Properties

Label 1470.4.a.a.1.1
Level $1470$
Weight $4$
Character 1470.1
Self dual yes
Analytic conductor $86.733$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1470,4,Mod(1,1470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1470, 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("1470.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1470.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: \(86.7328077084\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1470.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} +10.0000 q^{10} -60.0000 q^{11} -12.0000 q^{12} +34.0000 q^{13} +15.0000 q^{15} +16.0000 q^{16} -42.0000 q^{17} -18.0000 q^{18} +76.0000 q^{19} -20.0000 q^{20} +120.000 q^{22} +24.0000 q^{24} +25.0000 q^{25} -68.0000 q^{26} -27.0000 q^{27} +6.00000 q^{29} -30.0000 q^{30} +232.000 q^{31} -32.0000 q^{32} +180.000 q^{33} +84.0000 q^{34} +36.0000 q^{36} +134.000 q^{37} -152.000 q^{38} -102.000 q^{39} +40.0000 q^{40} -234.000 q^{41} -412.000 q^{43} -240.000 q^{44} -45.0000 q^{45} +360.000 q^{47} -48.0000 q^{48} -50.0000 q^{50} +126.000 q^{51} +136.000 q^{52} +222.000 q^{53} +54.0000 q^{54} +300.000 q^{55} -228.000 q^{57} -12.0000 q^{58} -660.000 q^{59} +60.0000 q^{60} +490.000 q^{61} -464.000 q^{62} +64.0000 q^{64} -170.000 q^{65} -360.000 q^{66} +812.000 q^{67} -168.000 q^{68} +120.000 q^{71} -72.0000 q^{72} -746.000 q^{73} -268.000 q^{74} -75.0000 q^{75} +304.000 q^{76} +204.000 q^{78} +152.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} +468.000 q^{82} +804.000 q^{83} +210.000 q^{85} +824.000 q^{86} -18.0000 q^{87} +480.000 q^{88} +678.000 q^{89} +90.0000 q^{90} -696.000 q^{93} -720.000 q^{94} -380.000 q^{95} +96.0000 q^{96} -194.000 q^{97} -540.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) −5.00000 −0.447214
\(6\) 6.00000 0.408248
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 10.0000 0.316228
\(11\) −60.0000 −1.64461 −0.822304 0.569049i \(-0.807311\pi\)
−0.822304 + 0.569049i \(0.807311\pi\)
\(12\) −12.0000 −0.288675
\(13\) 34.0000 0.725377 0.362689 0.931910i \(-0.381859\pi\)
0.362689 + 0.931910i \(0.381859\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 16.0000 0.250000
\(17\) −42.0000 −0.599206 −0.299603 0.954064i \(-0.596854\pi\)
−0.299603 + 0.954064i \(0.596854\pi\)
\(18\) −18.0000 −0.235702
\(19\) 76.0000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −20.0000 −0.223607
\(21\) 0 0
\(22\) 120.000 1.16291
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 24.0000 0.204124
\(25\) 25.0000 0.200000
\(26\) −68.0000 −0.512919
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 6.00000 0.0384197 0.0192099 0.999815i \(-0.493885\pi\)
0.0192099 + 0.999815i \(0.493885\pi\)
\(30\) −30.0000 −0.182574
\(31\) 232.000 1.34414 0.672071 0.740486i \(-0.265405\pi\)
0.672071 + 0.740486i \(0.265405\pi\)
\(32\) −32.0000 −0.176777
\(33\) 180.000 0.949514
\(34\) 84.0000 0.423702
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) 134.000 0.595391 0.297695 0.954661i \(-0.403782\pi\)
0.297695 + 0.954661i \(0.403782\pi\)
\(38\) −152.000 −0.648886
\(39\) −102.000 −0.418797
\(40\) 40.0000 0.158114
\(41\) −234.000 −0.891333 −0.445667 0.895199i \(-0.647033\pi\)
−0.445667 + 0.895199i \(0.647033\pi\)
\(42\) 0 0
\(43\) −412.000 −1.46115 −0.730575 0.682833i \(-0.760748\pi\)
−0.730575 + 0.682833i \(0.760748\pi\)
\(44\) −240.000 −0.822304
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) 360.000 1.11726 0.558632 0.829416i \(-0.311326\pi\)
0.558632 + 0.829416i \(0.311326\pi\)
\(48\) −48.0000 −0.144338
\(49\) 0 0
\(50\) −50.0000 −0.141421
\(51\) 126.000 0.345952
\(52\) 136.000 0.362689
\(53\) 222.000 0.575359 0.287680 0.957727i \(-0.407116\pi\)
0.287680 + 0.957727i \(0.407116\pi\)
\(54\) 54.0000 0.136083
\(55\) 300.000 0.735491
\(56\) 0 0
\(57\) −228.000 −0.529813
\(58\) −12.0000 −0.0271668
\(59\) −660.000 −1.45635 −0.728175 0.685391i \(-0.759631\pi\)
−0.728175 + 0.685391i \(0.759631\pi\)
\(60\) 60.0000 0.129099
\(61\) 490.000 1.02849 0.514246 0.857642i \(-0.328072\pi\)
0.514246 + 0.857642i \(0.328072\pi\)
\(62\) −464.000 −0.950453
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −170.000 −0.324399
\(66\) −360.000 −0.671408
\(67\) 812.000 1.48062 0.740310 0.672265i \(-0.234679\pi\)
0.740310 + 0.672265i \(0.234679\pi\)
\(68\) −168.000 −0.299603
\(69\) 0 0
\(70\) 0 0
\(71\) 120.000 0.200583 0.100291 0.994958i \(-0.468022\pi\)
0.100291 + 0.994958i \(0.468022\pi\)
\(72\) −72.0000 −0.117851
\(73\) −746.000 −1.19606 −0.598032 0.801472i \(-0.704051\pi\)
−0.598032 + 0.801472i \(0.704051\pi\)
\(74\) −268.000 −0.421005
\(75\) −75.0000 −0.115470
\(76\) 304.000 0.458831
\(77\) 0 0
\(78\) 204.000 0.296134
\(79\) 152.000 0.216473 0.108236 0.994125i \(-0.465480\pi\)
0.108236 + 0.994125i \(0.465480\pi\)
\(80\) −80.0000 −0.111803
\(81\) 81.0000 0.111111
\(82\) 468.000 0.630268
