Properties

Label 315.4.a.f.1.1
Level $315$
Weight $4$
Character 315.1
Self dual yes
Analytic conductor $18.586$
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] = [315,4,Mod(1,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, 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("315.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.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: \(18.5856016518\)
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 \(\chi\) \(=\) 315.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.41421 q^{2} +21.3137 q^{4} +5.00000 q^{5} -7.00000 q^{7} -72.0833 q^{8} +O(q^{10})\) \(q-5.41421 q^{2} +21.3137 q^{4} +5.00000 q^{5} -7.00000 q^{7} -72.0833 q^{8} -27.0711 q^{10} +52.2548 q^{11} +30.6569 q^{13} +37.8995 q^{14} +219.765 q^{16} -37.2254 q^{17} +80.2254 q^{19} +106.569 q^{20} -282.919 q^{22} -25.8335 q^{23} +25.0000 q^{25} -165.983 q^{26} -149.196 q^{28} -20.9411 q^{29} -314.558 q^{31} -613.186 q^{32} +201.546 q^{34} -35.0000 q^{35} +197.147 q^{37} -434.357 q^{38} -360.416 q^{40} -11.3625 q^{41} -33.8335 q^{43} +1113.74 q^{44} +139.868 q^{46} +361.676 q^{47} +49.0000 q^{49} -135.355 q^{50} +653.411 q^{52} -153.019 q^{53} +261.274 q^{55} +504.583 q^{56} +113.380 q^{58} +616.000 q^{59} +15.2649 q^{61} +1703.09 q^{62} +1561.80 q^{64} +153.284 q^{65} -166.510 q^{67} -793.411 q^{68} +189.497 q^{70} +952.000 q^{71} -148.489 q^{73} -1067.40 q^{74} +1709.90 q^{76} -365.784 q^{77} +857.725 q^{79} +1098.82 q^{80} +61.5189 q^{82} -660.528 q^{83} -186.127 q^{85} +183.182 q^{86} -3766.70 q^{88} +45.7746 q^{89} -214.598 q^{91} -550.607 q^{92} -1958.19 q^{94} +401.127 q^{95} +1682.13 q^{97} -265.296 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 20 q^{4} + 10 q^{5} - 14 q^{7} - 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} + 20 q^{4} + 10 q^{5} - 14 q^{7} - 48 q^{8} - 40 q^{10} + 14 q^{11} + 50 q^{13} + 56 q^{14} + 168 q^{16} + 50 q^{17} + 36 q^{19} + 100 q^{20} - 184 q^{22} - 244 q^{23} + 50 q^{25} - 216 q^{26} - 140 q^{28} + 26 q^{29} - 120 q^{31} - 672 q^{32} - 24 q^{34} - 70 q^{35} + 564 q^{37} - 320 q^{38} - 240 q^{40} + 328 q^{41} - 260 q^{43} + 1164 q^{44} + 704 q^{46} + 350 q^{47} + 98 q^{49} - 200 q^{50} + 628 q^{52} + 56 q^{53} + 70 q^{55} + 336 q^{56} - 8 q^{58} + 1232 q^{59} + 336 q^{61} + 1200 q^{62} + 2128 q^{64} + 250 q^{65} - 152 q^{67} - 908 q^{68} + 280 q^{70} + 1904 q^{71} + 676 q^{73} - 2016 q^{74} + 1768 q^{76} - 98 q^{77} + 1014 q^{79} + 840 q^{80} - 816 q^{82} + 376 q^{83} + 250 q^{85} + 768 q^{86} - 4688 q^{88} + 216 q^{89} - 350 q^{91} - 264 q^{92} - 1928 q^{94} + 180 q^{95} + 2742 q^{97} - 392 q^{98}+O(q^{100}) \) 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
\(7\) −7.00000 −0.377964
\(8\) −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 \(-0.393953\pi\)
0.327026 + 0.945015i \(0.393953\pi\)
\(14\) 37.8995 0.723505
\(15\) 0 0
\(16\) 219.765 3.43382
\(17\) −37.2254 −0.531087 −0.265544 0.964099i \(-0.585551\pi\)
−0.265544 + 0.964099i \(0.585551\pi\)
\(18\) 0 0
\(19\) 80.2254 0.968683 0.484341 0.874879i \(-0.339059\pi\)
0.484341 + 0.874879i \(0.339059\pi\)
\(20\) 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
\(28\) −149.196 −1.00698
\(29\) −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 \(-0.864867\pi\)
−0.911232 + 0.411894i \(0.864867\pi\)
\(32\) −613.186 −3.38741
\(33\) 0 0
\(34\) 201.546 1.01661
\(35\) −35.0000 −0.169031
\(36\) 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 \(-0.506889\pi\)
