Properties

Label 1575.4.a.bi.1.3
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1575,4,Mod(1,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.3030748.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 21x^{2} + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.19617\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.19617 q^{2} -6.56918 q^{4} +7.00000 q^{7} -17.4272 q^{8} +O(q^{10})\) \(q+1.19617 q^{2} -6.56918 q^{4} +7.00000 q^{7} -17.4272 q^{8} +4.52701 q^{11} -14.5692 q^{13} +8.37319 q^{14} +31.7075 q^{16} -15.5502 q^{17} -14.8616 q^{19} +5.41507 q^{22} +166.672 q^{23} -17.4272 q^{26} -45.9843 q^{28} -194.865 q^{29} +104.538 q^{31} +177.345 q^{32} -18.6007 q^{34} +47.5377 q^{37} -17.7770 q^{38} +378.633 q^{41} -194.969 q^{43} -29.7387 q^{44} +199.368 q^{46} -488.661 q^{47} +49.0000 q^{49} +95.7075 q^{52} -316.432 q^{53} -121.990 q^{56} -233.091 q^{58} +350.256 q^{59} +115.613 q^{61} +125.045 q^{62} -41.5253 q^{64} +434.890 q^{67} +102.152 q^{68} -410.119 q^{71} -464.182 q^{73} +56.8631 q^{74} +97.6288 q^{76} +31.6891 q^{77} -290.399 q^{79} +452.909 q^{82} -578.005 q^{83} -233.215 q^{86} -78.8931 q^{88} -937.022 q^{89} -101.984 q^{91} -1094.90 q^{92} -584.522 q^{94} +839.629 q^{97} +58.6123 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4} + 28 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{4} + 28 q^{7} - 22 q^{13} + 18 q^{16} - 132 q^{19} - 196 q^{22} + 70 q^{28} - 126 q^{31} - 546 q^{34} - 354 q^{37} - 272 q^{43} - 182 q^{46} + 196 q^{49} + 274 q^{52} - 98 q^{58} - 1170 q^{61} - 1726 q^{64} - 38 q^{67} - 188 q^{73} - 988 q^{76} - 690 q^{79} + 2646 q^{82} - 896 q^{88} - 154 q^{91} - 1540 q^{94} + 1980 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.19617 0.422910 0.211455 0.977388i \(-0.432180\pi\)
0.211455 + 0.977388i \(0.432180\pi\)
\(3\) 0 0
\(4\) −6.56918 −0.821147
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) −17.4272 −0.770181
\(9\) 0 0
\(10\) 0 0
\(11\) 4.52701 0.124086 0.0620430 0.998073i \(-0.480238\pi\)
0.0620430 + 0.998073i \(0.480238\pi\)
\(12\) 0 0
\(13\) −14.5692 −0.310828 −0.155414 0.987849i \(-0.549671\pi\)
−0.155414 + 0.987849i \(0.549671\pi\)
\(14\) 8.37319 0.159845
\(15\) 0 0
\(16\) 31.7075 0.495430
\(17\) −15.5502 −0.221852 −0.110926 0.993829i \(-0.535382\pi\)
−0.110926 + 0.993829i \(0.535382\pi\)
\(18\) 0 0
\(19\) −14.8616 −0.179447 −0.0897235 0.995967i \(-0.528598\pi\)
−0.0897235 + 0.995967i \(0.528598\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.41507 0.0524771
\(23\) 166.672 1.51102 0.755511 0.655136i \(-0.227389\pi\)
0.755511 + 0.655136i \(0.227389\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −17.4272 −0.131452
\(27\) 0 0
\(28\) −45.9843 −0.310365
\(29\) −194.865 −1.24777 −0.623887 0.781514i \(-0.714447\pi\)
−0.623887 + 0.781514i \(0.714447\pi\)
\(30\) 0 0
\(31\) 104.538 0.605662 0.302831 0.953044i \(-0.402068\pi\)
0.302831 + 0.953044i \(0.402068\pi\)
\(32\) 177.345 0.979703
\(33\) 0 0
\(34\) −18.6007 −0.0938232
\(35\) 0 0
\(36\) 0 0
\(37\) 47.5377 0.211220 0.105610 0.994408i \(-0.466320\pi\)
0.105610 + 0.994408i \(0.466320\pi\)
\(38\) −17.7770 −0.0758899
\(39\) 0 0
\(40\) 0 0
\(41\) 378.633 1.44226 0.721128 0.692802i \(-0.243624\pi\)
0.721128 + 0.692802i \(0.243624\pi\)
\(42\) 0 0
\(43\) −194.969 −0.691452 −0.345726 0.938336i \(-0.612367\pi\)
−0.345726 + 0.938336i \(0.612367\pi\)
\(44\) −29.7387 −0.101893
\(45\) 0 0
\(46\) 199.368 0.639026
\(47\) −488.661 −1.51657 −0.758283 0.651926i \(-0.773961\pi\)
−0.758283 + 0.651926i \(0.773961\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 95.7075 0.255236
\(53\) −316.432 −0.820099 −0.410050 0.912063i \(-0.634489\pi\)
−0.410050 + 0.912063i \(0.634489\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −121.990 −0.291101
\(57\) 0 0
\(58\) −233.091 −0.527696
\(59\) 350.256 0.772871 0.386436 0.922316i \(-0.373706\pi\)
0.386436 + 0.922316i \(0.373706\pi\)
\(60\) 0 0
\(61\) 115.613 0.242668 0.121334 0.992612i \(-0.461283\pi\)
0.121334 + 0.992612i \(0.461283\pi\)
\(62\) 125.045 0.256140
\(63\) 0 0
\(64\) −41.5253 −0.0811041
\(65\) 0 0
\(66\) 0 0
\(67\) 434.890 0.792989 0.396494 0.918037i \(-0.370227\pi\)
0.396494 + 0.918037i \(0.370227\pi\)
\(68\) 102.152 0.182173
\(69\) 0 0
\(70\) 0 0
\(71\) −410.119 −0.685523 −0.342761 0.939422i \(-0.611362\pi\)
−0.342761 + 0.939422i \(0.611362\pi\)
