Properties

Label 2475.2.a.v.1.2
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
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 \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{7} +6.56155 q^{8} +O(q^{10})\) \(q+2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{7} +6.56155 q^{8} -1.00000 q^{11} -0.438447 q^{13} +2.56155 q^{14} +7.68466 q^{16} +1.43845 q^{17} +3.00000 q^{19} -2.56155 q^{22} +6.56155 q^{23} -1.12311 q^{26} +4.56155 q^{28} +5.12311 q^{29} +4.68466 q^{31} +6.56155 q^{32} +3.68466 q^{34} -10.8078 q^{37} +7.68466 q^{38} -7.68466 q^{41} +4.68466 q^{43} -4.56155 q^{44} +16.8078 q^{46} +5.43845 q^{47} -6.00000 q^{49} -2.00000 q^{52} -1.12311 q^{53} +6.56155 q^{56} +13.1231 q^{58} +10.5616 q^{59} -7.56155 q^{61} +12.0000 q^{62} +1.43845 q^{64} -6.68466 q^{67} +6.56155 q^{68} +8.80776 q^{71} +16.2462 q^{73} -27.6847 q^{74} +13.6847 q^{76} -1.00000 q^{77} -15.6847 q^{79} -19.6847 q^{82} +12.0000 q^{86} -6.56155 q^{88} -17.1231 q^{89} -0.438447 q^{91} +29.9309 q^{92} +13.9309 q^{94} +2.12311 q^{97} -15.3693 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{4} + 2 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 5 q^{4} + 2 q^{7} + 9 q^{8} - 2 q^{11} - 5 q^{13} + q^{14} + 3 q^{16} + 7 q^{17} + 6 q^{19} - q^{22} + 9 q^{23} + 6 q^{26} + 5 q^{28} + 2 q^{29} - 3 q^{31} + 9 q^{32} - 5 q^{34} - q^{37} + 3 q^{38} - 3 q^{41} - 3 q^{43} - 5 q^{44} + 13 q^{46} + 15 q^{47} - 12 q^{49} - 4 q^{52} + 6 q^{53} + 9 q^{56} + 18 q^{58} + 17 q^{59} - 11 q^{61} + 24 q^{62} + 7 q^{64} - q^{67} + 9 q^{68} - 3 q^{71} + 16 q^{73} - 43 q^{74} + 15 q^{76} - 2 q^{77} - 19 q^{79} - 27 q^{82} + 24 q^{86} - 9 q^{88} - 26 q^{89} - 5 q^{91} + 31 q^{92} - q^{94} - 4 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56155 1.81129 0.905646 0.424035i \(-0.139387\pi\)
0.905646 + 0.424035i \(0.139387\pi\)
\(3\) 0 0
\(4\) 4.56155 2.28078
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 6.56155 2.31986
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −0.438447 −0.121603 −0.0608017 0.998150i \(-0.519366\pi\)
−0.0608017 + 0.998150i \(0.519366\pi\)
\(14\) 2.56155 0.684604
\(15\) 0 0
\(16\) 7.68466 1.92116
\(17\) 1.43845 0.348875 0.174437 0.984668i \(-0.444189\pi\)
0.174437 + 0.984668i \(0.444189\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.56155 −0.546125
\(23\) 6.56155 1.36818 0.684089 0.729398i \(-0.260200\pi\)
0.684089 + 0.729398i \(0.260200\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.12311 −0.220259
\(27\) 0 0
\(28\) 4.56155 0.862052
\(29\) 5.12311 0.951337 0.475668 0.879625i \(-0.342206\pi\)
0.475668 + 0.879625i \(0.342206\pi\)
\(30\) 0 0
\(31\) 4.68466 0.841389 0.420695 0.907202i \(-0.361786\pi\)
0.420695 + 0.907202i \(0.361786\pi\)
\(32\) 6.56155 1.15993
\(33\) 0 0
\(34\) 3.68466 0.631914
\(35\) 0 0
\(36\) 0 0
\(37\) −10.8078 −1.77679 −0.888393 0.459084i \(-0.848178\pi\)
−0.888393 + 0.459084i \(0.848178\pi\)
\(38\) 7.68466 1.24662
\(39\) 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
\(43\) 4.68466 0.714404 0.357202 0.934027i \(-0.383731\pi\)
0.357202 + 0.934027i \(0.383731\pi\)
\(44\) −4.56155 −0.687680
\(45\) 0 0
\(46\) 16.8078 2.47817
\(47\) 5.43845 0.793279 0.396640 0.917974i \(-0.370176\pi\)
0.396640 + 0.917974i \(0.370176\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −1.12311 −0.154270 −0.0771352 0.997021i \(-0.524577\pi\)
−0.0771352 + 0.997021i \(0.524577\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 6.56155 0.876824
\(57\) 0 0
\(58\) 13.1231 1.72315
\(59\) 10.5616 1.37500 0.687499 0.726186i \(-0.258709\pi\)
0.687499 + 0.726186i \(0.258709\pi\)
\(60\) 0 0
\(61\) −7.56155 −0.968158 −0.484079 0.875024i \(-0.660845\pi\)
−0.484079 + 0.875024i \(0.660845\pi\)
\(62\) 12.0000 1.52400
\(63\) 0 0
\(64\) 1.43845 0.179806
\(65\) 0 0
\(66\) 0 0
\(67\) −6.68466 −0.816661 −0.408331 0.912834i \(-0.633889\pi\)
−0.408331 + 0.912834i \(0.633889\pi\)
\(68\) 6.56155 0.795705
\(69\) 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
\(73\) 16.2462 1.90148 0.950738 0.309997i \(-0.100328\pi\)
