Properties

Label 624.4.a.p.1.1
Level $624$
Weight $4$
Character 624.1
Self dual yes
Analytic conductor $36.817$
Analytic rank $0$
Dimension $2$
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] = [624,4,Mod(1,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("624.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{22}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 156)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.69042\) of defining polynomial
Character \(\chi\) \(=\) 624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -9.38083 q^{5} -32.1425 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -9.38083 q^{5} -32.1425 q^{7} +9.00000 q^{9} +48.7617 q^{11} -13.0000 q^{13} -28.1425 q^{15} -110.285 q^{17} +136.142 q^{19} -96.4275 q^{21} +32.9533 q^{23} -37.0000 q^{25} +27.0000 q^{27} +133.617 q^{29} -180.712 q^{31} +146.285 q^{33} +301.523 q^{35} -114.855 q^{37} -39.0000 q^{39} +116.712 q^{41} +502.565 q^{43} -84.4275 q^{45} +526.757 q^{47} +690.140 q^{49} -330.855 q^{51} +382.570 q^{53} -457.425 q^{55} +408.427 q^{57} +801.233 q^{59} +266.570 q^{61} -289.282 q^{63} +121.951 q^{65} -213.852 q^{67} +98.8600 q^{69} -655.228 q^{71} +766.285 q^{73} -111.000 q^{75} -1567.32 q^{77} +314.850 q^{79} +81.0000 q^{81} -730.570 q^{83} +1034.56 q^{85} +400.850 q^{87} -1298.99 q^{89} +417.852 q^{91} -542.137 q^{93} -1277.13 q^{95} +20.5751 q^{97} +438.855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 8 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 8 q^{7} + 18 q^{9} + 60 q^{11} - 26 q^{13} - 108 q^{17} + 216 q^{19} - 24 q^{21} + 216 q^{23} - 74 q^{25} + 54 q^{27} - 108 q^{29} - 80 q^{31} + 180 q^{33} + 528 q^{35} + 108 q^{37} - 78 q^{39} - 48 q^{41} - 8 q^{43} + 228 q^{47} + 930 q^{49} - 324 q^{51} + 540 q^{53} - 352 q^{55} + 648 q^{57} + 852 q^{59} + 308 q^{61} - 72 q^{63} + 304 q^{67} + 648 q^{69} + 228 q^{71} + 1420 q^{73} - 222 q^{75} - 1296 q^{77} - 496 q^{79} + 162 q^{81} - 1236 q^{83} + 1056 q^{85} - 324 q^{87} - 1416 q^{89} + 104 q^{91} - 240 q^{93} - 528 q^{95} + 604 q^{97} + 540 q^{99}+O(q^{100}) \) 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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −9.38083 −0.839047 −0.419524 0.907744i \(-0.637803\pi\)
−0.419524 + 0.907744i \(0.637803\pi\)
\(6\) 0 0
\(7\) −32.1425 −1.73553 −0.867766 0.496973i \(-0.834445\pi\)
−0.867766 + 0.496973i \(0.834445\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 48.7617 1.33656 0.668282 0.743908i \(-0.267030\pi\)
0.668282 + 0.743908i \(0.267030\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) −28.1425 −0.484424
\(16\) 0 0
\(17\) −110.285 −1.57341 −0.786707 0.617327i \(-0.788216\pi\)
−0.786707 + 0.617327i \(0.788216\pi\)
\(18\) 0 0
\(19\) 136.142 1.64385 0.821927 0.569593i \(-0.192899\pi\)
0.821927 + 0.569593i \(0.192899\pi\)
\(20\) 0 0
\(21\) −96.4275 −1.00201
\(22\) 0 0
\(23\) 32.9533 0.298750 0.149375 0.988781i \(-0.452274\pi\)
0.149375 + 0.988781i \(0.452274\pi\)
\(24\) 0 0
\(25\) −37.0000 −0.296000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 133.617 0.855586 0.427793 0.903877i \(-0.359291\pi\)
0.427793 + 0.903877i \(0.359291\pi\)
\(30\) 0 0
\(31\) −180.712 −1.04700 −0.523499 0.852026i \(-0.675374\pi\)
−0.523499 + 0.852026i \(0.675374\pi\)
\(32\) 0 0
\(33\) 146.285 0.771665
\(34\) 0 0
\(35\) 301.523 1.45619
\(36\) 0 0
\(37\) −114.855 −0.510325 −0.255163 0.966898i \(-0.582129\pi\)
−0.255163 + 0.966898i \(0.582129\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) 116.712 0.444571 0.222286 0.974982i \(-0.428648\pi\)
0.222286 + 0.974982i \(0.428648\pi\)
\(42\) 0 0
\(43\) 502.565 1.78234 0.891168 0.453674i \(-0.149887\pi\)
0.891168 + 0.453674i \(0.149887\pi\)
\(44\) 0 0
\(45\) −84.4275 −0.279682
\(46\) 0 0
\(47\) 526.757 1.63479 0.817397 0.576074i \(-0.195416\pi\)
0.817397 + 0.576074i \(0.195416\pi\)
\(48\) 0 0
\(49\) 690.140 2.01207
\(50\) 0 0
\(51\) −330.855 −0.908411
\(52\) 0 0
\(53\) 382.570 0.991510 0.495755 0.868462i \(-0.334891\pi\)
0.495755 + 0.868462i \(0.334891\pi\)
\(54\) 0 0
\(55\) −457.425 −1.12144
\(56\) 0 0
\(57\) 408.427 0.949080
\(58\) 0 0
\(59\) 801.233 1.76799 0.883997 0.467492i \(-0.154842\pi\)
0.883997 + 0.467492i \(0.154842\pi\)
\(60\) 0 0
