Properties

Label 2112.4.a.bh.1.2
Level $2112$
Weight $4$
Character 2112.1
Self dual yes
Analytic conductor $124.612$
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] = [2112,4,Mod(1,2112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2112, 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("2112.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2112.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: \(124.612033932\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 2112.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +3.48913 q^{5} +4.74456 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +3.48913 q^{5} +4.74456 q^{7} +9.00000 q^{9} +11.0000 q^{11} +15.0217 q^{13} +10.4674 q^{15} +73.1684 q^{17} -78.7011 q^{19} +14.2337 q^{21} -112.000 q^{23} -112.826 q^{25} +27.0000 q^{27} -243.125 q^{29} -278.717 q^{31} +33.0000 q^{33} +16.5544 q^{35} -102.380 q^{37} +45.0652 q^{39} -241.255 q^{41} -280.016 q^{43} +31.4021 q^{45} +169.870 q^{47} -320.489 q^{49} +219.505 q^{51} +409.652 q^{53} +38.3804 q^{55} -236.103 q^{57} +196.000 q^{59} +701.359 q^{61} +42.7011 q^{63} +52.4128 q^{65} +900.587 q^{67} -336.000 q^{69} -756.500 q^{71} -1019.81 q^{73} -338.478 q^{75} +52.1902 q^{77} +327.549 q^{79} +81.0000 q^{81} -756.619 q^{83} +255.294 q^{85} -729.375 q^{87} +508.978 q^{89} +71.2716 q^{91} -836.152 q^{93} -274.598 q^{95} +614.358 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 16 q^{5} - 2 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 16 q^{5} - 2 q^{7} + 18 q^{9} + 22 q^{11} + 76 q^{13} - 48 q^{15} - 26 q^{17} - 54 q^{19} - 6 q^{21} - 224 q^{23} + 142 q^{25} + 54 q^{27} - 222 q^{29} + 40 q^{31} + 66 q^{33} + 148 q^{35} + 48 q^{37} + 228 q^{39} - 494 q^{41} - 66 q^{43} - 144 q^{45} + 64 q^{47} - 618 q^{49} - 78 q^{51} + 84 q^{53} - 176 q^{55} - 162 q^{57} + 392 q^{59} + 1104 q^{61} - 18 q^{63} - 1136 q^{65} + 928 q^{67} - 672 q^{69} - 456 q^{71} - 592 q^{73} + 426 q^{75} - 22 q^{77} + 230 q^{79} + 162 q^{81} + 348 q^{83} + 2188 q^{85} - 666 q^{87} + 972 q^{89} - 340 q^{91} + 120 q^{93} - 756 q^{95} - 1184 q^{97} + 198 q^{99}+O(q^{100}) \) 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
\(5\) 3.48913 0.312077 0.156038 0.987751i \(-0.450128\pi\)
0.156038 + 0.987751i \(0.450128\pi\)
\(6\) 0 0
\(7\) 4.74456 0.256182 0.128091 0.991762i \(-0.459115\pi\)
0.128091 + 0.991762i \(0.459115\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 15.0217 0.320483 0.160242 0.987078i \(-0.448773\pi\)
0.160242 + 0.987078i \(0.448773\pi\)
\(14\) 0 0
\(15\) 10.4674 0.180178
\(16\) 0 0
\(17\) 73.1684 1.04388 0.521940 0.852982i \(-0.325209\pi\)
0.521940 + 0.852982i \(0.325209\pi\)
\(18\) 0 0
\(19\) −78.7011 −0.950277 −0.475138 0.879911i \(-0.657602\pi\)
−0.475138 + 0.879911i \(0.657602\pi\)
\(20\) 0 0
\(21\) 14.2337 0.147907
\(22\) 0 0
\(23\) −112.000 −1.01537 −0.507687 0.861541i \(-0.669499\pi\)
−0.507687 + 0.861541i \(0.669499\pi\)
\(24\) 0 0
\(25\) −112.826 −0.902608
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −243.125 −1.55680 −0.778399 0.627769i \(-0.783968\pi\)
−0.778399 + 0.627769i \(0.783968\pi\)
\(30\) 0 0
\(31\) −278.717 −1.61481 −0.807405 0.589998i \(-0.799129\pi\)
−0.807405 + 0.589998i \(0.799129\pi\)
\(32\) 0 0
\(33\) 33.0000 0.174078
\(34\) 0 0
\(35\) 16.5544 0.0799486
\(36\) 0 0
\(37\) −102.380 −0.454898 −0.227449 0.973790i \(-0.573039\pi\)
−0.227449 + 0.973790i \(0.573039\pi\)
\(38\) 0 0
\(39\) 45.0652 0.185031
\(40\) 0 0
\(41\) −241.255 −0.918970 −0.459485 0.888186i \(-0.651966\pi\)
−0.459485 + 0.888186i \(0.651966\pi\)
\(42\) 0 0
\(43\) −280.016 −0.993071 −0.496536 0.868016i \(-0.665395\pi\)
−0.496536 + 0.868016i \(0.665395\pi\)
\(44\) 0 0
\(45\) 31.4021 0.104026
\(46\) 0 0
\(47\) 169.870 0.527192 0.263596 0.964633i \(-0.415091\pi\)
0.263596 + 0.964633i \(0.415091\pi\)
\(48\) 0 0
\(49\) −320.489 −0.934371
\(50\) 0 0
\(51\) 219.505 0.602684
\(52\) 0 0
\(53\) 409.652 1.06170 0.530849 0.847466i \(-0.321873\pi\)
0.530849 + 0.847466i \(0.321873\pi\)
\(54\) 0 0
\(55\) 38.3804 0.0940947
\(56\) 0 0
\(57\) −236.103 −0.548643
\(58\) 0 0
\(59\) 196.000 0.432492 0.216246 0.976339i \(-0.430619\pi\)
0.216246 + 0.976339i \(0.430619\pi\)
\(60\) 0 0
\(61\) 701.359 1.47213 0.736064 0.676912i \(-0.236682\pi\)
0.736064 + 0.676912i \(0.236682\pi\)