\(72\) 0 0
\(73\) −464.182 −0.744225 −0.372112 0.928188i \(-0.621366\pi\)
−0.372112 + 0.928188i \(0.621366\pi\)
\(74\) 56.8631 0.0893271
\(75\) 0 0
\(76\) 97.6288 0.147352
\(77\) 31.6891 0.0469001
\(78\) 0 0
\(79\) −290.399 −0.413576 −0.206788 0.978386i \(-0.566301\pi\)
−0.206788 + 0.978386i \(0.566301\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 452.909 0.609944
\(83\) −578.005 −0.764390 −0.382195 0.924082i \(-0.624832\pi\)
−0.382195 + 0.924082i \(0.624832\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −233.215 −0.292422
\(87\) 0 0
\(88\) −78.8931 −0.0955686
\(89\) −937.022 −1.11600 −0.558001 0.829841i \(-0.688431\pi\)
−0.558001 + 0.829841i \(0.688431\pi\)
\(90\) 0 0
\(91\) −101.984 −0.117482
\(92\) −1094.90 −1.24077
\(93\) 0 0
\(94\) −584.522 −0.641371
\(95\) 0 0
\(96\) 0 0
\(97\) 839.629 0.878880 0.439440 0.898272i \(-0.355177\pi\)
0.439440 + 0.898272i \(0.355177\pi\)
\(98\) 58.6123 0.0604157
\(99\) 0 0
\(100\) 0 0
\(101\) 1012.69 0.997690 0.498845 0.866691i \(-0.333758\pi\)
0.498845 + 0.866691i \(0.333758\pi\)
\(102\) 0 0
\(103\) −758.506 −0.725610 −0.362805 0.931865i \(-0.618181\pi\)
−0.362805 + 0.931865i \(0.618181\pi\)
\(104\) 253.900 0.239394
\(105\) 0 0
\(106\) −378.506 −0.346828
\(107\) −264.372 −0.238858 −0.119429 0.992843i \(-0.538106\pi\)
−0.119429 + 0.992843i \(0.538106\pi\)
\(108\) 0 0
\(109\) −775.538 −0.681496 −0.340748 0.940155i \(-0.610680\pi\)
−0.340748 + 0.940155i \(0.610680\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 221.953 0.187255
\(113\) 437.649 0.364341 0.182171 0.983267i \(-0.441688\pi\)
0.182171 + 0.983267i \(0.441688\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1280.10 1.02461
\(117\) 0 0
\(118\) 418.965 0.326855
\(119\) −108.851 −0.0838520
\(120\) 0 0
\(121\) −1310.51 −0.984603
\(122\) 138.293 0.102627
\(123\) 0 0
\(124\) −686.727 −0.497338
\(125\) 0 0
\(126\) 0 0
\(127\) 991.412 0.692705 0.346353 0.938104i \(-0.387420\pi\)
0.346353 + 0.938104i \(0.387420\pi\)
\(128\) −1468.43 −1.01400
\(129\) 0 0
\(130\) 0 0
\(131\) −1529.62 −1.02018 −0.510089 0.860122i \(-0.670388\pi\)
−0.510089 + 0.860122i \(0.670388\pi\)
\(132\) 0 0
\(133\) −104.031 −0.0678246
\(134\) 520.202 0.335363
\(135\) 0 0
\(136\) 270.997 0.170866
\(137\) −220.095 −0.137255 −0.0686277 0.997642i \(-0.521862\pi\)
−0.0686277 + 0.997642i \(0.521862\pi\)
\(138\) 0 0
\(139\) −2041.16 −1.24553 −0.622767 0.782407i \(-0.713991\pi\)
−0.622767 + 0.782407i \(0.713991\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −490.571 −0.289914
\(143\) −65.9548 −0.0385694
\(144\) 0 0
\(145\) 0 0
\(146\) −555.241 −0.314740
\(147\) 0 0
\(148\) −312.283 −0.173443
\(149\) −591.993 −0.325490 −0.162745 0.986668i \(-0.552035\pi\)
−0.162745 + 0.986668i \(0.552035\pi\)
\(150\) 0 0
\(151\) −1962.54 −1.05768 −0.528839 0.848722i \(-0.677373\pi\)
−0.528839 + 0.848722i \(0.677373\pi\)
\(152\) 258.997 0.138207
\(153\) 0 0
\(154\) 37.9055 0.0198345
\(155\) 0 0
\(156\) 0 0
\(157\) −684.081 −0.347743 −0.173871 0.984768i \(-0.555628\pi\)
−0.173871 + 0.984768i \(0.555628\pi\)
\(158\) −347.367 −0.174905
\(159\) 0 0
\(160\) 0 0
\(161\) 1166.70 0.571112
\(162\) 0 0
\(163\) −1199.68 −0.576478 −0.288239 0.957559i \(-0.593070\pi\)
−0.288239 + 0.957559i \(0.593070\pi\)
\(164\) −2487.31 −1.18430
\(165\) 0 0
\(166\) −691.393 −0.323268
\(167\) −2211.33 −1.02466 −0.512328 0.858790i \(-0.671217\pi\)
−0.512328 + 0.858790i \(0.671217\pi\)
\(168\) 0 0
\(169\) −1984.74 −0.903386
\(170\) 0 0
\(171\) 0 0
\(172\) 1280.78 0.567784
\(173\) 1973.01 0.867082 0.433541 0.901134i \(-0.357264\pi\)
0.433541 + 0.901134i \(0.357264\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 143.540 0.0614759
\(177\) 0 0
\(178\) −1120.84 −0.471968
\(179\) 30.8784 0.0128936 0.00644681 0.999979i \(-0.497948\pi\)
0.00644681 + 0.999979i \(0.497948\pi\)
\(180\) 0 0
\(181\) −2459.16 −1.00988 −0.504938 0.863155i \(-0.668485\pi\)
−0.504938 + 0.863155i \(0.668485\pi\)
\(182\) −121.990 −0.0496843
\(183\) 0 0
\(184\) −2904.63 −1.16376
\(185\) 0 0
\(186\) 0 0
\(187\) −70.3959 −0.0275287
\(188\) 3210.10 1.24532
\(189\) 0 0
\(190\) 0 0
\(191\) 4163.26 1.57719 0.788595 0.614913i \(-0.210809\pi\)
0.788595 + 0.614913i \(0.210809\pi\)
\(192\) 0 0
\(193\) −2057.39 −0.767326 −0.383663 0.923473i \(-0.625338\pi\)
−0.383663 + 0.923473i \(0.625338\pi\)
\(194\) 1004.34 0.371687
