Properties

Label 1575.4.a.z.1.1
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.58579 q^{2} -1.31371 q^{4} +7.00000 q^{7} -24.0833 q^{8} +O(q^{10})\) \(q+2.58579 q^{2} -1.31371 q^{4} +7.00000 q^{7} -24.0833 q^{8} -38.2548 q^{11} -19.3431 q^{13} +18.1005 q^{14} -51.7645 q^{16} -87.2254 q^{17} -44.2254 q^{19} -98.9188 q^{22} +218.167 q^{23} -50.0172 q^{26} -9.19596 q^{28} +46.9411 q^{29} +194.558 q^{31} +58.8141 q^{32} -225.546 q^{34} -366.853 q^{37} -114.357 q^{38} +339.362 q^{41} +226.167 q^{43} +50.2557 q^{44} +564.132 q^{46} +11.6762 q^{47} +49.0000 q^{49} +25.4113 q^{52} -209.019 q^{53} -168.583 q^{56} +121.380 q^{58} +616.000 q^{59} +320.735 q^{61} +503.087 q^{62} +566.197 q^{64} -14.5097 q^{67} +114.589 q^{68} +952.000 q^{71} -824.489 q^{73} -948.603 q^{74} +58.0993 q^{76} -267.784 q^{77} +156.275 q^{79} +877.519 q^{82} -1036.53 q^{83} +584.818 q^{86} +921.301 q^{88} +170.225 q^{89} -135.402 q^{91} -286.607 q^{92} +30.1921 q^{94} -1059.87 q^{97} +126.704 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 20 q^{4} + 14 q^{7} + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} + 20 q^{4} + 14 q^{7} + 48 q^{8} + 14 q^{11} - 50 q^{13} + 56 q^{14} + 168 q^{16} - 50 q^{17} + 36 q^{19} + 184 q^{22} + 244 q^{23} - 216 q^{26} + 140 q^{28} + 26 q^{29} - 120 q^{31} + 672 q^{32} - 24 q^{34} - 564 q^{37} + 320 q^{38} + 328 q^{41} + 260 q^{43} + 1164 q^{44} + 704 q^{46} - 350 q^{47} + 98 q^{49} - 628 q^{52} - 56 q^{53} + 336 q^{56} + 8 q^{58} + 1232 q^{59} + 336 q^{61} - 1200 q^{62} + 2128 q^{64} + 152 q^{67} + 908 q^{68} + 1904 q^{71} - 676 q^{73} - 2016 q^{74} + 1768 q^{76} + 98 q^{77} + 1014 q^{79} + 816 q^{82} - 376 q^{83} + 768 q^{86} + 4688 q^{88} + 216 q^{89} - 350 q^{91} + 264 q^{92} - 1928 q^{94} - 2742 q^{97} + 392 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.58579 0.914214 0.457107 0.889412i \(-0.348886\pi\)
0.457107 + 0.889412i \(0.348886\pi\)
\(3\) 0 0
\(4\) −1.31371 −0.164214
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) −24.0833 −1.06434
\(9\) 0 0
\(10\) 0 0
\(11\) −38.2548 −1.04857 −0.524285 0.851543i \(-0.675667\pi\)
−0.524285 + 0.851543i \(0.675667\pi\)
\(12\) 0 0
\(13\) −19.3431 −0.412679 −0.206339 0.978480i \(-0.566155\pi\)
−0.206339 + 0.978480i \(0.566155\pi\)
\(14\) 18.1005 0.345540
\(15\) 0 0
\(16\) −51.7645 −0.808820
\(17\) −87.2254 −1.24443 −0.622214 0.782847i \(-0.713767\pi\)
−0.622214 + 0.782847i \(0.713767\pi\)
\(18\) 0 0
\(19\) −44.2254 −0.534000 −0.267000 0.963697i \(-0.586032\pi\)
−0.267000 + 0.963697i \(0.586032\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −98.9188 −0.958617
\(23\) 218.167 1.97786 0.988932 0.148371i \(-0.0474028\pi\)
0.988932 + 0.148371i \(0.0474028\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −50.0172 −0.377276
\(27\) 0 0
\(28\) −9.19596 −0.0620669
\(29\) 46.9411 0.300578 0.150289 0.988642i \(-0.451980\pi\)
0.150289 + 0.988642i \(0.451980\pi\)
\(30\) 0 0
\(31\) 194.558 1.12722 0.563609 0.826042i \(-0.309413\pi\)
0.563609 + 0.826042i \(0.309413\pi\)
\(32\) 58.8141 0.324905
\(33\) 0 0
\(34\) −225.546 −1.13767
\(35\) 0 0
\(36\) 0 0
\(37\) −366.853 −1.63001 −0.815003 0.579457i \(-0.803265\pi\)
−0.815003 + 0.579457i \(0.803265\pi\)
\(38\) −114.357 −0.488190
\(39\) 0 0
\(40\) 0 0
\(41\) 339.362 1.29267 0.646336 0.763053i \(-0.276301\pi\)
0.646336 + 0.763053i \(0.276301\pi\)
\(42\) 0 0
\(43\) 226.167 0.802095 0.401047 0.916057i \(-0.368646\pi\)
0.401047 + 0.916057i \(0.368646\pi\)
\(44\) 50.2557 0.172189
\(45\) 0 0
\(46\) 564.132 1.80819
\(47\) 11.6762 0.0362372 0.0181186 0.999836i \(-0.494232\pi\)
0.0181186 + 0.999836i \(0.494232\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 25.4113 0.0677674
\(53\) −209.019 −0.541717 −0.270859 0.962619i \(-0.587308\pi\)
−0.270859 + 0.962619i \(0.587308\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −168.583 −0.402283
\(57\) 0 0
\(58\) 121.380 0.274792
\(59\) 616.000 1.35926 0.679630 0.733555i \(-0.262140\pi\)
0.679630 + 0.733555i \(0.262140\pi\)
\(60\) 0 0
\(61\) 320.735 0.673212 0.336606 0.941646i \(-0.390721\pi\)
0.336606 + 0.941646i \(0.390721\pi\)
\(62\) 503.087 1.03052
\(63\) 0 0
\(64\) 566.197 1.10585
\(65\) 0 0
\(66\) 0 0
\(67\) −14.5097 −0.0264573 −0.0132286 0.999912i \(-0.504211\pi\)
−0.0132286 + 0.999912i \(0.504211\pi\)
\(68\) 114.589 0.204352
\(69\) 0 0
\(70\) 0 0
\(71\) 952.000 1.59129 0.795645 0.605763i \(-0.207132\pi\)
0.795645 + 0.605763i \(0.207132\pi\)
