Properties

Label 450.4.a.v.1.2
Level $450$
Weight $4$
Character 450.1
Self dual yes
Analytic conductor $26.551$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,4,Mod(1,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("450.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.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: \(26.5508595026\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{31}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 90)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(5.56776\) of defining polynomial
Character \(\chi\) \(=\) 450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +22.2711 q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} +22.2711 q^{7} +8.00000 q^{8} +22.2711 q^{11} -66.8132 q^{13} +44.5421 q^{14} +16.0000 q^{16} +62.0000 q^{17} +84.0000 q^{19} +44.5421 q^{22} +140.000 q^{23} -133.626 q^{26} +89.0842 q^{28} -200.440 q^{29} +16.0000 q^{31} +32.0000 q^{32} +124.000 q^{34} -244.982 q^{37} +168.000 q^{38} +222.711 q^{41} +356.337 q^{43} +89.0842 q^{44} +280.000 q^{46} +100.000 q^{47} +153.000 q^{49} -267.253 q^{52} +738.000 q^{53} +178.168 q^{56} -400.879 q^{58} +645.861 q^{59} -358.000 q^{61} +32.0000 q^{62} +64.0000 q^{64} +846.300 q^{67} +248.000 q^{68} -935.384 q^{71} -445.421 q^{73} -489.963 q^{74} +336.000 q^{76} +496.000 q^{77} -936.000 q^{79} +445.421 q^{82} +1304.00 q^{83} +712.674 q^{86} +178.168 q^{88} -712.674 q^{89} -1488.00 q^{91} +560.000 q^{92} +200.000 q^{94} -757.216 q^{97} +306.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 32 q^{16} + 124 q^{17} + 168 q^{19} + 280 q^{23} + 32 q^{31} + 64 q^{32} + 248 q^{34} + 336 q^{38} + 560 q^{46} + 200 q^{47} + 306 q^{49} + 1476 q^{53} - 716 q^{61} + 64 q^{62} + 128 q^{64} + 496 q^{68} + 672 q^{76} + 992 q^{77} - 1872 q^{79} + 2608 q^{83} - 2976 q^{91} + 1120 q^{92} + 400 q^{94} + 612 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 22.2711 1.20252 0.601262 0.799052i \(-0.294665\pi\)
0.601262 + 0.799052i \(0.294665\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 22.2711 0.610452 0.305226 0.952280i \(-0.401268\pi\)
0.305226 + 0.952280i \(0.401268\pi\)
\(12\) 0 0
\(13\) −66.8132 −1.42543 −0.712717 0.701452i \(-0.752536\pi\)
−0.712717 + 0.701452i \(0.752536\pi\)
\(14\) 44.5421 0.850313
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 62.0000 0.884542 0.442271 0.896882i \(-0.354173\pi\)
0.442271 + 0.896882i \(0.354173\pi\)
\(18\) 0 0
\(19\) 84.0000 1.01426 0.507130 0.861870i \(-0.330707\pi\)
0.507130 + 0.861870i \(0.330707\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 44.5421 0.431655
\(23\) 140.000 1.26922 0.634609 0.772833i \(-0.281161\pi\)
0.634609 + 0.772833i \(0.281161\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −133.626 −1.00793
\(27\) 0 0
\(28\) 89.0842 0.601262
\(29\) −200.440 −1.28347 −0.641736 0.766926i \(-0.721785\pi\)
−0.641736 + 0.766926i \(0.721785\pi\)
\(30\) 0 0
\(31\) 16.0000 0.0926995 0.0463498 0.998925i \(-0.485241\pi\)
0.0463498 + 0.998925i \(0.485241\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) 124.000 0.625465
\(35\) 0 0
\(36\) 0 0
\(37\) −244.982 −1.08851 −0.544253 0.838921i \(-0.683187\pi\)
−0.544253 + 0.838921i \(0.683187\pi\)
\(38\) 168.000 0.717189
\(39\) 0 0
\(40\) 0 0
\(41\) 222.711 0.848330 0.424165 0.905585i \(-0.360568\pi\)
0.424165 + 0.905585i \(0.360568\pi\)
\(42\) 0 0
\(43\) 356.337 1.26374 0.631871 0.775074i \(-0.282287\pi\)
0.631871 + 0.775074i \(0.282287\pi\)
\(44\) 89.0842 0.305226
\(45\) 0 0
\(46\) 280.000 0.897473
\(47\) 100.000 0.310351 0.155176 0.987887i \(-0.450406\pi\)
0.155176 + 0.987887i \(0.450406\pi\)
\(48\) 0 0
\(49\) 153.000 0.446064
\(50\) 0 0
\(51\) 0 0
\(52\) −267.253 −0.712717
\(53\) 738.000 1.91268 0.956341 0.292255i \(-0.0944055\pi\)
0.956341 + 0.292255i \(0.0944055\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 178.168 0.425156
\(57\) 0 0
\(58\) −400.879 −0.907552
\(59\) 645.861 1.42515 0.712575 0.701596i \(-0.247529\pi\)
0.712575 + 0.701596i \(0.247529\pi\)
\(60\) 0 0
\(61\) −358.000 −0.751430 −0.375715 0.926735i \(-0.622603\pi\)
−0.375715 + 0.926735i \(0.622603\pi\)
\(62\) 32.0000 0.0655485
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 846.300 1.54316 0.771582 0.636130i \(-0.219466\pi\)
0.771582 + 0.636130i \(0.219466\pi\)
\(68\) 248.000 0.442271
\(69\) 0 0
\(70\) 0 0
\(71\) −935.384 −1.56352 −0.781758 0.623581i \(-0.785677\pi\)
