Properties

Label 1856.4.a.bk.1.8
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1856,4,Mod(1,1856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, 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("1856.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.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: \(109.507544971\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 245x^{10} + 21294x^{8} - 755514x^{6} + 8955005x^{4} - 27099393x^{2} + 23710340 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{19} \)
Twist minimal: no (minimal twist has level 928)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(1.50033\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.50033 q^{3} -0.787262 q^{5} +27.3506 q^{7} -24.7490 q^{9} +O(q^{10})\) \(q+1.50033 q^{3} -0.787262 q^{5} +27.3506 q^{7} -24.7490 q^{9} +62.2529 q^{11} +27.7949 q^{13} -1.18115 q^{15} +102.604 q^{17} +55.9390 q^{19} +41.0348 q^{21} +175.715 q^{23} -124.380 q^{25} -77.6404 q^{27} +29.0000 q^{29} +199.792 q^{31} +93.3997 q^{33} -21.5321 q^{35} -403.647 q^{37} +41.7014 q^{39} -118.286 q^{41} +354.315 q^{43} +19.4840 q^{45} -148.783 q^{47} +405.055 q^{49} +153.939 q^{51} -376.090 q^{53} -49.0093 q^{55} +83.9268 q^{57} +329.095 q^{59} -789.113 q^{61} -676.900 q^{63} -21.8818 q^{65} +147.681 q^{67} +263.630 q^{69} +254.191 q^{71} -402.746 q^{73} -186.611 q^{75} +1702.65 q^{77} -429.232 q^{79} +551.738 q^{81} +78.8317 q^{83} -80.7759 q^{85} +43.5095 q^{87} +1034.55 q^{89} +760.207 q^{91} +299.753 q^{93} -44.0386 q^{95} +1790.86 q^{97} -1540.70 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{5} + 166 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 10 q^{5} + 166 q^{9} - 30 q^{13} + 336 q^{17} - 56 q^{21} + 294 q^{25} + 348 q^{29} + 214 q^{33} - 196 q^{37} + 940 q^{41} - 1672 q^{45} + 972 q^{49} + 406 q^{53} + 1224 q^{57} - 400 q^{61} + 2998 q^{65} - 1004 q^{69} + 2252 q^{73} - 3032 q^{77} + 3932 q^{81} - 5564 q^{85} + 5212 q^{89} - 3722 q^{93} + 6164 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.50033 0.288738 0.144369 0.989524i \(-0.453885\pi\)
0.144369 + 0.989524i \(0.453885\pi\)
\(4\) 0 0
\(5\) −0.787262 −0.0704148 −0.0352074 0.999380i \(-0.511209\pi\)
−0.0352074 + 0.999380i \(0.511209\pi\)
\(6\) 0 0
\(7\) 27.3506 1.47679 0.738396 0.674367i \(-0.235583\pi\)
0.738396 + 0.674367i \(0.235583\pi\)
\(8\) 0 0
\(9\) −24.7490 −0.916630
\(10\) 0 0
\(11\) 62.2529 1.70636 0.853180 0.521616i \(-0.174671\pi\)
0.853180 + 0.521616i \(0.174671\pi\)
\(12\) 0 0
\(13\) 27.7949 0.592993 0.296497 0.955034i \(-0.404182\pi\)
0.296497 + 0.955034i \(0.404182\pi\)
\(14\) 0 0
\(15\) −1.18115 −0.0203314
\(16\) 0 0
\(17\) 102.604 1.46383 0.731913 0.681398i \(-0.238628\pi\)
0.731913 + 0.681398i \(0.238628\pi\)
\(18\) 0 0
\(19\) 55.9390 0.675436 0.337718 0.941247i \(-0.390345\pi\)
0.337718 + 0.941247i \(0.390345\pi\)
\(20\) 0 0
\(21\) 41.0348 0.426406
\(22\) 0 0
\(23\) 175.715 1.59300 0.796502 0.604636i \(-0.206681\pi\)
0.796502 + 0.604636i \(0.206681\pi\)
\(24\) 0 0
\(25\) −124.380 −0.995042
\(26\) 0 0
\(27\) −77.6404 −0.553404
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) 199.792 1.15754 0.578769 0.815492i \(-0.303533\pi\)
0.578769 + 0.815492i \(0.303533\pi\)
\(32\) 0 0
\(33\) 93.3997 0.492691
\(34\) 0 0
\(35\) −21.5321 −0.103988
\(36\) 0 0
\(37\) −403.647 −1.79349 −0.896746 0.442546i \(-0.854075\pi\)
−0.896746 + 0.442546i \(0.854075\pi\)
\(38\) 0 0
\(39\) 41.7014 0.171220
\(40\) 0 0
\(41\) −118.286 −0.450565 −0.225282 0.974294i \(-0.572330\pi\)
−0.225282 + 0.974294i \(0.572330\pi\)
\(42\) 0 0
\(43\) 354.315 1.25657 0.628286 0.777982i \(-0.283757\pi\)
0.628286 + 0.777982i \(0.283757\pi\)
\(44\) 0 0
\(45\) 19.4840 0.0645444
\(46\) 0 0
\(47\) −148.783 −0.461750 −0.230875 0.972983i \(-0.574159\pi\)
−0.230875 + 0.972983i \(0.574159\pi\)
\(48\) 0 0
\(49\) 405.055 1.18092
\(50\) 0 0
\(51\) 153.939 0.422662
\(52\) 0 0
\(53\) −376.090 −0.974716 −0.487358 0.873202i \(-0.662039\pi\)
−0.487358 + 0.873202i \(0.662039\pi\)
\(54\) 0 0
\(55\) −49.0093 −0.120153
\(56\) 0 0
\(57\) 83.9268 0.195024
\(58\) 0 0
\(59\) 329.095 0.726178 0.363089 0.931754i \(-0.381722\pi\)
0.363089 + 0.931754i \(0.381722\pi\)
\(60\) 0 0
