377i − 2.05735i
\(304\) 0 0
\(305\) 310.556 1.01822
\(306\) 0 0
\(307\) −0.693177 −0.00225790 −0.00112895 0.999999i \(-0.500359\pi\)
−0.00112895 + 0.999999i \(0.500359\pi\)
\(308\) 0 0
\(309\) − 135.243i − 0.437680i
\(310\) 0 0
\(311\) 62.1583i 0.199866i 0.994994 + 0.0999330i \(0.0318628\pi\)
−0.994994 + 0.0999330i \(0.968137\pi\)
\(312\) 0 0
\(313\) 213.594 0.682408 0.341204 0.939989i \(-0.389165\pi\)
0.341204 + 0.939989i \(0.389165\pi\)
\(314\) 0 0
\(315\) 36.7204 0.116573
\(316\) 0 0
\(317\) − 23.4577i − 0.0739990i −0.999315 0.0369995i \(-0.988220\pi\)
0.999315 0.0369995i \(-0.0117800\pi\)
\(318\) 0 0
\(319\) 744.059i 2.33247i
\(320\) 0 0
\(321\) −132.317 −0.412201
\(322\) 0 0
\(323\) 1.07079 0.00331515
\(324\) 0 0
\(325\) − 15.1369i − 0.0465751i
\(326\) 0 0
\(327\) 95.9380i 0.293388i
\(328\) 0 0
\(329\) −191.385 −0.581718
\(330\) 0 0
\(331\) −507.406 −1.53295 −0.766474 0.642275i \(-0.777991\pi\)
−0.766474 + 0.642275i \(0.777991\pi\)
\(332\) 0 0
\(333\) − 154.163i − 0.462951i
\(334\) 0 0
\(335\) 90.0126i 0.268694i
\(336\) 0 0
\(337\) −342.726 −1.01699 −0.508495 0.861065i \(-0.669798\pi\)
−0.508495 + 0.861065i \(0.669798\pi\)
\(338\) 0 0
\(339\) 283.705 0.836889
\(340\) 0 0
\(341\) 732.555i 2.14825i
\(342\) 0 0
\(343\) − 18.5203i − 0.0539949i
\(344\) 0 0
\(345\) −184.035 −0.533434
\(346\) 0 0
\(347\) −136.745 −0.394079 −0.197039 0.980396i \(-0.563133\pi\)
−0.197039 + 0.980396i \(0.563133\pi\)
\(348\) 0 0
\(349\) − 82.0565i − 0.235119i −0.993066 0.117559i \(-0.962493\pi\)
0.993066 0.117559i \(-0.0375071\pi\)
\(350\) 0 0
\(351\) 277.081i 0.789406i
\(352\) 0 0
\(353\) 507.367 1.43730 0.718651 0.695371i \(-0.244760\pi\)
0.718651 + 0.695371i \(0.244760\pi\)
\(354\) 0 0
\(355\) 232.055 0.653677
\(356\) 0 0
\(357\) − 2.13861i − 0.00599051i
\(358\) 0 0
\(359\) − 560.809i − 1.56214i −0.624442 0.781071i \(-0.714674\pi\)
0.624442 0.781071i \(-0.285326\pi\)
\(360\) 0 0
\(361\) −340.218 −0.942432
\(362\) 0 0
\(363\) 1171.02 3.22595
\(364\) 0 0
\(365\) − 273.003i − 0.747952i
\(366\) 0 0
\(367\) − 26.9431i − 0.0734145i −0.999326 0.0367072i \(-0.988313\pi\)
0.999326 0.0367072i \(-0.0116869\pi\)
\(368\) 0 0
\(369\) −107.528 −0.291403
\(370\) 0 0
\(371\) −57.3352 −0.154542
\(372\) 0 0
\(373\) − 538.034i − 1.44245i −0.692701 0.721225i \(-0.743579\pi\)
0.692701 0.721225i \(-0.256421\pi\)
\(374\) 0 0
\(375\) − 439.534i − 1.17209i
\(376\) 0 0
\(377\) 453.000 1.20159
\(378\) 0 0
\(379\) 182.132 0.480560 0.240280 0.970704i \(-0.422761\pi\)
0.240280 + 0.970704i \(0.422761\pi\)
\(380\) 0 0
\(381\) − 86.5166i − 0.227078i
\(382\) 0 0
\(383\) 333.271i 0.870160i 0.900392 + 0.435080i \(0.143280\pi\)
−0.900392 + 0.435080i \(0.856720\pi\)
\(384\) 0 0
\(385\) 277.466 0.720690
\(386\) 0 0
\(387\) 13.7775 0.0356007
\(388\) 0 0
\(389\) 109.639i 0.281847i 0.990020 + 0.140924i \(0.0450072\pi\)
−0.990020 + 0.140924i \(0.954993\pi\)
\(390\) 0 0
\(391\) 2.57258i 0.00657948i
\(392\) 0 0
