Properties

Label 475.4.a.b.1.1
Level $475$
Weight $4$
Character 475.1
Self dual yes
Analytic conductor $28.026$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,4,Mod(1,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, 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("475.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 475.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: \(28.0259072527\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{2} -7.00000 q^{3} +1.00000 q^{4} +21.0000 q^{6} -11.0000 q^{7} +21.0000 q^{8} +22.0000 q^{9} +O(q^{10})\) \(q-3.00000 q^{2} -7.00000 q^{3} +1.00000 q^{4} +21.0000 q^{6} -11.0000 q^{7} +21.0000 q^{8} +22.0000 q^{9} -36.0000 q^{11} -7.00000 q^{12} -65.0000 q^{13} +33.0000 q^{14} -71.0000 q^{16} +87.0000 q^{17} -66.0000 q^{18} +19.0000 q^{19} +77.0000 q^{21} +108.000 q^{22} +129.000 q^{23} -147.000 q^{24} +195.000 q^{26} +35.0000 q^{27} -11.0000 q^{28} +231.000 q^{29} +110.000 q^{31} +45.0000 q^{32} +252.000 q^{33} -261.000 q^{34} +22.0000 q^{36} +142.000 q^{37} -57.0000 q^{38} +455.000 q^{39} -330.000 q^{41} -231.000 q^{42} -74.0000 q^{43} -36.0000 q^{44} -387.000 q^{46} +336.000 q^{47} +497.000 q^{48} -222.000 q^{49} -609.000 q^{51} -65.0000 q^{52} -501.000 q^{53} -105.000 q^{54} -231.000 q^{56} -133.000 q^{57} -693.000 q^{58} +633.000 q^{59} -88.0000 q^{61} -330.000 q^{62} -242.000 q^{63} +433.000 q^{64} -756.000 q^{66} -119.000 q^{67} +87.0000 q^{68} -903.000 q^{69} -204.000 q^{71} +462.000 q^{72} -407.000 q^{73} -426.000 q^{74} +19.0000 q^{76} +396.000 q^{77} -1365.00 q^{78} +1262.00 q^{79} -839.000 q^{81} +990.000 q^{82} -270.000 q^{83} +77.0000 q^{84} +222.000 q^{86} -1617.00 q^{87} -756.000 q^{88} -30.0000 q^{89} +715.000 q^{91} +129.000 q^{92} -770.000 q^{93} -1008.00 q^{94} -315.000 q^{96} -1406.00 q^{97} +666.000 q^{98} -792.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.00000 −1.06066 −0.530330 0.847791i \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) −7.00000 −1.34715 −0.673575 0.739119i \(-0.735242\pi\)
−0.673575 + 0.739119i \(0.735242\pi\)
\(4\) 1.00000 0.125000
\(5\) 0 0
\(6\) 21.0000 1.42887
\(7\) −11.0000 −0.593944 −0.296972 0.954886i \(-0.595977\pi\)
−0.296972 + 0.954886i \(0.595977\pi\)
\(8\) 21.0000 0.928078
\(9\) 22.0000 0.814815
\(10\) 0 0
\(11\) −36.0000 −0.986764 −0.493382 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) −7.00000 −0.168394
\(13\) −65.0000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 33.0000 0.629973
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) 87.0000 1.24121 0.620606 0.784123i \(-0.286887\pi\)
0.620606 + 0.784123i \(0.286887\pi\)
\(18\) −66.0000 −0.864242
\(19\) 19.0000 0.229416
\(20\) 0 0
\(21\) 77.0000 0.800132
\(22\) 108.000 1.04662
\(23\) 129.000 1.16949 0.584747 0.811216i \(-0.301194\pi\)
0.584747 + 0.811216i \(0.301194\pi\)
\(24\) −147.000 −1.25026
\(25\) 0 0
\(26\) 195.000 1.47087
\(27\) 35.0000 0.249472
\(28\) −11.0000 −0.0742430
\(29\) 231.000 1.47916 0.739580 0.673069i \(-0.235024\pi\)
0.739580 + 0.673069i \(0.235024\pi\)
\(30\) 0 0
\(31\) 110.000 0.637309 0.318655 0.947871i \(-0.396769\pi\)
0.318655 + 0.947871i \(0.396769\pi\)
\(32\) 45.0000 0.248592
\(33\) 252.000 1.32932
\(34\) −261.000 −1.31650
\(35\) 0 0
\(36\) 22.0000 0.101852
\(37\) 142.000 0.630937 0.315468 0.948936i \(-0.397838\pi\)
0.315468 + 0.948936i \(0.397838\pi\)
\(38\) −57.0000 −0.243332
\(39\) 455.000 1.86816
\(40\) 0 0
\(41\) −330.000 −1.25701 −0.628504 0.777806i \(-0.716332\pi\)
−0.628504 + 0.777806i \(0.716332\pi\)
\(42\) −231.000 −0.848668
\(43\) −74.0000 −0.262439 −0.131220 0.991353i \(-0.541889\pi\)
−0.131220 + 0.991353i \(0.541889\pi\)
\(44\) −36.0000 −0.123346
\(45\) 0 0
\(46\) −387.000 −1.24044
\(47\) 336.000 1.04278 0.521390 0.853319i \(-0.325414\pi\)
0.521390 + 0.853319i \(0.325414\pi\)
\(48\) 497.000 1.49450
\(49\) −222.000 −0.647230
\(50\) 0 0
\(51\) −609.000 −1.67210
\(52\) −65.0000 −0.173344
\(53\) −501.000 −1.29845 −0.649223 0.760598i \(-0.724906\pi\)
−0.649223 + 0.760598i \(0.724906\pi\)
\(54\) −105.000 −0.264605
\(55\) 0 0
\(56\) −231.000 −0.551226
\(57\) −133.000 −0.309058
\(58\) −693.000 −1.56889
\(59\) 633.000 1.39677 0.698386 0.715721i \(-0.253902\pi\)
0.698386 + 0.715721i \(0.253902\pi\)
\(60\) 0 0
\(61\) −88.0000 −0.184709 −0.0923545 0.995726i \(-0.529439\pi\)
−0.0923545 + 0.995726i \(0.529439\pi\)
\(62\) −330.000 −0.675968
\(63\) −242.000 −0.483955
\(64\) 433.000 0.845703
\(65\) 0 0
\(66\) −756.000 −1.40996
\(67\) −119.000 −0.216988 −0.108494 0.994097i \(-0.534603\pi\)
−0.108494 + 0.994097i \(0.534603\pi\)
\(68\) 87.0000 0.155151
\(69\) −903.000 −1.57548
\(70\) 0 0
\(71\) −204.000 −0.340991 −0.170495 0.985358i \(-0.554537\pi\)
−0.170495 + 0.985358i \(0.554537\pi\)
