Properties

Label 608.4.a.h.1.3
Level $608$
Weight $4$
Character 608.1
Self dual yes
Analytic conductor $35.873$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.3
Root \(-3.54663\) 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)
 
\(f(q)\) \(=\) \(q+2.55330 q^{3} +7.40075 q^{5} +10.4962 q^{7} -20.4807 q^{9} -48.7928 q^{11} +4.45376 q^{13} +18.8963 q^{15} -61.5346 q^{17} -19.0000 q^{19} +26.7999 q^{21} -34.8416 q^{23} -70.2288 q^{25} -121.232 q^{27} -179.528 q^{29} -132.845 q^{31} -124.583 q^{33} +77.6798 q^{35} -99.3918 q^{37} +11.3718 q^{39} +382.310 q^{41} +129.659 q^{43} -151.572 q^{45} +488.106 q^{47} -232.830 q^{49} -157.116 q^{51} -129.199 q^{53} -361.104 q^{55} -48.5127 q^{57} +479.346 q^{59} -394.879 q^{61} -214.969 q^{63} +32.9612 q^{65} +19.8601 q^{67} -88.9610 q^{69} -221.694 q^{71} +258.902 q^{73} -179.315 q^{75} -512.139 q^{77} -469.595 q^{79} +243.436 q^{81} +373.014 q^{83} -455.402 q^{85} -458.390 q^{87} -1609.29 q^{89} +46.7476 q^{91} -339.193 q^{93} -140.614 q^{95} -1654.62 q^{97} +999.310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{3} + 5 q^{5} - 7 q^{7} - 5 q^{9} + 13 q^{11} - 72 q^{13} - 72 q^{15} - 59 q^{17} - 95 q^{19} - 224 q^{21} - 52 q^{23} - 86 q^{25} + 54 q^{27} - 128 q^{29} + 110 q^{31} - 68 q^{33} - 45 q^{35}+ \cdots + 601 q^{99}+O(q^{100}) \) 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
\(3\) 2.55330 0.491382 0.245691 0.969348i \(-0.420985\pi\)
0.245691 + 0.969348i \(0.420985\pi\)
\(4\) 0 0
\(5\) 7.40075 0.661944 0.330972 0.943641i \(-0.392623\pi\)
0.330972 + 0.943641i \(0.392623\pi\)
\(6\) 0 0
\(7\) 10.4962 0.566741 0.283371 0.959010i \(-0.408547\pi\)
0.283371 + 0.959010i \(0.408547\pi\)
\(8\) 0 0
\(9\) −20.4807 −0.758543
\(10\) 0 0
\(11\) −48.7928 −1.33742 −0.668709 0.743524i \(-0.733153\pi\)
−0.668709 + 0.743524i \(0.733153\pi\)
\(12\) 0 0
\(13\) 4.45376 0.0950194 0.0475097 0.998871i \(-0.484872\pi\)
0.0475097 + 0.998871i \(0.484872\pi\)
\(14\) 0 0
\(15\) 18.8963 0.325267
\(16\) 0 0
\(17\) −61.5346 −0.877901 −0.438951 0.898511i \(-0.644650\pi\)
−0.438951 + 0.898511i \(0.644650\pi\)
\(18\) 0 0
\(19\) −19.0000 −0.229416
\(20\) 0 0
\(21\) 26.7999 0.278487
\(22\) 0 0
\(23\) −34.8416 −0.315869 −0.157934 0.987450i \(-0.550483\pi\)
−0.157934 + 0.987450i \(0.550483\pi\)
\(24\) 0 0
\(25\) −70.2288 −0.561831
\(26\) 0 0
\(27\) −121.232 −0.864117
\(28\) 0 0
\(29\) −179.528 −1.14957 −0.574786 0.818304i \(-0.694915\pi\)
−0.574786 + 0.818304i \(0.694915\pi\)
\(30\) 0 0
\(31\) −132.845 −0.769667 −0.384833 0.922986i \(-0.625741\pi\)
−0.384833 + 0.922986i \(0.625741\pi\)
\(32\) 0 0
\(33\) −124.583 −0.657184
\(34\) 0 0
\(35\) 77.6798 0.375151
\(36\) 0 0
\(37\) −99.3918 −0.441619 −0.220810 0.975317i \(-0.570870\pi\)
−0.220810 + 0.975317i \(0.570870\pi\)
\(38\) 0 0
\(39\) 11.3718 0.0466908
\(40\) 0 0
\(41\) 382.310 1.45626 0.728132 0.685437i \(-0.240389\pi\)
0.728132 + 0.685437i \(0.240389\pi\)
\(42\) 0 0
\(43\) 129.659 0.459834 0.229917 0.973210i \(-0.426154\pi\)
0.229917 + 0.973210i \(0.426154\pi\)
\(44\) 0 0
\(45\) −151.572 −0.502113
\(46\) 0 0
\(47\) 488.106 1.51484 0.757420 0.652928i \(-0.226459\pi\)
0.757420 + 0.652928i \(0.226459\pi\)
\(48\) 0 0
\(49\) −232.830 −0.678804
\(50\) 0 0
\(51\) −157.116 −0.431385
\(52\) 0 0
\(53\) −129.199 −0.334846 −0.167423 0.985885i \(-0.553545\pi\)
−0.167423 + 0.985885i \(0.553545\pi\)
\(54\) 0 0
\(55\) −361.104 −0.885295
\(56\) 0 0
\(57\) −48.5127 −0.112731
\(58\) 0 0
\(59\) 479.346 1.05772 0.528861 0.848709i \(-0.322619\pi\)
0.528861 + 0.848709i \(0.322619\pi\)
\(60\) 0 0
\(61\) −394.879 −0.828836 −0.414418 0.910087i \(-0.636015\pi\)
−0.414418 + 0.910087i \(0.636015\pi\)
\(62\) 0 0
\(63\) −214.969 −0.429898
\(64\) 0 0
\(65\) 32.9612 0.0628975
\(66\) 0 0
\(67\) 19.8601 0.0362135 0.0181067 0.999836i \(-0.494236\pi\)
0.0181067 + 0.999836i \(0.494236\pi\)
\(68\) 0 0
\(69\) −88.9610 −0.155212
\(70\) 0 0
\(71\) −221.694 −0.370567 −0.185283 0.982685i \(-0.559320\pi\)
−0.185283 + 0.982685i \(0.559320\pi\)
\(72\) 0 0
\(73\) 258.902 0.415099 0.207549 0.978225i \(-0.433451\pi\)
0.207549 + 0.978225i \(0.433451\pi\)
\(74\) 0 0
\(75\) −179.315 −0.276074
\(76\) 0 0