\(61\) −789.113 −1.65632 −0.828160 0.560492i \(-0.810612\pi\)
−0.828160 + 0.560492i \(0.810612\pi\)
\(62\) 0 0
\(63\) −676.900 −1.35367
\(64\) 0 0
\(65\) −21.8818 −0.0417555
\(66\) 0 0
\(67\) 147.681 0.269286 0.134643 0.990894i \(-0.457011\pi\)
0.134643 + 0.990894i \(0.457011\pi\)
\(68\) 0 0
\(69\) 263.630 0.459961
\(70\) 0 0
\(71\) 254.191 0.424887 0.212443 0.977173i \(-0.431858\pi\)
0.212443 + 0.977173i \(0.431858\pi\)
\(72\) 0 0
\(73\) −402.746 −0.645725 −0.322862 0.946446i \(-0.604645\pi\)
−0.322862 + 0.946446i \(0.604645\pi\)
\(74\) 0 0
\(75\) −186.611 −0.287306
\(76\) 0 0
\(77\) 1702.65 2.51994
\(78\) 0 0
\(79\) −429.232 −0.611295 −0.305648 0.952145i \(-0.598873\pi\)
−0.305648 + 0.952145i \(0.598873\pi\)
\(80\) 0 0
\(81\) 551.738 0.756842
\(82\) 0 0
\(83\) 78.8317 0.104252 0.0521259 0.998641i \(-0.483400\pi\)
0.0521259 + 0.998641i \(0.483400\pi\)
\(84\) 0 0
\(85\) −80.7759 −0.103075
\(86\) 0 0
\(87\) 43.5095 0.0536173
\(88\) 0 0
\(89\) 1034.55 1.23216 0.616081 0.787683i \(-0.288720\pi\)
0.616081 + 0.787683i \(0.288720\pi\)
\(90\) 0 0
\(91\) 760.207 0.875729
\(92\) 0 0
\(93\) 299.753 0.334225
\(94\) 0 0
\(95\) −44.0386 −0.0475607
\(96\) 0 0
\(97\) 1790.86 1.87458 0.937289 0.348552i \(-0.113327\pi\)
0.937289 + 0.348552i \(0.113327\pi\)
\(98\) 0 0
\(99\) −1540.70 −1.56410
\(100\) 0 0
\(101\) −724.751 −0.714014 −0.357007 0.934102i \(-0.616203\pi\)
−0.357007 + 0.934102i \(0.616203\pi\)
\(102\) 0 0
\(103\) 178.280 0.170548 0.0852739 0.996358i \(-0.472823\pi\)
0.0852739 + 0.996358i \(0.472823\pi\)
\(104\) 0 0
\(105\) −32.3051 −0.0300253
\(106\) 0 0
\(107\) −1815.82 −1.64058 −0.820290 0.571948i \(-0.806188\pi\)
−0.820290 + 0.571948i \(0.806188\pi\)
\(108\) 0 0
\(109\) −1002.06 −0.880548 −0.440274 0.897863i \(-0.645119\pi\)
−0.440274 + 0.897863i \(0.645119\pi\)
\(110\) 0 0
\(111\) −605.603 −0.517849
\(112\) 0 0
\(113\) 85.9758 0.0715745 0.0357873 0.999359i \(-0.488606\pi\)
0.0357873 + 0.999359i \(0.488606\pi\)
\(114\) 0 0
\(115\) −138.334 −0.112171
\(116\) 0 0
\(117\) −687.896 −0.543556
\(118\) 0 0
\(119\) 2806.27 2.16177
\(120\) 0 0
\(121\) 2544.43 1.91167
\(122\) 0 0
\(123\) −177.468 −0.130095
\(124\) 0 0
\(125\) 196.327 0.140480
\(126\) 0 0
\(127\) −1766.02 −1.23393 −0.616965 0.786991i \(-0.711638\pi\)
−0.616965 + 0.786991i \(0.711638\pi\)
\(128\) 0 0
\(129\) 531.589 0.362820
\(130\) 0 0
\(131\) −1040.00 −0.693628 −0.346814 0.937934i \(-0.612737\pi\)
−0.346814 + 0.937934i \(0.612737\pi\)
\(132\) 0 0
\(133\) 1529.97 0.997480
\(134\) 0 0
\(135\) 61.1233 0.0389678
\(136\) 0 0
\(137\) 1620.70 1.01070 0.505351 0.862914i \(-0.331363\pi\)
0.505351 + 0.862914i \(0.331363\pi\)
\(138\) 0 0
\(139\) −606.981 −0.370385 −0.185192 0.982702i \(-0.559291\pi\)
−0.185192 + 0.982702i \(0.559291\pi\)
\(140\) 0 0
\(141\) −223.223 −0.133325
\(142\) 0 0
\(143\) 1730.31 1.01186
\(144\) 0 0
\(145\) −22.8306 −0.0130757
\(146\) 0 0
\(147\) 607.714 0.340976
\(148\) 0 0
\(149\) −1203.11 −0.661494 −0.330747 0.943719i \(-0.607301\pi\)
−0.330747 + 0.943719i \(0.607301\pi\)
\(150\) 0 0
\(151\) 1644.41 0.886227 0.443114 0.896465i \(-0.353874\pi\)
0.443114 + 0.896465i \(0.353874\pi\)
\(152\) 0 0
\(153\) −2539.34 −1.34179
\(154\) 0 0
\(155\) −157.288 −0.0815078
\(156\) 0 0
\(157\) −1262.40 −0.641725 −0.320862 0.947126i \(-0.603973\pi\)
−0.320862 + 0.947126i \(0.603973\pi\)
\(158\) 0 0
\(159\) −564.258 −0.281437
\(160\) 0 0
\(161\) 4805.91 2.35254
\(162\) 0 0
\(163\) 2456.18 1.18026 0.590131 0.807307i \(-0.299076\pi\)
0.590131 + 0.807307i \(0.299076\pi\)
\(164\) 0 0
\(165\) −73.5300 −0.0346928
\(166\) 0 0
\(167\) −775.298 −0.359248 −0.179624 0.983735i \(-0.557488\pi\)
−0.179624 + 0.983735i \(0.557488\pi\)
\(168\) 0 0
\(169\) −1424.44 −0.648359
\(170\) 0 0
\(171\) −1384.44 −0.619125
\(172\) 0 0
\(173\) −823.000 −0.361685 −0.180842 0.983512i \(-0.557882\pi\)
−0.180842 + 0.983512i \(0.557882\pi\)
\(174\) 0 0
\(175\) −3401.87 −1.46947
\(176\) 0 0
\(177\) 493.750 0.209675
\(178\) 0 0
\(179\) 3570.45 1.49088 0.745441 0.666571i \(-0.232239\pi\)
0.745441 + 0.666571i \(0.232239\pi\)
\(180\) 0 0
\(181\) −671.175 −0.275624 −0.137812 0.990458i \(-0.544007\pi\)
−0.137812 + 0.990458i \(0.544007\pi\)
\(182\) 0 0
\(183\) −1183.93 −0.478242
\(184\) 0 0
\(185\) 317.776 0.126288
\(186\) 0 0
