Properties

Label 3640.2.a.x.1.1
Level $3640$
Weight $2$
Character 3640.1
Self dual yes
Analytic conductor $29.066$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.0655463357\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4913.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.90570\) of defining polynomial
Character \(\chi\) \(=\) 3640.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.90570 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.631706 q^{9} +O(q^{10})\) \(q-1.90570 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.631706 q^{9} +6.09896 q^{11} +1.00000 q^{13} +1.90570 q^{15} +7.44311 q^{17} +0.655849 q^{19} +1.90570 q^{21} -4.09896 q^{23} +1.00000 q^{25} +4.51327 q^{27} -8.00467 q^{29} -1.21740 q^{31} -11.6228 q^{33} +1.00000 q^{35} -1.75845 q^{37} -1.90570 q^{39} -8.81972 q^{41} +2.28756 q^{43} -0.631706 q^{45} +5.12311 q^{47} +1.00000 q^{49} -14.1844 q^{51} -1.12311 q^{53} -6.09896 q^{55} -1.24985 q^{57} +1.51919 q^{59} -8.24621 q^{61} -0.631706 q^{63} -1.00000 q^{65} +15.2545 q^{67} +7.81141 q^{69} -15.0818 q^{71} -4.49971 q^{73} -1.90570 q^{75} -6.09896 q^{77} +2.41658 q^{79} -10.4961 q^{81} +3.74650 q^{83} -7.44311 q^{85} +15.2545 q^{87} +16.1179 q^{89} -1.00000 q^{91} +2.32001 q^{93} -0.655849 q^{95} +13.1397 q^{97} +3.85275 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9} + 6 q^{11} + 4 q^{13} - 3 q^{15} + 9 q^{17} + 5 q^{19} - 3 q^{21} + 2 q^{23} + 4 q^{25} + 6 q^{27} - 3 q^{29} + q^{31} - 4 q^{33} + 4 q^{35} - 11 q^{37} + 3 q^{39} - 5 q^{41} + 12 q^{43} - 3 q^{45} + 4 q^{47} + 4 q^{49} - 6 q^{51} + 12 q^{53} - 6 q^{55} + 8 q^{57} + 11 q^{59} - 3 q^{63} - 4 q^{65} + 19 q^{67} + 10 q^{69} - 8 q^{71} + 8 q^{73} + 3 q^{75} - 6 q^{77} + 13 q^{79} + 4 q^{81} + 8 q^{83} - 9 q^{85} + 19 q^{87} + 25 q^{89} - 4 q^{91} + 5 q^{93} - 5 q^{95} + 18 q^{97} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.90570 −1.10026 −0.550129 0.835080i \(-0.685422\pi\)
−0.550129 + 0.835080i \(0.685422\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0.631706 0.210569
\(10\) 0 0
\(11\) 6.09896 1.83891 0.919453 0.393200i \(-0.128632\pi\)
0.919453 + 0.393200i \(0.128632\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.90570 0.492051
\(16\) 0 0
\(17\) 7.44311 1.80522 0.902610 0.430459i \(-0.141648\pi\)
0.902610 + 0.430459i \(0.141648\pi\)
\(18\) 0 0
\(19\) 0.655849 0.150462 0.0752311 0.997166i \(-0.476031\pi\)
0.0752311 + 0.997166i \(0.476031\pi\)
\(20\) 0 0
\(21\) 1.90570 0.415859
\(22\) 0 0
\(23\) −4.09896 −0.854693 −0.427346 0.904088i \(-0.640552\pi\)
−0.427346 + 0.904088i \(0.640552\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.51327 0.868578
\(28\) 0 0
\(29\) −8.00467 −1.48643 −0.743215 0.669053i \(-0.766700\pi\)
−0.743215 + 0.669053i \(0.766700\pi\)
\(30\) 0 0
\(31\) −1.21740 −0.218652 −0.109326 0.994006i \(-0.534869\pi\)
−0.109326 + 0.994006i \(0.534869\pi\)
\(32\) 0 0
\(33\) −11.6228 −2.02327
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −1.75845 −0.289088 −0.144544 0.989498i \(-0.546172\pi\)
−0.144544 + 0.989498i \(0.546172\pi\)
\(38\) 0 0
\(39\) −1.90570 −0.305157
\(40\) 0 0
\(41\) −8.81972 −1.37741 −0.688704 0.725043i \(-0.741820\pi\)
−0.688704 + 0.725043i \(0.741820\pi\)
\(42\) 0 0
\(43\) 2.28756 0.348849 0.174424 0.984671i \(-0.444194\pi\)
0.174424 + 0.984671i \(0.444194\pi\)
\(44\) 0 0
\(45\) −0.631706 −0.0941692
\(46\) 0 0
\(47\) 5.12311 0.747282 0.373641 0.927573i \(-0.378109\pi\)
0.373641 + 0.927573i \(0.378109\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −14.1844 −1.98621
\(52\) 0 0
\(53\) −1.12311 −0.154270 −0.0771352 0.997021i \(-0.524577\pi\)
−0.0771352 + 0.997021i \(0.524577\pi\)
\(54\) 0 0
\(55\) −6.09896 −0.822384
\(56\) 0 0
\(57\) −1.24985 −0.165547
\(58\) 0 0
\(59\) 1.51919 0.197781 0.0988906 0.995098i \(-0.468471\pi\)
0.0988906 + 0.995098i \(0.468471\pi\)
\(60\) 0 0
\(61\) −8.24621 −1.05582 −0.527910 0.849301i \(-0.677024\pi\)
−0.527910 + 0.849301i \(0.677024\pi\)
\(62\) 0 0
\(63\) −0.631706 −0.0795875
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 15.2545 1.86364 0.931818 0.362926i \(-0.118222\pi\)
0.931818 + 0.362926i \(0.118222\pi\)
\(68\) 0 0
