Properties

Label 312.4.a.d.1.1
Level $312$
Weight $4$
Character 312.1
Self dual yes
Analytic conductor $18.409$
Analytic rank $1$
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] = [312,4,Mod(1,312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(312, 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("312.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 312.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: \(18.4085959218\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 312.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -13.1231 q^{5} +15.6155 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -13.1231 q^{5} +15.6155 q^{7} +9.00000 q^{9} -55.7235 q^{11} +13.0000 q^{13} -39.3693 q^{15} +23.4773 q^{17} -25.6458 q^{19} +46.8466 q^{21} -189.477 q^{23} +47.2159 q^{25} +27.0000 q^{27} -236.739 q^{29} +47.0625 q^{31} -167.170 q^{33} -204.924 q^{35} -154.648 q^{37} +39.0000 q^{39} -34.6610 q^{41} -398.833 q^{43} -118.108 q^{45} +582.773 q^{47} -99.1553 q^{49} +70.4318 q^{51} -361.015 q^{53} +731.265 q^{55} -76.9375 q^{57} +396.985 q^{59} -211.322 q^{61} +140.540 q^{63} -170.600 q^{65} +85.8277 q^{67} -568.432 q^{69} -651.602 q^{71} +927.329 q^{73} +141.648 q^{75} -870.152 q^{77} -1117.98 q^{79} +81.0000 q^{81} +391.629 q^{83} -308.095 q^{85} -710.216 q^{87} -745.960 q^{89} +203.002 q^{91} +141.187 q^{93} +336.553 q^{95} -173.231 q^{97} -501.511 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 18 q^{5} - 10 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 18 q^{5} - 10 q^{7} + 18 q^{9} + 4 q^{11} + 26 q^{13} - 54 q^{15} - 52 q^{17} - 142 q^{19} - 30 q^{21} - 280 q^{23} - 54 q^{25} + 54 q^{27} - 424 q^{29} - 178 q^{31} + 12 q^{33} - 80 q^{35} + 136 q^{37} + 78 q^{39} - 226 q^{41} - 72 q^{43} - 162 q^{45} + 176 q^{47} + 214 q^{49} - 156 q^{51} - 788 q^{53} + 440 q^{55} - 426 q^{57} + 728 q^{59} - 736 q^{61} - 90 q^{63} - 234 q^{65} + 1054 q^{67} - 840 q^{69} - 660 q^{71} + 24 q^{73} - 162 q^{75} - 2400 q^{77} + 40 q^{79} + 162 q^{81} + 1344 q^{83} + 60 q^{85} - 1272 q^{87} + 314 q^{89} - 130 q^{91} - 534 q^{93} + 904 q^{95} - 264 q^{97} + 36 q^{99}+O(q^{100}) \) 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
\(5\) −13.1231 −1.17377 −0.586883 0.809672i \(-0.699645\pi\)
−0.586883 + 0.809672i \(0.699645\pi\)
\(6\) 0 0
\(7\) 15.6155 0.843159 0.421580 0.906791i \(-0.361476\pi\)
0.421580 + 0.906791i \(0.361476\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −55.7235 −1.52739 −0.763694 0.645579i \(-0.776616\pi\)
−0.763694 + 0.645579i \(0.776616\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) −39.3693 −0.677674
\(16\) 0 0
\(17\) 23.4773 0.334946 0.167473 0.985877i \(-0.446439\pi\)
0.167473 + 0.985877i \(0.446439\pi\)
\(18\) 0 0
\(19\) −25.6458 −0.309661 −0.154830 0.987941i \(-0.549483\pi\)
−0.154830 + 0.987941i \(0.549483\pi\)
\(20\) 0 0
\(21\) 46.8466 0.486798
\(22\) 0 0
\(23\) −189.477 −1.71777 −0.858886 0.512167i \(-0.828843\pi\)
−0.858886 + 0.512167i \(0.828843\pi\)
\(24\) 0 0
\(25\) 47.2159 0.377727
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −236.739 −1.51591 −0.757953 0.652309i \(-0.773800\pi\)
−0.757953 + 0.652309i \(0.773800\pi\)
\(30\) 0 0
\(31\) 47.0625 0.272667 0.136333 0.990663i \(-0.456468\pi\)
0.136333 + 0.990663i \(0.456468\pi\)
\(32\) 0 0
\(33\) −167.170 −0.881838
\(34\) 0 0
\(35\) −204.924 −0.989672
\(36\) 0 0
\(37\) −154.648 −0.687133 −0.343567 0.939128i \(-0.611635\pi\)
−0.343567 + 0.939128i \(0.611635\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) −34.6610 −0.132028 −0.0660139 0.997819i \(-0.521028\pi\)
−0.0660139 + 0.997819i \(0.521028\pi\)
\(42\) 0 0
\(43\) −398.833 −1.41445 −0.707227 0.706987i \(-0.750054\pi\)
−0.707227 + 0.706987i \(0.750054\pi\)
\(44\) 0 0
\(45\) −118.108 −0.391255
\(46\) 0 0
\(47\) 582.773 1.80864 0.904321 0.426854i \(-0.140378\pi\)
0.904321 + 0.426854i \(0.140378\pi\)
\(48\) 0 0
\(49\) −99.1553 −0.289082
\(50\) 0 0
\(51\) 70.4318 0.193381
\(52\) 0 0
\(53\) −361.015 −0.935646 −0.467823 0.883822i \(-0.654962\pi\)
−0.467823 + 0.883822i \(0.654962\pi\)
\(54\) 0 0
\(55\) 731.265 1.79280
\(56\) 0 0
\(57\) −76.9375 −0.178783
\(58\) 0 0
\(59\) 396.985 0.875983 0.437992 0.898979i \(-0.355690\pi\)
0.437992 + 0.898979i \(0.355690\pi\)