\(83\) 804.000 1.06326 0.531629 0.846977i \(-0.321580\pi\)
0.531629 + 0.846977i \(0.321580\pi\)
\(84\) 0 0
\(85\) 210.000 0.267973
\(86\) 824.000 1.03319
\(87\) −18.0000 −0.0221816
\(88\) 480.000 0.581456
\(89\) 678.000 0.807504 0.403752 0.914868i \(-0.367706\pi\)
0.403752 + 0.914868i \(0.367706\pi\)
\(90\) 90.0000 0.105409
\(91\) 0 0
\(92\) 0 0
\(93\) −696.000 −0.776041
\(94\) −720.000 −0.790025
\(95\) −380.000 −0.410391
\(96\) 96.0000 0.102062
\(97\) −194.000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −540.000 −0.548202
\(100\) 100.000 0.100000
\(101\) −798.000 −0.786178 −0.393089 0.919500i \(-0.628594\pi\)
−0.393089 + 0.919500i \(0.628594\pi\)
\(102\) −252.000 −0.244625
\(103\) −1088.00 −1.04081 −0.520407 0.853918i \(-0.674220\pi\)
−0.520407 + 0.853918i \(0.674220\pi\)
\(104\) −272.000 −0.256460
\(105\) 0 0
\(106\) −444.000 −0.406840
\(107\) 1716.00 1.55039 0.775196 0.631721i \(-0.217651\pi\)
0.775196 + 0.631721i \(0.217651\pi\)
\(108\) −108.000 −0.0962250
\(109\) −970.000 −0.852378 −0.426189 0.904634i \(-0.640144\pi\)
−0.426189 + 0.904634i \(0.640144\pi\)
\(110\) −600.000 −0.520071
\(111\) −402.000 −0.343749
\(112\) 0 0
\(113\) 426.000 0.354643 0.177322 0.984153i \(-0.443257\pi\)
0.177322 + 0.984153i \(0.443257\pi\)
\(114\) 456.000 0.374634
\(115\) 0 0
\(116\) 24.0000 0.0192099
\(117\) 306.000 0.241792
\(118\) 1320.00 1.02980
\(119\) 0 0
\(120\) −120.000 −0.0912871
\(121\) 2269.00 1.70473
\(122\) −980.000 −0.727254
\(123\) 702.000 0.514611
\(124\) 928.000 0.672071
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 200.000 0.139741 0.0698706 0.997556i \(-0.477741\pi\)
0.0698706 + 0.997556i \(0.477741\pi\)
\(128\) −128.000 −0.0883883
\(129\) 1236.00 0.843595
\(130\) 340.000 0.229384
\(131\) −60.0000 −0.0400170 −0.0200085 0.999800i \(-0.506369\pi\)
−0.0200085 + 0.999800i \(0.506369\pi\)
\(132\) 720.000 0.474757
\(133\) 0 0
\(134\) −1624.00 −1.04696
\(135\) 135.000 0.0860663
\(136\) 336.000 0.211851
\(137\) 642.000 0.400363 0.200182 0.979759i \(-0.435847\pi\)
0.200182 + 0.979759i \(0.435847\pi\)
\(138\) 0 0
\(139\) 2836.00 1.73055 0.865275 0.501298i \(-0.167144\pi\)
0.865275 + 0.501298i \(0.167144\pi\)
\(140\) 0 0
\(141\) −1080.00 −0.645053
\(142\) −240.000 −0.141833
\(143\) −2040.00 −1.19296
\(144\) 144.000 0.0833333
\(145\) −30.0000 −0.0171818
\(146\) 1492.00 0.845745
\(147\) 0 0
\(148\) 536.000 0.297695
\(149\) −1554.00 −0.854420 −0.427210 0.904152i \(-0.640504\pi\)
−0.427210 + 0.904152i \(0.640504\pi\)
\(150\) 150.000 0.0816497
\(151\) −2272.00 −1.22446 −0.612228 0.790682i \(-0.709726\pi\)
−0.612228 + 0.790682i \(0.709726\pi\)
\(152\) −608.000 −0.324443
\(153\) −378.000 −0.199735
\(154\) 0 0
\(155\) −1160.00 −0.601119
\(156\) −408.000 −0.209398
\(157\) −1694.00 −0.861120 −0.430560 0.902562i \(-0.641684\pi\)
−0.430560 + 0.902562i \(0.641684\pi\)
\(158\) −304.000 −0.153069
\(159\) −666.000 −0.332184
\(160\) 160.000 0.0790569
\(161\) 0 0
\(162\) −162.000 −0.0785674
\(163\) −52.0000 −0.0249874 −0.0124937 0.999922i \(-0.503977\pi\)
−0.0124937 + 0.999922i \(0.503977\pi\)
\(164\) −936.000 −0.445667
\(165\) −900.000 −0.424636
\(166\) −1608.00 −0.751837
\(167\) 1200.00 0.556041 0.278020 0.960575i \(-0.410322\pi\)
0.278020 + 0.960575i \(0.410322\pi\)
\(168\) 0 0
\(169\) −1041.00 −0.473828
\(170\) −420.000 −0.189485
\(171\) 684.000 0.305888
\(172\) −1648.00 −0.730575
\(173\) −54.0000 −0.0237315 −0.0118657 0.999930i \(-0.503777\pi\)
−0.0118657 + 0.999930i \(0.503777\pi\)
\(174\) 36.0000 0.0156848
\(175\) 0 0
\(176\) −960.000 −0.411152
\(177\) 1980.00 0.840824
\(178\) −1356.00 −0.570992
\(179\) 876.000 0.365784 0.182892 0.983133i \(-0.441454\pi\)
0.182892 + 0.983133i \(0.441454\pi\)
\(180\) −180.000 −0.0745356
\(181\) −3854.00 −1.58268 −0.791341 0.611375i \(-0.790617\pi\)
−0.791341 + 0.611375i \(0.790617\pi\)
\(182\) 0 0
\(183\) −1470.00 −0.593801
\(184\) 0 0
\(185\) −670.000 −0.266267
\(186\) 1392.00 0.548744
\(187\) 2520.00 0.985458
\(188\) 1440.00 0.558632
\(189\) 0 0
\(190\) 760.000 0.290191
\(191\) −2784.00 −1.05468 −0.527338 0.849656i \(-0.676810\pi\)
−0.527338 + 0.849656i \(0.676810\pi\)
\(192\) −192.000 −0.0721688
\(193\) 914.000 0.340887 0.170443 0.985367i \(-0.445480\pi\)
0.170443 + 0.985367i \(0.445480\pi\)
\(194\) 388.000 0.143592
\(195\) 510.000 0.187292
\(196\) 0 0
\(197\) −5202.00 −1.88136 −0.940678 0.339300i \(-0.889810\pi\)
−0.940678 + 0.339300i \(0.889810\pi\)
\(198\) 1080.00 0.387638
\(199\) −3152.00 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(200\) −200.000 −0.0707107
\(201\) −2436.00 −0.854837
\(202\) 1596.00 0.555912
\(203\) 0 0
\(204\) 504.000 0.172976
\(205\) 1170.00 0.398616
\(206\) 2176.00 0.735967
\(207\) 0 0
\(208\) 544.000 0.181344
\(209\) −4560.00 −1.50920
\(210\) 0 0
\(211\) 740.000 0.241439 0.120720 0.992687i \(-0.461480\pi\)
0.120720 + 0.992687i \(0.461480\pi\)
\(212\) 888.000 0.287680