\(77\) −512.139 −0.757970
\(78\) 0 0
\(79\) −469.595 −0.668780 −0.334390 0.942435i \(-0.608530\pi\)
−0.334390 + 0.942435i \(0.608530\pi\)
\(80\) 0 0
\(81\) 243.436 0.333932
\(82\) 0 0
\(83\) 373.014 0.493296 0.246648 0.969105i \(-0.420671\pi\)
0.246648 + 0.969105i \(0.420671\pi\)
\(84\) 0 0
\(85\) −455.402 −0.581121
\(86\) 0 0
\(87\) −458.390 −0.564880
\(88\) 0 0
\(89\) −1609.29 −1.91668 −0.958340 0.285631i \(-0.907797\pi\)
−0.958340 + 0.285631i \(0.907797\pi\)
\(90\) 0 0
\(91\) 46.7476 0.0538514
\(92\) 0 0
\(93\) −339.193 −0.378201
\(94\) 0 0
\(95\) −140.614 −0.151860
\(96\) 0 0
\(97\) −1654.62 −1.73198 −0.865988 0.500065i \(-0.833309\pi\)
−0.865988 + 0.500065i \(0.833309\pi\)
\(98\) 0 0
\(99\) 999.310 1.01449
\(100\) 0 0
\(101\) −660.681 −0.650893 −0.325447 0.945560i \(-0.605515\pi\)
−0.325447 + 0.945560i \(0.605515\pi\)
\(102\) 0 0
\(103\) 99.1397 0.0948401 0.0474200 0.998875i \(-0.484900\pi\)
0.0474200 + 0.998875i \(0.484900\pi\)
\(104\) 0 0
\(105\) 198.340 0.184342
\(106\) 0 0
\(107\) 1584.99 1.43202 0.716011 0.698089i \(-0.245966\pi\)
0.716011 + 0.698089i \(0.245966\pi\)
\(108\) 0 0
\(109\) −203.005 −0.178389 −0.0891943 0.996014i \(-0.528429\pi\)
−0.0891943 + 0.996014i \(0.528429\pi\)
\(110\) 0 0
\(111\) −253.777 −0.217004
\(112\) 0 0
\(113\) −1849.28 −1.53952 −0.769761 0.638332i \(-0.779625\pi\)
−0.769761 + 0.638332i \(0.779625\pi\)
\(114\) 0 0
\(115\) −257.854 −0.209087
\(116\) 0 0
\(117\) −91.2161 −0.0720763
\(118\) 0 0
\(119\) −645.879 −0.497543
\(120\) 0 0
\(121\) 1049.74 0.788687
\(122\) 0 0
\(123\) 976.151 0.715582
\(124\) 0 0
\(125\) −1444.84 −1.03384
\(126\) 0 0
\(127\) 1739.70 1.21554 0.607769 0.794114i \(-0.292065\pi\)
0.607769 + 0.794114i \(0.292065\pi\)
\(128\) 0 0
\(129\) 331.059 0.225954
\(130\) 0 0
\(131\) 1490.46 0.994060 0.497030 0.867733i \(-0.334424\pi\)
0.497030 + 0.867733i \(0.334424\pi\)
\(132\) 0 0
\(133\) −199.428 −0.130019
\(134\) 0 0
\(135\) −897.210 −0.571997
\(136\) 0 0
\(137\) 866.483 0.540355 0.270178 0.962810i \(-0.412918\pi\)
0.270178 + 0.962810i \(0.412918\pi\)
\(138\) 0 0
\(139\) −498.851 −0.304403 −0.152202 0.988349i \(-0.548636\pi\)
−0.152202 + 0.988349i \(0.548636\pi\)
\(140\) 0 0
\(141\) 1246.28 0.744366
\(142\) 0 0
\(143\) −217.312 −0.127081
\(144\) 0 0
\(145\) −1328.65 −0.760952
\(146\) 0 0
\(147\) −594.484 −0.333552
\(148\) 0 0
\(149\) 696.388 0.382888 0.191444 0.981504i \(-0.438683\pi\)
0.191444 + 0.981504i \(0.438683\pi\)
\(150\) 0 0
\(151\) −2834.12 −1.52740 −0.763699 0.645573i \(-0.776619\pi\)
−0.763699 + 0.645573i \(0.776619\pi\)
\(152\) 0 0
\(153\) 1260.27 0.665926
\(154\) 0 0
\(155\) −983.153 −0.509476
\(156\) 0 0
\(157\) 2337.71 1.18834 0.594172 0.804338i \(-0.297480\pi\)
0.594172 + 0.804338i \(0.297480\pi\)
\(158\) 0 0
\(159\) −329.883 −0.164537
\(160\) 0 0
\(161\) −365.704 −0.179016
\(162\) 0 0
\(163\) −649.785 −0.312240 −0.156120 0.987738i \(-0.549899\pi\)
−0.156120 + 0.987738i \(0.549899\pi\)
\(164\) 0 0
\(165\) −922.006 −0.435018
\(166\) 0 0
\(167\) −3101.68 −1.43722 −0.718609 0.695415i \(-0.755221\pi\)
−0.718609 + 0.695415i \(0.755221\pi\)
\(168\) 0 0
\(169\) −2177.16 −0.990971
\(170\) 0 0
\(171\) 389.133 0.174022
\(172\) 0 0
\(173\) −226.436 −0.0995123 −0.0497561 0.998761i \(-0.515844\pi\)
−0.0497561 + 0.998761i \(0.515844\pi\)
\(174\) 0 0
\(175\) −737.135 −0.318413
\(176\) 0 0
\(177\) 1223.91 0.519746
\(178\) 0 0
\(179\) 2519.69 1.05213 0.526063 0.850446i \(-0.323668\pi\)
0.526063 + 0.850446i \(0.323668\pi\)
\(180\) 0 0
\(181\) −1012.54 −0.415809 −0.207904 0.978149i \(-0.566664\pi\)
−0.207904 + 0.978149i \(0.566664\pi\)
\(182\) 0 0
\(183\) −1008.24 −0.407276
\(184\) 0 0
\(185\) −735.575 −0.292327
\(186\) 0 0
\(187\) 3002.45 1.17412
\(188\) 0 0
\(189\) −1272.48 −0.489731
\(190\) 0 0
\(191\) −1941.43 −0.735480 −0.367740 0.929929i \(-0.619868\pi\)
−0.367740 + 0.929929i \(0.619868\pi\)
\(192\) 0 0
\(193\) 2069.72 0.771926 0.385963 0.922514i \(-0.373869\pi\)
0.385963 + 0.922514i \(0.373869\pi\)
\(194\) 0 0
\(195\) 84.1598 0.0309067
\(196\) 0 0
\(197\) 4208.36 1.52200 0.760998 0.648754i \(-0.224710\pi\)
0.760998 + 0.648754i \(0.224710\pi\)
\(198\) 0 0
