Properties

Label 1232.4.a.w.1.3
Level $1232$
Weight $4$
Character 1232.1
Self dual yes
Analytic conductor $72.690$
Analytic rank $0$
Dimension $4$
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] = [1232,4,Mod(1,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, 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("1232.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1232.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: \(72.6903531271\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.509800.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 5x + 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.66444\) of defining polynomial
Character \(\chi\) \(=\) 1232.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.36360 q^{3} -15.4926 q^{5} +7.00000 q^{7} +27.2227 q^{9} +O(q^{10})\) \(q+7.36360 q^{3} -15.4926 q^{5} +7.00000 q^{7} +27.2227 q^{9} -11.0000 q^{11} +49.0777 q^{13} -114.081 q^{15} -34.1261 q^{17} +144.114 q^{19} +51.5452 q^{21} -118.906 q^{23} +115.020 q^{25} +1.63966 q^{27} -63.6830 q^{29} +212.912 q^{31} -80.9996 q^{33} -108.448 q^{35} -200.224 q^{37} +361.389 q^{39} +451.267 q^{41} +130.664 q^{43} -421.750 q^{45} +176.271 q^{47} +49.0000 q^{49} -251.291 q^{51} -629.988 q^{53} +170.419 q^{55} +1061.20 q^{57} +86.9318 q^{59} +644.248 q^{61} +190.559 q^{63} -760.340 q^{65} +400.974 q^{67} -875.578 q^{69} -507.611 q^{71} +176.392 q^{73} +846.965 q^{75} -77.0000 q^{77} +701.122 q^{79} -722.938 q^{81} +1259.27 q^{83} +528.702 q^{85} -468.936 q^{87} +788.394 q^{89} +343.544 q^{91} +1567.80 q^{93} -2232.70 q^{95} +185.039 q^{97} -299.449 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} - 18 q^{5} + 28 q^{7} + 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} - 18 q^{5} + 28 q^{7} + 66 q^{9} - 44 q^{11} - 134 q^{13} + 62 q^{15} - 74 q^{17} + 164 q^{19} + 84 q^{21} - 194 q^{23} + 38 q^{25} + 510 q^{27} - 108 q^{29} + 412 q^{31} - 132 q^{33} - 126 q^{35} + 286 q^{37} + 256 q^{39} - 18 q^{41} + 496 q^{43} + 580 q^{45} - 62 q^{47} + 196 q^{49} + 508 q^{51} - 828 q^{53} + 198 q^{55} + 700 q^{57} + 1224 q^{59} - 350 q^{61} + 462 q^{63} - 396 q^{65} + 1498 q^{67} - 386 q^{69} - 2326 q^{71} - 1630 q^{73} + 1362 q^{75} - 308 q^{77} + 1020 q^{79} + 1128 q^{81} + 1920 q^{83} + 2008 q^{85} - 1640 q^{87} + 1550 q^{89} - 938 q^{91} + 6046 q^{93} - 2332 q^{95} - 2202 q^{97} - 726 q^{99}+O(q^{100}) \) 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
\(3\) 7.36360 1.41713 0.708563 0.705647i \(-0.249344\pi\)
0.708563 + 0.705647i \(0.249344\pi\)
\(4\) 0 0
\(5\) −15.4926 −1.38570 −0.692850 0.721082i \(-0.743645\pi\)
−0.692850 + 0.721082i \(0.743645\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 27.2227 1.00825
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) 49.0777 1.04705 0.523527 0.852009i \(-0.324616\pi\)
0.523527 + 0.852009i \(0.324616\pi\)
\(14\) 0 0
\(15\) −114.081 −1.96371
\(16\) 0 0
\(17\) −34.1261 −0.486871 −0.243435 0.969917i \(-0.578274\pi\)
−0.243435 + 0.969917i \(0.578274\pi\)
\(18\) 0 0
\(19\) 144.114 1.74011 0.870053 0.492958i \(-0.164084\pi\)
0.870053 + 0.492958i \(0.164084\pi\)
\(20\) 0 0
\(21\) 51.5452 0.535623
\(22\) 0 0
\(23\) −118.906 −1.07798 −0.538992 0.842311i \(-0.681195\pi\)
−0.538992 + 0.842311i \(0.681195\pi\)
\(24\) 0 0
\(25\) 115.020 0.920164
\(26\) 0 0
\(27\) 1.63966 0.0116872
\(28\) 0 0
\(29\) −63.6830 −0.407780 −0.203890 0.978994i \(-0.565359\pi\)
−0.203890 + 0.978994i \(0.565359\pi\)
\(30\) 0 0
\(31\) 212.912 1.23355 0.616777 0.787138i \(-0.288438\pi\)
0.616777 + 0.787138i \(0.288438\pi\)
\(32\) 0 0
\(33\) −80.9996 −0.427280
\(34\) 0 0
\(35\) −108.448 −0.523745
\(36\) 0 0
\(37\) −200.224 −0.889638 −0.444819 0.895620i \(-0.646732\pi\)
−0.444819 + 0.895620i \(0.646732\pi\)
\(38\) 0 0
\(39\) 361.389 1.48381
\(40\) 0 0
\(41\) 451.267 1.71893 0.859464 0.511197i \(-0.170798\pi\)
0.859464 + 0.511197i \(0.170798\pi\)
\(42\) 0 0
\(43\) 130.664 0.463396 0.231698 0.972788i \(-0.425572\pi\)
0.231698 + 0.972788i \(0.425572\pi\)
\(44\) 0 0
\(45\) −421.750 −1.39713
\(46\) 0 0
\(47\) 176.271 0.547057 0.273529 0.961864i \(-0.411809\pi\)
0.273529 + 0.961864i \(0.411809\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −251.291 −0.689957
\(52\) 0 0
\(53\) −629.988 −1.63274 −0.816372 0.577526i \(-0.804018\pi\)
