Properties

Label 2475.4.a.n.1.1
Level $2475$
Weight $4$
Character 2475.1
Self dual yes
Analytic conductor $146.030$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(146.029727264\)
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: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.56155 q^{2} -1.43845 q^{4} -6.24621 q^{7} +24.1771 q^{8} +O(q^{10})\) \(q-2.56155 q^{2} -1.43845 q^{4} -6.24621 q^{7} +24.1771 q^{8} +11.0000 q^{11} +49.1231 q^{13} +16.0000 q^{14} -50.4233 q^{16} +82.7083 q^{17} -130.354 q^{19} -28.1771 q^{22} -185.693 q^{23} -125.831 q^{26} +8.98485 q^{28} +8.90720 q^{29} +5.26137 q^{31} -64.2547 q^{32} -211.862 q^{34} +416.894 q^{37} +333.909 q^{38} +298.479 q^{41} +513.633 q^{43} -15.8229 q^{44} +475.663 q^{46} +557.295 q^{47} -303.985 q^{49} -70.6610 q^{52} -168.064 q^{53} -151.015 q^{56} -22.8163 q^{58} -618.773 q^{59} +786.405 q^{61} -13.4773 q^{62} +567.978 q^{64} +339.015 q^{67} -118.972 q^{68} -1120.71 q^{71} +123.430 q^{73} -1067.90 q^{74} +187.508 q^{76} -68.7083 q^{77} -309.835 q^{79} -764.570 q^{82} -1021.22 q^{83} -1315.70 q^{86} +265.948 q^{88} +141.879 q^{89} -306.833 q^{91} +267.110 q^{92} -1427.54 q^{94} -798.345 q^{97} +778.673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 7 q^{4} + 4 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 7 q^{4} + 4 q^{7} + 3 q^{8} + 22 q^{11} + 90 q^{13} + 32 q^{14} - 39 q^{16} - 16 q^{17} - 170 q^{19} - 11 q^{22} - 124 q^{23} - 62 q^{26} - 48 q^{28} + 158 q^{29} + 60 q^{31} + 123 q^{32} - 366 q^{34} + 372 q^{37} + 272 q^{38} - 38 q^{41} + 516 q^{43} - 77 q^{44} + 572 q^{46} + 224 q^{47} - 542 q^{49} - 298 q^{52} + 472 q^{53} - 368 q^{56} + 210 q^{58} - 248 q^{59} + 72 q^{61} + 72 q^{62} + 769 q^{64} + 744 q^{67} + 430 q^{68} - 2060 q^{71} + 486 q^{73} - 1138 q^{74} + 408 q^{76} + 44 q^{77} + 642 q^{79} - 1290 q^{82} - 286 q^{83} - 1312 q^{86} + 33 q^{88} - 244 q^{89} + 112 q^{91} - 76 q^{92} - 1948 q^{94} + 168 q^{97} + 407 q^{98}+O(q^{100}) \) 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 −0.905646 −0.452823 0.891601i \(-0.649583\pi\)
−0.452823 + 0.891601i \(0.649583\pi\)
\(3\) 0 0
\(4\) −1.43845 −0.179806
\(5\) 0 0
\(6\) 0 0
\(7\) −6.24621 −0.337264 −0.168632 0.985679i \(-0.553935\pi\)
−0.168632 + 0.985679i \(0.553935\pi\)
\(8\) 24.1771 1.06849
\(9\) 0 0
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 49.1231 1.04802 0.524011 0.851711i \(-0.324435\pi\)
0.524011 + 0.851711i \(0.324435\pi\)
\(14\) 16.0000 0.305441
\(15\) 0 0
\(16\) −50.4233 −0.787864
\(17\) 82.7083 1.17998 0.589992 0.807409i \(-0.299131\pi\)
0.589992 + 0.807409i \(0.299131\pi\)
\(18\) 0 0
\(19\) −130.354 −1.57396 −0.786981 0.616977i \(-0.788357\pi\)
−0.786981 + 0.616977i \(0.788357\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −28.1771 −0.273062
\(23\) −185.693 −1.68347 −0.841733 0.539895i \(-0.818464\pi\)
−0.841733 + 0.539895i \(0.818464\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −125.831 −0.949137
\(27\) 0 0
\(28\) 8.98485 0.0606420
\(29\) 8.90720 0.0570354 0.0285177 0.999593i \(-0.490921\pi\)
0.0285177 + 0.999593i \(0.490921\pi\)
\(30\) 0 0
\(31\) 5.26137 0.0304829 0.0152414 0.999884i \(-0.495148\pi\)
0.0152414 + 0.999884i \(0.495148\pi\)
\(32\) −64.2547 −0.354961
\(33\) 0 0
\(34\) −211.862 −1.06865
\(35\) 0 0
\(36\) 0 0
\(37\) 416.894 1.85235 0.926175 0.377094i \(-0.123077\pi\)
0.926175 + 0.377094i \(0.123077\pi\)
\(38\) 333.909 1.42545
\(39\) 0 0
\(40\) 0 0
\(41\) 298.479 1.13694 0.568471 0.822703i \(-0.307535\pi\)
0.568471 + 0.822703i \(0.307535\pi\)
\(42\) 0 0
\(43\) 513.633 1.82159 0.910793 0.412863i \(-0.135471\pi\)
0.910793 + 0.412863i \(0.135471\pi\)
\(44\) −15.8229 −0.0542135
\(45\) 0 0
\(46\) 475.663 1.52462
\(47\) 557.295 1.72957 0.864786 0.502140i \(-0.167454\pi\)
0.864786 + 0.502140i \(0.167454\pi\)
\(48\) 0 0
\(49\) −303.985 −0.886253
\(50\) 0 0
\(51\) 0 0
\(52\) −70.6610 −0.188441
\(53\) −168.064 −0.435574 −0.217787 0.975996i \(-0.569884\pi\)
−0.217787 + 0.975996i \(0.569884\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −151.015 −0.360362
\(57\) 0 0
\(58\) −22.8163 −0.0516539
\(59\) −618.773 −1.36538 −0.682689 0.730709i \(-0.739190\pi\)
−0.682689 + 0.730709i \(0.739190\pi\)
\(60\) 0 0
\(61\) 786.405 1.65064 0.825319 0.564667i \(-0.190996\pi\)
0.825319 + 0.564667i \(0.190996\pi\)
\(62\) −13.4773 −0.0276067
\(63\) 0 0
\(64\) 567.978 1.10933