0.950738 + 0.309997i \(0.100328\pi\)
\(74\) −27.6847 −3.21828
\(75\) 0 0
\(76\) 13.6847 1.56974
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −15.6847 −1.76466 −0.882331 0.470629i \(-0.844027\pi\)
−0.882331 + 0.470629i \(0.844027\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −19.6847 −2.17381
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) −6.56155 −0.699464
\(89\) −17.1231 −1.81505 −0.907523 0.420003i \(-0.862029\pi\)
−0.907523 + 0.420003i \(0.862029\pi\)
\(90\) 0 0
\(91\) −0.438447 −0.0459618
\(92\) 29.9309 3.12051
\(93\) 0 0
\(94\) 13.9309 1.43686
\(95\) 0 0
\(96\) 0 0
\(97\) 2.12311 0.215569 0.107784 0.994174i \(-0.465624\pi\)
0.107784 + 0.994174i \(0.465624\pi\)
\(98\) −15.3693 −1.55254
\(99\) 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
\(103\) −9.12311 −0.898926 −0.449463 0.893299i \(-0.648385\pi\)
−0.449463 + 0.893299i \(0.648385\pi\)
\(104\) −2.87689 −0.282103
\(105\) 0 0
\(106\) −2.87689 −0.279429
\(107\) −9.12311 −0.881964 −0.440982 0.897516i \(-0.645370\pi\)
−0.440982 + 0.897516i \(0.645370\pi\)
\(108\) 0 0
\(109\) −7.80776 −0.747848 −0.373924 0.927459i \(-0.621988\pi\)
−0.373924 + 0.927459i \(0.621988\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 7.68466 0.726132
\(113\) −9.12311 −0.858230 −0.429115 0.903250i \(-0.641174\pi\)
−0.429115 + 0.903250i \(0.641174\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 23.3693 2.16979
\(117\) 0 0
\(118\) 27.0540 2.49052
\(119\) 1.43845 0.131862
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −19.3693 −1.75362
\(123\) 0 0
\(124\) 21.3693 1.91902
\(125\) 0 0
\(126\) 0 0
\(127\) −4.80776 −0.426620 −0.213310 0.976985i \(-0.568424\pi\)
−0.213310 + 0.976985i \(0.568424\pi\)
\(128\) −9.43845 −0.834249
\(129\) 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
\(133\) 3.00000 0.260133
\(134\) −17.1231 −1.47921
\(135\) 0 0
\(136\) 9.43845 0.809340
\(137\) −3.36932 −0.287860 −0.143930 0.989588i \(-0.545974\pi\)
−0.143930 + 0.989588i \(0.545974\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 22.5616 1.89332
\(143\) 0.438447 0.0366648
\(144\) 0 0
\(145\) 0 0
\(146\) 41.6155 3.44413
\(147\) 0 0
\(148\) −49.3002 −4.05245
\(149\) 8.80776 0.721560 0.360780 0.932651i \(-0.382510\pi\)
0.360780 + 0.932651i \(0.382510\pi\)
\(150\) 0 0
\(151\) −16.6847 −1.35778 −0.678889 0.734241i \(-0.737538\pi\)
−0.678889 + 0.734241i \(0.737538\pi\)
\(152\) 19.6847 1.59664
\(153\) 0 0
\(154\) −2.56155 −0.206416
\(155\) 0 0
\(156\) 0 0
\(157\) 2.68466 0.214259 0.107130 0.994245i \(-0.465834\pi\)
0.107130 + 0.994245i \(0.465834\pi\)
\(158\) −40.1771 −3.19632
\(159\) 0 0
\(160\) 0 0
\(161\) 6.56155 0.517123
\(162\) 0 0
\(163\) −5.31534 −0.416330 −0.208165 0.978094i \(-0.566749\pi\)
−0.208165 + 0.978094i \(0.566749\pi\)
\(164\) −35.0540 −2.73726
\(165\) 0 0
\(166\) 0 0
\(167\) −15.3693 −1.18931 −0.594657 0.803980i \(-0.702712\pi\)
−0.594657 + 0.803980i \(0.702712\pi\)
\(168\) 0 0
\(169\) −12.8078 −0.985213
\(170\) 0 0
\(171\) 0 0
\(172\) 21.3693 1.62940
\(173\) 13.4384 1.02171 0.510853 0.859668i \(-0.329330\pi\)
0.510853 + 0.859668i \(0.329330\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −7.68466 −0.579253
\(177\) 0 0
\(178\) −43.8617 −3.28758
\(179\) 14.5616 1.08838 0.544191 0.838961i \(-0.316837\pi\)
0.544191 + 0.838961i \(0.316837\pi\)
\(180\) 0 0
\(181\) 11.8769 0.882803 0.441401 0.897310i \(-0.354482\pi\)
0.441401 + 0.897310i \(0.354482\pi\)
\(182\) −1.12311 −0.0832501
\(183\) 0 0
\(184\) 43.0540 3.17398
\(185\) 0 0
\(186\) 0 0
\(187\) −1.43845 −0.105190
\(188\) 24.8078 1.80929
\(189\) 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
\(193\) 17.5616 1.26411 0.632054 0.774924i \(-0.282212\pi\)
0.632054 + 0.774924i \(0.282212\pi\)
\(194\) 5.43845 0.390458
\(195\) 0 0
\(196\) −27.3693 −1.95495
\(197\) 17.9309 1.27752 0.638761 0.769405i \(-0.279447\pi\)
0.638761 + 0.769405i \(0.279447\pi\)
\(198\) 0 0
\(199\) −13.8078 −0.978806 −0.489403 0.872058i \(-0.662785\pi\)
−0.489403 + 0.872058i \(0.662785\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 3.68466 0.259252