−0.816372 + 0.577526i \(0.804018\pi\)
\(54\) 0 0
\(55\) 170.419 0.417804
\(56\) 0 0
\(57\) 1061.20 2.46595
\(58\) 0 0
\(59\) 86.9318 0.191823 0.0959115 0.995390i \(-0.469423\pi\)
0.0959115 + 0.995390i \(0.469423\pi\)
\(60\) 0 0
\(61\) 644.248 1.35225 0.676127 0.736785i \(-0.263657\pi\)
0.676127 + 0.736785i \(0.263657\pi\)
\(62\) 0 0
\(63\) 190.559 0.381082
\(64\) 0 0
\(65\) −760.340 −1.45090
\(66\) 0 0
\(67\) 400.974 0.731147 0.365573 0.930783i \(-0.380873\pi\)
0.365573 + 0.930783i \(0.380873\pi\)
\(68\) 0 0
\(69\) −875.578 −1.52764
\(70\) 0 0
\(71\) −507.611 −0.848484 −0.424242 0.905549i \(-0.639459\pi\)
−0.424242 + 0.905549i \(0.639459\pi\)
\(72\) 0 0
\(73\) 176.392 0.282811 0.141405 0.989952i \(-0.454838\pi\)
0.141405 + 0.989952i \(0.454838\pi\)
\(74\) 0 0
\(75\) 846.965 1.30399
\(76\) 0 0
\(77\) −77.0000 −0.113961
\(78\) 0 0
\(79\) 701.122 0.998512 0.499256 0.866455i \(-0.333607\pi\)
0.499256 + 0.866455i \(0.333607\pi\)
\(80\) 0 0
\(81\) −722.938 −0.991685
\(82\) 0 0
\(83\) 1259.27 1.66534 0.832670 0.553770i \(-0.186811\pi\)
0.832670 + 0.553770i \(0.186811\pi\)
\(84\) 0 0
\(85\) 528.702 0.674656
\(86\) 0 0
\(87\) −468.936 −0.577876
\(88\) 0 0
\(89\) 788.394 0.938984 0.469492 0.882937i \(-0.344437\pi\)
0.469492 + 0.882937i \(0.344437\pi\)
\(90\) 0 0
\(91\) 343.544 0.395749
\(92\) 0 0
\(93\) 1567.80 1.74810
\(94\) 0 0
\(95\) −2232.70 −2.41126
\(96\) 0 0
\(97\) 185.039 0.193689 0.0968446 0.995300i \(-0.469125\pi\)
0.0968446 + 0.995300i \(0.469125\pi\)
\(98\) 0 0
\(99\) −299.449 −0.303998
\(100\) 0 0
\(101\) 1243.55 1.22513 0.612565 0.790420i \(-0.290138\pi\)
0.612565 + 0.790420i \(0.290138\pi\)
\(102\) 0 0
\(103\) 1555.21 1.48776 0.743879 0.668315i \(-0.232984\pi\)
0.743879 + 0.668315i \(0.232984\pi\)
\(104\) 0 0
\(105\) −798.569 −0.742213
\(106\) 0 0
\(107\) 247.062 0.223218 0.111609 0.993752i \(-0.464400\pi\)
0.111609 + 0.993752i \(0.464400\pi\)
\(108\) 0 0
\(109\) 1160.58 1.01985 0.509923 0.860220i \(-0.329674\pi\)
0.509923 + 0.860220i \(0.329674\pi\)
\(110\) 0 0
\(111\) −1474.37 −1.26073
\(112\) 0 0
\(113\) 1738.50 1.44729 0.723646 0.690171i \(-0.242465\pi\)
0.723646 + 0.690171i \(0.242465\pi\)
\(114\) 0 0
\(115\) 1842.16 1.49376
\(116\) 0 0
\(117\) 1336.03 1.05569
\(118\) 0 0
\(119\) −238.883 −0.184020
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) 3322.95 2.43594
\(124\) 0 0
\(125\) 154.609 0.110629
\(126\) 0 0
\(127\) 1507.65 1.05340 0.526701 0.850051i \(-0.323429\pi\)
0.526701 + 0.850051i \(0.323429\pi\)
\(128\) 0 0
\(129\) 962.157 0.656691
\(130\) 0 0
\(131\) −1327.55 −0.885408 −0.442704 0.896668i \(-0.645981\pi\)
−0.442704 + 0.896668i \(0.645981\pi\)
\(132\) 0 0
\(133\) 1008.80 0.657698
\(134\) 0 0
\(135\) −25.4026 −0.0161949
\(136\) 0 0
\(137\) −441.531 −0.275347 −0.137674 0.990478i \(-0.543962\pi\)
−0.137674 + 0.990478i \(0.543962\pi\)
\(138\) 0 0
\(139\) −179.162 −0.109326 −0.0546631 0.998505i \(-0.517408\pi\)
−0.0546631 + 0.998505i \(0.517408\pi\)
\(140\) 0 0
\(141\) 1297.99 0.775250
\(142\) 0 0
\(143\) −539.854 −0.315698
\(144\) 0 0
\(145\) 986.615 0.565061
\(146\) 0 0
\(147\) 360.817 0.202447
\(148\) 0 0
\(149\) −1147.07 −0.630680 −0.315340 0.948979i \(-0.602118\pi\)
−0.315340 + 0.948979i \(0.602118\pi\)
\(150\) 0 0
\(151\) −1873.03 −1.00944 −0.504719 0.863284i \(-0.668404\pi\)
−0.504719 + 0.863284i \(0.668404\pi\)
\(152\) 0 0
\(153\) −929.004 −0.490886
\(154\) 0 0
\(155\) −3298.56 −1.70933
\(156\) 0 0
\(157\) −643.927 −0.327331 −0.163665 0.986516i \(-0.552332\pi\)
−0.163665 + 0.986516i \(0.552332\pi\)
\(158\) 0 0
\(159\) −4638.98 −2.31381
\(160\) 0 0
\(161\) −832.343 −0.407440
\(162\) 0 0
\(163\) 3044.47 1.46295 0.731477 0.681866i \(-0.238831\pi\)
0.731477 + 0.681866i \(0.238831\pi\)
\(164\) 0 0
\(165\) 1254.89 0.592081
\(166\) 0 0
\(167\) 337.038 0.156172 0.0780861 0.996947i \(-0.475119\pi\)
0.0780861 + 0.996947i \(0.475119\pi\)
\(168\) 0 0
\(169\) 211.617 0.0963209
\(170\) 0 0
\(171\) 3923.17 1.75446
\(172\) 0 0
\(173\) −3504.80 −1.54026 −0.770131 0.637886i \(-0.779809\pi\)
−0.770131 + 0.637886i \(0.779809\pi\)
\(174\) 0 0
\(175\) 805.143 0.347789
\(176\) 0 0
\(177\) 640.132 0.271837
\(178\) 0 0
\(179\) −221.078 −0.0923135 −0.0461568 0.998934i \(-0.514697\pi\)
−0.0461568 + 0.998934i \(0.514697\pi\)