\(72\) 0 0
\(73\) −824.489 −1.32191 −0.660953 0.750427i \(-0.729848\pi\)
−0.660953 + 0.750427i \(0.729848\pi\)
\(74\) −948.603 −1.49017
\(75\) 0 0
\(76\) 58.0993 0.0876901
\(77\) −267.784 −0.396322
\(78\) 0 0
\(79\) 156.275 0.222561 0.111280 0.993789i \(-0.464505\pi\)
0.111280 + 0.993789i \(0.464505\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 877.519 1.18178
\(83\) −1036.53 −1.37077 −0.685384 0.728182i \(-0.740366\pi\)
−0.685384 + 0.728182i \(0.740366\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 584.818 0.733286
\(87\) 0 0
\(88\) 921.301 1.11603
\(89\) 170.225 0.202740 0.101370 0.994849i \(-0.467677\pi\)
0.101370 + 0.994849i \(0.467677\pi\)
\(90\) 0 0
\(91\) −135.402 −0.155978
\(92\) −286.607 −0.324792
\(93\) 0 0
\(94\) 30.1921 0.0331285
\(95\) 0 0
\(96\) 0 0
\(97\) −1059.87 −1.10942 −0.554710 0.832044i \(-0.687171\pi\)
−0.554710 + 0.832044i \(0.687171\pi\)
\(98\) 126.704 0.130602
\(99\) 0 0
\(100\) 0 0
\(101\) 241.833 0.238251 0.119125 0.992879i \(-0.461991\pi\)
0.119125 + 0.992879i \(0.461991\pi\)
\(102\) 0 0
\(103\) 1679.58 1.60673 0.803367 0.595484i \(-0.203040\pi\)
0.803367 + 0.595484i \(0.203040\pi\)
\(104\) 465.846 0.439230
\(105\) 0 0
\(106\) −540.479 −0.495245
\(107\) 1506.88 1.36146 0.680728 0.732537i \(-0.261664\pi\)
0.680728 + 0.732537i \(0.261664\pi\)
\(108\) 0 0
\(109\) −1252.41 −1.10054 −0.550271 0.834986i \(-0.685476\pi\)
−0.550271 + 0.834986i \(0.685476\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −362.352 −0.305705
\(113\) 1370.20 1.14069 0.570345 0.821405i \(-0.306810\pi\)
0.570345 + 0.821405i \(0.306810\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −61.6670 −0.0493589
\(117\) 0 0
\(118\) 1592.84 1.24265
\(119\) −610.578 −0.470349
\(120\) 0 0
\(121\) 132.432 0.0994984
\(122\) 829.352 0.615459
\(123\) 0 0
\(124\) −255.593 −0.185104
\(125\) 0 0
\(126\) 0 0
\(127\) −1213.49 −0.847873 −0.423936 0.905692i \(-0.639352\pi\)
−0.423936 + 0.905692i \(0.639352\pi\)
\(128\) 993.551 0.686081
\(129\) 0 0
\(130\) 0 0
\(131\) 1982.42 1.32217 0.661087 0.750309i \(-0.270096\pi\)
0.661087 + 0.750309i \(0.270096\pi\)
\(132\) 0 0
\(133\) −309.578 −0.201833
\(134\) −37.5189 −0.0241876
\(135\) 0 0
\(136\) 2100.67 1.32449
\(137\) 2210.95 1.37879 0.689394 0.724386i \(-0.257877\pi\)
0.689394 + 0.724386i \(0.257877\pi\)
\(138\) 0 0
\(139\) 528.039 0.322213 0.161107 0.986937i \(-0.448494\pi\)
0.161107 + 0.986937i \(0.448494\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2461.67 1.45478
\(143\) 739.969 0.432722
\(144\) 0 0
\(145\) 0 0
\(146\) −2131.95 −1.20851
\(147\) 0 0
\(148\) 481.938 0.267669
\(149\) 328.372 0.180545 0.0902727 0.995917i \(-0.471226\pi\)
0.0902727 + 0.995917i \(0.471226\pi\)
\(150\) 0 0
\(151\) 1029.43 0.554793 0.277396 0.960756i \(-0.410528\pi\)
0.277396 + 0.960756i \(0.410528\pi\)
\(152\) 1065.09 0.568358
\(153\) 0 0
\(154\) −692.432 −0.362323
\(155\) 0 0
\(156\) 0 0
\(157\) −525.098 −0.266926 −0.133463 0.991054i \(-0.542610\pi\)
−0.133463 + 0.991054i \(0.542610\pi\)
\(158\) 404.094 0.203468
\(159\) 0 0
\(160\) 0 0
\(161\) 1527.17 0.747562
\(162\) 0 0
\(163\) −1002.63 −0.481790 −0.240895 0.970551i \(-0.577441\pi\)
−0.240895 + 0.970551i \(0.577441\pi\)
\(164\) −445.823 −0.212274
\(165\) 0 0
\(166\) −2680.24 −1.25317
\(167\) −359.422 −0.166544 −0.0832722 0.996527i \(-0.526537\pi\)
−0.0832722 + 0.996527i \(0.526537\pi\)
\(168\) 0 0
\(169\) −1822.84 −0.829696
\(170\) 0 0
\(171\) 0 0
\(172\) −297.117 −0.131715
\(173\) 3293.65 1.44747 0.723733 0.690080i \(-0.242425\pi\)
0.723733 + 0.690080i \(0.242425\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1980.24 0.848104
\(177\) 0 0
\(178\) 440.167 0.185348
\(179\) −2978.82 −1.24384 −0.621921 0.783080i \(-0.713647\pi\)
−0.621921 + 0.783080i \(0.713647\pi\)
\(180\) 0 0
\(181\) 1462.31 0.600514 0.300257 0.953858i \(-0.402928\pi\)
0.300257 + 0.953858i \(0.402928\pi\)
\(182\) −350.121 −0.142597
\(183\) 0 0
\(184\) −5254.16 −2.10512
\(185\) 0 0
\(186\) 0 0
\(187\) 3336.79 1.30487
\(188\) −15.3391 −0.00595064
\(189\) 0 0
\(190\) 0 0
\(191\) 374.923 0.142034 0.0710169 0.997475i \(-0.477376\pi\)
0.0710169 + 0.997475i \(0.477376\pi\)
\(192\) 0 0
\(193\) −733.028 −0.273391 −0.136696 0.990613i \(-0.543648\pi\)
−0.136696 + 0.990613i \(0.543648\pi\)
\(194\) −2740.61 −1.01425
\(195\) 0 0
\(196\) −64.3717 −0.0234591
\(197\) −2093.24 −0.757043 −0.378521 0.925593i \(-0.623567\pi\)
−0.378521 + 0.925593i \(0.623567\pi\)
