Properties

Label 6024.2.a.q.1.10
Level $6024$
Weight $2$
Character 6024.1
Self dual yes
Analytic conductor $48.102$
Analytic rank $0$
Dimension $18$
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] = [6024,2,Mod(1,6024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6024, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6024.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: \(48.1018821776\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} - 57 x^{16} + 51 x^{15} + 1328 x^{14} - 1116 x^{13} - 16275 x^{12} + 13699 x^{11} + 112394 x^{10} - 101250 x^{9} - 432956 x^{8} + 439806 x^{7} + 844117 x^{6} + \cdots + 44032 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(0.693705\) of defining polynomial
Character \(\chi\) \(=\) 6024.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.693705 q^{5} -3.39358 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.693705 q^{5} -3.39358 q^{7} +1.00000 q^{9} +5.07073 q^{11} +3.83146 q^{13} +0.693705 q^{15} +3.10131 q^{17} +6.00606 q^{19} -3.39358 q^{21} +1.28131 q^{23} -4.51877 q^{25} +1.00000 q^{27} +4.22100 q^{29} -5.03025 q^{31} +5.07073 q^{33} -2.35414 q^{35} -6.18338 q^{37} +3.83146 q^{39} +4.22552 q^{41} +5.19268 q^{43} +0.693705 q^{45} -4.33431 q^{47} +4.51638 q^{49} +3.10131 q^{51} -6.40817 q^{53} +3.51759 q^{55} +6.00606 q^{57} +10.0793 q^{59} -9.50602 q^{61} -3.39358 q^{63} +2.65790 q^{65} -7.04614 q^{67} +1.28131 q^{69} +2.91100 q^{71} +2.05497 q^{73} -4.51877 q^{75} -17.2079 q^{77} +10.6914 q^{79} +1.00000 q^{81} +7.04092 q^{83} +2.15140 q^{85} +4.22100 q^{87} +1.33632 q^{89} -13.0024 q^{91} -5.03025 q^{93} +4.16643 q^{95} -8.67805 q^{97} +5.07073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9} + 8 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} + 21 q^{19} + 7 q^{21} - 2 q^{23} + 25 q^{25} + 18 q^{27} + 5 q^{29} + 23 q^{31} + 8 q^{33} + 17 q^{35} + 15 q^{37} + 2 q^{39} + 20 q^{41} + 37 q^{43} + q^{45} - q^{47} + 33 q^{49} + 2 q^{51} + 2 q^{53} + 26 q^{55} + 21 q^{57} + 24 q^{59} + 20 q^{61} + 7 q^{63} + 26 q^{65} + 49 q^{67} - 2 q^{69} + 15 q^{71} + 15 q^{73} + 25 q^{75} + 6 q^{77} + 23 q^{79} + 18 q^{81} + 38 q^{83} + 21 q^{85} + 5 q^{87} + 31 q^{89} + 48 q^{91} + 23 q^{93} + 13 q^{95} + 25 q^{97} + 8 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0.693705 0.310234 0.155117 0.987896i \(-0.450425\pi\)
0.155117 + 0.987896i \(0.450425\pi\)
\(6\) 0 0
\(7\) −3.39358 −1.28265 −0.641326 0.767268i \(-0.721615\pi\)
−0.641326 + 0.767268i \(0.721615\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.07073 1.52888 0.764441 0.644693i \(-0.223015\pi\)
0.764441 + 0.644693i \(0.223015\pi\)
\(12\) 0 0
\(13\) 3.83146 1.06266 0.531328 0.847166i \(-0.321693\pi\)
0.531328 + 0.847166i \(0.321693\pi\)
\(14\) 0 0
\(15\) 0.693705 0.179114
\(16\) 0 0
\(17\) 3.10131 0.752179 0.376089 0.926583i \(-0.377269\pi\)
0.376089 + 0.926583i \(0.377269\pi\)
\(18\) 0 0
\(19\) 6.00606 1.37788 0.688942 0.724817i \(-0.258076\pi\)
0.688942 + 0.724817i \(0.258076\pi\)
\(20\) 0 0
\(21\) −3.39358 −0.740540
\(22\) 0 0
\(23\) 1.28131 0.267172 0.133586 0.991037i \(-0.457351\pi\)
0.133586 + 0.991037i \(0.457351\pi\)
\(24\) 0 0
\(25\) −4.51877 −0.903755
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.22100 0.783821 0.391910 0.920003i \(-0.371814\pi\)
0.391910 + 0.920003i \(0.371814\pi\)
\(30\) 0 0
\(31\) −5.03025 −0.903459 −0.451729 0.892155i \(-0.649193\pi\)
−0.451729 + 0.892155i \(0.649193\pi\)
\(32\) 0 0
\(33\) 5.07073 0.882701
\(34\) 0 0
\(35\) −2.35414 −0.397923
\(36\) 0 0
\(37\) −6.18338 −1.01654 −0.508271 0.861197i \(-0.669715\pi\)
−0.508271 + 0.861197i \(0.669715\pi\)
\(38\) 0 0
\(39\) 3.83146 0.613525
\(40\) 0 0
\(41\) 4.22552 0.659916 0.329958 0.943996i \(-0.392965\pi\)
0.329958 + 0.943996i \(0.392965\pi\)
\(42\) 0 0
\(43\) 5.19268 0.791877 0.395938 0.918277i \(-0.370419\pi\)
0.395938 + 0.918277i \(0.370419\pi\)
\(44\) 0 0
\(45\) 0.693705 0.103411
\(46\) 0 0
\(47\) −4.33431 −0.632224 −0.316112 0.948722i \(-0.602378\pi\)
−0.316112 + 0.948722i \(0.602378\pi\)
\(48\) 0 0
\(49\) 4.51638 0.645197
\(50\) 0 0
\(51\) 3.10131 0.434271
\(52\) 0 0
\(53\) −6.40817 −0.880230 −0.440115 0.897942i \(-0.645062\pi\)
−0.440115 + 0.897942i \(0.645062\pi\)