−0.781758 + 0.623581i \(0.785677\pi\)
\(72\) 0 0
\(73\) −445.421 −0.714145 −0.357073 0.934077i \(-0.616225\pi\)
−0.357073 + 0.934077i \(0.616225\pi\)
\(74\) −489.963 −0.769690
\(75\) 0 0
\(76\) 336.000 0.507130
\(77\) 496.000 0.734084
\(78\) 0 0
\(79\) −936.000 −1.33302 −0.666508 0.745498i \(-0.732212\pi\)
−0.666508 + 0.745498i \(0.732212\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 445.421 0.599860
\(83\) 1304.00 1.72449 0.862245 0.506492i \(-0.169058\pi\)
0.862245 + 0.506492i \(0.169058\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 712.674 0.893600
\(87\) 0 0
\(88\) 178.168 0.215828
\(89\) −712.674 −0.848801 −0.424400 0.905475i \(-0.639515\pi\)
−0.424400 + 0.905475i \(0.639515\pi\)
\(90\) 0 0
\(91\) −1488.00 −1.71412
\(92\) 560.000 0.634609
\(93\) 0 0
\(94\) 200.000 0.219451
\(95\) 0 0
\(96\) 0 0
\(97\) −757.216 −0.792615 −0.396307 0.918118i \(-0.629709\pi\)
−0.396307 + 0.918118i \(0.629709\pi\)
\(98\) 306.000 0.315415
\(99\) 0 0
\(100\) 0 0
\(101\) 645.861 0.636292 0.318146 0.948042i \(-0.396940\pi\)
0.318146 + 0.948042i \(0.396940\pi\)
\(102\) 0 0
\(103\) −690.403 −0.660460 −0.330230 0.943900i \(-0.607126\pi\)
−0.330230 + 0.943900i \(0.607126\pi\)
\(104\) −534.505 −0.503967
\(105\) 0 0
\(106\) 1476.00 1.35247
\(107\) 696.000 0.628830 0.314415 0.949286i \(-0.398192\pi\)
0.314415 + 0.949286i \(0.398192\pi\)
\(108\) 0 0
\(109\) −1142.00 −1.00352 −0.501760 0.865007i \(-0.667314\pi\)
−0.501760 + 0.865007i \(0.667314\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 356.337 0.300631
\(113\) −78.0000 −0.0649347 −0.0324674 0.999473i \(-0.510336\pi\)
−0.0324674 + 0.999473i \(0.510336\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −801.758 −0.641736
\(117\) 0 0
\(118\) 1291.72 1.00773
\(119\) 1380.81 1.06368
\(120\) 0 0
\(121\) −835.000 −0.627348
\(122\) −716.000 −0.531341
\(123\) 0 0
\(124\) 64.0000 0.0463498
\(125\) 0 0
\(126\) 0 0
\(127\) −1313.99 −0.918094 −0.459047 0.888412i \(-0.651809\pi\)
−0.459047 + 0.888412i \(0.651809\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −467.692 −0.311927 −0.155964 0.987763i \(-0.549848\pi\)
−0.155964 + 0.987763i \(0.549848\pi\)
\(132\) 0 0
\(133\) 1870.77 1.21967
\(134\) 1692.60 1.09118
\(135\) 0 0
\(136\) 496.000 0.312733
\(137\) −1062.00 −0.662283 −0.331142 0.943581i \(-0.607434\pi\)
−0.331142 + 0.943581i \(0.607434\pi\)
\(138\) 0 0
\(139\) −860.000 −0.524779 −0.262389 0.964962i \(-0.584510\pi\)
−0.262389 + 0.964962i \(0.584510\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1870.77 −1.10557
\(143\) −1488.00 −0.870160
\(144\) 0 0
\(145\) 0 0
\(146\) −890.842 −0.504977
\(147\) 0 0
\(148\) −979.927 −0.544253
\(149\) −1358.53 −0.746950 −0.373475 0.927640i \(-0.621834\pi\)
−0.373475 + 0.927640i \(0.621834\pi\)
\(150\) 0 0
\(151\) −1376.00 −0.741571 −0.370786 0.928718i \(-0.620912\pi\)
−0.370786 + 0.928718i \(0.620912\pi\)
\(152\) 672.000 0.358595
\(153\) 0 0
\(154\) 992.000 0.519076
\(155\) 0 0
\(156\) 0 0
\(157\) 1848.50 0.939657 0.469829 0.882758i \(-0.344316\pi\)
0.469829 + 0.882758i \(0.344316\pi\)
\(158\) −1872.00 −0.942584
\(159\) 0 0
\(160\) 0 0
\(161\) 3117.95 1.52627
\(162\) 0 0
\(163\) −222.711 −0.107019 −0.0535093 0.998567i \(-0.517041\pi\)
−0.0535093 + 0.998567i \(0.517041\pi\)
\(164\) 890.842 0.424165
\(165\) 0 0
\(166\) 2608.00 1.21940
\(167\) −2172.00 −1.00643 −0.503217 0.864160i \(-0.667850\pi\)
−0.503217 + 0.864160i \(0.667850\pi\)
\(168\) 0 0
\(169\) 2267.00 1.03186
\(170\) 0 0
\(171\) 0 0
\(172\) 1425.35 0.631871
\(173\) −3434.00 −1.50915 −0.754573 0.656216i \(-0.772156\pi\)
−0.754573 + 0.656216i \(0.772156\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 356.337 0.152613
\(177\) 0 0
\(178\) −1425.35 −0.600193
\(179\) −3362.93 −1.40423 −0.702115 0.712064i \(-0.747761\pi\)
−0.702115 + 0.712064i \(0.747761\pi\)
\(180\) 0 0
\(181\) −1678.00 −0.689087 −0.344544 0.938770i \(-0.611966\pi\)
−0.344544 + 0.938770i \(0.611966\pi\)
\(182\) −2976.00 −1.21206
\(183\) 0 0
\(184\) 1120.00 0.448736
\(185\) 0 0
\(186\) 0 0
\(187\) 1380.81 0.539971
\(188\) 400.000 0.155176
\(189\) 0 0
\(190\) 0 0
\(191\) −1737.14 −0.658090 −0.329045 0.944314i \(-0.606727\pi\)
−0.329045 + 0.944314i \(0.606727\pi\)
\(192\) 0 0
\(193\) −2449.82 −0.913687 −0.456844 0.889547i \(-0.651020\pi\)
−0.456844 + 0.889547i \(0.651020\pi\)
\(194\) −1514.43 −0.560463
\(195\) 0 0
\(196\) 612.000 0.223032