\(65\) 0 0
\(66\) 0 0
\(67\) 339.015 0.618169 0.309084 0.951035i \(-0.399977\pi\)
0.309084 + 0.951035i \(0.399977\pi\)
\(68\) −118.972 −0.212168
\(69\) 0 0
\(70\) 0 0
\(71\) −1120.71 −1.87329 −0.936645 0.350280i \(-0.886087\pi\)
−0.936645 + 0.350280i \(0.886087\pi\)
\(72\) 0 0
\(73\) 123.430 0.197896 0.0989478 0.995093i \(-0.468452\pi\)
0.0989478 + 0.995093i \(0.468452\pi\)
\(74\) −1067.90 −1.67757
\(75\) 0 0
\(76\) 187.508 0.283008
\(77\) −68.7083 −0.101689
\(78\) 0 0
\(79\) −309.835 −0.441255 −0.220628 0.975358i \(-0.570811\pi\)
−0.220628 + 0.975358i \(0.570811\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −764.570 −1.02967
\(83\) −1021.22 −1.35053 −0.675263 0.737577i \(-0.735970\pi\)
−0.675263 + 0.737577i \(0.735970\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1315.70 −1.64971
\(87\) 0 0
\(88\) 265.948 0.322161
\(89\) 141.879 0.168979 0.0844894 0.996424i \(-0.473074\pi\)
0.0844894 + 0.996424i \(0.473074\pi\)
\(90\) 0 0
\(91\) −306.833 −0.353460
\(92\) 267.110 0.302697
\(93\) 0 0
\(94\) −1427.54 −1.56638
\(95\) 0 0
\(96\) 0 0
\(97\) −798.345 −0.835666 −0.417833 0.908524i \(-0.637210\pi\)
−0.417833 + 0.908524i \(0.637210\pi\)
\(98\) 778.673 0.802631
\(99\) 0 0
\(100\) 0 0
\(101\) −241.400 −0.237823 −0.118912 0.992905i \(-0.537941\pi\)
−0.118912 + 0.992905i \(0.537941\pi\)
\(102\) 0 0
\(103\) −1168.38 −1.11771 −0.558853 0.829267i \(-0.688758\pi\)
−0.558853 + 0.829267i \(0.688758\pi\)
\(104\) 1187.65 1.11980
\(105\) 0 0
\(106\) 430.506 0.394476
\(107\) −2106.82 −1.90350 −0.951748 0.306882i \(-0.900714\pi\)
−0.951748 + 0.306882i \(0.900714\pi\)
\(108\) 0 0
\(109\) 493.792 0.433914 0.216957 0.976181i \(-0.430387\pi\)
0.216957 + 0.976181i \(0.430387\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 314.955 0.265718
\(113\) 170.000 0.141524 0.0707622 0.997493i \(-0.477457\pi\)
0.0707622 + 0.997493i \(0.477457\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −12.8125 −0.0102553
\(117\) 0 0
\(118\) 1585.02 1.23655
\(119\) −516.614 −0.397966
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −2014.42 −1.49489
\(123\) 0 0
\(124\) −7.56820 −0.00548100
\(125\) 0 0
\(126\) 0 0
\(127\) 948.182 0.662500 0.331250 0.943543i \(-0.392530\pi\)
0.331250 + 0.943543i \(0.392530\pi\)
\(128\) −940.868 −0.649702
\(129\) 0 0
\(130\) 0 0
\(131\) 1484.84 0.990312 0.495156 0.868804i \(-0.335111\pi\)
0.495156 + 0.868804i \(0.335111\pi\)
\(132\) 0 0
\(133\) 814.220 0.530841
\(134\) −868.405 −0.559842
\(135\) 0 0
\(136\) 1999.65 1.26080
\(137\) 684.928 0.427134 0.213567 0.976928i \(-0.431492\pi\)
0.213567 + 0.976928i \(0.431492\pi\)
\(138\) 0 0
\(139\) 830.483 0.506767 0.253384 0.967366i \(-0.418457\pi\)
0.253384 + 0.967366i \(0.418457\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2870.75 1.69654
\(143\) 540.354 0.315991
\(144\) 0 0
\(145\) 0 0
\(146\) −316.172 −0.179223
\(147\) 0 0
\(148\) −599.680 −0.333063
\(149\) 1213.64 0.667285 0.333642 0.942700i \(-0.391722\pi\)
0.333642 + 0.942700i \(0.391722\pi\)
\(150\) 0 0
\(151\) 30.8466 0.0166242 0.00831212 0.999965i \(-0.497354\pi\)
0.00831212 + 0.999965i \(0.497354\pi\)
\(152\) −3151.58 −1.68176
\(153\) 0 0
\(154\) 176.000 0.0920941
\(155\) 0 0
\(156\) 0 0
\(157\) −345.239 −0.175497 −0.0877485 0.996143i \(-0.527967\pi\)
−0.0877485 + 0.996143i \(0.527967\pi\)
\(158\) 793.659 0.399621
\(159\) 0 0
\(160\) 0 0
\(161\) 1159.88 0.567772
\(162\) 0 0
\(163\) −1921.49 −0.923331 −0.461665 0.887054i \(-0.652748\pi\)
−0.461665 + 0.887054i \(0.652748\pi\)
\(164\) −429.346 −0.204429
\(165\) 0 0
\(166\) 2615.91 1.22310
\(167\) 172.297 0.0798369 0.0399185 0.999203i \(-0.487290\pi\)
0.0399185 + 0.999203i \(0.487290\pi\)
\(168\) 0 0
\(169\) 216.080 0.0983521
\(170\) 0 0
\(171\) 0 0
\(172\) −738.833 −0.327532
\(173\) −1025.29 −0.450587 −0.225293 0.974291i \(-0.572334\pi\)
−0.225293 + 0.974291i \(0.572334\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −554.656 −0.237550
\(177\) 0 0
\(178\) −363.430 −0.153035
\(179\) −1658.29 −0.692437 −0.346219 0.938154i \(-0.612534\pi\)
−0.346219 + 0.938154i \(0.612534\pi\)
\(180\) 0 0
\(181\) −2021.81 −0.830275 −0.415137 0.909759i \(-0.636266\pi\)
−0.415137 + 0.909759i \(0.636266\pi\)
\(182\) 785.970 0.320110
\(183\) 0 0
\(184\) −4489.52 −1.79876
\(185\) 0 0
\(186\) 0 0
\(187\) 909.792 0.355778
\(188\) −801.640 −0.310987
\(189\) 0 0
\(190\) 0 0
\(191\) −1440.22 −0.545607 −0.272803 0.962070i \(-0.587951\pi\)