\(393\) 434.971 1.10680
\(394\) 0 0
\(395\) 463.938 1.17453
\(396\) 0 0
\(397\) − 310.938i − 0.783219i −0.920131 0.391610i \(-0.871918\pi\)
0.920131 0.391610i \(-0.128082\pi\)
\(398\) 0 0
\(399\) − 41.5061i − 0.104025i
\(400\) 0 0
\(401\) −423.903 −1.05711 −0.528557 0.848898i \(-0.677267\pi\)
−0.528557 + 0.848898i \(0.677267\pi\)
\(402\) 0 0
\(403\) 445.995 1.10669
\(404\) 0 0
\(405\) 480.974i 1.18759i
\(406\) 0 0
\(407\) − 1164.88i − 2.86211i
\(408\) 0 0
\(409\) 444.543 1.08690 0.543451 0.839441i \(-0.317117\pi\)
0.543451 + 0.839441i \(0.317117\pi\)
\(410\) 0 0
\(411\) 120.258 0.292598
\(412\) 0 0
\(413\) − 92.3385i − 0.223580i
\(414\) 0 0
\(415\) 349.201i 0.841449i
\(416\) 0 0
\(417\) 410.020 0.983261
\(418\) 0 0
\(419\) −457.129 −1.09100 −0.545500 0.838111i \(-0.683660\pi\)
−0.545500 + 0.838111i \(0.683660\pi\)
\(420\) 0 0
\(421\) 25.4812i 0.0605255i 0.999542 + 0.0302628i \(0.00963441\pi\)
−0.999542 + 0.0302628i \(0.990366\pi\)
\(422\) 0 0
\(423\) 205.610i 0.486075i
\(424\) 0 0
\(425\) −0.271910 −0.000639787 0
\(426\) 0 0
\(427\) −168.273 −0.394082
\(428\) 0 0
\(429\) − 966.453i − 2.25280i
\(430\) 0 0
\(431\) 124.595i 0.289084i 0.989499 + 0.144542i \(0.0461709\pi\)
−0.989499 + 0.144542i \(0.953829\pi\)
\(432\) 0 0
\(433\) −272.271 −0.628802 −0.314401 0.949290i \(-0.601804\pi\)
−0.314401 + 0.949290i \(0.601804\pi\)
\(434\) 0 0
\(435\) 582.126 1.33822
\(436\) 0 0
\(437\) 49.9285i 0.114253i
\(438\) 0 0
\(439\) − 255.069i − 0.581023i −0.956871 0.290512i \(-0.906174\pi\)
0.956871 0.290512i \(-0.0938255\pi\)
\(440\) 0 0
\(441\) −19.8967 −0.0451173
\(442\) 0 0
\(443\) −131.274 −0.296330 −0.148165 0.988963i \(-0.547337\pi\)
−0.148165 + 0.988963i \(0.547337\pi\)
\(444\) 0 0
\(445\) 780.066i 1.75296i
\(446\) 0 0
\(447\) 419.601i 0.938704i
\(448\) 0 0
\(449\) −642.824 −1.43168 −0.715839 0.698265i \(-0.753956\pi\)
−0.715839 + 0.698265i \(0.753956\pi\)
\(450\) 0 0
\(451\) −812.496 −1.80154
\(452\) 0 0
\(453\) − 758.472i − 1.67433i
\(454\) 0 0
\(455\) − 168.927i − 0.371269i
\(456\) 0 0
\(457\) 693.088 1.51660 0.758302 0.651903i \(-0.226029\pi\)
0.758302 + 0.651903i \(0.226029\pi\)
\(458\) 0 0
\(459\) 4.97731 0.0108438
\(460\) 0 0
\(461\) − 258.699i − 0.561170i −0.959829 0.280585i \(-0.909472\pi\)
0.959829 0.280585i \(-0.0905285\pi\)
\(462\) 0 0
\(463\) − 637.226i − 1.37630i −0.725569 0.688150i \(-0.758423\pi\)
0.725569 0.688150i \(-0.241577\pi\)
\(464\) 0 0
\(465\) 573.125 1.23253
\(466\) 0 0
\(467\) 199.483 0.427159 0.213580 0.976926i \(-0.431488\pi\)
0.213580 + 0.976926i \(0.431488\pi\)
\(468\) 0 0
\(469\) − 48.7727i − 0.103993i
\(470\) 0 0
\(471\) 23.2979i 0.0494648i
\(472\) 0 0
\(473\) 104.105 0.220095
\(474\) 0 0
\(475\) −5.27721 −0.0111099
\(476\) 0 0
\(477\) 61.5966i 0.129133i
\(478\) 0 0
\(479\) 674.160i 1.40743i 0.710481 + 0.703716i \(0.248477\pi\)
−0.710481 + 0.703716i \(0.751523\pi\)
\(480\) 0 0
\(481\) −709.204 −1.47444
\(482\) 0 0
\(483\) 99.7182 0.206456