\(197\) 526.000 0.190233 0.0951166 0.995466i \(-0.469678\pi\)
0.0951166 + 0.995466i \(0.469678\pi\)
\(198\) 0 0
\(199\) −744.000 −0.265029 −0.132514 0.991181i \(-0.542305\pi\)
−0.132514 + 0.991181i \(0.542305\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1291.72 0.449927
\(203\) −4464.00 −1.54341
\(204\) 0 0
\(205\) 0 0
\(206\) −1380.81 −0.467016
\(207\) 0 0
\(208\) −1069.01 −0.356358
\(209\) 1870.77 0.619157
\(210\) 0 0
\(211\) 1476.00 0.481574 0.240787 0.970578i \(-0.422595\pi\)
0.240787 + 0.970578i \(0.422595\pi\)
\(212\) 2952.00 0.956341
\(213\) 0 0
\(214\) 1392.00 0.444650
\(215\) 0 0
\(216\) 0 0
\(217\) 356.337 0.111473
\(218\) −2284.00 −0.709596
\(219\) 0 0
\(220\) 0 0
\(221\) −4142.42 −1.26086
\(222\) 0 0
\(223\) 824.029 0.247449 0.123724 0.992317i \(-0.460516\pi\)
0.123724 + 0.992317i \(0.460516\pi\)
\(224\) 712.674 0.212578
\(225\) 0 0
\(226\) −156.000 −0.0459158
\(227\) −2200.00 −0.643256 −0.321628 0.946866i \(-0.604230\pi\)
−0.321628 + 0.946866i \(0.604230\pi\)
\(228\) 0 0
\(229\) 5246.00 1.51382 0.756911 0.653517i \(-0.226707\pi\)
0.756911 + 0.653517i \(0.226707\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1603.52 −0.453776
\(233\) −842.000 −0.236744 −0.118372 0.992969i \(-0.537767\pi\)
−0.118372 + 0.992969i \(0.537767\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 2583.44 0.712575
\(237\) 0 0
\(238\) 2761.61 0.752137
\(239\) 5835.02 1.57923 0.789615 0.613603i \(-0.210280\pi\)
0.789615 + 0.613603i \(0.210280\pi\)
\(240\) 0 0
\(241\) 2530.00 0.676231 0.338115 0.941105i \(-0.390211\pi\)
0.338115 + 0.941105i \(0.390211\pi\)
\(242\) −1670.00 −0.443602
\(243\) 0 0
\(244\) −1432.00 −0.375715
\(245\) 0 0
\(246\) 0 0
\(247\) −5612.31 −1.44576
\(248\) 128.000 0.0327742
\(249\) 0 0
\(250\) 0 0
\(251\) 6748.13 1.69696 0.848482 0.529223i \(-0.177517\pi\)
0.848482 + 0.529223i \(0.177517\pi\)
\(252\) 0 0
\(253\) 3117.95 0.774797
\(254\) −2627.98 −0.649191
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 258.000 0.0626210 0.0313105 0.999510i \(-0.490032\pi\)
0.0313105 + 0.999510i \(0.490032\pi\)
\(258\) 0 0
\(259\) −5456.00 −1.30895
\(260\) 0 0
\(261\) 0 0
\(262\) −935.384 −0.220566
\(263\) −772.000 −0.181002 −0.0905011 0.995896i \(-0.528847\pi\)
−0.0905011 + 0.995896i \(0.528847\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 3741.54 0.862438
\(267\) 0 0
\(268\) 3385.20 0.771582
\(269\) 2605.71 0.590607 0.295303 0.955404i \(-0.404579\pi\)
0.295303 + 0.955404i \(0.404579\pi\)
\(270\) 0 0
\(271\) −6392.00 −1.43279 −0.716395 0.697694i \(-0.754209\pi\)
−0.716395 + 0.697694i \(0.754209\pi\)
\(272\) 992.000 0.221135
\(273\) 0 0
\(274\) −2124.00 −0.468305
\(275\) 0 0
\(276\) 0 0
\(277\) 1759.41 0.381635 0.190818 0.981626i \(-0.438886\pi\)
0.190818 + 0.981626i \(0.438886\pi\)
\(278\) −1720.00 −0.371075
\(279\) 0 0
\(280\) 0 0
\(281\) −6414.06 −1.36168 −0.680838 0.732434i \(-0.738384\pi\)
−0.680838 + 0.732434i \(0.738384\pi\)
\(282\) 0 0
\(283\) 1380.81 0.290037 0.145018 0.989429i \(-0.453676\pi\)
0.145018 + 0.989429i \(0.453676\pi\)
\(284\) −3741.54 −0.781758
\(285\) 0 0
\(286\) −2976.00 −0.615296
\(287\) 4960.00 1.02014
\(288\) 0 0
\(289\) −1069.00 −0.217586
\(290\) 0 0
\(291\) 0 0
\(292\) −1781.68 −0.357073
\(293\) 1374.00 0.273959 0.136979 0.990574i \(-0.456261\pi\)
0.136979 + 0.990574i \(0.456261\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1959.85 −0.384845
\(297\) 0 0
\(298\) −2717.07 −0.528173
\(299\) −9353.84 −1.80919
\(300\) 0 0
\(301\) 7936.00 1.51968
\(302\) −2752.00 −0.524370
\(303\) 0 0
\(304\) 1344.00 0.253565
\(305\) 0 0
\(306\) 0 0
\(307\) 267.253 0.0496838 0.0248419 0.999691i \(-0.492092\pi\)
0.0248419 + 0.999691i \(0.492092\pi\)
\(308\) 1984.00 0.367042
\(309\) 0 0
\(310\) 0 0
\(311\) −3607.91 −0.657832 −0.328916 0.944359i \(-0.606683\pi\)
−0.328916 + 0.944359i \(0.606683\pi\)
\(312\) 0 0
\(313\) −7794.87 −1.40764 −0.703821 0.710377i \(-0.748524\pi\)
−0.703821 + 0.710377i \(0.748524\pi\)
\(314\) 3697.00 0.664438
\(315\) 0 0
\(316\) −3744.00 −0.666508
\(317\) −9334.00 −1.65378 −0.826892 0.562360i \(-0.809893\pi\)
−0.826892 + 0.562360i \(0.809893\pi\)
\(318\) 0 0
\(319\) −4464.00 −0.783498
\(320\) 0 0
\(321\) 0 0
\(322\) 6235.90 1.07923
\(323\) 5208.00 0.897154
\(324\) 0 0
\(325\) 0 0
\(326\) −445.421 −0.0756736
\(327\) 0 0
\(328\) 1781.68 0.299930
\(329\) 2227.11 0.373205
\(330\) 0 0