\(199\) 267.700 0.0953604 0.0476802 0.998863i \(-0.484817\pi\)
0.0476802 + 0.998863i \(0.484817\pi\)
\(200\) 0 0
\(201\) 50.7089 0.0177947
\(202\) 0 0
\(203\) −1884.37 −0.651510
\(204\) 0 0
\(205\) 2829.38 0.963964
\(206\) 0 0
\(207\) 713.579 0.239600
\(208\) 0 0
\(209\) 927.064 0.306825
\(210\) 0 0
\(211\) −1259.32 −0.410876 −0.205438 0.978670i \(-0.565862\pi\)
−0.205438 + 0.978670i \(0.565862\pi\)
\(212\) 0 0
\(213\) −566.051 −0.182090
\(214\) 0 0
\(215\) 959.578 0.304384
\(216\) 0 0
\(217\) −1394.37 −0.436202
\(218\) 0 0
\(219\) 661.054 0.203972
\(220\) 0 0
\(221\) −274.060 −0.0834176
\(222\) 0 0
\(223\) 3639.90 1.09303 0.546516 0.837449i \(-0.315954\pi\)
0.546516 + 0.837449i \(0.315954\pi\)
\(224\) 0 0
\(225\) 1438.33 0.426173
\(226\) 0 0
\(227\) 1873.77 0.547870 0.273935 0.961748i \(-0.411675\pi\)
0.273935 + 0.961748i \(0.411675\pi\)
\(228\) 0 0
\(229\) 4120.43 1.18902 0.594510 0.804088i \(-0.297346\pi\)
0.594510 + 0.804088i \(0.297346\pi\)
\(230\) 0 0
\(231\) −1307.64 −0.372453
\(232\) 0 0
\(233\) 562.027 0.158024 0.0790121 0.996874i \(-0.474823\pi\)
0.0790121 + 0.996874i \(0.474823\pi\)
\(234\) 0 0
\(235\) 3612.35 1.00274
\(236\) 0 0
\(237\) −1199.02 −0.328627
\(238\) 0 0
\(239\) −4754.65 −1.28683 −0.643415 0.765517i \(-0.722483\pi\)
−0.643415 + 0.765517i \(0.722483\pi\)
\(240\) 0 0
\(241\) 806.451 0.215552 0.107776 0.994175i \(-0.465627\pi\)
0.107776 + 0.994175i \(0.465627\pi\)
\(242\) 0 0
\(243\) 3894.84 1.02821
\(244\) 0 0
\(245\) −1723.12 −0.449330
\(246\) 0 0
\(247\) −84.6215 −0.0217989
\(248\) 0 0
\(249\) 952.415 0.242397
\(250\) 0 0
\(251\) −2484.87 −0.624876 −0.312438 0.949938i \(-0.601146\pi\)
−0.312438 + 0.949938i \(0.601146\pi\)
\(252\) 0 0
\(253\) 1700.02 0.422448
\(254\) 0 0
\(255\) −1162.78 −0.285553
\(256\) 0 0
\(257\) 3530.59 0.856933 0.428467 0.903558i \(-0.359054\pi\)
0.428467 + 0.903558i \(0.359054\pi\)
\(258\) 0 0
\(259\) −1043.24 −0.250284
\(260\) 0 0
\(261\) 3676.86 0.872001
\(262\) 0 0
\(263\) 3025.99 0.709470 0.354735 0.934967i \(-0.384571\pi\)
0.354735 + 0.934967i \(0.384571\pi\)
\(264\) 0 0
\(265\) −956.169 −0.221649
\(266\) 0 0
\(267\) −4109.00 −0.941822
\(268\) 0 0
\(269\) 3179.99 0.720772 0.360386 0.932803i \(-0.382645\pi\)
0.360386 + 0.932803i \(0.382645\pi\)
\(270\) 0 0
\(271\) −5796.15 −1.29923 −0.649614 0.760264i \(-0.725069\pi\)
−0.649614 + 0.760264i \(0.725069\pi\)
\(272\) 0 0
\(273\) 119.360 0.0264616
\(274\) 0 0
\(275\) 3426.66 0.751402
\(276\) 0 0
\(277\) −5143.01 −1.11557 −0.557786 0.829984i \(-0.688349\pi\)
−0.557786 + 0.829984i \(0.688349\pi\)
\(278\) 0 0
\(279\) 2720.75 0.583826
\(280\) 0 0
\(281\) 2121.08 0.450296 0.225148 0.974325i \(-0.427713\pi\)
0.225148 + 0.974325i \(0.427713\pi\)
\(282\) 0 0
\(283\) 5446.02 1.14393 0.571965 0.820278i \(-0.306181\pi\)
0.571965 + 0.820278i \(0.306181\pi\)
\(284\) 0 0
\(285\) −359.030 −0.0746215
\(286\) 0 0
\(287\) 4012.80 0.825324
\(288\) 0 0
\(289\) −1126.50 −0.229289
\(290\) 0 0
\(291\) −4224.75 −0.851062
\(292\) 0 0
\(293\) 6125.16 1.22128 0.610641 0.791908i \(-0.290912\pi\)
0.610641 + 0.791908i \(0.290912\pi\)
\(294\) 0 0
\(295\) 3547.53 0.700152
\(296\) 0 0
\(297\) 5915.27 1.15569
\(298\) 0 0
\(299\) −155.176 −0.0300136
\(300\) 0 0
\(301\) 1360.93 0.260607
\(302\) 0 0
\(303\) −1686.92 −0.319837
\(304\) 0 0
\(305\) −2922.40 −0.548643
\(306\) 0 0
\(307\) −1263.62 −0.234914 −0.117457 0.993078i \(-0.537474\pi\)
−0.117457 + 0.993078i \(0.537474\pi\)
\(308\) 0 0
\(309\) 253.133 0.0466027
\(310\) 0 0
\(311\) −4954.26 −0.903313 −0.451657 0.892192i \(-0.649167\pi\)
−0.451657 + 0.892192i \(0.649167\pi\)
\(312\) 0 0
\(313\) 4740.07 0.855989 0.427995 0.903781i \(-0.359220\pi\)
0.427995 + 0.903781i \(0.359220\pi\)
\(314\) 0 0
\(315\) −1590.93 −0.284568
\(316\) 0 0
\(317\) −4986.86 −0.883564 −0.441782 0.897122i \(-0.645654\pi\)
−0.441782 + 0.897122i \(0.645654\pi\)
\(318\) 0 0
\(319\) 8759.71 1.53746
\(320\) 0 0
\(321\) 4046.94 0.703671
\(322\) 0 0
\(323\) 1169.16 0.201404
\(324\) 0 0
\(325\) −312.783 −0.0533848
\(326\) 0 0
\(327\) −518.332 −0.0876570
\(328\) 0 0
\(329\) 5123.25 0.858523
\(330\) 0 0
\(331\) 4783.52 0.794339 0.397169 0.917745i \(-0.369993\pi\)
