Properties

Label 2112.4.a.ba.1.1
Level $2112$
Weight $4$
Character 2112.1
Self dual yes
Analytic conductor $124.612$
Analytic rank $0$
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: \(0\)
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 1.1
Root \(3.37228\) 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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -19.4891 q^{5} +6.74456 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -19.4891 q^{5} +6.74456 q^{7} +9.00000 q^{9} -11.0000 q^{11} +60.9783 q^{13} +58.4674 q^{15} -99.1684 q^{17} -24.7011 q^{19} -20.2337 q^{21} +112.000 q^{23} +254.826 q^{25} -27.0000 q^{27} +21.1249 q^{29} -318.717 q^{31} +33.0000 q^{33} -131.446 q^{35} +150.380 q^{37} -182.935 q^{39} -252.745 q^{41} -214.016 q^{43} -175.402 q^{45} +105.870 q^{47} -297.511 q^{49} +297.505 q^{51} -325.652 q^{53} +214.380 q^{55} +74.1032 q^{57} -196.000 q^{59} +402.641 q^{61} +60.7011 q^{63} -1188.41 q^{65} -27.4132 q^{67} -336.000 q^{69} -300.500 q^{71} +427.815 q^{73} -764.478 q^{75} -74.1902 q^{77} +97.5488 q^{79} +81.0000 q^{81} -1104.62 q^{83} +1932.71 q^{85} -63.3748 q^{87} +463.022 q^{89} +411.272 q^{91} +956.152 q^{93} +481.402 q^{95} -1798.36 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 16 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) 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
\(5\) −19.4891 −1.74316 −0.871580 0.490253i \(-0.836904\pi\)
−0.871580 + 0.490253i \(0.836904\pi\)
\(6\) 0 0
\(7\) 6.74456 0.364172 0.182086 0.983283i \(-0.441715\pi\)
0.182086 + 0.983283i \(0.441715\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) 60.9783 1.30095 0.650474 0.759529i \(-0.274570\pi\)
0.650474 + 0.759529i \(0.274570\pi\)
\(14\) 0 0
\(15\) 58.4674 1.00641
\(16\) 0 0
\(17\) −99.1684 −1.41482 −0.707408 0.706805i \(-0.750136\pi\)
−0.707408 + 0.706805i \(0.750136\pi\)
\(18\) 0 0
\(19\) −24.7011 −0.298253 −0.149127 0.988818i \(-0.547646\pi\)
−0.149127 + 0.988818i \(0.547646\pi\)
\(20\) 0 0
\(21\) −20.2337 −0.210255
\(22\) 0 0
\(23\) 112.000 1.01537 0.507687 0.861541i \(-0.330501\pi\)
0.507687 + 0.861541i \(0.330501\pi\)
\(24\) 0 0
\(25\) 254.826 2.03861
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 21.1249 0.135269 0.0676345 0.997710i \(-0.478455\pi\)
0.0676345 + 0.997710i \(0.478455\pi\)
\(30\) 0 0
\(31\) −318.717 −1.84656 −0.923279 0.384130i \(-0.874502\pi\)
−0.923279 + 0.384130i \(0.874502\pi\)
\(32\) 0 0
\(33\) 33.0000 0.174078
\(34\) 0 0
\(35\) −131.446 −0.634810
\(36\) 0 0
\(37\) 150.380 0.668172 0.334086 0.942543i \(-0.391572\pi\)
0.334086 + 0.942543i \(0.391572\pi\)
\(38\) 0 0
\(39\) −182.935 −0.751103
\(40\) 0 0
\(41\) −252.745 −0.962733 −0.481367 0.876519i \(-0.659859\pi\)
−0.481367 + 0.876519i \(0.659859\pi\)
\(42\) 0 0
\(43\) −214.016 −0.759004 −0.379502 0.925191i \(-0.623905\pi\)
−0.379502 + 0.925191i \(0.623905\pi\)
\(44\) 0 0
\(45\) −175.402 −0.581053
\(46\) 0 0
\(47\) 105.870 0.328567 0.164284 0.986413i \(-0.447469\pi\)
0.164284 + 0.986413i \(0.447469\pi\)
\(48\) 0 0
\(49\) −297.511 −0.867379
\(50\) 0 0
\(51\) 297.505 0.816845
\(52\) 0 0
\(53\) −325.652 −0.843995 −0.421998 0.906597i \(-0.638671\pi\)
−0.421998 + 0.906597i \(0.638671\pi\)
\(54\) 0 0
\(55\) 214.380 0.525583
\(56\) 0 0
\(57\) 74.1032 0.172197
\(58\) 0 0
\(59\) −196.000 −0.432492 −0.216246 0.976339i \(-0.569381\pi\)
−0.216246 + 0.976339i \(0.569381\pi\)
\(60\) 0 0
\(61\) 402.641 0.845130 0.422565 0.906333i \(-0.361130\pi\)
0.422565 + 0.906333i \(0.361130\pi\)
\(62\) 0 0
\(63\) 60.7011 0.121391
\(64\) 0 0
\(65\) −1188.41 −2.26776
\(66\) 0 0
\(67\) −27.4132 −0.0499860 −0.0249930 0.999688i \(-0.507956\pi\)
−0.0249930 + 0.999688i \(0.507956\pi\)
\(68\) 0 0
\(69\) −336.000 −0.586227
\(70\) 0 0
\(71\) −300.500 −0.502292 −0.251146 0.967949i \(-0.580807\pi\)
−0.251146 + 0.967949i \(0.580807\pi\)
\(72\) 0 0
\(73\) 427.815 0.685917 0.342959 0.939351i \(-0.388571\pi\)
0.342959 + 0.939351i \(0.388571\pi\)
\(74\) 0 0
\(75\) −764.478 −1.17699
\(76\) 0 0
\(77\) −74.1902 −0.109802
\(78\) 0 0
\(79\) 97.5488 0.138925 0.0694627 0.997585i \(-0.477872\pi\)
0.0694627 + 0.997585i \(0.477872\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1104.62 −1.46082 −0.730408 0.683011i \(-0.760670\pi\)