\(198\) 0 0
\(199\) 2865.04 1.02059 0.510295 0.860000i \(-0.329536\pi\)
0.510295 + 0.860000i \(0.329536\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 625.330 0.217812
\(203\) 328.588 0.113608
\(204\) 0 0
\(205\) 0 0
\(206\) 4343.03 1.46890
\(207\) 0 0
\(208\) 1001.29 0.333783
\(209\) 1691.84 0.559936
\(210\) 0 0
\(211\) 5643.65 1.84135 0.920674 0.390331i \(-0.127640\pi\)
0.920674 + 0.390331i \(0.127640\pi\)
\(212\) 274.590 0.0889573
\(213\) 0 0
\(214\) 3896.47 1.24466
\(215\) 0 0
\(216\) 0 0
\(217\) 1361.91 0.426048
\(218\) −3238.46 −1.00613
\(219\) 0 0
\(220\) 0 0
\(221\) 1687.21 0.513549
\(222\) 0 0
\(223\) 6369.16 1.91260 0.956302 0.292381i \(-0.0944477\pi\)
0.956302 + 0.292381i \(0.0944477\pi\)
\(224\) 411.699 0.122803
\(225\) 0 0
\(226\) 3543.05 1.04283
\(227\) −1015.67 −0.296972 −0.148486 0.988914i \(-0.547440\pi\)
−0.148486 + 0.988914i \(0.547440\pi\)
\(228\) 0 0
\(229\) 4108.35 1.18554 0.592768 0.805373i \(-0.298035\pi\)
0.592768 + 0.805373i \(0.298035\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1130.50 −0.319917
\(233\) 608.431 0.171071 0.0855357 0.996335i \(-0.472740\pi\)
0.0855357 + 0.996335i \(0.472740\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −809.244 −0.223209
\(237\) 0 0
\(238\) −1578.82 −0.430000
\(239\) 5054.44 1.36797 0.683985 0.729496i \(-0.260245\pi\)
0.683985 + 0.729496i \(0.260245\pi\)
\(240\) 0 0
\(241\) 4.86782 0.00130109 0.000650547 1.00000i \(-0.499793\pi\)
0.000650547 1.00000i \(0.499793\pi\)
\(242\) 342.442 0.0909628
\(243\) 0 0
\(244\) −421.352 −0.110551
\(245\) 0 0
\(246\) 0 0
\(247\) 855.458 0.220370
\(248\) −4685.60 −1.19974
\(249\) 0 0
\(250\) 0 0
\(251\) 547.921 0.137787 0.0688934 0.997624i \(-0.478053\pi\)
0.0688934 + 0.997624i \(0.478053\pi\)
\(252\) 0 0
\(253\) −8345.92 −2.07393
\(254\) −3137.83 −0.775137
\(255\) 0 0
\(256\) −1960.46 −0.478629
\(257\) −1774.61 −0.430729 −0.215364 0.976534i \(-0.569094\pi\)
−0.215364 + 0.976534i \(0.569094\pi\)
\(258\) 0 0
\(259\) −2567.97 −0.616084
\(260\) 0 0
\(261\) 0 0
\(262\) 5126.11 1.20875
\(263\) −1199.09 −0.281138 −0.140569 0.990071i \(-0.544893\pi\)
−0.140569 + 0.990071i \(0.544893\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −800.502 −0.184519
\(267\) 0 0
\(268\) 19.0615 0.00434464
\(269\) −3250.29 −0.736706 −0.368353 0.929686i \(-0.620078\pi\)
−0.368353 + 0.929686i \(0.620078\pi\)
\(270\) 0 0
\(271\) −896.143 −0.200874 −0.100437 0.994943i \(-0.532024\pi\)
−0.100437 + 0.994943i \(0.532024\pi\)
\(272\) 4515.18 1.00652
\(273\) 0 0
\(274\) 5717.04 1.26051
\(275\) 0 0
\(276\) 0 0
\(277\) 386.562 0.0838492 0.0419246 0.999121i \(-0.486651\pi\)
0.0419246 + 0.999121i \(0.486651\pi\)
\(278\) 1365.40 0.294572
\(279\) 0 0
\(280\) 0 0
\(281\) 3335.10 0.708025 0.354013 0.935241i \(-0.384817\pi\)
0.354013 + 0.935241i \(0.384817\pi\)
\(282\) 0 0
\(283\) −5412.26 −1.13684 −0.568419 0.822739i \(-0.692445\pi\)
−0.568419 + 0.822739i \(0.692445\pi\)
\(284\) −1250.65 −0.261311
\(285\) 0 0
\(286\) 1913.40 0.395601
\(287\) 2375.54 0.488584
\(288\) 0 0
\(289\) 2695.27 0.548600
\(290\) 0 0
\(291\) 0 0
\(292\) 1083.14 0.217075
\(293\) 282.211 0.0562695 0.0281347 0.999604i \(-0.491043\pi\)
0.0281347 + 0.999604i \(0.491043\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8835.01 1.73488
\(297\) 0 0
\(298\) 849.099 0.165057
\(299\) −4220.03 −0.816222
\(300\) 0 0
\(301\) 1583.17 0.303163
\(302\) 2661.88 0.507199
\(303\) 0 0
\(304\) 2289.31 0.431910
\(305\) 0 0
\(306\) 0 0
\(307\) −1919.67 −0.356878 −0.178439 0.983951i \(-0.557105\pi\)
−0.178439 + 0.983951i \(0.557105\pi\)
\(308\) 351.790 0.0650815
\(309\) 0 0
\(310\) 0 0
\(311\) −1213.31 −0.221223 −0.110612 0.993864i \(-0.535281\pi\)
−0.110612 + 0.993864i \(0.535281\pi\)
\(312\) 0 0
\(313\) 1434.00 0.258960 0.129480 0.991582i \(-0.458669\pi\)
0.129480 + 0.991582i \(0.458669\pi\)
\(314\) −1357.79 −0.244028
\(315\) 0 0
\(316\) −205.300 −0.0365475
\(317\) 6496.95 1.15112 0.575560 0.817760i \(-0.304784\pi\)
0.575560 + 0.817760i \(0.304784\pi\)
\(318\) 0 0
\(319\) −1795.72 −0.315176
\(320\) 0 0
\(321\) 0 0
\(322\) 3948.92 0.683432
\(323\) 3857.58 0.664524
\(324\) 0 0
\(325\) 0 0
\(326\) −2592.58 −0.440459
\(327\) 0 0
\(328\) −8172.96 −1.37584
\(329\) 81.7333 0.0136964
\(330\) 0 0
\(331\) −9683.88 −1.60808 −0.804039 0.594576i \(-0.797320\pi\)
−0.804039 + 0.594576i \(0.797320\pi\)
\(332\) 1361.70 0.225099
\(333\) 0 0
\(334\) −929.389 −0.152257
