Properties

Label 1575.4.a.bi.1.2
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 1.2
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 \(-0.567820\pi\)
−0.211455 + 0.977388i \(0.567820\pi\)
\(3\) 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 \(-0.519762\pi\)
−0.0620430 + 0.998073i \(0.519762\pi\)
\(12\) 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 \(-0.464618\pi\)
0.110926 + 0.993829i \(0.464618\pi\)
\(18\) 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 \(-0.772611\pi\)
−0.755511 + 0.655136i \(0.772611\pi\)
\(24\) 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 \(-0.285553\pi\)
0.623887 + 0.781514i \(0.285553\pi\)
\(30\) 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 \(-0.756376\pi\)
−0.721128 + 0.692802i \(0.756376\pi\)
\(42\) 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 \(-0.226039\pi\)
0.758283 + 0.651926i \(0.226039\pi\)
\(48\) 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 \(-0.365511\pi\)
0.410050 + 0.912063i \(0.365511\pi\)
\(54\) 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 \(-0.626294\pi\)
−0.386436 + 0.922316i \(0.626294\pi\)
\(60\) 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 \(-0.388638\pi\)
0.342761 + 0.939422i \(0.388638\pi\)
\(72\) 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 \(-0.375168\pi\)
0.382195 + 0.924082i \(0.375168\pi\)
\(84\) 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 \(-0.311569\pi\)
0.558001 + 0.829841i \(0.311569\pi\)
\(90\) 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 \(-0.666242\pi\)
−0.498845 + 0.866691i \(0.666242\pi\)
\(102\) 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 \(-0.461894\pi\)
0.119429 + 0.992843i \(0.461894\pi\)
\(108\) 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 \(-0.558312\pi\)
−0.182171 + 0.983267i \(0.558312\pi\)
\(114\) 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 \(-0.329612\pi\)
0.510089 + 0.860122i \(0.329612\pi\)
\(132\) 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 \(-0.478138\pi\)
0.0686277 + 0.997642i \(0.478138\pi\)
\(138\) 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 \(-0.447965\pi\)
0.162745 + 0.986668i \(0.447965\pi\)
\(150\) 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 \(-0.328783\pi\)
0.512328 + 0.858790i \(0.328783\pi\)
\(168\) 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 \(-0.642736\pi\)
−0.433541 + 0.901134i \(0.642736\pi\)
\(174\) 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 \(-0.502052\pi\)
−0.00644681 + 0.999979i \(0.502052\pi\)
\(180\) 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 \(-0.789191\pi\)
−0.788595 + 0.614913i \(0.789191\pi\)
\(192\) 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 \(-0.644910\pi\)
−0.439684 + 0.898153i \(0.644910\pi\)
\(198\) 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 \(-0.468545\pi\)
0.0986577 + 0.995121i \(0.468545\pi\)
\(228\) 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 \(-0.513678\pi\)
−0.0429568 + 0.999077i \(0.513678\pi\)
\(234\) 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 \(-0.354181\pi\)
0.442247 + 0.896893i \(0.354181\pi\)
\(240\) 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 \(-0.311484\pi\)
0.558221 + 0.829692i \(0.311484\pi\)
\(252\) 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 \(-0.416953\pi\)
0.257949 + 0.966159i \(0.416953\pi\)
\(258\) 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 \(-0.462854\pi\)
0.116432 + 0.993199i \(0.462854\pi\)
\(264\) 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 \(-0.761369\pi\)
−0.731905 + 0.681406i \(0.761369\pi\)
\(270\) 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 \(-0.431001\pi\)
0.215072 + 0.976598i \(0.431001\pi\)
\(282\) 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 \(-0.517203\pi\)
−0.0540191 + 0.998540i \(0.517203\pi\)
\(294\) 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 \(-0.584250\pi\)
−0.261599 + 0.965177i \(0.584250\pi\)
\(312\) 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 \(-0.613884\pi\)
−0.350194 + 0.936677i \(0.613884\pi\)
\(318\) 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 \(-0.718352\pi\)
−0.633427 + 0.773803i \(0.718352\pi\)
\(348\) 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 \(-0.672014\pi\)
−0.514477 + 0.857504i \(0.672014\pi\)
\(354\) 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 \(-0.524218\pi\)
−0.0760100 + 0.997107i \(0.524218\pi\)
\(360\) 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 \(-0.609014\pi\)
−0.335822 + 0.941926i \(0.609014\pi\)
\(384\) 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 \(-0.585362\pi\)
−0.264971 + 0.964256i \(0.585362\pi\)
\(390\) 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 \(-0.217346\pi\)
0.775800 + 0.630979i \(0.217346\pi\)
\(402\) 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 \(-0.757714\pi\)
−0.724033 + 0.689765i \(0.757714\pi\)
\(420\) 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 \(-0.657248\pi\)
−0.474160 + 0.880439i \(0.657248\pi\)
\(432\) 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 \(-0.540189\pi\)
−0.125922 + 0.992040i \(0.540189\pi\)
\(444\) 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 \(-0.315466\pi\)
0.547798 + 0.836610i \(0.315466\pi\)
