Properties

Label 3332.2.a.u.1.7
Level $3332$
Weight $2$
Character 3332.1
Self dual yes
Analytic conductor $26.606$
Analytic rank $0$
Dimension $8$
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] = [3332,2,Mod(1,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3332.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: \(26.6061539535\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 8x^{6} + 36x^{5} + 17x^{4} - 76x^{3} - 20x^{2} + 44x + 17 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(3.08774\) of defining polynomial
Character \(\chi\) \(=\) 3332.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.08774 q^{3} +3.25463 q^{5} +6.53416 q^{9} +O(q^{10})\) \(q+3.08774 q^{3} +3.25463 q^{5} +6.53416 q^{9} -5.12130 q^{11} +5.13386 q^{13} +10.0495 q^{15} -1.00000 q^{17} -5.25326 q^{19} +6.97171 q^{23} +5.59260 q^{25} +10.9126 q^{27} -3.39300 q^{29} +2.35553 q^{31} -15.8132 q^{33} +2.60498 q^{37} +15.8520 q^{39} -5.70525 q^{41} +1.18641 q^{43} +21.2662 q^{45} +4.70250 q^{47} -3.08774 q^{51} +10.0116 q^{53} -16.6679 q^{55} -16.2207 q^{57} -13.0991 q^{59} +1.78135 q^{61} +16.7088 q^{65} -11.1779 q^{67} +21.5269 q^{69} +9.21864 q^{71} +14.6608 q^{73} +17.2685 q^{75} +2.05802 q^{79} +14.0927 q^{81} -15.7006 q^{83} -3.25463 q^{85} -10.4767 q^{87} -6.91321 q^{89} +7.27326 q^{93} -17.0974 q^{95} +10.5244 q^{97} -33.4634 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 4 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 4 q^{5} + 8 q^{9} - 4 q^{11} + 20 q^{13} + 12 q^{15} - 8 q^{17} + 8 q^{19} + 4 q^{23} + 8 q^{25} + 28 q^{27} - 16 q^{29} - 8 q^{31} + 16 q^{33} + 8 q^{37} + 20 q^{39} + 12 q^{41} - 4 q^{43} + 20 q^{45} + 4 q^{47} - 4 q^{51} + 12 q^{55} - 16 q^{59} + 32 q^{61} + 44 q^{69} + 24 q^{73} + 24 q^{75} + 4 q^{79} + 36 q^{81} + 28 q^{83} - 4 q^{85} - 40 q^{87} + 20 q^{89} - 16 q^{93} - 20 q^{95} + 56 q^{97} - 8 q^{99}+O(q^{100}) \) 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
\(3\) 3.08774 1.78271 0.891355 0.453307i \(-0.149756\pi\)
0.891355 + 0.453307i \(0.149756\pi\)
\(4\) 0 0
\(5\) 3.25463 1.45551 0.727757 0.685835i \(-0.240563\pi\)
0.727757 + 0.685835i \(0.240563\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 6.53416 2.17805
\(10\) 0 0
\(11\) −5.12130 −1.54413 −0.772064 0.635544i \(-0.780776\pi\)
−0.772064 + 0.635544i \(0.780776\pi\)
\(12\) 0 0
\(13\) 5.13386 1.42388 0.711939 0.702242i \(-0.247817\pi\)
0.711939 + 0.702242i \(0.247817\pi\)
\(14\) 0 0
\(15\) 10.0495 2.59476
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −5.25326 −1.20518 −0.602590 0.798051i \(-0.705865\pi\)
−0.602590 + 0.798051i \(0.705865\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.97171 1.45370 0.726851 0.686795i \(-0.240983\pi\)
0.726851 + 0.686795i \(0.240983\pi\)
\(24\) 0 0
\(25\) 5.59260 1.11852
\(26\) 0 0
\(27\) 10.9126 2.10013
\(28\) 0 0
\(29\) −3.39300 −0.630064 −0.315032 0.949081i \(-0.602015\pi\)
−0.315032 + 0.949081i \(0.602015\pi\)
\(30\) 0 0
\(31\) 2.35553 0.423065 0.211533 0.977371i \(-0.432155\pi\)
0.211533 + 0.977371i \(0.432155\pi\)
\(32\) 0 0
\(33\) −15.8132 −2.75273
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.60498 0.428257 0.214128 0.976806i \(-0.431309\pi\)
0.214128 + 0.976806i \(0.431309\pi\)
\(38\) 0 0
\(39\) 15.8520 2.53836
\(40\) 0 0
\(41\) −5.70525 −0.891010 −0.445505 0.895280i \(-0.646976\pi\)
−0.445505 + 0.895280i \(0.646976\pi\)
\(42\) 0 0
\(43\) 1.18641 0.180926 0.0904631 0.995900i \(-0.471165\pi\)
0.0904631 + 0.995900i \(0.471165\pi\)
\(44\) 0 0
\(45\) 21.2662 3.17019
\(46\) 0 0
\(47\) 4.70250 0.685930 0.342965 0.939348i \(-0.388569\pi\)
0.342965 + 0.939348i \(0.388569\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −3.08774 −0.432371
\(52\) 0 0
\(53\) 10.0116 1.37521 0.687603 0.726087i \(-0.258663\pi\)
0.687603 + 0.726087i \(0.258663\pi\)
\(54\) 0 0
\(55\) −16.6679 −2.24750
\(56\) 0 0
\(57\) −16.2207 −2.14849
\(58\) 0 0
\(59\) −13.0991 −1.70536 −0.852678 0.522436i \(-0.825023\pi\)
−0.852678 + 0.522436i \(0.825023\pi\)
\(60\) 0 0
\(61\) 1.78135 0.228079 0.114039 0.993476i \(-0.463621\pi\)