\(62\) 0 0
\(63\) 42.7011 0.0853941
\(64\) 0 0
\(65\) 52.4128 0.100015
\(66\) 0 0
\(67\) 900.587 1.64215 0.821076 0.570819i \(-0.193374\pi\)
0.821076 + 0.570819i \(0.193374\pi\)
\(68\) 0 0
\(69\) −336.000 −0.586227
\(70\) 0 0
\(71\) −756.500 −1.26451 −0.632254 0.774762i \(-0.717870\pi\)
−0.632254 + 0.774762i \(0.717870\pi\)
\(72\) 0 0
\(73\) −1019.81 −1.63507 −0.817536 0.575877i \(-0.804661\pi\)
−0.817536 + 0.575877i \(0.804661\pi\)
\(74\) 0 0
\(75\) −338.478 −0.521121
\(76\) 0 0
\(77\) 52.1902 0.0772419
\(78\) 0 0
\(79\) 327.549 0.466483 0.233241 0.972419i \(-0.425067\pi\)
0.233241 + 0.972419i \(0.425067\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −756.619 −1.00060 −0.500300 0.865852i \(-0.666777\pi\)
−0.500300 + 0.865852i \(0.666777\pi\)
\(84\) 0 0
\(85\) 255.294 0.325771
\(86\) 0 0
\(87\) −729.375 −0.898818
\(88\) 0 0
\(89\) 508.978 0.606198 0.303099 0.952959i \(-0.401979\pi\)
0.303099 + 0.952959i \(0.401979\pi\)
\(90\) 0 0
\(91\) 71.2716 0.0821022
\(92\) 0 0
\(93\) −836.152 −0.932311
\(94\) 0 0
\(95\) −274.598 −0.296559
\(96\) 0 0
\(97\) 614.358 0.643079 0.321539 0.946896i \(-0.395800\pi\)
0.321539 + 0.946896i \(0.395800\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) 1015.92 1.00087 0.500434 0.865775i \(-0.333174\pi\)
0.500434 + 0.865775i \(0.333174\pi\)
\(102\) 0 0
\(103\) −1102.16 −1.05436 −0.527181 0.849753i \(-0.676751\pi\)
−0.527181 + 0.849753i \(0.676751\pi\)
\(104\) 0 0
\(105\) 49.6631 0.0461583
\(106\) 0 0
\(107\) 1377.58 1.24463 0.622315 0.782767i \(-0.286192\pi\)
0.622315 + 0.782767i \(0.286192\pi\)
\(108\) 0 0
\(109\) −320.217 −0.281388 −0.140694 0.990053i \(-0.544933\pi\)
−0.140694 + 0.990053i \(0.544933\pi\)
\(110\) 0 0
\(111\) −307.141 −0.262636
\(112\) 0 0
\(113\) −1629.45 −1.35651 −0.678254 0.734828i \(-0.737263\pi\)
−0.678254 + 0.734828i \(0.737263\pi\)
\(114\) 0 0
\(115\) −390.782 −0.316875
\(116\) 0 0
\(117\) 135.196 0.106828
\(118\) 0 0
\(119\) 347.152 0.267423
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) −723.766 −0.530568
\(124\) 0 0
\(125\) −829.805 −0.593760
\(126\) 0 0
\(127\) −2291.26 −1.60091 −0.800457 0.599390i \(-0.795410\pi\)
−0.800457 + 0.599390i \(0.795410\pi\)
\(128\) 0 0
\(129\) −840.049 −0.573350
\(130\) 0 0
\(131\) −1147.41 −0.765267 −0.382633 0.923900i \(-0.624983\pi\)
−0.382633 + 0.923900i \(0.624983\pi\)
\(132\) 0 0
\(133\) −373.402 −0.243444
\(134\) 0 0
\(135\) 94.2064 0.0600592
\(136\) 0 0
\(137\) 1268.60 0.791121 0.395561 0.918440i \(-0.370550\pi\)
0.395561 + 0.918440i \(0.370550\pi\)
\(138\) 0 0
\(139\) −486.288 −0.296737 −0.148368 0.988932i \(-0.547402\pi\)
−0.148368 + 0.988932i \(0.547402\pi\)
\(140\) 0 0
\(141\) 509.609 0.304374
\(142\) 0 0
\(143\) 165.239 0.0966294
\(144\) 0 0
\(145\) −848.293 −0.485841
\(146\) 0 0
\(147\) −961.467 −0.539459
\(148\) 0 0
\(149\) −2354.11 −1.29434 −0.647169 0.762346i \(-0.724047\pi\)
−0.647169 + 0.762346i \(0.724047\pi\)
\(150\) 0 0
\(151\) 570.070 0.307229 0.153615 0.988131i \(-0.450909\pi\)
0.153615 + 0.988131i \(0.450909\pi\)
\(152\) 0 0
\(153\) 658.516 0.347960
\(154\) 0 0
\(155\) −972.479 −0.503945
\(156\) 0 0
\(157\) 2072.67 1.05361 0.526807 0.849985i \(-0.323389\pi\)
0.526807 + 0.849985i \(0.323389\pi\)
\(158\) 0 0
\(159\) 1228.96 0.612972
\(160\) 0 0
\(161\) −531.391 −0.260121
\(162\) 0 0
\(163\) 2676.51 1.28614 0.643069 0.765808i \(-0.277661\pi\)
0.643069 + 0.765808i \(0.277661\pi\)
\(164\) 0 0
\(165\) 115.141 0.0543256
\(166\) 0 0
\(167\) 1188.12 0.550536 0.275268 0.961368i \(-0.411233\pi\)
0.275268 + 0.961368i \(0.411233\pi\)
\(168\) 0 0
\(169\) −1971.35 −0.897290
\(170\) 0 0
\(171\) −708.310 −0.316759
\(172\) 0 0
\(173\) −807.147 −0.354718 −0.177359 0.984146i \(-0.556755\pi\)
−0.177359 + 0.984146i \(0.556755\pi\)
\(174\) 0 0
\(175\) −535.310 −0.231232
\(176\) 0 0
\(177\) 588.000 0.249699
\(178\) 0 0
\(179\) −1950.39 −0.814408 −0.407204 0.913337i \(-0.633496\pi\)
−0.407204 + 0.913337i \(0.633496\pi\)
\(180\) 0 0
\(181\) −1061.61 −0.435959 −0.217980 0.975953i \(-0.569947\pi\)
−0.217980 + 0.975953i \(0.569947\pi\)
\(182\) 0 0
\(183\) 2104.08 0.849933
\(184\) 0 0
\(185\) −357.218 −0.141963
\(186\) 0 0
\(187\) 804.853 0.314742
\(188\) 0 0
\(189\) 128.103 0.0493023
\(190\) 0 0
\(191\) −2136.41 −0.809348 −0.404674 0.914461i \(-0.632615\pi\)
