Properties

Label 400.4.a.x.1.1
Level $400$
Weight $4$
Character 400.1
Self dual yes
Analytic conductor $23.601$
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] = [400,4,Mod(1,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, 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("400.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.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: \(23.6007640023\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.89898 q^{3} +16.6969 q^{7} -18.5959 q^{9} +19.1918 q^{11} -61.7980 q^{13} +30.3837 q^{17} -59.1918 q^{19} -48.4041 q^{21} +205.687 q^{23} +132.182 q^{27} +8.38367 q^{29} -331.151 q^{31} -55.6367 q^{33} -266.565 q^{37} +179.151 q^{39} -320.788 q^{41} +83.1214 q^{43} -276.434 q^{47} -64.2122 q^{49} -88.0816 q^{51} -390.888 q^{53} +171.596 q^{57} -779.110 q^{59} -483.171 q^{61} -310.495 q^{63} +123.707 q^{67} -596.282 q^{69} -187.233 q^{71} +778.706 q^{73} +320.445 q^{77} +446.384 q^{79} +118.898 q^{81} -1054.05 q^{83} -24.3041 q^{87} -94.8490 q^{89} -1031.84 q^{91} +960.000 q^{93} -252.041 q^{97} -356.890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 4 q^{7} + 2 q^{9} - 40 q^{11} - 104 q^{13} - 96 q^{17} - 40 q^{19} - 136 q^{21} + 284 q^{23} + 88 q^{27} - 140 q^{29} - 192 q^{31} - 464 q^{33} - 200 q^{37} - 112 q^{39} - 524 q^{41} + 372 q^{43}+ \cdots - 1576 q^{99}+O(q^{100}) \) 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
\(3\) −2.89898 −0.557909 −0.278954 0.960304i \(-0.589988\pi\)
−0.278954 + 0.960304i \(0.589988\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 16.6969 0.901550 0.450775 0.892638i \(-0.351148\pi\)
0.450775 + 0.892638i \(0.351148\pi\)
\(8\) 0 0
\(9\) −18.5959 −0.688738
\(10\) 0 0
\(11\) 19.1918 0.526051 0.263025 0.964789i \(-0.415280\pi\)
0.263025 + 0.964789i \(0.415280\pi\)
\(12\) 0 0
\(13\) −61.7980 −1.31844 −0.659218 0.751952i \(-0.729113\pi\)
−0.659218 + 0.751952i \(0.729113\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 30.3837 0.433478 0.216739 0.976230i \(-0.430458\pi\)
0.216739 + 0.976230i \(0.430458\pi\)
\(18\) 0 0
\(19\) −59.1918 −0.714713 −0.357356 0.933968i \(-0.616322\pi\)
−0.357356 + 0.933968i \(0.616322\pi\)
\(20\) 0 0
\(21\) −48.4041 −0.502983
\(22\) 0 0
\(23\) 205.687 1.86472 0.932362 0.361526i \(-0.117744\pi\)
0.932362 + 0.361526i \(0.117744\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 132.182 0.942162
\(28\) 0 0
\(29\) 8.38367 0.0536831 0.0268415 0.999640i \(-0.491455\pi\)
0.0268415 + 0.999640i \(0.491455\pi\)
\(30\) 0 0
\(31\) −331.151 −1.91860 −0.959298 0.282396i \(-0.908871\pi\)
−0.959298 + 0.282396i \(0.908871\pi\)
\(32\) 0 0
\(33\) −55.6367 −0.293488
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −266.565 −1.18441 −0.592204 0.805788i \(-0.701742\pi\)
−0.592204 + 0.805788i \(0.701742\pi\)
\(38\) 0 0
\(39\) 179.151 0.735567
\(40\) 0 0
\(41\) −320.788 −1.22192 −0.610959 0.791662i \(-0.709216\pi\)
−0.610959 + 0.791662i \(0.709216\pi\)
\(42\) 0 0
\(43\) 83.1214 0.294788 0.147394 0.989078i \(-0.452911\pi\)
0.147394 + 0.989078i \(0.452911\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −276.434 −0.857915 −0.428957 0.903325i \(-0.641119\pi\)
−0.428957 + 0.903325i \(0.641119\pi\)
\(48\) 0 0
\(49\) −64.2122 −0.187208
\(50\) 0 0
\(51\) −88.0816 −0.241841
\(52\) 0 0
\(53\) −390.888 −1.01307 −0.506534 0.862220i \(-0.669073\pi\)
−0.506534 + 0.862220i \(0.669073\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 171.596 0.398744
\(58\) 0 0
\(59\) −779.110 −1.71918 −0.859589 0.510986i \(-0.829280\pi\)
−0.859589 + 0.510986i \(0.829280\pi\)
\(60\) 0 0
\(61\) −483.171 −1.01416 −0.507080 0.861899i \(-0.669275\pi\)
−0.507080 + 0.861899i \(0.669275\pi\)
\(62\) 0 0
\(63\) −310.495 −0.620931
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 123.707 0.225571 0.112785 0.993619i \(-0.464023\pi\)
0.112785 + 0.993619i \(0.464023\pi\)
\(68\) 0 0
\(69\) −596.282 −1.04035
\(70\) 0 0
\(71\) −187.233 −0.312964 −0.156482 0.987681i \(-0.550015\pi\)
−0.156482 + 0.987681i \(0.550015\pi\)
\(72\) 0 0
\(73\) 778.706 1.24850 0.624251 0.781224i \(-0.285404\pi\)
0.624251 + 0.781224i \(0.285404\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 320.445 0.474261
\(78\) 0 0
