Properties

Label 525.4.a.h.1.1
Level $525$
Weight $4$
Character 525.1
Self dual yes
Analytic conductor $30.976$
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] = [525,4,Mod(1,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.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: \(30.9760027530\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +9.00000 q^{6} -7.00000 q^{7} -21.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +9.00000 q^{6} -7.00000 q^{7} -21.0000 q^{8} +9.00000 q^{9} -6.00000 q^{11} +3.00000 q^{12} -41.0000 q^{13} -21.0000 q^{14} -71.0000 q^{16} -27.0000 q^{17} +27.0000 q^{18} -4.00000 q^{19} -21.0000 q^{21} -18.0000 q^{22} -75.0000 q^{23} -63.0000 q^{24} -123.000 q^{26} +27.0000 q^{27} -7.00000 q^{28} -123.000 q^{29} -205.000 q^{31} -45.0000 q^{32} -18.0000 q^{33} -81.0000 q^{34} +9.00000 q^{36} +262.000 q^{37} -12.0000 q^{38} -123.000 q^{39} +57.0000 q^{41} -63.0000 q^{42} -407.000 q^{43} -6.00000 q^{44} -225.000 q^{46} +60.0000 q^{47} -213.000 q^{48} +49.0000 q^{49} -81.0000 q^{51} -41.0000 q^{52} -327.000 q^{53} +81.0000 q^{54} +147.000 q^{56} -12.0000 q^{57} -369.000 q^{58} +33.0000 q^{59} -427.000 q^{61} -615.000 q^{62} -63.0000 q^{63} +433.000 q^{64} -54.0000 q^{66} +628.000 q^{67} -27.0000 q^{68} -225.000 q^{69} +300.000 q^{71} -189.000 q^{72} -98.0000 q^{73} +786.000 q^{74} -4.00000 q^{76} +42.0000 q^{77} -369.000 q^{78} +686.000 q^{79} +81.0000 q^{81} +171.000 q^{82} -1401.00 q^{83} -21.0000 q^{84} -1221.00 q^{86} -369.000 q^{87} +126.000 q^{88} +714.000 q^{89} +287.000 q^{91} -75.0000 q^{92} -615.000 q^{93} +180.000 q^{94} -135.000 q^{96} -494.000 q^{97} +147.000 q^{98} -54.0000 q^{99} +O(q^{100})\)

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 \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) 3.00000 0.577350
\(4\) 1.00000 0.125000
\(5\) 0 0
\(6\) 9.00000 0.612372
\(7\) −7.00000 −0.377964
\(8\) −21.0000 −0.928078
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −6.00000 −0.164461 −0.0822304 0.996613i \(-0.526204\pi\)
−0.0822304 + 0.996613i \(0.526204\pi\)
\(12\) 3.00000 0.0721688
\(13\) −41.0000 −0.874720 −0.437360 0.899287i \(-0.644086\pi\)
−0.437360 + 0.899287i \(0.644086\pi\)
\(14\) −21.0000 −0.400892
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) −27.0000 −0.385204 −0.192602 0.981277i \(-0.561693\pi\)
−0.192602 + 0.981277i \(0.561693\pi\)
\(18\) 27.0000 0.353553
\(19\) −4.00000 −0.0482980 −0.0241490 0.999708i \(-0.507688\pi\)
−0.0241490 + 0.999708i \(0.507688\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) −18.0000 −0.174437
\(23\) −75.0000 −0.679938 −0.339969 0.940437i \(-0.610417\pi\)
−0.339969 + 0.940437i \(0.610417\pi\)
\(24\) −63.0000 −0.535826
\(25\) 0 0
\(26\) −123.000 −0.927780
\(27\) 27.0000 0.192450
\(28\) −7.00000 −0.0472456
\(29\) −123.000 −0.787604 −0.393802 0.919195i \(-0.628841\pi\)
−0.393802 + 0.919195i \(0.628841\pi\)
\(30\) 0 0
\(31\) −205.000 −1.18771 −0.593856 0.804571i \(-0.702395\pi\)
−0.593856 + 0.804571i \(0.702395\pi\)
\(32\) −45.0000 −0.248592
\(33\) −18.0000 −0.0949514
\(34\) −81.0000 −0.408570
\(35\) 0 0
\(36\) 9.00000 0.0416667
\(37\) 262.000 1.16412 0.582061 0.813145i \(-0.302246\pi\)
0.582061 + 0.813145i \(0.302246\pi\)
\(38\) −12.0000 −0.0512278
\(39\) −123.000 −0.505020
\(40\) 0 0
\(41\) 57.0000 0.217120 0.108560 0.994090i \(-0.465376\pi\)
0.108560 + 0.994090i \(0.465376\pi\)
\(42\) −63.0000 −0.231455
\(43\) −407.000 −1.44342 −0.721708 0.692197i \(-0.756643\pi\)
−0.721708 + 0.692197i \(0.756643\pi\)
\(44\) −6.00000 −0.0205576
\(45\) 0 0
\(46\) −225.000 −0.721183
\(47\) 60.0000 0.186211 0.0931053 0.995656i \(-0.470321\pi\)
0.0931053 + 0.995656i \(0.470321\pi\)
\(48\) −213.000 −0.640498
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −81.0000 −0.222397
\(52\) −41.0000 −0.109340
\(53\) −327.000 −0.847489 −0.423744 0.905782i \(-0.639285\pi\)
−0.423744 + 0.905782i \(0.639285\pi\)
\(54\) 81.0000 0.204124
\(55\) 0 0
\(56\) 147.000 0.350780
\(57\) −12.0000 −0.0278849
\(58\) −369.000 −0.835381
\(59\) 33.0000 0.0728175 0.0364088 0.999337i \(-0.488408\pi\)
0.0364088 + 0.999337i \(0.488408\pi\)
\(60\) 0 0
\(61\) −427.000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) −615.000 −1.25976
\(63\) −63.0000 −0.125988
\(64\) 433.000 0.845703
\(65\) 0 0
\(66\) −54.0000 −0.100711
\(67\) 628.000 1.14511 0.572555 0.819866i \(-0.305952\pi\)
0.572555 + 0.819866i \(0.305952\pi\)
\(68\) −27.0000 −0.0481505
\(69\) −225.000 −0.392563
\(70\) 0 0
\(71\) 300.000 0.501457 0.250729 0.968057i \(-0.419330\pi\)
0.250729 + 0.968057i \(0.419330\pi\)
\(72\) −189.000 −0.309359
\(73\) −98.0000 −0.157124 −0.0785619 0.996909i \(-0.525033\pi\)
−0.0785619 + 0.996909i \(0.525033\pi\)
\(74\) 786.000 1.23474
\(75\) 0 0