0.114039 + 0.993476i \(0.463621\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 16.7088 2.07247
\(66\) 0 0
\(67\) −11.1779 −1.36559 −0.682796 0.730609i \(-0.739236\pi\)
−0.682796 + 0.730609i \(0.739236\pi\)
\(68\) 0 0
\(69\) 21.5269 2.59153
\(70\) 0 0
\(71\) 9.21864 1.09405 0.547026 0.837116i \(-0.315760\pi\)
0.547026 + 0.837116i \(0.315760\pi\)
\(72\) 0 0
\(73\) 14.6608 1.71591 0.857957 0.513722i \(-0.171734\pi\)
0.857957 + 0.513722i \(0.171734\pi\)
\(74\) 0 0
\(75\) 17.2685 1.99400
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.05802 0.231545 0.115773 0.993276i \(-0.463066\pi\)
0.115773 + 0.993276i \(0.463066\pi\)
\(80\) 0 0
\(81\) 14.0927 1.56586
\(82\) 0 0
\(83\) −15.7006 −1.72336 −0.861681 0.507450i \(-0.830588\pi\)
−0.861681 + 0.507450i \(0.830588\pi\)
\(84\) 0 0
\(85\) −3.25463 −0.353014
\(86\) 0 0
\(87\) −10.4767 −1.12322
\(88\) 0 0
\(89\) −6.91321 −0.732799 −0.366399 0.930458i \(-0.619410\pi\)
−0.366399 + 0.930458i \(0.619410\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 7.27326 0.754202
\(94\) 0 0
\(95\) −17.0974 −1.75416
\(96\) 0 0
\(97\) 10.5244 1.06859 0.534295 0.845298i \(-0.320577\pi\)
0.534295 + 0.845298i \(0.320577\pi\)
\(98\) 0 0
\(99\) −33.4634 −3.36319
\(100\) 0 0
\(101\) −4.06349 −0.404333 −0.202166 0.979351i \(-0.564798\pi\)
−0.202166 + 0.979351i \(0.564798\pi\)
\(102\) 0 0
\(103\) −14.0639 −1.38576 −0.692880 0.721053i \(-0.743658\pi\)
−0.692880 + 0.721053i \(0.743658\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.0690 1.45678 0.728388 0.685165i \(-0.240270\pi\)
0.728388 + 0.685165i \(0.240270\pi\)
\(108\) 0 0
\(109\) 2.50389 0.239829 0.119914 0.992784i \(-0.461738\pi\)
0.119914 + 0.992784i \(0.461738\pi\)
\(110\) 0 0
\(111\) 8.04352 0.763457
\(112\) 0 0
\(113\) −16.3088 −1.53421 −0.767103 0.641524i \(-0.778302\pi\)
−0.767103 + 0.641524i \(0.778302\pi\)
\(114\) 0 0
\(115\) 22.6903 2.11588
\(116\) 0 0
\(117\) 33.5455 3.10128
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 15.2277 1.38433
\(122\) 0 0
\(123\) −17.6163 −1.58841
\(124\) 0 0
\(125\) 1.92869 0.172507
\(126\) 0 0
\(127\) −3.50639 −0.311142 −0.155571 0.987825i \(-0.549722\pi\)
−0.155571 + 0.987825i \(0.549722\pi\)
\(128\) 0 0
\(129\) 3.66334 0.322539
\(130\) 0 0
\(131\) 9.11641 0.796505 0.398252 0.917276i \(-0.369617\pi\)
0.398252 + 0.917276i \(0.369617\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 35.5164 3.05676
\(136\) 0 0
\(137\) −10.0279 −0.856738 −0.428369 0.903604i \(-0.640912\pi\)
−0.428369 + 0.903604i \(0.640912\pi\)
\(138\) 0 0
\(139\) 10.1157 0.858002 0.429001 0.903304i \(-0.358866\pi\)
0.429001 + 0.903304i \(0.358866\pi\)
\(140\) 0 0
\(141\) 14.5201 1.22281
\(142\) 0 0
\(143\) −26.2920 −2.19865
\(144\) 0 0
\(145\) −11.0429 −0.917067
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −13.5436 −1.10954 −0.554769 0.832005i \(-0.687193\pi\)
−0.554769 + 0.832005i \(0.687193\pi\)
\(150\) 0 0
\(151\) −16.8102 −1.36799 −0.683997 0.729485i \(-0.739760\pi\)
−0.683997 + 0.729485i \(0.739760\pi\)
\(152\) 0 0
\(153\) −6.53416 −0.528255
\(154\) 0 0
\(155\) 7.66636 0.615777
\(156\) 0 0
\(157\) 17.7407 1.41586 0.707932 0.706280i \(-0.249628\pi\)
0.707932 + 0.706280i \(0.249628\pi\)
\(158\) 0 0
\(159\) 30.9134 2.45159
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −16.3081 −1.27735 −0.638673 0.769478i \(-0.720516\pi\)
−0.638673 + 0.769478i \(0.720516\pi\)
\(164\) 0 0
\(165\) −51.4662 −4.00664
\(166\) 0 0
\(167\) −22.3427 −1.72893 −0.864463 0.502696i \(-0.832342\pi\)
−0.864463 + 0.502696i \(0.832342\pi\)
\(168\) 0 0
\(169\) 13.3565 1.02743
\(170\) 0 0
\(171\) −34.3256 −2.62495
\(172\) 0 0
\(173\) 5.13505 0.390410 0.195205 0.980762i \(-0.437463\pi\)
0.195205 + 0.980762i \(0.437463\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −40.4466 −3.04015
\(178\) 0 0
\(179\) 5.67240 0.423975 0.211987 0.977272i \(-0.432006\pi\)
0.211987 + 0.977272i \(0.432006\pi\)
\(180\) 0 0
\(181\) −11.7700 −0.874858 −0.437429 0.899253i \(-0.644111\pi\)
−0.437429 + 0.899253i \(0.644111\pi\)
\(182\) 0 0
\(183\) 5.50036 0.406598
\(184\) 0 0
\(185\) 8.47825 0.623333
\(186\) 0 0
\(187\) 5.12130 0.374506
