Properties

Label 608.4.a.h.1.1
Level $608$
Weight $4$
Character 608.1
Self dual yes
Analytic conductor $35.873$
Analytic rank $1$
Dimension $5$
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] = [608,4,Mod(1,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 608.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: \(35.8731612835\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 28x^{3} - 8x^{2} + 73x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(6.33779\) of defining polynomial
Character \(\chi\) \(=\) 608.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.31962 q^{3} +9.41363 q^{5} +16.2508 q^{7} +1.29831 q^{9} +8.71440 q^{11} -43.4140 q^{13} -50.0769 q^{15} -124.442 q^{17} -19.0000 q^{19} -86.4480 q^{21} +71.6374 q^{23} -36.3835 q^{25} +136.723 q^{27} +279.955 q^{29} +252.949 q^{31} -46.3573 q^{33} +152.979 q^{35} +17.9550 q^{37} +230.946 q^{39} -466.221 q^{41} -340.723 q^{43} +12.2218 q^{45} -305.434 q^{47} -78.9116 q^{49} +661.985 q^{51} -293.326 q^{53} +82.0342 q^{55} +101.073 q^{57} -408.311 q^{59} +334.310 q^{61} +21.0985 q^{63} -408.683 q^{65} -754.117 q^{67} -381.084 q^{69} -758.951 q^{71} -393.936 q^{73} +193.546 q^{75} +141.616 q^{77} +91.7062 q^{79} -762.369 q^{81} +1396.71 q^{83} -1171.45 q^{85} -1489.25 q^{87} +931.233 q^{89} -705.512 q^{91} -1345.59 q^{93} -178.859 q^{95} -282.166 q^{97} +11.3140 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{3} + 5 q^{5} - 7 q^{7} - 5 q^{9} + 13 q^{11} - 72 q^{13} - 72 q^{15} - 59 q^{17} - 95 q^{19} - 224 q^{21} - 52 q^{23} - 86 q^{25} + 54 q^{27} - 128 q^{29} + 110 q^{31} - 68 q^{33} - 45 q^{35}+ \cdots + 601 q^{99}+O(q^{100}) \) 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
\(3\) −5.31962 −1.02376 −0.511880 0.859057i \(-0.671051\pi\)
−0.511880 + 0.859057i \(0.671051\pi\)
\(4\) 0 0
\(5\) 9.41363 0.841981 0.420990 0.907065i \(-0.361683\pi\)
0.420990 + 0.907065i \(0.361683\pi\)
\(6\) 0 0
\(7\) 16.2508 0.877461 0.438730 0.898619i \(-0.355428\pi\)
0.438730 + 0.898619i \(0.355428\pi\)
\(8\) 0 0
\(9\) 1.29831 0.0480854
\(10\) 0 0
\(11\) 8.71440 0.238863 0.119431 0.992842i \(-0.461893\pi\)
0.119431 + 0.992842i \(0.461893\pi\)
\(12\) 0 0
\(13\) −43.4140 −0.926221 −0.463110 0.886301i \(-0.653267\pi\)
−0.463110 + 0.886301i \(0.653267\pi\)
\(14\) 0 0
\(15\) −50.0769 −0.861987
\(16\) 0 0
\(17\) −124.442 −1.77539 −0.887697 0.460428i \(-0.847696\pi\)
−0.887697 + 0.460428i \(0.847696\pi\)
\(18\) 0 0
\(19\) −19.0000 −0.229416
\(20\) 0 0
\(21\) −86.4480 −0.898309
\(22\) 0 0
\(23\) 71.6374 0.649454 0.324727 0.945808i \(-0.394728\pi\)
0.324727 + 0.945808i \(0.394728\pi\)
\(24\) 0 0
\(25\) −36.3835 −0.291068
\(26\) 0 0
\(27\) 136.723 0.974532
\(28\) 0 0
\(29\) 279.955 1.79263 0.896315 0.443418i \(-0.146234\pi\)
0.896315 + 0.443418i \(0.146234\pi\)
\(30\) 0 0
\(31\) 252.949 1.46551 0.732757 0.680490i \(-0.238233\pi\)
0.732757 + 0.680490i \(0.238233\pi\)
\(32\) 0 0
\(33\) −46.3573 −0.244538
\(34\) 0 0
\(35\) 152.979 0.738805
\(36\) 0 0
\(37\) 17.9550 0.0797778 0.0398889 0.999204i \(-0.487300\pi\)
0.0398889 + 0.999204i \(0.487300\pi\)
\(38\) 0 0
\(39\) 230.946 0.948228
\(40\) 0 0
\(41\) −466.221 −1.77589 −0.887944 0.459951i \(-0.847867\pi\)
−0.887944 + 0.459951i \(0.847867\pi\)
\(42\) 0 0
\(43\) −340.723 −1.20837 −0.604184 0.796845i \(-0.706501\pi\)
−0.604184 + 0.796845i \(0.706501\pi\)
\(44\) 0 0
\(45\) 12.2218 0.0404870
\(46\) 0 0
\(47\) −305.434 −0.947916 −0.473958 0.880547i \(-0.657175\pi\)
−0.473958 + 0.880547i \(0.657175\pi\)
\(48\) 0 0
\(49\) −78.9116 −0.230063
\(50\) 0 0
\(51\) 661.985 1.81758
\(52\) 0 0
\(53\) −293.326 −0.760216 −0.380108 0.924942i \(-0.624113\pi\)
−0.380108 + 0.924942i \(0.624113\pi\)
\(54\) 0 0
\(55\) 82.0342 0.201118
\(56\) 0 0
\(57\) 101.073 0.234867
\(58\) 0 0
\(59\) −408.311 −0.900976 −0.450488 0.892782i \(-0.648750\pi\)
−0.450488 + 0.892782i \(0.648750\pi\)
\(60\) 0 0
\(61\) 334.310 0.701704 0.350852 0.936431i \(-0.385892\pi\)
0.350852 + 0.936431i \(0.385892\pi\)
\(62\) 0 0
\(63\) 21.0985 0.0421930
\(64\) 0 0
\(65\) −408.683 −0.779860
\(66\) 0 0
\(67\) −754.117 −1.37507 −0.687537 0.726149i \(-0.741308\pi\)
−0.687537 + 0.726149i \(0.741308\pi\)
\(68\) 0 0