\(76\) −4.00000 −0.00603726
\(77\) 42.0000 0.0621603
\(78\) −369.000 −0.535654
\(79\) 686.000 0.976975 0.488488 0.872571i \(-0.337549\pi\)
0.488488 + 0.872571i \(0.337549\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 171.000 0.230290
\(83\) −1401.00 −1.85277 −0.926384 0.376580i \(-0.877100\pi\)
−0.926384 + 0.376580i \(0.877100\pi\)
\(84\) −21.0000 −0.0272772
\(85\) 0 0
\(86\) −1221.00 −1.53097
\(87\) −369.000 −0.454724
\(88\) 126.000 0.152632
\(89\) 714.000 0.850380 0.425190 0.905104i \(-0.360207\pi\)
0.425190 + 0.905104i \(0.360207\pi\)
\(90\) 0 0
\(91\) 287.000 0.330613
\(92\) −75.0000 −0.0849923
\(93\) −615.000 −0.685726
\(94\) 180.000 0.197506
\(95\) 0 0
\(96\) −135.000 −0.143525
\(97\) −494.000 −0.517094 −0.258547 0.965999i \(-0.583244\pi\)
−0.258547 + 0.965999i \(0.583244\pi\)
\(98\) 147.000 0.151523
\(99\) −54.0000 −0.0548202
\(100\) 0 0
\(101\) 624.000 0.614756 0.307378 0.951588i \(-0.400548\pi\)
0.307378 + 0.951588i \(0.400548\pi\)
\(102\) −243.000 −0.235888
\(103\) 769.000 0.735649 0.367824 0.929895i \(-0.380103\pi\)
0.367824 + 0.929895i \(0.380103\pi\)
\(104\) 861.000 0.811808
\(105\) 0 0
\(106\) −981.000 −0.898898
\(107\) −1662.00 −1.50160 −0.750802 0.660527i \(-0.770333\pi\)
−0.750802 + 0.660527i \(0.770333\pi\)
\(108\) 27.0000 0.0240563
\(109\) 188.000 0.165203 0.0826015 0.996583i \(-0.473677\pi\)
0.0826015 + 0.996583i \(0.473677\pi\)
\(110\) 0 0
\(111\) 786.000 0.672106
\(112\) 497.000 0.419304
\(113\) 18.0000 0.0149849 0.00749247 0.999972i \(-0.497615\pi\)
0.00749247 + 0.999972i \(0.497615\pi\)
\(114\) −36.0000 −0.0295764
\(115\) 0 0
\(116\) −123.000 −0.0984505
\(117\) −369.000 −0.291573
\(118\) 99.0000 0.0772347
\(119\) 189.000 0.145593
\(120\) 0 0
\(121\) −1295.00 −0.972953
\(122\) −1281.00 −0.950625
\(123\) 171.000 0.125354
\(124\) −205.000 −0.148464
\(125\) 0 0
\(126\) −189.000 −0.133631
\(127\) 1450.00 1.01312 0.506562 0.862204i \(-0.330916\pi\)
0.506562 + 0.862204i \(0.330916\pi\)
\(128\) 1659.00 1.14560
\(129\) −1221.00 −0.833357
\(130\) 0 0
\(131\) 2664.00 1.77675 0.888377 0.459115i \(-0.151833\pi\)
0.888377 + 0.459115i \(0.151833\pi\)
\(132\) −18.0000 −0.0118689
\(133\) 28.0000 0.0182549
\(134\) 1884.00 1.21457
\(135\) 0 0
\(136\) 567.000 0.357499
\(137\) −1692.00 −1.05516 −0.527581 0.849504i \(-0.676901\pi\)
−0.527581 + 0.849504i \(0.676901\pi\)
\(138\) −675.000 −0.416375
\(139\) 1268.00 0.773744 0.386872 0.922134i \(-0.373556\pi\)
0.386872 + 0.922134i \(0.373556\pi\)
\(140\) 0 0
\(141\) 180.000 0.107509
\(142\) 900.000 0.531876
\(143\) 246.000 0.143857
\(144\) −639.000 −0.369792
\(145\) 0 0
\(146\) −294.000 −0.166655
\(147\) 147.000 0.0824786
\(148\) 262.000 0.145515
\(149\) 2169.00 1.19256 0.596280 0.802777i \(-0.296645\pi\)
0.596280 + 0.802777i \(0.296645\pi\)
\(150\) 0 0
\(151\) 518.000 0.279167 0.139584 0.990210i \(-0.455424\pi\)
0.139584 + 0.990210i \(0.455424\pi\)
\(152\) 84.0000 0.0448243
\(153\) −243.000 −0.128401
\(154\) 126.000 0.0659310
\(155\) 0 0
\(156\) −123.000 −0.0631274
\(157\) 886.000 0.450385 0.225193 0.974314i \(-0.427699\pi\)
0.225193 + 0.974314i \(0.427699\pi\)
\(158\) 2058.00 1.03624
\(159\) −981.000 −0.489298
\(160\) 0 0
\(161\) 525.000 0.256993
\(162\) 243.000 0.117851
\(163\) −3893.00 −1.87070 −0.935348 0.353730i \(-0.884913\pi\)
−0.935348 + 0.353730i \(0.884913\pi\)
\(164\) 57.0000 0.0271400
\(165\) 0 0
\(166\) −4203.00 −1.96516
\(167\) −2046.00 −0.948049 −0.474025 0.880512i \(-0.657199\pi\)
−0.474025 + 0.880512i \(0.657199\pi\)
\(168\) 441.000 0.202523
\(169\) −516.000 −0.234866
\(170\) 0 0
\(171\) −36.0000 −0.0160993
\(172\) −407.000 −0.180427
\(173\) 3540.00 1.55573 0.777865 0.628432i \(-0.216303\pi\)
0.777865 + 0.628432i \(0.216303\pi\)
\(174\) −1107.00 −0.482307
\(175\) 0 0
\(176\) 426.000 0.182449
\(177\) 99.0000 0.0420412
\(178\) 2142.00 0.901965
\(179\) −3246.00 −1.35540 −0.677702 0.735336i \(-0.737024\pi\)
−0.677702 + 0.735336i \(0.737024\pi\)
\(180\) 0 0
\(181\) 1334.00 0.547820 0.273910 0.961755i \(-0.411683\pi\)
0.273910 + 0.961755i \(0.411683\pi\)
\(182\) 861.000 0.350668
\(183\) −1281.00 −0.517455
\(184\) 1575.00 0.631036
\(185\) 0 0
\(186\) −1845.00 −0.727322
\(187\) 162.000 0.0633509
\(188\) 60.0000 0.0232763
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) −873.000 −0.330723 −0.165361 0.986233i \(-0.552879\pi\)
−0.165361 + 0.986233i \(0.552879\pi\)
\(192\) 1299.00 0.488267
\(193\) 1006.00 0.375199 0.187600 0.982246i \(-0.439929\pi\)
0.187600 + 0.982246i \(0.439929\pi\)
\(194\) −1482.00 −0.548461
\(195\) 0 0
\(196\) 49.0000 0.0178571
\(197\) −591.000 −0.213741 −0.106871 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) −162.000 −0.0581456
\(199\) −2584.00 −0.920477 −0.460238 0.887795i \(-0.652236\pi\)