\(54\) 0 0
\(55\) 3.51759 0.474312
\(56\) 0 0
\(57\) 6.00606 0.795522
\(58\) 0 0
\(59\) 10.0793 1.31221 0.656105 0.754669i \(-0.272203\pi\)
0.656105 + 0.754669i \(0.272203\pi\)
\(60\) 0 0
\(61\) −9.50602 −1.21712 −0.608560 0.793508i \(-0.708253\pi\)
−0.608560 + 0.793508i \(0.708253\pi\)
\(62\) 0 0
\(63\) −3.39358 −0.427551
\(64\) 0 0
\(65\) 2.65790 0.329672
\(66\) 0 0
\(67\) −7.04614 −0.860823 −0.430412 0.902633i \(-0.641632\pi\)
−0.430412 + 0.902633i \(0.641632\pi\)
\(68\) 0 0
\(69\) 1.28131 0.154252
\(70\) 0 0
\(71\) 2.91100 0.345472 0.172736 0.984968i \(-0.444739\pi\)
0.172736 + 0.984968i \(0.444739\pi\)
\(72\) 0 0
\(73\) 2.05497 0.240516 0.120258 0.992743i \(-0.461628\pi\)
0.120258 + 0.992743i \(0.461628\pi\)
\(74\) 0 0
\(75\) −4.51877 −0.521783
\(76\) 0 0
\(77\) −17.2079 −1.96102
\(78\) 0 0
\(79\) 10.6914 1.20288 0.601441 0.798917i \(-0.294594\pi\)
0.601441 + 0.798917i \(0.294594\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.04092 0.772842 0.386421 0.922323i \(-0.373711\pi\)
0.386421 + 0.922323i \(0.373711\pi\)
\(84\) 0 0
\(85\) 2.15140 0.233352
\(86\) 0 0
\(87\) 4.22100 0.452539
\(88\) 0 0
\(89\) 1.33632 0.141650 0.0708251 0.997489i \(-0.477437\pi\)
0.0708251 + 0.997489i \(0.477437\pi\)
\(90\) 0 0
\(91\) −13.0024 −1.36302
\(92\) 0 0
\(93\) −5.03025 −0.521612
\(94\) 0 0
\(95\) 4.16643 0.427467
\(96\) 0 0
\(97\) −8.67805 −0.881122 −0.440561 0.897723i \(-0.645220\pi\)
−0.440561 + 0.897723i \(0.645220\pi\)
\(98\) 0 0
\(99\) 5.07073 0.509628
\(100\) 0 0
\(101\) 15.1912 1.51158 0.755791 0.654813i \(-0.227253\pi\)
0.755791 + 0.654813i \(0.227253\pi\)
\(102\) 0 0
\(103\) −9.67044 −0.952856 −0.476428 0.879213i \(-0.658069\pi\)
−0.476428 + 0.879213i \(0.658069\pi\)
\(104\) 0 0
\(105\) −2.35414 −0.229741
\(106\) 0 0
\(107\) −7.61924 −0.736580 −0.368290 0.929711i \(-0.620057\pi\)
−0.368290 + 0.929711i \(0.620057\pi\)
\(108\) 0 0
\(109\) −0.715961 −0.0685766 −0.0342883 0.999412i \(-0.510916\pi\)
−0.0342883 + 0.999412i \(0.510916\pi\)
\(110\) 0 0
\(111\) −6.18338 −0.586901
\(112\) 0 0
\(113\) −2.55725 −0.240566 −0.120283 0.992740i \(-0.538380\pi\)
−0.120283 + 0.992740i \(0.538380\pi\)
\(114\) 0 0
\(115\) 0.888851 0.0828858
\(116\) 0 0
\(117\) 3.83146 0.354219
\(118\) 0 0
\(119\) −10.5246 −0.964784
\(120\) 0 0
\(121\) 14.7123 1.33748
\(122\) 0 0
\(123\) 4.22552 0.381003
\(124\) 0 0
\(125\) −6.60322 −0.590610
\(126\) 0 0
\(127\) 15.5687 1.38150 0.690750 0.723094i \(-0.257280\pi\)
0.690750 + 0.723094i \(0.257280\pi\)
\(128\) 0 0
\(129\) 5.19268 0.457190
\(130\) 0 0
\(131\) −8.68807 −0.759081 −0.379540 0.925175i \(-0.623918\pi\)
−0.379540 + 0.925175i \(0.623918\pi\)
\(132\) 0 0
\(133\) −20.3820 −1.76735
\(134\) 0 0
\(135\) 0.693705 0.0597046
\(136\) 0 0
\(137\) 11.6471 0.995082 0.497541 0.867440i \(-0.334236\pi\)
0.497541 + 0.867440i \(0.334236\pi\)
\(138\) 0 0
\(139\) 20.8127 1.76531 0.882654 0.470023i \(-0.155754\pi\)
0.882654 + 0.470023i \(0.155754\pi\)
\(140\) 0 0
\(141\) −4.33431 −0.365015
\(142\) 0 0
\(143\) 19.4283 1.62468
\(144\) 0 0
\(145\) 2.92813 0.243168
\(146\) 0 0
\(147\) 4.51638 0.372505
\(148\) 0 0
\(149\) −7.23090 −0.592378 −0.296189 0.955129i \(-0.595716\pi\)
−0.296189 + 0.955129i \(0.595716\pi\)
\(150\) 0 0
\(151\) 5.71992 0.465480 0.232740 0.972539i \(-0.425231\pi\)
0.232740 + 0.972539i \(0.425231\pi\)
\(152\) 0 0
\(153\) 3.10131 0.250726
\(154\) 0 0
\(155\) −3.48951 −0.280284
\(156\) 0 0
\(157\) −23.9148 −1.90861 −0.954303 0.298840i \(-0.903400\pi\)
−0.954303 + 0.298840i \(0.903400\pi\)
\(158\) 0 0
\(159\) −6.40817 −0.508201
\(160\) 0 0
\(161\) −4.34823 −0.342689
\(162\) 0 0
\(163\) 3.25523 0.254969 0.127485 0.991841i \(-0.459310\pi\)
0.127485 + 0.991841i \(0.459310\pi\)
\(164\) 0 0
\(165\) 3.51759 0.273844
\(166\) 0 0
\(167\) 19.6189 1.51815 0.759077 0.651000i \(-0.225650\pi\)
0.759077 + 0.651000i \(0.225650\pi\)
\(168\) 0 0
\(169\) 1.68010 0.129239
\(170\) 0 0
\(171\) 6.00606 0.459295
\(172\) 0 0
\(173\) 12.4915 0.949709 0.474855 0.880064i \(-0.342501\pi\)
0.474855 + 0.880064i \(0.342501\pi\)
\(174\) 0 0
\(175\) 15.3348 1.15920
\(176\) 0 0
\(177\) 10.0793 0.757605
\(178\) 0 0
\(179\) 17.5394 1.31096 0.655479 0.755213i \(-0.272467\pi\)
0.655479 + 0.755213i \(0.272467\pi\)