−0.404674 + 0.914461i \(0.632615\pi\)
\(192\) 0 0
\(193\) 3947.76 1.47236 0.736181 0.676784i \(-0.236627\pi\)
0.736181 + 0.676784i \(0.236627\pi\)
\(194\) 0 0
\(195\) 157.238 0.0577439
\(196\) 0 0
\(197\) −923.886 −0.334133 −0.167066 0.985946i \(-0.553429\pi\)
−0.167066 + 0.985946i \(0.553429\pi\)
\(198\) 0 0
\(199\) 476.152 0.169616 0.0848078 0.996397i \(-0.472972\pi\)
0.0848078 + 0.996397i \(0.472972\pi\)
\(200\) 0 0
\(201\) 2701.76 0.948097
\(202\) 0 0
\(203\) −1153.52 −0.398824
\(204\) 0 0
\(205\) −841.770 −0.286789
\(206\) 0 0
\(207\) −1008.00 −0.338458
\(208\) 0 0
\(209\) −865.712 −0.286519
\(210\) 0 0
\(211\) −4918.24 −1.60467 −0.802336 0.596872i \(-0.796410\pi\)
−0.802336 + 0.596872i \(0.796410\pi\)
\(212\) 0 0
\(213\) −2269.50 −0.730064
\(214\) 0 0
\(215\) −977.012 −0.309915
\(216\) 0 0
\(217\) −1322.39 −0.413686
\(218\) 0 0
\(219\) −3059.44 −0.944010
\(220\) 0 0
\(221\) 1099.12 0.334546
\(222\) 0 0
\(223\) −2100.29 −0.630700 −0.315350 0.948975i \(-0.602122\pi\)
−0.315350 + 0.948975i \(0.602122\pi\)
\(224\) 0 0
\(225\) −1015.43 −0.300869
\(226\) 0 0
\(227\) −2257.16 −0.659970 −0.329985 0.943986i \(-0.607044\pi\)
−0.329985 + 0.943986i \(0.607044\pi\)
\(228\) 0 0
\(229\) 5311.07 1.53260 0.766301 0.642482i \(-0.222095\pi\)
0.766301 + 0.642482i \(0.222095\pi\)
\(230\) 0 0
\(231\) 156.571 0.0445956
\(232\) 0 0
\(233\) 2466.27 0.693435 0.346718 0.937970i \(-0.387296\pi\)
0.346718 + 0.937970i \(0.387296\pi\)
\(234\) 0 0
\(235\) 592.696 0.164524
\(236\) 0 0
\(237\) 982.646 0.269324
\(238\) 0 0
\(239\) −1429.40 −0.386863 −0.193432 0.981114i \(-0.561962\pi\)
−0.193432 + 0.981114i \(0.561962\pi\)
\(240\) 0 0
\(241\) −978.989 −0.261669 −0.130835 0.991404i \(-0.541766\pi\)
−0.130835 + 0.991404i \(0.541766\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −1118.23 −0.291595
\(246\) 0 0
\(247\) −1182.23 −0.304548
\(248\) 0 0
\(249\) −2269.86 −0.577696
\(250\) 0 0
\(251\) −6530.63 −1.64227 −0.821135 0.570734i \(-0.806659\pi\)
−0.821135 + 0.570734i \(0.806659\pi\)
\(252\) 0 0
\(253\) −1232.00 −0.306147
\(254\) 0 0
\(255\) 765.882 0.188084
\(256\) 0 0
\(257\) 8130.26 1.97335 0.986676 0.162696i \(-0.0520188\pi\)
0.986676 + 0.162696i \(0.0520188\pi\)
\(258\) 0 0
\(259\) −485.750 −0.116537
\(260\) 0 0
\(261\) −2188.12 −0.518933
\(262\) 0 0
\(263\) 4549.42 1.06665 0.533326 0.845910i \(-0.320942\pi\)
0.533326 + 0.845910i \(0.320942\pi\)
\(264\) 0 0
\(265\) 1429.33 0.331332
\(266\) 0 0
\(267\) 1526.93 0.349988
\(268\) 0 0
\(269\) 29.1522 0.00660760 0.00330380 0.999995i \(-0.498948\pi\)
0.00330380 + 0.999995i \(0.498948\pi\)
\(270\) 0 0
\(271\) −7711.22 −1.72850 −0.864250 0.503063i \(-0.832206\pi\)
−0.864250 + 0.503063i \(0.832206\pi\)
\(272\) 0 0
\(273\) 213.815 0.0474017
\(274\) 0 0
\(275\) −1241.09 −0.272147
\(276\) 0 0
\(277\) −1127.52 −0.244571 −0.122286 0.992495i \(-0.539022\pi\)
−0.122286 + 0.992495i \(0.539022\pi\)
\(278\) 0 0
\(279\) −2508.46 −0.538270
\(280\) 0 0
\(281\) −1872.47 −0.397517 −0.198758 0.980049i \(-0.563691\pi\)
−0.198758 + 0.980049i \(0.563691\pi\)
\(282\) 0 0
\(283\) 2124.48 0.446245 0.223123 0.974790i \(-0.428375\pi\)
0.223123 + 0.974790i \(0.428375\pi\)
\(284\) 0 0
\(285\) −823.794 −0.171219
\(286\) 0 0
\(287\) −1144.65 −0.235424
\(288\) 0 0
\(289\) 440.621 0.0896846
\(290\) 0 0
\(291\) 1843.07 0.371282
\(292\) 0 0
\(293\) 3324.19 0.662802 0.331401 0.943490i \(-0.392479\pi\)
0.331401 + 0.943490i \(0.392479\pi\)
\(294\) 0 0
\(295\) 683.869 0.134971
\(296\) 0 0
\(297\) 297.000 0.0580259
\(298\) 0 0
\(299\) −1682.44 −0.325411
\(300\) 0 0
\(301\) −1328.55 −0.254407
\(302\) 0 0
\(303\) 3047.75 0.577851
\(304\) 0 0
\(305\) 2447.13 0.459417
\(306\) 0 0
\(307\) −1698.94 −0.315843 −0.157921 0.987452i \(-0.550479\pi\)
−0.157921 + 0.987452i \(0.550479\pi\)
\(308\) 0 0
\(309\) −3306.49 −0.608736
\(310\) 0 0
\(311\) −6928.83 −1.26334 −0.631668 0.775239i \(-0.717630\pi\)
−0.631668 + 0.775239i \(0.717630\pi\)
\(312\) 0 0
\(313\) −3560.75 −0.643020 −0.321510 0.946906i \(-0.604190\pi\)
−0.321510 + 0.946906i \(0.604190\pi\)
\(314\) 0 0
\(315\) 148.989 0.0266495
\(316\) 0 0
\(317\) −332.750 −0.0589561 −0.0294780 0.999565i \(-0.509385\pi\)
−0.0294780 + 0.999565i \(0.509385\pi\)
\(318\) 0 0
\(319\) −2674.37 −0.469393
\(320\) 0 0
\(321\) 4132.73 0.718587
\(322\) 0 0
\(323\) −5758.43 −0.991975