−0.0216405 + 0.999766i \(0.506889\pi\)
\(42\) 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 \(-0.310327\pi\)
0.561233 + 0.827658i \(0.310327\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) −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
\(56\) 504.583 1.20407
\(57\) 0 0
\(58\) 113.380 0.256681
\(59\) 616.000 1.35926 0.679630 0.733555i \(-0.262140\pi\)
0.679630 + 0.733555i \(0.262140\pi\)
\(60\) 0 0
\(61\) 15.2649 0.0320406 0.0160203 0.999872i \(-0.494900\pi\)
0.0160203 + 0.999872i \(0.494900\pi\)
\(62\) 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
\(70\) 189.497 0.323561
\(71\) 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 \(-0.537981\pi\)
−0.119037 + 0.992890i \(0.537981\pi\)
\(74\) −1067.40 −1.67679
\(75\) 0 0
\(76\) 1709.90 2.58078
\(77\) −365.784 −0.541363
\(78\) 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 \(-0.643875\pi\)
−0.436761 + 0.899577i \(0.643875\pi\)
\(84\) 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 \(-0.491322\pi\)
0.0272590 + 0.999628i \(0.491322\pi\)
\(90\) 0 0
\(91\) −214.598 −0.247209
\(92\) −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 \(-0.157286\pi\)
0.880382 + 0.474265i \(0.157286\pi\)
\(98\) −265.296 −0.273459
\(99\) 0 0
\(100\) 532.843 0.532843
\(101\) 434.167 0.427734 0.213867 0.976863i \(-0.431394\pi\)
0.213867 + 0.976863i \(0.431394\pi\)
\(102\) 0 0
\(103\) 345.577 0.330589 0.165295 0.986244i \(-0.447142\pi\)
0.165295 + 0.986244i \(0.447142\pi\)
\(104\) −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
\(112\) −1538.35 −1.29786
\(113\) 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
\(119\) 260.578 0.200732
\(120\) 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 \(-0.699975\pi\)
−0.587722 + 0.809063i \(0.699975\pi\)
\(132\) 0 0
\(133\) −561.578 −0.366128
\(134\) 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 \(-0.519050\pi\)
−0.0598122 + 0.998210i \(0.519050\pi\)
\(140\) −745.980 −0.450334
\(141\) 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
\(154\) 1980.43 1.03628
\(155\) −1572.79 −0.815030
\(156\) 0 0
\(157\) 1022.90 0.519977 0.259989 0.965612i \(-0.416281\pi\)
0.259989 + 0.965612i \(0.416281\pi\)
\(158\) −4643.91 −2.33829
\(159\) 0 0
\(160\) −3065.93 −1.51489
\(161\) 180.834 0.0885201
\(162\) 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 \(-0.407972\pi\)
0.285105 + 0.958496i \(0.407972\pi\)
\(168\) 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 \(-0.315912\pi\)
0.546626 + 0.837377i \(0.315912\pi\)
\(174\) 0 0
\(175\) −175.000 −0.0755929
\(176\) 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 \(-0.321230\pi\)
0.532561 + 0.846392i \(0.321230\pi\)
\(182\) 1161.88 0.473210
\(183\) 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
\(196\) 1044.37 0.380602
\(197\) −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 \(-0.300361\pi\)
0.586868 + 0.809682i \(0.300361\pi\)
\(200\) −1802.08 −0.637132
\(201\) 0 0
\(202\) −2350.67 −0.818775
\(203\) 146.588 0.0506820
\(204\) 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
\(217\) 2201.91 0.688826
\(218\) −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 \(-0.464216\pi\)
0.112183 + 0.993688i \(0.464216\pi\)
\(224\) 4292.30 1.28032
\(225\) 0 0
\(226\) −10039.1 −2.95481
\(227\) −1665.67 −0.487025 −0.243513 0.969898i \(-0.578300\pi\)
−0.243513 + 0.969898i \(0.578300\pi\)
\(228\) 0 0
\(229\) −6628.35 −1.91272 −0.956362 0.292183i \(-0.905618\pi\)
−0.956362 + 0.292183i \(0.905618\pi\)
\(230\) 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