\(213\) −360.000 −0.115807
\(214\) −3432.00 −1.09629
\(215\) 2060.00 0.653446
\(216\) 216.000 0.0680414
\(217\) 0 0
\(218\) 1940.00 0.602722
\(219\) 2238.00 0.690548
\(220\) 1200.00 0.367745
\(221\) −1428.00 −0.434650
\(222\) 804.000 0.243067
\(223\) 520.000 0.156151 0.0780757 0.996947i \(-0.475122\pi\)
0.0780757 + 0.996947i \(0.475122\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) −852.000 −0.250771
\(227\) −396.000 −0.115786 −0.0578930 0.998323i \(-0.518438\pi\)
−0.0578930 + 0.998323i \(0.518438\pi\)
\(228\) −912.000 −0.264906
\(229\) 1330.00 0.383794 0.191897 0.981415i \(-0.438536\pi\)
0.191897 + 0.981415i \(0.438536\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −48.0000 −0.0135834
\(233\) 4866.00 1.36816 0.684082 0.729405i \(-0.260203\pi\)
0.684082 + 0.729405i \(0.260203\pi\)
\(234\) −612.000 −0.170973
\(235\) −1800.00 −0.499656
\(236\) −2640.00 −0.728175
\(237\) −456.000 −0.124981
\(238\) 0 0
\(239\) −1824.00 −0.493660 −0.246830 0.969059i \(-0.579389\pi\)
−0.246830 + 0.969059i \(0.579389\pi\)
\(240\) 240.000 0.0645497
\(241\) −6482.00 −1.73254 −0.866270 0.499575i \(-0.833489\pi\)
−0.866270 + 0.499575i \(0.833489\pi\)
\(242\) −4538.00 −1.20543
\(243\) −243.000 −0.0641500
\(244\) 1960.00 0.514246
\(245\) 0 0
\(246\) −1404.00 −0.363885
\(247\) 2584.00 0.665652
\(248\) −1856.00 −0.475226
\(249\) −2412.00 −0.613873
\(250\) 250.000 0.0632456
\(251\) −1476.00 −0.371172 −0.185586 0.982628i \(-0.559418\pi\)
−0.185586 + 0.982628i \(0.559418\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −400.000 −0.0988119
\(255\) −630.000 −0.154714
\(256\) 256.000 0.0625000
\(257\) −4314.00 −1.04708 −0.523541 0.852001i \(-0.675389\pi\)
−0.523541 + 0.852001i \(0.675389\pi\)
\(258\) −2472.00 −0.596512
\(259\) 0 0
\(260\) −680.000 −0.162199
\(261\) 54.0000 0.0128066
\(262\) 120.000 0.0282963
\(263\) −5280.00 −1.23794 −0.618971 0.785414i \(-0.712450\pi\)
−0.618971 + 0.785414i \(0.712450\pi\)
\(264\) −1440.00 −0.335704
\(265\) −1110.00 −0.257309
\(266\) 0 0
\(267\) −2034.00 −0.466213
\(268\) 3248.00 0.740310
\(269\) −5526.00 −1.25251 −0.626257 0.779617i \(-0.715414\pi\)
−0.626257 + 0.779617i \(0.715414\pi\)
\(270\) −270.000 −0.0608581
\(271\) −2024.00 −0.453687 −0.226844 0.973931i \(-0.572841\pi\)
−0.226844 + 0.973931i \(0.572841\pi\)
\(272\) −672.000 −0.149801
\(273\) 0 0
\(274\) −1284.00 −0.283100
\(275\) −1500.00 −0.328921
\(276\) 0 0
\(277\) 2054.00 0.445534 0.222767 0.974872i \(-0.428491\pi\)
0.222767 + 0.974872i \(0.428491\pi\)
\(278\) −5672.00 −1.22368
\(279\) 2088.00 0.448048
\(280\) 0 0
\(281\) −7302.00 −1.55018 −0.775090 0.631850i \(-0.782296\pi\)
−0.775090 + 0.631850i \(0.782296\pi\)
\(282\) 2160.00 0.456121
\(283\) 3724.00 0.782222 0.391111 0.920344i \(-0.372091\pi\)
0.391111 + 0.920344i \(0.372091\pi\)
\(284\) 480.000 0.100291
\(285\) 1140.00 0.236940
\(286\) 4080.00 0.843551
\(287\) 0 0
\(288\) −288.000 −0.0589256
\(289\) −3149.00 −0.640953
\(290\) 60.0000 0.0121494
\(291\) 582.000 0.117242
\(292\) −2984.00 −0.598032
\(293\) 7218.00 1.43918 0.719591 0.694399i \(-0.244330\pi\)
0.719591 + 0.694399i \(0.244330\pi\)
\(294\) 0 0
\(295\) 3300.00 0.651300
\(296\) −1072.00 −0.210502
\(297\) 1620.00 0.316505
\(298\) 3108.00 0.604166
\(299\) 0 0
\(300\) −300.000 −0.0577350
\(301\) 0 0
\(302\) 4544.00 0.865821
\(303\) 2394.00 0.453900
\(304\) 1216.00 0.229416
\(305\) −2450.00 −0.459956
\(306\) 756.000 0.141234
\(307\) −2540.00 −0.472200 −0.236100 0.971729i \(-0.575869\pi\)
−0.236100 + 0.971729i \(0.575869\pi\)
\(308\) 0 0
\(309\) 3264.00 0.600914
\(310\) 2320.00 0.425055
\(311\) −1560.00 −0.284436 −0.142218 0.989835i \(-0.545423\pi\)
−0.142218 + 0.989835i \(0.545423\pi\)
\(312\) 816.000 0.148067
\(313\) 934.000 0.168667 0.0843335 0.996438i \(-0.473124\pi\)
0.0843335 + 0.996438i \(0.473124\pi\)
\(314\) 3388.00 0.608904
\(315\) 0 0
\(316\) 608.000 0.108236
\(317\) −1674.00 −0.296597 −0.148298 0.988943i \(-0.547380\pi\)
−0.148298 + 0.988943i \(0.547380\pi\)
\(318\) 1332.00 0.234889
\(319\) −360.000 −0.0631854
\(320\) −320.000 −0.0559017
\(321\) −5148.00 −0.895119
\(322\) 0 0
\(323\) −3192.00 −0.549869
\(324\) 324.000 0.0555556
\(325\) 850.000 0.145075
\(326\) 104.000 0.0176688
\(327\) 2910.00 0.492120
\(328\) 1872.00 0.315134
\(329\) 0 0
\(330\) 1800.00 0.300263
\(331\) −3988.00 −0.662237 −0.331118 0.943589i \(-0.607426\pi\)
−0.331118 + 0.943589i \(0.607426\pi\)
\(332\) 3216.00 0.531629
\(333\) 1206.00 0.198464
\(334\) −2400.00 −0.393180
\(335\) −4060.00 −0.662154
\(336\) 0 0
\(337\) 2.00000 0.000323285 0 0.000161642 1.00000i \(-0.499949\pi\)
0.000161642 1.00000i \(0.499949\pi\)
\(338\) 2082.00 0.335047
\(339\) −1278.00 −0.204753
\(340\) 840.000 0.133986
\(341\) −13920.0 −2.21059
\(342\) −1368.00 −0.216295
\(343\) 0 0
\(344\) 3296.00 0.516594
\(345\) 0 0
\(346\) 108.000 0.0167807
\(347\) 1764.00 0.272901 0.136450 0.990647i \(-0.456431\pi\)
