Properties

Label 1575.4.a.bh.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 \(-0.480238\pi\)
0.0620430 + 0.998073i \(0.480238\pi\)
\(12\) 0 0
\(13\) 14.5692 0.310828 0.155414 0.987849i \(-0.450329\pi\)
0.155414 + 0.987849i \(0.450329\pi\)
\(14\) 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 \(-0.714447\pi\)
−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 \(-0.533680\pi\)
−0.105610 + 0.994408i \(0.533680\pi\)
\(38\) 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 \(-0.387633\pi\)
0.345726 + 0.938336i \(0.387633\pi\)
\(44\) −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 \(-0.373706\pi\)
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 \(-0.629773\pi\)
−0.396494 + 0.918037i \(0.629773\pi\)
\(68\) −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 \(-0.378634\pi\)
0.372112 + 0.928188i \(0.378634\pi\)
\(74\) 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 \(-0.688431\pi\)
−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 \(-0.644823\pi\)
−0.439440 + 0.898272i \(0.644823\pi\)
\(98\) −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 \(-0.381819\pi\)
0.362805 + 0.931865i \(0.381819\pi\)
\(104\) 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 \(-0.612580\pi\)
−0.346353 + 0.938104i \(0.612580\pi\)
\(128\) 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 \(-0.478138\pi\)
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 \(-0.552035\pi\)
−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 \(-0.444372\pi\)
0.173871 + 0.984768i \(0.444372\pi\)
\(158\) 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 \(-0.406930\pi\)
0.288239 + 0.957559i \(0.406930\pi\)
\(164\) −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 \(-0.497948\pi\)
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 \(-0.374662\pi\)
0.383663 + 0.923473i \(0.374662\pi\)
\(194\) 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 \(-0.445871\pi\)
0.169234 + 0.985576i \(0.445871\pi\)
\(224\) 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 \(-0.645819\pi\)
−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 \(-0.416953\pi\)
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 \(-0.238631\pi\)
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 \(-0.676834\pi\)
−0.527404 + 0.849615i \(0.676834\pi\)
\(278\) 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 \(-0.263287\pi\)
0.676983 + 0.735998i \(0.263287\pi\)
\(284\) 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 \(-0.637191\pi\)
−0.417779 + 0.908549i \(0.637191\pi\)
\(308\) 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 \(-0.547694\pi\)
−0.149275 + 0.988796i \(0.547694\pi\)
\(314\) −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 \(-0.826846\pi\)
−0.855656 + 0.517545i \(0.826846\pi\)
\(338\) 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 \(-0.475782\pi\)
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 \(-0.261616\pi\)
0.680838 + 0.732434i \(0.261616\pi\)
\(368\) −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 \(-0.374358\pi\)
0.384547 + 0.923105i \(0.374358\pi\)
\(374\) −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 \(-0.414638\pi\)
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 \(-0.505751\pi\)
−0.0180675 + 0.999837i \(0.505751\pi\)
\(398\) 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 \(-0.668464\pi\)
−0.504881 + 0.863189i \(0.668464\pi\)
\(434\) 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 \(-0.684534\pi\)
−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 \(-0.324758\pi\)
0.523148 + 0.852242i \(0.324758\pi\)
\(458\) 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 \(-0.651973\pi\)
−0.459506 + 0.888175i \(0.651973\pi\)
\(464\) −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 \(-0.615205\pi\)
−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 \(-0.383951\pi\)
0.356555 + 0.934274i \(0.383951\pi\)
\(488\) 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 \(-0.740331\pi\)
−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 \(-0.308915\pi\)
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 \(-0.923502\pi\)
−0.971261 + 0.238018i \(0.923502\pi\)
\(524\) 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 \(-0.443572\pi\)
0.176346 + 0.984328i \(0.443572\pi\)
\(548\) −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 \(-0.785512\pi\)
−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 \(-0.213379\pi\)
0.783604 + 0.621261i \(0.213379\pi\)
\(578\) 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 \(-0.598555\pi\)
−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 \(-0.441812\pi\)
0.181786 + 0.983338i \(0.441812\pi\)
\(608\) 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 \(-0.578744\pi\)
−0.244865 + 0.969557i \(0.578744\pi\)
\(614\) 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 \(-0.599504\pi\)
−0.307536 + 0.951536i \(0.599504\pi\)
\(642\) 0 0
\(643\) −24635.5 −1.51093 −0.755465 0.655188i \(-0.772589\pi\)
−0.755465 + 0.655188i \(0.772589\pi\)
\(644\) −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 \(-0.456138\pi\)
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 \(-0.672094\pi\)
−0.514693 + 0.857374i \(0.672094\pi\)
\(674\) 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 \(-0.266345\pi\)
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 \(-0.385764\pi\)
0.351228 + 0.936290i \(0.385764\pi\)
\(728\) −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 \(-0.825479\pi\)
−0.853425 + 0.521216i \(0.825479\pi\)
\(734\) −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 \(-0.427308\pi\)
0.226388 + 0.974037i \(0.427308\pi\)
\(758\) 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 \(-0.701770\pi\)
−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 \(-0.458819\pi\)
0.129013 + 0.991643i \(0.458819\pi\)
\(788\) 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 \(-0.352666\pi\)
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 \(-0.366916\pi\)
0.406021 + 0.913864i \(0.366916\pi\)
\(824\) 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 \(-0.216249\pi\)
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 \(-0.467242\pi\)
0.102731 + 0.994709i \(0.467242\pi\)
\(854\) 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 \(-0.416290\pi\)
0.259962 + 0.965619i \(0.416290\pi\)
\(878\) 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 \(-0.657198\pi\)
−0.474022 + 0.880513i \(0.657198\pi\)
\(884\) −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 \(-0.382831\pi\)
0.359840 + 0.933014i \(0.382831\pi\)
\(908\) −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 \(-0.479072\pi\)
0.0657009 + 0.997839i \(0.479072\pi\)
\(938\) −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 \(-0.564641\pi\)
−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 \(-0.385580\pi\)
0.351771 + 0.936086i \(0.385580\pi\)
\(968\) −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 \(-0.397957\pi\)
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 \(-0.664301\pi\)
−0.493550 + 0.869718i \(0.664301\pi\)
\(998\) −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.bh.1.2 4
3.2 odd 2 inner 1575.4.a.bh.1.3 yes 4
5.4 even 2 1575.4.a.bi.1.3 yes 4
15.14 odd 2 1575.4.a.bi.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1575.4.a.bh.1.2 4 1.1 even 1 trivial
1575.4.a.bh.1.3 yes 4 3.2 odd 2 inner
1575.4.a.bi.1.2 yes 4 15.14 odd 2
1575.4.a.bi.1.3 yes 4 5.4 even 2