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.741528 −0.0536551 −0.0268275 0.999640i \(-0.508540\pi\)
−0.0268275 + 0.999640i \(0.508540\pi\)
\(192\) 0 0
\(193\) −11.7110 −0.842979 −0.421490 0.906833i \(-0.638493\pi\)
−0.421490 + 0.906833i \(0.638493\pi\)
\(194\) 0 0
\(195\) 51.5925 3.69462
\(196\) 0 0
\(197\) −3.93898 −0.280641 −0.140320 0.990106i \(-0.544813\pi\)
−0.140320 + 0.990106i \(0.544813\pi\)
\(198\) 0 0
\(199\) −8.68257 −0.615491 −0.307746 0.951469i \(-0.599575\pi\)
−0.307746 + 0.951469i \(0.599575\pi\)
\(200\) 0 0
\(201\) −34.5143 −2.43445
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −18.5684 −1.29688
\(206\) 0 0
\(207\) 45.5543 3.16624
\(208\) 0 0
\(209\) 26.9035 1.86095
\(210\) 0 0
\(211\) −9.05236 −0.623190 −0.311595 0.950215i \(-0.600863\pi\)
−0.311595 + 0.950215i \(0.600863\pi\)
\(212\) 0 0
\(213\) 28.4648 1.95038
\(214\) 0 0
\(215\) 3.86133 0.263340
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 45.2687 3.05897
\(220\) 0 0
\(221\) −5.13386 −0.345341
\(222\) 0 0
\(223\) −12.2667 −0.821442 −0.410721 0.911761i \(-0.634723\pi\)
−0.410721 + 0.911761i \(0.634723\pi\)
\(224\) 0 0
\(225\) 36.5429 2.43619
\(226\) 0 0
\(227\) 17.1364 1.13739 0.568693 0.822550i \(-0.307449\pi\)
0.568693 + 0.822550i \(0.307449\pi\)
\(228\) 0 0
\(229\) 21.5237 1.42232 0.711162 0.703028i \(-0.248169\pi\)
0.711162 + 0.703028i \(0.248169\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −19.2290 −1.25974 −0.629868 0.776702i \(-0.716891\pi\)
−0.629868 + 0.776702i \(0.716891\pi\)
\(234\) 0 0
\(235\) 15.3049 0.998381
\(236\) 0 0
\(237\) 6.35463 0.412778
\(238\) 0 0
\(239\) 8.19767 0.530263 0.265132 0.964212i \(-0.414585\pi\)
0.265132 + 0.964212i \(0.414585\pi\)
\(240\) 0 0
\(241\) −25.4006 −1.63620 −0.818099 0.575078i \(-0.804972\pi\)
−0.818099 + 0.575078i \(0.804972\pi\)
\(242\) 0 0
\(243\) 10.7771 0.691349
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −26.9695 −1.71603
\(248\) 0 0
\(249\) −48.4794 −3.07226
\(250\) 0 0
\(251\) 4.15836 0.262473 0.131237 0.991351i \(-0.458105\pi\)
0.131237 + 0.991351i \(0.458105\pi\)
\(252\) 0 0
\(253\) −35.7042 −2.24470
\(254\) 0 0
\(255\) −10.0495 −0.629321
\(256\) 0 0
\(257\) 17.6639 1.10185 0.550923 0.834556i \(-0.314276\pi\)
0.550923 + 0.834556i \(0.314276\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −22.1704 −1.37231
\(262\) 0 0
\(263\) −24.6955 −1.52279 −0.761395 0.648288i \(-0.775485\pi\)
−0.761395 + 0.648288i \(0.775485\pi\)
\(264\) 0 0
\(265\) 32.5842 2.00163
\(266\) 0 0
\(267\) −21.3462 −1.30637
\(268\) 0 0
\(269\) 14.7507 0.899365 0.449683 0.893188i \(-0.351537\pi\)
0.449683 + 0.893188i \(0.351537\pi\)
\(270\) 0 0
\(271\) 8.00715 0.486400 0.243200 0.969976i \(-0.421803\pi\)
0.243200 + 0.969976i \(0.421803\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −28.6414 −1.72714
\(276\) 0 0
\(277\) −6.21306 −0.373307 −0.186653 0.982426i \(-0.559764\pi\)
−0.186653 + 0.982426i \(0.559764\pi\)
\(278\) 0 0
\(279\) 15.3914 0.921458
\(280\) 0 0
\(281\) −11.0264 −0.657777 −0.328889 0.944369i \(-0.606674\pi\)
−0.328889 + 0.944369i \(0.606674\pi\)
\(282\) 0 0
\(283\) −16.8967 −1.00440 −0.502201 0.864751i \(-0.667476\pi\)
−0.502201 + 0.864751i \(0.667476\pi\)
\(284\) 0 0
\(285\) −52.7924 −3.12715
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 32.4966 1.90498
\(292\) 0 0
\(293\) −22.3788 −1.30738 −0.653692 0.756761i \(-0.726781\pi\)
−0.653692 + 0.756761i \(0.726781\pi\)
\(294\) 0 0
\(295\) −42.6327 −2.48217
\(296\) 0 0
\(297\) −55.8865 −3.24286
\(298\) 0 0
\(299\) 35.7918 2.06989
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −12.5470 −0.720808
\(304\) 0 0
\(305\) 5.79764 0.331972
\(306\) 0 0
\(307\) 2.01060 0.114751 0.0573755 0.998353i \(-0.481727\pi\)
0.0573755 + 0.998353i \(0.481727\pi\)
\(308\) 0 0
\(309\) −43.4258 −2.47041
\(310\) 0 0
\(311\) −0.0533120 −0.00302305 −0.00151152 0.999999i \(-0.500481\pi\)
−0.00151152 + 0.999999i \(0.500481\pi\)
\(312\) 0 0
\(313\) 17.0370 0.962988 0.481494 0.876449i \(-0.340094\pi\)
0.481494 + 0.876449i \(0.340094\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 29.5722 1.66094 0.830471 0.557062i \(-0.188071\pi\)