\(484\) 0 0
\(485\) 441.799i 0.910926i
\(486\) 0 0
\(487\) 401.718i 0.824883i 0.910984 + 0.412442i \(0.135324\pi\)
−0.910984 + 0.412442i \(0.864676\pi\)
\(488\) 0 0
\(489\) 713.181 1.45845
\(490\) 0 0
\(491\) −428.880 −0.873482 −0.436741 0.899587i \(-0.643867\pi\)
−0.436741 + 0.899587i \(0.643867\pi\)
\(492\) 0 0
\(493\) − 8.13739i − 0.0165059i
\(494\) 0 0
\(495\) − 298.088i − 0.602198i
\(496\) 0 0
\(497\) −125.738 −0.252993
\(498\) 0 0
\(499\) −182.619 −0.365970 −0.182985 0.983116i \(-0.558576\pi\)
−0.182985 + 0.983116i \(0.558576\pi\)
\(500\) 0 0
\(501\) 569.632i 1.13699i
\(502\) 0 0
\(503\) 380.158i 0.755781i 0.925850 + 0.377891i \(0.123350\pi\)
−0.925850 + 0.377891i \(0.876650\pi\)
\(504\) 0 0
\(505\) −884.516 −1.75152
\(506\) 0 0
\(507\) −6.82188 −0.0134554
\(508\) 0 0
\(509\) − 289.538i − 0.568836i −0.958700 0.284418i \(-0.908200\pi\)
0.958700 0.284418i \(-0.0918004\pi\)
\(510\) 0 0
\(511\) 147.925i 0.289481i
\(512\) 0 0
\(513\) 96.5995 0.188303
\(514\) 0 0
\(515\) −191.898 −0.372617
\(516\) 0 0
\(517\) 1553.62i 3.00507i
\(518\) 0 0
\(519\) 305.768i 0.589148i
\(520\) 0 0
\(521\) −738.899 −1.41823 −0.709116 0.705092i \(-0.750906\pi\)
−0.709116 + 0.705092i \(0.750906\pi\)
\(522\) 0 0
\(523\) −647.126 −1.23734 −0.618668 0.785653i \(-0.712327\pi\)
−0.618668 + 0.785653i \(0.712327\pi\)
\(524\) 0 0
\(525\) 10.5397i 0.0200757i
\(526\) 0 0
\(527\) − 8.01157i − 0.0152022i
\(528\) 0 0
\(529\) 409.047 0.773246
\(530\) 0 0
\(531\) −99.2014 −0.186820
\(532\) 0 0
\(533\) 494.666i 0.928078i
\(534\) 0 0
\(535\) 187.745i 0.350926i
\(536\) 0 0
\(537\) −276.607 −0.515097
\(538\) 0 0
\(539\) −150.343 −0.278930
\(540\) 0 0
\(541\) 178.722i 0.330355i 0.986264 + 0.165178i \(0.0528198\pi\)
−0.986264 + 0.165178i \(0.947180\pi\)
\(542\) 0 0
\(543\) − 951.008i − 1.75140i
\(544\) 0 0
\(545\) 136.127 0.249775
\(546\) 0 0
\(547\) −452.236 −0.826758 −0.413379 0.910559i \(-0.635651\pi\)
−0.413379 + 0.910559i \(0.635651\pi\)
\(548\) 0 0
\(549\) 180.780i 0.329289i
\(550\) 0 0
\(551\) − 157.930i − 0.286625i
\(552\) 0 0
\(553\) −251.382 −0.454579
\(554\) 0 0
\(555\) −911.360 −1.64209
\(556\) 0 0
\(557\) − 854.108i − 1.53341i −0.642001 0.766704i \(-0.721895\pi\)
0.642001 0.766704i \(-0.278105\pi\)
\(558\) 0 0
\(559\) − 63.3814i − 0.113383i
\(560\) 0 0
\(561\) −17.3607 −0.0309460
\(562\) 0 0
\(563\) −249.654 −0.443436 −0.221718 0.975111i \(-0.571166\pi\)
−0.221718 + 0.975111i \(0.571166\pi\)
\(564\) 0 0
\(565\) − 402.553i − 0.712483i
\(566\) 0 0
\(567\) − 260.613i − 0.459634i
\(568\) 0 0
\(569\) 104.353 0.183396 0.0916982 0.995787i \(-0.470771\pi\)
0.0916982 + 0.995787i \(0.470771\pi\)
\(570\) 0 0
\(571\) 649.705 1.13784 0.568919 0.822394i \(-0.307362\pi\)
0.568919 + 0.822394i \(0.307362\pi\)
\(572\) 0 0
\(573\) 698.630i 1.21925i
\(574\) 0 0
\(575\) − 12.6785i − 0.0220495i
\(576\) 0 0
\(577\) −346.022 −0.599692 −0.299846 0.953988i \(-0.596935\pi\)
−0.299846 + 0.953988i \(0.596935\pi\)