\(60\) 0 0
\(61\) −211.322 −0.443558 −0.221779 0.975097i \(-0.571186\pi\)
−0.221779 + 0.975097i \(0.571186\pi\)
\(62\) 0 0
\(63\) 140.540 0.281053
\(64\) 0 0
\(65\) −170.600 −0.325544
\(66\) 0 0
\(67\) 85.8277 0.156500 0.0782502 0.996934i \(-0.475067\pi\)
0.0782502 + 0.996934i \(0.475067\pi\)
\(68\) 0 0
\(69\) −568.432 −0.991756
\(70\) 0 0
\(71\) −651.602 −1.08917 −0.544584 0.838706i \(-0.683313\pi\)
−0.544584 + 0.838706i \(0.683313\pi\)
\(72\) 0 0
\(73\) 927.329 1.48679 0.743395 0.668852i \(-0.233214\pi\)
0.743395 + 0.668852i \(0.233214\pi\)
\(74\) 0 0
\(75\) 141.648 0.218081
\(76\) 0 0
\(77\) −870.152 −1.28783
\(78\) 0 0
\(79\) −1117.98 −1.59218 −0.796090 0.605178i \(-0.793102\pi\)
−0.796090 + 0.605178i \(0.793102\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 391.629 0.517914 0.258957 0.965889i \(-0.416621\pi\)
0.258957 + 0.965889i \(0.416621\pi\)
\(84\) 0 0
\(85\) −308.095 −0.393148
\(86\) 0 0
\(87\) −710.216 −0.875208
\(88\) 0 0
\(89\) −745.960 −0.888445 −0.444223 0.895916i \(-0.646520\pi\)
−0.444223 + 0.895916i \(0.646520\pi\)
\(90\) 0 0
\(91\) 203.002 0.233850
\(92\) 0 0
\(93\) 141.187 0.157424
\(94\) 0 0
\(95\) 336.553 0.363470
\(96\) 0 0
\(97\) −173.231 −0.181329 −0.0906647 0.995881i \(-0.528899\pi\)
−0.0906647 + 0.995881i \(0.528899\pi\)
\(98\) 0 0
\(99\) −501.511 −0.509129
\(100\) 0 0
\(101\) −1263.45 −1.24473 −0.622367 0.782726i \(-0.713829\pi\)
−0.622367 + 0.782726i \(0.713829\pi\)
\(102\) 0 0
\(103\) 874.401 0.836479 0.418240 0.908337i \(-0.362647\pi\)
0.418240 + 0.908337i \(0.362647\pi\)
\(104\) 0 0
\(105\) −614.773 −0.571387
\(106\) 0 0
\(107\) 1591.89 1.43826 0.719128 0.694877i \(-0.244541\pi\)
0.719128 + 0.694877i \(0.244541\pi\)
\(108\) 0 0
\(109\) 371.788 0.326705 0.163352 0.986568i \(-0.447769\pi\)
0.163352 + 0.986568i \(0.447769\pi\)
\(110\) 0 0
\(111\) −463.943 −0.396716
\(112\) 0 0
\(113\) 1173.98 0.977335 0.488667 0.872470i \(-0.337483\pi\)
0.488667 + 0.872470i \(0.337483\pi\)
\(114\) 0 0
\(115\) 2486.53 2.01626
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) 366.610 0.282412
\(120\) 0 0
\(121\) 1774.11 1.33291
\(122\) 0 0
\(123\) −103.983 −0.0762263
\(124\) 0 0
\(125\) 1020.77 0.730403
\(126\) 0 0
\(127\) 739.360 0.516595 0.258298 0.966065i \(-0.416839\pi\)
0.258298 + 0.966065i \(0.416839\pi\)
\(128\) 0 0
\(129\) −1196.50 −0.816635
\(130\) 0 0
\(131\) 2131.58 1.42166 0.710829 0.703365i \(-0.248320\pi\)
0.710829 + 0.703365i \(0.248320\pi\)
\(132\) 0 0
\(133\) −400.473 −0.261094
\(134\) 0 0
\(135\) −354.324 −0.225891
\(136\) 0 0
\(137\) −256.934 −0.160229 −0.0801144 0.996786i \(-0.525529\pi\)
−0.0801144 + 0.996786i \(0.525529\pi\)
\(138\) 0 0
\(139\) 437.443 0.266931 0.133466 0.991053i \(-0.457389\pi\)
0.133466 + 0.991053i \(0.457389\pi\)
\(140\) 0 0
\(141\) 1748.32 1.04422
\(142\) 0 0
\(143\) −724.405 −0.423621
\(144\) 0 0
\(145\) 3106.75 1.77932
\(146\) 0 0
\(147\) −297.466 −0.166902
\(148\) 0 0
\(149\) −3406.65 −1.87304 −0.936522 0.350609i \(-0.885975\pi\)
−0.936522 + 0.350609i \(0.885975\pi\)
\(150\) 0 0
\(151\) 3275.77 1.76542 0.882710 0.469918i \(-0.155717\pi\)
0.882710 + 0.469918i \(0.155717\pi\)
\(152\) 0 0
\(153\) 211.295 0.111649
\(154\) 0 0
\(155\) −617.606 −0.320047
\(156\) 0 0
\(157\) 1074.34 0.546128 0.273064 0.961996i \(-0.411963\pi\)
0.273064 + 0.961996i \(0.411963\pi\)
\(158\) 0 0
\(159\) −1083.05 −0.540196
\(160\) 0 0
\(161\) −2958.79 −1.44835
\(162\) 0 0
\(163\) 2031.49 0.976186 0.488093 0.872792i \(-0.337693\pi\)
0.488093 + 0.872792i \(0.337693\pi\)
\(164\) 0 0
\(165\) 2193.80 1.03507
\(166\) 0 0
\(167\) 299.697 0.138870 0.0694349 0.997586i \(-0.477880\pi\)
0.0694349 + 0.997586i \(0.477880\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −230.813 −0.103220
\(172\) 0 0
\(173\) 2371.16 1.04206 0.521028 0.853540i \(-0.325549\pi\)
0.521028 + 0.853540i \(0.325549\pi\)
\(174\) 0 0
\(175\) 737.301 0.318484
\(176\) 0 0
\(177\) 1190.95 0.505749
\(178\) 0 0
\(179\) −249.613 −0.104229 −0.0521145 0.998641i \(-0.516596\pi\)
−0.0521145 + 0.998641i \(0.516596\pi\)
\(180\) 0 0
\(181\) −3151.45 −1.29417 −0.647087 0.762416i \(-0.724013\pi\)
−0.647087 + 0.762416i \(0.724013\pi\)
\(182\) 0 0
\(183\) −633.966 −0.256088
\(184\) 0 0
\(185\) 2029.46 0.806534
\(186\) 0 0