\(331\) 1636.00 0.271670 0.135835 0.990731i \(-0.456628\pi\)
0.135835 + 0.990731i \(0.456628\pi\)
\(332\) 5216.00 0.862245
\(333\) 0 0
\(334\) −4344.00 −0.711656
\(335\) 0 0
\(336\) 0 0
\(337\) −178.168 −0.0287996 −0.0143998 0.999896i \(-0.504584\pi\)
−0.0143998 + 0.999896i \(0.504584\pi\)
\(338\) 4534.00 0.729636
\(339\) 0 0
\(340\) 0 0
\(341\) 356.337 0.0565886
\(342\) 0 0
\(343\) −4231.50 −0.666121
\(344\) 2850.70 0.446800
\(345\) 0 0
\(346\) −6868.00 −1.06713
\(347\) 3840.00 0.594069 0.297035 0.954867i \(-0.404002\pi\)
0.297035 + 0.954867i \(0.404002\pi\)
\(348\) 0 0
\(349\) 6682.00 1.02487 0.512434 0.858726i \(-0.328744\pi\)
0.512434 + 0.858726i \(0.328744\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 712.674 0.107914
\(353\) 2110.00 0.318142 0.159071 0.987267i \(-0.449150\pi\)
0.159071 + 0.987267i \(0.449150\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2850.70 −0.424400
\(357\) 0 0
\(358\) −6725.86 −0.992941
\(359\) −1870.77 −0.275029 −0.137514 0.990500i \(-0.543911\pi\)
−0.137514 + 0.990500i \(0.543911\pi\)
\(360\) 0 0
\(361\) 197.000 0.0287214
\(362\) −3356.00 −0.487258
\(363\) 0 0
\(364\) −5952.00 −0.857059
\(365\) 0 0
\(366\) 0 0
\(367\) 9643.37 1.37161 0.685803 0.727787i \(-0.259451\pi\)
0.685803 + 0.727787i \(0.259451\pi\)
\(368\) 2240.00 0.317305
\(369\) 0 0
\(370\) 0 0
\(371\) 16436.0 2.30005
\(372\) 0 0
\(373\) 11558.7 1.60452 0.802260 0.596975i \(-0.203631\pi\)
0.802260 + 0.596975i \(0.203631\pi\)
\(374\) 2761.61 0.381817
\(375\) 0 0
\(376\) 800.000 0.109726
\(377\) 13392.0 1.82950
\(378\) 0 0
\(379\) −6220.00 −0.843008 −0.421504 0.906827i \(-0.638498\pi\)
−0.421504 + 0.906827i \(0.638498\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −3474.28 −0.465340
\(383\) 13900.0 1.85446 0.927228 0.374497i \(-0.122185\pi\)
0.927228 + 0.374497i \(0.122185\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4899.63 −0.646074
\(387\) 0 0
\(388\) −3028.86 −0.396307
\(389\) 3630.18 0.473156 0.236578 0.971613i \(-0.423974\pi\)
0.236578 + 0.971613i \(0.423974\pi\)
\(390\) 0 0
\(391\) 8680.00 1.12268
\(392\) 1224.00 0.157707
\(393\) 0 0
\(394\) 1052.00 0.134515
\(395\) 0 0
\(396\) 0 0
\(397\) −512.234 −0.0647564 −0.0323782 0.999476i \(-0.510308\pi\)
−0.0323782 + 0.999476i \(0.510308\pi\)
\(398\) −1488.00 −0.187404
\(399\) 0 0
\(400\) 0 0
\(401\) 2984.32 0.371646 0.185823 0.982583i \(-0.440505\pi\)
0.185823 + 0.982583i \(0.440505\pi\)
\(402\) 0 0
\(403\) −1069.01 −0.132137
\(404\) 2583.44 0.318146
\(405\) 0 0
\(406\) −8928.00 −1.09135
\(407\) −5456.00 −0.664481
\(408\) 0 0
\(409\) 14822.0 1.79193 0.895967 0.444121i \(-0.146484\pi\)
0.895967 + 0.444121i \(0.146484\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2761.61 −0.330230
\(413\) 14384.0 1.71378
\(414\) 0 0
\(415\) 0 0
\(416\) −2138.02 −0.251983
\(417\) 0 0
\(418\) 3741.54 0.437810
\(419\) −8663.44 −1.01011 −0.505056 0.863087i \(-0.668528\pi\)
−0.505056 + 0.863087i \(0.668528\pi\)
\(420\) 0 0
\(421\) 4894.00 0.566553 0.283277 0.959038i \(-0.408579\pi\)
0.283277 + 0.959038i \(0.408579\pi\)
\(422\) 2952.00 0.340524
\(423\) 0 0
\(424\) 5904.00 0.676235
\(425\) 0 0
\(426\) 0 0
\(427\) −7973.04 −0.903612
\(428\) 2784.00 0.314415
\(429\) 0 0
\(430\) 0 0
\(431\) −9799.27 −1.09516 −0.547580 0.836753i \(-0.684451\pi\)
−0.547580 + 0.836753i \(0.684451\pi\)
\(432\) 0 0
\(433\) 3697.00 0.410315 0.205157 0.978729i \(-0.434229\pi\)
0.205157 + 0.978729i \(0.434229\pi\)
\(434\) 712.674 0.0788236
\(435\) 0 0
\(436\) −4568.00 −0.501760
\(437\) 11760.0 1.28732
\(438\) 0 0
\(439\) −9288.00 −1.00978 −0.504888 0.863185i \(-0.668466\pi\)
−0.504888 + 0.863185i \(0.668466\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −8284.83 −0.891560
\(443\) −7368.00 −0.790213 −0.395106 0.918635i \(-0.629292\pi\)
−0.395106 + 0.918635i \(0.629292\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1648.06 0.174973
\(447\) 0 0
\(448\) 1425.35 0.150316
\(449\) −400.879 −0.0421351 −0.0210675 0.999778i \(-0.506707\pi\)
−0.0210675 + 0.999778i \(0.506707\pi\)
\(450\) 0 0
\(451\) 4960.00 0.517865
\(452\) −312.000 −0.0324674
\(453\) 0 0
\(454\) −4400.00 −0.454851
\(455\) 0 0
\(456\) 0 0
\(457\) −14743.4 −1.50912 −0.754561 0.656230i \(-0.772150\pi\)
−0.754561 + 0.656230i \(0.772150\pi\)
\(458\) 10492.0 1.07043
\(459\) 0 0
\(460\) 0 0
\(461\) −4743.74 −0.479258 −0.239629 0.970865i \(-0.577026\pi\)
−0.239629 + 0.970865i \(0.577026\pi\)
\(462\) 0 0