0.830471 + 0.557062i \(0.188071\pi\)
\(318\) 0 0
\(319\) 17.3765 0.972900
\(320\) 0 0
\(321\) 46.5292 2.59701
\(322\) 0 0
\(323\) 5.25326 0.292299
\(324\) 0 0
\(325\) 28.7116 1.59263
\(326\) 0 0
\(327\) 7.73136 0.427545
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 25.0575 1.37729 0.688643 0.725101i \(-0.258207\pi\)
0.688643 + 0.725101i \(0.258207\pi\)
\(332\) 0 0
\(333\) 17.0214 0.932766
\(334\) 0 0
\(335\) −36.3798 −1.98764
\(336\) 0 0
\(337\) 0.187963 0.0102390 0.00511949 0.999987i \(-0.498370\pi\)
0.00511949 + 0.999987i \(0.498370\pi\)
\(338\) 0 0
\(339\) −50.3575 −2.73504
\(340\) 0 0
\(341\) −12.0634 −0.653267
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 70.0619 3.77201
\(346\) 0 0
\(347\) 7.39627 0.397053 0.198526 0.980096i \(-0.436384\pi\)
0.198526 + 0.980096i \(0.436384\pi\)
\(348\) 0 0
\(349\) −30.1736 −1.61516 −0.807579 0.589760i \(-0.799222\pi\)
−0.807579 + 0.589760i \(0.799222\pi\)
\(350\) 0 0
\(351\) 56.0236 2.99032
\(352\) 0 0
\(353\) −13.5872 −0.723174 −0.361587 0.932338i \(-0.617765\pi\)
−0.361587 + 0.932338i \(0.617765\pi\)
\(354\) 0 0
\(355\) 30.0032 1.59241
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.6480 1.40643 0.703214 0.710978i \(-0.251748\pi\)
0.703214 + 0.710978i \(0.251748\pi\)
\(360\) 0 0
\(361\) 8.59676 0.452461
\(362\) 0 0
\(363\) 47.0192 2.46787
\(364\) 0 0
\(365\) 47.7153 2.49753
\(366\) 0 0
\(367\) −25.0287 −1.30649 −0.653244 0.757147i \(-0.726592\pi\)
−0.653244 + 0.757147i \(0.726592\pi\)
\(368\) 0 0
\(369\) −37.2790 −1.94067
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 17.6096 0.911790 0.455895 0.890034i \(-0.349319\pi\)
0.455895 + 0.890034i \(0.349319\pi\)
\(374\) 0 0
\(375\) 5.95529 0.307530
\(376\) 0 0
\(377\) −17.4192 −0.897134
\(378\) 0 0
\(379\) −11.6143 −0.596589 −0.298295 0.954474i \(-0.596418\pi\)
−0.298295 + 0.954474i \(0.596418\pi\)
\(380\) 0 0
\(381\) −10.8268 −0.554675
\(382\) 0 0
\(383\) −2.89518 −0.147937 −0.0739684 0.997261i \(-0.523566\pi\)
−0.0739684 + 0.997261i \(0.523566\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.75220 0.394067
\(388\) 0 0
\(389\) 26.7804 1.35782 0.678910 0.734221i \(-0.262452\pi\)
0.678910 + 0.734221i \(0.262452\pi\)
\(390\) 0 0
\(391\) −6.97171 −0.352575
\(392\) 0 0
\(393\) 28.1491 1.41994
\(394\) 0 0
\(395\) 6.69809 0.337017
\(396\) 0 0
\(397\) 27.1376 1.36200 0.680998 0.732285i \(-0.261546\pi\)
0.680998 + 0.732285i \(0.261546\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.9281 0.695534 0.347767 0.937581i \(-0.386940\pi\)
0.347767 + 0.937581i \(0.386940\pi\)
\(402\) 0 0
\(403\) 12.0930 0.602393
\(404\) 0 0
\(405\) 45.8666 2.27913
\(406\) 0 0
\(407\) −13.3409 −0.661284
\(408\) 0 0
\(409\) 10.3532 0.511933 0.255967 0.966686i \(-0.417606\pi\)
0.255967 + 0.966686i \(0.417606\pi\)
\(410\) 0 0
\(411\) −30.9635 −1.52732
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −51.0995 −2.50838
\(416\) 0 0
\(417\) 31.2347 1.52957
\(418\) 0 0
\(419\) −3.50968 −0.171459 −0.0857294 0.996318i \(-0.527322\pi\)
−0.0857294 + 0.996318i \(0.527322\pi\)
\(420\) 0 0
\(421\) 10.5824 0.515755 0.257877 0.966178i \(-0.416977\pi\)
0.257877 + 0.966178i \(0.416977\pi\)
\(422\) 0 0
\(423\) 30.7269 1.49399
\(424\) 0 0
\(425\) −5.59260 −0.271281
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −81.1830 −3.91955
\(430\) 0 0
\(431\) −0.665241 −0.0320435 −0.0160218 0.999872i \(-0.505100\pi\)
−0.0160218 + 0.999872i \(0.505100\pi\)
\(432\) 0 0
\(433\) 10.1221 0.486435 0.243217 0.969972i \(-0.421797\pi\)
0.243217 + 0.969972i \(0.421797\pi\)
\(434\) 0 0
\(435\) −34.0978 −1.63486
\(436\) 0 0
\(437\) −36.6242 −1.75197
\(438\) 0 0
\(439\) 15.6542 0.747134 0.373567 0.927603i \(-0.378135\pi\)
0.373567 + 0.927603i \(0.378135\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −34.1217 −1.62117 −0.810585 0.585621i \(-0.800851\pi\)
−0.810585 + 0.585621i \(0.800851\pi\)
\(444\) 0 0
\(445\) −22.4999 −1.06660
\(446\) 0 0
\(447\) −41.8192 −1.97798
\(448\) 0 0
\(449\) 9.06845 0.427967 0.213983 0.976837i \(-0.431356\pi\)
0.213983 + 0.976837i \(0.431356\pi\)
\(450\) 0 0
\(451\) 29.2183 1.37583
\(452\) 0 0
