Properties

Label 2205.4.a.u.1.1
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
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] = [2205,4,Mod(1,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.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: \(130.099211563\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.41421 q^{2} +21.3137 q^{4} -5.00000 q^{5} -72.0833 q^{8} +O(q^{10})\) \(q-5.41421 q^{2} +21.3137 q^{4} -5.00000 q^{5} -72.0833 q^{8} +27.0711 q^{10} +52.2548 q^{11} -30.6569 q^{13} +219.765 q^{16} +37.2254 q^{17} -80.2254 q^{19} -106.569 q^{20} -282.919 q^{22} -25.8335 q^{23} +25.0000 q^{25} +165.983 q^{26} -20.9411 q^{29} +314.558 q^{31} -613.186 q^{32} -201.546 q^{34} +197.147 q^{37} +434.357 q^{38} +360.416 q^{40} +11.3625 q^{41} -33.8335 q^{43} +1113.74 q^{44} +139.868 q^{46} -361.676 q^{47} -135.355 q^{50} -653.411 q^{52} -153.019 q^{53} -261.274 q^{55} +113.380 q^{58} -616.000 q^{59} -15.2649 q^{61} -1703.09 q^{62} +1561.80 q^{64} +153.284 q^{65} -166.510 q^{67} +793.411 q^{68} +952.000 q^{71} +148.489 q^{73} -1067.40 q^{74} -1709.90 q^{76} +857.725 q^{79} -1098.82 q^{80} -61.5189 q^{82} +660.528 q^{83} -186.127 q^{85} +183.182 q^{86} -3766.70 q^{88} -45.7746 q^{89} -550.607 q^{92} +1958.19 q^{94} +401.127 q^{95} -1682.13 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 20 q^{4} - 10 q^{5} - 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} + 20 q^{4} - 10 q^{5} - 48 q^{8} + 40 q^{10} + 14 q^{11} - 50 q^{13} + 168 q^{16} - 50 q^{17} - 36 q^{19} - 100 q^{20} - 184 q^{22} - 244 q^{23} + 50 q^{25} + 216 q^{26} + 26 q^{29} + 120 q^{31} - 672 q^{32} + 24 q^{34} + 564 q^{37} + 320 q^{38} + 240 q^{40} - 328 q^{41} - 260 q^{43} + 1164 q^{44} + 704 q^{46} - 350 q^{47} - 200 q^{50} - 628 q^{52} + 56 q^{53} - 70 q^{55} - 8 q^{58} - 1232 q^{59} - 336 q^{61} - 1200 q^{62} + 2128 q^{64} + 250 q^{65} - 152 q^{67} + 908 q^{68} + 1904 q^{71} - 676 q^{73} - 2016 q^{74} - 1768 q^{76} + 1014 q^{79} - 840 q^{80} + 816 q^{82} - 376 q^{83} + 250 q^{85} + 768 q^{86} - 4688 q^{88} - 216 q^{89} - 264 q^{92} + 1928 q^{94} + 180 q^{95} - 2742 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.41421 −1.91421 −0.957107 0.289735i \(-0.906433\pi\)
−0.957107 + 0.289735i \(0.906433\pi\)
\(3\) 0 0
\(4\) 21.3137 2.66421
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) −72.0833 −3.18566
\(9\) 0 0
\(10\) 27.0711 0.856062
\(11\) 52.2548 1.43231 0.716156 0.697941i \(-0.245900\pi\)
0.716156 + 0.697941i \(0.245900\pi\)
\(12\) 0 0
\(13\) −30.6569 −0.654052 −0.327026 0.945015i \(-0.606047\pi\)
−0.327026 + 0.945015i \(0.606047\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 219.765 3.43382
\(17\) 37.2254 0.531087 0.265544 0.964099i \(-0.414449\pi\)
0.265544 + 0.964099i \(0.414449\pi\)
\(18\) 0 0
\(19\) −80.2254 −0.968683 −0.484341 0.874879i \(-0.660941\pi\)
−0.484341 + 0.874879i \(0.660941\pi\)
\(20\) −106.569 −1.19147
\(21\) 0 0
\(22\) −282.919 −2.74175
\(23\) −25.8335 −0.234202 −0.117101 0.993120i \(-0.537360\pi\)
−0.117101 + 0.993120i \(0.537360\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 165.983 1.25200
\(27\) 0 0
\(28\) 0 0
\(29\) −20.9411 −0.134092 −0.0670460 0.997750i \(-0.521357\pi\)
−0.0670460 + 0.997750i \(0.521357\pi\)
\(30\) 0 0
\(31\) 314.558 1.82246 0.911232 0.411894i \(-0.135133\pi\)
0.911232 + 0.411894i \(0.135133\pi\)
\(32\) −613.186 −3.38741
\(33\) 0 0
\(34\) −201.546 −1.01661
\(35\) 0 0
\(36\) 0 0
\(37\) 197.147 0.875968 0.437984 0.898983i \(-0.355693\pi\)
0.437984 + 0.898983i \(0.355693\pi\)
\(38\) 434.357 1.85427
\(39\) 0 0
\(40\) 360.416 1.42467
\(41\) 11.3625 0.0432810 0.0216405 0.999766i \(-0.493111\pi\)
0.0216405 + 0.999766i \(0.493111\pi\)
\(42\) 0 0
\(43\) −33.8335 −0.119990 −0.0599948 0.998199i \(-0.519108\pi\)
−0.0599948 + 0.998199i \(0.519108\pi\)
\(44\) 1113.74 3.81598
\(45\) 0 0
\(46\) 139.868 0.448313
\(47\) −361.676 −1.12247 −0.561233 0.827658i \(-0.689673\pi\)
−0.561233 + 0.827658i \(0.689673\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −135.355 −0.382843
\(51\) 0 0
\(52\) −653.411 −1.74254
\(53\) −153.019 −0.396582 −0.198291 0.980143i \(-0.563539\pi\)
−0.198291 + 0.980143i \(0.563539\pi\)
\(54\) 0 0
\(55\) −261.274 −0.640549
\(56\) 0 0
\(57\) 0 0
\(58\) 113.380 0.256681
\(59\) −616.000 −1.35926 −0.679630 0.733555i \(-0.737860\pi\)
−0.679630 + 0.733555i \(0.737860\pi\)
\(60\) 0 0
\(61\) −15.2649 −0.0320406 −0.0160203 0.999872i \(-0.505100\pi\)
−0.0160203 + 0.999872i \(0.505100\pi\)