\(72\) 462.000 0.756211
\(73\) −407.000 −0.652544 −0.326272 0.945276i \(-0.605793\pi\)
−0.326272 + 0.945276i \(0.605793\pi\)
\(74\) −426.000 −0.669209
\(75\) 0 0
\(76\) 19.0000 0.0286770
\(77\) 396.000 0.586083
\(78\) −1365.00 −1.98148
\(79\) 1262.00 1.79729 0.898646 0.438674i \(-0.144552\pi\)
0.898646 + 0.438674i \(0.144552\pi\)
\(80\) 0 0
\(81\) −839.000 −1.15089
\(82\) 990.000 1.33326
\(83\) −270.000 −0.357064 −0.178532 0.983934i \(-0.557135\pi\)
−0.178532 + 0.983934i \(0.557135\pi\)
\(84\) 77.0000 0.100017
\(85\) 0 0
\(86\) 222.000 0.278359
\(87\) −1617.00 −1.99265
\(88\) −756.000 −0.915794
\(89\) −30.0000 −0.0357303 −0.0178651 0.999840i \(-0.505687\pi\)
−0.0178651 + 0.999840i \(0.505687\pi\)
\(90\) 0 0
\(91\) 715.000 0.823652
\(92\) 129.000 0.146187
\(93\) −770.000 −0.858551
\(94\) −1008.00 −1.10603
\(95\) 0 0
\(96\) −315.000 −0.334891
\(97\) −1406.00 −1.47173 −0.735864 0.677129i \(-0.763224\pi\)
−0.735864 + 0.677129i \(0.763224\pi\)
\(98\) 666.000 0.686491
\(99\) −792.000 −0.804030
\(100\) 0 0
\(101\) 1032.00 1.01671 0.508356 0.861147i \(-0.330254\pi\)
0.508356 + 0.861147i \(0.330254\pi\)
\(102\) 1827.00 1.77353
\(103\) 484.000 0.463009 0.231505 0.972834i \(-0.425635\pi\)
0.231505 + 0.972834i \(0.425635\pi\)
\(104\) −1365.00 −1.28701
\(105\) 0 0
\(106\) 1503.00 1.37721
\(107\) −1881.00 −1.69947 −0.849734 0.527211i \(-0.823238\pi\)
−0.849734 + 0.527211i \(0.823238\pi\)
\(108\) 35.0000 0.0311840
\(109\) 281.000 0.246926 0.123463 0.992349i \(-0.460600\pi\)
0.123463 + 0.992349i \(0.460600\pi\)
\(110\) 0 0
\(111\) −994.000 −0.849967
\(112\) 781.000 0.658907
\(113\) −612.000 −0.509488 −0.254744 0.967009i \(-0.581991\pi\)
−0.254744 + 0.967009i \(0.581991\pi\)
\(114\) 399.000 0.327805
\(115\) 0 0
\(116\) 231.000 0.184895
\(117\) −1430.00 −1.12994
\(118\) −1899.00 −1.48150
\(119\) −957.000 −0.737210
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 264.000 0.195913
\(123\) 2310.00 1.69338
\(124\) 110.000 0.0796636
\(125\) 0 0
\(126\) 726.000 0.513311
\(127\) 1582.00 1.10535 0.552676 0.833396i \(-0.313607\pi\)
0.552676 + 0.833396i \(0.313607\pi\)
\(128\) −1659.00 −1.14560
\(129\) 518.000 0.353545
\(130\) 0 0
\(131\) −1932.00 −1.28855 −0.644273 0.764795i \(-0.722840\pi\)
−0.644273 + 0.764795i \(0.722840\pi\)
\(132\) 252.000 0.166165
\(133\) −209.000 −0.136260
\(134\) 357.000 0.230150
\(135\) 0 0
\(136\) 1827.00 1.15194
\(137\) 1515.00 0.944782 0.472391 0.881389i \(-0.343391\pi\)
0.472391 + 0.881389i \(0.343391\pi\)
\(138\) 2709.00 1.67105
\(139\) −2284.00 −1.39371 −0.696857 0.717210i \(-0.745419\pi\)
−0.696857 + 0.717210i \(0.745419\pi\)
\(140\) 0 0
\(141\) −2352.00 −1.40478
\(142\) 612.000 0.361675
\(143\) 2340.00 1.36840
\(144\) −1562.00 −0.903935
\(145\) 0 0
\(146\) 1221.00 0.692128
\(147\) 1554.00 0.871917
\(148\) 142.000 0.0788671
\(149\) −1356.00 −0.745556 −0.372778 0.927921i \(-0.621595\pi\)
−0.372778 + 0.927921i \(0.621595\pi\)
\(150\) 0 0
\(151\) 2234.00 1.20398 0.601988 0.798505i \(-0.294376\pi\)
0.601988 + 0.798505i \(0.294376\pi\)
\(152\) 399.000 0.212916
\(153\) 1914.00 1.01136
\(154\) −1188.00 −0.621635
\(155\) 0 0
\(156\) 455.000 0.233520
\(157\) −146.000 −0.0742170 −0.0371085 0.999311i \(-0.511815\pi\)
−0.0371085 + 0.999311i \(0.511815\pi\)
\(158\) −3786.00 −1.90632
\(159\) 3507.00 1.74920
\(160\) 0 0
\(161\) −1419.00 −0.694614
\(162\) 2517.00 1.22070
\(163\) −1154.00 −0.554529 −0.277265 0.960794i \(-0.589428\pi\)
−0.277265 + 0.960794i \(0.589428\pi\)
\(164\) −330.000 −0.157126
\(165\) 0 0
\(166\) 810.000 0.378724
\(167\) 3558.00 1.64866 0.824330 0.566109i \(-0.191552\pi\)
0.824330 + 0.566109i \(0.191552\pi\)
\(168\) 1617.00 0.742585
\(169\) 2028.00 0.923077
\(170\) 0 0
\(171\) 418.000 0.186931
\(172\) −74.0000 −0.0328049
\(173\) 1482.00 0.651297 0.325648 0.945491i \(-0.394417\pi\)
0.325648 + 0.945491i \(0.394417\pi\)
\(174\) 4851.00 2.11353
\(175\) 0 0
\(176\) 2556.00 1.09469
\(177\) −4431.00 −1.88166
\(178\) 90.0000 0.0378977
\(179\) −2880.00 −1.20258 −0.601289 0.799032i \(-0.705346\pi\)
−0.601289 + 0.799032i \(0.705346\pi\)
\(180\) 0 0
\(181\) 470.000 0.193010 0.0965050 0.995332i \(-0.469234\pi\)
0.0965050 + 0.995332i \(0.469234\pi\)
\(182\) −2145.00 −0.873615
\(183\) 616.000 0.248831
\(184\) 2709.00 1.08538
\(185\) 0 0
\(186\) 2310.00 0.910631
\(187\) −3132.00 −1.22478
\(188\) 336.000 0.130347
\(189\) −385.000 −0.148173
\(190\) 0 0
\(191\) −2475.00 −0.937616 −0.468808 0.883300i \(-0.655316\pi\)
−0.468808 + 0.883300i \(0.655316\pi\)
\(192\) −3031.00 −1.13929
\(193\) −1982.00 −0.739210 −0.369605 0.929189i \(-0.620507\pi\)
−0.369605 + 0.929189i \(0.620507\pi\)
\(194\) 4218.00 1.56100
\(195\) 0 0
\(196\) −222.000 −0.0809038
\(197\) −4122.00 −1.49076 −0.745382 0.666638i \(-0.767733\pi\)
