Properties

Label 1680.4.a.bg.1.2
Level $1680$
Weight $4$
Character 1680.1
Self dual yes
Analytic conductor $99.123$
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] = [1680,4,Mod(1,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1680.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1680.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: \(99.1232088096\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.53113\) of defining polynomial
Character \(\chi\) \(=\) 1680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +19.0623 q^{11} -2.93774 q^{13} -15.0000 q^{15} -6.49806 q^{17} +5.43580 q^{19} -21.0000 q^{21} -49.3774 q^{23} +25.0000 q^{25} -27.0000 q^{27} -291.494 q^{29} -244.307 q^{31} -57.1868 q^{33} +35.0000 q^{35} -193.121 q^{37} +8.81323 q^{39} +315.113 q^{41} +300.996 q^{43} +45.0000 q^{45} -86.5058 q^{47} +49.0000 q^{49} +19.4942 q^{51} +509.677 q^{53} +95.3113 q^{55} -16.3074 q^{57} +83.3852 q^{59} -5.25291 q^{61} +63.0000 q^{63} -14.6887 q^{65} -205.992 q^{67} +148.132 q^{69} -1004.31 q^{71} -1007.29 q^{73} -75.0000 q^{75} +133.436 q^{77} +863.237 q^{79} +81.0000 q^{81} -1334.72 q^{83} -32.4903 q^{85} +874.483 q^{87} +326.249 q^{89} -20.5642 q^{91} +732.922 q^{93} +27.1790 q^{95} +1526.77 q^{97} +171.560 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 22 q^{11} - 22 q^{13} - 30 q^{15} + 116 q^{17} - 102 q^{19} - 42 q^{21} - 260 q^{23} + 50 q^{25} - 54 q^{27} - 196 q^{29} - 150 q^{31} - 66 q^{33} + 70 q^{35} - 96 q^{37} + 66 q^{39} - 176 q^{41} + 344 q^{43} + 90 q^{45} - 560 q^{47} + 98 q^{49} - 348 q^{51} + 326 q^{53} + 110 q^{55} + 306 q^{57} + 844 q^{59} - 204 q^{61} + 126 q^{63} - 110 q^{65} + 104 q^{67} + 780 q^{69} - 1670 q^{71} - 386 q^{73} - 150 q^{75} + 154 q^{77} + 888 q^{79} + 162 q^{81} - 928 q^{83} + 580 q^{85} + 588 q^{87} + 588 q^{89} - 154 q^{91} + 450 q^{93} - 510 q^{95} + 522 q^{97} + 198 q^{99}+O(q^{100}) \) 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
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 19.0623 0.522499 0.261249 0.965271i \(-0.415865\pi\)
0.261249 + 0.965271i \(0.415865\pi\)
\(12\) 0 0
\(13\) −2.93774 −0.0626756 −0.0313378 0.999509i \(-0.509977\pi\)
−0.0313378 + 0.999509i \(0.509977\pi\)
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) 0 0
\(17\) −6.49806 −0.0927066 −0.0463533 0.998925i \(-0.514760\pi\)
−0.0463533 + 0.998925i \(0.514760\pi\)
\(18\) 0 0
\(19\) 5.43580 0.0656347 0.0328173 0.999461i \(-0.489552\pi\)
0.0328173 + 0.999461i \(0.489552\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) 0 0
\(23\) −49.3774 −0.447648 −0.223824 0.974630i \(-0.571854\pi\)
−0.223824 + 0.974630i \(0.571854\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −291.494 −1.86652 −0.933261 0.359200i \(-0.883050\pi\)
−0.933261 + 0.359200i \(0.883050\pi\)
\(30\) 0 0
\(31\) −244.307 −1.41545 −0.707724 0.706489i \(-0.750278\pi\)
−0.707724 + 0.706489i \(0.750278\pi\)
\(32\) 0 0
\(33\) −57.1868 −0.301665
\(34\) 0 0
\(35\) 35.0000 0.169031
\(36\) 0 0
\(37\) −193.121 −0.858077 −0.429038 0.903286i \(-0.641147\pi\)
−0.429038 + 0.903286i \(0.641147\pi\)
\(38\) 0 0
\(39\) 8.81323 0.0361858
\(40\) 0 0
\(41\) 315.113 1.20030 0.600151 0.799887i \(-0.295107\pi\)
0.600151 + 0.799887i \(0.295107\pi\)
\(42\) 0 0
\(43\) 300.996 1.06748 0.533738 0.845650i \(-0.320787\pi\)
0.533738 + 0.845650i \(0.320787\pi\)
\(44\) 0 0
\(45\) 45.0000 0.149071
\(46\) 0 0
\(47\) −86.5058 −0.268472 −0.134236 0.990949i \(-0.542858\pi\)
−0.134236 + 0.990949i \(0.542858\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 19.4942 0.0535242
\(52\) 0 0
\(53\) 509.677 1.32093 0.660467 0.750855i \(-0.270358\pi\)
0.660467 + 0.750855i \(0.270358\pi\)
\(54\) 0 0
\(55\) 95.3113 0.233669
\(56\) 0 0
\(57\) −16.3074 −0.0378942
\(58\) 0 0
\(59\) 83.3852 0.183997 0.0919985 0.995759i \(-0.470674\pi\)
0.0919985 + 0.995759i \(0.470674\pi\)
\(60\) 0 0
\(61\) −5.25291 −0.0110257 −0.00551283 0.999985i \(-0.501755\pi\)
−0.00551283 + 0.999985i \(0.501755\pi\)
\(62\) 0 0
\(63\) 63.0000 0.125988
\(64\) 0 0
\(65\) −14.6887 −0.0280294
\(66\) 0 0
\(67\) −205.992 −0.375611 −0.187806 0.982206i \(-0.560138\pi\)
−0.187806 + 0.982206i \(0.560138\pi\)
\(68\) 0 0
\(69\) 148.132 0.258450
\(70\) 0 0