−0.272803 + 0.962070i \(0.587951\pi\)
\(192\) 0 0
\(193\) 2798.05 1.04356 0.521782 0.853079i \(-0.325267\pi\)
0.521782 + 0.853079i \(0.325267\pi\)
\(194\) 2045.00 0.756817
\(195\) 0 0
\(196\) 437.266 0.159354
\(197\) 458.943 0.165982 0.0829908 0.996550i \(-0.473553\pi\)
0.0829908 + 0.996550i \(0.473553\pi\)
\(198\) 0 0
\(199\) −2371.04 −0.844615 −0.422308 0.906453i \(-0.638780\pi\)
−0.422308 + 0.906453i \(0.638780\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 618.358 0.215384
\(203\) −55.6363 −0.0192360
\(204\) 0 0
\(205\) 0 0
\(206\) 2992.86 1.01225
\(207\) 0 0
\(208\) −2476.95 −0.825699
\(209\) −1433.90 −0.474568
\(210\) 0 0
\(211\) 4319.87 1.40944 0.704721 0.709484i \(-0.251072\pi\)
0.704721 + 0.709484i \(0.251072\pi\)
\(212\) 241.752 0.0783187
\(213\) 0 0
\(214\) 5396.73 1.72389
\(215\) 0 0
\(216\) 0 0
\(217\) −32.8636 −0.0102808
\(218\) −1264.87 −0.392973
\(219\) 0 0
\(220\) 0 0
\(221\) 4062.89 1.23665
\(222\) 0 0
\(223\) 3837.73 1.15244 0.576219 0.817295i \(-0.304527\pi\)
0.576219 + 0.817295i \(0.304527\pi\)
\(224\) 401.349 0.119715
\(225\) 0 0
\(226\) −435.464 −0.128171
\(227\) −5003.71 −1.46303 −0.731515 0.681825i \(-0.761187\pi\)
−0.731515 + 0.681825i \(0.761187\pi\)
\(228\) 0 0
\(229\) −277.375 −0.0800412 −0.0400206 0.999199i \(-0.512742\pi\)
−0.0400206 + 0.999199i \(0.512742\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 215.350 0.0609415
\(233\) 2269.91 0.638225 0.319113 0.947717i \(-0.396615\pi\)
0.319113 + 0.947717i \(0.396615\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 890.072 0.245503
\(237\) 0 0
\(238\) 1323.33 0.360416
\(239\) 1617.11 0.437665 0.218832 0.975762i \(-0.429775\pi\)
0.218832 + 0.975762i \(0.429775\pi\)
\(240\) 0 0
\(241\) 5646.63 1.50926 0.754629 0.656151i \(-0.227817\pi\)
0.754629 + 0.656151i \(0.227817\pi\)
\(242\) −309.948 −0.0823314
\(243\) 0 0
\(244\) −1131.20 −0.296794
\(245\) 0 0
\(246\) 0 0
\(247\) −6403.40 −1.64955
\(248\) 127.204 0.0325705
\(249\) 0 0
\(250\) 0 0
\(251\) 6217.61 1.56355 0.781777 0.623558i \(-0.214313\pi\)
0.781777 + 0.623558i \(0.214313\pi\)
\(252\) 0 0
\(253\) −2042.62 −0.507584
\(254\) −2428.82 −0.599991
\(255\) 0 0
\(256\) −2133.74 −0.520933
\(257\) 7712.75 1.87202 0.936008 0.351980i \(-0.114491\pi\)
0.936008 + 0.351980i \(0.114491\pi\)
\(258\) 0 0
\(259\) −2604.01 −0.624730
\(260\) 0 0
\(261\) 0 0
\(262\) −3803.49 −0.896871
\(263\) 206.347 0.0483798 0.0241899 0.999707i \(-0.492299\pi\)
0.0241899 + 0.999707i \(0.492299\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2085.67 −0.480753
\(267\) 0 0
\(268\) −487.655 −0.111150
\(269\) −1712.47 −0.388146 −0.194073 0.980987i \(-0.562170\pi\)
−0.194073 + 0.980987i \(0.562170\pi\)
\(270\) 0 0
\(271\) −477.081 −0.106940 −0.0534698 0.998569i \(-0.517028\pi\)
−0.0534698 + 0.998569i \(0.517028\pi\)
\(272\) −4170.43 −0.929666
\(273\) 0 0
\(274\) −1754.48 −0.386832
\(275\) 0 0
\(276\) 0 0
\(277\) −4283.48 −0.929130 −0.464565 0.885539i \(-0.653789\pi\)
−0.464565 + 0.885539i \(0.653789\pi\)
\(278\) −2127.33 −0.458952
\(279\) 0 0
\(280\) 0 0
\(281\) −3477.79 −0.738319 −0.369160 0.929366i \(-0.620354\pi\)
−0.369160 + 0.929366i \(0.620354\pi\)
\(282\) 0 0
\(283\) 6568.27 1.37966 0.689829 0.723973i \(-0.257686\pi\)
0.689829 + 0.723973i \(0.257686\pi\)
\(284\) 1612.08 0.336829
\(285\) 0 0
\(286\) −1384.15 −0.286176
\(287\) −1864.36 −0.383449
\(288\) 0 0
\(289\) 1927.67 0.392360
\(290\) 0 0
\(291\) 0 0
\(292\) −177.547 −0.0355828
\(293\) 8352.29 1.66534 0.832672 0.553766i \(-0.186810\pi\)
0.832672 + 0.553766i \(0.186810\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 10079.3 1.97921
\(297\) 0 0
\(298\) −3108.81 −0.604324
\(299\) −9121.83 −1.76431
\(300\) 0 0
\(301\) −3208.26 −0.614355
\(302\) −79.0152 −0.0150557
\(303\) 0 0
\(304\) 6572.89 1.24007
\(305\) 0 0
\(306\) 0 0
\(307\) −5383.89 −1.00090 −0.500448 0.865767i \(-0.666831\pi\)
−0.500448 + 0.865767i \(0.666831\pi\)
\(308\) 98.8333 0.0182843
\(309\) 0 0
\(310\) 0 0
\(311\) 1790.41 0.326447 0.163223 0.986589i \(-0.447811\pi\)
0.163223 + 0.986589i \(0.447811\pi\)
\(312\) 0 0
\(313\) 809.076 0.146108 0.0730538 0.997328i \(-0.476726\pi\)
0.0730538 + 0.997328i \(0.476726\pi\)
\(314\) 884.347 0.158938
\(315\) 0 0
\(316\) 445.682 0.0793403
\(317\) 10744.5 1.90370 0.951849 0.306567i \(-0.0991804\pi\)
0.951849 + 0.306567i \(0.0991804\pi\)
\(318\) 0 0