\(187\) −1308.24 −0.511592
\(188\) 0 0
\(189\) 421.619 0.162266
\(190\) 0 0
\(191\) −4739.45 −1.79547 −0.897735 0.440535i \(-0.854789\pi\)
−0.897735 + 0.440535i \(0.854789\pi\)
\(192\) 0 0
\(193\) 636.818 0.237509 0.118754 0.992924i \(-0.462110\pi\)
0.118754 + 0.992924i \(0.462110\pi\)
\(194\) 0 0
\(195\) −511.801 −0.187953
\(196\) 0 0
\(197\) −3001.13 −1.08539 −0.542695 0.839930i \(-0.682596\pi\)
−0.542695 + 0.839930i \(0.682596\pi\)
\(198\) 0 0
\(199\) −648.958 −0.231173 −0.115587 0.993297i \(-0.536875\pi\)
−0.115587 + 0.993297i \(0.536875\pi\)
\(200\) 0 0
\(201\) 257.483 0.0903555
\(202\) 0 0
\(203\) −3696.80 −1.27815
\(204\) 0 0
\(205\) 454.860 0.154970
\(206\) 0 0
\(207\) −1705.30 −0.572590
\(208\) 0 0
\(209\) 1429.08 0.472972
\(210\) 0 0
\(211\) −240.114 −0.0783418 −0.0391709 0.999233i \(-0.512472\pi\)
−0.0391709 + 0.999233i \(0.512472\pi\)
\(212\) 0 0
\(213\) −1954.81 −0.628832
\(214\) 0 0
\(215\) 5233.93 1.66024
\(216\) 0 0
\(217\) 734.906 0.229902
\(218\) 0 0
\(219\) 2781.99 0.858399
\(220\) 0 0
\(221\) 305.204 0.0928972
\(222\) 0 0
\(223\) −1104.37 −0.331633 −0.165816 0.986157i \(-0.553026\pi\)
−0.165816 + 0.986157i \(0.553026\pi\)
\(224\) 0 0
\(225\) 424.943 0.125909
\(226\) 0 0
\(227\) 497.034 0.145327 0.0726636 0.997357i \(-0.476850\pi\)
0.0726636 + 0.997357i \(0.476850\pi\)
\(228\) 0 0
\(229\) 2879.05 0.830800 0.415400 0.909639i \(-0.363642\pi\)
0.415400 + 0.909639i \(0.363642\pi\)
\(230\) 0 0
\(231\) −2610.45 −0.743529
\(232\) 0 0
\(233\) −3846.91 −1.08163 −0.540814 0.841142i \(-0.681884\pi\)
−0.540814 + 0.841142i \(0.681884\pi\)
\(234\) 0 0
\(235\) −7647.79 −2.12292
\(236\) 0 0
\(237\) −3353.93 −0.919246
\(238\) 0 0
\(239\) 2903.25 0.785755 0.392877 0.919591i \(-0.371480\pi\)
0.392877 + 0.919591i \(0.371480\pi\)
\(240\) 0 0
\(241\) −4801.06 −1.28325 −0.641626 0.767018i \(-0.721740\pi\)
−0.641626 + 0.767018i \(0.721740\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 1301.23 0.339315
\(246\) 0 0
\(247\) −333.396 −0.0858845
\(248\) 0 0
\(249\) 1174.89 0.299018
\(250\) 0 0
\(251\) 4768.20 1.19907 0.599534 0.800350i \(-0.295353\pi\)
0.599534 + 0.800350i \(0.295353\pi\)
\(252\) 0 0
\(253\) 10558.3 2.62370
\(254\) 0 0
\(255\) −924.284 −0.226984
\(256\) 0 0
\(257\) 6002.18 1.45683 0.728415 0.685136i \(-0.240257\pi\)
0.728415 + 0.685136i \(0.240257\pi\)
\(258\) 0 0
\(259\) −2414.91 −0.579363
\(260\) 0 0
\(261\) −2130.65 −0.505302
\(262\) 0 0
\(263\) −3268.33 −0.766289 −0.383145 0.923688i \(-0.625159\pi\)
−0.383145 + 0.923688i \(0.625159\pi\)
\(264\) 0 0
\(265\) 4737.64 1.09823
\(266\) 0 0
\(267\) −2237.88 −0.512944
\(268\) 0 0
\(269\) −8690.14 −1.96969 −0.984846 0.173431i \(-0.944515\pi\)
−0.984846 + 0.173431i \(0.944515\pi\)
\(270\) 0 0
\(271\) −3239.17 −0.726073 −0.363036 0.931775i \(-0.618260\pi\)
−0.363036 + 0.931775i \(0.618260\pi\)
\(272\) 0 0
\(273\) 609.006 0.135014
\(274\) 0 0
\(275\) −2631.03 −0.576936
\(276\) 0 0
\(277\) −220.091 −0.0477400 −0.0238700 0.999715i \(-0.507599\pi\)
−0.0238700 + 0.999715i \(0.507599\pi\)
\(278\) 0 0
\(279\) 423.562 0.0908889
\(280\) 0 0
\(281\) 587.233 0.124667 0.0623334 0.998055i \(-0.480146\pi\)
0.0623334 + 0.998055i \(0.480146\pi\)
\(282\) 0 0
\(283\) −8274.09 −1.73796 −0.868981 0.494845i \(-0.835225\pi\)
−0.868981 + 0.494845i \(0.835225\pi\)
\(284\) 0 0
\(285\) 1009.66 0.209849
\(286\) 0 0
\(287\) −541.250 −0.111320
\(288\) 0 0
\(289\) −4361.82 −0.887812
\(290\) 0 0
\(291\) −519.693 −0.104691
\(292\) 0 0
\(293\) −5382.07 −1.07312 −0.536559 0.843863i \(-0.680276\pi\)
−0.536559 + 0.843863i \(0.680276\pi\)
\(294\) 0 0
\(295\) −5209.67 −1.02820
\(296\) 0 0
\(297\) −1504.53 −0.293946
\(298\) 0 0
\(299\) −2463.20 −0.476424
\(300\) 0 0
\(301\) −6227.99 −1.19261
\(302\) 0 0
\(303\) −3790.35 −0.718647
\(304\) 0 0
\(305\) 2773.20 0.520633
\(306\) 0 0
\(307\) −3095.92 −0.575549 −0.287774 0.957698i \(-0.592915\pi\)
−0.287774 + 0.957698i \(0.592915\pi\)
\(308\) 0 0
\(309\) 2623.20 0.482941
\(310\) 0 0
\(311\) 3474.58 0.633521 0.316761 0.948506i \(-0.397405\pi\)
0.316761 + 0.948506i \(0.397405\pi\)
\(312\) 0 0
\(313\) −3523.92 −0.636370 −0.318185 0.948029i \(-0.603073\pi\)
−0.318185 + 0.948029i \(0.603073\pi\)
\(314\) 0 0
\(315\) −1844.32 −0.329891
\(316\) 0 0
