Properties

Label 2475.2.c.i.199.4
Level $2475$
Weight $2$
Character 2475.199
Analytic conductor $19.763$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 2475.199
Dual form 2475.2.c.i.199.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155i q^{2} -4.56155 q^{4} +1.00000i q^{7} -6.56155i q^{8} +O(q^{10})\) \(q+2.56155i q^{2} -4.56155 q^{4} +1.00000i q^{7} -6.56155i q^{8} -1.00000 q^{11} +0.438447i q^{13} -2.56155 q^{14} +7.68466 q^{16} +1.43845i q^{17} -3.00000 q^{19} -2.56155i q^{22} -6.56155i q^{23} -1.12311 q^{26} -4.56155i q^{28} -5.12311 q^{29} +4.68466 q^{31} +6.56155i q^{32} -3.68466 q^{34} -10.8078i q^{37} -7.68466i q^{38} -7.68466 q^{41} -4.68466i q^{43} +4.56155 q^{44} +16.8078 q^{46} +5.43845i q^{47} +6.00000 q^{49} -2.00000i q^{52} +1.12311i q^{53} +6.56155 q^{56} -13.1231i q^{58} -10.5616 q^{59} -7.56155 q^{61} +12.0000i q^{62} -1.43845 q^{64} -6.68466i q^{67} -6.56155i q^{68} +8.80776 q^{71} -16.2462i q^{73} +27.6847 q^{74} +13.6847 q^{76} -1.00000i q^{77} +15.6847 q^{79} -19.6847i q^{82} +12.0000 q^{86} +6.56155i q^{88} +17.1231 q^{89} -0.438447 q^{91} +29.9309i q^{92} -13.9309 q^{94} +2.12311i q^{97} +15.3693i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4} + O(q^{10}) \) \( 4 q - 10 q^{4} - 4 q^{11} - 2 q^{14} + 6 q^{16} - 12 q^{19} + 12 q^{26} - 4 q^{29} - 6 q^{31} + 10 q^{34} - 6 q^{41} + 10 q^{44} + 26 q^{46} + 24 q^{49} + 18 q^{56} - 34 q^{59} - 22 q^{61} - 14 q^{64} - 6 q^{71} + 86 q^{74} + 30 q^{76} + 38 q^{79} + 48 q^{86} + 52 q^{89} - 10 q^{91} + 2 q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56155i 1.81129i 0.424035 + 0.905646i \(0.360613\pi\)
−0.424035 + 0.905646i \(0.639387\pi\)
\(3\) 0 0
\(4\) −4.56155 −2.28078
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) − 6.56155i − 2.31986i
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.438447i 0.121603i 0.998150 + 0.0608017i \(0.0193657\pi\)
−0.998150 + 0.0608017i \(0.980634\pi\)
\(14\) −2.56155 −0.684604
\(15\) 0 0
\(16\) 7.68466 1.92116
\(17\) 1.43845i 0.348875i 0.984668 + 0.174437i \(0.0558106\pi\)
−0.984668 + 0.174437i \(0.944189\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 2.56155i − 0.546125i
\(23\) − 6.56155i − 1.36818i −0.729398 0.684089i \(-0.760200\pi\)
0.729398 0.684089i \(-0.239800\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.12311 −0.220259
\(27\) 0 0
\(28\) − 4.56155i − 0.862052i
\(29\) −5.12311 −0.951337 −0.475668 0.879625i \(-0.657794\pi\)
−0.475668 + 0.879625i \(0.657794\pi\)
\(30\) 0 0
\(31\) 4.68466 0.841389 0.420695 0.907202i \(-0.361786\pi\)
0.420695 + 0.907202i \(0.361786\pi\)
\(32\) 6.56155i 1.15993i
\(33\) 0 0
\(34\) −3.68466 −0.631914
\(35\) 0 0
\(36\) 0 0
\(37\) − 10.8078i − 1.77679i −0.459084 0.888393i \(-0.651822\pi\)
0.459084 0.888393i \(-0.348178\pi\)
\(38\) − 7.68466i − 1.24662i
\(39\) 0 0
\(40\) 0 0
\(41\) −7.68466 −1.20014 −0.600071 0.799947i \(-0.704861\pi\)
−0.600071 + 0.799947i \(0.704861\pi\)
\(42\) 0 0
\(43\) − 4.68466i − 0.714404i −0.934027 0.357202i \(-0.883731\pi\)
0.934027 0.357202i \(-0.116269\pi\)
\(44\) 4.56155 0.687680
\(45\) 0 0
\(46\) 16.8078 2.47817
\(47\) 5.43845i 0.793279i 0.917974 + 0.396640i \(0.129824\pi\)
−0.917974 + 0.396640i \(0.870176\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) 1.12311i 0.154270i 0.997021 + 0.0771352i \(0.0245773\pi\)
−0.997021 + 0.0771352i \(0.975423\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 6.56155 0.876824
\(57\) 0 0
\(58\) − 13.1231i − 1.72315i
\(59\) −10.5616 −1.37500 −0.687499 0.726186i \(-0.741291\pi\)
−0.687499 + 0.726186i \(0.741291\pi\)
\(60\) 0 0
\(61\) −7.56155 −0.968158 −0.484079 0.875024i \(-0.660845\pi\)
−0.484079 + 0.875024i \(0.660845\pi\)
\(62\) 12.0000i 1.52400i
\(63\) 0 0
\(64\) −1.43845 −0.179806
\(65\) 0 0
\(66\) 0 0
\(67\) − 6.68466i − 0.816661i −0.912834 0.408331i \(-0.866111\pi\)
0.912834 0.408331i \(-0.133889\pi\)
\(68\) − 6.56155i − 0.795705i
\(69\) 0 0
\(70\) 0 0
\(71\) 8.80776 1.04529 0.522645 0.852551i \(-0.324945\pi\)
0.522645 + 0.852551i \(0.324945\pi\)
\(72\) 0 0
\(73\) − 16.2462i − 1.90148i −0.309997 0.950738i \(-0.600328\pi\)
0.309997 0.950738i \(-0.399672\pi\)
\(74\) 27.6847 3.21828
\(75\) 0 0
\(76\) 13.6847 1.56974
\(77\) − 1.00000i − 0.113961i
\(78\) 0 0
\(79\) 15.6847 1.76466 0.882331 0.470629i \(-0.155973\pi\)