\(69\) −381.084 −0.664885
\(70\) 0 0
\(71\) −758.951 −1.26861 −0.634303 0.773085i \(-0.718713\pi\)
−0.634303 + 0.773085i \(0.718713\pi\)
\(72\) 0 0
\(73\) −393.936 −0.631599 −0.315800 0.948826i \(-0.602273\pi\)
−0.315800 + 0.948826i \(0.602273\pi\)
\(74\) 0 0
\(75\) 193.546 0.297984
\(76\) 0 0
\(77\) 141.616 0.209593
\(78\) 0 0
\(79\) 91.7062 0.130605 0.0653023 0.997866i \(-0.479199\pi\)
0.0653023 + 0.997866i \(0.479199\pi\)
\(80\) 0 0
\(81\) −762.369 −1.04577
\(82\) 0 0
\(83\) 1396.71 1.84710 0.923550 0.383478i \(-0.125274\pi\)
0.923550 + 0.383478i \(0.125274\pi\)
\(84\) 0 0
\(85\) −1171.45 −1.49485
\(86\) 0 0
\(87\) −1489.25 −1.83522
\(88\) 0 0
\(89\) 931.233 1.10911 0.554553 0.832148i \(-0.312889\pi\)
0.554553 + 0.832148i \(0.312889\pi\)
\(90\) 0 0
\(91\) −705.512 −0.812722
\(92\) 0 0
\(93\) −1345.59 −1.50034
\(94\) 0 0
\(95\) −178.859 −0.193164
\(96\) 0 0
\(97\) −282.166 −0.295357 −0.147679 0.989035i \(-0.547180\pi\)
−0.147679 + 0.989035i \(0.547180\pi\)
\(98\) 0 0
\(99\) 11.3140 0.0114858
\(100\) 0 0
\(101\) 57.3942 0.0565440 0.0282720 0.999600i \(-0.491000\pi\)
0.0282720 + 0.999600i \(0.491000\pi\)
\(102\) 0 0
\(103\) −1320.03 −1.26278 −0.631390 0.775466i \(-0.717515\pi\)
−0.631390 + 0.775466i \(0.717515\pi\)
\(104\) 0 0
\(105\) −813.789 −0.756359
\(106\) 0 0
\(107\) 749.733 0.677377 0.338689 0.940898i \(-0.390017\pi\)
0.338689 + 0.940898i \(0.390017\pi\)
\(108\) 0 0
\(109\) 645.208 0.566970 0.283485 0.958977i \(-0.408509\pi\)
0.283485 + 0.958977i \(0.408509\pi\)
\(110\) 0 0
\(111\) −95.5136 −0.0816734
\(112\) 0 0
\(113\) −2308.79 −1.92206 −0.961029 0.276448i \(-0.910843\pi\)
−0.961029 + 0.276448i \(0.910843\pi\)
\(114\) 0 0
\(115\) 674.369 0.546828
\(116\) 0 0
\(117\) −56.3646 −0.0445377
\(118\) 0 0
\(119\) −2022.29 −1.55784
\(120\) 0 0
\(121\) −1255.06 −0.942945
\(122\) 0 0
\(123\) 2480.11 1.81808
\(124\) 0 0
\(125\) −1519.21 −1.08705
\(126\) 0 0
\(127\) 311.645 0.217749 0.108874 0.994056i \(-0.465275\pi\)
0.108874 + 0.994056i \(0.465275\pi\)
\(128\) 0 0
\(129\) 1812.52 1.23708
\(130\) 0 0
\(131\) −514.521 −0.343160 −0.171580 0.985170i \(-0.554887\pi\)
−0.171580 + 0.985170i \(0.554887\pi\)
\(132\) 0 0
\(133\) −308.765 −0.201303
\(134\) 0 0
\(135\) 1287.06 0.820538
\(136\) 0 0
\(137\) −2201.49 −1.37289 −0.686446 0.727181i \(-0.740830\pi\)
−0.686446 + 0.727181i \(0.740830\pi\)
\(138\) 0 0
\(139\) −188.406 −0.114967 −0.0574835 0.998346i \(-0.518308\pi\)
−0.0574835 + 0.998346i \(0.518308\pi\)
\(140\) 0 0
\(141\) 1624.79 0.970439
\(142\) 0 0
\(143\) −378.327 −0.221240
\(144\) 0 0
\(145\) 2635.39 1.50936
\(146\) 0 0
\(147\) 419.779 0.235529
\(148\) 0 0
\(149\) −2002.07 −1.10078 −0.550390 0.834908i \(-0.685521\pi\)
−0.550390 + 0.834908i \(0.685521\pi\)
\(150\) 0 0
\(151\) −1324.14 −0.713621 −0.356810 0.934177i \(-0.616136\pi\)
−0.356810 + 0.934177i \(0.616136\pi\)
\(152\) 0 0
\(153\) −161.564 −0.0853705
\(154\) 0 0
\(155\) 2381.17 1.23394
\(156\) 0 0
\(157\) 360.790 0.183403 0.0917013 0.995787i \(-0.470770\pi\)
0.0917013 + 0.995787i \(0.470770\pi\)
\(158\) 0 0
\(159\) 1560.38 0.778279
\(160\) 0 0
\(161\) 1164.17 0.569870
\(162\) 0 0
\(163\) −634.639 −0.304962 −0.152481 0.988306i \(-0.548726\pi\)
−0.152481 + 0.988306i \(0.548726\pi\)
\(164\) 0 0
\(165\) −436.390 −0.205897
\(166\) 0 0
\(167\) 3740.13 1.73305 0.866526 0.499131i \(-0.166348\pi\)
0.866526 + 0.499131i \(0.166348\pi\)
\(168\) 0 0
\(169\) −312.227 −0.142115
\(170\) 0 0
\(171\) −24.6678 −0.0110315
\(172\) 0 0
\(173\) 978.817 0.430162 0.215081 0.976596i \(-0.430998\pi\)
0.215081 + 0.976596i \(0.430998\pi\)
\(174\) 0 0
\(175\) −591.262 −0.255401
\(176\) 0 0
\(177\) 2172.06 0.922384
\(178\) 0 0
\(179\) −259.331 −0.108287 −0.0541434 0.998533i \(-0.517243\pi\)
−0.0541434 + 0.998533i \(0.517243\pi\)
\(180\) 0 0
\(181\) 892.412 0.366478 0.183239 0.983068i \(-0.441342\pi\)
0.183239 + 0.983068i \(0.441342\pi\)
\(182\) 0 0
\(183\) −1778.40 −0.718377
\(184\) 0 0
\(185\) 169.022 0.0671714
\(186\) 0 0
\(187\) −1084.44 −0.424076
\(188\) 0 0
\(189\) 2221.86 0.855114
\(190\) 0 0
\(191\) −4020.76 −1.52320 −0.761601 0.648046i \(-0.775587\pi\)
−0.761601 + 0.648046i \(0.775587\pi\)
\(192\) 0 0
\(193\) 407.021 0.151803 0.0759015 0.997115i \(-0.475817\pi\)