\(324\) 0 0
\(325\) −1694.84 −0.289271
\(326\) 0 0
\(327\) −960.652 −0.162459
\(328\) 0 0
\(329\) 805.957 0.135057
\(330\) 0 0
\(331\) −541.445 −0.0899108 −0.0449554 0.998989i \(-0.514315\pi\)
−0.0449554 + 0.998989i \(0.514315\pi\)
\(332\) 0 0
\(333\) −921.423 −0.151633
\(334\) 0 0
\(335\) 3142.26 0.512478
\(336\) 0 0
\(337\) 816.531 0.131986 0.0659930 0.997820i \(-0.478978\pi\)
0.0659930 + 0.997820i \(0.478978\pi\)
\(338\) 0 0
\(339\) −4888.34 −0.783180
\(340\) 0 0
\(341\) −3065.89 −0.486883
\(342\) 0 0
\(343\) −3147.97 −0.495552
\(344\) 0 0
\(345\) −1172.35 −0.182948
\(346\) 0 0
\(347\) 6260.53 0.968539 0.484269 0.874919i \(-0.339086\pi\)
0.484269 + 0.874919i \(0.339086\pi\)
\(348\) 0 0
\(349\) 12768.5 1.95840 0.979198 0.202906i \(-0.0650386\pi\)
0.979198 + 0.202906i \(0.0650386\pi\)
\(350\) 0 0
\(351\) 405.587 0.0616771
\(352\) 0 0
\(353\) −2649.28 −0.399453 −0.199727 0.979852i \(-0.564005\pi\)
−0.199727 + 0.979852i \(0.564005\pi\)
\(354\) 0 0
\(355\) −2639.52 −0.394623
\(356\) 0 0
\(357\) 1041.46 0.154397
\(358\) 0 0
\(359\) 3203.91 0.471020 0.235510 0.971872i \(-0.424324\pi\)
0.235510 + 0.971872i \(0.424324\pi\)
\(360\) 0 0
\(361\) −665.143 −0.0969737
\(362\) 0 0
\(363\) 363.000 0.0524864
\(364\) 0 0
\(365\) −3558.26 −0.510268
\(366\) 0 0
\(367\) 8429.40 1.19894 0.599470 0.800397i \(-0.295378\pi\)
0.599470 + 0.800397i \(0.295378\pi\)
\(368\) 0 0
\(369\) −2171.30 −0.306323
\(370\) 0 0
\(371\) 1943.62 0.271988
\(372\) 0 0
\(373\) 9388.53 1.30327 0.651635 0.758533i \(-0.274083\pi\)
0.651635 + 0.758533i \(0.274083\pi\)
\(374\) 0 0
\(375\) −2489.41 −0.342807
\(376\) 0 0
\(377\) −3652.16 −0.498928
\(378\) 0 0
\(379\) −14264.5 −1.93329 −0.966647 0.256112i \(-0.917558\pi\)
−0.966647 + 0.256112i \(0.917558\pi\)
\(380\) 0 0
\(381\) −6873.77 −0.924288
\(382\) 0 0
\(383\) −13462.2 −1.79605 −0.898026 0.439942i \(-0.854999\pi\)
−0.898026 + 0.439942i \(0.854999\pi\)
\(384\) 0 0
\(385\) 182.098 0.0241054
\(386\) 0 0
\(387\) −2520.15 −0.331024
\(388\) 0 0
\(389\) 941.881 0.122764 0.0613821 0.998114i \(-0.480449\pi\)
0.0613821 + 0.998114i \(0.480449\pi\)
\(390\) 0 0
\(391\) −8194.87 −1.05993
\(392\) 0 0
\(393\) −3442.24 −0.441827
\(394\) 0 0
\(395\) 1142.86 0.145578
\(396\) 0 0
\(397\) 847.839 0.107183 0.0535917 0.998563i \(-0.482933\pi\)
0.0535917 + 0.998563i \(0.482933\pi\)
\(398\) 0 0
\(399\) −1120.21 −0.140553
\(400\) 0 0
\(401\) 12203.6 1.51975 0.759875 0.650069i \(-0.225260\pi\)
0.759875 + 0.650069i \(0.225260\pi\)
\(402\) 0 0
\(403\) −4186.82 −0.517520
\(404\) 0 0
\(405\) 282.619 0.0346752
\(406\) 0 0
\(407\) −1126.18 −0.137157
\(408\) 0 0
\(409\) 8759.53 1.05900 0.529500 0.848310i \(-0.322380\pi\)
0.529500 + 0.848310i \(0.322380\pi\)
\(410\) 0 0
\(411\) 3805.79 0.456754
\(412\) 0 0
\(413\) 929.934 0.110797
\(414\) 0 0
\(415\) −2639.94 −0.312264
\(416\) 0 0
\(417\) −1458.86 −0.171321
\(418\) 0 0
\(419\) −11188.4 −1.30451 −0.652256 0.757999i \(-0.726177\pi\)
−0.652256 + 0.757999i \(0.726177\pi\)
\(420\) 0 0
\(421\) 14082.3 1.63023 0.815116 0.579298i \(-0.196673\pi\)
0.815116 + 0.579298i \(0.196673\pi\)
\(422\) 0 0
\(423\) 1528.83 0.175731
\(424\) 0 0
\(425\) −8255.30 −0.942214
\(426\) 0 0
\(427\) 3327.64 0.377133
\(428\) 0 0
\(429\) 495.718 0.0557890
\(430\) 0 0
\(431\) 5616.05 0.627647 0.313823 0.949481i \(-0.398390\pi\)
0.313823 + 0.949481i \(0.398390\pi\)
\(432\) 0 0
\(433\) 7195.75 0.798627 0.399314 0.916814i \(-0.369248\pi\)
0.399314 + 0.916814i \(0.369248\pi\)
\(434\) 0 0
\(435\) −2544.88 −0.280500
\(436\) 0 0
\(437\) 8814.52 0.964887
\(438\) 0 0
\(439\) −101.959 −0.0110848 −0.00554240 0.999985i \(-0.501764\pi\)
−0.00554240 + 0.999985i \(0.501764\pi\)
\(440\) 0 0
\(441\) −2884.40 −0.311457
\(442\) 0 0
\(443\) 4953.74 0.531285 0.265642 0.964072i \(-0.414416\pi\)
0.265642 + 0.964072i \(0.414416\pi\)
\(444\) 0 0
\(445\) 1775.89 0.189180
\(446\) 0 0
\(447\) −7062.34 −0.747287
\(448\) 0 0
\(449\) −11602.0 −1.21945 −0.609723 0.792615i \(-0.708719\pi\)
−0.609723 + 0.792615i \(0.708719\pi\)
\(450\) 0 0
\(451\) −2653.81 −0.277080
\(452\) 0 0
\(453\) 1710.21 0.177379
\(454\) 0 0
\(455\) 248.676 0.0256222
\(456\) 0 0
\(457\) −3530.68 −0.361397 −0.180698 0.983539i \(-0.557836\pi\)
−0.180698 + 0.983539i \(0.557836\pi\)
\(458\) 0 0
\(459\) 1975.55 0.200895
\(460\) 0 0