\(79\) 446.384 0.635723 0.317861 0.948137i \(-0.397035\pi\)
0.317861 + 0.948137i \(0.397035\pi\)
\(80\) 0 0
\(81\) 118.898 0.163097
\(82\) 0 0
\(83\) −1054.05 −1.39394 −0.696970 0.717100i \(-0.745469\pi\)
−0.696970 + 0.717100i \(0.745469\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −24.3041 −0.0299503
\(88\) 0 0
\(89\) −94.8490 −0.112966 −0.0564830 0.998404i \(-0.517989\pi\)
−0.0564830 + 0.998404i \(0.517989\pi\)
\(90\) 0 0
\(91\) −1031.84 −1.18864
\(92\) 0 0
\(93\) 960.000 1.07040
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −252.041 −0.263823 −0.131912 0.991261i \(-0.542112\pi\)
−0.131912 + 0.991261i \(0.542112\pi\)
\(98\) 0 0
\(99\) −356.890 −0.362311
\(100\) 0 0
\(101\) 37.9184 0.0373566 0.0186783 0.999826i \(-0.494054\pi\)
0.0186783 + 0.999826i \(0.494054\pi\)
\(102\) 0 0
\(103\) 94.4133 0.0903186 0.0451593 0.998980i \(-0.485620\pi\)
0.0451593 + 0.998980i \(0.485620\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −901.464 −0.814466 −0.407233 0.913324i \(-0.633506\pi\)
−0.407233 + 0.913324i \(0.633506\pi\)
\(108\) 0 0
\(109\) 1415.69 1.24403 0.622013 0.783007i \(-0.286315\pi\)
0.622013 + 0.783007i \(0.286315\pi\)
\(110\) 0 0
\(111\) 772.767 0.660791
\(112\) 0 0
\(113\) 293.576 0.244401 0.122200 0.992505i \(-0.461005\pi\)
0.122200 + 0.992505i \(0.461005\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1149.19 0.908057
\(118\) 0 0
\(119\) 507.314 0.390802
\(120\) 0 0
\(121\) −962.673 −0.723271
\(122\) 0 0
\(123\) 929.957 0.681719
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 774.717 0.541300 0.270650 0.962678i \(-0.412761\pi\)
0.270650 + 0.962678i \(0.412761\pi\)
\(128\) 0 0
\(129\) −240.967 −0.164465
\(130\) 0 0
\(131\) −334.343 −0.222990 −0.111495 0.993765i \(-0.535564\pi\)
−0.111495 + 0.993765i \(0.535564\pi\)
\(132\) 0 0
\(133\) −988.322 −0.644349
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −323.514 −0.201750 −0.100875 0.994899i \(-0.532164\pi\)
−0.100875 + 0.994899i \(0.532164\pi\)
\(138\) 0 0
\(139\) 396.482 0.241936 0.120968 0.992656i \(-0.461400\pi\)
0.120968 + 0.992656i \(0.461400\pi\)
\(140\) 0 0
\(141\) 801.376 0.478638
\(142\) 0 0
\(143\) −1186.02 −0.693564
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 186.150 0.104445
\(148\) 0 0
\(149\) −1682.89 −0.925284 −0.462642 0.886545i \(-0.653099\pi\)
−0.462642 + 0.886545i \(0.653099\pi\)
\(150\) 0 0
\(151\) 2924.52 1.57612 0.788060 0.615598i \(-0.211085\pi\)
0.788060 + 0.615598i \(0.211085\pi\)
\(152\) 0 0
\(153\) −565.012 −0.298553
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2768.42 −1.40729 −0.703644 0.710553i \(-0.748445\pi\)
−0.703644 + 0.710553i \(0.748445\pi\)
\(158\) 0 0
\(159\) 1133.18 0.565199
\(160\) 0 0
\(161\) 3434.34 1.68114
\(162\) 0 0
\(163\) 2816.33 1.35333 0.676663 0.736292i \(-0.263425\pi\)
0.676663 + 0.736292i \(0.263425\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1836.41 0.850933 0.425466 0.904974i \(-0.360110\pi\)
0.425466 + 0.904974i \(0.360110\pi\)
\(168\) 0 0
\(169\) 1621.99 0.738274
\(170\) 0 0
\(171\) 1100.73 0.492249
\(172\) 0 0
\(173\) −1224.22 −0.538011 −0.269006 0.963139i \(-0.586695\pi\)
−0.269006 + 0.963139i \(0.586695\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2258.62 0.959145
\(178\) 0 0
\(179\) −2729.58 −1.13977 −0.569883 0.821726i \(-0.693011\pi\)
−0.569883 + 0.821726i \(0.693011\pi\)
\(180\) 0 0
\(181\) 2642.36 1.08511 0.542555 0.840020i \(-0.317457\pi\)
0.542555 + 0.840020i \(0.317457\pi\)
\(182\) 0 0
\(183\) 1400.70 0.565809
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 583.118 0.228031
\(188\) 0 0
\(189\) 2207.03 0.849406
\(190\) 0 0
\(191\) 2339.23 0.886183 0.443091 0.896476i \(-0.353882\pi\)
0.443091 + 0.896476i \(0.353882\pi\)
\(192\) 0 0
\(193\) −4601.61 −1.71622 −0.858111 0.513464i \(-0.828362\pi\)
−0.858111 + 0.513464i \(0.828362\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 823.941 0.297987 0.148993 0.988838i \(-0.452397\pi\)
0.148993 + 0.988838i \(0.452397\pi\)
\(198\) 0 0
\(199\) 3329.70 1.18611 0.593055 0.805162i \(-0.297922\pi\)
0.593055 + 0.805162i \(0.297922\pi\)
\(200\) 0 0
\(201\) −358.624 −0.125848
\(202\) 0 0
\(203\) 139.982 0.0483980
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3824.93 −1.28431