\(180\) 0 0
\(181\) −4561.63 −1.87328 −0.936640 0.350295i \(-0.886081\pi\)
−0.936640 + 0.350295i \(0.886081\pi\)
\(182\) 0 0
\(183\) 4743.99 1.91631
\(184\) 0 0
\(185\) 3101.99 1.23277
\(186\) 0 0
\(187\) 375.387 0.146797
\(188\) 0 0
\(189\) 11.4776 0.00441733
\(190\) 0 0
\(191\) −4180.40 −1.58368 −0.791841 0.610727i \(-0.790877\pi\)
−0.791841 + 0.610727i \(0.790877\pi\)
\(192\) 0 0
\(193\) −1942.59 −0.724510 −0.362255 0.932079i \(-0.617993\pi\)
−0.362255 + 0.932079i \(0.617993\pi\)
\(194\) 0 0
\(195\) −5598.85 −2.05611
\(196\) 0 0
\(197\) 3775.15 1.36532 0.682660 0.730736i \(-0.260823\pi\)
0.682660 + 0.730736i \(0.260823\pi\)
\(198\) 0 0
\(199\) 993.760 0.353999 0.176999 0.984211i \(-0.443361\pi\)
0.176999 + 0.984211i \(0.443361\pi\)
\(200\) 0 0
\(201\) 2952.62 1.03613
\(202\) 0 0
\(203\) −445.781 −0.154127
\(204\) 0 0
\(205\) −6991.29 −2.38192
\(206\) 0 0
\(207\) −3236.94 −1.08687
\(208\) 0 0
\(209\) −1585.25 −0.524662
\(210\) 0 0
\(211\) 2912.24 0.950176 0.475088 0.879938i \(-0.342416\pi\)
0.475088 + 0.879938i \(0.342416\pi\)
\(212\) 0 0
\(213\) −3737.85 −1.20241
\(214\) 0 0
\(215\) −2024.32 −0.642128
\(216\) 0 0
\(217\) 1490.39 0.466239
\(218\) 0 0
\(219\) 1298.88 0.400778
\(220\) 0 0
\(221\) −1674.83 −0.509780
\(222\) 0 0
\(223\) −5069.26 −1.52225 −0.761127 0.648603i \(-0.775353\pi\)
−0.761127 + 0.648603i \(0.775353\pi\)
\(224\) 0 0
\(225\) 3131.16 0.927753
\(226\) 0 0
\(227\) 5653.25 1.65295 0.826475 0.562974i \(-0.190343\pi\)
0.826475 + 0.562974i \(0.190343\pi\)
\(228\) 0 0
\(229\) −5141.63 −1.48371 −0.741853 0.670563i \(-0.766053\pi\)
−0.741853 + 0.670563i \(0.766053\pi\)
\(230\) 0 0
\(231\) −566.998 −0.161497
\(232\) 0 0
\(233\) 312.296 0.0878077 0.0439039 0.999036i \(-0.486020\pi\)
0.0439039 + 0.999036i \(0.486020\pi\)
\(234\) 0 0
\(235\) −2730.89 −0.758057
\(236\) 0 0
\(237\) 5162.79 1.41502
\(238\) 0 0
\(239\) 4012.12 1.08587 0.542935 0.839775i \(-0.317313\pi\)
0.542935 + 0.839775i \(0.317313\pi\)
\(240\) 0 0
\(241\) 499.662 0.133552 0.0667761 0.997768i \(-0.478729\pi\)
0.0667761 + 0.997768i \(0.478729\pi\)
\(242\) 0 0
\(243\) −5367.70 −1.41703
\(244\) 0 0
\(245\) −759.137 −0.197957
\(246\) 0 0
\(247\) 7072.78 1.82198
\(248\) 0 0
\(249\) 9272.79 2.36000
\(250\) 0 0
\(251\) 6112.34 1.53708 0.768541 0.639801i \(-0.220983\pi\)
0.768541 + 0.639801i \(0.220983\pi\)
\(252\) 0 0
\(253\) 1307.97 0.325025
\(254\) 0 0
\(255\) 3893.15 0.956073
\(256\) 0 0
\(257\) −1910.22 −0.463642 −0.231821 0.972758i \(-0.574468\pi\)
−0.231821 + 0.972758i \(0.574468\pi\)
\(258\) 0 0
\(259\) −1401.57 −0.336252
\(260\) 0 0
\(261\) −1733.62 −0.411143
\(262\) 0 0
\(263\) −1749.49 −0.410184 −0.205092 0.978743i \(-0.565749\pi\)
−0.205092 + 0.978743i \(0.565749\pi\)
\(264\) 0 0
\(265\) 9760.14 2.26249
\(266\) 0 0
\(267\) 5805.42 1.33066
\(268\) 0 0
\(269\) −2884.89 −0.653883 −0.326942 0.945045i \(-0.606018\pi\)
−0.326942 + 0.945045i \(0.606018\pi\)
\(270\) 0 0
\(271\) −6065.04 −1.35950 −0.679751 0.733443i \(-0.737912\pi\)
−0.679751 + 0.733443i \(0.737912\pi\)
\(272\) 0 0
\(273\) 2529.72 0.560826
\(274\) 0 0
\(275\) −1265.23 −0.277440
\(276\) 0 0
\(277\) −3331.31 −0.722595 −0.361297 0.932451i \(-0.617666\pi\)
−0.361297 + 0.932451i \(0.617666\pi\)
\(278\) 0 0
\(279\) 5796.04 1.24373
\(280\) 0 0
\(281\) −384.608 −0.0816505 −0.0408252 0.999166i \(-0.512999\pi\)
−0.0408252 + 0.999166i \(0.512999\pi\)
\(282\) 0 0
\(283\) −1768.02 −0.371371 −0.185685 0.982609i \(-0.559451\pi\)
−0.185685 + 0.982609i \(0.559451\pi\)
\(284\) 0 0
\(285\) −16440.7 −3.41707
\(286\) 0 0
\(287\) 3158.87 0.649693
\(288\) 0 0
\(289\) −3748.41 −0.762957
\(290\) 0 0
\(291\) 1362.55 0.274482
\(292\) 0 0
\(293\) 3801.87 0.758047 0.379023 0.925387i \(-0.376260\pi\)
0.379023 + 0.925387i \(0.376260\pi\)
\(294\) 0 0
\(295\) −1346.80 −0.265809
\(296\) 0 0
\(297\) −18.0363 −0.00352381
\(298\) 0 0
\(299\) −5835.64 −1.12871
\(300\) 0 0
\(301\) 914.647 0.175147
\(302\) 0 0
\(303\) 9157.03 1.73616
\(304\) 0 0
\(305\) −9981.07 −1.87382
\(306\) 0 0
\(307\) 687.619 0.127832 0.0639161 0.997955i \(-0.479641\pi\)
0.0639161 + 0.997955i \(0.479641\pi\)
\(308\) 0 0
\(309\) 11451.9 2.10834
\(310\) 0 0
\(311\) 2517.73 0.459059 0.229530 0.973302i \(-0.426281\pi\)
0.229530 + 0.973302i \(0.426281\pi\)
\(312\) 0 0