−0.730408 + 0.683011i \(0.760670\pi\)
\(84\) 0 0
\(85\) 1932.71 2.46625
\(86\) 0 0
\(87\) −63.3748 −0.0780976
\(88\) 0 0
\(89\) 463.022 0.551463 0.275732 0.961235i \(-0.411080\pi\)
0.275732 + 0.961235i \(0.411080\pi\)
\(90\) 0 0
\(91\) 411.272 0.473769
\(92\) 0 0
\(93\) 956.152 1.06611
\(94\) 0 0
\(95\) 481.402 0.519903
\(96\) 0 0
\(97\) −1798.36 −1.88243 −0.941214 0.337810i \(-0.890314\pi\)
−0.941214 + 0.337810i \(0.890314\pi\)
\(98\) 0 0
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) −741.918 −0.730927 −0.365463 0.930826i \(-0.619089\pi\)
−0.365463 + 0.930826i \(0.619089\pi\)
\(102\) 0 0
\(103\) 389.837 0.372930 0.186465 0.982462i \(-0.440297\pi\)
0.186465 + 0.982462i \(0.440297\pi\)
\(104\) 0 0
\(105\) 394.337 0.366508
\(106\) 0 0
\(107\) −1538.42 −1.38995 −0.694977 0.719032i \(-0.744585\pi\)
−0.694977 + 0.719032i \(0.744585\pi\)
\(108\) 0 0
\(109\) −779.783 −0.685226 −0.342613 0.939477i \(-0.611312\pi\)
−0.342613 + 0.939477i \(0.611312\pi\)
\(110\) 0 0
\(111\) −451.141 −0.385770
\(112\) 0 0
\(113\) −1514.55 −1.26086 −0.630430 0.776246i \(-0.717122\pi\)
−0.630430 + 0.776246i \(0.717122\pi\)
\(114\) 0 0
\(115\) −2182.78 −1.76996
\(116\) 0 0
\(117\) 548.804 0.433649
\(118\) 0 0
\(119\) −668.848 −0.515237
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) 758.234 0.555834
\(124\) 0 0
\(125\) −2530.20 −1.81046
\(126\) 0 0
\(127\) 2302.74 1.60894 0.804471 0.593992i \(-0.202449\pi\)
0.804471 + 0.593992i \(0.202449\pi\)
\(128\) 0 0
\(129\) 642.049 0.438211
\(130\) 0 0
\(131\) 2020.59 1.34763 0.673815 0.738900i \(-0.264655\pi\)
0.673815 + 0.738900i \(0.264655\pi\)
\(132\) 0 0
\(133\) −166.598 −0.108616
\(134\) 0 0
\(135\) 526.206 0.335471
\(136\) 0 0
\(137\) 1475.40 0.920088 0.460044 0.887896i \(-0.347834\pi\)
0.460044 + 0.887896i \(0.347834\pi\)
\(138\) 0 0
\(139\) 1623.71 0.990802 0.495401 0.868665i \(-0.335021\pi\)
0.495401 + 0.868665i \(0.335021\pi\)
\(140\) 0 0
\(141\) −317.609 −0.189698
\(142\) 0 0
\(143\) −670.761 −0.392251
\(144\) 0 0
\(145\) −411.707 −0.235796
\(146\) 0 0
\(147\) 892.533 0.500781
\(148\) 0 0
\(149\) 1104.11 0.607064 0.303532 0.952821i \(-0.401834\pi\)
0.303532 + 0.952821i \(0.401834\pi\)
\(150\) 0 0
\(151\) 2980.07 1.60606 0.803029 0.595940i \(-0.203221\pi\)
0.803029 + 0.595940i \(0.203221\pi\)
\(152\) 0 0
\(153\) −892.516 −0.471605
\(154\) 0 0
\(155\) 6211.52 3.21885
\(156\) 0 0
\(157\) −2844.67 −1.44605 −0.723024 0.690823i \(-0.757249\pi\)
−0.723024 + 0.690823i \(0.757249\pi\)
\(158\) 0 0
\(159\) 976.956 0.487281
\(160\) 0 0
\(161\) 755.391 0.369771
\(162\) 0 0
\(163\) 1528.51 0.734492 0.367246 0.930124i \(-0.380301\pi\)
0.367246 + 0.930124i \(0.380301\pi\)
\(164\) 0 0
\(165\) −643.141 −0.303445
\(166\) 0 0
\(167\) −383.881 −0.177878 −0.0889388 0.996037i \(-0.528348\pi\)
−0.0889388 + 0.996037i \(0.528348\pi\)
\(168\) 0 0
\(169\) 1521.35 0.692466
\(170\) 0 0
\(171\) −222.310 −0.0994178
\(172\) 0 0
\(173\) −2702.85 −1.18783 −0.593914 0.804529i \(-0.702418\pi\)
−0.593914 + 0.804529i \(0.702418\pi\)
\(174\) 0 0
\(175\) 1718.69 0.742404
\(176\) 0 0
\(177\) 588.000 0.249699
\(178\) 0 0
\(179\) 2777.61 1.15982 0.579911 0.814680i \(-0.303087\pi\)
0.579911 + 0.814680i \(0.303087\pi\)
\(180\) 0 0
\(181\) 3993.61 1.64001 0.820007 0.572354i \(-0.193970\pi\)
0.820007 + 0.572354i \(0.193970\pi\)
\(182\) 0 0
\(183\) −1207.92 −0.487936
\(184\) 0 0
\(185\) −2930.78 −1.16473
\(186\) 0 0
\(187\) 1090.85 0.426583
\(188\) 0 0
\(189\) −182.103 −0.0700850
\(190\) 0 0
\(191\) 895.587 0.339280 0.169640 0.985506i \(-0.445740\pi\)
0.169640 + 0.985506i \(0.445740\pi\)
\(192\) 0 0
\(193\) 1328.24 0.495382 0.247691 0.968839i \(-0.420328\pi\)
0.247691 + 0.968839i \(0.420328\pi\)
\(194\) 0 0
\(195\) 3565.24 1.30929
\(196\) 0 0
\(197\) −154.114 −0.0557370 −0.0278685 0.999612i \(-0.508872\pi\)
−0.0278685 + 0.999612i \(0.508872\pi\)
\(198\) 0 0
\(199\) 1316.15 0.468842 0.234421 0.972135i \(-0.424681\pi\)
0.234421 + 0.972135i \(0.424681\pi\)
\(200\) 0 0
\(201\) 82.2397 0.0288594
\(202\) 0 0
\(203\) 142.478 0.0492612
\(204\) 0 0
\(205\) 4925.77 1.67820
\(206\) 0 0
\(207\) 1008.00 0.338458
\(208\) 0 0
\(209\) 271.712 0.0899268