\(208\) 0 0
\(209\) −1136.00 −0.375975
\(210\) 0 0
\(211\) −1018.78 −0.332398 −0.166199 0.986092i \(-0.553149\pi\)
−0.166199 + 0.986092i \(0.553149\pi\)
\(212\) 0 0
\(213\) 542.784 0.174605
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −5529.21 −1.72971
\(218\) 0 0
\(219\) −2257.45 −0.696550
\(220\) 0 0
\(221\) −1877.65 −0.571513
\(222\) 0 0
\(223\) −99.1581 −0.0297763 −0.0148882 0.999889i \(-0.504739\pi\)
−0.0148882 + 0.999889i \(0.504739\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1197.59 −0.350163 −0.175081 0.984554i \(-0.556019\pi\)
−0.175081 + 0.984554i \(0.556019\pi\)
\(228\) 0 0
\(229\) −453.592 −0.130892 −0.0654458 0.997856i \(-0.520847\pi\)
−0.0654458 + 0.997856i \(0.520847\pi\)
\(230\) 0 0
\(231\) −928.963 −0.264594
\(232\) 0 0
\(233\) 3788.49 1.06520 0.532601 0.846367i \(-0.321215\pi\)
0.532601 + 0.846367i \(0.321215\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1294.06 −0.354675
\(238\) 0 0
\(239\) 6000.47 1.62401 0.812005 0.583651i \(-0.198376\pi\)
0.812005 + 0.583651i \(0.198376\pi\)
\(240\) 0 0
\(241\) 1842.53 0.492480 0.246240 0.969209i \(-0.420805\pi\)
0.246240 + 0.969209i \(0.420805\pi\)
\(242\) 0 0
\(243\) −3913.59 −1.03316
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3657.93 0.942303
\(248\) 0 0
\(249\) 3055.67 0.777691
\(250\) 0 0
\(251\) −1149.46 −0.289057 −0.144529 0.989501i \(-0.546167\pi\)
−0.144529 + 0.989501i \(0.546167\pi\)
\(252\) 0 0
\(253\) 3947.51 0.980939
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5407.67 1.31253 0.656267 0.754528i \(-0.272134\pi\)
0.656267 + 0.754528i \(0.272134\pi\)
\(258\) 0 0
\(259\) −4450.82 −1.06780
\(260\) 0 0
\(261\) −155.902 −0.0369735
\(262\) 0 0
\(263\) 2067.34 0.484705 0.242352 0.970188i \(-0.422081\pi\)
0.242352 + 0.970188i \(0.422081\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 274.965 0.0630247
\(268\) 0 0
\(269\) −592.388 −0.134270 −0.0671348 0.997744i \(-0.521386\pi\)
−0.0671348 + 0.997744i \(0.521386\pi\)
\(270\) 0 0
\(271\) 2583.27 0.579049 0.289524 0.957171i \(-0.406503\pi\)
0.289524 + 0.957171i \(0.406503\pi\)
\(272\) 0 0
\(273\) 2991.27 0.663151
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4488.09 0.973513 0.486756 0.873538i \(-0.338180\pi\)
0.486756 + 0.873538i \(0.338180\pi\)
\(278\) 0 0
\(279\) 6158.06 1.32141
\(280\) 0 0
\(281\) −6280.54 −1.33333 −0.666665 0.745357i \(-0.732279\pi\)
−0.666665 + 0.745357i \(0.732279\pi\)
\(282\) 0 0
\(283\) −5233.40 −1.09927 −0.549635 0.835405i \(-0.685233\pi\)
−0.549635 + 0.835405i \(0.685233\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5356.17 −1.10162
\(288\) 0 0
\(289\) −3989.83 −0.812097
\(290\) 0 0
\(291\) 730.661 0.147189
\(292\) 0 0
\(293\) 2438.21 0.486149 0.243074 0.970008i \(-0.421844\pi\)
0.243074 + 0.970008i \(0.421844\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2536.81 0.495625
\(298\) 0 0
\(299\) −12711.0 −2.45852
\(300\) 0 0
\(301\) 1387.87 0.265766
\(302\) 0 0
\(303\) −109.925 −0.0208416
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7910.44 1.47060 0.735298 0.677744i \(-0.237042\pi\)
0.735298 + 0.677744i \(0.237042\pi\)
\(308\) 0 0
\(309\) −273.702 −0.0503895
\(310\) 0 0
\(311\) −5419.71 −0.988178 −0.494089 0.869411i \(-0.664498\pi\)
−0.494089 + 0.869411i \(0.664498\pi\)
\(312\) 0 0
\(313\) −5570.86 −1.00602 −0.503009 0.864281i \(-0.667774\pi\)
−0.503009 + 0.864281i \(0.667774\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3724.09 0.659829 0.329915 0.944011i \(-0.392980\pi\)
0.329915 + 0.944011i \(0.392980\pi\)
\(318\) 0 0
\(319\) 160.898 0.0282400
\(320\) 0 0
\(321\) 2613.33 0.454398
\(322\) 0 0
\(323\) −1798.47 −0.309812
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −4104.07 −0.694053
\(328\) 0 0
\(329\) −4615.60 −0.773453
\(330\) 0 0
\(331\) −3223.96 −0.535362 −0.267681 0.963508i \(-0.586257\pi\)
−0.267681 + 0.963508i \(0.586257\pi\)
\(332\) 0 0
\(333\) 4957.03 0.815746
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −9524.43 −1.53955 −0.769776 0.638314i \(-0.779632\pi\)
−0.769776 + 0.638314i \(0.779632\pi\)
\(338\) 0 0
\(339\) −851.069 −0.136353
\(340\) 0 0
\(341\) −6355.40 −1.00928
\(342\) 0 0
\(343\) −6799.20 −1.07033