−0.460238 + 0.887795i \(0.652236\pi\)
\(200\) 0 0
\(201\) 1884.00 0.661130
\(202\) 1872.00 0.652047
\(203\) 861.000 0.297686
\(204\) −81.0000 −0.0277997
\(205\) 0 0
\(206\) 2307.00 0.780273
\(207\) −675.000 −0.226646
\(208\) 2911.00 0.970392
\(209\) 24.0000 0.00794313
\(210\) 0 0
\(211\) −1441.00 −0.470154 −0.235077 0.971977i \(-0.575534\pi\)
−0.235077 + 0.971977i \(0.575534\pi\)
\(212\) −327.000 −0.105936
\(213\) 900.000 0.289516
\(214\) −4986.00 −1.59269
\(215\) 0 0
\(216\) −567.000 −0.178609
\(217\) 1435.00 0.448913
\(218\) 564.000 0.175224
\(219\) −294.000 −0.0907154
\(220\) 0 0
\(221\) 1107.00 0.336945
\(222\) 2358.00 0.712877
\(223\) −3827.00 −1.14921 −0.574607 0.818429i \(-0.694845\pi\)
−0.574607 + 0.818429i \(0.694845\pi\)
\(224\) 315.000 0.0939590
\(225\) 0 0
\(226\) 54.0000 0.0158939
\(227\) 5421.00 1.58504 0.792521 0.609845i \(-0.208768\pi\)
0.792521 + 0.609845i \(0.208768\pi\)
\(228\) −12.0000 −0.00348561
\(229\) −5290.00 −1.52652 −0.763260 0.646092i \(-0.776402\pi\)
−0.763260 + 0.646092i \(0.776402\pi\)
\(230\) 0 0
\(231\) 126.000 0.0358883
\(232\) 2583.00 0.730958
\(233\) −4908.00 −1.37997 −0.689987 0.723822i \(-0.742384\pi\)
−0.689987 + 0.723822i \(0.742384\pi\)
\(234\) −1107.00 −0.309260
\(235\) 0 0
\(236\) 33.0000 0.00910219
\(237\) 2058.00 0.564057
\(238\) 567.000 0.154425
\(239\) 1056.00 0.285803 0.142902 0.989737i \(-0.454357\pi\)
0.142902 + 0.989737i \(0.454357\pi\)
\(240\) 0 0
\(241\) 5342.00 1.42784 0.713918 0.700229i \(-0.246919\pi\)
0.713918 + 0.700229i \(0.246919\pi\)
\(242\) −3885.00 −1.03197
\(243\) 243.000 0.0641500
\(244\) −427.000 −0.112032
\(245\) 0 0
\(246\) 513.000 0.132958
\(247\) 164.000 0.0422472
\(248\) 4305.00 1.10229
\(249\) −4203.00 −1.06970
\(250\) 0 0
\(251\) 5805.00 1.45979 0.729897 0.683557i \(-0.239568\pi\)
0.729897 + 0.683557i \(0.239568\pi\)
\(252\) −63.0000 −0.0157485
\(253\) 450.000 0.111823
\(254\) 4350.00 1.07458
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) −543.000 −0.131795 −0.0658977 0.997826i \(-0.520991\pi\)
−0.0658977 + 0.997826i \(0.520991\pi\)
\(258\) −3663.00 −0.883909
\(259\) −1834.00 −0.439997
\(260\) 0 0
\(261\) −1107.00 −0.262535
\(262\) 7992.00 1.88453
\(263\) 5193.00 1.21754 0.608772 0.793345i \(-0.291662\pi\)
0.608772 + 0.793345i \(0.291662\pi\)
\(264\) 378.000 0.0881223
\(265\) 0 0
\(266\) 84.0000 0.0193623
\(267\) 2142.00 0.490967
\(268\) 628.000 0.143139
\(269\) 2154.00 0.488222 0.244111 0.969747i \(-0.421504\pi\)
0.244111 + 0.969747i \(0.421504\pi\)
\(270\) 0 0
\(271\) 2396.00 0.537072 0.268536 0.963270i \(-0.413460\pi\)
0.268536 + 0.963270i \(0.413460\pi\)
\(272\) 1917.00 0.427335
\(273\) 861.000 0.190879
\(274\) −5076.00 −1.11917
\(275\) 0 0
\(276\) −225.000 −0.0490703
\(277\) −4286.00 −0.929678 −0.464839 0.885395i \(-0.653888\pi\)
−0.464839 + 0.885395i \(0.653888\pi\)
\(278\) 3804.00 0.820679
\(279\) −1845.00 −0.395904
\(280\) 0 0
\(281\) 2208.00 0.468748 0.234374 0.972146i \(-0.424696\pi\)
0.234374 + 0.972146i \(0.424696\pi\)
\(282\) 540.000 0.114030
\(283\) −3620.00 −0.760377 −0.380188 0.924909i \(-0.624141\pi\)
−0.380188 + 0.924909i \(0.624141\pi\)
\(284\) 300.000 0.0626821
\(285\) 0 0
\(286\) 738.000 0.152583
\(287\) −399.000 −0.0820635
\(288\) −405.000 −0.0828641
\(289\) −4184.00 −0.851618
\(290\) 0 0
\(291\) −1482.00 −0.298544
\(292\) −98.0000 −0.0196405
\(293\) 1392.00 0.277548 0.138774 0.990324i \(-0.455684\pi\)
0.138774 + 0.990324i \(0.455684\pi\)
\(294\) 441.000 0.0874818
\(295\) 0 0
\(296\) −5502.00 −1.08040
\(297\) −162.000 −0.0316505
\(298\) 6507.00 1.26490
\(299\) 3075.00 0.594755
\(300\) 0 0
\(301\) 2849.00 0.545560
\(302\) 1554.00 0.296101
\(303\) 1872.00 0.354929
\(304\) 284.000 0.0535806
\(305\) 0 0
\(306\) −729.000 −0.136190
\(307\) −9002.00 −1.67352 −0.836761 0.547568i \(-0.815554\pi\)
−0.836761 + 0.547568i \(0.815554\pi\)
\(308\) 42.0000 0.00777004
\(309\) 2307.00 0.424727
\(310\) 0 0
\(311\) 6666.00 1.21542 0.607708 0.794161i \(-0.292089\pi\)
0.607708 + 0.794161i \(0.292089\pi\)
\(312\) 2583.00 0.468697
\(313\) 2878.00 0.519726 0.259863 0.965646i \(-0.416323\pi\)
0.259863 + 0.965646i \(0.416323\pi\)
\(314\) 2658.00 0.477706
\(315\) 0 0
\(316\) 686.000 0.122122
\(317\) −4611.00 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(318\) −2943.00 −0.518979
\(319\) 738.000 0.129530
\(320\) 0 0
\(321\) −4986.00 −0.866951
\(322\) 1575.00 0.272582
\(323\) 108.000 0.0186046
\(324\) 81.0000 0.0138889
\(325\) 0 0
\(326\) −11679.0 −1.98417
\(327\) 564.000 0.0953800
\(328\) −1197.00 −0.201504
\(329\) −420.000 −0.0703810
\(330\) 0 0
\(331\) −7459.00 −1.23862 −0.619311 0.785146i \(-0.712588\pi\)
−0.619311 + 0.785146i \(0.712588\pi\)
\(332\) −1401.00 −0.231596
\(333\) 2358.00 0.388041
\(334\) −6138.00 −1.00556