\(71\) −1004.31 −1.67872 −0.839362 0.543573i \(-0.817071\pi\)
−0.839362 + 0.543573i \(0.817071\pi\)
\(72\) 0 0
\(73\) −1007.29 −1.61499 −0.807494 0.589876i \(-0.799177\pi\)
−0.807494 + 0.589876i \(0.799177\pi\)
\(74\) 0 0
\(75\) −75.0000 −0.115470
\(76\) 0 0
\(77\) 133.436 0.197486
\(78\) 0 0
\(79\) 863.237 1.22939 0.614695 0.788765i \(-0.289279\pi\)
0.614695 + 0.788765i \(0.289279\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1334.72 −1.76512 −0.882560 0.470200i \(-0.844182\pi\)
−0.882560 + 0.470200i \(0.844182\pi\)
\(84\) 0 0
\(85\) −32.4903 −0.0414596
\(86\) 0 0
\(87\) 874.483 1.07764
\(88\) 0 0
\(89\) 326.249 0.388565 0.194283 0.980946i \(-0.437762\pi\)
0.194283 + 0.980946i \(0.437762\pi\)
\(90\) 0 0
\(91\) −20.5642 −0.0236892
\(92\) 0 0
\(93\) 732.922 0.817210
\(94\) 0 0
\(95\) 27.1790 0.0293527
\(96\) 0 0
\(97\) 1526.77 1.59815 0.799075 0.601232i \(-0.205323\pi\)
0.799075 + 0.601232i \(0.205323\pi\)
\(98\) 0 0
\(99\) 171.560 0.174166
\(100\) 0 0
\(101\) 96.8716 0.0954365 0.0477182 0.998861i \(-0.484805\pi\)
0.0477182 + 0.998861i \(0.484805\pi\)
\(102\) 0 0
\(103\) −1321.99 −1.26466 −0.632329 0.774700i \(-0.717901\pi\)
−0.632329 + 0.774700i \(0.717901\pi\)
\(104\) 0 0
\(105\) −105.000 −0.0975900
\(106\) 0 0
\(107\) −1745.71 −1.57724 −0.788619 0.614883i \(-0.789203\pi\)
−0.788619 + 0.614883i \(0.789203\pi\)
\(108\) 0 0
\(109\) 476.856 0.419032 0.209516 0.977805i \(-0.432811\pi\)
0.209516 + 0.977805i \(0.432811\pi\)
\(110\) 0 0
\(111\) 579.362 0.495411
\(112\) 0 0
\(113\) 1641.65 1.36666 0.683332 0.730108i \(-0.260530\pi\)
0.683332 + 0.730108i \(0.260530\pi\)
\(114\) 0 0
\(115\) −246.887 −0.200194
\(116\) 0 0
\(117\) −26.4397 −0.0208919
\(118\) 0 0
\(119\) −45.4864 −0.0350398
\(120\) 0 0
\(121\) −967.630 −0.726995
\(122\) 0 0
\(123\) −945.339 −0.692994
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 844.016 0.589719 0.294859 0.955541i \(-0.404727\pi\)
0.294859 + 0.955541i \(0.404727\pi\)
\(128\) 0 0
\(129\) −902.988 −0.616308
\(130\) 0 0
\(131\) 2796.20 1.86493 0.932463 0.361265i \(-0.117655\pi\)
0.932463 + 0.361265i \(0.117655\pi\)
\(132\) 0 0
\(133\) 38.0506 0.0248076
\(134\) 0 0
\(135\) −135.000 −0.0860663
\(136\) 0 0
\(137\) −2057.13 −1.28287 −0.641433 0.767179i \(-0.721660\pi\)
−0.641433 + 0.767179i \(0.721660\pi\)
\(138\) 0 0
\(139\) 1745.12 1.06489 0.532444 0.846465i \(-0.321274\pi\)
0.532444 + 0.846465i \(0.321274\pi\)
\(140\) 0 0
\(141\) 259.517 0.155002
\(142\) 0 0
\(143\) −56.0000 −0.0327479
\(144\) 0 0
\(145\) −1457.47 −0.834734
\(146\) 0 0
\(147\) −147.000 −0.0824786
\(148\) 0 0
\(149\) −1173.57 −0.645254 −0.322627 0.946526i \(-0.604566\pi\)
−0.322627 + 0.946526i \(0.604566\pi\)
\(150\) 0 0
\(151\) −1540.07 −0.829994 −0.414997 0.909823i \(-0.636217\pi\)
−0.414997 + 0.909823i \(0.636217\pi\)
\(152\) 0 0
\(153\) −58.4826 −0.0309022
\(154\) 0 0
\(155\) −1221.54 −0.633008
\(156\) 0 0
\(157\) −2544.53 −1.29348 −0.646738 0.762712i \(-0.723867\pi\)
−0.646738 + 0.762712i \(0.723867\pi\)
\(158\) 0 0
\(159\) −1529.03 −0.762642
\(160\) 0 0
\(161\) −345.642 −0.169195
\(162\) 0 0
\(163\) 594.708 0.285774 0.142887 0.989739i \(-0.454361\pi\)
0.142887 + 0.989739i \(0.454361\pi\)
\(164\) 0 0
\(165\) −285.934 −0.134909
\(166\) 0 0
\(167\) −928.498 −0.430236 −0.215118 0.976588i \(-0.569014\pi\)
−0.215118 + 0.976588i \(0.569014\pi\)
\(168\) 0 0
\(169\) −2188.37 −0.996072
\(170\) 0 0
\(171\) 48.9222 0.0218782
\(172\) 0 0
\(173\) −315.642 −0.138716 −0.0693578 0.997592i \(-0.522095\pi\)
−0.0693578 + 0.997592i \(0.522095\pi\)
\(174\) 0 0
\(175\) 175.000 0.0755929
\(176\) 0 0
\(177\) −250.156 −0.106231
\(178\) 0 0
\(179\) 1445.49 0.603581 0.301791 0.953374i \(-0.402416\pi\)
0.301791 + 0.953374i \(0.402416\pi\)
\(180\) 0 0
\(181\) −1843.81 −0.757180 −0.378590 0.925564i \(-0.623591\pi\)
−0.378590 + 0.925564i \(0.623591\pi\)
\(182\) 0 0
\(183\) 15.7587 0.00636567
\(184\) 0 0
\(185\) −965.603 −0.383744
\(186\) 0 0
\(187\) −123.868 −0.0484391
\(188\) 0 0
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) 244.074 0.0924637 0.0462318 0.998931i \(-0.485279\pi\)
0.0462318 + 0.998931i \(0.485279\pi\)
\(192\) 0 0
\(193\) −1733.03 −0.646355 −0.323178 0.946338i \(-0.604751\pi\)
−0.323178 + 0.946338i \(0.604751\pi\)
\(194\) 0 0