0.136450 + 0.990647i \(0.456431\pi\)
\(348\) −72.0000 −0.0110908
\(349\) −4310.00 −0.661057 −0.330529 0.943796i \(-0.607227\pi\)
−0.330529 + 0.943796i \(0.607227\pi\)
\(350\) 0 0
\(351\) −918.000 −0.139599
\(352\) 1920.00 0.290728
\(353\) −138.000 −0.0208074 −0.0104037 0.999946i \(-0.503312\pi\)
−0.0104037 + 0.999946i \(0.503312\pi\)
\(354\) −3960.00 −0.594553
\(355\) −600.000 −0.0897034
\(356\) 2712.00 0.403752
\(357\) 0 0
\(358\) −1752.00 −0.258648
\(359\) −11976.0 −1.76064 −0.880319 0.474382i \(-0.842672\pi\)
−0.880319 + 0.474382i \(0.842672\pi\)
\(360\) 360.000 0.0527046
\(361\) −1083.00 −0.157895
\(362\) 7708.00 1.11913
\(363\) −6807.00 −0.984228
\(364\) 0 0
\(365\) 3730.00 0.534896
\(366\) 2940.00 0.419880
\(367\) −9704.00 −1.38023 −0.690115 0.723699i \(-0.742440\pi\)
−0.690115 + 0.723699i \(0.742440\pi\)
\(368\) 0 0
\(369\) −2106.00 −0.297111
\(370\) 1340.00 0.188279
\(371\) 0 0
\(372\) −2784.00 −0.388021
\(373\) −8122.00 −1.12746 −0.563728 0.825960i \(-0.690633\pi\)
−0.563728 + 0.825960i \(0.690633\pi\)
\(374\) −5040.00 −0.696824
\(375\) 375.000 0.0516398
\(376\) −2880.00 −0.395012
\(377\) 204.000 0.0278688
\(378\) 0 0
\(379\) 3404.00 0.461350 0.230675 0.973031i \(-0.425907\pi\)
0.230675 + 0.973031i \(0.425907\pi\)
\(380\) −1520.00 −0.205196
\(381\) −600.000 −0.0806796
\(382\) 5568.00 0.745769
\(383\) 2520.00 0.336204 0.168102 0.985770i \(-0.446236\pi\)
0.168102 + 0.985770i \(0.446236\pi\)
\(384\) 384.000 0.0510310
\(385\) 0 0
\(386\) −1828.00 −0.241043
\(387\) −3708.00 −0.487050
\(388\) −776.000 −0.101535
\(389\) 1566.00 0.204111 0.102056 0.994779i \(-0.467458\pi\)
0.102056 + 0.994779i \(0.467458\pi\)
\(390\) −1020.00 −0.132435
\(391\) 0 0
\(392\) 0 0
\(393\) 180.000 0.0231038
\(394\) 10404.0 1.33032
\(395\) −760.000 −0.0968095
\(396\) −2160.00 −0.274101
\(397\) 4354.00 0.550431 0.275215 0.961383i \(-0.411251\pi\)
0.275215 + 0.961383i \(0.411251\pi\)
\(398\) 6304.00 0.793947
\(399\) 0 0
\(400\) 400.000 0.0500000
\(401\) −8046.00 −1.00199 −0.500995 0.865450i \(-0.667033\pi\)
−0.500995 + 0.865450i \(0.667033\pi\)
\(402\) 4872.00 0.604461
\(403\) 7888.00 0.975011
\(404\) −3192.00 −0.393089
\(405\) −405.000 −0.0496904
\(406\) 0 0
\(407\) −8040.00 −0.979184
\(408\) −1008.00 −0.122312
\(409\) 2806.00 0.339237 0.169618 0.985510i \(-0.445747\pi\)
0.169618 + 0.985510i \(0.445747\pi\)
\(410\) −2340.00 −0.281864
\(411\) −1926.00 −0.231150
\(412\) −4352.00 −0.520407
\(413\) 0 0
\(414\) 0 0
\(415\) −4020.00 −0.475504
\(416\) −1088.00 −0.128230
\(417\) −8508.00 −0.999133
\(418\) 9120.00 1.06716
\(419\) −11580.0 −1.35017 −0.675084 0.737741i \(-0.735892\pi\)
−0.675084 + 0.737741i \(0.735892\pi\)
\(420\) 0 0
\(421\) −370.000 −0.0428330 −0.0214165 0.999771i \(-0.506818\pi\)
−0.0214165 + 0.999771i \(0.506818\pi\)
\(422\) −1480.00 −0.170723
\(423\) 3240.00 0.372421
\(424\) −1776.00 −0.203420
\(425\) −1050.00 −0.119841
\(426\) 720.000 0.0818876
\(427\) 0 0
\(428\) 6864.00 0.775196
\(429\) 6120.00 0.688756
\(430\) −4120.00 −0.462056
\(431\) 5040.00 0.563267 0.281634 0.959522i \(-0.409124\pi\)
0.281634 + 0.959522i \(0.409124\pi\)
\(432\) −432.000 −0.0481125
\(433\) 3742.00 0.415310 0.207655 0.978202i \(-0.433417\pi\)
0.207655 + 0.978202i \(0.433417\pi\)
\(434\) 0 0
\(435\) 90.0000 0.00991993
\(436\) −3880.00 −0.426189
\(437\) 0 0
\(438\) −4476.00 −0.488291
\(439\) 6208.00 0.674924 0.337462 0.941339i \(-0.390432\pi\)
0.337462 + 0.941339i \(0.390432\pi\)
\(440\) −2400.00 −0.260035
\(441\) 0 0
\(442\) 2856.00 0.307344
\(443\) −15564.0 −1.66923 −0.834614 0.550835i \(-0.814309\pi\)
−0.834614 + 0.550835i \(0.814309\pi\)
\(444\) −1608.00 −0.171875
\(445\) −3390.00 −0.361127
\(446\) −1040.00 −0.110416
\(447\) 4662.00 0.493300
\(448\) 0 0
\(449\) −15774.0 −1.65795 −0.828977 0.559283i \(-0.811076\pi\)
−0.828977 + 0.559283i \(0.811076\pi\)
\(450\) −450.000 −0.0471405
\(451\) 14040.0 1.46589
\(452\) 1704.00 0.177322
\(453\) 6816.00 0.706940
\(454\) 792.000 0.0818731
\(455\) 0 0
\(456\) 1824.00 0.187317
\(457\) 9722.00 0.995133 0.497567 0.867426i \(-0.334227\pi\)
0.497567 + 0.867426i \(0.334227\pi\)
\(458\) −2660.00 −0.271383
\(459\) 1134.00 0.115317
\(460\) 0 0
\(461\) 10890.0 1.10021 0.550106 0.835095i \(-0.314587\pi\)
0.550106 + 0.835095i \(0.314587\pi\)
\(462\) 0 0
\(463\) 15128.0 1.51848 0.759242 0.650809i \(-0.225570\pi\)
0.759242 + 0.650809i \(0.225570\pi\)
\(464\) 96.0000 0.00960493
\(465\) 3480.00 0.347056
\(466\) −9732.00 −0.967438
\(467\) −10668.0 −1.05708 −0.528540 0.848909i \(-0.677260\pi\)
−0.528540 + 0.848909i \(0.677260\pi\)
\(468\) 1224.00 0.120896
\(469\) 0 0
\(470\) 3600.00 0.353310
\(471\) 5082.00 0.497168
\(472\) 5280.00 0.514898
\(473\) 24720.0 2.40302
\(474\) 912.000 0.0883746
\(475\) 1900.00 0.183533
\(476\) 0 0
\(477\) 1998.00 0.191786
\(478\) 3648.00 0.349070
\(479\) −15264.0 −1.45601 −0.728006 0.685571i \(-0.759553\pi\)
