Properties

Label 153.4.a.f.1.2
Level $153$
Weight $4$
Character 153.1
Self dual yes
Analytic conductor $9.027$
Analytic rank $1$
Dimension $3$
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] = [153,4,Mod(1,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, 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("153.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 153.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: \(9.02729223088\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.5912.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 14x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.795427\) of defining polynomial
Character \(\chi\) \(=\) 153.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.79543 q^{2} -0.185590 q^{4} -7.18559 q^{5} +19.9241 q^{7} +22.8822 q^{8} +O(q^{10})\) \(q-2.79543 q^{2} -0.185590 q^{4} -7.18559 q^{5} +19.9241 q^{7} +22.8822 q^{8} +20.0868 q^{10} +20.2836 q^{11} -69.3212 q^{13} -55.6963 q^{14} -62.4808 q^{16} -17.0000 q^{17} -2.16352 q^{19} +1.33357 q^{20} -56.7013 q^{22} +13.0415 q^{23} -73.3673 q^{25} +193.782 q^{26} -3.69771 q^{28} -246.831 q^{29} -166.779 q^{31} -8.39714 q^{32} +47.5223 q^{34} -143.166 q^{35} -162.065 q^{37} +6.04795 q^{38} -164.422 q^{40} -253.572 q^{41} +556.702 q^{43} -3.76443 q^{44} -36.4565 q^{46} -198.218 q^{47} +53.9684 q^{49} +205.093 q^{50} +12.8653 q^{52} -496.904 q^{53} -145.750 q^{55} +455.907 q^{56} +689.997 q^{58} +343.050 q^{59} +332.776 q^{61} +466.218 q^{62} +523.320 q^{64} +498.114 q^{65} -169.821 q^{67} +3.15503 q^{68} +400.211 q^{70} +509.616 q^{71} -621.448 q^{73} +453.041 q^{74} +0.401527 q^{76} +404.132 q^{77} -1122.10 q^{79} +448.962 q^{80} +708.843 q^{82} +485.798 q^{83} +122.155 q^{85} -1556.22 q^{86} +464.134 q^{88} -1038.49 q^{89} -1381.16 q^{91} -2.42037 q^{92} +554.103 q^{94} +15.5462 q^{95} +1014.67 q^{97} -150.865 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 5 q^{2} + 13 q^{4} - 8 q^{5} - 8 q^{7} - 33 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 5 q^{2} + 13 q^{4} - 8 q^{5} - 8 q^{7} - 33 q^{8} - 38 q^{10} - 34 q^{11} + 36 q^{13} + 104 q^{14} - 79 q^{16} - 51 q^{17} - 142 q^{19} + 126 q^{20} - 248 q^{22} - 110 q^{23} - 193 q^{25} - 154 q^{26} - 472 q^{28} - 90 q^{29} - 148 q^{31} + 151 q^{32} + 85 q^{34} - 416 q^{35} + 110 q^{37} - 80 q^{38} - 202 q^{40} - 720 q^{41} - 146 q^{43} + 192 q^{44} + 748 q^{46} - 500 q^{47} + 379 q^{49} + 385 q^{50} + 1218 q^{52} - 610 q^{53} + 430 q^{55} + 1368 q^{56} + 1006 q^{58} + 216 q^{59} - 18 q^{61} + 904 q^{62} + 377 q^{64} + 966 q^{65} - 1404 q^{67} - 221 q^{68} + 1472 q^{70} + 960 q^{71} - 794 q^{73} + 1874 q^{74} - 392 q^{76} - 48 q^{77} - 276 q^{79} + 1130 q^{80} + 382 q^{82} + 1552 q^{83} + 136 q^{85} - 16 q^{86} + 1724 q^{88} - 1394 q^{89} - 3968 q^{91} - 2244 q^{92} + 3960 q^{94} + 602 q^{95} + 402 q^{97} - 4109 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.79543 −0.988333 −0.494166 0.869367i \(-0.664527\pi\)
−0.494166 + 0.869367i \(0.664527\pi\)
\(3\) 0 0
\(4\) −0.185590 −0.0231987
\(5\) −7.18559 −0.642699 −0.321349 0.946961i \(-0.604136\pi\)
−0.321349 + 0.946961i \(0.604136\pi\)
\(6\) 0 0
\(7\) 19.9241 1.07580 0.537899 0.843009i \(-0.319218\pi\)
0.537899 + 0.843009i \(0.319218\pi\)
\(8\) 22.8822 1.01126
\(9\) 0 0
\(10\) 20.0868 0.635200
\(11\) 20.2836 0.555976 0.277988 0.960585i \(-0.410332\pi\)
0.277988 + 0.960585i \(0.410332\pi\)
\(12\) 0 0
\(13\) −69.3212 −1.47894 −0.739471 0.673189i \(-0.764924\pi\)
−0.739471 + 0.673189i \(0.764924\pi\)
\(14\) −55.6963 −1.06325
\(15\) 0 0
\(16\) −62.4808 −0.976263
\(17\) −17.0000 −0.242536
\(18\) 0 0
\(19\) −2.16352 −0.0261234 −0.0130617 0.999915i \(-0.504158\pi\)
−0.0130617 + 0.999915i \(0.504158\pi\)
\(20\) 1.33357 0.0149098
\(21\) 0 0
\(22\) −56.7013 −0.549489
\(23\) 13.0415 0.118232 0.0591161 0.998251i \(-0.481172\pi\)
0.0591161 + 0.998251i \(0.481172\pi\)
\(24\) 0 0
\(25\) −73.3673 −0.586938
\(26\) 193.782 1.46169
\(27\) 0 0
\(28\) −3.69771 −0.0249572
\(29\) −246.831 −1.58053 −0.790264 0.612767i \(-0.790057\pi\)
−0.790264 + 0.612767i \(0.790057\pi\)
\(30\) 0 0
\(31\) −166.779 −0.966269 −0.483135 0.875546i \(-0.660502\pi\)
−0.483135 + 0.875546i \(0.660502\pi\)
\(32\) −8.39714 −0.0463881
\(33\) 0 0
\(34\) 47.5223 0.239706
\(35\) −143.166 −0.691414
\(36\) 0 0
\(37\) −162.065 −0.720090 −0.360045 0.932935i \(-0.617239\pi\)
−0.360045 + 0.932935i \(0.617239\pi\)
\(38\) 6.04795 0.0258186
\(39\) 0 0
\(40\) −164.422 −0.649936
\(41\) −253.572 −0.965886 −0.482943 0.875652i \(-0.660432\pi\)
−0.482943 + 0.875652i \(0.660432\pi\)
\(42\) 0 0
\(43\) 556.702 1.97433 0.987166 0.159698i \(-0.0510520\pi\)