\(195\) 44.0661 0.0161828
\(196\) 0 0
\(197\) 358.230 0.129557 0.0647787 0.997900i \(-0.479366\pi\)
0.0647787 + 0.997900i \(0.479366\pi\)
\(198\) 0 0
\(199\) 3203.63 1.14120 0.570601 0.821227i \(-0.306710\pi\)
0.570601 + 0.821227i \(0.306710\pi\)
\(200\) 0 0
\(201\) 617.977 0.216859
\(202\) 0 0
\(203\) −2040.46 −0.705479
\(204\) 0 0
\(205\) 1575.56 0.536791
\(206\) 0 0
\(207\) −444.397 −0.149216
\(208\) 0 0
\(209\) 103.619 0.0342940
\(210\) 0 0
\(211\) −4943.16 −1.61280 −0.806401 0.591369i \(-0.798588\pi\)
−0.806401 + 0.591369i \(0.798588\pi\)
\(212\) 0 0
\(213\) 3012.92 0.969211
\(214\) 0 0
\(215\) 1504.98 0.477390
\(216\) 0 0
\(217\) −1710.15 −0.534989
\(218\) 0 0
\(219\) 3021.86 0.932414
\(220\) 0 0
\(221\) 19.0896 0.00581044
\(222\) 0 0
\(223\) −3160.15 −0.948965 −0.474482 0.880265i \(-0.657365\pi\)
−0.474482 + 0.880265i \(0.657365\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) 3651.11 1.06755 0.533773 0.845628i \(-0.320774\pi\)
0.533773 + 0.845628i \(0.320774\pi\)
\(228\) 0 0
\(229\) −4083.70 −1.17842 −0.589210 0.807980i \(-0.700561\pi\)
−0.589210 + 0.807980i \(0.700561\pi\)
\(230\) 0 0
\(231\) −400.307 −0.114019
\(232\) 0 0
\(233\) −3682.51 −1.03540 −0.517702 0.855561i \(-0.673212\pi\)
−0.517702 + 0.855561i \(0.673212\pi\)
\(234\) 0 0
\(235\) −432.529 −0.120064
\(236\) 0 0
\(237\) −2589.71 −0.709789
\(238\) 0 0
\(239\) −2658.78 −0.719591 −0.359796 0.933031i \(-0.617154\pi\)
−0.359796 + 0.933031i \(0.617154\pi\)
\(240\) 0 0
\(241\) −4820.39 −1.28842 −0.644209 0.764850i \(-0.722813\pi\)
−0.644209 + 0.764850i \(0.722813\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 245.000 0.0638877
\(246\) 0 0
\(247\) −15.9690 −0.00411369
\(248\) 0 0
\(249\) 4004.17 1.01909
\(250\) 0 0
\(251\) −1672.27 −0.420530 −0.210265 0.977644i \(-0.567433\pi\)
−0.210265 + 0.977644i \(0.567433\pi\)
\(252\) 0 0
\(253\) −941.245 −0.233896
\(254\) 0 0
\(255\) 97.4709 0.0239367
\(256\) 0 0
\(257\) −3697.74 −0.897506 −0.448753 0.893656i \(-0.648132\pi\)
−0.448753 + 0.893656i \(0.648132\pi\)
\(258\) 0 0
\(259\) −1351.84 −0.324323
\(260\) 0 0
\(261\) −2623.45 −0.622174
\(262\) 0 0
\(263\) −7319.00 −1.71600 −0.858002 0.513646i \(-0.828294\pi\)
−0.858002 + 0.513646i \(0.828294\pi\)
\(264\) 0 0
\(265\) 2548.39 0.590740
\(266\) 0 0
\(267\) −978.747 −0.224338
\(268\) 0 0
\(269\) −815.097 −0.184749 −0.0923743 0.995724i \(-0.529446\pi\)
−0.0923743 + 0.995724i \(0.529446\pi\)
\(270\) 0 0
\(271\) 5106.02 1.14453 0.572267 0.820068i \(-0.306064\pi\)
0.572267 + 0.820068i \(0.306064\pi\)
\(272\) 0 0
\(273\) 61.6926 0.0136769
\(274\) 0 0
\(275\) 476.556 0.104500
\(276\) 0 0
\(277\) 1398.72 0.303398 0.151699 0.988427i \(-0.451526\pi\)
0.151699 + 0.988427i \(0.451526\pi\)
\(278\) 0 0
\(279\) −2198.77 −0.471816
\(280\) 0 0
\(281\) −7102.38 −1.50780 −0.753901 0.656988i \(-0.771830\pi\)
−0.753901 + 0.656988i \(0.771830\pi\)
\(282\) 0 0
\(283\) −4465.18 −0.937907 −0.468953 0.883223i \(-0.655369\pi\)
−0.468953 + 0.883223i \(0.655369\pi\)
\(284\) 0 0
\(285\) −81.5371 −0.0169468
\(286\) 0 0
\(287\) 2205.79 0.453671
\(288\) 0 0
\(289\) −4870.78 −0.991405
\(290\) 0 0
\(291\) −4580.32 −0.922692
\(292\) 0 0
\(293\) 7590.61 1.51348 0.756738 0.653718i \(-0.226792\pi\)
0.756738 + 0.653718i \(0.226792\pi\)
\(294\) 0 0
\(295\) 416.926 0.0822860
\(296\) 0 0
\(297\) −514.681 −0.100555
\(298\) 0 0
\(299\) 145.058 0.0280566
\(300\) 0 0
\(301\) 2106.97 0.403468
\(302\) 0 0
\(303\) −290.615 −0.0551003
\(304\) 0 0
\(305\) −26.2645 −0.00493083
\(306\) 0 0
\(307\) −9480.12 −1.76241 −0.881203 0.472737i \(-0.843266\pi\)
−0.881203 + 0.472737i \(0.843266\pi\)
\(308\) 0 0
\(309\) 3965.98 0.730151
\(310\) 0 0
\(311\) −7078.01 −1.29054 −0.645268 0.763956i \(-0.723254\pi\)
−0.645268 + 0.763956i \(0.723254\pi\)
\(312\) 0 0
\(313\) 5593.84 1.01017 0.505084 0.863070i \(-0.331461\pi\)
0.505084 + 0.863070i \(0.331461\pi\)
\(314\) 0 0
\(315\) 315.000 0.0563436
\(316\) 0 0
\(317\) 3567.81 0.632139 0.316070 0.948736i \(-0.397637\pi\)
0.316070 + 0.948736i \(0.397637\pi\)
\(318\) 0 0
\(319\) −5556.54 −0.975255
\(320\) 0 0
\(321\) 5237.14 0.910618
\(322\) 0 0
\(323\) −35.3222 −0.00608477
\(324\) 0 0
\(325\) −73.4436 −0.0125351
\(326\) 0 0
\(327\) −1430.57 −0.241928
\(328\) 0 0