0.882331 + 0.470629i \(0.155973\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 19.6847i − 2.17381i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) 6.56155i 0.699464i
\(89\) 17.1231 1.81505 0.907523 0.420003i \(-0.137971\pi\)
0.907523 + 0.420003i \(0.137971\pi\)
\(90\) 0 0
\(91\) −0.438447 −0.0459618
\(92\) 29.9309i 3.12051i
\(93\) 0 0
\(94\) −13.9309 −1.43686
\(95\) 0 0
\(96\) 0 0
\(97\) 2.12311i 0.215569i 0.994174 + 0.107784i \(0.0343756\pi\)
−0.994174 + 0.107784i \(0.965624\pi\)
\(98\) 15.3693i 1.55254i
\(99\) 0 0
\(100\) 0 0
\(101\) 1.43845 0.143131 0.0715654 0.997436i \(-0.477201\pi\)
0.0715654 + 0.997436i \(0.477201\pi\)
\(102\) 0 0
\(103\) 9.12311i 0.898926i 0.893299 + 0.449463i \(0.148385\pi\)
−0.893299 + 0.449463i \(0.851615\pi\)
\(104\) 2.87689 0.282103
\(105\) 0 0
\(106\) −2.87689 −0.279429
\(107\) − 9.12311i − 0.881964i −0.897516 0.440982i \(-0.854630\pi\)
0.897516 0.440982i \(-0.145370\pi\)
\(108\) 0 0
\(109\) 7.80776 0.747848 0.373924 0.927459i \(-0.378012\pi\)
0.373924 + 0.927459i \(0.378012\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 7.68466i 0.726132i
\(113\) 9.12311i 0.858230i 0.903250 + 0.429115i \(0.141174\pi\)
−0.903250 + 0.429115i \(0.858826\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 23.3693 2.16979
\(117\) 0 0
\(118\) − 27.0540i − 2.49052i
\(119\) −1.43845 −0.131862
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 19.3693i − 1.75362i
\(123\) 0 0
\(124\) −21.3693 −1.91902
\(125\) 0 0
\(126\) 0 0
\(127\) − 4.80776i − 0.426620i −0.976985 0.213310i \(-0.931576\pi\)
0.976985 0.213310i \(-0.0684244\pi\)
\(128\) 9.43845i 0.834249i
\(129\) 0 0
\(130\) 0 0
\(131\) −16.4924 −1.44095 −0.720475 0.693481i \(-0.756076\pi\)
−0.720475 + 0.693481i \(0.756076\pi\)
\(132\) 0 0
\(133\) − 3.00000i − 0.260133i
\(134\) 17.1231 1.47921
\(135\) 0 0
\(136\) 9.43845 0.809340
\(137\) − 3.36932i − 0.287860i −0.989588 0.143930i \(-0.954026\pi\)
0.989588 0.143930i \(-0.0459740\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 22.5616i 1.89332i
\(143\) − 0.438447i − 0.0366648i
\(144\) 0 0
\(145\) 0 0
\(146\) 41.6155 3.44413
\(147\) 0 0
\(148\) 49.3002i 4.05245i
\(149\) −8.80776 −0.721560 −0.360780 0.932651i \(-0.617490\pi\)
−0.360780 + 0.932651i \(0.617490\pi\)
\(150\) 0 0
\(151\) −16.6847 −1.35778 −0.678889 0.734241i \(-0.737538\pi\)
−0.678889 + 0.734241i \(0.737538\pi\)
\(152\) 19.6847i 1.59664i
\(153\) 0 0
\(154\) 2.56155 0.206416
\(155\) 0 0
\(156\) 0 0
\(157\) 2.68466i 0.214259i 0.994245 + 0.107130i \(0.0341660\pi\)
−0.994245 + 0.107130i \(0.965834\pi\)
\(158\) 40.1771i 3.19632i
\(159\) 0 0
\(160\) 0 0
\(161\) 6.56155 0.517123
\(162\) 0 0
\(163\) 5.31534i 0.416330i 0.978094 + 0.208165i \(0.0667490\pi\)
−0.978094 + 0.208165i \(0.933251\pi\)
\(164\) 35.0540 2.73726
\(165\) 0 0
\(166\) 0 0
\(167\) − 15.3693i − 1.18931i −0.803980 0.594657i \(-0.797288\pi\)
0.803980 0.594657i \(-0.202712\pi\)
\(168\) 0 0
\(169\) 12.8078 0.985213
\(170\) 0 0
\(171\) 0 0
\(172\) 21.3693i 1.62940i
\(173\) − 13.4384i − 1.02171i −0.859668 0.510853i \(-0.829330\pi\)
0.859668 0.510853i \(-0.170670\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −7.68466 −0.579253
\(177\) 0 0
\(178\) 43.8617i 3.28758i
\(179\) −14.5616 −1.08838 −0.544191 0.838961i \(-0.683163\pi\)
−0.544191 + 0.838961i \(0.683163\pi\)
\(180\) 0 0
\(181\) 11.8769 0.882803 0.441401 0.897310i \(-0.354482\pi\)
0.441401 + 0.897310i \(0.354482\pi\)
\(182\) − 1.12311i − 0.0832501i
\(183\) 0 0
\(184\) −43.0540 −3.17398
\(185\) 0 0
\(186\) 0 0
\(187\) − 1.43845i − 0.105190i
\(188\) − 24.8078i − 1.80929i
\(189\) 0 0
\(190\) 0 0
\(191\) −1.93087 −0.139713 −0.0698564 0.997557i \(-0.522254\pi\)
−0.0698564 + 0.997557i \(0.522254\pi\)
\(192\) 0 0
\(193\) − 17.5616i − 1.26411i −0.774924 0.632054i \(-0.782212\pi\)
0.774924 0.632054i \(-0.217788\pi\)
\(194\) −5.43845 −0.390458
\(195\) 0 0
\(196\) −27.3693 −1.95495
\(197\) 17.9309i 1.27752i 0.769405 + 0.638761i \(0.220553\pi\)
−0.769405 + 0.638761i \(0.779447\pi\)
\(198\) 0 0
\(199\) 13.8078 0.978806 0.489403 0.872058i \(-0.337215\pi\)
0.489403 + 0.872058i \(0.337215\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 3.68466i 0.259252i
\(203\) − 5.12311i − 0.359572i
\(204\) 0 0
\(205\) 0 0
\(206\) −23.3693 −1.62822
\(207\) 0 0
\(208\) 3.36932i 0.233620i
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) 3.31534 0.228238 0.114119 0.993467i \(-0.463596\pi\)