\(62\) −1703.09 −3.48858
\(63\) 0 0
\(64\) 1561.80 3.05040
\(65\) 153.284 0.292501
\(66\) 0 0
\(67\) −166.510 −0.303618 −0.151809 0.988410i \(-0.548510\pi\)
−0.151809 + 0.988410i \(0.548510\pi\)
\(68\) 793.411 1.41493
\(69\) 0 0
\(70\) 0 0
\(71\) 952.000 1.59129 0.795645 0.605763i \(-0.207132\pi\)
0.795645 + 0.605763i \(0.207132\pi\)
\(72\) 0 0
\(73\) 148.489 0.238074 0.119037 0.992890i \(-0.462019\pi\)
0.119037 + 0.992890i \(0.462019\pi\)
\(74\) −1067.40 −1.67679
\(75\) 0 0
\(76\) −1709.90 −2.58078
\(77\) 0 0
\(78\) 0 0
\(79\) 857.725 1.22154 0.610770 0.791808i \(-0.290860\pi\)
0.610770 + 0.791808i \(0.290860\pi\)
\(80\) −1098.82 −1.53565
\(81\) 0 0
\(82\) −61.5189 −0.0828491
\(83\) 660.528 0.873523 0.436761 0.899577i \(-0.356125\pi\)
0.436761 + 0.899577i \(0.356125\pi\)
\(84\) 0 0
\(85\) −186.127 −0.237509
\(86\) 183.182 0.229686
\(87\) 0 0
\(88\) −3766.70 −4.56286
\(89\) −45.7746 −0.0545180 −0.0272590 0.999628i \(-0.508678\pi\)
−0.0272590 + 0.999628i \(0.508678\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −550.607 −0.623965
\(93\) 0 0
\(94\) 1958.19 2.14864
\(95\) 401.127 0.433208
\(96\) 0 0
\(97\) −1682.13 −1.76076 −0.880382 0.474265i \(-0.842714\pi\)
−0.880382 + 0.474265i \(0.842714\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 532.843 0.532843
\(101\) −434.167 −0.427734 −0.213867 0.976863i \(-0.568606\pi\)
−0.213867 + 0.976863i \(0.568606\pi\)
\(102\) 0 0
\(103\) −345.577 −0.330589 −0.165295 0.986244i \(-0.552858\pi\)
−0.165295 + 0.986244i \(0.552858\pi\)
\(104\) 2209.85 2.08359
\(105\) 0 0
\(106\) 828.479 0.759142
\(107\) −217.119 −0.196165 −0.0980825 0.995178i \(-0.531271\pi\)
−0.0980825 + 0.995178i \(0.531271\pi\)
\(108\) 0 0
\(109\) 1734.41 1.52409 0.762047 0.647521i \(-0.224194\pi\)
0.762047 + 0.647521i \(0.224194\pi\)
\(110\) 1414.59 1.22615
\(111\) 0 0
\(112\) 0 0
\(113\) 1854.20 1.54362 0.771809 0.635855i \(-0.219352\pi\)
0.771809 + 0.635855i \(0.219352\pi\)
\(114\) 0 0
\(115\) 129.167 0.104738
\(116\) −446.333 −0.357250
\(117\) 0 0
\(118\) 3335.16 2.60191
\(119\) 0 0
\(120\) 0 0
\(121\) 1399.57 1.05152
\(122\) 82.6476 0.0613325
\(123\) 0 0
\(124\) 6704.41 4.85543
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1394.51 0.974352 0.487176 0.873304i \(-0.338027\pi\)
0.487176 + 0.873304i \(0.338027\pi\)
\(128\) −3550.45 −2.45171
\(129\) 0 0
\(130\) −829.914 −0.559910
\(131\) 1762.42 1.17544 0.587722 0.809063i \(-0.300025\pi\)
0.587722 + 0.809063i \(0.300025\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 901.519 0.581189
\(135\) 0 0
\(136\) −2683.33 −1.69186
\(137\) 922.949 0.575568 0.287784 0.957695i \(-0.407081\pi\)
0.287784 + 0.957695i \(0.407081\pi\)
\(138\) 0 0
\(139\) 196.039 0.119624 0.0598122 0.998210i \(-0.480950\pi\)
0.0598122 + 0.998210i \(0.480950\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −5154.33 −3.04607
\(143\) −1601.97 −0.936807
\(144\) 0 0
\(145\) 104.706 0.0599678
\(146\) −803.954 −0.455724
\(147\) 0 0
\(148\) 4201.94 2.33376
\(149\) −780.372 −0.429064 −0.214532 0.976717i \(-0.568823\pi\)
−0.214532 + 0.976717i \(0.568823\pi\)
\(150\) 0 0
\(151\) −2319.43 −1.25002 −0.625008 0.780618i \(-0.714904\pi\)
−0.625008 + 0.780618i \(0.714904\pi\)
\(152\) 5782.91 3.08589
\(153\) 0 0
\(154\) 0 0
\(155\) −1572.79 −0.815030
\(156\) 0 0
\(157\) −1022.90 −0.519977 −0.259989 0.965612i \(-0.583719\pi\)
−0.259989 + 0.965612i \(0.583719\pi\)
\(158\) −4643.91 −2.33829
\(159\) 0 0
\(160\) 3065.93 1.51489
\(161\) 0 0
\(162\) 0 0
\(163\) −1350.63 −0.649013 −0.324507 0.945883i \(-0.605198\pi\)
−0.324507 + 0.945883i \(0.605198\pi\)
\(164\) 242.177 0.115310
\(165\) 0 0
\(166\) −3576.24 −1.67211
\(167\) −1230.58 −0.570209 −0.285105 0.958496i \(-0.592028\pi\)
−0.285105 + 0.958496i \(0.592028\pi\)
\(168\) 0 0
\(169\) −1257.16 −0.572215
\(170\) 1007.73 0.454644
\(171\) 0 0
\(172\) −721.117 −0.319678
\(173\) −2487.65 −1.09325 −0.546626 0.837377i \(-0.684088\pi\)
−0.546626 + 0.837377i \(0.684088\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 11483.8 4.91830
\(177\) 0 0
\(178\) 247.833 0.104359
\(179\) −1621.18 −0.676941 −0.338471 0.940977i \(-0.609910\pi\)
−0.338471 + 0.940977i \(0.609910\pi\)
\(180\) 0 0
\(181\) −2593.69 −1.06512 −0.532561 0.846392i \(-0.678770\pi\)
−0.532561 + 0.846392i \(0.678770\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1862.16 0.746089
\(185\) −985.736 −0.391745
\(186\) 0 0
\(187\) 1945.21 0.760682
\(188\) −7708.66 −2.99049
\(189\) 0 0
\(190\) −2171.79 −0.829253
\(191\) 1823.08 0.690645 0.345323 0.938484i \(-0.387769\pi\)