\(450\) 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 \(-0.735400\pi\)
−0.673941 + 0.738786i \(0.735400\pi\)
\(462\) 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 \(-0.738763\pi\)
−0.681710 + 0.731623i \(0.738763\pi\)
\(468\) 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 \(-0.384795\pi\)
0.354079 + 0.935216i \(0.384795\pi\)
\(480\) 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 \(-0.481599\pi\)
0.0577760 + 0.998330i \(0.481599\pi\)
\(492\) 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 \(-0.740331\pi\)
−0.685304 + 0.728257i \(0.740331\pi\)
\(504\) 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 \(-0.691085\pi\)
−0.564899 + 0.825160i \(0.691085\pi\)
\(510\) 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 \(-0.442258\pi\)
0.180409 + 0.983592i \(0.442258\pi\)
\(522\) 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 \(-0.174555\pi\)
0.853370 + 0.521306i \(0.174555\pi\)
\(558\) 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 \(-0.442824\pi\)
0.178658 + 0.983911i \(0.442824\pi\)
\(564\) 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 \(-0.214488\pi\)
0.781435 + 0.623987i \(0.214488\pi\)
\(570\) 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 \(-0.353662\pi\)
0.443710 + 0.896170i \(0.353662\pi\)
\(588\) 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 \(-0.153426\pi\)
0.886069 + 0.463554i \(0.153426\pi\)
\(594\) 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 \(-0.401445\pi\)
0.304695 + 0.952450i \(0.401445\pi\)
\(600\) 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 \(-0.555366\pi\)
−0.173063 + 0.984911i \(0.555366\pi\)
\(618\) 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 \(-0.400496\pi\)
0.307536 + 0.951536i \(0.400496\pi\)
\(642\) 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 \(-0.312895\pi\)
0.554538 + 0.832158i \(0.312895\pi\)
\(648\) 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 \(-0.388472\pi\)
0.343251 + 0.939244i \(0.388472\pi\)
\(654\) 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 \(-0.543862\pi\)
−0.137360 + 0.990521i \(0.543862\pi\)
\(660\) 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 \(-0.652352\pi\)
−0.460562 + 0.887627i \(0.652352\pi\)
\(678\) 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 \(-0.868149\pi\)
−0.915430 + 0.402478i \(0.868149\pi\)
\(684\) 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 \(-0.733655\pi\)
−0.669881 + 0.742469i \(0.733655\pi\)
\(702\) 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 \(-0.173074\pi\)
0.855786 + 0.517329i \(0.173074\pi\)
\(720\) 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 \(-0.314558\pi\)
0.550182 + 0.835045i \(0.314558\pi\)
\(744\) 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 \(-0.701234\pi\)
−0.590918 + 0.806732i \(0.701234\pi\)
\(762\) 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 \(-0.701770\pi\)
−0.592274 + 0.805737i \(0.701770\pi\)
\(774\) 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 \(-0.406389\pi\)
0.289868 + 0.957067i \(0.406389\pi\)
\(798\) 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 \(-0.647334\pi\)
−0.446513 + 0.894777i \(0.647334\pi\)
\(810\) 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 \(-0.280095\pi\)
0.637193 + 0.770704i \(0.280095\pi\)
\(822\) 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 \(-0.591525\pi\)
−0.283588 + 0.958946i \(0.591525\pi\)
\(828\) 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 \(-0.783751\pi\)
−0.777971 + 0.628301i \(0.783751\pi\)
\(840\) 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 \(-0.742335\pi\)
−0.689877 + 0.723927i \(0.742335\pi\)
\(858\) 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 \(-0.439128\pi\)
0.190072 + 0.981770i \(0.439128\pi\)
\(864\) 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 \(-0.816072\pi\)
−0.837652 + 0.546205i \(0.816072\pi\)
\(882\) 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 \(-0.234619\pi\)
0.740435 + 0.672128i \(0.234619\pi\)
\(888\) 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 \(-0.244284\pi\)
0.719690 + 0.694295i \(0.244284\pi\)
\(912\) 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 \(-0.474871\pi\)
0.0788619 + 0.996886i \(0.474871\pi\)
\(930\) 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 \(-0.413916\pi\)
0.267156 + 0.963653i \(0.413916\pi\)
\(942\) 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 \(-0.564641\pi\)
−0.201682 + 0.979451i \(0.564641\pi\)
\(948\) 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 \(-0.539709\pi\)
−0.124425 + 0.992229i \(0.539709\pi\)
\(954\) 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 \(-0.0481095\pi\)
0.988600 + 0.150566i \(0.0481095\pi\)
\(972\) 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 \(-0.397957\pi\)
0.315113 + 0.949054i \(0.397957\pi\)
\(978\) 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 \(-0.173962\pi\)
0.854339 + 0.519716i \(0.173962\pi\)
\(984\) 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.1.2 yes 4
3.2 odd 2 inner 1575.4.a.bi.1.3 yes 4
5.4 even 2 1575.4.a.bh.1.3 yes 4
15.14 odd 2 1575.4.a.bh.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1575.4.a.bh.1.2 4 15.14 odd 2
1575.4.a.bh.1.3 yes 4 5.4 even 2
1575.4.a.bi.1.2 yes 4 1.1 even 1 trivial
1575.4.a.bi.1.3 yes 4 3.2 odd 2 inner