\(61\) 266.570 0.559521 0.279761 0.960070i \(-0.409745\pi\)
0.279761 + 0.960070i \(0.409745\pi\)
\(62\) 0 0
\(63\) −289.282 −0.578511
\(64\) 0 0
\(65\) 121.951 0.232710
\(66\) 0 0
\(67\) −213.852 −0.389944 −0.194972 0.980809i \(-0.562462\pi\)
−0.194972 + 0.980809i \(0.562462\pi\)
\(68\) 0 0
\(69\) 98.8600 0.172483
\(70\) 0 0
\(71\) −655.228 −1.09523 −0.547615 0.836731i \(-0.684464\pi\)
−0.547615 + 0.836731i \(0.684464\pi\)
\(72\) 0 0
\(73\) 766.285 1.22859 0.614294 0.789077i \(-0.289441\pi\)
0.614294 + 0.789077i \(0.289441\pi\)
\(74\) 0 0
\(75\) −111.000 −0.170896
\(76\) 0 0
\(77\) −1567.32 −2.31965
\(78\) 0 0
\(79\) 314.850 0.448397 0.224199 0.974543i \(-0.428024\pi\)
0.224199 + 0.974543i \(0.428024\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −730.570 −0.966150 −0.483075 0.875579i \(-0.660480\pi\)
−0.483075 + 0.875579i \(0.660480\pi\)
\(84\) 0 0
\(85\) 1034.56 1.32017
\(86\) 0 0
\(87\) 400.850 0.493973
\(88\) 0 0
\(89\) −1298.99 −1.54711 −0.773556 0.633728i \(-0.781524\pi\)
−0.773556 + 0.633728i \(0.781524\pi\)
\(90\) 0 0
\(91\) 417.852 0.481350
\(92\) 0 0
\(93\) −542.137 −0.604484
\(94\) 0 0
\(95\) −1277.13 −1.37927
\(96\) 0 0
\(97\) 20.5751 0.0215369 0.0107685 0.999942i \(-0.496572\pi\)
0.0107685 + 0.999942i \(0.496572\pi\)
\(98\) 0 0
\(99\) 438.855 0.445521
\(100\) 0 0
\(101\) −273.902 −0.269844 −0.134922 0.990856i \(-0.543078\pi\)
−0.134922 + 0.990856i \(0.543078\pi\)
\(102\) 0 0
\(103\) 760.285 0.727312 0.363656 0.931533i \(-0.381528\pi\)
0.363656 + 0.931533i \(0.381528\pi\)
\(104\) 0 0
\(105\) 904.570 0.840733
\(106\) 0 0
\(107\) −728.176 −0.657902 −0.328951 0.944347i \(-0.606695\pi\)
−0.328951 + 0.944347i \(0.606695\pi\)
\(108\) 0 0
\(109\) 1058.57 0.930207 0.465104 0.885256i \(-0.346017\pi\)
0.465104 + 0.885256i \(0.346017\pi\)
\(110\) 0 0
\(111\) −344.565 −0.294637
\(112\) 0 0
\(113\) 845.036 0.703490 0.351745 0.936096i \(-0.385588\pi\)
0.351745 + 0.936096i \(0.385588\pi\)
\(114\) 0 0
\(115\) −309.130 −0.250665
\(116\) 0 0
\(117\) −117.000 −0.0924500
\(118\) 0 0
\(119\) 3544.83 2.73071
\(120\) 0 0
\(121\) 1046.70 0.786401
\(122\) 0 0
\(123\) 350.137 0.256673
\(124\) 0 0
\(125\) 1519.69 1.08741
\(126\) 0 0
\(127\) 1071.69 0.748799 0.374400 0.927267i \(-0.377849\pi\)
0.374400 + 0.927267i \(0.377849\pi\)
\(128\) 0 0
\(129\) 1507.69 1.02903
\(130\) 0 0
\(131\) 777.513 0.518562 0.259281 0.965802i \(-0.416514\pi\)
0.259281 + 0.965802i \(0.416514\pi\)
\(132\) 0 0
\(133\) −4375.96 −2.85296
\(134\) 0 0
\(135\) −253.282 −0.161475
\(136\) 0 0
\(137\) −1558.69 −0.972026 −0.486013 0.873951i \(-0.661549\pi\)
−0.486013 + 0.873951i \(0.661549\pi\)
\(138\) 0 0
\(139\) 2514.29 1.53424 0.767120 0.641504i \(-0.221689\pi\)
0.767120 + 0.641504i \(0.221689\pi\)
\(140\) 0 0
\(141\) 1580.27 0.943849
\(142\) 0 0
\(143\) −633.902 −0.370696
\(144\) 0 0
\(145\) −1253.44 −0.717877
\(146\) 0 0
\(147\) 2070.42 1.16167
\(148\) 0 0
\(149\) −1172.15 −0.644473 −0.322237 0.946659i \(-0.604435\pi\)
−0.322237 + 0.946659i \(0.604435\pi\)
\(150\) 0 0
\(151\) −446.137 −0.240438 −0.120219 0.992747i \(-0.538360\pi\)
−0.120219 + 0.992747i \(0.538360\pi\)
\(152\) 0 0
\(153\) −992.565 −0.524471
\(154\) 0 0
\(155\) 1695.23 0.878480
\(156\) 0 0
\(157\) 2258.56 1.14811 0.574053 0.818818i \(-0.305370\pi\)
0.574053 + 0.818818i \(0.305370\pi\)
\(158\) 0 0
\(159\) 1147.71 0.572449
\(160\) 0 0
\(161\) −1059.20 −0.518490
\(162\) 0 0
\(163\) −2290.40 −1.10060 −0.550301 0.834966i \(-0.685487\pi\)
−0.550301 + 0.834966i \(0.685487\pi\)
\(164\) 0 0
\(165\) −1372.27 −0.647463
\(166\) 0 0
\(167\) −3323.13 −1.53983 −0.769915 0.638147i \(-0.779701\pi\)
−0.769915 + 0.638147i \(0.779701\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 1225.28 0.547951
\(172\) 0 0
\(173\) 1691.58 0.743402 0.371701 0.928353i \(-0.378775\pi\)
0.371701 + 0.928353i \(0.378775\pi\)
\(174\) 0 0
\(175\) 1189.27 0.513717
\(176\) 0 0
\(177\) 2403.70 1.02075
\(178\) 0 0
\(179\) −2490.84 −1.04008 −0.520039 0.854142i \(-0.674083\pi\)
−0.520039 + 0.854142i \(0.674083\pi\)
\(180\) 0 0
\(181\) −776.860 −0.319025 −0.159513 0.987196i \(-0.550992\pi\)
−0.159513 + 0.987196i \(0.550992\pi\)
\(182\) 0 0
\(183\) 799.710 0.323040
\(184\) 0 0
\(185\) 1077.44 0.428187
\(186\) 0 0
\(187\) −5377.68 −2.10297
\(188\) 0 0
\(189\) −867.847 −0.334003