\(69\) 7.81141 0.940383
\(70\) 0 0
\(71\) −15.0818 −1.78988 −0.894938 0.446191i \(-0.852780\pi\)
−0.894938 + 0.446191i \(0.852780\pi\)
\(72\) 0 0
\(73\) −4.49971 −0.526651 −0.263326 0.964707i \(-0.584819\pi\)
−0.263326 + 0.964707i \(0.584819\pi\)
\(74\) 0 0
\(75\) −1.90570 −0.220052
\(76\) 0 0
\(77\) −6.09896 −0.695041
\(78\) 0 0
\(79\) 2.41658 0.271887 0.135943 0.990717i \(-0.456594\pi\)
0.135943 + 0.990717i \(0.456594\pi\)
\(80\) 0 0
\(81\) −10.4961 −1.16623
\(82\) 0 0
\(83\) 3.74650 0.411232 0.205616 0.978633i \(-0.434080\pi\)
0.205616 + 0.978633i \(0.434080\pi\)
\(84\) 0 0
\(85\) −7.44311 −0.807319
\(86\) 0 0
\(87\) 15.2545 1.63546
\(88\) 0 0
\(89\) 16.1179 1.70849 0.854245 0.519871i \(-0.174020\pi\)
0.854245 + 0.519871i \(0.174020\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 2.32001 0.240574
\(94\) 0 0
\(95\) −0.655849 −0.0672887
\(96\) 0 0
\(97\) 13.1397 1.33414 0.667068 0.744996i \(-0.267549\pi\)
0.667068 + 0.744996i \(0.267549\pi\)
\(98\) 0 0
\(99\) 3.85275 0.387216
\(100\) 0 0
\(101\) 9.17139 0.912588 0.456294 0.889829i \(-0.349177\pi\)
0.456294 + 0.889829i \(0.349177\pi\)
\(102\) 0 0
\(103\) 11.8439 1.16701 0.583505 0.812109i \(-0.301681\pi\)
0.583505 + 0.812109i \(0.301681\pi\)
\(104\) 0 0
\(105\) −1.90570 −0.185978
\(106\) 0 0
\(107\) −1.36237 −0.131706 −0.0658528 0.997829i \(-0.520977\pi\)
−0.0658528 + 0.997829i \(0.520977\pi\)
\(108\) 0 0
\(109\) −11.8763 −1.13754 −0.568772 0.822495i \(-0.692581\pi\)
−0.568772 + 0.822495i \(0.692581\pi\)
\(110\) 0 0
\(111\) 3.35109 0.318072
\(112\) 0 0
\(113\) 14.3865 1.35337 0.676685 0.736273i \(-0.263416\pi\)
0.676685 + 0.736273i \(0.263416\pi\)
\(114\) 0 0
\(115\) 4.09896 0.382230
\(116\) 0 0
\(117\) 0.631706 0.0584012
\(118\) 0 0
\(119\) −7.44311 −0.682309
\(120\) 0 0
\(121\) 26.1973 2.38158
\(122\) 0 0
\(123\) 16.8078 1.51551
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 6.72236 0.596513 0.298256 0.954486i \(-0.403595\pi\)
0.298256 + 0.954486i \(0.403595\pi\)
\(128\) 0 0
\(129\) −4.35940 −0.383824
\(130\) 0 0
\(131\) 9.12311 0.797089 0.398545 0.917149i \(-0.369515\pi\)
0.398545 + 0.917149i \(0.369515\pi\)
\(132\) 0 0
\(133\) −0.655849 −0.0568693
\(134\) 0 0
\(135\) −4.51327 −0.388440
\(136\) 0 0
\(137\) −0.193259 −0.0165112 −0.00825561 0.999966i \(-0.502628\pi\)
−0.00825561 + 0.999966i \(0.502628\pi\)
\(138\) 0 0
\(139\) −2.11319 −0.179239 −0.0896193 0.995976i \(-0.528565\pi\)
−0.0896193 + 0.995976i \(0.528565\pi\)
\(140\) 0 0
\(141\) −9.76312 −0.822203
\(142\) 0 0
\(143\) 6.09896 0.510021
\(144\) 0 0
\(145\) 8.00467 0.664751
\(146\) 0 0
\(147\) −1.90570 −0.157180
\(148\) 0 0
\(149\) 1.12311 0.0920084 0.0460042 0.998941i \(-0.485351\pi\)
0.0460042 + 0.998941i \(0.485351\pi\)
\(150\) 0 0
\(151\) 16.4584 1.33936 0.669681 0.742649i \(-0.266431\pi\)
0.669681 + 0.742649i \(0.266431\pi\)
\(152\) 0 0
\(153\) 4.70186 0.380123
\(154\) 0 0
\(155\) 1.21740 0.0977841
\(156\) 0 0
\(157\) −17.0041 −1.35707 −0.678537 0.734566i \(-0.737386\pi\)
−0.678537 + 0.734566i \(0.737386\pi\)
\(158\) 0 0
\(159\) 2.14031 0.169737
\(160\) 0 0
\(161\) 4.09896 0.323043
\(162\) 0 0
\(163\) 9.68305 0.758435 0.379218 0.925308i \(-0.376193\pi\)
0.379218 + 0.925308i \(0.376193\pi\)
\(164\) 0 0
\(165\) 11.6228 0.904835
\(166\) 0 0
\(167\) 19.3376 1.49639 0.748196 0.663478i \(-0.230920\pi\)
0.748196 + 0.663478i \(0.230920\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0.414304 0.0316826
\(172\) 0 0
\(173\) −15.2410 −1.15875 −0.579374 0.815061i \(-0.696703\pi\)
−0.579374 + 0.815061i \(0.696703\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −2.89512 −0.217610
\(178\) 0 0
\(179\) −6.81243 −0.509185 −0.254592 0.967048i \(-0.581941\pi\)
−0.254592 + 0.967048i \(0.581941\pi\)
\(180\) 0 0
\(181\) −0.236879 −0.0176071 −0.00880356 0.999961i \(-0.502802\pi\)
−0.00880356 + 0.999961i \(0.502802\pi\)
\(182\) 0 0
\(183\) 15.7148 1.16167
\(184\) 0 0
\(185\) 1.75845 0.129284
\(186\) 0 0
\(187\) 45.3953 3.31963
\(188\) 0 0
\(189\) −4.51327 −0.328292
\(190\) 0 0
\(191\) −6.92985 −0.501426 −0.250713 0.968061i \(-0.580665\pi\)
−0.250713 + 0.968061i \(0.580665\pi\)
\(192\) 0 0
\(193\) −16.5179 −1.18899 −0.594493 0.804100i \(-0.702647\pi\)