0.345323 + 0.938484i \(0.387769\pi\)
\(192\) 0 0
\(193\) −1541.03 −0.574744 −0.287372 0.957819i \(-0.592782\pi\)
−0.287372 + 0.957819i \(0.592782\pi\)
\(194\) 9107.39 3.37048
\(195\) 0 0
\(196\) 0 0
\(197\) −701.243 −0.253612 −0.126806 0.991928i \(-0.540473\pi\)
−0.126806 + 0.991928i \(0.540473\pi\)
\(198\) 0 0
\(199\) −3294.96 −1.17374 −0.586868 0.809682i \(-0.699639\pi\)
−0.586868 + 0.809682i \(0.699639\pi\)
\(200\) −1802.08 −0.637132
\(201\) 0 0
\(202\) 2350.67 0.818775
\(203\) 0 0
\(204\) 0 0
\(205\) −56.8124 −0.0193559
\(206\) 1871.03 0.632819
\(207\) 0 0
\(208\) −6737.29 −2.24590
\(209\) −4192.16 −1.38746
\(210\) 0 0
\(211\) 4082.35 1.33195 0.665974 0.745975i \(-0.268016\pi\)
0.665974 + 0.745975i \(0.268016\pi\)
\(212\) −3261.41 −1.05658
\(213\) 0 0
\(214\) 1175.53 0.375502
\(215\) 169.167 0.0536610
\(216\) 0 0
\(217\) 0 0
\(218\) −9390.46 −2.91744
\(219\) 0 0
\(220\) −5568.72 −1.70656
\(221\) −1141.21 −0.347359
\(222\) 0 0
\(223\) −747.161 −0.224366 −0.112183 0.993688i \(-0.535784\pi\)
−0.112183 + 0.993688i \(0.535784\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −10039.1 −2.95481
\(227\) 1665.67 0.487025 0.243513 0.969898i \(-0.421700\pi\)
0.243513 + 0.969898i \(0.421700\pi\)
\(228\) 0 0
\(229\) 6628.35 1.91272 0.956362 0.292183i \(-0.0943816\pi\)
0.956362 + 0.292183i \(0.0943816\pi\)
\(230\) −699.340 −0.200492
\(231\) 0 0
\(232\) 1509.50 0.427172
\(233\) 432.431 0.121586 0.0607929 0.998150i \(-0.480637\pi\)
0.0607929 + 0.998150i \(0.480637\pi\)
\(234\) 0 0
\(235\) 1808.38 0.501982
\(236\) −13129.2 −3.62136
\(237\) 0 0
\(238\) 0 0
\(239\) −5580.44 −1.51033 −0.755165 0.655535i \(-0.772443\pi\)
−0.755165 + 0.655535i \(0.772443\pi\)
\(240\) 0 0
\(241\) 6296.87 1.68306 0.841529 0.540212i \(-0.181656\pi\)
0.841529 + 0.540212i \(0.181656\pi\)
\(242\) −7577.56 −2.01283
\(243\) 0 0
\(244\) −325.352 −0.0853629
\(245\) 0 0
\(246\) 0 0
\(247\) 2459.46 0.633569
\(248\) −22674.4 −5.80575
\(249\) 0 0
\(250\) 676.777 0.171212
\(251\) 311.921 0.0784393 0.0392197 0.999231i \(-0.487513\pi\)
0.0392197 + 0.999231i \(0.487513\pi\)
\(252\) 0 0
\(253\) −1349.92 −0.335451
\(254\) −7550.17 −1.86512
\(255\) 0 0
\(256\) 6728.46 1.64269
\(257\) −7861.39 −1.90809 −0.954046 0.299659i \(-0.903127\pi\)
−0.954046 + 0.299659i \(0.903127\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 3267.06 0.779285
\(261\) 0 0
\(262\) −9542.11 −2.25005
\(263\) −5227.09 −1.22554 −0.612769 0.790262i \(-0.709944\pi\)
−0.612769 + 0.790262i \(0.709944\pi\)
\(264\) 0 0
\(265\) 765.097 0.177357
\(266\) 0 0
\(267\) 0 0
\(268\) −3548.94 −0.808903
\(269\) 1281.71 0.290510 0.145255 0.989394i \(-0.453600\pi\)
0.145255 + 0.989394i \(0.453600\pi\)
\(270\) 0 0
\(271\) −4704.14 −1.05445 −0.527226 0.849725i \(-0.676768\pi\)
−0.527226 + 0.849725i \(0.676768\pi\)
\(272\) 8180.82 1.82366
\(273\) 0 0
\(274\) −4997.04 −1.10176
\(275\) 1306.37 0.286462
\(276\) 0 0
\(277\) 8958.56 1.94321 0.971603 0.236619i \(-0.0760393\pi\)
0.971603 + 0.236619i \(0.0760393\pi\)
\(278\) −1061.40 −0.228987
\(279\) 0 0
\(280\) 0 0
\(281\) 370.904 0.0787412 0.0393706 0.999225i \(-0.487465\pi\)
0.0393706 + 0.999225i \(0.487465\pi\)
\(282\) 0 0
\(283\) 5822.26 1.22296 0.611479 0.791261i \(-0.290575\pi\)
0.611479 + 0.791261i \(0.290575\pi\)
\(284\) 20290.7 4.23954
\(285\) 0 0
\(286\) 8673.40 1.79325
\(287\) 0 0
\(288\) 0 0
\(289\) −3527.27 −0.717946
\(290\) −566.899 −0.114791
\(291\) 0 0
\(292\) 3164.86 0.634279
\(293\) 7443.79 1.48420 0.742100 0.670289i \(-0.233830\pi\)
0.742100 + 0.670289i \(0.233830\pi\)
\(294\) 0 0
\(295\) 3080.00 0.607880
\(296\) −14211.0 −2.79053
\(297\) 0 0
\(298\) 4225.10 0.821320
\(299\) 791.973 0.153181
\(300\) 0 0
\(301\) 0 0
\(302\) 12557.9 2.39280
\(303\) 0 0
\(304\) −17630.7 −3.32628
\(305\) 76.3247 0.0143290
\(306\) 0 0
\(307\) 761.674 0.141600 0.0707998 0.997491i \(-0.477445\pi\)
0.0707998 + 0.997491i \(0.477445\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 8515.43 1.56014
\(311\) 7718.69 1.40735 0.703677 0.710520i \(-0.251540\pi\)
0.703677 + 0.710520i \(0.251540\pi\)
\(312\) 0 0
\(313\) −8556.00 −1.54509 −0.772546 0.634959i \(-0.781017\pi\)
−0.772546 + 0.634959i \(0.781017\pi\)
\(314\) 5538.21 0.995348
\(315\) 0 0
\(316\) 18281.3 3.25444
\(317\) 7780.95 1.37862 0.689309 0.724468i \(-0.257914\pi\)
0.689309 + 0.724468i \(0.257914\pi\)
\(318\) 0 0
\(319\) −1094.28 −0.192062
\(320\) −7809.02 −1.36418
\(321\) 0 0
\(322\) 0 0
\(323\) −2986.42 −0.514455
\(324\) 0 0
\(325\) −766.421 −0.130810