\(453\) −51.9056 −2.43874
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −23.7883 −1.11277 −0.556384 0.830925i \(-0.687812\pi\)
−0.556384 + 0.830925i \(0.687812\pi\)
\(458\) 0 0
\(459\) −10.9126 −0.509355
\(460\) 0 0
\(461\) −21.4966 −1.00119 −0.500597 0.865680i \(-0.666886\pi\)
−0.500597 + 0.865680i \(0.666886\pi\)
\(462\) 0 0
\(463\) −2.55322 −0.118658 −0.0593292 0.998238i \(-0.518896\pi\)
−0.0593292 + 0.998238i \(0.518896\pi\)
\(464\) 0 0
\(465\) 23.6718 1.09775
\(466\) 0 0
\(467\) −20.1634 −0.933053 −0.466526 0.884507i \(-0.654495\pi\)
−0.466526 + 0.884507i \(0.654495\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 54.7788 2.52407
\(472\) 0 0
\(473\) −6.07597 −0.279373
\(474\) 0 0
\(475\) −29.3794 −1.34802
\(476\) 0 0
\(477\) 65.4177 2.99527
\(478\) 0 0
\(479\) 17.4499 0.797305 0.398652 0.917102i \(-0.369478\pi\)
0.398652 + 0.917102i \(0.369478\pi\)
\(480\) 0 0
\(481\) 13.3736 0.609785
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 34.2529 1.55535
\(486\) 0 0
\(487\) 39.1143 1.77244 0.886219 0.463266i \(-0.153323\pi\)
0.886219 + 0.463266i \(0.153323\pi\)
\(488\) 0 0
\(489\) −50.3551 −2.27714
\(490\) 0 0
\(491\) 21.3590 0.963917 0.481958 0.876194i \(-0.339926\pi\)
0.481958 + 0.876194i \(0.339926\pi\)
\(492\) 0 0
\(493\) 3.39300 0.152813
\(494\) 0 0
\(495\) −108.911 −4.89517
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 31.4317 1.40708 0.703538 0.710657i \(-0.251602\pi\)
0.703538 + 0.710657i \(0.251602\pi\)
\(500\) 0 0
\(501\) −68.9884 −3.08217
\(502\) 0 0
\(503\) 20.1087 0.896603 0.448302 0.893882i \(-0.352029\pi\)
0.448302 + 0.893882i \(0.352029\pi\)
\(504\) 0 0
\(505\) −13.2252 −0.588512
\(506\) 0 0
\(507\) 41.2416 1.83160
\(508\) 0 0
\(509\) 17.0898 0.757490 0.378745 0.925501i \(-0.376356\pi\)
0.378745 + 0.925501i \(0.376356\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −57.3266 −2.53103
\(514\) 0 0
\(515\) −45.7728 −2.01699
\(516\) 0 0
\(517\) −24.0829 −1.05916
\(518\) 0 0
\(519\) 15.8557 0.695988
\(520\) 0 0
\(521\) −7.41261 −0.324752 −0.162376 0.986729i \(-0.551916\pi\)
−0.162376 + 0.986729i \(0.551916\pi\)
\(522\) 0 0
\(523\) −0.240819 −0.0105303 −0.00526515 0.999986i \(-0.501676\pi\)
−0.00526515 + 0.999986i \(0.501676\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.35553 −0.102608
\(528\) 0 0
\(529\) 25.6048 1.11325
\(530\) 0 0
\(531\) −85.5915 −3.71436
\(532\) 0 0
\(533\) −29.2899 −1.26869
\(534\) 0 0
\(535\) 49.0440 2.12036
\(536\) 0 0
\(537\) 17.5149 0.755824
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 26.9932 1.16053 0.580265 0.814428i \(-0.302949\pi\)
0.580265 + 0.814428i \(0.302949\pi\)
\(542\) 0 0
\(543\) −36.3428 −1.55962
\(544\) 0 0
\(545\) 8.14922 0.349074
\(546\) 0 0
\(547\) 20.0657 0.857947 0.428973 0.903317i \(-0.358875\pi\)
0.428973 + 0.903317i \(0.358875\pi\)
\(548\) 0 0
\(549\) 11.6396 0.496768
\(550\) 0 0
\(551\) 17.8243 0.759341
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 26.1787 1.11122
\(556\) 0 0
\(557\) 2.84986 0.120753 0.0603763 0.998176i \(-0.480770\pi\)
0.0603763 + 0.998176i \(0.480770\pi\)
\(558\) 0 0
\(559\) 6.09088 0.257617
\(560\) 0 0
\(561\) 15.8132 0.667636
\(562\) 0 0
\(563\) 29.4232 1.24004 0.620021 0.784586i \(-0.287124\pi\)
0.620021 + 0.784586i \(0.287124\pi\)
\(564\) 0 0
\(565\) −53.0792 −2.23306
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −35.4141 −1.48463 −0.742317 0.670048i \(-0.766273\pi\)
−0.742317 + 0.670048i \(0.766273\pi\)
\(570\) 0 0
\(571\) −25.5965 −1.07118 −0.535590 0.844478i \(-0.679911\pi\)
−0.535590 + 0.844478i \(0.679911\pi\)
\(572\) 0 0
\(573\) −2.28965 −0.0956514
\(574\) 0 0
\(575\) 38.9900 1.62599
\(576\) 0 0
\(577\) 39.7244 1.65375 0.826874 0.562388i \(-0.190117\pi\)
0.826874 + 0.562388i \(0.190117\pi\)
\(578\) 0 0
\(579\) −36.1607 −1.50279
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −51.2726 −2.12349
\(584\) 0 0
\(585\) 109.178 4.51395
\(586\) 0 0
\(587\) −0.00154296 −6.36846e−5 0 −3.18423e−5 1.00000i \(-0.500010\pi\)
−3.18423e−5 1.00000i \(0.500010\pi\)
\(588\) 0 0
\(589\) −12.3742 −0.509870
\(590\) 0 0
\(591\) −12.1626 −0.500301
\(592\) 0 0
