Properties

Label 1452.4.a.t.1.4
Level $1452$
Weight $4$
Character 1452.1
Self dual yes
Analytic conductor $85.671$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1452,4,Mod(1,1452)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1452, 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("1452.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1452.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: \(85.6707733283\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 174x^{4} + 63x^{3} + 7614x^{2} + 1579x - 12611 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 11 \)
Twist minimal: no (minimal twist has level 132)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-8.43670\) of defining polynomial
Character \(\chi\) \(=\) 1452.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +3.10899 q^{5} -29.8524 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +3.10899 q^{5} -29.8524 q^{7} +9.00000 q^{9} +32.2030 q^{13} +9.32697 q^{15} +70.3884 q^{17} -47.1971 q^{19} -89.5572 q^{21} +56.3381 q^{23} -115.334 q^{25} +27.0000 q^{27} -194.765 q^{29} +146.250 q^{31} -92.8108 q^{35} +310.527 q^{37} +96.6091 q^{39} -332.509 q^{41} +87.1866 q^{43} +27.9809 q^{45} -101.101 q^{47} +548.166 q^{49} +211.165 q^{51} +365.931 q^{53} -141.591 q^{57} -641.892 q^{59} -758.230 q^{61} -268.672 q^{63} +100.119 q^{65} +123.262 q^{67} +169.014 q^{69} -1066.33 q^{71} +525.946 q^{73} -346.003 q^{75} -1155.63 q^{79} +81.0000 q^{81} -330.379 q^{83} +218.837 q^{85} -584.295 q^{87} -300.630 q^{89} -961.338 q^{91} +438.749 q^{93} -146.735 q^{95} -1548.21 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9} - 66 q^{13} + 39 q^{15} - 44 q^{17} - 270 q^{19} - 69 q^{21} - 124 q^{23} + 131 q^{25} + 162 q^{27} - 141 q^{29} - 253 q^{31} - 884 q^{35} + 288 q^{37} - 198 q^{39} - 428 q^{41} - 1006 q^{43} + 117 q^{45} - 674 q^{47} + 181 q^{49} - 132 q^{51} - 773 q^{53} - 810 q^{57} - 17 q^{59} - 1016 q^{61} - 207 q^{63} - 1220 q^{65} + 1836 q^{67} - 372 q^{69} - 208 q^{71} - 1521 q^{73} + 393 q^{75} - 1425 q^{79} + 486 q^{81} - 3065 q^{83} - 1304 q^{85} - 423 q^{87} + 1444 q^{89} + 1328 q^{91} - 759 q^{93} - 1760 q^{95} - 3887 q^{97}+O(q^{100}) \) 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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 3.10899 0.278076 0.139038 0.990287i \(-0.455599\pi\)
0.139038 + 0.990287i \(0.455599\pi\)
\(6\) 0 0
\(7\) −29.8524 −1.61188 −0.805939 0.591999i \(-0.798339\pi\)
−0.805939 + 0.591999i \(0.798339\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 32.2030 0.687040 0.343520 0.939145i \(-0.388381\pi\)
0.343520 + 0.939145i \(0.388381\pi\)
\(14\) 0 0
\(15\) 9.32697 0.160548
\(16\) 0 0
\(17\) 70.3884 1.00422 0.502109 0.864805i \(-0.332558\pi\)
0.502109 + 0.864805i \(0.332558\pi\)
\(18\) 0 0
\(19\) −47.1971 −0.569882 −0.284941 0.958545i \(-0.591974\pi\)
−0.284941 + 0.958545i \(0.591974\pi\)
\(20\) 0 0
\(21\) −89.5572 −0.930618
\(22\) 0 0
\(23\) 56.3381 0.510753 0.255376 0.966842i \(-0.417801\pi\)
0.255376 + 0.966842i \(0.417801\pi\)
\(24\) 0 0
\(25\) −115.334 −0.922673
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −194.765 −1.24714 −0.623568 0.781769i \(-0.714318\pi\)
−0.623568 + 0.781769i \(0.714318\pi\)
\(30\) 0 0
\(31\) 146.250 0.847330 0.423665 0.905819i \(-0.360743\pi\)
0.423665 + 0.905819i \(0.360743\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −92.8108 −0.448225
\(36\) 0 0
\(37\) 310.527 1.37974 0.689869 0.723935i \(-0.257668\pi\)
0.689869 + 0.723935i \(0.257668\pi\)
\(38\) 0 0
\(39\) 96.6091 0.396663
\(40\) 0 0
\(41\) −332.509 −1.26657 −0.633284 0.773920i \(-0.718293\pi\)
−0.633284 + 0.773920i \(0.718293\pi\)
\(42\) 0 0
\(43\) 87.1866 0.309205 0.154603 0.987977i \(-0.450590\pi\)
0.154603 + 0.987977i \(0.450590\pi\)
\(44\) 0 0
\(45\) 27.9809 0.0926922
\(46\) 0 0
\(47\) −101.101 −0.313769 −0.156885 0.987617i \(-0.550145\pi\)
−0.156885 + 0.987617i \(0.550145\pi\)
\(48\) 0 0
\(49\) 548.166 1.59815
\(50\) 0 0
\(51\) 211.165 0.579785
\(52\) 0 0
\(53\) 365.931 0.948388 0.474194 0.880420i \(-0.342740\pi\)
0.474194 + 0.880420i \(0.342740\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −141.591 −0.329021
\(58\) 0 0
\(59\) −641.892 −1.41639 −0.708197 0.706015i \(-0.750491\pi\)