\(578\) 0 0
\(579\) 300.537 0.519061
\(580\) 0 0
\(581\) − 189.213i − 0.325667i
\(582\) 0 0
\(583\) 465.434i 0.798342i
\(584\) 0 0
\(585\) −181.482 −0.310226
\(586\) 0 0
\(587\) 1153.54 1.96514 0.982572 0.185885i \(-0.0595150\pi\)
0.982572 + 0.185885i \(0.0595150\pi\)
\(588\) 0 0
\(589\) − 155.488i − 0.263987i
\(590\) 0 0
\(591\) − 74.5515i − 0.126145i
\(592\) 0 0
\(593\) 880.135 1.48421 0.742104 0.670285i \(-0.233828\pi\)
0.742104 + 0.670285i \(0.233828\pi\)
\(594\) 0 0
\(595\) −3.03450 −0.00510000
\(596\) 0 0
\(597\) − 626.081i − 1.04871i
\(598\) 0 0
\(599\) 554.939i 0.926442i 0.886243 + 0.463221i \(0.153306\pi\)
−0.886243 + 0.463221i \(0.846694\pi\)
\(600\) 0 0
\(601\) −666.057 −1.10825 −0.554124 0.832434i \(-0.686946\pi\)
−0.554124 + 0.832434i \(0.686946\pi\)
\(602\) 0 0
\(603\) −52.3977 −0.0868950
\(604\) 0 0
\(605\) − 1661.57i − 2.74640i
\(606\) 0 0
\(607\) 192.927i 0.317836i 0.987292 + 0.158918i \(0.0508006\pi\)
−0.987292 + 0.158918i \(0.949199\pi\)
\(608\) 0 0
\(609\) −315.421 −0.517933
\(610\) 0 0
\(611\) 945.878 1.54808
\(612\) 0 0
\(613\) 608.234i 0.992226i 0.868258 + 0.496113i \(0.165240\pi\)
−0.868258 + 0.496113i \(0.834760\pi\)
\(614\) 0 0
\(615\) 635.668i 1.03361i
\(616\) 0 0
\(617\) 0.884056 0.00143283 0.000716415 1.00000i \(-0.499772\pi\)
0.000716415 1.00000i \(0.499772\pi\)
\(618\) 0 0
\(619\) −358.525 −0.579200 −0.289600 0.957148i \(-0.593522\pi\)
−0.289600 + 0.957148i \(0.593522\pi\)
\(620\) 0 0
\(621\) 232.080i 0.373719i
\(622\) 0 0
\(623\) − 422.674i − 0.678449i
\(624\) 0 0
\(625\) −594.720 −0.951552
\(626\) 0 0
\(627\) −336.937 −0.537379
\(628\) 0 0
\(629\) 12.7397i 0.0202539i
\(630\) 0 0
\(631\) 390.515i 0.618883i 0.950918 + 0.309442i \(0.100142\pi\)
−0.950918 + 0.309442i \(0.899858\pi\)
\(632\) 0 0
\(633\) 73.7137 0.116451
\(634\) 0 0
\(635\) −122.759 −0.193322
\(636\) 0 0
\(637\) 91.5322i 0.143693i
\(638\) 0 0
\(639\) 135.083i 0.211397i
\(640\) 0 0
\(641\) −431.936 −0.673848 −0.336924 0.941532i \(-0.609386\pi\)
−0.336924 + 0.941532i \(0.609386\pi\)
\(642\) 0 0
\(643\) −49.9370 −0.0776625 −0.0388313 0.999246i \(-0.512363\pi\)
−0.0388313 + 0.999246i \(0.512363\pi\)
\(644\) 0 0
\(645\) − 81.4480i − 0.126276i
\(646\) 0 0
\(647\) 224.141i 0.346431i 0.984884 + 0.173216i \(0.0554157\pi\)
−0.984884 + 0.173216i \(0.944584\pi\)
\(648\) 0 0
\(649\) −749.582 −1.15498
\(650\) 0 0
\(651\) −310.544 −0.477027
\(652\) 0 0
\(653\) 80.7637i 0.123681i 0.998086 + 0.0618405i \(0.0196970\pi\)
−0.998086 + 0.0618405i \(0.980303\pi\)
\(654\) 0 0
\(655\) − 617.186i − 0.942268i
\(656\) 0 0
\(657\) 158.919 0.241886
\(658\) 0 0
\(659\) 940.466 1.42711 0.713555 0.700599i \(-0.247084\pi\)
0.713555 + 0.700599i \(0.247084\pi\)
\(660\) 0 0
\(661\) 119.930i 0.181437i 0.995877 + 0.0907184i \(0.0289163\pi\)
−0.995877 + 0.0907184i \(0.971084\pi\)
\(662\) 0 0
\(663\) 10.5696i 0.0159421i
\(664\) 0 0
\(665\) −58.8935 −0.0885616
\(666\) 0 0
\(667\) 379.426 0.568855