0.987166 + 0.159698i \(0.0510520\pi\)
\(44\) −3.76443 −0.0128979
\(45\) 0 0
\(46\) −36.4565 −0.116853
\(47\) −198.218 −0.615170 −0.307585 0.951521i \(-0.599521\pi\)
−0.307585 + 0.951521i \(0.599521\pi\)
\(48\) 0 0
\(49\) 53.9684 0.157342
\(50\) 205.093 0.580090
\(51\) 0 0
\(52\) 12.8653 0.0343096
\(53\) −496.904 −1.28783 −0.643915 0.765097i \(-0.722691\pi\)
−0.643915 + 0.765097i \(0.722691\pi\)
\(54\) 0 0
\(55\) −145.750 −0.357325
\(56\) 455.907 1.08791
\(57\) 0 0
\(58\) 689.997 1.56209
\(59\) 343.050 0.756971 0.378486 0.925607i \(-0.376445\pi\)
0.378486 + 0.925607i \(0.376445\pi\)
\(60\) 0 0
\(61\) 332.776 0.698484 0.349242 0.937033i \(-0.386439\pi\)
0.349242 + 0.937033i \(0.386439\pi\)
\(62\) 466.218 0.954995
\(63\) 0 0
\(64\) 523.320 1.02211
\(65\) 498.114 0.950514
\(66\) 0 0
\(67\) −169.821 −0.309656 −0.154828 0.987941i \(-0.549482\pi\)
−0.154828 + 0.987941i \(0.549482\pi\)
\(68\) 3.15503 0.00562652
\(69\) 0 0
\(70\) 400.211 0.683347
\(71\) 509.616 0.851835 0.425918 0.904762i \(-0.359951\pi\)
0.425918 + 0.904762i \(0.359951\pi\)
\(72\) 0 0
\(73\) −621.448 −0.996370 −0.498185 0.867071i \(-0.666000\pi\)
−0.498185 + 0.867071i \(0.666000\pi\)
\(74\) 453.041 0.711688
\(75\) 0 0
\(76\) 0.401527 0.000606031 0
\(77\) 404.132 0.598118
\(78\) 0 0
\(79\) −1122.10 −1.59806 −0.799029 0.601292i \(-0.794653\pi\)
−0.799029 + 0.601292i \(0.794653\pi\)
\(80\) 448.962 0.627443
\(81\) 0 0
\(82\) 708.843 0.954617
\(83\) 485.798 0.642449 0.321224 0.947003i \(-0.395906\pi\)
0.321224 + 0.947003i \(0.395906\pi\)
\(84\) 0 0
\(85\) 122.155 0.155877
\(86\) −1556.22 −1.95130
\(87\) 0 0
\(88\) 464.134 0.562236
\(89\) −1038.49 −1.23685 −0.618424 0.785844i \(-0.712229\pi\)
−0.618424 + 0.785844i \(0.712229\pi\)
\(90\) 0 0
\(91\) −1381.16 −1.59104
\(92\) −2.42037 −0.00274284
\(93\) 0 0
\(94\) 554.103 0.607993
\(95\) 15.5462 0.0167895
\(96\) 0 0
\(97\) 1014.67 1.06211 0.531054 0.847338i \(-0.321796\pi\)
0.531054 + 0.847338i \(0.321796\pi\)
\(98\) −150.865 −0.155507
\(99\) 0 0
\(100\) 13.6162 0.0136162
\(101\) 653.602 0.643919 0.321959 0.946753i \(-0.395659\pi\)
0.321959 + 0.946753i \(0.395659\pi\)
\(102\) 0 0
\(103\) 489.213 0.467996 0.233998 0.972237i \(-0.424819\pi\)
0.233998 + 0.972237i \(0.424819\pi\)
\(104\) −1586.22 −1.49560
\(105\) 0 0
\(106\) 1389.06 1.27281
\(107\) −486.534 −0.439580 −0.219790 0.975547i \(-0.570537\pi\)
−0.219790 + 0.975547i \(0.570537\pi\)
\(108\) 0 0
\(109\) 88.6813 0.0779278 0.0389639 0.999241i \(-0.487594\pi\)
0.0389639 + 0.999241i \(0.487594\pi\)
\(110\) 407.432 0.353156
\(111\) 0 0
\(112\) −1244.87 −1.05026
\(113\) −1814.48 −1.51055 −0.755274 0.655409i \(-0.772496\pi\)
−0.755274 + 0.655409i \(0.772496\pi\)
\(114\) 0 0
\(115\) −93.7108 −0.0759876
\(116\) 45.8093 0.0366663
\(117\) 0 0
\(118\) −958.971 −0.748139
\(119\) −338.709 −0.260919
\(120\) 0 0
\(121\) −919.576 −0.690891
\(122\) −930.250 −0.690335
\(123\) 0 0
\(124\) 30.9525 0.0224162
\(125\) 1425.39 1.01992
\(126\) 0 0
\(127\) −2775.34 −1.93915 −0.969573 0.244803i \(-0.921277\pi\)
−0.969573 + 0.244803i \(0.921277\pi\)
\(128\) −1395.73 −0.963796
\(129\) 0 0
\(130\) −1392.44 −0.939424
\(131\) 1856.34 1.23808 0.619041 0.785359i \(-0.287521\pi\)
0.619041 + 0.785359i \(0.287521\pi\)
\(132\) 0 0
\(133\) −43.1061 −0.0281035
\(134\) 474.722 0.306043
\(135\) 0 0
\(136\) −388.998 −0.245267
\(137\) 1532.70 0.955823 0.477912 0.878408i \(-0.341394\pi\)
0.477912 + 0.878408i \(0.341394\pi\)
\(138\) 0 0
\(139\) 518.039 0.316111 0.158056 0.987430i \(-0.449477\pi\)
0.158056 + 0.987430i \(0.449477\pi\)
\(140\) 26.5702 0.0160399
\(141\) 0 0
\(142\) −1424.59 −0.841897
\(143\) −1406.08 −0.822256
\(144\) 0 0
\(145\) 1773.62 1.01580
\(146\) 1737.21 0.984745
\(147\) 0 0
\(148\) 30.0776 0.0167052
\(149\) −3401.05 −1.86997 −0.934983 0.354693i \(-0.884585\pi\)
−0.934983 + 0.354693i \(0.884585\pi\)
\(150\) 0 0
\(151\) 108.797 0.0586345 0.0293173 0.999570i \(-0.490667\pi\)
0.0293173 + 0.999570i \(0.490667\pi\)
\(152\) −49.5061 −0.0264176
\(153\) 0 0
\(154\) −1129.72 −0.591139
\(155\) 1198.40 0.621020
\(156\) 0 0
\(157\) 3553.34 1.80629 0.903145 0.429335i \(-0.141252\pi\)
0.903145 + 0.429335i \(0.141252\pi\)
\(158\) 3136.76 1.57941
\(159\) 0 0
\(160\) 60.3384 0.0298135
\(161\) 259.840 0.127194
\(162\) 0 0
\(163\) −3875.94 −1.86250 −0.931249 0.364384i \(-0.881280\pi\)
−0.931249 + 0.364384i \(0.881280\pi\)
\(164\) 47.0605 0.0224074
\(165\) 0 0
\(166\) −1358.01 −0.634953
\(167\) −730.820 −0.338638 −0.169319 0.985561i \(-0.554157\pi\)
−0.169319 + 0.985561i \(0.554157\pi\)
\(168\) 0 0
\(169\) 2608.43 1.18727
\(170\) −341.475 −0.154059
\(171\) 0 0
\(172\) −103.318 −0.0458020
\(173\) −2979.89 −1.30958 −0.654789 0.755812i \(-0.727242\pi\)