\(329\) −605.541 −0.101473
\(330\) 0 0
\(331\) 4389.67 0.728936 0.364468 0.931216i \(-0.381251\pi\)
0.364468 + 0.931216i \(0.381251\pi\)
\(332\) 0 0
\(333\) −1738.09 −0.286026
\(334\) 0 0
\(335\) −1029.96 −0.167978
\(336\) 0 0
\(337\) 2348.83 0.379671 0.189835 0.981816i \(-0.439205\pi\)
0.189835 + 0.981816i \(0.439205\pi\)
\(338\) 0 0
\(339\) −4924.94 −0.789044
\(340\) 0 0
\(341\) −4657.05 −0.739570
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) 740.661 0.115582
\(346\) 0 0
\(347\) −558.436 −0.0863931 −0.0431965 0.999067i \(-0.513754\pi\)
−0.0431965 + 0.999067i \(0.513754\pi\)
\(348\) 0 0
\(349\) 3233.89 0.496006 0.248003 0.968759i \(-0.420226\pi\)
0.248003 + 0.968759i \(0.420226\pi\)
\(350\) 0 0
\(351\) 79.3190 0.0120619
\(352\) 0 0
\(353\) −7516.35 −1.13330 −0.566650 0.823959i \(-0.691761\pi\)
−0.566650 + 0.823959i \(0.691761\pi\)
\(354\) 0 0
\(355\) −5021.54 −0.750748
\(356\) 0 0
\(357\) 136.459 0.0202302
\(358\) 0 0
\(359\) 6577.76 0.967021 0.483511 0.875338i \(-0.339361\pi\)
0.483511 + 0.875338i \(0.339361\pi\)
\(360\) 0 0
\(361\) −6829.45 −0.995692
\(362\) 0 0
\(363\) 2902.89 0.419731
\(364\) 0 0
\(365\) −5036.44 −0.722245
\(366\) 0 0
\(367\) −8307.17 −1.18155 −0.590777 0.806835i \(-0.701179\pi\)
−0.590777 + 0.806835i \(0.701179\pi\)
\(368\) 0 0
\(369\) 2836.02 0.400101
\(370\) 0 0
\(371\) 3567.74 0.499266
\(372\) 0 0
\(373\) −4551.09 −0.631760 −0.315880 0.948799i \(-0.602300\pi\)
−0.315880 + 0.948799i \(0.602300\pi\)
\(374\) 0 0
\(375\) −375.000 −0.0516398
\(376\) 0 0
\(377\) 856.335 0.116985
\(378\) 0 0
\(379\) 1788.29 0.242370 0.121185 0.992630i \(-0.461331\pi\)
0.121185 + 0.992630i \(0.461331\pi\)
\(380\) 0 0
\(381\) −2532.05 −0.340474
\(382\) 0 0
\(383\) 1358.47 0.181240 0.0906199 0.995886i \(-0.471115\pi\)
0.0906199 + 0.995886i \(0.471115\pi\)
\(384\) 0 0
\(385\) 667.179 0.0883184
\(386\) 0 0
\(387\) 2708.97 0.355825
\(388\) 0 0
\(389\) 9722.54 1.26723 0.633615 0.773649i \(-0.281570\pi\)
0.633615 + 0.773649i \(0.281570\pi\)
\(390\) 0 0
\(391\) 320.858 0.0414999
\(392\) 0 0
\(393\) −8388.61 −1.07672
\(394\) 0 0
\(395\) 4316.19 0.549800
\(396\) 0 0
\(397\) −4788.04 −0.605302 −0.302651 0.953101i \(-0.597872\pi\)
−0.302651 + 0.953101i \(0.597872\pi\)
\(398\) 0 0
\(399\) −114.152 −0.0143227
\(400\) 0 0
\(401\) 9681.41 1.20565 0.602826 0.797873i \(-0.294041\pi\)
0.602826 + 0.797873i \(0.294041\pi\)
\(402\) 0 0
\(403\) 717.712 0.0887141
\(404\) 0 0
\(405\) 405.000 0.0496904
\(406\) 0 0
\(407\) −3681.32 −0.448344
\(408\) 0 0
\(409\) 11113.1 1.34353 0.671767 0.740763i \(-0.265536\pi\)
0.671767 + 0.740763i \(0.265536\pi\)
\(410\) 0 0
\(411\) 6171.40 0.740663
\(412\) 0 0
\(413\) 583.696 0.0695443
\(414\) 0 0
\(415\) −6673.62 −0.789386
\(416\) 0 0
\(417\) −5235.37 −0.614814
\(418\) 0 0
\(419\) 1230.09 0.143421 0.0717107 0.997425i \(-0.477154\pi\)
0.0717107 + 0.997425i \(0.477154\pi\)
\(420\) 0 0
\(421\) −12356.5 −1.43044 −0.715222 0.698897i \(-0.753674\pi\)
−0.715222 + 0.698897i \(0.753674\pi\)
\(422\) 0 0
\(423\) −778.552 −0.0894906
\(424\) 0 0
\(425\) −162.452 −0.0185413
\(426\) 0 0
\(427\) −36.7703 −0.00416731
\(428\) 0 0
\(429\) 168.000 0.0189070
\(430\) 0 0
\(431\) 7375.27 0.824256 0.412128 0.911126i \(-0.364786\pi\)
0.412128 + 0.911126i \(0.364786\pi\)
\(432\) 0 0
\(433\) −690.067 −0.0765877 −0.0382939 0.999267i \(-0.512192\pi\)
−0.0382939 + 0.999267i \(0.512192\pi\)
\(434\) 0 0
\(435\) 4372.41 0.481934
\(436\) 0 0
\(437\) −268.406 −0.0293812
\(438\) 0 0
\(439\) −8408.79 −0.914191 −0.457095 0.889418i \(-0.651110\pi\)
−0.457095 + 0.889418i \(0.651110\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) −6568.55 −0.704473 −0.352236 0.935911i \(-0.614579\pi\)
−0.352236 + 0.935911i \(0.614579\pi\)
\(444\) 0 0
\(445\) 1631.25 0.173772
\(446\) 0 0
\(447\) 3520.72 0.372537
\(448\) 0 0
\(449\) 2954.55 0.310543 0.155271 0.987872i \(-0.450375\pi\)
0.155271 + 0.987872i \(0.450375\pi\)
\(450\) 0 0
\(451\) 6006.76 0.627156
\(452\) 0 0
\(453\) 4620.21 0.479197
\(454\) 0 0
\(455\) −102.821 −0.0105941
\(456\) 0 0
\(457\) −8144.84 −0.833697 −0.416849 0.908976i \(-0.636865\pi\)
−0.416849 + 0.908976i \(0.636865\pi\)
\(458\) 0 0
\(459\) 175.448 0.0178414
\(460\) 0 0
\(461\) 2495.26 0.252095 0.126048 0.992024i \(-0.459771\pi\)