\(668\) 0 0
\(669\) − 674.346i − 1.00799i
\(670\) 0 0
\(671\) 1366.00i 2.03577i
\(672\) 0 0
\(673\) 1085.06 1.61227 0.806136 0.591731i \(-0.201555\pi\)
0.806136 + 0.591731i \(0.201555\pi\)
\(674\) 0 0
\(675\) −24.5298 −0.0363404
\(676\) 0 0
\(677\) 949.901i 1.40310i 0.712618 + 0.701552i \(0.247509\pi\)
−0.712618 + 0.701552i \(0.752491\pi\)
\(678\) 0 0
\(679\) − 239.386i − 0.352557i
\(680\) 0 0
\(681\) 93.8270 0.137778
\(682\) 0 0
\(683\) 893.785 1.30862 0.654308 0.756228i \(-0.272960\pi\)
0.654308 + 0.756228i \(0.272960\pi\)
\(684\) 0 0
\(685\) − 170.635i − 0.249102i
\(686\) 0 0
\(687\) − 606.861i − 0.883349i
\(688\) 0 0
\(689\) 283.366 0.411272
\(690\) 0 0
\(691\) −1208.56 −1.74901 −0.874504 0.485019i \(-0.838813\pi\)
−0.874504 + 0.485019i \(0.838813\pi\)
\(692\) 0 0
\(693\) 161.517i 0.233069i
\(694\) 0 0
\(695\) − 581.782i − 0.837096i
\(696\) 0 0
\(697\) 8.88585 0.0127487
\(698\) 0 0
\(699\) 247.210 0.353662
\(700\) 0 0
\(701\) 219.477i 0.313091i 0.987671 + 0.156546i \(0.0500358\pi\)
−0.987671 + 0.156546i \(0.949964\pi\)
\(702\) 0 0
\(703\) 247.251i 0.351709i
\(704\) 0 0
\(705\) 1215.50 1.72411
\(706\) 0 0
\(707\) 479.270 0.677892
\(708\) 0 0
\(709\) 1265.13i 1.78439i 0.451651 + 0.892195i \(0.350835\pi\)
−0.451651 + 0.892195i \(0.649165\pi\)
\(710\) 0 0
\(711\) 270.066i 0.379839i
\(712\) 0 0
\(713\) 373.560 0.523927
\(714\) 0 0
\(715\) −1371.31 −1.91792
\(716\) 0 0
\(717\) 244.650i 0.341213i
\(718\) 0 0
\(719\) − 1163.47i − 1.61818i −0.587687 0.809089i \(-0.699961\pi\)
0.587687 0.809089i \(-0.300039\pi\)
\(720\) 0 0
\(721\) 103.979 0.144215
\(722\) 0 0
\(723\) 193.094 0.267074
\(724\) 0 0
\(725\) 40.1036i 0.0553153i
\(726\) 0 0
\(727\) − 1303.68i − 1.79324i −0.442803 0.896619i \(-0.646016\pi\)
0.442803 0.896619i \(-0.353984\pi\)
\(728\) 0 0
\(729\) 376.305 0.516193
\(730\) 0 0
\(731\) −1.13854 −0.00155751
\(732\) 0 0
\(733\) − 1256.12i − 1.71367i −0.515589 0.856836i \(-0.672427\pi\)
0.515589 0.856836i \(-0.327573\pi\)
\(734\) 0 0
\(735\) 117.623i 0.160031i
\(736\) 0 0
\(737\) −395.925 −0.537212
\(738\) 0 0
\(739\) −687.168 −0.929862 −0.464931 0.885347i \(-0.653921\pi\)
−0.464931 + 0.885347i \(0.653921\pi\)
\(740\) 0 0
\(741\) 205.134i 0.276835i
\(742\) 0 0
\(743\) 362.628i 0.488059i 0.969768 + 0.244030i \(0.0784694\pi\)
−0.969768 + 0.244030i \(0.921531\pi\)
\(744\) 0 0
\(745\) 595.376 0.799163
\(746\) 0 0
\(747\) −203.275 −0.272122
\(748\) 0 0
\(749\) − 101.729i − 0.135819i
\(750\) 0 0
\(751\) − 261.366i − 0.348024i −0.984744 0.174012i \(-0.944327\pi\)
0.984744 0.174012i \(-0.0556732\pi\)
\(752\) 0 0
\(753\) −1269.67 −1.68615
\(754\) 0 0
\(755\) −1076.20 −1.42544
\(756\) 0 0
\(757\) − 1395.34i − 1.84325i −0.388081 0.921625i \(-0.626862\pi\)
0.388081 0.921625i \(-0.373138\pi\)
\(758\) 0 0
\(759\) − 809.488i − 1.06652i
\(760\) 0 0
\(761\) −319.500 −0.419843 −0.209921 0.977718i \(-0.567321\pi\)
−0.209921 + 0.977718i \(0.567321\pi\)
\(762\) 0 0