\(210\) 0 0
\(211\) 1735.76 0.566324 0.283162 0.959072i \(-0.408617\pi\)
0.283162 + 0.959072i \(0.408617\pi\)
\(212\) 0 0
\(213\) 901.499 0.289999
\(214\) 0 0
\(215\) 4170.99 1.32307
\(216\) 0 0
\(217\) −2149.61 −0.672465
\(218\) 0 0
\(219\) −1283.44 −0.396014
\(220\) 0 0
\(221\) −6047.12 −1.84060
\(222\) 0 0
\(223\) 1663.71 0.499597 0.249798 0.968298i \(-0.419636\pi\)
0.249798 + 0.968298i \(0.419636\pi\)
\(224\) 0 0
\(225\) 2293.43 0.679536
\(226\) 0 0
\(227\) 3658.84 1.06980 0.534902 0.844914i \(-0.320349\pi\)
0.534902 + 0.844914i \(0.320349\pi\)
\(228\) 0 0
\(229\) −1927.07 −0.556090 −0.278045 0.960568i \(-0.589686\pi\)
−0.278045 + 0.960568i \(0.589686\pi\)
\(230\) 0 0
\(231\) 222.571 0.0633942
\(232\) 0 0
\(233\) −2784.27 −0.782847 −0.391423 0.920211i \(-0.628017\pi\)
−0.391423 + 0.920211i \(0.628017\pi\)
\(234\) 0 0
\(235\) −2063.30 −0.572745
\(236\) 0 0
\(237\) −292.646 −0.0802086
\(238\) 0 0
\(239\) 1222.60 0.330892 0.165446 0.986219i \(-0.447094\pi\)
0.165446 + 0.986219i \(0.447094\pi\)
\(240\) 0 0
\(241\) −2013.01 −0.538047 −0.269024 0.963134i \(-0.586701\pi\)
−0.269024 + 0.963134i \(0.586701\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 5798.23 1.51198
\(246\) 0 0
\(247\) −1506.23 −0.388012
\(248\) 0 0
\(249\) 3313.86 0.843402
\(250\) 0 0
\(251\) −2706.63 −0.680641 −0.340320 0.940310i \(-0.610536\pi\)
−0.340320 + 0.940310i \(0.610536\pi\)
\(252\) 0 0
\(253\) −1232.00 −0.306147
\(254\) 0 0
\(255\) −5798.12 −1.42389
\(256\) 0 0
\(257\) 225.741 0.0547912 0.0273956 0.999625i \(-0.491279\pi\)
0.0273956 + 0.999625i \(0.491279\pi\)
\(258\) 0 0
\(259\) 1014.25 0.243330
\(260\) 0 0
\(261\) 190.124 0.0450897
\(262\) 0 0
\(263\) 1953.42 0.457997 0.228998 0.973427i \(-0.426455\pi\)
0.228998 + 0.973427i \(0.426455\pi\)
\(264\) 0 0
\(265\) 6346.67 1.47122
\(266\) 0 0
\(267\) −1389.07 −0.318387
\(268\) 0 0
\(269\) 350.848 0.0795225 0.0397613 0.999209i \(-0.487340\pi\)
0.0397613 + 0.999209i \(0.487340\pi\)
\(270\) 0 0
\(271\) 254.779 0.0571096 0.0285548 0.999592i \(-0.490909\pi\)
0.0285548 + 0.999592i \(0.490909\pi\)
\(272\) 0 0
\(273\) −1233.81 −0.273531
\(274\) 0 0
\(275\) −2803.09 −0.614663
\(276\) 0 0
\(277\) −116.478 −0.0252654 −0.0126327 0.999920i \(-0.504021\pi\)
−0.0126327 + 0.999920i \(0.504021\pi\)
\(278\) 0 0
\(279\) −2868.46 −0.615519
\(280\) 0 0
\(281\) 8226.47 1.74644 0.873221 0.487325i \(-0.162027\pi\)
0.873221 + 0.487325i \(0.162027\pi\)
\(282\) 0 0
\(283\) −1561.52 −0.327995 −0.163997 0.986461i \(-0.552439\pi\)
−0.163997 + 0.986461i \(0.552439\pi\)
\(284\) 0 0
\(285\) −1444.21 −0.300166
\(286\) 0 0
\(287\) −1704.65 −0.350601
\(288\) 0 0
\(289\) 4921.38 1.00171
\(290\) 0 0
\(291\) 5395.07 1.08682
\(292\) 0 0
\(293\) −9486.19 −1.89143 −0.945715 0.324997i \(-0.894637\pi\)
−0.945715 + 0.324997i \(0.894637\pi\)
\(294\) 0 0
\(295\) 3819.87 0.753903
\(296\) 0 0
\(297\) 297.000 0.0580259
\(298\) 0 0
\(299\) 6829.56 1.32095
\(300\) 0 0
\(301\) −1443.45 −0.276408
\(302\) 0 0
\(303\) 2225.75 0.422001
\(304\) 0 0
\(305\) −7847.13 −1.47320
\(306\) 0 0
\(307\) 8443.06 1.56961 0.784806 0.619742i \(-0.212763\pi\)
0.784806 + 0.619742i \(0.212763\pi\)
\(308\) 0 0
\(309\) −1169.51 −0.215311
\(310\) 0 0
\(311\) 4447.17 0.810856 0.405428 0.914127i \(-0.367123\pi\)
0.405428 + 0.914127i \(0.367123\pi\)
\(312\) 0 0
\(313\) 6480.75 1.17033 0.585165 0.810914i \(-0.301030\pi\)
0.585165 + 0.810914i \(0.301030\pi\)
\(314\) 0 0
\(315\) −1183.01 −0.211603
\(316\) 0 0
\(317\) 1252.75 0.221960 0.110980 0.993823i \(-0.464601\pi\)
0.110980 + 0.993823i \(0.464601\pi\)
\(318\) 0 0
\(319\) −232.374 −0.0407852
\(320\) 0 0
\(321\) 4615.27 0.802490
\(322\) 0 0
\(323\) 2449.57 0.421974
\(324\) 0 0
\(325\) 15538.8 2.65212
\(326\) 0 0
\(327\) 2339.35 0.395615
\(328\) 0 0
\(329\) 714.043 0.119655
\(330\) 0 0
\(331\) −3801.44 −0.631258 −0.315629 0.948883i \(-0.602215\pi\)
−0.315629 + 0.948883i \(0.602215\pi\)
\(332\) 0 0
\(333\) 1353.42 0.222724
\(334\) 0 0
\(335\) 534.260 0.0871336
\(336\) 0 0
\(337\) −5456.53 −0.882007 −0.441003 0.897506i \(-0.645377\pi\)
−0.441003 + 0.897506i \(0.645377\pi\)
\(338\) 0 0
\(339\) 4543.66 0.727958
\(340\) 0 0
\(341\) 3505.89 0.556758
\(342\) 0 0