\(335\) 0 0
\(336\) 1491.00 0.242085
\(337\) −5843.00 −0.944476 −0.472238 0.881471i \(-0.656554\pi\)
−0.472238 + 0.881471i \(0.656554\pi\)
\(338\) −1548.00 −0.249113
\(339\) 54.0000 0.00865156
\(340\) 0 0
\(341\) 1230.00 0.195332
\(342\) −108.000 −0.0170759
\(343\) −343.000 −0.0539949
\(344\) 8547.00 1.33960
\(345\) 0 0
\(346\) 10620.0 1.65010
\(347\) −2346.00 −0.362939 −0.181470 0.983397i \(-0.558085\pi\)
−0.181470 + 0.983397i \(0.558085\pi\)
\(348\) −369.000 −0.0568404
\(349\) 5807.00 0.890664 0.445332 0.895366i \(-0.353086\pi\)
0.445332 + 0.895366i \(0.353086\pi\)
\(350\) 0 0
\(351\) −1107.00 −0.168340
\(352\) 270.000 0.0408837
\(353\) 5190.00 0.782538 0.391269 0.920276i \(-0.372036\pi\)
0.391269 + 0.920276i \(0.372036\pi\)
\(354\) 297.000 0.0445914
\(355\) 0 0
\(356\) 714.000 0.106298
\(357\) 567.000 0.0840583
\(358\) −9738.00 −1.43762
\(359\) 8883.00 1.30592 0.652962 0.757391i \(-0.273526\pi\)
0.652962 + 0.757391i \(0.273526\pi\)
\(360\) 0 0
\(361\) −6843.00 −0.997667
\(362\) 4002.00 0.581051
\(363\) −3885.00 −0.561734
\(364\) 287.000 0.0413266
\(365\) 0 0
\(366\) −3843.00 −0.548844
\(367\) −9965.00 −1.41735 −0.708677 0.705533i \(-0.750708\pi\)
−0.708677 + 0.705533i \(0.750708\pi\)
\(368\) 5325.00 0.754307
\(369\) 513.000 0.0723732
\(370\) 0 0
\(371\) 2289.00 0.320321
\(372\) −615.000 −0.0857158
\(373\) −11660.0 −1.61858 −0.809292 0.587406i \(-0.800149\pi\)
−0.809292 + 0.587406i \(0.800149\pi\)
\(374\) 486.000 0.0671937
\(375\) 0 0
\(376\) −1260.00 −0.172818
\(377\) 5043.00 0.688933
\(378\) −567.000 −0.0771517
\(379\) 3203.00 0.434108 0.217054 0.976160i \(-0.430355\pi\)
0.217054 + 0.976160i \(0.430355\pi\)
\(380\) 0 0
\(381\) 4350.00 0.584927
\(382\) −2619.00 −0.350785
\(383\) 8220.00 1.09666 0.548332 0.836261i \(-0.315263\pi\)
0.548332 + 0.836261i \(0.315263\pi\)
\(384\) 4977.00 0.661410
\(385\) 0 0
\(386\) 3018.00 0.397959
\(387\) −3663.00 −0.481139
\(388\) −494.000 −0.0646367
\(389\) −2226.00 −0.290135 −0.145068 0.989422i \(-0.546340\pi\)
−0.145068 + 0.989422i \(0.546340\pi\)
\(390\) 0 0
\(391\) 2025.00 0.261915
\(392\) −1029.00 −0.132583
\(393\) 7992.00 1.02581
\(394\) −1773.00 −0.226707
\(395\) 0 0
\(396\) −54.0000 −0.00685253
\(397\) −10451.0 −1.32121 −0.660605 0.750733i \(-0.729700\pi\)
−0.660605 + 0.750733i \(0.729700\pi\)
\(398\) −7752.00 −0.976313
\(399\) 84.0000 0.0105395
\(400\) 0 0
\(401\) −1320.00 −0.164383 −0.0821916 0.996617i \(-0.526192\pi\)
−0.0821916 + 0.996617i \(0.526192\pi\)
\(402\) 5652.00 0.701234
\(403\) 8405.00 1.03892
\(404\) 624.000 0.0768445
\(405\) 0 0
\(406\) 2583.00 0.315744
\(407\) −1572.00 −0.191452
\(408\) 1701.00 0.206402
\(409\) 2402.00 0.290394 0.145197 0.989403i \(-0.453618\pi\)
0.145197 + 0.989403i \(0.453618\pi\)
\(410\) 0 0
\(411\) −5076.00 −0.609199
\(412\) 769.000 0.0919561
\(413\) −231.000 −0.0275224
\(414\) −2025.00 −0.240394
\(415\) 0 0
\(416\) 1845.00 0.217448
\(417\) 3804.00 0.446721
\(418\) 72.0000 0.00842496
\(419\) −3333.00 −0.388610 −0.194305 0.980941i \(-0.562245\pi\)
−0.194305 + 0.980941i \(0.562245\pi\)
\(420\) 0 0
\(421\) −1462.00 −0.169248 −0.0846241 0.996413i \(-0.526969\pi\)
−0.0846241 + 0.996413i \(0.526969\pi\)
\(422\) −4323.00 −0.498674
\(423\) 540.000 0.0620702
\(424\) 6867.00 0.786535
\(425\) 0 0
\(426\) 2700.00 0.307078
\(427\) 2989.00 0.338754
\(428\) −1662.00 −0.187700
\(429\) 738.000 0.0830559
\(430\) 0 0
\(431\) −10089.0 −1.12754 −0.563770 0.825932i \(-0.690650\pi\)
−0.563770 + 0.825932i \(0.690650\pi\)
\(432\) −1917.00 −0.213499
\(433\) −3242.00 −0.359817 −0.179908 0.983683i \(-0.557580\pi\)
−0.179908 + 0.983683i \(0.557580\pi\)
\(434\) 4305.00 0.476144
\(435\) 0 0
\(436\) 188.000 0.0206504
\(437\) 300.000 0.0328397
\(438\) −882.000 −0.0962182
\(439\) 1799.00 0.195584 0.0977922 0.995207i \(-0.468822\pi\)
0.0977922 + 0.995207i \(0.468822\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 3321.00 0.357384
\(443\) −8772.00 −0.940791 −0.470395 0.882456i \(-0.655889\pi\)
−0.470395 + 0.882456i \(0.655889\pi\)
\(444\) 786.000 0.0840133
\(445\) 0 0
\(446\) −11481.0 −1.21893
\(447\) 6507.00 0.688525
\(448\) −3031.00 −0.319646
\(449\) 1560.00 0.163966 0.0819832 0.996634i \(-0.473875\pi\)
0.0819832 + 0.996634i \(0.473875\pi\)
\(450\) 0 0
\(451\) −342.000 −0.0357077
\(452\) 18.0000 0.00187312
\(453\) 1554.00 0.161177
\(454\) 16263.0 1.68119
\(455\) 0 0
\(456\) 252.000 0.0258793
\(457\) −11615.0 −1.18890 −0.594449 0.804133i \(-0.702630\pi\)
−0.594449 + 0.804133i \(0.702630\pi\)
\(458\) −15870.0 −1.61912
\(459\) −729.000 −0.0741325
\(460\) 0 0
\(461\) 6960.00 0.703166 0.351583 0.936157i \(-0.385644\pi\)
0.351583 + 0.936157i \(0.385644\pi\)
\(462\) 378.000 0.0380653
\(463\) −13052.0 −1.31010 −0.655052 0.755584i \(-0.727353\pi\)