−0.654789 + 0.755812i \(0.727242\pi\)
\(174\) 0 0
\(175\) −1461.77 −0.631427
\(176\) −1267.34 −0.542779
\(177\) 0 0
\(178\) 2903.02 1.22242
\(179\) 3112.68 1.29974 0.649868 0.760047i \(-0.274824\pi\)
0.649868 + 0.760047i \(0.274824\pi\)
\(180\) 0 0
\(181\) 2454.92 1.00814 0.504068 0.863664i \(-0.331836\pi\)
0.504068 + 0.863664i \(0.331836\pi\)
\(182\) 3860.93 1.57248
\(183\) 0 0
\(184\) 298.418 0.119563
\(185\) 1164.53 0.462801
\(186\) 0 0
\(187\) −344.821 −0.134844
\(188\) 36.7872 0.0142712
\(189\) 0 0
\(190\) −43.4581 −0.0165936
\(191\) 1806.74 0.684457 0.342228 0.939617i \(-0.388818\pi\)
0.342228 + 0.939617i \(0.388818\pi\)
\(192\) 0 0
\(193\) −3252.39 −1.21302 −0.606508 0.795077i \(-0.707430\pi\)
−0.606508 + 0.795077i \(0.707430\pi\)
\(194\) −2836.45 −1.04972
\(195\) 0 0
\(196\) −10.0160 −0.00365015
\(197\) 2227.38 0.805554 0.402777 0.915298i \(-0.368045\pi\)
0.402777 + 0.915298i \(0.368045\pi\)
\(198\) 0 0
\(199\) 4309.13 1.53500 0.767502 0.641046i \(-0.221499\pi\)
0.767502 + 0.641046i \(0.221499\pi\)
\(200\) −1678.81 −0.593548
\(201\) 0 0
\(202\) −1827.10 −0.636406
\(203\) −4917.87 −1.70033
\(204\) 0 0
\(205\) 1822.07 0.620774
\(206\) −1367.56 −0.462536
\(207\) 0 0
\(208\) 4331.25 1.44384
\(209\) −43.8839 −0.0145240
\(210\) 0 0
\(211\) 4219.39 1.37666 0.688330 0.725398i \(-0.258344\pi\)
0.688330 + 0.725398i \(0.258344\pi\)
\(212\) 92.2204 0.0298761
\(213\) 0 0
\(214\) 1360.07 0.434451
\(215\) −4000.23 −1.26890
\(216\) 0 0
\(217\) −3322.91 −1.03951
\(218\) −247.902 −0.0770185
\(219\) 0 0
\(220\) 27.0497 0.00828949
\(221\) 1178.46 0.358696
\(222\) 0 0
\(223\) −4825.85 −1.44916 −0.724580 0.689191i \(-0.757966\pi\)
−0.724580 + 0.689191i \(0.757966\pi\)
\(224\) −167.305 −0.0499042
\(225\) 0 0
\(226\) 5072.25 1.49292
\(227\) 1665.18 0.486880 0.243440 0.969916i \(-0.421724\pi\)
0.243440 + 0.969916i \(0.421724\pi\)
\(228\) 0 0
\(229\) 2726.00 0.786633 0.393317 0.919403i \(-0.371328\pi\)
0.393317 + 0.919403i \(0.371328\pi\)
\(230\) 261.962 0.0751010
\(231\) 0 0
\(232\) −5648.03 −1.59833
\(233\) 4836.51 1.35987 0.679937 0.733271i \(-0.262007\pi\)
0.679937 + 0.733271i \(0.262007\pi\)
\(234\) 0 0
\(235\) 1424.31 0.395369
\(236\) −63.6666 −0.0175608
\(237\) 0 0
\(238\) 946.837 0.257875
\(239\) −5200.13 −1.40740 −0.703700 0.710497i \(-0.748470\pi\)
−0.703700 + 0.710497i \(0.748470\pi\)
\(240\) 0 0
\(241\) −372.453 −0.0995512 −0.0497756 0.998760i \(-0.515851\pi\)
−0.0497756 + 0.998760i \(0.515851\pi\)
\(242\) 2570.61 0.682830
\(243\) 0 0
\(244\) −61.7598 −0.0162040
\(245\) −387.795 −0.101124
\(246\) 0 0
\(247\) 149.978 0.0386350
\(248\) −3816.27 −0.977150
\(249\) 0 0
\(250\) −3984.56 −1.00802
\(251\) 902.694 0.227002 0.113501 0.993538i \(-0.463793\pi\)
0.113501 + 0.993538i \(0.463793\pi\)
\(252\) 0 0
\(253\) 264.528 0.0657342
\(254\) 7758.26 1.91652
\(255\) 0 0
\(256\) −284.912 −0.0695585
\(257\) 1361.43 0.330442 0.165221 0.986257i \(-0.447166\pi\)
0.165221 + 0.986257i \(0.447166\pi\)
\(258\) 0 0
\(259\) −3228.99 −0.774671
\(260\) −92.4449 −0.0220507
\(261\) 0 0
\(262\) −5189.25 −1.22364
\(263\) 1065.57 0.249833 0.124916 0.992167i \(-0.460134\pi\)
0.124916 + 0.992167i \(0.460134\pi\)
\(264\) 0 0
\(265\) 3570.55 0.827687
\(266\) 120.500 0.0277756
\(267\) 0 0
\(268\) 31.5171 0.00718363
\(269\) 5516.38 1.25033 0.625166 0.780491i \(-0.285031\pi\)
0.625166 + 0.780491i \(0.285031\pi\)
\(270\) 0 0
\(271\) 6271.52 1.40578 0.702892 0.711296i \(-0.251892\pi\)
0.702892 + 0.711296i \(0.251892\pi\)
\(272\) 1062.17 0.236779
\(273\) 0 0
\(274\) −4284.56 −0.944671
\(275\) −1488.15 −0.326324
\(276\) 0 0
\(277\) −612.909 −0.132946 −0.0664732 0.997788i \(-0.521175\pi\)
−0.0664732 + 0.997788i \(0.521175\pi\)
\(278\) −1448.14 −0.312423
\(279\) 0 0
\(280\) −3275.96 −0.699200
\(281\) 8957.63 1.90166 0.950831 0.309709i \(-0.100232\pi\)
0.950831 + 0.309709i \(0.100232\pi\)
\(282\) 0 0
\(283\) −366.649 −0.0770143 −0.0385072 0.999258i \(-0.512260\pi\)
−0.0385072 + 0.999258i \(0.512260\pi\)
\(284\) −94.5797 −0.0197615
\(285\) 0 0
\(286\) 3930.60 0.812662
\(287\) −5052.19 −1.03910
\(288\) 0 0
\(289\) 289.000 0.0588235
\(290\) −4958.04 −1.00395
\(291\) 0 0
\(292\) 115.335 0.0231145
\(293\) −544.260 −0.108519 −0.0542594 0.998527i \(-0.517280\pi\)
−0.0542594 + 0.998527i \(0.517280\pi\)
\(294\) 0 0
\(295\) −2465.02 −0.486504
\(296\) −3708.41 −0.728198
\(297\) 0 0
\(298\) 9507.39 1.84815
\(299\) −904.052 −0.174858
\(300\) 0 0
\(301\) 11091.8 2.12398
\(302\) −304.135 −0.0579504
\(303\) 0 0
\(304\) 135.178 0.0255033
\(305\) −2391.19 −0.448915
\(306\) 0 0
\(307\) −6140.08 −1.14148 −0.570738 0.821132i \(-0.693343\pi\)
−0.570738 + 0.821132i \(0.693343\pi\)