\(203\) 5.12311 0.359572
\(204\) 0 0
\(205\) 0 0
\(206\) −23.3693 −1.62822
\(207\) 0 0
\(208\) −3.36932 −0.233620
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 3.31534 0.228238 0.114119 0.993467i \(-0.463596\pi\)
0.114119 + 0.993467i \(0.463596\pi\)
\(212\) −5.12311 −0.351856
\(213\) 0 0
\(214\) −23.3693 −1.59749
\(215\) 0 0
\(216\) 0 0
\(217\) 4.68466 0.318015
\(218\) −20.0000 −1.35457
\(219\) 0 0
\(220\) 0 0
\(221\) −0.630683 −0.0424243
\(222\) 0 0
\(223\) 21.8078 1.46036 0.730178 0.683257i \(-0.239437\pi\)
0.730178 + 0.683257i \(0.239437\pi\)
\(224\) 6.56155 0.438412
\(225\) 0 0
\(226\) −23.3693 −1.55450
\(227\) 20.4924 1.36013 0.680065 0.733152i \(-0.261952\pi\)
0.680065 + 0.733152i \(0.261952\pi\)
\(228\) 0 0
\(229\) −8.12311 −0.536790 −0.268395 0.963309i \(-0.586493\pi\)
−0.268395 + 0.963309i \(0.586493\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 33.6155 2.20697
\(233\) 24.8078 1.62521 0.812605 0.582814i \(-0.198049\pi\)
0.812605 + 0.582814i \(0.198049\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 48.1771 3.13606
\(237\) 0 0
\(238\) 3.68466 0.238841
\(239\) −16.4924 −1.06681 −0.533403 0.845861i \(-0.679087\pi\)
−0.533403 + 0.845861i \(0.679087\pi\)
\(240\) 0 0
\(241\) −14.0540 −0.905296 −0.452648 0.891689i \(-0.649521\pi\)
−0.452648 + 0.891689i \(0.649521\pi\)
\(242\) 2.56155 0.164663
\(243\) 0 0
\(244\) −34.4924 −2.20815
\(245\) 0 0
\(246\) 0 0
\(247\) −1.31534 −0.0836932
\(248\) 30.7386 1.95191
\(249\) 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
\(253\) −6.56155 −0.412521
\(254\) −12.3153 −0.772733
\(255\) 0 0
\(256\) −27.0540 −1.69087
\(257\) 18.2462 1.13817 0.569084 0.822280i \(-0.307298\pi\)
0.569084 + 0.822280i \(0.307298\pi\)
\(258\) 0 0
\(259\) −10.8078 −0.671562
\(260\) 0 0
\(261\) 0 0
\(262\) −42.2462 −2.60998
\(263\) −3.36932 −0.207761 −0.103880 0.994590i \(-0.533126\pi\)
−0.103880 + 0.994590i \(0.533126\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 7.68466 0.471177
\(267\) 0 0
\(268\) −30.4924 −1.86262
\(269\) 28.4924 1.73721 0.868607 0.495502i \(-0.165016\pi\)
0.868607 + 0.495502i \(0.165016\pi\)
\(270\) 0 0
\(271\) −28.1771 −1.71164 −0.855818 0.517277i \(-0.826946\pi\)
−0.855818 + 0.517277i \(0.826946\pi\)
\(272\) 11.0540 0.670246
\(273\) 0 0
\(274\) −8.63068 −0.521399
\(275\) 0 0
\(276\) 0 0
\(277\) −23.1771 −1.39258 −0.696288 0.717763i \(-0.745166\pi\)
−0.696288 + 0.717763i \(0.745166\pi\)
\(278\) 20.4924 1.22905
\(279\) 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
\(283\) 32.1231 1.90952 0.954760 0.297377i \(-0.0961117\pi\)
0.954760 + 0.297377i \(0.0961117\pi\)
\(284\) 40.1771 2.38407
\(285\) 0 0
\(286\) 1.12311 0.0664106
\(287\) −7.68466 −0.453611
\(288\) 0 0
\(289\) −14.9309 −0.878286
\(290\) 0 0
\(291\) 0 0
\(292\) 74.1080 4.33684
\(293\) 2.06913 0.120880 0.0604399 0.998172i \(-0.480750\pi\)
0.0604399 + 0.998172i \(0.480750\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −70.9157 −4.12189
\(297\) 0 0
\(298\) 22.5616 1.30696
\(299\) −2.87689 −0.166375
\(300\) 0 0
\(301\) 4.68466 0.270019
\(302\) −42.7386 −2.45933
\(303\) 0 0
\(304\) 23.0540 1.32224
\(305\) 0 0
\(306\) 0 0
\(307\) −22.9309 −1.30873 −0.654367 0.756177i \(-0.727065\pi\)
−0.654367 + 0.756177i \(0.727065\pi\)
\(308\) −4.56155 −0.259919
\(309\) 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
\(313\) −31.2462 −1.76614 −0.883070 0.469241i \(-0.844528\pi\)
−0.883070 + 0.469241i \(0.844528\pi\)
\(314\) 6.87689 0.388086
\(315\) 0 0
\(316\) −71.5464 −4.02480
\(317\) 6.87689 0.386245 0.193122 0.981175i \(-0.438139\pi\)
0.193122 + 0.981175i \(0.438139\pi\)
\(318\) 0 0
\(319\) −5.12311 −0.286839
\(320\) 0 0
\(321\) 0 0
\(322\) 16.8078 0.936660
\(323\) 4.31534 0.240112
\(324\) 0 0
\(325\) 0 0
\(326\) −13.6155 −0.754094
\(327\) 0 0
\(328\) −50.4233 −2.78416
\(329\) 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
\(337\) −10.6847 −0.582030 −0.291015 0.956718i \(-0.593993\pi\)
−0.291015 + 0.956718i \(0.593993\pi\)
\(338\) −32.8078 −1.78451