\(190\) 0 0
\(191\) 283.026 0.107220 0.0536101 0.998562i \(-0.482927\pi\)
0.0536101 + 0.998562i \(0.482927\pi\)
\(192\) 0 0
\(193\) −1747.11 −0.651605 −0.325802 0.945438i \(-0.605634\pi\)
−0.325802 + 0.945438i \(0.605634\pi\)
\(194\) 0 0
\(195\) 365.852 0.134355
\(196\) 0 0
\(197\) 328.702 0.118879 0.0594393 0.998232i \(-0.481069\pi\)
0.0594393 + 0.998232i \(0.481069\pi\)
\(198\) 0 0
\(199\) 5205.39 1.85427 0.927137 0.374722i \(-0.122262\pi\)
0.927137 + 0.374722i \(0.122262\pi\)
\(200\) 0 0
\(201\) −641.557 −0.225134
\(202\) 0 0
\(203\) −4294.77 −1.48490
\(204\) 0 0
\(205\) −1094.86 −0.373016
\(206\) 0 0
\(207\) 296.580 0.0995833
\(208\) 0 0
\(209\) 6638.53 2.19711
\(210\) 0 0
\(211\) 3763.98 1.22807 0.614036 0.789278i \(-0.289545\pi\)
0.614036 + 0.789278i \(0.289545\pi\)
\(212\) 0 0
\(213\) −1965.68 −0.632331
\(214\) 0 0
\(215\) −4714.48 −1.49546
\(216\) 0 0
\(217\) 5808.55 1.81710
\(218\) 0 0
\(219\) 2298.85 0.709325
\(220\) 0 0
\(221\) 1433.70 0.436387
\(222\) 0 0
\(223\) −900.102 −0.270293 −0.135146 0.990826i \(-0.543150\pi\)
−0.135146 + 0.990826i \(0.543150\pi\)
\(224\) 0 0
\(225\) −333.000 −0.0986667
\(226\) 0 0
\(227\) −3193.73 −0.933811 −0.466905 0.884307i \(-0.654631\pi\)
−0.466905 + 0.884307i \(0.654631\pi\)
\(228\) 0 0
\(229\) −1261.43 −0.364007 −0.182004 0.983298i \(-0.558258\pi\)
−0.182004 + 0.983298i \(0.558258\pi\)
\(230\) 0 0
\(231\) −4701.96 −1.33925
\(232\) 0 0
\(233\) −4125.90 −1.16007 −0.580036 0.814591i \(-0.696962\pi\)
−0.580036 + 0.814591i \(0.696962\pi\)
\(234\) 0 0
\(235\) −4941.41 −1.37167
\(236\) 0 0
\(237\) 944.550 0.258882
\(238\) 0 0
\(239\) −3282.30 −0.888343 −0.444172 0.895942i \(-0.646502\pi\)
−0.444172 + 0.895942i \(0.646502\pi\)
\(240\) 0 0
\(241\) 1768.59 0.472716 0.236358 0.971666i \(-0.424046\pi\)
0.236358 + 0.971666i \(0.424046\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −6474.09 −1.68822
\(246\) 0 0
\(247\) −1769.85 −0.455923
\(248\) 0 0
\(249\) −2191.71 −0.557807
\(250\) 0 0
\(251\) 6097.65 1.53339 0.766694 0.642013i \(-0.221901\pi\)
0.766694 + 0.642013i \(0.221901\pi\)
\(252\) 0 0
\(253\) 1606.86 0.399298
\(254\) 0 0
\(255\) 3103.69 0.762200
\(256\) 0 0
\(257\) −1612.59 −0.391402 −0.195701 0.980664i \(-0.562698\pi\)
−0.195701 + 0.980664i \(0.562698\pi\)
\(258\) 0 0
\(259\) 3691.73 0.885686
\(260\) 0 0
\(261\) 1202.55 0.285195
\(262\) 0 0
\(263\) −3219.39 −0.754814 −0.377407 0.926048i \(-0.623184\pi\)
−0.377407 + 0.926048i \(0.623184\pi\)
\(264\) 0 0
\(265\) −3588.82 −0.831924
\(266\) 0 0
\(267\) −3896.98 −0.893225
\(268\) 0 0
\(269\) −2235.17 −0.506620 −0.253310 0.967385i \(-0.581519\pi\)
−0.253310 + 0.967385i \(0.581519\pi\)
\(270\) 0 0
\(271\) 2652.38 0.594540 0.297270 0.954793i \(-0.403924\pi\)
0.297270 + 0.954793i \(0.403924\pi\)
\(272\) 0 0
\(273\) 1253.56 0.277907
\(274\) 0 0
\(275\) −1804.18 −0.395623
\(276\) 0 0
\(277\) −2450.02 −0.531435 −0.265717 0.964051i \(-0.585609\pi\)
−0.265717 + 0.964051i \(0.585609\pi\)
\(278\) 0 0
\(279\) −1626.41 −0.348999
\(280\) 0 0
\(281\) 1999.41 0.424465 0.212232 0.977219i \(-0.431927\pi\)
0.212232 + 0.977219i \(0.431927\pi\)
\(282\) 0 0
\(283\) 2846.54 0.597913 0.298957 0.954267i \(-0.403361\pi\)
0.298957 + 0.954267i \(0.403361\pi\)
\(284\) 0 0
\(285\) −3831.39 −0.796323
\(286\) 0 0
\(287\) −3751.43 −0.771568
\(288\) 0 0
\(289\) 7249.78 1.47563
\(290\) 0 0
\(291\) 61.7252 0.0124343
\(292\) 0 0
\(293\) 1737.35 0.346406 0.173203 0.984886i \(-0.444588\pi\)
0.173203 + 0.984886i \(0.444588\pi\)
\(294\) 0 0
\(295\) −7516.23 −1.48343
\(296\) 0 0
\(297\) 1316.56 0.257222
\(298\) 0 0
\(299\) −428.394 −0.0828583
\(300\) 0 0
\(301\) −16153.7 −3.09330
\(302\) 0 0
\(303\) −821.705 −0.155794
\(304\) 0 0
\(305\) −2500.65 −0.469465
\(306\) 0 0
\(307\) 2523.51 0.469135 0.234567 0.972100i \(-0.424633\pi\)
0.234567 + 0.972100i \(0.424633\pi\)
\(308\) 0 0
\(309\) 2280.85 0.419914
\(310\) 0 0
\(311\) 2697.76 0.491884 0.245942 0.969285i \(-0.420903\pi\)
0.245942 + 0.969285i \(0.420903\pi\)
\(312\) 0 0
\(313\) −2589.38 −0.467605 −0.233803 0.972284i \(-0.575117\pi\)
−0.233803 + 0.972284i \(0.575117\pi\)
\(314\) 0 0
\(315\) 2713.71 0.485398
\(316\) 0 0
\(317\) 3692.06 0.654154 0.327077 0.944998i \(-0.393936\pi\)
0.327077 + 0.944998i \(0.393936\pi\)
\(318\) 0 0