\(308\) −75.0028 −0.0138756
\(309\) 0 0
\(310\) −3350.05 −0.613774
\(311\) 2449.54 0.446626 0.223313 0.974747i \(-0.428313\pi\)
0.223313 + 0.974747i \(0.428313\pi\)
\(312\) 0 0
\(313\) −7062.20 −1.27533 −0.637666 0.770313i \(-0.720100\pi\)
−0.637666 + 0.770313i \(0.720100\pi\)
\(314\) −9933.11 −1.78522
\(315\) 0 0
\(316\) 208.251 0.0370730
\(317\) −4075.00 −0.722002 −0.361001 0.932565i \(-0.617565\pi\)
−0.361001 + 0.932565i \(0.617565\pi\)
\(318\) 0 0
\(319\) −5006.61 −0.878735
\(320\) −3760.36 −0.656909
\(321\) 0 0
\(322\) −726.362 −0.125710
\(323\) 36.7798 0.00633586
\(324\) 0 0
\(325\) 5085.91 0.868048
\(326\) 10834.9 1.84077
\(327\) 0 0
\(328\) −5802.30 −0.976763
\(329\) −3949.30 −0.661799
\(330\) 0 0
\(331\) 3869.63 0.642580 0.321290 0.946981i \(-0.395883\pi\)
0.321290 + 0.946981i \(0.395883\pi\)
\(332\) −90.1592 −0.0149040
\(333\) 0 0
\(334\) 2042.95 0.334687
\(335\) 1220.26 0.199015
\(336\) 0 0
\(337\) −6253.84 −1.01089 −0.505443 0.862860i \(-0.668671\pi\)
−0.505443 + 0.862860i \(0.668671\pi\)
\(338\) −7291.67 −1.17342
\(339\) 0 0
\(340\) −22.6708 −0.00361616
\(341\) −3382.87 −0.537222
\(342\) 0 0
\(343\) −5758.68 −0.906530
\(344\) 12738.6 1.99656
\(345\) 0 0
\(346\) 8330.06 1.29430
\(347\) 6632.55 1.02609 0.513046 0.858361i \(-0.328517\pi\)
0.513046 + 0.858361i \(0.328517\pi\)
\(348\) 0 0
\(349\) −3485.68 −0.534625 −0.267313 0.963610i \(-0.586136\pi\)
−0.267313 + 0.963610i \(0.586136\pi\)
\(350\) 4086.28 0.624060
\(351\) 0 0
\(352\) −170.324 −0.0257906
\(353\) −6439.69 −0.970963 −0.485481 0.874247i \(-0.661356\pi\)
−0.485481 + 0.874247i \(0.661356\pi\)
\(354\) 0 0
\(355\) −3661.89 −0.547473
\(356\) 192.733 0.0286933
\(357\) 0 0
\(358\) −8701.28 −1.28457
\(359\) −7104.81 −1.04451 −0.522253 0.852791i \(-0.674908\pi\)
−0.522253 + 0.852791i \(0.674908\pi\)
\(360\) 0 0
\(361\) −6854.32 −0.999318
\(362\) −6862.54 −0.996374
\(363\) 0 0
\(364\) 256.330 0.0369102
\(365\) 4465.47 0.640366
\(366\) 0 0
\(367\) 7921.51 1.12670 0.563350 0.826218i \(-0.309512\pi\)
0.563350 + 0.826218i \(0.309512\pi\)
\(368\) −814.843 −0.115426
\(369\) 0 0
\(370\) −3255.37 −0.457401
\(371\) −9900.35 −1.38545
\(372\) 0 0
\(373\) −5560.92 −0.771940 −0.385970 0.922511i \(-0.626133\pi\)
−0.385970 + 0.922511i \(0.626133\pi\)
\(374\) 963.922 0.133271
\(375\) 0 0
\(376\) −4535.66 −0.622098
\(377\) 17110.6 2.33751
\(378\) 0 0
\(379\) 3393.45 0.459920 0.229960 0.973200i \(-0.426140\pi\)
0.229960 + 0.973200i \(0.426140\pi\)
\(380\) −2.88521 −0.000389495 0
\(381\) 0 0
\(382\) −5050.61 −0.676471
\(383\) 2057.57 0.274509 0.137255 0.990536i \(-0.456172\pi\)
0.137255 + 0.990536i \(0.456172\pi\)
\(384\) 0 0
\(385\) −2903.92 −0.384410
\(386\) 9091.82 1.19886
\(387\) 0 0
\(388\) −188.313 −0.0246396
\(389\) −5909.30 −0.770215 −0.385107 0.922872i \(-0.625836\pi\)
−0.385107 + 0.922872i \(0.625836\pi\)
\(390\) 0 0
\(391\) −221.705 −0.0286755
\(392\) 1234.92 0.159114
\(393\) 0 0
\(394\) −6226.47 −0.796155
\(395\) 8062.99 1.02707
\(396\) 0 0
\(397\) 8871.61 1.12154 0.560772 0.827970i \(-0.310504\pi\)
0.560772 + 0.827970i \(0.310504\pi\)
\(398\) −12045.8 −1.51710
\(399\) 0 0
\(400\) 4584.05 0.573006
\(401\) −5958.38 −0.742013 −0.371007 0.928630i \(-0.620987\pi\)
−0.371007 + 0.928630i \(0.620987\pi\)
\(402\) 0 0
\(403\) 11561.3 1.42906
\(404\) −121.302 −0.0149381
\(405\) 0 0
\(406\) 13747.5 1.68049
\(407\) −3287.26 −0.400352
\(408\) 0 0
\(409\) 7517.10 0.908793 0.454397 0.890799i \(-0.349855\pi\)
0.454397 + 0.890799i \(0.349855\pi\)
\(410\) −5093.45 −0.613531
\(411\) 0 0
\(412\) −90.7931 −0.0108569
\(413\) 6834.95 0.814348
\(414\) 0 0
\(415\) −3490.74 −0.412901
\(416\) 582.100 0.0686052
\(417\) 0 0
\(418\) 122.674 0.0143545
\(419\) −2102.77 −0.245172 −0.122586 0.992458i \(-0.539119\pi\)
−0.122586 + 0.992458i \(0.539119\pi\)
\(420\) 0 0
\(421\) 3480.77 0.402951 0.201476 0.979494i \(-0.435426\pi\)
0.201476 + 0.979494i \(0.435426\pi\)
\(422\) −11795.0 −1.36060
\(423\) 0 0
\(424\) −11370.3 −1.30233
\(425\) 1247.24 0.142353
\(426\) 0 0
\(427\) 6630.24 0.751428
\(428\) 90.2958 0.0101977
\(429\) 0 0
\(430\) 11182.4 1.25410
\(431\) 6815.12 0.761654 0.380827 0.924646i \(-0.375639\pi\)
0.380827 + 0.924646i \(0.375639\pi\)
\(432\) 0 0
\(433\) 2867.85 0.318292 0.159146 0.987255i \(-0.449126\pi\)
0.159146 + 0.987255i \(0.449126\pi\)
\(434\) 9288.95 1.02738
\(435\) 0 0
\(436\) −16.4584 −0.00180783
\(437\) −28.2155 −0.00308863
\(438\) 0 0
\(439\) −3268.85 −0.355384 −0.177692 0.984086i \(-0.556863\pi\)
−0.177692 + 0.984086i \(0.556863\pi\)
\(440\) −3335.07 −0.361349
\(441\) 0 0
\(442\) −3294.30 −0.354511
\(443\) −12387.8 −1.32858 −0.664289 0.747476i \(-0.731265\pi\)
−0.664289 + 0.747476i \(0.731265\pi\)