0.126048 + 0.992024i \(0.459771\pi\)
\(462\) 0 0
\(463\) 5755.66 0.577728 0.288864 0.957370i \(-0.406722\pi\)
0.288864 + 0.957370i \(0.406722\pi\)
\(464\) 0 0
\(465\) 3664.61 0.365467
\(466\) 0 0
\(467\) 4143.73 0.410598 0.205299 0.978699i \(-0.434183\pi\)
0.205299 + 0.978699i \(0.434183\pi\)
\(468\) 0 0
\(469\) −1441.95 −0.141968
\(470\) 0 0
\(471\) 7633.60 0.746789
\(472\) 0 0
\(473\) 5737.67 0.557755
\(474\) 0 0
\(475\) 135.895 0.0131269
\(476\) 0 0
\(477\) 4587.09 0.440312
\(478\) 0 0
\(479\) 6765.96 0.645396 0.322698 0.946502i \(-0.395410\pi\)
0.322698 + 0.946502i \(0.395410\pi\)
\(480\) 0 0
\(481\) 567.339 0.0537805
\(482\) 0 0
\(483\) 1036.93 0.0976848
\(484\) 0 0
\(485\) 7633.87 0.714714
\(486\) 0 0
\(487\) −6360.42 −0.591824 −0.295912 0.955215i \(-0.595623\pi\)
−0.295912 + 0.955215i \(0.595623\pi\)
\(488\) 0 0
\(489\) −1784.12 −0.164992
\(490\) 0 0
\(491\) 7072.54 0.650060 0.325030 0.945704i \(-0.394626\pi\)
0.325030 + 0.945704i \(0.394626\pi\)
\(492\) 0 0
\(493\) 1894.15 0.173039
\(494\) 0 0
\(495\) 857.802 0.0778895
\(496\) 0 0
\(497\) −7030.15 −0.634498
\(498\) 0 0
\(499\) 18473.9 1.65732 0.828661 0.559751i \(-0.189103\pi\)
0.828661 + 0.559751i \(0.189103\pi\)
\(500\) 0 0
\(501\) 2785.49 0.248397
\(502\) 0 0
\(503\) −11379.2 −1.00869 −0.504347 0.863501i \(-0.668267\pi\)
−0.504347 + 0.863501i \(0.668267\pi\)
\(504\) 0 0
\(505\) 484.358 0.0426805
\(506\) 0 0
\(507\) 6565.11 0.575082
\(508\) 0 0
\(509\) 6064.48 0.528101 0.264051 0.964509i \(-0.414941\pi\)
0.264051 + 0.964509i \(0.414941\pi\)
\(510\) 0 0
\(511\) −7051.02 −0.610408
\(512\) 0 0
\(513\) −146.767 −0.0126314
\(514\) 0 0
\(515\) −6609.96 −0.565572
\(516\) 0 0
\(517\) −1649.00 −0.140276
\(518\) 0 0
\(519\) 946.926 0.0800875
\(520\) 0 0
\(521\) 2682.88 0.225603 0.112801 0.993618i \(-0.464018\pi\)
0.112801 + 0.993618i \(0.464018\pi\)
\(522\) 0 0
\(523\) 4309.02 0.360268 0.180134 0.983642i \(-0.442347\pi\)
0.180134 + 0.983642i \(0.442347\pi\)
\(524\) 0 0
\(525\) −525.000 −0.0436436
\(526\) 0 0
\(527\) 1587.52 0.131221
\(528\) 0 0
\(529\) −9728.87 −0.799611
\(530\) 0 0
\(531\) 750.467 0.0613323
\(532\) 0 0
\(533\) −925.720 −0.0752296
\(534\) 0 0
\(535\) −8728.56 −0.705362
\(536\) 0 0
\(537\) −4336.47 −0.348478
\(538\) 0 0
\(539\) 934.051 0.0746427
\(540\) 0 0
\(541\) 4081.47 0.324355 0.162178 0.986762i \(-0.448148\pi\)
0.162178 + 0.986762i \(0.448148\pi\)
\(542\) 0 0
\(543\) 5531.44 0.437158
\(544\) 0 0
\(545\) 2384.28 0.187397
\(546\) 0 0
\(547\) −8844.82 −0.691366 −0.345683 0.938351i \(-0.612353\pi\)
−0.345683 + 0.938351i \(0.612353\pi\)
\(548\) 0 0
\(549\) −47.2762 −0.00367522
\(550\) 0 0
\(551\) −1584.51 −0.122509
\(552\) 0 0
\(553\) 6042.66 0.464666
\(554\) 0 0
\(555\) 2896.81 0.221554
\(556\) 0 0
\(557\) −11144.7 −0.847787 −0.423894 0.905712i \(-0.639337\pi\)
−0.423894 + 0.905712i \(0.639337\pi\)
\(558\) 0 0
\(559\) −884.249 −0.0669047
\(560\) 0 0
\(561\) 371.603 0.0279663
\(562\) 0 0
\(563\) 21857.5 1.63621 0.818104 0.575071i \(-0.195025\pi\)
0.818104 + 0.575071i \(0.195025\pi\)
\(564\) 0 0
\(565\) 8208.23 0.611191
\(566\) 0 0
\(567\) 567.000 0.0419961
\(568\) 0 0
\(569\) 23496.4 1.73115 0.865573 0.500783i \(-0.166954\pi\)
0.865573 + 0.500783i \(0.166954\pi\)
\(570\) 0 0
\(571\) −11067.8 −0.811164 −0.405582 0.914059i \(-0.632931\pi\)
−0.405582 + 0.914059i \(0.632931\pi\)
\(572\) 0 0
\(573\) −732.222 −0.0533839
\(574\) 0 0
\(575\) −1234.44 −0.0895296
\(576\) 0 0
\(577\) 20482.9 1.47784 0.738922 0.673791i \(-0.235335\pi\)
0.738922 + 0.673791i \(0.235335\pi\)
\(578\) 0 0
\(579\) 5199.10 0.373173
\(580\) 0 0
\(581\) −9343.07 −0.667153
\(582\) 0 0
\(583\) 9715.60 0.690187
\(584\) 0 0
\(585\) −132.198 −0.00934313
\(586\) 0 0
\(587\) 23444.3 1.64847 0.824235 0.566248i \(-0.191606\pi\)
0.824235 + 0.566248i \(0.191606\pi\)
\(588\) 0 0
\(589\) −1328.01 −0.0929025
\(590\) 0 0
\(591\) −1074.69 −0.0748000
\(592\) 0 0
\(593\) 4404.69 0.305024 0.152512 0.988302i \(-0.451264\pi\)
0.152512 + 0.988302i \(0.451264\pi\)
\(594\) 0 0
\(595\) −227.432 −0.0156703
\(596\) 0 0
\(597\) −9610.89 −0.658874
\(598\) 0 0
\(599\) −3327.05 −0.226945 −0.113472 0.993541i \(-0.536197\pi\)
−0.113472 + 0.993541i \(0.536197\pi\)
\(600\) 0 0
\(601\) −14244.8 −0.966818 −0.483409 0.875395i \(-0.660602\pi\)