0.397169 + 0.917745i \(0.369993\pi\)
\(332\) 0 0
\(333\) 2035.61 0.334987
\(334\) 0 0
\(335\) 146.980 0.0239713
\(336\) 0 0
\(337\) 5653.91 0.913912 0.456956 0.889489i \(-0.348940\pi\)
0.456956 + 0.889489i \(0.348940\pi\)
\(338\) 0 0
\(339\) −4721.77 −0.756494
\(340\) 0 0
\(341\) 6481.88 1.02937
\(342\) 0 0
\(343\) −6044.02 −0.951448
\(344\) 0 0
\(345\) −658.378 −0.102742
\(346\) 0 0
\(347\) 12278.2 1.89951 0.949753 0.312999i \(-0.101334\pi\)
0.949753 + 0.312999i \(0.101334\pi\)
\(348\) 0 0
\(349\) −12930.5 −1.98324 −0.991621 0.129180i \(-0.958765\pi\)
−0.991621 + 0.129180i \(0.958765\pi\)
\(350\) 0 0
\(351\) −539.940 −0.0821079
\(352\) 0 0
\(353\) 1583.46 0.238751 0.119375 0.992849i \(-0.461911\pi\)
0.119375 + 0.992849i \(0.461911\pi\)
\(354\) 0 0
\(355\) −1640.70 −0.245294
\(356\) 0 0
\(357\) −1649.12 −0.244484
\(358\) 0 0
\(359\) −8248.96 −1.21271 −0.606355 0.795194i \(-0.707369\pi\)
−0.606355 + 0.795194i \(0.707369\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) 2680.30 0.387547
\(364\) 0 0
\(365\) 1916.07 0.274772
\(366\) 0 0
\(367\) 1330.75 0.189277 0.0946387 0.995512i \(-0.469830\pi\)
0.0946387 + 0.995512i \(0.469830\pi\)
\(368\) 0 0
\(369\) −7829.97 −1.10464
\(370\) 0 0
\(371\) −1356.10 −0.189771
\(372\) 0 0
\(373\) −262.288 −0.0364095 −0.0182048 0.999834i \(-0.505795\pi\)
−0.0182048 + 0.999834i \(0.505795\pi\)
\(374\) 0 0
\(375\) −3689.11 −0.508013
\(376\) 0 0
\(377\) −799.577 −0.109232
\(378\) 0 0
\(379\) −808.151 −0.109530 −0.0547651 0.998499i \(-0.517441\pi\)
−0.0547651 + 0.998499i \(0.517441\pi\)
\(380\) 0 0
\(381\) 4441.97 0.597294
\(382\) 0 0
\(383\) −1379.74 −0.184076 −0.0920382 0.995755i \(-0.529338\pi\)
−0.0920382 + 0.995755i \(0.529338\pi\)
\(384\) 0 0
\(385\) −3790.22 −0.501733
\(386\) 0 0
\(387\) −2655.51 −0.348804
\(388\) 0 0
\(389\) 15154.6 1.97524 0.987618 0.156880i \(-0.0501436\pi\)
0.987618 + 0.156880i \(0.0501436\pi\)
\(390\) 0 0
\(391\) 2143.96 0.277301
\(392\) 0 0
\(393\) 3805.58 0.488463
\(394\) 0 0
\(395\) −3475.36 −0.442695
\(396\) 0 0
\(397\) −3851.15 −0.486861 −0.243431 0.969918i \(-0.578273\pi\)
−0.243431 + 0.969918i \(0.578273\pi\)
\(398\) 0 0
\(399\) −509.198 −0.0638892
\(400\) 0 0
\(401\) 539.933 0.0672394 0.0336197 0.999435i \(-0.489297\pi\)
0.0336197 + 0.999435i \(0.489297\pi\)
\(402\) 0 0
\(403\) −591.660 −0.0731332
\(404\) 0 0
\(405\) 1801.61 0.221044
\(406\) 0 0
\(407\) 4849.61 0.590630
\(408\) 0 0
\(409\) −8778.60 −1.06130 −0.530652 0.847590i \(-0.678053\pi\)
−0.530652 + 0.847590i \(0.678053\pi\)
\(410\) 0 0
\(411\) 2212.39 0.265521
\(412\) 0 0
\(413\) 5031.31 0.599455
\(414\) 0 0
\(415\) 2760.58 0.326534
\(416\) 0 0
\(417\) −1273.72 −0.149578
\(418\) 0 0
\(419\) −13534.2 −1.57802 −0.789010 0.614380i \(-0.789406\pi\)
−0.789010 + 0.614380i \(0.789406\pi\)
\(420\) 0 0
\(421\) −4055.19 −0.469448 −0.234724 0.972062i \(-0.575419\pi\)
−0.234724 + 0.972062i \(0.575419\pi\)
\(422\) 0 0
\(423\) −9996.73 −1.14907
\(424\) 0 0
\(425\) 4321.50 0.493232
\(426\) 0 0
\(427\) −4144.72 −0.469736
\(428\) 0 0
\(429\) −554.862 −0.0624452
\(430\) 0 0
\(431\) 14601.5 1.63186 0.815929 0.578153i \(-0.196226\pi\)
0.815929 + 0.578153i \(0.196226\pi\)
\(432\) 0 0
\(433\) −3546.74 −0.393638 −0.196819 0.980440i \(-0.563061\pi\)
−0.196819 + 0.980440i \(0.563061\pi\)
\(434\) 0 0
\(435\) −3392.43 −0.373918
\(436\) 0 0
\(437\) 661.990 0.0724652
\(438\) 0 0
\(439\) 4947.47 0.537881 0.268941 0.963157i \(-0.413326\pi\)
0.268941 + 0.963157i \(0.413326\pi\)
\(440\) 0 0
\(441\) 4768.51 0.514903
\(442\) 0 0
\(443\) 3858.07 0.413776 0.206888 0.978365i \(-0.433666\pi\)
0.206888 + 0.978365i \(0.433666\pi\)
\(444\) 0 0
\(445\) −11910.0 −1.26873
\(446\) 0 0
\(447\) 1778.08 0.188144
\(448\) 0 0
\(449\) −5045.14 −0.530278 −0.265139 0.964210i \(-0.585418\pi\)
−0.265139 + 0.964210i \(0.585418\pi\)
\(450\) 0 0
\(451\) −18654.0 −1.94763
\(452\) 0 0
\(453\) −7236.34 −0.750536
\(454\) 0 0
\(455\) 345.967 0.0356466
\(456\) 0 0
\(457\) −8779.52 −0.898662 −0.449331 0.893365i \(-0.648338\pi\)
−0.449331 + 0.893365i \(0.648338\pi\)
\(458\) 0 0
\(459\) 7459.97 0.758610
\(460\) 0 0
\(461\) −16829.5 −1.70028 −0.850138 0.526561i \(-0.823481\pi\)