\(319\) 6515.37 1.14354
\(320\) 0 0
\(321\) −2184.53 −0.379840
\(322\) 0 0
\(323\) −15014.5 −2.58646
\(324\) 0 0
\(325\) 481.000 0.0820956
\(326\) 0 0
\(327\) 3175.71 0.537056
\(328\) 0 0
\(329\) −16931.3 −2.83724
\(330\) 0 0
\(331\) −6100.12 −1.01297 −0.506485 0.862249i \(-0.669055\pi\)
−0.506485 + 0.862249i \(0.669055\pi\)
\(332\) 0 0
\(333\) −1033.69 −0.170108
\(334\) 0 0
\(335\) 2006.11 0.327181
\(336\) 0 0
\(337\) −1105.39 −0.178678 −0.0893389 0.996001i \(-0.528475\pi\)
−0.0893389 + 0.996001i \(0.528475\pi\)
\(338\) 0 0
\(339\) 2535.11 0.406160
\(340\) 0 0
\(341\) −8811.84 −1.39938
\(342\) 0 0
\(343\) −11157.9 −1.75648
\(344\) 0 0
\(345\) −927.389 −0.144722
\(346\) 0 0
\(347\) 11014.7 1.70404 0.852020 0.523510i \(-0.175378\pi\)
0.852020 + 0.523510i \(0.175378\pi\)
\(348\) 0 0
\(349\) −1450.35 −0.222451 −0.111225 0.993795i \(-0.535478\pi\)
−0.111225 + 0.993795i \(0.535478\pi\)
\(350\) 0 0
\(351\) −351.000 −0.0533761
\(352\) 0 0
\(353\) −4589.21 −0.691953 −0.345976 0.938243i \(-0.612452\pi\)
−0.345976 + 0.938243i \(0.612452\pi\)
\(354\) 0 0
\(355\) 6146.59 0.918949
\(356\) 0 0
\(357\) 10634.5 1.57658
\(358\) 0 0
\(359\) 879.170 0.129250 0.0646251 0.997910i \(-0.479415\pi\)
0.0646251 + 0.997910i \(0.479415\pi\)
\(360\) 0 0
\(361\) 11675.8 1.70226
\(362\) 0 0
\(363\) 3140.10 0.454029
\(364\) 0 0
\(365\) −7188.39 −1.03084
\(366\) 0 0
\(367\) 9883.42 1.40575 0.702875 0.711313i \(-0.251899\pi\)
0.702875 + 0.711313i \(0.251899\pi\)
\(368\) 0 0
\(369\) 1050.41 0.148190
\(370\) 0 0
\(371\) −12296.8 −1.72080
\(372\) 0 0
\(373\) −4981.98 −0.691574 −0.345787 0.938313i \(-0.612388\pi\)
−0.345787 + 0.938313i \(0.612388\pi\)
\(374\) 0 0
\(375\) 4559.08 0.627814
\(376\) 0 0
\(377\) −1737.02 −0.237297
\(378\) 0 0
\(379\) −2095.29 −0.283978 −0.141989 0.989868i \(-0.545350\pi\)
−0.141989 + 0.989868i \(0.545350\pi\)
\(380\) 0 0
\(381\) 3215.08 0.432320
\(382\) 0 0
\(383\) 11211.7 1.49580 0.747900 0.663812i \(-0.231062\pi\)
0.747900 + 0.663812i \(0.231062\pi\)
\(384\) 0 0
\(385\) 14702.8 1.94629
\(386\) 0 0
\(387\) 4523.08 0.594112
\(388\) 0 0
\(389\) −7697.74 −1.00332 −0.501659 0.865065i \(-0.667277\pi\)
−0.501659 + 0.865065i \(0.667277\pi\)
\(390\) 0 0
\(391\) −3634.26 −0.470057
\(392\) 0 0
\(393\) 2332.54 0.299392
\(394\) 0 0
\(395\) −2953.55 −0.376226
\(396\) 0 0
\(397\) 9240.59 1.16819 0.584095 0.811685i \(-0.301449\pi\)
0.584095 + 0.811685i \(0.301449\pi\)
\(398\) 0 0
\(399\) −13127.9 −1.64716
\(400\) 0 0
\(401\) 12268.3 1.52781 0.763903 0.645331i \(-0.223280\pi\)
0.763903 + 0.645331i \(0.223280\pi\)
\(402\) 0 0
\(403\) 2349.26 0.290385
\(404\) 0 0
\(405\) −759.847 −0.0932275
\(406\) 0 0
\(407\) −5600.52 −0.682082
\(408\) 0 0
\(409\) 7748.49 0.936769 0.468384 0.883525i \(-0.344836\pi\)
0.468384 + 0.883525i \(0.344836\pi\)
\(410\) 0 0
\(411\) −4676.06 −0.561200
\(412\) 0 0
\(413\) −25753.6 −3.06841
\(414\) 0 0
\(415\) 6853.35 0.810646
\(416\) 0 0
\(417\) 7542.87 0.885794
\(418\) 0 0
\(419\) 6153.08 0.717417 0.358708 0.933450i \(-0.383217\pi\)
0.358708 + 0.933450i \(0.383217\pi\)
\(420\) 0 0
\(421\) 1163.77 0.134724 0.0673619 0.997729i \(-0.478542\pi\)
0.0673619 + 0.997729i \(0.478542\pi\)
\(422\) 0 0
\(423\) 4740.81 0.544932
\(424\) 0 0
\(425\) 4080.54 0.465731
\(426\) 0 0
\(427\) −8568.22 −0.971067
\(428\) 0 0
\(429\) −1901.70 −0.214021
\(430\) 0 0
\(431\) 8945.32 0.999723 0.499862 0.866105i \(-0.333384\pi\)
0.499862 + 0.866105i \(0.333384\pi\)
\(432\) 0 0
\(433\) 3602.01 0.399773 0.199886 0.979819i \(-0.435943\pi\)
0.199886 + 0.979819i \(0.435943\pi\)
\(434\) 0 0
\(435\) −3760.31 −0.414466
\(436\) 0 0
\(437\) 4486.35 0.491101
\(438\) 0 0
\(439\) 9684.72 1.05291 0.526454 0.850204i \(-0.323521\pi\)
0.526454 + 0.850204i \(0.323521\pi\)
\(440\) 0 0
\(441\) 6211.26 0.670690
\(442\) 0 0
\(443\) 2294.27 0.246059 0.123029 0.992403i \(-0.460739\pi\)
0.123029 + 0.992403i \(0.460739\pi\)
\(444\) 0 0
\(445\) 12185.6 1.29810
\(446\) 0 0
\(447\) −3516.46 −0.372087
\(448\) 0 0
\(449\) 12453.3 1.30893 0.654464 0.756093i \(-0.272894\pi\)
0.654464 + 0.756093i \(0.272894\pi\)
\(450\) 0 0
\(451\) 5691.09 0.594198
\(452\) 0 0
\(453\) −1338.41 −0.138817
\(454\) 0 0
\(455\) −3919.80 −0.403875
\(456\) 0 0
\(457\) −13922.7 −1.42511 −0.712555 0.701616i \(-0.752462\pi\)