\(343\) −4319.97 −0.680047
\(344\) 0 0
\(345\) 6548.35 1.02189
\(346\) 0 0
\(347\) 4240.53 0.656033 0.328017 0.944672i \(-0.393620\pi\)
0.328017 + 0.944672i \(0.393620\pi\)
\(348\) 0 0
\(349\) 8471.53 1.29934 0.649672 0.760215i \(-0.274906\pi\)
0.649672 + 0.760215i \(0.274906\pi\)
\(350\) 0 0
\(351\) −1646.41 −0.250368
\(352\) 0 0
\(353\) 981.282 0.147956 0.0739778 0.997260i \(-0.476431\pi\)
0.0739778 + 0.997260i \(0.476431\pi\)
\(354\) 0 0
\(355\) 5856.48 0.875576
\(356\) 0 0
\(357\) 2006.54 0.297472
\(358\) 0 0
\(359\) −3020.09 −0.443995 −0.221997 0.975047i \(-0.571258\pi\)
−0.221997 + 0.975047i \(0.571258\pi\)
\(360\) 0 0
\(361\) −6248.86 −0.911045
\(362\) 0 0
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) −8337.74 −1.19566
\(366\) 0 0
\(367\) 8689.40 1.23592 0.617960 0.786209i \(-0.287959\pi\)
0.617960 + 0.786209i \(0.287959\pi\)
\(368\) 0 0
\(369\) −2274.70 −0.320911
\(370\) 0 0
\(371\) −2196.38 −0.307360
\(372\) 0 0
\(373\) −5340.53 −0.741346 −0.370673 0.928763i \(-0.620873\pi\)
−0.370673 + 0.928763i \(0.620873\pi\)
\(374\) 0 0
\(375\) 7590.59 1.04527
\(376\) 0 0
\(377\) 1288.16 0.175978
\(378\) 0 0
\(379\) 1603.49 0.217324 0.108662 0.994079i \(-0.465343\pi\)
0.108662 + 0.994079i \(0.465343\pi\)
\(380\) 0 0
\(381\) −6908.23 −0.928923
\(382\) 0 0
\(383\) −830.236 −0.110765 −0.0553826 0.998465i \(-0.517638\pi\)
−0.0553826 + 0.998465i \(0.517638\pi\)
\(384\) 0 0
\(385\) 1445.90 0.191403
\(386\) 0 0
\(387\) −1926.15 −0.253001
\(388\) 0 0
\(389\) 1746.12 0.227588 0.113794 0.993504i \(-0.463700\pi\)
0.113794 + 0.993504i \(0.463700\pi\)
\(390\) 0 0
\(391\) −11106.9 −1.43657
\(392\) 0 0
\(393\) −6061.76 −0.778054
\(394\) 0 0
\(395\) −1901.14 −0.242169
\(396\) 0 0
\(397\) 10016.2 1.26624 0.633119 0.774054i \(-0.281774\pi\)
0.633119 + 0.774054i \(0.281774\pi\)
\(398\) 0 0
\(399\) 499.794 0.0627092
\(400\) 0 0
\(401\) 8228.38 1.02470 0.512351 0.858776i \(-0.328775\pi\)
0.512351 + 0.858776i \(0.328775\pi\)
\(402\) 0 0
\(403\) −19434.8 −2.40228
\(404\) 0 0
\(405\) −1578.62 −0.193684
\(406\) 0 0
\(407\) −1654.18 −0.201462
\(408\) 0 0
\(409\) −12311.5 −1.48843 −0.744213 0.667943i \(-0.767175\pi\)
−0.744213 + 0.667943i \(0.767175\pi\)
\(410\) 0 0
\(411\) −4426.21 −0.531213
\(412\) 0 0
\(413\) −1321.93 −0.157502
\(414\) 0 0
\(415\) 21528.1 2.54644
\(416\) 0 0
\(417\) −4871.14 −0.572040
\(418\) 0 0
\(419\) −13260.4 −1.54610 −0.773048 0.634347i \(-0.781269\pi\)
−0.773048 + 0.634347i \(0.781269\pi\)
\(420\) 0 0
\(421\) 6177.74 0.715165 0.357583 0.933881i \(-0.383601\pi\)
0.357583 + 0.933881i \(0.383601\pi\)
\(422\) 0 0
\(423\) 952.826 0.109522
\(424\) 0 0
\(425\) −25270.7 −2.88426
\(426\) 0 0
\(427\) 2715.64 0.307773
\(428\) 0 0
\(429\) 2012.28 0.226466
\(430\) 0 0
\(431\) 1668.05 0.186421 0.0932103 0.995646i \(-0.470287\pi\)
0.0932103 + 0.995646i \(0.470287\pi\)
\(432\) 0 0
\(433\) −731.748 −0.0812138 −0.0406069 0.999175i \(-0.512929\pi\)
−0.0406069 + 0.999175i \(0.512929\pi\)
\(434\) 0 0
\(435\) 1235.12 0.136137
\(436\) 0 0
\(437\) −2766.52 −0.302839
\(438\) 0 0
\(439\) −14248.0 −1.54902 −0.774508 0.632564i \(-0.782002\pi\)
−0.774508 + 0.632564i \(0.782002\pi\)
\(440\) 0 0
\(441\) −2677.60 −0.289126
\(442\) 0 0
\(443\) −174.262 −0.0186895 −0.00934473 0.999956i \(-0.502975\pi\)
−0.00934473 + 0.999956i \(0.502975\pi\)
\(444\) 0 0
\(445\) −9023.89 −0.961288
\(446\) 0 0
\(447\) −3312.34 −0.350488
\(448\) 0 0
\(449\) 7469.97 0.785144 0.392572 0.919721i \(-0.371585\pi\)
0.392572 + 0.919721i \(0.371585\pi\)
\(450\) 0 0
\(451\) 2780.19 0.290275
\(452\) 0 0
\(453\) −8940.21 −0.927258
\(454\) 0 0
\(455\) −8015.32 −0.825855
\(456\) 0 0
\(457\) 8762.68 0.896939 0.448469 0.893798i \(-0.351969\pi\)
0.448469 + 0.893798i \(0.351969\pi\)
\(458\) 0 0
\(459\) 2677.55 0.272282
\(460\) 0 0
\(461\) −4339.67 −0.438435 −0.219218 0.975676i \(-0.570350\pi\)
−0.219218 + 0.975676i \(0.570350\pi\)
\(462\) 0 0
\(463\) −3932.49 −0.394726 −0.197363 0.980330i \(-0.563238\pi\)
−0.197363 + 0.980330i \(0.563238\pi\)
\(464\) 0 0
\(465\) −18634.6 −1.85840
\(466\) 0 0
\(467\) 8383.97 0.830758 0.415379 0.909648i \(-0.363649\pi\)
0.415379 + 0.909648i \(0.363649\pi\)
\(468\) 0 0
\(469\) −184.890 −0.0182035
\(470\) 0 0