−0.483409 + 0.875395i \(0.660602\pi\)
\(602\) 0 0
\(603\) −1853.93 −0.125204
\(604\) 0 0
\(605\) −4838.15 −0.325122
\(606\) 0 0
\(607\) −11446.5 −0.765402 −0.382701 0.923872i \(-0.625006\pi\)
−0.382701 + 0.923872i \(0.625006\pi\)
\(608\) 0 0
\(609\) 6121.38 0.407308
\(610\) 0 0
\(611\) 254.132 0.0168266
\(612\) 0 0
\(613\) −19436.4 −1.28063 −0.640316 0.768111i \(-0.721197\pi\)
−0.640316 + 0.768111i \(0.721197\pi\)
\(614\) 0 0
\(615\) −4726.69 −0.309917
\(616\) 0 0
\(617\) −20530.1 −1.33956 −0.669781 0.742558i \(-0.733612\pi\)
−0.669781 + 0.742558i \(0.733612\pi\)
\(618\) 0 0
\(619\) −5833.35 −0.378776 −0.189388 0.981902i \(-0.560650\pi\)
−0.189388 + 0.981902i \(0.560650\pi\)
\(620\) 0 0
\(621\) 1333.19 0.0861499
\(622\) 0 0
\(623\) 2283.74 0.146864
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −310.856 −0.0197997
\(628\) 0 0
\(629\) 1254.91 0.0795493
\(630\) 0 0
\(631\) −24776.6 −1.56314 −0.781568 0.623820i \(-0.785580\pi\)
−0.781568 + 0.623820i \(0.785580\pi\)
\(632\) 0 0
\(633\) 14829.5 0.931152
\(634\) 0 0
\(635\) 4220.08 0.263730
\(636\) 0 0
\(637\) −143.949 −0.00895366
\(638\) 0 0
\(639\) −9038.77 −0.559574
\(640\) 0 0
\(641\) 27219.4 1.67723 0.838613 0.544728i \(-0.183367\pi\)
0.838613 + 0.544728i \(0.183367\pi\)
\(642\) 0 0
\(643\) −7091.79 −0.434950 −0.217475 0.976066i \(-0.569782\pi\)
−0.217475 + 0.976066i \(0.569782\pi\)
\(644\) 0 0
\(645\) −4514.94 −0.275621
\(646\) 0 0
\(647\) −27773.0 −1.68758 −0.843792 0.536670i \(-0.819682\pi\)
−0.843792 + 0.536670i \(0.819682\pi\)
\(648\) 0 0
\(649\) 1589.51 0.0961382
\(650\) 0 0
\(651\) 5130.46 0.308876
\(652\) 0 0
\(653\) 21380.4 1.28129 0.640643 0.767839i \(-0.278668\pi\)
0.640643 + 0.767839i \(0.278668\pi\)
\(654\) 0 0
\(655\) 13981.0 0.834020
\(656\) 0 0
\(657\) −9065.59 −0.538329
\(658\) 0 0
\(659\) 17232.3 1.01863 0.509315 0.860580i \(-0.329899\pi\)
0.509315 + 0.860580i \(0.329899\pi\)
\(660\) 0 0
\(661\) 26577.7 1.56392 0.781962 0.623326i \(-0.214219\pi\)
0.781962 + 0.623326i \(0.214219\pi\)
\(662\) 0 0
\(663\) −57.2689 −0.00335466
\(664\) 0 0
\(665\) 190.253 0.0110943
\(666\) 0 0
\(667\) 14393.2 0.835544
\(668\) 0 0
\(669\) 9480.44 0.547885
\(670\) 0 0
\(671\) −100.132 −0.00576090
\(672\) 0 0
\(673\) −31695.2 −1.81540 −0.907698 0.419624i \(-0.862162\pi\)
−0.907698 + 0.419624i \(0.862162\pi\)
\(674\) 0 0
\(675\) −675.000 −0.0384900
\(676\) 0 0
\(677\) 20440.3 1.16039 0.580195 0.814477i \(-0.302976\pi\)
0.580195 + 0.814477i \(0.302976\pi\)
\(678\) 0 0
\(679\) 10687.4 0.604044
\(680\) 0 0
\(681\) −10953.3 −0.616348
\(682\) 0 0
\(683\) 22896.9 1.28276 0.641381 0.767223i \(-0.278362\pi\)
0.641381 + 0.767223i \(0.278362\pi\)
\(684\) 0 0
\(685\) −10285.7 −0.573715
\(686\) 0 0
\(687\) 12251.1 0.680361
\(688\) 0 0
\(689\) −1497.30 −0.0827904
\(690\) 0 0
\(691\) 23764.0 1.30829 0.654143 0.756371i \(-0.273029\pi\)
0.654143 + 0.756371i \(0.273029\pi\)
\(692\) 0 0
\(693\) 1200.92 0.0658287
\(694\) 0 0
\(695\) 8725.62 0.476233
\(696\) 0 0
\(697\) −2047.62 −0.111276
\(698\) 0 0
\(699\) 11047.5 0.597791
\(700\) 0 0
\(701\) 26259.5 1.41485 0.707423 0.706791i \(-0.249858\pi\)
0.707423 + 0.706791i \(0.249858\pi\)
\(702\) 0 0
\(703\) −1049.77 −0.0563196
\(704\) 0 0
\(705\) 1297.59 0.0693191
\(706\) 0 0
\(707\) 678.101 0.0360716
\(708\) 0 0
\(709\) 12783.0 0.677116 0.338558 0.940945i \(-0.390061\pi\)
0.338558 + 0.940945i \(0.390061\pi\)
\(710\) 0 0
\(711\) 7769.14 0.409797
\(712\) 0 0
\(713\) 12063.3 0.633623
\(714\) 0 0
\(715\) −280.000 −0.0146453
\(716\) 0 0
\(717\) 7976.35 0.415456
\(718\) 0 0
\(719\) 27609.0 1.43205 0.716025 0.698075i \(-0.245960\pi\)
0.716025 + 0.698075i \(0.245960\pi\)
\(720\) 0 0
\(721\) −9253.95 −0.477996
\(722\) 0 0
\(723\) 14461.2 0.743868
\(724\) 0 0
\(725\) −7287.35 −0.373304
\(726\) 0 0
\(727\) −31306.2 −1.59709 −0.798544 0.601937i \(-0.794396\pi\)
−0.798544 + 0.601937i \(0.794396\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −1955.89 −0.0989621
\(732\) 0 0
\(733\) −15765.8 −0.794441 −0.397220 0.917723i \(-0.630025\pi\)
−0.397220 + 0.917723i \(0.630025\pi\)
\(734\) 0 0
\(735\) −735.000 −0.0368856
\(736\) 0 0
\(737\) −3926.68 −0.196256
\(738\) 0 0
\(739\) −3966.51 −0.197443 −0.0987216 0.995115i \(-0.531475\pi\)