\(463\) 7282.64 0.731000 0.365500 0.930811i \(-0.380898\pi\)
0.365500 + 0.930811i \(0.380898\pi\)
\(464\) −3207.03 −0.320868
\(465\) 0 0
\(466\) −1684.00 −0.167403
\(467\) 5664.00 0.561239 0.280620 0.959819i \(-0.409460\pi\)
0.280620 + 0.959819i \(0.409460\pi\)
\(468\) 0 0
\(469\) 18848.0 1.85569
\(470\) 0 0
\(471\) 0 0
\(472\) 5166.89 0.503867
\(473\) 7936.00 0.771454
\(474\) 0 0
\(475\) 0 0
\(476\) 5523.22 0.531841
\(477\) 0 0
\(478\) 11670.0 1.11668
\(479\) 10111.1 0.964480 0.482240 0.876039i \(-0.339823\pi\)
0.482240 + 0.876039i \(0.339823\pi\)
\(480\) 0 0
\(481\) 16368.0 1.55159
\(482\) 5060.00 0.478167
\(483\) 0 0
\(484\) −3340.00 −0.313674
\(485\) 0 0
\(486\) 0 0
\(487\) 11558.7 1.07551 0.537755 0.843101i \(-0.319272\pi\)
0.537755 + 0.843101i \(0.319272\pi\)
\(488\) −2864.00 −0.265670
\(489\) 0 0
\(490\) 0 0
\(491\) −467.692 −0.0429871 −0.0214935 0.999769i \(-0.506842\pi\)
−0.0214935 + 0.999769i \(0.506842\pi\)
\(492\) 0 0
\(493\) −12427.3 −1.13528
\(494\) −11224.6 −1.02231
\(495\) 0 0
\(496\) 256.000 0.0231749
\(497\) −20832.0 −1.88017
\(498\) 0 0
\(499\) 9204.00 0.825707 0.412853 0.910798i \(-0.364532\pi\)
0.412853 + 0.910798i \(0.364532\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 13496.3 1.19994
\(503\) −1260.00 −0.111691 −0.0558455 0.998439i \(-0.517785\pi\)
−0.0558455 + 0.998439i \(0.517785\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6235.90 0.547864
\(507\) 0 0
\(508\) −5255.97 −0.459047
\(509\) −2071.21 −0.180363 −0.0901814 0.995925i \(-0.528745\pi\)
−0.0901814 + 0.995925i \(0.528745\pi\)
\(510\) 0 0
\(511\) −9920.00 −0.858777
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 516.000 0.0442797
\(515\) 0 0
\(516\) 0 0
\(517\) 2227.11 0.189455
\(518\) −10912.0 −0.925571
\(519\) 0 0
\(520\) 0 0
\(521\) −5300.51 −0.445719 −0.222860 0.974851i \(-0.571539\pi\)
−0.222860 + 0.974851i \(0.571539\pi\)
\(522\) 0 0
\(523\) −9309.30 −0.778331 −0.389166 0.921168i \(-0.627237\pi\)
−0.389166 + 0.921168i \(0.627237\pi\)
\(524\) −1870.77 −0.155964
\(525\) 0 0
\(526\) −1544.00 −0.127988
\(527\) 992.000 0.0819966
\(528\) 0 0
\(529\) 7433.00 0.610915
\(530\) 0 0
\(531\) 0 0
\(532\) 7483.08 0.609835
\(533\) −14880.0 −1.20924
\(534\) 0 0
\(535\) 0 0
\(536\) 6770.40 0.545591
\(537\) 0 0
\(538\) 5211.43 0.417622
\(539\) 3407.47 0.272301
\(540\) 0 0
\(541\) 9658.00 0.767523 0.383761 0.923432i \(-0.374629\pi\)
0.383761 + 0.923432i \(0.374629\pi\)
\(542\) −12784.0 −1.01314
\(543\) 0 0
\(544\) 1984.00 0.156366
\(545\) 0 0
\(546\) 0 0
\(547\) 890.842 0.0696338 0.0348169 0.999394i \(-0.488915\pi\)
0.0348169 + 0.999394i \(0.488915\pi\)
\(548\) −4248.00 −0.331142
\(549\) 0 0
\(550\) 0 0
\(551\) −16836.9 −1.30177
\(552\) 0 0
\(553\) −20845.7 −1.60298
\(554\) 3518.83 0.269857
\(555\) 0 0
\(556\) −3440.00 −0.262389
\(557\) 9066.00 0.689657 0.344828 0.938666i \(-0.387937\pi\)
0.344828 + 0.938666i \(0.387937\pi\)
\(558\) 0 0
\(559\) −23808.0 −1.80138
\(560\) 0 0
\(561\) 0 0
\(562\) −12828.1 −0.962850
\(563\) −1568.00 −0.117377 −0.0586886 0.998276i \(-0.518692\pi\)
−0.0586886 + 0.998276i \(0.518692\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2761.61 0.205087
\(567\) 0 0
\(568\) −7483.08 −0.552787
\(569\) 2093.48 0.154241 0.0771206 0.997022i \(-0.475427\pi\)
0.0771206 + 0.997022i \(0.475427\pi\)
\(570\) 0 0
\(571\) −13916.0 −1.01991 −0.509953 0.860202i \(-0.670337\pi\)
−0.509953 + 0.860202i \(0.670337\pi\)
\(572\) −5952.00 −0.435080
\(573\) 0 0
\(574\) 9920.00 0.721346
\(575\) 0 0
\(576\) 0 0
\(577\) 8106.66 0.584896 0.292448 0.956281i \(-0.405530\pi\)
0.292448 + 0.956281i \(0.405530\pi\)
\(578\) −2138.00 −0.153857
\(579\) 0 0
\(580\) 0 0
\(581\) 29041.5 2.07374
\(582\) 0 0
\(583\) 16436.0 1.16760
\(584\) −3563.37 −0.252488
\(585\) 0 0
\(586\) 2748.00 0.193718
\(587\) −20808.0 −1.46310 −0.731549 0.681789i \(-0.761202\pi\)
−0.731549 + 0.681789i \(0.761202\pi\)
\(588\) 0 0
\(589\) 1344.00 0.0940213
\(590\) 0 0
\(591\) 0 0
\(592\) −3919.71 −0.272127
\(593\) −21486.0 −1.48790 −0.743950 0.668236i \(-0.767050\pi\)
−0.743950 + 0.668236i \(0.767050\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −5434.14 −0.373475
\(597\) 0 0
\(598\) −18707.7 −1.27929
\(599\) 25923.5 1.76829 0.884145 0.467212i \(-0.154742\pi\)
0.884145 + 0.467212i \(0.154742\pi\)
\(600\) 0 0
\(601\) −250.000 −0.0169679 −0.00848395 0.999964i \(-0.502701\pi\)
−0.00848395 + 0.999964i \(0.502701\pi\)