−0.594493 + 0.804100i \(0.702647\pi\)
\(194\) 0 0
\(195\) 1.90570 0.136470
\(196\) 0 0
\(197\) −22.6374 −1.61285 −0.806424 0.591338i \(-0.798600\pi\)
−0.806424 + 0.591338i \(0.798600\pi\)
\(198\) 0 0
\(199\) 17.2846 1.22527 0.612636 0.790365i \(-0.290109\pi\)
0.612636 + 0.790365i \(0.290109\pi\)
\(200\) 0 0
\(201\) −29.0706 −2.05048
\(202\) 0 0
\(203\) 8.00467 0.561817
\(204\) 0 0
\(205\) 8.81972 0.615996
\(206\) 0 0
\(207\) −2.58934 −0.179972
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −18.0077 −1.23970 −0.619851 0.784719i \(-0.712807\pi\)
−0.619851 + 0.784719i \(0.712807\pi\)
\(212\) 0 0
\(213\) 28.7414 1.96933
\(214\) 0 0
\(215\) −2.28756 −0.156010
\(216\) 0 0
\(217\) 1.21740 0.0826426
\(218\) 0 0
\(219\) 8.57511 0.579452
\(220\) 0 0
\(221\) 7.44311 0.500678
\(222\) 0 0
\(223\) 14.2145 0.951876 0.475938 0.879479i \(-0.342109\pi\)
0.475938 + 0.879479i \(0.342109\pi\)
\(224\) 0 0
\(225\) 0.631706 0.0421137
\(226\) 0 0
\(227\) 5.21968 0.346442 0.173221 0.984883i \(-0.444582\pi\)
0.173221 + 0.984883i \(0.444582\pi\)
\(228\) 0 0
\(229\) 5.45244 0.360308 0.180154 0.983638i \(-0.442340\pi\)
0.180154 + 0.983638i \(0.442340\pi\)
\(230\) 0 0
\(231\) 11.6228 0.764725
\(232\) 0 0
\(233\) −2.25350 −0.147632 −0.0738158 0.997272i \(-0.523518\pi\)
−0.0738158 + 0.997272i \(0.523518\pi\)
\(234\) 0 0
\(235\) −5.12311 −0.334195
\(236\) 0 0
\(237\) −4.60529 −0.299145
\(238\) 0 0
\(239\) 0.476148 0.0307995 0.0153997 0.999881i \(-0.495098\pi\)
0.0153997 + 0.999881i \(0.495098\pi\)
\(240\) 0 0
\(241\) −23.8025 −1.53325 −0.766627 0.642092i \(-0.778067\pi\)
−0.766627 + 0.642092i \(0.778067\pi\)
\(242\) 0 0
\(243\) 6.46259 0.414575
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 0.655849 0.0417307
\(248\) 0 0
\(249\) −7.13972 −0.452462
\(250\) 0 0
\(251\) 18.1874 1.14798 0.573990 0.818862i \(-0.305395\pi\)
0.573990 + 0.818862i \(0.305395\pi\)
\(252\) 0 0
\(253\) −24.9994 −1.57170
\(254\) 0 0
\(255\) 14.1844 0.888260
\(256\) 0 0
\(257\) 29.6155 1.84737 0.923683 0.383158i \(-0.125163\pi\)
0.923683 + 0.383158i \(0.125163\pi\)
\(258\) 0 0
\(259\) 1.75845 0.109265
\(260\) 0 0
\(261\) −5.05660 −0.312995
\(262\) 0 0
\(263\) −12.2003 −0.752304 −0.376152 0.926558i \(-0.622753\pi\)
−0.376152 + 0.926558i \(0.622753\pi\)
\(264\) 0 0
\(265\) 1.12311 0.0689918
\(266\) 0 0
\(267\) −30.7159 −1.87978
\(268\) 0 0
\(269\) 2.92518 0.178351 0.0891757 0.996016i \(-0.471577\pi\)
0.0891757 + 0.996016i \(0.471577\pi\)
\(270\) 0 0
\(271\) 19.8714 1.20710 0.603551 0.797324i \(-0.293752\pi\)
0.603551 + 0.797324i \(0.293752\pi\)
\(272\) 0 0
\(273\) 1.90570 0.115338
\(274\) 0 0
\(275\) 6.09896 0.367781
\(276\) 0 0
\(277\) 20.6222 1.23907 0.619535 0.784969i \(-0.287321\pi\)
0.619535 + 0.784969i \(0.287321\pi\)
\(278\) 0 0
\(279\) −0.769040 −0.0460412
\(280\) 0 0
\(281\) −6.51021 −0.388366 −0.194183 0.980965i \(-0.562206\pi\)
−0.194183 + 0.980965i \(0.562206\pi\)
\(282\) 0 0
\(283\) −13.6917 −0.813888 −0.406944 0.913453i \(-0.633406\pi\)
−0.406944 + 0.913453i \(0.633406\pi\)
\(284\) 0 0
\(285\) 1.24985 0.0740350
\(286\) 0 0
\(287\) 8.81972 0.520611
\(288\) 0 0
\(289\) 38.3999 2.25882
\(290\) 0 0
\(291\) −25.0404 −1.46790
\(292\) 0 0
\(293\) 31.2338 1.82470 0.912349 0.409414i \(-0.134267\pi\)
0.912349 + 0.409414i \(0.134267\pi\)
\(294\) 0 0
\(295\) −1.51919 −0.0884504
\(296\) 0 0
\(297\) 27.5262 1.59723
\(298\) 0 0
\(299\) −4.09896 −0.237049
\(300\) 0 0
\(301\) −2.28756 −0.131852
\(302\) 0 0
\(303\) −17.4780 −1.00408
\(304\) 0 0
\(305\) 8.24621 0.472177
\(306\) 0 0
\(307\) 8.54800 0.487860 0.243930 0.969793i \(-0.421563\pi\)
0.243930 + 0.969793i \(0.421563\pi\)
\(308\) 0 0
\(309\) −22.5709 −1.28401
\(310\) 0 0
\(311\) 15.8524 0.898908 0.449454 0.893304i \(-0.351619\pi\)
0.449454 + 0.893304i \(0.351619\pi\)
\(312\) 0 0
\(313\) −25.2774 −1.42876 −0.714382 0.699756i \(-0.753292\pi\)
−0.714382 + 0.699756i \(0.753292\pi\)
\(314\) 0 0
\(315\) 0.631706 0.0355926
\(316\) 0 0
\(317\) −12.0198 −0.675101 −0.337550 0.941307i \(-0.609598\pi\)
−0.337550 + 0.941307i \(0.609598\pi\)
\(318\) 0 0
\(319\) −48.8202 −2.73340
\(320\) 0 0
\(321\) 2.59628 0.144910
\(322\) 0 0
\(323\) 4.88156 0.271617