\(763\) −73.7598 −0.0966708
\(764\) 0 0
\(765\) 3.26003i 0.00426148i
\(766\) 0 0
\(767\) 456.362i 0.594996i
\(768\) 0 0
\(769\) 634.936 0.825664 0.412832 0.910807i \(-0.364540\pi\)
0.412832 + 0.910807i \(0.364540\pi\)
\(770\) 0 0
\(771\) 81.7056 0.105974
\(772\) 0 0
\(773\) 96.1663i 0.124407i 0.998063 + 0.0622033i \(0.0198127\pi\)
−0.998063 + 0.0622033i \(0.980187\pi\)
\(774\) 0 0
\(775\) 39.4836i 0.0509465i
\(776\) 0 0
\(777\) 493.815 0.635540
\(778\) 0 0
\(779\) 172.456 0.221382
\(780\) 0 0
\(781\) 1020.71i 1.30693i
\(782\) 0 0
\(783\) − 734.098i − 0.937545i
\(784\) 0 0
\(785\) 33.0577 0.0421117
\(786\) 0 0
\(787\) 1319.25 1.67630 0.838148 0.545442i \(-0.183638\pi\)
0.838148 + 0.545442i \(0.183638\pi\)
\(788\) 0 0
\(789\) 254.554i 0.322628i
\(790\) 0 0
\(791\) 218.121i 0.275753i
\(792\) 0 0
\(793\) 831.651 1.04874
\(794\) 0 0
\(795\) 364.139 0.458036
\(796\) 0 0
\(797\) − 818.575i − 1.02707i −0.858068 0.513535i \(-0.828336\pi\)
0.858068 0.513535i \(-0.171664\pi\)
\(798\) 0 0
\(799\) − 16.9911i − 0.0212655i
\(800\) 0 0
\(801\) −454.088 −0.566902
\(802\) 0 0
\(803\) 1200.82 1.49541
\(804\) 0 0
\(805\) − 141.491i − 0.175765i
\(806\) 0 0
\(807\) 1154.87i 1.43106i
\(808\) 0 0
\(809\) 1232.72 1.52376 0.761881 0.647717i \(-0.224276\pi\)
0.761881 + 0.647717i \(0.224276\pi\)
\(810\) 0 0
\(811\) 1009.05 1.24421 0.622103 0.782935i \(-0.286278\pi\)
0.622103 + 0.782935i \(0.286278\pi\)
\(812\) 0 0
\(813\) − 644.468i − 0.792703i
\(814\) 0 0
\(815\) − 1011.94i − 1.24164i
\(816\) 0 0
\(817\) −22.0968 −0.0270462
\(818\) 0 0
\(819\) 98.3351 0.120067
\(820\) 0 0
\(821\) − 939.093i − 1.14384i −0.820309 0.571920i \(-0.806199\pi\)
0.820309 0.571920i \(-0.193801\pi\)
\(822\) 0 0
\(823\) 911.100i 1.10705i 0.832833 + 0.553524i \(0.186717\pi\)
−0.832833 + 0.553524i \(0.813283\pi\)
\(824\) 0 0
\(825\) 85.5591 0.103708
\(826\) 0 0
\(827\) 65.6564 0.0793910 0.0396955 0.999212i \(-0.487361\pi\)
0.0396955 + 0.999212i \(0.487361\pi\)
\(828\) 0 0
\(829\) 1515.94i 1.82864i 0.404997 + 0.914318i \(0.367272\pi\)
−0.404997 + 0.914318i \(0.632728\pi\)
\(830\) 0 0
\(831\) 456.285i 0.549079i
\(832\) 0 0
\(833\) 1.64422 0.00197386
\(834\) 0 0
\(835\) 808.257 0.967972
\(836\) 0 0
\(837\) − 722.747i − 0.863497i
\(838\) 0 0
\(839\) − 869.972i − 1.03692i −0.855103 0.518458i \(-0.826506\pi\)
0.855103 0.518458i \(-0.173494\pi\)
\(840\) 0 0
\(841\) −359.174 −0.427080
\(842\) 0 0
\(843\) 1140.85 1.35333
\(844\) 0 0
\(845\) 9.67964i 0.0114552i
\(846\) 0 0
\(847\) 900.313i 1.06294i
\(848\) 0 0
\(849\) 229.588 0.270421
\(850\) 0 0
\(851\) −594.020 −0.698026
\(852\) 0 0
\(853\) − 1643.91i − 1.92721i −0.267322 0.963607i \(-0.586139\pi\)
0.267322 0.963607i \(-0.413861\pi\)
\(854\) 0 0
\(855\) 63.2706i 0.0740007i
\(856\) 0 0
\(857\) 286.059 0.333791 0.166895 0.985975i \(-0.446626\pi\)
0.166895 + 0.985975i \(0.446626\pi\)
\(858\) 0 0
\(859\) −719.782 −0.837930 −0.418965 0.908002i \(-0.637607\pi\)