\(180\) 0 0
\(181\) 2.88595 0.214511 0.107255 0.994232i \(-0.465794\pi\)
0.107255 + 0.994232i \(0.465794\pi\)
\(182\) 0 0
\(183\) −9.50602 −0.702705
\(184\) 0 0
\(185\) −4.28944 −0.315366
\(186\) 0 0
\(187\) 15.7259 1.14999
\(188\) 0 0
\(189\) −3.39358 −0.246847
\(190\) 0 0
\(191\) −3.22169 −0.233113 −0.116557 0.993184i \(-0.537186\pi\)
−0.116557 + 0.993184i \(0.537186\pi\)
\(192\) 0 0
\(193\) 12.2520 0.881918 0.440959 0.897527i \(-0.354638\pi\)
0.440959 + 0.897527i \(0.354638\pi\)
\(194\) 0 0
\(195\) 2.65790 0.190336
\(196\) 0 0
\(197\) −7.06249 −0.503182 −0.251591 0.967834i \(-0.580954\pi\)
−0.251591 + 0.967834i \(0.580954\pi\)
\(198\) 0 0
\(199\) 19.8918 1.41009 0.705046 0.709162i \(-0.250926\pi\)
0.705046 + 0.709162i \(0.250926\pi\)
\(200\) 0 0
\(201\) −7.04614 −0.496996
\(202\) 0 0
\(203\) −14.3243 −1.00537
\(204\) 0 0
\(205\) 2.93127 0.204728
\(206\) 0 0
\(207\) 1.28131 0.0890573
\(208\) 0 0
\(209\) 30.4551 2.10662
\(210\) 0 0
\(211\) 7.40796 0.509985 0.254992 0.966943i \(-0.417927\pi\)
0.254992 + 0.966943i \(0.417927\pi\)
\(212\) 0 0
\(213\) 2.91100 0.199458
\(214\) 0 0
\(215\) 3.60219 0.245667
\(216\) 0 0
\(217\) 17.0705 1.15882
\(218\) 0 0
\(219\) 2.05497 0.138862
\(220\) 0 0
\(221\) 11.8826 0.799308
\(222\) 0 0
\(223\) 0.474828 0.0317969 0.0158984 0.999874i \(-0.494939\pi\)
0.0158984 + 0.999874i \(0.494939\pi\)
\(224\) 0 0
\(225\) −4.51877 −0.301252
\(226\) 0 0
\(227\) −16.8443 −1.11799 −0.558996 0.829170i \(-0.688813\pi\)
−0.558996 + 0.829170i \(0.688813\pi\)
\(228\) 0 0
\(229\) 2.13575 0.141134 0.0705670 0.997507i \(-0.477519\pi\)
0.0705670 + 0.997507i \(0.477519\pi\)
\(230\) 0 0
\(231\) −17.2079 −1.13220
\(232\) 0 0
\(233\) 22.3369 1.46334 0.731670 0.681659i \(-0.238741\pi\)
0.731670 + 0.681659i \(0.238741\pi\)
\(234\) 0 0
\(235\) −3.00673 −0.196138
\(236\) 0 0
\(237\) 10.6914 0.694484
\(238\) 0 0
\(239\) 16.3907 1.06023 0.530114 0.847926i \(-0.322149\pi\)
0.530114 + 0.847926i \(0.322149\pi\)
\(240\) 0 0
\(241\) −14.1952 −0.914396 −0.457198 0.889365i \(-0.651147\pi\)
−0.457198 + 0.889365i \(0.651147\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.13303 0.200162
\(246\) 0 0
\(247\) 23.0120 1.46422
\(248\) 0 0
\(249\) 7.04092 0.446200
\(250\) 0 0
\(251\) 1.00000 0.0631194
\(252\) 0 0
\(253\) 6.49718 0.408474
\(254\) 0 0
\(255\) 2.15140 0.134726
\(256\) 0 0
\(257\) −11.2880 −0.704127 −0.352063 0.935976i \(-0.614520\pi\)
−0.352063 + 0.935976i \(0.614520\pi\)
\(258\) 0 0
\(259\) 20.9838 1.30387
\(260\) 0 0
\(261\) 4.22100 0.261274
\(262\) 0 0
\(263\) −15.0221 −0.926302 −0.463151 0.886279i \(-0.653281\pi\)
−0.463151 + 0.886279i \(0.653281\pi\)
\(264\) 0 0
\(265\) −4.44538 −0.273077
\(266\) 0 0
\(267\) 1.33632 0.0817817
\(268\) 0 0
\(269\) 13.2636 0.808694 0.404347 0.914606i \(-0.367499\pi\)
0.404347 + 0.914606i \(0.367499\pi\)
\(270\) 0 0
\(271\) 13.2457 0.804618 0.402309 0.915504i \(-0.368208\pi\)
0.402309 + 0.915504i \(0.368208\pi\)
\(272\) 0 0
\(273\) −13.0024 −0.786939
\(274\) 0 0
\(275\) −22.9135 −1.38173
\(276\) 0 0
\(277\) −15.4428 −0.927868 −0.463934 0.885870i \(-0.653563\pi\)
−0.463934 + 0.885870i \(0.653563\pi\)
\(278\) 0 0
\(279\) −5.03025 −0.301153
\(280\) 0 0
\(281\) 14.0074 0.835612 0.417806 0.908536i \(-0.362799\pi\)
0.417806 + 0.908536i \(0.362799\pi\)
\(282\) 0 0
\(283\) −19.8561 −1.18033 −0.590163 0.807284i \(-0.700936\pi\)
−0.590163 + 0.807284i \(0.700936\pi\)
\(284\) 0 0
\(285\) 4.16643 0.246798
\(286\) 0 0
\(287\) −14.3397 −0.846443
\(288\) 0 0
\(289\) −7.38186 −0.434227
\(290\) 0 0
\(291\) −8.67805 −0.508716
\(292\) 0 0
\(293\) 24.3819 1.42441 0.712204 0.701973i \(-0.247697\pi\)
0.712204 + 0.701973i \(0.247697\pi\)
\(294\) 0 0
\(295\) 6.99204 0.407093
\(296\) 0 0
\(297\) 5.07073 0.294234
\(298\) 0 0
\(299\) 4.90929 0.283912
\(300\) 0 0
\(301\) −17.6218 −1.01570
\(302\) 0 0
\(303\) 15.1912 0.872712
\(304\) 0 0
\(305\) −6.59437 −0.377592
\(306\) 0 0
\(307\) −9.02581 −0.515130 −0.257565 0.966261i \(-0.582920\pi\)
−0.257565 + 0.966261i \(0.582920\pi\)
\(308\) 0 0
\(309\) −9.67044 −0.550132
\(310\) 0 0
\(311\) 3.83059 0.217213 0.108606 0.994085i \(-0.465361\pi\)
0.108606 + 0.994085i \(0.465361\pi\)
\(312\) 0 0
\(313\) 10.6014 0.599227 0.299613 0.954061i \(-0.403142\pi\)