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 22.6327 1.25159
\(328\) 0 0
\(329\) −5.12311 −0.282446
\(330\) 0 0
\(331\) 12.1472 0.667673 0.333837 0.942631i \(-0.391657\pi\)
0.333837 + 0.942631i \(0.391657\pi\)
\(332\) 0 0
\(333\) −1.11083 −0.0608729
\(334\) 0 0
\(335\) −15.2545 −0.833443
\(336\) 0 0
\(337\) 17.1491 0.934169 0.467084 0.884213i \(-0.345304\pi\)
0.467084 + 0.884213i \(0.345304\pi\)
\(338\) 0 0
\(339\) −27.4164 −1.48906
\(340\) 0 0
\(341\) −7.42489 −0.402080
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −7.81141 −0.420552
\(346\) 0 0
\(347\) −26.8449 −1.44111 −0.720554 0.693398i \(-0.756113\pi\)
−0.720554 + 0.693398i \(0.756113\pi\)
\(348\) 0 0
\(349\) −20.3819 −1.09102 −0.545508 0.838106i \(-0.683663\pi\)
−0.545508 + 0.838106i \(0.683663\pi\)
\(350\) 0 0
\(351\) 4.51327 0.240900
\(352\) 0 0
\(353\) 2.37719 0.126525 0.0632624 0.997997i \(-0.479849\pi\)
0.0632624 + 0.997997i \(0.479849\pi\)
\(354\) 0 0
\(355\) 15.0818 0.800457
\(356\) 0 0
\(357\) 14.1844 0.750716
\(358\) 0 0
\(359\) 21.6642 1.14339 0.571695 0.820466i \(-0.306286\pi\)
0.571695 + 0.820466i \(0.306286\pi\)
\(360\) 0 0
\(361\) −18.5699 −0.977361
\(362\) 0 0
\(363\) −49.9244 −2.62035
\(364\) 0 0
\(365\) 4.49971 0.235526
\(366\) 0 0
\(367\) 24.1805 1.26221 0.631106 0.775697i \(-0.282601\pi\)
0.631106 + 0.775697i \(0.282601\pi\)
\(368\) 0 0
\(369\) −5.57147 −0.290039
\(370\) 0 0
\(371\) 1.12311 0.0583087
\(372\) 0 0
\(373\) 13.1324 0.679972 0.339986 0.940431i \(-0.389578\pi\)
0.339986 + 0.940431i \(0.389578\pi\)
\(374\) 0 0
\(375\) 1.90570 0.0984101
\(376\) 0 0
\(377\) −8.00467 −0.412261
\(378\) 0 0
\(379\) 1.44962 0.0744618 0.0372309 0.999307i \(-0.488146\pi\)
0.0372309 + 0.999307i \(0.488146\pi\)
\(380\) 0 0
\(381\) −12.8108 −0.656318
\(382\) 0 0
\(383\) 31.3105 1.59989 0.799947 0.600071i \(-0.204861\pi\)
0.799947 + 0.600071i \(0.204861\pi\)
\(384\) 0 0
\(385\) 6.09896 0.310832
\(386\) 0 0
\(387\) 1.44506 0.0734566
\(388\) 0 0
\(389\) 16.6193 0.842631 0.421316 0.906914i \(-0.361568\pi\)
0.421316 + 0.906914i \(0.361568\pi\)
\(390\) 0 0
\(391\) −30.5090 −1.54291
\(392\) 0 0
\(393\) −17.3859 −0.877004
\(394\) 0 0
\(395\) −2.41658 −0.121591
\(396\) 0 0
\(397\) 34.1801 1.71545 0.857726 0.514107i \(-0.171877\pi\)
0.857726 + 0.514107i \(0.171877\pi\)
\(398\) 0 0
\(399\) 1.24985 0.0625710
\(400\) 0 0
\(401\) 22.6587 1.13152 0.565760 0.824570i \(-0.308583\pi\)
0.565760 + 0.824570i \(0.308583\pi\)
\(402\) 0 0
\(403\) −1.21740 −0.0606431
\(404\) 0 0
\(405\) 10.4961 0.521554
\(406\) 0 0
\(407\) −10.7247 −0.531606
\(408\) 0 0
\(409\) 17.3495 0.857877 0.428938 0.903334i \(-0.358888\pi\)
0.428938 + 0.903334i \(0.358888\pi\)
\(410\) 0 0
\(411\) 0.368294 0.0181666
\(412\) 0 0
\(413\) −1.51919 −0.0747543
\(414\) 0 0
\(415\) −3.74650 −0.183909
\(416\) 0 0
\(417\) 4.02712 0.197209
\(418\) 0 0
\(419\) 14.0166 0.684757 0.342378 0.939562i \(-0.388768\pi\)
0.342378 + 0.939562i \(0.388768\pi\)
\(420\) 0 0
\(421\) 21.2383 1.03509 0.517547 0.855655i \(-0.326845\pi\)
0.517547 + 0.855655i \(0.326845\pi\)
\(422\) 0 0
\(423\) 3.23630 0.157354
\(424\) 0 0
\(425\) 7.44311 0.361044
\(426\) 0 0
\(427\) 8.24621 0.399062
\(428\) 0 0
\(429\) −11.6228 −0.561155
\(430\) 0 0
\(431\) 28.3062 1.36346 0.681731 0.731603i \(-0.261227\pi\)
0.681731 + 0.731603i \(0.261227\pi\)
\(432\) 0 0
\(433\) −10.6100 −0.509882 −0.254941 0.966957i \(-0.582056\pi\)
−0.254941 + 0.966957i \(0.582056\pi\)
\(434\) 0 0
\(435\) −15.2545 −0.731398
\(436\) 0 0
\(437\) −2.68830 −0.128599
\(438\) 0 0
\(439\) 14.8213 0.707383 0.353692 0.935362i \(-0.384926\pi\)
0.353692 + 0.935362i \(0.384926\pi\)
\(440\) 0 0
\(441\) 0.631706 0.0300812
\(442\) 0 0
\(443\) 25.0911 1.19211 0.596057 0.802942i \(-0.296733\pi\)
0.596057 + 0.802942i \(0.296733\pi\)
\(444\) 0 0
\(445\) −16.1179 −0.764060
\(446\) 0 0
\(447\) −2.14031 −0.101233
\(448\) 0 0
\(449\) −9.51691 −0.449131 −0.224565 0.974459i \(-0.572096\pi\)
−0.224565 + 0.974459i \(0.572096\pi\)
\(450\) 0 0
\(451\) −53.7911 −2.53292
\(452\) 0 0
\(453\) −31.3648 −1.47365
\(454\) 0 0
\(455\) 1.00000 0.0468807
\(456\) 0 0
\(457\) −33.5885 −1.57120 −0.785602 0.618732i \(-0.787647\pi\)