\(187\) 6387.38 2.49782
\(188\) 0 0
\(189\) −2123.51 −0.817263
\(190\) 0 0
\(191\) 5081.79 1.92516 0.962579 0.271002i \(-0.0873550\pi\)
0.962579 + 0.271002i \(0.0873550\pi\)
\(192\) 0 0
\(193\) −2000.69 −0.746179 −0.373089 0.927795i \(-0.621702\pi\)
−0.373089 + 0.927795i \(0.621702\pi\)
\(194\) 0 0
\(195\) −32.8299 −0.0120564
\(196\) 0 0
\(197\) 1144.50 0.413919 0.206960 0.978349i \(-0.433643\pi\)
0.206960 + 0.978349i \(0.433643\pi\)
\(198\) 0 0
\(199\) −1651.91 −0.588445 −0.294222 0.955737i \(-0.595061\pi\)
−0.294222 + 0.955737i \(0.595061\pi\)
\(200\) 0 0
\(201\) 221.570 0.0777530
\(202\) 0 0
\(203\) 793.167 0.274234
\(204\) 0 0
\(205\) 93.1220 0.0317264
\(206\) 0 0
\(207\) −4348.77 −1.46020
\(208\) 0 0
\(209\) 3482.37 1.15254
\(210\) 0 0
\(211\) −2761.98 −0.901149 −0.450574 0.892739i \(-0.648781\pi\)
−0.450574 + 0.892739i \(0.648781\pi\)
\(212\) 0 0
\(213\) 381.370 0.122681
\(214\) 0 0
\(215\) −278.939 −0.0884813
\(216\) 0 0
\(217\) 5464.42 1.70944
\(218\) 0 0
\(219\) −604.251 −0.186445
\(220\) 0 0
\(221\) 2851.86 0.868039
\(222\) 0 0
\(223\) −701.622 −0.210691 −0.105345 0.994436i \(-0.533595\pi\)
−0.105345 + 0.994436i \(0.533595\pi\)
\(224\) 0 0
\(225\) 3078.29 0.912085
\(226\) 0 0
\(227\) −5281.91 −1.54437 −0.772187 0.635395i \(-0.780837\pi\)
−0.772187 + 0.635395i \(0.780837\pi\)
\(228\) 0 0
\(229\) 5162.80 1.48982 0.744908 0.667168i \(-0.232493\pi\)
0.744908 + 0.667168i \(0.232493\pi\)
\(230\) 0 0
\(231\) 2554.54 0.727603
\(232\) 0 0
\(233\) 5656.98 1.59056 0.795281 0.606241i \(-0.207323\pi\)
0.795281 + 0.606241i \(0.207323\pi\)
\(234\) 0 0
\(235\) 117.131 0.0325140
\(236\) 0 0
\(237\) −643.988 −0.176504
\(238\) 0 0
\(239\) 4688.50 1.26893 0.634464 0.772953i \(-0.281221\pi\)
0.634464 + 0.772953i \(0.281221\pi\)
\(240\) 0 0
\(241\) −2254.32 −0.602545 −0.301273 0.953538i \(-0.597411\pi\)
−0.301273 + 0.953538i \(0.597411\pi\)
\(242\) 0 0
\(243\) 2924.08 0.771933
\(244\) 0 0
\(245\) −318.884 −0.0831541
\(246\) 0 0
\(247\) 1554.82 0.400529
\(248\) 0 0
\(249\) 118.273 0.0301015
\(250\) 0 0
\(251\) 906.394 0.227933 0.113966 0.993485i \(-0.463644\pi\)
0.113966 + 0.993485i \(0.463644\pi\)
\(252\) 0 0
\(253\) 10938.8 2.71824
\(254\) 0 0
\(255\) −121.190 −0.0297617
\(256\) 0 0
\(257\) −5953.47 −1.44501 −0.722504 0.691367i \(-0.757009\pi\)
−0.722504 + 0.691367i \(0.757009\pi\)
\(258\) 0 0
\(259\) −11040.0 −2.64862
\(260\) 0 0
\(261\) −717.722 −0.170214
\(262\) 0 0
\(263\) −2339.06 −0.548414 −0.274207 0.961671i \(-0.588415\pi\)
−0.274207 + 0.961671i \(0.588415\pi\)
\(264\) 0 0
\(265\) 296.081 0.0686344
\(266\) 0 0
\(267\) 1552.17 0.355772
\(268\) 0 0
\(269\) 574.684 0.130257 0.0651285 0.997877i \(-0.479254\pi\)
0.0651285 + 0.997877i \(0.479254\pi\)
\(270\) 0 0
\(271\) 4985.61 1.11754 0.558772 0.829321i \(-0.311273\pi\)
0.558772 + 0.829321i \(0.311273\pi\)
\(272\) 0 0
\(273\) 1140.56 0.252856
\(274\) 0 0
\(275\) −7743.03 −1.69790
\(276\) 0 0
\(277\) −2383.28 −0.516959 −0.258480 0.966017i \(-0.583221\pi\)
−0.258480 + 0.966017i \(0.583221\pi\)
\(278\) 0 0
\(279\) −4944.65 −1.06103
\(280\) 0 0
\(281\) −1020.93 −0.216739 −0.108370 0.994111i \(-0.534563\pi\)
−0.108370 + 0.994111i \(0.534563\pi\)
\(282\) 0 0
\(283\) 2873.16 0.603504 0.301752 0.953387i \(-0.402429\pi\)
0.301752 + 0.953387i \(0.402429\pi\)
\(284\) 0 0
\(285\) −66.0723 −0.0137326
\(286\) 0 0
\(287\) −3235.19 −0.665391
\(288\) 0 0
\(289\) 5614.51 1.14279
\(290\) 0 0
\(291\) 2686.87 0.541262
\(292\) 0 0
\(293\) −2250.38 −0.448698 −0.224349 0.974509i \(-0.572026\pi\)
−0.224349 + 0.974509i \(0.572026\pi\)
\(294\) 0 0
\(295\) −259.084 −0.0511337
\(296\) 0 0
\(297\) −4833.35 −0.944307
\(298\) 0 0
\(299\) 4883.98 0.944641
\(300\) 0 0
\(301\) 9690.74 1.85570
\(302\) 0 0
\(303\) −1087.36 −0.206163
\(304\) 0 0
\(305\) 621.238 0.116629
\(306\) 0 0
\(307\) 4353.67 0.809372 0.404686 0.914456i \(-0.367381\pi\)
0.404686 + 0.914456i \(0.367381\pi\)
\(308\) 0 0
\(309\) 267.478 0.0492436
\(310\) 0 0
\(311\) 995.466 0.181504 0.0907519 0.995874i \(-0.471073\pi\)
0.0907519 + 0.995874i \(0.471073\pi\)
\(312\) 0 0
\(313\) 10687.1 1.92993 0.964966 0.262375i \(-0.0845059\pi\)
0.964966 + 0.262375i \(0.0845059\pi\)
\(314\) 0 0
\(315\) 532.898 0.0953187
\(316\) 0 0