\(471\) 8534.02 0.834876
\(472\) 0 0
\(473\) 2354.18 0.228848
\(474\) 0 0
\(475\) −6294.47 −0.608022
\(476\) 0 0
\(477\) −2930.87 −0.281332
\(478\) 0 0
\(479\) 9534.10 0.909445 0.454722 0.890633i \(-0.349739\pi\)
0.454722 + 0.890633i \(0.349739\pi\)
\(480\) 0 0
\(481\) 9169.93 0.869258
\(482\) 0 0
\(483\) −2266.17 −0.213487
\(484\) 0 0
\(485\) 35048.4 3.28138
\(486\) 0 0
\(487\) 4451.09 0.414164 0.207082 0.978324i \(-0.433603\pi\)
0.207082 + 0.978324i \(0.433603\pi\)
\(488\) 0 0
\(489\) −4585.53 −0.424059
\(490\) 0 0
\(491\) 9757.40 0.896833 0.448417 0.893825i \(-0.351988\pi\)
0.448417 + 0.893825i \(0.351988\pi\)
\(492\) 0 0
\(493\) −2094.93 −0.191381
\(494\) 0 0
\(495\) 1929.42 0.175194
\(496\) 0 0
\(497\) −2026.74 −0.182921
\(498\) 0 0
\(499\) 7173.69 0.643564 0.321782 0.946814i \(-0.395718\pi\)
0.321782 + 0.946814i \(0.395718\pi\)
\(500\) 0 0
\(501\) 1151.64 0.102698
\(502\) 0 0
\(503\) −15617.0 −1.38435 −0.692177 0.721728i \(-0.743348\pi\)
−0.692177 + 0.721728i \(0.743348\pi\)
\(504\) 0 0
\(505\) 14459.3 1.27412
\(506\) 0 0
\(507\) −4564.04 −0.399795
\(508\) 0 0
\(509\) 8789.23 0.765375 0.382688 0.923878i \(-0.374999\pi\)
0.382688 + 0.923878i \(0.374999\pi\)
\(510\) 0 0
\(511\) 2885.42 0.249792
\(512\) 0 0
\(513\) 666.929 0.0573989
\(514\) 0 0
\(515\) −7597.58 −0.650077
\(516\) 0 0
\(517\) −1164.56 −0.0990667
\(518\) 0 0
\(519\) 8108.56 0.685792
\(520\) 0 0
\(521\) 13099.3 1.10152 0.550760 0.834664i \(-0.314338\pi\)
0.550760 + 0.834664i \(0.314338\pi\)
\(522\) 0 0
\(523\) −16824.2 −1.40664 −0.703318 0.710876i \(-0.748299\pi\)
−0.703318 + 0.710876i \(0.748299\pi\)
\(524\) 0 0
\(525\) −5156.07 −0.428627
\(526\) 0 0
\(527\) 31606.7 2.61254
\(528\) 0 0
\(529\) 377.000 0.0309855
\(530\) 0 0
\(531\) −1764.00 −0.144164
\(532\) 0 0
\(533\) −15411.9 −1.25247
\(534\) 0 0
\(535\) 29982.5 2.42291
\(536\) 0 0
\(537\) −8332.83 −0.669624
\(538\) 0 0
\(539\) 3272.62 0.261525
\(540\) 0 0
\(541\) −18863.1 −1.49905 −0.749527 0.661974i \(-0.769719\pi\)
−0.749527 + 0.661974i \(0.769719\pi\)
\(542\) 0 0
\(543\) −11980.8 −0.946862
\(544\) 0 0
\(545\) 15197.3 1.19446
\(546\) 0 0
\(547\) 12283.0 0.960119 0.480059 0.877236i \(-0.340615\pi\)
0.480059 + 0.877236i \(0.340615\pi\)
\(548\) 0 0
\(549\) 3623.77 0.281710
\(550\) 0 0
\(551\) −521.809 −0.0403444
\(552\) 0 0
\(553\) 657.924 0.0505927
\(554\) 0 0
\(555\) 8792.35 0.672458
\(556\) 0 0
\(557\) 9752.05 0.741845 0.370923 0.928664i \(-0.379042\pi\)
0.370923 + 0.928664i \(0.379042\pi\)
\(558\) 0 0
\(559\) −13050.3 −0.987424
\(560\) 0 0
\(561\) −3272.56 −0.246288
\(562\) 0 0
\(563\) 3447.50 0.258072 0.129036 0.991640i \(-0.458812\pi\)
0.129036 + 0.991640i \(0.458812\pi\)
\(564\) 0 0
\(565\) 29517.3 2.19788
\(566\) 0 0
\(567\) 546.310 0.0404636
\(568\) 0 0
\(569\) −3371.02 −0.248366 −0.124183 0.992259i \(-0.539631\pi\)
−0.124183 + 0.992259i \(0.539631\pi\)
\(570\) 0 0
\(571\) 15852.5 1.16183 0.580916 0.813964i \(-0.302695\pi\)
0.580916 + 0.813964i \(0.302695\pi\)
\(572\) 0 0
\(573\) −2686.76 −0.195883
\(574\) 0 0
\(575\) 28540.5 2.06995
\(576\) 0 0
\(577\) 1376.35 0.0993036 0.0496518 0.998767i \(-0.484189\pi\)
0.0496518 + 0.998767i \(0.484189\pi\)
\(578\) 0 0
\(579\) −3984.72 −0.286009
\(580\) 0 0
\(581\) −7450.17 −0.531988
\(582\) 0 0
\(583\) 3582.17 0.254474
\(584\) 0 0
\(585\) −10695.7 −0.755920
\(586\) 0 0
\(587\) 23021.4 1.61873 0.809366 0.587304i \(-0.199811\pi\)
0.809366 + 0.587304i \(0.199811\pi\)
\(588\) 0 0
\(589\) 7872.66 0.550742
\(590\) 0 0
\(591\) 462.343 0.0321798
\(592\) 0 0
\(593\) 2818.69 0.195194 0.0975968 0.995226i \(-0.468884\pi\)
0.0975968 + 0.995226i \(0.468884\pi\)
\(594\) 0 0
\(595\) 13035.3 0.898140
\(596\) 0 0
\(597\) −3948.46 −0.270686
\(598\) 0 0
\(599\) −17691.4 −1.20676 −0.603380 0.797454i \(-0.706180\pi\)
−0.603380 + 0.797454i \(0.706180\pi\)
\(600\) 0 0
\(601\) −24516.1 −1.66394 −0.831972 0.554817i \(-0.812788\pi\)
−0.831972 + 0.554817i \(0.812788\pi\)
\(602\) 0 0
\(603\) −246.719 −0.0166620
\(604\) 0 0
\(605\) −2358.18 −0.158469
\(606\) 0 0
\(607\) 4288.59 0.286768 0.143384 0.989667i \(-0.454202\pi\)
0.143384 + 0.989667i \(0.454202\pi\)
\(608\) 0 0
\(609\) −427.435 −0.0284410
\(610\) 0 0
\(611\) 6455.74 0.427449