−0.712555 + 0.701616i \(0.752462\pi\)
\(458\) 0 0
\(459\) −2977.69 −0.302804
\(460\) 0 0
\(461\) −14345.0 −1.44927 −0.724633 0.689135i \(-0.757991\pi\)
−0.724633 + 0.689135i \(0.757991\pi\)
\(462\) 0 0
\(463\) −125.252 −0.0125723 −0.00628613 0.999980i \(-0.502001\pi\)
−0.00628613 + 0.999980i \(0.502001\pi\)
\(464\) 0 0
\(465\) 5085.70 0.507191
\(466\) 0 0
\(467\) −12764.6 −1.26482 −0.632412 0.774632i \(-0.717935\pi\)
−0.632412 + 0.774632i \(0.717935\pi\)
\(468\) 0 0
\(469\) 6873.75 0.676760
\(470\) 0 0
\(471\) 6775.68 0.662860
\(472\) 0 0
\(473\) 24505.9 2.38220
\(474\) 0 0
\(475\) −5037.27 −0.486581
\(476\) 0 0
\(477\) 3443.13 0.330503
\(478\) 0 0
\(479\) 5904.54 0.563227 0.281613 0.959528i \(-0.409130\pi\)
0.281613 + 0.959528i \(0.409130\pi\)
\(480\) 0 0
\(481\) 1493.11 0.141539
\(482\) 0 0
\(483\) −3177.61 −0.299350
\(484\) 0 0
\(485\) −193.011 −0.0180705
\(486\) 0 0
\(487\) −20385.3 −1.89680 −0.948402 0.317069i \(-0.897301\pi\)
−0.948402 + 0.317069i \(0.897301\pi\)
\(488\) 0 0
\(489\) −6871.21 −0.635433
\(490\) 0 0
\(491\) −9155.16 −0.841480 −0.420740 0.907181i \(-0.638230\pi\)
−0.420740 + 0.907181i \(0.638230\pi\)
\(492\) 0 0
\(493\) −14735.9 −1.34619
\(494\) 0 0
\(495\) −4116.82 −0.373813
\(496\) 0 0
\(497\) 21060.7 1.90081
\(498\) 0 0
\(499\) 20308.9 1.82195 0.910974 0.412464i \(-0.135332\pi\)
0.910974 + 0.412464i \(0.135332\pi\)
\(500\) 0 0
\(501\) −9969.39 −0.889021
\(502\) 0 0
\(503\) 5341.99 0.473534 0.236767 0.971566i \(-0.423912\pi\)
0.236767 + 0.971566i \(0.423912\pi\)
\(504\) 0 0
\(505\) 2569.42 0.226412
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 0 0
\(509\) 8142.10 0.709022 0.354511 0.935052i \(-0.384647\pi\)
0.354511 + 0.935052i \(0.384647\pi\)
\(510\) 0 0
\(511\) −24630.3 −2.13225
\(512\) 0 0
\(513\) 3675.85 0.316360
\(514\) 0 0
\(515\) −7132.11 −0.610249
\(516\) 0 0
\(517\) 25685.5 2.18501
\(518\) 0 0
\(519\) 5074.74 0.429203
\(520\) 0 0
\(521\) 3103.65 0.260985 0.130493 0.991449i \(-0.458344\pi\)
0.130493 + 0.991449i \(0.458344\pi\)
\(522\) 0 0
\(523\) 17409.0 1.45553 0.727767 0.685824i \(-0.240558\pi\)
0.727767 + 0.685824i \(0.240558\pi\)
\(524\) 0 0
\(525\) 3567.82 0.296595
\(526\) 0 0
\(527\) 19929.9 1.64736
\(528\) 0 0
\(529\) −11081.1 −0.910748
\(530\) 0 0
\(531\) 7211.10 0.589332
\(532\) 0 0
\(533\) −1517.26 −0.123302
\(534\) 0 0
\(535\) 6830.90 0.552010
\(536\) 0 0
\(537\) −7472.52 −0.600490
\(538\) 0 0
\(539\) 33652.4 2.68926
\(540\) 0 0
\(541\) −8794.50 −0.698900 −0.349450 0.936955i \(-0.613632\pi\)
−0.349450 + 0.936955i \(0.613632\pi\)
\(542\) 0 0
\(543\) −2330.58 −0.184189
\(544\) 0 0
\(545\) −9930.27 −0.780488
\(546\) 0 0
\(547\) −3153.07 −0.246463 −0.123232 0.992378i \(-0.539326\pi\)
−0.123232 + 0.992378i \(0.539326\pi\)
\(548\) 0 0
\(549\) 2399.13 0.186507
\(550\) 0 0
\(551\) 18190.9 1.40646
\(552\) 0 0
\(553\) −10120.1 −0.778208
\(554\) 0 0
\(555\) 3232.31 0.247214
\(556\) 0 0
\(557\) −15234.8 −1.15892 −0.579460 0.815000i \(-0.696737\pi\)
−0.579460 + 0.815000i \(0.696737\pi\)
\(558\) 0 0
\(559\) −6533.34 −0.494331
\(560\) 0 0
\(561\) −16133.0 −1.21415
\(562\) 0 0
\(563\) 3473.44 0.260014 0.130007 0.991513i \(-0.458500\pi\)
0.130007 + 0.991513i \(0.458500\pi\)
\(564\) 0 0
\(565\) −7927.15 −0.590261
\(566\) 0 0
\(567\) −2603.54 −0.192837
\(568\) 0 0
\(569\) −15167.6 −1.11750 −0.558752 0.829335i \(-0.688719\pi\)
−0.558752 + 0.829335i \(0.688719\pi\)
\(570\) 0 0
\(571\) −16058.9 −1.17696 −0.588479 0.808512i \(-0.700273\pi\)
−0.588479 + 0.808512i \(0.700273\pi\)
\(572\) 0 0
\(573\) 849.079 0.0619036
\(574\) 0 0
\(575\) −1219.27 −0.0884300
\(576\) 0 0
\(577\) −9238.39 −0.666550 −0.333275 0.942830i \(-0.608154\pi\)
−0.333275 + 0.942830i \(0.608154\pi\)
\(578\) 0 0
\(579\) −5241.33 −0.376204
\(580\) 0 0
\(581\) 23482.3 1.67678
\(582\) 0 0
\(583\) 18654.7 1.32522
\(584\) 0 0
\(585\) 1097.56 0.0775699
\(586\) 0 0
\(587\) −25895.9 −1.82085 −0.910424 0.413677i \(-0.864244\pi\)
−0.910424 + 0.413677i \(0.864244\pi\)
\(588\) 0 0
\(589\) −24602.6 −1.72111
\(590\) 0 0
\(591\) 986.107 0.0686346
\(592\) 0 0
\(593\) 17572.0 1.21686 0.608428 0.793609i \(-0.291800\pi\)
0.608428 + 0.793609i \(0.291800\pi\)
\(594\) 0 0
\(595\) −33253.5 −2.29119
\(596\) 0 0
\(597\) 15616.2 1.07057
\(598\) 0 0