0.299613 + 0.954061i \(0.403142\pi\)
\(314\) 0 0
\(315\) −2.35414 −0.132641
\(316\) 0 0
\(317\) 7.40231 0.415755 0.207878 0.978155i \(-0.433344\pi\)
0.207878 + 0.978155i \(0.433344\pi\)
\(318\) 0 0
\(319\) 21.4036 1.19837
\(320\) 0 0
\(321\) −7.61924 −0.425264
\(322\) 0 0
\(323\) 18.6267 1.03642
\(324\) 0 0
\(325\) −17.3135 −0.960381
\(326\) 0 0
\(327\) −0.715961 −0.0395927
\(328\) 0 0
\(329\) 14.7088 0.810924
\(330\) 0 0
\(331\) −3.57955 −0.196750 −0.0983749 0.995149i \(-0.531364\pi\)
−0.0983749 + 0.995149i \(0.531364\pi\)
\(332\) 0 0
\(333\) −6.18338 −0.338847
\(334\) 0 0
\(335\) −4.88794 −0.267057
\(336\) 0 0
\(337\) 1.48567 0.0809297 0.0404649 0.999181i \(-0.487116\pi\)
0.0404649 + 0.999181i \(0.487116\pi\)
\(338\) 0 0
\(339\) −2.55725 −0.138891
\(340\) 0 0
\(341\) −25.5070 −1.38128
\(342\) 0 0
\(343\) 8.42836 0.455088
\(344\) 0 0
\(345\) 0.888851 0.0478541
\(346\) 0 0
\(347\) 1.06849 0.0573595 0.0286797 0.999589i \(-0.490870\pi\)
0.0286797 + 0.999589i \(0.490870\pi\)
\(348\) 0 0
\(349\) 24.5151 1.31226 0.656131 0.754647i \(-0.272192\pi\)
0.656131 + 0.754647i \(0.272192\pi\)
\(350\) 0 0
\(351\) 3.83146 0.204508
\(352\) 0 0
\(353\) 5.78487 0.307897 0.153949 0.988079i \(-0.450801\pi\)
0.153949 + 0.988079i \(0.450801\pi\)
\(354\) 0 0
\(355\) 2.01937 0.107177
\(356\) 0 0
\(357\) −10.5246 −0.557018
\(358\) 0 0
\(359\) −26.3651 −1.39150 −0.695749 0.718285i \(-0.744927\pi\)
−0.695749 + 0.718285i \(0.744927\pi\)
\(360\) 0 0
\(361\) 17.0727 0.898563
\(362\) 0 0
\(363\) 14.7123 0.772195
\(364\) 0 0
\(365\) 1.42554 0.0746161
\(366\) 0 0
\(367\) −19.4810 −1.01690 −0.508450 0.861091i \(-0.669781\pi\)
−0.508450 + 0.861091i \(0.669781\pi\)
\(368\) 0 0
\(369\) 4.22552 0.219972
\(370\) 0 0
\(371\) 21.7466 1.12903
\(372\) 0 0
\(373\) 1.24977 0.0647105 0.0323552 0.999476i \(-0.489699\pi\)
0.0323552 + 0.999476i \(0.489699\pi\)
\(374\) 0 0
\(375\) −6.60322 −0.340989
\(376\) 0 0
\(377\) 16.1726 0.832932
\(378\) 0 0
\(379\) −18.7550 −0.963379 −0.481690 0.876342i \(-0.659977\pi\)
−0.481690 + 0.876342i \(0.659977\pi\)
\(380\) 0 0
\(381\) 15.5687 0.797610
\(382\) 0 0
\(383\) −37.7851 −1.93073 −0.965365 0.260904i \(-0.915979\pi\)
−0.965365 + 0.260904i \(0.915979\pi\)
\(384\) 0 0
\(385\) −11.9372 −0.608377
\(386\) 0 0
\(387\) 5.19268 0.263959
\(388\) 0 0
\(389\) −17.0969 −0.866848 −0.433424 0.901190i \(-0.642695\pi\)
−0.433424 + 0.901190i \(0.642695\pi\)
\(390\) 0 0
\(391\) 3.97375 0.200961
\(392\) 0 0
\(393\) −8.68807 −0.438255
\(394\) 0 0
\(395\) 7.41670 0.373175
\(396\) 0 0
\(397\) 7.16723 0.359713 0.179856 0.983693i \(-0.442437\pi\)
0.179856 + 0.983693i \(0.442437\pi\)
\(398\) 0 0
\(399\) −20.3820 −1.02038
\(400\) 0 0
\(401\) −38.1232 −1.90378 −0.951891 0.306437i \(-0.900863\pi\)
−0.951891 + 0.306437i \(0.900863\pi\)
\(402\) 0 0
\(403\) −19.2732 −0.960066
\(404\) 0 0
\(405\) 0.693705 0.0344705
\(406\) 0 0
\(407\) −31.3543 −1.55417
\(408\) 0 0
\(409\) 33.8724 1.67488 0.837441 0.546528i \(-0.184051\pi\)
0.837441 + 0.546528i \(0.184051\pi\)
\(410\) 0 0
\(411\) 11.6471 0.574511
\(412\) 0 0
\(413\) −34.2048 −1.68311
\(414\) 0 0
\(415\) 4.88432 0.239762
\(416\) 0 0
\(417\) 20.8127 1.01920
\(418\) 0 0
\(419\) −11.2717 −0.550660 −0.275330 0.961350i \(-0.588787\pi\)
−0.275330 + 0.961350i \(0.588787\pi\)
\(420\) 0 0
\(421\) −34.3100 −1.67217 −0.836085 0.548601i \(-0.815161\pi\)
−0.836085 + 0.548601i \(0.815161\pi\)
\(422\) 0 0
\(423\) −4.33431 −0.210741
\(424\) 0 0
\(425\) −14.0141 −0.679785
\(426\) 0 0
\(427\) 32.2594 1.56114
\(428\) 0 0
\(429\) 19.4283 0.938008
\(430\) 0 0
\(431\) 25.8027 1.24287 0.621436 0.783465i \(-0.286550\pi\)
0.621436 + 0.783465i \(0.286550\pi\)
\(432\) 0 0
\(433\) −13.4503 −0.646381 −0.323190 0.946334i \(-0.604755\pi\)
−0.323190 + 0.946334i \(0.604755\pi\)
\(434\) 0 0
\(435\) 2.92813 0.140393
\(436\) 0 0
\(437\) 7.69562 0.368132
\(438\) 0 0
\(439\) −29.7102 −1.41799 −0.708995 0.705214i \(-0.750851\pi\)
−0.708995 + 0.705214i \(0.750851\pi\)
\(440\) 0 0
\(441\) 4.51638 0.215066
\(442\) 0 0
\(443\) 11.3143 0.537560 0.268780 0.963202i \(-0.413380\pi\)
0.268780 + 0.963202i \(0.413380\pi\)
\(444\) 0 0
\(445\) 0.927015 0.0439447
\(446\) 0 0
\(447\) −7.23090 −0.342010
\(448\) 0 0