−0.708197 + 0.706015i \(0.750491\pi\)
\(60\) 0 0
\(61\) −758.230 −1.59150 −0.795750 0.605626i \(-0.792923\pi\)
−0.795750 + 0.605626i \(0.792923\pi\)
\(62\) 0 0
\(63\) −268.672 −0.537293
\(64\) 0 0
\(65\) 100.119 0.191050
\(66\) 0 0
\(67\) 123.262 0.224759 0.112380 0.993665i \(-0.464153\pi\)
0.112380 + 0.993665i \(0.464153\pi\)
\(68\) 0 0
\(69\) 169.014 0.294883
\(70\) 0 0
\(71\) −1066.33 −1.78240 −0.891200 0.453611i \(-0.850135\pi\)
−0.891200 + 0.453611i \(0.850135\pi\)
\(72\) 0 0
\(73\) 525.946 0.843251 0.421625 0.906770i \(-0.361460\pi\)
0.421625 + 0.906770i \(0.361460\pi\)
\(74\) 0 0
\(75\) −346.003 −0.532706
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1155.63 −1.64580 −0.822899 0.568187i \(-0.807645\pi\)
−0.822899 + 0.568187i \(0.807645\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −330.379 −0.436913 −0.218456 0.975847i \(-0.570102\pi\)
−0.218456 + 0.975847i \(0.570102\pi\)
\(84\) 0 0
\(85\) 218.837 0.279249
\(86\) 0 0
\(87\) −584.295 −0.720034
\(88\) 0 0
\(89\) −300.630 −0.358053 −0.179027 0.983844i \(-0.557295\pi\)
−0.179027 + 0.983844i \(0.557295\pi\)
\(90\) 0 0
\(91\) −961.338 −1.10742
\(92\) 0 0
\(93\) 438.749 0.489206
\(94\) 0 0
\(95\) −146.735 −0.158471
\(96\) 0 0
\(97\) −1548.21 −1.62058 −0.810292 0.586027i \(-0.800691\pi\)
−0.810292 + 0.586027i \(0.800691\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 715.641 0.705039 0.352520 0.935804i \(-0.385325\pi\)
0.352520 + 0.935804i \(0.385325\pi\)
\(102\) 0 0
\(103\) −245.027 −0.234400 −0.117200 0.993108i \(-0.537392\pi\)
−0.117200 + 0.993108i \(0.537392\pi\)
\(104\) 0 0
\(105\) −278.432 −0.258783
\(106\) 0 0
\(107\) −1511.96 −1.36604 −0.683020 0.730400i \(-0.739334\pi\)
−0.683020 + 0.730400i \(0.739334\pi\)
\(108\) 0 0
\(109\) −1213.16 −1.06605 −0.533024 0.846100i \(-0.678945\pi\)
−0.533024 + 0.846100i \(0.678945\pi\)
\(110\) 0 0
\(111\) 931.580 0.796592
\(112\) 0 0
\(113\) 370.312 0.308284 0.154142 0.988049i \(-0.450739\pi\)
0.154142 + 0.988049i \(0.450739\pi\)
\(114\) 0 0
\(115\) 175.155 0.142028
\(116\) 0 0
\(117\) 289.827 0.229013
\(118\) 0 0
\(119\) −2101.26 −1.61868
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −997.528 −0.731253
\(124\) 0 0
\(125\) −747.196 −0.534650
\(126\) 0 0
\(127\) 179.811 0.125635 0.0628175 0.998025i \(-0.479991\pi\)
0.0628175 + 0.998025i \(0.479991\pi\)
\(128\) 0 0
\(129\) 261.560 0.178520
\(130\) 0 0
\(131\) 1146.39 0.764583 0.382291 0.924042i \(-0.375135\pi\)
0.382291 + 0.924042i \(0.375135\pi\)
\(132\) 0 0
\(133\) 1408.95 0.918580
\(134\) 0 0
\(135\) 83.9427 0.0535158
\(136\) 0 0
\(137\) 2053.05 1.28032 0.640160 0.768242i \(-0.278868\pi\)
0.640160 + 0.768242i \(0.278868\pi\)
\(138\) 0 0
\(139\) 215.546 0.131528 0.0657641 0.997835i \(-0.479052\pi\)
0.0657641 + 0.997835i \(0.479052\pi\)
\(140\) 0 0
\(141\) −303.304 −0.181155
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −605.522 −0.346799
\(146\) 0 0
\(147\) 1644.50 0.922693
\(148\) 0 0
\(149\) −1159.51 −0.637522 −0.318761 0.947835i \(-0.603267\pi\)
−0.318761 + 0.947835i \(0.603267\pi\)
\(150\) 0 0
\(151\) −2723.16 −1.46760 −0.733799 0.679367i \(-0.762255\pi\)
−0.733799 + 0.679367i \(0.762255\pi\)
\(152\) 0 0
\(153\) 633.496 0.334739
\(154\) 0 0
\(155\) 454.689 0.235622
\(156\) 0 0
\(157\) 2513.02 1.27746 0.638728 0.769433i \(-0.279461\pi\)
0.638728 + 0.769433i \(0.279461\pi\)
\(158\) 0 0
\(159\) 1097.79 0.547552
\(160\) 0 0
\(161\) −1681.83 −0.823271
\(162\) 0 0
\(163\) 976.748 0.469355 0.234677 0.972073i \(-0.424597\pi\)
0.234677 + 0.972073i \(0.424597\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1117.72 −0.517915 −0.258957 0.965889i \(-0.583379\pi\)
−0.258957 + 0.965889i \(0.583379\pi\)
\(168\) 0 0
\(169\) −1159.96 −0.527976
\(170\) 0 0
\(171\) −424.774 −0.189961
\(172\) 0 0
\(173\) −4055.13 −1.78211 −0.891057 0.453891i \(-0.850036\pi\)
−0.891057 + 0.453891i \(0.850036\pi\)
\(174\) 0 0
\(175\) 3443.00 1.48724
\(176\) 0 0
\(177\) −1925.68 −0.817755
\(178\) 0 0
\(179\) −3518.15 −1.46904 −0.734522 0.678585i \(-0.762593\pi\)
−0.734522 + 0.678585i \(0.762593\pi\)
\(180\) 0 0
\(181\) 1824.02 0.749051 0.374526 0.927217i \(-0.377806\pi\)
0.374526 + 0.927217i \(0.377806\pi\)
\(182\) 0 0