\(313\) −4725.03 −0.853274 −0.426637 0.904423i \(-0.640302\pi\)
−0.426637 + 0.904423i \(0.640302\pi\)
\(314\) 0 0
\(315\) −2952.25 −0.528065
\(316\) 0 0
\(317\) 2870.09 0.508519 0.254259 0.967136i \(-0.418168\pi\)
0.254259 + 0.967136i \(0.418168\pi\)
\(318\) 0 0
\(319\) 700.513 0.122950
\(320\) 0 0
\(321\) 1819.26 0.316328
\(322\) 0 0
\(323\) −4918.05 −0.847207
\(324\) 0 0
\(325\) 5644.94 0.963461
\(326\) 0 0
\(327\) 8546.03 1.44525
\(328\) 0 0
\(329\) 1233.89 0.206768
\(330\) 0 0
\(331\) −365.467 −0.0606885 −0.0303443 0.999540i \(-0.509660\pi\)
−0.0303443 + 0.999540i \(0.509660\pi\)
\(332\) 0 0
\(333\) −5450.63 −0.896975
\(334\) 0 0
\(335\) −6212.13 −1.01315
\(336\) 0 0
\(337\) −4292.33 −0.693822 −0.346911 0.937898i \(-0.612769\pi\)
−0.346911 + 0.937898i \(0.612769\pi\)
\(338\) 0 0
\(339\) 12801.6 2.05100
\(340\) 0 0
\(341\) −2342.03 −0.371930
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) 13565.0 2.11685
\(346\) 0 0
\(347\) 6090.40 0.942218 0.471109 0.882075i \(-0.343854\pi\)
0.471109 + 0.882075i \(0.343854\pi\)
\(348\) 0 0
\(349\) 6338.89 0.972244 0.486122 0.873891i \(-0.338411\pi\)
0.486122 + 0.873891i \(0.338411\pi\)
\(350\) 0 0
\(351\) 80.4709 0.0122371
\(352\) 0 0
\(353\) 5754.44 0.867643 0.433821 0.900999i \(-0.357165\pi\)
0.433821 + 0.900999i \(0.357165\pi\)
\(354\) 0 0
\(355\) 7864.21 1.17574
\(356\) 0 0
\(357\) −1759.04 −0.260779
\(358\) 0 0
\(359\) −7396.98 −1.08746 −0.543729 0.839261i \(-0.682988\pi\)
−0.543729 + 0.839261i \(0.682988\pi\)
\(360\) 0 0
\(361\) 13909.8 2.02797
\(362\) 0 0
\(363\) 890.996 0.128830
\(364\) 0 0
\(365\) −2732.78 −0.391891
\(366\) 0 0
\(367\) 4613.59 0.656205 0.328102 0.944642i \(-0.393591\pi\)
0.328102 + 0.944642i \(0.393591\pi\)
\(368\) 0 0
\(369\) 12284.7 1.73310
\(370\) 0 0
\(371\) −4409.91 −0.617119
\(372\) 0 0
\(373\) 4603.15 0.638986 0.319493 0.947589i \(-0.396487\pi\)
0.319493 + 0.947589i \(0.396487\pi\)
\(374\) 0 0
\(375\) 1138.48 0.156775
\(376\) 0 0
\(377\) −3125.41 −0.426968
\(378\) 0 0
\(379\) −6982.50 −0.946350 −0.473175 0.880968i \(-0.656892\pi\)
−0.473175 + 0.880968i \(0.656892\pi\)
\(380\) 0 0
\(381\) 11101.7 1.49280
\(382\) 0 0
\(383\) −12743.5 −1.70016 −0.850079 0.526655i \(-0.823446\pi\)
−0.850079 + 0.526655i \(0.823446\pi\)
\(384\) 0 0
\(385\) 1192.93 0.157915
\(386\) 0 0
\(387\) 3557.02 0.467218
\(388\) 0 0
\(389\) 1568.76 0.204471 0.102235 0.994760i \(-0.467400\pi\)
0.102235 + 0.994760i \(0.467400\pi\)
\(390\) 0 0
\(391\) 4057.81 0.524839
\(392\) 0 0
\(393\) −9775.54 −1.25474
\(394\) 0 0
\(395\) −10862.2 −1.38364
\(396\) 0 0
\(397\) 9841.87 1.24420 0.622102 0.782936i \(-0.286279\pi\)
0.622102 + 0.782936i \(0.286279\pi\)
\(398\) 0 0
\(399\) 7428.39 0.932042
\(400\) 0 0
\(401\) 5128.60 0.638679 0.319339 0.947640i \(-0.396539\pi\)
0.319339 + 0.947640i \(0.396539\pi\)
\(402\) 0 0
\(403\) 10449.2 1.29160
\(404\) 0 0
\(405\) 11200.2 1.37418
\(406\) 0 0
\(407\) 2202.46 0.268236
\(408\) 0 0
\(409\) −87.6006 −0.0105906 −0.00529532 0.999986i \(-0.501686\pi\)
−0.00529532 + 0.999986i \(0.501686\pi\)
\(410\) 0 0
\(411\) −3251.26 −0.390202
\(412\) 0 0
\(413\) 608.523 0.0725023
\(414\) 0 0
\(415\) −19509.4 −2.30766
\(416\) 0 0
\(417\) −1319.28 −0.154929
\(418\) 0 0
\(419\) 7130.86 0.831420 0.415710 0.909497i \(-0.363533\pi\)
0.415710 + 0.909497i \(0.363533\pi\)
\(420\) 0 0
\(421\) 6159.77 0.713085 0.356543 0.934279i \(-0.383955\pi\)
0.356543 + 0.934279i \(0.383955\pi\)
\(422\) 0 0
\(423\) 4798.55 0.551569
\(424\) 0 0
\(425\) −3925.20 −0.448001
\(426\) 0 0
\(427\) 4509.74 0.511104
\(428\) 0 0
\(429\) −3975.27 −0.447385
\(430\) 0 0
\(431\) −12044.1 −1.34604 −0.673019 0.739625i \(-0.735003\pi\)
−0.673019 + 0.739625i \(0.735003\pi\)
\(432\) 0 0
\(433\) −9609.67 −1.06654 −0.533270 0.845945i \(-0.679037\pi\)
−0.533270 + 0.845945i \(0.679037\pi\)
\(434\) 0 0
\(435\) 7265.04 0.800763
\(436\) 0 0
\(437\) −17136.0 −1.87581
\(438\) 0 0
\(439\) 3122.62 0.339486 0.169743 0.985488i \(-0.445706\pi\)
0.169743 + 0.985488i \(0.445706\pi\)
\(440\) 0 0
\(441\) 1333.91 0.144035
\(442\) 0 0
\(443\) −9381.76 −1.00619 −0.503093 0.864232i \(-0.667805\pi\)
−0.503093 + 0.864232i \(0.667805\pi\)
\(444\) 0 0
\(445\) −12214.3 −1.30115
\(446\) 0 0
\(447\) −8446.54 −0.893753
\(448\) 0 0
\(449\) 3407.36 0.358136 0.179068 0.983837i \(-0.442692\pi\)