0.0759015 + 0.997115i \(0.475817\pi\)
\(194\) 0 0
\(195\) 2174.04 0.798390
\(196\) 0 0
\(197\) −3699.86 −1.33809 −0.669046 0.743221i \(-0.733297\pi\)
−0.669046 + 0.743221i \(0.733297\pi\)
\(198\) 0 0
\(199\) −1802.62 −0.642131 −0.321066 0.947057i \(-0.604041\pi\)
−0.321066 + 0.947057i \(0.604041\pi\)
\(200\) 0 0
\(201\) 4011.61 1.40775
\(202\) 0 0
\(203\) 4549.49 1.57296
\(204\) 0 0
\(205\) −4388.83 −1.49526
\(206\) 0 0
\(207\) 93.0073 0.0312292
\(208\) 0 0
\(209\) −165.574 −0.0547989
\(210\) 0 0
\(211\) 5032.87 1.64207 0.821035 0.570878i \(-0.193397\pi\)
0.821035 + 0.570878i \(0.193397\pi\)
\(212\) 0 0
\(213\) 4037.33 1.29875
\(214\) 0 0
\(215\) −3207.45 −1.01742
\(216\) 0 0
\(217\) 4110.62 1.28593
\(218\) 0 0
\(219\) 2095.59 0.646606
\(220\) 0 0
\(221\) 5402.54 1.64441
\(222\) 0 0
\(223\) 417.179 0.125275 0.0626376 0.998036i \(-0.480049\pi\)
0.0626376 + 0.998036i \(0.480049\pi\)
\(224\) 0 0
\(225\) −47.2370 −0.0139961
\(226\) 0 0
\(227\) −3166.92 −0.925974 −0.462987 0.886365i \(-0.653222\pi\)
−0.462987 + 0.886365i \(0.653222\pi\)
\(228\) 0 0
\(229\) −2198.00 −0.634271 −0.317135 0.948380i \(-0.602721\pi\)
−0.317135 + 0.948380i \(0.602721\pi\)
\(230\) 0 0
\(231\) −753.343 −0.214573
\(232\) 0 0
\(233\) 1956.70 0.550161 0.275081 0.961421i \(-0.411295\pi\)
0.275081 + 0.961421i \(0.411295\pi\)
\(234\) 0 0
\(235\) −2875.24 −0.798127
\(236\) 0 0
\(237\) −487.842 −0.133708
\(238\) 0 0
\(239\) 358.707 0.0970829 0.0485415 0.998821i \(-0.484543\pi\)
0.0485415 + 0.998821i \(0.484543\pi\)
\(240\) 0 0
\(241\) 51.0894 0.0136554 0.00682771 0.999977i \(-0.497827\pi\)
0.00682771 + 0.999977i \(0.497827\pi\)
\(242\) 0 0
\(243\) 363.984 0.0960887
\(244\) 0 0
\(245\) −742.845 −0.193709
\(246\) 0 0
\(247\) 824.865 0.212490
\(248\) 0 0
\(249\) −7429.98 −1.89099
\(250\) 0 0
\(251\) −460.611 −0.115831 −0.0579153 0.998322i \(-0.518445\pi\)
−0.0579153 + 0.998322i \(0.518445\pi\)
\(252\) 0 0
\(253\) 624.278 0.155130
\(254\) 0 0
\(255\) 6231.69 1.53037
\(256\) 0 0
\(257\) 1349.26 0.327489 0.163744 0.986503i \(-0.447643\pi\)
0.163744 + 0.986503i \(0.447643\pi\)
\(258\) 0 0
\(259\) 291.783 0.0700019
\(260\) 0 0
\(261\) 363.467 0.0861993
\(262\) 0 0
\(263\) 312.198 0.0731977 0.0365988 0.999330i \(-0.488348\pi\)
0.0365988 + 0.999330i \(0.488348\pi\)
\(264\) 0 0
\(265\) −2761.26 −0.640087
\(266\) 0 0
\(267\) −4953.80 −1.13546
\(268\) 0 0
\(269\) 4178.51 0.947093 0.473546 0.880769i \(-0.342974\pi\)
0.473546 + 0.880769i \(0.342974\pi\)
\(270\) 0 0
\(271\) 2140.56 0.479815 0.239907 0.970796i \(-0.422883\pi\)
0.239907 + 0.970796i \(0.422883\pi\)
\(272\) 0 0
\(273\) 3753.05 0.832033
\(274\) 0 0
\(275\) −317.061 −0.0695254
\(276\) 0 0
\(277\) −6290.18 −1.36440 −0.682202 0.731164i \(-0.738978\pi\)
−0.682202 + 0.731164i \(0.738978\pi\)
\(278\) 0 0
\(279\) 328.405 0.0704698
\(280\) 0 0
\(281\) 5960.18 1.26532 0.632659 0.774430i \(-0.281963\pi\)
0.632659 + 0.774430i \(0.281963\pi\)
\(282\) 0 0
\(283\) 2151.83 0.451990 0.225995 0.974128i \(-0.427437\pi\)
0.225995 + 0.974128i \(0.427437\pi\)
\(284\) 0 0
\(285\) 951.461 0.197753
\(286\) 0 0
\(287\) −7576.46 −1.55827
\(288\) 0 0
\(289\) 10572.9 2.15203
\(290\) 0 0
\(291\) 1501.02 0.302375
\(292\) 0 0
\(293\) 6159.82 1.22819 0.614096 0.789231i \(-0.289521\pi\)
0.614096 + 0.789231i \(0.289521\pi\)
\(294\) 0 0
\(295\) −3843.69 −0.758605
\(296\) 0 0
\(297\) 1191.46 0.232780
\(298\) 0 0
\(299\) −3110.07 −0.601538
\(300\) 0 0
\(301\) −5537.03 −1.06030
\(302\) 0 0
\(303\) −305.315 −0.0578875
\(304\) 0 0
\(305\) 3147.07 0.590821
\(306\) 0 0
\(307\) −9672.80 −1.79823 −0.899114 0.437714i \(-0.855788\pi\)
−0.899114 + 0.437714i \(0.855788\pi\)
\(308\) 0 0
\(309\) 7022.04 1.29278
\(310\) 0 0
\(311\) −6369.85 −1.16142 −0.580709 0.814111i \(-0.697224\pi\)
−0.580709 + 0.814111i \(0.697224\pi\)
\(312\) 0 0
\(313\) −3346.98 −0.604417 −0.302209 0.953242i \(-0.597724\pi\)
−0.302209 + 0.953242i \(0.597724\pi\)
\(314\) 0 0
\(315\) 198.614 0.0355257
\(316\) 0 0
\(317\) 10720.3 1.89941 0.949706 0.313142i \(-0.101382\pi\)
0.949706 + 0.313142i \(0.101382\pi\)
\(318\) 0 0
\(319\) 2439.64 0.428193
\(320\) 0 0
\(321\) −3988.29 −0.693472
\(322\) 0 0
\(323\) 2364.40 0.407303