\(461\) −11566.3 −1.16854 −0.584271 0.811559i \(-0.698619\pi\)
−0.584271 + 0.811559i \(0.698619\pi\)
\(462\) 0 0
\(463\) −10888.5 −1.09294 −0.546470 0.837479i \(-0.684029\pi\)
−0.546470 + 0.837479i \(0.684029\pi\)
\(464\) 0 0
\(465\) −2917.44 −0.290953
\(466\) 0 0
\(467\) 10688.0 1.05906 0.529529 0.848292i \(-0.322369\pi\)
0.529529 + 0.848292i \(0.322369\pi\)
\(468\) 0 0
\(469\) 4272.89 0.420690
\(470\) 0 0
\(471\) 6218.02 0.608304
\(472\) 0 0
\(473\) −3080.18 −0.299422
\(474\) 0 0
\(475\) 8879.53 0.857728
\(476\) 0 0
\(477\) 3686.87 0.353900
\(478\) 0 0
\(479\) −2341.90 −0.223391 −0.111696 0.993742i \(-0.535628\pi\)
−0.111696 + 0.993742i \(0.535628\pi\)
\(480\) 0 0
\(481\) −1537.93 −0.145787
\(482\) 0 0
\(483\) −1594.17 −0.150181
\(484\) 0 0
\(485\) 2143.57 0.200690
\(486\) 0 0
\(487\) −6748.91 −0.627972 −0.313986 0.949428i \(-0.601665\pi\)
−0.313986 + 0.949428i \(0.601665\pi\)
\(488\) 0 0
\(489\) 8029.53 0.742552
\(490\) 0 0
\(491\) 7361.40 0.676609 0.338305 0.941037i \(-0.390147\pi\)
0.338305 + 0.941037i \(0.390147\pi\)
\(492\) 0 0
\(493\) −17789.1 −1.62511
\(494\) 0 0
\(495\) 345.423 0.0313649
\(496\) 0 0
\(497\) −3589.26 −0.323944
\(498\) 0 0
\(499\) 10381.7 0.931359 0.465680 0.884953i \(-0.345810\pi\)
0.465680 + 0.884953i \(0.345810\pi\)
\(500\) 0 0
\(501\) 3564.36 0.317852
\(502\) 0 0
\(503\) −19149.0 −1.69744 −0.848721 0.528840i \(-0.822627\pi\)
−0.848721 + 0.528840i \(0.822627\pi\)
\(504\) 0 0
\(505\) 3544.67 0.312348
\(506\) 0 0
\(507\) −5914.04 −0.518051
\(508\) 0 0
\(509\) −16073.2 −1.39967 −0.699836 0.714303i \(-0.746744\pi\)
−0.699836 + 0.714303i \(0.746744\pi\)
\(510\) 0 0
\(511\) −4838.58 −0.418877
\(512\) 0 0
\(513\) −2124.93 −0.182881
\(514\) 0 0
\(515\) −3845.58 −0.329042
\(516\) 0 0
\(517\) 1868.56 0.158954
\(518\) 0 0
\(519\) −2421.44 −0.204797
\(520\) 0 0
\(521\) −18955.3 −1.59395 −0.796975 0.604012i \(-0.793568\pi\)
−0.796975 + 0.604012i \(0.793568\pi\)
\(522\) 0 0
\(523\) −4442.19 −0.371402 −0.185701 0.982606i \(-0.559456\pi\)
−0.185701 + 0.982606i \(0.559456\pi\)
\(524\) 0 0
\(525\) −1605.93 −0.133502
\(526\) 0 0
\(527\) −20393.3 −1.68567
\(528\) 0 0
\(529\) 377.000 0.0309855
\(530\) 0 0
\(531\) 1764.00 0.144164
\(532\) 0 0
\(533\) −3624.08 −0.294515
\(534\) 0 0
\(535\) 4806.54 0.388420
\(536\) 0 0
\(537\) −5851.17 −0.470199
\(538\) 0 0
\(539\) −3525.38 −0.281723
\(540\) 0 0
\(541\) −2180.90 −0.173316 −0.0866580 0.996238i \(-0.527619\pi\)
−0.0866580 + 0.996238i \(0.527619\pi\)
\(542\) 0 0
\(543\) −3184.82 −0.251701
\(544\) 0 0
\(545\) −1117.28 −0.0878146
\(546\) 0 0
\(547\) 8225.04 0.642920 0.321460 0.946923i \(-0.395826\pi\)
0.321460 + 0.946923i \(0.395826\pi\)
\(548\) 0 0
\(549\) 6312.23 0.490709
\(550\) 0 0
\(551\) 19134.2 1.47939
\(552\) 0 0
\(553\) 1554.08 0.119505
\(554\) 0 0
\(555\) −1071.65 −0.0819625
\(556\) 0 0
\(557\) 25181.9 1.91561 0.957804 0.287423i \(-0.0927986\pi\)
0.957804 + 0.287423i \(0.0927986\pi\)
\(558\) 0 0
\(559\) −4206.33 −0.318263
\(560\) 0 0
\(561\) 2414.56 0.181716
\(562\) 0 0
\(563\) −4504.50 −0.337197 −0.168599 0.985685i \(-0.553924\pi\)
−0.168599 + 0.985685i \(0.553924\pi\)
\(564\) 0 0
\(565\) −5685.34 −0.423335
\(566\) 0 0
\(567\) 384.310 0.0284647
\(568\) 0 0
\(569\) −13447.0 −0.990732 −0.495366 0.868684i \(-0.664966\pi\)
−0.495366 + 0.868684i \(0.664966\pi\)
\(570\) 0 0
\(571\) −2605.52 −0.190959 −0.0954795 0.995431i \(-0.530438\pi\)
−0.0954795 + 0.995431i \(0.530438\pi\)
\(572\) 0 0
\(573\) −6409.24 −0.467277
\(574\) 0 0
\(575\) 12636.5 0.916485
\(576\) 0 0
\(577\) 6339.65 0.457406 0.228703 0.973496i \(-0.426552\pi\)
0.228703 + 0.973496i \(0.426552\pi\)
\(578\) 0 0
\(579\) 11843.3 0.850069
\(580\) 0 0
\(581\) −3589.83 −0.256336
\(582\) 0 0
\(583\) 4506.17 0.320114
\(584\) 0 0
\(585\) 471.715 0.0333385
\(586\) 0 0
\(587\) −13370.6 −0.940140 −0.470070 0.882629i \(-0.655771\pi\)
−0.470070 + 0.882629i \(0.655771\pi\)
\(588\) 0 0
\(589\) 21935.3 1.53452
\(590\) 0 0
\(591\) −2771.66 −0.192912
\(592\) 0 0
\(593\) 14319.3 0.991608 0.495804 0.868434i \(-0.334873\pi\)
0.495804 + 0.868434i \(0.334873\pi\)
\(594\) 0 0
\(595\) 1211.26 0.0834567
\(596\) 0 0
\(597\) 1428.46 0.0979276
\(598\) 0 0
\(599\) 5788.63 0.394853 0.197427 0.980318i \(-0.436742\pi\)
0.197427 + 0.980318i \(0.436742\pi\)
\(600\) 0 0
\(601\) 23968.1 1.62675 0.813375 0.581739i \(-0.197628\pi\)