\(449\) −38.6462 −1.82383 −0.911913 0.410384i \(-0.865395\pi\)
−0.911913 + 0.410384i \(0.865395\pi\)
\(450\) 0 0
\(451\) 21.4265 1.00893
\(452\) 0 0
\(453\) 5.71992 0.268745
\(454\) 0 0
\(455\) −9.01980 −0.422855
\(456\) 0 0
\(457\) 17.6469 0.825486 0.412743 0.910847i \(-0.364571\pi\)
0.412743 + 0.910847i \(0.364571\pi\)
\(458\) 0 0
\(459\) 3.10131 0.144757
\(460\) 0 0
\(461\) −36.4542 −1.69784 −0.848921 0.528519i \(-0.822747\pi\)
−0.848921 + 0.528519i \(0.822747\pi\)
\(462\) 0 0
\(463\) 24.2739 1.12811 0.564053 0.825739i \(-0.309241\pi\)
0.564053 + 0.825739i \(0.309241\pi\)
\(464\) 0 0
\(465\) −3.48951 −0.161822
\(466\) 0 0
\(467\) −0.187624 −0.00868221 −0.00434110 0.999991i \(-0.501382\pi\)
−0.00434110 + 0.999991i \(0.501382\pi\)
\(468\) 0 0
\(469\) 23.9116 1.10414
\(470\) 0 0
\(471\) −23.9148 −1.10193
\(472\) 0 0
\(473\) 26.3307 1.21069
\(474\) 0 0
\(475\) −27.1400 −1.24527
\(476\) 0 0
\(477\) −6.40817 −0.293410
\(478\) 0 0
\(479\) −33.1735 −1.51573 −0.757867 0.652409i \(-0.773759\pi\)
−0.757867 + 0.652409i \(0.773759\pi\)
\(480\) 0 0
\(481\) −23.6914 −1.08023
\(482\) 0 0
\(483\) −4.34823 −0.197851
\(484\) 0 0
\(485\) −6.02000 −0.273354
\(486\) 0 0
\(487\) 42.5776 1.92938 0.964688 0.263396i \(-0.0848427\pi\)
0.964688 + 0.263396i \(0.0848427\pi\)
\(488\) 0 0
\(489\) 3.25523 0.147207
\(490\) 0 0
\(491\) 13.2273 0.596942 0.298471 0.954419i \(-0.403523\pi\)
0.298471 + 0.954419i \(0.403523\pi\)
\(492\) 0 0
\(493\) 13.0907 0.589574
\(494\) 0 0
\(495\) 3.51759 0.158104
\(496\) 0 0
\(497\) −9.87870 −0.443120
\(498\) 0 0
\(499\) 17.0630 0.763846 0.381923 0.924194i \(-0.375262\pi\)
0.381923 + 0.924194i \(0.375262\pi\)
\(500\) 0 0
\(501\) 19.6189 0.876507
\(502\) 0 0
\(503\) −2.53644 −0.113094 −0.0565471 0.998400i \(-0.518009\pi\)
−0.0565471 + 0.998400i \(0.518009\pi\)
\(504\) 0 0
\(505\) 10.5382 0.468944
\(506\) 0 0
\(507\) 1.68010 0.0746160
\(508\) 0 0
\(509\) −40.8897 −1.81240 −0.906201 0.422847i \(-0.861031\pi\)
−0.906201 + 0.422847i \(0.861031\pi\)
\(510\) 0 0
\(511\) −6.97369 −0.308498
\(512\) 0 0
\(513\) 6.00606 0.265174
\(514\) 0 0
\(515\) −6.70843 −0.295609
\(516\) 0 0
\(517\) −21.9781 −0.966597
\(518\) 0 0
\(519\) 12.4915 0.548315
\(520\) 0 0
\(521\) −25.2017 −1.10411 −0.552054 0.833808i \(-0.686156\pi\)
−0.552054 + 0.833808i \(0.686156\pi\)
\(522\) 0 0
\(523\) −6.15204 −0.269010 −0.134505 0.990913i \(-0.542944\pi\)
−0.134505 + 0.990913i \(0.542944\pi\)
\(524\) 0 0
\(525\) 15.3348 0.669266
\(526\) 0 0
\(527\) −15.6004 −0.679563
\(528\) 0 0
\(529\) −21.3582 −0.928619
\(530\) 0 0
\(531\) 10.0793 0.437404
\(532\) 0 0
\(533\) 16.1899 0.701264
\(534\) 0 0
\(535\) −5.28550 −0.228512
\(536\) 0 0
\(537\) 17.5394 0.756882
\(538\) 0 0
\(539\) 22.9014 0.986431
\(540\) 0 0
\(541\) −5.50258 −0.236574 −0.118287 0.992979i \(-0.537740\pi\)
−0.118287 + 0.992979i \(0.537740\pi\)
\(542\) 0 0
\(543\) 2.88595 0.123848
\(544\) 0 0
\(545\) −0.496665 −0.0212748
\(546\) 0 0
\(547\) 22.2024 0.949306 0.474653 0.880173i \(-0.342573\pi\)
0.474653 + 0.880173i \(0.342573\pi\)
\(548\) 0 0
\(549\) −9.50602 −0.405707
\(550\) 0 0
\(551\) 25.3516 1.08001
\(552\) 0 0
\(553\) −36.2823 −1.54288
\(554\) 0 0
\(555\) −4.28944 −0.182077
\(556\) 0 0
\(557\) −41.5670 −1.76125 −0.880626 0.473813i \(-0.842877\pi\)
−0.880626 + 0.473813i \(0.842877\pi\)
\(558\) 0 0
\(559\) 19.8956 0.841493
\(560\) 0 0
\(561\) 15.7259 0.663949
\(562\) 0 0
\(563\) 17.6524 0.743959 0.371979 0.928241i \(-0.378679\pi\)
0.371979 + 0.928241i \(0.378679\pi\)
\(564\) 0 0
\(565\) −1.77398 −0.0746318
\(566\) 0 0
\(567\) −3.39358 −0.142517
\(568\) 0 0
\(569\) −33.3115 −1.39649 −0.698246 0.715858i \(-0.746036\pi\)
−0.698246 + 0.715858i \(0.746036\pi\)
\(570\) 0 0
\(571\) 44.4041 1.85825 0.929127 0.369760i \(-0.120560\pi\)
0.929127 + 0.369760i \(0.120560\pi\)
\(572\) 0 0
\(573\) −3.22169 −0.134588
\(574\) 0 0
\(575\) −5.78995 −0.241458
\(576\) 0 0
\(577\) 27.7325 1.15452 0.577260 0.816560i \(-0.304122\pi\)
0.577260 + 0.816560i \(0.304122\pi\)
\(578\) 0 0
\(579\) 12.2520 0.509176
\(580\) 0 0
\(581\) −23.8939 −0.991287
\(582\) 0 0
\(583\) −32.4941 −1.34577
\(584\) 0 0
\(585\) 2.65790 0.109891
\(586\) 0 0
\(587\) −17.1948 −0.709706 −0.354853 0.934922i \(-0.615469\pi\)