\(238\) −1410.82 −0.384244
\(239\) −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 \(-0.818344\pi\)
−0.841529 + 0.540212i \(0.818344\pi\)
\(242\) −7577.56 −2.01283
\(243\) 0 0
\(244\) 325.352 0.0853629
\(245\) 245.000 0.0638877
\(246\) 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 \(-0.512487\pi\)
−0.0392197 + 0.999231i \(0.512487\pi\)
\(252\) 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 \(-0.0968728\pi\)
0.954046 + 0.299659i \(0.0968728\pi\)
\(258\) 0 0
\(259\) −1380.03 −0.331085
\(260\) 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
\(266\) 3040.50 0.700846
\(267\) 0 0
\(268\) −3548.94 −0.808903
\(269\) −1281.71 −0.290510 −0.145255 0.989394i \(-0.546400\pi\)
−0.145255 + 0.989394i \(0.546400\pi\)
\(270\) 0 0
\(271\) 4704.14 1.05445 0.527226 0.849725i \(-0.323232\pi\)
0.527226 + 0.849725i \(0.323232\pi\)
\(272\) −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
\(280\) 2522.91 0.538475
\(281\) 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 \(-0.709425\pi\)
−0.611479 + 0.791261i \(0.709425\pi\)
\(284\) 20290.7 4.23954
\(285\) 0 0
\(286\) −8673.40 −1.79325
\(287\) 79.5374 0.0163587
\(288\) 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 \(-0.766170\pi\)
−0.742100 + 0.670289i \(0.766170\pi\)
\(294\) 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
\(301\) 236.834 0.0453518
\(302\) 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 \(-0.522555\pi\)
−0.0707998 + 0.997491i \(0.522555\pi\)
\(308\) −7796.21 −1.44231
\(309\) 0 0
\(310\) 8515.43 1.56014
\(311\) −7718.69 −1.40735 −0.703677 0.710520i \(-0.748460\pi\)
−0.703677 + 0.710520i \(0.748460\pi\)
\(312\) 0 0
\(313\) 8556.00 1.54509 0.772546 0.634959i \(-0.218983\pi\)
0.772546 + 0.634959i \(0.218983\pi\)
\(314\) −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
\(322\) −979.076 −0.169446
\(323\) −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
\(329\) −2531.73 −0.424252
\(330\) 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
\(343\) −343.000 −0.0539949
\(344\) 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 \(-0.468663\pi\)
0.0982880 + 0.995158i \(0.468663\pi\)
\(350\) 947.487 0.144701
\(351\) 0 0
\(352\) −32041.9 −4.85182
\(353\) −5798.07 −0.874221 −0.437110 0.899408i \(-0.643998\pi\)
−0.437110 + 0.899408i \(0.643998\pi\)
\(354\) 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
\(364\) −4573.88 −0.658616
\(365\) −742.447 −0.106470
\(366\) 0 0
\(367\) −7372.85 −1.04866 −0.524332 0.851514i \(-0.675685\pi\)
−0.524332 + 0.851514i \(0.675685\pi\)
\(368\) −5677.28 −0.804209
\(369\) 0 0
\(370\) −5336.98 −0.749883
\(371\) 1071.14 0.149894
\(372\) 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 \(-0.352945\pi\)
0.445727 + 0.895169i \(0.352945\pi\)
\(384\) 0 0
\(385\) −1828.92 −0.242105
\(386\) 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
\(392\) −3532.08 −0.455094
\(393\) 0 0
\(394\) 3796.68 0.485467
\(395\) 4288.62 0.546289
\(396\) 0 0
\(397\) 4247.93 0.537021 0.268510 0.963277i \(-0.413469\pi\)
0.268510 + 0.963277i \(0.413469\pi\)
\(398\) −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
\(406\) −793.658 −0.0970162
\(407\) 10301.9 1.25466
\(408\) 0 0
\(409\) −319.205 −0.0385908 −0.0192954 0.999814i \(-0.506142\pi\)
−0.0192954 + 0.999814i \(0.506142\pi\)
\(410\) 307.595 0.0370512
\(411\) 0 0
\(412\) 7365.53 0.880761
\(413\) −4312.00 −0.513752
\(414\) 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 \(-0.232174\pi\)
0.745577 + 0.666420i \(0.232174\pi\)