0.813375 + 0.581739i \(0.197628\pi\)
\(602\) 0 0
\(603\) 8105.28 0.547384
\(604\) 0 0
\(605\) 422.184 0.0283706
\(606\) 0 0
\(607\) 23526.6 1.57317 0.786585 0.617482i \(-0.211847\pi\)
0.786585 + 0.617482i \(0.211847\pi\)
\(608\) 0 0
\(609\) −3460.56 −0.230261
\(610\) 0 0
\(611\) 2551.74 0.168956
\(612\) 0 0
\(613\) −1228.07 −0.0809159 −0.0404579 0.999181i \(-0.512882\pi\)
−0.0404579 + 0.999181i \(0.512882\pi\)
\(614\) 0 0
\(615\) −2525.31 −0.165578
\(616\) 0 0
\(617\) −9844.90 −0.642368 −0.321184 0.947017i \(-0.604081\pi\)
−0.321184 + 0.947017i \(0.604081\pi\)
\(618\) 0 0
\(619\) −6551.68 −0.425419 −0.212709 0.977115i \(-0.568229\pi\)
−0.212709 + 0.977115i \(0.568229\pi\)
\(620\) 0 0
\(621\) −3024.00 −0.195409
\(622\) 0 0
\(623\) 2414.88 0.155297
\(624\) 0 0
\(625\) 11208.0 0.717309
\(626\) 0 0
\(627\) −2597.14 −0.165422
\(628\) 0 0
\(629\) −7491.01 −0.474859
\(630\) 0 0
\(631\) 26440.5 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(632\) 0 0
\(633\) −14754.7 −0.926458
\(634\) 0 0
\(635\) −7994.48 −0.499608
\(636\) 0 0
\(637\) −4814.31 −0.299450
\(638\) 0 0
\(639\) −6808.50 −0.421502
\(640\) 0 0
\(641\) −27927.2 −1.72084 −0.860421 0.509584i \(-0.829799\pi\)
−0.860421 + 0.509584i \(0.829799\pi\)
\(642\) 0 0
\(643\) −16737.7 −1.02655 −0.513274 0.858225i \(-0.671568\pi\)
−0.513274 + 0.858225i \(0.671568\pi\)
\(644\) 0 0
\(645\) −2931.03 −0.178929
\(646\) 0 0
\(647\) −7818.70 −0.475092 −0.237546 0.971376i \(-0.576343\pi\)
−0.237546 + 0.971376i \(0.576343\pi\)
\(648\) 0 0
\(649\) 2156.00 0.130401
\(650\) 0 0
\(651\) −3967.17 −0.238842
\(652\) 0 0
\(653\) −19747.6 −1.18344 −0.591719 0.806144i \(-0.701550\pi\)
−0.591719 + 0.806144i \(0.701550\pi\)
\(654\) 0 0
\(655\) −4003.47 −0.238822
\(656\) 0 0
\(657\) −9178.33 −0.545024
\(658\) 0 0
\(659\) 7867.72 0.465072 0.232536 0.972588i \(-0.425298\pi\)
0.232536 + 0.972588i \(0.425298\pi\)
\(660\) 0 0
\(661\) −4227.41 −0.248755 −0.124378 0.992235i \(-0.539693\pi\)
−0.124378 + 0.992235i \(0.539693\pi\)
\(662\) 0 0
\(663\) 3297.35 0.193150
\(664\) 0 0
\(665\) −1302.85 −0.0759733
\(666\) 0 0
\(667\) 27230.0 1.58073
\(668\) 0 0
\(669\) −6300.88 −0.364135
\(670\) 0 0
\(671\) 7714.94 0.443863
\(672\) 0 0
\(673\) 29397.6 1.68379 0.841897 0.539638i \(-0.181439\pi\)
0.841897 + 0.539638i \(0.181439\pi\)
\(674\) 0 0
\(675\) −3046.30 −0.173707
\(676\) 0 0
\(677\) −5737.14 −0.325696 −0.162848 0.986651i \(-0.552068\pi\)
−0.162848 + 0.986651i \(0.552068\pi\)
\(678\) 0 0
\(679\) 2914.86 0.164745
\(680\) 0 0
\(681\) −6771.49 −0.381034
\(682\) 0 0
\(683\) 32097.6 1.79821 0.899107 0.437729i \(-0.144217\pi\)
0.899107 + 0.437729i \(0.144217\pi\)
\(684\) 0 0
\(685\) 4426.30 0.246891
\(686\) 0 0
\(687\) 15933.2 0.884848
\(688\) 0 0
\(689\) 6153.69 0.340257
\(690\) 0 0
\(691\) −16456.2 −0.905965 −0.452983 0.891519i \(-0.649640\pi\)
−0.452983 + 0.891519i \(0.649640\pi\)
\(692\) 0 0
\(693\) 469.712 0.0257473
\(694\) 0 0
\(695\) −1696.72 −0.0926047
\(696\) 0 0
\(697\) −17652.3 −0.959294
\(698\) 0 0
\(699\) 7398.80 0.400355
\(700\) 0 0
\(701\) −27238.1 −1.46758 −0.733788 0.679379i \(-0.762249\pi\)
−0.733788 + 0.679379i \(0.762249\pi\)
\(702\) 0 0
\(703\) 8057.44 0.432279
\(704\) 0 0
\(705\) 1778.09 0.0949882
\(706\) 0 0
\(707\) 4820.09 0.256405
\(708\) 0 0
\(709\) −28761.4 −1.52349 −0.761747 0.647875i \(-0.775658\pi\)
−0.761747 + 0.647875i \(0.775658\pi\)
\(710\) 0 0
\(711\) 2947.94 0.155494
\(712\) 0 0
\(713\) 31216.3 1.63964
\(714\) 0 0
\(715\) 576.540 0.0301558
\(716\) 0 0
\(717\) −4288.21 −0.223356
\(718\) 0 0
\(719\) 27272.0 1.41456 0.707282 0.706931i \(-0.249921\pi\)
0.707282 + 0.706931i \(0.249921\pi\)
\(720\) 0 0
\(721\) −5229.28 −0.270109
\(722\) 0 0
\(723\) −2936.97 −0.151075
\(724\) 0 0
\(725\) 27430.8 1.40518
\(726\) 0 0
\(727\) −3979.75 −0.203027 −0.101514 0.994834i \(-0.532369\pi\)
−0.101514 + 0.994834i \(0.532369\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −20488.3 −1.03665
\(732\) 0 0
\(733\) −9342.48 −0.470767 −0.235384 0.971903i \(-0.575635\pi\)
−0.235384 + 0.971903i \(0.575635\pi\)
\(734\) 0 0
\(735\) −3354.68 −0.168353
\(736\) 0 0
\(737\) 9906.45 0.495127
\(738\) 0 0
\(739\) −28928.0 −1.43997 −0.719983 0.693992i \(-0.755850\pi\)
−0.719983 + 0.693992i \(0.755850\pi\)
\(740\) 0 0
\(741\) −3546.68 −0.175831