\(317\) −323.396 −0.0572988 −0.0286494 0.999590i \(-0.509121\pi\)
−0.0286494 + 0.999590i \(0.509121\pi\)
\(318\) 0 0
\(319\) 13191.9 2.31537
\(320\) 0 0
\(321\) 4775.66 0.830378
\(322\) 0 0
\(323\) −602.094 −0.103720
\(324\) 0 0
\(325\) 613.807 0.104763
\(326\) 0 0
\(327\) 1115.36 0.188623
\(328\) 0 0
\(329\) 9100.30 1.52497
\(330\) 0 0
\(331\) −1057.74 −0.175646 −0.0878228 0.996136i \(-0.527991\pi\)
−0.0878228 + 0.996136i \(0.527991\pi\)
\(332\) 0 0
\(333\) −1391.83 −0.229044
\(334\) 0 0
\(335\) −1126.33 −0.183695
\(336\) 0 0
\(337\) 11538.4 1.86510 0.932550 0.361041i \(-0.117578\pi\)
0.932550 + 0.361041i \(0.117578\pi\)
\(338\) 0 0
\(339\) 3521.94 0.564265
\(340\) 0 0
\(341\) −2622.49 −0.416468
\(342\) 0 0
\(343\) −6904.49 −1.08690
\(344\) 0 0
\(345\) 7459.59 1.16409
\(346\) 0 0
\(347\) 5211.75 0.806286 0.403143 0.915137i \(-0.367918\pi\)
0.403143 + 0.915137i \(0.367918\pi\)
\(348\) 0 0
\(349\) −2369.81 −0.363476 −0.181738 0.983347i \(-0.558172\pi\)
−0.181738 + 0.983347i \(0.558172\pi\)
\(350\) 0 0
\(351\) 351.000 0.0533761
\(352\) 0 0
\(353\) −8997.40 −1.35661 −0.678305 0.734780i \(-0.737285\pi\)
−0.678305 + 0.734780i \(0.737285\pi\)
\(354\) 0 0
\(355\) 8551.05 1.27843
\(356\) 0 0
\(357\) 1099.83 0.163051
\(358\) 0 0
\(359\) 10863.4 1.59707 0.798537 0.601946i \(-0.205608\pi\)
0.798537 + 0.601946i \(0.205608\pi\)
\(360\) 0 0
\(361\) −6201.29 −0.904110
\(362\) 0 0
\(363\) 5322.32 0.769557
\(364\) 0 0
\(365\) −12169.4 −1.74514
\(366\) 0 0
\(367\) −13552.2 −1.92757 −0.963785 0.266682i \(-0.914073\pi\)
−0.963785 + 0.266682i \(0.914073\pi\)
\(368\) 0 0
\(369\) −311.949 −0.0440092
\(370\) 0 0
\(371\) −5637.44 −0.788899
\(372\) 0 0
\(373\) 2255.64 0.313118 0.156559 0.987669i \(-0.449960\pi\)
0.156559 + 0.987669i \(0.449960\pi\)
\(374\) 0 0
\(375\) 3062.31 0.421698
\(376\) 0 0
\(377\) −3077.60 −0.420437
\(378\) 0 0
\(379\) −7698.92 −1.04345 −0.521724 0.853114i \(-0.674711\pi\)
−0.521724 + 0.853114i \(0.674711\pi\)
\(380\) 0 0
\(381\) 2218.08 0.298256
\(382\) 0 0
\(383\) 211.235 0.0281818 0.0140909 0.999901i \(-0.495515\pi\)
0.0140909 + 0.999901i \(0.495515\pi\)
\(384\) 0 0
\(385\) 11419.1 1.51161
\(386\) 0 0
\(387\) −3589.50 −0.471485
\(388\) 0 0
\(389\) 3515.25 0.458175 0.229088 0.973406i \(-0.426426\pi\)
0.229088 + 0.973406i \(0.426426\pi\)
\(390\) 0 0
\(391\) −4448.41 −0.575360
\(392\) 0 0
\(393\) 6394.75 0.820795
\(394\) 0 0
\(395\) 14671.3 1.86885
\(396\) 0 0
\(397\) −10968.8 −1.38667 −0.693337 0.720613i \(-0.743860\pi\)
−0.693337 + 0.720613i \(0.743860\pi\)
\(398\) 0 0
\(399\) −1201.42 −0.150742
\(400\) 0 0
\(401\) 6508.24 0.810488 0.405244 0.914208i \(-0.367187\pi\)
0.405244 + 0.914208i \(0.367187\pi\)
\(402\) 0 0
\(403\) 611.812 0.0756242
\(404\) 0 0
\(405\) −1062.97 −0.130418
\(406\) 0 0
\(407\) 8617.51 1.04952
\(408\) 0 0
\(409\) 9363.46 1.13201 0.566006 0.824401i \(-0.308488\pi\)
0.566006 + 0.824401i \(0.308488\pi\)
\(410\) 0 0
\(411\) −770.801 −0.0925081
\(412\) 0 0
\(413\) 6199.13 0.738594
\(414\) 0 0
\(415\) −5139.39 −0.607910
\(416\) 0 0
\(417\) 1312.33 0.154113
\(418\) 0 0
\(419\) −12057.8 −1.40588 −0.702940 0.711249i \(-0.748130\pi\)
−0.702940 + 0.711249i \(0.748130\pi\)
\(420\) 0 0
\(421\) −15937.1 −1.84496 −0.922479 0.386047i \(-0.873840\pi\)
−0.922479 + 0.386047i \(0.873840\pi\)
\(422\) 0 0
\(423\) 5244.95 0.602880
\(424\) 0 0
\(425\) 1108.50 0.126518
\(426\) 0 0
\(427\) −3299.90 −0.373990
\(428\) 0 0
\(429\) −2173.22 −0.244578
\(430\) 0 0
\(431\) 6792.37 0.759111 0.379556 0.925169i \(-0.376077\pi\)
0.379556 + 0.925169i \(0.376077\pi\)
\(432\) 0 0
\(433\) −3338.00 −0.370471 −0.185235 0.982694i \(-0.559305\pi\)
−0.185235 + 0.982694i \(0.559305\pi\)
\(434\) 0 0
\(435\) 9320.24 1.02729
\(436\) 0 0
\(437\) 4859.30 0.531927
\(438\) 0 0
\(439\) 13313.1 1.44738 0.723688 0.690127i \(-0.242445\pi\)
0.723688 + 0.690127i \(0.242445\pi\)
\(440\) 0 0
\(441\) −892.398 −0.0963608
\(442\) 0 0
\(443\) 5175.32 0.555050 0.277525 0.960718i \(-0.410486\pi\)
0.277525 + 0.960718i \(0.410486\pi\)
\(444\) 0 0
\(445\) 9789.31 1.04283
\(446\) 0 0
\(447\) −10219.9 −1.08140
\(448\) 0 0
\(449\) −4090.57 −0.429947 −0.214973 0.976620i \(-0.568966\pi\)
−0.214973 + 0.976620i \(0.568966\pi\)
\(450\) 0 0
\(451\) 1931.43 0.201658
\(452\) 0 0
\(453\) 9827.31 1.01927