0.114119 + 0.993467i \(0.463596\pi\)
\(212\) − 5.12311i − 0.351856i
\(213\) 0 0
\(214\) 23.3693 1.59749
\(215\) 0 0
\(216\) 0 0
\(217\) 4.68466i 0.318015i
\(218\) 20.0000i 1.35457i
\(219\) 0 0
\(220\) 0 0
\(221\) −0.630683 −0.0424243
\(222\) 0 0
\(223\) − 21.8078i − 1.46036i −0.683257 0.730178i \(-0.739437\pi\)
0.683257 0.730178i \(-0.260563\pi\)
\(224\) −6.56155 −0.438412
\(225\) 0 0
\(226\) −23.3693 −1.55450
\(227\) 20.4924i 1.36013i 0.733152 + 0.680065i \(0.238048\pi\)
−0.733152 + 0.680065i \(0.761952\pi\)
\(228\) 0 0
\(229\) 8.12311 0.536790 0.268395 0.963309i \(-0.413507\pi\)
0.268395 + 0.963309i \(0.413507\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 33.6155i 2.20697i
\(233\) − 24.8078i − 1.62521i −0.582814 0.812605i \(-0.698049\pi\)
0.582814 0.812605i \(-0.301951\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 48.1771 3.13606
\(237\) 0 0
\(238\) − 3.68466i − 0.238841i
\(239\) 16.4924 1.06681 0.533403 0.845861i \(-0.320913\pi\)
0.533403 + 0.845861i \(0.320913\pi\)
\(240\) 0 0
\(241\) −14.0540 −0.905296 −0.452648 0.891689i \(-0.649521\pi\)
−0.452648 + 0.891689i \(0.649521\pi\)
\(242\) 2.56155i 0.164663i
\(243\) 0 0
\(244\) 34.4924 2.20815
\(245\) 0 0
\(246\) 0 0
\(247\) − 1.31534i − 0.0836932i
\(248\) − 30.7386i − 1.95191i
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 6.56155i 0.412521i
\(254\) 12.3153 0.772733
\(255\) 0 0
\(256\) −27.0540 −1.69087
\(257\) 18.2462i 1.13817i 0.822280 + 0.569084i \(0.192702\pi\)
−0.822280 + 0.569084i \(0.807298\pi\)
\(258\) 0 0
\(259\) 10.8078 0.671562
\(260\) 0 0
\(261\) 0 0
\(262\) − 42.2462i − 2.60998i
\(263\) 3.36932i 0.207761i 0.994590 + 0.103880i \(0.0331259\pi\)
−0.994590 + 0.103880i \(0.966874\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 7.68466 0.471177
\(267\) 0 0
\(268\) 30.4924i 1.86262i
\(269\) −28.4924 −1.73721 −0.868607 0.495502i \(-0.834984\pi\)
−0.868607 + 0.495502i \(0.834984\pi\)
\(270\) 0 0
\(271\) −28.1771 −1.71164 −0.855818 0.517277i \(-0.826946\pi\)
−0.855818 + 0.517277i \(0.826946\pi\)
\(272\) 11.0540i 0.670246i
\(273\) 0 0
\(274\) 8.63068 0.521399
\(275\) 0 0
\(276\) 0 0
\(277\) − 23.1771i − 1.39258i −0.717763 0.696288i \(-0.754834\pi\)
0.717763 0.696288i \(-0.245166\pi\)
\(278\) − 20.4924i − 1.22905i
\(279\) 0 0
\(280\) 0 0
\(281\) −3.19224 −0.190433 −0.0952164 0.995457i \(-0.530354\pi\)
−0.0952164 + 0.995457i \(0.530354\pi\)
\(282\) 0 0
\(283\) − 32.1231i − 1.90952i −0.297377 0.954760i \(-0.596112\pi\)
0.297377 0.954760i \(-0.403888\pi\)
\(284\) −40.1771 −2.38407
\(285\) 0 0
\(286\) 1.12311 0.0664106
\(287\) − 7.68466i − 0.453611i
\(288\) 0 0
\(289\) 14.9309 0.878286
\(290\) 0 0
\(291\) 0 0
\(292\) 74.1080i 4.33684i
\(293\) − 2.06913i − 0.120880i −0.998172 0.0604399i \(-0.980750\pi\)
0.998172 0.0604399i \(-0.0192504\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −70.9157 −4.12189
\(297\) 0 0
\(298\) − 22.5616i − 1.30696i
\(299\) 2.87689 0.166375
\(300\) 0 0
\(301\) 4.68466 0.270019
\(302\) − 42.7386i − 2.45933i
\(303\) 0 0
\(304\) −23.0540 −1.32224
\(305\) 0 0
\(306\) 0 0
\(307\) − 22.9309i − 1.30873i −0.756177 0.654367i \(-0.772935\pi\)
0.756177 0.654367i \(-0.227065\pi\)
\(308\) 4.56155i 0.259919i
\(309\) 0 0
\(310\) 0 0
\(311\) −13.7538 −0.779906 −0.389953 0.920835i \(-0.627509\pi\)
−0.389953 + 0.920835i \(0.627509\pi\)
\(312\) 0 0
\(313\) 31.2462i 1.76614i 0.469241 + 0.883070i \(0.344528\pi\)
−0.469241 + 0.883070i \(0.655472\pi\)
\(314\) −6.87689 −0.388086
\(315\) 0 0
\(316\) −71.5464 −4.02480
\(317\) 6.87689i 0.386245i 0.981175 + 0.193122i \(0.0618615\pi\)
−0.981175 + 0.193122i \(0.938139\pi\)
\(318\) 0 0
\(319\) 5.12311 0.286839
\(320\) 0 0
\(321\) 0 0
\(322\) 16.8078i 0.936660i
\(323\) − 4.31534i − 0.240112i
\(324\) 0 0
\(325\) 0 0
\(326\) −13.6155 −0.754094
\(327\) 0 0
\(328\) 50.4233i 2.78416i
\(329\) −5.43845 −0.299831
\(330\) 0 0
\(331\) −10.2462 −0.563183 −0.281591 0.959534i \(-0.590862\pi\)
−0.281591 + 0.959534i \(0.590862\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 39.3693 2.15419
\(335\) 0 0
\(336\) 0 0
\(337\) − 10.6847i − 0.582030i −0.956718 0.291015i \(-0.906007\pi\)
0.956718 0.291015i \(-0.0939930\pi\)
\(338\) 32.8078i 1.78451i
\(339\) 0 0
\(340\) 0 0
\(341\) −4.68466 −0.253688
\(342\) 0 0
\(343\) 13.0000i 0.701934i
\(344\) −30.7386 −1.65732
\(345\) 0 0
\(346\) 34.4233 1.85061