0.179068 + 0.983837i \(0.442692\pi\)
\(450\) 0 0
\(451\) −4963.93 −0.518276
\(452\) 0 0
\(453\) −13792.3 −1.43050
\(454\) 0 0
\(455\) −5322.38 −0.548389
\(456\) 0 0
\(457\) −13714.0 −1.40375 −0.701876 0.712299i \(-0.747654\pi\)
−0.701876 + 0.712299i \(0.747654\pi\)
\(458\) 0 0
\(459\) −55.9554 −0.00569014
\(460\) 0 0
\(461\) −8864.54 −0.895581 −0.447790 0.894139i \(-0.647789\pi\)
−0.447790 + 0.894139i \(0.647789\pi\)
\(462\) 0 0
\(463\) 14753.3 1.48087 0.740437 0.672126i \(-0.234619\pi\)
0.740437 + 0.672126i \(0.234619\pi\)
\(464\) 0 0
\(465\) −24289.3 −2.42234
\(466\) 0 0
\(467\) −488.428 −0.0483977 −0.0241989 0.999707i \(-0.507703\pi\)
−0.0241989 + 0.999707i \(0.507703\pi\)
\(468\) 0 0
\(469\) 2806.82 0.276347
\(470\) 0 0
\(471\) −4741.62 −0.463869
\(472\) 0 0
\(473\) −1437.30 −0.139719
\(474\) 0 0
\(475\) 16576.1 1.60118
\(476\) 0 0
\(477\) −17149.9 −1.64621
\(478\) 0 0
\(479\) 1233.56 0.117668 0.0588338 0.998268i \(-0.481262\pi\)
0.0588338 + 0.998268i \(0.481262\pi\)
\(480\) 0 0
\(481\) −9826.52 −0.931499
\(482\) 0 0
\(483\) −6129.04 −0.577394
\(484\) 0 0
\(485\) −2866.73 −0.268395
\(486\) 0 0
\(487\) −16700.6 −1.55395 −0.776977 0.629529i \(-0.783248\pi\)
−0.776977 + 0.629529i \(0.783248\pi\)
\(488\) 0 0
\(489\) 22418.3 2.07319
\(490\) 0 0
\(491\) 2127.96 0.195587 0.0977937 0.995207i \(-0.468821\pi\)
0.0977937 + 0.995207i \(0.468821\pi\)
\(492\) 0 0
\(493\) 2173.25 0.198536
\(494\) 0 0
\(495\) 4639.25 0.421250
\(496\) 0 0
\(497\) −3553.28 −0.320697
\(498\) 0 0
\(499\) −12341.5 −1.10718 −0.553590 0.832789i \(-0.686742\pi\)
−0.553590 + 0.832789i \(0.686742\pi\)
\(500\) 0 0
\(501\) 2481.81 0.221316
\(502\) 0 0
\(503\) 3433.67 0.304374 0.152187 0.988352i \(-0.451368\pi\)
0.152187 + 0.988352i \(0.451368\pi\)
\(504\) 0 0
\(505\) −19265.9 −1.69766
\(506\) 0 0
\(507\) 1558.26 0.136499
\(508\) 0 0
\(509\) −1842.48 −0.160445 −0.0802223 0.996777i \(-0.525563\pi\)
−0.0802223 + 0.996777i \(0.525563\pi\)
\(510\) 0 0
\(511\) 1234.75 0.106892
\(512\) 0 0
\(513\) 236.298 0.0203369
\(514\) 0 0
\(515\) −24094.2 −2.06158
\(516\) 0 0
\(517\) −1938.98 −0.164944
\(518\) 0 0
\(519\) −25808.0 −2.18275
\(520\) 0 0
\(521\) 3532.65 0.297060 0.148530 0.988908i \(-0.452546\pi\)
0.148530 + 0.988908i \(0.452546\pi\)
\(522\) 0 0
\(523\) 4497.20 0.376001 0.188001 0.982169i \(-0.439799\pi\)
0.188001 + 0.982169i \(0.439799\pi\)
\(524\) 0 0
\(525\) 5928.76 0.492861
\(526\) 0 0
\(527\) −7265.87 −0.600581
\(528\) 0 0
\(529\) 1971.67 0.162050
\(530\) 0 0
\(531\) 2366.52 0.193405
\(532\) 0 0
\(533\) 22147.1 1.79981
\(534\) 0 0
\(535\) −3827.62 −0.309313
\(536\) 0 0
\(537\) −1627.93 −0.130820
\(538\) 0 0
\(539\) −539.000 −0.0430730
\(540\) 0 0
\(541\) −11426.0 −0.908023 −0.454011 0.890996i \(-0.650007\pi\)
−0.454011 + 0.890996i \(0.650007\pi\)
\(542\) 0 0
\(543\) −33590.1 −2.65467
\(544\) 0 0
\(545\) −17980.4 −1.41320
\(546\) 0 0
\(547\) −17875.3 −1.39725 −0.698623 0.715490i \(-0.746204\pi\)
−0.698623 + 0.715490i \(0.746204\pi\)
\(548\) 0 0
\(549\) 17538.2 1.36341
\(550\) 0 0
\(551\) −9177.61 −0.709581
\(552\) 0 0
\(553\) 4907.86 0.377402
\(554\) 0 0
\(555\) 22841.8 1.74699
\(556\) 0 0
\(557\) 25343.4 1.92789 0.963946 0.266097i \(-0.0857341\pi\)
0.963946 + 0.266097i \(0.0857341\pi\)
\(558\) 0 0
\(559\) 6412.68 0.485201
\(560\) 0 0
\(561\) 2764.20 0.208030
\(562\) 0 0
\(563\) 1597.92 0.119617 0.0598084 0.998210i \(-0.480951\pi\)
0.0598084 + 0.998210i \(0.480951\pi\)
\(564\) 0 0
\(565\) −26933.8 −2.00551
\(566\) 0 0
\(567\) −5060.57 −0.374822
\(568\) 0 0
\(569\) −11487.7 −0.846377 −0.423188 0.906042i \(-0.639089\pi\)
−0.423188 + 0.906042i \(0.639089\pi\)
\(570\) 0 0
\(571\) −17027.0 −1.24791 −0.623957 0.781459i \(-0.714476\pi\)
−0.623957 + 0.781459i \(0.714476\pi\)
\(572\) 0 0
\(573\) −30782.8 −2.24428
\(574\) 0 0
\(575\) −13676.6 −0.991922
\(576\) 0 0
\(577\) 19368.0 1.39740 0.698701 0.715414i \(-0.253762\pi\)
0.698701 + 0.715414i \(0.253762\pi\)
\(578\) 0 0
\(579\) −14304.4 −1.02672
\(580\) 0 0
\(581\) 8814.91 0.629439
\(582\) 0 0
\(583\) 6929.86 0.492291
\(584\) 0 0
\(585\) −20698.5 −1.46287
\(586\) 0 0
\(587\) 27035.0 1.90094 0.950472 0.310809i \(-0.100600\pi\)
0.950472 + 0.310809i \(0.100600\pi\)
\(588\) 0 0
\(589\) 30683.6 2.14651
\(590\) 0 0
\(591\) 27798.7 1.93483