\(317\) −8547.86 −1.51450 −0.757249 0.653127i \(-0.773457\pi\)
−0.757249 + 0.653127i \(0.773457\pi\)
\(318\) 0 0
\(319\) 1805.34 0.316863
\(320\) 0 0
\(321\) −2724.32 −0.473698
\(322\) 0 0
\(323\) 5739.55 0.988721
\(324\) 0 0
\(325\) −3457.13 −0.590053
\(326\) 0 0
\(327\) −1503.41 −0.254248
\(328\) 0 0
\(329\) −4069.30 −0.681908
\(330\) 0 0
\(331\) −4474.69 −0.743055 −0.371528 0.928422i \(-0.621166\pi\)
−0.371528 + 0.928422i \(0.621166\pi\)
\(332\) 0 0
\(333\) 9989.87 1.64397
\(334\) 0 0
\(335\) −116.264 −0.0189617
\(336\) 0 0
\(337\) 7158.57 1.15713 0.578564 0.815637i \(-0.303613\pi\)
0.578564 + 0.815637i \(0.303613\pi\)
\(338\) 0 0
\(339\) 128.992 0.0206663
\(340\) 0 0
\(341\) 12437.6 1.97518
\(342\) 0 0
\(343\) 1697.23 0.267178
\(344\) 0 0
\(345\) −207.546 −0.0323881
\(346\) 0 0
\(347\) 1529.77 0.236663 0.118332 0.992974i \(-0.462245\pi\)
0.118332 + 0.992974i \(0.462245\pi\)
\(348\) 0 0
\(349\) −1291.38 −0.198068 −0.0990340 0.995084i \(-0.531575\pi\)
−0.0990340 + 0.995084i \(0.531575\pi\)
\(350\) 0 0
\(351\) −2158.01 −0.328165
\(352\) 0 0
\(353\) −1268.27 −0.191228 −0.0956139 0.995418i \(-0.530481\pi\)
−0.0956139 + 0.995418i \(0.530481\pi\)
\(354\) 0 0
\(355\) −200.115 −0.0299183
\(356\) 0 0
\(357\) 4210.32 0.624185
\(358\) 0 0
\(359\) −11996.6 −1.76366 −0.881832 0.471564i \(-0.843690\pi\)
−0.881832 + 0.471564i \(0.843690\pi\)
\(360\) 0 0
\(361\) −3729.83 −0.543786
\(362\) 0 0
\(363\) 3817.47 0.551971
\(364\) 0 0
\(365\) 317.067 0.0454686
\(366\) 0 0
\(367\) 11662.6 1.65881 0.829406 0.558646i \(-0.188679\pi\)
0.829406 + 0.558646i \(0.188679\pi\)
\(368\) 0 0
\(369\) 2927.46 0.413001
\(370\) 0 0
\(371\) −10286.3 −1.43945
\(372\) 0 0
\(373\) 9970.91 1.38411 0.692057 0.721843i \(-0.256705\pi\)
0.692057 + 0.721843i \(0.256705\pi\)
\(374\) 0 0
\(375\) 294.555 0.0405621
\(376\) 0 0
\(377\) 806.052 0.110116
\(378\) 0 0
\(379\) 7518.92 1.01905 0.509526 0.860455i \(-0.329821\pi\)
0.509526 + 0.860455i \(0.329821\pi\)
\(380\) 0 0
\(381\) −2649.61 −0.356282
\(382\) 0 0
\(383\) −6929.25 −0.924460 −0.462230 0.886760i \(-0.652951\pi\)
−0.462230 + 0.886760i \(0.652951\pi\)
\(384\) 0 0
\(385\) −1340.43 −0.177441
\(386\) 0 0
\(387\) −8768.96 −1.15181
\(388\) 0 0
\(389\) 296.982 0.0387084 0.0193542 0.999813i \(-0.493839\pi\)
0.0193542 + 0.999813i \(0.493839\pi\)
\(390\) 0 0
\(391\) 18029.0 2.33188
\(392\) 0 0
\(393\) −1560.34 −0.200277
\(394\) 0 0
\(395\) 337.918 0.0430442
\(396\) 0 0
\(397\) 14349.1 1.81401 0.907004 0.421123i \(-0.138364\pi\)
0.907004 + 0.421123i \(0.138364\pi\)
\(398\) 0 0
\(399\) 2295.45 0.288010
\(400\) 0 0
\(401\) −250.164 −0.0311536 −0.0155768 0.999879i \(-0.504958\pi\)
−0.0155768 + 0.999879i \(0.504958\pi\)
\(402\) 0 0
\(403\) 5553.19 0.686412
\(404\) 0 0
\(405\) −434.362 −0.0532929
\(406\) 0 0
\(407\) −25128.2 −3.06034
\(408\) 0 0
\(409\) −13128.8 −1.58723 −0.793616 0.608419i \(-0.791804\pi\)
−0.793616 + 0.608419i \(0.791804\pi\)
\(410\) 0 0
\(411\) 2431.59 0.291828
\(412\) 0 0
\(413\) 9000.94 1.07241
\(414\) 0 0
\(415\) −62.0612 −0.00734088
\(416\) 0 0
\(417\) −910.670 −0.106944
\(418\) 0 0
\(419\) 11430.9 1.33278 0.666389 0.745604i \(-0.267839\pi\)
0.666389 + 0.745604i \(0.267839\pi\)
\(420\) 0 0
\(421\) −13092.7 −1.51567 −0.757835 0.652446i \(-0.773743\pi\)
−0.757835 + 0.652446i \(0.773743\pi\)
\(422\) 0 0
\(423\) 3682.23 0.423254
\(424\) 0 0
\(425\) −12761.9 −1.45657
\(426\) 0 0
\(427\) −21582.7 −2.44604
\(428\) 0 0
\(429\) 2596.04 0.292163
\(430\) 0 0
\(431\) 2761.82 0.308660 0.154330 0.988019i \(-0.450678\pi\)
0.154330 + 0.988019i \(0.450678\pi\)
\(432\) 0 0
\(433\) −12593.9 −1.39774 −0.698872 0.715247i \(-0.746314\pi\)
−0.698872 + 0.715247i \(0.746314\pi\)
\(434\) 0 0
\(435\) −34.2533 −0.00377545
\(436\) 0 0
\(437\) 9829.32 1.07597
\(438\) 0 0
\(439\) −15492.1 −1.68428 −0.842138 0.539261i \(-0.818703\pi\)
−0.842138 + 0.539261i \(0.818703\pi\)
\(440\) 0 0
\(441\) −10024.7 −1.08246
\(442\) 0 0
\(443\) −8097.69 −0.868472 −0.434236 0.900799i \(-0.642982\pi\)
−0.434236 + 0.900799i \(0.642982\pi\)
\(444\) 0 0
\(445\) −814.463 −0.0867624
\(446\) 0 0
\(447\) −1805.06 −0.190998
\(448\) 0 0
\(449\) 7625.66 0.801508 0.400754 0.916186i \(-0.368748\pi\)
0.400754 + 0.916186i \(0.368748\pi\)
\(450\) 0 0
\(451\) −7363.65 −0.768826