\(347\) − 27.3693i − 1.46926i −0.678467 0.734631i \(-0.737355\pi\)
0.678467 0.734631i \(-0.262645\pi\)
\(348\) 0 0
\(349\) −6.63068 −0.354932 −0.177466 0.984127i \(-0.556790\pi\)
−0.177466 + 0.984127i \(0.556790\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 6.56155i − 0.349732i
\(353\) − 27.8617i − 1.48293i −0.670991 0.741465i \(-0.734131\pi\)
0.670991 0.741465i \(-0.265869\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −78.1080 −4.13971
\(357\) 0 0
\(358\) − 37.3002i − 1.97138i
\(359\) −26.2462 −1.38522 −0.692611 0.721311i \(-0.743540\pi\)
−0.692611 + 0.721311i \(0.743540\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 30.4233i 1.59901i
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) 14.0540i 0.733612i 0.930298 + 0.366806i \(0.119549\pi\)
−0.930298 + 0.366806i \(0.880451\pi\)
\(368\) − 50.4233i − 2.62850i
\(369\) 0 0
\(370\) 0 0
\(371\) −1.12311 −0.0583087
\(372\) 0 0
\(373\) 13.8078i 0.714939i 0.933925 + 0.357469i \(0.116360\pi\)
−0.933925 + 0.357469i \(0.883640\pi\)
\(374\) 3.68466 0.190529
\(375\) 0 0
\(376\) 35.6847 1.84030
\(377\) − 2.24621i − 0.115686i
\(378\) 0 0
\(379\) 24.3002 1.24822 0.624108 0.781338i \(-0.285462\pi\)
0.624108 + 0.781338i \(0.285462\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 4.94602i − 0.253061i
\(383\) − 22.2462i − 1.13673i −0.822777 0.568364i \(-0.807576\pi\)
0.822777 0.568364i \(-0.192424\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 44.9848 2.28967
\(387\) 0 0
\(388\) − 9.68466i − 0.491664i
\(389\) 23.8617 1.20984 0.604919 0.796287i \(-0.293205\pi\)
0.604919 + 0.796287i \(0.293205\pi\)
\(390\) 0 0
\(391\) 9.43845 0.477323
\(392\) − 39.3693i − 1.98845i
\(393\) 0 0
\(394\) −45.9309 −2.31396
\(395\) 0 0
\(396\) 0 0
\(397\) 28.4384i 1.42728i 0.700510 + 0.713642i \(0.252956\pi\)
−0.700510 + 0.713642i \(0.747044\pi\)
\(398\) 35.3693i 1.77290i
\(399\) 0 0
\(400\) 0 0
\(401\) 5.61553 0.280426 0.140213 0.990121i \(-0.455221\pi\)
0.140213 + 0.990121i \(0.455221\pi\)
\(402\) 0 0
\(403\) 2.05398i 0.102316i
\(404\) −6.56155 −0.326449
\(405\) 0 0
\(406\) 13.1231 0.651289
\(407\) 10.8078i 0.535721i
\(408\) 0 0
\(409\) −8.05398 −0.398243 −0.199122 0.979975i \(-0.563809\pi\)
−0.199122 + 0.979975i \(0.563809\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 41.6155i − 2.05025i
\(413\) − 10.5616i − 0.519700i
\(414\) 0 0
\(415\) 0 0
\(416\) −2.87689 −0.141051
\(417\) 0 0
\(418\) 7.68466i 0.375869i
\(419\) −25.4384 −1.24275 −0.621375 0.783514i \(-0.713426\pi\)
−0.621375 + 0.783514i \(0.713426\pi\)
\(420\) 0 0
\(421\) −17.0540 −0.831160 −0.415580 0.909557i \(-0.636421\pi\)
−0.415580 + 0.909557i \(0.636421\pi\)
\(422\) 8.49242i 0.413405i
\(423\) 0 0
\(424\) 7.36932 0.357886
\(425\) 0 0
\(426\) 0 0
\(427\) − 7.56155i − 0.365929i
\(428\) 41.6155i 2.01156i
\(429\) 0 0
\(430\) 0 0
\(431\) −21.7538 −1.04784 −0.523922 0.851767i \(-0.675532\pi\)
−0.523922 + 0.851767i \(0.675532\pi\)
\(432\) 0 0
\(433\) − 16.9309i − 0.813646i −0.913507 0.406823i \(-0.866637\pi\)
0.913507 0.406823i \(-0.133363\pi\)
\(434\) −12.0000 −0.576018
\(435\) 0 0
\(436\) −35.6155 −1.70567
\(437\) 19.6847i 0.941645i
\(438\) 0 0
\(439\) −11.7386 −0.560254 −0.280127 0.959963i \(-0.590377\pi\)
−0.280127 + 0.959963i \(0.590377\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 1.61553i − 0.0768428i
\(443\) − 25.3002i − 1.20205i −0.799231 0.601024i \(-0.794760\pi\)
0.799231 0.601024i \(-0.205240\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 55.8617 2.64513
\(447\) 0 0
\(448\) − 1.43845i − 0.0679602i
\(449\) 28.4924 1.34464 0.672320 0.740260i \(-0.265298\pi\)
0.672320 + 0.740260i \(0.265298\pi\)
\(450\) 0 0
\(451\) 7.68466 0.361856
\(452\) − 41.6155i − 1.95743i
\(453\) 0 0
\(454\) −52.4924 −2.46359
\(455\) 0 0
\(456\) 0 0
\(457\) 13.3693i 0.625390i 0.949854 + 0.312695i \(0.101232\pi\)
−0.949854 + 0.312695i \(0.898768\pi\)
\(458\) 20.8078i 0.972283i
\(459\) 0 0
\(460\) 0 0
\(461\) −15.8617 −0.738755 −0.369377 0.929279i \(-0.620429\pi\)
−0.369377 + 0.929279i \(0.620429\pi\)
\(462\) 0 0
\(463\) 1.75379i 0.0815055i 0.999169 + 0.0407527i \(0.0129756\pi\)
−0.999169 + 0.0407527i \(0.987024\pi\)
\(464\) −39.3693 −1.82767
\(465\) 0 0
\(466\) 63.5464 2.94373
\(467\) 2.24621i 0.103942i 0.998649 + 0.0519711i \(0.0165504\pi\)
−0.998649 + 0.0519711i \(0.983450\pi\)
\(468\) 0 0
\(469\) 6.68466 0.308669
\(470\) 0 0
\(471\) 0 0