\(599\) −12752.8 −0.869894 −0.434947 0.900456i \(-0.643233\pi\)
−0.434947 + 0.900456i \(0.643233\pi\)
\(600\) 0 0
\(601\) −951.801 −0.0646003 −0.0323002 0.999478i \(-0.510283\pi\)
−0.0323002 + 0.999478i \(0.510283\pi\)
\(602\) 0 0
\(603\) −1924.67 −0.129981
\(604\) 0 0
\(605\) −9818.91 −0.659827
\(606\) 0 0
\(607\) 18701.1 1.25050 0.625252 0.780423i \(-0.284996\pi\)
0.625252 + 0.780423i \(0.284996\pi\)
\(608\) 0 0
\(609\) −12884.3 −0.857305
\(610\) 0 0
\(611\) −6847.84 −0.453410
\(612\) 0 0
\(613\) 25600.3 1.68676 0.843381 0.537316i \(-0.180562\pi\)
0.843381 + 0.537316i \(0.180562\pi\)
\(614\) 0 0
\(615\) −3284.58 −0.215361
\(616\) 0 0
\(617\) 7742.12 0.505164 0.252582 0.967575i \(-0.418720\pi\)
0.252582 + 0.967575i \(0.418720\pi\)
\(618\) 0 0
\(619\) 17409.3 1.13044 0.565218 0.824942i \(-0.308792\pi\)
0.565218 + 0.824942i \(0.308792\pi\)
\(620\) 0 0
\(621\) 889.740 0.0574944
\(622\) 0 0
\(623\) 41752.9 2.68506
\(624\) 0 0
\(625\) −9631.00 −0.616384
\(626\) 0 0
\(627\) 19915.6 1.26850
\(628\) 0 0
\(629\) 12666.8 0.802953
\(630\) 0 0
\(631\) 4994.82 0.315120 0.157560 0.987509i \(-0.449637\pi\)
0.157560 + 0.987509i \(0.449637\pi\)
\(632\) 0 0
\(633\) 11291.9 0.709027
\(634\) 0 0
\(635\) −10053.4 −0.628278
\(636\) 0 0
\(637\) −8971.82 −0.558048
\(638\) 0 0
\(639\) −5897.05 −0.365076
\(640\) 0 0
\(641\) 14610.1 0.900257 0.450128 0.892964i \(-0.351378\pi\)
0.450128 + 0.892964i \(0.351378\pi\)
\(642\) 0 0
\(643\) 23102.5 1.41691 0.708456 0.705755i \(-0.249392\pi\)
0.708456 + 0.705755i \(0.249392\pi\)
\(644\) 0 0
\(645\) −14143.4 −0.863406
\(646\) 0 0
\(647\) −1830.50 −0.111228 −0.0556139 0.998452i \(-0.517712\pi\)
−0.0556139 + 0.998452i \(0.517712\pi\)
\(648\) 0 0
\(649\) 39069.5 2.36304
\(650\) 0 0
\(651\) 17425.6 1.04910
\(652\) 0 0
\(653\) −2834.46 −0.169864 −0.0849318 0.996387i \(-0.527067\pi\)
−0.0849318 + 0.996387i \(0.527067\pi\)
\(654\) 0 0
\(655\) −7293.72 −0.435098
\(656\) 0 0
\(657\) 6896.56 0.409529
\(658\) 0 0
\(659\) 24787.3 1.46521 0.732607 0.680652i \(-0.238303\pi\)
0.732607 + 0.680652i \(0.238303\pi\)
\(660\) 0 0
\(661\) −9370.39 −0.551385 −0.275693 0.961246i \(-0.588907\pi\)
−0.275693 + 0.961246i \(0.588907\pi\)
\(662\) 0 0
\(663\) 4301.11 0.251948
\(664\) 0 0
\(665\) 41050.1 2.39377
\(666\) 0 0
\(667\) 4403.12 0.255606
\(668\) 0 0
\(669\) −2700.31 −0.156054
\(670\) 0 0
\(671\) 12998.4 0.747835
\(672\) 0 0
\(673\) −19925.8 −1.14128 −0.570641 0.821200i \(-0.693305\pi\)
−0.570641 + 0.821200i \(0.693305\pi\)
\(674\) 0 0
\(675\) −999.000 −0.0569652
\(676\) 0 0
\(677\) 10634.5 0.603719 0.301860 0.953352i \(-0.402393\pi\)
0.301860 + 0.953352i \(0.402393\pi\)
\(678\) 0 0
\(679\) −661.334 −0.0373780
\(680\) 0 0
\(681\) −9581.18 −0.539136
\(682\) 0 0
\(683\) −8250.11 −0.462199 −0.231099 0.972930i \(-0.574232\pi\)
−0.231099 + 0.972930i \(0.574232\pi\)
\(684\) 0 0
\(685\) 14621.8 0.815576
\(686\) 0 0
\(687\) −3784.29 −0.210160
\(688\) 0 0
\(689\) −4973.41 −0.274995
\(690\) 0 0
\(691\) 33965.8 1.86993 0.934964 0.354744i \(-0.115432\pi\)
0.934964 + 0.354744i \(0.115432\pi\)
\(692\) 0 0
\(693\) −14105.9 −0.773216
\(694\) 0 0
\(695\) −23586.1 −1.28730
\(696\) 0 0
\(697\) −12871.6 −0.699495
\(698\) 0 0
\(699\) −12377.7 −0.669768
\(700\) 0 0
\(701\) 25354.8 1.36610 0.683051 0.730371i \(-0.260653\pi\)
0.683051 + 0.730371i \(0.260653\pi\)
\(702\) 0 0
\(703\) −15636.6 −0.838901
\(704\) 0 0
\(705\) −14824.2 −0.791934
\(706\) 0 0
\(707\) 8803.88 0.468323
\(708\) 0 0
\(709\) −6783.28 −0.359311 −0.179656 0.983730i \(-0.557498\pi\)
−0.179656 + 0.983730i \(0.557498\pi\)
\(710\) 0 0
\(711\) 2833.65 0.149466
\(712\) 0 0
\(713\) −5955.08 −0.312790
\(714\) 0 0
\(715\) 5946.52 0.311031
\(716\) 0 0
\(717\) −9846.89 −0.512885
\(718\) 0 0
\(719\) −21704.4 −1.12578 −0.562892 0.826530i \(-0.690311\pi\)
−0.562892 + 0.826530i \(0.690311\pi\)
\(720\) 0 0
\(721\) −24437.5 −1.26227
\(722\) 0 0
\(723\) 5305.76 0.272923
\(724\) 0 0
\(725\) −4943.82 −0.253253
\(726\) 0 0
\(727\) −30734.0 −1.56790 −0.783949 0.620825i \(-0.786798\pi\)
−0.783949 + 0.620825i \(0.786798\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −55425.4 −2.80435
\(732\) 0 0
\(733\) 1005.22 0.0506528 0.0253264 0.999679i \(-0.491937\pi\)
0.0253264 + 0.999679i \(0.491937\pi\)
\(734\) 0 0
\(735\) −19422.3 −0.974695