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6042.30 −0.934777 −0.467388 0.884052i \(-0.654805\pi\)
−0.467388 + 0.884052i \(0.654805\pi\)
\(348\) 0 0
\(349\) −1626.20 −0.249422 −0.124711 0.992193i \(-0.539800\pi\)
−0.124711 + 0.992193i \(0.539800\pi\)
\(350\) 0 0
\(351\) −8168.55 −1.24218
\(352\) 0 0
\(353\) −886.955 −0.133733 −0.0668667 0.997762i \(-0.521300\pi\)
−0.0668667 + 0.997762i \(0.521300\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1470.69 −0.218032
\(358\) 0 0
\(359\) −1722.27 −0.253198 −0.126599 0.991954i \(-0.540406\pi\)
−0.126599 + 0.991954i \(0.540406\pi\)
\(360\) 0 0
\(361\) −3355.33 −0.489186
\(362\) 0 0
\(363\) 2790.77 0.403519
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 9271.77 1.31875 0.659377 0.751813i \(-0.270820\pi\)
0.659377 + 0.751813i \(0.270820\pi\)
\(368\) 0 0
\(369\) 5965.34 0.841581
\(370\) 0 0
\(371\) −6526.63 −0.913331
\(372\) 0 0
\(373\) −5697.15 −0.790850 −0.395425 0.918498i \(-0.629403\pi\)
−0.395425 + 0.918498i \(0.629403\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −518.094 −0.0707777
\(378\) 0 0
\(379\) 8526.24 1.15558 0.577789 0.816186i \(-0.303916\pi\)
0.577789 + 0.816186i \(0.303916\pi\)
\(380\) 0 0
\(381\) −2245.89 −0.301996
\(382\) 0 0
\(383\) 4069.23 0.542893 0.271447 0.962453i \(-0.412498\pi\)
0.271447 + 0.962453i \(0.412498\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1545.72 −0.203032
\(388\) 0 0
\(389\) −2394.17 −0.312054 −0.156027 0.987753i \(-0.549869\pi\)
−0.156027 + 0.987753i \(0.549869\pi\)
\(390\) 0 0
\(391\) 6249.52 0.808316
\(392\) 0 0
\(393\) 969.253 0.124408
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −3497.79 −0.442190 −0.221095 0.975252i \(-0.570963\pi\)
−0.221095 + 0.975252i \(0.570963\pi\)
\(398\) 0 0
\(399\) 2865.13 0.359488
\(400\) 0 0
\(401\) −8608.89 −1.07209 −0.536044 0.844190i \(-0.680082\pi\)
−0.536044 + 0.844190i \(0.680082\pi\)
\(402\) 0 0
\(403\) 20464.5 2.52955
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5115.88 −0.623058
\(408\) 0 0
\(409\) 1385.67 0.167523 0.0837615 0.996486i \(-0.473307\pi\)
0.0837615 + 0.996486i \(0.473307\pi\)
\(410\) 0 0
\(411\) 937.861 0.112558
\(412\) 0 0
\(413\) −13008.8 −1.54992
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1149.39 −0.134978
\(418\) 0 0
\(419\) −3738.23 −0.435858 −0.217929 0.975965i \(-0.569930\pi\)
−0.217929 + 0.975965i \(0.569930\pi\)
\(420\) 0 0
\(421\) −8993.95 −1.04118 −0.520592 0.853806i \(-0.674289\pi\)
−0.520592 + 0.853806i \(0.674289\pi\)
\(422\) 0 0
\(423\) 5140.54 0.590878
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −8067.48 −0.914316
\(428\) 0 0
\(429\) 3438.24 0.386946
\(430\) 0 0
\(431\) −462.547 −0.0516940 −0.0258470 0.999666i \(-0.508228\pi\)
−0.0258470 + 0.999666i \(0.508228\pi\)
\(432\) 0 0
\(433\) 5231.82 0.580659 0.290329 0.956927i \(-0.406235\pi\)
0.290329 + 0.956927i \(0.406235\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12175.0 −1.33274
\(438\) 0 0
\(439\) −8995.77 −0.978006 −0.489003 0.872282i \(-0.662639\pi\)
−0.489003 + 0.872282i \(0.662639\pi\)
\(440\) 0 0
\(441\) 1194.09 0.128937
\(442\) 0 0
\(443\) −5549.52 −0.595182 −0.297591 0.954694i \(-0.596183\pi\)
−0.297591 + 0.954694i \(0.596183\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4878.65 0.516224
\(448\) 0 0
\(449\) −5951.29 −0.625521 −0.312760 0.949832i \(-0.601254\pi\)
−0.312760 + 0.949832i \(0.601254\pi\)
\(450\) 0 0
\(451\) −6156.51 −0.642791
\(452\) 0 0
\(453\) −8478.13 −0.879332
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 13912.7 1.42409 0.712047 0.702132i \(-0.247768\pi\)
0.712047 + 0.702132i \(0.247768\pi\)
\(458\) 0 0
\(459\) 4016.16 0.408406
\(460\) 0 0
\(461\) 17467.6 1.76475 0.882374 0.470549i \(-0.155944\pi\)
0.882374 + 0.470549i \(0.155944\pi\)
\(462\) 0 0
\(463\) 1575.36 0.158127 0.0790637 0.996870i \(-0.474807\pi\)
0.0790637 + 0.996870i \(0.474807\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15618.3 1.54760 0.773800 0.633430i \(-0.218354\pi\)
0.773800 + 0.633430i \(0.218354\pi\)
\(468\) 0 0
\(469\) 2065.53 0.203363
\(470\) 0 0
\(471\) 8025.60 0.785138
\(472\) 0 0
\(473\) 1595.25 0.155074
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 7268.92 0.697738
\(478\) 0 0