\(472\) 69.3002i 3.18980i
\(473\) 4.68466i 0.215401i
\(474\) 0 0
\(475\) 0 0
\(476\) 6.56155 0.300748
\(477\) 0 0
\(478\) 42.2462i 1.93230i
\(479\) −3.36932 −0.153948 −0.0769740 0.997033i \(-0.524526\pi\)
−0.0769740 + 0.997033i \(0.524526\pi\)
\(480\) 0 0
\(481\) 4.73863 0.216063
\(482\) − 36.0000i − 1.63976i
\(483\) 0 0
\(484\) −4.56155 −0.207343
\(485\) 0 0
\(486\) 0 0
\(487\) − 12.0540i − 0.546218i −0.961983 0.273109i \(-0.911948\pi\)
0.961983 0.273109i \(-0.0880519\pi\)
\(488\) 49.6155i 2.24599i
\(489\) 0 0
\(490\) 0 0
\(491\) 16.4924 0.744293 0.372146 0.928174i \(-0.378622\pi\)
0.372146 + 0.928174i \(0.378622\pi\)
\(492\) 0 0
\(493\) − 7.36932i − 0.331897i
\(494\) 3.36932 0.151593
\(495\) 0 0
\(496\) 36.0000 1.61645
\(497\) 8.80776i 0.395082i
\(498\) 0 0
\(499\) −3.80776 −0.170459 −0.0852295 0.996361i \(-0.527162\pi\)
−0.0852295 + 0.996361i \(0.527162\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 30.7386i − 1.37193i
\(503\) 16.4924i 0.735361i 0.929952 + 0.367680i \(0.119848\pi\)
−0.929952 + 0.367680i \(0.880152\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −16.8078 −0.747196
\(507\) 0 0
\(508\) 21.9309i 0.973025i
\(509\) 17.1231 0.758968 0.379484 0.925198i \(-0.376101\pi\)
0.379484 + 0.925198i \(0.376101\pi\)
\(510\) 0 0
\(511\) 16.2462 0.718690
\(512\) − 50.4233i − 2.22842i
\(513\) 0 0
\(514\) −46.7386 −2.06155
\(515\) 0 0
\(516\) 0 0
\(517\) − 5.43845i − 0.239183i
\(518\) 27.6847i 1.21639i
\(519\) 0 0
\(520\) 0 0
\(521\) −25.6155 −1.12224 −0.561118 0.827736i \(-0.689629\pi\)
−0.561118 + 0.827736i \(0.689629\pi\)
\(522\) 0 0
\(523\) − 9.24621i − 0.404309i −0.979354 0.202154i \(-0.935206\pi\)
0.979354 0.202154i \(-0.0647942\pi\)
\(524\) 75.2311 3.28648
\(525\) 0 0
\(526\) −8.63068 −0.376316
\(527\) 6.73863i 0.293539i
\(528\) 0 0
\(529\) −20.0540 −0.871912
\(530\) 0 0
\(531\) 0 0
\(532\) 13.6847i 0.593305i
\(533\) − 3.36932i − 0.145941i
\(534\) 0 0
\(535\) 0 0
\(536\) −43.8617 −1.89454
\(537\) 0 0
\(538\) − 72.9848i − 3.14660i
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −7.31534 −0.314511 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(542\) − 72.1771i − 3.10027i
\(543\) 0 0
\(544\) −9.43845 −0.404670
\(545\) 0 0
\(546\) 0 0
\(547\) 3.68466i 0.157545i 0.996893 + 0.0787723i \(0.0251000\pi\)
−0.996893 + 0.0787723i \(0.974900\pi\)
\(548\) 15.3693i 0.656545i
\(549\) 0 0
\(550\) 0 0
\(551\) 15.3693 0.654755
\(552\) 0 0
\(553\) 15.6847i 0.666980i
\(554\) 59.3693 2.52236
\(555\) 0 0
\(556\) 36.4924 1.54762
\(557\) 1.12311i 0.0475875i 0.999717 + 0.0237938i \(0.00757450\pi\)
−0.999717 + 0.0237938i \(0.992425\pi\)
\(558\) 0 0
\(559\) 2.05398 0.0868739
\(560\) 0 0
\(561\) 0 0
\(562\) − 8.17708i − 0.344929i
\(563\) 10.8769i 0.458406i 0.973379 + 0.229203i \(0.0736120\pi\)
−0.973379 + 0.229203i \(0.926388\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 82.2850 3.45870
\(567\) 0 0
\(568\) − 57.7926i − 2.42492i
\(569\) −37.9309 −1.59014 −0.795072 0.606515i \(-0.792567\pi\)
−0.795072 + 0.606515i \(0.792567\pi\)
\(570\) 0 0
\(571\) 38.0540 1.59251 0.796255 0.604962i \(-0.206812\pi\)
0.796255 + 0.604962i \(0.206812\pi\)
\(572\) 2.00000i 0.0836242i
\(573\) 0 0
\(574\) 19.6847 0.821622
\(575\) 0 0
\(576\) 0 0
\(577\) − 28.3693i − 1.18103i −0.807027 0.590515i \(-0.798925\pi\)
0.807027 0.590515i \(-0.201075\pi\)
\(578\) 38.2462i 1.59083i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 1.12311i − 0.0465143i
\(584\) −106.600 −4.41115
\(585\) 0 0
\(586\) 5.30019 0.218949
\(587\) − 27.1922i − 1.12234i −0.827699 0.561172i \(-0.810351\pi\)
0.827699 0.561172i \(-0.189649\pi\)
\(588\) 0 0
\(589\) −14.0540 −0.579084
\(590\) 0 0
\(591\) 0 0
\(592\) − 83.0540i − 3.41350i
\(593\) 34.1080i 1.40065i 0.713826 + 0.700323i \(0.246961\pi\)
−0.713826 + 0.700323i \(0.753039\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 40.1771 1.64572
\(597\) 0 0
\(598\) 7.36932i 0.301354i
\(599\) −24.1771 −0.987849 −0.493924 0.869505i \(-0.664438\pi\)
−0.493924 + 0.869505i \(0.664438\pi\)
\(600\) 0 0
\(601\) 37.4233 1.52653 0.763264 0.646087i \(-0.223596\pi\)
0.763264 + 0.646087i \(0.223596\pi\)
\(602\) 12.0000i 0.489083i
\(603\) 0 0
\(604\) 76.1080 3.09679
\(605\) 0 0
\(606\) 0 0
\(607\) 6.24621i 0.253526i 0.991933 + 0.126763i \(0.0404588\pi\)
−0.991933 + 0.126763i \(0.959541\pi\)
\(608\) − 19.6847i − 0.798318i
\(609\) 0 0
\(610\) 0 0