\(593\) −22.3965 −0.919714 −0.459857 0.887993i \(-0.652099\pi\)
−0.459857 + 0.887993i \(0.652099\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −26.8096 −1.09724
\(598\) 0 0
\(599\) 10.8107 0.441713 0.220857 0.975306i \(-0.429115\pi\)
0.220857 + 0.975306i \(0.429115\pi\)
\(600\) 0 0
\(601\) 19.5231 0.796364 0.398182 0.917306i \(-0.369641\pi\)
0.398182 + 0.917306i \(0.369641\pi\)
\(602\) 0 0
\(603\) −73.0379 −2.97433
\(604\) 0 0
\(605\) 49.5604 2.01492
\(606\) 0 0
\(607\) 14.1715 0.575203 0.287601 0.957750i \(-0.407142\pi\)
0.287601 + 0.957750i \(0.407142\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.1420 0.976681
\(612\) 0 0
\(613\) −7.61930 −0.307741 −0.153870 0.988091i \(-0.549174\pi\)
−0.153870 + 0.988091i \(0.549174\pi\)
\(614\) 0 0
\(615\) −57.3346 −2.31195
\(616\) 0 0
\(617\) −4.52135 −0.182023 −0.0910113 0.995850i \(-0.529010\pi\)
−0.0910113 + 0.995850i \(0.529010\pi\)
\(618\) 0 0
\(619\) −27.8625 −1.11989 −0.559943 0.828531i \(-0.689177\pi\)
−0.559943 + 0.828531i \(0.689177\pi\)
\(620\) 0 0
\(621\) 76.0793 3.05296
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −21.6858 −0.867433
\(626\) 0 0
\(627\) 83.0711 3.31754
\(628\) 0 0
\(629\) −2.60498 −0.103868
\(630\) 0 0
\(631\) 33.1123 1.31818 0.659090 0.752064i \(-0.270942\pi\)
0.659090 + 0.752064i \(0.270942\pi\)
\(632\) 0 0
\(633\) −27.9513 −1.11097
\(634\) 0 0
\(635\) −11.4120 −0.452871
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 60.2361 2.38290
\(640\) 0 0
\(641\) −20.2853 −0.801221 −0.400610 0.916249i \(-0.631202\pi\)
−0.400610 + 0.916249i \(0.631202\pi\)
\(642\) 0 0
\(643\) 11.6066 0.457718 0.228859 0.973460i \(-0.426500\pi\)
0.228859 + 0.973460i \(0.426500\pi\)
\(644\) 0 0
\(645\) 11.9228 0.469459
\(646\) 0 0
\(647\) 17.4179 0.684770 0.342385 0.939560i \(-0.388765\pi\)
0.342385 + 0.939560i \(0.388765\pi\)
\(648\) 0 0
\(649\) 67.0843 2.63329
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −31.1391 −1.21857 −0.609284 0.792952i \(-0.708543\pi\)
−0.609284 + 0.792952i \(0.708543\pi\)
\(654\) 0 0
\(655\) 29.6705 1.15932
\(656\) 0 0
\(657\) 95.7958 3.73735
\(658\) 0 0
\(659\) −32.6601 −1.27226 −0.636128 0.771583i \(-0.719465\pi\)
−0.636128 + 0.771583i \(0.719465\pi\)
\(660\) 0 0
\(661\) −19.1904 −0.746420 −0.373210 0.927747i \(-0.621743\pi\)
−0.373210 + 0.927747i \(0.621743\pi\)
\(662\) 0 0
\(663\) −15.8520 −0.615643
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −23.6550 −0.915925
\(668\) 0 0
\(669\) −37.8766 −1.46439
\(670\) 0 0
\(671\) −9.12283 −0.352183
\(672\) 0 0
\(673\) 5.61344 0.216382 0.108191 0.994130i \(-0.465494\pi\)
0.108191 + 0.994130i \(0.465494\pi\)
\(674\) 0 0
\(675\) 61.0296 2.34903
\(676\) 0 0
\(677\) 37.4047 1.43758 0.718789 0.695228i \(-0.244697\pi\)
0.718789 + 0.695228i \(0.244697\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 52.9129 2.02763
\(682\) 0 0
\(683\) −22.0019 −0.841880 −0.420940 0.907089i \(-0.638300\pi\)
−0.420940 + 0.907089i \(0.638300\pi\)
\(684\) 0 0
\(685\) −32.6370 −1.24699
\(686\) 0 0
\(687\) 66.4596 2.53559
\(688\) 0 0
\(689\) 51.3984 1.95812
\(690\) 0 0
\(691\) −25.7800 −0.980717 −0.490359 0.871521i \(-0.663134\pi\)
−0.490359 + 0.871521i \(0.663134\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 32.9228 1.24883
\(696\) 0 0
\(697\) 5.70525 0.216102
\(698\) 0 0
\(699\) −59.3744 −2.24574
\(700\) 0 0
\(701\) 8.82347 0.333258 0.166629 0.986020i \(-0.446712\pi\)
0.166629 + 0.986020i \(0.446712\pi\)
\(702\) 0 0
\(703\) −13.6847 −0.516127
\(704\) 0 0
\(705\) 47.2576 1.77982
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −11.4925 −0.431609 −0.215805 0.976437i \(-0.569237\pi\)
−0.215805 + 0.976437i \(0.569237\pi\)
\(710\) 0 0
\(711\) 13.4474 0.504318
\(712\) 0 0
\(713\) 16.4221 0.615011
\(714\) 0 0
\(715\) −85.5708 −3.20017
\(716\) 0 0
\(717\) 25.3123 0.945305
\(718\) 0 0
\(719\) 9.74362 0.363376 0.181688 0.983356i \(-0.441844\pi\)
0.181688 + 0.983356i \(0.441844\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −78.4306 −2.91686
\(724\) 0 0
\(725\) −18.9757 −0.704739
\(726\) 0 0
\(727\) 22.2530 0.825316 0.412658 0.910886i \(-0.364600\pi\)
0.412658 + 0.910886i \(0.364600\pi\)