\(339\) 0 0
\(340\) 0 0
\(341\) −4.68466 −0.253688
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 30.7386 1.65732
\(345\) 0 0
\(346\) 34.4233 1.85061
\(347\) −27.3693 −1.46926 −0.734631 0.678467i \(-0.762645\pi\)
−0.734631 + 0.678467i \(0.762645\pi\)
\(348\) 0 0
\(349\) 6.63068 0.354932 0.177466 0.984127i \(-0.443210\pi\)
0.177466 + 0.984127i \(0.443210\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6.56155 −0.349732
\(353\) 27.8617 1.48293 0.741465 0.670991i \(-0.234131\pi\)
0.741465 + 0.670991i \(0.234131\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −78.1080 −4.13971
\(357\) 0 0
\(358\) 37.3002 1.97138
\(359\) 26.2462 1.38522 0.692611 0.721311i \(-0.256460\pi\)
0.692611 + 0.721311i \(0.256460\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 30.4233 1.59901
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) 14.0540 0.733612 0.366806 0.930298i \(-0.380451\pi\)
0.366806 + 0.930298i \(0.380451\pi\)
\(368\) 50.4233 2.62850
\(369\) 0 0
\(370\) 0 0
\(371\) −1.12311 −0.0583087
\(372\) 0 0
\(373\) −13.8078 −0.714939 −0.357469 0.933925i \(-0.616360\pi\)
−0.357469 + 0.933925i \(0.616360\pi\)
\(374\) −3.68466 −0.190529
\(375\) 0 0
\(376\) 35.6847 1.84030
\(377\) −2.24621 −0.115686
\(378\) 0 0
\(379\) −24.3002 −1.24822 −0.624108 0.781338i \(-0.714538\pi\)
−0.624108 + 0.781338i \(0.714538\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4.94602 −0.253061
\(383\) 22.2462 1.13673 0.568364 0.822777i \(-0.307576\pi\)
0.568364 + 0.822777i \(0.307576\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 44.9848 2.28967
\(387\) 0 0
\(388\) 9.68466 0.491664
\(389\) −23.8617 −1.20984 −0.604919 0.796287i \(-0.706795\pi\)
−0.604919 + 0.796287i \(0.706795\pi\)
\(390\) 0 0
\(391\) 9.43845 0.477323
\(392\) −39.3693 −1.98845
\(393\) 0 0
\(394\) 45.9309 2.31396
\(395\) 0 0
\(396\) 0 0
\(397\) 28.4384 1.42728 0.713642 0.700510i \(-0.247044\pi\)
0.713642 + 0.700510i \(0.247044\pi\)
\(398\) −35.3693 −1.77290
\(399\) 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
\(403\) −2.05398 −0.102316
\(404\) 6.56155 0.326449
\(405\) 0 0
\(406\) 13.1231 0.651289
\(407\) 10.8078 0.535721
\(408\) 0 0
\(409\) 8.05398 0.398243 0.199122 0.979975i \(-0.436191\pi\)
0.199122 + 0.979975i \(0.436191\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −41.6155 −2.05025
\(413\) 10.5616 0.519700
\(414\) 0 0
\(415\) 0 0
\(416\) −2.87689 −0.141051
\(417\) 0 0
\(418\) −7.68466 −0.375869
\(419\) 25.4384 1.24275 0.621375 0.783514i \(-0.286574\pi\)
0.621375 + 0.783514i \(0.286574\pi\)
\(420\) 0 0
\(421\) −17.0540 −0.831160 −0.415580 0.909557i \(-0.636421\pi\)
−0.415580 + 0.909557i \(0.636421\pi\)
\(422\) 8.49242 0.413405
\(423\) 0 0
\(424\) −7.36932 −0.357886
\(425\) 0 0
\(426\) 0 0
\(427\) −7.56155 −0.365929
\(428\) −41.6155 −2.01156
\(429\) 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
\(433\) 16.9309 0.813646 0.406823 0.913507i \(-0.366637\pi\)
0.406823 + 0.913507i \(0.366637\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) −35.6155 −1.70567
\(437\) 19.6847 0.941645
\(438\) 0 0
\(439\) 11.7386 0.560254 0.280127 0.959963i \(-0.409623\pi\)
0.280127 + 0.959963i \(0.409623\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.61553 −0.0768428
\(443\) 25.3002 1.20205 0.601024 0.799231i \(-0.294760\pi\)
0.601024 + 0.799231i \(0.294760\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 55.8617 2.64513
\(447\) 0 0
\(448\) 1.43845 0.0679602
\(449\) −28.4924 −1.34464 −0.672320 0.740260i \(-0.734702\pi\)
−0.672320 + 0.740260i \(0.734702\pi\)
\(450\) 0 0
\(451\) 7.68466 0.361856
\(452\) −41.6155 −1.95743
\(453\) 0 0
\(454\) 52.4924 2.46359
\(455\) 0 0
\(456\) 0 0
\(457\) 13.3693 0.625390 0.312695 0.949854i \(-0.398768\pi\)
0.312695 + 0.949854i \(0.398768\pi\)
\(458\) −20.8078 −0.972283
\(459\) 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
\(463\) −1.75379 −0.0815055 −0.0407527 0.999169i \(-0.512976\pi\)
−0.0407527 + 0.999169i \(0.512976\pi\)
\(464\) 39.3693 1.82767
\(465\) 0 0
\(466\) 63.5464 2.94373