\(454\) 0 0
\(455\) −2664.01 −0.274486
\(456\) 0 0
\(457\) −4300.78 −0.440224 −0.220112 0.975475i \(-0.570642\pi\)
−0.220112 + 0.975475i \(0.570642\pi\)
\(458\) 0 0
\(459\) 633.886 0.0644603
\(460\) 0 0
\(461\) −2578.07 −0.260461 −0.130231 0.991484i \(-0.541572\pi\)
−0.130231 + 0.991484i \(0.541572\pi\)
\(462\) 0 0
\(463\) −4529.58 −0.454660 −0.227330 0.973818i \(-0.573000\pi\)
−0.227330 + 0.973818i \(0.573000\pi\)
\(464\) 0 0
\(465\) −1852.82 −0.184779
\(466\) 0 0
\(467\) 4761.89 0.471850 0.235925 0.971771i \(-0.424188\pi\)
0.235925 + 0.971771i \(0.424188\pi\)
\(468\) 0 0
\(469\) 1340.24 0.131955
\(470\) 0 0
\(471\) 3223.03 0.315307
\(472\) 0 0
\(473\) 22224.4 2.16042
\(474\) 0 0
\(475\) −1210.89 −0.116967
\(476\) 0 0
\(477\) −3249.14 −0.311882
\(478\) 0 0
\(479\) 11621.4 1.10855 0.554275 0.832333i \(-0.312995\pi\)
0.554275 + 0.832333i \(0.312995\pi\)
\(480\) 0 0
\(481\) −2010.42 −0.190576
\(482\) 0 0
\(483\) −8876.36 −0.836208
\(484\) 0 0
\(485\) 2273.33 0.212838
\(486\) 0 0
\(487\) 7836.62 0.729181 0.364590 0.931168i \(-0.381209\pi\)
0.364590 + 0.931168i \(0.381209\pi\)
\(488\) 0 0
\(489\) 6094.46 0.563601
\(490\) 0 0
\(491\) −14163.9 −1.30185 −0.650925 0.759142i \(-0.725619\pi\)
−0.650925 + 0.759142i \(0.725619\pi\)
\(492\) 0 0
\(493\) −5557.98 −0.507746
\(494\) 0 0
\(495\) 6581.39 0.597599
\(496\) 0 0
\(497\) −10175.1 −0.918342
\(498\) 0 0
\(499\) −811.953 −0.0728417 −0.0364208 0.999337i \(-0.511596\pi\)
−0.0364208 + 0.999337i \(0.511596\pi\)
\(500\) 0 0
\(501\) 899.091 0.0801765
\(502\) 0 0
\(503\) 9718.41 0.861476 0.430738 0.902477i \(-0.358253\pi\)
0.430738 + 0.902477i \(0.358253\pi\)
\(504\) 0 0
\(505\) 16580.4 1.46103
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 0 0
\(509\) −10546.5 −0.918403 −0.459201 0.888332i \(-0.651864\pi\)
−0.459201 + 0.888332i \(0.651864\pi\)
\(510\) 0 0
\(511\) 14480.7 1.25360
\(512\) 0 0
\(513\) −692.438 −0.0595943
\(514\) 0 0
\(515\) −11474.9 −0.981831
\(516\) 0 0
\(517\) −32474.1 −2.76250
\(518\) 0 0
\(519\) 7113.47 0.601631
\(520\) 0 0
\(521\) 1157.10 0.0973005 0.0486503 0.998816i \(-0.484508\pi\)
0.0486503 + 0.998816i \(0.484508\pi\)
\(522\) 0 0
\(523\) 19193.0 1.60468 0.802342 0.596865i \(-0.203587\pi\)
0.802342 + 0.596865i \(0.203587\pi\)
\(524\) 0 0
\(525\) 2211.90 0.183877
\(526\) 0 0
\(527\) 1104.90 0.0913285
\(528\) 0 0
\(529\) 23734.6 1.95074
\(530\) 0 0
\(531\) 3572.86 0.291994
\(532\) 0 0
\(533\) −450.593 −0.0366179
\(534\) 0 0
\(535\) −20890.5 −1.68818
\(536\) 0 0
\(537\) −748.840 −0.0601766
\(538\) 0 0
\(539\) 5525.28 0.441541
\(540\) 0 0
\(541\) −7809.48 −0.620620 −0.310310 0.950635i \(-0.600433\pi\)
−0.310310 + 0.950635i \(0.600433\pi\)
\(542\) 0 0
\(543\) −9454.35 −0.747192
\(544\) 0 0
\(545\) −4879.01 −0.383475
\(546\) 0 0
\(547\) 4979.12 0.389199 0.194599 0.980883i \(-0.437659\pi\)
0.194599 + 0.980883i \(0.437659\pi\)
\(548\) 0 0
\(549\) −1901.90 −0.147853
\(550\) 0 0
\(551\) 6071.36 0.469417
\(552\) 0 0
\(553\) −17457.8 −1.34246
\(554\) 0 0
\(555\) 6088.37 0.465652
\(556\) 0 0
\(557\) 24508.2 1.86435 0.932176 0.362005i \(-0.117908\pi\)
0.932176 + 0.362005i \(0.117908\pi\)
\(558\) 0 0
\(559\) −5184.83 −0.392299
\(560\) 0 0
\(561\) −3924.71 −0.295368
\(562\) 0 0
\(563\) 14957.3 1.11967 0.559836 0.828604i \(-0.310864\pi\)
0.559836 + 0.828604i \(0.310864\pi\)
\(564\) 0 0
\(565\) −15406.3 −1.14716
\(566\) 0 0
\(567\) 1264.86 0.0936844
\(568\) 0 0
\(569\) 5756.83 0.424146 0.212073 0.977254i \(-0.431979\pi\)
0.212073 + 0.977254i \(0.431979\pi\)
\(570\) 0 0
\(571\) −7292.96 −0.534502 −0.267251 0.963627i \(-0.586115\pi\)
−0.267251 + 0.963627i \(0.586115\pi\)
\(572\) 0 0
\(573\) −14218.4 −1.03662
\(574\) 0 0
\(575\) −8946.34 −0.648849
\(576\) 0 0
\(577\) 20749.7 1.49709 0.748544 0.663085i \(-0.230753\pi\)
0.748544 + 0.663085i \(0.230753\pi\)
\(578\) 0 0
\(579\) 1910.46 0.137126
\(580\) 0 0
\(581\) 6115.49 0.436684
\(582\) 0 0
\(583\) 20117.0 1.42909
\(584\) 0 0
\(585\) −1535.40 −0.108515
\(586\) 0 0
\(587\) 2046.37 0.143889 0.0719444 0.997409i \(-0.477080\pi\)
0.0719444 + 0.997409i \(0.477080\pi\)
\(588\) 0 0
\(589\) −1206.96 −0.0844343
\(590\) 0 0
\(591\) −9003.39 −0.626650
\(592\) 0 0
\(593\) 12664.1 0.876988 0.438494 0.898734i \(-0.355512\pi\)