\(195\) 0 0
\(196\) −321.890 −0.117307
\(197\) 2431.48 0.879368 0.439684 0.898153i \(-0.355090\pi\)
0.439684 + 0.898153i \(0.355090\pi\)
\(198\) 0 0
\(199\) −5291.18 −1.88483 −0.942417 0.334441i \(-0.891453\pi\)
−0.942417 + 0.334441i \(0.891453\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1211.35 0.421933
\(203\) −1364.05 −0.471614
\(204\) 0 0
\(205\) 0 0
\(206\) −907.302 −0.306868
\(207\) 0 0
\(208\) −461.953 −0.153994
\(209\) −67.2788 −0.0222669
\(210\) 0 0
\(211\) −3586.37 −1.17012 −0.585061 0.810989i \(-0.698929\pi\)
−0.585061 + 0.810989i \(0.698929\pi\)
\(212\) 2078.70 0.673422
\(213\) 0 0
\(214\) −316.234 −0.101015
\(215\) 0 0
\(216\) 0 0
\(217\) 731.764 0.228919
\(218\) −927.675 −0.288211
\(219\) 0 0
\(220\) 0 0
\(221\) 226.554 0.0689577
\(222\) 0 0
\(223\) −1127.13 −0.338467 −0.169234 0.985576i \(-0.554129\pi\)
−0.169234 + 0.985576i \(0.554129\pi\)
\(224\) 1241.42 0.370293
\(225\) 0 0
\(226\) 523.503 0.154084
\(227\) −674.839 −0.197315 −0.0986577 0.995121i \(-0.531455\pi\)
−0.0986577 + 0.995121i \(0.531455\pi\)
\(228\) 0 0
\(229\) −5906.65 −1.70447 −0.852233 0.523163i \(-0.824752\pi\)
−0.852233 + 0.523163i \(0.824752\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3395.95 0.961012
\(233\) 305.559 0.0859136 0.0429568 0.999077i \(-0.486322\pi\)
0.0429568 + 0.999077i \(0.486322\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2300.89 −0.634641
\(237\) 0 0
\(238\) −130.205 −0.0354619
\(239\) −3268.07 −0.884494 −0.442247 0.896893i \(-0.645819\pi\)
−0.442247 + 0.896893i \(0.645819\pi\)
\(240\) 0 0
\(241\) 477.426 0.127609 0.0638044 0.997962i \(-0.479677\pi\)
0.0638044 + 0.997962i \(0.479677\pi\)
\(242\) −1567.59 −0.416398
\(243\) 0 0
\(244\) −759.483 −0.199266
\(245\) 0 0
\(246\) 0 0
\(247\) 216.522 0.0557772
\(248\) −1821.80 −0.466469
\(249\) 0 0
\(250\) 0 0
\(251\) −4439.63 −1.11644 −0.558221 0.829692i \(-0.688516\pi\)
−0.558221 + 0.829692i \(0.688516\pi\)
\(252\) 0 0
\(253\) 754.525 0.187496
\(254\) 1185.90 0.292952
\(255\) 0 0
\(256\) −1424.29 −0.347728
\(257\) −2125.51 −0.515897 −0.257949 0.966159i \(-0.583047\pi\)
−0.257949 + 0.966159i \(0.583047\pi\)
\(258\) 0 0
\(259\) 332.764 0.0798337
\(260\) 0 0
\(261\) 0 0
\(262\) −1829.68 −0.431443
\(263\) −993.198 −0.232864 −0.116432 0.993199i \(-0.537146\pi\)
−0.116432 + 0.993199i \(0.537146\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −124.439 −0.0286837
\(267\) 0 0
\(268\) −2856.87 −0.651161
\(269\) 6458.23 1.46381 0.731905 0.681406i \(-0.238631\pi\)
0.731905 + 0.681406i \(0.238631\pi\)
\(270\) 0 0
\(271\) 417.093 0.0934930 0.0467465 0.998907i \(-0.485115\pi\)
0.0467465 + 0.998907i \(0.485115\pi\)
\(272\) −493.059 −0.109912
\(273\) 0 0
\(274\) −263.271 −0.0580467
\(275\) 0 0
\(276\) 0 0
\(277\) 4862.87 1.05481 0.527404 0.849615i \(-0.323166\pi\)
0.527404 + 0.849615i \(0.323166\pi\)
\(278\) −2441.58 −0.526748
\(279\) 0 0
\(280\) 0 0
\(281\) −2026.16 −0.430144 −0.215072 0.976598i \(-0.568999\pi\)
−0.215072 + 0.976598i \(0.568999\pi\)
\(282\) 0 0
\(283\) −6445.96 −1.35397 −0.676983 0.735998i \(-0.736713\pi\)
−0.676983 + 0.735998i \(0.736713\pi\)
\(284\) 2694.14 0.562915
\(285\) 0 0
\(286\) −78.8931 −0.0163114
\(287\) 2650.43 0.545122
\(288\) 0 0
\(289\) −4671.19 −0.950782
\(290\) 0 0
\(291\) 0 0
\(292\) 3049.30 0.611118
\(293\) 541.850 0.108038 0.0540191 0.998540i \(-0.482797\pi\)
0.0540191 + 0.998540i \(0.482797\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −828.449 −0.162678
\(297\) 0 0
\(298\) −708.124 −0.137653
\(299\) −2428.27 −0.469668
\(300\) 0 0
\(301\) −1364.78 −0.261344
\(302\) −2347.53 −0.447303
\(303\) 0 0
\(304\) −471.226 −0.0889035
\(305\) 0 0
\(306\) 0 0
\(307\) 4494.53 0.835558 0.417779 0.908549i \(-0.362809\pi\)
0.417779 + 0.908549i \(0.362809\pi\)
\(308\) −208.171 −0.0385119
\(309\) 0 0
\(310\) 0 0
\(311\) 2869.51 0.523199 0.261599 0.965177i \(-0.415750\pi\)
0.261599 + 0.965177i \(0.415750\pi\)
\(312\) 0 0
\(313\) 1653.23 0.298550 0.149275 0.988796i \(-0.452306\pi\)
0.149275 + 0.988796i \(0.452306\pi\)
\(314\) −818.277 −0.147064
\(315\) 0 0
\(316\) 1907.69 0.339607
\(317\) 3953.00 0.700387 0.350194 0.936677i \(-0.386116\pi\)
0.350194 + 0.936677i \(0.386116\pi\)
\(318\) 0 0
\(319\) −882.154 −0.154831
\(320\) 0 0
\(321\) 0 0
\(322\) 1395.57 0.241529
\(323\) 231.102 0.0398106
\(324\) 0 0
\(325\) 0 0
\(326\) −1435.02 −0.243798