\(444\) 0 0
\(445\) 7462.15 0.794921
\(446\) 13490.3 1.43225
\(447\) 0 0
\(448\) 10426.7 1.09958
\(449\) 13922.0 1.46329 0.731646 0.681685i \(-0.238753\pi\)
0.731646 + 0.681685i \(0.238753\pi\)
\(450\) 0 0
\(451\) −5143.36 −0.537009
\(452\) 336.749 0.0350428
\(453\) 0 0
\(454\) −4654.88 −0.481199
\(455\) 9924.45 1.02256
\(456\) 0 0
\(457\) 7503.09 0.768008 0.384004 0.923331i \(-0.374545\pi\)
0.384004 + 0.923331i \(0.374545\pi\)
\(458\) −7620.32 −0.777455
\(459\) 0 0
\(460\) 17.3918 0.00176282
\(461\) 1804.23 0.182280 0.0911402 0.995838i \(-0.470949\pi\)
0.0911402 + 0.995838i \(0.470949\pi\)
\(462\) 0 0
\(463\) 1843.06 0.184998 0.0924992 0.995713i \(-0.470514\pi\)
0.0924992 + 0.995713i \(0.470514\pi\)
\(464\) 15422.2 1.54301
\(465\) 0 0
\(466\) −13520.1 −1.34401
\(467\) −11884.3 −1.17760 −0.588799 0.808280i \(-0.700399\pi\)
−0.588799 + 0.808280i \(0.700399\pi\)
\(468\) 0 0
\(469\) −3383.52 −0.333127
\(470\) −3981.55 −0.390756
\(471\) 0 0
\(472\) 7849.74 0.765495
\(473\) 11291.9 1.09768
\(474\) 0 0
\(475\) 158.731 0.0153328
\(476\) 62.8610 0.00605300
\(477\) 0 0
\(478\) 14536.6 1.39098
\(479\) −4591.44 −0.437971 −0.218986 0.975728i \(-0.570275\pi\)
−0.218986 + 0.975728i \(0.570275\pi\)
\(480\) 0 0
\(481\) 11234.5 1.06497
\(482\) 1041.17 0.0983897
\(483\) 0 0
\(484\) 170.664 0.0160278
\(485\) −7291.03 −0.682616
\(486\) 0 0
\(487\) 3075.97 0.286213 0.143106 0.989707i \(-0.454291\pi\)
0.143106 + 0.989707i \(0.454291\pi\)
\(488\) 7614.64 0.706350
\(489\) 0 0
\(490\) 1084.05 0.0999439
\(491\) 2556.76 0.235000 0.117500 0.993073i \(-0.462512\pi\)
0.117500 + 0.993073i \(0.462512\pi\)
\(492\) 0 0
\(493\) 4196.12 0.383334
\(494\) −419.251 −0.0381842
\(495\) 0 0
\(496\) 10420.5 0.943333
\(497\) 10153.6 0.916403
\(498\) 0 0
\(499\) 3551.49 0.318610 0.159305 0.987229i \(-0.449075\pi\)
0.159305 + 0.987229i \(0.449075\pi\)
\(500\) −264.537 −0.0236609
\(501\) 0 0
\(502\) −2523.42 −0.224354
\(503\) 96.3146 0.00853768 0.00426884 0.999991i \(-0.498641\pi\)
0.00426884 + 0.999991i \(0.498641\pi\)
\(504\) 0 0
\(505\) −4696.51 −0.413846
\(506\) −739.469 −0.0649672
\(507\) 0 0
\(508\) 515.075 0.0449858
\(509\) −19081.4 −1.66163 −0.830815 0.556549i \(-0.812125\pi\)
−0.830815 + 0.556549i \(0.812125\pi\)
\(510\) 0 0
\(511\) −12381.8 −1.07189
\(512\) 11962.3 1.03254
\(513\) 0 0
\(514\) −3805.77 −0.326587
\(515\) −3515.29 −0.300781
\(516\) 0 0
\(517\) −4020.57 −0.342020
\(518\) 9026.41 0.765633
\(519\) 0 0
\(520\) 11397.9 0.961217
\(521\) −9570.95 −0.804820 −0.402410 0.915460i \(-0.631827\pi\)
−0.402410 + 0.915460i \(0.631827\pi\)
\(522\) 0 0
\(523\) −16979.2 −1.41959 −0.709797 0.704406i \(-0.751214\pi\)
−0.709797 + 0.704406i \(0.751214\pi\)
\(524\) −344.517 −0.0287220
\(525\) 0 0
\(526\) −2978.73 −0.246918
\(527\) 2835.24 0.234355
\(528\) 0 0
\(529\) −11996.9 −0.986021
\(530\) −9981.21 −0.818030
\(531\) 0 0
\(532\) 8.00005 0.000651967 0
\(533\) 17577.9 1.42849
\(534\) 0 0
\(535\) 3496.03 0.282517
\(536\) −3885.88 −0.313143
\(537\) 0 0
\(538\) −15420.6 −1.23574
\(539\) 1094.67 0.0874785
\(540\) 0 0
\(541\) −9339.39 −0.742203 −0.371101 0.928592i \(-0.621020\pi\)
−0.371101 + 0.928592i \(0.621020\pi\)
\(542\) −17531.6 −1.38938
\(543\) 0 0
\(544\) 142.751 0.0112508
\(545\) −637.227 −0.0500841
\(546\) 0 0
\(547\) 3347.48 0.261660 0.130830 0.991405i \(-0.458236\pi\)
0.130830 + 0.991405i \(0.458236\pi\)
\(548\) −284.455 −0.0221739
\(549\) 0 0
\(550\) 4160.02 0.322516
\(551\) 534.022 0.0412888
\(552\) 0 0
\(553\) −22356.9 −1.71919
\(554\) 1713.34 0.131395
\(555\) 0 0
\(556\) −96.1428 −0.00733339
\(557\) −20271.9 −1.54209 −0.771047 0.636778i \(-0.780267\pi\)
−0.771047 + 0.636778i \(0.780267\pi\)
\(558\) 0 0
\(559\) −38591.3 −2.91992
\(560\) 8945.14 0.675002
\(561\) 0 0
\(562\) −25040.4 −1.87947
\(563\) 6144.58 0.459970 0.229985 0.973194i \(-0.426132\pi\)
0.229985 + 0.973194i \(0.426132\pi\)
\(564\) 0 0
\(565\) 13038.1 0.970828
\(566\) 1024.94 0.0761158
\(567\) 0 0
\(568\) 11661.1 0.861428
\(569\) −19198.3 −1.41447 −0.707237 0.706977i \(-0.750058\pi\)
−0.707237 + 0.706977i \(0.750058\pi\)
\(570\) 0 0
\(571\) −2944.07 −0.215772 −0.107886 0.994163i \(-0.534408\pi\)
−0.107886 + 0.994163i \(0.534408\pi\)
\(572\) 260.955 0.0190753
\(573\) 0 0
\(574\) 14123.0 1.02698
\(575\) −956.819 −0.0693950
\(576\) 0 0
\(577\) −1802.77 −0.130070 −0.0650351 0.997883i \(-0.520716\pi\)
−0.0650351 + 0.997883i \(0.520716\pi\)
\(578\) −807.878 −0.0581372
\(579\) 0 0
\(580\) −329.167 −0.0235654
\(581\) 9679.07 0.691145
\(582\) 0 0
\(583\) −10079.0 −0.716003
\(584\) −14220.1 −1.00759
\(585\) 0 0
\(586\) 1521.44 0.107253
\(587\) −24517.6 −1.72394 −0.861969 0.506961i \(-0.830769\pi\)