0.438494 + 0.898734i \(0.355512\pi\)
\(594\) 0 0
\(595\) −4811.06 −0.331486
\(596\) 0 0
\(597\) −1946.87 −0.133468
\(598\) 0 0
\(599\) 5788.48 0.394843 0.197421 0.980319i \(-0.436743\pi\)
0.197421 + 0.980319i \(0.436743\pi\)
\(600\) 0 0
\(601\) −19622.2 −1.33179 −0.665897 0.746044i \(-0.731951\pi\)
−0.665897 + 0.746044i \(0.731951\pi\)
\(602\) 0 0
\(603\) 772.449 0.0521668
\(604\) 0 0
\(605\) −23281.8 −1.56453
\(606\) 0 0
\(607\) −11386.5 −0.761389 −0.380695 0.924701i \(-0.624315\pi\)
−0.380695 + 0.924701i \(0.624315\pi\)
\(608\) 0 0
\(609\) −11090.4 −0.737940
\(610\) 0 0
\(611\) 7576.04 0.501627
\(612\) 0 0
\(613\) −16046.0 −1.05725 −0.528623 0.848857i \(-0.677291\pi\)
−0.528623 + 0.848857i \(0.677291\pi\)
\(614\) 0 0
\(615\) 1364.58 0.0894718
\(616\) 0 0
\(617\) −25943.8 −1.69280 −0.846402 0.532545i \(-0.821236\pi\)
−0.846402 + 0.532545i \(0.821236\pi\)
\(618\) 0 0
\(619\) 22044.1 1.43139 0.715693 0.698415i \(-0.246111\pi\)
0.715693 + 0.698415i \(0.246111\pi\)
\(620\) 0 0
\(621\) −5115.89 −0.330585
\(622\) 0 0
\(623\) −11648.6 −0.749101
\(624\) 0 0
\(625\) −19297.6 −1.23505
\(626\) 0 0
\(627\) 4287.23 0.273071
\(628\) 0 0
\(629\) −3630.71 −0.230152
\(630\) 0 0
\(631\) −6948.74 −0.438392 −0.219196 0.975681i \(-0.570343\pi\)
−0.219196 + 0.975681i \(0.570343\pi\)
\(632\) 0 0
\(633\) −720.341 −0.0452306
\(634\) 0 0
\(635\) −9702.70 −0.606362
\(636\) 0 0
\(637\) −1289.02 −0.0801770
\(638\) 0 0
\(639\) −5864.42 −0.363056
\(640\) 0 0
\(641\) −17916.9 −1.10402 −0.552009 0.833838i \(-0.686139\pi\)
−0.552009 + 0.833838i \(0.686139\pi\)
\(642\) 0 0
\(643\) 10248.9 0.628581 0.314290 0.949327i \(-0.398233\pi\)
0.314290 + 0.949327i \(0.398233\pi\)
\(644\) 0 0
\(645\) 15701.8 0.958539
\(646\) 0 0
\(647\) −13034.7 −0.792035 −0.396018 0.918243i \(-0.629608\pi\)
−0.396018 + 0.918243i \(0.629608\pi\)
\(648\) 0 0
\(649\) −22121.4 −1.33797
\(650\) 0 0
\(651\) 2204.72 0.132734
\(652\) 0 0
\(653\) 5517.54 0.330656 0.165328 0.986239i \(-0.447132\pi\)
0.165328 + 0.986239i \(0.447132\pi\)
\(654\) 0 0
\(655\) −27973.0 −1.66869
\(656\) 0 0
\(657\) 8345.97 0.495597
\(658\) 0 0
\(659\) −31919.9 −1.88683 −0.943416 0.331610i \(-0.892408\pi\)
−0.943416 + 0.331610i \(0.892408\pi\)
\(660\) 0 0
\(661\) −3517.33 −0.206972 −0.103486 0.994631i \(-0.533000\pi\)
−0.103486 + 0.994631i \(0.533000\pi\)
\(662\) 0 0
\(663\) 915.613 0.0536342
\(664\) 0 0
\(665\) 5255.45 0.306463
\(666\) 0 0
\(667\) 44856.6 2.60398
\(668\) 0 0
\(669\) −3313.11 −0.191468
\(670\) 0 0
\(671\) 11775.6 0.677484
\(672\) 0 0
\(673\) −33399.0 −1.91298 −0.956490 0.291766i \(-0.905757\pi\)
−0.956490 + 0.291766i \(0.905757\pi\)
\(674\) 0 0
\(675\) 1274.83 0.0726936
\(676\) 0 0
\(677\) −15843.8 −0.899450 −0.449725 0.893167i \(-0.648478\pi\)
−0.449725 + 0.893167i \(0.648478\pi\)
\(678\) 0 0
\(679\) −2705.09 −0.152890
\(680\) 0 0
\(681\) 1491.10 0.0839047
\(682\) 0 0
\(683\) −10145.6 −0.568392 −0.284196 0.958766i \(-0.591727\pi\)
−0.284196 + 0.958766i \(0.591727\pi\)
\(684\) 0 0
\(685\) 3371.77 0.188071
\(686\) 0 0
\(687\) 8637.16 0.479663
\(688\) 0 0
\(689\) −4693.20 −0.259502
\(690\) 0 0
\(691\) 18650.9 1.02679 0.513396 0.858152i \(-0.328387\pi\)
0.513396 + 0.858152i \(0.328387\pi\)
\(692\) 0 0
\(693\) −7831.36 −0.429277
\(694\) 0 0
\(695\) −5740.61 −0.313315
\(696\) 0 0
\(697\) −813.745 −0.0442221
\(698\) 0 0
\(699\) −11540.7 −0.624478
\(700\) 0 0
\(701\) 25182.9 1.35684 0.678420 0.734675i \(-0.262665\pi\)
0.678420 + 0.734675i \(0.262665\pi\)
\(702\) 0 0
\(703\) 3966.07 0.212778
\(704\) 0 0
\(705\) −22943.4 −1.22567
\(706\) 0 0
\(707\) −19729.4 −1.04951
\(708\) 0 0
\(709\) 16697.6 0.884473 0.442236 0.896899i \(-0.354185\pi\)
0.442236 + 0.896899i \(0.354185\pi\)
\(710\) 0 0
\(711\) −10061.8 −0.530727
\(712\) 0 0
\(713\) −8917.27 −0.468379
\(714\) 0 0
\(715\) 9506.45 0.497232
\(716\) 0 0
\(717\) 8709.74 0.453656
\(718\) 0 0
\(719\) 22657.4 1.17521 0.587606 0.809147i \(-0.300070\pi\)
0.587606 + 0.809147i \(0.300070\pi\)
\(720\) 0 0
\(721\) 13654.2 0.705285
\(722\) 0 0
\(723\) −14403.2 −0.740886
\(724\) 0 0
\(725\) −11177.8 −0.572599
\(726\) 0 0
\(727\) −32052.8 −1.63518 −0.817588 0.575803i \(-0.804690\pi\)
−0.817588 + 0.575803i \(0.804690\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −9363.52 −0.473765