\(327\) 0 0
\(328\) −6598.51 −1.11080
\(329\) −3420.63 −0.573208
\(330\) 0 0
\(331\) 2464.07 0.409177 0.204589 0.978848i \(-0.434414\pi\)
0.204589 + 0.978848i \(0.434414\pi\)
\(332\) 3797.02 0.627677
\(333\) 0 0
\(334\) −2645.12 −0.433337
\(335\) 0 0
\(336\) 0 0
\(337\) 10587.0 1.71131 0.855656 0.517545i \(-0.173154\pi\)
0.855656 + 0.517545i \(0.173154\pi\)
\(338\) −2374.08 −0.382051
\(339\) 0 0
\(340\) 0 0
\(341\) 473.243 0.0751541
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 3397.76 0.532543
\(345\) 0 0
\(346\) 2360.05 0.366697
\(347\) 8188.81 1.26685 0.633427 0.773803i \(-0.281648\pi\)
0.633427 + 0.773803i \(0.281648\pi\)
\(348\) 0 0
\(349\) 2023.68 0.310387 0.155194 0.987884i \(-0.450400\pi\)
0.155194 + 0.987884i \(0.450400\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 802.844 0.121567
\(353\) 6824.30 1.02895 0.514477 0.857504i \(-0.327986\pi\)
0.514477 + 0.857504i \(0.327986\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6155.46 0.916401
\(357\) 0 0
\(358\) 36.9358 0.00545284
\(359\) 1034.05 0.152020 0.0760100 0.997107i \(-0.475782\pi\)
0.0760100 + 0.997107i \(0.475782\pi\)
\(360\) 0 0
\(361\) −6638.13 −0.967799
\(362\) −2941.57 −0.427087
\(363\) 0 0
\(364\) 669.953 0.0964700
\(365\) 0 0
\(366\) 0 0
\(367\) −9573.55 −1.36168 −0.680838 0.732434i \(-0.738384\pi\)
−0.680838 + 0.732434i \(0.738384\pi\)
\(368\) 5284.75 0.748606
\(369\) 0 0
\(370\) 0 0
\(371\) −2215.02 −0.309968
\(372\) 0 0
\(373\) −5540.42 −0.769094 −0.384547 0.923105i \(-0.625642\pi\)
−0.384547 + 0.923105i \(0.625642\pi\)
\(374\) −84.2055 −0.0116421
\(375\) 0 0
\(376\) 8516.00 1.16803
\(377\) 2839.02 0.387843
\(378\) 0 0
\(379\) −1601.43 −0.217045 −0.108522 0.994094i \(-0.534612\pi\)
−0.108522 + 0.994094i \(0.534612\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 4979.97 0.667009
\(383\) 5034.28 0.671644 0.335822 0.941926i \(-0.390986\pi\)
0.335822 + 0.941926i \(0.390986\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2460.98 −0.324510
\(387\) 0 0
\(388\) −5515.67 −0.721690
\(389\) 4065.87 0.529943 0.264971 0.964256i \(-0.414638\pi\)
0.264971 + 0.964256i \(0.414638\pi\)
\(390\) 0 0
\(391\) −2591.78 −0.335223
\(392\) −853.933 −0.110026
\(393\) 0 0
\(394\) 2908.46 0.371893
\(395\) 0 0
\(396\) 0 0
\(397\) 285.834 0.0361349 0.0180675 0.999837i \(-0.494249\pi\)
0.0180675 + 0.999837i \(0.494249\pi\)
\(398\) −6329.15 −0.797115
\(399\) 0 0
\(400\) 0 0
\(401\) −12459.4 −1.55160 −0.775800 0.630979i \(-0.782654\pi\)
−0.775800 + 0.630979i \(0.782654\pi\)
\(402\) 0 0
\(403\) −1523.03 −0.188257
\(404\) −6652.56 −0.819250
\(405\) 0 0
\(406\) −1631.64 −0.199450
\(407\) 215.204 0.0262094
\(408\) 0 0
\(409\) −8756.93 −1.05868 −0.529342 0.848408i \(-0.677561\pi\)
−0.529342 + 0.848408i \(0.677561\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4982.76 0.595833
\(413\) 2451.79 0.292118
\(414\) 0 0
\(415\) 0 0
\(416\) −2583.77 −0.304519
\(417\) 0 0
\(418\) −80.4769 −0.00941687
\(419\) 12419.7 1.44807 0.724033 0.689765i \(-0.242286\pi\)
0.724033 + 0.689765i \(0.242286\pi\)
\(420\) 0 0
\(421\) −793.783 −0.0918922 −0.0459461 0.998944i \(-0.514630\pi\)
−0.0459461 + 0.998944i \(0.514630\pi\)
\(422\) −4289.90 −0.494856
\(423\) 0 0
\(424\) 5514.52 0.631625
\(425\) 0 0
\(426\) 0 0
\(427\) 809.291 0.0917198
\(428\) 1736.71 0.196138
\(429\) 0 0
\(430\) 0 0
\(431\) 8485.37 0.948320 0.474160 0.880439i \(-0.342752\pi\)
0.474160 + 0.880439i \(0.342752\pi\)
\(432\) 0 0
\(433\) 9098.11 1.00976 0.504881 0.863189i \(-0.331536\pi\)
0.504881 + 0.863189i \(0.331536\pi\)
\(434\) 875.313 0.0968120
\(435\) 0 0
\(436\) 5094.65 0.559608
\(437\) −2477.02 −0.271148
\(438\) 0 0
\(439\) −4232.27 −0.460126 −0.230063 0.973176i \(-0.573893\pi\)
−0.230063 + 0.973176i \(0.573893\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 270.997 0.0291629
\(443\) 2348.20 0.251843 0.125922 0.992040i \(-0.459811\pi\)
0.125922 + 0.992040i \(0.459811\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −1348.24 −0.143141
\(447\) 0 0
\(448\) −290.677 −0.0306545
\(449\) −10423.7 −1.09560 −0.547798 0.836610i \(-0.684534\pi\)
−0.547798 + 0.836610i \(0.684534\pi\)
\(450\) 0 0
\(451\) 1714.07 0.178964
\(452\) −2875.00 −0.299178
\(453\) 0 0
\(454\) −807.221 −0.0834467
\(455\) 0 0
\(456\) 0 0
\(457\) −10221.8 −1.04630 −0.523148 0.852242i \(-0.675242\pi\)
−0.523148 + 0.852242i \(0.675242\pi\)
