Properties

Label 1452.4.a.q.1.2
Level $1452$
Weight $4$
Character 1452.1
Self dual yes
Analytic conductor $85.671$
Analytic rank $0$
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] = [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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.20959101.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 90x^{2} + 91x + 2026 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(6.75283\) 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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -10.3041 q^{5} +18.4086 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -10.3041 q^{5} +18.4086 q^{7} +9.00000 q^{9} +93.4426 q^{13} -30.9124 q^{15} -75.0340 q^{17} +38.2168 q^{19} +55.2258 q^{21} -107.345 q^{23} -18.8248 q^{25} +27.0000 q^{27} +185.486 q^{29} +285.474 q^{31} -189.685 q^{35} +35.8248 q^{37} +280.328 q^{39} -446.005 q^{41} +295.937 q^{43} -92.7372 q^{45} -66.3868 q^{47} -4.12404 q^{49} -225.102 q^{51} +430.562 q^{53} +114.651 q^{57} -382.691 q^{59} +277.528 q^{61} +165.677 q^{63} -962.845 q^{65} -363.299 q^{67} -322.036 q^{69} +991.954 q^{71} -223.702 q^{73} -56.4744 q^{75} -310.147 q^{79} +81.0000 q^{81} -1073.30 q^{83} +773.161 q^{85} +556.457 q^{87} -519.917 q^{89} +1720.15 q^{91} +856.423 q^{93} -393.791 q^{95} +1091.12 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} + 12 q^{5} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} + 12 q^{5} + 36 q^{9} + 36 q^{15} + 156 q^{23} + 244 q^{25} + 108 q^{27} + 184 q^{31} - 176 q^{37} + 108 q^{45} + 852 q^{47} + 1580 q^{49} + 924 q^{53} - 360 q^{59} - 176 q^{67} + 468 q^{69} + 3276 q^{71} + 732 q^{75} + 324 q^{81} - 3144 q^{89} - 144 q^{91} + 552 q^{93} + 2768 q^{97}+O(q^{100}) \) 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
\(5\) −10.3041 −0.921630 −0.460815 0.887496i \(-0.652443\pi\)
−0.460815 + 0.887496i \(0.652443\pi\)
\(6\) 0 0
\(7\) 18.4086 0.993970 0.496985 0.867759i \(-0.334440\pi\)
0.496985 + 0.867759i \(0.334440\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 93.4426 1.99356 0.996781 0.0801697i \(-0.0255462\pi\)
0.996781 + 0.0801697i \(0.0255462\pi\)
\(14\) 0 0
\(15\) −30.9124 −0.532103
\(16\) 0 0
\(17\) −75.0340 −1.07050 −0.535248 0.844695i \(-0.679782\pi\)
−0.535248 + 0.844695i \(0.679782\pi\)
\(18\) 0 0
\(19\) 38.2168 0.461450 0.230725 0.973019i \(-0.425890\pi\)
0.230725 + 0.973019i \(0.425890\pi\)
\(20\) 0 0
\(21\) 55.2258 0.573869
\(22\) 0 0
\(23\) −107.345 −0.973177 −0.486589 0.873631i \(-0.661759\pi\)
−0.486589 + 0.873631i \(0.661759\pi\)
\(24\) 0 0
\(25\) −18.8248 −0.150598
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 185.486 1.18772 0.593859 0.804569i \(-0.297604\pi\)
0.593859 + 0.804569i \(0.297604\pi\)
\(30\) 0 0
\(31\) 285.474 1.65396 0.826979 0.562232i \(-0.190057\pi\)
0.826979 + 0.562232i \(0.190057\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −189.685 −0.916072
\(36\) 0 0
\(37\) 35.8248 0.159177 0.0795887 0.996828i \(-0.474639\pi\)
0.0795887 + 0.996828i \(0.474639\pi\)
\(38\) 0 0
\(39\) 280.328 1.15098
\(40\) 0 0
\(41\) −446.005 −1.69888 −0.849442 0.527681i \(-0.823062\pi\)
−0.849442 + 0.527681i \(0.823062\pi\)
\(42\) 0 0
\(43\) 295.937 1.04953 0.524767 0.851246i \(-0.324152\pi\)
0.524767 + 0.851246i \(0.324152\pi\)
\(44\) 0 0
\(45\) −92.7372 −0.307210
\(46\) 0 0
\(47\) −66.3868 −0.206032 −0.103016 0.994680i \(-0.532849\pi\)
−0.103016 + 0.994680i \(0.532849\pi\)
\(48\) 0 0
\(49\) −4.12404 −0.0120234
\(50\) 0 0
\(51\) −225.102 −0.618051
\(52\) 0 0
\(53\) 430.562 1.11589 0.557946 0.829877i \(-0.311590\pi\)
0.557946 + 0.829877i \(0.311590\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 114.651 0.266418
\(58\) 0 0
\(59\) −382.691 −0.844443 −0.422221 0.906493i \(-0.638750\pi\)
−0.422221 + 0.906493i \(0.638750\pi\)
\(60\) 0 0
\(61\) 277.528 0.582523 0.291261 0.956644i \(-0.405925\pi\)
0.291261 + 0.956644i \(0.405925\pi\)
\(62\) 0 0
\(63\) 165.677 0.331323
\(64\) 0 0
\(65\) −962.845 −1.83733
\(66\) 0 0
\(67\) −363.299 −0.662449 −0.331224 0.943552i \(-0.607462\pi\)
−0.331224 + 0.943552i \(0.607462\pi\)
\(68\) 0 0
\(69\) −322.036 −0.561864