\(335\) 0 0
\(336\) 0 0
\(337\) −29.1319 −0.00470895 −0.00235447 0.999997i \(-0.500749\pi\)
−0.00235447 + 0.999997i \(0.500749\pi\)
\(338\) −4713.48 −0.758520
\(339\) 0 0
\(340\) 0 0
\(341\) −7442.80 −1.18197
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) −5446.83 −0.853701
\(345\) 0 0
\(346\) 8516.68 1.32329
\(347\) −7848.58 −1.21422 −0.607110 0.794618i \(-0.707671\pi\)
−0.607110 + 0.794618i \(0.707671\pi\)
\(348\) 0 0
\(349\) −10269.6 −1.57513 −0.787567 0.616229i \(-0.788659\pi\)
−0.787567 + 0.616229i \(0.788659\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2249.93 −0.340686
\(353\) 2799.93 0.422168 0.211084 0.977468i \(-0.432301\pi\)
0.211084 + 0.977468i \(0.432301\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −223.627 −0.0332927
\(357\) 0 0
\(358\) −7702.60 −1.13714
\(359\) 3163.29 0.465048 0.232524 0.972591i \(-0.425302\pi\)
0.232524 + 0.972591i \(0.425302\pi\)
\(360\) 0 0
\(361\) −4903.11 −0.714844
\(362\) 3781.23 0.548998
\(363\) 0 0
\(364\) 177.879 0.0256137
\(365\) 0 0
\(366\) 0 0
\(367\) −3182.85 −0.452706 −0.226353 0.974045i \(-0.572680\pi\)
−0.226353 + 0.974045i \(0.572680\pi\)
\(368\) −11293.3 −1.59974
\(369\) 0 0
\(370\) 0 0
\(371\) −1463.14 −0.204750
\(372\) 0 0
\(373\) 2615.14 0.363021 0.181510 0.983389i \(-0.441901\pi\)
0.181510 + 0.983389i \(0.441901\pi\)
\(374\) 8628.23 1.19293
\(375\) 0 0
\(376\) −281.201 −0.0385687
\(377\) −907.989 −0.124042
\(378\) 0 0
\(379\) −672.434 −0.0911362 −0.0455681 0.998961i \(-0.514510\pi\)
−0.0455681 + 0.998961i \(0.514510\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 969.470 0.129849
\(383\) 1169.86 0.156075 0.0780377 0.996950i \(-0.475135\pi\)
0.0780377 + 0.996950i \(0.475135\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1895.45 −0.249938
\(387\) 0 0
\(388\) 1392.36 0.182182
\(389\) 1122.22 0.146269 0.0731347 0.997322i \(-0.476700\pi\)
0.0731347 + 0.997322i \(0.476700\pi\)
\(390\) 0 0
\(391\) −19029.7 −2.46131
\(392\) −1180.08 −0.152049
\(393\) 0 0
\(394\) −5412.68 −0.692099
\(395\) 0 0
\(396\) 0 0
\(397\) 1985.93 0.251060 0.125530 0.992090i \(-0.459937\pi\)
0.125530 + 0.992090i \(0.459937\pi\)
\(398\) 7408.38 0.933037
\(399\) 0 0
\(400\) 0 0
\(401\) 4172.38 0.519597 0.259799 0.965663i \(-0.416344\pi\)
0.259799 + 0.965663i \(0.416344\pi\)
\(402\) 0 0
\(403\) −3763.37 −0.465178
\(404\) −317.699 −0.0391240
\(405\) 0 0
\(406\) 849.658 0.103862
\(407\) 14033.9 1.70918
\(408\) 0 0
\(409\) −11700.8 −1.41459 −0.707295 0.706919i \(-0.750085\pi\)
−0.707295 + 0.706919i \(0.750085\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2206.47 −0.263848
\(413\) 4312.00 0.513752
\(414\) 0 0
\(415\) 0 0
\(416\) −1137.65 −0.134082
\(417\) 0 0
\(418\) 4374.72 0.511901
\(419\) −2733.20 −0.318677 −0.159339 0.987224i \(-0.550936\pi\)
−0.159339 + 0.987224i \(0.550936\pi\)
\(420\) 0 0
\(421\) 13549.4 1.56854 0.784272 0.620417i \(-0.213037\pi\)
0.784272 + 0.620417i \(0.213037\pi\)
\(422\) 14593.3 1.68339
\(423\) 0 0
\(424\) 5033.87 0.576571
\(425\) 0 0
\(426\) 0 0
\(427\) 2245.15 0.254450
\(428\) −1979.60 −0.223569
\(429\) 0 0
\(430\) 0 0
\(431\) 6429.25 0.718530 0.359265 0.933236i \(-0.383027\pi\)
0.359265 + 0.933236i \(0.383027\pi\)
\(432\) 0 0
\(433\) −8022.03 −0.890333 −0.445166 0.895448i \(-0.646855\pi\)
−0.445166 + 0.895448i \(0.646855\pi\)
\(434\) 3521.61 0.389499
\(435\) 0 0
\(436\) 1645.30 0.180724
\(437\) −9648.50 −1.05618
\(438\) 0 0
\(439\) −5569.88 −0.605549 −0.302774 0.953062i \(-0.597913\pi\)
−0.302774 + 0.953062i \(0.597913\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 4362.77 0.469493
\(443\) −5486.21 −0.588392 −0.294196 0.955745i \(-0.595052\pi\)
−0.294196 + 0.955745i \(0.595052\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 16469.3 1.74853
\(447\) 0 0
\(448\) 3963.38 0.417973
\(449\) 7232.67 0.760203 0.380101 0.924945i \(-0.375889\pi\)
0.380101 + 0.924945i \(0.375889\pi\)
\(450\) 0 0
\(451\) −12982.3 −1.35546
\(452\) −1800.05 −0.187317
\(453\) 0 0
\(454\) −2626.32 −0.271496
\(455\) 0 0
\(456\) 0 0
\(457\) 2900.51 0.296893 0.148446 0.988920i \(-0.452573\pi\)
0.148446 + 0.988920i \(0.452573\pi\)
\(458\) 10623.3 1.08383
\(459\) 0 0
\(460\) 0 0
\(461\) −6073.57 −0.613611 −0.306805 0.951772i \(-0.599260\pi\)
−0.306805 + 0.951772i \(0.599260\pi\)
\(462\) 0 0
\(463\) 18922.8 1.89939 0.949693 0.313183i \(-0.101395\pi\)
0.949693 + 0.313183i \(0.101395\pi\)
\(464\) −2429.88 −0.243113
\(465\) 0 0
\(466\) 1573.27 0.156396