\(458\) −7065.36 −0.720835
\(459\) 0 0
\(460\) 0 0
\(461\) 13341.4 1.34788 0.673941 0.738786i \(-0.264600\pi\)
0.673941 + 0.738786i \(0.264600\pi\)
\(462\) 0 0
\(463\) 9155.72 0.919012 0.459506 0.888175i \(-0.348027\pi\)
0.459506 + 0.888175i \(0.348027\pi\)
\(464\) −6178.68 −0.618185
\(465\) 0 0
\(466\) 365.501 0.0363337
\(467\) 13759.6 1.36342 0.681710 0.731623i \(-0.261237\pi\)
0.681710 + 0.731623i \(0.261237\pi\)
\(468\) 0 0
\(469\) 3044.23 0.299722
\(470\) 0 0
\(471\) 0 0
\(472\) −6103.98 −0.595251
\(473\) −882.624 −0.0857994
\(474\) 0 0
\(475\) 0 0
\(476\) 715.064 0.0688549
\(477\) 0 0
\(478\) −3909.17 −0.374061
\(479\) −7423.91 −0.708157 −0.354079 0.935216i \(-0.615205\pi\)
−0.354079 + 0.935216i \(0.615205\pi\)
\(480\) 0 0
\(481\) −692.585 −0.0656531
\(482\) 571.083 0.0539670
\(483\) 0 0
\(484\) 8608.95 0.808504
\(485\) 0 0
\(486\) 0 0
\(487\) −7663.90 −0.713110 −0.356555 0.934274i \(-0.616049\pi\)
−0.356555 + 0.934274i \(0.616049\pi\)
\(488\) −2014.81 −0.186898
\(489\) 0 0
\(490\) 0 0
\(491\) −1257.19 −0.115552 −0.0577760 0.998330i \(-0.518401\pi\)
−0.0577760 + 0.998330i \(0.518401\pi\)
\(492\) 0 0
\(493\) 3030.18 0.276821
\(494\) 258.997 0.0235887
\(495\) 0 0
\(496\) 3314.63 0.300063
\(497\) −2870.83 −0.259103
\(498\) 0 0
\(499\) 402.799 0.0361358 0.0180679 0.999837i \(-0.494248\pi\)
0.0180679 + 0.999837i \(0.494248\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −5310.55 −0.472154
\(503\) 15462.0 1.37061 0.685304 0.728257i \(-0.259669\pi\)
0.685304 + 0.728257i \(0.259669\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 902.540 0.0792941
\(507\) 0 0
\(508\) −6512.76 −0.568813
\(509\) 12974.1 1.12980 0.564899 0.825160i \(-0.308915\pi\)
0.564899 + 0.825160i \(0.308915\pi\)
\(510\) 0 0
\(511\) −3249.28 −0.281291
\(512\) 10043.8 0.866946
\(513\) 0 0
\(514\) −2542.47 −0.218178
\(515\) 0 0
\(516\) 0 0
\(517\) −2212.18 −0.188184
\(518\) 398.042 0.0337625
\(519\) 0 0
\(520\) 0 0
\(521\) −4290.85 −0.360817 −0.180409 0.983592i \(-0.557742\pi\)
−0.180409 + 0.983592i \(0.557742\pi\)
\(522\) 0 0
\(523\) 23233.7 1.94252 0.971261 0.238018i \(-0.0764977\pi\)
0.971261 + 0.238018i \(0.0764977\pi\)
\(524\) 10048.3 0.837717
\(525\) 0 0
\(526\) −1188.03 −0.0984804
\(527\) −1625.58 −0.134367
\(528\) 0 0
\(529\) 15612.5 1.28319
\(530\) 0 0
\(531\) 0 0
\(532\) 683.402 0.0556940
\(533\) −5516.37 −0.448293
\(534\) 0 0
\(535\) 0 0
\(536\) −7578.91 −0.610745
\(537\) 0 0
\(538\) 7725.13 0.619060
\(539\) 221.823 0.0177266
\(540\) 0 0
\(541\) −10788.7 −0.857378 −0.428689 0.903452i \(-0.641024\pi\)
−0.428689 + 0.903452i \(0.641024\pi\)
\(542\) 498.914 0.0395391
\(543\) 0 0
\(544\) −2757.75 −0.217349
\(545\) 0 0
\(546\) 0 0
\(547\) −4512.07 −0.352691 −0.176346 0.984328i \(-0.556428\pi\)
−0.176346 + 0.984328i \(0.556428\pi\)
\(548\) 1445.84 0.112707
\(549\) 0 0
\(550\) 0 0
\(551\) 2896.01 0.223909
\(552\) 0 0
\(553\) −2032.80 −0.156317
\(554\) 5816.82 0.446088
\(555\) 0 0
\(556\) 13408.8 1.02277
\(557\) −22436.2 −1.70674 −0.853370 0.521306i \(-0.825445\pi\)
−0.853370 + 0.521306i \(0.825445\pi\)
\(558\) 0 0
\(559\) 2840.53 0.214922
\(560\) 0 0
\(561\) 0 0
\(562\) −2423.63 −0.181912
\(563\) −4773.27 −0.357317 −0.178658 0.983911i \(-0.557176\pi\)
−0.178658 + 0.983911i \(0.557176\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −7710.46 −0.572606
\(567\) 0 0
\(568\) 7147.22 0.527977
\(569\) −21212.5 −1.56287 −0.781435 0.623987i \(-0.785512\pi\)
−0.781435 + 0.623987i \(0.785512\pi\)
\(570\) 0 0
\(571\) −16931.8 −1.24093 −0.620467 0.784232i \(-0.713057\pi\)
−0.620467 + 0.784232i \(0.713057\pi\)
\(572\) 433.269 0.0316711
\(573\) 0 0
\(574\) 3170.36 0.230537
\(575\) 0 0
\(576\) 0 0
\(577\) −21721.5 −1.56721 −0.783604 0.621261i \(-0.786621\pi\)
−0.783604 + 0.621261i \(0.786621\pi\)
\(578\) −5587.54 −0.402095
\(579\) 0 0
\(580\) 0 0
\(581\) −4046.04 −0.288912
\(582\) 0 0
\(583\) −1432.49 −0.101763
\(584\) 8089.40 0.573188
\(585\) 0 0
\(586\) 648.144 0.0456904
\(587\) −12620.8 −0.887421 −0.443710 0.896170i \(-0.646338\pi\)
−0.443710 + 0.896170i \(0.646338\pi\)
\(588\) 0 0
\(589\) −1553.60 −0.108684
\(590\) 0 0
\(591\) 0 0
\(592\) 1507.30 0.104645
\(593\) −25590.5 −1.77214 −0.886069 0.463554i \(-0.846574\pi\)
−0.886069 + 0.463554i \(0.846574\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3888.91 0.267275
\(597\) 0 0