\(324\) 0 0
\(325\) 1579.55 0.269594
\(326\) 0 0
\(327\) −3432.26 −0.580442
\(328\) 0 0
\(329\) −4963.54 −0.831759
\(330\) 0 0
\(331\) 6992.48 1.16115 0.580576 0.814206i \(-0.302827\pi\)
0.580576 + 0.814206i \(0.302827\pi\)
\(332\) 0 0
\(333\) 23.3110 0.00383615
\(334\) 0 0
\(335\) −7098.98 −1.15779
\(336\) 0 0
\(337\) −7780.91 −1.25772 −0.628862 0.777517i \(-0.716479\pi\)
−0.628862 + 0.777517i \(0.716479\pi\)
\(338\) 0 0
\(339\) 12281.9 1.96773
\(340\) 0 0
\(341\) 2204.30 0.350057
\(342\) 0 0
\(343\) −6856.40 −1.07933
\(344\) 0 0
\(345\) −3587.38 −0.559821
\(346\) 0 0
\(347\) −2574.64 −0.398311 −0.199156 0.979968i \(-0.563820\pi\)
−0.199156 + 0.979968i \(0.563820\pi\)
\(348\) 0 0
\(349\) 4736.45 0.726465 0.363232 0.931699i \(-0.381673\pi\)
0.363232 + 0.931699i \(0.381673\pi\)
\(350\) 0 0
\(351\) −5935.69 −0.902632
\(352\) 0 0
\(353\) 8513.83 1.28370 0.641849 0.766831i \(-0.278167\pi\)
0.641849 + 0.766831i \(0.278167\pi\)
\(354\) 0 0
\(355\) −7144.49 −1.06814
\(356\) 0 0
\(357\) 10757.8 1.59485
\(358\) 0 0
\(359\) 7961.99 1.17052 0.585261 0.810845i \(-0.300992\pi\)
0.585261 + 0.810845i \(0.300992\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) 6676.43 0.965349
\(364\) 0 0
\(365\) −3708.37 −0.531794
\(366\) 0 0
\(367\) 2283.26 0.324755 0.162377 0.986729i \(-0.448084\pi\)
0.162377 + 0.986729i \(0.448084\pi\)
\(368\) 0 0
\(369\) −605.297 −0.0853943
\(370\) 0 0
\(371\) −4766.78 −0.667060
\(372\) 0 0
\(373\) 7740.69 1.07452 0.537262 0.843415i \(-0.319458\pi\)
0.537262 + 0.843415i \(0.319458\pi\)
\(374\) 0 0
\(375\) 8081.59 1.11288
\(376\) 0 0
\(377\) −12153.9 −1.66037
\(378\) 0 0
\(379\) 8461.28 1.14677 0.573386 0.819285i \(-0.305629\pi\)
0.573386 + 0.819285i \(0.305629\pi\)
\(380\) 0 0
\(381\) −1657.83 −0.222922
\(382\) 0 0
\(383\) 11065.4 1.47628 0.738139 0.674649i \(-0.235705\pi\)
0.738139 + 0.674649i \(0.235705\pi\)
\(384\) 0 0
\(385\) 1333.12 0.176473
\(386\) 0 0
\(387\) −442.363 −0.0581049
\(388\) 0 0
\(389\) −9051.42 −1.17976 −0.589878 0.807492i \(-0.700824\pi\)
−0.589878 + 0.807492i \(0.700824\pi\)
\(390\) 0 0
\(391\) −8914.73 −1.15304
\(392\) 0 0
\(393\) 2737.06 0.351313
\(394\) 0 0
\(395\) 863.289 0.109966
\(396\) 0 0
\(397\) 10777.0 1.36243 0.681213 0.732085i \(-0.261453\pi\)
0.681213 + 0.732085i \(0.261453\pi\)
\(398\) 0 0
\(399\) 1642.51 0.206086
\(400\) 0 0
\(401\) 610.869 0.0760732 0.0380366 0.999276i \(-0.487890\pi\)
0.0380366 + 0.999276i \(0.487890\pi\)
\(402\) 0 0
\(403\) −10981.5 −1.35739
\(404\) 0 0
\(405\) −7176.66 −0.880521
\(406\) 0 0
\(407\) 156.467 0.0190560
\(408\) 0 0
\(409\) −218.808 −0.0264532 −0.0132266 0.999913i \(-0.504210\pi\)
−0.0132266 + 0.999913i \(0.504210\pi\)
\(410\) 0 0
\(411\) 11711.1 1.40551
\(412\) 0 0
\(413\) −6635.38 −0.790571
\(414\) 0 0
\(415\) 13148.2 1.55522
\(416\) 0 0
\(417\) 1002.25 0.117699
\(418\) 0 0
\(419\) −4826.13 −0.562702 −0.281351 0.959605i \(-0.590782\pi\)
−0.281351 + 0.959605i \(0.590782\pi\)
\(420\) 0 0
\(421\) −9386.61 −1.08664 −0.543320 0.839526i \(-0.682833\pi\)
−0.543320 + 0.839526i \(0.682833\pi\)
\(422\) 0 0
\(423\) −396.546 −0.0455809
\(424\) 0 0
\(425\) 4527.65 0.516761
\(426\) 0 0
\(427\) 5432.80 0.615718
\(428\) 0 0
\(429\) 2012.55 0.226496
\(430\) 0 0
\(431\) 8452.72 0.944671 0.472336 0.881419i \(-0.343411\pi\)
0.472336 + 0.881419i \(0.343411\pi\)
\(432\) 0 0
\(433\) −10404.1 −1.15471 −0.577355 0.816494i \(-0.695915\pi\)
−0.577355 + 0.816494i \(0.695915\pi\)
\(434\) 0 0
\(435\) −14019.3 −1.54522
\(436\) 0 0
\(437\) −1361.11 −0.148995
\(438\) 0 0
\(439\) 9961.31 1.08298 0.541489 0.840708i \(-0.317861\pi\)
0.541489 + 0.840708i \(0.317861\pi\)
\(440\) 0 0
\(441\) −102.451 −0.0110627
\(442\) 0 0
\(443\) 8962.09 0.961178 0.480589 0.876946i \(-0.340423\pi\)
0.480589 + 0.876946i \(0.340423\pi\)
\(444\) 0 0
\(445\) 8766.28 0.933846
\(446\) 0 0
\(447\) 10650.3 1.12694
\(448\) 0 0
\(449\) 8431.24 0.886180 0.443090 0.896477i \(-0.353882\pi\)
0.443090 + 0.896477i \(0.353882\pi\)
\(450\) 0 0
\(451\) −4062.83 −0.424194
\(452\) 0 0
\(453\) 7043.90 0.730577
\(454\) 0 0
\(455\) −6641.43 −0.684296
\(456\) 0 0
\(457\) −13669.0 −1.39915 −0.699574 0.714560i \(-0.746627\pi\)
−0.699574 + 0.714560i \(0.746627\pi\)