\(728\) 0 0
\(729\) −9.00142 −0.333386
\(730\) 0 0
\(731\) −1.18641 −0.0438810
\(732\) 0 0
\(733\) 16.2214 0.599150 0.299575 0.954073i \(-0.403155\pi\)
0.299575 + 0.954073i \(0.403155\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 57.2451 2.10865
\(738\) 0 0
\(739\) 27.6708 1.01789 0.508943 0.860800i \(-0.330036\pi\)
0.508943 + 0.860800i \(0.330036\pi\)
\(740\) 0 0
\(741\) −83.2750 −3.05918
\(742\) 0 0
\(743\) −16.5594 −0.607507 −0.303754 0.952751i \(-0.598240\pi\)
−0.303754 + 0.952751i \(0.598240\pi\)
\(744\) 0 0
\(745\) −44.0795 −1.61495
\(746\) 0 0
\(747\) −102.590 −3.75358
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −31.2453 −1.14016 −0.570078 0.821591i \(-0.693087\pi\)
−0.570078 + 0.821591i \(0.693087\pi\)
\(752\) 0 0
\(753\) 12.8399 0.467913
\(754\) 0 0
\(755\) −54.7109 −1.99113
\(756\) 0 0
\(757\) −11.7582 −0.427359 −0.213680 0.976904i \(-0.568545\pi\)
−0.213680 + 0.976904i \(0.568545\pi\)
\(758\) 0 0
\(759\) −110.245 −4.00165
\(760\) 0 0
\(761\) −4.17953 −0.151508 −0.0757540 0.997127i \(-0.524136\pi\)
−0.0757540 + 0.997127i \(0.524136\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −21.2662 −0.768883
\(766\) 0 0
\(767\) −67.2489 −2.42822
\(768\) 0 0
\(769\) 0.863383 0.0311344 0.0155672 0.999879i \(-0.495045\pi\)
0.0155672 + 0.999879i \(0.495045\pi\)
\(770\) 0 0
\(771\) 54.5417 1.96427
\(772\) 0 0
\(773\) 38.0952 1.37019 0.685094 0.728454i \(-0.259761\pi\)
0.685094 + 0.728454i \(0.259761\pi\)
\(774\) 0 0
\(775\) 13.1735 0.473207
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 29.9711 1.07383
\(780\) 0 0
\(781\) −47.2114 −1.68936
\(782\) 0 0
\(783\) −37.0263 −1.32321
\(784\) 0 0
\(785\) 57.7395 2.06081
\(786\) 0 0
\(787\) 48.9204 1.74383 0.871913 0.489661i \(-0.162880\pi\)
0.871913 + 0.489661i \(0.162880\pi\)
\(788\) 0 0
\(789\) −76.2534 −2.71469
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 9.14522 0.324756
\(794\) 0 0
\(795\) 100.612 3.56833
\(796\) 0 0
\(797\) 41.4404 1.46789 0.733946 0.679208i \(-0.237676\pi\)
0.733946 + 0.679208i \(0.237676\pi\)
\(798\) 0 0
\(799\) −4.70250 −0.166363
\(800\) 0 0
\(801\) −45.1720 −1.59607
\(802\) 0 0
\(803\) −75.0821 −2.64959
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 45.5463 1.60331
\(808\) 0 0
\(809\) 12.3338 0.433633 0.216816 0.976212i \(-0.430433\pi\)
0.216816 + 0.976212i \(0.430433\pi\)
\(810\) 0 0
\(811\) −6.92765 −0.243263 −0.121631 0.992575i \(-0.538813\pi\)
−0.121631 + 0.992575i \(0.538813\pi\)
\(812\) 0 0
\(813\) 24.7240 0.867109
\(814\) 0 0
\(815\) −53.0767 −1.85920
\(816\) 0 0
\(817\) −6.23253 −0.218049
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −6.22857 −0.217379 −0.108689 0.994076i \(-0.534665\pi\)
−0.108689 + 0.994076i \(0.534665\pi\)
\(822\) 0 0
\(823\) 25.2597 0.880497 0.440249 0.897876i \(-0.354890\pi\)
0.440249 + 0.897876i \(0.354890\pi\)
\(824\) 0 0
\(825\) −88.4371 −3.07899
\(826\) 0 0
\(827\) −21.2451 −0.738764 −0.369382 0.929278i \(-0.620431\pi\)
−0.369382 + 0.929278i \(0.620431\pi\)
\(828\) 0 0
\(829\) 15.9143 0.552725 0.276362 0.961053i \(-0.410871\pi\)
0.276362 + 0.961053i \(0.410871\pi\)
\(830\) 0 0
\(831\) −19.1843 −0.665497
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −72.7170 −2.51648
\(836\) 0 0
\(837\) 25.7049 0.888490
\(838\) 0 0
\(839\) −4.58809 −0.158398 −0.0791992 0.996859i \(-0.525236\pi\)
−0.0791992 + 0.996859i \(0.525236\pi\)
\(840\) 0 0
\(841\) −17.4876 −0.603020
\(842\) 0 0
\(843\) −34.0466 −1.17263
\(844\) 0 0
\(845\) 43.4706 1.49543
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −52.1725 −1.79056
\(850\) 0 0
\(851\) 18.1612 0.622558
\(852\) 0 0
\(853\) −40.8442 −1.39848 −0.699240 0.714887i \(-0.746478\pi\)
−0.699240 + 0.714887i \(0.746478\pi\)
\(854\) 0 0
\(855\) −111.717 −3.82065
\(856\) 0 0
\(857\) −0.0503896 −0.00172128 −0.000860638 1.00000i \(-0.500274\pi\)
−0.000860638 1.00000i \(0.500274\pi\)
\(858\) 0 0
\(859\) 14.7256 0.502431 0.251215 0.967931i \(-0.419170\pi\)
0.251215 + 0.967931i \(0.419170\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −25.1275 −0.855349 −0.427675 0.903933i \(-0.640667\pi\)
−0.427675 + 0.903933i \(0.640667\pi\)