\(467\) 2.24621 0.103942 0.0519711 0.998649i \(-0.483450\pi\)
0.0519711 + 0.998649i \(0.483450\pi\)
\(468\) 0 0
\(469\) −6.68466 −0.308669
\(470\) 0 0
\(471\) 0 0
\(472\) 69.3002 3.18980
\(473\) −4.68466 −0.215401
\(474\) 0 0
\(475\) 0 0
\(476\) 6.56155 0.300748
\(477\) 0 0
\(478\) −42.2462 −1.93230
\(479\) 3.36932 0.153948 0.0769740 0.997033i \(-0.475474\pi\)
0.0769740 + 0.997033i \(0.475474\pi\)
\(480\) 0 0
\(481\) 4.73863 0.216063
\(482\) −36.0000 −1.63976
\(483\) 0 0
\(484\) 4.56155 0.207343
\(485\) 0 0
\(486\) 0 0
\(487\) −12.0540 −0.546218 −0.273109 0.961983i \(-0.588052\pi\)
−0.273109 + 0.961983i \(0.588052\pi\)
\(488\) −49.6155 −2.24599
\(489\) 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
\(493\) 7.36932 0.331897
\(494\) −3.36932 −0.151593
\(495\) 0 0
\(496\) 36.0000 1.61645
\(497\) 8.80776 0.395082
\(498\) 0 0
\(499\) 3.80776 0.170459 0.0852295 0.996361i \(-0.472838\pi\)
0.0852295 + 0.996361i \(0.472838\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −30.7386 −1.37193
\(503\) −16.4924 −0.735361 −0.367680 0.929952i \(-0.619848\pi\)
−0.367680 + 0.929952i \(0.619848\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −16.8078 −0.747196
\(507\) 0 0
\(508\) −21.9309 −0.973025
\(509\) −17.1231 −0.758968 −0.379484 0.925198i \(-0.623899\pi\)
−0.379484 + 0.925198i \(0.623899\pi\)
\(510\) 0 0
\(511\) 16.2462 0.718690
\(512\) −50.4233 −2.22842
\(513\) 0 0
\(514\) 46.7386 2.06155
\(515\) 0 0
\(516\) 0 0
\(517\) −5.43845 −0.239183
\(518\) −27.6847 −1.21639
\(519\) 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
\(523\) 9.24621 0.404309 0.202154 0.979354i \(-0.435206\pi\)
0.202154 + 0.979354i \(0.435206\pi\)
\(524\) −75.2311 −3.28648
\(525\) 0 0
\(526\) −8.63068 −0.376316
\(527\) 6.73863 0.293539
\(528\) 0 0
\(529\) 20.0540 0.871912
\(530\) 0 0
\(531\) 0 0
\(532\) 13.6847 0.593305
\(533\) 3.36932 0.145941
\(534\) 0 0
\(535\) 0 0
\(536\) −43.8617 −1.89454
\(537\) 0 0
\(538\) 72.9848 3.14660
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −7.31534 −0.314511 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(542\) −72.1771 −3.10027
\(543\) 0 0
\(544\) 9.43845 0.404670
\(545\) 0 0
\(546\) 0 0
\(547\) 3.68466 0.157545 0.0787723 0.996893i \(-0.474900\pi\)
0.0787723 + 0.996893i \(0.474900\pi\)
\(548\) −15.3693 −0.656545
\(549\) 0 0
\(550\) 0 0
\(551\) 15.3693 0.654755
\(552\) 0 0
\(553\) −15.6847 −0.666980
\(554\) −59.3693 −2.52236
\(555\) 0 0
\(556\) 36.4924 1.54762
\(557\) 1.12311 0.0475875 0.0237938 0.999717i \(-0.492425\pi\)
0.0237938 + 0.999717i \(0.492425\pi\)
\(558\) 0 0
\(559\) −2.05398 −0.0868739
\(560\) 0 0
\(561\) 0 0
\(562\) −8.17708 −0.344929
\(563\) −10.8769 −0.458406 −0.229203 0.973379i \(-0.573612\pi\)
−0.229203 + 0.973379i \(0.573612\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 82.2850 3.45870
\(567\) 0 0
\(568\) 57.7926 2.42492
\(569\) 37.9309 1.59014 0.795072 0.606515i \(-0.207433\pi\)
0.795072 + 0.606515i \(0.207433\pi\)
\(570\) 0 0
\(571\) 38.0540 1.59251 0.796255 0.604962i \(-0.206812\pi\)
0.796255 + 0.604962i \(0.206812\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) −19.6847 −0.821622
\(575\) 0 0
\(576\) 0 0
\(577\) −28.3693 −1.18103 −0.590515 0.807027i \(-0.701075\pi\)
−0.590515 + 0.807027i \(0.701075\pi\)
\(578\) −38.2462 −1.59083
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1.12311 0.0465143
\(584\) 106.600 4.41115
\(585\) 0 0
\(586\) 5.30019 0.218949
\(587\) −27.1922 −1.12234 −0.561172 0.827699i \(-0.689649\pi\)
−0.561172 + 0.827699i \(0.689649\pi\)
\(588\) 0 0
\(589\) 14.0540 0.579084
\(590\) 0 0
\(591\) 0 0
\(592\) −83.0540 −3.41350
\(593\) −34.1080 −1.40065 −0.700323 0.713826i \(-0.746961\pi\)
−0.700323 + 0.713826i \(0.746961\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 40.1771 1.64572
\(597\) 0 0
\(598\) −7.36932 −0.301354
\(599\) 24.1771 0.987849 0.493924 0.869505i \(-0.335562\pi\)
0.493924 + 0.869505i \(0.335562\pi\)
\(600\) 0 0
\(601\) 37.4233 1.52653 0.763264 0.646087i \(-0.223596\pi\)
0.763264 + 0.646087i \(0.223596\pi\)
\(602\) 12.0000 0.489083
\(603\) 0 0