\(612\) 0 0
\(613\) 8124.07 0.535283 0.267641 0.963519i \(-0.413756\pi\)
0.267641 + 0.963519i \(0.413756\pi\)
\(614\) 0 0
\(615\) −14777.3 −0.968908
\(616\) 0 0
\(617\) −14923.1 −0.973714 −0.486857 0.873482i \(-0.661857\pi\)
−0.486857 + 0.873482i \(0.661857\pi\)
\(618\) 0 0
\(619\) −3627.68 −0.235556 −0.117778 0.993040i \(-0.537577\pi\)
−0.117778 + 0.993040i \(0.537577\pi\)
\(620\) 0 0
\(621\) −3024.00 −0.195409
\(622\) 0 0
\(623\) 3122.88 0.200827
\(624\) 0 0
\(625\) 17458.0 1.11731
\(626\) 0 0
\(627\) −815.135 −0.0519192
\(628\) 0 0
\(629\) −14913.0 −0.945341
\(630\) 0 0
\(631\) 12576.5 0.793445 0.396723 0.917939i \(-0.370147\pi\)
0.396723 + 0.917939i \(0.370147\pi\)
\(632\) 0 0
\(633\) −5207.27 −0.326967
\(634\) 0 0
\(635\) −44878.5 −2.80464
\(636\) 0 0
\(637\) −18141.7 −1.12841
\(638\) 0 0
\(639\) −2704.50 −0.167431
\(640\) 0 0
\(641\) −7292.77 −0.449371 −0.224686 0.974431i \(-0.572136\pi\)
−0.224686 + 0.974431i \(0.572136\pi\)
\(642\) 0 0
\(643\) 12946.3 0.794016 0.397008 0.917815i \(-0.370049\pi\)
0.397008 + 0.917815i \(0.370049\pi\)
\(644\) 0 0
\(645\) −12513.0 −0.763872
\(646\) 0 0
\(647\) 21973.3 1.33518 0.667589 0.744530i \(-0.267327\pi\)
0.667589 + 0.744530i \(0.267327\pi\)
\(648\) 0 0
\(649\) 2156.00 0.130401
\(650\) 0 0
\(651\) 6448.83 0.388248
\(652\) 0 0
\(653\) 27495.6 1.64776 0.823880 0.566764i \(-0.191805\pi\)
0.823880 + 0.566764i \(0.191805\pi\)
\(654\) 0 0
\(655\) −39379.5 −2.34913
\(656\) 0 0
\(657\) 3850.33 0.228639
\(658\) 0 0
\(659\) −5156.28 −0.304796 −0.152398 0.988319i \(-0.548699\pi\)
−0.152398 + 0.988319i \(0.548699\pi\)
\(660\) 0 0
\(661\) −9328.59 −0.548926 −0.274463 0.961598i \(-0.588500\pi\)
−0.274463 + 0.961598i \(0.588500\pi\)
\(662\) 0 0
\(663\) 18141.4 1.06267
\(664\) 0 0
\(665\) 3246.85 0.189334
\(666\) 0 0
\(667\) 2365.99 0.137349
\(668\) 0 0
\(669\) −4991.12 −0.288442
\(670\) 0 0
\(671\) −4429.06 −0.254816
\(672\) 0 0
\(673\) 22182.4 1.27053 0.635267 0.772293i \(-0.280890\pi\)
0.635267 + 0.772293i \(0.280890\pi\)
\(674\) 0 0
\(675\) −6880.30 −0.392330
\(676\) 0 0
\(677\) 13507.1 0.766797 0.383399 0.923583i \(-0.374754\pi\)
0.383399 + 0.923583i \(0.374754\pi\)
\(678\) 0 0
\(679\) −12129.1 −0.685528
\(680\) 0 0
\(681\) −10976.5 −0.617652
\(682\) 0 0
\(683\) 19465.6 1.09053 0.545264 0.838264i \(-0.316429\pi\)
0.545264 + 0.838264i \(0.316429\pi\)
\(684\) 0 0
\(685\) −28754.3 −1.60386
\(686\) 0 0
\(687\) 5781.22 0.321059
\(688\) 0 0
\(689\) −19857.7 −1.09799
\(690\) 0 0
\(691\) 19971.8 1.09951 0.549757 0.835324i \(-0.314720\pi\)
0.549757 + 0.835324i \(0.314720\pi\)
\(692\) 0 0
\(693\) −667.712 −0.0366007
\(694\) 0 0
\(695\) −31644.7 −1.72713
\(696\) 0 0
\(697\) 25064.3 1.36209
\(698\) 0 0
\(699\) 8352.80 0.451977
\(700\) 0 0
\(701\) 14180.1 0.764018 0.382009 0.924159i \(-0.375232\pi\)
0.382009 + 0.924159i \(0.375232\pi\)
\(702\) 0 0
\(703\) −3714.56 −0.199285
\(704\) 0 0
\(705\) 6189.91 0.330675
\(706\) 0 0
\(707\) −5003.91 −0.266183
\(708\) 0 0
\(709\) −15870.6 −0.840667 −0.420334 0.907370i \(-0.638087\pi\)
−0.420334 + 0.907370i \(0.638087\pi\)
\(710\) 0 0
\(711\) 877.939 0.0463084
\(712\) 0 0
\(713\) −35696.3 −1.87495
\(714\) 0 0
\(715\) 13072.5 0.683756
\(716\) 0 0
\(717\) −3667.79 −0.191041
\(718\) 0 0
\(719\) −6040.05 −0.313290 −0.156645 0.987655i \(-0.550068\pi\)
−0.156645 + 0.987655i \(0.550068\pi\)
\(720\) 0 0
\(721\) 2629.28 0.135811
\(722\) 0 0
\(723\) 6039.03 0.310642
\(724\) 0 0
\(725\) 5383.18 0.275761
\(726\) 0 0
\(727\) 37252.3 1.90043 0.950213 0.311601i \(-0.100865\pi\)
0.950213 + 0.311601i \(0.100865\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 21223.7 1.07385
\(732\) 0 0
\(733\) −6125.52 −0.308665 −0.154332 0.988019i \(-0.549323\pi\)
−0.154332 + 0.988019i \(0.549323\pi\)
\(734\) 0 0
\(735\) −17394.7 −0.872942
\(736\) 0 0
\(737\) 301.546 0.0150713
\(738\) 0 0
\(739\) 5225.97 0.260136 0.130068 0.991505i \(-0.458480\pi\)
0.130068 + 0.991505i \(0.458480\pi\)
\(740\) 0 0
\(741\) 4518.68 0.224019
\(742\) 0 0
\(743\) 7062.96 0.348741 0.174371 0.984680i \(-0.444211\pi\)
0.174371 + 0.984680i \(0.444211\pi\)
\(744\) 0 0
\(745\) −21518.2 −1.05821
\(746\) 0 0
\(747\) −9941.57 −0.486939
\(748\) 0 0