\(326\) 7312.58 1.24235
\(327\) 0 0
\(328\) −819.045 −0.137879
\(329\) 0 0
\(330\) 0 0
\(331\) −4932.12 −0.819015 −0.409507 0.912307i \(-0.634299\pi\)
−0.409507 + 0.912307i \(0.634299\pi\)
\(332\) 14078.3 2.32725
\(333\) 0 0
\(334\) 6662.61 1.09150
\(335\) 832.548 0.135782
\(336\) 0 0
\(337\) −7121.13 −1.15108 −0.575538 0.817775i \(-0.695207\pi\)
−0.575538 + 0.817775i \(0.695207\pi\)
\(338\) 6806.52 1.09534
\(339\) 0 0
\(340\) −3967.06 −0.632776
\(341\) 16437.2 2.61034
\(342\) 0 0
\(343\) 0 0
\(344\) 2438.83 0.382246
\(345\) 0 0
\(346\) 13468.7 2.09272
\(347\) −9540.58 −1.47598 −0.737991 0.674811i \(-0.764225\pi\)
−0.737991 + 0.674811i \(0.764225\pi\)
\(348\) 0 0
\(349\) −1281.65 −0.196576 −0.0982880 0.995158i \(-0.531337\pi\)
−0.0982880 + 0.995158i \(0.531337\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −32041.9 −4.85182
\(353\) 5798.07 0.874221 0.437110 0.899408i \(-0.356002\pi\)
0.437110 + 0.899408i \(0.356002\pi\)
\(354\) 0 0
\(355\) −4760.00 −0.711647
\(356\) −975.627 −0.145247
\(357\) 0 0
\(358\) 8777.40 1.29581
\(359\) −2267.29 −0.333323 −0.166662 0.986014i \(-0.553299\pi\)
−0.166662 + 0.986014i \(0.553299\pi\)
\(360\) 0 0
\(361\) −422.886 −0.0616541
\(362\) 14042.8 2.03887
\(363\) 0 0
\(364\) 0 0
\(365\) −742.447 −0.106470
\(366\) 0 0
\(367\) 7372.85 1.04866 0.524332 0.851514i \(-0.324315\pi\)
0.524332 + 0.851514i \(0.324315\pi\)
\(368\) −5677.28 −0.804209
\(369\) 0 0
\(370\) 5336.98 0.749883
\(371\) 0 0
\(372\) 0 0
\(373\) 6447.14 0.894961 0.447480 0.894294i \(-0.352321\pi\)
0.447480 + 0.894294i \(0.352321\pi\)
\(374\) −10531.8 −1.45611
\(375\) 0 0
\(376\) 26070.8 3.57579
\(377\) 641.989 0.0877032
\(378\) 0 0
\(379\) −4247.57 −0.575680 −0.287840 0.957678i \(-0.592937\pi\)
−0.287840 + 0.957678i \(0.592937\pi\)
\(380\) 8549.50 1.15416
\(381\) 0 0
\(382\) −9870.53 −1.32204
\(383\) −6681.86 −0.891454 −0.445727 0.895169i \(-0.647055\pi\)
−0.445727 + 0.895169i \(0.647055\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8343.45 1.10018
\(387\) 0 0
\(388\) −35852.4 −4.69105
\(389\) 6371.78 0.830494 0.415247 0.909709i \(-0.363695\pi\)
0.415247 + 0.909709i \(0.363695\pi\)
\(390\) 0 0
\(391\) −961.661 −0.124382
\(392\) 0 0
\(393\) 0 0
\(394\) 3796.68 0.485467
\(395\) −4288.62 −0.546289
\(396\) 0 0
\(397\) −4247.93 −0.537021 −0.268510 0.963277i \(-0.586531\pi\)
−0.268510 + 0.963277i \(0.586531\pi\)
\(398\) 17839.6 2.24678
\(399\) 0 0
\(400\) 5494.11 0.686764
\(401\) 8833.62 1.10008 0.550038 0.835140i \(-0.314613\pi\)
0.550038 + 0.835140i \(0.314613\pi\)
\(402\) 0 0
\(403\) −9643.37 −1.19199
\(404\) −9253.70 −1.13958
\(405\) 0 0
\(406\) 0 0
\(407\) 10301.9 1.25466
\(408\) 0 0
\(409\) 319.205 0.0385908 0.0192954 0.999814i \(-0.493858\pi\)
0.0192954 + 0.999814i \(0.493858\pi\)
\(410\) 307.595 0.0370512
\(411\) 0 0
\(412\) −7365.53 −0.880761
\(413\) 0 0
\(414\) 0 0
\(415\) −3302.64 −0.390651
\(416\) 18798.3 2.21554
\(417\) 0 0
\(418\) 22697.3 2.65589
\(419\) −12789.2 −1.49115 −0.745577 0.666420i \(-0.767826\pi\)
−0.745577 + 0.666420i \(0.767826\pi\)
\(420\) 0 0
\(421\) −6747.40 −0.781112 −0.390556 0.920579i \(-0.627717\pi\)
−0.390556 + 0.920579i \(0.627717\pi\)
\(422\) −22102.7 −2.54963
\(423\) 0 0
\(424\) 11030.1 1.26337
\(425\) 930.635 0.106217
\(426\) 0 0
\(427\) 0 0
\(428\) −4627.60 −0.522625
\(429\) 0 0
\(430\) −915.908 −0.102719
\(431\) 5184.75 0.579444 0.289722 0.957111i \(-0.406437\pi\)
0.289722 + 0.957111i \(0.406437\pi\)
\(432\) 0 0
\(433\) 4242.03 0.470806 0.235403 0.971898i \(-0.424359\pi\)
0.235403 + 0.971898i \(0.424359\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 36966.7 4.06051
\(437\) 2072.50 0.226868
\(438\) 0 0
\(439\) 5434.12 0.590789 0.295394 0.955375i \(-0.404549\pi\)
0.295394 + 0.955375i \(0.404549\pi\)
\(440\) 18833.5 2.04057
\(441\) 0 0
\(442\) 6178.77 0.664919
\(443\) 11493.8 1.23270 0.616350 0.787472i \(-0.288611\pi\)
0.616350 + 0.787472i \(0.288611\pi\)
\(444\) 0 0
\(445\) 228.873 0.0243812
\(446\) 4045.29 0.429484
\(447\) 0 0
\(448\) 0 0
\(449\) 16849.3 1.77098 0.885489 0.464661i \(-0.153824\pi\)
0.885489 + 0.464661i \(0.153824\pi\)
\(450\) 0 0
\(451\) 593.745 0.0619919
\(452\) 39520.0 4.11253
\(453\) 0 0
\(454\) −9018.32 −0.932270
\(455\) 0 0
\(456\) 0 0
\(457\) 15348.5 1.57106 0.785528 0.618826i \(-0.212391\pi\)
0.785528 + 0.618826i \(0.212391\pi\)
\(458\) −35887.3 −3.66136
\(459\) 0 0
\(460\) 2753.04 0.279046
\(461\) 14038.4 1.41830 0.709148 0.705059i \(-0.249080\pi\)
0.709148 + 0.705059i \(0.249080\pi\)