\(458\) 0 0
\(459\) −17014.1 −1.73018
\(460\) 0 0
\(461\) 270.681 0.0273468 0.0136734 0.999907i \(-0.495647\pi\)
0.0136734 + 0.999907i \(0.495647\pi\)
\(462\) 0 0
\(463\) 17732.8 1.77995 0.889973 0.456013i \(-0.150723\pi\)
0.889973 + 0.456013i \(0.150723\pi\)
\(464\) 0 0
\(465\) −12666.9 −1.26325
\(466\) 0 0
\(467\) 55.8896 0.00553803 0.00276902 0.999996i \(-0.499119\pi\)
0.00276902 + 0.999996i \(0.499119\pi\)
\(468\) 0 0
\(469\) −12255.0 −1.20657
\(470\) 0 0
\(471\) −1919.27 −0.187760
\(472\) 0 0
\(473\) −2969.20 −0.288634
\(474\) 0 0
\(475\) 691.287 0.0667757
\(476\) 0 0
\(477\) −380.827 −0.0365553
\(478\) 0 0
\(479\) 2012.67 0.191986 0.0959929 0.995382i \(-0.469397\pi\)
0.0959929 + 0.995382i \(0.469397\pi\)
\(480\) 0 0
\(481\) −779.497 −0.0738919
\(482\) 0 0
\(483\) −6192.91 −0.583411
\(484\) 0 0
\(485\) −2656.21 −0.248685
\(486\) 0 0
\(487\) 4313.69 0.401380 0.200690 0.979655i \(-0.435682\pi\)
0.200690 + 0.979655i \(0.435682\pi\)
\(488\) 0 0
\(489\) 3376.03 0.312208
\(490\) 0 0
\(491\) −3762.75 −0.345846 −0.172923 0.984935i \(-0.555321\pi\)
−0.172923 + 0.984935i \(0.555321\pi\)
\(492\) 0 0
\(493\) −34838.2 −3.18263
\(494\) 0 0
\(495\) 106.505 0.00967083
\(496\) 0 0
\(497\) −12333.6 −1.11315
\(498\) 0 0
\(499\) −4203.71 −0.377122 −0.188561 0.982061i \(-0.560382\pi\)
−0.188561 + 0.982061i \(0.560382\pi\)
\(500\) 0 0
\(501\) −19896.0 −1.77423
\(502\) 0 0
\(503\) 19614.2 1.73868 0.869340 0.494215i \(-0.164545\pi\)
0.869340 + 0.494215i \(0.164545\pi\)
\(504\) 0 0
\(505\) 540.288 0.0476089
\(506\) 0 0
\(507\) 1660.93 0.145492
\(508\) 0 0
\(509\) −18720.4 −1.63019 −0.815093 0.579330i \(-0.803314\pi\)
−0.815093 + 0.579330i \(0.803314\pi\)
\(510\) 0 0
\(511\) −6401.78 −0.554203
\(512\) 0 0
\(513\) −2597.74 −0.223573
\(514\) 0 0
\(515\) −12426.3 −1.06324
\(516\) 0 0
\(517\) −2661.67 −0.226422
\(518\) 0 0
\(519\) −5206.93 −0.440383
\(520\) 0 0
\(521\) 20279.4 1.70529 0.852646 0.522489i \(-0.174996\pi\)
0.852646 + 0.522489i \(0.174996\pi\)
\(522\) 0 0
\(523\) 12501.4 1.04522 0.522608 0.852573i \(-0.324959\pi\)
0.522608 + 0.852573i \(0.324959\pi\)
\(524\) 0 0
\(525\) 3145.28 0.261469
\(526\) 0 0
\(527\) −31477.5 −2.60187
\(528\) 0 0
\(529\) −7035.08 −0.578210
\(530\) 0 0
\(531\) −530.113 −0.0433238
\(532\) 0 0
\(533\) 20240.5 1.64486
\(534\) 0 0
\(535\) 7057.71 0.570339
\(536\) 0 0
\(537\) 1379.54 0.110860
\(538\) 0 0
\(539\) −687.667 −0.0549535
\(540\) 0 0
\(541\) −11028.5 −0.876437 −0.438219 0.898868i \(-0.644390\pi\)
−0.438219 + 0.898868i \(0.644390\pi\)
\(542\) 0 0
\(543\) −4747.29 −0.375185
\(544\) 0 0
\(545\) 6073.75 0.477378
\(546\) 0 0
\(547\) −20679.4 −1.61643 −0.808215 0.588887i \(-0.799566\pi\)
−0.808215 + 0.588887i \(0.799566\pi\)
\(548\) 0 0
\(549\) 434.036 0.0337417
\(550\) 0 0
\(551\) −5319.14 −0.411258
\(552\) 0 0
\(553\) 1490.30 0.114600
\(554\) 0 0
\(555\) −899.129 −0.0687674
\(556\) 0 0
\(557\) −6393.51 −0.486359 −0.243179 0.969981i \(-0.578190\pi\)
−0.243179 + 0.969981i \(0.578190\pi\)
\(558\) 0 0
\(559\) 14792.2 1.11922
\(560\) 0 0
\(561\) 5768.81 0.434152
\(562\) 0 0
\(563\) 2530.58 0.189434 0.0947170 0.995504i \(-0.469805\pi\)
0.0947170 + 0.995504i \(0.469805\pi\)
\(564\) 0 0
\(565\) −21734.1 −1.61834
\(566\) 0 0
\(567\) −12389.1 −0.917625
\(568\) 0 0
\(569\) 18965.6 1.39733 0.698664 0.715450i \(-0.253778\pi\)
0.698664 + 0.715450i \(0.253778\pi\)
\(570\) 0 0
\(571\) −19660.1 −1.44089 −0.720445 0.693512i \(-0.756062\pi\)
−0.720445 + 0.693512i \(0.756062\pi\)
\(572\) 0 0
\(573\) 21388.9 1.55939
\(574\) 0 0
\(575\) −2606.42 −0.189035
\(576\) 0 0
\(577\) −5446.15 −0.392940 −0.196470 0.980510i \(-0.562948\pi\)
−0.196470 + 0.980510i \(0.562948\pi\)
\(578\) 0 0
\(579\) −2165.19 −0.155410
\(580\) 0 0
\(581\) 22697.7 1.62076
\(582\) 0 0
\(583\) −2556.16 −0.181587
\(584\) 0 0
\(585\) −530.596 −0.0374999
\(586\) 0 0
\(587\) −3622.30 −0.254699 −0.127349 0.991858i \(-0.540647\pi\)
−0.127349 + 0.991858i \(0.540647\pi\)
\(588\) 0 0
\(589\) −4806.03 −0.336212
\(590\) 0 0
\(591\) 19681.8 1.36989
\(592\) 0 0
\(593\) 19137.7 1.32528 0.662641 0.748938i \(-0.269436\pi\)
0.662641 + 0.748938i \(0.269436\pi\)
\(594\) 0 0
\(595\) −19037.1 −1.31167
\(596\) 0 0