−0.728006 + 0.685571i \(0.759553\pi\)
\(480\) −480.000 −0.0456435
\(481\) 4556.00 0.431883
\(482\) 12964.0 1.22509
\(483\) 0 0
\(484\) 9076.00 0.852367
\(485\) 970.000 0.0908153
\(486\) 486.000 0.0453609
\(487\) −5776.00 −0.537445 −0.268722 0.963218i \(-0.586601\pi\)
−0.268722 + 0.963218i \(0.586601\pi\)
\(488\) −3920.00 −0.363627
\(489\) 156.000 0.0144265
\(490\) 0 0
\(491\) 14244.0 1.30921 0.654606 0.755971i \(-0.272835\pi\)
0.654606 + 0.755971i \(0.272835\pi\)
\(492\) 2808.00 0.257306
\(493\) −252.000 −0.0230213
\(494\) −5168.00 −0.470687
\(495\) 2700.00 0.245164
\(496\) 3712.00 0.336036
\(497\) 0 0
\(498\) 4824.00 0.434074
\(499\) −17116.0 −1.53551 −0.767753 0.640746i \(-0.778625\pi\)
−0.767753 + 0.640746i \(0.778625\pi\)
\(500\) −500.000 −0.0447214
\(501\) −3600.00 −0.321030
\(502\) 2952.00 0.262459
\(503\) 16848.0 1.49347 0.746735 0.665122i \(-0.231620\pi\)
0.746735 + 0.665122i \(0.231620\pi\)
\(504\) 0 0
\(505\) 3990.00 0.351589
\(506\) 0 0
\(507\) 3123.00 0.273565
\(508\) 800.000 0.0698706
\(509\) 3834.00 0.333868 0.166934 0.985968i \(-0.446613\pi\)
0.166934 + 0.985968i \(0.446613\pi\)
\(510\) 1260.00 0.109399
\(511\) 0 0
\(512\) −512.000 −0.0441942
\(513\) −2052.00 −0.176604
\(514\) 8628.00 0.740398
\(515\) 5440.00 0.465466
\(516\) 4944.00 0.421797
\(517\) −21600.0 −1.83746
\(518\) 0 0
\(519\) 162.000 0.0137014
\(520\) 1360.00 0.114692
\(521\) 18822.0 1.58274 0.791369 0.611338i \(-0.209369\pi\)
0.791369 + 0.611338i \(0.209369\pi\)
\(522\) −108.000 −0.00905562
\(523\) 15340.0 1.28255 0.641273 0.767313i \(-0.278407\pi\)
0.641273 + 0.767313i \(0.278407\pi\)
\(524\) −240.000 −0.0200085
\(525\) 0 0
\(526\) 10560.0 0.875357
\(527\) −9744.00 −0.805418
\(528\) 2880.00 0.237379
\(529\) −12167.0 −1.00000
\(530\) 2220.00 0.181945
\(531\) −5940.00 −0.485450
\(532\) 0 0
\(533\) −7956.00 −0.646553
\(534\) 4068.00 0.329662
\(535\) −8580.00 −0.693357
\(536\) −6496.00 −0.523478
\(537\) −2628.00 −0.211185
\(538\) 11052.0 0.885661
\(539\) 0 0
\(540\) 540.000 0.0430331
\(541\) 18950.0 1.50596 0.752980 0.658044i \(-0.228616\pi\)
0.752980 + 0.658044i \(0.228616\pi\)
\(542\) 4048.00 0.320805
\(543\) 11562.0 0.913762
\(544\) 1344.00 0.105926
\(545\) 4850.00 0.381195
\(546\) 0 0
\(547\) −10036.0 −0.784476 −0.392238 0.919864i \(-0.628299\pi\)
−0.392238 + 0.919864i \(0.628299\pi\)
\(548\) 2568.00 0.200182
\(549\) 4410.00 0.342831
\(550\) 3000.00 0.232583
\(551\) 456.000 0.0352564
\(552\) 0 0
\(553\) 0 0
\(554\) −4108.00 −0.315040
\(555\) 2010.00 0.153729
\(556\) 11344.0 0.865275
\(557\) 10326.0 0.785506 0.392753 0.919644i \(-0.371523\pi\)
0.392753 + 0.919644i \(0.371523\pi\)
\(558\) −4176.00 −0.316818
\(559\) −14008.0 −1.05988
\(560\) 0 0
\(561\) −7560.00 −0.568954
\(562\) 14604.0 1.09614
\(563\) −4524.00 −0.338657 −0.169328 0.985560i \(-0.554160\pi\)
−0.169328 + 0.985560i \(0.554160\pi\)
\(564\) −4320.00 −0.322526
\(565\) −2130.00 −0.158601
\(566\) −7448.00 −0.553114
\(567\) 0 0
\(568\) −960.000 −0.0709167
\(569\) 16362.0 1.20550 0.602751 0.797929i \(-0.294071\pi\)
0.602751 + 0.797929i \(0.294071\pi\)
\(570\) −2280.00 −0.167542
\(571\) 6620.00 0.485181 0.242591 0.970129i \(-0.422003\pi\)
0.242591 + 0.970129i \(0.422003\pi\)
\(572\) −8160.00 −0.596480
\(573\) 8352.00 0.608918
\(574\) 0 0
\(575\) 0 0
\(576\) 576.000 0.0416667
\(577\) −8834.00 −0.637373 −0.318687 0.947860i \(-0.603242\pi\)
−0.318687 + 0.947860i \(0.603242\pi\)
\(578\) 6298.00 0.453222
\(579\) −2742.00 −0.196811
\(580\) −120.000 −0.00859091
\(581\) 0 0
\(582\) −1164.00 −0.0829027
\(583\) −13320.0 −0.946240
\(584\) 5968.00 0.422873
\(585\) −1530.00 −0.108133
\(586\) −14436.0 −1.01765
\(587\) −3636.00 −0.255662 −0.127831 0.991796i \(-0.540802\pi\)
−0.127831 + 0.991796i \(0.540802\pi\)
\(588\) 0 0
\(589\) 17632.0 1.23347
\(590\) −6600.00 −0.460538
\(591\) 15606.0 1.08620
\(592\) 2144.00 0.148848
\(593\) −6570.00 −0.454971 −0.227485 0.973782i \(-0.573050\pi\)
−0.227485 + 0.973782i \(0.573050\pi\)
\(594\) −3240.00 −0.223803
\(595\) 0 0
\(596\) −6216.00 −0.427210
\(597\) 9456.00 0.648255
\(598\) 0 0
\(599\) 16584.0 1.13123 0.565613 0.824671i \(-0.308640\pi\)
0.565613 + 0.824671i \(0.308640\pi\)
\(600\) 600.000 0.0408248
\(601\) 502.000 0.0340716 0.0170358 0.999855i \(-0.494577\pi\)
0.0170358 + 0.999855i \(0.494577\pi\)
\(602\) 0 0
\(603\) 7308.00 0.493540
\(604\) −9088.00 −0.612228
\(605\) −11345.0 −0.762380
\(606\) −4788.00 −0.320956
\(607\) 18568.0 1.24160 0.620801 0.783969i \(-0.286808\pi\)
0.620801 + 0.783969i \(0.286808\pi\)
\(608\) −2432.00 −0.162221
\(609\) 0 0
\(610\) 4900.00 0.325238
\(611\) 12240.0 0.810438
\(612\) −1512.00 −0.0998676
\(613\) −13114.0 −0.864061 −0.432031 0.901859i \(-0.642203\pi\)
−0.432031 + 0.901859i \(0.642203\pi\)
\(614\) 5080.00 0.333896
\(615\) −3510.00 −0.230141
\(616\) 0 0
\(617\) 5250.00 0.342556 0.171278 0.985223i \(-0.445210\pi\)