\(462\) 0 0
\(463\) −8661.23 −0.869377 −0.434689 0.900581i \(-0.643142\pi\)
−0.434689 + 0.900581i \(0.643142\pi\)
\(464\) −4602.12 −0.460448
\(465\) 0 0
\(466\) −2341.27 −0.232741
\(467\) 7014.71 0.695079 0.347539 0.937665i \(-0.387017\pi\)
0.347539 + 0.937665i \(0.387017\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −9790.96 −0.960901
\(471\) 0 0
\(472\) 44403.3 4.33014
\(473\) −1767.96 −0.171863
\(474\) 0 0
\(475\) −2005.63 −0.193737
\(476\) 0 0
\(477\) 0 0
\(478\) 30213.7 2.89109
\(479\) 18134.7 1.72984 0.864922 0.501907i \(-0.167368\pi\)
0.864922 + 0.501907i \(0.167368\pi\)
\(480\) 0 0
\(481\) −6043.91 −0.572929
\(482\) −34092.6 −3.22173
\(483\) 0 0
\(484\) 29830.0 2.80146
\(485\) 8410.63 0.787438
\(486\) 0 0
\(487\) 16537.8 1.53881 0.769405 0.638761i \(-0.220553\pi\)
0.769405 + 0.638761i \(0.220553\pi\)
\(488\) 1100.35 0.102070
\(489\) 0 0
\(490\) 0 0
\(491\) −220.608 −0.0202768 −0.0101384 0.999949i \(-0.503227\pi\)
−0.0101384 + 0.999949i \(0.503227\pi\)
\(492\) 0 0
\(493\) −779.542 −0.0712146
\(494\) −13316.0 −1.21279
\(495\) 0 0
\(496\) 69128.8 6.25801
\(497\) 0 0
\(498\) 0 0
\(499\) 5939.04 0.532801 0.266401 0.963862i \(-0.414166\pi\)
0.266401 + 0.963862i \(0.414166\pi\)
\(500\) −2664.21 −0.238295
\(501\) 0 0
\(502\) −1688.81 −0.150150
\(503\) −11604.8 −1.02869 −0.514345 0.857584i \(-0.671965\pi\)
−0.514345 + 0.857584i \(0.671965\pi\)
\(504\) 0 0
\(505\) 2170.83 0.191289
\(506\) 7308.78 0.642124
\(507\) 0 0
\(508\) 29722.2 2.59588
\(509\) −1867.67 −0.162639 −0.0813193 0.996688i \(-0.525913\pi\)
−0.0813193 + 0.996688i \(0.525913\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −8025.75 −0.692757
\(513\) 0 0
\(514\) 42563.2 3.65250
\(515\) 1727.88 0.147844
\(516\) 0 0
\(517\) −18899.3 −1.60772
\(518\) 0 0
\(519\) 0 0
\(520\) −11049.2 −0.931809
\(521\) 6117.21 0.514395 0.257197 0.966359i \(-0.417201\pi\)
0.257197 + 0.966359i \(0.417201\pi\)
\(522\) 0 0
\(523\) 16685.6 1.39505 0.697524 0.716561i \(-0.254285\pi\)
0.697524 + 0.716561i \(0.254285\pi\)
\(524\) 37563.7 3.13164
\(525\) 0 0
\(526\) 28300.6 2.34594
\(527\) 11709.6 0.967887
\(528\) 0 0
\(529\) −11499.6 −0.945149
\(530\) −4142.40 −0.339499
\(531\) 0 0
\(532\) 0 0
\(533\) −348.338 −0.0283081
\(534\) 0 0
\(535\) 1085.59 0.0877276
\(536\) 12002.6 0.967223
\(537\) 0 0
\(538\) −6939.44 −0.556097
\(539\) 0 0
\(540\) 0 0
\(541\) 9309.03 0.739790 0.369895 0.929074i \(-0.379394\pi\)
0.369895 + 0.929074i \(0.379394\pi\)
\(542\) 25469.2 2.01845
\(543\) 0 0
\(544\) −22826.1 −1.79901
\(545\) −8672.05 −0.681596
\(546\) 0 0
\(547\) 10894.7 0.851598 0.425799 0.904818i \(-0.359993\pi\)
0.425799 + 0.904818i \(0.359993\pi\)
\(548\) 19671.5 1.53344
\(549\) 0 0
\(550\) −7072.97 −0.548350
\(551\) 1680.01 0.129893
\(552\) 0 0
\(553\) 0 0
\(554\) −48503.6 −3.71971
\(555\) 0 0
\(556\) 4178.31 0.318705
\(557\) 7873.90 0.598973 0.299486 0.954101i \(-0.403185\pi\)
0.299486 + 0.954101i \(0.403185\pi\)
\(558\) 0 0
\(559\) 1037.23 0.0784796
\(560\) 0 0
\(561\) 0 0
\(562\) −2008.15 −0.150728
\(563\) 21770.7 1.62971 0.814854 0.579666i \(-0.196817\pi\)
0.814854 + 0.579666i \(0.196817\pi\)
\(564\) 0 0
\(565\) −9271.02 −0.690327
\(566\) −31522.9 −2.34100
\(567\) 0 0
\(568\) −68623.3 −5.06931
\(569\) 12381.3 0.912213 0.456106 0.889925i \(-0.349244\pi\)
0.456106 + 0.889925i \(0.349244\pi\)
\(570\) 0 0
\(571\) −5768.38 −0.422765 −0.211383 0.977403i \(-0.567797\pi\)
−0.211383 + 0.977403i \(0.567797\pi\)
\(572\) −34143.9 −2.49585
\(573\) 0 0
\(574\) 0 0
\(575\) −645.837 −0.0468405
\(576\) 0 0
\(577\) −4733.38 −0.341513 −0.170757 0.985313i \(-0.554621\pi\)
−0.170757 + 0.985313i \(0.554621\pi\)
\(578\) 19097.4 1.37430
\(579\) 0 0
\(580\) 2231.67 0.159767
\(581\) 0 0
\(582\) 0 0
\(583\) −7996.00 −0.568028
\(584\) −10703.6 −0.758422
\(585\) 0 0
\(586\) −40302.3 −2.84108
\(587\) −8441.67 −0.593569 −0.296785 0.954944i \(-0.595914\pi\)
−0.296785 + 0.954944i \(0.595914\pi\)
\(588\) 0 0
\(589\) −25235.6 −1.76539
\(590\) −16675.8 −1.16361
\(591\) 0 0
\(592\) 43326.0 3.00792
\(593\) 18939.9 1.31158 0.655791 0.754943i \(-0.272335\pi\)
0.655791 + 0.754943i \(0.272335\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −16632.6 −1.14312
\(597\) 0 0
\(598\) −4287.91 −0.293220
\(599\) −22655.3 −1.54536 −0.772681 0.634794i \(-0.781085\pi\)
−0.772681 + 0.634794i \(0.781085\pi\)
\(600\) 0 0
\(601\) 15947.4 1.08237 0.541187 0.840902i \(-0.317975\pi\)
0.541187 + 0.840902i \(0.317975\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −49435.6 −3.33031