\(479\) −9527.90 −0.908854 −0.454427 0.890784i \(-0.650156\pi\)
−0.454427 + 0.890784i \(0.650156\pi\)
\(480\) 0 0
\(481\) 16473.2 1.56157
\(482\) 0 0
\(483\) −9956.08 −0.937924
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 15729.6 1.46361 0.731805 0.681514i \(-0.238678\pi\)
0.731805 + 0.681514i \(0.238678\pi\)
\(488\) 0 0
\(489\) −8164.49 −0.755033
\(490\) 0 0
\(491\) −2566.49 −0.235894 −0.117947 0.993020i \(-0.537631\pi\)
−0.117947 + 0.993020i \(0.537631\pi\)
\(492\) 0 0
\(493\) 254.727 0.0232704
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3126.21 −0.282152
\(498\) 0 0
\(499\) 13560.4 1.21652 0.608261 0.793737i \(-0.291867\pi\)
0.608261 + 0.793737i \(0.291867\pi\)
\(500\) 0 0
\(501\) −5323.72 −0.474743
\(502\) 0 0
\(503\) −5222.51 −0.462943 −0.231471 0.972842i \(-0.574354\pi\)
−0.231471 + 0.972842i \(0.574354\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −4702.11 −0.411890
\(508\) 0 0
\(509\) −11875.9 −1.03416 −0.517082 0.855936i \(-0.672982\pi\)
−0.517082 + 0.855936i \(0.672982\pi\)
\(510\) 0 0
\(511\) 13002.0 1.12559
\(512\) 0 0
\(513\) −7824.07 −0.673375
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −5305.27 −0.451307
\(518\) 0 0
\(519\) 3549.00 0.300161
\(520\) 0 0
\(521\) 2456.04 0.206528 0.103264 0.994654i \(-0.467071\pi\)
0.103264 + 0.994654i \(0.467071\pi\)
\(522\) 0 0
\(523\) 634.460 0.0530459 0.0265229 0.999648i \(-0.491556\pi\)
0.0265229 + 0.999648i \(0.491556\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10061.6 −0.831669
\(528\) 0 0
\(529\) 30140.0 2.47720
\(530\) 0 0
\(531\) 14488.3 1.18406
\(532\) 0 0
\(533\) 19824.0 1.61102
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 7912.98 0.635885
\(538\) 0 0
\(539\) −1232.35 −0.0984807
\(540\) 0 0
\(541\) 18078.4 1.43669 0.718347 0.695685i \(-0.244899\pi\)
0.718347 + 0.695685i \(0.244899\pi\)
\(542\) 0 0
\(543\) −7660.14 −0.605393
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −24815.3 −1.93972 −0.969860 0.243661i \(-0.921651\pi\)
−0.969860 + 0.243661i \(0.921651\pi\)
\(548\) 0 0
\(549\) 8985.02 0.698490
\(550\) 0 0
\(551\) −496.245 −0.0383680
\(552\) 0 0
\(553\) 7453.24 0.573136
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10073.7 0.766310 0.383155 0.923684i \(-0.374838\pi\)
0.383155 + 0.923684i \(0.374838\pi\)
\(558\) 0 0
\(559\) −5136.73 −0.388660
\(560\) 0 0
\(561\) −1690.45 −0.127221
\(562\) 0 0
\(563\) 7505.06 0.561813 0.280906 0.959735i \(-0.409365\pi\)
0.280906 + 0.959735i \(0.409365\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1985.23 0.147040
\(568\) 0 0
\(569\) −15251.4 −1.12368 −0.561838 0.827247i \(-0.689906\pi\)
−0.561838 + 0.827247i \(0.689906\pi\)
\(570\) 0 0
\(571\) 2683.78 0.196695 0.0983474 0.995152i \(-0.468644\pi\)
0.0983474 + 0.995152i \(0.468644\pi\)
\(572\) 0 0
\(573\) −6781.39 −0.494409
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 7369.88 0.531737 0.265868 0.964009i \(-0.414341\pi\)
0.265868 + 0.964009i \(0.414341\pi\)
\(578\) 0 0
\(579\) 13340.0 0.957496
\(580\) 0 0
\(581\) −17599.4 −1.25671
\(582\) 0 0
\(583\) −7501.85 −0.532925
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20865.4 1.46713 0.733567 0.679618i \(-0.237854\pi\)
0.733567 + 0.679618i \(0.237854\pi\)
\(588\) 0 0
\(589\) 19601.4 1.37124
\(590\) 0 0
\(591\) −2388.59 −0.166249
\(592\) 0 0
\(593\) 25894.0 1.79316 0.896578 0.442886i \(-0.146046\pi\)
0.896578 + 0.442886i \(0.146046\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −9652.73 −0.661742
\(598\) 0 0
\(599\) −5632.42 −0.384198 −0.192099 0.981376i \(-0.561529\pi\)
−0.192099 + 0.981376i \(0.561529\pi\)
\(600\) 0 0
\(601\) −13079.7 −0.887742 −0.443871 0.896091i \(-0.646395\pi\)
−0.443871 + 0.896091i \(0.646395\pi\)
\(602\) 0 0
\(603\) −2300.45 −0.155359
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −18890.2 −1.26314 −0.631572 0.775317i \(-0.717590\pi\)
−0.631572 + 0.775317i \(0.717590\pi\)
\(608\) 0 0
\(609\) −405.804 −0.0270017
\(610\) 0 0
\(611\) 17083.0 1.13111
\(612\) 0 0
\(613\) 13631.5 0.898155 0.449078 0.893493i \(-0.351753\pi\)
0.449078 + 0.893493i \(0.351753\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −14982.2 −0.977568 −0.488784 0.872405i \(-0.662559\pi\)
−0.488784 + 0.872405i \(0.662559\pi\)