\(602\) 15872.0 1.07458
\(603\) 0 0
\(604\) −5504.00 −0.370786
\(605\) 0 0
\(606\) 0 0
\(607\) 13340.4 0.892041 0.446020 0.895023i \(-0.352841\pi\)
0.446020 + 0.895023i \(0.352841\pi\)
\(608\) 2688.00 0.179297
\(609\) 0 0
\(610\) 0 0
\(611\) −6681.32 −0.442385
\(612\) 0 0
\(613\) 14721.2 0.969955 0.484977 0.874527i \(-0.338828\pi\)
0.484977 + 0.874527i \(0.338828\pi\)
\(614\) 534.505 0.0351317
\(615\) 0 0
\(616\) 3968.00 0.259538
\(617\) −14410.0 −0.940235 −0.470117 0.882604i \(-0.655788\pi\)
−0.470117 + 0.882604i \(0.655788\pi\)
\(618\) 0 0
\(619\) −21580.0 −1.40125 −0.700625 0.713530i \(-0.747095\pi\)
−0.700625 + 0.713530i \(0.747095\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −7215.82 −0.465158
\(623\) −15872.0 −1.02070
\(624\) 0 0
\(625\) 0 0
\(626\) −15589.7 −0.995354
\(627\) 0 0
\(628\) 7393.99 0.469829
\(629\) −15188.9 −0.962829
\(630\) 0 0
\(631\) −15160.0 −0.956434 −0.478217 0.878242i \(-0.658717\pi\)
−0.478217 + 0.878242i \(0.658717\pi\)
\(632\) −7488.00 −0.471292
\(633\) 0 0
\(634\) −18668.0 −1.16940
\(635\) 0 0
\(636\) 0 0
\(637\) −10222.4 −0.635835
\(638\) −8928.00 −0.554017
\(639\) 0 0
\(640\) 0 0
\(641\) 6102.27 0.376014 0.188007 0.982168i \(-0.439797\pi\)
0.188007 + 0.982168i \(0.439797\pi\)
\(642\) 0 0
\(643\) 14298.0 0.876919 0.438459 0.898751i \(-0.355524\pi\)
0.438459 + 0.898751i \(0.355524\pi\)
\(644\) 12471.8 0.763133
\(645\) 0 0
\(646\) 10416.0 0.634384
\(647\) 11916.0 0.724059 0.362030 0.932167i \(-0.382084\pi\)
0.362030 + 0.932167i \(0.382084\pi\)
\(648\) 0 0
\(649\) 14384.0 0.869987
\(650\) 0 0
\(651\) 0 0
\(652\) −890.842 −0.0535093
\(653\) −24646.0 −1.47699 −0.738493 0.674261i \(-0.764462\pi\)
−0.738493 + 0.674261i \(0.764462\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3563.37 0.212083
\(657\) 0 0
\(658\) 4454.21 0.263896
\(659\) −17928.2 −1.05976 −0.529881 0.848072i \(-0.677764\pi\)
−0.529881 + 0.848072i \(0.677764\pi\)
\(660\) 0 0
\(661\) −14638.0 −0.861350 −0.430675 0.902507i \(-0.641724\pi\)
−0.430675 + 0.902507i \(0.641724\pi\)
\(662\) 3272.00 0.192100
\(663\) 0 0
\(664\) 10432.0 0.609699
\(665\) 0 0
\(666\) 0 0
\(667\) −28061.5 −1.62901
\(668\) −8688.00 −0.503217
\(669\) 0 0
\(670\) 0 0
\(671\) −7973.04 −0.458712
\(672\) 0 0
\(673\) 31134.9 1.78330 0.891652 0.452721i \(-0.149547\pi\)
0.891652 + 0.452721i \(0.149547\pi\)
\(674\) −356.337 −0.0203644
\(675\) 0 0
\(676\) 9068.00 0.515931
\(677\) −6382.00 −0.362305 −0.181152 0.983455i \(-0.557983\pi\)
−0.181152 + 0.983455i \(0.557983\pi\)
\(678\) 0 0
\(679\) −16864.0 −0.953138
\(680\) 0 0
\(681\) 0 0
\(682\) 712.674 0.0400142
\(683\) 4544.00 0.254570 0.127285 0.991866i \(-0.459374\pi\)
0.127285 + 0.991866i \(0.459374\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −8463.00 −0.471019
\(687\) 0 0
\(688\) 5701.39 0.315935
\(689\) −49308.1 −2.72640
\(690\) 0 0
\(691\) 13564.0 0.746742 0.373371 0.927682i \(-0.378202\pi\)
0.373371 + 0.927682i \(0.378202\pi\)
\(692\) −13736.0 −0.754573
\(693\) 0 0
\(694\) 7680.00 0.420070
\(695\) 0 0
\(696\) 0 0
\(697\) 13808.1 0.750384
\(698\) 13364.0 0.724692
\(699\) 0 0
\(700\) 0 0
\(701\) 1492.16 0.0803968 0.0401984 0.999192i \(-0.487201\pi\)
0.0401984 + 0.999192i \(0.487201\pi\)
\(702\) 0 0
\(703\) −20578.5 −1.10403
\(704\) 1425.35 0.0763066
\(705\) 0 0
\(706\) 4220.00 0.224960
\(707\) 14384.0 0.765157
\(708\) 0 0
\(709\) −17826.0 −0.944245 −0.472122 0.881533i \(-0.656512\pi\)
−0.472122 + 0.881533i \(0.656512\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −5701.39 −0.300096
\(713\) 2240.00 0.117656
\(714\) 0 0
\(715\) 0 0
\(716\) −13451.7 −0.702115
\(717\) 0 0
\(718\) −3741.54 −0.194475
\(719\) 18796.8 0.974967 0.487484 0.873132i \(-0.337915\pi\)
0.487484 + 0.873132i \(0.337915\pi\)
\(720\) 0 0
\(721\) −15376.0 −0.794219
\(722\) 394.000 0.0203091
\(723\) 0 0
\(724\) −6712.00 −0.344544
\(725\) 0 0
\(726\) 0 0
\(727\) −33072.5 −1.68720 −0.843598 0.536975i \(-0.819567\pi\)
−0.843598 + 0.536975i \(0.819567\pi\)
\(728\) −11904.0 −0.606032
\(729\) 0 0
\(730\) 0 0
\(731\) 22092.9 1.11783
\(732\) 0 0
\(733\) −34765.1 −1.75181 −0.875907 0.482481i \(-0.839736\pi\)
−0.875907 + 0.482481i \(0.839736\pi\)
\(734\) 19286.7 0.969872
\(735\) 0 0
\(736\) 4480.00 0.224368
\(737\) 18848.0 0.942028
\(738\) 0 0
\(739\) −22940.0 −1.14190 −0.570948 0.820986i \(-0.693424\pi\)
−0.570948 + 0.820986i \(0.693424\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 32872.1 1.62638