\(70\) 0 0
\(71\) 991.954 1.65807 0.829037 0.559194i \(-0.188889\pi\)
0.829037 + 0.559194i \(0.188889\pi\)
\(72\) 0 0
\(73\) −223.702 −0.358663 −0.179331 0.983789i \(-0.557393\pi\)
−0.179331 + 0.983789i \(0.557393\pi\)
\(74\) 0 0
\(75\) −56.4744 −0.0869481
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −310.147 −0.441699 −0.220849 0.975308i \(-0.570883\pi\)
−0.220849 + 0.975308i \(0.570883\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1073.30 −1.41939 −0.709696 0.704508i \(-0.751168\pi\)
−0.709696 + 0.704508i \(0.751168\pi\)
\(84\) 0 0
\(85\) 773.161 0.986600
\(86\) 0 0
\(87\) 556.457 0.685729
\(88\) 0 0
\(89\) −519.917 −0.619226 −0.309613 0.950863i \(-0.600200\pi\)
−0.309613 + 0.950863i \(0.600200\pi\)
\(90\) 0 0
\(91\) 1720.15 1.98154
\(92\) 0 0
\(93\) 856.423 0.954914
\(94\) 0 0
\(95\) −393.791 −0.425286
\(96\) 0 0
\(97\) 1091.12 1.14213 0.571066 0.820904i \(-0.306530\pi\)
0.571066 + 0.820904i \(0.306530\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1187.95 1.17035 0.585174 0.810908i \(-0.301026\pi\)
0.585174 + 0.810908i \(0.301026\pi\)
\(102\) 0 0
\(103\) 833.124 0.796992 0.398496 0.917170i \(-0.369532\pi\)
0.398496 + 0.917170i \(0.369532\pi\)
\(104\) 0 0
\(105\) −569.054 −0.528895
\(106\) 0 0
\(107\) 829.573 0.749513 0.374756 0.927123i \(-0.377726\pi\)
0.374756 + 0.927123i \(0.377726\pi\)
\(108\) 0 0
\(109\) −201.095 −0.176710 −0.0883550 0.996089i \(-0.528161\pi\)
−0.0883550 + 0.996089i \(0.528161\pi\)
\(110\) 0 0
\(111\) 107.474 0.0919011
\(112\) 0 0
\(113\) 889.835 0.740784 0.370392 0.928876i \(-0.379223\pi\)
0.370392 + 0.928876i \(0.379223\pi\)
\(114\) 0 0
\(115\) 1106.10 0.896909
\(116\) 0 0
\(117\) 840.983 0.664521
\(118\) 0 0
\(119\) −1381.27 −1.06404
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −1338.02 −0.980852
\(124\) 0 0
\(125\) 1481.99 1.06043
\(126\) 0 0
\(127\) −15.6092 −0.0109063 −0.00545313 0.999985i \(-0.501736\pi\)
−0.00545313 + 0.999985i \(0.501736\pi\)
\(128\) 0 0
\(129\) 887.811 0.605949
\(130\) 0 0
\(131\) 418.986 0.279442 0.139721 0.990191i \(-0.455379\pi\)
0.139721 + 0.990191i \(0.455379\pi\)
\(132\) 0 0
\(133\) 703.518 0.458667
\(134\) 0 0
\(135\) −278.212 −0.177368
\(136\) 0 0
\(137\) 2197.54 1.37043 0.685213 0.728343i \(-0.259709\pi\)
0.685213 + 0.728343i \(0.259709\pi\)
\(138\) 0 0
\(139\) 32.6182 0.0199039 0.00995193 0.999950i \(-0.496832\pi\)
0.00995193 + 0.999950i \(0.496832\pi\)
\(140\) 0 0
\(141\) −199.160 −0.118953
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1911.27 −1.09464
\(146\) 0 0
\(147\) −12.3721 −0.00694174
\(148\) 0 0
\(149\) 3359.73 1.84725 0.923625 0.383298i \(-0.125212\pi\)
0.923625 + 0.383298i \(0.125212\pi\)
\(150\) 0 0
\(151\) −3073.81 −1.65658 −0.828288 0.560303i \(-0.810685\pi\)
−0.828288 + 0.560303i \(0.810685\pi\)
\(152\) 0 0
\(153\) −675.306 −0.356832
\(154\) 0 0
\(155\) −2941.57 −1.52434
\(156\) 0 0
\(157\) 2295.20 1.16673 0.583365 0.812210i \(-0.301736\pi\)
0.583365 + 0.812210i \(0.301736\pi\)
\(158\) 0 0
\(159\) 1291.69 0.644260
\(160\) 0 0
\(161\) −1976.08 −0.967309
\(162\) 0 0
\(163\) 1173.55 0.563922 0.281961 0.959426i \(-0.409015\pi\)
0.281961 + 0.959426i \(0.409015\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2874.11 1.33177 0.665885 0.746055i \(-0.268054\pi\)
0.665885 + 0.746055i \(0.268054\pi\)
\(168\) 0 0
\(169\) 6534.52 2.97429
\(170\) 0 0
\(171\) 343.952 0.153817
\(172\) 0 0
\(173\) −2640.61 −1.16047 −0.580237 0.814448i \(-0.697040\pi\)
−0.580237 + 0.814448i \(0.697040\pi\)
\(174\) 0 0
\(175\) −346.538 −0.149690
\(176\) 0 0
\(177\) −1148.07 −0.487539
\(178\) 0 0
\(179\) 3375.89 1.40964 0.704820 0.709386i \(-0.251028\pi\)
0.704820 + 0.709386i \(0.251028\pi\)
\(180\) 0 0
\(181\) 785.927 0.322749 0.161374 0.986893i \(-0.448407\pi\)
0.161374 + 0.986893i \(0.448407\pi\)
\(182\) 0 0
\(183\) 832.585 0.336320
\(184\) 0 0
\(185\) −369.144 −0.146703
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 497.032 0.191290
\(190\) 0 0
\(191\) −1039.91 −0.393956 −0.196978 0.980408i \(-0.563113\pi\)
−0.196978 + 0.980408i \(0.563113\pi\)
\(192\) 0 0
\(193\) −1245.76 −0.464620 −0.232310 0.972642i \(-0.574628\pi\)