\(452\) 0 0
\(453\) 2467.15 0.255887
\(454\) 0 0
\(455\) −598.481 −0.0616643
\(456\) 0 0
\(457\) 3035.58 0.310719 0.155360 0.987858i \(-0.450346\pi\)
0.155360 + 0.987858i \(0.450346\pi\)
\(458\) 0 0
\(459\) −7966.19 −0.810087
\(460\) 0 0
\(461\) −18036.1 −1.82218 −0.911088 0.412213i \(-0.864756\pi\)
−0.911088 + 0.412213i \(0.864756\pi\)
\(462\) 0 0
\(463\) 2284.08 0.229266 0.114633 0.993408i \(-0.463431\pi\)
0.114633 + 0.993408i \(0.463431\pi\)
\(464\) 0 0
\(465\) −235.984 −0.0235344
\(466\) 0 0
\(467\) 8813.46 0.873316 0.436658 0.899628i \(-0.356162\pi\)
0.436658 + 0.899628i \(0.356162\pi\)
\(468\) 0 0
\(469\) 4039.17 0.397679
\(470\) 0 0
\(471\) −1894.02 −0.185290
\(472\) 0 0
\(473\) 22057.2 2.14417
\(474\) 0 0
\(475\) −6957.71 −0.672087
\(476\) 0 0
\(477\) 9307.86 0.893454
\(478\) 0 0
\(479\) 18983.6 1.81082 0.905411 0.424537i \(-0.139563\pi\)
0.905411 + 0.424537i \(0.139563\pi\)
\(480\) 0 0
\(481\) −11219.3 −1.06353
\(482\) 0 0
\(483\) 7210.43 0.679267
\(484\) 0 0
\(485\) −1409.87 −0.131998
\(486\) 0 0
\(487\) 17208.6 1.60122 0.800610 0.599186i \(-0.204509\pi\)
0.800610 + 0.599186i \(0.204509\pi\)
\(488\) 0 0
\(489\) 3685.07 0.340787
\(490\) 0 0
\(491\) −20652.2 −1.89821 −0.949106 0.314957i \(-0.898010\pi\)
−0.949106 + 0.314957i \(0.898010\pi\)
\(492\) 0 0
\(493\) 2975.51 0.271826
\(494\) 0 0
\(495\) 1212.93 0.110136
\(496\) 0 0
\(497\) 6952.28 0.627469
\(498\) 0 0
\(499\) 19866.7 1.78228 0.891139 0.453730i \(-0.149907\pi\)
0.891139 + 0.453730i \(0.149907\pi\)
\(500\) 0 0
\(501\) −1163.20 −0.103729
\(502\) 0 0
\(503\) −5377.97 −0.476723 −0.238362 0.971176i \(-0.576610\pi\)
−0.238362 + 0.971176i \(0.576610\pi\)
\(504\) 0 0
\(505\) 570.568 0.0502771
\(506\) 0 0
\(507\) −2137.13 −0.187206
\(508\) 0 0
\(509\) 701.222 0.0610631 0.0305315 0.999534i \(-0.490280\pi\)
0.0305315 + 0.999534i \(0.490280\pi\)
\(510\) 0 0
\(511\) −11015.4 −0.953602
\(512\) 0 0
\(513\) −4343.13 −0.373789
\(514\) 0 0
\(515\) −140.353 −0.0120091
\(516\) 0 0
\(517\) −9262.18 −0.787911
\(518\) 0 0
\(519\) −1234.77 −0.104432
\(520\) 0 0
\(521\) −21334.1 −1.79398 −0.896990 0.442051i \(-0.854251\pi\)
−0.896990 + 0.442051i \(0.854251\pi\)
\(522\) 0 0
\(523\) 12288.4 1.02741 0.513703 0.857968i \(-0.328273\pi\)
0.513703 + 0.857968i \(0.328273\pi\)
\(524\) 0 0
\(525\) −5103.92 −0.424292
\(526\) 0 0
\(527\) 20499.4 1.69443
\(528\) 0 0
\(529\) 18708.7 1.53766
\(530\) 0 0
\(531\) −8144.78 −0.665637
\(532\) 0 0
\(533\) −3287.74 −0.267182
\(534\) 0 0
\(535\) 1429.53 0.115521
\(536\) 0 0
\(537\) 5356.84 0.430474
\(538\) 0 0
\(539\) 25215.8 2.01507
\(540\) 0 0
\(541\) −1517.00 −0.120556 −0.0602780 0.998182i \(-0.519199\pi\)
−0.0602780 + 0.998182i \(0.519199\pi\)
\(542\) 0 0
\(543\) −1006.98 −0.0795832
\(544\) 0 0
\(545\) 788.882 0.0620036
\(546\) 0 0
\(547\) 15366.1 1.20111 0.600554 0.799584i \(-0.294947\pi\)
0.600554 + 0.799584i \(0.294947\pi\)
\(548\) 0 0
\(549\) 19529.8 1.51823
\(550\) 0 0
\(551\) 1622.23 0.125425
\(552\) 0 0
\(553\) −11739.7 −0.902757
\(554\) 0 0
\(555\) 476.768 0.0364642
\(556\) 0 0
\(557\) 7717.53 0.587078 0.293539 0.955947i \(-0.405167\pi\)
0.293539 + 0.955947i \(0.405167\pi\)
\(558\) 0 0
\(559\) 9848.16 0.745139
\(560\) 0 0
\(561\) 9583.15 0.721214
\(562\) 0 0
\(563\) 7026.17 0.525964 0.262982 0.964801i \(-0.415294\pi\)
0.262982 + 0.964801i \(0.415294\pi\)
\(564\) 0 0
\(565\) −67.6854 −0.00503991
\(566\) 0 0
\(567\) 15090.3 1.11770
\(568\) 0 0
\(569\) −10743.0 −0.791510 −0.395755 0.918356i \(-0.629517\pi\)
−0.395755 + 0.918356i \(0.629517\pi\)
\(570\) 0 0
\(571\) −9274.65 −0.679741 −0.339871 0.940472i \(-0.610383\pi\)
−0.339871 + 0.940472i \(0.610383\pi\)
\(572\) 0 0
\(573\) 7624.34 0.555866
\(574\) 0 0
\(575\) −21855.5 −1.58511
\(576\) 0 0
\(577\) −14538.6 −1.04896 −0.524479 0.851423i \(-0.675740\pi\)
−0.524479 + 0.851423i \(0.675740\pi\)
\(578\) 0 0
\(579\) −3001.68 −0.215450
\(580\) 0 0
\(581\) 2156.09 0.153958
\(582\) 0 0
\(583\) −23412.7 −1.66322
\(584\) 0 0
\(585\) 541.554 0.0382744
\(586\) 0 0
\(587\) −15075.6 −1.06003 −0.530014 0.847989i \(-0.677813\pi\)
−0.530014 + 0.847989i \(0.677813\pi\)
\(588\) 0 0
\(589\) 11176.2 0.781843
\(590\) 0 0
\(591\) 1717.12 0.119514
\(592\) 0 0
\(593\) 10204.6 0.706668 0.353334 0.935497i \(-0.385048\pi\)