\(420\) 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
\(427\) −106.855 −0.0121102
\(428\) −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 \(-0.575641\pi\)
−0.235403 + 0.971898i \(0.575641\pi\)
\(434\) −11921.6 −1.31856
\(435\) 0 0
\(436\) 36966.7 4.06051
\(437\) −2072.50 −0.226868
\(438\) 0 0
\(439\) −5434.12 −0.590789 −0.295394 0.955375i \(-0.595451\pi\)
−0.295394 + 0.955375i \(0.595451\pi\)
\(440\) −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
\(448\) −10932.6 −1.15294
\(449\) 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
\(455\) −1072.99 −0.110555
\(456\) 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 \(-0.750920\pi\)
−0.709148 + 0.705059i \(0.750920\pi\)
\(462\) 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 \(-0.612983\pi\)
−0.347539 + 0.937665i \(0.612983\pi\)
\(468\) 0 0
\(469\) 1165.57 0.114757
\(470\) −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
\(476\) 5553.88 0.534793
\(477\) 0 0
\(478\) 30213.7 2.89109
\(479\) −18134.7 −1.72984 −0.864922 0.501907i \(-0.832632\pi\)
−0.864922 + 0.501907i \(0.832632\pi\)
\(480\) 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
\(490\) −1326.48 −0.122295
\(491\) −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
\(497\) −6664.00 −0.601451
\(498\) 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 \(-0.328035\pi\)
0.514345 + 0.857584i \(0.328035\pi\)
\(504\) 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 \(-0.474087\pi\)
0.0813193 + 0.996688i \(0.474087\pi\)
\(510\) 0 0
\(511\) 1039.43 0.0899834
\(512\) −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
\(518\) 7471.78 0.633767
\(519\) 0 0
\(520\) −11049.2 −0.931809
\(521\) −6117.21 −0.514395 −0.257197 0.966359i \(-0.582799\pi\)
−0.257197 + 0.966359i \(0.582799\pi\)
\(522\) 0 0
\(523\) −16685.6 −1.39505 −0.697524 0.716561i \(-0.745715\pi\)
−0.697524 + 0.716561i \(0.745715\pi\)
\(524\) −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
\(532\) −11969.3 −0.975442
\(533\) −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
\(539\) 2560.49 0.204616
\(540\) 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
\(553\) −6004.07 −0.461698
\(554\) −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
\(560\) −7691.76 −0.580422
\(561\) 0 0
\(562\) −2008.15 −0.150728
\(563\) −21770.7 −1.62971 −0.814854 0.579666i \(-0.803183\pi\)
−0.814854 + 0.579666i \(0.803183\pi\)
\(564\) 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
\(574\) −430.632 −0.0313140
\(575\) −645.837 −0.0468405
\(576\) 0 0
\(577\) 4733.38 0.341513 0.170757 0.985313i \(-0.445379\pi\)
0.170757 + 0.985313i \(0.445379\pi\)
\(578\) 19097.4 1.37430
\(579\) 0 0
\(580\) −2231.67 −0.159767
\(581\) 4623.70 0.330161
\(582\) 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 \(-0.404086\pi\)
0.296785 + 0.954944i \(0.404086\pi\)
\(588\) 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 \(-0.727665\pi\)
−0.655791 + 0.754943i \(0.727665\pi\)
\(594\) 0 0
\(595\) 1302.89 0.0897701
\(596\) −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 \(-0.682025\pi\)
−0.541187 + 0.840902i \(0.682025\pi\)
\(602\) −1282.27 −0.0868131
\(603\) 0 0
\(604\) −49435.6 −3.33031
\(605\) 6997.84 0.470252
\(606\) 0 0
\(607\) −25993.2 −1.73811 −0.869053 0.494719i \(-0.835271\pi\)
−0.869053 + 0.494719i \(0.835271\pi\)
\(608\) −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
\(616\) 26366.9 1.72460
\(617\) −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 \(-0.617913\pi\)
−0.362020 + 0.932170i \(0.617913\pi\)
\(620\) −33522.0 −2.17142