\(183\) −2274.69 −0.918853
\(184\) 0 0
\(185\) 965.424 0.383672
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −806.015 −0.310206
\(190\) 0 0
\(191\) −1270.69 −0.481382 −0.240691 0.970602i \(-0.577374\pi\)
−0.240691 + 0.970602i \(0.577374\pi\)
\(192\) 0 0
\(193\) 1028.31 0.383520 0.191760 0.981442i \(-0.438581\pi\)
0.191760 + 0.981442i \(0.438581\pi\)
\(194\) 0 0
\(195\) 300.357 0.110303
\(196\) 0 0
\(197\) 3541.77 1.28092 0.640459 0.767992i \(-0.278744\pi\)
0.640459 + 0.767992i \(0.278744\pi\)
\(198\) 0 0
\(199\) 2616.56 0.932074 0.466037 0.884765i \(-0.345681\pi\)
0.466037 + 0.884765i \(0.345681\pi\)
\(200\) 0 0
\(201\) 369.786 0.129765
\(202\) 0 0
\(203\) 5814.20 2.01023
\(204\) 0 0
\(205\) −1033.77 −0.352203
\(206\) 0 0
\(207\) 507.043 0.170251
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −3030.15 −0.988646 −0.494323 0.869278i \(-0.664584\pi\)
−0.494323 + 0.869278i \(0.664584\pi\)
\(212\) 0 0
\(213\) −3199.00 −1.02907
\(214\) 0 0
\(215\) 271.062 0.0859828
\(216\) 0 0
\(217\) −4365.90 −1.36579
\(218\) 0 0
\(219\) 1577.84 0.486851
\(220\) 0 0
\(221\) 2266.72 0.689937
\(222\) 0 0
\(223\) 446.011 0.133933 0.0669666 0.997755i \(-0.478668\pi\)
0.0669666 + 0.997755i \(0.478668\pi\)
\(224\) 0 0
\(225\) −1038.01 −0.307558
\(226\) 0 0
\(227\) 3770.73 1.10252 0.551260 0.834334i \(-0.314147\pi\)
0.551260 + 0.834334i \(0.314147\pi\)
\(228\) 0 0
\(229\) −5645.24 −1.62903 −0.814514 0.580143i \(-0.802997\pi\)
−0.814514 + 0.580143i \(0.802997\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −684.619 −0.192493 −0.0962466 0.995358i \(-0.530684\pi\)
−0.0962466 + 0.995358i \(0.530684\pi\)
\(234\) 0 0
\(235\) −314.323 −0.0872519
\(236\) 0 0
\(237\) −3466.88 −0.950202
\(238\) 0 0
\(239\) 2955.66 0.799941 0.399971 0.916528i \(-0.369020\pi\)
0.399971 + 0.916528i \(0.369020\pi\)
\(240\) 0 0
\(241\) 481.604 0.128725 0.0643627 0.997927i \(-0.479499\pi\)
0.0643627 + 0.997927i \(0.479499\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 1704.24 0.444408
\(246\) 0 0
\(247\) −1519.89 −0.391531
\(248\) 0 0
\(249\) −991.136 −0.252252
\(250\) 0 0
\(251\) −7561.16 −1.90142 −0.950710 0.310083i \(-0.899643\pi\)
−0.950710 + 0.310083i \(0.899643\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 656.510 0.161225
\(256\) 0 0
\(257\) −2562.02 −0.621845 −0.310922 0.950435i \(-0.600638\pi\)
−0.310922 + 0.950435i \(0.600638\pi\)
\(258\) 0 0
\(259\) −9269.97 −2.22397
\(260\) 0 0
\(261\) −1752.88 −0.415712
\(262\) 0 0
\(263\) −7518.10 −1.76268 −0.881342 0.472480i \(-0.843359\pi\)
−0.881342 + 0.472480i \(0.843359\pi\)
\(264\) 0 0
\(265\) 1137.68 0.263724
\(266\) 0 0
\(267\) −901.890 −0.206722
\(268\) 0 0
\(269\) 8339.96 1.89032 0.945160 0.326608i \(-0.105906\pi\)
0.945160 + 0.326608i \(0.105906\pi\)
\(270\) 0 0
\(271\) −2005.14 −0.449460 −0.224730 0.974421i \(-0.572150\pi\)
−0.224730 + 0.974421i \(0.572150\pi\)
\(272\) 0 0
\(273\) −2884.01 −0.639372
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1146.94 0.248784 0.124392 0.992233i \(-0.460302\pi\)
0.124392 + 0.992233i \(0.460302\pi\)
\(278\) 0 0
\(279\) 1316.25 0.282443
\(280\) 0 0
\(281\) 3845.96 0.816480 0.408240 0.912875i \(-0.366143\pi\)
0.408240 + 0.912875i \(0.366143\pi\)
\(282\) 0 0
\(283\) −1078.38 −0.226513 −0.113256 0.993566i \(-0.536128\pi\)
−0.113256 + 0.993566i \(0.536128\pi\)
\(284\) 0 0
\(285\) −440.206 −0.0914931
\(286\) 0 0
\(287\) 9926.21 2.04155
\(288\) 0 0
\(289\) 41.5253 0.00845214
\(290\) 0 0
\(291\) −4644.62 −0.935644
\(292\) 0 0
\(293\) 3978.34 0.793233 0.396616 0.917985i \(-0.370184\pi\)
0.396616 + 0.917985i \(0.370184\pi\)
\(294\) 0 0
\(295\) −1995.64 −0.393866
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1814.26 0.350907
\(300\) 0 0
\(301\) −2602.73 −0.498401
\(302\) 0 0
\(303\) 2146.92 0.407055
\(304\) 0 0
\(305\) −2357.33 −0.442559
\(306\) 0 0
\(307\) 2166.24 0.402716 0.201358 0.979518i \(-0.435465\pi\)
0.201358 + 0.979518i \(0.435465\pi\)
\(308\) 0 0
\(309\) −735.081 −0.135331
\(310\) 0 0
\(311\) 1978.32 0.360709 0.180355 0.983602i \(-0.442276\pi\)
0.180355 + 0.983602i \(0.442276\pi\)
\(312\) 0 0
\(313\) −4766.98 −0.860848 −0.430424 0.902627i \(-0.641636\pi\)
−0.430424 + 0.902627i \(0.641636\pi\)
\(314\) 0 0
\(315\) −835.297 −0.149408