\(618\) 0 0
\(619\) 25049.9 1.62656 0.813279 0.581873i \(-0.197680\pi\)
0.813279 + 0.581873i \(0.197680\pi\)
\(620\) 0 0
\(621\) 27188.0 1.75687
\(622\) 0 0
\(623\) −1583.69 −0.101844
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 3293.24 0.209760
\(628\) 0 0
\(629\) −8099.23 −0.513414
\(630\) 0 0
\(631\) −18711.0 −1.18046 −0.590232 0.807233i \(-0.700964\pi\)
−0.590232 + 0.807233i \(0.700964\pi\)
\(632\) 0 0
\(633\) 2953.43 0.185448
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3968.19 0.246821
\(638\) 0 0
\(639\) 3481.76 0.215550
\(640\) 0 0
\(641\) 25792.2 1.58929 0.794643 0.607077i \(-0.207658\pi\)
0.794643 + 0.607077i \(0.207658\pi\)
\(642\) 0 0
\(643\) −20256.7 −1.24237 −0.621186 0.783663i \(-0.713349\pi\)
−0.621186 + 0.783663i \(0.713349\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6655.43 −0.404408 −0.202204 0.979343i \(-0.564810\pi\)
−0.202204 + 0.979343i \(0.564810\pi\)
\(648\) 0 0
\(649\) −14952.6 −0.904375
\(650\) 0 0
\(651\) 16029.1 0.965021
\(652\) 0 0
\(653\) −8490.91 −0.508844 −0.254422 0.967093i \(-0.581885\pi\)
−0.254422 + 0.967093i \(0.581885\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −14480.8 −0.859890
\(658\) 0 0
\(659\) −15543.8 −0.918816 −0.459408 0.888225i \(-0.651938\pi\)
−0.459408 + 0.888225i \(0.651938\pi\)
\(660\) 0 0
\(661\) −13519.8 −0.795553 −0.397777 0.917482i \(-0.630218\pi\)
−0.397777 + 0.917482i \(0.630218\pi\)
\(662\) 0 0
\(663\) 5443.27 0.318852
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1724.41 0.100104
\(668\) 0 0
\(669\) 287.457 0.0166125
\(670\) 0 0
\(671\) −9272.95 −0.533499
\(672\) 0 0
\(673\) −11565.3 −0.662421 −0.331211 0.943557i \(-0.607457\pi\)
−0.331211 + 0.943557i \(0.607457\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28227.5 1.60247 0.801235 0.598350i \(-0.204177\pi\)
0.801235 + 0.598350i \(0.204177\pi\)
\(678\) 0 0
\(679\) −4208.31 −0.237850
\(680\) 0 0
\(681\) 3471.79 0.195359
\(682\) 0 0
\(683\) −6425.99 −0.360005 −0.180003 0.983666i \(-0.557611\pi\)
−0.180003 + 0.983666i \(0.557611\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1314.95 0.0730256
\(688\) 0 0
\(689\) 24156.1 1.33566
\(690\) 0 0
\(691\) 19066.9 1.04969 0.524846 0.851197i \(-0.324123\pi\)
0.524846 + 0.851197i \(0.324123\pi\)
\(692\) 0 0
\(693\) −5958.97 −0.326641
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −9746.71 −0.529674
\(698\) 0 0
\(699\) −10982.7 −0.594285
\(700\) 0 0
\(701\) 4796.00 0.258406 0.129203 0.991618i \(-0.458758\pi\)
0.129203 + 0.991618i \(0.458758\pi\)
\(702\) 0 0
\(703\) 15778.5 0.846511
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 633.121 0.0336789
\(708\) 0 0
\(709\) −9805.33 −0.519389 −0.259695 0.965691i \(-0.583622\pi\)
−0.259695 + 0.965691i \(0.583622\pi\)
\(710\) 0 0
\(711\) −8300.91 −0.437846
\(712\) 0 0
\(713\) −68113.4 −3.57765
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −17395.2 −0.906049
\(718\) 0 0
\(719\) −27539.7 −1.42845 −0.714227 0.699915i \(-0.753221\pi\)
−0.714227 + 0.699915i \(0.753221\pi\)
\(720\) 0 0
\(721\) 1576.41 0.0814267
\(722\) 0 0
\(723\) −5341.45 −0.274759
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −16543.8 −0.843985 −0.421993 0.906599i \(-0.638669\pi\)
−0.421993 + 0.906599i \(0.638669\pi\)
\(728\) 0 0
\(729\) 8135.16 0.413309
\(730\) 0 0
\(731\) 2525.53 0.127784
\(732\) 0 0
\(733\) 16718.6 0.842450 0.421225 0.906956i \(-0.361600\pi\)
0.421225 + 0.906956i \(0.361600\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2374.17 0.118662
\(738\) 0 0
\(739\) 24022.6 1.19579 0.597893 0.801576i \(-0.296005\pi\)
0.597893 + 0.801576i \(0.296005\pi\)
\(740\) 0 0
\(741\) −10604.3 −0.525719
\(742\) 0 0
\(743\) 26201.8 1.29374 0.646871 0.762600i \(-0.276077\pi\)
0.646871 + 0.762600i \(0.276077\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 19601.0 0.960059
\(748\) 0 0
\(749\) −15051.7 −0.734282
\(750\) 0 0
\(751\) −24451.7 −1.18809 −0.594045 0.804432i \(-0.702470\pi\)
−0.594045 + 0.804432i \(0.702470\pi\)
\(752\) 0 0
\(753\) 3332.26 0.161268
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 21403.8 1.02765 0.513827 0.857894i \(-0.328227\pi\)
0.513827 + 0.857894i \(0.328227\pi\)
\(758\) 0 0
\(759\) −11443.7 −0.547275
\(760\) 0 0