\(621\) 0 0
\(622\) 41790.6 2.69397
\(623\) −320.422 −0.0206059
\(624\) 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
\(637\) 1502.19 0.0934361
\(638\) 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 \(-0.648811\pi\)
−0.450658 + 0.892696i \(0.648811\pi\)
\(644\) 3854.25 0.235837
\(645\) 0 0
\(646\) 16169.1 0.984777
\(647\) 12694.8 0.771383 0.385691 0.922628i \(-0.373963\pi\)
0.385691 + 0.922628i \(0.373963\pi\)
\(648\) 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
\(658\) 13707.3 0.812109
\(659\) 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 \(-0.489937\pi\)
0.0316096 + 0.999500i \(0.489937\pi\)
\(662\) 26703.6 1.56777
\(663\) 0 0
\(664\) 47613.0 2.78275
\(665\) −2807.89 −0.163737
\(666\) 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 \(-0.461703\pi\)
0.120024 + 0.992771i \(0.461703\pi\)
\(678\) 0 0
\(679\) −11774.9 −0.665506
\(680\) 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
\(686\) 1857.08 0.103358
\(687\) 0 0
\(688\) −7435.40 −0.412023
\(689\) −4691.09 −0.259385
\(690\) 0 0
\(691\) −33324.4 −1.83462 −0.917309 0.398177i \(-0.869643\pi\)
−0.917309 + 0.398177i \(0.869643\pi\)
\(692\) 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
\(700\) −3729.90 −0.201396
\(701\) 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
\(707\) −3039.17 −0.161668
\(708\) 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 \(-0.432843\pi\)
0.209419 + 0.977826i \(0.432843\pi\)
\(720\) 0 0
\(721\) −2419.04 −0.124951
\(722\) 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 \(-0.529831\pi\)
−0.0935794 + 0.995612i \(0.529831\pi\)
\(728\) 15468.9 0.787523
\(729\) 0 0
\(730\) 4019.77 0.203806
\(731\) 1259.46 0.0637250
\(732\) 0 0
\(733\) −14980.3 −0.754857 −0.377428 0.926039i \(-0.623192\pi\)
−0.377428 + 0.926039i \(0.623192\pi\)
\(734\) 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
\(742\) −5799.36 −0.286929
\(743\) −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
\(749\) 1519.83 0.0741434
\(750\) 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 \(-0.516831\pi\)
−0.0528527 + 0.998602i \(0.516831\pi\)
\(762\) 0 0
\(763\) −12140.9 −0.576054
\(764\) 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 \(-0.676592\pi\)
−0.526758 + 0.850015i \(0.676592\pi\)
\(770\) 9902.16 0.463440
\(771\) 0 0
\(772\) −32845.0 −1.53124
\(773\) −9674.79 −0.450165 −0.225083 0.974340i \(-0.572265\pi\)
−0.225083 + 0.974340i \(0.572265\pi\)
\(774\) 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
\(784\) 10768.5 0.490546
\(785\) 5114.51 0.232541
\(786\) 0 0
\(787\) −20942.8 −0.948577 −0.474288 0.880370i \(-0.657295\pi\)
−0.474288 + 0.880370i \(0.657295\pi\)
\(788\) −14946.1 −0.675676
\(789\) 0 0
\(790\) −23219.5 −1.04571
\(791\) −12979.4 −0.583433
\(792\) 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 \(-0.675115\pi\)
−0.522807 + 0.852451i \(0.675115\pi\)
\(798\) 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
\(805\) 904.172 0.0395874
\(806\) 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 \(-0.517307\pi\)
−0.0543443 + 0.998522i \(0.517307\pi\)
\(812\) 3124.33 0.135028
\(813\) 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
\(826\) 23346.1 0.983431
\(827\) −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 \(-0.347646\pi\)
0.460568 + 0.887624i \(0.347646\pi\)
\(830\) 17881.2 0.747790
\(831\) 0 0
\(832\) 47880.0 1.99512
\(833\) −1824.04 −0.0758696
\(834\) 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 \(-0.485083\pi\)
0.0468457 + 0.998902i \(0.485083\pi\)