−0.232310 + 0.972642i \(0.574628\pi\)
\(194\) 0 0
\(195\) −2888.54 −1.06078
\(196\) 0 0
\(197\) 2892.73 1.04619 0.523093 0.852275i \(-0.324778\pi\)
0.523093 + 0.852275i \(0.324778\pi\)
\(198\) 0 0
\(199\) −5130.39 −1.82756 −0.913779 0.406212i \(-0.866849\pi\)
−0.913779 + 0.406212i \(0.866849\pi\)
\(200\) 0 0
\(201\) −1089.90 −0.382465
\(202\) 0 0
\(203\) 3414.53 1.18056
\(204\) 0 0
\(205\) 4595.70 1.56574
\(206\) 0 0
\(207\) −966.109 −0.324392
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 2977.57 0.971489 0.485744 0.874101i \(-0.338549\pi\)
0.485744 + 0.874101i \(0.338549\pi\)
\(212\) 0 0
\(213\) 2975.86 0.957289
\(214\) 0 0
\(215\) −3049.37 −0.967282
\(216\) 0 0
\(217\) 5255.18 1.64399
\(218\) 0 0
\(219\) −671.107 −0.207074
\(220\) 0 0
\(221\) −7011.37 −2.13410
\(222\) 0 0
\(223\) −1865.55 −0.560208 −0.280104 0.959970i \(-0.590369\pi\)
−0.280104 + 0.959970i \(0.590369\pi\)
\(224\) 0 0
\(225\) −169.423 −0.0501995
\(226\) 0 0
\(227\) 1950.88 0.570417 0.285209 0.958465i \(-0.407937\pi\)
0.285209 + 0.958465i \(0.407937\pi\)
\(228\) 0 0
\(229\) 443.547 0.127993 0.0639966 0.997950i \(-0.479615\pi\)
0.0639966 + 0.997950i \(0.479615\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5814.86 −1.63495 −0.817477 0.575961i \(-0.804628\pi\)
−0.817477 + 0.575961i \(0.804628\pi\)
\(234\) 0 0
\(235\) 684.059 0.189885
\(236\) 0 0
\(237\) −930.440 −0.255015
\(238\) 0 0
\(239\) 5208.02 1.40953 0.704767 0.709439i \(-0.251052\pi\)
0.704767 + 0.709439i \(0.251052\pi\)
\(240\) 0 0
\(241\) 1871.23 0.500150 0.250075 0.968226i \(-0.419545\pi\)
0.250075 + 0.968226i \(0.419545\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 42.4947 0.0110812
\(246\) 0 0
\(247\) 3571.08 0.919929
\(248\) 0 0
\(249\) −3219.89 −0.819487
\(250\) 0 0
\(251\) 501.085 0.126009 0.0630044 0.998013i \(-0.479932\pi\)
0.0630044 + 0.998013i \(0.479932\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 2319.48 0.569614
\(256\) 0 0
\(257\) 3578.85 0.868647 0.434324 0.900757i \(-0.356987\pi\)
0.434324 + 0.900757i \(0.356987\pi\)
\(258\) 0 0
\(259\) 659.484 0.158218
\(260\) 0 0
\(261\) 1669.37 0.395906
\(262\) 0 0
\(263\) −1769.60 −0.414898 −0.207449 0.978246i \(-0.566516\pi\)
−0.207449 + 0.978246i \(0.566516\pi\)
\(264\) 0 0
\(265\) −4436.57 −1.02844
\(266\) 0 0
\(267\) −1559.75 −0.357510
\(268\) 0 0
\(269\) −3356.68 −0.760819 −0.380409 0.924818i \(-0.624217\pi\)
−0.380409 + 0.924818i \(0.624217\pi\)
\(270\) 0 0
\(271\) −7222.10 −1.61886 −0.809430 0.587216i \(-0.800224\pi\)
−0.809430 + 0.587216i \(0.800224\pi\)
\(272\) 0 0
\(273\) 5160.44 1.14404
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5161.74 1.11963 0.559817 0.828616i \(-0.310871\pi\)
0.559817 + 0.828616i \(0.310871\pi\)
\(278\) 0 0
\(279\) 2569.27 0.551320
\(280\) 0 0
\(281\) 958.646 0.203516 0.101758 0.994809i \(-0.467553\pi\)
0.101758 + 0.994809i \(0.467553\pi\)
\(282\) 0 0
\(283\) −1283.98 −0.269698 −0.134849 0.990866i \(-0.543055\pi\)
−0.134849 + 0.990866i \(0.543055\pi\)
\(284\) 0 0
\(285\) −1181.37 −0.245539
\(286\) 0 0
\(287\) −8210.32 −1.68864
\(288\) 0 0
\(289\) 717.102 0.145960
\(290\) 0 0
\(291\) 3273.37 0.659411
\(292\) 0 0
\(293\) 6400.71 1.27622 0.638112 0.769944i \(-0.279716\pi\)
0.638112 + 0.769944i \(0.279716\pi\)
\(294\) 0 0
\(295\) 3943.30 0.778264
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −10030.6 −1.94009
\(300\) 0 0
\(301\) 5447.78 1.04321
\(302\) 0 0
\(303\) 3563.84 0.675701
\(304\) 0 0
\(305\) −2859.69 −0.536870
\(306\) 0 0
\(307\) −5637.23 −1.04799 −0.523996 0.851721i \(-0.675559\pi\)
−0.523996 + 0.851721i \(0.675559\pi\)
\(308\) 0 0
\(309\) 2499.37 0.460143
\(310\) 0 0
\(311\) 8785.63 1.60189 0.800944 0.598739i \(-0.204331\pi\)
0.800944 + 0.598739i \(0.204331\pi\)
\(312\) 0 0
\(313\) 5970.23 1.07814 0.539069 0.842261i \(-0.318776\pi\)
0.539069 + 0.842261i \(0.318776\pi\)
\(314\) 0 0
\(315\) −1707.16 −0.305357
\(316\) 0 0
\(317\) −2490.40 −0.441245 −0.220622 0.975359i \(-0.570809\pi\)
−0.220622 + 0.975359i \(0.570809\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 2488.72 0.432731
\(322\) 0 0
\(323\) −2867.56 −0.493980
\(324\) 0 0