\(611\) −2.38447 −0.0964654
\(612\) 0 0
\(613\) − 28.1080i − 1.13527i −0.823281 0.567635i \(-0.807859\pi\)
0.823281 0.567635i \(-0.192141\pi\)
\(614\) 58.7386 2.37050
\(615\) 0 0
\(616\) −6.56155 −0.264372
\(617\) 21.7538i 0.875775i 0.899030 + 0.437887i \(0.144273\pi\)
−0.899030 + 0.437887i \(0.855727\pi\)
\(618\) 0 0
\(619\) −7.17708 −0.288471 −0.144236 0.989543i \(-0.546072\pi\)
−0.144236 + 0.989543i \(0.546072\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 35.2311i − 1.41264i
\(623\) 17.1231i 0.686023i
\(624\) 0 0
\(625\) 0 0
\(626\) −80.0388 −3.19899
\(627\) 0 0
\(628\) − 12.2462i − 0.488677i
\(629\) 15.5464 0.619875
\(630\) 0 0
\(631\) −22.6847 −0.903062 −0.451531 0.892255i \(-0.649122\pi\)
−0.451531 + 0.892255i \(0.649122\pi\)
\(632\) − 102.916i − 4.09377i
\(633\) 0 0
\(634\) −17.6155 −0.699602
\(635\) 0 0
\(636\) 0 0
\(637\) 2.63068i 0.104231i
\(638\) 13.1231i 0.519549i
\(639\) 0 0
\(640\) 0 0
\(641\) 42.7386 1.68807 0.844037 0.536285i \(-0.180173\pi\)
0.844037 + 0.536285i \(0.180173\pi\)
\(642\) 0 0
\(643\) − 15.3693i − 0.606107i −0.952974 0.303053i \(-0.901994\pi\)
0.952974 0.303053i \(-0.0980060\pi\)
\(644\) −29.9309 −1.17944
\(645\) 0 0
\(646\) 11.0540 0.434913
\(647\) − 40.8078i − 1.60432i −0.597110 0.802159i \(-0.703684\pi\)
0.597110 0.802159i \(-0.296316\pi\)
\(648\) 0 0
\(649\) 10.5616 0.414577
\(650\) 0 0
\(651\) 0 0
\(652\) − 24.2462i − 0.949555i
\(653\) − 7.50758i − 0.293794i −0.989152 0.146897i \(-0.953071\pi\)
0.989152 0.146897i \(-0.0469286\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −59.0540 −2.30567
\(657\) 0 0
\(658\) − 13.9309i − 0.543082i
\(659\) −8.63068 −0.336204 −0.168102 0.985770i \(-0.553764\pi\)
−0.168102 + 0.985770i \(0.553764\pi\)
\(660\) 0 0
\(661\) 6.80776 0.264791 0.132396 0.991197i \(-0.457733\pi\)
0.132396 + 0.991197i \(0.457733\pi\)
\(662\) − 26.2462i − 1.02009i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 33.6155i 1.30160i
\(668\) 70.1080i 2.71256i
\(669\) 0 0
\(670\) 0 0
\(671\) 7.56155 0.291911
\(672\) 0 0
\(673\) − 10.6307i − 0.409783i −0.978785 0.204891i \(-0.934316\pi\)
0.978785 0.204891i \(-0.0656841\pi\)
\(674\) 27.3693 1.05423
\(675\) 0 0
\(676\) −58.4233 −2.24705
\(677\) 29.1231i 1.11929i 0.828732 + 0.559646i \(0.189063\pi\)
−0.828732 + 0.559646i \(0.810937\pi\)
\(678\) 0 0
\(679\) −2.12311 −0.0814773
\(680\) 0 0
\(681\) 0 0
\(682\) − 12.0000i − 0.459504i
\(683\) 20.8078i 0.796187i 0.917345 + 0.398093i \(0.130328\pi\)
−0.917345 + 0.398093i \(0.869672\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −33.3002 −1.27141
\(687\) 0 0
\(688\) − 36.0000i − 1.37249i
\(689\) −0.492423 −0.0187598
\(690\) 0 0
\(691\) 12.4924 0.475234 0.237617 0.971359i \(-0.423634\pi\)
0.237617 + 0.971359i \(0.423634\pi\)
\(692\) 61.3002i 2.33028i
\(693\) 0 0
\(694\) 70.1080 2.66126
\(695\) 0 0
\(696\) 0 0
\(697\) − 11.0540i − 0.418699i
\(698\) − 16.9848i − 0.642886i
\(699\) 0 0
\(700\) 0 0
\(701\) −1.93087 −0.0729279 −0.0364640 0.999335i \(-0.511609\pi\)
−0.0364640 + 0.999335i \(0.511609\pi\)
\(702\) 0 0
\(703\) 32.4233i 1.22287i
\(704\) 1.43845 0.0542135
\(705\) 0 0
\(706\) 71.3693 2.68602
\(707\) 1.43845i 0.0540984i
\(708\) 0 0
\(709\) −6.12311 −0.229958 −0.114979 0.993368i \(-0.536680\pi\)
−0.114979 + 0.993368i \(0.536680\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 112.354i − 4.21065i
\(713\) − 30.7386i − 1.15117i
\(714\) 0 0
\(715\) 0 0
\(716\) 66.4233 2.48235
\(717\) 0 0
\(718\) − 67.2311i − 2.50904i
\(719\) 32.9848 1.23013 0.615064 0.788478i \(-0.289130\pi\)
0.615064 + 0.788478i \(0.289130\pi\)
\(720\) 0 0
\(721\) −9.12311 −0.339762
\(722\) − 25.6155i − 0.953311i
\(723\) 0 0
\(724\) −54.1771 −2.01348
\(725\) 0 0
\(726\) 0 0
\(727\) − 30.0540i − 1.11464i −0.830298 0.557320i \(-0.811830\pi\)
0.830298 0.557320i \(-0.188170\pi\)
\(728\) 2.87689i 0.106625i
\(729\) 0 0
\(730\) 0 0
\(731\) 6.73863 0.249237
\(732\) 0 0
\(733\) − 2.00000i − 0.0738717i −0.999318 0.0369358i \(-0.988240\pi\)
0.999318 0.0369358i \(-0.0117597\pi\)
\(734\) −36.0000 −1.32878
\(735\) 0 0
\(736\) 43.0540 1.58699
\(737\) 6.68466i 0.246233i
\(738\) 0 0
\(739\) 11.6847 0.429827 0.214914 0.976633i \(-0.431053\pi\)
0.214914 + 0.976633i \(0.431053\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 2.87689i − 0.105614i
\(743\) − 47.8617i − 1.75588i −0.478773 0.877938i \(-0.658918\pi\)