\(592\) 0 0
\(593\) −9717.62 −0.672943 −0.336471 0.941694i \(-0.609234\pi\)
−0.336471 + 0.941694i \(0.609234\pi\)
\(594\) 0 0
\(595\) 3700.92 0.254996
\(596\) 0 0
\(597\) 7317.65 0.501661
\(598\) 0 0
\(599\) 18386.6 1.25418 0.627092 0.778945i \(-0.284245\pi\)
0.627092 + 0.778945i \(0.284245\pi\)
\(600\) 0 0
\(601\) −3288.31 −0.223183 −0.111591 0.993754i \(-0.535595\pi\)
−0.111591 + 0.993754i \(0.535595\pi\)
\(602\) 0 0
\(603\) 10915.6 0.737176
\(604\) 0 0
\(605\) −1874.60 −0.125973
\(606\) 0 0
\(607\) 27356.2 1.82925 0.914623 0.404307i \(-0.132487\pi\)
0.914623 + 0.404307i \(0.132487\pi\)
\(608\) 0 0
\(609\) −3282.55 −0.218417
\(610\) 0 0
\(611\) 8650.95 0.572798
\(612\) 0 0
\(613\) −9573.59 −0.630789 −0.315395 0.948961i \(-0.602137\pi\)
−0.315395 + 0.948961i \(0.602137\pi\)
\(614\) 0 0
\(615\) −51481.1 −3.37548
\(616\) 0 0
\(617\) −7506.76 −0.489807 −0.244903 0.969547i \(-0.578756\pi\)
−0.244903 + 0.969547i \(0.578756\pi\)
\(618\) 0 0
\(619\) 9294.64 0.603527 0.301763 0.953383i \(-0.402425\pi\)
0.301763 + 0.953383i \(0.402425\pi\)
\(620\) 0 0
\(621\) −194.966 −0.0125986
\(622\) 0 0
\(623\) 5518.75 0.354902
\(624\) 0 0
\(625\) −16772.8 −1.07346
\(626\) 0 0
\(627\) −11673.2 −0.743512
\(628\) 0 0
\(629\) 6832.87 0.433139
\(630\) 0 0
\(631\) −4254.20 −0.268395 −0.134197 0.990955i \(-0.542846\pi\)
−0.134197 + 0.990955i \(0.542846\pi\)
\(632\) 0 0
\(633\) 21444.6 1.34652
\(634\) 0 0
\(635\) −23357.4 −1.45970
\(636\) 0 0
\(637\) 2404.81 0.149579
\(638\) 0 0
\(639\) −13818.5 −0.855482
\(640\) 0 0
\(641\) 5112.91 0.315051 0.157526 0.987515i \(-0.449648\pi\)
0.157526 + 0.987515i \(0.449648\pi\)
\(642\) 0 0
\(643\) −7296.68 −0.447516 −0.223758 0.974645i \(-0.571833\pi\)
−0.223758 + 0.974645i \(0.571833\pi\)
\(644\) 0 0
\(645\) −14906.3 −0.909977
\(646\) 0 0
\(647\) −15612.1 −0.948649 −0.474324 0.880350i \(-0.657308\pi\)
−0.474324 + 0.880350i \(0.657308\pi\)
\(648\) 0 0
\(649\) −956.250 −0.0578368
\(650\) 0 0
\(651\) 10974.6 0.660720
\(652\) 0 0
\(653\) −23160.6 −1.38797 −0.693986 0.719988i \(-0.744147\pi\)
−0.693986 + 0.719988i \(0.744147\pi\)
\(654\) 0 0
\(655\) 20567.2 1.22691
\(656\) 0 0
\(657\) 4801.87 0.285143
\(658\) 0 0
\(659\) 5707.65 0.337388 0.168694 0.985668i \(-0.446045\pi\)
0.168694 + 0.985668i \(0.446045\pi\)
\(660\) 0 0
\(661\) −16202.5 −0.953411 −0.476706 0.879063i \(-0.658169\pi\)
−0.476706 + 0.879063i \(0.658169\pi\)
\(662\) 0 0
\(663\) −12332.8 −0.722422
\(664\) 0 0
\(665\) −15628.9 −0.911372
\(666\) 0 0
\(667\) 7572.30 0.439581
\(668\) 0 0
\(669\) −37328.0 −2.15723
\(670\) 0 0
\(671\) −7086.73 −0.407720
\(672\) 0 0
\(673\) −16956.2 −0.971192 −0.485596 0.874183i \(-0.661397\pi\)
−0.485596 + 0.874183i \(0.661397\pi\)
\(674\) 0 0
\(675\) 188.595 0.0107541
\(676\) 0 0
\(677\) 21259.0 1.20687 0.603433 0.797414i \(-0.293799\pi\)
0.603433 + 0.797414i \(0.293799\pi\)
\(678\) 0 0
\(679\) 1295.27 0.0732076
\(680\) 0 0
\(681\) 41628.3 2.34244
\(682\) 0 0
\(683\) −5867.98 −0.328744 −0.164372 0.986398i \(-0.552560\pi\)
−0.164372 + 0.986398i \(0.552560\pi\)
\(684\) 0 0
\(685\) 6840.46 0.381548
\(686\) 0 0
\(687\) −37860.9 −2.10260
\(688\) 0 0
\(689\) −30918.3 −1.70957
\(690\) 0 0
\(691\) −15583.0 −0.857892 −0.428946 0.903330i \(-0.641115\pi\)
−0.428946 + 0.903330i \(0.641115\pi\)
\(692\) 0 0
\(693\) −2096.15 −0.114900
\(694\) 0 0
\(695\) 2775.69 0.151493
\(696\) 0 0
\(697\) −15400.0 −0.836895
\(698\) 0 0
\(699\) 2299.63 0.124435
\(700\) 0 0
\(701\) −19210.5 −1.03505 −0.517525 0.855668i \(-0.673147\pi\)
−0.517525 + 0.855668i \(0.673147\pi\)
\(702\) 0 0
\(703\) −28855.1 −1.54806
\(704\) 0 0
\(705\) −20109.2 −1.07426
\(706\) 0 0
\(707\) 8704.87 0.463055
\(708\) 0 0
\(709\) −1760.77 −0.0932683 −0.0466342 0.998912i \(-0.514850\pi\)
−0.0466342 + 0.998912i \(0.514850\pi\)
\(710\) 0 0
\(711\) 19086.4 1.00675
\(712\) 0 0
\(713\) −25316.6 −1.32975
\(714\) 0 0
\(715\) 8363.74 0.437463
\(716\) 0 0
\(717\) 29543.7 1.53881
\(718\) 0 0
\(719\) −10517.7 −0.545541 −0.272770 0.962079i \(-0.587940\pi\)
−0.272770 + 0.962079i \(0.587940\pi\)
\(720\) 0 0
\(721\) 10886.4 0.562319
\(722\) 0 0
\(723\) 3679.31 0.189260
\(724\) 0 0
\(725\) −7324.85 −0.375225
\(726\) 0 0
\(727\) 10483.4 0.534814 0.267407 0.963584i \(-0.413833\pi\)
0.267407 + 0.963584i \(0.413833\pi\)
\(728\) 0 0