−0.850138 + 0.526561i \(0.823481\pi\)
\(462\) 0 0
\(463\) 6166.41 0.618957 0.309479 0.950906i \(-0.399845\pi\)
0.309479 + 0.950906i \(0.399845\pi\)
\(464\) 0 0
\(465\) −2510.28 −0.250347
\(466\) 0 0
\(467\) −2117.84 −0.209854 −0.104927 0.994480i \(-0.533461\pi\)
−0.104927 + 0.994480i \(0.533461\pi\)
\(468\) 0 0
\(469\) 208.456 0.0205237
\(470\) 0 0
\(471\) 5968.88 0.583931
\(472\) 0 0
\(473\) −6326.45 −0.614991
\(474\) 0 0
\(475\) 1334.35 0.128893
\(476\) 0 0
\(477\) 2646.08 0.253995
\(478\) 0 0
\(479\) 3832.94 0.365619 0.182809 0.983148i \(-0.441481\pi\)
0.182809 + 0.983148i \(0.441481\pi\)
\(480\) 0 0
\(481\) −442.668 −0.0419624
\(482\) 0 0
\(483\) −933.752 −0.0879652
\(484\) 0 0
\(485\) −12245.5 −1.14647
\(486\) 0 0
\(487\) −3080.97 −0.286678 −0.143339 0.989674i \(-0.545784\pi\)
−0.143339 + 0.989674i \(0.545784\pi\)
\(488\) 0 0
\(489\) −1659.10 −0.153429
\(490\) 0 0
\(491\) 9947.19 0.914278 0.457139 0.889395i \(-0.348874\pi\)
0.457139 + 0.889395i \(0.348874\pi\)
\(492\) 0 0
\(493\) 11047.2 1.00921
\(494\) 0 0
\(495\) 7395.65 0.671535
\(496\) 0 0
\(497\) −2326.94 −0.210015
\(498\) 0 0
\(499\) −13670.7 −1.22642 −0.613211 0.789919i \(-0.710122\pi\)
−0.613211 + 0.789919i \(0.710122\pi\)
\(500\) 0 0
\(501\) −7919.51 −0.706223
\(502\) 0 0
\(503\) −2851.63 −0.252779 −0.126390 0.991981i \(-0.540339\pi\)
−0.126390 + 0.991981i \(0.540339\pi\)
\(504\) 0 0
\(505\) −4889.54 −0.430855
\(506\) 0 0
\(507\) −5558.95 −0.486946
\(508\) 0 0
\(509\) 2228.47 0.194057 0.0970286 0.995282i \(-0.469066\pi\)
0.0970286 + 0.995282i \(0.469066\pi\)
\(510\) 0 0
\(511\) 2717.49 0.235254
\(512\) 0 0
\(513\) 2303.41 0.198242
\(514\) 0 0
\(515\) 733.709 0.0627788
\(516\) 0 0
\(517\) −23816.1 −2.02597
\(518\) 0 0
\(519\) −578.159 −0.0488986
\(520\) 0 0
\(521\) −12531.3 −1.05375 −0.526877 0.849942i \(-0.676637\pi\)
−0.526877 + 0.849942i \(0.676637\pi\)
\(522\) 0 0
\(523\) −11402.8 −0.953362 −0.476681 0.879076i \(-0.658160\pi\)
−0.476681 + 0.879076i \(0.658160\pi\)
\(524\) 0 0
\(525\) −1882.13 −0.156462
\(526\) 0 0
\(527\) 8174.56 0.675691
\(528\) 0 0
\(529\) −10953.1 −0.900227
\(530\) 0 0
\(531\) −9817.34 −0.802328
\(532\) 0 0
\(533\) 1702.72 0.138373
\(534\) 0 0
\(535\) 11730.1 0.947918
\(536\) 0 0
\(537\) 6433.52 0.516996
\(538\) 0 0
\(539\) 11360.4 0.907845
\(540\) 0 0
\(541\) −1217.02 −0.0967171 −0.0483585 0.998830i \(-0.515399\pi\)
−0.0483585 + 0.998830i \(0.515399\pi\)
\(542\) 0 0
\(543\) −2585.31 −0.204321
\(544\) 0 0
\(545\) −1502.39 −0.118083
\(546\) 0 0
\(547\) −13940.4 −1.08967 −0.544833 0.838545i \(-0.683407\pi\)
−0.544833 + 0.838545i \(0.683407\pi\)
\(548\) 0 0
\(549\) 8087.38 0.628708
\(550\) 0 0
\(551\) 3411.04 0.263730
\(552\) 0 0
\(553\) −4928.97 −0.379025
\(554\) 0 0
\(555\) −1878.14 −0.143644
\(556\) 0 0
\(557\) 10002.4 0.760889 0.380445 0.924804i \(-0.375771\pi\)
0.380445 + 0.924804i \(0.375771\pi\)
\(558\) 0 0
\(559\) 577.473 0.0436932
\(560\) 0 0
\(561\) 7666.14 0.576942
\(562\) 0 0
\(563\) −8869.58 −0.663958 −0.331979 0.943287i \(-0.607716\pi\)
−0.331979 + 0.943287i \(0.607716\pi\)
\(564\) 0 0
\(565\) −13686.1 −1.01908
\(566\) 0 0
\(567\) 2555.15 0.189253
\(568\) 0 0
\(569\) 8649.42 0.637263 0.318631 0.947879i \(-0.396777\pi\)
0.318631 + 0.947879i \(0.396777\pi\)
\(570\) 0 0
\(571\) 6544.85 0.479673 0.239837 0.970813i \(-0.422906\pi\)
0.239837 + 0.970813i \(0.422906\pi\)
\(572\) 0 0
\(573\) −4957.04 −0.361402
\(574\) 0 0
\(575\) 2446.88 0.177465
\(576\) 0 0
\(577\) 4202.01 0.303175 0.151587 0.988444i \(-0.451561\pi\)
0.151587 + 0.988444i \(0.451561\pi\)
\(578\) 0 0
\(579\) 5284.61 0.379311
\(580\) 0 0
\(581\) 3915.23 0.279571
\(582\) 0 0
\(583\) 6303.98 0.447829
\(584\) 0 0
\(585\) −675.068 −0.0477105
\(586\) 0 0
\(587\) −21992.1 −1.54635 −0.773177 0.634190i \(-0.781334\pi\)
−0.773177 + 0.634190i \(0.781334\pi\)
\(588\) 0 0
\(589\) 2524.05 0.176574
\(590\) 0 0
\(591\) 10745.2 0.747882
\(592\) 0 0
\(593\) −1596.92 −0.110586 −0.0552932 0.998470i \(-0.517609\pi\)
−0.0552932 + 0.998470i \(0.517609\pi\)
\(594\) 0 0
\(595\) −4779.99 −0.329345
\(596\) 0 0
\(597\) 683.517 0.0468584
\(598\) 0 0
\(599\) −9965.77 −0.679783 −0.339892 0.940465i \(-0.610390\pi\)