\(325\) −1759.04 −0.300227
\(326\) 0 0
\(327\) −603.284 −0.102024
\(328\) 0 0
\(329\) −1222.09 −0.204790
\(330\) 0 0
\(331\) 297.606 0.0494197 0.0247098 0.999695i \(-0.492134\pi\)
0.0247098 + 0.999695i \(0.492134\pi\)
\(332\) 0 0
\(333\) 322.423 0.0530591
\(334\) 0 0
\(335\) 3743.48 0.610533
\(336\) 0 0
\(337\) −11726.6 −1.89552 −0.947758 0.318991i \(-0.896656\pi\)
−0.947758 + 0.318991i \(0.896656\pi\)
\(338\) 0 0
\(339\) 2669.50 0.427692
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −6390.06 −1.00592
\(344\) 0 0
\(345\) 3318.31 0.517831
\(346\) 0 0
\(347\) 1627.93 0.251849 0.125925 0.992040i \(-0.459810\pi\)
0.125925 + 0.992040i \(0.459810\pi\)
\(348\) 0 0
\(349\) −11665.9 −1.78929 −0.894644 0.446780i \(-0.852571\pi\)
−0.894644 + 0.446780i \(0.852571\pi\)
\(350\) 0 0
\(351\) 2522.95 0.383661
\(352\) 0 0
\(353\) −2269.34 −0.342166 −0.171083 0.985257i \(-0.554727\pi\)
−0.171083 + 0.985257i \(0.554727\pi\)
\(354\) 0 0
\(355\) −10221.2 −1.52813
\(356\) 0 0
\(357\) −4143.81 −0.614324
\(358\) 0 0
\(359\) −2809.30 −0.413006 −0.206503 0.978446i \(-0.566208\pi\)
−0.206503 + 0.978446i \(0.566208\pi\)
\(360\) 0 0
\(361\) −5398.47 −0.787064
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2305.06 0.330554
\(366\) 0 0
\(367\) −3366.60 −0.478842 −0.239421 0.970916i \(-0.576958\pi\)
−0.239421 + 0.970916i \(0.576958\pi\)
\(368\) 0 0
\(369\) −4014.05 −0.566295
\(370\) 0 0
\(371\) 7926.04 1.10916
\(372\) 0 0
\(373\) −2071.89 −0.287610 −0.143805 0.989606i \(-0.545934\pi\)
−0.143805 + 0.989606i \(0.545934\pi\)
\(374\) 0 0
\(375\) 4445.97 0.612237
\(376\) 0 0
\(377\) 17332.2 2.36779
\(378\) 0 0
\(379\) 9563.59 1.29617 0.648085 0.761568i \(-0.275570\pi\)
0.648085 + 0.761568i \(0.275570\pi\)
\(380\) 0 0
\(381\) −46.8277 −0.00629674
\(382\) 0 0
\(383\) 956.509 0.127612 0.0638059 0.997962i \(-0.479676\pi\)
0.0638059 + 0.997962i \(0.479676\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2663.43 0.349845
\(388\) 0 0
\(389\) −14004.3 −1.82531 −0.912655 0.408731i \(-0.865972\pi\)
−0.912655 + 0.408731i \(0.865972\pi\)
\(390\) 0 0
\(391\) 8054.56 1.04178
\(392\) 0 0
\(393\) 1256.96 0.161336
\(394\) 0 0
\(395\) 3195.79 0.407083
\(396\) 0 0
\(397\) 11361.4 1.43630 0.718152 0.695887i \(-0.244988\pi\)
0.718152 + 0.695887i \(0.244988\pi\)
\(398\) 0 0
\(399\) 2110.55 0.264812
\(400\) 0 0
\(401\) −6752.88 −0.840954 −0.420477 0.907303i \(-0.638137\pi\)
−0.420477 + 0.907303i \(0.638137\pi\)
\(402\) 0 0
\(403\) 26675.5 3.29727
\(404\) 0 0
\(405\) −834.635 −0.102403
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1362.24 0.164690 0.0823450 0.996604i \(-0.473759\pi\)
0.0823450 + 0.996604i \(0.473759\pi\)
\(410\) 0 0
\(411\) 6592.61 0.791216
\(412\) 0 0
\(413\) −7044.80 −0.839351
\(414\) 0 0
\(415\) 11059.4 1.30815
\(416\) 0 0
\(417\) 97.8545 0.0114915
\(418\) 0 0
\(419\) 6496.80 0.757493 0.378746 0.925500i \(-0.376355\pi\)
0.378746 + 0.925500i \(0.376355\pi\)
\(420\) 0 0
\(421\) −2898.48 −0.335542 −0.167771 0.985826i \(-0.553657\pi\)
−0.167771 + 0.985826i \(0.553657\pi\)
\(422\) 0 0
\(423\) −597.481 −0.0686774
\(424\) 0 0
\(425\) 1412.50 0.161215
\(426\) 0 0
\(427\) 5108.91 0.579010
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16041.8 1.79282 0.896409 0.443228i \(-0.146167\pi\)
0.896409 + 0.443228i \(0.146167\pi\)
\(432\) 0 0
\(433\) 8413.94 0.933830 0.466915 0.884302i \(-0.345365\pi\)
0.466915 + 0.884302i \(0.345365\pi\)
\(434\) 0 0
\(435\) −5733.80 −0.631988
\(436\) 0 0
\(437\) −4102.41 −0.449072
\(438\) 0 0
\(439\) 15686.4 1.70540 0.852700 0.522400i \(-0.174963\pi\)
0.852700 + 0.522400i \(0.174963\pi\)
\(440\) 0 0
\(441\) −37.1164 −0.00400781
\(442\) 0 0
\(443\) 8628.26 0.925374 0.462687 0.886522i \(-0.346885\pi\)
0.462687 + 0.886522i \(0.346885\pi\)
\(444\) 0 0
\(445\) 5357.30 0.570697
\(446\) 0 0
\(447\) 10079.2 1.06651
\(448\) 0 0
\(449\) −5832.05 −0.612988 −0.306494 0.951873i \(-0.599156\pi\)
−0.306494 + 0.951873i \(0.599156\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −9221.42 −0.956424
\(454\) 0 0
\(455\) −17724.6 −1.82625
\(456\) 0 0
\(457\) −4795.06 −0.490817 −0.245408 0.969420i \(-0.578922\pi\)