0.171278 + 0.985223i \(0.445210\pi\)
\(618\) −6528.00 −0.424910
\(619\) 10804.0 0.701534 0.350767 0.936463i \(-0.385921\pi\)
0.350767 + 0.936463i \(0.385921\pi\)
\(620\) −4640.00 −0.300559
\(621\) 0 0
\(622\) 3120.00 0.201126
\(623\) 0 0
\(624\) −1632.00 −0.104699
\(625\) 625.000 0.0400000
\(626\) −1868.00 −0.119266
\(627\) 13680.0 0.871334
\(628\) −6776.00 −0.430560
\(629\) −5628.00 −0.356762
\(630\) 0 0
\(631\) −27088.0 −1.70896 −0.854482 0.519481i \(-0.826125\pi\)
−0.854482 + 0.519481i \(0.826125\pi\)
\(632\) −1216.00 −0.0765346
\(633\) −2220.00 −0.139395
\(634\) 3348.00 0.209726
\(635\) −1000.00 −0.0624942
\(636\) −2664.00 −0.166092
\(637\) 0 0
\(638\) 720.000 0.0446788
\(639\) 1080.00 0.0668609
\(640\) 640.000 0.0395285
\(641\) 18930.0 1.16644 0.583222 0.812313i \(-0.301792\pi\)
0.583222 + 0.812313i \(0.301792\pi\)
\(642\) 10296.0 0.632945
\(643\) −20108.0 −1.23325 −0.616627 0.787256i \(-0.711501\pi\)
−0.616627 + 0.787256i \(0.711501\pi\)
\(644\) 0 0
\(645\) −6180.00 −0.377267
\(646\) 6384.00 0.388816
\(647\) 7152.00 0.434581 0.217291 0.976107i \(-0.430278\pi\)
0.217291 + 0.976107i \(0.430278\pi\)
\(648\) −648.000 −0.0392837
\(649\) 39600.0 2.39512
\(650\) −1700.00 −0.102584
\(651\) 0 0
\(652\) −208.000 −0.0124937
\(653\) −31626.0 −1.89528 −0.947642 0.319333i \(-0.896541\pi\)
−0.947642 + 0.319333i \(0.896541\pi\)
\(654\) −5820.00 −0.347982
\(655\) 300.000 0.0178961
\(656\) −3744.00 −0.222833
\(657\) −6714.00 −0.398688
\(658\) 0 0
\(659\) 28092.0 1.66056 0.830280 0.557347i \(-0.188181\pi\)
0.830280 + 0.557347i \(0.188181\pi\)
\(660\) −3600.00 −0.212318
\(661\) 13186.0 0.775909 0.387955 0.921678i \(-0.373182\pi\)
0.387955 + 0.921678i \(0.373182\pi\)
\(662\) 7976.00 0.468272
\(663\) 4284.00 0.250945
\(664\) −6432.00 −0.375919
\(665\) 0 0
\(666\) −2412.00 −0.140335
\(667\) 0 0
\(668\) 4800.00 0.278020
\(669\) −1560.00 −0.0901541
\(670\) 8120.00 0.468213
\(671\) −29400.0 −1.69147
\(672\) 0 0
\(673\) 5138.00 0.294287 0.147144 0.989115i \(-0.452992\pi\)
0.147144 + 0.989115i \(0.452992\pi\)
\(674\) −4.00000 −0.000228597 0
\(675\) −675.000 −0.0384900
\(676\) −4164.00 −0.236914
\(677\) −6078.00 −0.345047 −0.172523 0.985005i \(-0.555192\pi\)
−0.172523 + 0.985005i \(0.555192\pi\)
\(678\) 2556.00 0.144783
\(679\) 0 0
\(680\) −1680.00 −0.0947427
\(681\) 1188.00 0.0668491
\(682\) 27840.0 1.56312
\(683\) 32244.0 1.80642 0.903208 0.429203i \(-0.141205\pi\)
0.903208 + 0.429203i \(0.141205\pi\)
\(684\) 2736.00 0.152944
\(685\) −3210.00 −0.179048
\(686\) 0 0
\(687\) −3990.00 −0.221584
\(688\) −6592.00 −0.365287
\(689\) 7548.00 0.417353
\(690\) 0 0
\(691\) −4484.00 −0.246859 −0.123429 0.992353i \(-0.539389\pi\)
−0.123429 + 0.992353i \(0.539389\pi\)
\(692\) −216.000 −0.0118657
\(693\) 0 0
\(694\) −3528.00 −0.192970
\(695\) −14180.0 −0.773925
\(696\) 144.000 0.00784239
\(697\) 9828.00 0.534092
\(698\) 8620.00 0.467438
\(699\) −14598.0 −0.789910
\(700\) 0 0
\(701\) −30426.0 −1.63934 −0.819668 0.572839i \(-0.805842\pi\)
−0.819668 + 0.572839i \(0.805842\pi\)
\(702\) 1836.00 0.0987113
\(703\) 10184.0 0.546368
\(704\) −3840.00 −0.205576
\(705\) 5400.00 0.288476
\(706\) 276.000 0.0147130
\(707\) 0 0
\(708\) 7920.00 0.420412
\(709\) 13262.0 0.702489 0.351245 0.936284i \(-0.385759\pi\)
0.351245 + 0.936284i \(0.385759\pi\)
\(710\) 1200.00 0.0634299
\(711\) 1368.00 0.0721575
\(712\) −5424.00 −0.285496
\(713\) 0 0
\(714\) 0 0
\(715\) 10200.0 0.533508
\(716\) 3504.00 0.182892
\(717\) 5472.00 0.285015
\(718\) 23952.0 1.24496
\(719\) −13920.0 −0.722014 −0.361007 0.932563i \(-0.617567\pi\)
−0.361007 + 0.932563i \(0.617567\pi\)
\(720\) −720.000 −0.0372678
\(721\) 0 0
\(722\) 2166.00 0.111648
\(723\) 19446.0 1.00028
\(724\) −15416.0 −0.791341
\(725\) 150.000 0.00768395
\(726\) 13614.0 0.695954
\(727\) 9376.00 0.478317 0.239159 0.970981i \(-0.423128\pi\)
0.239159 + 0.970981i \(0.423128\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −7460.00 −0.378229
\(731\) 17304.0 0.875529
\(732\) −5880.00 −0.296900
\(733\) −6014.00 −0.303045 −0.151523 0.988454i \(-0.548418\pi\)
−0.151523 + 0.988454i \(0.548418\pi\)
\(734\) 19408.0 0.975971
\(735\) 0 0
\(736\) 0 0
\(737\) −48720.0 −2.43504
\(738\) 4212.00 0.210089
\(739\) −7468.00 −0.371739 −0.185869 0.982574i \(-0.559510\pi\)
−0.185869 + 0.982574i \(0.559510\pi\)
\(740\) −2680.00 −0.133133
\(741\) −7752.00 −0.384314
\(742\) 0 0
\(743\) 31248.0 1.54290 0.771452 0.636287i \(-0.219531\pi\)
0.771452 + 0.636287i \(0.219531\pi\)
\(744\) 5568.00 0.274372
\(745\) 7770.00 0.382108
\(746\) 16244.0 0.797232
\(747\) 7236.00 0.354420
\(748\) 10080.0 0.492729
\(749\) 0 0
\(750\) −750.000 −0.0365148
\(751\) 32840.0 1.59567 0.797835 0.602875i \(-0.205978\pi\)
0.797835 + 0.602875i \(0.205978\pi\)
\(752\) 5760.00 0.279316
\(753\) 4428.00 0.214297
\(754\) −408.000 −0.0197062
\(755\) 11360.0 0.547593
\(756\) 0 0