\(316\) 0 0
\(317\) −3540.69 −0.627333 −0.313667 0.949533i \(-0.601557\pi\)
−0.313667 + 0.949533i \(0.601557\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −4535.87 −0.788683
\(322\) 0 0
\(323\) −3322.13 −0.572285
\(324\) 0 0
\(325\) −3714.11 −0.633913
\(326\) 0 0
\(327\) −3639.47 −0.615483
\(328\) 0 0
\(329\) 3018.12 0.505758
\(330\) 0 0
\(331\) −10927.7 −1.81463 −0.907315 0.420451i \(-0.861872\pi\)
−0.907315 + 0.420451i \(0.861872\pi\)
\(332\) 0 0
\(333\) 2794.74 0.459912
\(334\) 0 0
\(335\) 383.220 0.0625002
\(336\) 0 0
\(337\) −9153.38 −1.47957 −0.739787 0.672842i \(-0.765074\pi\)
−0.739787 + 0.672842i \(0.765074\pi\)
\(338\) 0 0
\(339\) 1110.94 0.177988
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −6124.69 −0.964146
\(344\) 0 0
\(345\) 525.464 0.0820001
\(346\) 0 0
\(347\) −7626.15 −1.17981 −0.589903 0.807474i \(-0.700834\pi\)
−0.589903 + 0.807474i \(0.700834\pi\)
\(348\) 0 0
\(349\) 3708.63 0.568821 0.284410 0.958703i \(-0.408202\pi\)
0.284410 + 0.958703i \(0.408202\pi\)
\(350\) 0 0
\(351\) 869.482 0.132221
\(352\) 0 0
\(353\) 7111.77 1.07230 0.536149 0.844123i \(-0.319878\pi\)
0.536149 + 0.844123i \(0.319878\pi\)
\(354\) 0 0
\(355\) −3315.22 −0.495643
\(356\) 0 0
\(357\) −6303.79 −0.934543
\(358\) 0 0
\(359\) 12035.1 1.76933 0.884665 0.466228i \(-0.154387\pi\)
0.884665 + 0.466228i \(0.154387\pi\)
\(360\) 0 0
\(361\) −4631.43 −0.675235
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1635.16 0.234488
\(366\) 0 0
\(367\) 2187.73 0.311167 0.155584 0.987823i \(-0.450274\pi\)
0.155584 + 0.987823i \(0.450274\pi\)
\(368\) 0 0
\(369\) −2992.59 −0.422189
\(370\) 0 0
\(371\) −10923.9 −1.52869
\(372\) 0 0
\(373\) 2599.10 0.360794 0.180397 0.983594i \(-0.442262\pi\)
0.180397 + 0.983594i \(0.442262\pi\)
\(374\) 0 0
\(375\) −2241.59 −0.308680
\(376\) 0 0
\(377\) −6272.02 −0.856832
\(378\) 0 0
\(379\) 9772.05 1.32442 0.662212 0.749317i \(-0.269618\pi\)
0.662212 + 0.749317i \(0.269618\pi\)
\(380\) 0 0
\(381\) 539.433 0.0725354
\(382\) 0 0
\(383\) 10067.7 1.34317 0.671586 0.740927i \(-0.265614\pi\)
0.671586 + 0.740927i \(0.265614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 784.680 0.103068
\(388\) 0 0
\(389\) −1025.63 −0.133680 −0.0668401 0.997764i \(-0.521292\pi\)
−0.0668401 + 0.997764i \(0.521292\pi\)
\(390\) 0 0
\(391\) 3965.55 0.512907
\(392\) 0 0
\(393\) 3439.16 0.441432
\(394\) 0 0
\(395\) −3592.83 −0.457658
\(396\) 0 0
\(397\) −6069.83 −0.767346 −0.383673 0.923469i \(-0.625341\pi\)
−0.383673 + 0.923469i \(0.625341\pi\)
\(398\) 0 0
\(399\) 4226.84 0.530342
\(400\) 0 0
\(401\) 14193.3 1.76753 0.883767 0.467927i \(-0.154999\pi\)
0.883767 + 0.467927i \(0.154999\pi\)
\(402\) 0 0
\(403\) 4709.69 0.582149
\(404\) 0 0
\(405\) 251.828 0.0308974
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −11547.7 −1.39608 −0.698039 0.716060i \(-0.745944\pi\)
−0.698039 + 0.716060i \(0.745944\pi\)
\(410\) 0 0
\(411\) 6159.14 0.739193
\(412\) 0 0
\(413\) 19162.0 2.28305
\(414\) 0 0
\(415\) −1027.14 −0.121495
\(416\) 0 0
\(417\) 646.639 0.0759378
\(418\) 0 0
\(419\) −10748.2 −1.25318 −0.626590 0.779349i \(-0.715550\pi\)
−0.626590 + 0.779349i \(0.715550\pi\)
\(420\) 0 0
\(421\) −3124.08 −0.361658 −0.180829 0.983515i \(-0.557878\pi\)
−0.180829 + 0.983515i \(0.557878\pi\)
\(422\) 0 0
\(423\) −909.913 −0.104590
\(424\) 0 0
\(425\) −8118.19 −0.926565
\(426\) 0 0
\(427\) 22635.0 2.56530
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7448.50 0.832440 0.416220 0.909264i \(-0.363355\pi\)
0.416220 + 0.909264i \(0.363355\pi\)
\(432\) 0 0
\(433\) 9124.75 1.01272 0.506360 0.862322i \(-0.330991\pi\)
0.506360 + 0.862322i \(0.330991\pi\)
\(434\) 0 0
\(435\) −1816.57 −0.200225
\(436\) 0 0
\(437\) −2659.00 −0.291069
\(438\) 0 0
\(439\) −14044.5 −1.52689 −0.763447 0.645870i \(-0.776495\pi\)
−0.763447 + 0.645870i \(0.776495\pi\)
\(440\) 0 0
\(441\) 4933.49 0.532717
\(442\) 0 0
\(443\) −7537.00 −0.808338 −0.404169 0.914684i \(-0.632439\pi\)
−0.404169 + 0.914684i \(0.632439\pi\)
\(444\) 0 0
\(445\) −934.656 −0.0995661
\(446\) 0 0
\(447\) −3478.53 −0.368073
\(448\) 0 0
\(449\) 7767.34 0.816400 0.408200 0.912893i \(-0.366157\pi\)
0.408200 + 0.912893i \(0.366157\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −8169.47 −0.847318