−0.785602 + 0.618732i \(0.787647\pi\)
\(458\) 0 0
\(459\) 33.5928 1.56798
\(460\) 0 0
\(461\) −0.768369 −0.0357865 −0.0178933 0.999840i \(-0.505696\pi\)
−0.0178933 + 0.999840i \(0.505696\pi\)
\(462\) 0 0
\(463\) −11.3194 −0.526058 −0.263029 0.964788i \(-0.584722\pi\)
−0.263029 + 0.964788i \(0.584722\pi\)
\(464\) 0 0
\(465\) −2.32001 −0.107588
\(466\) 0 0
\(467\) 4.27866 0.197993 0.0989965 0.995088i \(-0.468437\pi\)
0.0989965 + 0.995088i \(0.468437\pi\)
\(468\) 0 0
\(469\) −15.2545 −0.704388
\(470\) 0 0
\(471\) 32.4047 1.49313
\(472\) 0 0
\(473\) 13.9517 0.641500
\(474\) 0 0
\(475\) 0.655849 0.0300924
\(476\) 0 0
\(477\) −0.709473 −0.0324845
\(478\) 0 0
\(479\) −24.3665 −1.11333 −0.556666 0.830736i \(-0.687920\pi\)
−0.556666 + 0.830736i \(0.687920\pi\)
\(480\) 0 0
\(481\) −1.75845 −0.0801786
\(482\) 0 0
\(483\) −7.81141 −0.355431
\(484\) 0 0
\(485\) −13.1397 −0.596644
\(486\) 0 0
\(487\) −3.11844 −0.141310 −0.0706550 0.997501i \(-0.522509\pi\)
−0.0706550 + 0.997501i \(0.522509\pi\)
\(488\) 0 0
\(489\) −18.4530 −0.834475
\(490\) 0 0
\(491\) −13.8690 −0.625900 −0.312950 0.949770i \(-0.601317\pi\)
−0.312950 + 0.949770i \(0.601317\pi\)
\(492\) 0 0
\(493\) −59.5796 −2.68333
\(494\) 0 0
\(495\) −3.85275 −0.173168
\(496\) 0 0
\(497\) 15.0818 0.676509
\(498\) 0 0
\(499\) −24.7695 −1.10883 −0.554417 0.832239i \(-0.687059\pi\)
−0.554417 + 0.832239i \(0.687059\pi\)
\(500\) 0 0
\(501\) −36.8518 −1.64642
\(502\) 0 0
\(503\) −11.6735 −0.520495 −0.260248 0.965542i \(-0.583804\pi\)
−0.260248 + 0.965542i \(0.583804\pi\)
\(504\) 0 0
\(505\) −9.17139 −0.408122
\(506\) 0 0
\(507\) −1.90570 −0.0846353
\(508\) 0 0
\(509\) 18.0560 0.800319 0.400159 0.916446i \(-0.368955\pi\)
0.400159 + 0.916446i \(0.368955\pi\)
\(510\) 0 0
\(511\) 4.49971 0.199055
\(512\) 0 0
\(513\) 2.96002 0.130688
\(514\) 0 0
\(515\) −11.8439 −0.521903
\(516\) 0 0
\(517\) 31.2456 1.37418
\(518\) 0 0
\(519\) 29.0448 1.27492
\(520\) 0 0
\(521\) 33.8763 1.48415 0.742074 0.670318i \(-0.233842\pi\)
0.742074 + 0.670318i \(0.233842\pi\)
\(522\) 0 0
\(523\) −0.550384 −0.0240666 −0.0120333 0.999928i \(-0.503830\pi\)
−0.0120333 + 0.999928i \(0.503830\pi\)
\(524\) 0 0
\(525\) 1.90570 0.0831717
\(526\) 0 0
\(527\) −9.06126 −0.394715
\(528\) 0 0
\(529\) −6.19851 −0.269500
\(530\) 0 0
\(531\) 0.959679 0.0416465
\(532\) 0 0
\(533\) −8.81972 −0.382024
\(534\) 0 0
\(535\) 1.36237 0.0589006
\(536\) 0 0
\(537\) 12.9825 0.560235
\(538\) 0 0
\(539\) 6.09896 0.262701
\(540\) 0 0
\(541\) 19.1584 0.823683 0.411842 0.911255i \(-0.364886\pi\)
0.411842 + 0.911255i \(0.364886\pi\)
\(542\) 0 0
\(543\) 0.451422 0.0193724
\(544\) 0 0
\(545\) 11.8763 0.508725
\(546\) 0 0
\(547\) −14.3525 −0.613667 −0.306833 0.951763i \(-0.599269\pi\)
−0.306833 + 0.951763i \(0.599269\pi\)
\(548\) 0 0
\(549\) −5.20918 −0.222322
\(550\) 0 0
\(551\) −5.24985 −0.223651
\(552\) 0 0
\(553\) −2.41658 −0.102763
\(554\) 0 0
\(555\) −3.35109 −0.142246
\(556\) 0 0
\(557\) 25.2261 1.06886 0.534431 0.845212i \(-0.320526\pi\)
0.534431 + 0.845212i \(0.320526\pi\)
\(558\) 0 0
\(559\) 2.28756 0.0967533
\(560\) 0 0
\(561\) −86.5099 −3.65245
\(562\) 0 0
\(563\) 39.1697 1.65080 0.825402 0.564545i \(-0.190948\pi\)
0.825402 + 0.564545i \(0.190948\pi\)
\(564\) 0 0
\(565\) −14.3865 −0.605245
\(566\) 0 0
\(567\) 10.4961 0.440793
\(568\) 0 0
\(569\) −1.56986 −0.0658120 −0.0329060 0.999458i \(-0.510476\pi\)
−0.0329060 + 0.999458i \(0.510476\pi\)
\(570\) 0 0
\(571\) −3.21501 −0.134544 −0.0672720 0.997735i \(-0.521430\pi\)
−0.0672720 + 0.997735i \(0.521430\pi\)
\(572\) 0 0
\(573\) 13.2062 0.551698
\(574\) 0 0
\(575\) −4.09896 −0.170939
\(576\) 0 0
\(577\) 9.84308 0.409773 0.204886 0.978786i \(-0.434318\pi\)
0.204886 + 0.978786i \(0.434318\pi\)
\(578\) 0 0
\(579\) 31.4783 1.30819
\(580\) 0 0
\(581\) −3.74650 −0.155431
\(582\) 0 0
\(583\) −6.84978 −0.283689
\(584\) 0 0
\(585\) −0.631706 −0.0261178
\(586\) 0 0
\(587\) −43.4119 −1.79180 −0.895900 0.444256i \(-0.853468\pi\)
−0.895900 + 0.444256i \(0.853468\pi\)
\(588\) 0 0
\(589\) −0.798432 −0.0328988
\(590\) 0 0
\(591\) 43.1402 1.77455
\(592\) 0 0
\(593\) 9.97288 0.409537 0.204769 0.978810i \(-0.434356\pi\)
0.204769 + 0.978810i \(0.434356\pi\)