−0.745382 + 0.666638i \(0.767733\pi\)
\(198\) 2376.00 0.852803
\(199\) −4075.00 −1.45160 −0.725802 0.687904i \(-0.758531\pi\)
−0.725802 + 0.687904i \(0.758531\pi\)
\(200\) 0 0
\(201\) 833.000 0.292315
\(202\) −3096.00 −1.07839
\(203\) −2541.00 −0.878538
\(204\) −609.000 −0.209012
\(205\) 0 0
\(206\) −1452.00 −0.491095
\(207\) 2838.00 0.952921
\(208\) 4615.00 1.53843
\(209\) −684.000 −0.226379
\(210\) 0 0
\(211\) 2765.00 0.902135 0.451067 0.892490i \(-0.351043\pi\)
0.451067 + 0.892490i \(0.351043\pi\)
\(212\) −501.000 −0.162306
\(213\) 1428.00 0.459366
\(214\) 5643.00 1.80256
\(215\) 0 0
\(216\) 735.000 0.231530
\(217\) −1210.00 −0.378526
\(218\) −843.000 −0.261904
\(219\) 2849.00 0.879076
\(220\) 0 0
\(221\) −5655.00 −1.72125
\(222\) 2982.00 0.901526
\(223\) 5326.00 1.59935 0.799676 0.600432i \(-0.205005\pi\)
0.799676 + 0.600432i \(0.205005\pi\)
\(224\) −495.000 −0.147650
\(225\) 0 0
\(226\) 1836.00 0.540393
\(227\) 1881.00 0.549984 0.274992 0.961447i \(-0.411325\pi\)
0.274992 + 0.961447i \(0.411325\pi\)
\(228\) −133.000 −0.0386322
\(229\) 758.000 0.218734 0.109367 0.994001i \(-0.465118\pi\)
0.109367 + 0.994001i \(0.465118\pi\)
\(230\) 0 0
\(231\) −2772.00 −0.789542
\(232\) 4851.00 1.37277
\(233\) 5334.00 1.49975 0.749875 0.661579i \(-0.230113\pi\)
0.749875 + 0.661579i \(0.230113\pi\)
\(234\) 4290.00 1.19849
\(235\) 0 0
\(236\) 633.000 0.174597
\(237\) −8834.00 −2.42122
\(238\) 2871.00 0.781930
\(239\) −4671.00 −1.26419 −0.632096 0.774890i \(-0.717805\pi\)
−0.632096 + 0.774890i \(0.717805\pi\)
\(240\) 0 0
\(241\) 1748.00 0.467214 0.233607 0.972331i \(-0.424947\pi\)
0.233607 + 0.972331i \(0.424947\pi\)
\(242\) 105.000 0.0278911
\(243\) 4928.00 1.30095
\(244\) −88.0000 −0.0230886
\(245\) 0 0
\(246\) −6930.00 −1.79610
\(247\) −1235.00 −0.318142
\(248\) 2310.00 0.591472
\(249\) 1890.00 0.481020
\(250\) 0 0
\(251\) 3984.00 1.00186 0.500932 0.865487i \(-0.332991\pi\)
0.500932 + 0.865487i \(0.332991\pi\)
\(252\) −242.000 −0.0604943
\(253\) −4644.00 −1.15401
\(254\) −4746.00 −1.17240
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) −6126.00 −1.48688 −0.743442 0.668800i \(-0.766808\pi\)
−0.743442 + 0.668800i \(0.766808\pi\)
\(258\) −1554.00 −0.374992
\(259\) −1562.00 −0.374741
\(260\) 0 0
\(261\) 5082.00 1.20524
\(262\) 5796.00 1.36671
\(263\) 6576.00 1.54180 0.770900 0.636956i \(-0.219807\pi\)
0.770900 + 0.636956i \(0.219807\pi\)
\(264\) 5292.00 1.23371
\(265\) 0 0
\(266\) 627.000 0.144526
\(267\) 210.000 0.0481340
\(268\) −119.000 −0.0271234
\(269\) 6078.00 1.37763 0.688814 0.724938i \(-0.258131\pi\)
0.688814 + 0.724938i \(0.258131\pi\)
\(270\) 0 0
\(271\) −1015.00 −0.227516 −0.113758 0.993508i \(-0.536289\pi\)
−0.113758 + 0.993508i \(0.536289\pi\)
\(272\) −6177.00 −1.37697
\(273\) −5005.00 −1.10958
\(274\) −4545.00 −1.00209
\(275\) 0 0
\(276\) −903.000 −0.196936
\(277\) 1924.00 0.417336 0.208668 0.977987i \(-0.433087\pi\)
0.208668 + 0.977987i \(0.433087\pi\)
\(278\) 6852.00 1.47826
\(279\) 2420.00 0.519289
\(280\) 0 0
\(281\) −4134.00 −0.877629 −0.438815 0.898578i \(-0.644601\pi\)
−0.438815 + 0.898578i \(0.644601\pi\)
\(282\) 7056.00 1.49000
\(283\) −5798.00 −1.21786 −0.608932 0.793223i \(-0.708402\pi\)
−0.608932 + 0.793223i \(0.708402\pi\)
\(284\) −204.000 −0.0426238
\(285\) 0 0
\(286\) −7020.00 −1.45140
\(287\) 3630.00 0.746593
\(288\) 990.000 0.202557
\(289\) 2656.00 0.540607
\(290\) 0 0
\(291\) 9842.00 1.98264
\(292\) −407.000 −0.0815681
\(293\) −9315.00 −1.85730 −0.928649 0.370960i \(-0.879029\pi\)
−0.928649 + 0.370960i \(0.879029\pi\)
\(294\) −4662.00 −0.924807
\(295\) 0 0
\(296\) 2982.00 0.585558
\(297\) −1260.00 −0.246170
\(298\) 4068.00 0.790782
\(299\) −8385.00 −1.62180
\(300\) 0 0
\(301\) 814.000 0.155874
\(302\) −6702.00 −1.27701
\(303\) −7224.00 −1.36966
\(304\) −1349.00 −0.254508
\(305\) 0 0
\(306\) −5742.00 −1.07271
\(307\) 7036.00 1.30803 0.654016 0.756481i \(-0.273083\pi\)
0.654016 + 0.756481i \(0.273083\pi\)
\(308\) 396.000 0.0732604
\(309\) −3388.00 −0.623743
\(310\) 0 0
\(311\) −2691.00 −0.490651 −0.245326 0.969441i \(-0.578895\pi\)
−0.245326 + 0.969441i \(0.578895\pi\)
\(312\) 9555.00 1.73380
\(313\) 565.000 0.102031 0.0510155 0.998698i \(-0.483754\pi\)
0.0510155 + 0.998698i \(0.483754\pi\)
\(314\) 438.000 0.0787190
\(315\) 0 0
\(316\) 1262.00 0.224662
\(317\) −2463.00 −0.436391 −0.218195 0.975905i \(-0.570017\pi\)
−0.218195 + 0.975905i \(0.570017\pi\)
\(318\) −10521.0 −1.85531
\(319\) −8316.00 −1.45958
\(320\) 0 0
\(321\) 13167.0 2.28944
\(322\) 4257.00 0.736749
\(323\) 1653.00 0.284753
\(324\) −839.000 −0.143861
\(325\) 0 0
\(326\) 3462.00 0.588167
\(327\) −1967.00 −0.332646
\(328\) −6930.00 −1.16660
\(329\) −3696.00 −0.619353
\(330\) 0 0