\(454\) 0 0
\(455\) −2988.79 −0.307949
\(456\) 0 0
\(457\) 169.378 0.0173373 0.00866866 0.999962i \(-0.497241\pi\)
0.00866866 + 0.999962i \(0.497241\pi\)
\(458\) 0 0
\(459\) 1900.49 0.193262
\(460\) 0 0
\(461\) 5648.85 0.570701 0.285351 0.958423i \(-0.407890\pi\)
0.285351 + 0.958423i \(0.407890\pi\)
\(462\) 0 0
\(463\) 16305.5 1.63668 0.818338 0.574738i \(-0.194896\pi\)
0.818338 + 0.574738i \(0.194896\pi\)
\(464\) 0 0
\(465\) 1364.07 0.136037
\(466\) 0 0
\(467\) −15725.6 −1.55823 −0.779117 0.626879i \(-0.784332\pi\)
−0.779117 + 0.626879i \(0.784332\pi\)
\(468\) 0 0
\(469\) −3679.67 −0.362284
\(470\) 0 0
\(471\) 7539.05 0.737539
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 5443.44 0.525815
\(476\) 0 0
\(477\) 3293.38 0.316129
\(478\) 0 0
\(479\) 13243.8 1.26331 0.631656 0.775249i \(-0.282376\pi\)
0.631656 + 0.775249i \(0.282376\pi\)
\(480\) 0 0
\(481\) 9999.90 0.947934
\(482\) 0 0
\(483\) −5045.48 −0.475316
\(484\) 0 0
\(485\) −4813.36 −0.450646
\(486\) 0 0
\(487\) 6558.26 0.610232 0.305116 0.952315i \(-0.401305\pi\)
0.305116 + 0.952315i \(0.401305\pi\)
\(488\) 0 0
\(489\) 2930.24 0.270982
\(490\) 0 0
\(491\) 6925.61 0.636554 0.318277 0.947998i \(-0.396896\pi\)
0.318277 + 0.947998i \(0.396896\pi\)
\(492\) 0 0
\(493\) −13709.2 −1.25240
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 31832.6 2.87301
\(498\) 0 0
\(499\) 15286.3 1.37136 0.685680 0.727903i \(-0.259505\pi\)
0.685680 + 0.727903i \(0.259505\pi\)
\(500\) 0 0
\(501\) −3353.16 −0.299018
\(502\) 0 0
\(503\) 3998.72 0.354461 0.177231 0.984169i \(-0.443286\pi\)
0.177231 + 0.984169i \(0.443286\pi\)
\(504\) 0 0
\(505\) 2224.92 0.196055
\(506\) 0 0
\(507\) −3479.89 −0.304827
\(508\) 0 0
\(509\) −18157.1 −1.58114 −0.790570 0.612371i \(-0.790216\pi\)
−0.790570 + 0.612371i \(0.790216\pi\)
\(510\) 0 0
\(511\) −15700.7 −1.35922
\(512\) 0 0
\(513\) −1274.32 −0.109674
\(514\) 0 0
\(515\) −761.786 −0.0651812
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −12165.4 −1.02890
\(520\) 0 0
\(521\) −8815.08 −0.741258 −0.370629 0.928781i \(-0.620858\pi\)
−0.370629 + 0.928781i \(0.620858\pi\)
\(522\) 0 0
\(523\) 5258.60 0.439660 0.219830 0.975538i \(-0.429450\pi\)
0.219830 + 0.975538i \(0.429450\pi\)
\(524\) 0 0
\(525\) 10329.0 0.858657
\(526\) 0 0
\(527\) 10294.3 0.850903
\(528\) 0 0
\(529\) −8993.02 −0.739132
\(530\) 0 0
\(531\) −5777.03 −0.472131
\(532\) 0 0
\(533\) −10707.8 −0.870182
\(534\) 0 0
\(535\) −4700.65 −0.379864
\(536\) 0 0
\(537\) −10554.4 −0.848153
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2737.07 0.217515 0.108758 0.994068i \(-0.465313\pi\)
0.108758 + 0.994068i \(0.465313\pi\)
\(542\) 0 0
\(543\) 5472.06 0.432465
\(544\) 0 0
\(545\) −3771.69 −0.296443
\(546\) 0 0
\(547\) −10663.8 −0.833549 −0.416775 0.909010i \(-0.636840\pi\)
−0.416775 + 0.909010i \(0.636840\pi\)
\(548\) 0 0
\(549\) −6824.07 −0.530500
\(550\) 0 0
\(551\) 9192.34 0.710720
\(552\) 0 0
\(553\) 34498.2 2.65283
\(554\) 0 0
\(555\) 2896.27 0.221513
\(556\) 0 0
\(557\) −5987.45 −0.455469 −0.227735 0.973723i \(-0.573132\pi\)
−0.227735 + 0.973723i \(0.573132\pi\)
\(558\) 0 0
\(559\) 2807.67 0.212436
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18636.9 1.39512 0.697559 0.716528i \(-0.254270\pi\)
0.697559 + 0.716528i \(0.254270\pi\)
\(564\) 0 0
\(565\) 1151.30 0.0857264
\(566\) 0 0
\(567\) −2418.04 −0.179098
\(568\) 0 0
\(569\) 18099.5 1.33352 0.666759 0.745273i \(-0.267681\pi\)
0.666759 + 0.745273i \(0.267681\pi\)
\(570\) 0 0
\(571\) −21590.0 −1.58234 −0.791168 0.611599i \(-0.790526\pi\)
−0.791168 + 0.611599i \(0.790526\pi\)
\(572\) 0 0
\(573\) −3812.08 −0.277926
\(574\) 0 0
\(575\) −6497.71 −0.471258
\(576\) 0 0
\(577\) −22439.3 −1.61900 −0.809499 0.587121i \(-0.800261\pi\)
−0.809499 + 0.587121i \(0.800261\pi\)
\(578\) 0 0
\(579\) 3084.93 0.221425
\(580\) 0 0
\(581\) 9862.60 0.704250
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 901.070 0.0636832
\(586\) 0 0
\(587\) −24440.1 −1.71849 −0.859244 0.511566i \(-0.829066\pi\)
−0.859244 + 0.511566i \(0.829066\pi\)
\(588\) 0 0
\(589\) −6902.56 −0.482878
\(590\) 0 0
\(591\) 10625.3 0.739539
\(592\) 0 0
\(593\) −2703.41 −0.187211 −0.0936053 0.995609i \(-0.529839\pi\)