\(749\) −10376.0 −0.506182
\(750\) 0 0
\(751\) −20755.4 −1.00849 −0.504244 0.863561i \(-0.668229\pi\)
−0.504244 + 0.863561i \(0.668229\pi\)
\(752\) 0 0
\(753\) 8119.89 0.392968
\(754\) 0 0
\(755\) −58079.0 −2.79962
\(756\) 0 0
\(757\) −31182.9 −1.49717 −0.748587 0.663037i \(-0.769267\pi\)
−0.748587 + 0.663037i \(0.769267\pi\)
\(758\) 0 0
\(759\) 3696.00 0.176754
\(760\) 0 0
\(761\) 32047.8 1.52659 0.763293 0.646052i \(-0.223581\pi\)
0.763293 + 0.646052i \(0.223581\pi\)
\(762\) 0 0
\(763\) −5259.29 −0.249540
\(764\) 0 0
\(765\) 17394.4 0.822084
\(766\) 0 0
\(767\) −11951.7 −0.562650
\(768\) 0 0
\(769\) 2215.88 0.103910 0.0519548 0.998649i \(-0.483455\pi\)
0.0519548 + 0.998649i \(0.483455\pi\)
\(770\) 0 0
\(771\) −677.223 −0.0316337
\(772\) 0 0
\(773\) −5300.56 −0.246634 −0.123317 0.992367i \(-0.539353\pi\)
−0.123317 + 0.992367i \(0.539353\pi\)
\(774\) 0 0
\(775\) −81217.4 −3.76441
\(776\) 0 0
\(777\) −3042.75 −0.140487
\(778\) 0 0
\(779\) 6243.06 0.287138
\(780\) 0 0
\(781\) 3305.50 0.151447
\(782\) 0 0
\(783\) −570.373 −0.0260325
\(784\) 0 0
\(785\) 55440.2 2.52069
\(786\) 0 0
\(787\) −34374.1 −1.55693 −0.778466 0.627687i \(-0.784002\pi\)
−0.778466 + 0.627687i \(0.784002\pi\)
\(788\) 0 0
\(789\) −5860.27 −0.264425
\(790\) 0 0
\(791\) −10215.0 −0.459170
\(792\) 0 0
\(793\) 24552.4 1.09947
\(794\) 0 0
\(795\) −19040.0 −0.849409
\(796\) 0 0
\(797\) −29867.1 −1.32741 −0.663707 0.747993i \(-0.731018\pi\)
−0.663707 + 0.747993i \(0.731018\pi\)
\(798\) 0 0
\(799\) −10498.9 −0.464862
\(800\) 0 0
\(801\) 4167.20 0.183821
\(802\) 0 0
\(803\) −4705.96 −0.206812
\(804\) 0 0
\(805\) −14721.9 −0.644570
\(806\) 0 0
\(807\) −1052.54 −0.0459124
\(808\) 0 0
\(809\) −15857.2 −0.689133 −0.344566 0.938762i \(-0.611974\pi\)
−0.344566 + 0.938762i \(0.611974\pi\)
\(810\) 0 0
\(811\) −36122.6 −1.56404 −0.782020 0.623253i \(-0.785811\pi\)
−0.782020 + 0.623253i \(0.785811\pi\)
\(812\) 0 0
\(813\) −764.337 −0.0329723
\(814\) 0 0
\(815\) −29789.3 −1.28034
\(816\) 0 0
\(817\) 5286.43 0.226375
\(818\) 0 0
\(819\) 3701.44 0.157923
\(820\) 0 0
\(821\) −25779.6 −1.09588 −0.547938 0.836519i \(-0.684587\pi\)
−0.547938 + 0.836519i \(0.684587\pi\)
\(822\) 0 0
\(823\) −22130.2 −0.937315 −0.468658 0.883380i \(-0.655262\pi\)
−0.468658 + 0.883380i \(0.655262\pi\)
\(824\) 0 0
\(825\) 8409.26 0.354876
\(826\) 0 0
\(827\) −18288.6 −0.768994 −0.384497 0.923126i \(-0.625625\pi\)
−0.384497 + 0.923126i \(0.625625\pi\)
\(828\) 0 0
\(829\) 34956.6 1.46453 0.732263 0.681022i \(-0.238464\pi\)
0.732263 + 0.681022i \(0.238464\pi\)
\(830\) 0 0
\(831\) 349.435 0.0145870
\(832\) 0 0
\(833\) 29503.7 1.22718
\(834\) 0 0
\(835\) 7481.50 0.310069
\(836\) 0 0
\(837\) 8605.37 0.355370
\(838\) 0 0
\(839\) −14507.5 −0.596965 −0.298482 0.954415i \(-0.596480\pi\)
−0.298482 + 0.954415i \(0.596480\pi\)
\(840\) 0 0
\(841\) −23942.7 −0.981702
\(842\) 0 0
\(843\) −24679.4 −1.00831
\(844\) 0 0
\(845\) −29649.7 −1.20708
\(846\) 0 0
\(847\) 816.092 0.0331066
\(848\) 0 0
\(849\) 4684.55 0.189368
\(850\) 0 0
\(851\) 16842.6 0.678445
\(852\) 0 0
\(853\) 44146.9 1.77205 0.886027 0.463634i \(-0.153455\pi\)
0.886027 + 0.463634i \(0.153455\pi\)
\(854\) 0 0
\(855\) 4332.62 0.173301
\(856\) 0 0
\(857\) −47679.3 −1.90046 −0.950230 0.311549i \(-0.899152\pi\)
−0.950230 + 0.311549i \(0.899152\pi\)
\(858\) 0 0
\(859\) −7525.09 −0.298897 −0.149449 0.988769i \(-0.547750\pi\)
−0.149449 + 0.988769i \(0.547750\pi\)
\(860\) 0 0
\(861\) 5113.95 0.202419
\(862\) 0 0
\(863\) 45816.1 1.80718 0.903591 0.428396i \(-0.140921\pi\)
0.903591 + 0.428396i \(0.140921\pi\)
\(864\) 0 0
\(865\) 52676.2 2.07057
\(866\) 0 0
\(867\) −14764.1 −0.578335
\(868\) 0 0
\(869\) −1073.04 −0.0418876
\(870\) 0 0
\(871\) −1671.61 −0.0650292
\(872\) 0 0
\(873\) −16185.2 −0.627476
\(874\) 0 0
\(875\) −17065.1 −0.659319
\(876\) 0 0
\(877\) 34168.3 1.31560 0.657801 0.753192i \(-0.271487\pi\)
0.657801 + 0.753192i \(0.271487\pi\)
\(878\) 0 0
\(879\) 28458.6 1.09202
\(880\) 0 0
\(881\) −100.796 −0.00385460 −0.00192730 0.999998i \(-0.500613\pi\)
−0.00192730 + 0.999998i \(0.500613\pi\)
\(882\) 0 0
\(883\) 12346.1 0.470531 0.235266 0.971931i \(-0.424404\pi\)
0.235266 + 0.971931i \(0.424404\pi\)
\(884\) 0 0