−0.339892 + 0.940465i \(0.610390\pi\)
\(600\) 0 0
\(601\) −12790.5 −0.868110 −0.434055 0.900886i \(-0.642918\pi\)
−0.434055 + 0.900886i \(0.642918\pi\)
\(602\) 0 0
\(603\) −406.749 −0.0274695
\(604\) 0 0
\(605\) 7768.88 0.522066
\(606\) 0 0
\(607\) 25391.9 1.69790 0.848949 0.528474i \(-0.177236\pi\)
0.848949 + 0.528474i \(0.177236\pi\)
\(608\) 0 0
\(609\) −4811.35 −0.320141
\(610\) 0 0
\(611\) 2173.91 0.143939
\(612\) 0 0
\(613\) −9439.11 −0.621929 −0.310964 0.950422i \(-0.600652\pi\)
−0.310964 + 0.950422i \(0.600652\pi\)
\(614\) 0 0
\(615\) 7224.26 0.473675
\(616\) 0 0
\(617\) 3247.93 0.211923 0.105962 0.994370i \(-0.466208\pi\)
0.105962 + 0.994370i \(0.466208\pi\)
\(618\) 0 0
\(619\) −2781.32 −0.180599 −0.0902995 0.995915i \(-0.528782\pi\)
−0.0902995 + 0.995915i \(0.528782\pi\)
\(620\) 0 0
\(621\) 4223.93 0.272947
\(622\) 0 0
\(623\) −16891.4 −1.08626
\(624\) 0 0
\(625\) −1914.31 −0.122516
\(626\) 0 0
\(627\) 2367.07 0.150768
\(628\) 0 0
\(629\) 6116.03 0.387698
\(630\) 0 0
\(631\) −877.112 −0.0553364 −0.0276682 0.999617i \(-0.508808\pi\)
−0.0276682 + 0.999617i \(0.508808\pi\)
\(632\) 0 0
\(633\) −3215.41 −0.201897
\(634\) 0 0
\(635\) 12875.1 0.804618
\(636\) 0 0
\(637\) −1036.97 −0.0644996
\(638\) 0 0
\(639\) 4540.44 0.281091
\(640\) 0 0
\(641\) 18518.8 1.14111 0.570554 0.821260i \(-0.306728\pi\)
0.570554 + 0.821260i \(0.306728\pi\)
\(642\) 0 0
\(643\) 13493.0 0.827548 0.413774 0.910380i \(-0.364210\pi\)
0.413774 + 0.910380i \(0.364210\pi\)
\(644\) 0 0
\(645\) 2450.09 0.149569
\(646\) 0 0
\(647\) −9832.93 −0.597485 −0.298742 0.954334i \(-0.596567\pi\)
−0.298742 + 0.954334i \(0.596567\pi\)
\(648\) 0 0
\(649\) −23388.7 −1.41462
\(650\) 0 0
\(651\) −3560.23 −0.214342
\(652\) 0 0
\(653\) 4752.63 0.284816 0.142408 0.989808i \(-0.454515\pi\)
0.142408 + 0.989808i \(0.454515\pi\)
\(654\) 0 0
\(655\) 11030.5 0.658012
\(656\) 0 0
\(657\) −5302.49 −0.314870
\(658\) 0 0
\(659\) −2061.33 −0.121848 −0.0609242 0.998142i \(-0.519405\pi\)
−0.0609242 + 0.998142i \(0.519405\pi\)
\(660\) 0 0
\(661\) −2408.74 −0.141739 −0.0708693 0.997486i \(-0.522577\pi\)
−0.0708693 + 0.997486i \(0.522577\pi\)
\(662\) 0 0
\(663\) −699.758 −0.0409900
\(664\) 0 0
\(665\) −1475.92 −0.0860655
\(666\) 0 0
\(667\) 6255.06 0.363114
\(668\) 0 0
\(669\) 9293.76 0.537096
\(670\) 0 0
\(671\) 19267.2 1.10850
\(672\) 0 0
\(673\) −22916.0 −1.31255 −0.656275 0.754522i \(-0.727869\pi\)
−0.656275 + 0.754522i \(0.727869\pi\)
\(674\) 0 0
\(675\) 8514.00 0.485487
\(676\) 0 0
\(677\) −11902.9 −0.675724 −0.337862 0.941196i \(-0.609704\pi\)
−0.337862 + 0.941196i \(0.609704\pi\)
\(678\) 0 0
\(679\) −17367.3 −0.981582
\(680\) 0 0
\(681\) 4784.29 0.269214
\(682\) 0 0
\(683\) 25810.3 1.44598 0.722989 0.690860i \(-0.242768\pi\)
0.722989 + 0.690860i \(0.242768\pi\)
\(684\) 0 0
\(685\) 6412.63 0.357685
\(686\) 0 0
\(687\) 10520.7 0.584264
\(688\) 0 0
\(689\) −575.421 −0.0318168
\(690\) 0 0
\(691\) −1966.02 −0.108236 −0.0541180 0.998535i \(-0.517235\pi\)
−0.0541180 + 0.998535i \(0.517235\pi\)
\(692\) 0 0
\(693\) 10489.0 0.574953
\(694\) 0 0
\(695\) −3691.88 −0.201498
\(696\) 0 0
\(697\) −23525.3 −1.27846
\(698\) 0 0
\(699\) 1435.02 0.0776503
\(700\) 0 0
\(701\) 29611.2 1.59543 0.797717 0.603032i \(-0.206041\pi\)
0.797717 + 0.603032i \(0.206041\pi\)
\(702\) 0 0
\(703\) 1888.44 0.101314
\(704\) 0 0
\(705\) 9223.40 0.492728
\(706\) 0 0
\(707\) −6934.64 −0.368888
\(708\) 0 0
\(709\) −27618.7 −1.46297 −0.731483 0.681860i \(-0.761171\pi\)
−0.731483 + 0.681860i \(0.761171\pi\)
\(710\) 0 0
\(711\) 9617.63 0.507299
\(712\) 0 0
\(713\) 4628.53 0.243113
\(714\) 0 0
\(715\) −1608.27 −0.0841202
\(716\) 0 0
\(717\) −12140.0 −0.632326
\(718\) 0 0
\(719\) −32303.4 −1.67554 −0.837770 0.546023i \(-0.816141\pi\)
−0.837770 + 0.546023i \(0.816141\pi\)
\(720\) 0 0
\(721\) 1040.59 0.0537498
\(722\) 0 0
\(723\) 2059.11 0.105919
\(724\) 0 0
\(725\) 12608.1 0.645865
\(726\) 0 0
\(727\) −30854.5 −1.57405 −0.787023 0.616924i \(-0.788379\pi\)
−0.787023 + 0.616924i \(0.788379\pi\)
\(728\) 0 0
\(729\) 3371.90 0.171310
\(730\) 0 0
\(731\) −7978.54 −0.403689
\(732\) 0 0
\(733\) −22692.4 −1.14347 −0.571735 0.820438i \(-0.693729\pi\)