\(742\) 0 0
\(743\) −4857.04 −0.239822 −0.119911 0.992785i \(-0.538261\pi\)
−0.119911 + 0.992785i \(0.538261\pi\)
\(744\) 0 0
\(745\) −8213.80 −0.403933
\(746\) 0 0
\(747\) −6809.57 −0.333533
\(748\) 0 0
\(749\) 6536.00 0.318852
\(750\) 0 0
\(751\) −14355.4 −0.697517 −0.348759 0.937213i \(-0.613397\pi\)
−0.348759 + 0.937213i \(0.613397\pi\)
\(752\) 0 0
\(753\) −19591.9 −0.948165
\(754\) 0 0
\(755\) 1989.05 0.0958792
\(756\) 0 0
\(757\) 17714.9 0.850538 0.425269 0.905067i \(-0.360179\pi\)
0.425269 + 0.905067i \(0.360179\pi\)
\(758\) 0 0
\(759\) −3696.00 −0.176754
\(760\) 0 0
\(761\) −7945.82 −0.378497 −0.189248 0.981929i \(-0.560605\pi\)
−0.189248 + 0.981929i \(0.560605\pi\)
\(762\) 0 0
\(763\) −1519.29 −0.0720866
\(764\) 0 0
\(765\) 2297.64 0.108590
\(766\) 0 0
\(767\) 2944.26 0.138606
\(768\) 0 0
\(769\) 27308.1 1.28057 0.640284 0.768139i \(-0.278817\pi\)
0.640284 + 0.768139i \(0.278817\pi\)
\(770\) 0 0
\(771\) 24390.8 1.13932
\(772\) 0 0
\(773\) 18872.6 0.878136 0.439068 0.898454i \(-0.355309\pi\)
0.439068 + 0.898454i \(0.355309\pi\)
\(774\) 0 0
\(775\) 31446.6 1.45754
\(776\) 0 0
\(777\) −1457.25 −0.0672826
\(778\) 0 0
\(779\) 18987.1 0.873276
\(780\) 0 0
\(781\) −8321.50 −0.381263
\(782\) 0 0
\(783\) −6564.37 −0.299606
\(784\) 0 0
\(785\) 7231.82 0.328808
\(786\) 0 0
\(787\) −14512.1 −0.657307 −0.328654 0.944450i \(-0.606595\pi\)
−0.328654 + 0.944450i \(0.606595\pi\)
\(788\) 0 0
\(789\) 13648.3 0.615832
\(790\) 0 0
\(791\) −7731.01 −0.347513
\(792\) 0 0
\(793\) 10535.6 0.471792
\(794\) 0 0
\(795\) 4287.98 0.191294
\(796\) 0 0
\(797\) −29108.9 −1.29371 −0.646856 0.762612i \(-0.723917\pi\)
−0.646856 + 0.762612i \(0.723917\pi\)
\(798\) 0 0
\(799\) 12429.1 0.550325
\(800\) 0 0
\(801\) 4580.80 0.202066
\(802\) 0 0
\(803\) −11218.0 −0.492993
\(804\) 0 0
\(805\) −1854.09 −0.0811777
\(806\) 0 0
\(807\) 87.4567 0.00381490
\(808\) 0 0
\(809\) −3000.83 −0.130413 −0.0652063 0.997872i \(-0.520771\pi\)
−0.0652063 + 0.997872i \(0.520771\pi\)
\(810\) 0 0
\(811\) 6239.39 0.270154 0.135077 0.990835i \(-0.456872\pi\)
0.135077 + 0.990835i \(0.456872\pi\)
\(812\) 0 0
\(813\) −23133.7 −0.997950
\(814\) 0 0
\(815\) 9338.68 0.401374
\(816\) 0 0
\(817\) 22037.6 0.943693
\(818\) 0 0
\(819\) 641.445 0.0273674
\(820\) 0 0
\(821\) −14922.4 −0.634342 −0.317171 0.948368i \(-0.602733\pi\)
−0.317171 + 0.948368i \(0.602733\pi\)
\(822\) 0 0
\(823\) 25737.8 1.09011 0.545057 0.838399i \(-0.316508\pi\)
0.545057 + 0.838399i \(0.316508\pi\)
\(824\) 0 0
\(825\) −3723.26 −0.157124
\(826\) 0 0
\(827\) 27043.4 1.13711 0.568555 0.822645i \(-0.307503\pi\)
0.568555 + 0.822645i \(0.307503\pi\)
\(828\) 0 0
\(829\) 9795.41 0.410384 0.205192 0.978722i \(-0.434218\pi\)
0.205192 + 0.978722i \(0.434218\pi\)
\(830\) 0 0
\(831\) −3382.56 −0.141203
\(832\) 0 0
\(833\) −23449.7 −0.975370
\(834\) 0 0
\(835\) 4145.50 0.171809
\(836\) 0 0
\(837\) −7525.37 −0.310770
\(838\) 0 0
\(839\) −28875.5 −1.18819 −0.594095 0.804395i \(-0.702490\pi\)
−0.594095 + 0.804395i \(0.702490\pi\)
\(840\) 0 0
\(841\) 34720.7 1.42362
\(842\) 0 0
\(843\) −5617.41 −0.229506
\(844\) 0 0
\(845\) −6878.28 −0.280024
\(846\) 0 0
\(847\) 574.092 0.0232893
\(848\) 0 0
\(849\) 6373.45 0.257640
\(850\) 0 0
\(851\) 11466.6 0.461892
\(852\) 0 0
\(853\) 47157.1 1.89288 0.946441 0.322878i \(-0.104650\pi\)
0.946441 + 0.322878i \(0.104650\pi\)
\(854\) 0 0
\(855\) −2471.38 −0.0988531
\(856\) 0 0
\(857\) 5021.31 0.200145 0.100073 0.994980i \(-0.468092\pi\)
0.100073 + 0.994980i \(0.468092\pi\)
\(858\) 0 0
\(859\) −22921.1 −0.910428 −0.455214 0.890382i \(-0.650437\pi\)
−0.455214 + 0.890382i \(0.650437\pi\)
\(860\) 0 0
\(861\) −3433.95 −0.135922
\(862\) 0 0
\(863\) 19488.1 0.768693 0.384347 0.923189i \(-0.374427\pi\)
0.384347 + 0.923189i \(0.374427\pi\)
\(864\) 0 0
\(865\) −2816.24 −0.110699
\(866\) 0 0
\(867\) 1321.86 0.0517794
\(868\) 0 0
\(869\) 3603.04 0.140650
\(870\) 0 0
\(871\) 13528.4 0.526282
\(872\) 0 0
\(873\) 5529.22 0.214360
\(874\) 0 0
\(875\) −3937.06 −0.152111
\(876\) 0 0
\(877\) 8455.67 0.325573 0.162787 0.986661i \(-0.447952\pi\)
0.162787 + 0.986661i \(0.447952\pi\)
\(878\) 0 0
\(879\) 9972.56 0.382669
\(880\) 0 0
\(881\) −11291.2 −0.431794 −0.215897 0.976416i \(-0.569268\pi\)
−0.215897 + 0.976416i \(0.569268\pi\)
\(882\) 0 0