\(331\) 7463.00 1.23929 0.619643 0.784884i \(-0.287277\pi\)
0.619643 + 0.784884i \(0.287277\pi\)
\(332\) −270.000 −0.0446331
\(333\) 3124.00 0.514097
\(334\) −10674.0 −1.74867
\(335\) 0 0
\(336\) −5467.00 −0.887647
\(337\) 11194.0 1.80942 0.904712 0.426023i \(-0.140086\pi\)
0.904712 + 0.426023i \(0.140086\pi\)
\(338\) −6084.00 −0.979071
\(339\) 4284.00 0.686357
\(340\) 0 0
\(341\) −3960.00 −0.628874
\(342\) −1254.00 −0.198271
\(343\) 6215.00 0.978363
\(344\) −1554.00 −0.243564
\(345\) 0 0
\(346\) −4446.00 −0.690805
\(347\) −7806.00 −1.20763 −0.603816 0.797124i \(-0.706354\pi\)
−0.603816 + 0.797124i \(0.706354\pi\)
\(348\) −1617.00 −0.249081
\(349\) −8278.00 −1.26966 −0.634830 0.772652i \(-0.718930\pi\)
−0.634830 + 0.772652i \(0.718930\pi\)
\(350\) 0 0
\(351\) −2275.00 −0.345956
\(352\) −1620.00 −0.245302
\(353\) −7443.00 −1.12224 −0.561120 0.827734i \(-0.689629\pi\)
−0.561120 + 0.827734i \(0.689629\pi\)
\(354\) 13293.0 1.99581
\(355\) 0 0
\(356\) −30.0000 −0.00446628
\(357\) 6699.00 0.993134
\(358\) 8640.00 1.27553
\(359\) −3705.00 −0.544686 −0.272343 0.962200i \(-0.587799\pi\)
−0.272343 + 0.962200i \(0.587799\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) −1410.00 −0.204718
\(363\) 245.000 0.0354247
\(364\) 715.000 0.102957
\(365\) 0 0
\(366\) −1848.00 −0.263925
\(367\) −992.000 −0.141095 −0.0705477 0.997508i \(-0.522475\pi\)
−0.0705477 + 0.997508i \(0.522475\pi\)
\(368\) −9159.00 −1.29741
\(369\) −7260.00 −1.02423
\(370\) 0 0
\(371\) 5511.00 0.771204
\(372\) −770.000 −0.107319
\(373\) −11135.0 −1.54571 −0.772853 0.634585i \(-0.781171\pi\)
−0.772853 + 0.634585i \(0.781171\pi\)
\(374\) 9396.00 1.29908
\(375\) 0 0
\(376\) 7056.00 0.967780
\(377\) −15015.0 −2.05123
\(378\) 1155.00 0.157161
\(379\) −7135.00 −0.967019 −0.483510 0.875339i \(-0.660638\pi\)
−0.483510 + 0.875339i \(0.660638\pi\)
\(380\) 0 0
\(381\) −11074.0 −1.48908
\(382\) 7425.00 0.994492
\(383\) −5124.00 −0.683614 −0.341807 0.939770i \(-0.611039\pi\)
−0.341807 + 0.939770i \(0.611039\pi\)
\(384\) 11613.0 1.54329
\(385\) 0 0
\(386\) 5946.00 0.784050
\(387\) −1628.00 −0.213840
\(388\) −1406.00 −0.183966
\(389\) 11004.0 1.43425 0.717127 0.696942i \(-0.245457\pi\)
0.717127 + 0.696942i \(0.245457\pi\)
\(390\) 0 0
\(391\) 11223.0 1.45159
\(392\) −4662.00 −0.600680
\(393\) 13524.0 1.73587
\(394\) 12366.0 1.58119
\(395\) 0 0
\(396\) −792.000 −0.100504
\(397\) 2428.00 0.306947 0.153473 0.988153i \(-0.450954\pi\)
0.153473 + 0.988153i \(0.450954\pi\)
\(398\) 12225.0 1.53966
\(399\) 1463.00 0.183563
\(400\) 0 0
\(401\) −4146.00 −0.516313 −0.258156 0.966103i \(-0.583115\pi\)
−0.258156 + 0.966103i \(0.583115\pi\)
\(402\) −2499.00 −0.310047
\(403\) −7150.00 −0.883789
\(404\) 1032.00 0.127089
\(405\) 0 0
\(406\) 7623.00 0.931830
\(407\) −5112.00 −0.622586
\(408\) −12789.0 −1.55184
\(409\) 11630.0 1.40603 0.703015 0.711175i \(-0.251837\pi\)
0.703015 + 0.711175i \(0.251837\pi\)
\(410\) 0 0
\(411\) −10605.0 −1.27276
\(412\) 484.000 0.0578761
\(413\) −6963.00 −0.829605
\(414\) −8514.00 −1.01073
\(415\) 0 0
\(416\) −2925.00 −0.344735
\(417\) 15988.0 1.87754
\(418\) 2052.00 0.240111
\(419\) −14670.0 −1.71044 −0.855222 0.518261i \(-0.826580\pi\)
−0.855222 + 0.518261i \(0.826580\pi\)
\(420\) 0 0
\(421\) −8989.00 −1.04061 −0.520305 0.853980i \(-0.674182\pi\)
−0.520305 + 0.853980i \(0.674182\pi\)
\(422\) −8295.00 −0.956858
\(423\) 7392.00 0.849672
\(424\) −10521.0 −1.20506
\(425\) 0 0
\(426\) −4284.00 −0.487231
\(427\) 968.000 0.109707
\(428\) −1881.00 −0.212434
\(429\) −16380.0 −1.84344
\(430\) 0 0
\(431\) −10344.0 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) −2485.00 −0.276758
\(433\) 4048.00 0.449271 0.224636 0.974443i \(-0.427881\pi\)
0.224636 + 0.974443i \(0.427881\pi\)
\(434\) 3630.00 0.401488
\(435\) 0 0
\(436\) 281.000 0.0308657
\(437\) 2451.00 0.268300
\(438\) −8547.00 −0.932401
\(439\) 7526.00 0.818215 0.409107 0.912486i \(-0.365840\pi\)
0.409107 + 0.912486i \(0.365840\pi\)
\(440\) 0 0
\(441\) −4884.00 −0.527373
\(442\) 16965.0 1.82566
\(443\) 8796.00 0.943365 0.471682 0.881769i \(-0.343647\pi\)
0.471682 + 0.881769i \(0.343647\pi\)
\(444\) −994.000 −0.106246
\(445\) 0 0
\(446\) −15978.0 −1.69637
\(447\) 9492.00 1.00438
\(448\) −4763.00 −0.502300
\(449\) 1638.00 0.172165 0.0860824 0.996288i \(-0.472565\pi\)
0.0860824 + 0.996288i \(0.472565\pi\)
\(450\) 0 0
\(451\) 11880.0 1.24037
\(452\) −612.000 −0.0636860
\(453\) −15638.0 −1.62194
\(454\) −5643.00 −0.583346
\(455\) 0 0
\(456\) −2793.00 −0.286829
\(457\) −10703.0 −1.09555 −0.547774 0.836627i \(-0.684525\pi\)
−0.547774 + 0.836627i \(0.684525\pi\)
\(458\) −2274.00 −0.232002
\(459\) 3045.00 0.309648
\(460\) 0 0
\(461\) −7350.00 −0.742568 −0.371284 0.928519i \(-0.621082\pi\)
−0.371284 + 0.928519i \(0.621082\pi\)