0.353334 + 0.935497i \(0.385048\pi\)
\(594\) 0 0
\(595\) −2209.27 −0.152220
\(596\) 0 0
\(597\) −2478.40 −0.169906
\(598\) 0 0
\(599\) 12544.9 0.855713 0.427856 0.903847i \(-0.359269\pi\)
0.427856 + 0.903847i \(0.359269\pi\)
\(600\) 0 0
\(601\) −18702.6 −1.26937 −0.634687 0.772770i \(-0.718871\pi\)
−0.634687 + 0.772770i \(0.718871\pi\)
\(602\) 0 0
\(603\) −3654.97 −0.246835
\(604\) 0 0
\(605\) −2003.13 −0.134610
\(606\) 0 0
\(607\) −5121.68 −0.342475 −0.171238 0.985230i \(-0.554777\pi\)
−0.171238 + 0.985230i \(0.554777\pi\)
\(608\) 0 0
\(609\) 1190.01 0.0791817
\(610\) 0 0
\(611\) −4135.41 −0.273814
\(612\) 0 0
\(613\) −3382.24 −0.222850 −0.111425 0.993773i \(-0.535542\pi\)
−0.111425 + 0.993773i \(0.535542\pi\)
\(614\) 0 0
\(615\) 139.713 0.00916063
\(616\) 0 0
\(617\) −571.111 −0.0372643 −0.0186321 0.999826i \(-0.505931\pi\)
−0.0186321 + 0.999826i \(0.505931\pi\)
\(618\) 0 0
\(619\) 10978.5 0.712867 0.356433 0.934321i \(-0.383993\pi\)
0.356433 + 0.934321i \(0.383993\pi\)
\(620\) 0 0
\(621\) −13642.6 −0.881575
\(622\) 0 0
\(623\) 28295.6 1.81965
\(624\) 0 0
\(625\) 15393.0 0.985150
\(626\) 0 0
\(627\) 5224.69 0.332782
\(628\) 0 0
\(629\) −41415.7 −2.62536
\(630\) 0 0
\(631\) 2688.48 0.169615 0.0848073 0.996397i \(-0.472973\pi\)
0.0848073 + 0.996397i \(0.472973\pi\)
\(632\) 0 0
\(633\) −4143.87 −0.260196
\(634\) 0 0
\(635\) 1390.32 0.0868869
\(636\) 0 0
\(637\) 11258.4 0.700276
\(638\) 0 0
\(639\) −6290.98 −0.389464
\(640\) 0 0
\(641\) −12783.8 −0.787724 −0.393862 0.919170i \(-0.628861\pi\)
−0.393862 + 0.919170i \(0.628861\pi\)
\(642\) 0 0
\(643\) 21418.6 1.31363 0.656816 0.754051i \(-0.271903\pi\)
0.656816 + 0.754051i \(0.271903\pi\)
\(644\) 0 0
\(645\) −418.500 −0.0255479
\(646\) 0 0
\(647\) 1439.68 0.0874799 0.0437400 0.999043i \(-0.486073\pi\)
0.0437400 + 0.999043i \(0.486073\pi\)
\(648\) 0 0
\(649\) 20487.1 1.23912
\(650\) 0 0
\(651\) 8198.42 0.493581
\(652\) 0 0
\(653\) 11505.7 0.689515 0.344757 0.938692i \(-0.387961\pi\)
0.344757 + 0.938692i \(0.387961\pi\)
\(654\) 0 0
\(655\) 818.753 0.0488417
\(656\) 0 0
\(657\) 9967.58 0.591891
\(658\) 0 0
\(659\) −3275.96 −0.193647 −0.0968234 0.995302i \(-0.530868\pi\)
−0.0968234 + 0.995302i \(0.530868\pi\)
\(660\) 0 0
\(661\) 5455.49 0.321019 0.160510 0.987034i \(-0.448686\pi\)
0.160510 + 0.987034i \(0.448686\pi\)
\(662\) 0 0
\(663\) 4278.72 0.250636
\(664\) 0 0
\(665\) −1204.48 −0.0702373
\(666\) 0 0
\(667\) 5095.73 0.295813
\(668\) 0 0
\(669\) −1052.66 −0.0608345
\(670\) 0 0
\(671\) −49124.6 −2.82628
\(672\) 0 0
\(673\) 535.453 0.0306689 0.0153345 0.999882i \(-0.495119\pi\)
0.0153345 + 0.999882i \(0.495119\pi\)
\(674\) 0 0
\(675\) 9656.93 0.550660
\(676\) 0 0
\(677\) −16698.6 −0.947976 −0.473988 0.880531i \(-0.657186\pi\)
−0.473988 + 0.880531i \(0.657186\pi\)
\(678\) 0 0
\(679\) 48981.0 2.76836
\(680\) 0 0
\(681\) −7924.59 −0.445919
\(682\) 0 0
\(683\) −1082.40 −0.0606396 −0.0303198 0.999540i \(-0.509653\pi\)
−0.0303198 + 0.999540i \(0.509653\pi\)
\(684\) 0 0
\(685\) −1275.92 −0.0711683
\(686\) 0 0
\(687\) 7745.89 0.430166
\(688\) 0 0
\(689\) −10453.4 −0.578000
\(690\) 0 0
\(691\) 9761.34 0.537393 0.268697 0.963225i \(-0.413407\pi\)
0.268697 + 0.963225i \(0.413407\pi\)
\(692\) 0 0
\(693\) −42139.0 −2.30985
\(694\) 0 0
\(695\) 477.853 0.0260806
\(696\) 0 0
\(697\) −12136.6 −0.659549
\(698\) 0 0
\(699\) 8487.31 0.459256
\(700\) 0 0
\(701\) 3032.05 0.163365 0.0816827 0.996658i \(-0.473971\pi\)
0.0816827 + 0.996658i \(0.473971\pi\)
\(702\) 0 0
\(703\) −22579.6 −1.21139
\(704\) 0 0
\(705\) 175.735 0.00938803
\(706\) 0 0
\(707\) −19822.4 −1.05445
\(708\) 0 0
\(709\) 4640.54 0.245810 0.122905 0.992418i \(-0.460779\pi\)
0.122905 + 0.992418i \(0.460779\pi\)
\(710\) 0 0
\(711\) 10623.1 0.560332
\(712\) 0 0
\(713\) 35106.4 1.84396
\(714\) 0 0
\(715\) −1362.21 −0.0712500
\(716\) 0 0
\(717\) 7034.27 0.366387
\(718\) 0 0
\(719\) −20575.7 −1.06724 −0.533619 0.845725i \(-0.679168\pi\)
−0.533619 + 0.845725i \(0.679168\pi\)
\(720\) 0 0
\(721\) 4876.06 0.251864
\(722\) 0 0
\(723\) −3382.21 −0.173978
\(724\) 0 0
\(725\) −3607.03 −0.184775
\(726\) 0 0
\(727\) −13890.0 −0.708600 −0.354300 0.935132i \(-0.615281\pi\)
−0.354300 + 0.935132i \(0.615281\pi\)
\(728\) 0 0