\(757\) −19066.0 −0.915410 −0.457705 0.889104i \(-0.651328\pi\)
−0.457705 + 0.889104i \(0.651328\pi\)
\(758\) −6808.00 −0.326224
\(759\) 0 0
\(760\) 3040.00 0.145095
\(761\) −6858.00 −0.326678 −0.163339 0.986570i \(-0.552227\pi\)
−0.163339 + 0.986570i \(0.552227\pi\)
\(762\) 1200.00 0.0570491
\(763\) 0 0
\(764\) −11136.0 −0.527338
\(765\) 1890.00 0.0893243
\(766\) −5040.00 −0.237732
\(767\) −22440.0 −1.05640
\(768\) −768.000 −0.0360844
\(769\) −22178.0 −1.04000 −0.519999 0.854167i \(-0.674068\pi\)
−0.519999 + 0.854167i \(0.674068\pi\)
\(770\) 0 0
\(771\) 12942.0 0.604533
\(772\) 3656.00 0.170443
\(773\) −14286.0 −0.664724 −0.332362 0.943152i \(-0.607846\pi\)
−0.332362 + 0.943152i \(0.607846\pi\)
\(774\) 7416.00 0.344396
\(775\) 5800.00 0.268829
\(776\) 1552.00 0.0717958
\(777\) 0 0
\(778\) −3132.00 −0.144329
\(779\) −17784.0 −0.817943
\(780\) 2040.00 0.0936458
\(781\) −7200.00 −0.329880
\(782\) 0 0
\(783\) −162.000 −0.00739388
\(784\) 0 0
\(785\) 8470.00 0.385105
\(786\) −360.000 −0.0163369
\(787\) 18868.0 0.854602 0.427301 0.904109i \(-0.359465\pi\)
0.427301 + 0.904109i \(0.359465\pi\)
\(788\) −20808.0 −0.940678
\(789\) 15840.0 0.714726
\(790\) 1520.00 0.0684546
\(791\) 0 0
\(792\) 4320.00 0.193819
\(793\) 16660.0 0.746045
\(794\) −8708.00 −0.389213
\(795\) 3330.00 0.148557
\(796\) −12608.0 −0.561405
\(797\) 21690.0 0.963989 0.481994 0.876174i \(-0.339913\pi\)
0.481994 + 0.876174i \(0.339913\pi\)
\(798\) 0 0
\(799\) −15120.0 −0.669471
\(800\) −800.000 −0.0353553
\(801\) 6102.00 0.269168
\(802\) 16092.0 0.708514
\(803\) 44760.0 1.96706
\(804\) −9744.00 −0.427418
\(805\) 0 0
\(806\) −15776.0 −0.689437
\(807\) 16578.0 0.723139
\(808\) 6384.00 0.277956
\(809\) −24726.0 −1.07456 −0.537281 0.843404i \(-0.680548\pi\)
−0.537281 + 0.843404i \(0.680548\pi\)
\(810\) 810.000 0.0351364
\(811\) 2644.00 0.114480 0.0572401 0.998360i \(-0.481770\pi\)
0.0572401 + 0.998360i \(0.481770\pi\)
\(812\) 0 0
\(813\) 6072.00 0.261936
\(814\) 16080.0 0.692388
\(815\) 260.000 0.0111747
\(816\) 2016.00 0.0864879
\(817\) −31312.0 −1.34084
\(818\) −5612.00 −0.239877
\(819\) 0 0
\(820\) 4680.00 0.199308
\(821\) −37842.0 −1.60864 −0.804321 0.594195i \(-0.797471\pi\)
−0.804321 + 0.594195i \(0.797471\pi\)
\(822\) 3852.00 0.163448
\(823\) −880.000 −0.0372720 −0.0186360 0.999826i \(-0.505932\pi\)
−0.0186360 + 0.999826i \(0.505932\pi\)
\(824\) 8704.00 0.367983
\(825\) 4500.00 0.189903
\(826\) 0 0
\(827\) −12876.0 −0.541406 −0.270703 0.962663i \(-0.587256\pi\)
−0.270703 + 0.962663i \(0.587256\pi\)
\(828\) 0 0
\(829\) 25498.0 1.06825 0.534127 0.845404i \(-0.320641\pi\)
0.534127 + 0.845404i \(0.320641\pi\)
\(830\) 8040.00 0.336232
\(831\) −6162.00 −0.257229
\(832\) 2176.00 0.0906721
\(833\) 0 0
\(834\) 17016.0 0.706494
\(835\) −6000.00 −0.248669
\(836\) −18240.0 −0.754598
\(837\) −6264.00 −0.258680
\(838\) 23160.0 0.954712
\(839\) 40584.0 1.66998 0.834991 0.550263i \(-0.185473\pi\)
0.834991 + 0.550263i \(0.185473\pi\)
\(840\) 0 0
\(841\) −24353.0 −0.998524
\(842\) 740.000 0.0302875
\(843\) 21906.0 0.894997
\(844\) 2960.00 0.120720
\(845\) 5205.00 0.211902
\(846\) −6480.00 −0.263342
\(847\) 0 0
\(848\) 3552.00 0.143840
\(849\) −11172.0 −0.451616
\(850\) 2100.00 0.0847405
\(851\) 0 0
\(852\) −1440.00 −0.0579033
\(853\) 25738.0 1.03312 0.516561 0.856251i \(-0.327212\pi\)
0.516561 + 0.856251i \(0.327212\pi\)
\(854\) 0 0
\(855\) −3420.00 −0.136797
\(856\) −13728.0 −0.548146
\(857\) −13314.0 −0.530686 −0.265343 0.964154i \(-0.585485\pi\)
−0.265343 + 0.964154i \(0.585485\pi\)
\(858\) −12240.0 −0.487024
\(859\) −24524.0 −0.974096 −0.487048 0.873375i \(-0.661926\pi\)
−0.487048 + 0.873375i \(0.661926\pi\)
\(860\) 8240.00 0.326723
\(861\) 0 0
\(862\) −10080.0 −0.398290
\(863\) 5592.00 0.220572 0.110286 0.993900i \(-0.464823\pi\)
0.110286 + 0.993900i \(0.464823\pi\)
\(864\) 864.000 0.0340207
\(865\) 270.000 0.0106130
\(866\) −7484.00 −0.293668
\(867\) 9447.00 0.370054
\(868\) 0 0
\(869\) −9120.00 −0.356012
\(870\) −180.000 −0.00701445
\(871\) 27608.0 1.07401
\(872\) 7760.00 0.301361
\(873\) −1746.00 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 8952.00 0.345274
\(877\) −14386.0 −0.553912 −0.276956 0.960883i \(-0.589326\pi\)
−0.276956 + 0.960883i \(0.589326\pi\)
\(878\) −12416.0 −0.477243
\(879\) −21654.0 −0.830912
\(880\) 4800.00 0.183873
\(881\) −47106.0 −1.80141 −0.900705 0.434432i \(-0.856949\pi\)
−0.900705 + 0.434432i \(0.856949\pi\)
\(882\) 0 0
\(883\) 51548.0 1.96458 0.982292 0.187354i \(-0.0599913\pi\)
0.982292 + 0.187354i \(0.0599913\pi\)
\(884\) −5712.00 −0.217325
\(885\) −9900.00 −0.376028
\(886\) 31128.0 1.18032
\(887\) −34080.0 −1.29007 −0.645036 0.764152i \(-0.723158\pi\)
−0.645036 + 0.764152i \(0.723158\pi\)
\(888\) 3216.00 0.121534
\(889\) 0 0
\(890\) 6780.00 0.255355
\(891\) −4860.00 −0.182734
\(892\) 2080.00 0.0780757
\(893\) 27360.0 1.02527
\(894\) −9324.00 −0.348816