\(467\) −6776.71 −0.671496 −0.335748 0.941952i \(-0.608989\pi\)
−0.335748 + 0.941952i \(0.608989\pi\)
\(468\) 0 0
\(469\) −101.568 −0.00999991
\(470\) 0 0
\(471\) 0 0
\(472\) −14835.3 −1.44672
\(473\) −8651.96 −0.841052
\(474\) 0 0
\(475\) 0 0
\(476\) 802.121 0.0772377
\(477\) 0 0
\(478\) 13069.7 1.25062
\(479\) −2397.32 −0.228677 −0.114338 0.993442i \(-0.536475\pi\)
−0.114338 + 0.993442i \(0.536475\pi\)
\(480\) 0 0
\(481\) 7096.09 0.672669
\(482\) 12.5871 0.00118948
\(483\) 0 0
\(484\) −173.977 −0.0163390
\(485\) 0 0
\(486\) 0 0
\(487\) −5586.17 −0.519781 −0.259890 0.965638i \(-0.583686\pi\)
−0.259890 + 0.965638i \(0.583686\pi\)
\(488\) −7724.35 −0.716526
\(489\) 0 0
\(490\) 0 0
\(491\) −537.392 −0.0493934 −0.0246967 0.999695i \(-0.507862\pi\)
−0.0246967 + 0.999695i \(0.507862\pi\)
\(492\) 0 0
\(493\) −4094.46 −0.374047
\(494\) 2212.03 0.201466
\(495\) 0 0
\(496\) −10071.2 −0.911716
\(497\) 6664.00 0.601451
\(498\) 0 0
\(499\) 598.965 0.0537342 0.0268671 0.999639i \(-0.491447\pi\)
0.0268671 + 0.999639i \(0.491447\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1416.81 0.125966
\(503\) 4426.76 0.392405 0.196202 0.980563i \(-0.437139\pi\)
0.196202 + 0.980563i \(0.437139\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −21580.8 −1.89601
\(507\) 0 0
\(508\) 1594.17 0.139232
\(509\) −17727.7 −1.54374 −0.771872 0.635779i \(-0.780679\pi\)
−0.771872 + 0.635779i \(0.780679\pi\)
\(510\) 0 0
\(511\) −5771.43 −0.499634
\(512\) −13017.7 −1.12365
\(513\) 0 0
\(514\) −4588.77 −0.393778
\(515\) 0 0
\(516\) 0 0
\(517\) −446.671 −0.0379972
\(518\) −6640.22 −0.563233
\(519\) 0 0
\(520\) 0 0
\(521\) −8662.79 −0.728453 −0.364226 0.931310i \(-0.618667\pi\)
−0.364226 + 0.931310i \(0.618667\pi\)
\(522\) 0 0
\(523\) 7770.40 0.649667 0.324833 0.945771i \(-0.394692\pi\)
0.324833 + 0.945771i \(0.394692\pi\)
\(524\) −2604.32 −0.217119
\(525\) 0 0
\(526\) −3100.60 −0.257020
\(527\) −16970.4 −1.40274
\(528\) 0 0
\(529\) 35429.6 2.91194
\(530\) 0 0
\(531\) 0 0
\(532\) 406.695 0.0331437
\(533\) −6564.34 −0.533458
\(534\) 0 0
\(535\) 0 0
\(536\) 349.440 0.0281595
\(537\) 0 0
\(538\) −8404.56 −0.673506
\(539\) −1874.49 −0.149796
\(540\) 0 0
\(541\) 21641.0 1.71981 0.859906 0.510453i \(-0.170522\pi\)
0.859906 + 0.510453i \(0.170522\pi\)
\(542\) −2317.23 −0.183642
\(543\) 0 0
\(544\) −5130.09 −0.404321
\(545\) 0 0
\(546\) 0 0
\(547\) −7489.29 −0.585409 −0.292705 0.956203i \(-0.594555\pi\)
−0.292705 + 0.956203i \(0.594555\pi\)
\(548\) −2904.54 −0.226416
\(549\) 0 0
\(550\) 0 0
\(551\) −2075.99 −0.160508
\(552\) 0 0
\(553\) 1093.93 0.0841201
\(554\) 999.566 0.0766561
\(555\) 0 0
\(556\) −693.689 −0.0529118
\(557\) 25297.9 1.92443 0.962214 0.272295i \(-0.0877826\pi\)
0.962214 + 0.272295i \(0.0877826\pi\)
\(558\) 0 0
\(559\) −4374.77 −0.331007
\(560\) 0 0
\(561\) 0 0
\(562\) 8623.85 0.647286
\(563\) 15661.3 1.17237 0.586186 0.810177i \(-0.300629\pi\)
0.586186 + 0.810177i \(0.300629\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −13994.9 −1.03931
\(567\) 0 0
\(568\) −22927.3 −1.69367
\(569\) 9982.75 0.735498 0.367749 0.929925i \(-0.380129\pi\)
0.367749 + 0.929925i \(0.380129\pi\)
\(570\) 0 0
\(571\) −11583.6 −0.848966 −0.424483 0.905436i \(-0.639544\pi\)
−0.424483 + 0.905436i \(0.639544\pi\)
\(572\) −972.103 −0.0710589
\(573\) 0 0
\(574\) 6142.63 0.446670
\(575\) 0 0
\(576\) 0 0
\(577\) 595.378 0.0429565 0.0214783 0.999769i \(-0.493163\pi\)
0.0214783 + 0.999769i \(0.493163\pi\)
\(578\) 6969.39 0.501537
\(579\) 0 0
\(580\) 0 0
\(581\) −7255.70 −0.518102
\(582\) 0 0
\(583\) 7996.00 0.568028
\(584\) 19856.4 1.40696
\(585\) 0 0
\(586\) 729.738 0.0514423
\(587\) −15750.3 −1.10747 −0.553736 0.832693i \(-0.686798\pi\)
−0.553736 + 0.832693i \(0.686798\pi\)
\(588\) 0 0
\(589\) −8604.42 −0.601934
\(590\) 0 0
\(591\) 0 0
\(592\) 18990.0 1.31838
\(593\) −417.878 −0.0289379 −0.0144690 0.999895i \(-0.504606\pi\)
−0.0144690 + 0.999895i \(0.504606\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −431.385 −0.0296480
\(597\) 0 0
\(598\) −10912.1 −0.746201
\(599\) 19997.3 1.36406 0.682028 0.731326i \(-0.261098\pi\)
0.682028 + 0.731326i \(0.261098\pi\)
\(600\) 0 0
\(601\) −15992.6 −1.08545 −0.542723 0.839912i \(-0.682607\pi\)
−0.542723 + 0.839912i \(0.682607\pi\)
\(602\) 4093.73 0.277156
\(603\) 0 0
\(604\) −1352.37 −0.0911045
\(605\) 0 0
\(606\) 0 0
\(607\) −14159.2 −0.946793 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(608\) −2601.08 −0.173499