0.478773 0.877938i \(-0.341082\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −35.3693 −1.29496
\(747\) 0 0
\(748\) 6.56155i 0.239914i
\(749\) 9.12311 0.333351
\(750\) 0 0
\(751\) 26.7386 0.975707 0.487853 0.872926i \(-0.337780\pi\)
0.487853 + 0.872926i \(0.337780\pi\)
\(752\) 41.7926i 1.52402i
\(753\) 0 0
\(754\) 5.75379 0.209541
\(755\) 0 0
\(756\) 0 0
\(757\) 6.30019i 0.228984i 0.993424 + 0.114492i \(0.0365241\pi\)
−0.993424 + 0.114492i \(0.963476\pi\)
\(758\) 62.2462i 2.26088i
\(759\) 0 0
\(760\) 0 0
\(761\) 20.6307 0.747862 0.373931 0.927457i \(-0.378010\pi\)
0.373931 + 0.927457i \(0.378010\pi\)
\(762\) 0 0
\(763\) 7.80776i 0.282660i
\(764\) 8.80776 0.318654
\(765\) 0 0
\(766\) 56.9848 2.05895
\(767\) − 4.63068i − 0.167204i
\(768\) 0 0
\(769\) −6.05398 −0.218312 −0.109156 0.994025i \(-0.534815\pi\)
−0.109156 + 0.994025i \(0.534815\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 80.1080i 2.88315i
\(773\) − 48.4924i − 1.74415i −0.489371 0.872076i \(-0.662774\pi\)
0.489371 0.872076i \(-0.337226\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 13.9309 0.500089
\(777\) 0 0
\(778\) 61.1231i 2.19137i
\(779\) 23.0540 0.825994
\(780\) 0 0
\(781\) −8.80776 −0.315167
\(782\) 24.1771i 0.864571i
\(783\) 0 0
\(784\) 46.1080 1.64671
\(785\) 0 0
\(786\) 0 0
\(787\) 4.26137i 0.151901i 0.997112 + 0.0759507i \(0.0241991\pi\)
−0.997112 + 0.0759507i \(0.975801\pi\)
\(788\) − 81.7926i − 2.91374i
\(789\) 0 0
\(790\) 0 0
\(791\) −9.12311 −0.324380
\(792\) 0 0
\(793\) − 3.31534i − 0.117731i
\(794\) −72.8466 −2.58523
\(795\) 0 0
\(796\) −62.9848 −2.23244
\(797\) 42.8769i 1.51878i 0.650637 + 0.759389i \(0.274502\pi\)
−0.650637 + 0.759389i \(0.725498\pi\)
\(798\) 0 0
\(799\) −7.82292 −0.276755
\(800\) 0 0
\(801\) 0 0
\(802\) 14.3845i 0.507933i
\(803\) 16.2462i 0.573316i
\(804\) 0 0
\(805\) 0 0
\(806\) −5.26137 −0.185324
\(807\) 0 0
\(808\) − 9.43845i − 0.332043i
\(809\) −19.6847 −0.692076 −0.346038 0.938221i \(-0.612473\pi\)
−0.346038 + 0.938221i \(0.612473\pi\)
\(810\) 0 0
\(811\) −1.00000 −0.0351147 −0.0175574 0.999846i \(-0.505589\pi\)
−0.0175574 + 0.999846i \(0.505589\pi\)
\(812\) 23.3693i 0.820102i
\(813\) 0 0
\(814\) −27.6847 −0.970347
\(815\) 0 0
\(816\) 0 0
\(817\) 14.0540i 0.491686i
\(818\) − 20.6307i − 0.721335i
\(819\) 0 0
\(820\) 0 0
\(821\) −25.1231 −0.876802 −0.438401 0.898779i \(-0.644455\pi\)
−0.438401 + 0.898779i \(0.644455\pi\)
\(822\) 0 0
\(823\) − 0.0539753i − 0.00188146i −1.00000 0.000940731i \(-0.999701\pi\)
1.00000 0.000940731i \(-0.000299444\pi\)
\(824\) 59.8617 2.08538
\(825\) 0 0
\(826\) 27.0540 0.941328
\(827\) 9.75379i 0.339172i 0.985515 + 0.169586i \(0.0542431\pi\)
−0.985515 + 0.169586i \(0.945757\pi\)
\(828\) 0 0
\(829\) −48.7386 −1.69276 −0.846381 0.532577i \(-0.821224\pi\)
−0.846381 + 0.532577i \(0.821224\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 0.630683i − 0.0218650i
\(833\) 8.63068i 0.299035i
\(834\) 0 0
\(835\) 0 0
\(836\) −13.6847 −0.473294
\(837\) 0 0
\(838\) − 65.1619i − 2.25098i
\(839\) −20.4924 −0.707477 −0.353738 0.935344i \(-0.615090\pi\)
−0.353738 + 0.935344i \(0.615090\pi\)
\(840\) 0 0
\(841\) −2.75379 −0.0949582
\(842\) − 43.6847i − 1.50547i
\(843\) 0 0
\(844\) −15.1231 −0.520559
\(845\) 0 0
\(846\) 0 0
\(847\) 1.00000i 0.0343604i
\(848\) 8.63068i 0.296379i
\(849\) 0 0
\(850\) 0 0
\(851\) −70.9157 −2.43096
\(852\) 0 0
\(853\) 35.4233i 1.21287i 0.795133 + 0.606435i \(0.207401\pi\)
−0.795133 + 0.606435i \(0.792599\pi\)
\(854\) 19.3693 0.662804
\(855\) 0 0
\(856\) −59.8617 −2.04603
\(857\) 5.43845i 0.185774i 0.995677 + 0.0928869i \(0.0296095\pi\)
−0.995677 + 0.0928869i \(0.970390\pi\)
\(858\) 0 0
\(859\) −56.3542 −1.92278 −0.961390 0.275191i \(-0.911259\pi\)
−0.961390 + 0.275191i \(0.911259\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 55.7235i − 1.89795i
\(863\) 2.24621i 0.0764619i 0.999269 + 0.0382310i \(0.0121723\pi\)
−0.999269 + 0.0382310i \(0.987828\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 43.3693 1.47375
\(867\) 0 0
\(868\) − 21.3693i − 0.725322i
\(869\) −15.6847 −0.532066
\(870\) 0 0
\(871\) 2.93087 0.0993087
\(872\) − 51.2311i − 1.73490i
\(873\) 0 0
\(874\) −50.4233 −1.70559
\(875\) 0 0
\(876\) 0 0
\(877\) 14.0540i 0.474569i 0.971440 + 0.237285i \(0.0762574\pi\)
−0.971440 + 0.237285i \(0.923743\pi\)
\(878\) − 30.0691i − 1.01478i
\(879\) 0 0
\(880\) 0 0