\(605\) −6997.84 −0.470252
\(606\) 0 0
\(607\) 25993.2 1.73811 0.869053 0.494719i \(-0.164729\pi\)
0.869053 + 0.494719i \(0.164729\pi\)
\(608\) 49193.1 3.28132
\(609\) 0 0
\(610\) −413.238 −0.0274287
\(611\) 11087.9 0.734152
\(612\) 0 0
\(613\) 665.408 0.0438427 0.0219213 0.999760i \(-0.493022\pi\)
0.0219213 + 0.999760i \(0.493022\pi\)
\(614\) −4123.87 −0.271052
\(615\) 0 0
\(616\) 0 0
\(617\) −18401.3 −1.20066 −0.600330 0.799752i \(-0.704964\pi\)
−0.600330 + 0.799752i \(0.704964\pi\)
\(618\) 0 0
\(619\) 11150.6 0.724040 0.362020 0.932170i \(-0.382087\pi\)
0.362020 + 0.932170i \(0.382087\pi\)
\(620\) −33522.0 −2.17142
\(621\) 0 0
\(622\) −41790.6 −2.69397
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 46324.0 2.95764
\(627\) 0 0
\(628\) −21801.8 −1.38533
\(629\) 7338.88 0.465215
\(630\) 0 0
\(631\) 5381.79 0.339534 0.169767 0.985484i \(-0.445699\pi\)
0.169767 + 0.985484i \(0.445699\pi\)
\(632\) −61827.6 −3.89141
\(633\) 0 0
\(634\) −42127.7 −2.63897
\(635\) −6972.55 −0.435744
\(636\) 0 0
\(637\) 0 0
\(638\) 5924.64 0.367647
\(639\) 0 0
\(640\) 17752.2 1.09644
\(641\) 19455.1 1.19880 0.599398 0.800451i \(-0.295407\pi\)
0.599398 + 0.800451i \(0.295407\pi\)
\(642\) 0 0
\(643\) 14695.8 0.901317 0.450658 0.892696i \(-0.351189\pi\)
0.450658 + 0.892696i \(0.351189\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 16169.1 0.984777
\(647\) −12694.8 −0.771383 −0.385691 0.922628i \(-0.626037\pi\)
−0.385691 + 0.922628i \(0.626037\pi\)
\(648\) 0 0
\(649\) −32189.0 −1.94688
\(650\) 4149.57 0.250399
\(651\) 0 0
\(652\) −28786.8 −1.72911
\(653\) 12385.6 0.742247 0.371124 0.928583i \(-0.378973\pi\)
0.371124 + 0.928583i \(0.378973\pi\)
\(654\) 0 0
\(655\) −8812.09 −0.525675
\(656\) 2497.07 0.148619
\(657\) 0 0
\(658\) 0 0
\(659\) 2072.18 0.122489 0.0612447 0.998123i \(-0.480493\pi\)
0.0612447 + 0.998123i \(0.480493\pi\)
\(660\) 0 0
\(661\) −1074.36 −0.0632193 −0.0316096 0.999500i \(-0.510063\pi\)
−0.0316096 + 0.999500i \(0.510063\pi\)
\(662\) 26703.6 1.56777
\(663\) 0 0
\(664\) −47613.0 −2.78275
\(665\) 0 0
\(666\) 0 0
\(667\) 540.982 0.0314047
\(668\) −26228.2 −1.51916
\(669\) 0 0
\(670\) −4507.59 −0.259916
\(671\) −797.667 −0.0458921
\(672\) 0 0
\(673\) 26195.2 1.50037 0.750186 0.661226i \(-0.229964\pi\)
0.750186 + 0.661226i \(0.229964\pi\)
\(674\) 38555.3 2.20341
\(675\) 0 0
\(676\) −26794.7 −1.52450
\(677\) −4228.44 −0.240047 −0.120024 0.992771i \(-0.538297\pi\)
−0.120024 + 0.992771i \(0.538297\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 13416.6 0.756624
\(681\) 0 0
\(682\) −88994.5 −4.99674
\(683\) −27525.5 −1.54207 −0.771036 0.636792i \(-0.780261\pi\)
−0.771036 + 0.636792i \(0.780261\pi\)
\(684\) 0 0
\(685\) −4614.74 −0.257402
\(686\) 0 0
\(687\) 0 0
\(688\) −7435.40 −0.412023
\(689\) 4691.09 0.259385
\(690\) 0 0
\(691\) 33324.4 1.83462 0.917309 0.398177i \(-0.130357\pi\)
0.917309 + 0.398177i \(0.130357\pi\)
\(692\) −53021.1 −2.91266
\(693\) 0 0
\(694\) 51654.8 2.82534
\(695\) −980.193 −0.0534976
\(696\) 0 0
\(697\) 422.973 0.0229860
\(698\) 6939.12 0.376289
\(699\) 0 0
\(700\) 0 0
\(701\) 33262.9 1.79219 0.896094 0.443864i \(-0.146393\pi\)
0.896094 + 0.443864i \(0.146393\pi\)
\(702\) 0 0
\(703\) −15816.2 −0.848534
\(704\) 81611.8 4.36912
\(705\) 0 0
\(706\) −31392.0 −1.67345
\(707\) 0 0
\(708\) 0 0
\(709\) 13703.0 0.725851 0.362926 0.931818i \(-0.381778\pi\)
0.362926 + 0.931818i \(0.381778\pi\)
\(710\) 25771.7 1.36224
\(711\) 0 0
\(712\) 3299.58 0.173676
\(713\) −8126.14 −0.426825
\(714\) 0 0
\(715\) 8009.84 0.418953
\(716\) −34553.3 −1.80352
\(717\) 0 0
\(718\) 12275.6 0.638052
\(719\) −8074.93 −0.418838 −0.209419 0.977826i \(-0.567157\pi\)
−0.209419 + 0.977826i \(0.567157\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 2289.59 0.118019
\(723\) 0 0
\(724\) −55281.1 −2.83771
\(725\) −523.528 −0.0268184
\(726\) 0 0
\(727\) 3668.70 0.187159 0.0935794 0.995612i \(-0.470169\pi\)
0.0935794 + 0.995612i \(0.470169\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 4019.77 0.203806
\(731\) −1259.46 −0.0637250
\(732\) 0 0
\(733\) 14980.3 0.754857 0.377428 0.926039i \(-0.376808\pi\)
0.377428 + 0.926039i \(0.376808\pi\)
\(734\) −39918.2 −2.00737
\(735\) 0 0
\(736\) 15840.7 0.793338
\(737\) −8700.94 −0.434875
\(738\) 0 0
\(739\) 6530.59 0.325077 0.162538 0.986702i \(-0.448032\pi\)
0.162538 + 0.986702i \(0.448032\pi\)
\(740\) −21009.7 −1.04369
\(741\) 0 0
\(742\) 0 0
\(743\) −25952.0 −1.28141 −0.640704 0.767788i \(-0.721357\pi\)
−0.640704 + 0.767788i \(0.721357\pi\)