\(729\) −20006.3 −1.01643
\(730\) 0 0
\(731\) −4459.05 −0.225614
\(732\) 0 0
\(733\) −8568.83 −0.431783 −0.215892 0.976417i \(-0.569266\pi\)
−0.215892 + 0.976417i \(0.569266\pi\)
\(734\) 0 0
\(735\) −5589.99 −0.280530
\(736\) 0 0
\(737\) −4410.72 −0.220449
\(738\) 0 0
\(739\) 3089.07 0.153766 0.0768832 0.997040i \(-0.475503\pi\)
0.0768832 + 0.997040i \(0.475503\pi\)
\(740\) 0 0
\(741\) 52081.1 2.58198
\(742\) 0 0
\(743\) 28558.8 1.41012 0.705060 0.709148i \(-0.250920\pi\)
0.705060 + 0.709148i \(0.250920\pi\)
\(744\) 0 0
\(745\) 17771.0 0.873933
\(746\) 0 0
\(747\) 34280.8 1.67907
\(748\) 0 0
\(749\) 1729.43 0.0843685
\(750\) 0 0
\(751\) 2062.51 0.100216 0.0501078 0.998744i \(-0.484044\pi\)
0.0501078 + 0.998744i \(0.484044\pi\)
\(752\) 0 0
\(753\) 45008.9 2.17824
\(754\) 0 0
\(755\) 29018.1 1.39878
\(756\) 0 0
\(757\) 39089.1 1.87678 0.938388 0.345584i \(-0.112319\pi\)
0.938388 + 0.345584i \(0.112319\pi\)
\(758\) 0 0
\(759\) 9631.35 0.460601
\(760\) 0 0
\(761\) −13820.5 −0.658337 −0.329169 0.944271i \(-0.606768\pi\)
−0.329169 + 0.944271i \(0.606768\pi\)
\(762\) 0 0
\(763\) 8124.04 0.385465
\(764\) 0 0
\(765\) 14392.7 0.680220
\(766\) 0 0
\(767\) 4266.41 0.200849
\(768\) 0 0
\(769\) −28495.1 −1.33623 −0.668115 0.744058i \(-0.732899\pi\)
−0.668115 + 0.744058i \(0.732899\pi\)
\(770\) 0 0
\(771\) −14066.1 −0.657040
\(772\) 0 0
\(773\) 1874.14 0.0872032 0.0436016 0.999049i \(-0.486117\pi\)
0.0436016 + 0.999049i \(0.486117\pi\)
\(774\) 0 0
\(775\) 24489.3 1.13507
\(776\) 0 0
\(777\) −10320.6 −0.476511
\(778\) 0 0
\(779\) 65033.8 2.99112
\(780\) 0 0
\(781\) 5583.72 0.255828
\(782\) 0 0
\(783\) −104.419 −0.00476580
\(784\) 0 0
\(785\) 9976.10 0.453582
\(786\) 0 0
\(787\) −16614.9 −0.752551 −0.376275 0.926508i \(-0.622795\pi\)
−0.376275 + 0.926508i \(0.622795\pi\)
\(788\) 0 0
\(789\) −12882.6 −0.581282
\(790\) 0 0
\(791\) 12169.5 0.547025
\(792\) 0 0
\(793\) 31618.2 1.41588
\(794\) 0 0
\(795\) 71869.8 3.20624
\(796\) 0 0
\(797\) 19323.7 0.858823 0.429411 0.903109i \(-0.358721\pi\)
0.429411 + 0.903109i \(0.358721\pi\)
\(798\) 0 0
\(799\) −6015.43 −0.266346
\(800\) 0 0
\(801\) 21462.2 0.946728
\(802\) 0 0
\(803\) −1940.32 −0.0852706
\(804\) 0 0
\(805\) 12895.2 0.564589
\(806\) 0 0
\(807\) −21243.2 −0.926635
\(808\) 0 0
\(809\) −15363.0 −0.667657 −0.333829 0.942634i \(-0.608341\pi\)
−0.333829 + 0.942634i \(0.608341\pi\)
\(810\) 0 0
\(811\) −11404.0 −0.493771 −0.246886 0.969045i \(-0.579407\pi\)
−0.246886 + 0.969045i \(0.579407\pi\)
\(812\) 0 0
\(813\) −44660.6 −1.92659
\(814\) 0 0
\(815\) −47166.8 −2.02722
\(816\) 0 0
\(817\) 18830.5 0.806359
\(818\) 0 0
\(819\) 9352.18 0.399013
\(820\) 0 0
\(821\) 5529.00 0.235035 0.117517 0.993071i \(-0.462506\pi\)
0.117517 + 0.993071i \(0.462506\pi\)
\(822\) 0 0
\(823\) 12216.1 0.517407 0.258704 0.965957i \(-0.416705\pi\)
0.258704 + 0.965957i \(0.416705\pi\)
\(824\) 0 0
\(825\) −9316.62 −0.393167
\(826\) 0 0
\(827\) 29123.0 1.22455 0.612277 0.790643i \(-0.290254\pi\)
0.612277 + 0.790643i \(0.290254\pi\)
\(828\) 0 0
\(829\) 25679.6 1.07586 0.537930 0.842990i \(-0.319207\pi\)
0.537930 + 0.842990i \(0.319207\pi\)
\(830\) 0 0
\(831\) −24530.4 −1.02401
\(832\) 0 0
\(833\) −1672.18 −0.0695529
\(834\) 0 0
\(835\) −5221.59 −0.216408
\(836\) 0 0
\(837\) 349.104 0.0144167
\(838\) 0 0
\(839\) 7009.97 0.288452 0.144226 0.989545i \(-0.453931\pi\)
0.144226 + 0.989545i \(0.453931\pi\)
\(840\) 0 0
\(841\) −20333.5 −0.833715
\(842\) 0 0
\(843\) −2832.10 −0.115709
\(844\) 0 0
\(845\) −3278.50 −0.133472
\(846\) 0 0
\(847\) 847.000 0.0343604
\(848\) 0 0
\(849\) −13019.0 −0.526279
\(850\) 0 0
\(851\) 23807.9 0.959016
\(852\) 0 0
\(853\) 7430.54 0.298261 0.149131 0.988818i \(-0.452353\pi\)
0.149131 + 0.988818i \(0.452353\pi\)
\(854\) 0 0
\(855\) −60780.0 −2.43115
\(856\) 0 0
\(857\) −16112.3 −0.642225 −0.321113 0.947041i \(-0.604057\pi\)
−0.321113 + 0.947041i \(0.604057\pi\)
\(858\) 0 0
\(859\) −46836.9 −1.86037 −0.930183 0.367097i \(-0.880352\pi\)
−0.930183 + 0.367097i \(0.880352\pi\)
\(860\) 0 0
\(861\) 23260.6 0.920698
\(862\) 0 0
\(863\) 2453.00 0.0967568 0.0483784 0.998829i \(-0.484595\pi\)
0.0483784 + 0.998829i \(0.484595\pi\)
\(864\) 0 0
\(865\) 54298.5 2.13434
\(866\) 0 0
\(867\) −27601.8 −1.08121
\(868\) 0 0
\(869\) −7712.34 −0.301063