\(604\) −76.1080 −3.09679
\(605\) 0 0
\(606\) 0 0
\(607\) 6.24621 0.253526 0.126763 0.991933i \(-0.459541\pi\)
0.126763 + 0.991933i \(0.459541\pi\)
\(608\) 19.6847 0.798318
\(609\) 0 0
\(610\) 0 0
\(611\) −2.38447 −0.0964654
\(612\) 0 0
\(613\) 28.1080 1.13527 0.567635 0.823281i \(-0.307859\pi\)
0.567635 + 0.823281i \(0.307859\pi\)
\(614\) −58.7386 −2.37050
\(615\) 0 0
\(616\) −6.56155 −0.264372
\(617\) 21.7538 0.875775 0.437887 0.899030i \(-0.355727\pi\)
0.437887 + 0.899030i \(0.355727\pi\)
\(618\) 0 0
\(619\) 7.17708 0.288471 0.144236 0.989543i \(-0.453928\pi\)
0.144236 + 0.989543i \(0.453928\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −35.2311 −1.41264
\(623\) −17.1231 −0.686023
\(624\) 0 0
\(625\) 0 0
\(626\) −80.0388 −3.19899
\(627\) 0 0
\(628\) 12.2462 0.488677
\(629\) −15.5464 −0.619875
\(630\) 0 0
\(631\) −22.6847 −0.903062 −0.451531 0.892255i \(-0.649122\pi\)
−0.451531 + 0.892255i \(0.649122\pi\)
\(632\) −102.916 −4.09377
\(633\) 0 0
\(634\) 17.6155 0.699602
\(635\) 0 0
\(636\) 0 0
\(637\) 2.63068 0.104231
\(638\) −13.1231 −0.519549
\(639\) 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
\(643\) 15.3693 0.606107 0.303053 0.952974i \(-0.401994\pi\)
0.303053 + 0.952974i \(0.401994\pi\)
\(644\) 29.9309 1.17944
\(645\) 0 0
\(646\) 11.0540 0.434913
\(647\) −40.8078 −1.60432 −0.802159 0.597110i \(-0.796316\pi\)
−0.802159 + 0.597110i \(0.796316\pi\)
\(648\) 0 0
\(649\) −10.5616 −0.414577
\(650\) 0 0
\(651\) 0 0
\(652\) −24.2462 −0.949555
\(653\) 7.50758 0.293794 0.146897 0.989152i \(-0.453071\pi\)
0.146897 + 0.989152i \(0.453071\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −59.0540 −2.30567
\(657\) 0 0
\(658\) 13.9309 0.543082
\(659\) 8.63068 0.336204 0.168102 0.985770i \(-0.446236\pi\)
0.168102 + 0.985770i \(0.446236\pi\)
\(660\) 0 0
\(661\) 6.80776 0.264791 0.132396 0.991197i \(-0.457733\pi\)
0.132396 + 0.991197i \(0.457733\pi\)
\(662\) −26.2462 −1.02009
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 33.6155 1.30160
\(668\) −70.1080 −2.71256
\(669\) 0 0
\(670\) 0 0
\(671\) 7.56155 0.291911
\(672\) 0 0
\(673\) 10.6307 0.409783 0.204891 0.978785i \(-0.434316\pi\)
0.204891 + 0.978785i \(0.434316\pi\)
\(674\) −27.3693 −1.05423
\(675\) 0 0
\(676\) −58.4233 −2.24705
\(677\) 29.1231 1.11929 0.559646 0.828732i \(-0.310937\pi\)
0.559646 + 0.828732i \(0.310937\pi\)
\(678\) 0 0
\(679\) 2.12311 0.0814773
\(680\) 0 0
\(681\) 0 0
\(682\) −12.0000 −0.459504
\(683\) −20.8078 −0.796187 −0.398093 0.917345i \(-0.630328\pi\)
−0.398093 + 0.917345i \(0.630328\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −33.3002 −1.27141
\(687\) 0 0
\(688\) 36.0000 1.37249
\(689\) 0.492423 0.0187598
\(690\) 0 0
\(691\) 12.4924 0.475234 0.237617 0.971359i \(-0.423634\pi\)
0.237617 + 0.971359i \(0.423634\pi\)
\(692\) 61.3002 2.33028
\(693\) 0 0
\(694\) −70.1080 −2.66126
\(695\) 0 0
\(696\) 0 0
\(697\) −11.0540 −0.418699
\(698\) 16.9848 0.642886
\(699\) 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
\(703\) −32.4233 −1.22287
\(704\) −1.43845 −0.0542135
\(705\) 0 0
\(706\) 71.3693 2.68602
\(707\) 1.43845 0.0540984
\(708\) 0 0
\(709\) 6.12311 0.229958 0.114979 0.993368i \(-0.463320\pi\)
0.114979 + 0.993368i \(0.463320\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −112.354 −4.21065
\(713\) 30.7386 1.15117
\(714\) 0 0
\(715\) 0 0
\(716\) 66.4233 2.48235
\(717\) 0 0
\(718\) 67.2311 2.50904
\(719\) −32.9848 −1.23013 −0.615064 0.788478i \(-0.710870\pi\)
−0.615064 + 0.788478i \(0.710870\pi\)
\(720\) 0 0
\(721\) −9.12311 −0.339762
\(722\) −25.6155 −0.953311
\(723\) 0 0
\(724\) 54.1771 2.01348
\(725\) 0 0
\(726\) 0 0
\(727\) −30.0540 −1.11464 −0.557320 0.830298i \(-0.688170\pi\)
−0.557320 + 0.830298i \(0.688170\pi\)
\(728\) −2.87689 −0.106625
\(729\) 0 0
\(730\) 0 0
\(731\) 6.73863 0.249237
\(732\) 0 0
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 36.0000 1.32878
\(735\) 0 0
\(736\) 43.0540 1.58699
\(737\) 6.68466 0.246233
\(738\) 0 0
\(739\) −11.6847 −0.429827 −0.214914 0.976633i \(-0.568947\pi\)