\(462\) 8316.00 0.837436
\(463\) −11288.0 −1.13304 −0.566520 0.824048i \(-0.691711\pi\)
−0.566520 + 0.824048i \(0.691711\pi\)
\(464\) −16401.0 −1.64094
\(465\) 0 0
\(466\) −16002.0 −1.59073
\(467\) −13644.0 −1.35197 −0.675984 0.736916i \(-0.736281\pi\)
−0.675984 + 0.736916i \(0.736281\pi\)
\(468\) −1430.00 −0.141243
\(469\) 1309.00 0.128878
\(470\) 0 0
\(471\) 1022.00 0.0999815
\(472\) 13293.0 1.29631
\(473\) 2664.00 0.258966
\(474\) 26502.0 2.56810
\(475\) 0 0
\(476\) −957.000 −0.0921513
\(477\) −11022.0 −1.05799
\(478\) 14013.0 1.34088
\(479\) 11472.0 1.09430 0.547149 0.837035i \(-0.315713\pi\)
0.547149 + 0.837035i \(0.315713\pi\)
\(480\) 0 0
\(481\) −9230.00 −0.874952
\(482\) −5244.00 −0.495555
\(483\) 9933.00 0.935750
\(484\) −35.0000 −0.00328700
\(485\) 0 0
\(486\) −14784.0 −1.37987
\(487\) 9394.00 0.874092 0.437046 0.899439i \(-0.356025\pi\)
0.437046 + 0.899439i \(0.356025\pi\)
\(488\) −1848.00 −0.171424
\(489\) 8078.00 0.747034
\(490\) 0 0
\(491\) −6.00000 −0.000551479 0 −0.000275740 1.00000i \(-0.500088\pi\)
−0.000275740 1.00000i \(0.500088\pi\)
\(492\) 2310.00 0.211672
\(493\) 20097.0 1.83595
\(494\) 3705.00 0.337441
\(495\) 0 0
\(496\) −7810.00 −0.707015
\(497\) 2244.00 0.202529
\(498\) −5670.00 −0.510198
\(499\) 8894.00 0.797896 0.398948 0.916974i \(-0.369375\pi\)
0.398948 + 0.916974i \(0.369375\pi\)
\(500\) 0 0
\(501\) −24906.0 −2.22099
\(502\) −11952.0 −1.06264
\(503\) −7107.00 −0.629991 −0.314995 0.949093i \(-0.602003\pi\)
−0.314995 + 0.949093i \(0.602003\pi\)
\(504\) −5082.00 −0.449147
\(505\) 0 0
\(506\) 13932.0 1.22402
\(507\) −14196.0 −1.24352
\(508\) 1582.00 0.138169
\(509\) −20946.0 −1.82400 −0.911999 0.410192i \(-0.865462\pi\)
−0.911999 + 0.410192i \(0.865462\pi\)
\(510\) 0 0
\(511\) 4477.00 0.387575
\(512\) 8733.00 0.753804
\(513\) 665.000 0.0572329
\(514\) 18378.0 1.57708
\(515\) 0 0
\(516\) 518.000 0.0441932
\(517\) −12096.0 −1.02898
\(518\) 4686.00 0.397473
\(519\) −10374.0 −0.877395
\(520\) 0 0
\(521\) 15696.0 1.31987 0.659937 0.751321i \(-0.270583\pi\)
0.659937 + 0.751321i \(0.270583\pi\)
\(522\) −15246.0 −1.27835
\(523\) 6667.00 0.557414 0.278707 0.960376i \(-0.410094\pi\)
0.278707 + 0.960376i \(0.410094\pi\)
\(524\) −1932.00 −0.161068
\(525\) 0 0
\(526\) −19728.0 −1.63533
\(527\) 9570.00 0.791036
\(528\) −17892.0 −1.47471
\(529\) 4474.00 0.367716
\(530\) 0 0
\(531\) 13926.0 1.13811
\(532\) −209.000 −0.0170325
\(533\) 21450.0 1.74316
\(534\) −630.000 −0.0510539
\(535\) 0 0
\(536\) −2499.00 −0.201381
\(537\) 20160.0 1.62005
\(538\) −18234.0 −1.46120
\(539\) 7992.00 0.638664
\(540\) 0 0
\(541\) −16792.0 −1.33446 −0.667231 0.744850i \(-0.732521\pi\)
−0.667231 + 0.744850i \(0.732521\pi\)
\(542\) 3045.00 0.241317
\(543\) −3290.00 −0.260014
\(544\) 3915.00 0.308556
\(545\) 0 0
\(546\) 15015.0 1.17689
\(547\) 1276.00 0.0997401 0.0498700 0.998756i \(-0.484119\pi\)
0.0498700 + 0.998756i \(0.484119\pi\)
\(548\) 1515.00 0.118098
\(549\) −1936.00 −0.150504
\(550\) 0 0
\(551\) 4389.00 0.339342
\(552\) −18963.0 −1.46217
\(553\) −13882.0 −1.06749
\(554\) −5772.00 −0.442651
\(555\) 0 0
\(556\) −2284.00 −0.174214
\(557\) 21144.0 1.60844 0.804219 0.594333i \(-0.202584\pi\)
0.804219 + 0.594333i \(0.202584\pi\)
\(558\) −7260.00 −0.550789
\(559\) 4810.00 0.363938
\(560\) 0 0
\(561\) 21924.0 1.64997
\(562\) 12402.0 0.930866
\(563\) 14652.0 1.09682 0.548409 0.836210i \(-0.315234\pi\)
0.548409 + 0.836210i \(0.315234\pi\)
\(564\) −2352.00 −0.175598
\(565\) 0 0
\(566\) 17394.0 1.29174
\(567\) 9229.00 0.683565
\(568\) −4284.00 −0.316466
\(569\) 18288.0 1.34740 0.673702 0.739003i \(-0.264703\pi\)
0.673702 + 0.739003i \(0.264703\pi\)
\(570\) 0 0
\(571\) 19766.0 1.44865 0.724327 0.689457i \(-0.242151\pi\)
0.724327 + 0.689457i \(0.242151\pi\)
\(572\) 2340.00 0.171050
\(573\) 17325.0 1.26311
\(574\) −10890.0 −0.791881
\(575\) 0 0
\(576\) 9526.00 0.689091
\(577\) −16229.0 −1.17092 −0.585461 0.810700i \(-0.699087\pi\)
−0.585461 + 0.810700i \(0.699087\pi\)
\(578\) −7968.00 −0.573400
\(579\) 13874.0 0.995827
\(580\) 0 0
\(581\) 2970.00 0.212076
\(582\) −29526.0 −2.10291
\(583\) 18036.0 1.28126
\(584\) −8547.00 −0.605612
\(585\) 0 0
\(586\) 27945.0 1.96996
\(587\) 3750.00 0.263678 0.131839 0.991271i \(-0.457912\pi\)
0.131839 + 0.991271i \(0.457912\pi\)
\(588\) 1554.00 0.108990
\(589\) 2090.00 0.146209
\(590\) 0 0
\(591\) 28854.0 2.00828
\(592\) −10082.0 −0.699945
\(593\) 22530.0 1.56020 0.780098 0.625657i \(-0.215169\pi\)
0.780098 + 0.625657i \(0.215169\pi\)
\(594\) 3780.00 0.261103
\(595\) 0 0
\(596\) −1356.00 −0.0931945
\(597\) 28525.0 1.95553
\(598\) 25155.0 1.72017
\(599\) 11850.0 0.808310 0.404155 0.914690i \(-0.367566\pi\)
0.404155 + 0.914690i \(0.367566\pi\)
\(600\) 0 0