\(732\) 0 0
\(733\) 22728.1 1.14527 0.572634 0.819811i \(-0.305922\pi\)
0.572634 + 0.819811i \(0.305922\pi\)
\(734\) 0 0
\(735\) 3903.68 0.195904
\(736\) 0 0
\(737\) −4782.62 −0.239037
\(738\) 0 0
\(739\) 3741.58 0.186247 0.0931234 0.995655i \(-0.470315\pi\)
0.0931234 + 0.995655i \(0.470315\pi\)
\(740\) 0 0
\(741\) −1000.19 −0.0495854
\(742\) 0 0
\(743\) −23932.4 −1.18169 −0.590844 0.806786i \(-0.701205\pi\)
−0.590844 + 0.806786i \(0.701205\pi\)
\(744\) 0 0
\(745\) 44705.8 2.19852
\(746\) 0 0
\(747\) 3524.66 0.172638
\(748\) 0 0
\(749\) 24858.1 1.21268
\(750\) 0 0
\(751\) −15777.6 −0.766623 −0.383311 0.923619i \(-0.625216\pi\)
−0.383311 + 0.923619i \(0.625216\pi\)
\(752\) 0 0
\(753\) 14304.6 0.692282
\(754\) 0 0
\(755\) −42988.3 −2.07219
\(756\) 0 0
\(757\) 32474.4 1.55919 0.779593 0.626287i \(-0.215426\pi\)
0.779593 + 0.626287i \(0.215426\pi\)
\(758\) 0 0
\(759\) 31675.0 1.51480
\(760\) 0 0
\(761\) 24471.4 1.16569 0.582844 0.812584i \(-0.301940\pi\)
0.582844 + 0.812584i \(0.301940\pi\)
\(762\) 0 0
\(763\) 5805.66 0.275464
\(764\) 0 0
\(765\) −2772.85 −0.131049
\(766\) 0 0
\(767\) 5160.80 0.242954
\(768\) 0 0
\(769\) 5185.33 0.243157 0.121578 0.992582i \(-0.461204\pi\)
0.121578 + 0.992582i \(0.461204\pi\)
\(770\) 0 0
\(771\) 18006.5 0.841102
\(772\) 0 0
\(773\) 6218.64 0.289352 0.144676 0.989479i \(-0.453786\pi\)
0.144676 + 0.989479i \(0.453786\pi\)
\(774\) 0 0
\(775\) 2222.10 0.102994
\(776\) 0 0
\(777\) −7244.72 −0.334495
\(778\) 0 0
\(779\) 888.910 0.0408838
\(780\) 0 0
\(781\) 36309.5 1.66358
\(782\) 0 0
\(783\) −6391.94 −0.291736
\(784\) 0 0
\(785\) −14098.7 −0.641026
\(786\) 0 0
\(787\) −15933.8 −0.721702 −0.360851 0.932623i \(-0.617514\pi\)
−0.360851 + 0.932623i \(0.617514\pi\)
\(788\) 0 0
\(789\) −9805.00 −0.442417
\(790\) 0 0
\(791\) 18332.3 0.824049
\(792\) 0 0
\(793\) −2747.19 −0.123021
\(794\) 0 0
\(795\) 14212.9 0.634063
\(796\) 0 0
\(797\) −825.379 −0.0366831 −0.0183415 0.999832i \(-0.505839\pi\)
−0.0183415 + 0.999832i \(0.505839\pi\)
\(798\) 0 0
\(799\) 13681.9 0.605796
\(800\) 0 0
\(801\) −6713.64 −0.296148
\(802\) 0 0
\(803\) −51674.0 −2.27090
\(804\) 0 0
\(805\) 38828.5 1.70003
\(806\) 0 0
\(807\) −26070.4 −1.13720
\(808\) 0 0
\(809\) 23981.8 1.04222 0.521110 0.853490i \(-0.325518\pi\)
0.521110 + 0.853490i \(0.325518\pi\)
\(810\) 0 0
\(811\) −1359.77 −0.0588753 −0.0294376 0.999567i \(-0.509372\pi\)
−0.0294376 + 0.999567i \(0.509372\pi\)
\(812\) 0 0
\(813\) −9717.52 −0.419198
\(814\) 0 0
\(815\) −26659.4 −1.14581
\(816\) 0 0
\(817\) 10228.4 0.438001
\(818\) 0 0
\(819\) 1827.02 0.0779501
\(820\) 0 0
\(821\) −29276.9 −1.24455 −0.622273 0.782801i \(-0.713791\pi\)
−0.622273 + 0.782801i \(0.713791\pi\)
\(822\) 0 0
\(823\) −3971.15 −0.168196 −0.0840982 0.996457i \(-0.526801\pi\)
−0.0840982 + 0.996457i \(0.526801\pi\)
\(824\) 0 0
\(825\) −7893.10 −0.333094
\(826\) 0 0
\(827\) −18920.5 −0.795564 −0.397782 0.917480i \(-0.630220\pi\)
−0.397782 + 0.917480i \(0.630220\pi\)
\(828\) 0 0
\(829\) 29902.3 1.25277 0.626387 0.779512i \(-0.284533\pi\)
0.626387 + 0.779512i \(0.284533\pi\)
\(830\) 0 0
\(831\) −660.273 −0.0275627
\(832\) 0 0
\(833\) −2327.90 −0.0968269
\(834\) 0 0
\(835\) −3932.95 −0.163001
\(836\) 0 0
\(837\) 1270.69 0.0524748
\(838\) 0 0
\(839\) 41125.0 1.69224 0.846121 0.532991i \(-0.178932\pi\)
0.846121 + 0.532991i \(0.178932\pi\)
\(840\) 0 0
\(841\) 31656.2 1.29797
\(842\) 0 0
\(843\) 1761.70 0.0719765
\(844\) 0 0
\(845\) −2217.80 −0.0902897
\(846\) 0 0
\(847\) 27703.6 1.12386
\(848\) 0 0
\(849\) −24822.3 −1.00341
\(850\) 0 0
\(851\) 29302.2 1.18034
\(852\) 0 0
\(853\) 2252.79 0.0904266 0.0452133 0.998977i \(-0.485603\pi\)
0.0452133 + 0.998977i \(0.485603\pi\)
\(854\) 0 0
\(855\) 3028.98 0.121157
\(856\) 0 0
\(857\) 16608.6 0.662005 0.331002 0.943630i \(-0.392613\pi\)
0.331002 + 0.943630i \(0.392613\pi\)
\(858\) 0 0
\(859\) −14112.0 −0.560529 −0.280265 0.959923i \(-0.590422\pi\)
−0.280265 + 0.959923i \(0.590422\pi\)
\(860\) 0 0
\(861\) −1623.75 −0.0642709
\(862\) 0 0
\(863\) 40154.6 1.58387 0.791934 0.610607i \(-0.209075\pi\)
0.791934 + 0.610607i \(0.209075\pi\)
\(864\) 0 0
\(865\) −31116.9 −1.22313
\(866\) 0 0
\(867\) −13085.5 −0.512578
\(868\) 0 0
\(869\) 62297.6 2.43188
\(870\) 0 0
\(871\) 1115.76 0.0434054