\(883\) 31818.1 1.21264 0.606322 0.795219i \(-0.292644\pi\)
0.606322 + 0.795219i \(0.292644\pi\)
\(884\) 0 0
\(885\) 2051.61 0.0779254
\(886\) 0 0
\(887\) −17481.1 −0.661732 −0.330866 0.943678i \(-0.607341\pi\)
−0.330866 + 0.943678i \(0.607341\pi\)
\(888\) 0 0
\(889\) −10871.0 −0.410126
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) 0 0
\(893\) −13368.9 −0.500978
\(894\) 0 0
\(895\) −6805.16 −0.254158
\(896\) 0 0
\(897\) −5047.31 −0.187876
\(898\) 0 0
\(899\) 67763.1 2.51393
\(900\) 0 0
\(901\) 29973.6 1.10829
\(902\) 0 0
\(903\) −3985.66 −0.146882
\(904\) 0 0
\(905\) −3704.08 −0.136053
\(906\) 0 0
\(907\) 10607.4 0.388326 0.194163 0.980969i \(-0.437801\pi\)
0.194163 + 0.980969i \(0.437801\pi\)
\(908\) 0 0
\(909\) 9143.26 0.333623
\(910\) 0 0
\(911\) 41249.2 1.50016 0.750080 0.661347i \(-0.230015\pi\)
0.750080 + 0.661347i \(0.230015\pi\)
\(912\) 0 0
\(913\) −8322.81 −0.301692
\(914\) 0 0
\(915\) 7341.38 0.265244
\(916\) 0 0
\(917\) −5443.97 −0.196048
\(918\) 0 0
\(919\) 13858.1 0.497429 0.248714 0.968577i \(-0.419992\pi\)
0.248714 + 0.968577i \(0.419992\pi\)
\(920\) 0 0
\(921\) −5096.83 −0.182352
\(922\) 0 0
\(923\) −11363.9 −0.405253
\(924\) 0 0
\(925\) 11551.2 0.410595
\(926\) 0 0
\(927\) −9919.47 −0.351454
\(928\) 0 0
\(929\) 20893.7 0.737890 0.368945 0.929451i \(-0.379719\pi\)
0.368945 + 0.929451i \(0.379719\pi\)
\(930\) 0 0
\(931\) 25222.8 0.887911
\(932\) 0 0
\(933\) −20786.5 −0.729388
\(934\) 0 0
\(935\) 2808.23 0.0982236
\(936\) 0 0
\(937\) 3203.52 0.111691 0.0558454 0.998439i \(-0.482215\pi\)
0.0558454 + 0.998439i \(0.482215\pi\)
\(938\) 0 0
\(939\) −10682.2 −0.371248
\(940\) 0 0
\(941\) −19951.6 −0.691182 −0.345591 0.938385i \(-0.612322\pi\)
−0.345591 + 0.938385i \(0.612322\pi\)
\(942\) 0 0
\(943\) 27020.6 0.933099
\(944\) 0 0
\(945\) 446.968 0.0153861
\(946\) 0 0
\(947\) −38216.7 −1.31138 −0.655689 0.755031i \(-0.727622\pi\)
−0.655689 + 0.755031i \(0.727622\pi\)
\(948\) 0 0
\(949\) −15319.4 −0.524014
\(950\) 0 0
\(951\) −998.249 −0.0340383
\(952\) 0 0
\(953\) −47661.4 −1.62004 −0.810022 0.586399i \(-0.800545\pi\)
−0.810022 + 0.586399i \(0.800545\pi\)
\(954\) 0 0
\(955\) −7454.21 −0.252579
\(956\) 0 0
\(957\) −8023.12 −0.271004
\(958\) 0 0
\(959\) 6018.94 0.202671
\(960\) 0 0
\(961\) 47892.3 1.60761
\(962\) 0 0
\(963\) 12398.2 0.414876
\(964\) 0 0
\(965\) 13774.2 0.459490
\(966\) 0 0
\(967\) −18933.2 −0.629628 −0.314814 0.949153i \(-0.601942\pi\)
−0.314814 + 0.949153i \(0.601942\pi\)
\(968\) 0 0
\(969\) −17275.3 −0.572717
\(970\) 0 0
\(971\) −40660.3 −1.34382 −0.671911 0.740632i \(-0.734526\pi\)
−0.671911 + 0.740632i \(0.734526\pi\)
\(972\) 0 0
\(973\) −2307.23 −0.0760188
\(974\) 0 0
\(975\) −5084.53 −0.167011
\(976\) 0 0
\(977\) −22502.8 −0.736876 −0.368438 0.929652i \(-0.620107\pi\)
−0.368438 + 0.929652i \(0.620107\pi\)
\(978\) 0 0
\(979\) 5598.76 0.182775
\(980\) 0 0
\(981\) −2881.96 −0.0937959
\(982\) 0 0
\(983\) 4435.20 0.143907 0.0719536 0.997408i \(-0.477077\pi\)
0.0719536 + 0.997408i \(0.477077\pi\)
\(984\) 0 0
\(985\) −3223.55 −0.104275
\(986\) 0 0
\(987\) 2417.87 0.0779753
\(988\) 0 0
\(989\) 31361.8 1.00834
\(990\) 0 0
\(991\) −7362.76 −0.236010 −0.118005 0.993013i \(-0.537650\pi\)
−0.118005 + 0.993013i \(0.537650\pi\)
\(992\) 0 0
\(993\) −1624.33 −0.0519101
\(994\) 0 0
\(995\) 1661.35 0.0529331
\(996\) 0 0
\(997\) 53480.1 1.69883 0.849413 0.527728i \(-0.176956\pi\)
0.849413 + 0.527728i \(0.176956\pi\)
\(998\) 0 0
\(999\) −2764.27 −0.0875452
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 2112.4.a.bh.1.2 2
4.3 odd 2 2112.4.a.ba.1.2 2
8.3 odd 2 33.4.a.d.1.2 2
8.5 even 2 528.4.a.o.1.1 2
24.5 odd 2 1584.4.a.x.1.2 2
24.11 even 2 99.4.a.e.1.1 2
40.3 even 4 825.4.c.i.199.1 4
40.19 odd 2 825.4.a.k.1.1 2
40.27 even 4 825.4.c.i.199.4 4
56.27 even 2 1617.4.a.j.1.2 2
88.43 even 2 363.4.a.j.1.1 2
120.59 even 2 2475.4.a.o.1.2 2
264.131 odd 2 1089.4.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.2 2 8.3 odd 2
99.4.a.e.1.1 2 24.11 even 2
363.4.a.j.1.1 2 88.43 even 2
528.4.a.o.1.1 2 8.5 even 2
825.4.a.k.1.1 2 40.19 odd 2
825.4.c.i.199.1 4 40.3 even 4
825.4.c.i.199.4 4 40.27 even 4
1089.4.a.t.1.2 2 264.131 odd 2
1584.4.a.x.1.2 2 24.5 odd 2
1617.4.a.j.1.2 2 56.27 even 2
2112.4.a.ba.1.2 2 4.3 odd 2
2112.4.a.bh.1.2 2 1.1 even 1 trivial
2475.4.a.o.1.2 2 120.59 even 2