\(736\) 0 0
\(737\) −10427.8 −0.521184
\(738\) 0 0
\(739\) 5216.68 0.259674 0.129837 0.991535i \(-0.458555\pi\)
0.129837 + 0.991535i \(0.458555\pi\)
\(740\) 0 0
\(741\) −5309.56 −0.263227
\(742\) 0 0
\(743\) −26994.7 −1.33289 −0.666447 0.745552i \(-0.732186\pi\)
−0.666447 + 0.745552i \(0.732186\pi\)
\(744\) 0 0
\(745\) 10995.8 0.540743
\(746\) 0 0
\(747\) −6575.13 −0.322050
\(748\) 0 0
\(749\) 23405.4 1.14181
\(750\) 0 0
\(751\) 14488.7 0.703996 0.351998 0.936001i \(-0.385502\pi\)
0.351998 + 0.936001i \(0.385502\pi\)
\(752\) 0 0
\(753\) 18292.9 0.885302
\(754\) 0 0
\(755\) 4185.14 0.201739
\(756\) 0 0
\(757\) 21724.8 1.04307 0.521534 0.853230i \(-0.325360\pi\)
0.521534 + 0.853230i \(0.325360\pi\)
\(758\) 0 0
\(759\) 4820.58 0.230535
\(760\) 0 0
\(761\) −11449.5 −0.545391 −0.272695 0.962100i \(-0.587915\pi\)
−0.272695 + 0.962100i \(0.587915\pi\)
\(762\) 0 0
\(763\) −34025.1 −1.61440
\(764\) 0 0
\(765\) 9311.08 0.440056
\(766\) 0 0
\(767\) −10416.0 −0.490353
\(768\) 0 0
\(769\) −19602.4 −0.919220 −0.459610 0.888121i \(-0.652011\pi\)
−0.459610 + 0.888121i \(0.652011\pi\)
\(770\) 0 0
\(771\) −4837.76 −0.225976
\(772\) 0 0
\(773\) −31311.3 −1.45691 −0.728454 0.685094i \(-0.759761\pi\)
−0.728454 + 0.685094i \(0.759761\pi\)
\(774\) 0 0
\(775\) 6686.36 0.309911
\(776\) 0 0
\(777\) 11075.2 0.511351
\(778\) 0 0
\(779\) 15889.5 0.730811
\(780\) 0 0
\(781\) −31950.0 −1.46384
\(782\) 0 0
\(783\) 3607.65 0.164658
\(784\) 0 0
\(785\) −21187.2 −0.963315
\(786\) 0 0
\(787\) 1739.06 0.0787686 0.0393843 0.999224i \(-0.487460\pi\)
0.0393843 + 0.999224i \(0.487460\pi\)
\(788\) 0 0
\(789\) −9658.17 −0.435792
\(790\) 0 0
\(791\) −27161.6 −1.22093
\(792\) 0 0
\(793\) −3465.41 −0.155183
\(794\) 0 0
\(795\) −10766.5 −0.480311
\(796\) 0 0
\(797\) 34119.8 1.51642 0.758209 0.652011i \(-0.226074\pi\)
0.758209 + 0.652011i \(0.226074\pi\)
\(798\) 0 0
\(799\) −58093.3 −2.57221
\(800\) 0 0
\(801\) −11690.9 −0.515704
\(802\) 0 0
\(803\) 37365.3 1.64208
\(804\) 0 0
\(805\) 9936.20 0.435037
\(806\) 0 0
\(807\) −6705.51 −0.292497
\(808\) 0 0
\(809\) 26108.4 1.13464 0.567319 0.823498i \(-0.307981\pi\)
0.567319 + 0.823498i \(0.307981\pi\)
\(810\) 0 0
\(811\) −13917.9 −0.602620 −0.301310 0.953526i \(-0.597424\pi\)
−0.301310 + 0.953526i \(0.597424\pi\)
\(812\) 0 0
\(813\) 7957.13 0.343258
\(814\) 0 0
\(815\) 21485.9 0.923457
\(816\) 0 0
\(817\) 68420.4 2.92990
\(818\) 0 0
\(819\) 3760.67 0.160450
\(820\) 0 0
\(821\) 1594.62 0.0677863 0.0338932 0.999425i \(-0.489209\pi\)
0.0338932 + 0.999425i \(0.489209\pi\)
\(822\) 0 0
\(823\) −17656.3 −0.747825 −0.373913 0.927464i \(-0.621984\pi\)
−0.373913 + 0.927464i \(0.621984\pi\)
\(824\) 0 0
\(825\) −5412.54 −0.228413
\(826\) 0 0
\(827\) −3246.70 −0.136516 −0.0682581 0.997668i \(-0.521744\pi\)
−0.0682581 + 0.997668i \(0.521744\pi\)
\(828\) 0 0
\(829\) 6464.17 0.270820 0.135410 0.990790i \(-0.456765\pi\)
0.135410 + 0.990790i \(0.456765\pi\)
\(830\) 0 0
\(831\) −7350.06 −0.306824
\(832\) 0 0
\(833\) −76112.1 −3.16582
\(834\) 0 0
\(835\) 31173.7 1.29199
\(836\) 0 0
\(837\) −4879.24 −0.201495
\(838\) 0 0
\(839\) 38238.9 1.57348 0.786741 0.617283i \(-0.211767\pi\)
0.786741 + 0.617283i \(0.211767\pi\)
\(840\) 0 0
\(841\) −6535.60 −0.267973
\(842\) 0 0
\(843\) 5998.22 0.245065
\(844\) 0 0
\(845\) −1585.36 −0.0645421
\(846\) 0 0
\(847\) −33643.5 −1.36482
\(848\) 0 0
\(849\) 8539.63 0.345206
\(850\) 0 0
\(851\) −3784.86 −0.152460
\(852\) 0 0
\(853\) 32957.3 1.32290 0.661451 0.749988i \(-0.269941\pi\)
0.661451 + 0.749988i \(0.269941\pi\)
\(854\) 0 0
\(855\) −11494.2 −0.459757
\(856\) 0 0
\(857\) −46853.1 −1.86753 −0.933765 0.357887i \(-0.883497\pi\)
−0.933765 + 0.357887i \(0.883497\pi\)
\(858\) 0 0
\(859\) 44005.5 1.74790 0.873951 0.486014i \(-0.161550\pi\)
0.873951 + 0.486014i \(0.161550\pi\)
\(860\) 0 0
\(861\) −11254.3 −0.445465
\(862\) 0 0
\(863\) 27483.9 1.08408 0.542041 0.840352i \(-0.317652\pi\)
0.542041 + 0.840352i \(0.317652\pi\)
\(864\) 0 0
\(865\) −15868.4 −0.623749
\(866\) 0 0
\(867\) 21749.3 0.851956
\(868\) 0 0
\(869\) 15352.6 0.599311
\(870\) 0 0
\(871\) 2780.08 0.108151
\(872\) 0 0
\(873\) 185.175 0.00717897
\(874\) 0 0
\(875\) −48846.8 −1.88723
\(876\) 0 0
\(877\) 23933.7 0.921531 0.460766 0.887522i \(-0.347575\pi\)