\(761\) −28935.9 −1.37835 −0.689176 0.724594i \(-0.742027\pi\)
−0.689176 + 0.724594i \(0.742027\pi\)
\(762\) 0 0
\(763\) 23637.8 1.12155
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 48147.4 2.26663
\(768\) 0 0
\(769\) −11479.3 −0.538301 −0.269151 0.963098i \(-0.586743\pi\)
−0.269151 + 0.963098i \(0.586743\pi\)
\(770\) 0 0
\(771\) −15676.7 −0.732275
\(772\) 0 0
\(773\) 2512.27 0.116895 0.0584477 0.998290i \(-0.481385\pi\)
0.0584477 + 0.998290i \(0.481385\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 12902.8 0.595736
\(778\) 0 0
\(779\) 18988.0 0.873320
\(780\) 0 0
\(781\) −3593.34 −0.164635
\(782\) 0 0
\(783\) 1108.17 0.0505781
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −2650.18 −0.120036 −0.0600182 0.998197i \(-0.519116\pi\)
−0.0600182 + 0.998197i \(0.519116\pi\)
\(788\) 0 0
\(789\) −5993.16 −0.270421
\(790\) 0 0
\(791\) 4901.81 0.220339
\(792\) 0 0
\(793\) 29859.0 1.33711
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −24516.7 −1.08962 −0.544810 0.838560i \(-0.683398\pi\)
−0.544810 + 0.838560i \(0.683398\pi\)
\(798\) 0 0
\(799\) −8399.07 −0.371887
\(800\) 0 0
\(801\) 1763.80 0.0778039
\(802\) 0 0
\(803\) 14944.8 0.656775
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1717.32 0.0749102
\(808\) 0 0
\(809\) −29725.5 −1.29183 −0.645917 0.763408i \(-0.723525\pi\)
−0.645917 + 0.763408i \(0.723525\pi\)
\(810\) 0 0
\(811\) 2050.38 0.0887773 0.0443887 0.999014i \(-0.485866\pi\)
0.0443887 + 0.999014i \(0.485866\pi\)
\(812\) 0 0
\(813\) −7488.83 −0.323056
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −4920.11 −0.210689
\(818\) 0 0
\(819\) 19188.0 0.818658
\(820\) 0 0
\(821\) −33863.4 −1.43952 −0.719758 0.694225i \(-0.755747\pi\)
−0.719758 + 0.694225i \(0.755747\pi\)
\(822\) 0 0
\(823\) −1216.52 −0.0515251 −0.0257625 0.999668i \(-0.508201\pi\)
−0.0257625 + 0.999668i \(0.508201\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −25654.4 −1.07871 −0.539355 0.842079i \(-0.681332\pi\)
−0.539355 + 0.842079i \(0.681332\pi\)
\(828\) 0 0
\(829\) 24544.6 1.02831 0.514154 0.857698i \(-0.328106\pi\)
0.514154 + 0.857698i \(0.328106\pi\)
\(830\) 0 0
\(831\) −13010.9 −0.543131
\(832\) 0 0
\(833\) −1951.00 −0.0811504
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −43772.1 −1.80763
\(838\) 0 0
\(839\) −16548.6 −0.680954 −0.340477 0.940253i \(-0.610589\pi\)
−0.340477 + 0.940253i \(0.610589\pi\)
\(840\) 0 0
\(841\) −24318.7 −0.997118
\(842\) 0 0
\(843\) 18207.2 0.743877
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −16073.7 −0.652065
\(848\) 0 0
\(849\) 15171.5 0.613292
\(850\) 0 0
\(851\) −54828.9 −2.20859
\(852\) 0 0
\(853\) −24363.5 −0.977950 −0.488975 0.872298i \(-0.662629\pi\)
−0.488975 + 0.872298i \(0.662629\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −575.718 −0.0229477 −0.0114738 0.999934i \(-0.503652\pi\)
−0.0114738 + 0.999934i \(0.503652\pi\)
\(858\) 0 0
\(859\) −1531.31 −0.0608236 −0.0304118 0.999537i \(-0.509682\pi\)
−0.0304118 + 0.999537i \(0.509682\pi\)
\(860\) 0 0
\(861\) 15527.4 0.614604
\(862\) 0 0
\(863\) −7706.51 −0.303978 −0.151989 0.988382i \(-0.548568\pi\)
−0.151989 + 0.988382i \(0.548568\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 11566.4 0.453076
\(868\) 0 0
\(869\) 8566.92 0.334422
\(870\) 0 0
\(871\) −7644.85 −0.297400
\(872\) 0 0
\(873\) 4686.93 0.181705
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 43618.9 1.67948 0.839741 0.542988i \(-0.182707\pi\)
0.839741 + 0.542988i \(0.182707\pi\)
\(878\) 0 0
\(879\) −7068.31 −0.271227
\(880\) 0 0
\(881\) −13416.4 −0.513066 −0.256533 0.966536i \(-0.582580\pi\)
−0.256533 + 0.966536i \(0.582580\pi\)
\(882\) 0 0
\(883\) −29538.2 −1.12575 −0.562875 0.826542i \(-0.690305\pi\)
−0.562875 + 0.826542i \(0.690305\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1950.88 0.0738490 0.0369245 0.999318i \(-0.488244\pi\)
0.0369245 + 0.999318i \(0.488244\pi\)
\(888\) 0 0
\(889\) 12935.4 0.488009
\(890\) 0 0
\(891\) 2281.87 0.0857974
\(892\) 0 0
\(893\) 16362.6 0.613162
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 36849.0 1.37163
\(898\) 0 0
\(899\) −2776.26 −0.102996
\(900\) 0 0
\(901\) −11876.6 −0.439142
\(902\) 0 0
\(903\) −4023.42 −0.148273