\(594\) 0 0
\(595\) 7.44311 0.305138
\(596\) 0 0
\(597\) −32.9393 −1.34812
\(598\) 0 0
\(599\) 25.0960 1.02539 0.512697 0.858570i \(-0.328646\pi\)
0.512697 + 0.858570i \(0.328646\pi\)
\(600\) 0 0
\(601\) −31.3859 −1.28026 −0.640129 0.768267i \(-0.721119\pi\)
−0.640129 + 0.768267i \(0.721119\pi\)
\(602\) 0 0
\(603\) 9.63637 0.392423
\(604\) 0 0
\(605\) −26.1973 −1.06507
\(606\) 0 0
\(607\) −29.7192 −1.20626 −0.603132 0.797642i \(-0.706081\pi\)
−0.603132 + 0.797642i \(0.706081\pi\)
\(608\) 0 0
\(609\) −15.2545 −0.618144
\(610\) 0 0
\(611\) 5.12311 0.207259
\(612\) 0 0
\(613\) −39.7725 −1.60639 −0.803197 0.595713i \(-0.796869\pi\)
−0.803197 + 0.595713i \(0.796869\pi\)
\(614\) 0 0
\(615\) −16.8078 −0.677754
\(616\) 0 0
\(617\) 47.2566 1.90248 0.951239 0.308455i \(-0.0998120\pi\)
0.951239 + 0.308455i \(0.0998120\pi\)
\(618\) 0 0
\(619\) −43.7253 −1.75747 −0.878734 0.477312i \(-0.841611\pi\)
−0.878734 + 0.477312i \(0.841611\pi\)
\(620\) 0 0
\(621\) −18.4997 −0.742368
\(622\) 0 0
\(623\) −16.1179 −0.645748
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −7.62281 −0.304426
\(628\) 0 0
\(629\) −13.0884 −0.521868
\(630\) 0 0
\(631\) 15.7457 0.626826 0.313413 0.949617i \(-0.398528\pi\)
0.313413 + 0.949617i \(0.398528\pi\)
\(632\) 0 0
\(633\) 34.3174 1.36399
\(634\) 0 0
\(635\) −6.72236 −0.266769
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −9.52724 −0.376892
\(640\) 0 0
\(641\) 42.6467 1.68444 0.842222 0.539131i \(-0.181247\pi\)
0.842222 + 0.539131i \(0.181247\pi\)
\(642\) 0 0
\(643\) 21.7996 0.859691 0.429846 0.902902i \(-0.358568\pi\)
0.429846 + 0.902902i \(0.358568\pi\)
\(644\) 0 0
\(645\) 4.35940 0.171651
\(646\) 0 0
\(647\) −6.02823 −0.236994 −0.118497 0.992954i \(-0.537808\pi\)
−0.118497 + 0.992954i \(0.537808\pi\)
\(648\) 0 0
\(649\) 9.26546 0.363701
\(650\) 0 0
\(651\) −2.32001 −0.0909283
\(652\) 0 0
\(653\) 23.1397 0.905527 0.452764 0.891631i \(-0.350438\pi\)
0.452764 + 0.891631i \(0.350438\pi\)
\(654\) 0 0
\(655\) −9.12311 −0.356469
\(656\) 0 0
\(657\) −2.84249 −0.110896
\(658\) 0 0
\(659\) 4.64891 0.181096 0.0905478 0.995892i \(-0.471138\pi\)
0.0905478 + 0.995892i \(0.471138\pi\)
\(660\) 0 0
\(661\) 35.4148 1.37748 0.688739 0.725010i \(-0.258165\pi\)
0.688739 + 0.725010i \(0.258165\pi\)
\(662\) 0 0
\(663\) −14.1844 −0.550875
\(664\) 0 0
\(665\) 0.655849 0.0254327
\(666\) 0 0
\(667\) 32.8108 1.27044
\(668\) 0 0
\(669\) −27.0887 −1.04731
\(670\) 0 0
\(671\) −50.2933 −1.94155
\(672\) 0 0
\(673\) −32.3886 −1.24849 −0.624244 0.781230i \(-0.714593\pi\)
−0.624244 + 0.781230i \(0.714593\pi\)
\(674\) 0 0
\(675\) 4.51327 0.173716
\(676\) 0 0
\(677\) −5.21239 −0.200329 −0.100164 0.994971i \(-0.531937\pi\)
−0.100164 + 0.994971i \(0.531937\pi\)
\(678\) 0 0
\(679\) −13.1397 −0.504256
\(680\) 0 0
\(681\) −9.94716 −0.381176
\(682\) 0 0
\(683\) −6.92358 −0.264923 −0.132462 0.991188i \(-0.542288\pi\)
−0.132462 + 0.991188i \(0.542288\pi\)
\(684\) 0 0
\(685\) 0.193259 0.00738404
\(686\) 0 0
\(687\) −10.3907 −0.396432
\(688\) 0 0
\(689\) −1.12311 −0.0427869
\(690\) 0 0
\(691\) −44.3592 −1.68750 −0.843751 0.536734i \(-0.819658\pi\)
−0.843751 + 0.536734i \(0.819658\pi\)
\(692\) 0 0
\(693\) −3.85275 −0.146354
\(694\) 0 0
\(695\) 2.11319 0.0801579
\(696\) 0 0
\(697\) −65.6461 −2.48652
\(698\) 0 0
\(699\) 4.29450 0.162433
\(700\) 0 0
\(701\) −38.6960 −1.46153 −0.730764 0.682630i \(-0.760836\pi\)
−0.730764 + 0.682630i \(0.760836\pi\)
\(702\) 0 0
\(703\) −1.15328 −0.0434968
\(704\) 0 0
\(705\) 9.76312 0.367700
\(706\) 0 0
\(707\) −9.17139 −0.344926
\(708\) 0 0
\(709\) 0.299276 0.0112396 0.00561978 0.999984i \(-0.498211\pi\)
0.00561978 + 0.999984i \(0.498211\pi\)
\(710\) 0 0
\(711\) 1.52657 0.0572508
\(712\) 0 0
\(713\) 4.99009 0.186880
\(714\) 0 0
\(715\) −6.09896 −0.228088
\(716\) 0 0
\(717\) −0.907397 −0.0338874
\(718\) 0 0
\(719\) −7.26341 −0.270880 −0.135440 0.990786i \(-0.543245\pi\)
−0.135440 + 0.990786i \(0.543245\pi\)
\(720\) 0 0
\(721\) −11.8439 −0.441088
\(722\) 0 0
\(723\) 45.3605 1.68698
\(724\) 0 0
\(725\) −8.00467 −0.297286
\(726\) 0 0
\(727\) −22.6954 −0.841724 −0.420862 0.907125i \(-0.638272\pi\)
−0.420862 + 0.907125i \(0.638272\pi\)