\(881\) −9.75379 −0.328613 −0.164307 0.986409i \(-0.552539\pi\)
−0.164307 + 0.986409i \(0.552539\pi\)
\(882\) 0 0
\(883\) 44.9309i 1.51204i 0.654546 + 0.756022i \(0.272860\pi\)
−0.654546 + 0.756022i \(0.727140\pi\)
\(884\) 2.87689 0.0967604
\(885\) 0 0
\(886\) 64.8078 2.17726
\(887\) 27.8617i 0.935506i 0.883859 + 0.467753i \(0.154936\pi\)
−0.883859 + 0.467753i \(0.845064\pi\)
\(888\) 0 0
\(889\) 4.80776 0.161247
\(890\) 0 0
\(891\) 0 0
\(892\) 99.4773i 3.33075i
\(893\) − 16.3153i − 0.545972i
\(894\) 0 0
\(895\) 0 0
\(896\) −9.43845 −0.315316
\(897\) 0 0
\(898\) 72.9848i 2.43554i
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) −1.61553 −0.0538210
\(902\) 19.6847i 0.655427i
\(903\) 0 0
\(904\) 59.8617 1.99097
\(905\) 0 0
\(906\) 0 0
\(907\) 46.1080i 1.53099i 0.643442 + 0.765495i \(0.277506\pi\)
−0.643442 + 0.765495i \(0.722494\pi\)
\(908\) − 93.4773i − 3.10215i
\(909\) 0 0
\(910\) 0 0
\(911\) −57.1619 −1.89386 −0.946930 0.321441i \(-0.895833\pi\)
−0.946930 + 0.321441i \(0.895833\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −34.2462 −1.13276
\(915\) 0 0
\(916\) −37.0540 −1.22430
\(917\) − 16.4924i − 0.544628i
\(918\) 0 0
\(919\) −41.1080 −1.35603 −0.678013 0.735050i \(-0.737159\pi\)
−0.678013 + 0.735050i \(0.737159\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 40.6307i − 1.33810i
\(923\) 3.86174i 0.127111i
\(924\) 0 0
\(925\) 0 0
\(926\) −4.49242 −0.147630
\(927\) 0 0
\(928\) − 33.6155i − 1.10348i
\(929\) −23.2311 −0.762186 −0.381093 0.924537i \(-0.624452\pi\)
−0.381093 + 0.924537i \(0.624452\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 113.162i 3.70674i
\(933\) 0 0
\(934\) −5.75379 −0.188270
\(935\) 0 0
\(936\) 0 0
\(937\) 34.6847i 1.13310i 0.824028 + 0.566549i \(0.191722\pi\)
−0.824028 + 0.566549i \(0.808278\pi\)
\(938\) 17.1231i 0.559089i
\(939\) 0 0
\(940\) 0 0
\(941\) 60.1771 1.96172 0.980858 0.194722i \(-0.0623806\pi\)
0.980858 + 0.194722i \(0.0623806\pi\)
\(942\) 0 0
\(943\) 50.4233i 1.64201i
\(944\) −81.1619 −2.64160
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) − 32.1771i − 1.04561i −0.852451 0.522807i \(-0.824885\pi\)
0.852451 0.522807i \(-0.175115\pi\)
\(948\) 0 0
\(949\) 7.12311 0.231226
\(950\) 0 0
\(951\) 0 0
\(952\) 9.43845i 0.305902i
\(953\) 4.80776i 0.155739i 0.996964 + 0.0778694i \(0.0248117\pi\)
−0.996964 + 0.0778694i \(0.975188\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −75.2311 −2.43315
\(957\) 0 0
\(958\) − 8.63068i − 0.278845i
\(959\) 3.36932 0.108801
\(960\) 0 0
\(961\) −9.05398 −0.292064
\(962\) 12.1383i 0.391353i
\(963\) 0 0
\(964\) 64.1080 2.06478
\(965\) 0 0
\(966\) 0 0
\(967\) 44.0000i 1.41494i 0.706741 + 0.707472i \(0.250165\pi\)
−0.706741 + 0.707472i \(0.749835\pi\)
\(968\) − 6.56155i − 0.210896i
\(969\) 0 0
\(970\) 0 0
\(971\) 8.17708 0.262415 0.131208 0.991355i \(-0.458115\pi\)
0.131208 + 0.991355i \(0.458115\pi\)
\(972\) 0 0
\(973\) − 8.00000i − 0.256468i
\(974\) 30.8769 0.989360
\(975\) 0 0
\(976\) −58.1080 −1.85999
\(977\) − 25.6155i − 0.819513i −0.912195 0.409757i \(-0.865614\pi\)
0.912195 0.409757i \(-0.134386\pi\)
\(978\) 0 0
\(979\) −17.1231 −0.547257
\(980\) 0 0
\(981\) 0 0
\(982\) 42.2462i 1.34813i
\(983\) − 21.9309i − 0.699486i −0.936846 0.349743i \(-0.886269\pi\)
0.936846 0.349743i \(-0.113731\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 18.8769 0.601163
\(987\) 0 0
\(988\) 6.00000i 0.190885i
\(989\) −30.7386 −0.977432
\(990\) 0 0
\(991\) 37.3153 1.18536 0.592680 0.805438i \(-0.298070\pi\)
0.592680 + 0.805438i \(0.298070\pi\)
\(992\) 30.7386i 0.975953i
\(993\) 0 0
\(994\) −22.5616 −0.715609
\(995\) 0 0
\(996\) 0 0
\(997\) 20.7386i 0.656799i 0.944539 + 0.328400i \(0.106509\pi\)
−0.944539 + 0.328400i \(0.893491\pi\)
\(998\) − 9.75379i − 0.308751i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2475.2.c.i.199.4 4
3.2 odd 2 2475.2.c.j.199.1 4
5.2 odd 4 2475.2.a.p.1.1 2
5.3 odd 4 2475.2.a.v.1.2 yes 2
5.4 even 2 inner 2475.2.c.i.199.1 4
15.2 even 4 2475.2.a.u.1.2 yes 2
15.8 even 4 2475.2.a.q.1.1 yes 2
15.14 odd 2 2475.2.c.j.199.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2475.2.a.p.1.1 2 5.2 odd 4
2475.2.a.q.1.1 yes 2 15.8 even 4
2475.2.a.u.1.2 yes 2 15.2 even 4
2475.2.a.v.1.2 yes 2 5.3 odd 4
2475.2.c.i.199.1 4 5.4 even 2 inner
2475.2.c.i.199.4 4 1.1 even 1 trivial
2475.2.c.j.199.1 4 3.2 odd 2
2475.2.c.j.199.4 4 15.14 odd 2