−0.655052 + 0.755584i \(0.727353\pi\)
\(464\) 8733.00 0.873749
\(465\) 0 0
\(466\) −14724.0 −1.46368
\(467\) −14013.0 −1.38853 −0.694266 0.719719i \(-0.744271\pi\)
−0.694266 + 0.719719i \(0.744271\pi\)
\(468\) −369.000 −0.0364466
\(469\) −4396.00 −0.432811
\(470\) 0 0
\(471\) 2658.00 0.260030
\(472\) −693.000 −0.0675803
\(473\) 2442.00 0.237385
\(474\) 6174.00 0.598273
\(475\) 0 0
\(476\) 189.000 0.0181992
\(477\) −2943.00 −0.282496
\(478\) 3168.00 0.303140
\(479\) 1056.00 0.100730 0.0503652 0.998731i \(-0.483961\pi\)
0.0503652 + 0.998731i \(0.483961\pi\)
\(480\) 0 0
\(481\) −10742.0 −1.01828
\(482\) 16026.0 1.51445
\(483\) 1575.00 0.148375
\(484\) −1295.00 −0.121619
\(485\) 0 0
\(486\) 729.000 0.0680414
\(487\) −7886.00 −0.733776 −0.366888 0.930265i \(-0.619577\pi\)
−0.366888 + 0.930265i \(0.619577\pi\)
\(488\) 8967.00 0.831797
\(489\) −11679.0 −1.08005
\(490\) 0 0
\(491\) 10590.0 0.973361 0.486680 0.873580i \(-0.338208\pi\)
0.486680 + 0.873580i \(0.338208\pi\)
\(492\) 171.000 0.0156693
\(493\) 3321.00 0.303388
\(494\) 492.000 0.0448100
\(495\) 0 0
\(496\) 14555.0 1.31762
\(497\) −2100.00 −0.189533
\(498\) −12609.0 −1.13458
\(499\) −14899.0 −1.33661 −0.668307 0.743885i \(-0.732981\pi\)
−0.668307 + 0.743885i \(0.732981\pi\)
\(500\) 0 0
\(501\) −6138.00 −0.547357
\(502\) 17415.0 1.54835
\(503\) −21558.0 −1.91098 −0.955491 0.295021i \(-0.904673\pi\)
−0.955491 + 0.295021i \(0.904673\pi\)
\(504\) 1323.00 0.116927
\(505\) 0 0
\(506\) 1350.00 0.118606
\(507\) −1548.00 −0.135600
\(508\) 1450.00 0.126640
\(509\) −15240.0 −1.32711 −0.663557 0.748126i \(-0.730954\pi\)
−0.663557 + 0.748126i \(0.730954\pi\)
\(510\) 0 0
\(511\) 686.000 0.0593872
\(512\) −8733.00 −0.753804
\(513\) −108.000 −0.00929496
\(514\) −1629.00 −0.139790
\(515\) 0 0
\(516\) −1221.00 −0.104170
\(517\) −360.000 −0.0306243
\(518\) −5502.00 −0.466687
\(519\) 10620.0 0.898201
\(520\) 0 0
\(521\) −16797.0 −1.41246 −0.706228 0.707984i \(-0.749605\pi\)
−0.706228 + 0.707984i \(0.749605\pi\)
\(522\) −3321.00 −0.278460
\(523\) −22520.0 −1.88285 −0.941425 0.337222i \(-0.890513\pi\)
−0.941425 + 0.337222i \(0.890513\pi\)
\(524\) 2664.00 0.222094
\(525\) 0 0
\(526\) 15579.0 1.29140
\(527\) 5535.00 0.457511
\(528\) 1278.00 0.105337
\(529\) −6542.00 −0.537684
\(530\) 0 0
\(531\) 297.000 0.0242725
\(532\) 28.0000 0.00228187
\(533\) −2337.00 −0.189919
\(534\) 6426.00 0.520749
\(535\) 0 0
\(536\) −13188.0 −1.06275
\(537\) −9738.00 −0.782543
\(538\) 6462.00 0.517838
\(539\) −294.000 −0.0234944
\(540\) 0 0
\(541\) −12004.0 −0.953960 −0.476980 0.878914i \(-0.658269\pi\)
−0.476980 + 0.878914i \(0.658269\pi\)
\(542\) 7188.00 0.569651
\(543\) 4002.00 0.316284
\(544\) 1215.00 0.0957586
\(545\) 0 0
\(546\) 2583.00 0.202458
\(547\) 16423.0 1.28372 0.641862 0.766820i \(-0.278162\pi\)
0.641862 + 0.766820i \(0.278162\pi\)
\(548\) −1692.00 −0.131895
\(549\) −3843.00 −0.298753
\(550\) 0 0
\(551\) 492.000 0.0380398
\(552\) 4725.00 0.364329
\(553\) −4802.00 −0.369262
\(554\) −12858.0 −0.986072
\(555\) 0 0
\(556\) 1268.00 0.0967179
\(557\) −714.000 −0.0543145 −0.0271572 0.999631i \(-0.508645\pi\)
−0.0271572 + 0.999631i \(0.508645\pi\)
\(558\) −5535.00 −0.419920
\(559\) 16687.0 1.26258
\(560\) 0 0
\(561\) 486.000 0.0365756
\(562\) 6624.00 0.497183
\(563\) −16293.0 −1.21966 −0.609830 0.792533i \(-0.708762\pi\)
−0.609830 + 0.792533i \(0.708762\pi\)
\(564\) 180.000 0.0134386
\(565\) 0 0
\(566\) −10860.0 −0.806501
\(567\) −567.000 −0.0419961
\(568\) −6300.00 −0.465391
\(569\) −17370.0 −1.27977 −0.639884 0.768471i \(-0.721018\pi\)
−0.639884 + 0.768471i \(0.721018\pi\)
\(570\) 0 0
\(571\) −12589.0 −0.922650 −0.461325 0.887231i \(-0.652626\pi\)
−0.461325 + 0.887231i \(0.652626\pi\)
\(572\) 246.000 0.0179821
\(573\) −2619.00 −0.190943
\(574\) −1197.00 −0.0870415
\(575\) 0 0
\(576\) 3897.00 0.281901
\(577\) 13318.0 0.960894 0.480447 0.877024i \(-0.340475\pi\)
0.480447 + 0.877024i \(0.340475\pi\)
\(578\) −12552.0 −0.903277
\(579\) 3018.00 0.216621
\(580\) 0 0
\(581\) 9807.00 0.700280
\(582\) −4446.00 −0.316654
\(583\) 1962.00 0.139379
\(584\) 2058.00 0.145823
\(585\) 0 0
\(586\) 4176.00 0.294384
\(587\) 7551.00 0.530942 0.265471 0.964119i \(-0.414472\pi\)
0.265471 + 0.964119i \(0.414472\pi\)
\(588\) 147.000 0.0103098
\(589\) 820.000 0.0573642
\(590\) 0 0
\(591\) −1773.00 −0.123404
\(592\) −18602.0 −1.29145
\(593\) 6402.00 0.443337 0.221668 0.975122i \(-0.428850\pi\)
0.221668 + 0.975122i \(0.428850\pi\)
\(594\) −486.000 −0.0335704
\(595\) 0 0
\(596\) 2169.00 0.149070
\(597\) −7752.00 −0.531438
\(598\) 9225.00 0.630833
\(599\) −15753.0 −1.07454 −0.537271 0.843410i \(-0.680545\pi\)
−0.537271 + 0.843410i \(0.680545\pi\)
\(600\) 0 0
\(601\) 9764.00 0.662699 0.331349 0.943508i \(-0.392496\pi\)