\(609\) 0 0
\(610\) 0 0
\(611\) −225.854 −0.0149543
\(612\) 0 0
\(613\) 4629.41 0.305025 0.152512 0.988302i \(-0.451264\pi\)
0.152512 + 0.988302i \(0.451264\pi\)
\(614\) −4963.87 −0.326263
\(615\) 0 0
\(616\) 6449.11 0.421821
\(617\) −23165.3 −1.51151 −0.755753 0.654857i \(-0.772729\pi\)
−0.755753 + 0.654857i \(0.772729\pi\)
\(618\) 0 0
\(619\) 12370.6 0.803258 0.401629 0.915803i \(-0.368444\pi\)
0.401629 + 0.915803i \(0.368444\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −3137.36 −0.202245
\(623\) 1191.58 0.0766285
\(624\) 0 0
\(625\) 0 0
\(626\) 3708.02 0.236745
\(627\) 0 0
\(628\) 689.826 0.0438329
\(629\) 31998.9 2.02842
\(630\) 0 0
\(631\) 13980.2 0.882002 0.441001 0.897507i \(-0.354623\pi\)
0.441001 + 0.897507i \(0.354623\pi\)
\(632\) −3763.61 −0.236880
\(633\) 0 0
\(634\) 16799.7 1.05237
\(635\) 0 0
\(636\) 0 0
\(637\) −947.814 −0.0589541
\(638\) −4643.36 −0.288139
\(639\) 0 0
\(640\) 0 0
\(641\) 16060.9 0.989655 0.494828 0.868991i \(-0.335231\pi\)
0.494828 + 0.868991i \(0.335231\pi\)
\(642\) 0 0
\(643\) 4502.17 0.276125 0.138063 0.990424i \(-0.455913\pi\)
0.138063 + 0.990424i \(0.455913\pi\)
\(644\) −2006.25 −0.122760
\(645\) 0 0
\(646\) 9974.87 0.607517
\(647\) 29414.8 1.78735 0.893675 0.448715i \(-0.148118\pi\)
0.893675 + 0.448715i \(0.148118\pi\)
\(648\) 0 0
\(649\) −23565.0 −1.42528
\(650\) 0 0
\(651\) 0 0
\(652\) 1317.16 0.0791164
\(653\) 13013.6 0.779882 0.389941 0.920840i \(-0.372495\pi\)
0.389941 + 0.920840i \(0.372495\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −17566.9 −1.04554
\(657\) 0 0
\(658\) 211.345 0.0125214
\(659\) −23474.2 −1.38759 −0.693797 0.720171i \(-0.744063\pi\)
−0.693797 + 0.720171i \(0.744063\pi\)
\(660\) 0 0
\(661\) −9266.36 −0.545264 −0.272632 0.962118i \(-0.587894\pi\)
−0.272632 + 0.962118i \(0.587894\pi\)
\(662\) −25040.4 −1.47013
\(663\) 0 0
\(664\) 24963.0 1.45896
\(665\) 0 0
\(666\) 0 0
\(667\) 10241.0 0.594501
\(668\) 472.176 0.0273489
\(669\) 0 0
\(670\) 0 0
\(671\) −12269.7 −0.705909
\(672\) 0 0
\(673\) 25067.2 1.43576 0.717882 0.696164i \(-0.245112\pi\)
0.717882 + 0.696164i \(0.245112\pi\)
\(674\) −75.3288 −0.00430498
\(675\) 0 0
\(676\) 2394.68 0.136247
\(677\) −22409.6 −1.27219 −0.636093 0.771613i \(-0.719450\pi\)
−0.636093 + 0.771613i \(0.719450\pi\)
\(678\) 0 0
\(679\) −7419.11 −0.419322
\(680\) 0 0
\(681\) 0 0
\(682\) −19245.5 −1.08057
\(683\) −8757.53 −0.490626 −0.245313 0.969444i \(-0.578891\pi\)
−0.245313 + 0.969444i \(0.578891\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 886.925 0.0493629
\(687\) 0 0
\(688\) −11707.4 −0.648750
\(689\) 4043.09 0.223555
\(690\) 0 0
\(691\) 8468.42 0.466214 0.233107 0.972451i \(-0.425111\pi\)
0.233107 + 0.972451i \(0.425111\pi\)
\(692\) −4326.90 −0.237694
\(693\) 0 0
\(694\) −20294.8 −1.11006
\(695\) 0 0
\(696\) 0 0
\(697\) −29601.0 −1.60864
\(698\) −26555.1 −1.44001
\(699\) 0 0
\(700\) 0 0
\(701\) −15996.9 −0.861906 −0.430953 0.902374i \(-0.641823\pi\)
−0.430953 + 0.902374i \(0.641823\pi\)
\(702\) 0 0
\(703\) 16224.2 0.870423
\(704\) −21659.8 −1.15956
\(705\) 0 0
\(706\) 7240.03 0.385952
\(707\) 1692.83 0.0900503
\(708\) 0 0
\(709\) 19903.0 1.05426 0.527131 0.849784i \(-0.323268\pi\)
0.527131 + 0.849784i \(0.323268\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −4099.58 −0.215784
\(713\) 42446.1 2.22948
\(714\) 0 0
\(715\) 0 0
\(716\) 3913.30 0.204256
\(717\) 0 0
\(718\) 8179.59 0.425153
\(719\) 11073.1 0.574347 0.287174 0.957879i \(-0.407284\pi\)
0.287174 + 0.957879i \(0.407284\pi\)
\(720\) 0 0
\(721\) 11757.0 0.607288
\(722\) −12678.4 −0.653520
\(723\) 0 0
\(724\) −1921.06 −0.0986125
\(725\) 0 0
\(726\) 0 0
\(727\) −31652.7 −1.61476 −0.807382 0.590029i \(-0.799116\pi\)
−0.807382 + 0.590029i \(0.799116\pi\)
\(728\) 3260.92 0.166013
\(729\) 0 0
\(730\) 0 0
\(731\) −19727.5 −0.998149
\(732\) 0 0
\(733\) −16958.3 −0.854528 −0.427264 0.904127i \(-0.640523\pi\)
−0.427264 + 0.904127i \(0.640523\pi\)
\(734\) −8230.16 −0.413870
\(735\) 0 0
\(736\) 12831.3 0.642618
\(737\) 555.065 0.0277423
\(738\) 0 0
\(739\) −11616.6 −0.578245 −0.289123 0.957292i \(-0.593364\pi\)
−0.289123 + 0.957292i \(0.593364\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −3783.36 −0.187185
\(743\) 15928.0 0.786464 0.393232 0.919439i \(-0.371357\pi\)
0.393232 + 0.919439i \(0.371357\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 6762.19 0.331879
\(747\) 0 0