−0.214914 + 0.976633i \(0.568947\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.87689 −0.105614
\(743\) 47.8617 1.75588 0.877938 0.478773i \(-0.158918\pi\)
0.877938 + 0.478773i \(0.158918\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −35.3693 −1.29496
\(747\) 0 0
\(748\) −6.56155 −0.239914
\(749\) −9.12311 −0.333351
\(750\) 0 0
\(751\) 26.7386 0.975707 0.487853 0.872926i \(-0.337780\pi\)
0.487853 + 0.872926i \(0.337780\pi\)
\(752\) 41.7926 1.52402
\(753\) 0 0
\(754\) −5.75379 −0.209541
\(755\) 0 0
\(756\) 0 0
\(757\) 6.30019 0.228984 0.114492 0.993424i \(-0.463476\pi\)
0.114492 + 0.993424i \(0.463476\pi\)
\(758\) −62.2462 −2.26088
\(759\) 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
\(763\) −7.80776 −0.282660
\(764\) −8.80776 −0.318654
\(765\) 0 0
\(766\) 56.9848 2.05895
\(767\) −4.63068 −0.167204
\(768\) 0 0
\(769\) 6.05398 0.218312 0.109156 0.994025i \(-0.465185\pi\)
0.109156 + 0.994025i \(0.465185\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 80.1080 2.88315
\(773\) 48.4924 1.74415 0.872076 0.489371i \(-0.162774\pi\)
0.872076 + 0.489371i \(0.162774\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 13.9309 0.500089
\(777\) 0 0
\(778\) −61.1231 −2.19137
\(779\) −23.0540 −0.825994
\(780\) 0 0
\(781\) −8.80776 −0.315167
\(782\) 24.1771 0.864571
\(783\) 0 0
\(784\) −46.1080 −1.64671
\(785\) 0 0
\(786\) 0 0
\(787\) 4.26137 0.151901 0.0759507 0.997112i \(-0.475801\pi\)
0.0759507 + 0.997112i \(0.475801\pi\)
\(788\) 81.7926 2.91374
\(789\) 0 0
\(790\) 0 0
\(791\) −9.12311 −0.324380
\(792\) 0 0
\(793\) 3.31534 0.117731
\(794\) 72.8466 2.58523
\(795\) 0 0
\(796\) −62.9848 −2.23244
\(797\) 42.8769 1.51878 0.759389 0.650637i \(-0.225498\pi\)
0.759389 + 0.650637i \(0.225498\pi\)
\(798\) 0 0
\(799\) 7.82292 0.276755
\(800\) 0 0
\(801\) 0 0
\(802\) 14.3845 0.507933
\(803\) −16.2462 −0.573316
\(804\) 0 0
\(805\) 0 0
\(806\) −5.26137 −0.185324
\(807\) 0 0
\(808\) 9.43845 0.332043
\(809\) 19.6847 0.692076 0.346038 0.938221i \(-0.387527\pi\)
0.346038 + 0.938221i \(0.387527\pi\)
\(810\) 0 0
\(811\) −1.00000 −0.0351147 −0.0175574 0.999846i \(-0.505589\pi\)
−0.0175574 + 0.999846i \(0.505589\pi\)
\(812\) 23.3693 0.820102
\(813\) 0 0
\(814\) 27.6847 0.970347
\(815\) 0 0
\(816\) 0 0
\(817\) 14.0540 0.491686
\(818\) 20.6307 0.721335
\(819\) 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
\(823\) 0.0539753 0.00188146 0.000940731 1.00000i \(-0.499701\pi\)
0.000940731 1.00000i \(0.499701\pi\)
\(824\) −59.8617 −2.08538
\(825\) 0 0
\(826\) 27.0540 0.941328
\(827\) 9.75379 0.339172 0.169586 0.985515i \(-0.445757\pi\)
0.169586 + 0.985515i \(0.445757\pi\)
\(828\) 0 0
\(829\) 48.7386 1.69276 0.846381 0.532577i \(-0.178776\pi\)
0.846381 + 0.532577i \(0.178776\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −0.630683 −0.0218650
\(833\) −8.63068 −0.299035
\(834\) 0 0
\(835\) 0 0
\(836\) −13.6847 −0.473294
\(837\) 0 0
\(838\) 65.1619 2.25098
\(839\) 20.4924 0.707477 0.353738 0.935344i \(-0.384910\pi\)
0.353738 + 0.935344i \(0.384910\pi\)
\(840\) 0 0
\(841\) −2.75379 −0.0949582
\(842\) −43.6847 −1.50547
\(843\) 0 0
\(844\) 15.1231 0.520559
\(845\) 0 0
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) −8.63068 −0.296379
\(849\) 0 0
\(850\) 0 0
\(851\) −70.9157 −2.43096
\(852\) 0 0
\(853\) −35.4233 −1.21287 −0.606435 0.795133i \(-0.707401\pi\)
−0.606435 + 0.795133i \(0.707401\pi\)
\(854\) −19.3693 −0.662804
\(855\) 0 0
\(856\) −59.8617 −2.04603
\(857\) 5.43845 0.185774 0.0928869 0.995677i \(-0.470390\pi\)
0.0928869 + 0.995677i \(0.470390\pi\)
\(858\) 0 0
\(859\) 56.3542 1.92278 0.961390 0.275191i \(-0.0887411\pi\)
0.961390 + 0.275191i \(0.0887411\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −55.7235 −1.89795
\(863\) −2.24621 −0.0764619 −0.0382310 0.999269i \(-0.512172\pi\)
−0.0382310 + 0.999269i \(0.512172\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 43.3693 1.47375
\(867\) 0 0
\(868\) 21.3693 0.725322
\(869\) 15.6847 0.532066
\(870\) 0 0
\(871\) 2.93087 0.0993087
\(872\) −51.2311 −1.73490
\(873\) 0 0