\(597\) 9589.23 0.657388
\(598\) 0 0
\(599\) −4350.42 −0.296750 −0.148375 0.988931i \(-0.547404\pi\)
−0.148375 + 0.988931i \(0.547404\pi\)
\(600\) 0 0
\(601\) −10657.8 −0.723362 −0.361681 0.932302i \(-0.617797\pi\)
−0.361681 + 0.932302i \(0.617797\pi\)
\(602\) 0 0
\(603\) −979.074 −0.0661210
\(604\) 0 0
\(605\) −11814.7 −0.793941
\(606\) 0 0
\(607\) −5515.67 −0.368821 −0.184410 0.982849i \(-0.559038\pi\)
−0.184410 + 0.982849i \(0.559038\pi\)
\(608\) 0 0
\(609\) −24201.5 −1.61034
\(610\) 0 0
\(611\) 13260.1 0.877980
\(612\) 0 0
\(613\) −26144.0 −1.72259 −0.861293 0.508109i \(-0.830345\pi\)
−0.861293 + 0.508109i \(0.830345\pi\)
\(614\) 0 0
\(615\) 23346.9 1.53079
\(616\) 0 0
\(617\) −4104.01 −0.267781 −0.133891 0.990996i \(-0.542747\pi\)
−0.133891 + 0.990996i \(0.542747\pi\)
\(618\) 0 0
\(619\) 265.000 0.0172072 0.00860360 0.999963i \(-0.497261\pi\)
0.00860360 + 0.999963i \(0.497261\pi\)
\(620\) 0 0
\(621\) 9794.50 0.632914
\(622\) 0 0
\(623\) 15133.3 0.973197
\(624\) 0 0
\(625\) −9753.29 −0.624211
\(626\) 0 0
\(627\) 880.788 0.0561009
\(628\) 0 0
\(629\) −2234.36 −0.141637
\(630\) 0 0
\(631\) 23088.2 1.45662 0.728309 0.685249i \(-0.240307\pi\)
0.728309 + 0.685249i \(0.240307\pi\)
\(632\) 0 0
\(633\) −26772.9 −1.68109
\(634\) 0 0
\(635\) 2933.72 0.183340
\(636\) 0 0
\(637\) 3425.87 0.213089
\(638\) 0 0
\(639\) −985.351 −0.0610014
\(640\) 0 0
\(641\) 3574.05 0.220229 0.110114 0.993919i \(-0.464878\pi\)
0.110114 + 0.993919i \(0.464878\pi\)
\(642\) 0 0
\(643\) −25705.2 −1.57654 −0.788270 0.615329i \(-0.789023\pi\)
−0.788270 + 0.615329i \(0.789023\pi\)
\(644\) 0 0
\(645\) 17062.4 1.04160
\(646\) 0 0
\(647\) 1822.68 0.110752 0.0553762 0.998466i \(-0.482364\pi\)
0.0553762 + 0.998466i \(0.482364\pi\)
\(648\) 0 0
\(649\) −3558.19 −0.215210
\(650\) 0 0
\(651\) −21866.9 −1.31649
\(652\) 0 0
\(653\) −15678.9 −0.939609 −0.469804 0.882771i \(-0.655676\pi\)
−0.469804 + 0.882771i \(0.655676\pi\)
\(654\) 0 0
\(655\) −4843.51 −0.288934
\(656\) 0 0
\(657\) −511.450 −0.0303707
\(658\) 0 0
\(659\) 18817.0 1.11230 0.556151 0.831081i \(-0.312277\pi\)
0.556151 + 0.831081i \(0.312277\pi\)
\(660\) 0 0
\(661\) −2463.08 −0.144936 −0.0724680 0.997371i \(-0.523088\pi\)
−0.0724680 + 0.997371i \(0.523088\pi\)
\(662\) 0 0
\(663\) −28739.4 −1.68348
\(664\) 0 0
\(665\) −2906.60 −0.169493
\(666\) 0 0
\(667\) 20055.2 1.16423
\(668\) 0 0
\(669\) −2219.23 −0.128252
\(670\) 0 0
\(671\) 2913.31 0.167611
\(672\) 0 0
\(673\) −27177.3 −1.55662 −0.778312 0.627877i \(-0.783924\pi\)
−0.778312 + 0.627877i \(0.783924\pi\)
\(674\) 0 0
\(675\) −4974.47 −0.283656
\(676\) 0 0
\(677\) −3111.27 −0.176626 −0.0883131 0.996093i \(-0.528148\pi\)
−0.0883131 + 0.996093i \(0.528148\pi\)
\(678\) 0 0
\(679\) −4585.43 −0.259164
\(680\) 0 0
\(681\) 16846.8 0.947976
\(682\) 0 0
\(683\) 16068.7 0.900221 0.450110 0.892973i \(-0.351385\pi\)
0.450110 + 0.892973i \(0.351385\pi\)
\(684\) 0 0
\(685\) −20724.0 −1.15595
\(686\) 0 0
\(687\) 11692.5 0.649341
\(688\) 0 0
\(689\) 12734.5 0.704128
\(690\) 0 0
\(691\) −11854.9 −0.652653 −0.326327 0.945257i \(-0.605811\pi\)
−0.326327 + 0.945257i \(0.605811\pi\)
\(692\) 0 0
\(693\) 183.861 0.0100783
\(694\) 0 0
\(695\) −1773.59 −0.0968000
\(696\) 0 0
\(697\) 58017.6 3.15290
\(698\) 0 0
\(699\) −10408.9 −0.563233
\(700\) 0 0
\(701\) 19270.9 1.03830 0.519151 0.854682i \(-0.326248\pi\)
0.519151 + 0.854682i \(0.326248\pi\)
\(702\) 0 0
\(703\) −341.145 −0.0183023
\(704\) 0 0
\(705\) 15295.2 0.817091
\(706\) 0 0
\(707\) 932.702 0.0496151
\(708\) 0 0
\(709\) −952.832 −0.0504716 −0.0252358 0.999682i \(-0.508034\pi\)
−0.0252358 + 0.999682i \(0.508034\pi\)
\(710\) 0 0
\(711\) 119.063 0.00628017
\(712\) 0 0
\(713\) 18120.6 0.951784
\(714\) 0 0
\(715\) −3561.43 −0.186280
\(716\) 0 0
\(717\) −1908.18 −0.0993897
\(718\) 0 0
\(719\) 26859.5 1.39317 0.696585 0.717474i \(-0.254702\pi\)
0.696585 + 0.717474i \(0.254702\pi\)
\(720\) 0 0
\(721\) −21451.5 −1.10804
\(722\) 0 0
\(723\) −271.776 −0.0139799
\(724\) 0 0
\(725\) −10185.7 −0.521778
\(726\) 0 0
\(727\) −26684.9 −1.36133 −0.680667 0.732593i \(-0.738310\pi\)
−0.680667 + 0.732593i \(0.738310\pi\)
\(728\) 0 0
\(729\) 18647.7 0.947401
\(730\) 0 0
\(731\) 42400.4 2.14533
\(732\) 0 0