\(748\) −4383.57 −0.214277
\(749\) 10548.2 0.514582
\(750\) 0 0
\(751\) 25571.9 1.24252 0.621260 0.783604i \(-0.286621\pi\)
0.621260 + 0.783604i \(0.286621\pi\)
\(752\) −604.412 −0.0293094
\(753\) 0 0
\(754\) −2347.87 −0.113401
\(755\) 0 0
\(756\) 0 0
\(757\) −6202.41 −0.297794 −0.148897 0.988853i \(-0.547572\pi\)
−0.148897 + 0.988853i \(0.547572\pi\)
\(758\) −1738.77 −0.0833179
\(759\) 0 0
\(760\) 0 0
\(761\) 29199.1 1.39089 0.695444 0.718580i \(-0.255208\pi\)
0.695444 + 0.718580i \(0.255208\pi\)
\(762\) 0 0
\(763\) −8766.87 −0.415966
\(764\) −492.539 −0.0233239
\(765\) 0 0
\(766\) 3025.00 0.142686
\(767\) −11915.4 −0.560938
\(768\) 0 0
\(769\) 21838.2 1.02407 0.512033 0.858966i \(-0.328892\pi\)
0.512033 + 0.858966i \(0.328892\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 962.985 0.0448945
\(773\) −25544.8 −1.18859 −0.594296 0.804246i \(-0.702569\pi\)
−0.594296 + 0.804246i \(0.702569\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 25525.2 1.18080
\(777\) 0 0
\(778\) 2901.82 0.133721
\(779\) −15008.4 −0.690286
\(780\) 0 0
\(781\) −36418.6 −1.66858
\(782\) −49206.6 −2.25016
\(783\) 0 0
\(784\) −2536.46 −0.115546
\(785\) 0 0
\(786\) 0 0
\(787\) 37223.2 1.68598 0.842989 0.537931i \(-0.180794\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(788\) 2749.91 0.124317
\(789\) 0 0
\(790\) 0 0
\(791\) 9591.42 0.431140
\(792\) 0 0
\(793\) −6204.03 −0.277820
\(794\) 5135.18 0.229522
\(795\) 0 0
\(796\) −3763.83 −0.167595
\(797\) −40384.6 −1.79485 −0.897425 0.441168i \(-0.854564\pi\)
−0.897425 + 0.441168i \(0.854564\pi\)
\(798\) 0 0
\(799\) −1018.46 −0.0450945
\(800\) 0 0
\(801\) 0 0
\(802\) 10788.9 0.475023
\(803\) 31540.7 1.38611
\(804\) 0 0
\(805\) 0 0
\(806\) −9731.28 −0.425272
\(807\) 0 0
\(808\) −5824.14 −0.253580
\(809\) 1955.76 0.0849948 0.0424974 0.999097i \(-0.486469\pi\)
0.0424974 + 0.999097i \(0.486469\pi\)
\(810\) 0 0
\(811\) −34301.8 −1.48520 −0.742600 0.669735i \(-0.766408\pi\)
−0.742600 + 0.669735i \(0.766408\pi\)
\(812\) −431.669 −0.0186559
\(813\) 0 0
\(814\) 36288.7 1.56255
\(815\) 0 0
\(816\) 0 0
\(817\) −10002.3 −0.428319
\(818\) −30255.8 −1.29324
\(819\) 0 0
\(820\) 0 0
\(821\) 13665.6 0.580918 0.290459 0.956887i \(-0.406192\pi\)
0.290459 + 0.956887i \(0.406192\pi\)
\(822\) 0 0
\(823\) 21519.5 0.911449 0.455724 0.890121i \(-0.349380\pi\)
0.455724 + 0.890121i \(0.349380\pi\)
\(824\) −40449.7 −1.71011
\(825\) 0 0
\(826\) 11149.9 0.469679
\(827\) 35220.6 1.48094 0.740471 0.672088i \(-0.234602\pi\)
0.740471 + 0.672088i \(0.234602\pi\)
\(828\) 0 0
\(829\) 31365.5 1.31408 0.657039 0.753857i \(-0.271809\pi\)
0.657039 + 0.753857i \(0.271809\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −10952.0 −0.456362
\(833\) −4274.04 −0.177775
\(834\) 0 0
\(835\) 0 0
\(836\) −2222.58 −0.0919491
\(837\) 0 0
\(838\) −7067.48 −0.291339
\(839\) 28287.1 1.16398 0.581990 0.813196i \(-0.302274\pi\)
0.581990 + 0.813196i \(0.302274\pi\)
\(840\) 0 0
\(841\) −22185.5 −0.909653
\(842\) 35035.8 1.43398
\(843\) 0 0
\(844\) −7414.11 −0.302374
\(845\) 0 0
\(846\) 0 0
\(847\) 927.026 0.0376068
\(848\) 10819.8 0.438152
\(849\) 0 0
\(850\) 0 0
\(851\) −80035.0 −3.22393
\(852\) 0 0
\(853\) −9405.41 −0.377533 −0.188766 0.982022i \(-0.560449\pi\)
−0.188766 + 0.982022i \(0.560449\pi\)
\(854\) 5805.47 0.232622
\(855\) 0 0
\(856\) −36290.6 −1.44905
\(857\) −27966.9 −1.11474 −0.557369 0.830265i \(-0.688189\pi\)
−0.557369 + 0.830265i \(0.688189\pi\)
\(858\) 0 0
\(859\) 6281.11 0.249486 0.124743 0.992189i \(-0.460189\pi\)
0.124743 + 0.992189i \(0.460189\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 16624.7 0.656890
\(863\) −4757.13 −0.187642 −0.0938208 0.995589i \(-0.529908\pi\)
−0.0938208 + 0.995589i \(0.529908\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −20743.3 −0.813954
\(867\) 0 0
\(868\) −1789.15 −0.0699629
\(869\) −5978.28 −0.233371
\(870\) 0 0
\(871\) 280.663 0.0109184
\(872\) 30162.1 1.17135
\(873\) 0 0
\(874\) −24949.0 −0.965574
\(875\) 0 0
\(876\) 0 0
\(877\) 30240.5 1.16437 0.582184 0.813057i \(-0.302198\pi\)
0.582184 + 0.813057i \(0.302198\pi\)
\(878\) −14402.5 −0.553601
\(879\) 0 0
\(880\) 0 0
\(881\) −44875.5 −1.71611 −0.858056 0.513556i \(-0.828328\pi\)
−0.858056 + 0.513556i \(0.828328\pi\)
\(882\) 0 0
\(883\) −4892.13 −0.186448 −0.0932238 0.995645i \(-0.529717\pi\)
−0.0932238 + 0.995645i \(0.529717\pi\)
\(884\) −2216.51 −0.0843317
\(885\) 0 0
\(886\) −14186.2 −0.537916