−0.571735 + 0.820438i \(0.693729\pi\)
\(734\) 0 0
\(735\) −4399.63 −0.220793
\(736\) 0 0
\(737\) −969.033 −0.0484325
\(738\) 0 0
\(739\) −34965.3 −1.74048 −0.870242 0.492624i \(-0.836038\pi\)
−0.870242 + 0.492624i \(0.836038\pi\)
\(740\) 0 0
\(741\) −216.064 −0.0107116
\(742\) 0 0
\(743\) −18699.8 −0.923325 −0.461663 0.887056i \(-0.652747\pi\)
−0.461663 + 0.887056i \(0.652747\pi\)
\(744\) 0 0
\(745\) 5153.79 0.253450
\(746\) 0 0
\(747\) −7639.57 −0.374187
\(748\) 0 0
\(749\) 16636.3 0.811586
\(750\) 0 0
\(751\) 20650.7 1.00340 0.501700 0.865041i \(-0.332708\pi\)
0.501700 + 0.865041i \(0.332708\pi\)
\(752\) 0 0
\(753\) −6344.62 −0.307053
\(754\) 0 0
\(755\) −20974.6 −1.01105
\(756\) 0 0
\(757\) 33118.8 1.59012 0.795062 0.606528i \(-0.207438\pi\)
0.795062 + 0.606528i \(0.207438\pi\)
\(758\) 0 0
\(759\) 4340.66 0.207584
\(760\) 0 0
\(761\) 16513.6 0.786618 0.393309 0.919406i \(-0.371330\pi\)
0.393309 + 0.919406i \(0.371330\pi\)
\(762\) 0 0
\(763\) −2130.78 −0.101100
\(764\) 0 0
\(765\) 9326.94 0.440806
\(766\) 0 0
\(767\) 2134.90 0.100504
\(768\) 0 0
\(769\) 16599.5 0.778406 0.389203 0.921152i \(-0.372750\pi\)
0.389203 + 0.921152i \(0.372750\pi\)
\(770\) 0 0
\(771\) 9014.64 0.421082
\(772\) 0 0
\(773\) −22940.9 −1.06743 −0.533717 0.845663i \(-0.679205\pi\)
−0.533717 + 0.845663i \(0.679205\pi\)
\(774\) 0 0
\(775\) 9329.55 0.432422
\(776\) 0 0
\(777\) −2663.69 −0.122985
\(778\) 0 0
\(779\) −7263.89 −0.334090
\(780\) 0 0
\(781\) 10817.1 0.495603
\(782\) 0 0
\(783\) 21764.6 0.993365
\(784\) 0 0
\(785\) 17300.8 0.786616
\(786\) 0 0
\(787\) 15086.1 0.683304 0.341652 0.939826i \(-0.389014\pi\)
0.341652 + 0.939826i \(0.389014\pi\)
\(788\) 0 0
\(789\) 7726.25 0.348621
\(790\) 0 0
\(791\) −19410.4 −0.872511
\(792\) 0 0
\(793\) −1758.70 −0.0787555
\(794\) 0 0
\(795\) −2441.38 −0.108914
\(796\) 0 0
\(797\) −36309.4 −1.61373 −0.806867 0.590733i \(-0.798839\pi\)
−0.806867 + 0.590733i \(0.798839\pi\)
\(798\) 0 0
\(799\) −30035.4 −1.32988
\(800\) 0 0
\(801\) 32959.4 1.45388
\(802\) 0 0
\(803\) −12632.6 −0.555161
\(804\) 0 0
\(805\) −2706.49 −0.118498
\(806\) 0 0
\(807\) 8119.47 0.354175
\(808\) 0 0
\(809\) 32700.4 1.42112 0.710560 0.703637i \(-0.248442\pi\)
0.710560 + 0.703637i \(0.248442\pi\)
\(810\) 0 0
\(811\) 6484.51 0.280767 0.140383 0.990097i \(-0.455166\pi\)
0.140383 + 0.990097i \(0.455166\pi\)
\(812\) 0 0
\(813\) −14799.3 −0.638418
\(814\) 0 0
\(815\) −4808.90 −0.206685
\(816\) 0 0
\(817\) −2463.53 −0.105493
\(818\) 0 0
\(819\) −957.422 −0.0408486
\(820\) 0 0
\(821\) −8864.19 −0.376812 −0.188406 0.982091i \(-0.560332\pi\)
−0.188406 + 0.982091i \(0.560332\pi\)
\(822\) 0 0
\(823\) −24083.4 −1.02004 −0.510022 0.860161i \(-0.670363\pi\)
−0.510022 + 0.860161i \(0.670363\pi\)
\(824\) 0 0
\(825\) 8749.29 0.369226
\(826\) 0 0
\(827\) 35052.0 1.47385 0.736927 0.675972i \(-0.236276\pi\)
0.736927 + 0.675972i \(0.236276\pi\)
\(828\) 0 0
\(829\) 1914.79 0.0802214 0.0401107 0.999195i \(-0.487229\pi\)
0.0401107 + 0.999195i \(0.487229\pi\)
\(830\) 0 0
\(831\) −13131.6 −0.548173
\(832\) 0 0
\(833\) 14327.1 0.595923
\(834\) 0 0
\(835\) −22954.8 −0.951357
\(836\) 0 0
\(837\) 16105.1 0.665082
\(838\) 0 0
\(839\) 19785.9 0.814164 0.407082 0.913392i \(-0.366546\pi\)
0.407082 + 0.913392i \(0.366546\pi\)
\(840\) 0 0
\(841\) 7841.48 0.321517
\(842\) 0 0
\(843\) 5415.75 0.221267
\(844\) 0 0
\(845\) −16112.7 −0.655967
\(846\) 0 0
\(847\) 11018.3 0.446981
\(848\) 0 0
\(849\) 13905.3 0.562107
\(850\) 0 0
\(851\) 3462.97 0.139494
\(852\) 0 0
\(853\) −287.455 −0.0115384 −0.00576920 0.999983i \(-0.501836\pi\)
−0.00576920 + 0.999983i \(0.501836\pi\)
\(854\) 0 0
\(855\) 2879.88 0.115193
\(856\) 0 0
\(857\) 2352.88 0.0937839 0.0468919 0.998900i \(-0.485068\pi\)
0.0468919 + 0.998900i \(0.485068\pi\)
\(858\) 0 0
\(859\) 29976.3 1.19066 0.595331 0.803480i \(-0.297021\pi\)
0.595331 + 0.803480i \(0.297021\pi\)
\(860\) 0 0
\(861\) 10245.9 0.405550
\(862\) 0 0
\(863\) 28858.3 1.13829 0.569147 0.822236i \(-0.307274\pi\)
0.569147 + 0.822236i \(0.307274\pi\)
\(864\) 0 0
\(865\) −1675.80 −0.0658715
\(866\) 0 0
\(867\) −2876.28 −0.112669
\(868\) 0 0
\(869\) 22912.9 0.894438
\(870\) 0 0