−0.245408 + 0.969420i \(0.578922\pi\)
\(458\) 0 0
\(459\) −2025.92 −0.206017
\(460\) 0 0
\(461\) 11452.6 1.15706 0.578528 0.815663i \(-0.303628\pi\)
0.578528 + 0.815663i \(0.303628\pi\)
\(462\) 0 0
\(463\) 13765.5 1.38172 0.690859 0.722989i \(-0.257232\pi\)
0.690859 + 0.722989i \(0.257232\pi\)
\(464\) 0 0
\(465\) −8824.70 −0.880077
\(466\) 0 0
\(467\) 8670.67 0.859166 0.429583 0.903027i \(-0.358661\pi\)
0.429583 + 0.903027i \(0.358661\pi\)
\(468\) 0 0
\(469\) −6687.82 −0.658454
\(470\) 0 0
\(471\) 6885.59 0.673612
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −719.425 −0.0694936
\(476\) 0 0
\(477\) 3875.06 0.371964
\(478\) 0 0
\(479\) 12507.3 1.19306 0.596528 0.802592i \(-0.296547\pi\)
0.596528 + 0.802592i \(0.296547\pi\)
\(480\) 0 0
\(481\) 3347.56 0.317330
\(482\) 0 0
\(483\) −5928.23 −0.558476
\(484\) 0 0
\(485\) −11243.1 −1.05262
\(486\) 0 0
\(487\) −17148.5 −1.59563 −0.797817 0.602900i \(-0.794012\pi\)
−0.797817 + 0.602900i \(0.794012\pi\)
\(488\) 0 0
\(489\) 3520.64 0.325581
\(490\) 0 0
\(491\) −15140.1 −1.39157 −0.695786 0.718249i \(-0.744944\pi\)
−0.695786 + 0.718249i \(0.744944\pi\)
\(492\) 0 0
\(493\) −13917.7 −1.27145
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 18260.5 1.64808
\(498\) 0 0
\(499\) −11101.7 −0.995952 −0.497976 0.867191i \(-0.665923\pi\)
−0.497976 + 0.867191i \(0.665923\pi\)
\(500\) 0 0
\(501\) 8622.34 0.768898
\(502\) 0 0
\(503\) −18622.9 −1.65080 −0.825400 0.564549i \(-0.809050\pi\)
−0.825400 + 0.564549i \(0.809050\pi\)
\(504\) 0 0
\(505\) −12240.8 −1.07863
\(506\) 0 0
\(507\) 19603.6 1.71721
\(508\) 0 0
\(509\) 2403.28 0.209280 0.104640 0.994510i \(-0.466631\pi\)
0.104640 + 0.994510i \(0.466631\pi\)
\(510\) 0 0
\(511\) −4118.04 −0.356500
\(512\) 0 0
\(513\) 1031.85 0.0888060
\(514\) 0 0
\(515\) −8584.62 −0.734531
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −7921.84 −0.670000
\(520\) 0 0
\(521\) −7624.25 −0.641122 −0.320561 0.947228i \(-0.603871\pi\)
−0.320561 + 0.947228i \(0.603871\pi\)
\(522\) 0 0
\(523\) −12095.3 −1.01126 −0.505632 0.862749i \(-0.668741\pi\)
−0.505632 + 0.862749i \(0.668741\pi\)
\(524\) 0 0
\(525\) −1039.61 −0.0864238
\(526\) 0 0
\(527\) −21420.3 −1.77056
\(528\) 0 0
\(529\) −643.948 −0.0529257
\(530\) 0 0
\(531\) −3444.22 −0.281481
\(532\) 0 0
\(533\) −41675.9 −3.38683
\(534\) 0 0
\(535\) −8548.03 −0.690773
\(536\) 0 0
\(537\) 10127.7 0.813857
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −16614.3 −1.32034 −0.660169 0.751117i \(-0.729515\pi\)
−0.660169 + 0.751117i \(0.729515\pi\)
\(542\) 0 0
\(543\) 2357.78 0.186339
\(544\) 0 0
\(545\) 2072.11 0.162861
\(546\) 0 0
\(547\) −12871.3 −1.00610 −0.503050 0.864258i \(-0.667789\pi\)
−0.503050 + 0.864258i \(0.667789\pi\)
\(548\) 0 0
\(549\) 2497.76 0.194174
\(550\) 0 0
\(551\) 7088.67 0.548072
\(552\) 0 0
\(553\) −5709.36 −0.439036
\(554\) 0 0
\(555\) −1107.43 −0.0846988
\(556\) 0 0
\(557\) −5691.26 −0.432938 −0.216469 0.976289i \(-0.569454\pi\)
−0.216469 + 0.976289i \(0.569454\pi\)
\(558\) 0 0
\(559\) 27653.1 2.09231
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −17890.6 −1.33925 −0.669626 0.742699i \(-0.733545\pi\)
−0.669626 + 0.742699i \(0.733545\pi\)
\(564\) 0 0
\(565\) −9168.98 −0.682729
\(566\) 0 0
\(567\) 1491.10 0.110441
\(568\) 0 0
\(569\) 9099.56 0.670428 0.335214 0.942142i \(-0.391191\pi\)
0.335214 + 0.942142i \(0.391191\pi\)
\(570\) 0 0
\(571\) 3320.97 0.243394 0.121697 0.992567i \(-0.461166\pi\)
0.121697 + 0.992567i \(0.461166\pi\)
\(572\) 0 0
\(573\) −3119.74 −0.227451
\(574\) 0 0
\(575\) 2020.76 0.146559
\(576\) 0 0
\(577\) −24181.1 −1.74467 −0.872333 0.488912i \(-0.837394\pi\)
−0.872333 + 0.488912i \(0.837394\pi\)
\(578\) 0 0
\(579\) −3737.28 −0.268249
\(580\) 0 0
\(581\) −19757.9 −1.41083
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −8665.61 −0.612442
\(586\) 0 0
\(587\) −26367.9 −1.85403 −0.927017 0.375018i \(-0.877636\pi\)
−0.927017 + 0.375018i \(0.877636\pi\)
\(588\) 0 0
\(589\) 10909.9 0.763219
\(590\) 0 0
\(591\) 8678.20 0.604016
\(592\) 0 0
\(593\) −3262.61 −0.225935 −0.112967 0.993599i \(-0.536036\pi\)
−0.112967 + 0.993599i \(0.536036\pi\)