\(601\) −16918.0 −1.14825 −0.574126 0.818767i \(-0.694658\pi\)
−0.574126 + 0.818767i \(0.694658\pi\)
\(602\) −2442.00 −0.165330
\(603\) −2618.00 −0.176805
\(604\) 2234.00 0.150497
\(605\) 0 0
\(606\) 21672.0 1.45275
\(607\) −2702.00 −0.180677 −0.0903384 0.995911i \(-0.528795\pi\)
−0.0903384 + 0.995911i \(0.528795\pi\)
\(608\) 855.000 0.0570310
\(609\) 17787.0 1.18352
\(610\) 0 0
\(611\) −21840.0 −1.44608
\(612\) 1914.00 0.126420
\(613\) −18884.0 −1.24424 −0.622119 0.782923i \(-0.713728\pi\)
−0.622119 + 0.782923i \(0.713728\pi\)
\(614\) −21108.0 −1.38738
\(615\) 0 0
\(616\) 8316.00 0.543931
\(617\) −9798.00 −0.639307 −0.319654 0.947534i \(-0.603567\pi\)
−0.319654 + 0.947534i \(0.603567\pi\)
\(618\) 10164.0 0.661579
\(619\) 1172.00 0.0761012 0.0380506 0.999276i \(-0.487885\pi\)
0.0380506 + 0.999276i \(0.487885\pi\)
\(620\) 0 0
\(621\) 4515.00 0.291756
\(622\) 8073.00 0.520414
\(623\) 330.000 0.0212218
\(624\) −32305.0 −2.07249
\(625\) 0 0
\(626\) −1695.00 −0.108220
\(627\) 4788.00 0.304967
\(628\) −146.000 −0.00927712
\(629\) 12354.0 0.783126
\(630\) 0 0
\(631\) 920.000 0.0580422 0.0290211 0.999579i \(-0.490761\pi\)
0.0290211 + 0.999579i \(0.490761\pi\)
\(632\) 26502.0 1.66803
\(633\) −19355.0 −1.21531
\(634\) 7389.00 0.462862
\(635\) 0 0
\(636\) 3507.00 0.218650
\(637\) 14430.0 0.897547
\(638\) 24948.0 1.54812
\(639\) −4488.00 −0.277844
\(640\) 0 0
\(641\) −9240.00 −0.569357 −0.284679 0.958623i \(-0.591887\pi\)
−0.284679 + 0.958623i \(0.591887\pi\)
\(642\) −39501.0 −2.42832
\(643\) −8246.00 −0.505739 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(644\) −1419.00 −0.0868268
\(645\) 0 0
\(646\) −4959.00 −0.302027
\(647\) −11475.0 −0.697262 −0.348631 0.937260i \(-0.613353\pi\)
−0.348631 + 0.937260i \(0.613353\pi\)
\(648\) −17619.0 −1.06812
\(649\) −22788.0 −1.37829
\(650\) 0 0
\(651\) 8470.00 0.509932
\(652\) −1154.00 −0.0693161
\(653\) −2196.00 −0.131602 −0.0658010 0.997833i \(-0.520960\pi\)
−0.0658010 + 0.997833i \(0.520960\pi\)
\(654\) 5901.00 0.352825
\(655\) 0 0
\(656\) 23430.0 1.39449
\(657\) −8954.00 −0.531703
\(658\) 11088.0 0.656923
\(659\) −21513.0 −1.27167 −0.635833 0.771827i \(-0.719343\pi\)
−0.635833 + 0.771827i \(0.719343\pi\)
\(660\) 0 0
\(661\) 31853.0 1.87434 0.937170 0.348874i \(-0.113436\pi\)
0.937170 + 0.348874i \(0.113436\pi\)
\(662\) −22389.0 −1.31446
\(663\) 39585.0 2.31878
\(664\) −5670.00 −0.331384
\(665\) 0 0
\(666\) −9372.00 −0.545282
\(667\) 29799.0 1.72987
\(668\) 3558.00 0.206083
\(669\) −37282.0 −2.15457
\(670\) 0 0
\(671\) 3168.00 0.182264
\(672\) 3465.00 0.198907
\(673\) −27308.0 −1.56411 −0.782055 0.623209i \(-0.785828\pi\)
−0.782055 + 0.623209i \(0.785828\pi\)
\(674\) −33582.0 −1.91918
\(675\) 0 0
\(676\) 2028.00 0.115385
\(677\) 12609.0 0.715810 0.357905 0.933758i \(-0.383491\pi\)
0.357905 + 0.933758i \(0.383491\pi\)
\(678\) −12852.0 −0.727991
\(679\) 15466.0 0.874125
\(680\) 0 0
\(681\) −13167.0 −0.740911
\(682\) 11880.0 0.667022
\(683\) −32292.0 −1.80911 −0.904553 0.426362i \(-0.859795\pi\)
−0.904553 + 0.426362i \(0.859795\pi\)
\(684\) 418.000 0.0233664
\(685\) 0 0
\(686\) −18645.0 −1.03771
\(687\) −5306.00 −0.294667
\(688\) 5254.00 0.291144
\(689\) 32565.0 1.80062
\(690\) 0 0
\(691\) −16558.0 −0.911572 −0.455786 0.890089i \(-0.650642\pi\)
−0.455786 + 0.890089i \(0.650642\pi\)
\(692\) 1482.00 0.0814121
\(693\) 8712.00 0.477549
\(694\) 23418.0 1.28089
\(695\) 0 0
\(696\) −33957.0 −1.84933
\(697\) −28710.0 −1.56021
\(698\) 24834.0 1.34668
\(699\) −37338.0 −2.02039
\(700\) 0 0
\(701\) −23712.0 −1.27759 −0.638794 0.769377i \(-0.720567\pi\)
−0.638794 + 0.769377i \(0.720567\pi\)
\(702\) 6825.00 0.366942
\(703\) 2698.00 0.144747
\(704\) −15588.0 −0.834510
\(705\) 0 0
\(706\) 22329.0 1.19032
\(707\) −11352.0 −0.603870
\(708\) −4431.00 −0.235208
\(709\) 10334.0 0.547393 0.273696 0.961816i \(-0.411754\pi\)
0.273696 + 0.961816i \(0.411754\pi\)
\(710\) 0 0
\(711\) 27764.0 1.46446
\(712\) −630.000 −0.0331605
\(713\) 14190.0 0.745329
\(714\) −20097.0 −1.05338
\(715\) 0 0
\(716\) −2880.00 −0.150322
\(717\) 32697.0 1.70306
\(718\) 11115.0 0.577727
\(719\) 1353.00 0.0701786 0.0350893 0.999384i \(-0.488828\pi\)
0.0350893 + 0.999384i \(0.488828\pi\)
\(720\) 0 0
\(721\) −5324.00 −0.275002
\(722\) −1083.00 −0.0558242
\(723\) −12236.0 −0.629408
\(724\) 470.000 0.0241263
\(725\) 0 0
\(726\) −735.000 −0.0375736
\(727\) −8687.00 −0.443168 −0.221584 0.975141i \(-0.571123\pi\)
−0.221584 + 0.975141i \(0.571123\pi\)
\(728\) 15015.0 0.764413
\(729\) −11843.0 −0.601687
\(730\) 0 0
\(731\) −6438.00 −0.325743
\(732\) 616.000 0.0311038
\(733\) −29144.0 −1.46857 −0.734283 0.678844i \(-0.762481\pi\)
−0.734283 + 0.678844i \(0.762481\pi\)
\(734\) 2976.00 0.149654
\(735\) 0 0
\(736\) 5805.00 0.290727