\(319\) 97.9792 0.0171968
\(320\) 0 0
\(321\) 0 0
\(322\) −2971.09 −0.514200
\(323\) −10781.4 −1.85725
\(324\) 0 0
\(325\) 0 0
\(326\) 4922.00 0.836210
\(327\) 0 0
\(328\) 7216.35 1.21481
\(329\) −3480.98 −0.583322
\(330\) 0 0
\(331\) 3399.12 0.564449 0.282224 0.959348i \(-0.408928\pi\)
0.282224 + 0.959348i \(0.408928\pi\)
\(332\) 1468.97 0.242832
\(333\) 0 0
\(334\) −441.349 −0.0723039
\(335\) 0 0
\(336\) 0 0
\(337\) 11840.0 1.91384 0.956919 0.290356i \(-0.0937736\pi\)
0.956919 + 0.290356i \(0.0937736\pi\)
\(338\) −553.499 −0.0890721
\(339\) 0 0
\(340\) 0 0
\(341\) 57.8750 0.00919093
\(342\) 0 0
\(343\) 4041.20 0.636165
\(344\) 12418.1 1.94634
\(345\) 0 0
\(346\) 2626.34 0.408072
\(347\) −2076.67 −0.321272 −0.160636 0.987014i \(-0.551355\pi\)
−0.160636 + 0.987014i \(0.551355\pi\)
\(348\) 0 0
\(349\) 5837.37 0.895322 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −706.802 −0.107025
\(353\) −2423.64 −0.365431 −0.182715 0.983166i \(-0.558489\pi\)
−0.182715 + 0.983166i \(0.558489\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −204.085 −0.0303834
\(357\) 0 0
\(358\) 4247.79 0.627103
\(359\) 3882.22 0.570740 0.285370 0.958417i \(-0.407884\pi\)
0.285370 + 0.958417i \(0.407884\pi\)
\(360\) 0 0
\(361\) 10133.2 1.47736
\(362\) 5178.96 0.751935
\(363\) 0 0
\(364\) 441.363 0.0635542
\(365\) 0 0
\(366\) 0 0
\(367\) −5666.65 −0.805986 −0.402993 0.915203i \(-0.632030\pi\)
−0.402993 + 0.915203i \(0.632030\pi\)
\(368\) 9363.26 1.32634
\(369\) 0 0
\(370\) 0 0
\(371\) 1049.77 0.146903
\(372\) 0 0
\(373\) −174.771 −0.0242608 −0.0121304 0.999926i \(-0.503861\pi\)
−0.0121304 + 0.999926i \(0.503861\pi\)
\(374\) −2330.48 −0.322209
\(375\) 0 0
\(376\) 13473.8 1.84802
\(377\) 437.550 0.0597744
\(378\) 0 0
\(379\) 252.686 0.0342470 0.0171235 0.999853i \(-0.494549\pi\)
0.0171235 + 0.999853i \(0.494549\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 3689.21 0.494126
\(383\) −11014.5 −1.46950 −0.734748 0.678340i \(-0.762700\pi\)
−0.734748 + 0.678340i \(0.762700\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −7167.35 −0.945099
\(387\) 0 0
\(388\) 1148.38 0.150258
\(389\) −8099.40 −1.05567 −0.527835 0.849347i \(-0.676996\pi\)
−0.527835 + 0.849347i \(0.676996\pi\)
\(390\) 0 0
\(391\) −15358.4 −1.98646
\(392\) −7349.47 −0.946949
\(393\) 0 0
\(394\) −1175.61 −0.150320
\(395\) 0 0
\(396\) 0 0
\(397\) 424.353 0.0536465 0.0268232 0.999640i \(-0.491461\pi\)
0.0268232 + 0.999640i \(0.491461\pi\)
\(398\) 6073.54 0.764922
\(399\) 0 0
\(400\) 0 0
\(401\) 5904.18 0.735263 0.367632 0.929972i \(-0.380169\pi\)
0.367632 + 0.929972i \(0.380169\pi\)
\(402\) 0 0
\(403\) 258.455 0.0319468
\(404\) 347.241 0.0427620
\(405\) 0 0
\(406\) 142.515 0.0174210
\(407\) 4585.83 0.558504
\(408\) 0 0
\(409\) 1370.47 0.165686 0.0828430 0.996563i \(-0.473600\pi\)
0.0828430 + 0.996563i \(0.473600\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1680.65 0.200970
\(413\) 3864.98 0.460493
\(414\) 0 0
\(415\) 0 0
\(416\) −3156.39 −0.372007
\(417\) 0 0
\(418\) 3673.00 0.429790
\(419\) −1268.73 −0.147927 −0.0739635 0.997261i \(-0.523565\pi\)
−0.0739635 + 0.997261i \(0.523565\pi\)
\(420\) 0 0
\(421\) −12241.9 −1.41719 −0.708594 0.705617i \(-0.750670\pi\)
−0.708594 + 0.705617i \(0.750670\pi\)
\(422\) −11065.6 −1.27646
\(423\) 0 0
\(424\) −4063.31 −0.465405
\(425\) 0 0
\(426\) 0 0
\(427\) −4912.05 −0.556700
\(428\) 3030.55 0.342260
\(429\) 0 0
\(430\) 0 0
\(431\) −8050.11 −0.899675 −0.449838 0.893110i \(-0.648518\pi\)
−0.449838 + 0.893110i \(0.648518\pi\)
\(432\) 0 0
\(433\) 16565.7 1.83856 0.919282 0.393600i \(-0.128770\pi\)
0.919282 + 0.393600i \(0.128770\pi\)
\(434\) 84.1819 0.00931073
\(435\) 0 0
\(436\) −710.293 −0.0780203
\(437\) 24205.9 2.64971
\(438\) 0 0
\(439\) 4705.80 0.511607 0.255804 0.966729i \(-0.417660\pi\)
0.255804 + 0.966729i \(0.417660\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −10407.3 −1.11997
\(443\) 15094.0 1.61882 0.809408 0.587246i \(-0.199788\pi\)
0.809408 + 0.587246i \(0.199788\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −9830.56 −1.04370
\(447\) 0 0
\(448\) −3547.71 −0.374138
\(449\) −973.478 −0.102319 −0.0511595 0.998690i \(-0.516292\pi\)
−0.0511595 + 0.998690i \(0.516292\pi\)
\(450\) 0 0
\(451\) 3283.27 0.342801
\(452\) −244.536 −0.0254469
\(453\) 0 0
\(454\) 12817.3 1.32499
\(455\) 0 0
\(456\) 0 0
\(457\) −62.6577 −0.00641358 −0.00320679 0.999995i \(-0.501021\pi\)