\(594\) 0 0
\(595\) 14232.8 0.980651
\(596\) 0 0
\(597\) −15391.2 −1.05514
\(598\) 0 0
\(599\) −10205.7 −0.696148 −0.348074 0.937467i \(-0.613164\pi\)
−0.348074 + 0.937467i \(0.613164\pi\)
\(600\) 0 0
\(601\) 19629.0 1.33225 0.666125 0.745840i \(-0.267952\pi\)
0.666125 + 0.745840i \(0.267952\pi\)
\(602\) 0 0
\(603\) −3269.69 −0.220816
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 18047.2 1.20678 0.603388 0.797448i \(-0.293817\pi\)
0.603388 + 0.797448i \(0.293817\pi\)
\(608\) 0 0
\(609\) 10243.6 0.681594
\(610\) 0 0
\(611\) −6203.36 −0.410738
\(612\) 0 0
\(613\) 21199.2 1.39678 0.698390 0.715717i \(-0.253900\pi\)
0.698390 + 0.715717i \(0.253900\pi\)
\(614\) 0 0
\(615\) 13787.1 0.903982
\(616\) 0 0
\(617\) −24424.5 −1.59367 −0.796835 0.604196i \(-0.793494\pi\)
−0.796835 + 0.604196i \(0.793494\pi\)
\(618\) 0 0
\(619\) −6954.04 −0.451545 −0.225773 0.974180i \(-0.572491\pi\)
−0.225773 + 0.974180i \(0.572491\pi\)
\(620\) 0 0
\(621\) −2898.33 −0.187288
\(622\) 0 0
\(623\) −9570.94 −0.615492
\(624\) 0 0
\(625\) −12917.5 −0.826722
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2688.08 −0.170399
\(630\) 0 0
\(631\) 10767.5 0.679314 0.339657 0.940549i \(-0.389689\pi\)
0.339657 + 0.940549i \(0.389689\pi\)
\(632\) 0 0
\(633\) 8932.70 0.560889
\(634\) 0 0
\(635\) 160.840 0.0100515
\(636\) 0 0
\(637\) −385.361 −0.0239695
\(638\) 0 0
\(639\) 8927.58 0.552691
\(640\) 0 0
\(641\) −4627.73 −0.285155 −0.142577 0.989784i \(-0.545539\pi\)
−0.142577 + 0.989784i \(0.545539\pi\)
\(642\) 0 0
\(643\) −21050.2 −1.29104 −0.645521 0.763743i \(-0.723360\pi\)
−0.645521 + 0.763743i \(0.723360\pi\)
\(644\) 0 0
\(645\) −9148.12 −0.558461
\(646\) 0 0
\(647\) 4869.46 0.295886 0.147943 0.988996i \(-0.452735\pi\)
0.147943 + 0.988996i \(0.452735\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 15765.5 0.949155
\(652\) 0 0
\(653\) −30026.5 −1.79943 −0.899715 0.436477i \(-0.856226\pi\)
−0.899715 + 0.436477i \(0.856226\pi\)
\(654\) 0 0
\(655\) −4317.28 −0.257542
\(656\) 0 0
\(657\) −2013.32 −0.119554
\(658\) 0 0
\(659\) −10255.0 −0.606189 −0.303095 0.952960i \(-0.598020\pi\)
−0.303095 + 0.952960i \(0.598020\pi\)
\(660\) 0 0
\(661\) 15754.2 0.927031 0.463515 0.886089i \(-0.346588\pi\)
0.463515 + 0.886089i \(0.346588\pi\)
\(662\) 0 0
\(663\) −21034.1 −1.23212
\(664\) 0 0
\(665\) −7249.14 −0.422721
\(666\) 0 0
\(667\) −19911.0 −1.15586
\(668\) 0 0
\(669\) −5596.64 −0.323436
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −6016.90 −0.344628 −0.172314 0.985042i \(-0.555124\pi\)
−0.172314 + 0.985042i \(0.555124\pi\)
\(674\) 0 0
\(675\) −508.270 −0.0289827
\(676\) 0 0
\(677\) −25373.0 −1.44042 −0.720209 0.693757i \(-0.755954\pi\)
−0.720209 + 0.693757i \(0.755954\pi\)
\(678\) 0 0
\(679\) 20086.0 1.13525
\(680\) 0 0
\(681\) 5852.65 0.329331
\(682\) 0 0
\(683\) −3803.49 −0.213084 −0.106542 0.994308i \(-0.533978\pi\)
−0.106542 + 0.994308i \(0.533978\pi\)
\(684\) 0 0
\(685\) −22643.7 −1.26303
\(686\) 0 0
\(687\) 1330.64 0.0738969
\(688\) 0 0
\(689\) 40232.8 2.22460
\(690\) 0 0
\(691\) 15703.7 0.864540 0.432270 0.901744i \(-0.357713\pi\)
0.432270 + 0.901744i \(0.357713\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −336.102 −0.0183440
\(696\) 0 0
\(697\) 33465.5 1.81865
\(698\) 0 0
\(699\) −17444.6 −0.943941
\(700\) 0 0
\(701\) 16367.6 0.881879 0.440939 0.897537i \(-0.354645\pi\)
0.440939 + 0.897537i \(0.354645\pi\)
\(702\) 0 0
\(703\) 1369.11 0.0734523
\(704\) 0 0
\(705\) 2052.18 0.109630
\(706\) 0 0
\(707\) 21868.4 1.16329
\(708\) 0 0
\(709\) −31262.6 −1.65598 −0.827992 0.560739i \(-0.810517\pi\)
−0.827992 + 0.560739i \(0.810517\pi\)
\(710\) 0 0
\(711\) −2791.32 −0.147233
\(712\) 0 0
\(713\) −30644.4 −1.60960
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 15624.1 0.813795
\(718\) 0 0
\(719\) −29448.9 −1.52748 −0.763741 0.645523i \(-0.776640\pi\)
−0.763741 + 0.645523i \(0.776640\pi\)
\(720\) 0 0
\(721\) 15336.6 0.792186
\(722\) 0 0
\(723\) 5613.68 0.288762
\(724\) 0 0
\(725\) −3491.73 −0.178868
\(726\) 0 0
\(727\) 23185.3 1.18280 0.591399 0.806379i \(-0.298576\pi\)
0.591399 + 0.806379i \(0.298576\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −22205.3 −1.12352