\(872\) 0 0
\(873\) −1559.08 −0.0604431
\(874\) 0 0
\(875\) 15939.8 0.615846
\(876\) 0 0
\(877\) −22495.6 −0.866161 −0.433080 0.901355i \(-0.642573\pi\)
−0.433080 + 0.901355i \(0.642573\pi\)
\(878\) 0 0
\(879\) −16146.2 −0.619565
\(880\) 0 0
\(881\) −42296.3 −1.61748 −0.808740 0.588166i \(-0.799850\pi\)
−0.808740 + 0.588166i \(0.799850\pi\)
\(882\) 0 0
\(883\) −37440.9 −1.42694 −0.713469 0.700687i \(-0.752877\pi\)
−0.713469 + 0.700687i \(0.752877\pi\)
\(884\) 0 0
\(885\) −15629.0 −0.593631
\(886\) 0 0
\(887\) 3372.82 0.127675 0.0638377 0.997960i \(-0.479666\pi\)
0.0638377 + 0.997960i \(0.479666\pi\)
\(888\) 0 0
\(889\) 11545.5 0.435572
\(890\) 0 0
\(891\) −4513.60 −0.169710
\(892\) 0 0
\(893\) −14945.7 −0.560066
\(894\) 0 0
\(895\) 3275.70 0.122340
\(896\) 0 0
\(897\) −7389.61 −0.275064
\(898\) 0 0
\(899\) −11141.5 −0.413337
\(900\) 0 0
\(901\) −8475.65 −0.313390
\(902\) 0 0
\(903\) −18684.0 −0.688554
\(904\) 0 0
\(905\) 41356.8 1.51906
\(906\) 0 0
\(907\) 30130.3 1.10304 0.551522 0.834160i \(-0.314047\pi\)
0.551522 + 0.834160i \(0.314047\pi\)
\(908\) 0 0
\(909\) −11371.1 −0.414911
\(910\) 0 0
\(911\) −26089.9 −0.948844 −0.474422 0.880297i \(-0.657343\pi\)
−0.474422 + 0.880297i \(0.657343\pi\)
\(912\) 0 0
\(913\) −21822.9 −0.791055
\(914\) 0 0
\(915\) 8319.60 0.300588
\(916\) 0 0
\(917\) 33285.8 1.19868
\(918\) 0 0
\(919\) −5142.55 −0.184589 −0.0922944 0.995732i \(-0.529420\pi\)
−0.0922944 + 0.995732i \(0.529420\pi\)
\(920\) 0 0
\(921\) −9287.76 −0.332293
\(922\) 0 0
\(923\) −8470.83 −0.302081
\(924\) 0 0
\(925\) −7301.83 −0.259549
\(926\) 0 0
\(927\) 7869.61 0.278826
\(928\) 0 0
\(929\) 23311.6 0.823282 0.411641 0.911346i \(-0.364956\pi\)
0.411641 + 0.911346i \(0.364956\pi\)
\(930\) 0 0
\(931\) 2542.92 0.0895176
\(932\) 0 0
\(933\) 10423.7 0.365764
\(934\) 0 0
\(935\) 17168.1 0.600489
\(936\) 0 0
\(937\) −17214.9 −0.600199 −0.300099 0.953908i \(-0.597020\pi\)
−0.300099 + 0.953908i \(0.597020\pi\)
\(938\) 0 0
\(939\) −10571.8 −0.367409
\(940\) 0 0
\(941\) −27871.2 −0.965541 −0.482770 0.875747i \(-0.660369\pi\)
−0.482770 + 0.875747i \(0.660369\pi\)
\(942\) 0 0
\(943\) 6567.47 0.226793
\(944\) 0 0
\(945\) −5532.95 −0.190462
\(946\) 0 0
\(947\) −14956.7 −0.513227 −0.256614 0.966514i \(-0.582607\pi\)
−0.256614 + 0.966514i \(0.582607\pi\)
\(948\) 0 0
\(949\) 12055.3 0.412361
\(950\) 0 0
\(951\) −970.188 −0.0330815
\(952\) 0 0
\(953\) −2056.14 −0.0698898 −0.0349449 0.999389i \(-0.511126\pi\)
−0.0349449 + 0.999389i \(0.511126\pi\)
\(954\) 0 0
\(955\) 62196.4 2.10746
\(956\) 0 0
\(957\) 39575.7 1.33678
\(958\) 0 0
\(959\) −4012.16 −0.135098
\(960\) 0 0
\(961\) −27576.1 −0.925653
\(962\) 0 0
\(963\) 14327.0 0.479419
\(964\) 0 0
\(965\) −8357.03 −0.278780
\(966\) 0 0
\(967\) 25282.1 0.840764 0.420382 0.907347i \(-0.361896\pi\)
0.420382 + 0.907347i \(0.361896\pi\)
\(968\) 0 0
\(969\) −1806.28 −0.0598825
\(970\) 0 0
\(971\) 22919.6 0.757493 0.378747 0.925500i \(-0.376355\pi\)
0.378747 + 0.925500i \(0.376355\pi\)
\(972\) 0 0
\(973\) 6830.91 0.225066
\(974\) 0 0
\(975\) 1841.42 0.0604848
\(976\) 0 0
\(977\) 42964.8 1.40693 0.703463 0.710732i \(-0.251636\pi\)
0.703463 + 0.710732i \(0.251636\pi\)
\(978\) 0 0
\(979\) 41567.5 1.35700
\(980\) 0 0
\(981\) 3346.09 0.108902
\(982\) 0 0
\(983\) −53757.9 −1.74426 −0.872132 0.489271i \(-0.837263\pi\)
−0.872132 + 0.489271i \(0.837263\pi\)
\(984\) 0 0
\(985\) 39384.2 1.27399
\(986\) 0 0
\(987\) 27300.9 0.880443
\(988\) 0 0
\(989\) 75569.8 2.42971
\(990\) 0 0
\(991\) −28021.7 −0.898224 −0.449112 0.893475i \(-0.648260\pi\)
−0.449112 + 0.893475i \(0.648260\pi\)
\(992\) 0 0
\(993\) −3173.22 −0.101409
\(994\) 0 0
\(995\) 8516.35 0.271343
\(996\) 0 0
\(997\) 7057.01 0.224170 0.112085 0.993699i \(-0.464247\pi\)
0.112085 + 0.993699i \(0.464247\pi\)
\(998\) 0 0
\(999\) −4175.49 −0.132239
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 312.4.a.d.1.1 2
3.2 odd 2 936.4.a.j.1.2 2
4.3 odd 2 624.4.a.j.1.1 2
8.3 odd 2 2496.4.a.bj.1.2 2
8.5 even 2 2496.4.a.ba.1.2 2
12.11 even 2 1872.4.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.d.1.1 2 1.1 even 1 trivial
624.4.a.j.1.1 2 4.3 odd 2
936.4.a.j.1.2 2 3.2 odd 2
1872.4.a.bi.1.2 2 12.11 even 2
2496.4.a.ba.1.2 2 8.5 even 2
2496.4.a.bj.1.2 2 8.3 odd 2