\(743\) −26460.0 −1.30649 −0.653246 0.757146i \(-0.726593\pi\)
−0.653246 + 0.757146i \(0.726593\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 23117.4 1.13457
\(747\) 0 0
\(748\) 5523.22 0.269985
\(749\) 15500.7 0.756184
\(750\) 0 0
\(751\) −30160.0 −1.46545 −0.732726 0.680524i \(-0.761752\pi\)
−0.732726 + 0.680524i \(0.761752\pi\)
\(752\) 1600.00 0.0775878
\(753\) 0 0
\(754\) 26784.0 1.29365
\(755\) 0 0
\(756\) 0 0
\(757\) 18596.3 0.892860 0.446430 0.894818i \(-0.352695\pi\)
0.446430 + 0.894818i \(0.352695\pi\)
\(758\) −12440.0 −0.596097
\(759\) 0 0
\(760\) 0 0
\(761\) 5478.68 0.260975 0.130488 0.991450i \(-0.458346\pi\)
0.130488 + 0.991450i \(0.458346\pi\)
\(762\) 0 0
\(763\) −25433.5 −1.20676
\(764\) −6948.57 −0.329045
\(765\) 0 0
\(766\) 27800.0 1.31130
\(767\) −43152.0 −2.03146
\(768\) 0 0
\(769\) 29406.0 1.37894 0.689472 0.724313i \(-0.257843\pi\)
0.689472 + 0.724313i \(0.257843\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −9799.27 −0.456844
\(773\) 31122.0 1.44810 0.724050 0.689748i \(-0.242279\pi\)
0.724050 + 0.689748i \(0.242279\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −6057.73 −0.280232
\(777\) 0 0
\(778\) 7260.36 0.334572
\(779\) 18707.7 0.860427
\(780\) 0 0
\(781\) −20832.0 −0.954453
\(782\) 17360.0 0.793852
\(783\) 0 0
\(784\) 2448.00 0.111516
\(785\) 0 0
\(786\) 0 0
\(787\) 34431.1 1.55951 0.779755 0.626085i \(-0.215344\pi\)
0.779755 + 0.626085i \(0.215344\pi\)
\(788\) 2104.00 0.0951166
\(789\) 0 0
\(790\) 0 0
\(791\) −1737.14 −0.0780856
\(792\) 0 0
\(793\) 23919.1 1.07111
\(794\) −1024.47 −0.0457897
\(795\) 0 0
\(796\) −2976.00 −0.132514
\(797\) −13066.0 −0.580704 −0.290352 0.956920i \(-0.593772\pi\)
−0.290352 + 0.956920i \(0.593772\pi\)
\(798\) 0 0
\(799\) 6200.00 0.274518
\(800\) 0 0
\(801\) 0 0
\(802\) 5968.64 0.262793
\(803\) −9920.00 −0.435952
\(804\) 0 0
\(805\) 0 0
\(806\) −2138.02 −0.0934350
\(807\) 0 0
\(808\) 5166.89 0.224963
\(809\) 39954.3 1.73636 0.868181 0.496247i \(-0.165289\pi\)
0.868181 + 0.496247i \(0.165289\pi\)
\(810\) 0 0
\(811\) 14428.0 0.624705 0.312352 0.949966i \(-0.398883\pi\)
0.312352 + 0.949966i \(0.398883\pi\)
\(812\) −17856.0 −0.771703
\(813\) 0 0
\(814\) −10912.0 −0.469859
\(815\) 0 0
\(816\) 0 0
\(817\) 29932.3 1.28176
\(818\) 29644.0 1.26709
\(819\) 0 0
\(820\) 0 0
\(821\) −16591.9 −0.705314 −0.352657 0.935753i \(-0.614722\pi\)
−0.352657 + 0.935753i \(0.614722\pi\)
\(822\) 0 0
\(823\) −25322.2 −1.07251 −0.536255 0.844056i \(-0.680162\pi\)
−0.536255 + 0.844056i \(0.680162\pi\)
\(824\) −5523.22 −0.233508
\(825\) 0 0
\(826\) 28768.0 1.21182
\(827\) 16592.0 0.697655 0.348827 0.937187i \(-0.386580\pi\)
0.348827 + 0.937187i \(0.386580\pi\)
\(828\) 0 0
\(829\) 18234.0 0.763924 0.381962 0.924178i \(-0.375249\pi\)
0.381962 + 0.924178i \(0.375249\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4276.04 −0.178179
\(833\) 9486.00 0.394562
\(834\) 0 0
\(835\) 0 0
\(836\) 7483.08 0.309578
\(837\) 0 0
\(838\) −17326.9 −0.714257
\(839\) −31357.6 −1.29033 −0.645165 0.764044i \(-0.723211\pi\)
−0.645165 + 0.764044i \(0.723211\pi\)
\(840\) 0 0
\(841\) 15787.0 0.647300
\(842\) 9788.00 0.400614
\(843\) 0 0
\(844\) 5904.00 0.240787
\(845\) 0 0
\(846\) 0 0
\(847\) −18596.3 −0.754401
\(848\) 11808.0 0.478170
\(849\) 0 0
\(850\) 0 0
\(851\) −34297.4 −1.38155
\(852\) 0 0
\(853\) 2249.38 0.0902898 0.0451449 0.998980i \(-0.485625\pi\)
0.0451449 + 0.998980i \(0.485625\pi\)
\(854\) −15946.1 −0.638950
\(855\) 0 0
\(856\) 5568.00 0.222325
\(857\) 25206.0 1.00469 0.502346 0.864667i \(-0.332470\pi\)
0.502346 + 0.864667i \(0.332470\pi\)
\(858\) 0 0
\(859\) 17540.0 0.696690 0.348345 0.937366i \(-0.386744\pi\)
0.348345 + 0.937366i \(0.386744\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −19598.5 −0.774395
\(863\) 33108.0 1.30592 0.652960 0.757392i \(-0.273527\pi\)
0.652960 + 0.757392i \(0.273527\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 7393.99 0.290136
\(867\) 0 0
\(868\) 1425.35 0.0557367
\(869\) −20845.7 −0.813743
\(870\) 0 0
\(871\) −56544.0 −2.19968
\(872\) −9136.00 −0.354798
\(873\) 0 0
\(874\) 23520.0 0.910270
\(875\) 0 0
\(876\) 0 0
\(877\) −29152.8 −1.12249 −0.561243 0.827651i \(-0.689677\pi\)
−0.561243 + 0.827651i \(0.689677\pi\)
\(878\) −18576.0 −0.714020
\(879\) 0 0
\(880\) 0 0
\(881\) −8819.34 −0.337266 −0.168633 0.985679i \(-0.553935\pi\)