0.331349 + 0.943508i \(0.392496\pi\)
\(602\) 8547.00 0.578654
\(603\) 5652.00 0.381704
\(604\) 518.000 0.0348959
\(605\) 0 0
\(606\) 5616.00 0.376459
\(607\) −7772.00 −0.519696 −0.259848 0.965649i \(-0.583673\pi\)
−0.259848 + 0.965649i \(0.583673\pi\)
\(608\) 180.000 0.0120065
\(609\) 2583.00 0.171869
\(610\) 0 0
\(611\) −2460.00 −0.162882
\(612\) −243.000 −0.0160502
\(613\) 24262.0 1.59859 0.799293 0.600942i \(-0.205208\pi\)
0.799293 + 0.600942i \(0.205208\pi\)
\(614\) −27006.0 −1.77504
\(615\) 0 0
\(616\) −882.000 −0.0576896
\(617\) 25818.0 1.68459 0.842296 0.539015i \(-0.181203\pi\)
0.842296 + 0.539015i \(0.181203\pi\)
\(618\) 6921.00 0.450491
\(619\) 22430.0 1.45644 0.728221 0.685342i \(-0.240347\pi\)
0.728221 + 0.685342i \(0.240347\pi\)
\(620\) 0 0
\(621\) −2025.00 −0.130854
\(622\) 19998.0 1.28914
\(623\) −4998.00 −0.321414
\(624\) 8733.00 0.560256
\(625\) 0 0
\(626\) 8634.00 0.551252
\(627\) 72.0000 0.00458597
\(628\) 886.000 0.0562982
\(629\) −7074.00 −0.448424
\(630\) 0 0
\(631\) −2830.00 −0.178543 −0.0892714 0.996007i \(-0.528454\pi\)
−0.0892714 + 0.996007i \(0.528454\pi\)
\(632\) −14406.0 −0.906709
\(633\) −4323.00 −0.271444
\(634\) −13833.0 −0.866528
\(635\) 0 0
\(636\) −981.000 −0.0611622
\(637\) −2009.00 −0.124960
\(638\) 2214.00 0.137387
\(639\) 2700.00 0.167152
\(640\) 0 0
\(641\) −5202.00 −0.320541 −0.160270 0.987073i \(-0.551237\pi\)
−0.160270 + 0.987073i \(0.551237\pi\)
\(642\) −14958.0 −0.919541
\(643\) 22030.0 1.35113 0.675566 0.737299i \(-0.263899\pi\)
0.675566 + 0.737299i \(0.263899\pi\)
\(644\) 525.000 0.0321241
\(645\) 0 0
\(646\) 324.000 0.0197331
\(647\) 20370.0 1.23775 0.618877 0.785488i \(-0.287588\pi\)
0.618877 + 0.785488i \(0.287588\pi\)
\(648\) −1701.00 −0.103120
\(649\) −198.000 −0.0119756
\(650\) 0 0
\(651\) 4305.00 0.259180
\(652\) −3893.00 −0.233837
\(653\) 31626.0 1.89528 0.947642 0.319333i \(-0.103459\pi\)
0.947642 + 0.319333i \(0.103459\pi\)
\(654\) 1692.00 0.101166
\(655\) 0 0
\(656\) −4047.00 −0.240867
\(657\) −882.000 −0.0523746
\(658\) −1260.00 −0.0746503
\(659\) −11142.0 −0.658620 −0.329310 0.944222i \(-0.606816\pi\)
−0.329310 + 0.944222i \(0.606816\pi\)
\(660\) 0 0
\(661\) −5518.00 −0.324698 −0.162349 0.986733i \(-0.551907\pi\)
−0.162349 + 0.986733i \(0.551907\pi\)
\(662\) −22377.0 −1.31376
\(663\) 3321.00 0.194535
\(664\) 29421.0 1.71951
\(665\) 0 0
\(666\) 7074.00 0.411579
\(667\) 9225.00 0.535522
\(668\) −2046.00 −0.118506
\(669\) −11481.0 −0.663499
\(670\) 0 0
\(671\) 2562.00 0.147399
\(672\) 945.000 0.0542473
\(673\) 12517.0 0.716931 0.358466 0.933543i \(-0.383300\pi\)
0.358466 + 0.933543i \(0.383300\pi\)
\(674\) −17529.0 −1.00177
\(675\) 0 0
\(676\) −516.000 −0.0293582
\(677\) 2604.00 0.147828 0.0739142 0.997265i \(-0.476451\pi\)
0.0739142 + 0.997265i \(0.476451\pi\)
\(678\) 162.000 0.00917636
\(679\) 3458.00 0.195443
\(680\) 0 0
\(681\) 16263.0 0.915124
\(682\) 3690.00 0.207181
\(683\) −25986.0 −1.45582 −0.727911 0.685671i \(-0.759509\pi\)
−0.727911 + 0.685671i \(0.759509\pi\)
\(684\) −36.0000 −0.00201242
\(685\) 0 0
\(686\) −1029.00 −0.0572703
\(687\) −15870.0 −0.881337
\(688\) 28897.0 1.60129
\(689\) 13407.0 0.741315
\(690\) 0 0
\(691\) −24610.0 −1.35486 −0.677430 0.735587i \(-0.736906\pi\)
−0.677430 + 0.735587i \(0.736906\pi\)
\(692\) 3540.00 0.194466
\(693\) 378.000 0.0207201
\(694\) −7038.00 −0.384955
\(695\) 0 0
\(696\) 7749.00 0.422019
\(697\) −1539.00 −0.0836353
\(698\) 17421.0 0.944691
\(699\) −14724.0 −0.796728
\(700\) 0 0
\(701\) 10725.0 0.577857 0.288928 0.957351i \(-0.406701\pi\)
0.288928 + 0.957351i \(0.406701\pi\)
\(702\) −3321.00 −0.178551
\(703\) −1048.00 −0.0562248
\(704\) −2598.00 −0.139085
\(705\) 0 0
\(706\) 15570.0 0.830007
\(707\) −4368.00 −0.232356
\(708\) 99.0000 0.00525515
\(709\) −34516.0 −1.82832 −0.914158 0.405359i \(-0.867147\pi\)
−0.914158 + 0.405359i \(0.867147\pi\)
\(710\) 0 0
\(711\) 6174.00 0.325658
\(712\) −14994.0 −0.789219
\(713\) 15375.0 0.807571
\(714\) 1701.00 0.0891573
\(715\) 0 0
\(716\) −3246.00 −0.169426
\(717\) 3168.00 0.165009
\(718\) 26649.0 1.38514
\(719\) 17958.0 0.931461 0.465730 0.884927i \(-0.345792\pi\)
0.465730 + 0.884927i \(0.345792\pi\)
\(720\) 0 0
\(721\) −5383.00 −0.278049
\(722\) −20529.0 −1.05819
\(723\) 16026.0 0.824361
\(724\) 1334.00 0.0684775
\(725\) 0 0
\(726\) −11655.0 −0.595809
\(727\) −2255.00 −0.115039 −0.0575195 0.998344i \(-0.518319\pi\)
−0.0575195 + 0.998344i \(0.518319\pi\)
\(728\) −6027.00 −0.306834
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 10989.0 0.556009
\(732\) −1281.00 −0.0646819
\(733\) −9539.00 −0.480670 −0.240335 0.970690i \(-0.577257\pi\)
−0.240335 + 0.970690i \(0.577257\pi\)
\(734\) −29895.0 −1.50333
\(735\) 0 0
\(736\) 3375.00 0.169027