\(874\) 50.4233 1.70559
\(875\) 0 0
\(876\) 0 0
\(877\) 14.0540 0.474569 0.237285 0.971440i \(-0.423743\pi\)
0.237285 + 0.971440i \(0.423743\pi\)
\(878\) 30.0691 1.01478
\(879\) 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
\(883\) −44.9309 −1.51204 −0.756022 0.654546i \(-0.772860\pi\)
−0.756022 + 0.654546i \(0.772860\pi\)
\(884\) −2.87689 −0.0967604
\(885\) 0 0
\(886\) 64.8078 2.17726
\(887\) 27.8617 0.935506 0.467753 0.883859i \(-0.345064\pi\)
0.467753 + 0.883859i \(0.345064\pi\)
\(888\) 0 0
\(889\) −4.80776 −0.161247
\(890\) 0 0
\(891\) 0 0
\(892\) 99.4773 3.33075
\(893\) 16.3153 0.545972
\(894\) 0 0
\(895\) 0 0
\(896\) −9.43845 −0.315316
\(897\) 0 0
\(898\) −72.9848 −2.43554
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −1.61553 −0.0538210
\(902\) 19.6847 0.655427
\(903\) 0 0
\(904\) −59.8617 −1.99097
\(905\) 0 0
\(906\) 0 0
\(907\) 46.1080 1.53099 0.765495 0.643442i \(-0.222494\pi\)
0.765495 + 0.643442i \(0.222494\pi\)
\(908\) 93.4773 3.10215
\(909\) 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
\(917\) −16.4924 −0.544628
\(918\) 0 0
\(919\) 41.1080 1.35603 0.678013 0.735050i \(-0.262841\pi\)
0.678013 + 0.735050i \(0.262841\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −40.6307 −1.33810
\(923\) −3.86174 −0.127111
\(924\) 0 0
\(925\) 0 0
\(926\) −4.49242 −0.147630
\(927\) 0 0
\(928\) 33.6155 1.10348
\(929\) 23.2311 0.762186 0.381093 0.924537i \(-0.375548\pi\)
0.381093 + 0.924537i \(0.375548\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 113.162 3.70674
\(933\) 0 0
\(934\) 5.75379 0.188270
\(935\) 0 0
\(936\) 0 0
\(937\) 34.6847 1.13310 0.566549 0.824028i \(-0.308278\pi\)
0.566549 + 0.824028i \(0.308278\pi\)
\(938\) −17.1231 −0.559089
\(939\) 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
\(943\) −50.4233 −1.64201
\(944\) 81.1619 2.64160
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) −32.1771 −1.04561 −0.522807 0.852451i \(-0.675115\pi\)
−0.522807 + 0.852451i \(0.675115\pi\)
\(948\) 0 0
\(949\) −7.12311 −0.231226
\(950\) 0 0
\(951\) 0 0
\(952\) 9.43845 0.305902
\(953\) −4.80776 −0.155739 −0.0778694 0.996964i \(-0.524812\pi\)
−0.0778694 + 0.996964i \(0.524812\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −75.2311 −2.43315
\(957\) 0 0
\(958\) 8.63068 0.278845
\(959\) −3.36932 −0.108801
\(960\) 0 0
\(961\) −9.05398 −0.292064
\(962\) 12.1383 0.391353
\(963\) 0 0
\(964\) −64.1080 −2.06478
\(965\) 0 0
\(966\) 0 0
\(967\) 44.0000 1.41494 0.707472 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(968\) 6.56155 0.210896
\(969\) 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
\(973\) 8.00000 0.256468
\(974\) −30.8769 −0.989360
\(975\) 0 0
\(976\) −58.1080 −1.85999
\(977\) −25.6155 −0.819513 −0.409757 0.912195i \(-0.634386\pi\)
−0.409757 + 0.912195i \(0.634386\pi\)
\(978\) 0 0
\(979\) 17.1231 0.547257
\(980\) 0 0
\(981\) 0 0
\(982\) 42.2462 1.34813
\(983\) 21.9309 0.699486 0.349743 0.936846i \(-0.386269\pi\)
0.349743 + 0.936846i \(0.386269\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 18.8769 0.601163
\(987\) 0 0
\(988\) −6.00000 −0.190885
\(989\) 30.7386 0.977432
\(990\) 0 0
\(991\) 37.3153 1.18536 0.592680 0.805438i \(-0.298070\pi\)
0.592680 + 0.805438i \(0.298070\pi\)
\(992\) 30.7386 0.975953
\(993\) 0 0
\(994\) 22.5616 0.715609
\(995\) 0 0
\(996\) 0 0
\(997\) 20.7386 0.656799 0.328400 0.944539i \(-0.393491\pi\)
0.328400 + 0.944539i \(0.393491\pi\)
\(998\) 9.75379 0.308751
\(999\) 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.a.v.1.2 yes 2
3.2 odd 2 2475.2.a.q.1.1 yes 2
5.2 odd 4 2475.2.c.i.199.4 4
5.3 odd 4 2475.2.c.i.199.1 4
5.4 even 2 2475.2.a.p.1.1 2
15.2 even 4 2475.2.c.j.199.1 4
15.8 even 4 2475.2.c.j.199.4 4
15.14 odd 2 2475.2.a.u.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2475.2.a.p.1.1 2 5.4 even 2
2475.2.a.q.1.1 yes 2 3.2 odd 2
2475.2.a.u.1.2 yes 2 15.14 odd 2
2475.2.a.v.1.2 yes 2 1.1 even 1 trivial
2475.2.c.i.199.1 4 5.3 odd 4
2475.2.c.i.199.4 4 5.2 odd 4
2475.2.c.j.199.1 4 15.2 even 4
2475.2.c.j.199.4 4 15.8 even 4