−0.418965 + 0.908002i \(0.637607\pi\)
\(860\) 0 0
\(861\) − 344.433i − 0.400038i
\(862\) 0 0
\(863\) − 1120.47i − 1.29835i −0.760641 0.649173i \(-0.775115\pi\)
0.760641 0.649173i \(-0.224885\pi\)
\(864\) 0 0
\(865\) 433.857 0.501569
\(866\) 0 0
\(867\) −994.339 −1.14687
\(868\) 0 0
\(869\) 2040.66i 2.34828i
\(870\) 0 0
\(871\) 241.048i 0.276749i
\(872\) 0 0
\(873\) −257.178 −0.294591
\(874\) 0 0
\(875\) 337.926 0.386201
\(876\) 0 0
\(877\) 145.400i 0.165792i 0.996558 + 0.0828960i \(0.0264169\pi\)
−0.996558 + 0.0828960i \(0.973583\pi\)
\(878\) 0 0
\(879\) − 995.269i − 1.13227i
\(880\) 0 0
\(881\) 476.080 0.540386 0.270193 0.962806i \(-0.412913\pi\)
0.270193 + 0.962806i \(0.412913\pi\)
\(882\) 0 0
\(883\) 1101.22 1.24714 0.623568 0.781769i \(-0.285682\pi\)
0.623568 + 0.781769i \(0.285682\pi\)
\(884\) 0 0
\(885\) 586.447i 0.662652i
\(886\) 0 0
\(887\) 1491.49i 1.68150i 0.541427 + 0.840748i \(0.317884\pi\)
−0.541427 + 0.840748i \(0.682116\pi\)
\(888\) 0 0
\(889\) 66.5164 0.0748216
\(890\) 0 0
\(891\) −2115.59 −2.37440
\(892\) 0 0
\(893\) − 329.764i − 0.369276i
\(894\) 0 0
\(895\) 392.481i 0.438526i
\(896\) 0 0
\(897\) −492.834 −0.549425
\(898\) 0 0
\(899\) −1181.62 −1.31437
\(900\) 0 0
\(901\) − 5.09021i − 0.00564951i
\(902\) 0 0
\(903\) 44.1321i 0.0488728i
\(904\) 0 0
\(905\) −1349.40 −1.49104
\(906\) 0 0
\(907\) −1155.46 −1.27394 −0.636969 0.770889i \(-0.719812\pi\)
−0.636969 + 0.770889i \(0.719812\pi\)
\(908\) 0 0
\(909\) − 514.891i − 0.566436i
\(910\) 0 0
\(911\) 944.690i 1.03698i 0.855083 + 0.518491i \(0.173506\pi\)
−0.855083 + 0.518491i \(0.826494\pi\)
\(912\) 0 0
\(913\) −1535.98 −1.68235
\(914\) 0 0
\(915\) 1068.71 1.16799
\(916\) 0 0
\(917\) 334.418i 0.364687i
\(918\) 0 0
\(919\) − 149.150i − 0.162296i −0.996702 0.0811478i \(-0.974141\pi\)
0.996702 0.0811478i \(-0.0258586\pi\)
\(920\) 0 0
\(921\) −2.38541 −0.00259003
\(922\) 0 0
\(923\) 621.430 0.673272
\(924\) 0 0
\(925\) − 62.7851i − 0.0678758i
\(926\) 0 0
\(927\) − 111.707i − 0.120503i
\(928\) 0 0
\(929\) −58.8399 −0.0633368 −0.0316684 0.999498i \(-0.510082\pi\)
−0.0316684 + 0.999498i \(0.510082\pi\)
\(930\) 0 0
\(931\) 31.9111 0.0342761
\(932\) 0 0
\(933\) 213.904i 0.229265i
\(934\) 0 0
\(935\) 24.6333i 0.0263458i
\(936\) 0 0
\(937\) −1700.18 −1.81449 −0.907246 0.420601i \(-0.861819\pi\)
−0.907246 + 0.420601i \(0.861819\pi\)
\(938\) 0 0
\(939\) 735.036 0.782786
\(940\) 0 0
\(941\) − 56.6116i − 0.0601611i −0.999547 0.0300805i \(-0.990424\pi\)
0.999547 0.0300805i \(-0.00957637\pi\)
\(942\) 0 0
\(943\) 414.325i 0.439369i
\(944\) 0 0
\(945\) −273.751 −0.289684
\(946\) 0 0
\(947\) −242.533 −0.256107 −0.128053 0.991767i \(-0.540873\pi\)
−0.128053 + 0.991767i \(0.540873\pi\)
\(948\) 0 0
\(949\) − 731.084i − 0.770373i
\(950\) 0 0
\(951\) − 80.7244i − 0.0848837i
\(952\) 0 0
\(953\) −364.070 −0.382025 −0.191013 0.981588i \(-0.561177\pi\)
−0.191013 + 0.981588i \(0.561177\pi\)
\(954\) 0 0
\(955\) 991.294 1.03800