−0.0936053 + 0.995609i \(0.529839\pi\)
\(594\) 0 0
\(595\) −6532.80 −0.450116
\(596\) 0 0
\(597\) 7849.67 0.538133
\(598\) 0 0
\(599\) 14555.5 0.992856 0.496428 0.868078i \(-0.334645\pi\)
0.496428 + 0.868078i \(0.334645\pi\)
\(600\) 0 0
\(601\) 25383.3 1.72281 0.861403 0.507922i \(-0.169586\pi\)
0.861403 + 0.507922i \(0.169586\pi\)
\(602\) 0 0
\(603\) 1109.36 0.0749197
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −17684.1 −1.18250 −0.591249 0.806489i \(-0.701365\pi\)
−0.591249 + 0.806489i \(0.701365\pi\)
\(608\) 0 0
\(609\) 17442.6 1.16061
\(610\) 0 0
\(611\) −3255.77 −0.215572
\(612\) 0 0
\(613\) 20806.2 1.37089 0.685444 0.728125i \(-0.259608\pi\)
0.685444 + 0.728125i \(0.259608\pi\)
\(614\) 0 0
\(615\) −3101.31 −0.203344
\(616\) 0 0
\(617\) −25625.0 −1.67200 −0.835999 0.548731i \(-0.815111\pi\)
−0.835999 + 0.548731i \(0.815111\pi\)
\(618\) 0 0
\(619\) −14375.1 −0.933417 −0.466709 0.884411i \(-0.654560\pi\)
−0.466709 + 0.884411i \(0.654560\pi\)
\(620\) 0 0
\(621\) 1521.13 0.0982944
\(622\) 0 0
\(623\) 8974.53 0.577138
\(624\) 0 0
\(625\) 12093.7 0.774000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 21857.5 1.38556
\(630\) 0 0
\(631\) 6824.66 0.430563 0.215282 0.976552i \(-0.430933\pi\)
0.215282 + 0.976552i \(0.430933\pi\)
\(632\) 0 0
\(633\) −9090.46 −0.570795
\(634\) 0 0
\(635\) 559.030 0.0349361
\(636\) 0 0
\(637\) 17652.6 1.09799
\(638\) 0 0
\(639\) −9596.99 −0.594133
\(640\) 0 0
\(641\) 24879.1 1.53302 0.766509 0.642233i \(-0.221992\pi\)
0.766509 + 0.642233i \(0.221992\pi\)
\(642\) 0 0
\(643\) −20537.3 −1.25959 −0.629793 0.776763i \(-0.716860\pi\)
−0.629793 + 0.776763i \(0.716860\pi\)
\(644\) 0 0
\(645\) 813.187 0.0496422
\(646\) 0 0
\(647\) 6296.05 0.382571 0.191285 0.981534i \(-0.438734\pi\)
0.191285 + 0.981534i \(0.438734\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −13097.7 −0.788541
\(652\) 0 0
\(653\) −16220.2 −0.972047 −0.486024 0.873946i \(-0.661553\pi\)
−0.486024 + 0.873946i \(0.661553\pi\)
\(654\) 0 0
\(655\) 3564.11 0.212613
\(656\) 0 0
\(657\) 4733.51 0.281084
\(658\) 0 0
\(659\) −7204.55 −0.425872 −0.212936 0.977066i \(-0.568303\pi\)
−0.212936 + 0.977066i \(0.568303\pi\)
\(660\) 0 0
\(661\) 10969.0 0.645456 0.322728 0.946492i \(-0.395400\pi\)
0.322728 + 0.946492i \(0.395400\pi\)
\(662\) 0 0
\(663\) 6800.16 0.398335
\(664\) 0 0
\(665\) 4380.40 0.255436
\(666\) 0 0
\(667\) −10972.7 −0.636978
\(668\) 0 0
\(669\) 1338.03 0.0773264
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 11005.3 0.630345 0.315172 0.949034i \(-0.397938\pi\)
0.315172 + 0.949034i \(0.397938\pi\)
\(674\) 0 0
\(675\) −3114.02 −0.177569
\(676\) 0 0
\(677\) 24640.3 1.39882 0.699411 0.714720i \(-0.253446\pi\)
0.699411 + 0.714720i \(0.253446\pi\)
\(678\) 0 0
\(679\) 46217.7 2.61218
\(680\) 0 0
\(681\) 11312.2 0.636540
\(682\) 0 0
\(683\) −1240.24 −0.0694825 −0.0347413 0.999396i \(-0.511061\pi\)
−0.0347413 + 0.999396i \(0.511061\pi\)
\(684\) 0 0
\(685\) 6382.90 0.356027
\(686\) 0 0
\(687\) −16935.7 −0.940520
\(688\) 0 0
\(689\) 11784.1 0.651580
\(690\) 0 0
\(691\) −13126.4 −0.722648 −0.361324 0.932440i \(-0.617675\pi\)
−0.361324 + 0.932440i \(0.617675\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 670.132 0.0365749
\(696\) 0 0
\(697\) −23404.8 −1.27191
\(698\) 0 0
\(699\) −2053.86 −0.111136
\(700\) 0 0
\(701\) 15373.2 0.828299 0.414149 0.910209i \(-0.364079\pi\)
0.414149 + 0.910209i \(0.364079\pi\)
\(702\) 0 0
\(703\) −14656.0 −0.786287
\(704\) 0 0
\(705\) −942.970 −0.0503749
\(706\) 0 0
\(707\) −21363.6 −1.13644
\(708\) 0 0
\(709\) 20921.9 1.10823 0.554117 0.832439i \(-0.313056\pi\)
0.554117 + 0.832439i \(0.313056\pi\)
\(710\) 0 0
\(711\) −10400.6 −0.548600
\(712\) 0 0
\(713\) 8239.43 0.432776
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 8866.99 0.461846
\(718\) 0 0
\(719\) 11994.3 0.622131 0.311065 0.950389i \(-0.399314\pi\)
0.311065 + 0.950389i \(0.399314\pi\)
\(720\) 0 0
\(721\) 7314.64 0.377824
\(722\) 0 0
\(723\) 1444.81 0.0743197
\(724\) 0 0
\(725\) 22463.1 1.15070
\(726\) 0 0
\(727\) −35755.6 −1.82408 −0.912038 0.410106i \(-0.865492\pi\)
−0.912038 + 0.410106i \(0.865492\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 6136.93 0.310509
\(732\) 0 0