\(737\) 4284.00 0.214116
\(738\) 21780.0 1.08636
\(739\) −2050.00 −0.102044 −0.0510220 0.998698i \(-0.516248\pi\)
−0.0510220 + 0.998698i \(0.516248\pi\)
\(740\) 0 0
\(741\) 8645.00 0.428586
\(742\) −16533.0 −0.817986
\(743\) −5142.00 −0.253892 −0.126946 0.991910i \(-0.540517\pi\)
−0.126946 + 0.991910i \(0.540517\pi\)
\(744\) −16170.0 −0.796802
\(745\) 0 0
\(746\) 33405.0 1.63947
\(747\) −5940.00 −0.290941
\(748\) −3132.00 −0.153098
\(749\) 20691.0 1.00939
\(750\) 0 0
\(751\) −27070.0 −1.31531 −0.657655 0.753319i \(-0.728452\pi\)
−0.657655 + 0.753319i \(0.728452\pi\)
\(752\) −23856.0 −1.15683
\(753\) −27888.0 −1.34966
\(754\) 45045.0 2.17565
\(755\) 0 0
\(756\) −385.000 −0.0185216
\(757\) −10694.0 −0.513448 −0.256724 0.966485i \(-0.582643\pi\)
−0.256724 + 0.966485i \(0.582643\pi\)
\(758\) 21405.0 1.02568
\(759\) 32508.0 1.55463
\(760\) 0 0
\(761\) −21285.0 −1.01390 −0.506952 0.861974i \(-0.669228\pi\)
−0.506952 + 0.861974i \(0.669228\pi\)
\(762\) 33222.0 1.57940
\(763\) −3091.00 −0.146660
\(764\) −2475.00 −0.117202
\(765\) 0 0
\(766\) 15372.0 0.725082
\(767\) −41145.0 −1.93698
\(768\) −10591.0 −0.497617
\(769\) 12809.0 0.600656 0.300328 0.953836i \(-0.402904\pi\)
0.300328 + 0.953836i \(0.402904\pi\)
\(770\) 0 0
\(771\) 42882.0 2.00306
\(772\) −1982.00 −0.0924012
\(773\) 1893.00 0.0880808 0.0440404 0.999030i \(-0.485977\pi\)
0.0440404 + 0.999030i \(0.485977\pi\)
\(774\) 4884.00 0.226811
\(775\) 0 0
\(776\) −29526.0 −1.36588
\(777\) 10934.0 0.504833
\(778\) −33012.0 −1.52126
\(779\) −6270.00 −0.288377
\(780\) 0 0
\(781\) 7344.00 0.336478
\(782\) −33669.0 −1.53964
\(783\) 8085.00 0.369009
\(784\) 15762.0 0.718021
\(785\) 0 0
\(786\) −40572.0 −1.84116
\(787\) 32101.0 1.45397 0.726987 0.686652i \(-0.240920\pi\)
0.726987 + 0.686652i \(0.240920\pi\)
\(788\) −4122.00 −0.186345
\(789\) −46032.0 −2.07704
\(790\) 0 0
\(791\) 6732.00 0.302607
\(792\) −16632.0 −0.746203
\(793\) 5720.00 0.256145
\(794\) −7284.00 −0.325566
\(795\) 0 0
\(796\) −4075.00 −0.181450
\(797\) −21753.0 −0.966789 −0.483394 0.875403i \(-0.660596\pi\)
−0.483394 + 0.875403i \(0.660596\pi\)
\(798\) −4389.00 −0.194698
\(799\) 29232.0 1.29431
\(800\) 0 0
\(801\) −660.000 −0.0291135
\(802\) 12438.0 0.547632
\(803\) 14652.0 0.643908
\(804\) 833.000 0.0365394
\(805\) 0 0
\(806\) 21450.0 0.937400
\(807\) −42546.0 −1.85587
\(808\) 21672.0 0.943587
\(809\) −14247.0 −0.619157 −0.309578 0.950874i \(-0.600188\pi\)
−0.309578 + 0.950874i \(0.600188\pi\)
\(810\) 0 0
\(811\) −25045.0 −1.08440 −0.542200 0.840249i \(-0.682409\pi\)
−0.542200 + 0.840249i \(0.682409\pi\)
\(812\) −2541.00 −0.109817
\(813\) 7105.00 0.306498
\(814\) 15336.0 0.660352
\(815\) 0 0
\(816\) 43239.0 1.85499
\(817\) −1406.00 −0.0602077
\(818\) −34890.0 −1.49132
\(819\) 15730.0 0.671124
\(820\) 0 0
\(821\) −22224.0 −0.944730 −0.472365 0.881403i \(-0.656599\pi\)
−0.472365 + 0.881403i \(0.656599\pi\)
\(822\) 31815.0 1.34997
\(823\) 31867.0 1.34971 0.674856 0.737949i \(-0.264206\pi\)
0.674856 + 0.737949i \(0.264206\pi\)
\(824\) 10164.0 0.429708
\(825\) 0 0
\(826\) 20889.0 0.879929
\(827\) 4233.00 0.177988 0.0889939 0.996032i \(-0.471635\pi\)
0.0889939 + 0.996032i \(0.471635\pi\)
\(828\) 2838.00 0.119115
\(829\) 26561.0 1.11279 0.556394 0.830918i \(-0.312184\pi\)
0.556394 + 0.830918i \(0.312184\pi\)
\(830\) 0 0
\(831\) −13468.0 −0.562214
\(832\) −28145.0 −1.17278
\(833\) −19314.0 −0.803350
\(834\) −47964.0 −1.99144
\(835\) 0 0
\(836\) −684.000 −0.0282974
\(837\) 3850.00 0.158991
\(838\) 44010.0 1.81420
\(839\) 3792.00 0.156036 0.0780181 0.996952i \(-0.475141\pi\)
0.0780181 + 0.996952i \(0.475141\pi\)
\(840\) 0 0
\(841\) 28972.0 1.18791
\(842\) 26967.0 1.10373
\(843\) 28938.0 1.18230
\(844\) 2765.00 0.112767
\(845\) 0 0
\(846\) −22176.0 −0.901213
\(847\) 385.000 0.0156184
\(848\) 35571.0 1.44046
\(849\) 40586.0 1.64065
\(850\) 0 0
\(851\) 18318.0 0.737877
\(852\) 1428.00 0.0574207
\(853\) −26858.0 −1.07808 −0.539039 0.842281i \(-0.681212\pi\)
−0.539039 + 0.842281i \(0.681212\pi\)
\(854\) −2904.00 −0.116362
\(855\) 0 0
\(856\) −39501.0 −1.57724
\(857\) −6672.00 −0.265941 −0.132970 0.991120i \(-0.542452\pi\)
−0.132970 + 0.991120i \(0.542452\pi\)
\(858\) 49140.0 1.95526
\(859\) −8764.00 −0.348107 −0.174053 0.984736i \(-0.555687\pi\)
−0.174053 + 0.984736i \(0.555687\pi\)
\(860\) 0 0
\(861\) −25410.0 −1.00577
\(862\) 31032.0 1.22616
\(863\) 6012.00 0.237139 0.118569 0.992946i \(-0.462169\pi\)
0.118569 + 0.992946i \(0.462169\pi\)
\(864\) 1575.00 0.0620169
\(865\) 0 0
\(866\) −12144.0 −0.476524
\(867\) −18592.0 −0.728278
\(868\) −1210.00 −0.0473158
\(869\) −45432.0 −1.77350
\(870\) 0 0
\(871\) 7735.00 0.300908
\(872\) 5901.00 0.229166
\(873\) −30932.0 −1.19919
\(874\) −7353.00 −0.284575
\(875\) 0 0