\(956\) 0 0
\(957\) 2560.51i 2.67556i
\(958\) 0 0
\(959\) 92.4575i 0.0964103i
\(960\) 0 0
\(961\) −202.348 −0.210560
\(962\) 0 0
\(963\) −109.290 −0.113489
\(964\) 0 0
\(965\) − 426.435i − 0.441901i
\(966\) 0 0
\(967\) − 1221.99i − 1.26369i −0.775093 0.631847i \(-0.782297\pi\)
0.775093 0.631847i \(-0.217703\pi\)
\(968\) 0 0
\(969\) 3.68490 0.00380279
\(970\) 0 0
\(971\) 1088.53 1.12104 0.560521 0.828140i \(-0.310601\pi\)
0.560521 + 0.828140i \(0.310601\pi\)
\(972\) 0 0
\(973\) 315.235i 0.323982i
\(974\) 0 0
\(975\) − 52.0903i − 0.0534259i
\(976\) 0 0
\(977\) −1061.51 −1.08650 −0.543252 0.839570i \(-0.682807\pi\)
−0.543252 + 0.839570i \(0.682807\pi\)
\(978\) 0 0
\(979\) −3431.17 −3.50477
\(980\) 0 0
\(981\) 79.2419i 0.0807766i
\(982\) 0 0
\(983\) − 1322.61i − 1.34549i −0.739877 0.672743i \(-0.765116\pi\)
0.739877 0.672743i \(-0.234884\pi\)
\(984\) 0 0
\(985\) −105.782 −0.107393
\(986\) 0 0
\(987\) −658.610 −0.667285
\(988\) 0 0
\(989\) − 53.0874i − 0.0536778i
\(990\) 0 0
\(991\) 675.806i 0.681944i 0.940073 + 0.340972i \(0.110756\pi\)
−0.940073 + 0.340972i \(0.889244\pi\)
\(992\) 0 0
\(993\) −1746.12 −1.75843
\(994\) 0 0
\(995\) −888.354 −0.892818
\(996\) 0 0
\(997\) − 409.220i − 0.410452i −0.978715 0.205226i \(-0.934207\pi\)
0.978715 0.205226i \(-0.0657929\pi\)
\(998\) 0 0
\(999\) 1149.28i 1.15043i
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 224.3.g.b.15.5 8
3.2 odd 2 2016.3.g.b.1135.6 8
4.3 odd 2 56.3.g.b.43.4 yes 8
7.6 odd 2 1568.3.g.m.687.4 8
8.3 odd 2 inner 224.3.g.b.15.6 8
8.5 even 2 56.3.g.b.43.3 8
12.11 even 2 504.3.g.b.379.5 8
16.3 odd 4 1792.3.d.j.1023.5 16
16.5 even 4 1792.3.d.j.1023.6 16
16.11 odd 4 1792.3.d.j.1023.12 16
16.13 even 4 1792.3.d.j.1023.11 16
24.5 odd 2 504.3.g.b.379.6 8
24.11 even 2 2016.3.g.b.1135.3 8
28.3 even 6 392.3.k.n.275.3 16
28.11 odd 6 392.3.k.o.275.3 16
28.19 even 6 392.3.k.n.67.8 16
28.23 odd 6 392.3.k.o.67.8 16
28.27 even 2 392.3.g.m.99.4 8
56.5 odd 6 392.3.k.n.67.3 16
56.13 odd 2 392.3.g.m.99.3 8
56.27 even 2 1568.3.g.m.687.3 8
56.37 even 6 392.3.k.o.67.3 16
56.45 odd 6 392.3.k.n.275.8 16
56.53 even 6 392.3.k.o.275.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.g.b.43.3 8 8.5 even 2
56.3.g.b.43.4 yes 8 4.3 odd 2
224.3.g.b.15.5 8 1.1 even 1 trivial
224.3.g.b.15.6 8 8.3 odd 2 inner
392.3.g.m.99.3 8 56.13 odd 2
392.3.g.m.99.4 8 28.27 even 2
392.3.k.n.67.3 16 56.5 odd 6
392.3.k.n.67.8 16 28.19 even 6
392.3.k.n.275.3 16 28.3 even 6
392.3.k.n.275.8 16 56.45 odd 6
392.3.k.o.67.3 16 56.37 even 6
392.3.k.o.67.8 16 28.23 odd 6
392.3.k.o.275.3 16 28.11 odd 6
392.3.k.o.275.8 16 56.53 even 6
504.3.g.b.379.5 8 12.11 even 2
504.3.g.b.379.6 8 24.5 odd 2
1568.3.g.m.687.3 8 56.27 even 2
1568.3.g.m.687.4 8 7.6 odd 2
1792.3.d.j.1023.5 16 16.3 odd 4
1792.3.d.j.1023.6 16 16.5 even 4
1792.3.d.j.1023.11 16 16.13 even 4
1792.3.d.j.1023.12 16 16.11 odd 4
2016.3.g.b.1135.3 8 24.11 even 2
2016.3.g.b.1135.6 8 3.2 odd 2