\(733\) −13984.4 −0.704672 −0.352336 0.935873i \(-0.614613\pi\)
−0.352336 + 0.935873i \(0.614613\pi\)
\(734\) 0 0
\(735\) 5112.72 0.256579
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 13255.7 0.659837 0.329919 0.944009i \(-0.392979\pi\)
0.329919 + 0.944009i \(0.392979\pi\)
\(740\) 0 0
\(741\) −4559.67 −0.226051
\(742\) 0 0
\(743\) 27444.5 1.35510 0.677552 0.735475i \(-0.263041\pi\)
0.677552 + 0.735475i \(0.263041\pi\)
\(744\) 0 0
\(745\) −3604.90 −0.177280
\(746\) 0 0
\(747\) −2973.41 −0.145638
\(748\) 0 0
\(749\) 45135.5 2.20189
\(750\) 0 0
\(751\) −10600.1 −0.515053 −0.257527 0.966271i \(-0.582907\pi\)
−0.257527 + 0.966271i \(0.582907\pi\)
\(752\) 0 0
\(753\) −22683.5 −1.09778
\(754\) 0 0
\(755\) −8466.26 −0.408104
\(756\) 0 0
\(757\) 22837.7 1.09650 0.548250 0.836314i \(-0.315294\pi\)
0.548250 + 0.836314i \(0.315294\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 35381.9 1.68540 0.842701 0.538381i \(-0.180964\pi\)
0.842701 + 0.538381i \(0.180964\pi\)
\(762\) 0 0
\(763\) 36215.6 1.71834
\(764\) 0 0
\(765\) 1969.53 0.0930831
\(766\) 0 0
\(767\) −20670.9 −0.973119
\(768\) 0 0
\(769\) 21448.1 1.00577 0.502885 0.864353i \(-0.332272\pi\)
0.502885 + 0.864353i \(0.332272\pi\)
\(770\) 0 0
\(771\) −7686.05 −0.359022
\(772\) 0 0
\(773\) 16900.6 0.786380 0.393190 0.919457i \(-0.371372\pi\)
0.393190 + 0.919457i \(0.371372\pi\)
\(774\) 0 0
\(775\) −16867.6 −0.781809
\(776\) 0 0
\(777\) −27809.9 −1.28401
\(778\) 0 0
\(779\) 15693.5 0.721794
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −5258.65 −0.240011
\(784\) 0 0
\(785\) 7812.94 0.355230
\(786\) 0 0
\(787\) 7633.16 0.345734 0.172867 0.984945i \(-0.444697\pi\)
0.172867 + 0.984945i \(0.444697\pi\)
\(788\) 0 0
\(789\) −22554.3 −1.01769
\(790\) 0 0
\(791\) −11054.7 −0.496916
\(792\) 0 0
\(793\) −24417.3 −1.09342
\(794\) 0 0
\(795\) 3413.03 0.152261
\(796\) 0 0
\(797\) 488.565 0.0217138 0.0108569 0.999941i \(-0.496544\pi\)
0.0108569 + 0.999941i \(0.496544\pi\)
\(798\) 0 0
\(799\) −7116.37 −0.315093
\(800\) 0 0
\(801\) −2705.67 −0.119351
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −5228.79 −0.228932
\(806\) 0 0
\(807\) 25019.9 1.09138
\(808\) 0 0
\(809\) 27235.7 1.18363 0.591814 0.806075i \(-0.298412\pi\)
0.591814 + 0.806075i \(0.298412\pi\)
\(810\) 0 0
\(811\) −7346.61 −0.318094 −0.159047 0.987271i \(-0.550842\pi\)
−0.159047 + 0.987271i \(0.550842\pi\)
\(812\) 0 0
\(813\) −6015.42 −0.259496
\(814\) 0 0
\(815\) 3036.70 0.130516
\(816\) 0 0
\(817\) −4114.95 −0.176211
\(818\) 0 0
\(819\) −8652.04 −0.369141
\(820\) 0 0
\(821\) 30190.3 1.28337 0.641686 0.766967i \(-0.278235\pi\)
0.641686 + 0.766967i \(0.278235\pi\)
\(822\) 0 0
\(823\) 21592.1 0.914524 0.457262 0.889332i \(-0.348830\pi\)
0.457262 + 0.889332i \(0.348830\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20421.1 0.858659 0.429330 0.903148i \(-0.358750\pi\)
0.429330 + 0.903148i \(0.358750\pi\)
\(828\) 0 0
\(829\) 28570.2 1.19696 0.598482 0.801136i \(-0.295771\pi\)
0.598482 + 0.801136i \(0.295771\pi\)
\(830\) 0 0
\(831\) 3440.83 0.143635
\(832\) 0 0
\(833\) 38584.5 1.60489
\(834\) 0 0
\(835\) −3474.98 −0.144020
\(836\) 0 0
\(837\) 3948.74 0.163069
\(838\) 0 0
\(839\) −23875.5 −0.982447 −0.491224 0.871034i \(-0.663450\pi\)
−0.491224 + 0.871034i \(0.663450\pi\)
\(840\) 0 0
\(841\) 13544.4 0.555348
\(842\) 0 0
\(843\) 11537.9 0.471395
\(844\) 0 0
\(845\) −3606.32 −0.146818
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −3235.14 −0.130777
\(850\) 0 0
\(851\) 17494.5 0.704704
\(852\) 0 0
\(853\) −28777.2 −1.15511 −0.577557 0.816351i \(-0.695994\pi\)
−0.577557 + 0.816351i \(0.695994\pi\)
\(854\) 0 0
\(855\) −1320.62 −0.0528236
\(856\) 0 0
\(857\) −25020.3 −0.997291 −0.498646 0.866806i \(-0.666169\pi\)
−0.498646 + 0.866806i \(0.666169\pi\)
\(858\) 0 0
\(859\) 40749.7 1.61858 0.809290 0.587409i \(-0.199852\pi\)
0.809290 + 0.587409i \(0.199852\pi\)
\(860\) 0 0
\(861\) 29778.6 1.17869
\(862\) 0 0
\(863\) 43635.0 1.72115 0.860575 0.509324i \(-0.170104\pi\)
0.860575 + 0.509324i \(0.170104\pi\)
\(864\) 0 0
\(865\) −12607.4 −0.495564
\(866\) 0 0
\(867\) 124.576 0.00487984
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 3969.41 0.154418
\(872\) 0 0
\(873\) −13933.9 −0.540195