\(864\) 0 0
\(865\) 16.7127 0.568247
\(866\) 0 0
\(867\) 3.08774 0.104865
\(868\) 0 0
\(869\) −10.5397 −0.357536
\(870\) 0 0
\(871\) −57.3856 −1.94444
\(872\) 0 0
\(873\) 68.7680 2.32744
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19.1059 0.645161 0.322581 0.946542i \(-0.395450\pi\)
0.322581 + 0.946542i \(0.395450\pi\)
\(878\) 0 0
\(879\) −69.1000 −2.33069
\(880\) 0 0
\(881\) 22.2649 0.750124 0.375062 0.927000i \(-0.377621\pi\)
0.375062 + 0.927000i \(0.377621\pi\)
\(882\) 0 0
\(883\) 20.2839 0.682607 0.341303 0.939953i \(-0.389132\pi\)
0.341303 + 0.939953i \(0.389132\pi\)
\(884\) 0 0
\(885\) −131.639 −4.42499
\(886\) 0 0
\(887\) −39.3402 −1.32092 −0.660458 0.750863i \(-0.729638\pi\)
−0.660458 + 0.750863i \(0.729638\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −72.1731 −2.41789
\(892\) 0 0
\(893\) −24.7035 −0.826670
\(894\) 0 0
\(895\) 18.4615 0.617101
\(896\) 0 0
\(897\) 110.516 3.69002
\(898\) 0 0
\(899\) −7.99230 −0.266558
\(900\) 0 0
\(901\) −10.0116 −0.333536
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −38.3070 −1.27337
\(906\) 0 0
\(907\) −17.2290 −0.572080 −0.286040 0.958218i \(-0.592339\pi\)
−0.286040 + 0.958218i \(0.592339\pi\)
\(908\) 0 0
\(909\) −26.5515 −0.880658
\(910\) 0 0
\(911\) −27.5456 −0.912628 −0.456314 0.889819i \(-0.650831\pi\)
−0.456314 + 0.889819i \(0.650831\pi\)
\(912\) 0 0
\(913\) 80.4073 2.66109
\(914\) 0 0
\(915\) 17.9016 0.591809
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 4.05113 0.133634 0.0668172 0.997765i \(-0.478716\pi\)
0.0668172 + 0.997765i \(0.478716\pi\)
\(920\) 0 0
\(921\) 6.20821 0.204568
\(922\) 0 0
\(923\) 47.3272 1.55780
\(924\) 0 0
\(925\) 14.5686 0.479014
\(926\) 0 0
\(927\) −91.8959 −3.01826
\(928\) 0 0
\(929\) −41.3767 −1.35753 −0.678763 0.734358i \(-0.737484\pi\)
−0.678763 + 0.734358i \(0.737484\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −0.164614 −0.00538922
\(934\) 0 0
\(935\) 16.6679 0.545099
\(936\) 0 0
\(937\) 9.85113 0.321823 0.160911 0.986969i \(-0.448557\pi\)
0.160911 + 0.986969i \(0.448557\pi\)
\(938\) 0 0
\(939\) 52.6059 1.71673
\(940\) 0 0
\(941\) −31.0786 −1.01313 −0.506567 0.862200i \(-0.669086\pi\)
−0.506567 + 0.862200i \(0.669086\pi\)
\(942\) 0 0
\(943\) −39.7753 −1.29526
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −38.8811 −1.26347 −0.631733 0.775186i \(-0.717656\pi\)
−0.631733 + 0.775186i \(0.717656\pi\)
\(948\) 0 0
\(949\) 75.2664 2.44325
\(950\) 0 0
\(951\) 91.3115 2.96098
\(952\) 0 0
\(953\) 56.4465 1.82848 0.914242 0.405170i \(-0.132788\pi\)
0.914242 + 0.405170i \(0.132788\pi\)
\(954\) 0 0
\(955\) −2.41340 −0.0780957
\(956\) 0 0
\(957\) 53.6543 1.73440
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −25.4515 −0.821016
\(962\) 0 0
\(963\) 98.4632 3.17293
\(964\) 0 0
\(965\) −38.1151 −1.22697
\(966\) 0 0
\(967\) −14.1217 −0.454122 −0.227061 0.973881i \(-0.572912\pi\)
−0.227061 + 0.973881i \(0.572912\pi\)
\(968\) 0 0
\(969\) 16.2207 0.521085
\(970\) 0 0
\(971\) 12.5180 0.401723 0.200861 0.979620i \(-0.435626\pi\)
0.200861 + 0.979620i \(0.435626\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 88.6541 2.83920
\(976\) 0 0
\(977\) −26.0496 −0.833402 −0.416701 0.909044i \(-0.636814\pi\)
−0.416701 + 0.909044i \(0.636814\pi\)
\(978\) 0 0
\(979\) 35.4046 1.13154
\(980\) 0 0
\(981\) 16.3608 0.522360
\(982\) 0 0
\(983\) −13.4621 −0.429374 −0.214687 0.976683i \(-0.568873\pi\)
−0.214687 + 0.976683i \(0.568873\pi\)
\(984\) 0 0
\(985\) −12.8199 −0.408476
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.27132 0.263013
\(990\) 0 0
\(991\) −20.8991 −0.663882 −0.331941 0.943300i \(-0.607703\pi\)
−0.331941 + 0.943300i \(0.607703\pi\)
\(992\) 0 0
\(993\) 77.3712 2.45530
\(994\) 0 0
\(995\) −28.2585 −0.895856
\(996\) 0 0
\(997\) 35.9728 1.13927 0.569636 0.821897i \(-0.307084\pi\)
0.569636 + 0.821897i \(0.307084\pi\)
\(998\) 0 0
\(999\) 28.4271 0.899393
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 3332.2.a.u.1.7 yes 8
7.6 odd 2 3332.2.a.t.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3332.2.a.t.1.2 8 7.6 odd 2
3332.2.a.u.1.7 yes 8 1.1 even 1 trivial