\(598\) −2904.63 −0.198627
\(599\) −8933.79 −0.609390 −0.304695 0.952450i \(-0.598555\pi\)
−0.304695 + 0.952450i \(0.598555\pi\)
\(600\) 0 0
\(601\) 2249.31 0.152665 0.0763323 0.997082i \(-0.475679\pi\)
0.0763323 + 0.997082i \(0.475679\pi\)
\(602\) −1632.51 −0.110525
\(603\) 0 0
\(604\) 12892.3 0.868510
\(605\) 0 0
\(606\) 0 0
\(607\) −5437.17 −0.363572 −0.181786 0.983338i \(-0.558188\pi\)
−0.181786 + 0.983338i \(0.558188\pi\)
\(608\) −2635.64 −0.175805
\(609\) 0 0
\(610\) 0 0
\(611\) 7119.40 0.471391
\(612\) 0 0
\(613\) 7432.72 0.489731 0.244865 0.969557i \(-0.421256\pi\)
0.244865 + 0.969557i \(0.421256\pi\)
\(614\) 5376.22 0.353365
\(615\) 0 0
\(616\) −552.252 −0.0361215
\(617\) 5304.71 0.346126 0.173063 0.984911i \(-0.444634\pi\)
0.173063 + 0.984911i \(0.444634\pi\)
\(618\) 0 0
\(619\) 16716.9 1.08548 0.542738 0.839902i \(-0.317388\pi\)
0.542738 + 0.839902i \(0.317388\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 3432.42 0.221266
\(623\) −6559.15 −0.421809
\(624\) 0 0
\(625\) 0 0
\(626\) 1977.55 0.126260
\(627\) 0 0
\(628\) 4493.85 0.285548
\(629\) −739.221 −0.0468595
\(630\) 0 0
\(631\) −7013.44 −0.442473 −0.221237 0.975220i \(-0.571009\pi\)
−0.221237 + 0.975220i \(0.571009\pi\)
\(632\) 5060.85 0.318528
\(633\) 0 0
\(634\) 4728.46 0.296201
\(635\) 0 0
\(636\) 0 0
\(637\) −713.890 −0.0444040
\(638\) −1055.21 −0.0654796
\(639\) 0 0
\(640\) 0 0
\(641\) −9981.90 −0.615072 −0.307536 0.951536i \(-0.599504\pi\)
−0.307536 + 0.951536i \(0.599504\pi\)
\(642\) 0 0
\(643\) 24635.5 1.51093 0.755465 0.655188i \(-0.227411\pi\)
0.755465 + 0.655188i \(0.227411\pi\)
\(644\) −7664.28 −0.468967
\(645\) 0 0
\(646\) 276.437 0.0168363
\(647\) −18252.3 −1.10908 −0.554538 0.832158i \(-0.687105\pi\)
−0.554538 + 0.832158i \(0.687105\pi\)
\(648\) 0 0
\(649\) 1585.61 0.0959024
\(650\) 0 0
\(651\) 0 0
\(652\) 7880.89 0.473373
\(653\) −11455.4 −0.686501 −0.343251 0.939244i \(-0.611528\pi\)
−0.343251 + 0.939244i \(0.611528\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 12005.5 0.714537
\(657\) 0 0
\(658\) −4091.65 −0.242415
\(659\) 4647.50 0.274721 0.137360 0.990521i \(-0.456138\pi\)
0.137360 + 0.990521i \(0.456138\pi\)
\(660\) 0 0
\(661\) −11966.6 −0.704158 −0.352079 0.935970i \(-0.614525\pi\)
−0.352079 + 0.935970i \(0.614525\pi\)
\(662\) 2947.45 0.173045
\(663\) 0 0
\(664\) 10073.0 0.588718
\(665\) 0 0
\(666\) 0 0
\(667\) −32478.5 −1.88541
\(668\) 14526.6 0.841393
\(669\) 0 0
\(670\) 0 0
\(671\) 523.381 0.0301116
\(672\) 0 0
\(673\) 17972.2 1.02939 0.514693 0.857374i \(-0.327906\pi\)
0.514693 + 0.857374i \(0.327906\pi\)
\(674\) 12663.9 0.723731
\(675\) 0 0
\(676\) 13038.1 0.741813
\(677\) 16225.6 0.921125 0.460562 0.887627i \(-0.347648\pi\)
0.460562 + 0.887627i \(0.347648\pi\)
\(678\) 0 0
\(679\) 5877.40 0.332186
\(680\) 0 0
\(681\) 0 0
\(682\) 566.079 0.0317834
\(683\) 32680.3 1.83086 0.915430 0.402478i \(-0.131851\pi\)
0.915430 + 0.402478i \(0.131851\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 410.286 0.0228350
\(687\) 0 0
\(688\) −6181.97 −0.342566
\(689\) 4610.15 0.254910
\(690\) 0 0
\(691\) 30421.0 1.67477 0.837386 0.546611i \(-0.184083\pi\)
0.837386 + 0.546611i \(0.184083\pi\)
\(692\) −12961.1 −0.712002
\(693\) 0 0
\(694\) 9795.20 0.535765
\(695\) 0 0
\(696\) 0 0
\(697\) −5887.82 −0.319967
\(698\) 2420.66 0.131266
\(699\) 0 0
\(700\) 0 0
\(701\) 24865.9 1.33976 0.669881 0.742469i \(-0.266345\pi\)
0.669881 + 0.742469i \(0.266345\pi\)
\(702\) 0 0
\(703\) −706.488 −0.0379028
\(704\) −187.986 −0.0100639
\(705\) 0 0
\(706\) 8163.02 0.435155
\(707\) 7088.85 0.377091
\(708\) 0 0
\(709\) 22833.5 1.20949 0.604746 0.796418i \(-0.293275\pi\)
0.604746 + 0.796418i \(0.293275\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 16329.7 0.859523
\(713\) 17423.5 0.915168
\(714\) 0 0
\(715\) 0 0
\(716\) −202.846 −0.0105876
\(717\) 0 0
\(718\) 1236.90 0.0642907
\(719\) −32998.1 −1.71157 −0.855786 0.517329i \(-0.826926\pi\)
−0.855786 + 0.517329i \(0.826926\pi\)
\(720\) 0 0
\(721\) −5309.54 −0.274255
\(722\) −7940.33 −0.409292
\(723\) 0 0
\(724\) 16154.6 0.829257
\(725\) 0 0
\(726\) 0 0
\(727\) −13769.6 −0.702455 −0.351228 0.936290i \(-0.614236\pi\)
−0.351228 + 0.936290i \(0.614236\pi\)
\(728\) 1777.30 0.0904823
\(729\) 0 0
\(730\) 0 0
\(731\) 3031.80 0.153400
\(732\) 0 0
\(733\) 33872.8 1.70685 0.853425 0.521216i \(-0.174521\pi\)