\(728\) 0 0
\(729\) 19.1724 0.710089
\(730\) 0 0
\(731\) 17.0265 0.629749
\(732\) 0 0
\(733\) −30.7188 −1.13462 −0.567312 0.823503i \(-0.692017\pi\)
−0.567312 + 0.823503i \(0.692017\pi\)
\(734\) 0 0
\(735\) 1.90570 0.0702929
\(736\) 0 0
\(737\) 93.0367 3.42705
\(738\) 0 0
\(739\) −30.9084 −1.13699 −0.568493 0.822688i \(-0.692473\pi\)
−0.568493 + 0.822688i \(0.692473\pi\)
\(740\) 0 0
\(741\) −1.24985 −0.0459145
\(742\) 0 0
\(743\) −40.5168 −1.48642 −0.743208 0.669060i \(-0.766697\pi\)
−0.743208 + 0.669060i \(0.766697\pi\)
\(744\) 0 0
\(745\) −1.12311 −0.0411474
\(746\) 0 0
\(747\) 2.36669 0.0865926
\(748\) 0 0
\(749\) 1.36237 0.0497801
\(750\) 0 0
\(751\) 24.2238 0.883938 0.441969 0.897030i \(-0.354280\pi\)
0.441969 + 0.897030i \(0.354280\pi\)
\(752\) 0 0
\(753\) −34.6598 −1.26307
\(754\) 0 0
\(755\) −16.4584 −0.598981
\(756\) 0 0
\(757\) 20.5528 0.747003 0.373502 0.927630i \(-0.378157\pi\)
0.373502 + 0.927630i \(0.378157\pi\)
\(758\) 0 0
\(759\) 47.6415 1.72928
\(760\) 0 0
\(761\) −38.2768 −1.38753 −0.693767 0.720200i \(-0.744050\pi\)
−0.693767 + 0.720200i \(0.744050\pi\)
\(762\) 0 0
\(763\) 11.8763 0.429951
\(764\) 0 0
\(765\) −4.70186 −0.169996
\(766\) 0 0
\(767\) 1.51919 0.0548546
\(768\) 0 0
\(769\) 7.49913 0.270425 0.135213 0.990817i \(-0.456828\pi\)
0.135213 + 0.990817i \(0.456828\pi\)
\(770\) 0 0
\(771\) −56.4384 −2.03258
\(772\) 0 0
\(773\) 6.57511 0.236490 0.118245 0.992984i \(-0.462273\pi\)
0.118245 + 0.992984i \(0.462273\pi\)
\(774\) 0 0
\(775\) −1.21740 −0.0437304
\(776\) 0 0
\(777\) −3.35109 −0.120220
\(778\) 0 0
\(779\) −5.78440 −0.207248
\(780\) 0 0
\(781\) −91.9831 −3.29141
\(782\) 0 0
\(783\) −36.1272 −1.29108
\(784\) 0 0
\(785\) 17.0041 0.606902
\(786\) 0 0
\(787\) 18.1635 0.647460 0.323730 0.946150i \(-0.395063\pi\)
0.323730 + 0.946150i \(0.395063\pi\)
\(788\) 0 0
\(789\) 23.2502 0.827728
\(790\) 0 0
\(791\) −14.3865 −0.511526
\(792\) 0 0
\(793\) −8.24621 −0.292832
\(794\) 0 0
\(795\) −2.14031 −0.0759088
\(796\) 0 0
\(797\) 41.2066 1.45961 0.729806 0.683655i \(-0.239611\pi\)
0.729806 + 0.683655i \(0.239611\pi\)
\(798\) 0 0
\(799\) 38.1319 1.34901
\(800\) 0 0
\(801\) 10.1817 0.359754
\(802\) 0 0
\(803\) −27.4436 −0.968462
\(804\) 0 0
\(805\) −4.09896 −0.144469
\(806\) 0 0
\(807\) −5.57453 −0.196233
\(808\) 0 0
\(809\) 35.8779 1.26140 0.630700 0.776027i \(-0.282768\pi\)
0.630700 + 0.776027i \(0.282768\pi\)
\(810\) 0 0
\(811\) −50.4875 −1.77286 −0.886428 0.462866i \(-0.846821\pi\)
−0.886428 + 0.462866i \(0.846821\pi\)
\(812\) 0 0
\(813\) −37.8690 −1.32812
\(814\) 0 0
\(815\) −9.68305 −0.339183
\(816\) 0 0
\(817\) 1.50029 0.0524886
\(818\) 0 0
\(819\) −0.631706 −0.0220736
\(820\) 0 0
\(821\) −18.6327 −0.650287 −0.325143 0.945665i \(-0.605413\pi\)
−0.325143 + 0.945665i \(0.605413\pi\)
\(822\) 0 0
\(823\) 51.0416 1.77920 0.889600 0.456741i \(-0.150983\pi\)
0.889600 + 0.456741i \(0.150983\pi\)
\(824\) 0 0
\(825\) −11.6228 −0.404654
\(826\) 0 0
\(827\) −31.7923 −1.10553 −0.552763 0.833339i \(-0.686427\pi\)
−0.552763 + 0.833339i \(0.686427\pi\)
\(828\) 0 0
\(829\) −45.1074 −1.56664 −0.783322 0.621616i \(-0.786476\pi\)
−0.783322 + 0.621616i \(0.786476\pi\)
\(830\) 0 0
\(831\) −39.2999 −1.36330
\(832\) 0 0
\(833\) 7.44311 0.257889
\(834\) 0 0
\(835\) −19.3376 −0.669207
\(836\) 0 0
\(837\) −5.49446 −0.189916
\(838\) 0 0
\(839\) 18.4267 0.636160 0.318080 0.948064i \(-0.396962\pi\)
0.318080 + 0.948064i \(0.396962\pi\)
\(840\) 0 0
\(841\) 35.0747 1.20947
\(842\) 0 0
\(843\) 12.4065 0.427303
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −26.1973 −0.900151
\(848\) 0 0
\(849\) 26.0923 0.895487
\(850\) 0 0
\(851\) 7.20784 0.247082
\(852\) 0 0
\(853\) −3.52875 −0.120822 −0.0604110 0.998174i \(-0.519241\pi\)
−0.0604110 + 0.998174i \(0.519241\pi\)
\(854\) 0 0
\(855\) −0.414304 −0.0141689
\(856\) 0 0
\(857\) 17.4945 0.597599 0.298800 0.954316i \(-0.403414\pi\)
0.298800 + 0.954316i \(0.403414\pi\)
\(858\) 0 0
\(859\) −49.1040 −1.67541 −0.837703 0.546126i \(-0.816102\pi\)
−0.837703 + 0.546126i \(0.816102\pi\)
\(860\) 0 0
\(861\) −16.8078 −0.572807
\(862\) 0 0
\(863\) 51.6610 1.75856 0.879281 0.476303i \(-0.158023\pi\)
0.879281 + 0.476303i \(0.158023\pi\)