\(733\) 30013.3 1.51237 0.756183 0.654360i \(-0.227062\pi\)
0.756183 + 0.654360i \(0.227062\pi\)
\(734\) 0 0
\(735\) 3951.65 0.198311
\(736\) 0 0
\(737\) −6571.68 −0.328454
\(738\) 0 0
\(739\) 12404.1 0.617444 0.308722 0.951152i \(-0.400099\pi\)
0.308722 + 0.951152i \(0.400099\pi\)
\(740\) 0 0
\(741\) −4387.97 −0.217538
\(742\) 0 0
\(743\) −35368.0 −1.74634 −0.873168 0.487420i \(-0.837938\pi\)
−0.873168 + 0.487420i \(0.837938\pi\)
\(744\) 0 0
\(745\) −18846.8 −0.926836
\(746\) 0 0
\(747\) 1813.36 0.0888185
\(748\) 0 0
\(749\) 12183.8 0.594372
\(750\) 0 0
\(751\) −1974.41 −0.0959348 −0.0479674 0.998849i \(-0.515274\pi\)
−0.0479674 + 0.998849i \(0.515274\pi\)
\(752\) 0 0
\(753\) 2450.27 0.118583
\(754\) 0 0
\(755\) −12464.9 −0.600855
\(756\) 0 0
\(757\) 5328.85 0.255852 0.127926 0.991784i \(-0.459168\pi\)
0.127926 + 0.991784i \(0.459168\pi\)
\(758\) 0 0
\(759\) −3320.92 −0.158816
\(760\) 0 0
\(761\) −13374.6 −0.637092 −0.318546 0.947907i \(-0.603195\pi\)
−0.318546 + 0.947907i \(0.603195\pi\)
\(762\) 0 0
\(763\) 10485.2 0.497494
\(764\) 0 0
\(765\) −1520.91 −0.0718804
\(766\) 0 0
\(767\) 17726.4 0.834503
\(768\) 0 0
\(769\) −41658.8 −1.95352 −0.976760 0.214337i \(-0.931241\pi\)
−0.976760 + 0.214337i \(0.931241\pi\)
\(770\) 0 0
\(771\) −7177.55 −0.335270
\(772\) 0 0
\(773\) 24505.2 1.14022 0.570110 0.821569i \(-0.306901\pi\)
0.570110 + 0.821569i \(0.306901\pi\)
\(774\) 0 0
\(775\) −9203.17 −0.426565
\(776\) 0 0
\(777\) −1552.17 −0.0716652
\(778\) 0 0
\(779\) 8858.19 0.407417
\(780\) 0 0
\(781\) −6613.81 −0.303023
\(782\) 0 0
\(783\) 38276.3 1.74698
\(784\) 0 0
\(785\) 3396.35 0.154421
\(786\) 0 0
\(787\) 24681.7 1.11793 0.558963 0.829192i \(-0.311199\pi\)
0.558963 + 0.829192i \(0.311199\pi\)
\(788\) 0 0
\(789\) −1660.78 −0.0749369
\(790\) 0 0
\(791\) −37519.7 −1.68653
\(792\) 0 0
\(793\) −14513.7 −0.649933
\(794\) 0 0
\(795\) 14688.9 0.655296
\(796\) 0 0
\(797\) 7727.34 0.343434 0.171717 0.985146i \(-0.445069\pi\)
0.171717 + 0.985146i \(0.445069\pi\)
\(798\) 0 0
\(799\) 38008.9 1.68293
\(800\) 0 0
\(801\) 1209.02 0.0533318
\(802\) 0 0
\(803\) −3432.92 −0.150866
\(804\) 0 0
\(805\) 10959.0 0.479820
\(806\) 0 0
\(807\) −22228.0 −0.969596
\(808\) 0 0
\(809\) 4220.96 0.183438 0.0917188 0.995785i \(-0.470764\pi\)
0.0917188 + 0.995785i \(0.470764\pi\)
\(810\) 0 0
\(811\) −26026.3 −1.12689 −0.563445 0.826154i \(-0.690524\pi\)
−0.563445 + 0.826154i \(0.690524\pi\)
\(812\) 0 0
\(813\) −11387.0 −0.491215
\(814\) 0 0
\(815\) −5974.25 −0.256772
\(816\) 0 0
\(817\) 6473.75 0.277219
\(818\) 0 0
\(819\) −915.970 −0.0390801
\(820\) 0 0
\(821\) 13319.5 0.566204 0.283102 0.959090i \(-0.408636\pi\)
0.283102 + 0.959090i \(0.408636\pi\)
\(822\) 0 0
\(823\) −10436.6 −0.442038 −0.221019 0.975270i \(-0.570938\pi\)
−0.221019 + 0.975270i \(0.570938\pi\)
\(824\) 0 0
\(825\) 1686.64 0.0711774
\(826\) 0 0
\(827\) 8612.06 0.362117 0.181058 0.983472i \(-0.442048\pi\)
0.181058 + 0.983472i \(0.442048\pi\)
\(828\) 0 0
\(829\) 10298.1 0.431444 0.215722 0.976455i \(-0.430789\pi\)
0.215722 + 0.976455i \(0.430789\pi\)
\(830\) 0 0
\(831\) 33461.3 1.39682
\(832\) 0 0
\(833\) 9819.94 0.408452
\(834\) 0 0
\(835\) 35208.2 1.45920
\(836\) 0 0
\(837\) 34584.0 1.42819
\(838\) 0 0
\(839\) 14169.2 0.583045 0.291522 0.956564i \(-0.405838\pi\)
0.291522 + 0.956564i \(0.405838\pi\)
\(840\) 0 0
\(841\) 53985.6 2.21352
\(842\) 0 0
\(843\) −31705.9 −1.29538
\(844\) 0 0
\(845\) −2939.19 −0.119658
\(846\) 0 0
\(847\) −20395.7 −0.827397
\(848\) 0 0
\(849\) −11446.9 −0.462730
\(850\) 0 0
\(851\) 1286.25 0.0518120
\(852\) 0 0
\(853\) 9822.77 0.394285 0.197143 0.980375i \(-0.436834\pi\)
0.197143 + 0.980375i \(0.436834\pi\)
\(854\) 0 0
\(855\) −232.214 −0.00928835
\(856\) 0 0
\(857\) −11841.3 −0.471983 −0.235992 0.971755i \(-0.575834\pi\)
−0.235992 + 0.971755i \(0.575834\pi\)
\(858\) 0 0
\(859\) 44204.6 1.75581 0.877905 0.478835i \(-0.158941\pi\)
0.877905 + 0.478835i \(0.158941\pi\)
\(860\) 0 0
\(861\) 40303.8 1.59530
\(862\) 0 0
\(863\) 16756.6 0.660952 0.330476 0.943814i \(-0.392791\pi\)
0.330476 + 0.943814i \(0.392791\pi\)
\(864\) 0 0
\(865\) 9214.22 0.362188
\(866\) 0 0
\(867\) −56243.8 −2.20316
\(868\) 0 0