\(744\) 0 0
\(745\) 3901.86 0.191883
\(746\) −34906.2 −1.71315
\(747\) 0 0
\(748\) 41459.6 2.02662
\(749\) 0 0
\(750\) 0 0
\(751\) −14093.9 −0.684813 −0.342407 0.939552i \(-0.611242\pi\)
−0.342407 + 0.939552i \(0.611242\pi\)
\(752\) −79483.6 −3.85435
\(753\) 0 0
\(754\) −3475.87 −0.167883
\(755\) 11597.1 0.559024
\(756\) 0 0
\(757\) −2554.41 −0.122644 −0.0613220 0.998118i \(-0.519532\pi\)
−0.0613220 + 0.998118i \(0.519532\pi\)
\(758\) 22997.2 1.10197
\(759\) 0 0
\(760\) −28914.5 −1.38005
\(761\) 2219.08 0.105705 0.0528527 0.998602i \(-0.483169\pi\)
0.0528527 + 0.998602i \(0.483169\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 38856.5 1.84003
\(765\) 0 0
\(766\) 36177.0 1.70643
\(767\) 18884.6 0.889028
\(768\) 0 0
\(769\) 22466.2 1.05352 0.526758 0.850015i \(-0.323408\pi\)
0.526758 + 0.850015i \(0.323408\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −32845.0 −1.53124
\(773\) 9674.79 0.450165 0.225083 0.974340i \(-0.427735\pi\)
0.225083 + 0.974340i \(0.427735\pi\)
\(774\) 0 0
\(775\) 7863.96 0.364493
\(776\) 121253. 5.60920
\(777\) 0 0
\(778\) −34498.2 −1.58974
\(779\) −911.560 −0.0419256
\(780\) 0 0
\(781\) 49746.6 2.27922
\(782\) 5206.64 0.238093
\(783\) 0 0
\(784\) 0 0
\(785\) 5114.51 0.232541
\(786\) 0 0
\(787\) 20942.8 0.948577 0.474288 0.880370i \(-0.342705\pi\)
0.474288 + 0.880370i \(0.342705\pi\)
\(788\) −14946.1 −0.675676
\(789\) 0 0
\(790\) 23219.5 1.04571
\(791\) 0 0
\(792\) 0 0
\(793\) 467.975 0.0209562
\(794\) 22999.2 1.02797
\(795\) 0 0
\(796\) −70227.8 −3.12708
\(797\) 23526.6 1.04561 0.522807 0.852451i \(-0.324885\pi\)
0.522807 + 0.852451i \(0.324885\pi\)
\(798\) 0 0
\(799\) −13463.5 −0.596127
\(800\) −15329.6 −0.677481
\(801\) 0 0
\(802\) −47827.1 −2.10578
\(803\) 7759.29 0.340996
\(804\) 0 0
\(805\) 0 0
\(806\) 52211.3 2.28172
\(807\) 0 0
\(808\) 31296.1 1.36262
\(809\) 18202.2 0.791047 0.395523 0.918456i \(-0.370563\pi\)
0.395523 + 0.918456i \(0.370563\pi\)
\(810\) 0 0
\(811\) 2510.24 0.108689 0.0543443 0.998522i \(-0.482693\pi\)
0.0543443 + 0.998522i \(0.482693\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −55776.7 −2.40168
\(815\) 6753.13 0.290248
\(816\) 0 0
\(817\) 2714.30 0.116232
\(818\) −1728.24 −0.0738711
\(819\) 0 0
\(820\) −1210.88 −0.0515681
\(821\) −17899.6 −0.760903 −0.380451 0.924801i \(-0.624231\pi\)
−0.380451 + 0.924801i \(0.624231\pi\)
\(822\) 0 0
\(823\) 14039.5 0.594637 0.297318 0.954778i \(-0.403908\pi\)
0.297318 + 0.954778i \(0.403908\pi\)
\(824\) 24910.3 1.05315
\(825\) 0 0
\(826\) 0 0
\(827\) −15127.4 −0.636073 −0.318036 0.948079i \(-0.603023\pi\)
−0.318036 + 0.948079i \(0.603023\pi\)
\(828\) 0 0
\(829\) −21986.5 −0.921136 −0.460568 0.887624i \(-0.652354\pi\)
−0.460568 + 0.887624i \(0.652354\pi\)
\(830\) 17881.2 0.747790
\(831\) 0 0
\(832\) −47880.0 −1.99512
\(833\) 0 0
\(834\) 0 0
\(835\) 6152.89 0.255005
\(836\) −89350.6 −3.69648
\(837\) 0 0
\(838\) 69243.5 2.85439
\(839\) −2276.89 −0.0936914 −0.0468457 0.998902i \(-0.514917\pi\)
−0.0468457 + 0.998902i \(0.514917\pi\)
\(840\) 0 0
\(841\) −23950.5 −0.982019
\(842\) 36531.8 1.49521
\(843\) 0 0
\(844\) 87010.1 3.54859
\(845\) 6285.79 0.255903
\(846\) 0 0
\(847\) 0 0
\(848\) −33628.2 −1.36179
\(849\) 0 0
\(850\) −5038.66 −0.203323
\(851\) −5093.00 −0.205154
\(852\) 0 0
\(853\) −13342.6 −0.535570 −0.267785 0.963479i \(-0.586292\pi\)
−0.267785 + 0.963479i \(0.586292\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 15650.6 0.624915
\(857\) 18690.9 0.745003 0.372502 0.928032i \(-0.378500\pi\)
0.372502 + 0.928032i \(0.378500\pi\)
\(858\) 0 0
\(859\) −18318.9 −0.727628 −0.363814 0.931472i \(-0.618526\pi\)
−0.363814 + 0.931472i \(0.618526\pi\)
\(860\) 3605.58 0.142964
\(861\) 0 0
\(862\) −28071.3 −1.10918
\(863\) −38133.1 −1.50413 −0.752067 0.659087i \(-0.770943\pi\)
−0.752067 + 0.659087i \(0.770943\pi\)
\(864\) 0 0
\(865\) 12438.3 0.488917
\(866\) −22967.3 −0.901223
\(867\) 0 0
\(868\) 0 0
\(869\) 44820.3 1.74962
\(870\) 0 0
\(871\) 5104.66 0.198582
\(872\) −125022. −4.85525
\(873\) 0 0
\(874\) −11221.0 −0.434273
\(875\) 0 0
\(876\) 0 0
\(877\) −19707.5 −0.758807 −0.379404 0.925231i \(-0.623871\pi\)
−0.379404 + 0.925231i \(0.623871\pi\)
\(878\) −29421.5 −1.13090
\(879\) 0 0
\(880\) −57418.8 −2.19953
\(881\) −14091.5 −0.538883 −0.269441 0.963017i \(-0.586839\pi\)
−0.269441 + 0.963017i \(0.586839\pi\)
\(882\) 0 0
\(883\) 3115.87 0.118751 0.0593757 0.998236i \(-0.481089\pi\)
0.0593757 + 0.998236i \(0.481089\pi\)
\(884\) −24323.5 −0.925438
\(885\) 0 0