\(732\) 0 0
\(733\) −19688.4 −0.992097 −0.496049 0.868295i \(-0.665216\pi\)
−0.496049 + 0.868295i \(0.665216\pi\)
\(734\) 0 0
\(735\) 127.484 0.00639771
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 3261.15 0.162332 0.0811660 0.996701i \(-0.474136\pi\)
0.0811660 + 0.996701i \(0.474136\pi\)
\(740\) 0 0
\(741\) 10713.2 0.531121
\(742\) 0 0
\(743\) −20001.0 −0.987573 −0.493787 0.869583i \(-0.664388\pi\)
−0.493787 + 0.869583i \(0.664388\pi\)
\(744\) 0 0
\(745\) −34619.2 −1.70248
\(746\) 0 0
\(747\) −9659.67 −0.473131
\(748\) 0 0
\(749\) 15271.3 0.744993
\(750\) 0 0
\(751\) −10624.2 −0.516224 −0.258112 0.966115i \(-0.583100\pi\)
−0.258112 + 0.966115i \(0.583100\pi\)
\(752\) 0 0
\(753\) 1503.25 0.0727512
\(754\) 0 0
\(755\) 31672.9 1.52675
\(756\) 0 0
\(757\) −14839.5 −0.712486 −0.356243 0.934393i \(-0.615942\pi\)
−0.356243 + 0.934393i \(0.615942\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4401.81 −0.209679 −0.104839 0.994489i \(-0.533433\pi\)
−0.104839 + 0.994489i \(0.533433\pi\)
\(762\) 0 0
\(763\) −3701.87 −0.175644
\(764\) 0 0
\(765\) 6958.44 0.328867
\(766\) 0 0
\(767\) −35759.6 −1.68345
\(768\) 0 0
\(769\) 34100.9 1.59910 0.799552 0.600597i \(-0.205071\pi\)
0.799552 + 0.600597i \(0.205071\pi\)
\(770\) 0 0
\(771\) 10736.5 0.501514
\(772\) 0 0
\(773\) −15004.5 −0.698157 −0.349079 0.937093i \(-0.613505\pi\)
−0.349079 + 0.937093i \(0.613505\pi\)
\(774\) 0 0
\(775\) −5374.00 −0.249084
\(776\) 0 0
\(777\) 1978.45 0.0913469
\(778\) 0 0
\(779\) −17044.9 −0.783950
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 5008.11 0.228576
\(784\) 0 0
\(785\) −23650.0 −1.07529
\(786\) 0 0
\(787\) 15766.9 0.714143 0.357072 0.934077i \(-0.383775\pi\)
0.357072 + 0.934077i \(0.383775\pi\)
\(788\) 0 0
\(789\) −5308.79 −0.239541
\(790\) 0 0
\(791\) 16380.6 0.736317
\(792\) 0 0
\(793\) 25933.0 1.16130
\(794\) 0 0
\(795\) −13309.7 −0.593769
\(796\) 0 0
\(797\) −8490.24 −0.377340 −0.188670 0.982041i \(-0.560418\pi\)
−0.188670 + 0.982041i \(0.560418\pi\)
\(798\) 0 0
\(799\) 4981.27 0.220557
\(800\) 0 0
\(801\) −4679.26 −0.206409
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 20361.8 0.891501
\(806\) 0 0
\(807\) −10070.0 −0.439259
\(808\) 0 0
\(809\) 41661.0 1.81054 0.905268 0.424841i \(-0.139670\pi\)
0.905268 + 0.424841i \(0.139670\pi\)
\(810\) 0 0
\(811\) −38286.1 −1.65772 −0.828858 0.559458i \(-0.811009\pi\)
−0.828858 + 0.559458i \(0.811009\pi\)
\(812\) 0 0
\(813\) −21666.3 −0.934650
\(814\) 0 0
\(815\) −12092.4 −0.519728
\(816\) 0 0
\(817\) 11309.8 0.484307
\(818\) 0 0
\(819\) 15481.3 0.660514
\(820\) 0 0
\(821\) −18942.7 −0.805244 −0.402622 0.915366i \(-0.631901\pi\)
−0.402622 + 0.915366i \(0.631901\pi\)
\(822\) 0 0
\(823\) 22445.5 0.950668 0.475334 0.879805i \(-0.342327\pi\)
0.475334 + 0.879805i \(0.342327\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −31648.8 −1.33076 −0.665380 0.746505i \(-0.731730\pi\)
−0.665380 + 0.746505i \(0.731730\pi\)
\(828\) 0 0
\(829\) 13695.0 0.573760 0.286880 0.957967i \(-0.407382\pi\)
0.286880 + 0.957967i \(0.407382\pi\)
\(830\) 0 0
\(831\) 15485.2 0.646421
\(832\) 0 0
\(833\) 309.443 0.0128710
\(834\) 0 0
\(835\) −29615.2 −1.22740
\(836\) 0 0
\(837\) 7707.81 0.318305
\(838\) 0 0
\(839\) 34249.7 1.40934 0.704668 0.709537i \(-0.251096\pi\)
0.704668 + 0.709537i \(0.251096\pi\)
\(840\) 0 0
\(841\) 10015.9 0.410672
\(842\) 0 0
\(843\) 2875.94 0.117500
\(844\) 0 0
\(845\) −67332.6 −2.74120
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −3851.93 −0.155710
\(850\) 0 0
\(851\) −3845.63 −0.154908
\(852\) 0 0
\(853\) −2046.03 −0.0821275 −0.0410637 0.999157i \(-0.513075\pi\)
−0.0410637 + 0.999157i \(0.513075\pi\)
\(854\) 0 0
\(855\) −3544.12 −0.141762
\(856\) 0 0
\(857\) −281.695 −0.0112281 −0.00561407 0.999984i \(-0.501787\pi\)
−0.00561407 + 0.999984i \(0.501787\pi\)
\(858\) 0 0
\(859\) 19534.2 0.775901 0.387951 0.921680i \(-0.373183\pi\)
0.387951 + 0.921680i \(0.373183\pi\)
\(860\) 0 0
\(861\) −24631.0 −0.974937
\(862\) 0 0
\(863\) 25836.9 1.01912 0.509559 0.860435i \(-0.329808\pi\)
0.509559 + 0.860435i \(0.329808\pi\)