0.460766 + 0.887522i \(0.347575\pi\)
\(878\) 0 0
\(879\) 5212.05 0.199998
\(880\) 0 0
\(881\) 4109.49 0.157154 0.0785768 0.996908i \(-0.474962\pi\)
0.0785768 + 0.996908i \(0.474962\pi\)
\(882\) 0 0
\(883\) −30422.0 −1.15944 −0.579718 0.814817i \(-0.696837\pi\)
−0.579718 + 0.814817i \(0.696837\pi\)
\(884\) 0 0
\(885\) −22548.7 −0.856459
\(886\) 0 0
\(887\) −32471.3 −1.22918 −0.614588 0.788848i \(-0.710678\pi\)
−0.614588 + 0.788848i \(0.710678\pi\)
\(888\) 0 0
\(889\) −34446.9 −1.29957
\(890\) 0 0
\(891\) 3949.69 0.148507
\(892\) 0 0
\(893\) 71714.0 2.68736
\(894\) 0 0
\(895\) 23366.1 0.872675
\(896\) 0 0
\(897\) −1285.18 −0.0478383
\(898\) 0 0
\(899\) −24146.2 −0.895796
\(900\) 0 0
\(901\) −42191.7 −1.56006
\(902\) 0 0
\(903\) −48461.1 −1.78592
\(904\) 0 0
\(905\) 7287.59 0.267677
\(906\) 0 0
\(907\) 37788.4 1.38340 0.691699 0.722186i \(-0.256862\pi\)
0.691699 + 0.722186i \(0.256862\pi\)
\(908\) 0 0
\(909\) −2465.11 −0.0899480
\(910\) 0 0
\(911\) 47254.0 1.71855 0.859274 0.511516i \(-0.170916\pi\)
0.859274 + 0.511516i \(0.170916\pi\)
\(912\) 0 0
\(913\) −35623.8 −1.29132
\(914\) 0 0
\(915\) −7501.94 −0.271045
\(916\) 0 0
\(917\) −24991.2 −0.899981
\(918\) 0 0
\(919\) −27178.7 −0.975562 −0.487781 0.872966i \(-0.662194\pi\)
−0.487781 + 0.872966i \(0.662194\pi\)
\(920\) 0 0
\(921\) 7570.53 0.270855
\(922\) 0 0
\(923\) 8517.97 0.303762
\(924\) 0 0
\(925\) 4249.63 0.151056
\(926\) 0 0
\(927\) 6842.56 0.242437
\(928\) 0 0
\(929\) −32561.1 −1.14994 −0.574971 0.818174i \(-0.694987\pi\)
−0.574971 + 0.818174i \(0.694987\pi\)
\(930\) 0 0
\(931\) 93957.4 3.30755
\(932\) 0 0
\(933\) 8093.28 0.283989
\(934\) 0 0
\(935\) 50447.1 1.76449
\(936\) 0 0
\(937\) 38995.8 1.35959 0.679797 0.733401i \(-0.262068\pi\)
0.679797 + 0.733401i \(0.262068\pi\)
\(938\) 0 0
\(939\) −7768.14 −0.269972
\(940\) 0 0
\(941\) −8625.11 −0.298800 −0.149400 0.988777i \(-0.547734\pi\)
−0.149400 + 0.988777i \(0.547734\pi\)
\(942\) 0 0
\(943\) 3846.07 0.132816
\(944\) 0 0
\(945\) 8141.13 0.280244
\(946\) 0 0
\(947\) 5147.81 0.176644 0.0883218 0.996092i \(-0.471850\pi\)
0.0883218 + 0.996092i \(0.471850\pi\)
\(948\) 0 0
\(949\) −9961.70 −0.340749
\(950\) 0 0
\(951\) 11076.2 0.377676
\(952\) 0 0
\(953\) −33910.8 −1.15265 −0.576326 0.817220i \(-0.695514\pi\)
−0.576326 + 0.817220i \(0.695514\pi\)
\(954\) 0 0
\(955\) −2655.02 −0.0899628
\(956\) 0 0
\(957\) 19546.1 0.660226
\(958\) 0 0
\(959\) 50100.1 1.68698
\(960\) 0 0
\(961\) 2866.00 0.0962035
\(962\) 0 0
\(963\) −6553.59 −0.219301
\(964\) 0 0
\(965\) 16389.3 0.546727
\(966\) 0 0
\(967\) −12891.9 −0.428725 −0.214362 0.976754i \(-0.568767\pi\)
−0.214362 + 0.976754i \(0.568767\pi\)
\(968\) 0 0
\(969\) −45043.4 −1.49330
\(970\) 0 0
\(971\) 19035.1 0.629109 0.314555 0.949239i \(-0.398145\pi\)
0.314555 + 0.949239i \(0.398145\pi\)
\(972\) 0 0
\(973\) −80815.6 −2.66272
\(974\) 0 0
\(975\) 1443.00 0.0473979
\(976\) 0 0
\(977\) 7411.48 0.242696 0.121348 0.992610i \(-0.461278\pi\)
0.121348 + 0.992610i \(0.461278\pi\)
\(978\) 0 0
\(979\) −63341.0 −2.06781
\(980\) 0 0
\(981\) 9527.13 0.310069
\(982\) 0 0
\(983\) 43006.5 1.39542 0.697708 0.716382i \(-0.254203\pi\)
0.697708 + 0.716382i \(0.254203\pi\)
\(984\) 0 0
\(985\) −3083.50 −0.0997447
\(986\) 0 0
\(987\) −50793.8 −1.63808
\(988\) 0 0
\(989\) 16561.2 0.532473
\(990\) 0 0
\(991\) −49948.6 −1.60108 −0.800539 0.599280i \(-0.795454\pi\)
−0.800539 + 0.599280i \(0.795454\pi\)
\(992\) 0 0
\(993\) −18300.4 −0.584838
\(994\) 0 0
\(995\) −48830.9 −1.55582
\(996\) 0 0
\(997\) −10657.9 −0.338555 −0.169278 0.985568i \(-0.554143\pi\)
−0.169278 + 0.985568i \(0.554143\pi\)
\(998\) 0 0
\(999\) −3101.08 −0.0982122
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 624.4.a.p.1.1 2
3.2 odd 2 1872.4.a.y.1.2 2
4.3 odd 2 156.4.a.c.1.1 2
8.3 odd 2 2496.4.a.bf.1.2 2
8.5 even 2 2496.4.a.w.1.2 2
12.11 even 2 468.4.a.g.1.2 2
52.31 even 4 2028.4.b.e.337.4 4
52.47 even 4 2028.4.b.e.337.1 4
52.51 odd 2 2028.4.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
156.4.a.c.1.1 2 4.3 odd 2
468.4.a.g.1.2 2 12.11 even 2
624.4.a.p.1.1 2 1.1 even 1 trivial
1872.4.a.y.1.2 2 3.2 odd 2
2028.4.a.d.1.2 2 52.51 odd 2
2028.4.b.e.337.1 4 52.47 even 4
2028.4.b.e.337.4 4 52.31 even 4
2496.4.a.w.1.2 2 8.5 even 2
2496.4.a.bf.1.2 2 8.3 odd 2