\(876\) 2849.00 0.109884
\(877\) −47207.0 −1.81764 −0.908818 0.417192i \(-0.863014\pi\)
−0.908818 + 0.417192i \(0.863014\pi\)
\(878\) −22578.0 −0.867848
\(879\) 65205.0 2.50206
\(880\) 0 0
\(881\) 21618.0 0.826707 0.413354 0.910571i \(-0.364357\pi\)
0.413354 + 0.910571i \(0.364357\pi\)
\(882\) 14652.0 0.559363
\(883\) −25706.0 −0.979701 −0.489850 0.871807i \(-0.662949\pi\)
−0.489850 + 0.871807i \(0.662949\pi\)
\(884\) −5655.00 −0.215156
\(885\) 0 0
\(886\) −26388.0 −1.00059
\(887\) −3336.00 −0.126282 −0.0631409 0.998005i \(-0.520112\pi\)
−0.0631409 + 0.998005i \(0.520112\pi\)
\(888\) −20874.0 −0.788835
\(889\) −17402.0 −0.656518
\(890\) 0 0
\(891\) 30204.0 1.13566
\(892\) 5326.00 0.199919
\(893\) 6384.00 0.239230
\(894\) −28476.0 −1.06530
\(895\) 0 0
\(896\) 18249.0 0.680420
\(897\) 58695.0 2.18480
\(898\) −4914.00 −0.182608
\(899\) 25410.0 0.942682
\(900\) 0 0
\(901\) −43587.0 −1.61165
\(902\) −35640.0 −1.31561
\(903\) −5698.00 −0.209986
\(904\) −12852.0 −0.472844
\(905\) 0 0
\(906\) 46914.0 1.72032
\(907\) 18979.0 0.694804 0.347402 0.937716i \(-0.387064\pi\)
0.347402 + 0.937716i \(0.387064\pi\)
\(908\) 1881.00 0.0687480
\(909\) 22704.0 0.828431
\(910\) 0 0
\(911\) 3600.00 0.130926 0.0654629 0.997855i \(-0.479148\pi\)
0.0654629 + 0.997855i \(0.479148\pi\)
\(912\) 9443.00 0.342861
\(913\) 9720.00 0.352338
\(914\) 32109.0 1.16200
\(915\) 0 0
\(916\) 758.000 0.0273417
\(917\) 21252.0 0.765325
\(918\) −9135.00 −0.328431
\(919\) 13655.0 0.490138 0.245069 0.969506i \(-0.421189\pi\)
0.245069 + 0.969506i \(0.421189\pi\)
\(920\) 0 0
\(921\) −49252.0 −1.76212
\(922\) 22050.0 0.787612
\(923\) 13260.0 0.472869
\(924\) −2772.00 −0.0986928
\(925\) 0 0
\(926\) 33864.0 1.20177
\(927\) 10648.0 0.377267
\(928\) 10395.0 0.367708
\(929\) −35043.0 −1.23759 −0.618796 0.785551i \(-0.712379\pi\)
−0.618796 + 0.785551i \(0.712379\pi\)
\(930\) 0 0
\(931\) −4218.00 −0.148485
\(932\) 5334.00 0.187469
\(933\) 18837.0 0.660981
\(934\) 40932.0 1.43398
\(935\) 0 0
\(936\) −30030.0 −1.04868
\(937\) 7045.00 0.245624 0.122812 0.992430i \(-0.460809\pi\)
0.122812 + 0.992430i \(0.460809\pi\)
\(938\) −3927.00 −0.136696
\(939\) −3955.00 −0.137451
\(940\) 0 0
\(941\) 3399.00 0.117752 0.0588758 0.998265i \(-0.481248\pi\)
0.0588758 + 0.998265i \(0.481248\pi\)
\(942\) −3066.00 −0.106046
\(943\) −42570.0 −1.47006
\(944\) −44943.0 −1.54954
\(945\) 0 0
\(946\) −7992.00 −0.274675
\(947\) 22884.0 0.785248 0.392624 0.919699i \(-0.371567\pi\)
0.392624 + 0.919699i \(0.371567\pi\)
\(948\) −8834.00 −0.302653
\(949\) 26455.0 0.904916
\(950\) 0 0
\(951\) 17241.0 0.587884
\(952\) −20097.0 −0.684189
\(953\) −50520.0 −1.71721 −0.858606 0.512636i \(-0.828669\pi\)
−0.858606 + 0.512636i \(0.828669\pi\)
\(954\) 33066.0 1.12217
\(955\) 0 0
\(956\) −4671.00 −0.158024
\(957\) 58212.0 1.96628
\(958\) −34416.0 −1.16068
\(959\) −16665.0 −0.561148
\(960\) 0 0
\(961\) −17691.0 −0.593837
\(962\) 27690.0 0.928026
\(963\) −41382.0 −1.38475
\(964\) 1748.00 0.0584018
\(965\) 0 0
\(966\) −29799.0 −0.992513
\(967\) 7468.00 0.248350 0.124175 0.992260i \(-0.460372\pi\)
0.124175 + 0.992260i \(0.460372\pi\)
\(968\) −735.000 −0.0244047
\(969\) −11571.0 −0.383606
\(970\) 0 0
\(971\) 11172.0 0.369234 0.184617 0.982811i \(-0.440895\pi\)
0.184617 + 0.982811i \(0.440895\pi\)
\(972\) 4928.00 0.162619
\(973\) 25124.0 0.827789
\(974\) −28182.0 −0.927115
\(975\) 0 0
\(976\) 6248.00 0.204911
\(977\) −35586.0 −1.16530 −0.582649 0.812724i \(-0.697984\pi\)
−0.582649 + 0.812724i \(0.697984\pi\)
\(978\) −24234.0 −0.792350
\(979\) 1080.00 0.0352574
\(980\) 0 0
\(981\) 6182.00 0.201199
\(982\) 18.0000 0.000584932 0
\(983\) −29694.0 −0.963471 −0.481735 0.876317i \(-0.659993\pi\)
−0.481735 + 0.876317i \(0.659993\pi\)
\(984\) 48510.0 1.57159
\(985\) 0 0
\(986\) −60291.0 −1.94732
\(987\) 25872.0 0.834362
\(988\) −1235.00 −0.0397678
\(989\) −9546.00 −0.306921
\(990\) 0 0
\(991\) −46546.0 −1.49201 −0.746005 0.665940i \(-0.768031\pi\)
−0.746005 + 0.665940i \(0.768031\pi\)
\(992\) 4950.00 0.158430
\(993\) −52241.0 −1.66950
\(994\) −6732.00 −0.214815
\(995\) 0 0
\(996\) 1890.00 0.0601275
\(997\) −6338.00 −0.201330 −0.100665 0.994920i \(-0.532097\pi\)
−0.100665 + 0.994920i \(0.532097\pi\)
\(998\) −26682.0 −0.846297
\(999\) 4970.00 0.157401
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 475.4.a.b.1.1 1
5.2 odd 4 475.4.b.d.324.1 2
5.3 odd 4 475.4.b.d.324.2 2
5.4 even 2 95.4.a.c.1.1 1
15.14 odd 2 855.4.a.c.1.1 1
20.19 odd 2 1520.4.a.a.1.1 1
95.94 odd 2 1805.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.a.c.1.1 1 5.4 even 2
475.4.a.b.1.1 1 1.1 even 1 trivial
475.4.b.d.324.1 2 5.2 odd 4
475.4.b.d.324.2 2 5.3 odd 4
855.4.a.c.1.1 1 15.14 odd 2
1520.4.a.a.1.1 1 20.19 odd 2
1805.4.a.c.1.1 1 95.94 odd 2