−0.354853 + 0.934922i \(0.615469\pi\)
\(588\) 0 0
\(589\) −30.2119 −1.24486
\(590\) 0 0
\(591\) −7.06249 −0.290512
\(592\) 0 0
\(593\) −21.2007 −0.870610 −0.435305 0.900283i \(-0.643359\pi\)
−0.435305 + 0.900283i \(0.643359\pi\)
\(594\) 0 0
\(595\) −7.30093 −0.299309
\(596\) 0 0
\(597\) 19.8918 0.814117
\(598\) 0 0
\(599\) 27.1790 1.11050 0.555252 0.831682i \(-0.312622\pi\)
0.555252 + 0.831682i \(0.312622\pi\)
\(600\) 0 0
\(601\) −10.6950 −0.436259 −0.218130 0.975920i \(-0.569996\pi\)
−0.218130 + 0.975920i \(0.569996\pi\)
\(602\) 0 0
\(603\) −7.04614 −0.286941
\(604\) 0 0
\(605\) 10.2060 0.414932
\(606\) 0 0
\(607\) 28.3530 1.15081 0.575407 0.817867i \(-0.304844\pi\)
0.575407 + 0.817867i \(0.304844\pi\)
\(608\) 0 0
\(609\) −14.3243 −0.580451
\(610\) 0 0
\(611\) −16.6068 −0.671837
\(612\) 0 0
\(613\) −3.75349 −0.151602 −0.0758010 0.997123i \(-0.524151\pi\)
−0.0758010 + 0.997123i \(0.524151\pi\)
\(614\) 0 0
\(615\) 2.93127 0.118200
\(616\) 0 0
\(617\) 40.9051 1.64678 0.823389 0.567478i \(-0.192081\pi\)
0.823389 + 0.567478i \(0.192081\pi\)
\(618\) 0 0
\(619\) 1.84390 0.0741127 0.0370563 0.999313i \(-0.488202\pi\)
0.0370563 + 0.999313i \(0.488202\pi\)
\(620\) 0 0
\(621\) 1.28131 0.0514172
\(622\) 0 0
\(623\) −4.53492 −0.181688
\(624\) 0 0
\(625\) 18.0132 0.720528
\(626\) 0 0
\(627\) 30.4551 1.21626
\(628\) 0 0
\(629\) −19.1766 −0.764621
\(630\) 0 0
\(631\) 25.1801 1.00240 0.501202 0.865330i \(-0.332891\pi\)
0.501202 + 0.865330i \(0.332891\pi\)
\(632\) 0 0
\(633\) 7.40796 0.294440
\(634\) 0 0
\(635\) 10.8001 0.428589
\(636\) 0 0
\(637\) 17.3043 0.685623
\(638\) 0 0
\(639\) 2.91100 0.115157
\(640\) 0 0
\(641\) 20.7359 0.819017 0.409508 0.912306i \(-0.365700\pi\)
0.409508 + 0.912306i \(0.365700\pi\)
\(642\) 0 0
\(643\) −30.4230 −1.19976 −0.599882 0.800088i \(-0.704786\pi\)
−0.599882 + 0.800088i \(0.704786\pi\)
\(644\) 0 0
\(645\) 3.60219 0.141836
\(646\) 0 0
\(647\) −34.3132 −1.34899 −0.674496 0.738279i \(-0.735639\pi\)
−0.674496 + 0.738279i \(0.735639\pi\)
\(648\) 0 0
\(649\) 51.1093 2.00622
\(650\) 0 0
\(651\) 17.0705 0.669047
\(652\) 0 0
\(653\) −15.6874 −0.613894 −0.306947 0.951727i \(-0.599307\pi\)
−0.306947 + 0.951727i \(0.599307\pi\)
\(654\) 0 0
\(655\) −6.02696 −0.235493
\(656\) 0 0
\(657\) 2.05497 0.0801718
\(658\) 0 0
\(659\) −31.4806 −1.22631 −0.613155 0.789962i \(-0.710100\pi\)
−0.613155 + 0.789962i \(0.710100\pi\)
\(660\) 0 0
\(661\) 27.3891 1.06531 0.532657 0.846331i \(-0.321194\pi\)
0.532657 + 0.846331i \(0.321194\pi\)
\(662\) 0 0
\(663\) 11.8826 0.461481
\(664\) 0 0
\(665\) −14.1391 −0.548291
\(666\) 0 0
\(667\) 5.40842 0.209415
\(668\) 0 0
\(669\) 0.474828 0.0183579
\(670\) 0 0
\(671\) −48.2024 −1.86083
\(672\) 0 0
\(673\) −23.8967 −0.921149 −0.460575 0.887621i \(-0.652357\pi\)
−0.460575 + 0.887621i \(0.652357\pi\)
\(674\) 0 0
\(675\) −4.51877 −0.173928
\(676\) 0 0
\(677\) 1.16212 0.0446637 0.0223319 0.999751i \(-0.492891\pi\)
0.0223319 + 0.999751i \(0.492891\pi\)
\(678\) 0 0
\(679\) 29.4496 1.13017
\(680\) 0 0
\(681\) −16.8443 −0.645473
\(682\) 0 0
\(683\) −16.6893 −0.638598 −0.319299 0.947654i \(-0.603447\pi\)
−0.319299 + 0.947654i \(0.603447\pi\)
\(684\) 0 0
\(685\) 8.07967 0.308708
\(686\) 0 0
\(687\) 2.13575 0.0814838
\(688\) 0 0
\(689\) −24.5527 −0.935382
\(690\) 0 0
\(691\) 5.93496 0.225777 0.112888 0.993608i \(-0.463990\pi\)
0.112888 + 0.993608i \(0.463990\pi\)
\(692\) 0 0
\(693\) −17.2079 −0.653675
\(694\) 0 0
\(695\) 14.4379 0.547659
\(696\) 0 0
\(697\) 13.1047 0.496375
\(698\) 0 0
\(699\) 22.3369 0.844860
\(700\) 0 0
\(701\) −16.5746 −0.626013 −0.313006 0.949751i \(-0.601336\pi\)
−0.313006 + 0.949751i \(0.601336\pi\)
\(702\) 0 0
\(703\) −37.1377 −1.40068
\(704\) 0 0
\(705\) −3.00673 −0.113240
\(706\) 0 0
\(707\) −51.5526 −1.93883
\(708\) 0 0
\(709\) −6.34699 −0.238366 −0.119183 0.992872i \(-0.538028\pi\)
−0.119183 + 0.992872i \(0.538028\pi\)
\(710\) 0 0
\(711\) 10.6914 0.400960
\(712\) 0 0
\(713\) −6.44531 −0.241379
\(714\) 0 0
\(715\) 13.4775 0.504030
\(716\) 0 0
\(717\) 16.3907 0.612123
\(718\) 0 0
\(719\) 11.0405 0.411742 0.205871 0.978579i \(-0.433997\pi\)
0.205871 + 0.978579i \(0.433997\pi\)
\(720\) 0 0
\(721\) 32.8174 1.22218
\(722\) 0 0
\(723\) −14.1952 −0.527927