−0.00320679 + 0.999995i \(0.501021\pi\)
\(458\) 710.510 0.0724890
\(459\) 0 0
\(460\) 0 0
\(461\) 11866.2 1.19884 0.599419 0.800436i \(-0.295398\pi\)
0.599419 + 0.800436i \(0.295398\pi\)
\(462\) 0 0
\(463\) 13144.8 1.31942 0.659711 0.751519i \(-0.270679\pi\)
0.659711 + 0.751519i \(0.270679\pi\)
\(464\) −449.131 −0.0449361
\(465\) 0 0
\(466\) −5814.48 −0.578006
\(467\) −10176.8 −1.00840 −0.504201 0.863586i \(-0.668213\pi\)
−0.504201 + 0.863586i \(0.668213\pi\)
\(468\) 0 0
\(469\) −2117.56 −0.208486
\(470\) 0 0
\(471\) 0 0
\(472\) −14960.1 −1.45889
\(473\) 5649.96 0.549229
\(474\) 0 0
\(475\) 0 0
\(476\) 743.121 0.0715565
\(477\) 0 0
\(478\) −4142.30 −0.396369
\(479\) −3431.25 −0.327302 −0.163651 0.986518i \(-0.552327\pi\)
−0.163651 + 0.986518i \(0.552327\pi\)
\(480\) 0 0
\(481\) 20479.1 1.94130
\(482\) −14464.1 −1.36685
\(483\) 0 0
\(484\) −174.052 −0.0163460
\(485\) 0 0
\(486\) 0 0
\(487\) 2833.20 0.263624 0.131812 0.991275i \(-0.457921\pi\)
0.131812 + 0.991275i \(0.457921\pi\)
\(488\) 19013.0 1.76368
\(489\) 0 0
\(490\) 0 0
\(491\) 2667.29 0.245159 0.122580 0.992459i \(-0.460883\pi\)
0.122580 + 0.992459i \(0.460883\pi\)
\(492\) 0 0
\(493\) 736.700 0.0673008
\(494\) 16402.7 1.49391
\(495\) 0 0
\(496\) −265.295 −0.0240164
\(497\) 7000.18 0.631793
\(498\) 0 0
\(499\) −11137.0 −0.999120 −0.499560 0.866279i \(-0.666505\pi\)
−0.499560 + 0.866279i \(0.666505\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −15926.7 −1.41603
\(503\) 8780.30 0.778319 0.389159 0.921170i \(-0.372766\pi\)
0.389159 + 0.921170i \(0.372766\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 5232.29 0.459691
\(507\) 0 0
\(508\) −1363.91 −0.119121
\(509\) −13597.4 −1.18408 −0.592039 0.805910i \(-0.701677\pi\)
−0.592039 + 0.805910i \(0.701677\pi\)
\(510\) 0 0
\(511\) −770.969 −0.0667430
\(512\) 12992.6 1.12148
\(513\) 0 0
\(514\) −19756.6 −1.69538
\(515\) 0 0
\(516\) 0 0
\(517\) 6130.25 0.521486
\(518\) 6670.30 0.565784
\(519\) 0 0
\(520\) 0 0
\(521\) −14001.3 −1.17736 −0.588682 0.808364i \(-0.700353\pi\)
−0.588682 + 0.808364i \(0.700353\pi\)
\(522\) 0 0
\(523\) 14749.8 1.23320 0.616602 0.787275i \(-0.288509\pi\)
0.616602 + 0.787275i \(0.288509\pi\)
\(524\) −2135.86 −0.178064
\(525\) 0 0
\(526\) −528.568 −0.0438149
\(527\) 435.159 0.0359693
\(528\) 0 0
\(529\) 22315.0 1.83406
\(530\) 0 0
\(531\) 0 0
\(532\) −1171.21 −0.0954483
\(533\) 14662.2 1.19154
\(534\) 0 0
\(535\) 0 0
\(536\) 8196.40 0.660505
\(537\) 0 0
\(538\) 4386.59 0.351523
\(539\) −3343.83 −0.267215
\(540\) 0 0
\(541\) 1484.06 0.117939 0.0589694 0.998260i \(-0.481219\pi\)
0.0589694 + 0.998260i \(0.481219\pi\)
\(542\) 1222.07 0.0968493
\(543\) 0 0
\(544\) −5314.40 −0.418847
\(545\) 0 0
\(546\) 0 0
\(547\) 16562.2 1.29460 0.647302 0.762234i \(-0.275897\pi\)
0.647302 + 0.762234i \(0.275897\pi\)
\(548\) −985.233 −0.0768012
\(549\) 0 0
\(550\) 0 0
\(551\) −1161.09 −0.0897716
\(552\) 0 0
\(553\) 1935.30 0.148819
\(554\) 10972.3 0.841463
\(555\) 0 0
\(556\) −1194.61 −0.0911197
\(557\) −8821.52 −0.671059 −0.335529 0.942030i \(-0.608915\pi\)
−0.335529 + 0.942030i \(0.608915\pi\)
\(558\) 0 0
\(559\) 25231.2 1.90906
\(560\) 0 0
\(561\) 0 0
\(562\) 8908.55 0.668656
\(563\) −5985.53 −0.448064 −0.224032 0.974582i \(-0.571922\pi\)
−0.224032 + 0.974582i \(0.571922\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −16825.0 −1.24948
\(567\) 0 0
\(568\) −27095.5 −2.00158
\(569\) 3453.08 0.254413 0.127206 0.991876i \(-0.459399\pi\)
0.127206 + 0.991876i \(0.459399\pi\)
\(570\) 0 0
\(571\) −21484.5 −1.57460 −0.787302 0.616568i \(-0.788523\pi\)
−0.787302 + 0.616568i \(0.788523\pi\)
\(572\) −777.271 −0.0568170
\(573\) 0 0
\(574\) 4775.67 0.347269
\(575\) 0 0
\(576\) 0 0
\(577\) 13294.4 0.959189 0.479594 0.877490i \(-0.340784\pi\)
0.479594 + 0.877490i \(0.340784\pi\)
\(578\) −4937.82 −0.355340
\(579\) 0 0
\(580\) 0 0
\(581\) 6378.77 0.455483
\(582\) 0 0
\(583\) −1848.71 −0.131330
\(584\) 2984.18 0.211449
\(585\) 0 0
\(586\) −21394.8 −1.50821
\(587\) 6695.73 0.470805 0.235402 0.971898i \(-0.424359\pi\)
0.235402 + 0.971898i \(0.424359\pi\)
\(588\) 0 0
\(589\) −685.841 −0.0479789
\(590\) 0 0
\(591\) 0 0
\(592\) −21021.2 −1.45940
\(593\) −10239.6 −0.709088 −0.354544 0.935039i \(-0.615364\pi\)
−0.354544 + 0.935039i \(0.615364\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1745.76 −0.119982
\(597\) 0 0