\(729\) −10509.8 −0.533955
\(730\) 0 0
\(731\) 36354.1 1.83940
\(732\) 0 0
\(733\) −6253.36 −0.315107 −0.157553 0.987510i \(-0.550361\pi\)
−0.157553 + 0.987510i \(0.550361\pi\)
\(734\) 0 0
\(735\) −478.430 −0.0240097
\(736\) 0 0
\(737\) 9193.59 0.459498
\(738\) 0 0
\(739\) −4938.33 −0.245818 −0.122909 0.992418i \(-0.539222\pi\)
−0.122909 + 0.992418i \(0.539222\pi\)
\(740\) 0 0
\(741\) 2332.74 0.115648
\(742\) 0 0
\(743\) −17443.8 −0.861307 −0.430653 0.902517i \(-0.641717\pi\)
−0.430653 + 0.902517i \(0.641717\pi\)
\(744\) 0 0
\(745\) 947.162 0.0465790
\(746\) 0 0
\(747\) −1951.01 −0.0955605
\(748\) 0 0
\(749\) −49663.8 −2.42280
\(750\) 0 0
\(751\) −13989.1 −0.679718 −0.339859 0.940476i \(-0.610379\pi\)
−0.339859 + 0.940476i \(0.610379\pi\)
\(752\) 0 0
\(753\) 1359.89 0.0658128
\(754\) 0 0
\(755\) −1294.58 −0.0624035
\(756\) 0 0
\(757\) −35278.7 −1.69382 −0.846912 0.531733i \(-0.821541\pi\)
−0.846912 + 0.531733i \(0.821541\pi\)
\(758\) 0 0
\(759\) 16411.7 0.784859
\(760\) 0 0
\(761\) −15090.1 −0.718814 −0.359407 0.933181i \(-0.617021\pi\)
−0.359407 + 0.933181i \(0.617021\pi\)
\(762\) 0 0
\(763\) −27406.9 −1.30039
\(764\) 0 0
\(765\) 1999.12 0.0944817
\(766\) 0 0
\(767\) 9147.16 0.430619
\(768\) 0 0
\(769\) −4238.61 −0.198762 −0.0993812 0.995049i \(-0.531686\pi\)
−0.0993812 + 0.995049i \(0.531686\pi\)
\(770\) 0 0
\(771\) −8932.14 −0.417229
\(772\) 0 0
\(773\) −36539.4 −1.70017 −0.850084 0.526647i \(-0.823449\pi\)
−0.850084 + 0.526647i \(0.823449\pi\)
\(774\) 0 0
\(775\) −24850.1 −1.15180
\(776\) 0 0
\(777\) −16563.6 −0.764756
\(778\) 0 0
\(779\) −6616.80 −0.304328
\(780\) 0 0
\(781\) 15824.1 0.725010
\(782\) 0 0
\(783\) −2251.57 −0.102765
\(784\) 0 0
\(785\) 993.842 0.0451869
\(786\) 0 0
\(787\) −25556.5 −1.15755 −0.578774 0.815488i \(-0.696469\pi\)
−0.578774 + 0.815488i \(0.696469\pi\)
\(788\) 0 0
\(789\) −3509.36 −0.158348
\(790\) 0 0
\(791\) 2351.49 0.105701
\(792\) 0 0
\(793\) −21933.3 −0.982187
\(794\) 0 0
\(795\) 444.218 0.0198174
\(796\) 0 0
\(797\) −16208.1 −0.720352 −0.360176 0.932884i \(-0.617283\pi\)
−0.360176 + 0.932884i \(0.617283\pi\)
\(798\) 0 0
\(799\) −15265.7 −0.675921
\(800\) 0 0
\(801\) −25604.2 −1.12944
\(802\) 0 0
\(803\) −25072.2 −1.10184
\(804\) 0 0
\(805\) −3783.51 −0.165653
\(806\) 0 0
\(807\) 862.214 0.0376101
\(808\) 0 0
\(809\) 24649.6 1.07124 0.535621 0.844459i \(-0.320078\pi\)
0.535621 + 0.844459i \(0.320078\pi\)
\(810\) 0 0
\(811\) −43969.2 −1.90378 −0.951892 0.306435i \(-0.900864\pi\)
−0.951892 + 0.306435i \(0.900864\pi\)
\(812\) 0 0
\(813\) 7480.05 0.322677
\(814\) 0 0
\(815\) −1933.66 −0.0831080
\(816\) 0 0
\(817\) 19820.1 0.848735
\(818\) 0 0
\(819\) −18814.4 −0.802719
\(820\) 0 0
\(821\) −14759.0 −0.627398 −0.313699 0.949523i \(-0.601568\pi\)
−0.313699 + 0.949523i \(0.601568\pi\)
\(822\) 0 0
\(823\) −36849.9 −1.56076 −0.780381 0.625304i \(-0.784975\pi\)
−0.780381 + 0.625304i \(0.784975\pi\)
\(824\) 0 0
\(825\) −11617.1 −0.490248
\(826\) 0 0
\(827\) 7780.68 0.327159 0.163580 0.986530i \(-0.447696\pi\)
0.163580 + 0.986530i \(0.447696\pi\)
\(828\) 0 0
\(829\) −40014.8 −1.67644 −0.838221 0.545330i \(-0.816404\pi\)
−0.838221 + 0.545330i \(0.816404\pi\)
\(830\) 0 0
\(831\) −3575.70 −0.149266
\(832\) 0 0
\(833\) 41560.1 1.72866
\(834\) 0 0
\(835\) 610.363 0.0252964
\(836\) 0 0
\(837\) −15511.9 −0.640586
\(838\) 0 0
\(839\) −2466.60 −0.101498 −0.0507488 0.998711i \(-0.516161\pi\)
−0.0507488 + 0.998711i \(0.516161\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −1531.73 −0.0625809
\(844\) 0 0
\(845\) 1121.41 0.0456541
\(846\) 0 0
\(847\) 69591.6 2.82314
\(848\) 0 0
\(849\) 4310.68 0.174254
\(850\) 0 0
\(851\) −70926.8 −2.85704
\(852\) 0 0
\(853\) 7269.34 0.291791 0.145895 0.989300i \(-0.453394\pi\)
0.145895 + 0.989300i \(0.453394\pi\)
\(854\) 0 0
\(855\) 1089.91 0.0435956
\(856\) 0 0
\(857\) −13378.5 −0.533256 −0.266628 0.963800i \(-0.585910\pi\)
−0.266628 + 0.963800i \(0.585910\pi\)
\(858\) 0 0
\(859\) −35532.4 −1.41135 −0.705675 0.708536i \(-0.749356\pi\)
−0.705675 + 0.708536i \(0.749356\pi\)
\(860\) 0 0
\(861\) −4853.84 −0.192124
\(862\) 0 0
\(863\) 9556.27 0.376940 0.188470 0.982079i \(-0.439647\pi\)
0.188470 + 0.982079i \(0.439647\pi\)
\(864\) 0 0
\(865\) 647.916 0.0254680
\(866\) 0 0