−0.861969 + 0.506961i \(0.830769\pi\)
\(588\) 0 0
\(589\) 360.829 0.0252423
\(590\) 6890.77 0.480828
\(591\) 0 0
\(592\) 10126.0 0.702997
\(593\) 14269.5 0.988161 0.494081 0.869416i \(-0.335505\pi\)
0.494081 + 0.869416i \(0.335505\pi\)
\(594\) 0 0
\(595\) 2433.82 0.167693
\(596\) 631.201 0.0433809
\(597\) 0 0
\(598\) 2527.21 0.172818
\(599\) −15362.8 −1.04792 −0.523961 0.851742i \(-0.675546\pi\)
−0.523961 + 0.851742i \(0.675546\pi\)
\(600\) 0 0
\(601\) 19852.3 1.34741 0.673704 0.739001i \(-0.264702\pi\)
0.673704 + 0.739001i \(0.264702\pi\)
\(602\) −31006.2 −2.09920
\(603\) 0 0
\(604\) −20.1917 −0.00136025
\(605\) 6607.69 0.444035
\(606\) 0 0
\(607\) −19495.1 −1.30359 −0.651796 0.758394i \(-0.725984\pi\)
−0.651796 + 0.758394i \(0.725984\pi\)
\(608\) 18.1674 0.00121181
\(609\) 0 0
\(610\) 6684.39 0.443677
\(611\) 13740.7 0.909801
\(612\) 0 0
\(613\) 16796.4 1.10669 0.553343 0.832954i \(-0.313352\pi\)
0.553343 + 0.832954i \(0.313352\pi\)
\(614\) 17164.2 1.12816
\(615\) 0 0
\(616\) 9247.43 0.604853
\(617\) −1185.68 −0.0773642 −0.0386821 0.999252i \(-0.512316\pi\)
−0.0386821 + 0.999252i \(0.512316\pi\)
\(618\) 0 0
\(619\) 11747.1 0.762771 0.381385 0.924416i \(-0.375447\pi\)
0.381385 + 0.924416i \(0.375447\pi\)
\(620\) −222.412 −0.0144069
\(621\) 0 0
\(622\) −6847.51 −0.441415
\(623\) −20690.9 −1.33060
\(624\) 0 0
\(625\) −1071.33 −0.0685650
\(626\) 19741.9 1.26045
\(627\) 0 0
\(628\) −659.465 −0.0419037
\(629\) 2755.10 0.174647
\(630\) 0 0
\(631\) −11744.3 −0.740943 −0.370471 0.928844i \(-0.620804\pi\)
−0.370471 + 0.928844i \(0.620804\pi\)
\(632\) −25676.2 −1.61605
\(633\) 0 0
\(634\) 11391.4 0.713578
\(635\) 19942.4 1.24629
\(636\) 0 0
\(637\) −3741.16 −0.232700
\(638\) 13995.6 0.868483
\(639\) 0 0
\(640\) 10029.1 0.619431
\(641\) −4318.07 −0.266074 −0.133037 0.991111i \(-0.542473\pi\)
−0.133037 + 0.991111i \(0.542473\pi\)
\(642\) 0 0
\(643\) −16920.9 −1.03779 −0.518893 0.854839i \(-0.673656\pi\)
−0.518893 + 0.854839i \(0.673656\pi\)
\(644\) −48.2236 −0.00295074
\(645\) 0 0
\(646\) −102.815 −0.00626194
\(647\) −12459.5 −0.757084 −0.378542 0.925584i \(-0.623574\pi\)
−0.378542 + 0.925584i \(0.623574\pi\)
\(648\) 0 0
\(649\) 6958.29 0.420858
\(650\) −14217.3 −0.857920
\(651\) 0 0
\(652\) 719.336 0.0432076
\(653\) −12781.8 −0.765990 −0.382995 0.923750i \(-0.625107\pi\)
−0.382995 + 0.923750i \(0.625107\pi\)
\(654\) 0 0
\(655\) −13338.9 −0.795714
\(656\) 15843.4 0.942959
\(657\) 0 0
\(658\) 11040.0 0.654078
\(659\) 7400.62 0.437461 0.218731 0.975785i \(-0.429808\pi\)
0.218731 + 0.975785i \(0.429808\pi\)
\(660\) 0 0
\(661\) −21906.0 −1.28902 −0.644512 0.764594i \(-0.722939\pi\)
−0.644512 + 0.764594i \(0.722939\pi\)
\(662\) −10817.3 −0.635083
\(663\) 0 0
\(664\) 11116.1 0.649683
\(665\) 309.743 0.0180621
\(666\) 0 0
\(667\) −3219.04 −0.186869
\(668\) 135.633 0.00785598
\(669\) 0 0
\(670\) −3411.16 −0.196693
\(671\) 6749.89 0.388340
\(672\) 0 0
\(673\) −440.817 −0.0252485 −0.0126243 0.999920i \(-0.504019\pi\)
−0.0126243 + 0.999920i \(0.504019\pi\)
\(674\) 17482.2 0.999091
\(675\) 0 0
\(676\) −484.098 −0.0275431
\(677\) −17950.8 −1.01906 −0.509532 0.860452i \(-0.670181\pi\)
−0.509532 + 0.860452i \(0.670181\pi\)
\(678\) 0 0
\(679\) 20216.4 1.14261
\(680\) 2795.18 0.157633
\(681\) 0 0
\(682\) 9456.57 0.530954
\(683\) 32522.6 1.82203 0.911013 0.412377i \(-0.135301\pi\)
0.911013 + 0.412377i \(0.135301\pi\)
\(684\) 0 0
\(685\) −11013.4 −0.614306
\(686\) 16098.0 0.895953
\(687\) 0 0
\(688\) −34783.2 −1.92747
\(689\) 34446.0 1.90463
\(690\) 0 0
\(691\) 29316.7 1.61398 0.806991 0.590564i \(-0.201095\pi\)
0.806991 + 0.590564i \(0.201095\pi\)
\(692\) 553.038 0.0303805
\(693\) 0 0
\(694\) −18540.8 −1.01412
\(695\) −3722.42 −0.203164
\(696\) 0 0
\(697\) 4310.73 0.234262
\(698\) 9743.96 0.528387
\(699\) 0 0
\(700\) 271.291 0.0146483
\(701\) −25948.5 −1.39809 −0.699045 0.715077i \(-0.746392\pi\)
−0.699045 + 0.715077i \(0.746392\pi\)
\(702\) 0 0
\(703\) 350.630 0.0188112
\(704\) 10614.8 0.568268
\(705\) 0 0
\(706\) 18001.7 0.959634
\(707\) 13022.4 0.692727
\(708\) 0 0
\(709\) −22102.6 −1.17077 −0.585387 0.810754i \(-0.699057\pi\)
−0.585387 + 0.810754i \(0.699057\pi\)
\(710\) 10236.6 0.541086
\(711\) 0 0
\(712\) −23762.9 −1.25078
\(713\) −2175.04 −0.114244
\(714\) 0 0
\(715\) 10103.5 0.528463
\(716\) −577.683 −0.0301523
\(717\) 0 0
\(718\) 19861.0 1.03232
\(719\) 9876.18 0.512266 0.256133 0.966642i \(-0.417551\pi\)
0.256133 + 0.966642i \(0.417551\pi\)
\(720\) 0 0
\(721\) 9747.12 0.503470
\(722\) 19160.7 0.987658
\(723\) 0 0
\(724\) −455.608 −0.0233875
\(725\) 18109.3 0.927672
\(726\) 0 0
\(727\) 7187.21 0.366656 0.183328 0.983052i \(-0.441313\pi\)
0.183328 + 0.983052i \(0.441313\pi\)