\(598\) 23366.0 1.59784
\(599\) 23890.8 1.62963 0.814817 0.579719i \(-0.196838\pi\)
0.814817 + 0.579719i \(0.196838\pi\)
\(600\) 0 0
\(601\) −11343.8 −0.769920 −0.384960 0.922933i \(-0.625785\pi\)
−0.384960 + 0.922933i \(0.625785\pi\)
\(602\) 8218.12 0.556388
\(603\) 0 0
\(604\) −44.3712 −0.00298914
\(605\) 0 0
\(606\) 0 0
\(607\) 26032.5 1.74074 0.870369 0.492399i \(-0.163880\pi\)
0.870369 + 0.492399i \(0.163880\pi\)
\(608\) 8375.87 0.558695
\(609\) 0 0
\(610\) 0 0
\(611\) 27376.1 1.81263
\(612\) 0 0
\(613\) 4568.13 0.300987 0.150493 0.988611i \(-0.451914\pi\)
0.150493 + 0.988611i \(0.451914\pi\)
\(614\) 13791.1 0.906457
\(615\) 0 0
\(616\) −1661.17 −0.108653
\(617\) −12755.9 −0.832308 −0.416154 0.909294i \(-0.636622\pi\)
−0.416154 + 0.909294i \(0.636622\pi\)
\(618\) 0 0
\(619\) −1138.94 −0.0739545 −0.0369772 0.999316i \(-0.511773\pi\)
−0.0369772 + 0.999316i \(0.511773\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −4586.24 −0.295645
\(623\) −886.205 −0.0569904
\(624\) 0 0
\(625\) 0 0
\(626\) −2072.49 −0.132322
\(627\) 0 0
\(628\) 496.607 0.0315554
\(629\) 34480.6 2.18574
\(630\) 0 0
\(631\) 7997.36 0.504548 0.252274 0.967656i \(-0.418822\pi\)
0.252274 + 0.967656i \(0.418822\pi\)
\(632\) −7490.91 −0.471475
\(633\) 0 0
\(634\) −27522.7 −1.72408
\(635\) 0 0
\(636\) 0 0
\(637\) −14932.7 −0.928814
\(638\) −250.979 −0.0155742
\(639\) 0 0
\(640\) 0 0
\(641\) 573.115 0.0353146 0.0176573 0.999844i \(-0.494379\pi\)
0.0176573 + 0.999844i \(0.494379\pi\)
\(642\) 0 0
\(643\) 16027.8 0.983009 0.491504 0.870875i \(-0.336447\pi\)
0.491504 + 0.870875i \(0.336447\pi\)
\(644\) −1668.42 −0.102089
\(645\) 0 0
\(646\) 27617.1 1.68201
\(647\) −2622.74 −0.159367 −0.0796837 0.996820i \(-0.525391\pi\)
−0.0796837 + 0.996820i \(0.525391\pi\)
\(648\) 0 0
\(649\) −6806.50 −0.411677
\(650\) 0 0
\(651\) 0 0
\(652\) 2763.97 0.166020
\(653\) 3102.00 0.185897 0.0929484 0.995671i \(-0.470371\pi\)
0.0929484 + 0.995671i \(0.470371\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −15050.3 −0.895755
\(657\) 0 0
\(658\) 8916.73 0.528283
\(659\) 20840.2 1.23190 0.615948 0.787787i \(-0.288773\pi\)
0.615948 + 0.787787i \(0.288773\pi\)
\(660\) 0 0
\(661\) 18242.9 1.07348 0.536738 0.843749i \(-0.319657\pi\)
0.536738 + 0.843749i \(0.319657\pi\)
\(662\) −8707.03 −0.511191
\(663\) 0 0
\(664\) −24690.2 −1.44302
\(665\) 0 0
\(666\) 0 0
\(667\) −1654.01 −0.0960171
\(668\) −247.841 −0.0143551
\(669\) 0 0
\(670\) 0 0
\(671\) 8650.46 0.497686
\(672\) 0 0
\(673\) 12746.6 0.730084 0.365042 0.930991i \(-0.381055\pi\)
0.365042 + 0.930991i \(0.381055\pi\)
\(674\) −30328.7 −1.73326
\(675\) 0 0
\(676\) −310.819 −0.0176843
\(677\) −7683.11 −0.436168 −0.218084 0.975930i \(-0.569981\pi\)
−0.218084 + 0.975930i \(0.569981\pi\)
\(678\) 0 0
\(679\) 4986.63 0.281840
\(680\) 0 0
\(681\) 0 0
\(682\) −148.250 −0.00832373
\(683\) −21397.1 −1.19874 −0.599368 0.800473i \(-0.704582\pi\)
−0.599368 + 0.800473i \(0.704582\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −10351.8 −0.576140
\(687\) 0 0
\(688\) −25899.0 −1.43516
\(689\) −8255.84 −0.456491
\(690\) 0 0
\(691\) 26137.5 1.43895 0.719477 0.694516i \(-0.244381\pi\)
0.719477 + 0.694516i \(0.244381\pi\)
\(692\) 1474.83 0.0810181
\(693\) 0 0
\(694\) 5319.50 0.290959
\(695\) 0 0
\(696\) 0 0
\(697\) 24686.7 1.34157
\(698\) −14952.7 −0.810845
\(699\) 0 0
\(700\) 0 0
\(701\) −13382.4 −0.721036 −0.360518 0.932752i \(-0.617400\pi\)
−0.360518 + 0.932752i \(0.617400\pi\)
\(702\) 0 0
\(703\) −54343.9 −2.91553
\(704\) 6247.76 0.334476
\(705\) 0 0
\(706\) 6208.27 0.330951
\(707\) 1507.83 0.0802092
\(708\) 0 0
\(709\) 18164.6 0.962179 0.481090 0.876671i \(-0.340241\pi\)
0.481090 + 0.876671i \(0.340241\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 3430.21 0.180552
\(713\) −977.000 −0.0513169
\(714\) 0 0
\(715\) 0 0
\(716\) 2385.36 0.124504
\(717\) 0 0
\(718\) −9944.50 −0.516888
\(719\) −9665.62 −0.501344 −0.250672 0.968072i \(-0.580652\pi\)
−0.250672 + 0.968072i \(0.580652\pi\)
\(720\) 0 0
\(721\) 7297.94 0.376962
\(722\) −25956.7 −1.33796
\(723\) 0 0
\(724\) 2908.26 0.149288
\(725\) 0 0
\(726\) 0 0
\(727\) 29779.6 1.51921 0.759605 0.650385i \(-0.225392\pi\)
0.759605 + 0.650385i \(0.225392\pi\)
\(728\) −7418.33 −0.377667
\(729\) 0 0
\(730\) 0 0
\(731\) 42481.7 2.14944
\(732\) 0 0
\(733\) −35029.5 −1.76513 −0.882567 0.470187i \(-0.844186\pi\)
−0.882567 + 0.470187i \(0.844186\pi\)