\(867\) 8423.60 0.329966
\(868\) 0 0
\(869\) −26720.9 −1.04309
\(870\) 0 0
\(871\) 4104.78 0.159685
\(872\) 0 0
\(873\) −44322.0 −1.71830
\(874\) 0 0
\(875\) 5369.67 0.207461
\(876\) 0 0
\(877\) 18138.4 0.698394 0.349197 0.937049i \(-0.386454\pi\)
0.349197 + 0.937049i \(0.386454\pi\)
\(878\) 0 0
\(879\) −3376.30 −0.129556
\(880\) 0 0
\(881\) 38547.5 1.47412 0.737058 0.675829i \(-0.236214\pi\)
0.737058 + 0.675829i \(0.236214\pi\)
\(882\) 0 0
\(883\) 12228.3 0.466040 0.233020 0.972472i \(-0.425139\pi\)
0.233020 + 0.972472i \(0.425139\pi\)
\(884\) 0 0
\(885\) −388.710 −0.0147642
\(886\) 0 0
\(887\) −35387.4 −1.33956 −0.669782 0.742558i \(-0.733613\pi\)
−0.669782 + 0.742558i \(0.733613\pi\)
\(888\) 0 0
\(889\) −48301.7 −1.82226
\(890\) 0 0
\(891\) 34347.3 1.29144
\(892\) 0 0
\(893\) −8322.77 −0.311882
\(894\) 0 0
\(895\) −2810.88 −0.104980
\(896\) 0 0
\(897\) 7327.56 0.272754
\(898\) 0 0
\(899\) 5793.96 0.214949
\(900\) 0 0
\(901\) −38588.2 −1.42681
\(902\) 0 0
\(903\) 14539.3 0.535810
\(904\) 0 0
\(905\) 528.390 0.0194080
\(906\) 0 0
\(907\) 25552.8 0.935465 0.467733 0.883870i \(-0.345071\pi\)
0.467733 + 0.883870i \(0.345071\pi\)
\(908\) 0 0
\(909\) 17936.9 0.654487
\(910\) 0 0
\(911\) 11127.5 0.404688 0.202344 0.979314i \(-0.435144\pi\)
0.202344 + 0.979314i \(0.435144\pi\)
\(912\) 0 0
\(913\) 4907.51 0.177891
\(914\) 0 0
\(915\) 932.060 0.0336754
\(916\) 0 0
\(917\) −28444.6 −1.02435
\(918\) 0 0
\(919\) 34243.6 1.22915 0.614577 0.788857i \(-0.289327\pi\)
0.614577 + 0.788857i \(0.289327\pi\)
\(920\) 0 0
\(921\) 6531.93 0.233697
\(922\) 0 0
\(923\) 7065.22 0.251955
\(924\) 0 0
\(925\) 50205.7 1.78460
\(926\) 0 0
\(927\) −4412.25 −0.156329
\(928\) 0 0
\(929\) 5315.50 0.187724 0.0938621 0.995585i \(-0.470079\pi\)
0.0938621 + 0.995585i \(0.470079\pi\)
\(930\) 0 0
\(931\) 22658.4 0.797635
\(932\) 0 0
\(933\) 1493.52 0.0524071
\(934\) 0 0
\(935\) −5028.54 −0.175883
\(936\) 0 0
\(937\) −31475.4 −1.09739 −0.548697 0.836021i \(-0.684876\pi\)
−0.548697 + 0.836021i \(0.684876\pi\)
\(938\) 0 0
\(939\) 16034.1 0.557245
\(940\) 0 0
\(941\) 16014.6 0.554794 0.277397 0.960755i \(-0.410528\pi\)
0.277397 + 0.960755i \(0.410528\pi\)
\(942\) 0 0
\(943\) −20784.6 −0.717752
\(944\) 0 0
\(945\) 1671.76 0.0575474
\(946\) 0 0
\(947\) −28388.8 −0.974140 −0.487070 0.873363i \(-0.661934\pi\)
−0.487070 + 0.873363i \(0.661934\pi\)
\(948\) 0 0
\(949\) −11194.3 −0.382911
\(950\) 0 0
\(951\) −12824.6 −0.437293
\(952\) 0 0
\(953\) 32633.4 1.10924 0.554618 0.832105i \(-0.312865\pi\)
0.554618 + 0.832105i \(0.312865\pi\)
\(954\) 0 0
\(955\) −4000.69 −0.135560
\(956\) 0 0
\(957\) 2708.59 0.0914905
\(958\) 0 0
\(959\) 44327.2 1.49260
\(960\) 0 0
\(961\) 10125.7 0.339893
\(962\) 0 0
\(963\) 44939.8 1.50380
\(964\) 0 0
\(965\) 1575.06 0.0525421
\(966\) 0 0
\(967\) −36126.7 −1.20140 −0.600702 0.799473i \(-0.705112\pi\)
−0.600702 + 0.799473i \(0.705112\pi\)
\(968\) 0 0
\(969\) 8611.20 0.285481
\(970\) 0 0
\(971\) 45941.5 1.51836 0.759182 0.650878i \(-0.225599\pi\)
0.759182 + 0.650878i \(0.225599\pi\)
\(972\) 0 0
\(973\) −16601.3 −0.546982
\(974\) 0 0
\(975\) −5186.83 −0.170371
\(976\) 0 0
\(977\) −37251.1 −1.21982 −0.609911 0.792470i \(-0.708795\pi\)
−0.609911 + 0.792470i \(0.708795\pi\)
\(978\) 0 0
\(979\) 64403.9 2.10251
\(980\) 0 0
\(981\) 24800.0 0.807137
\(982\) 0 0
\(983\) 52551.7 1.70513 0.852563 0.522624i \(-0.175047\pi\)
0.852563 + 0.522624i \(0.175047\pi\)
\(984\) 0 0
\(985\) −901.020 −0.0291461
\(986\) 0 0
\(987\) −6105.28 −0.196893
\(988\) 0 0
\(989\) 62258.5 2.00172
\(990\) 0 0
\(991\) −10063.1 −0.322569 −0.161284 0.986908i \(-0.551564\pi\)
−0.161284 + 0.986908i \(0.551564\pi\)
\(992\) 0 0
\(993\) −6713.50 −0.214548
\(994\) 0 0
\(995\) 1300.48 0.0414352
\(996\) 0 0
\(997\) −2111.64 −0.0670776 −0.0335388 0.999437i \(-0.510678\pi\)
−0.0335388 + 0.999437i \(0.510678\pi\)
\(998\) 0 0
\(999\) 31339.3 0.992525
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 1856.4.a.bk.1.8 12
4.3 odd 2 inner 1856.4.a.bk.1.5 12
8.3 odd 2 928.4.a.i.1.8 yes 12
8.5 even 2 928.4.a.i.1.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
928.4.a.i.1.5 12 8.5 even 2
928.4.a.i.1.8 yes 12 8.3 odd 2
1856.4.a.bk.1.5 12 4.3 odd 2 inner
1856.4.a.bk.1.8 12 1.1 even 1 trivial