−0.168633 + 0.985679i \(0.553935\pi\)
\(882\) 0 0
\(883\) 19643.1 0.748632 0.374316 0.927301i \(-0.377878\pi\)
0.374316 + 0.927301i \(0.377878\pi\)
\(884\) −16569.7 −0.630428
\(885\) 0 0
\(886\) −14736.0 −0.558765
\(887\) 8004.00 0.302985 0.151493 0.988458i \(-0.451592\pi\)
0.151493 + 0.988458i \(0.451592\pi\)
\(888\) 0 0
\(889\) −29264.0 −1.10403
\(890\) 0 0
\(891\) 0 0
\(892\) 3296.12 0.123724
\(893\) 8400.00 0.314776
\(894\) 0 0
\(895\) 0 0
\(896\) 2850.70 0.106289
\(897\) 0 0
\(898\) −801.758 −0.0297940
\(899\) −3207.03 −0.118977
\(900\) 0 0
\(901\) 45756.0 1.69185
\(902\) 9920.00 0.366186
\(903\) 0 0
\(904\) −624.000 −0.0229579
\(905\) 0 0
\(906\) 0 0
\(907\) 1336.26 0.0489194 0.0244597 0.999701i \(-0.492213\pi\)
0.0244597 + 0.999701i \(0.492213\pi\)
\(908\) −8800.00 −0.321628
\(909\) 0 0
\(910\) 0 0
\(911\) 32070.3 1.16634 0.583171 0.812350i \(-0.301812\pi\)
0.583171 + 0.812350i \(0.301812\pi\)
\(912\) 0 0
\(913\) 29041.5 1.05272
\(914\) −29486.9 −1.06711
\(915\) 0 0
\(916\) 20984.0 0.756911
\(917\) −10416.0 −0.375100
\(918\) 0 0
\(919\) 2832.00 0.101653 0.0508265 0.998707i \(-0.483814\pi\)
0.0508265 + 0.998707i \(0.483814\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −9487.47 −0.338886
\(923\) 62496.0 2.22869
\(924\) 0 0
\(925\) 0 0
\(926\) 14565.3 0.516895
\(927\) 0 0
\(928\) −6414.06 −0.226888
\(929\) 31936.7 1.12789 0.563945 0.825813i \(-0.309283\pi\)
0.563945 + 0.825813i \(0.309283\pi\)
\(930\) 0 0
\(931\) 12852.0 0.452425
\(932\) −3368.00 −0.118372
\(933\) 0 0
\(934\) 11328.0 0.396856
\(935\) 0 0
\(936\) 0 0
\(937\) −25745.3 −0.897613 −0.448807 0.893629i \(-0.648151\pi\)
−0.448807 + 0.893629i \(0.648151\pi\)
\(938\) 37696.0 1.31217
\(939\) 0 0
\(940\) 0 0
\(941\) −7059.93 −0.244577 −0.122289 0.992495i \(-0.539023\pi\)
−0.122289 + 0.992495i \(0.539023\pi\)
\(942\) 0 0
\(943\) 31179.5 1.07672
\(944\) 10333.8 0.356288
\(945\) 0 0
\(946\) 15872.0 0.545500
\(947\) 1752.00 0.0601186 0.0300593 0.999548i \(-0.490430\pi\)
0.0300593 + 0.999548i \(0.490430\pi\)
\(948\) 0 0
\(949\) 29760.0 1.01797
\(950\) 0 0
\(951\) 0 0
\(952\) 11046.4 0.376069
\(953\) 3850.00 0.130864 0.0654322 0.997857i \(-0.479157\pi\)
0.0654322 + 0.997857i \(0.479157\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 23340.1 0.789615
\(957\) 0 0
\(958\) 20222.1 0.681991
\(959\) −23651.9 −0.796411
\(960\) 0 0
\(961\) −29535.0 −0.991407
\(962\) 32736.0 1.09714
\(963\) 0 0
\(964\) 10120.0 0.338115
\(965\) 0 0
\(966\) 0 0
\(967\) −24475.9 −0.813952 −0.406976 0.913439i \(-0.633417\pi\)
−0.406976 + 0.913439i \(0.633417\pi\)
\(968\) −6680.00 −0.221801
\(969\) 0 0
\(970\) 0 0
\(971\) 2204.83 0.0728697 0.0364349 0.999336i \(-0.488400\pi\)
0.0364349 + 0.999336i \(0.488400\pi\)
\(972\) 0 0
\(973\) −19153.1 −0.631059
\(974\) 23117.4 0.760501
\(975\) 0 0
\(976\) −5728.00 −0.187857
\(977\) 29982.0 0.981790 0.490895 0.871219i \(-0.336670\pi\)
0.490895 + 0.871219i \(0.336670\pi\)
\(978\) 0 0
\(979\) −15872.0 −0.518153
\(980\) 0 0
\(981\) 0 0
\(982\) −935.384 −0.0303965
\(983\) 35284.0 1.14485 0.572424 0.819958i \(-0.306003\pi\)
0.572424 + 0.819958i \(0.306003\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −24854.5 −0.802767
\(987\) 0 0
\(988\) −22449.2 −0.722880
\(989\) 49887.2 1.60396
\(990\) 0 0
\(991\) −11528.0 −0.369525 −0.184762 0.982783i \(-0.559152\pi\)
−0.184762 + 0.982783i \(0.559152\pi\)
\(992\) 512.000 0.0163871
\(993\) 0 0
\(994\) −41664.0 −1.32948
\(995\) 0 0
\(996\) 0 0
\(997\) 9955.16 0.316232 0.158116 0.987421i \(-0.449458\pi\)
0.158116 + 0.987421i \(0.449458\pi\)
\(998\) 18408.0 0.583863
\(999\) 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 450.4.a.v.1.2 2
3.2 odd 2 450.4.a.u.1.2 2
5.2 odd 4 90.4.c.c.19.4 yes 4
5.3 odd 4 90.4.c.c.19.2 yes 4
5.4 even 2 450.4.a.u.1.1 2
15.2 even 4 90.4.c.c.19.1 4
15.8 even 4 90.4.c.c.19.3 yes 4
15.14 odd 2 inner 450.4.a.v.1.1 2
20.3 even 4 720.4.f.l.289.4 4
20.7 even 4 720.4.f.l.289.3 4
60.23 odd 4 720.4.f.l.289.1 4
60.47 odd 4 720.4.f.l.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.4.c.c.19.1 4 15.2 even 4
90.4.c.c.19.2 yes 4 5.3 odd 4
90.4.c.c.19.3 yes 4 15.8 even 4
90.4.c.c.19.4 yes 4 5.2 odd 4
450.4.a.u.1.1 2 5.4 even 2
450.4.a.u.1.2 2 3.2 odd 2
450.4.a.v.1.1 2 15.14 odd 2 inner
450.4.a.v.1.2 2 1.1 even 1 trivial
720.4.f.l.289.1 4 60.23 odd 4
720.4.f.l.289.2 4 60.47 odd 4
720.4.f.l.289.3 4 20.7 even 4
720.4.f.l.289.4 4 20.3 even 4