\(874\) 0 0
\(875\) 22305.6 0.861791
\(876\) 0 0
\(877\) 35002.0 1.34770 0.673850 0.738868i \(-0.264639\pi\)
0.673850 + 0.738868i \(0.264639\pi\)
\(878\) 0 0
\(879\) 11935.0 0.457973
\(880\) 0 0
\(881\) 5760.33 0.220284 0.110142 0.993916i \(-0.464869\pi\)
0.110142 + 0.993916i \(0.464869\pi\)
\(882\) 0 0
\(883\) −14286.6 −0.544487 −0.272244 0.962228i \(-0.587766\pi\)
−0.272244 + 0.962228i \(0.587766\pi\)
\(884\) 0 0
\(885\) −5986.91 −0.227398
\(886\) 0 0
\(887\) 9440.24 0.357353 0.178677 0.983908i \(-0.442818\pi\)
0.178677 + 0.983908i \(0.442818\pi\)
\(888\) 0 0
\(889\) −5367.79 −0.202508
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4771.69 0.178811
\(894\) 0 0
\(895\) −10937.9 −0.408506
\(896\) 0 0
\(897\) 5442.78 0.202596
\(898\) 0 0
\(899\) −28484.3 −1.05674
\(900\) 0 0
\(901\) 25757.3 0.952387
\(902\) 0 0
\(903\) −7808.19 −0.287752
\(904\) 0 0
\(905\) 5670.86 0.208294
\(906\) 0 0
\(907\) −49416.3 −1.80909 −0.904543 0.426383i \(-0.859788\pi\)
−0.904543 + 0.426383i \(0.859788\pi\)
\(908\) 0 0
\(909\) 6440.77 0.235013
\(910\) 0 0
\(911\) −5531.63 −0.201176 −0.100588 0.994928i \(-0.532072\pi\)
−0.100588 + 0.994928i \(0.532072\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −7071.99 −0.255511
\(916\) 0 0
\(917\) −34222.4 −1.23241
\(918\) 0 0
\(919\) −23529.6 −0.844582 −0.422291 0.906460i \(-0.638774\pi\)
−0.422291 + 0.906460i \(0.638774\pi\)
\(920\) 0 0
\(921\) 6498.72 0.232508
\(922\) 0 0
\(923\) −34339.1 −1.22458
\(924\) 0 0
\(925\) −35814.3 −1.27305
\(926\) 0 0
\(927\) −2205.24 −0.0781334
\(928\) 0 0
\(929\) 32996.3 1.16531 0.582656 0.812719i \(-0.302014\pi\)
0.582656 + 0.812719i \(0.302014\pi\)
\(930\) 0 0
\(931\) −25871.8 −0.910757
\(932\) 0 0
\(933\) 5934.97 0.208255
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −28998.0 −1.01102 −0.505508 0.862822i \(-0.668695\pi\)
−0.505508 + 0.862822i \(0.668695\pi\)
\(938\) 0 0
\(939\) −14300.9 −0.497011
\(940\) 0 0
\(941\) −7317.71 −0.253507 −0.126754 0.991934i \(-0.540456\pi\)
−0.126754 + 0.991934i \(0.540456\pi\)
\(942\) 0 0
\(943\) −18733.0 −0.646903
\(944\) 0 0
\(945\) −2505.89 −0.0862610
\(946\) 0 0
\(947\) −11124.6 −0.381732 −0.190866 0.981616i \(-0.561130\pi\)
−0.190866 + 0.981616i \(0.561130\pi\)
\(948\) 0 0
\(949\) 16937.1 0.579347
\(950\) 0 0
\(951\) −10622.1 −0.362191
\(952\) 0 0
\(953\) 43980.0 1.49491 0.747457 0.664310i \(-0.231275\pi\)
0.747457 + 0.664310i \(0.231275\pi\)
\(954\) 0 0
\(955\) −3950.57 −0.133861
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −61288.4 −2.06372
\(960\) 0 0
\(961\) −8402.02 −0.282032
\(962\) 0 0
\(963\) −13607.6 −0.455347
\(964\) 0 0
\(965\) 3197.00 0.106648
\(966\) 0 0
\(967\) 15009.5 0.499143 0.249572 0.968356i \(-0.419710\pi\)
0.249572 + 0.968356i \(0.419710\pi\)
\(968\) 0 0
\(969\) −9966.38 −0.330409
\(970\) 0 0
\(971\) −3141.24 −0.103818 −0.0519089 0.998652i \(-0.516531\pi\)
−0.0519089 + 0.998652i \(0.516531\pi\)
\(972\) 0 0
\(973\) −6434.58 −0.212007
\(974\) 0 0
\(975\) −11142.3 −0.365990
\(976\) 0 0
\(977\) −46209.5 −1.51318 −0.756588 0.653892i \(-0.773135\pi\)
−0.756588 + 0.653892i \(0.773135\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −10918.4 −0.355349
\(982\) 0 0
\(983\) 11471.1 0.372198 0.186099 0.982531i \(-0.440415\pi\)
0.186099 + 0.982531i \(0.440415\pi\)
\(984\) 0 0
\(985\) 11011.3 0.356193
\(986\) 0 0
\(987\) 9054.36 0.292000
\(988\) 0 0
\(989\) 4911.93 0.157928
\(990\) 0 0
\(991\) −39758.4 −1.27444 −0.637218 0.770684i \(-0.719915\pi\)
−0.637218 + 0.770684i \(0.719915\pi\)
\(992\) 0 0
\(993\) −32783.2 −1.04768
\(994\) 0 0
\(995\) 8134.85 0.259188
\(996\) 0 0
\(997\) 15019.4 0.477099 0.238550 0.971130i \(-0.423328\pi\)
0.238550 + 0.971130i \(0.423328\pi\)
\(998\) 0 0
\(999\) 8384.22 0.265531
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 1452.4.a.t.1.4 6
11.2 odd 10 132.4.i.c.37.2 yes 12
11.6 odd 10 132.4.i.c.25.2 12
11.10 odd 2 1452.4.a.u.1.4 6
33.2 even 10 396.4.j.c.37.2 12
33.17 even 10 396.4.j.c.289.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
132.4.i.c.25.2 12 11.6 odd 10
132.4.i.c.37.2 yes 12 11.2 odd 10
396.4.j.c.37.2 12 33.2 even 10
396.4.j.c.289.2 12 33.17 even 10
1452.4.a.t.1.4 6 1.1 even 1 trivial
1452.4.a.u.1.4 6 11.10 odd 2