\(870\) 0 0
\(871\) 19678.9 0.765549
\(872\) 0 0
\(873\) 5037.25 0.195287
\(874\) 0 0
\(875\) 1082.26 0.0418138
\(876\) 0 0
\(877\) 23733.6 0.913829 0.456914 0.889511i \(-0.348955\pi\)
0.456914 + 0.889511i \(0.348955\pi\)
\(878\) 0 0
\(879\) 27995.5 1.07425
\(880\) 0 0
\(881\) 10868.6 0.415634 0.207817 0.978168i \(-0.433364\pi\)
0.207817 + 0.978168i \(0.433364\pi\)
\(882\) 0 0
\(883\) 43075.4 1.64168 0.820840 0.571158i \(-0.193505\pi\)
0.820840 + 0.571158i \(0.193505\pi\)
\(884\) 0 0
\(885\) −9917.30 −0.376685
\(886\) 0 0
\(887\) 16224.0 0.614149 0.307075 0.951685i \(-0.400650\pi\)
0.307075 + 0.951685i \(0.400650\pi\)
\(888\) 0 0
\(889\) 10553.5 0.398149
\(890\) 0 0
\(891\) 7952.32 0.299004
\(892\) 0 0
\(893\) 25403.1 0.951938
\(894\) 0 0
\(895\) 3425.07 0.127919
\(896\) 0 0
\(897\) −42971.3 −1.59952
\(898\) 0 0
\(899\) −13558.9 −0.503019
\(900\) 0 0
\(901\) 21499.0 0.794935
\(902\) 0 0
\(903\) 6735.10 0.248206
\(904\) 0 0
\(905\) 70671.5 2.59580
\(906\) 0 0
\(907\) −39985.6 −1.46384 −0.731918 0.681392i \(-0.761375\pi\)
−0.731918 + 0.681392i \(0.761375\pi\)
\(908\) 0 0
\(909\) 33852.8 1.23523
\(910\) 0 0
\(911\) −20330.9 −0.739398 −0.369699 0.929152i \(-0.620539\pi\)
−0.369699 + 0.929152i \(0.620539\pi\)
\(912\) 0 0
\(913\) −13852.0 −0.502119
\(914\) 0 0
\(915\) −73496.7 −2.65544
\(916\) 0 0
\(917\) −9292.84 −0.334653
\(918\) 0 0
\(919\) 3734.71 0.134055 0.0670275 0.997751i \(-0.478648\pi\)
0.0670275 + 0.997751i \(0.478648\pi\)
\(920\) 0 0
\(921\) 5063.36 0.181154
\(922\) 0 0
\(923\) −24912.4 −0.888408
\(924\) 0 0
\(925\) −23029.9 −0.818613
\(926\) 0 0
\(927\) 42336.9 1.50003
\(928\) 0 0
\(929\) 30122.2 1.06381 0.531904 0.846805i \(-0.321477\pi\)
0.531904 + 0.846805i \(0.321477\pi\)
\(930\) 0 0
\(931\) 7061.59 0.248587
\(932\) 0 0
\(933\) 18539.6 0.650545
\(934\) 0 0
\(935\) −5815.72 −0.203417
\(936\) 0 0
\(937\) −33701.9 −1.17502 −0.587509 0.809217i \(-0.699891\pi\)
−0.587509 + 0.809217i \(0.699891\pi\)
\(938\) 0 0
\(939\) −34793.3 −1.20920
\(940\) 0 0
\(941\) 20897.3 0.723944 0.361972 0.932189i \(-0.382104\pi\)
0.361972 + 0.932189i \(0.382104\pi\)
\(942\) 0 0
\(943\) −53658.4 −1.85298
\(944\) 0 0
\(945\) −177.819 −0.00612110
\(946\) 0 0
\(947\) 43378.9 1.48852 0.744258 0.667892i \(-0.232803\pi\)
0.744258 + 0.667892i \(0.232803\pi\)
\(948\) 0 0
\(949\) 8656.93 0.296118
\(950\) 0 0
\(951\) 21134.2 0.720635
\(952\) 0 0
\(953\) −36362.5 −1.23599 −0.617994 0.786183i \(-0.712054\pi\)
−0.617994 + 0.786183i \(0.712054\pi\)
\(954\) 0 0
\(955\) 64765.3 2.19451
\(956\) 0 0
\(957\) 5158.30 0.174236
\(958\) 0 0
\(959\) −3090.72 −0.104071
\(960\) 0 0
\(961\) 15540.6 0.521654
\(962\) 0 0
\(963\) 6725.67 0.225059
\(964\) 0 0
\(965\) 30095.7 1.00395
\(966\) 0 0
\(967\) −23769.2 −0.790451 −0.395225 0.918584i \(-0.629333\pi\)
−0.395225 + 0.918584i \(0.629333\pi\)
\(968\) 0 0
\(969\) −36214.6 −1.20060
\(970\) 0 0
\(971\) 27125.4 0.896494 0.448247 0.893910i \(-0.352048\pi\)
0.448247 + 0.893910i \(0.352048\pi\)
\(972\) 0 0
\(973\) −1254.14 −0.0413214
\(974\) 0 0
\(975\) 41567.1 1.36535
\(976\) 0 0
\(977\) −30145.5 −0.987146 −0.493573 0.869704i \(-0.664309\pi\)
−0.493573 + 0.869704i \(0.664309\pi\)
\(978\) 0 0
\(979\) −8672.33 −0.283114
\(980\) 0 0
\(981\) 31594.0 1.02826
\(982\) 0 0
\(983\) −23037.5 −0.747489 −0.373745 0.927532i \(-0.621926\pi\)
−0.373745 + 0.927532i \(0.621926\pi\)
\(984\) 0 0
\(985\) −58486.8 −1.89192
\(986\) 0 0
\(987\) 9085.91 0.293017
\(988\) 0 0
\(989\) −15536.7 −0.499534
\(990\) 0 0
\(991\) −13580.1 −0.435303 −0.217651 0.976027i \(-0.569840\pi\)
−0.217651 + 0.976027i \(0.569840\pi\)
\(992\) 0 0
\(993\) −2691.16 −0.0860033
\(994\) 0 0
\(995\) −15395.9 −0.490536
\(996\) 0 0
\(997\) −26627.3 −0.845832 −0.422916 0.906169i \(-0.638993\pi\)
−0.422916 + 0.906169i \(0.638993\pi\)
\(998\) 0 0
\(999\) −328.300 −0.0103973
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 1232.4.a.w.1.3 4
4.3 odd 2 77.4.a.c.1.4 4
12.11 even 2 693.4.a.m.1.1 4
20.19 odd 2 1925.4.a.q.1.1 4
28.27 even 2 539.4.a.f.1.4 4
44.43 even 2 847.4.a.e.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.4.a.c.1.4 4 4.3 odd 2
539.4.a.f.1.4 4 28.27 even 2
693.4.a.m.1.1 4 12.11 even 2
847.4.a.e.1.1 4 44.43 even 2
1232.4.a.w.1.3 4 1.1 even 1 trivial
1925.4.a.q.1.1 4 20.19 odd 2