\(864\) 0 0
\(865\) 15.2410 0.518208
\(866\) 0 0
\(867\) −73.1789 −2.48529
\(868\) 0 0
\(869\) 14.7386 0.499974
\(870\) 0 0
\(871\) 15.2545 0.516880
\(872\) 0 0
\(873\) 8.30044 0.280927
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 4.79377 0.161874 0.0809370 0.996719i \(-0.474209\pi\)
0.0809370 + 0.996719i \(0.474209\pi\)
\(878\) 0 0
\(879\) −59.5223 −2.00764
\(880\) 0 0
\(881\) 4.55073 0.153318 0.0766590 0.997057i \(-0.475575\pi\)
0.0766590 + 0.997057i \(0.475575\pi\)
\(882\) 0 0
\(883\) −7.19495 −0.242129 −0.121065 0.992645i \(-0.538631\pi\)
−0.121065 + 0.992645i \(0.538631\pi\)
\(884\) 0 0
\(885\) 2.89512 0.0973183
\(886\) 0 0
\(887\) 13.6627 0.458749 0.229374 0.973338i \(-0.426332\pi\)
0.229374 + 0.973338i \(0.426332\pi\)
\(888\) 0 0
\(889\) −6.72236 −0.225461
\(890\) 0 0
\(891\) −64.0151 −2.14459
\(892\) 0 0
\(893\) 3.35999 0.112438
\(894\) 0 0
\(895\) 6.81243 0.227714
\(896\) 0 0
\(897\) 7.81141 0.260815
\(898\) 0 0
\(899\) 9.74490 0.325011
\(900\) 0 0
\(901\) −8.35940 −0.278492
\(902\) 0 0
\(903\) 4.35940 0.145072
\(904\) 0 0
\(905\) 0.236879 0.00787414
\(906\) 0 0
\(907\) −31.6193 −1.04990 −0.524950 0.851133i \(-0.675916\pi\)
−0.524950 + 0.851133i \(0.675916\pi\)
\(908\) 0 0
\(909\) 5.79362 0.192162
\(910\) 0 0
\(911\) 16.2255 0.537574 0.268787 0.963200i \(-0.413377\pi\)
0.268787 + 0.963200i \(0.413377\pi\)
\(912\) 0 0
\(913\) 22.8498 0.756217
\(914\) 0 0
\(915\) −15.7148 −0.519516
\(916\) 0 0
\(917\) −9.12311 −0.301271
\(918\) 0 0
\(919\) 26.1661 0.863141 0.431571 0.902079i \(-0.357960\pi\)
0.431571 + 0.902079i \(0.357960\pi\)
\(920\) 0 0
\(921\) −16.2899 −0.536772
\(922\) 0 0
\(923\) −15.0818 −0.496422
\(924\) 0 0
\(925\) −1.75845 −0.0578176
\(926\) 0 0
\(927\) 7.48184 0.245736
\(928\) 0 0
\(929\) −40.6342 −1.33316 −0.666582 0.745432i \(-0.732243\pi\)
−0.666582 + 0.745432i \(0.732243\pi\)
\(930\) 0 0
\(931\) 0.655849 0.0214946
\(932\) 0 0
\(933\) −30.2100 −0.989031
\(934\) 0 0
\(935\) −45.3953 −1.48458
\(936\) 0 0
\(937\) 23.9169 0.781330 0.390665 0.920533i \(-0.372245\pi\)
0.390665 + 0.920533i \(0.372245\pi\)
\(938\) 0 0
\(939\) 48.1712 1.57201
\(940\) 0 0
\(941\) −4.51315 −0.147125 −0.0735623 0.997291i \(-0.523437\pi\)
−0.0735623 + 0.997291i \(0.523437\pi\)
\(942\) 0 0
\(943\) 36.1517 1.17726
\(944\) 0 0
\(945\) 4.51327 0.146817
\(946\) 0 0
\(947\) 10.8999 0.354199 0.177100 0.984193i \(-0.443329\pi\)
0.177100 + 0.984193i \(0.443329\pi\)
\(948\) 0 0
\(949\) −4.49971 −0.146067
\(950\) 0 0
\(951\) 22.9062 0.742785
\(952\) 0 0
\(953\) 17.7797 0.575942 0.287971 0.957639i \(-0.407019\pi\)
0.287971 + 0.957639i \(0.407019\pi\)
\(954\) 0 0
\(955\) 6.92985 0.224245
\(956\) 0 0
\(957\) 93.0367 3.00745
\(958\) 0 0
\(959\) 0.193259 0.00624065
\(960\) 0 0
\(961\) −29.5179 −0.952191
\(962\) 0 0
\(963\) −0.860620 −0.0277331
\(964\) 0 0
\(965\) 16.5179 0.531731
\(966\) 0 0
\(967\) −12.1085 −0.389384 −0.194692 0.980864i \(-0.562371\pi\)
−0.194692 + 0.980864i \(0.562371\pi\)
\(968\) 0 0
\(969\) −9.30281 −0.298849
\(970\) 0 0
\(971\) −31.8540 −1.02224 −0.511121 0.859509i \(-0.670770\pi\)
−0.511121 + 0.859509i \(0.670770\pi\)
\(972\) 0 0
\(973\) 2.11319 0.0677458
\(974\) 0 0
\(975\) −1.90570 −0.0610314
\(976\) 0 0
\(977\) 21.3372 0.682638 0.341319 0.939948i \(-0.389126\pi\)
0.341319 + 0.939948i \(0.389126\pi\)
\(978\) 0 0
\(979\) 98.3022 3.14175
\(980\) 0 0
\(981\) −7.50234 −0.239531
\(982\) 0 0
\(983\) 32.8618 1.04813 0.524065 0.851678i \(-0.324415\pi\)
0.524065 + 0.851678i \(0.324415\pi\)
\(984\) 0 0
\(985\) 22.6374 0.721287
\(986\) 0 0
\(987\) 9.76312 0.310764
\(988\) 0 0
\(989\) −9.37660 −0.298159
\(990\) 0 0
\(991\) 1.71644 0.0545245 0.0272623 0.999628i \(-0.491321\pi\)
0.0272623 + 0.999628i \(0.491321\pi\)
\(992\) 0 0
\(993\) −23.1491 −0.734613
\(994\) 0 0
\(995\) −17.2846 −0.547958
\(996\) 0 0
\(997\) 40.6616 1.28777 0.643883 0.765124i \(-0.277322\pi\)
0.643883 + 0.765124i \(0.277322\pi\)
\(998\) 0 0
\(999\) −7.93637 −0.251096
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 3640.2.a.x.1.1 4
4.3 odd 2 7280.2.a.br.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3640.2.a.x.1.1 4 1.1 even 1 trivial
7280.2.a.br.1.4 4 4.3 odd 2