\(886\) −62229.8 −2.35965
\(887\) 38734.6 1.46627 0.733134 0.680084i \(-0.238057\pi\)
0.733134 + 0.680084i \(0.238057\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −1239.17 −0.0466708
\(891\) 0 0
\(892\) −15924.8 −0.597759
\(893\) 29015.6 1.08731
\(894\) 0 0
\(895\) 8105.89 0.302737
\(896\) 0 0
\(897\) 0 0
\(898\) −91225.8 −3.39003
\(899\) −6587.21 −0.244378
\(900\) 0 0
\(901\) −5696.21 −0.210619
\(902\) −3214.66 −0.118666
\(903\) 0 0
\(904\) −133657. −4.91744
\(905\) 12968.4 0.476337
\(906\) 0 0
\(907\) 19242.9 0.704464 0.352232 0.935913i \(-0.385423\pi\)
0.352232 + 0.935913i \(0.385423\pi\)
\(908\) 35501.7 1.29754
\(909\) 0 0
\(910\) 0 0
\(911\) −34613.3 −1.25882 −0.629412 0.777072i \(-0.716704\pi\)
−0.629412 + 0.777072i \(0.716704\pi\)
\(912\) 0 0
\(913\) 34515.8 1.25116
\(914\) −83100.1 −3.00734
\(915\) 0 0
\(916\) 141275. 5.09591
\(917\) 0 0
\(918\) 0 0
\(919\) 25826.4 0.927022 0.463511 0.886091i \(-0.346589\pi\)
0.463511 + 0.886091i \(0.346589\pi\)
\(920\) −9310.81 −0.333661
\(921\) 0 0
\(922\) −76007.0 −2.71492
\(923\) −29185.3 −1.04079
\(924\) 0 0
\(925\) 4928.68 0.175194
\(926\) 46893.8 1.66417
\(927\) 0 0
\(928\) 12840.8 0.454224
\(929\) 19451.6 0.686960 0.343480 0.939160i \(-0.388394\pi\)
0.343480 + 0.939160i \(0.388394\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 9216.70 0.323930
\(933\) 0 0
\(934\) −37979.1 −1.33053
\(935\) −9726.03 −0.340188
\(936\) 0 0
\(937\) −34469.1 −1.20177 −0.600884 0.799336i \(-0.705185\pi\)
−0.600884 + 0.799336i \(0.705185\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 38543.3 1.33739
\(941\) 14156.4 0.490419 0.245209 0.969470i \(-0.421143\pi\)
0.245209 + 0.969470i \(0.421143\pi\)
\(942\) 0 0
\(943\) −293.532 −0.0101365
\(944\) −135375. −4.66746
\(945\) 0 0
\(946\) 9572.13 0.328982
\(947\) 38092.4 1.30711 0.653557 0.756877i \(-0.273276\pi\)
0.653557 + 0.756877i \(0.273276\pi\)
\(948\) 0 0
\(949\) −4552.22 −0.155713
\(950\) 10858.9 0.370853
\(951\) 0 0
\(952\) 0 0
\(953\) −5037.40 −0.171225 −0.0856126 0.996329i \(-0.527285\pi\)
−0.0856126 + 0.996329i \(0.527285\pi\)
\(954\) 0 0
\(955\) −9115.39 −0.308866
\(956\) −118940. −4.02384
\(957\) 0 0
\(958\) −98185.1 −3.31129
\(959\) 0 0
\(960\) 0 0
\(961\) 69156.0 2.32137
\(962\) 32723.0 1.09671
\(963\) 0 0
\(964\) 134210. 4.48403
\(965\) 7705.14 0.257033
\(966\) 0 0
\(967\) 11495.3 0.382278 0.191139 0.981563i \(-0.438782\pi\)
0.191139 + 0.981563i \(0.438782\pi\)
\(968\) −100885. −3.34977
\(969\) 0 0
\(970\) −45537.0 −1.50732
\(971\) 22352.7 0.738757 0.369379 0.929279i \(-0.379571\pi\)
0.369379 + 0.929279i \(0.379571\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −89539.4 −2.94561
\(975\) 0 0
\(976\) −3354.69 −0.110022
\(977\) −14345.7 −0.469765 −0.234882 0.972024i \(-0.575470\pi\)
−0.234882 + 0.972024i \(0.575470\pi\)
\(978\) 0 0
\(979\) −2391.94 −0.0780867
\(980\) 0 0
\(981\) 0 0
\(982\) 1194.42 0.0388141
\(983\) 34460.9 1.11814 0.559070 0.829120i \(-0.311158\pi\)
0.559070 + 0.829120i \(0.311158\pi\)
\(984\) 0 0
\(985\) 3506.21 0.113419
\(986\) 4220.61 0.136320
\(987\) 0 0
\(988\) 52420.2 1.68796
\(989\) 874.036 0.0281019
\(990\) 0 0
\(991\) −35189.6 −1.12799 −0.563993 0.825780i \(-0.690735\pi\)
−0.563993 + 0.825780i \(0.690735\pi\)
\(992\) −192883. −6.17342
\(993\) 0 0
\(994\) 0 0
\(995\) 16474.8 0.524911
\(996\) 0 0
\(997\) 50730.0 1.61147 0.805734 0.592277i \(-0.201771\pi\)
0.805734 + 0.592277i \(0.201771\pi\)
\(998\) −32155.2 −1.01990
\(999\) 0 0
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 2205.4.a.u.1.1 2
3.2 odd 2 245.4.a.k.1.2 2
7.6 odd 2 315.4.a.f.1.1 2
15.14 odd 2 1225.4.a.m.1.1 2
21.2 odd 6 245.4.e.i.116.1 4
21.5 even 6 245.4.e.h.116.1 4
21.11 odd 6 245.4.e.i.226.1 4
21.17 even 6 245.4.e.h.226.1 4
21.20 even 2 35.4.a.b.1.2 2
35.34 odd 2 1575.4.a.z.1.2 2
84.83 odd 2 560.4.a.r.1.2 2
105.62 odd 4 175.4.b.c.99.4 4
105.83 odd 4 175.4.b.c.99.1 4
105.104 even 2 175.4.a.c.1.1 2
168.83 odd 2 2240.4.a.bo.1.1 2
168.125 even 2 2240.4.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 2 21.20 even 2
175.4.a.c.1.1 2 105.104 even 2
175.4.b.c.99.1 4 105.83 odd 4
175.4.b.c.99.4 4 105.62 odd 4
245.4.a.k.1.2 2 3.2 odd 2
245.4.e.h.116.1 4 21.5 even 6
245.4.e.h.226.1 4 21.17 even 6
245.4.e.i.116.1 4 21.2 odd 6
245.4.e.i.226.1 4 21.11 odd 6
315.4.a.f.1.1 2 7.6 odd 2
560.4.a.r.1.2 2 84.83 odd 2
1225.4.a.m.1.1 2 15.14 odd 2
1575.4.a.z.1.2 2 35.34 odd 2
2205.4.a.u.1.1 2 1.1 even 1 trivial
2240.4.a.bn.1.2 2 168.125 even 2
2240.4.a.bo.1.1 2 168.83 odd 2