\(887\) 1761.40 0.0666765 0.0333382 0.999444i \(-0.489386\pi\)
0.0333382 + 0.999444i \(0.489386\pi\)
\(888\) 0 0
\(889\) −8494.43 −0.320466
\(890\) 0 0
\(891\) 0 0
\(892\) −8367.22 −0.314075
\(893\) −516.384 −0.0193507
\(894\) 0 0
\(895\) 0 0
\(896\) 6954.86 0.259314
\(897\) 0 0
\(898\) 18702.2 0.694988
\(899\) 9132.79 0.338816
\(900\) 0 0
\(901\) 18231.8 0.674128
\(902\) −33569.3 −1.23918
\(903\) 0 0
\(904\) −32999.0 −1.21408
\(905\) 0 0
\(906\) 0 0
\(907\) −23689.1 −0.867238 −0.433619 0.901096i \(-0.642764\pi\)
−0.433619 + 0.901096i \(0.642764\pi\)
\(908\) 1334.30 0.0487669
\(909\) 0 0
\(910\) 0 0
\(911\) 13877.3 0.504692 0.252346 0.967637i \(-0.418798\pi\)
0.252346 + 0.967637i \(0.418798\pi\)
\(912\) 0 0
\(913\) 39652.2 1.43735
\(914\) 7500.09 0.271423
\(915\) 0 0
\(916\) −5397.18 −0.194681
\(917\) 13876.9 0.499735
\(918\) 0 0
\(919\) 14331.6 0.514426 0.257213 0.966355i \(-0.417196\pi\)
0.257213 + 0.966355i \(0.417196\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −15705.0 −0.560971
\(923\) −18414.7 −0.656692
\(924\) 0 0
\(925\) 0 0
\(926\) 48930.2 1.73644
\(927\) 0 0
\(928\) 2760.80 0.0976592
\(929\) −16668.4 −0.588668 −0.294334 0.955703i \(-0.595098\pi\)
−0.294334 + 0.955703i \(0.595098\pi\)
\(930\) 0 0
\(931\) −2167.04 −0.0762857
\(932\) −799.300 −0.0280922
\(933\) 0 0
\(934\) −17523.1 −0.613891
\(935\) 0 0
\(936\) 0 0
\(937\) −30384.9 −1.05937 −0.529685 0.848194i \(-0.677690\pi\)
−0.529685 + 0.848194i \(0.677690\pi\)
\(938\) −262.632 −0.00914206
\(939\) 0 0
\(940\) 0 0
\(941\) 1196.35 0.0414452 0.0207226 0.999785i \(-0.493403\pi\)
0.0207226 + 0.999785i \(0.493403\pi\)
\(942\) 0 0
\(943\) 74037.5 2.55673
\(944\) −31886.9 −1.09940
\(945\) 0 0
\(946\) −22372.1 −0.768901
\(947\) 1788.41 0.0613681 0.0306840 0.999529i \(-0.490231\pi\)
0.0306840 + 0.999529i \(0.490231\pi\)
\(948\) 0 0
\(949\) 15948.2 0.545523
\(950\) 0 0
\(951\) 0 0
\(952\) 14704.7 0.500612
\(953\) 8578.60 0.291593 0.145796 0.989315i \(-0.453426\pi\)
0.145796 + 0.989315i \(0.453426\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −6640.06 −0.224639
\(957\) 0 0
\(958\) −6198.95 −0.209059
\(959\) 15476.6 0.521133
\(960\) 0 0
\(961\) 8061.99 0.270618
\(962\) 18349.0 0.614963
\(963\) 0 0
\(964\) −6.39489 −0.000213657 0
\(965\) 0 0
\(966\) 0 0
\(967\) 55459.3 1.84431 0.922156 0.386818i \(-0.126426\pi\)
0.922156 + 0.386818i \(0.126426\pi\)
\(968\) −3189.40 −0.105900
\(969\) 0 0
\(970\) 0 0
\(971\) −22047.3 −0.728662 −0.364331 0.931270i \(-0.618702\pi\)
−0.364331 + 0.931270i \(0.618702\pi\)
\(972\) 0 0
\(973\) 3696.27 0.121785
\(974\) −14444.6 −0.475191
\(975\) 0 0
\(976\) −16602.7 −0.544507
\(977\) 14402.3 0.471617 0.235809 0.971800i \(-0.424226\pi\)
0.235809 + 0.971800i \(0.424226\pi\)
\(978\) 0 0
\(979\) −6511.94 −0.212587
\(980\) 0 0
\(981\) 0 0
\(982\) −1389.58 −0.0451561
\(983\) 7817.11 0.253639 0.126819 0.991926i \(-0.459523\pi\)
0.126819 + 0.991926i \(0.459523\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −10587.4 −0.341959
\(987\) 0 0
\(988\) −1123.82 −0.0361878
\(989\) 49342.0 1.58643
\(990\) 0 0
\(991\) 24501.6 0.785386 0.392693 0.919670i \(-0.371543\pi\)
0.392693 + 0.919670i \(0.371543\pi\)
\(992\) 11442.8 0.366239
\(993\) 0 0
\(994\) 17231.7 0.549855
\(995\) 0 0
\(996\) 0 0
\(997\) 50696.0 1.61039 0.805195 0.593010i \(-0.202060\pi\)
0.805195 + 0.593010i \(0.202060\pi\)
\(998\) 1548.80 0.0491245
\(999\) 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 1575.4.a.z.1.1 2
3.2 odd 2 175.4.a.c.1.2 2
5.4 even 2 315.4.a.f.1.2 2
15.2 even 4 175.4.b.c.99.2 4
15.8 even 4 175.4.b.c.99.3 4
15.14 odd 2 35.4.a.b.1.1 2
21.20 even 2 1225.4.a.m.1.2 2
35.34 odd 2 2205.4.a.u.1.2 2
60.59 even 2 560.4.a.r.1.1 2
105.44 odd 6 245.4.e.h.116.2 4
105.59 even 6 245.4.e.i.226.2 4
105.74 odd 6 245.4.e.h.226.2 4
105.89 even 6 245.4.e.i.116.2 4
105.104 even 2 245.4.a.k.1.1 2
120.29 odd 2 2240.4.a.bn.1.1 2
120.59 even 2 2240.4.a.bo.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.1 2 15.14 odd 2
175.4.a.c.1.2 2 3.2 odd 2
175.4.b.c.99.2 4 15.2 even 4
175.4.b.c.99.3 4 15.8 even 4
245.4.a.k.1.1 2 105.104 even 2
245.4.e.h.116.2 4 105.44 odd 6
245.4.e.h.226.2 4 105.74 odd 6
245.4.e.i.116.2 4 105.89 even 6
245.4.e.i.226.2 4 105.59 even 6
315.4.a.f.1.2 2 5.4 even 2
560.4.a.r.1.1 2 60.59 even 2
1225.4.a.m.1.2 2 21.20 even 2
1575.4.a.z.1.1 2 1.1 even 1 trivial
2205.4.a.u.1.2 2 35.34 odd 2
2240.4.a.bn.1.1 2 120.29 odd 2
2240.4.a.bo.1.2 2 120.59 even 2