\(734\) 14515.4 0.729937
\(735\) 0 0
\(736\) 11931.7 0.597564
\(737\) 3729.17 0.186385
\(738\) 0 0
\(739\) 23297.3 1.15968 0.579842 0.814729i \(-0.303114\pi\)
0.579842 + 0.814729i \(0.303114\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2689.03 −0.133042
\(743\) 21570.4 1.06506 0.532530 0.846411i \(-0.321241\pi\)
0.532530 + 0.846411i \(0.321241\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 447.684 0.0219717
\(747\) 0 0
\(748\) −1308.69 −0.0639710
\(749\) 13159.6 0.641980
\(750\) 0 0
\(751\) 28554.8 1.38746 0.693729 0.720236i \(-0.255966\pi\)
0.693729 + 0.720236i \(0.255966\pi\)
\(752\) −28100.7 −1.36267
\(753\) 0 0
\(754\) −1120.81 −0.0541344
\(755\) 0 0
\(756\) 0 0
\(757\) −7812.81 −0.375114 −0.187557 0.982254i \(-0.560057\pi\)
−0.187557 + 0.982254i \(0.560057\pi\)
\(758\) −647.268 −0.0310156
\(759\) 0 0
\(760\) 0 0
\(761\) −2875.13 −0.136956 −0.0684778 0.997653i \(-0.521814\pi\)
−0.0684778 + 0.997653i \(0.521814\pi\)
\(762\) 0 0
\(763\) −3084.33 −0.146344
\(764\) 2071.69 0.0981033
\(765\) 0 0
\(766\) 28214.3 1.33084
\(767\) −30396.0 −1.43095
\(768\) 0 0
\(769\) −27657.7 −1.29696 −0.648479 0.761233i \(-0.724594\pi\)
−0.648479 + 0.761233i \(0.724594\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4024.84 −0.187639
\(773\) −3929.35 −0.182832 −0.0914160 0.995813i \(-0.529139\pi\)
−0.0914160 + 0.995813i \(0.529139\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −19301.6 −0.892898
\(777\) 0 0
\(778\) 20747.0 0.956064
\(779\) −38908.0 −1.78950
\(780\) 0 0
\(781\) −12327.8 −0.564818
\(782\) 39341.3 1.79903
\(783\) 0 0
\(784\) 15327.9 0.698247
\(785\) 0 0
\(786\) 0 0
\(787\) 21125.7 0.956860 0.478430 0.878126i \(-0.341206\pi\)
0.478430 + 0.878126i \(0.341206\pi\)
\(788\) −660.166 −0.0298445
\(789\) 0 0
\(790\) 0 0
\(791\) −1061.86 −0.0477310
\(792\) 0 0
\(793\) 38630.7 1.72991
\(794\) −1087.00 −0.0485847
\(795\) 0 0
\(796\) 3410.61 0.151867
\(797\) 11696.3 0.519828 0.259914 0.965632i \(-0.416306\pi\)
0.259914 + 0.965632i \(0.416306\pi\)
\(798\) 0 0
\(799\) 46093.0 2.04087
\(800\) 0 0
\(801\) 0 0
\(802\) −15123.9 −0.665888
\(803\) 1357.73 0.0596678
\(804\) 0 0
\(805\) 0 0
\(806\) −662.045 −0.0289324
\(807\) 0 0
\(808\) −5836.34 −0.254111
\(809\) −14310.2 −0.621902 −0.310951 0.950426i \(-0.600648\pi\)
−0.310951 + 0.950426i \(0.600648\pi\)
\(810\) 0 0
\(811\) 21697.9 0.939477 0.469739 0.882806i \(-0.344348\pi\)
0.469739 + 0.882806i \(0.344348\pi\)
\(812\) 80.0299 0.00345874
\(813\) 0 0
\(814\) −11746.9 −0.505807
\(815\) 0 0
\(816\) 0 0
\(817\) −66954.1 −2.86711
\(818\) −3510.54 −0.150053
\(819\) 0 0
\(820\) 0 0
\(821\) −3613.00 −0.153587 −0.0767934 0.997047i \(-0.524468\pi\)
−0.0767934 + 0.997047i \(0.524468\pi\)
\(822\) 0 0
\(823\) 4763.98 0.201776 0.100888 0.994898i \(-0.467832\pi\)
0.100888 + 0.994898i \(0.467832\pi\)
\(824\) −28248.0 −1.19425
\(825\) 0 0
\(826\) −9900.36 −0.417043
\(827\) −33571.7 −1.41161 −0.705806 0.708405i \(-0.749415\pi\)
−0.705806 + 0.708405i \(0.749415\pi\)
\(828\) 0 0
\(829\) 17980.5 0.753303 0.376652 0.926355i \(-0.377075\pi\)
0.376652 + 0.926355i \(0.377075\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 27900.9 1.16261
\(833\) −25142.1 −1.04576
\(834\) 0 0
\(835\) 0 0
\(836\) 2062.58 0.0853301
\(837\) 0 0
\(838\) 3249.91 0.133969
\(839\) 40139.5 1.65169 0.825847 0.563895i \(-0.190698\pi\)
0.825847 + 0.563895i \(0.190698\pi\)
\(840\) 0 0
\(841\) −24309.7 −0.996747
\(842\) 31358.4 1.28347
\(843\) 0 0
\(844\) −6213.91 −0.253426
\(845\) 0 0
\(846\) 0 0
\(847\) −755.792 −0.0306603
\(848\) 8474.36 0.343173
\(849\) 0 0
\(850\) 0 0
\(851\) −77414.4 −3.11837
\(852\) 0 0
\(853\) 15369.2 0.616919 0.308459 0.951237i \(-0.400187\pi\)
0.308459 + 0.951237i \(0.400187\pi\)
\(854\) 12582.5 0.504173
\(855\) 0 0
\(856\) −50936.8 −2.03386
\(857\) 10324.9 0.411541 0.205770 0.978600i \(-0.434030\pi\)
0.205770 + 0.978600i \(0.434030\pi\)
\(858\) 0 0
\(859\) −27112.5 −1.07691 −0.538455 0.842655i \(-0.680992\pi\)
−0.538455 + 0.842655i \(0.680992\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 20620.8 0.814787
\(863\) 30463.6 1.20161 0.600807 0.799394i \(-0.294846\pi\)
0.600807 + 0.799394i \(0.294846\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −42434.0 −1.66509
\(867\) 0 0
\(868\) 47.2726 0.00184854
\(869\) −3408.19 −0.133044
\(870\) 0 0
\(871\) 16653.5 0.647855
\(872\) 11938.4 0.463631
\(873\) 0 0