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −3507.55 −0.128408 −0.0642042 0.997937i \(-0.520451\pi\)
−0.0642042 + 0.997937i \(0.520451\pi\)
\(908\) 0 0
\(909\) −705.127 −0.0257289
\(910\) 0 0
\(911\) 33841.4 1.23075 0.615377 0.788233i \(-0.289004\pi\)
0.615377 + 0.788233i \(0.289004\pi\)
\(912\) 0 0
\(913\) −20229.2 −0.733283
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −5582.50 −0.201036
\(918\) 0 0
\(919\) −25440.7 −0.913178 −0.456589 0.889678i \(-0.650929\pi\)
−0.456589 + 0.889678i \(0.650929\pi\)
\(920\) 0 0
\(921\) −22932.2 −0.820458
\(922\) 0 0
\(923\) 11570.6 0.412623
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1755.70 −0.0622058
\(928\) 0 0
\(929\) −26416.7 −0.932941 −0.466471 0.884537i \(-0.654475\pi\)
−0.466471 + 0.884537i \(0.654475\pi\)
\(930\) 0 0
\(931\) 3800.84 0.133800
\(932\) 0 0
\(933\) 15711.6 0.551313
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1936.70 −0.0675231 −0.0337616 0.999430i \(-0.510749\pi\)
−0.0337616 + 0.999430i \(0.510749\pi\)
\(938\) 0 0
\(939\) 16149.8 0.561266
\(940\) 0 0
\(941\) 23459.1 0.812694 0.406347 0.913719i \(-0.366802\pi\)
0.406347 + 0.913719i \(0.366802\pi\)
\(942\) 0 0
\(943\) −65981.8 −2.27854
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23606.8 −0.810049 −0.405025 0.914306i \(-0.632737\pi\)
−0.405025 + 0.914306i \(0.632737\pi\)
\(948\) 0 0
\(949\) −48122.4 −1.64607
\(950\) 0 0
\(951\) −10796.1 −0.368125
\(952\) 0 0
\(953\) 30164.2 1.02530 0.512652 0.858596i \(-0.328663\pi\)
0.512652 + 0.858596i \(0.328663\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −466.440 −0.0157553
\(958\) 0 0
\(959\) −5401.70 −0.181887
\(960\) 0 0
\(961\) 79870.0 2.68101
\(962\) 0 0
\(963\) 16763.6 0.560953
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −9034.04 −0.300429 −0.150215 0.988653i \(-0.547996\pi\)
−0.150215 + 0.988653i \(0.547996\pi\)
\(968\) 0 0
\(969\) 5213.71 0.172847
\(970\) 0 0
\(971\) −36159.0 −1.19506 −0.597528 0.801848i \(-0.703850\pi\)
−0.597528 + 0.801848i \(0.703850\pi\)
\(972\) 0 0
\(973\) 6620.03 0.218118
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −38584.0 −1.26347 −0.631736 0.775184i \(-0.717657\pi\)
−0.631736 + 0.775184i \(0.717657\pi\)
\(978\) 0 0
\(979\) −1820.33 −0.0594258
\(980\) 0 0
\(981\) −26326.1 −0.856808
\(982\) 0 0
\(983\) −53046.3 −1.72117 −0.860587 0.509303i \(-0.829903\pi\)
−0.860587 + 0.509303i \(0.829903\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 13380.5 0.431516
\(988\) 0 0
\(989\) 17097.0 0.549699
\(990\) 0 0
\(991\) 16425.7 0.526517 0.263259 0.964725i \(-0.415203\pi\)
0.263259 + 0.964725i \(0.415203\pi\)
\(992\) 0 0
\(993\) 9346.19 0.298683
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 20024.6 0.636092 0.318046 0.948075i \(-0.396973\pi\)
0.318046 + 0.948075i \(0.396973\pi\)
\(998\) 0 0
\(999\) −35235.0 −1.11590
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 400.4.a.x.1.1 2
4.3 odd 2 200.4.a.k.1.2 2
5.2 odd 4 80.4.c.c.49.3 4
5.3 odd 4 80.4.c.c.49.2 4
5.4 even 2 400.4.a.v.1.2 2
8.3 odd 2 1600.4.a.cl.1.1 2
8.5 even 2 1600.4.a.cf.1.2 2
12.11 even 2 1800.4.a.bk.1.1 2
15.2 even 4 720.4.f.m.289.2 4
15.8 even 4 720.4.f.m.289.1 4
20.3 even 4 40.4.c.a.9.3 yes 4
20.7 even 4 40.4.c.a.9.2 4
20.19 odd 2 200.4.a.l.1.1 2
40.3 even 4 320.4.c.g.129.2 4
40.13 odd 4 320.4.c.h.129.3 4
40.19 odd 2 1600.4.a.ce.1.2 2
40.27 even 4 320.4.c.g.129.3 4
40.29 even 2 1600.4.a.cm.1.1 2
40.37 odd 4 320.4.c.h.129.2 4
60.23 odd 4 360.4.f.e.289.1 4
60.47 odd 4 360.4.f.e.289.2 4
60.59 even 2 1800.4.a.bp.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.c.a.9.2 4 20.7 even 4
40.4.c.a.9.3 yes 4 20.3 even 4
80.4.c.c.49.2 4 5.3 odd 4
80.4.c.c.49.3 4 5.2 odd 4
200.4.a.k.1.2 2 4.3 odd 2
200.4.a.l.1.1 2 20.19 odd 2
320.4.c.g.129.2 4 40.3 even 4
320.4.c.g.129.3 4 40.27 even 4
320.4.c.h.129.2 4 40.37 odd 4
320.4.c.h.129.3 4 40.13 odd 4
360.4.f.e.289.1 4 60.23 odd 4
360.4.f.e.289.2 4 60.47 odd 4
400.4.a.v.1.2 2 5.4 even 2
400.4.a.x.1.1 2 1.1 even 1 trivial
720.4.f.m.289.1 4 15.8 even 4
720.4.f.m.289.2 4 15.2 even 4
1600.4.a.ce.1.2 2 40.19 odd 2
1600.4.a.cf.1.2 2 8.5 even 2
1600.4.a.cl.1.1 2 8.3 odd 2
1600.4.a.cm.1.1 2 40.29 even 2
1800.4.a.bk.1.1 2 12.11 even 2
1800.4.a.bp.1.2 2 60.59 even 2