\(871\) 88.4524 0.00344098
\(872\) 0 0
\(873\) 33887.8 1.31378
\(874\) 0 0
\(875\) −15165.3 −0.585922
\(876\) 0 0
\(877\) 45124.0 1.73743 0.868716 0.495310i \(-0.164946\pi\)
0.868716 + 0.495310i \(0.164946\pi\)
\(878\) 0 0
\(879\) 15639.4 0.600116
\(880\) 0 0
\(881\) 46543.5 1.77990 0.889949 0.456060i \(-0.150740\pi\)
0.889949 + 0.456060i \(0.150740\pi\)
\(882\) 0 0
\(883\) −10533.3 −0.401442 −0.200721 0.979648i \(-0.564329\pi\)
−0.200721 + 0.979648i \(0.564329\pi\)
\(884\) 0 0
\(885\) 9057.89 0.344042
\(886\) 0 0
\(887\) −28070.3 −1.06258 −0.531289 0.847191i \(-0.678292\pi\)
−0.531289 + 0.847191i \(0.678292\pi\)
\(888\) 0 0
\(889\) 18260.2 0.688896
\(890\) 0 0
\(891\) −11877.9 −0.446606
\(892\) 0 0
\(893\) −9274.01 −0.347528
\(894\) 0 0
\(895\) 18647.6 0.696448
\(896\) 0 0
\(897\) −396.211 −0.0147482
\(898\) 0 0
\(899\) 23849.5 0.884788
\(900\) 0 0
\(901\) 7950.19 0.293962
\(902\) 0 0
\(903\) 3474.86 0.128058
\(904\) 0 0
\(905\) −7493.55 −0.275242
\(906\) 0 0
\(907\) 26589.1 0.973404 0.486702 0.873568i \(-0.338200\pi\)
0.486702 + 0.873568i \(0.338200\pi\)
\(908\) 0 0
\(909\) 13531.2 0.493731
\(910\) 0 0
\(911\) 32296.8 1.17458 0.587290 0.809377i \(-0.300195\pi\)
0.587290 + 0.809377i \(0.300195\pi\)
\(912\) 0 0
\(913\) −18200.4 −0.659743
\(914\) 0 0
\(915\) −7461.76 −0.269593
\(916\) 0 0
\(917\) 15644.1 0.563375
\(918\) 0 0
\(919\) −35127.6 −1.26089 −0.630443 0.776236i \(-0.717127\pi\)
−0.630443 + 0.776236i \(0.717127\pi\)
\(920\) 0 0
\(921\) −3226.40 −0.115433
\(922\) 0 0
\(923\) −987.373 −0.0352110
\(924\) 0 0
\(925\) 6980.17 0.248115
\(926\) 0 0
\(927\) −2030.45 −0.0719403
\(928\) 0 0
\(929\) −14258.8 −0.503570 −0.251785 0.967783i \(-0.581018\pi\)
−0.251785 + 0.967783i \(0.581018\pi\)
\(930\) 0 0
\(931\) 4423.77 0.155728
\(932\) 0 0
\(933\) −12649.7 −0.443872
\(934\) 0 0
\(935\) 22220.4 0.777202
\(936\) 0 0
\(937\) −40641.2 −1.41696 −0.708479 0.705732i \(-0.750618\pi\)
−0.708479 + 0.705732i \(0.750618\pi\)
\(938\) 0 0
\(939\) 12102.8 0.420618
\(940\) 0 0
\(941\) −22364.8 −0.774782 −0.387391 0.921915i \(-0.626624\pi\)
−0.387391 + 0.921915i \(0.626624\pi\)
\(942\) 0 0
\(943\) −13320.3 −0.459988
\(944\) 0 0
\(945\) −9417.30 −0.324174
\(946\) 0 0
\(947\) 27719.2 0.951166 0.475583 0.879671i \(-0.342237\pi\)
0.475583 + 0.879671i \(0.342237\pi\)
\(948\) 0 0
\(949\) 1153.09 0.0394424
\(950\) 0 0
\(951\) −12732.9 −0.434168
\(952\) 0 0
\(953\) −709.901 −0.0241300 −0.0120650 0.999927i \(-0.503841\pi\)
−0.0120650 + 0.999927i \(0.503841\pi\)
\(954\) 0 0
\(955\) −14368.0 −0.486846
\(956\) 0 0
\(957\) 22366.1 0.755480
\(958\) 0 0
\(959\) 9094.78 0.306242
\(960\) 0 0
\(961\) −12143.2 −0.407613
\(962\) 0 0
\(963\) −32461.6 −1.08625
\(964\) 0 0
\(965\) 15317.5 0.510972
\(966\) 0 0
\(967\) −2175.29 −0.0723400 −0.0361700 0.999346i \(-0.511516\pi\)
−0.0361700 + 0.999346i \(0.511516\pi\)
\(968\) 0 0
\(969\) 2985.20 0.0989666
\(970\) 0 0
\(971\) 13964.7 0.461533 0.230766 0.973009i \(-0.425877\pi\)
0.230766 + 0.973009i \(0.425877\pi\)
\(972\) 0 0
\(973\) −5236.04 −0.172518
\(974\) 0 0
\(975\) −798.627 −0.0262323
\(976\) 0 0
\(977\) −8452.26 −0.276778 −0.138389 0.990378i \(-0.544192\pi\)
−0.138389 + 0.990378i \(0.544192\pi\)
\(978\) 0 0
\(979\) 78521.9 2.56340
\(980\) 0 0
\(981\) 4157.68 0.135316
\(982\) 0 0
\(983\) −55328.2 −1.79521 −0.897607 0.440798i \(-0.854696\pi\)
−0.897607 + 0.440798i \(0.854696\pi\)
\(984\) 0 0
\(985\) 31145.0 1.00748
\(986\) 0 0
\(987\) 13081.2 0.421863
\(988\) 0 0
\(989\) −4517.54 −0.145247
\(990\) 0 0
\(991\) −33520.4 −1.07448 −0.537240 0.843429i \(-0.680533\pi\)
−0.537240 + 0.843429i \(0.680533\pi\)
\(992\) 0 0
\(993\) 12213.8 0.390324
\(994\) 0 0
\(995\) 1981.18 0.0631232
\(996\) 0 0
\(997\) −36267.7 −1.15206 −0.576032 0.817427i \(-0.695400\pi\)
−0.576032 + 0.817427i \(0.695400\pi\)
\(998\) 0 0
\(999\) 12049.5 0.381611
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.1.3 yes 5
4.3 odd 2 608.4.a.e.1.3 5
8.3 odd 2 1216.4.a.bd.1.3 5
8.5 even 2 1216.4.a.y.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.4.a.e.1.3 5 4.3 odd 2
608.4.a.h.1.3 yes 5 1.1 even 1 trivial
1216.4.a.y.1.3 5 8.5 even 2
1216.4.a.bd.1.3 5 8.3 odd 2