\(737\) −3768.00 −0.188326
\(738\) 1539.00 0.0767634
\(739\) 37979.0 1.89050 0.945250 0.326346i \(-0.105817\pi\)
0.945250 + 0.326346i \(0.105817\pi\)
\(740\) 0 0
\(741\) 492.000 0.0243915
\(742\) 6867.00 0.339751
\(743\) 20535.0 1.01394 0.506969 0.861964i \(-0.330766\pi\)
0.506969 + 0.861964i \(0.330766\pi\)
\(744\) 12915.0 0.636407
\(745\) 0 0
\(746\) −34980.0 −1.71677
\(747\) −12609.0 −0.617589
\(748\) 162.000 0.00791886
\(749\) 11634.0 0.567553
\(750\) 0 0
\(751\) −2452.00 −0.119141 −0.0595704 0.998224i \(-0.518973\pi\)
−0.0595704 + 0.998224i \(0.518973\pi\)
\(752\) −4260.00 −0.206577
\(753\) 17415.0 0.842813
\(754\) 15129.0 0.730724
\(755\) 0 0
\(756\) −189.000 −0.00909241
\(757\) 36850.0 1.76927 0.884634 0.466286i \(-0.154408\pi\)
0.884634 + 0.466286i \(0.154408\pi\)
\(758\) 9609.00 0.460441
\(759\) 1350.00 0.0645611
\(760\) 0 0
\(761\) 30258.0 1.44133 0.720665 0.693284i \(-0.243837\pi\)
0.720665 + 0.693284i \(0.243837\pi\)
\(762\) 13050.0 0.620409
\(763\) −1316.00 −0.0624409
\(764\) −873.000 −0.0413404
\(765\) 0 0
\(766\) 24660.0 1.16319
\(767\) −1353.00 −0.0636949
\(768\) 4539.00 0.213264
\(769\) 29288.0 1.37341 0.686705 0.726936i \(-0.259056\pi\)
0.686705 + 0.726936i \(0.259056\pi\)
\(770\) 0 0
\(771\) −1629.00 −0.0760921
\(772\) 1006.00 0.0468999
\(773\) −40668.0 −1.89227 −0.946136 0.323769i \(-0.895050\pi\)
−0.946136 + 0.323769i \(0.895050\pi\)
\(774\) −10989.0 −0.510325
\(775\) 0 0
\(776\) 10374.0 0.479903
\(777\) −5502.00 −0.254032
\(778\) −6678.00 −0.307735
\(779\) −228.000 −0.0104865
\(780\) 0 0
\(781\) −1800.00 −0.0824700
\(782\) 6075.00 0.277803
\(783\) −3321.00 −0.151575
\(784\) −3479.00 −0.158482
\(785\) 0 0
\(786\) 23976.0 1.08804
\(787\) −13118.0 −0.594163 −0.297081 0.954852i \(-0.596013\pi\)
−0.297081 + 0.954852i \(0.596013\pi\)
\(788\) −591.000 −0.0267176
\(789\) 15579.0 0.702949
\(790\) 0 0
\(791\) −126.000 −0.00566377
\(792\) 1134.00 0.0508774
\(793\) 17507.0 0.783975
\(794\) −31353.0 −1.40136
\(795\) 0 0
\(796\) −2584.00 −0.115060
\(797\) 37278.0 1.65678 0.828391 0.560151i \(-0.189257\pi\)
0.828391 + 0.560151i \(0.189257\pi\)
\(798\) 252.000 0.0111788
\(799\) −1620.00 −0.0717290
\(800\) 0 0
\(801\) 6426.00 0.283460
\(802\) −3960.00 −0.174355
\(803\) 588.000 0.0258407
\(804\) 1884.00 0.0826412
\(805\) 0 0
\(806\) 25215.0 1.10194
\(807\) 6462.00 0.281875
\(808\) −13104.0 −0.570541
\(809\) 5268.00 0.228941 0.114470 0.993427i \(-0.463483\pi\)
0.114470 + 0.993427i \(0.463483\pi\)
\(810\) 0 0
\(811\) 34994.0 1.51517 0.757587 0.652735i \(-0.226378\pi\)
0.757587 + 0.652735i \(0.226378\pi\)
\(812\) 861.000 0.0372108
\(813\) 7188.00 0.310079
\(814\) −4716.00 −0.203066
\(815\) 0 0
\(816\) 5751.00 0.246722
\(817\) 1628.00 0.0697142
\(818\) 7206.00 0.308010
\(819\) 2583.00 0.110204
\(820\) 0 0
\(821\) 28818.0 1.22504 0.612518 0.790456i \(-0.290157\pi\)
0.612518 + 0.790456i \(0.290157\pi\)
\(822\) −15228.0 −0.646153
\(823\) 20962.0 0.887836 0.443918 0.896067i \(-0.353588\pi\)
0.443918 + 0.896067i \(0.353588\pi\)
\(824\) −16149.0 −0.682739
\(825\) 0 0
\(826\) −693.000 −0.0291920
\(827\) 28266.0 1.18852 0.594259 0.804273i \(-0.297445\pi\)
0.594259 + 0.804273i \(0.297445\pi\)
\(828\) −675.000 −0.0283308
\(829\) 42491.0 1.78019 0.890093 0.455780i \(-0.150640\pi\)
0.890093 + 0.455780i \(0.150640\pi\)
\(830\) 0 0
\(831\) −12858.0 −0.536750
\(832\) −17753.0 −0.739753
\(833\) −1323.00 −0.0550291
\(834\) 11412.0 0.473819
\(835\) 0 0
\(836\) 24.0000 0.000992892 0
\(837\) −5535.00 −0.228575
\(838\) −9999.00 −0.412183
\(839\) −40512.0 −1.66702 −0.833510 0.552505i \(-0.813672\pi\)
−0.833510 + 0.552505i \(0.813672\pi\)
\(840\) 0 0
\(841\) −9260.00 −0.379679
\(842\) −4386.00 −0.179515
\(843\) 6624.00 0.270632
\(844\) −1441.00 −0.0587693
\(845\) 0 0
\(846\) 1620.00 0.0658354
\(847\) 9065.00 0.367742
\(848\) 23217.0 0.940183
\(849\) −10860.0 −0.439004
\(850\) 0 0
\(851\) −19650.0 −0.791532
\(852\) 900.000 0.0361895
\(853\) 40525.0 1.62667 0.813335 0.581796i \(-0.197650\pi\)
0.813335 + 0.581796i \(0.197650\pi\)
\(854\) 8967.00 0.359303
\(855\) 0 0
\(856\) 34902.0 1.39360
\(857\) 7182.00 0.286269 0.143134 0.989703i \(-0.454282\pi\)
0.143134 + 0.989703i \(0.454282\pi\)
\(858\) 2214.00 0.0880941
\(859\) −7864.00 −0.312359 −0.156179 0.987729i \(-0.549918\pi\)
−0.156179 + 0.987729i \(0.549918\pi\)
\(860\) 0 0
\(861\) −1197.00 −0.0473794
\(862\) −30267.0 −1.19594
\(863\) −28872.0 −1.13883 −0.569417 0.822049i \(-0.692831\pi\)
−0.569417 + 0.822049i \(0.692831\pi\)
\(864\) −1215.00 −0.0478416
\(865\) 0 0
\(866\) −9726.00 −0.381643
\(867\) −12552.0 −0.491682
\(868\) 1435.00 0.0561141
\(869\) −4116.00 −0.160674
\(870\) 0 0
\(871\) −25748.0 −1.00165
\(872\) −3948.00 −0.153321
\(873\) −4446.00 −0.172365
\(874\) 900.000 0.0348318