\(864\) 0 0
\(865\) 27209.2 1.06953
\(866\) 0 0
\(867\) 2151.31 0.0842701
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −33947.6 −1.32063
\(872\) 0 0
\(873\) 9820.12 0.380711
\(874\) 0 0
\(875\) 27281.3 1.05403
\(876\) 0 0
\(877\) 7433.88 0.286230 0.143115 0.989706i \(-0.454288\pi\)
0.143115 + 0.989706i \(0.454288\pi\)
\(878\) 0 0
\(879\) 19202.1 0.736828
\(880\) 0 0
\(881\) −26580.5 −1.01648 −0.508240 0.861215i \(-0.669704\pi\)
−0.508240 + 0.861215i \(0.669704\pi\)
\(882\) 0 0
\(883\) 47009.3 1.79161 0.895804 0.444449i \(-0.146600\pi\)
0.895804 + 0.444449i \(0.146600\pi\)
\(884\) 0 0
\(885\) 11829.9 0.449331
\(886\) 0 0
\(887\) −23948.4 −0.906550 −0.453275 0.891371i \(-0.649744\pi\)
−0.453275 + 0.891371i \(0.649744\pi\)
\(888\) 0 0
\(889\) −287.344 −0.0108405
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2537.09 −0.0950735
\(894\) 0 0
\(895\) −34785.6 −1.29917
\(896\) 0 0
\(897\) −30091.9 −1.12011
\(898\) 0 0
\(899\) 52951.4 1.96443
\(900\) 0 0
\(901\) −32306.8 −1.19456
\(902\) 0 0
\(903\) 16343.3 0.602295
\(904\) 0 0
\(905\) −8098.30 −0.297455
\(906\) 0 0
\(907\) 8876.24 0.324951 0.162475 0.986713i \(-0.448052\pi\)
0.162475 + 0.986713i \(0.448052\pi\)
\(908\) 0 0
\(909\) 10691.5 0.390116
\(910\) 0 0
\(911\) 29594.4 1.07630 0.538148 0.842851i \(-0.319124\pi\)
0.538148 + 0.842851i \(0.319124\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −8579.07 −0.309962
\(916\) 0 0
\(917\) 7712.93 0.277757
\(918\) 0 0
\(919\) −37066.9 −1.33049 −0.665247 0.746623i \(-0.731674\pi\)
−0.665247 + 0.746623i \(0.731674\pi\)
\(920\) 0 0
\(921\) −16911.7 −0.605058
\(922\) 0 0
\(923\) 92690.7 3.30547
\(924\) 0 0
\(925\) −674.395 −0.0239719
\(926\) 0 0
\(927\) 7498.12 0.265664
\(928\) 0 0
\(929\) −39465.4 −1.39378 −0.696888 0.717180i \(-0.745432\pi\)
−0.696888 + 0.717180i \(0.745432\pi\)
\(930\) 0 0
\(931\) −157.608 −0.00554821
\(932\) 0 0
\(933\) 26356.9 0.924851
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 29964.1 1.04470 0.522350 0.852731i \(-0.325056\pi\)
0.522350 + 0.852731i \(0.325056\pi\)
\(938\) 0 0
\(939\) 17910.7 0.622464
\(940\) 0 0
\(941\) −13533.3 −0.468835 −0.234417 0.972136i \(-0.575318\pi\)
−0.234417 + 0.972136i \(0.575318\pi\)
\(942\) 0 0
\(943\) 47876.6 1.65332
\(944\) 0 0
\(945\) −5121.48 −0.176298
\(946\) 0 0
\(947\) 23764.6 0.815466 0.407733 0.913101i \(-0.366319\pi\)
0.407733 + 0.913101i \(0.366319\pi\)
\(948\) 0 0
\(949\) −20903.3 −0.715017
\(950\) 0 0
\(951\) −7471.19 −0.254753
\(952\) 0 0
\(953\) −44164.0 −1.50117 −0.750583 0.660776i \(-0.770227\pi\)
−0.750583 + 0.660776i \(0.770227\pi\)
\(954\) 0 0
\(955\) 10715.4 0.363081
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 40453.6 1.36216
\(960\) 0 0
\(961\) 51704.6 1.73558
\(962\) 0 0
\(963\) 7466.16 0.249838
\(964\) 0 0
\(965\) 12836.5 0.428208
\(966\) 0 0
\(967\) −12183.8 −0.405174 −0.202587 0.979264i \(-0.564935\pi\)
−0.202587 + 0.979264i \(0.564935\pi\)
\(968\) 0 0
\(969\) −8602.69 −0.285199
\(970\) 0 0
\(971\) 53903.0 1.78149 0.890747 0.454500i \(-0.150182\pi\)
0.890747 + 0.454500i \(0.150182\pi\)
\(972\) 0 0
\(973\) 600.454 0.0197838
\(974\) 0 0
\(975\) −5277.12 −0.173336
\(976\) 0 0
\(977\) 23888.9 0.782264 0.391132 0.920334i \(-0.372084\pi\)
0.391132 + 0.920334i \(0.372084\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1809.85 −0.0589033
\(982\) 0 0
\(983\) −54001.2 −1.75216 −0.876079 0.482167i \(-0.839850\pi\)
−0.876079 + 0.482167i \(0.839850\pi\)
\(984\) 0 0
\(985\) −29807.1 −0.964197
\(986\) 0 0
\(987\) −3666.26 −0.118235
\(988\) 0 0
\(989\) −31767.5 −1.02138
\(990\) 0 0
\(991\) −8358.61 −0.267931 −0.133966 0.990986i \(-0.542771\pi\)
−0.133966 + 0.990986i \(0.542771\pi\)
\(992\) 0 0
\(993\) 892.818 0.0285325
\(994\) 0 0
\(995\) 52864.3 1.68433
\(996\) 0 0
\(997\) 11623.2 0.369219 0.184610 0.982812i \(-0.440898\pi\)
0.184610 + 0.982812i \(0.440898\pi\)
\(998\) 0 0
\(999\) 967.270 0.0306337
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.q.1.2 yes 4
11.10 odd 2 inner 1452.4.a.q.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1452.4.a.q.1.1 4 11.10 odd 2 inner
1452.4.a.q.1.2 yes 4 1.1 even 1 trivial