\(728\) −31604.0 −1.60896
\(729\) 0 0
\(730\) −12482.9 −0.632894
\(731\) −9463.94 −0.478846
\(732\) 0 0
\(733\) 28840.8 1.45328 0.726642 0.687016i \(-0.241080\pi\)
0.726642 + 0.687016i \(0.241080\pi\)
\(734\) −22144.0 −1.11356
\(735\) 0 0
\(736\) −109.511 −0.00548456
\(737\) −3444.58 −0.172161
\(738\) 0 0
\(739\) 13008.1 0.647514 0.323757 0.946140i \(-0.395054\pi\)
0.323757 + 0.946140i \(0.395054\pi\)
\(740\) −216.126 −0.0107364
\(741\) 0 0
\(742\) 27675.7 1.36928
\(743\) 29485.3 1.45587 0.727934 0.685647i \(-0.240481\pi\)
0.727934 + 0.685647i \(0.240481\pi\)
\(744\) 0 0
\(745\) 24438.6 1.20182
\(746\) 15545.2 0.762934
\(747\) 0 0
\(748\) 63.9954 0.00312821
\(749\) −9693.73 −0.472899
\(750\) 0 0
\(751\) 17194.3 0.835458 0.417729 0.908572i \(-0.362826\pi\)
0.417729 + 0.908572i \(0.362826\pi\)
\(752\) 12384.8 0.600568
\(753\) 0 0
\(754\) −47831.4 −2.31024
\(755\) −781.774 −0.0376843
\(756\) 0 0
\(757\) 7578.25 0.363852 0.181926 0.983312i \(-0.441767\pi\)
0.181926 + 0.983312i \(0.441767\pi\)
\(758\) −9486.14 −0.454554
\(759\) 0 0
\(760\) 355.730 0.0169785
\(761\) −12964.9 −0.617577 −0.308789 0.951131i \(-0.599924\pi\)
−0.308789 + 0.951131i \(0.599924\pi\)
\(762\) 0 0
\(763\) 1766.89 0.0838346
\(764\) −335.313 −0.0158785
\(765\) 0 0
\(766\) −5751.79 −0.271306
\(767\) −23780.6 −1.11952
\(768\) 0 0
\(769\) −5997.37 −0.281236 −0.140618 0.990064i \(-0.544909\pi\)
−0.140618 + 0.990064i \(0.544909\pi\)
\(770\) 8117.71 0.379925
\(771\) 0 0
\(772\) 603.611 0.0281405
\(773\) 457.468 0.0212859 0.0106429 0.999943i \(-0.496612\pi\)
0.0106429 + 0.999943i \(0.496612\pi\)
\(774\) 0 0
\(775\) 12236.1 0.567140
\(776\) 23218.0 1.07407
\(777\) 0 0
\(778\) 16519.0 0.761228
\(779\) 548.608 0.0252323
\(780\) 0 0
\(781\) 10336.8 0.473600
\(782\) 619.761 0.0283409
\(783\) 0 0
\(784\) −3371.99 −0.153608
\(785\) −25532.9 −1.16090
\(786\) 0 0
\(787\) −837.185 −0.0379192 −0.0189596 0.999820i \(-0.506035\pi\)
−0.0189596 + 0.999820i \(0.506035\pi\)
\(788\) −413.379 −0.0186878
\(789\) 0 0
\(790\) −22539.5 −1.01509
\(791\) −36151.8 −1.62505
\(792\) 0 0
\(793\) −23068.4 −1.03302
\(794\) −24799.9 −1.10846
\(795\) 0 0
\(796\) −799.731 −0.0356102
\(797\) −21061.7 −0.936063 −0.468032 0.883712i \(-0.655037\pi\)
−0.468032 + 0.883712i \(0.655037\pi\)
\(798\) 0 0
\(799\) 3369.70 0.149201
\(800\) 616.075 0.0272269
\(801\) 0 0
\(802\) 16656.2 0.733356
\(803\) −12605.2 −0.553957
\(804\) 0 0
\(805\) −1867.10 −0.0817474
\(806\) −32318.8 −1.41238
\(807\) 0 0
\(808\) 14955.9 0.651170
\(809\) 13593.4 0.590752 0.295376 0.955381i \(-0.404555\pi\)
0.295376 + 0.955381i \(0.404555\pi\)
\(810\) 0 0
\(811\) 21346.1 0.924243 0.462122 0.886817i \(-0.347088\pi\)
0.462122 + 0.886817i \(0.347088\pi\)
\(812\) 912.707 0.0394455
\(813\) 0 0
\(814\) 9189.30 0.395681
\(815\) 27850.9 1.19702
\(816\) 0 0
\(817\) −1204.43 −0.0515763
\(818\) −21013.5 −0.898190
\(819\) 0 0
\(820\) −338.157 −0.0144012
\(821\) −10609.9 −0.451019 −0.225510 0.974241i \(-0.572405\pi\)
−0.225510 + 0.974241i \(0.572405\pi\)
\(822\) 0 0
\(823\) −16152.8 −0.684145 −0.342072 0.939674i \(-0.611129\pi\)
−0.342072 + 0.939674i \(0.611129\pi\)
\(824\) 11194.3 0.473266
\(825\) 0 0
\(826\) −19106.6 −0.804847
\(827\) 17755.3 0.746570 0.373285 0.927717i \(-0.378231\pi\)
0.373285 + 0.927717i \(0.378231\pi\)
\(828\) 0 0
\(829\) 16425.3 0.688149 0.344075 0.938942i \(-0.388193\pi\)
0.344075 + 0.938942i \(0.388193\pi\)
\(830\) 9758.12 0.408084
\(831\) 0 0
\(832\) −36277.2 −1.51164
\(833\) −917.463 −0.0381611
\(834\) 0 0
\(835\) 5251.37 0.217642
\(836\) 8.14442 0.000336938 0
\(837\) 0 0
\(838\) 5878.15 0.242312
\(839\) 1680.00 0.0691298 0.0345649 0.999402i \(-0.488995\pi\)
0.0345649 + 0.999402i \(0.488995\pi\)
\(840\) 0 0
\(841\) 36536.4 1.49807
\(842\) −9730.24 −0.398250
\(843\) 0 0
\(844\) −783.077 −0.0319368
\(845\) −18743.1 −0.763056
\(846\) 0 0
\(847\) −18321.7 −0.743259
\(848\) 31047.0 1.25726
\(849\) 0 0
\(850\) −3486.58 −0.140693
\(851\) −2113.57 −0.0851377
\(852\) 0 0
\(853\) −14637.8 −0.587560 −0.293780 0.955873i \(-0.594913\pi\)
−0.293780 + 0.955873i \(0.594913\pi\)
\(854\) −18534.4 −0.742661
\(855\) 0 0
\(856\) −11133.0 −0.444529
\(857\) 28175.6 1.12306 0.561528 0.827458i \(-0.310214\pi\)
0.561528 + 0.827458i \(0.310214\pi\)
\(858\) 0 0
\(859\) −941.593 −0.0374002 −0.0187001 0.999825i \(-0.505953\pi\)
−0.0187001 + 0.999825i \(0.505953\pi\)
\(860\) 742.403 0.0294369
\(861\) 0 0
\(862\) −19051.2 −0.752768
\(863\) 17106.4 0.674750 0.337375 0.941370i \(-0.390461\pi\)
0.337375 + 0.941370i \(0.390461\pi\)
\(864\) 0 0
\(865\) 21412.3 0.841663
\(866\) −8016.87 −0.314578
\(867\) 0 0
\(868\) 616.699 0.0241154
\(869\) −22760.3 −0.888482