\(724\) 0 0
\(725\) −19.0738 −0.708382
\(726\) 0 0
\(727\) 14.3769 0.533211 0.266605 0.963806i \(-0.414098\pi\)
0.266605 + 0.963806i \(0.414098\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.1041 0.595633
\(732\) 0 0
\(733\) 36.7241 1.35644 0.678218 0.734861i \(-0.262753\pi\)
0.678218 + 0.734861i \(0.262753\pi\)
\(734\) 0 0
\(735\) 3.13303 0.115564
\(736\) 0 0
\(737\) −35.7291 −1.31610
\(738\) 0 0
\(739\) 53.8580 1.98120 0.990600 0.136789i \(-0.0436784\pi\)
0.990600 + 0.136789i \(0.0436784\pi\)
\(740\) 0 0
\(741\) 23.0120 0.845366
\(742\) 0 0
\(743\) 22.5704 0.828029 0.414014 0.910270i \(-0.364126\pi\)
0.414014 + 0.910270i \(0.364126\pi\)
\(744\) 0 0
\(745\) −5.01611 −0.183776
\(746\) 0 0
\(747\) 7.04092 0.257614
\(748\) 0 0
\(749\) 25.8565 0.944776
\(750\) 0 0
\(751\) 9.93159 0.362409 0.181204 0.983445i \(-0.442000\pi\)
0.181204 + 0.983445i \(0.442000\pi\)
\(752\) 0 0
\(753\) 1.00000 0.0364420
\(754\) 0 0
\(755\) 3.96793 0.144408
\(756\) 0 0
\(757\) −15.4794 −0.562609 −0.281304 0.959619i \(-0.590767\pi\)
−0.281304 + 0.959619i \(0.590767\pi\)
\(758\) 0 0
\(759\) 6.49718 0.235833
\(760\) 0 0
\(761\) −5.66788 −0.205460 −0.102730 0.994709i \(-0.532758\pi\)
−0.102730 + 0.994709i \(0.532758\pi\)
\(762\) 0 0
\(763\) 2.42967 0.0879599
\(764\) 0 0
\(765\) 2.15140 0.0777839
\(766\) 0 0
\(767\) 38.6184 1.39443
\(768\) 0 0
\(769\) −38.3596 −1.38328 −0.691641 0.722242i \(-0.743112\pi\)
−0.691641 + 0.722242i \(0.743112\pi\)
\(770\) 0 0
\(771\) −11.2880 −0.406528
\(772\) 0 0
\(773\) 18.9003 0.679796 0.339898 0.940462i \(-0.389607\pi\)
0.339898 + 0.940462i \(0.389607\pi\)
\(774\) 0 0
\(775\) 22.7305 0.816505
\(776\) 0 0
\(777\) 20.9838 0.752789
\(778\) 0 0
\(779\) 25.3787 0.909288
\(780\) 0 0
\(781\) 14.7609 0.528186
\(782\) 0 0
\(783\) 4.22100 0.150846
\(784\) 0 0
\(785\) −16.5898 −0.592115
\(786\) 0 0
\(787\) −29.5875 −1.05468 −0.527340 0.849654i \(-0.676811\pi\)
−0.527340 + 0.849654i \(0.676811\pi\)
\(788\) 0 0
\(789\) −15.0221 −0.534800
\(790\) 0 0
\(791\) 8.67824 0.308563
\(792\) 0 0
\(793\) −36.4219 −1.29338
\(794\) 0 0
\(795\) −4.44538 −0.157661
\(796\) 0 0
\(797\) 25.1718 0.891633 0.445816 0.895124i \(-0.352913\pi\)
0.445816 + 0.895124i \(0.352913\pi\)
\(798\) 0 0
\(799\) −13.4421 −0.475546
\(800\) 0 0
\(801\) 1.33632 0.0472167
\(802\) 0 0
\(803\) 10.4202 0.367720
\(804\) 0 0
\(805\) −3.01639 −0.106314
\(806\) 0 0
\(807\) 13.2636 0.466900
\(808\) 0 0
\(809\) 31.8237 1.11886 0.559431 0.828877i \(-0.311020\pi\)
0.559431 + 0.828877i \(0.311020\pi\)
\(810\) 0 0
\(811\) 5.80329 0.203781 0.101891 0.994796i \(-0.467511\pi\)
0.101891 + 0.994796i \(0.467511\pi\)
\(812\) 0 0
\(813\) 13.2457 0.464546
\(814\) 0 0
\(815\) 2.25817 0.0791001
\(816\) 0 0
\(817\) 31.1875 1.09111
\(818\) 0 0
\(819\) −13.0024 −0.454340
\(820\) 0 0
\(821\) −9.45101 −0.329842 −0.164921 0.986307i \(-0.552737\pi\)
−0.164921 + 0.986307i \(0.552737\pi\)
\(822\) 0 0
\(823\) −46.2383 −1.61176 −0.805882 0.592076i \(-0.798309\pi\)
−0.805882 + 0.592076i \(0.798309\pi\)
\(824\) 0 0
\(825\) −22.9135 −0.797745
\(826\) 0 0
\(827\) −6.87014 −0.238898 −0.119449 0.992840i \(-0.538113\pi\)
−0.119449 + 0.992840i \(0.538113\pi\)
\(828\) 0 0
\(829\) 3.15593 0.109610 0.0548050 0.998497i \(-0.482546\pi\)
0.0548050 + 0.998497i \(0.482546\pi\)
\(830\) 0 0
\(831\) −15.4428 −0.535705
\(832\) 0 0
\(833\) 14.0067 0.485304
\(834\) 0 0
\(835\) 13.6097 0.470983
\(836\) 0 0
\(837\) −5.03025 −0.173871
\(838\) 0 0
\(839\) 21.2050 0.732077 0.366039 0.930600i \(-0.380714\pi\)
0.366039 + 0.930600i \(0.380714\pi\)
\(840\) 0 0
\(841\) −11.1831 −0.385625
\(842\) 0 0
\(843\) 14.0074 0.482441
\(844\) 0 0
\(845\) 1.16549 0.0400942
\(846\) 0 0
\(847\) −49.9274 −1.71552
\(848\) 0 0
\(849\) −19.8561 −0.681461
\(850\) 0 0
\(851\) −7.92283 −0.271591
\(852\) 0 0
\(853\) −31.6002 −1.08197 −0.540985 0.841032i \(-0.681948\pi\)
−0.540985 + 0.841032i \(0.681948\pi\)
\(854\) 0 0
\(855\) 4.16643 0.142489
\(856\) 0 0
\(857\) −10.2412 −0.349832 −0.174916 0.984583i \(-0.555965\pi\)
−0.174916 + 0.984583i \(0.555965\pi\)
\(858\) 0 0
\(859\) −20.3519 −0.694399 −0.347200 0.937791i \(-0.612867\pi\)
−0.347200 + 0.937791i \(0.612867\pi\)
\(860\) 0 0
\(861\) −14.3397 −0.488694
\(862\) 0 0