\(885\) −11459.6 −0.435266
\(886\) 0 0
\(887\) −37345.1 −1.41367 −0.706834 0.707379i \(-0.749877\pi\)
−0.706834 + 0.707379i \(0.749877\pi\)
\(888\) 0 0
\(889\) 15531.0 0.585932
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) 0 0
\(893\) −2615.09 −0.0979962
\(894\) 0 0
\(895\) −54133.2 −2.02176
\(896\) 0 0
\(897\) −20488.7 −0.762651
\(898\) 0 0
\(899\) −6732.88 −0.249782
\(900\) 0 0
\(901\) 32294.4 1.19410
\(902\) 0 0
\(903\) 4330.34 0.159584
\(904\) 0 0
\(905\) −77831.9 −2.85881
\(906\) 0 0
\(907\) −10308.6 −0.377390 −0.188695 0.982036i \(-0.560426\pi\)
−0.188695 + 0.982036i \(0.560426\pi\)
\(908\) 0 0
\(909\) −6677.26 −0.243642
\(910\) 0 0
\(911\) −22590.8 −0.821589 −0.410795 0.911728i \(-0.634749\pi\)
−0.410795 + 0.911728i \(0.634749\pi\)
\(912\) 0 0
\(913\) 12150.8 0.440453
\(914\) 0 0
\(915\) 23541.4 0.850551
\(916\) 0 0
\(917\) 13628.0 0.490769
\(918\) 0 0
\(919\) 47712.1 1.71260 0.856299 0.516481i \(-0.172758\pi\)
0.856299 + 0.516481i \(0.172758\pi\)
\(920\) 0 0
\(921\) −25329.2 −0.906216
\(922\) 0 0
\(923\) −18323.9 −0.653456
\(924\) 0 0
\(925\) 38320.8 1.36214
\(926\) 0 0
\(927\) 3508.53 0.124310
\(928\) 0 0
\(929\) 10714.3 0.378391 0.189196 0.981939i \(-0.439412\pi\)
0.189196 + 0.981939i \(0.439412\pi\)
\(930\) 0 0
\(931\) 7348.84 0.258699
\(932\) 0 0
\(933\) −13341.5 −0.468148
\(934\) 0 0
\(935\) −21259.8 −0.743603
\(936\) 0 0
\(937\) −14719.5 −0.513197 −0.256599 0.966518i \(-0.582602\pi\)
−0.256599 + 0.966518i \(0.582602\pi\)
\(938\) 0 0
\(939\) −19442.2 −0.675691
\(940\) 0 0
\(941\) −3694.44 −0.127987 −0.0639933 0.997950i \(-0.520384\pi\)
−0.0639933 + 0.997950i \(0.520384\pi\)
\(942\) 0 0
\(943\) −28307.4 −0.977535
\(944\) 0 0
\(945\) 3549.03 0.122169
\(946\) 0 0
\(947\) −36416.7 −1.24961 −0.624806 0.780780i \(-0.714822\pi\)
−0.624806 + 0.780780i \(0.714822\pi\)
\(948\) 0 0
\(949\) 26087.4 0.892342
\(950\) 0 0
\(951\) −3758.25 −0.128149
\(952\) 0 0
\(953\) 20779.4 0.706306 0.353153 0.935566i \(-0.385110\pi\)
0.353153 + 0.935566i \(0.385110\pi\)
\(954\) 0 0
\(955\) −17454.2 −0.591419
\(956\) 0 0
\(957\) 697.123 0.0235473
\(958\) 0 0
\(959\) 9950.94 0.335071
\(960\) 0 0
\(961\) 71789.7 2.40978
\(962\) 0 0
\(963\) −13845.8 −0.463318
\(964\) 0 0
\(965\) −25886.2 −0.863531
\(966\) 0 0
\(967\) 56812.8 1.88932 0.944662 0.328044i \(-0.106389\pi\)
0.944662 + 0.328044i \(0.106389\pi\)
\(968\) 0 0
\(969\) −7348.70 −0.243627
\(970\) 0 0
\(971\) 26459.7 0.874493 0.437247 0.899342i \(-0.355954\pi\)
0.437247 + 0.899342i \(0.355954\pi\)
\(972\) 0 0
\(973\) 10951.2 0.360822
\(974\) 0 0
\(975\) −46616.5 −1.53120
\(976\) 0 0
\(977\) −21009.2 −0.687967 −0.343984 0.938976i \(-0.611776\pi\)
−0.343984 + 0.938976i \(0.611776\pi\)
\(978\) 0 0
\(979\) −5093.24 −0.166272
\(980\) 0 0
\(981\) −7018.04 −0.228409
\(982\) 0 0
\(983\) −9076.80 −0.294512 −0.147256 0.989098i \(-0.547044\pi\)
−0.147256 + 0.989098i \(0.547044\pi\)
\(984\) 0 0
\(985\) 3003.55 0.0971585
\(986\) 0 0
\(987\) −2142.13 −0.0690828
\(988\) 0 0
\(989\) −23969.8 −0.770673
\(990\) 0 0
\(991\) 2629.24 0.0842791 0.0421395 0.999112i \(-0.486583\pi\)
0.0421395 + 0.999112i \(0.486583\pi\)
\(992\) 0 0
\(993\) 11404.3 0.364457
\(994\) 0 0
\(995\) −25650.6 −0.817267
\(996\) 0 0
\(997\) 50423.9 1.60175 0.800874 0.598833i \(-0.204369\pi\)
0.800874 + 0.598833i \(0.204369\pi\)
\(998\) 0 0
\(999\) −4060.27 −0.128590
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.ba.1.1 2
4.3 odd 2 2112.4.a.bh.1.1 2
8.3 odd 2 528.4.a.o.1.2 2
8.5 even 2 33.4.a.d.1.1 2
24.5 odd 2 99.4.a.e.1.2 2
24.11 even 2 1584.4.a.x.1.1 2
40.13 odd 4 825.4.c.i.199.3 4
40.29 even 2 825.4.a.k.1.2 2
40.37 odd 4 825.4.c.i.199.2 4
56.13 odd 2 1617.4.a.j.1.1 2
88.21 odd 2 363.4.a.j.1.2 2
120.29 odd 2 2475.4.a.o.1.1 2
264.197 even 2 1089.4.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.1 2 8.5 even 2
99.4.a.e.1.2 2 24.5 odd 2
363.4.a.j.1.2 2 88.21 odd 2
528.4.a.o.1.2 2 8.3 odd 2
825.4.a.k.1.2 2 40.29 even 2
825.4.c.i.199.2 4 40.37 odd 4
825.4.c.i.199.3 4 40.13 odd 4
1089.4.a.t.1.1 2 264.197 even 2
1584.4.a.x.1.1 2 24.11 even 2
1617.4.a.j.1.1 2 56.13 odd 2
2112.4.a.ba.1.1 2 1.1 even 1 trivial
2112.4.a.bh.1.1 2 4.3 odd 2
2475.4.a.o.1.1 2 120.29 odd 2