\(870\) 0 0
\(871\) 11772.2 0.457963
\(872\) 2029.22 0.0788053
\(873\) 0 0
\(874\) 78.8743 0.00305259
\(875\) 28399.5 1.09723
\(876\) 0 0
\(877\) −16734.4 −0.644332 −0.322166 0.946683i \(-0.604411\pi\)
−0.322166 + 0.946683i \(0.604411\pi\)
\(878\) 9137.82 0.351237
\(879\) 0 0
\(880\) 9106.56 0.348843
\(881\) 19999.2 0.764802 0.382401 0.923996i \(-0.375097\pi\)
0.382401 + 0.923996i \(0.375097\pi\)
\(882\) 0 0
\(883\) 10029.6 0.382248 0.191124 0.981566i \(-0.438787\pi\)
0.191124 + 0.981566i \(0.438787\pi\)
\(884\) −218.710 −0.00832130
\(885\) 0 0
\(886\) 34629.1 1.31308
\(887\) 40481.7 1.53240 0.766202 0.642600i \(-0.222144\pi\)
0.766202 + 0.642600i \(0.222144\pi\)
\(888\) 0 0
\(889\) −55296.0 −2.08613
\(890\) −20859.9 −0.785646
\(891\) 0 0
\(892\) 895.629 0.0336187
\(893\) 428.847 0.0160704
\(894\) 0 0
\(895\) −22366.5 −0.835339
\(896\) −27808.5 −1.03685
\(897\) 0 0
\(898\) −38917.8 −1.44622
\(899\) 41166.1 1.52722
\(900\) 0 0
\(901\) 8447.37 0.312345
\(902\) 14377.9 0.530744
\(903\) 0 0
\(904\) −41519.3 −1.52756
\(905\) −17640.0 −0.647928
\(906\) 0 0
\(907\) −22918.7 −0.839034 −0.419517 0.907747i \(-0.637801\pi\)
−0.419517 + 0.907747i \(0.637801\pi\)
\(908\) −309.040 −0.0112950
\(909\) 0 0
\(910\) −27743.1 −1.01063
\(911\) −20326.2 −0.739230 −0.369615 0.929185i \(-0.620510\pi\)
−0.369615 + 0.929185i \(0.620510\pi\)
\(912\) 0 0
\(913\) 9853.73 0.357186
\(914\) −20974.3 −0.759048
\(915\) 0 0
\(916\) −505.918 −0.0182489
\(917\) 36985.7 1.33193
\(918\) 0 0
\(919\) 29975.7 1.07596 0.537980 0.842958i \(-0.319188\pi\)
0.537980 + 0.842958i \(0.319188\pi\)
\(920\) −2144.31 −0.0768433
\(921\) 0 0
\(922\) −5043.59 −0.180154
\(923\) −35327.2 −1.25981
\(924\) 0 0
\(925\) 11890.3 0.422648
\(926\) −5152.14 −0.182840
\(927\) 0 0
\(928\) 2072.67 0.0733176
\(929\) −18708.2 −0.660706 −0.330353 0.943857i \(-0.607168\pi\)
−0.330353 + 0.943857i \(0.607168\pi\)
\(930\) 0 0
\(931\) −116.762 −0.00411032
\(932\) −897.608 −0.0315474
\(933\) 0 0
\(934\) 33221.6 1.16386
\(935\) 2477.74 0.0866640
\(936\) 0 0
\(937\) −23835.6 −0.831032 −0.415516 0.909586i \(-0.636399\pi\)
−0.415516 + 0.909586i \(0.636399\pi\)
\(938\) 9458.39 0.329240
\(939\) 0 0
\(940\) −264.338 −0.00917207
\(941\) 29086.9 1.00766 0.503829 0.863803i \(-0.331924\pi\)
0.503829 + 0.863803i \(0.331924\pi\)
\(942\) 0 0
\(943\) −3306.96 −0.114199
\(944\) −21434.0 −0.739003
\(945\) 0 0
\(946\) −31565.7 −1.08487
\(947\) −4232.84 −0.145247 −0.0726234 0.997359i \(-0.523137\pi\)
−0.0726234 + 0.997359i \(0.523137\pi\)
\(948\) 0 0
\(949\) 43079.5 1.47357
\(950\) −443.722 −0.0151539
\(951\) 0 0
\(952\) −7750.42 −0.263858
\(953\) 34662.0 1.17819 0.589093 0.808065i \(-0.299485\pi\)
0.589093 + 0.808065i \(0.299485\pi\)
\(954\) 0 0
\(955\) −12982.5 −0.439899
\(956\) 965.093 0.0326499
\(957\) 0 0
\(958\) 12835.0 0.432861
\(959\) 30537.7 1.02827
\(960\) 0 0
\(961\) −1975.85 −0.0663238
\(962\) −31405.3 −1.05255
\(963\) 0 0
\(964\) 69.1236 0.00230946
\(965\) 23370.3 0.779604
\(966\) 0 0
\(967\) 21658.2 0.720248 0.360124 0.932904i \(-0.382734\pi\)
0.360124 + 0.932904i \(0.382734\pi\)
\(968\) −21041.9 −0.698671
\(969\) 0 0
\(970\) 20381.5 0.674651
\(971\) −4143.58 −0.136945 −0.0684727 0.997653i \(-0.521813\pi\)
−0.0684727 + 0.997653i \(0.521813\pi\)
\(972\) 0 0
\(973\) 10321.4 0.340072
\(974\) −8598.65 −0.282873
\(975\) 0 0
\(976\) −20792.1 −0.681904
\(977\) 9390.27 0.307494 0.153747 0.988110i \(-0.450866\pi\)
0.153747 + 0.988110i \(0.450866\pi\)
\(978\) 0 0
\(979\) −21064.3 −0.687658
\(980\) 71.9709 0.00234594
\(981\) 0 0
\(982\) −7147.22 −0.232258
\(983\) 42182.9 1.36869 0.684346 0.729157i \(-0.260088\pi\)
0.684346 + 0.729157i \(0.260088\pi\)
\(984\) 0 0
\(985\) −16005.0 −0.517728
\(986\) −11729.9 −0.378862
\(987\) 0 0
\(988\) −27.8343 −0.000896284 0
\(989\) 7260.22 0.233429
\(990\) 0 0
\(991\) −9493.78 −0.304319 −0.152159 0.988356i \(-0.548623\pi\)
−0.152159 + 0.988356i \(0.548623\pi\)
\(992\) 1400.46 0.0448234
\(993\) 0 0
\(994\) −28383.7 −0.905711
\(995\) −30963.6 −0.986546
\(996\) 0 0
\(997\) −53688.9 −1.70546 −0.852731 0.522350i \(-0.825055\pi\)
−0.852731 + 0.522350i \(0.825055\pi\)
\(998\) −9927.93 −0.314893
\(999\) 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 153.4.a.f.1.2 3
3.2 odd 2 51.4.a.e.1.2 3
4.3 odd 2 2448.4.a.bd.1.2 3
12.11 even 2 816.4.a.s.1.2 3
15.14 odd 2 1275.4.a.q.1.2 3
21.20 even 2 2499.4.a.n.1.2 3
51.50 odd 2 867.4.a.k.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
51.4.a.e.1.2 3 3.2 odd 2
153.4.a.f.1.2 3 1.1 even 1 trivial
816.4.a.s.1.2 3 12.11 even 2
867.4.a.k.1.2 3 51.50 odd 2
1275.4.a.q.1.2 3 15.14 odd 2
2448.4.a.bd.1.2 3 4.3 odd 2
2499.4.a.n.1.2 3 21.20 even 2