\(895\) −4380.00 −0.163584
\(896\) 0 0
\(897\) 0 0
\(898\) 31548.0 1.17235
\(899\) 1392.00 0.0516416
\(900\) 900.000 0.0333333
\(901\) −9324.00 −0.344759
\(902\) −28080.0 −1.03654
\(903\) 0 0
\(904\) −3408.00 −0.125385
\(905\) 19270.0 0.707797
\(906\) −13632.0 −0.499882
\(907\) 25748.0 0.942611 0.471306 0.881970i \(-0.343783\pi\)
0.471306 + 0.881970i \(0.343783\pi\)
\(908\) −1584.00 −0.0578930
\(909\) −7182.00 −0.262059
\(910\) 0 0
\(911\) −24768.0 −0.900769 −0.450384 0.892835i \(-0.648713\pi\)
−0.450384 + 0.892835i \(0.648713\pi\)
\(912\) −3648.00 −0.132453
\(913\) −48240.0 −1.74864
\(914\) −19444.0 −0.703666
\(915\) 7350.00 0.265556
\(916\) 5320.00 0.191897
\(917\) 0 0
\(918\) −2268.00 −0.0815416
\(919\) −31264.0 −1.12220 −0.561101 0.827747i \(-0.689622\pi\)
−0.561101 + 0.827747i \(0.689622\pi\)
\(920\) 0 0
\(921\) 7620.00 0.272625
\(922\) −21780.0 −0.777968
\(923\) 4080.00 0.145498
\(924\) 0 0
\(925\) 3350.00 0.119078
\(926\) −30256.0 −1.07373
\(927\) −9792.00 −0.346938
\(928\) −192.000 −0.00679171
\(929\) 6174.00 0.218043 0.109022 0.994039i \(-0.465228\pi\)
0.109022 + 0.994039i \(0.465228\pi\)
\(930\) −6960.00 −0.245406
\(931\) 0 0
\(932\) 19464.0 0.684082
\(933\) 4680.00 0.164219
\(934\) 21336.0 0.747468
\(935\) −12600.0 −0.440710
\(936\) −2448.00 −0.0854865
\(937\) −28922.0 −1.00837 −0.504184 0.863596i \(-0.668207\pi\)
−0.504184 + 0.863596i \(0.668207\pi\)
\(938\) 0 0
\(939\) −2802.00 −0.0973800
\(940\) −7200.00 −0.249828
\(941\) −29238.0 −1.01289 −0.506446 0.862272i \(-0.669041\pi\)
−0.506446 + 0.862272i \(0.669041\pi\)
\(942\) −10164.0 −0.351551
\(943\) 0 0
\(944\) −10560.0 −0.364088
\(945\) 0 0
\(946\) −49440.0 −1.69919
\(947\) −2868.00 −0.0984134 −0.0492067 0.998789i \(-0.515669\pi\)
−0.0492067 + 0.998789i \(0.515669\pi\)
\(948\) −1824.00 −0.0624903
\(949\) −25364.0 −0.867598
\(950\) −3800.00 −0.129777
\(951\) 5022.00 0.171240
\(952\) 0 0
\(953\) 24018.0 0.816390 0.408195 0.912895i \(-0.366158\pi\)
0.408195 + 0.912895i \(0.366158\pi\)
\(954\) −3996.00 −0.135613
\(955\) 13920.0 0.471666
\(956\) −7296.00 −0.246830
\(957\) 1080.00 0.0364801
\(958\) 30528.0 1.02956
\(959\) 0 0
\(960\) 960.000 0.0322749
\(961\) 24033.0 0.806720
\(962\) −9112.00 −0.305387
\(963\) 15444.0 0.516797
\(964\) −25928.0 −0.866270
\(965\) −4570.00 −0.152449
\(966\) 0 0
\(967\) 25712.0 0.855059 0.427530 0.904001i \(-0.359384\pi\)
0.427530 + 0.904001i \(0.359384\pi\)
\(968\) −18152.0 −0.602714
\(969\) 9576.00 0.317467
\(970\) −1940.00 −0.0642161
\(971\) 12396.0 0.409688 0.204844 0.978795i \(-0.434331\pi\)
0.204844 + 0.978795i \(0.434331\pi\)
\(972\) −972.000 −0.0320750
\(973\) 0 0
\(974\) 11552.0 0.380031
\(975\) −2550.00 −0.0837593
\(976\) 7840.00 0.257123
\(977\) −46614.0 −1.52642 −0.763211 0.646150i \(-0.776378\pi\)
−0.763211 + 0.646150i \(0.776378\pi\)
\(978\) −312.000 −0.0102011
\(979\) −40680.0 −1.32803
\(980\) 0 0
\(981\) −8730.00 −0.284126
\(982\) −28488.0 −0.925752
\(983\) 672.000 0.0218041 0.0109021 0.999941i \(-0.496530\pi\)
0.0109021 + 0.999941i \(0.496530\pi\)
\(984\) −5616.00 −0.181943
\(985\) 26010.0 0.841368
\(986\) 504.000 0.0162785
\(987\) 0 0
\(988\) 10336.0 0.332826
\(989\) 0 0
\(990\) −5400.00 −0.173357
\(991\) −38776.0 −1.24295 −0.621473 0.783435i \(-0.713466\pi\)
−0.621473 + 0.783435i \(0.713466\pi\)
\(992\) −7424.00 −0.237613
\(993\) 11964.0 0.382342
\(994\) 0 0
\(995\) 15760.0 0.502136
\(996\) −9648.00 −0.306936
\(997\) −30422.0 −0.966374 −0.483187 0.875517i \(-0.660521\pi\)
−0.483187 + 0.875517i \(0.660521\pi\)
\(998\) 34232.0 1.08577
\(999\) −3618.00 −0.114583
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 1470.4.a.a.1.1 1
7.6 odd 2 30.4.a.a.1.1 1
21.20 even 2 90.4.a.d.1.1 1
28.27 even 2 240.4.a.c.1.1 1
35.13 even 4 150.4.c.a.49.2 2
35.27 even 4 150.4.c.a.49.1 2
35.34 odd 2 150.4.a.e.1.1 1
56.13 odd 2 960.4.a.j.1.1 1
56.27 even 2 960.4.a.s.1.1 1
63.13 odd 6 810.4.e.m.541.1 2
63.20 even 6 810.4.e.e.271.1 2
63.34 odd 6 810.4.e.m.271.1 2
63.41 even 6 810.4.e.e.541.1 2
84.83 odd 2 720.4.a.b.1.1 1
105.62 odd 4 450.4.c.k.199.2 2
105.83 odd 4 450.4.c.k.199.1 2
105.104 even 2 450.4.a.b.1.1 1
140.27 odd 4 1200.4.f.u.49.2 2
140.83 odd 4 1200.4.f.u.49.1 2
140.139 even 2 1200.4.a.bk.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.4.a.a.1.1 1 7.6 odd 2
90.4.a.d.1.1 1 21.20 even 2
150.4.a.e.1.1 1 35.34 odd 2
150.4.c.a.49.1 2 35.27 even 4
150.4.c.a.49.2 2 35.13 even 4
240.4.a.c.1.1 1 28.27 even 2
450.4.a.b.1.1 1 105.104 even 2
450.4.c.k.199.1 2 105.83 odd 4
450.4.c.k.199.2 2 105.62 odd 4
720.4.a.b.1.1 1 84.83 odd 2
810.4.e.e.271.1 2 63.20 even 6
810.4.e.e.541.1 2 63.41 even 6
810.4.e.m.271.1 2 63.34 odd 6
810.4.e.m.541.1 2 63.13 odd 6
960.4.a.j.1.1 1 56.13 odd 2
960.4.a.s.1.1 1 56.27 even 2
1200.4.a.bk.1.1 1 140.139 even 2
1200.4.f.u.49.1 2 140.83 odd 4
1200.4.f.u.49.2 2 140.27 odd 4
1470.4.a.a.1.1 1 1.1 even 1 trivial