0.853425 + 0.521216i \(0.174521\pi\)
\(734\) −11451.6 −0.575866
\(735\) 0 0
\(736\) 29558.5 1.48035
\(737\) 1968.75 0.0983987
\(738\) 0 0
\(739\) 19929.6 0.992048 0.496024 0.868309i \(-0.334793\pi\)
0.496024 + 0.868309i \(0.334793\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2649.54 −0.131089
\(743\) −22285.3 −1.10036 −0.550182 0.835045i \(-0.685442\pi\)
−0.550182 + 0.835045i \(0.685442\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −6627.28 −0.325257
\(747\) 0 0
\(748\) 462.443 0.0226051
\(749\) −1850.61 −0.0902799
\(750\) 0 0
\(751\) −9443.37 −0.458846 −0.229423 0.973327i \(-0.573684\pi\)
−0.229423 + 0.973327i \(0.573684\pi\)
\(752\) −15494.3 −0.751353
\(753\) 0 0
\(754\) 3395.95 0.164023
\(755\) 0 0
\(756\) 0 0
\(757\) −9430.33 −0.452776 −0.226388 0.974037i \(-0.572692\pi\)
−0.226388 + 0.974037i \(0.572692\pi\)
\(758\) −1915.58 −0.0917903
\(759\) 0 0
\(760\) 0 0
\(761\) 24810.4 1.18184 0.590918 0.806732i \(-0.298766\pi\)
0.590918 + 0.806732i \(0.298766\pi\)
\(762\) 0 0
\(763\) −5428.76 −0.257581
\(764\) −27349.2 −1.29510
\(765\) 0 0
\(766\) 6021.85 0.284045
\(767\) −5102.94 −0.240230
\(768\) 0 0
\(769\) 4174.16 0.195740 0.0978699 0.995199i \(-0.468797\pi\)
0.0978699 + 0.995199i \(0.468797\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 13515.3 0.630088
\(773\) 25457.8 1.18455 0.592274 0.805737i \(-0.298230\pi\)
0.592274 + 0.805737i \(0.298230\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −14632.4 −0.676897
\(777\) 0 0
\(778\) 4863.47 0.224118
\(779\) −5627.10 −0.258809
\(780\) 0 0
\(781\) −1856.61 −0.0850637
\(782\) −3100.21 −0.141769
\(783\) 0 0
\(784\) 1553.67 0.0707757
\(785\) 0 0
\(786\) 0 0
\(787\) −5696.75 −0.258027 −0.129013 0.991643i \(-0.541181\pi\)
−0.129013 + 0.991643i \(0.541181\pi\)
\(788\) −15972.8 −0.722091
\(789\) 0 0
\(790\) 0 0
\(791\) 3063.55 0.137708
\(792\) 0 0
\(793\) −1684.39 −0.0754279
\(794\) 341.905 0.0152818
\(795\) 0 0
\(796\) 34758.7 1.54773
\(797\) −13044.2 −0.579736 −0.289868 0.957067i \(-0.593611\pi\)
−0.289868 + 0.957067i \(0.593611\pi\)
\(798\) 0 0
\(799\) 7598.79 0.336453
\(800\) 0 0
\(801\) 0 0
\(802\) −14903.5 −0.656187
\(803\) −2101.36 −0.0923478
\(804\) 0 0
\(805\) 0 0
\(806\) −1821.80 −0.0796156
\(807\) 0 0
\(808\) −17648.4 −0.768402
\(809\) 20548.8 0.893026 0.446513 0.894777i \(-0.352666\pi\)
0.446513 + 0.894777i \(0.352666\pi\)
\(810\) 0 0
\(811\) −1266.15 −0.0548220 −0.0274110 0.999624i \(-0.508726\pi\)
−0.0274110 + 0.999624i \(0.508726\pi\)
\(812\) 8960.70 0.387265
\(813\) 0 0
\(814\) 257.420 0.0110842
\(815\) 0 0
\(816\) 0 0
\(817\) 2897.55 0.124079
\(818\) −10474.8 −0.447728
\(819\) 0 0
\(820\) 0 0
\(821\) −29978.9 −1.27439 −0.637193 0.770704i \(-0.719905\pi\)
−0.637193 + 0.770704i \(0.719905\pi\)
\(822\) 0 0
\(823\) −19172.5 −0.812043 −0.406021 0.913864i \(-0.633084\pi\)
−0.406021 + 0.913864i \(0.633084\pi\)
\(824\) 13218.6 0.558851
\(825\) 0 0
\(826\) 2932.76 0.123539
\(827\) 13488.9 0.567175 0.283588 0.958946i \(-0.408475\pi\)
0.283588 + 0.958946i \(0.408475\pi\)
\(828\) 0 0
\(829\) 16585.9 0.694876 0.347438 0.937703i \(-0.387052\pi\)
0.347438 + 0.937703i \(0.387052\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 604.990 0.0252094
\(833\) −761.960 −0.0316931
\(834\) 0 0
\(835\) 0 0
\(836\) 441.967 0.0182844
\(837\) 0 0
\(838\) 14856.0 0.612401
\(839\) 37812.6 1.55594 0.777971 0.628301i \(-0.216249\pi\)
0.777971 + 0.628301i \(0.216249\pi\)
\(840\) 0 0
\(841\) 13583.2 0.556940
\(842\) −949.499 −0.0388621
\(843\) 0 0
\(844\) 23559.5 0.960842
\(845\) 0 0
\(846\) 0 0
\(847\) −9173.54 −0.372145
\(848\) −10033.3 −0.406302
\(849\) 0 0
\(850\) 0 0
\(851\) 7923.19 0.319158
\(852\) 0 0
\(853\) −5118.65 −0.205462 −0.102731 0.994709i \(-0.532758\pi\)
−0.102731 + 0.994709i \(0.532758\pi\)
\(854\) 968.049 0.0387892
\(855\) 0 0
\(856\) 4607.27 0.183964
\(857\) 34615.7 1.37975 0.689877 0.723927i \(-0.257665\pi\)
0.689877 + 0.723927i \(0.257665\pi\)
\(858\) 0 0
\(859\) −24854.2 −0.987210 −0.493605 0.869686i \(-0.664321\pi\)
−0.493605 + 0.869686i \(0.664321\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 10149.9 0.401054
\(863\) −9637.52 −0.380145 −0.190072 0.981770i \(-0.560872\pi\)
−0.190072 + 0.981770i \(0.560872\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 10882.9 0.427038
\(867\) 0 0
\(868\) −4807.09 −0.187976