−0.0987216 + 0.995115i \(0.531475\pi\)
\(740\) 0 0
\(741\) 47.9070 0.00237504
\(742\) 0 0
\(743\) −8224.50 −0.406094 −0.203047 0.979169i \(-0.565084\pi\)
−0.203047 + 0.979169i \(0.565084\pi\)
\(744\) 0 0
\(745\) −5867.86 −0.288566
\(746\) 0 0
\(747\) −12012.5 −0.588373
\(748\) 0 0
\(749\) −12220.0 −0.596140
\(750\) 0 0
\(751\) −18929.2 −0.919754 −0.459877 0.887983i \(-0.652106\pi\)
−0.459877 + 0.887983i \(0.652106\pi\)
\(752\) 0 0
\(753\) 5016.82 0.242793
\(754\) 0 0
\(755\) −7700.35 −0.371185
\(756\) 0 0
\(757\) −34906.8 −1.67597 −0.837984 0.545695i \(-0.816266\pi\)
−0.837984 + 0.545695i \(0.816266\pi\)
\(758\) 0 0
\(759\) 2823.74 0.135040
\(760\) 0 0
\(761\) −13683.4 −0.651803 −0.325902 0.945404i \(-0.605668\pi\)
−0.325902 + 0.945404i \(0.605668\pi\)
\(762\) 0 0
\(763\) 3337.99 0.158379
\(764\) 0 0
\(765\) −292.413 −0.0138199
\(766\) 0 0
\(767\) −244.964 −0.0115321
\(768\) 0 0
\(769\) 41837.3 1.96189 0.980943 0.194294i \(-0.0622416\pi\)
0.980943 + 0.194294i \(0.0622416\pi\)
\(770\) 0 0
\(771\) 11093.2 0.518175
\(772\) 0 0
\(773\) 19640.0 0.913843 0.456921 0.889507i \(-0.348952\pi\)
0.456921 + 0.889507i \(0.348952\pi\)
\(774\) 0 0
\(775\) −6107.69 −0.283090
\(776\) 0 0
\(777\) 4055.53 0.187248
\(778\) 0 0
\(779\) 1712.89 0.0787814
\(780\) 0 0
\(781\) −19144.4 −0.877131
\(782\) 0 0
\(783\) 7870.34 0.359212
\(784\) 0 0
\(785\) −12722.7 −0.578460
\(786\) 0 0
\(787\) −24935.3 −1.12941 −0.564705 0.825293i \(-0.691010\pi\)
−0.564705 + 0.825293i \(0.691010\pi\)
\(788\) 0 0
\(789\) 21957.0 0.990736
\(790\) 0 0
\(791\) 11491.5 0.516551
\(792\) 0 0
\(793\) 15.4317 0.000691041 0
\(794\) 0 0
\(795\) −7645.16 −0.341064
\(796\) 0 0
\(797\) −1168.33 −0.0519251 −0.0259625 0.999663i \(-0.508265\pi\)
−0.0259625 + 0.999663i \(0.508265\pi\)
\(798\) 0 0
\(799\) 562.120 0.0248891
\(800\) 0 0
\(801\) 2936.24 0.129522
\(802\) 0 0
\(803\) −19201.2 −0.843829
\(804\) 0 0
\(805\) −1728.21 −0.0756663
\(806\) 0 0
\(807\) 2445.29 0.106665
\(808\) 0 0
\(809\) −35175.7 −1.52869 −0.764345 0.644807i \(-0.776938\pi\)
−0.764345 + 0.644807i \(0.776938\pi\)
\(810\) 0 0
\(811\) 15256.5 0.660577 0.330288 0.943880i \(-0.392854\pi\)
0.330288 + 0.943880i \(0.392854\pi\)
\(812\) 0 0
\(813\) −15318.1 −0.660797
\(814\) 0 0
\(815\) 2973.54 0.127802
\(816\) 0 0
\(817\) 1636.16 0.0700635
\(818\) 0 0
\(819\) −185.078 −0.00789639
\(820\) 0 0
\(821\) −15971.9 −0.678956 −0.339478 0.940614i \(-0.610250\pi\)
−0.339478 + 0.940614i \(0.610250\pi\)
\(822\) 0 0
\(823\) −2312.41 −0.0979409 −0.0489705 0.998800i \(-0.515594\pi\)
−0.0489705 + 0.998800i \(0.515594\pi\)
\(824\) 0 0
\(825\) −1429.67 −0.0603330
\(826\) 0 0
\(827\) −10422.4 −0.438238 −0.219119 0.975698i \(-0.570318\pi\)
−0.219119 + 0.975698i \(0.570318\pi\)
\(828\) 0 0
\(829\) −13213.4 −0.553584 −0.276792 0.960930i \(-0.589271\pi\)
−0.276792 + 0.960930i \(0.589271\pi\)
\(830\) 0 0
\(831\) −4196.17 −0.175167
\(832\) 0 0
\(833\) −318.405 −0.0132438
\(834\) 0 0
\(835\) −4642.49 −0.192407
\(836\) 0 0
\(837\) 6596.30 0.272403
\(838\) 0 0
\(839\) 10119.6 0.416409 0.208205 0.978085i \(-0.433238\pi\)
0.208205 + 0.978085i \(0.433238\pi\)
\(840\) 0 0
\(841\) 60579.9 2.48390
\(842\) 0 0
\(843\) 21307.1 0.870530
\(844\) 0 0
\(845\) −10941.8 −0.445457
\(846\) 0 0
\(847\) −6773.41 −0.274778
\(848\) 0 0
\(849\) 13395.5 0.541501
\(850\) 0 0
\(851\) 9535.80 0.384116
\(852\) 0 0
\(853\) 35378.1 1.42007 0.710037 0.704165i \(-0.248678\pi\)
0.710037 + 0.704165i \(0.248678\pi\)
\(854\) 0 0
\(855\) 244.611 0.00978424
\(856\) 0 0
\(857\) 6697.57 0.266960 0.133480 0.991052i \(-0.457385\pi\)
0.133480 + 0.991052i \(0.457385\pi\)
\(858\) 0 0
\(859\) 24298.4 0.965135 0.482568 0.875859i \(-0.339704\pi\)
0.482568 + 0.875859i \(0.339704\pi\)
\(860\) 0 0
\(861\) −6617.37 −0.261927
\(862\) 0 0
\(863\) −24942.9 −0.983853 −0.491926 0.870637i \(-0.663707\pi\)
−0.491926 + 0.870637i \(0.663707\pi\)
\(864\) 0 0
\(865\) −1578.21 −0.0620355
\(866\) 0 0
\(867\) 14612.3 0.572388
\(868\) 0 0
\(869\) 16455.3 0.642355
\(870\) 0 0
\(871\) 605.152 0.0235417
\(872\) 0 0
\(873\) 13741.0 0.532716
\(874\) 0 0
\(875\) 875.000 0.0338062
\(876\) 0 0
\(877\) −16276.6 −0.626705 −0.313353 0.949637i \(-0.601452\pi\)
−0.313353 + 0.949637i \(0.601452\pi\)