\(875\) 0 0
\(876\) −294.000 −0.0113394
\(877\) 47392.0 1.82476 0.912380 0.409345i \(-0.134243\pi\)
0.912380 + 0.409345i \(0.134243\pi\)
\(878\) 5397.00 0.207449
\(879\) 4176.00 0.160242
\(880\) 0 0
\(881\) 17769.0 0.679515 0.339758 0.940513i \(-0.389655\pi\)
0.339758 + 0.940513i \(0.389655\pi\)
\(882\) 1323.00 0.0505076
\(883\) −32549.0 −1.24050 −0.620250 0.784404i \(-0.712969\pi\)
−0.620250 + 0.784404i \(0.712969\pi\)
\(884\) 1107.00 0.0421181
\(885\) 0 0
\(886\) −26316.0 −0.997859
\(887\) −6714.00 −0.254153 −0.127077 0.991893i \(-0.540559\pi\)
−0.127077 + 0.991893i \(0.540559\pi\)
\(888\) −16506.0 −0.623767
\(889\) −10150.0 −0.382925
\(890\) 0 0
\(891\) −486.000 −0.0182734
\(892\) −3827.00 −0.143652
\(893\) −240.000 −0.00899361
\(894\) 19521.0 0.730291
\(895\) 0 0
\(896\) −11613.0 −0.432995
\(897\) 9225.00 0.343382
\(898\) 4680.00 0.173913
\(899\) 25215.0 0.935448
\(900\) 0 0
\(901\) 8829.00 0.326456
\(902\) −1026.00 −0.0378737
\(903\) 8547.00 0.314979
\(904\) −378.000 −0.0139072
\(905\) 0 0
\(906\) 4662.00 0.170954
\(907\) −29693.0 −1.08703 −0.543517 0.839398i \(-0.682908\pi\)
−0.543517 + 0.839398i \(0.682908\pi\)
\(908\) 5421.00 0.198130
\(909\) 5616.00 0.204919
\(910\) 0 0
\(911\) −8199.00 −0.298183 −0.149092 0.988823i \(-0.547635\pi\)
−0.149092 + 0.988823i \(0.547635\pi\)
\(912\) 852.000 0.0309348
\(913\) 8406.00 0.304708
\(914\) −34845.0 −1.26102
\(915\) 0 0
\(916\) −5290.00 −0.190815
\(917\) −18648.0 −0.671550
\(918\) −2187.00 −0.0786294
\(919\) −15388.0 −0.552343 −0.276172 0.961108i \(-0.589066\pi\)
−0.276172 + 0.961108i \(0.589066\pi\)
\(920\) 0 0
\(921\) −27006.0 −0.966208
\(922\) 20880.0 0.745820
\(923\) −12300.0 −0.438634
\(924\) 126.000 0.00448603
\(925\) 0 0
\(926\) −39156.0 −1.38957
\(927\) 6921.00 0.245216
\(928\) 5535.00 0.195792
\(929\) −411.000 −0.0145150 −0.00725752 0.999974i \(-0.502310\pi\)
−0.00725752 + 0.999974i \(0.502310\pi\)
\(930\) 0 0
\(931\) −196.000 −0.00689972
\(932\) −4908.00 −0.172497
\(933\) 19998.0 0.701720
\(934\) −42039.0 −1.47276
\(935\) 0 0
\(936\) 7749.00 0.270603
\(937\) −26066.0 −0.908793 −0.454397 0.890800i \(-0.650145\pi\)
−0.454397 + 0.890800i \(0.650145\pi\)
\(938\) −13188.0 −0.459066
\(939\) 8634.00 0.300064
\(940\) 0 0
\(941\) 24048.0 0.833095 0.416548 0.909114i \(-0.363240\pi\)
0.416548 + 0.909114i \(0.363240\pi\)
\(942\) 7974.00 0.275804
\(943\) −4275.00 −0.147628
\(944\) −2343.00 −0.0807819
\(945\) 0 0
\(946\) 7326.00 0.251785
\(947\) 22968.0 0.788131 0.394065 0.919082i \(-0.371068\pi\)
0.394065 + 0.919082i \(0.371068\pi\)
\(948\) 2058.00 0.0705071
\(949\) 4018.00 0.137439
\(950\) 0 0
\(951\) −13833.0 −0.471678
\(952\) −3969.00 −0.135122
\(953\) 10248.0 0.348337 0.174169 0.984716i \(-0.444276\pi\)
0.174169 + 0.984716i \(0.444276\pi\)
\(954\) −8829.00 −0.299633
\(955\) 0 0
\(956\) 1056.00 0.0357254
\(957\) 2214.00 0.0747842
\(958\) 3168.00 0.106841
\(959\) 11844.0 0.398814
\(960\) 0 0
\(961\) 12234.0 0.410661
\(962\) −32226.0 −1.08005
\(963\) −14958.0 −0.500535
\(964\) 5342.00 0.178479
\(965\) 0 0
\(966\) 4725.00 0.157375
\(967\) 43060.0 1.43197 0.715986 0.698115i \(-0.245978\pi\)
0.715986 + 0.698115i \(0.245978\pi\)
\(968\) 27195.0 0.902976
\(969\) 324.000 0.0107414
\(970\) 0 0
\(971\) −12348.0 −0.408101 −0.204051 0.978960i \(-0.565411\pi\)
−0.204051 + 0.978960i \(0.565411\pi\)
\(972\) 243.000 0.00801875
\(973\) −8876.00 −0.292448
\(974\) −23658.0 −0.778287
\(975\) 0 0
\(976\) 30317.0 0.994286
\(977\) 44190.0 1.44705 0.723523 0.690301i \(-0.242522\pi\)
0.723523 + 0.690301i \(0.242522\pi\)
\(978\) −35037.0 −1.14556
\(979\) −4284.00 −0.139854
\(980\) 0 0
\(981\) 1692.00 0.0550677
\(982\) 31770.0 1.03240
\(983\) −60168.0 −1.95225 −0.976125 0.217211i \(-0.930304\pi\)
−0.976125 + 0.217211i \(0.930304\pi\)
\(984\) −3591.00 −0.116338
\(985\) 0 0
\(986\) 9963.00 0.321792
\(987\) −1260.00 −0.0406345
\(988\) 164.000 0.00528091
\(989\) 30525.0 0.981434
\(990\) 0 0
\(991\) 8804.00 0.282208 0.141104 0.989995i \(-0.454935\pi\)
0.141104 + 0.989995i \(0.454935\pi\)
\(992\) 9225.00 0.295256
\(993\) −22377.0 −0.715118
\(994\) −6300.00 −0.201030
\(995\) 0 0
\(996\) −4203.00 −0.133712
\(997\) −15302.0 −0.486077 −0.243039 0.970017i \(-0.578144\pi\)
−0.243039 + 0.970017i \(0.578144\pi\)
\(998\) −44697.0 −1.41769
\(999\) 7074.00 0.224035
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 525.4.a.h.1.1 yes 1
3.2 odd 2 1575.4.a.a.1.1 1
5.2 odd 4 525.4.d.d.274.2 2
5.3 odd 4 525.4.d.d.274.1 2
5.4 even 2 525.4.a.c.1.1 1
15.14 odd 2 1575.4.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.c.1.1 1 5.4 even 2
525.4.a.h.1.1 yes 1 1.1 even 1 trivial
525.4.d.d.274.1 2 5.3 odd 4
525.4.d.d.274.2 2 5.2 odd 4
1575.4.a.a.1.1 1 3.2 odd 2
1575.4.a.j.1.1 1 15.14 odd 2