\(869\) −1314.64 −0.0513189
\(870\) 0 0
\(871\) −6335.99 −0.246483
\(872\) 13515.5 0.524875
\(873\) 0 0
\(874\) −2962.93 −0.114671
\(875\) 0 0
\(876\) 0 0
\(877\) −13503.3 −0.519924 −0.259962 0.965619i \(-0.583710\pi\)
−0.259962 + 0.965619i \(0.583710\pi\)
\(878\) −5062.52 −0.194592
\(879\) 0 0
\(880\) 0 0
\(881\) 43808.4 1.67530 0.837652 0.546205i \(-0.183928\pi\)
0.837652 + 0.546205i \(0.183928\pi\)
\(882\) 0 0
\(883\) 24875.4 0.948044 0.474022 0.880513i \(-0.342802\pi\)
0.474022 + 0.880513i \(0.342802\pi\)
\(884\) −1488.27 −0.0566244
\(885\) 0 0
\(886\) 2808.85 0.106507
\(887\) −39120.3 −1.48087 −0.740435 0.672128i \(-0.765381\pi\)
−0.740435 + 0.672128i \(0.765381\pi\)
\(888\) 0 0
\(889\) 6939.88 0.261818
\(890\) 0 0
\(891\) 0 0
\(892\) 7404.31 0.277931
\(893\) 7262.31 0.272143
\(894\) 0 0
\(895\) 0 0
\(896\) −10279.0 −0.383257
\(897\) 0 0
\(898\) −12468.5 −0.463338
\(899\) −20370.7 −0.755729
\(900\) 0 0
\(901\) 4920.58 0.181940
\(902\) 2050.32 0.0756855
\(903\) 0 0
\(904\) −7627.01 −0.280609
\(905\) 0 0
\(906\) 0 0
\(907\) −19658.5 −0.719681 −0.359840 0.933014i \(-0.617169\pi\)
−0.359840 + 0.933014i \(0.617169\pi\)
\(908\) 4433.14 0.162025
\(909\) 0 0
\(910\) 0 0
\(911\) −39577.9 −1.43938 −0.719690 0.694295i \(-0.755716\pi\)
−0.719690 + 0.694295i \(0.755716\pi\)
\(912\) 0 0
\(913\) −2616.64 −0.0948500
\(914\) −12227.0 −0.442489
\(915\) 0 0
\(916\) 38801.9 1.39962
\(917\) −10707.3 −0.385591
\(918\) 0 0
\(919\) 24855.9 0.892188 0.446094 0.894986i \(-0.352815\pi\)
0.446094 + 0.894986i \(0.352815\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 15958.6 0.570032
\(923\) 5975.09 0.213080
\(924\) 0 0
\(925\) 0 0
\(926\) 10951.8 0.388659
\(927\) 0 0
\(928\) −34558.3 −1.22245
\(929\) −4466.02 −0.157724 −0.0788619 0.996886i \(-0.525129\pi\)
−0.0788619 + 0.996886i \(0.525129\pi\)
\(930\) 0 0
\(931\) −728.220 −0.0256353
\(932\) −2007.27 −0.0705477
\(933\) 0 0
\(934\) 16458.8 0.576604
\(935\) 0 0
\(936\) 0 0
\(937\) −3768.86 −0.131402 −0.0657009 0.997839i \(-0.520928\pi\)
−0.0657009 + 0.997839i \(0.520928\pi\)
\(938\) 3641.41 0.126755
\(939\) 0 0
\(940\) 0 0
\(941\) −15423.4 −0.534312 −0.267156 0.963653i \(-0.586084\pi\)
−0.267156 + 0.963653i \(0.586084\pi\)
\(942\) 0 0
\(943\) 63107.4 2.17928
\(944\) 11105.7 0.382904
\(945\) 0 0
\(946\) −1055.77 −0.0362854
\(947\) 11755.0 0.403365 0.201682 0.979451i \(-0.435359\pi\)
0.201682 + 0.979451i \(0.435359\pi\)
\(948\) 0 0
\(949\) 6762.75 0.231326
\(950\) 0 0
\(951\) 0 0
\(952\) 1896.98 0.0645813
\(953\) 7321.10 0.248850 0.124425 0.992229i \(-0.460291\pi\)
0.124425 + 0.992229i \(0.460291\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 21468.6 0.726300
\(957\) 0 0
\(958\) −8880.26 −0.299487
\(959\) −1540.67 −0.0518777
\(960\) 0 0
\(961\) −18862.9 −0.633174
\(962\) −828.449 −0.0277653
\(963\) 0 0
\(964\) −3136.30 −0.104786
\(965\) 0 0
\(966\) 0 0
\(967\) −21155.8 −0.703542 −0.351771 0.936086i \(-0.614420\pi\)
−0.351771 + 0.936086i \(0.614420\pi\)
\(968\) 22838.5 0.758322
\(969\) 0 0
\(970\) 0 0
\(971\) −59824.6 −1.97720 −0.988600 0.150566i \(-0.951891\pi\)
−0.988600 + 0.150566i \(0.951891\pi\)
\(972\) 0 0
\(973\) −14288.1 −0.470768
\(974\) −9167.33 −0.301581
\(975\) 0 0
\(976\) 3665.80 0.120225
\(977\) −19245.9 −0.630227 −0.315113 0.949054i \(-0.602043\pi\)
−0.315113 + 0.949054i \(0.602043\pi\)
\(978\) 0 0
\(979\) −4241.91 −0.138480
\(980\) 0 0
\(981\) 0 0
\(982\) −1503.81 −0.0488681
\(983\) −52661.2 −1.70868 −0.854339 0.519716i \(-0.826038\pi\)
−0.854339 + 0.519716i \(0.826038\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 3624.61 0.117070
\(987\) 0 0
\(988\) −1422.37 −0.0458013
\(989\) −32495.8 −1.04480
\(990\) 0 0
\(991\) −39065.5 −1.25223 −0.626114 0.779732i \(-0.715356\pi\)
−0.626114 + 0.779732i \(0.715356\pi\)
\(992\) 18539.3 0.593369
\(993\) 0 0
\(994\) −3434.00 −0.109577
\(995\) 0 0
\(996\) 0 0
\(997\) 31074.5 0.987099 0.493550 0.869718i \(-0.335699\pi\)
0.493550 + 0.869718i \(0.335699\pi\)
\(998\) 481.816 0.0152822
\(999\) 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.bi.1.3 yes 4
3.2 odd 2 inner 1575.4.a.bi.1.2 yes 4
5.4 even 2 1575.4.a.bh.1.2 4
15.14 odd 2 1575.4.a.bh.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1575.4.a.bh.1.2 4 5.4 even 2
1575.4.a.bh.1.3 yes 4 15.14 odd 2
1575.4.a.bi.1.2 yes 4 3.2 odd 2 inner
1575.4.a.bi.1.3 yes 4 1.1 even 1 trivial