\(878\) 0 0
\(879\) −22771.8 −0.873806
\(880\) 0 0
\(881\) 26636.5 1.01862 0.509311 0.860582i \(-0.329900\pi\)
0.509311 + 0.860582i \(0.329900\pi\)
\(882\) 0 0
\(883\) 21788.3 0.830392 0.415196 0.909732i \(-0.363713\pi\)
0.415196 + 0.909732i \(0.363713\pi\)
\(884\) 0 0
\(885\) −1250.78 −0.0475078
\(886\) 0 0
\(887\) 26813.2 1.01499 0.507496 0.861654i \(-0.330571\pi\)
0.507496 + 0.861654i \(0.330571\pi\)
\(888\) 0 0
\(889\) 5908.11 0.222893
\(890\) 0 0
\(891\) 1544.04 0.0580554
\(892\) 0 0
\(893\) −470.229 −0.0176211
\(894\) 0 0
\(895\) 7227.45 0.269930
\(896\) 0 0
\(897\) −435.174 −0.0161985
\(898\) 0 0
\(899\) 71214.2 2.64196
\(900\) 0 0
\(901\) −3311.91 −0.122459
\(902\) 0 0
\(903\) −6320.92 −0.232942
\(904\) 0 0
\(905\) −9219.07 −0.338621
\(906\) 0 0
\(907\) −15543.0 −0.569014 −0.284507 0.958674i \(-0.591830\pi\)
−0.284507 + 0.958674i \(0.591830\pi\)
\(908\) 0 0
\(909\) 871.844 0.0318122
\(910\) 0 0
\(911\) −48711.1 −1.77154 −0.885768 0.464128i \(-0.846368\pi\)
−0.885768 + 0.464128i \(0.846368\pi\)
\(912\) 0 0
\(913\) −25442.8 −0.922273
\(914\) 0 0
\(915\) 78.7936 0.00284682
\(916\) 0 0
\(917\) 19573.4 0.704876
\(918\) 0 0
\(919\) −1030.47 −0.0369883 −0.0184941 0.999829i \(-0.505887\pi\)
−0.0184941 + 0.999829i \(0.505887\pi\)
\(920\) 0 0
\(921\) 28440.4 1.01753
\(922\) 0 0
\(923\) 2950.40 0.105215
\(924\) 0 0
\(925\) −4828.02 −0.171615
\(926\) 0 0
\(927\) −11897.9 −0.421553
\(928\) 0 0
\(929\) 879.756 0.0310698 0.0155349 0.999879i \(-0.495055\pi\)
0.0155349 + 0.999879i \(0.495055\pi\)
\(930\) 0 0
\(931\) 266.354 0.00937638
\(932\) 0 0
\(933\) 21234.0 0.745092
\(934\) 0 0
\(935\) −619.339 −0.0216626
\(936\) 0 0
\(937\) −18668.1 −0.650864 −0.325432 0.945565i \(-0.605510\pi\)
−0.325432 + 0.945565i \(0.605510\pi\)
\(938\) 0 0
\(939\) −16781.5 −0.583221
\(940\) 0 0
\(941\) 29613.4 1.02590 0.512948 0.858420i \(-0.328553\pi\)
0.512948 + 0.858420i \(0.328553\pi\)
\(942\) 0 0
\(943\) −15559.5 −0.537313
\(944\) 0 0
\(945\) −945.000 −0.0325300
\(946\) 0 0
\(947\) −20738.9 −0.711640 −0.355820 0.934554i \(-0.615798\pi\)
−0.355820 + 0.934554i \(0.615798\pi\)
\(948\) 0 0
\(949\) 2959.15 0.101220
\(950\) 0 0
\(951\) −10703.4 −0.364966
\(952\) 0 0
\(953\) −45776.5 −1.55598 −0.777988 0.628279i \(-0.783760\pi\)
−0.777988 + 0.628279i \(0.783760\pi\)
\(954\) 0 0
\(955\) 1220.37 0.0413510
\(956\) 0 0
\(957\) 16669.6 0.563064
\(958\) 0 0
\(959\) −14399.9 −0.484878
\(960\) 0 0
\(961\) 29895.1 1.00349
\(962\) 0 0
\(963\) −15711.4 −0.525746
\(964\) 0 0
\(965\) −8665.17 −0.289059
\(966\) 0 0
\(967\) −34461.0 −1.14601 −0.573005 0.819552i \(-0.694222\pi\)
−0.573005 + 0.819552i \(0.694222\pi\)
\(968\) 0 0
\(969\) 105.967 0.00351304
\(970\) 0 0
\(971\) 22762.8 0.752309 0.376154 0.926557i \(-0.377246\pi\)
0.376154 + 0.926557i \(0.377246\pi\)
\(972\) 0 0
\(973\) 12215.9 0.402490
\(974\) 0 0
\(975\) 220.331 0.00723716
\(976\) 0 0
\(977\) 4809.57 0.157494 0.0787470 0.996895i \(-0.474908\pi\)
0.0787470 + 0.996895i \(0.474908\pi\)
\(978\) 0 0
\(979\) 6219.04 0.203025
\(980\) 0 0
\(981\) 4291.70 0.139677
\(982\) 0 0
\(983\) 27591.6 0.895256 0.447628 0.894220i \(-0.352269\pi\)
0.447628 + 0.894220i \(0.352269\pi\)
\(984\) 0 0
\(985\) 1791.15 0.0579399
\(986\) 0 0
\(987\) 1816.62 0.0585853
\(988\) 0 0
\(989\) −14862.4 −0.477854
\(990\) 0 0
\(991\) 22263.4 0.713643 0.356822 0.934173i \(-0.383860\pi\)
0.356822 + 0.934173i \(0.383860\pi\)
\(992\) 0 0
\(993\) −13169.0 −0.420851
\(994\) 0 0
\(995\) 16018.2 0.510361
\(996\) 0 0
\(997\) 30378.2 0.964983 0.482491 0.875901i \(-0.339732\pi\)
0.482491 + 0.875901i \(0.339732\pi\)
\(998\) 0 0
\(999\) 5214.26 0.165137
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 1680.4.a.bg.1.2 2
4.3 odd 2 105.4.a.f.1.2 2
12.11 even 2 315.4.a.i.1.1 2
20.3 even 4 525.4.d.h.274.1 4
20.7 even 4 525.4.d.h.274.4 4
20.19 odd 2 525.4.a.k.1.1 2
28.27 even 2 735.4.a.p.1.2 2
60.59 even 2 1575.4.a.w.1.2 2
84.83 odd 2 2205.4.a.z.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.f.1.2 2 4.3 odd 2
315.4.a.i.1.1 2 12.11 even 2
525.4.a.k.1.1 2 20.19 odd 2
525.4.d.h.274.1 4 20.3 even 4
525.4.d.h.274.4 4 20.7 even 4
735.4.a.p.1.2 2 28.27 even 2
1575.4.a.w.1.2 2 60.59 even 2
1680.4.a.bg.1.2 2 1.1 even 1 trivial
2205.4.a.z.1.1 2 84.83 odd 2