\(874\) −62004.6 −2.39970
\(875\) 0 0
\(876\) 0 0
\(877\) 5086.12 0.195833 0.0979167 0.995195i \(-0.468782\pi\)
0.0979167 + 0.995195i \(0.468782\pi\)
\(878\) −12054.2 −0.463335
\(879\) 0 0
\(880\) 0 0
\(881\) 10625.5 0.406338 0.203169 0.979144i \(-0.434876\pi\)
0.203169 + 0.979144i \(0.434876\pi\)
\(882\) 0 0
\(883\) −13112.2 −0.499728 −0.249864 0.968281i \(-0.580386\pi\)
−0.249864 + 0.968281i \(0.580386\pi\)
\(884\) −5844.25 −0.222357
\(885\) 0 0
\(886\) −38664.0 −1.46607
\(887\) −14442.8 −0.546719 −0.273360 0.961912i \(-0.588135\pi\)
−0.273360 + 0.961912i \(0.588135\pi\)
\(888\) 0 0
\(889\) −5922.54 −0.223437
\(890\) 0 0
\(891\) 0 0
\(892\) −5520.38 −0.207215
\(893\) −72645.8 −2.72228
\(894\) 0 0
\(895\) 0 0
\(896\) 5876.86 0.219121
\(897\) 0 0
\(898\) 2493.61 0.0926648
\(899\) 46.8641 0.00173860
\(900\) 0 0
\(901\) −13900.3 −0.513970
\(902\) −8410.27 −0.310456
\(903\) 0 0
\(904\) 4110.10 0.151217
\(905\) 0 0
\(906\) 0 0
\(907\) 44981.9 1.64675 0.823374 0.567499i \(-0.192089\pi\)
0.823374 + 0.567499i \(0.192089\pi\)
\(908\) 7197.57 0.263062
\(909\) 0 0
\(910\) 0 0
\(911\) −6841.96 −0.248830 −0.124415 0.992230i \(-0.539705\pi\)
−0.124415 + 0.992230i \(0.539705\pi\)
\(912\) 0 0
\(913\) −11233.4 −0.407199
\(914\) 160.501 0.00580843
\(915\) 0 0
\(916\) 398.989 0.0143919
\(917\) −9274.61 −0.333996
\(918\) 0 0
\(919\) −4753.54 −0.170625 −0.0853127 0.996354i \(-0.527189\pi\)
−0.0853127 + 0.996354i \(0.527189\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −30395.9 −1.08572
\(923\) −55052.7 −1.96325
\(924\) 0 0
\(925\) 0 0
\(926\) −33671.2 −1.19493
\(927\) 0 0
\(928\) −572.330 −0.0202453
\(929\) 7507.93 0.265153 0.132576 0.991173i \(-0.457675\pi\)
0.132576 + 0.991173i \(0.457675\pi\)
\(930\) 0 0
\(931\) 39625.7 1.39493
\(932\) −3265.14 −0.114757
\(933\) 0 0
\(934\) 26068.3 0.913256
\(935\) 0 0
\(936\) 0 0
\(937\) −8540.47 −0.297764 −0.148882 0.988855i \(-0.547567\pi\)
−0.148882 + 0.988855i \(0.547567\pi\)
\(938\) 5424.24 0.188814
\(939\) 0 0
\(940\) 0 0
\(941\) −9101.13 −0.315290 −0.157645 0.987496i \(-0.550390\pi\)
−0.157645 + 0.987496i \(0.550390\pi\)
\(942\) 0 0
\(943\) −55425.5 −1.91400
\(944\) 31200.6 1.07573
\(945\) 0 0
\(946\) −14472.7 −0.497407
\(947\) 47540.0 1.63130 0.815650 0.578546i \(-0.196380\pi\)
0.815650 + 0.578546i \(0.196380\pi\)
\(948\) 0 0
\(949\) 6063.26 0.207399
\(950\) 0 0
\(951\) 0 0
\(952\) −12490.2 −0.425221
\(953\) 47370.7 1.61016 0.805082 0.593164i \(-0.202121\pi\)
0.805082 + 0.593164i \(0.202121\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −2326.12 −0.0786947
\(957\) 0 0
\(958\) 8789.33 0.296420
\(959\) −4278.20 −0.144057
\(960\) 0 0
\(961\) −29763.3 −0.999071
\(962\) −52458.4 −1.75813
\(963\) 0 0
\(964\) −8122.38 −0.271374
\(965\) 0 0
\(966\) 0 0
\(967\) 36171.6 1.20290 0.601448 0.798912i \(-0.294591\pi\)
0.601448 + 0.798912i \(0.294591\pi\)
\(968\) 2925.43 0.0971351
\(969\) 0 0
\(970\) 0 0
\(971\) 31713.2 1.04812 0.524060 0.851681i \(-0.324417\pi\)
0.524060 + 0.851681i \(0.324417\pi\)
\(972\) 0 0
\(973\) −5187.37 −0.170914
\(974\) −7257.40 −0.238750
\(975\) 0 0
\(976\) −39653.1 −1.30048
\(977\) 22800.5 0.746626 0.373313 0.927706i \(-0.378222\pi\)
0.373313 + 0.927706i \(0.378222\pi\)
\(978\) 0 0
\(979\) 1560.67 0.0509490
\(980\) 0 0
\(981\) 0 0
\(982\) −6832.41 −0.222027
\(983\) −44597.4 −1.44704 −0.723518 0.690305i \(-0.757476\pi\)
−0.723518 + 0.690305i \(0.757476\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −1887.10 −0.0609507
\(987\) 0 0
\(988\) 9210.95 0.296599
\(989\) −95378.1 −3.06658
\(990\) 0 0
\(991\) −34788.1 −1.11512 −0.557558 0.830138i \(-0.688261\pi\)
−0.557558 + 0.830138i \(0.688261\pi\)
\(992\) −338.068 −0.0108202
\(993\) 0 0
\(994\) −17931.3 −0.572180
\(995\) 0 0
\(996\) 0 0
\(997\) 20360.5 0.646765 0.323383 0.946268i \(-0.395180\pi\)
0.323383 + 0.946268i \(0.395180\pi\)
\(998\) 28528.0 0.904849
\(999\) 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 2475.4.a.n.1.1 2
3.2 odd 2 825.4.a.m.1.2 2
5.4 even 2 495.4.a.d.1.2 2
15.2 even 4 825.4.c.j.199.4 4
15.8 even 4 825.4.c.j.199.1 4
15.14 odd 2 165.4.a.c.1.1 2
165.164 even 2 1815.4.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.c.1.1 2 15.14 odd 2
495.4.a.d.1.2 2 5.4 even 2
825.4.a.m.1.2 2 3.2 odd 2
825.4.c.j.199.1 4 15.8 even 4
825.4.c.j.199.4 4 15.2 even 4
1815.4.a.n.1.2 2 165.164 even 2
2475.4.a.n.1.1 2 1.1 even 1 trivial