\(840\) 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
\(847\) −9796.97 −0.397436
\(848\) −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 \(-0.413708\pi\)
0.267785 + 0.963479i \(0.413708\pi\)
\(854\) 578.533 0.0231815
\(855\) 0 0
\(856\) 15650.6 0.624915
\(857\) −18690.9 −0.745003 −0.372502 0.928032i \(-0.621500\pi\)
−0.372502 + 0.928032i \(0.621500\pi\)
\(858\) 0 0
\(859\) 18318.9 0.727628 0.363814 0.931472i \(-0.381474\pi\)
0.363814 + 0.931472i \(0.381474\pi\)
\(860\) −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
\(868\) 46930.8 1.83518
\(869\) 44820.3 1.74962
\(870\) 0 0
\(871\) −5104.66 −0.198582
\(872\) −125022. −4.85525
\(873\) 0 0
\(874\) 11221.0 0.434273
\(875\) −875.000 −0.0338062
\(876\) 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 \(-0.413161\pi\)
0.269441 + 0.963017i \(0.413161\pi\)
\(882\) 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 \(-0.761943\pi\)
−0.733134 + 0.680084i \(0.761943\pi\)
\(888\) 0 0
\(889\) −9761.57 −0.368270
\(890\) −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
\(896\) 24853.1 0.926658
\(897\) 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
\(910\) 5809.40 0.211626
\(911\) −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
\(917\) 12336.9 0.444276
\(918\) 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 \(-0.611606\pi\)
−0.343480 + 0.939160i \(0.611606\pi\)
\(930\) 0 0
\(931\) 3931.04 0.138383
\(932\) 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 \(-0.294815\pi\)
0.600884 + 0.799336i \(0.294815\pi\)
\(938\) −6310.63 −0.219669
\(939\) 0 0
\(940\) 38543.3 1.33739
\(941\) −14156.4 −0.490419 −0.245209 0.969470i \(-0.578857\pi\)
−0.245209 + 0.969470i \(0.578857\pi\)
\(942\) 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
\(952\) −18783.3 −0.639464
\(953\) −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
\(959\) −6460.64 −0.217544
\(960\) 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 \(-0.620429\pi\)
−0.369379 + 0.929279i \(0.620429\pi\)
\(972\) 0 0
\(973\) 1372.27 0.0452138
\(974\) −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
\(980\) 5221.86 0.170210
\(981\) 0 0
\(982\) 1194.42 0.0388141
\(983\) −34460.9 −1.11814 −0.559070 0.829120i \(-0.688842\pi\)
−0.559070 + 0.829120i \(0.688842\pi\)
\(984\) 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
\(994\) 36080.3 1.15131
\(995\) 16474.8 0.524911
\(996\) 0 0
\(997\) −50730.0 −1.61147 −0.805734 0.592277i \(-0.798229\pi\)
−0.805734 + 0.592277i \(0.798229\pi\)
\(998\) −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 315.4.a.f.1.1 2
3.2 odd 2 35.4.a.b.1.2 2
5.4 even 2 1575.4.a.z.1.2 2
7.6 odd 2 2205.4.a.u.1.1 2
12.11 even 2 560.4.a.r.1.2 2
15.2 even 4 175.4.b.c.99.4 4
15.8 even 4 175.4.b.c.99.1 4
15.14 odd 2 175.4.a.c.1.1 2
21.2 odd 6 245.4.e.h.116.1 4
21.5 even 6 245.4.e.i.116.1 4
21.11 odd 6 245.4.e.h.226.1 4
21.17 even 6 245.4.e.i.226.1 4
21.20 even 2 245.4.a.k.1.2 2
24.5 odd 2 2240.4.a.bn.1.2 2
24.11 even 2 2240.4.a.bo.1.1 2
105.104 even 2 1225.4.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 2 3.2 odd 2
175.4.a.c.1.1 2 15.14 odd 2
175.4.b.c.99.1 4 15.8 even 4
175.4.b.c.99.4 4 15.2 even 4
245.4.a.k.1.2 2 21.20 even 2
245.4.e.h.116.1 4 21.2 odd 6
245.4.e.h.226.1 4 21.11 odd 6
245.4.e.i.116.1 4 21.5 even 6
245.4.e.i.226.1 4 21.17 even 6
315.4.a.f.1.1 2 1.1 even 1 trivial
560.4.a.r.1.2 2 12.11 even 2
1225.4.a.m.1.1 2 105.104 even 2
1575.4.a.z.1.2 2 5.4 even 2
2205.4.a.u.1.1 2 7.6 odd 2
2240.4.a.bn.1.2 2 24.5 odd 2
2240.4.a.bo.1.1 2 24.11 even 2