\(863\) 0.843947 0.0287283 0.0143641 0.999897i \(-0.495428\pi\)
0.0143641 + 0.999897i \(0.495428\pi\)
\(864\) 0 0
\(865\) 8.66539 0.294632
\(866\) 0 0
\(867\) −7.38186 −0.250701
\(868\) 0 0
\(869\) 54.2134 1.83906
\(870\) 0 0
\(871\) −26.9970 −0.914759
\(872\) 0 0
\(873\) −8.67805 −0.293707
\(874\) 0 0
\(875\) 22.4085 0.757547
\(876\) 0 0
\(877\) −11.0910 −0.374517 −0.187258 0.982311i \(-0.559960\pi\)
−0.187258 + 0.982311i \(0.559960\pi\)
\(878\) 0 0
\(879\) 24.3819 0.822382
\(880\) 0 0
\(881\) −9.37418 −0.315824 −0.157912 0.987453i \(-0.550476\pi\)
−0.157912 + 0.987453i \(0.550476\pi\)
\(882\) 0 0
\(883\) −12.4958 −0.420518 −0.210259 0.977646i \(-0.567431\pi\)
−0.210259 + 0.977646i \(0.567431\pi\)
\(884\) 0 0
\(885\) 6.99204 0.235035
\(886\) 0 0
\(887\) 29.4508 0.988862 0.494431 0.869217i \(-0.335376\pi\)
0.494431 + 0.869217i \(0.335376\pi\)
\(888\) 0 0
\(889\) −52.8337 −1.77198
\(890\) 0 0
\(891\) 5.07073 0.169876
\(892\) 0 0
\(893\) −26.0321 −0.871132
\(894\) 0 0
\(895\) 12.1672 0.406704
\(896\) 0 0
\(897\) 4.90929 0.163917
\(898\) 0 0
\(899\) −21.2327 −0.708150
\(900\) 0 0
\(901\) −19.8737 −0.662090
\(902\) 0 0
\(903\) −17.6218 −0.586416
\(904\) 0 0
\(905\) 2.00199 0.0665485
\(906\) 0 0
\(907\) −38.6496 −1.28334 −0.641670 0.766981i \(-0.721758\pi\)
−0.641670 + 0.766981i \(0.721758\pi\)
\(908\) 0 0
\(909\) 15.1912 0.503860
\(910\) 0 0
\(911\) 43.3850 1.43741 0.718705 0.695316i \(-0.244735\pi\)
0.718705 + 0.695316i \(0.244735\pi\)
\(912\) 0 0
\(913\) 35.7026 1.18158
\(914\) 0 0
\(915\) −6.59437 −0.218003
\(916\) 0 0
\(917\) 29.4837 0.973637
\(918\) 0 0
\(919\) 35.8276 1.18184 0.590922 0.806729i \(-0.298764\pi\)
0.590922 + 0.806729i \(0.298764\pi\)
\(920\) 0 0
\(921\) −9.02581 −0.297411
\(922\) 0 0
\(923\) 11.1534 0.367118
\(924\) 0 0
\(925\) 27.9413 0.918704
\(926\) 0 0
\(927\) −9.67044 −0.317619
\(928\) 0 0
\(929\) 27.8382 0.913343 0.456671 0.889636i \(-0.349042\pi\)
0.456671 + 0.889636i \(0.349042\pi\)
\(930\) 0 0
\(931\) 27.1256 0.889007
\(932\) 0 0
\(933\) 3.83059 0.125408
\(934\) 0 0
\(935\) 10.9091 0.356767
\(936\) 0 0
\(937\) −12.4256 −0.405928 −0.202964 0.979186i \(-0.565057\pi\)
−0.202964 + 0.979186i \(0.565057\pi\)
\(938\) 0 0
\(939\) 10.6014 0.345964
\(940\) 0 0
\(941\) −14.0448 −0.457848 −0.228924 0.973444i \(-0.573521\pi\)
−0.228924 + 0.973444i \(0.573521\pi\)
\(942\) 0 0
\(943\) 5.41421 0.176311
\(944\) 0 0
\(945\) −2.35414 −0.0765802
\(946\) 0 0
\(947\) 41.1617 1.33758 0.668788 0.743453i \(-0.266813\pi\)
0.668788 + 0.743453i \(0.266813\pi\)
\(948\) 0 0
\(949\) 7.87352 0.255585
\(950\) 0 0
\(951\) 7.40231 0.240036
\(952\) 0 0
\(953\) −28.2204 −0.914149 −0.457074 0.889428i \(-0.651103\pi\)
−0.457074 + 0.889428i \(0.651103\pi\)
\(954\) 0 0
\(955\) −2.23490 −0.0723197
\(956\) 0 0
\(957\) 21.4036 0.691879
\(958\) 0 0
\(959\) −39.5255 −1.27634
\(960\) 0 0
\(961\) −5.69662 −0.183762
\(962\) 0 0
\(963\) −7.61924 −0.245527
\(964\) 0 0
\(965\) 8.49927 0.273601
\(966\) 0 0
\(967\) 48.6825 1.56552 0.782761 0.622322i \(-0.213810\pi\)
0.782761 + 0.622322i \(0.213810\pi\)
\(968\) 0 0
\(969\) 18.6267 0.598375
\(970\) 0 0
\(971\) 43.4086 1.39305 0.696524 0.717534i \(-0.254729\pi\)
0.696524 + 0.717534i \(0.254729\pi\)
\(972\) 0 0
\(973\) −70.6295 −2.26428
\(974\) 0 0
\(975\) −17.3135 −0.554476
\(976\) 0 0
\(977\) −34.7387 −1.11139 −0.555695 0.831386i \(-0.687548\pi\)
−0.555695 + 0.831386i \(0.687548\pi\)
\(978\) 0 0
\(979\) 6.77614 0.216566
\(980\) 0 0
\(981\) −0.715961 −0.0228589
\(982\) 0 0
\(983\) −14.1559 −0.451504 −0.225752 0.974185i \(-0.572484\pi\)
−0.225752 + 0.974185i \(0.572484\pi\)
\(984\) 0 0
\(985\) −4.89928 −0.156104
\(986\) 0 0
\(987\) 14.7088 0.468187
\(988\) 0 0
\(989\) 6.65344 0.211567
\(990\) 0 0
\(991\) −22.0177 −0.699416 −0.349708 0.936859i \(-0.613719\pi\)
−0.349708 + 0.936859i \(0.613719\pi\)
\(992\) 0 0
\(993\) −3.57955 −0.113594
\(994\) 0 0
\(995\) 13.7990 0.437459
\(996\) 0 0
\(997\) 16.9683 0.537393 0.268696 0.963225i \(-0.413407\pi\)
0.268696 + 0.963225i \(0.413407\pi\)
\(998\) 0 0
\(999\) −6.18338 −0.195634
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 6024.2.a.q.1.10 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6024.2.a.q.1.10 18 1.1 even 1 trivial