\(869\) 799.165 0.0311966
\(870\) 0 0
\(871\) 32739.2 1.27362
\(872\) 0 0
\(873\) −366.338 −0.0142024
\(874\) 0 0
\(875\) −24688.3 −0.953848
\(876\) 0 0
\(877\) −40606.0 −1.56347 −0.781737 0.623609i \(-0.785666\pi\)
−0.781737 + 0.623609i \(0.785666\pi\)
\(878\) 0 0
\(879\) −32767.9 −1.25737
\(880\) 0 0
\(881\) 18464.8 0.706124 0.353062 0.935600i \(-0.385140\pi\)
0.353062 + 0.935600i \(0.385140\pi\)
\(882\) 0 0
\(883\) −20592.3 −0.784809 −0.392404 0.919793i \(-0.628357\pi\)
−0.392404 + 0.919793i \(0.628357\pi\)
\(884\) 0 0
\(885\) 20447.0 0.776630
\(886\) 0 0
\(887\) −17682.8 −0.669370 −0.334685 0.942330i \(-0.608630\pi\)
−0.334685 + 0.942330i \(0.608630\pi\)
\(888\) 0 0
\(889\) 5064.49 0.191066
\(890\) 0 0
\(891\) −6643.59 −0.249796
\(892\) 0 0
\(893\) 5803.24 0.217467
\(894\) 0 0
\(895\) −2441.25 −0.0911753
\(896\) 0 0
\(897\) 16544.4 0.615830
\(898\) 0 0
\(899\) 70814.2 2.62713
\(900\) 0 0
\(901\) 36502.2 1.34968
\(902\) 0 0
\(903\) 29454.9 1.08549
\(904\) 0 0
\(905\) 8400.84 0.308567
\(906\) 0 0
\(907\) 19256.9 0.704978 0.352489 0.935816i \(-0.385335\pi\)
0.352489 + 0.935816i \(0.385335\pi\)
\(908\) 0 0
\(909\) 74.5153 0.00271894
\(910\) 0 0
\(911\) 24122.8 0.877306 0.438653 0.898657i \(-0.355456\pi\)
0.438653 + 0.898657i \(0.355456\pi\)
\(912\) 0 0
\(913\) 12171.5 0.441204
\(914\) 0 0
\(915\) −16741.2 −0.604860
\(916\) 0 0
\(917\) −8361.38 −0.301109
\(918\) 0 0
\(919\) −16780.0 −0.602309 −0.301154 0.953575i \(-0.597372\pi\)
−0.301154 + 0.953575i \(0.597372\pi\)
\(920\) 0 0
\(921\) 51455.6 1.84095
\(922\) 0 0
\(923\) 32949.1 1.17501
\(924\) 0 0
\(925\) −653.266 −0.0232208
\(926\) 0 0
\(927\) −1713.80 −0.0607212
\(928\) 0 0
\(929\) 32219.1 1.13786 0.568931 0.822386i \(-0.307357\pi\)
0.568931 + 0.822386i \(0.307357\pi\)
\(930\) 0 0
\(931\) 1499.32 0.0527801
\(932\) 0 0
\(933\) 33885.1 1.18901
\(934\) 0 0
\(935\) −10208.5 −0.357064
\(936\) 0 0
\(937\) −36861.2 −1.28517 −0.642584 0.766215i \(-0.722138\pi\)
−0.642584 + 0.766215i \(0.722138\pi\)
\(938\) 0 0
\(939\) 17804.7 0.618779
\(940\) 0 0
\(941\) 1000.08 0.0346460 0.0173230 0.999850i \(-0.494486\pi\)
0.0173230 + 0.999850i \(0.494486\pi\)
\(942\) 0 0
\(943\) −33398.8 −1.15336
\(944\) 0 0
\(945\) 20915.8 0.719989
\(946\) 0 0
\(947\) −50954.9 −1.74848 −0.874240 0.485494i \(-0.838640\pi\)
−0.874240 + 0.485494i \(0.838640\pi\)
\(948\) 0 0
\(949\) 17102.3 0.585000
\(950\) 0 0
\(951\) −57028.0 −1.94454
\(952\) 0 0
\(953\) −34671.8 −1.17852 −0.589260 0.807943i \(-0.700581\pi\)
−0.589260 + 0.807943i \(0.700581\pi\)
\(954\) 0 0
\(955\) −37849.9 −1.28251
\(956\) 0 0
\(957\) −12977.9 −0.438367
\(958\) 0 0
\(959\) −35776.0 −1.20466
\(960\) 0 0
\(961\) 34192.1 1.14773
\(962\) 0 0
\(963\) 973.382 0.0325720
\(964\) 0 0
\(965\) 3831.54 0.127815
\(966\) 0 0
\(967\) −1487.32 −0.0494610 −0.0247305 0.999694i \(-0.507873\pi\)
−0.0247305 + 0.999694i \(0.507873\pi\)
\(968\) 0 0
\(969\) −12577.7 −0.416981
\(970\) 0 0
\(971\) −19161.1 −0.633273 −0.316636 0.948547i \(-0.602553\pi\)
−0.316636 + 0.948547i \(0.602553\pi\)
\(972\) 0 0
\(973\) −3061.75 −0.100879
\(974\) 0 0
\(975\) −8402.62 −0.275999
\(976\) 0 0
\(977\) −29747.5 −0.974110 −0.487055 0.873371i \(-0.661929\pi\)
−0.487055 + 0.873371i \(0.661929\pi\)
\(978\) 0 0
\(979\) 8115.14 0.264924
\(980\) 0 0
\(981\) 837.678 0.0272630
\(982\) 0 0
\(983\) −58843.1 −1.90926 −0.954631 0.297791i \(-0.903750\pi\)
−0.954631 + 0.297791i \(0.903750\pi\)
\(984\) 0 0
\(985\) −34829.1 −1.12665
\(986\) 0 0
\(987\) 26404.1 0.851522
\(988\) 0 0
\(989\) −24408.6 −0.784780
\(990\) 0 0
\(991\) 48869.6 1.56649 0.783247 0.621711i \(-0.213562\pi\)
0.783247 + 0.621711i \(0.213562\pi\)
\(992\) 0 0
\(993\) −37197.3 −1.18874
\(994\) 0 0
\(995\) −16969.2 −0.540662
\(996\) 0 0
\(997\) −2207.55 −0.0701243 −0.0350621 0.999385i \(-0.511163\pi\)
−0.0350621 + 0.999385i \(0.511163\pi\)
\(998\) 0 0
\(999\) 2454.86 0.0777461
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 608.4.a.h.1.1 yes 5
4.3 odd 2 608.4.a.e.1.5 5
8.3 odd 2 1216.4.a.bd.1.1 5
8.5 even 2 1216.4.a.y.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.4.a.e.1.5 5 4.3 odd 2
608.4.a.h.1.1 yes 5 1.1 even 1 trivial
1216.4.a.y.1.5 5 8.5 even 2
1216.4.a.bd.1.1 5 8.3 odd 2