Properties

Label 1815.4.a.a.1.1
Level $1815$
Weight $4$
Character 1815.1
Self dual yes
Analytic conductor $107.088$
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] = [1815,4,Mod(1,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, 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("1815.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1815.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: \(107.088466660\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1815.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -5.00000 q^{5} +9.00000 q^{6} -20.0000 q^{7} +21.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -5.00000 q^{5} +9.00000 q^{6} -20.0000 q^{7} +21.0000 q^{8} +9.00000 q^{9} +15.0000 q^{10} -3.00000 q^{12} -74.0000 q^{13} +60.0000 q^{14} +15.0000 q^{15} -71.0000 q^{16} -54.0000 q^{17} -27.0000 q^{18} +124.000 q^{19} -5.00000 q^{20} +60.0000 q^{21} -120.000 q^{23} -63.0000 q^{24} +25.0000 q^{25} +222.000 q^{26} -27.0000 q^{27} -20.0000 q^{28} +78.0000 q^{29} -45.0000 q^{30} +200.000 q^{31} +45.0000 q^{32} +162.000 q^{34} +100.000 q^{35} +9.00000 q^{36} -70.0000 q^{37} -372.000 q^{38} +222.000 q^{39} -105.000 q^{40} -330.000 q^{41} -180.000 q^{42} -92.0000 q^{43} -45.0000 q^{45} +360.000 q^{46} -24.0000 q^{47} +213.000 q^{48} +57.0000 q^{49} -75.0000 q^{50} +162.000 q^{51} -74.0000 q^{52} +450.000 q^{53} +81.0000 q^{54} -420.000 q^{56} -372.000 q^{57} -234.000 q^{58} +24.0000 q^{59} +15.0000 q^{60} +322.000 q^{61} -600.000 q^{62} -180.000 q^{63} +433.000 q^{64} +370.000 q^{65} -196.000 q^{67} -54.0000 q^{68} +360.000 q^{69} -300.000 q^{70} -288.000 q^{71} +189.000 q^{72} +430.000 q^{73} +210.000 q^{74} -75.0000 q^{75} +124.000 q^{76} -666.000 q^{78} +520.000 q^{79} +355.000 q^{80} +81.0000 q^{81} +990.000 q^{82} -156.000 q^{83} +60.0000 q^{84} +270.000 q^{85} +276.000 q^{86} -234.000 q^{87} +1026.00 q^{89} +135.000 q^{90} +1480.00 q^{91} -120.000 q^{92} -600.000 q^{93} +72.0000 q^{94} -620.000 q^{95} -135.000 q^{96} -286.000 q^{97} -171.000 q^{98} +O(q^{100})\)

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 \(0.677932\pi\)
\(3\) −3.00000 −0.577350
\(4\) 1.00000 0.125000
\(5\) −5.00000 −0.447214
\(6\) 9.00000 0.612372
\(7\) −20.0000 −1.07990 −0.539949 0.841698i \(-0.681557\pi\)
−0.539949 + 0.841698i \(0.681557\pi\)
\(8\) 21.0000 0.928078
\(9\) 9.00000 0.333333
\(10\) 15.0000 0.474342
\(11\) 0 0
\(12\) −3.00000 −0.0721688
\(13\) −74.0000 −1.57876 −0.789381 0.613904i \(-0.789598\pi\)
−0.789381 + 0.613904i \(0.789598\pi\)
\(14\) 60.0000 1.14541
\(15\) 15.0000 0.258199
\(16\) −71.0000 −1.10938
\(17\) −54.0000 −0.770407 −0.385204 0.922832i \(-0.625869\pi\)
−0.385204 + 0.922832i \(0.625869\pi\)
\(18\) −27.0000 −0.353553
\(19\) 124.000 1.49724 0.748620 0.663000i \(-0.230717\pi\)
0.748620 + 0.663000i \(0.230717\pi\)
\(20\) −5.00000 −0.0559017
\(21\) 60.0000 0.623480
\(22\) 0 0
\(23\) −120.000 −1.08790 −0.543951 0.839117i \(-0.683072\pi\)
−0.543951 + 0.839117i \(0.683072\pi\)
\(24\) −63.0000 −0.535826
\(25\) 25.0000 0.200000
\(26\) 222.000 1.67453
\(27\) −27.0000 −0.192450
\(28\) −20.0000 −0.134987
\(29\) 78.0000 0.499456 0.249728 0.968316i \(-0.419659\pi\)
0.249728 + 0.968316i \(0.419659\pi\)
\(30\) −45.0000 −0.273861
\(31\) 200.000 1.15874 0.579372 0.815063i \(-0.303298\pi\)
0.579372 + 0.815063i \(0.303298\pi\)
\(32\) 45.0000 0.248592
\(33\) 0 0
\(34\) 162.000 0.817140
\(35\) 100.000 0.482945
\(36\) 9.00000 0.0416667
\(37\) −70.0000 −0.311025 −0.155513 0.987834i \(-0.549703\pi\)
−0.155513 + 0.987834i \(0.549703\pi\)
\(38\) −372.000 −1.58806
\(39\) 222.000 0.911499
\(40\) −105.000 −0.415049
\(41\) −330.000 −1.25701 −0.628504 0.777806i \(-0.716332\pi\)
−0.628504 + 0.777806i \(0.716332\pi\)
\(42\) −180.000 −0.661300
\(43\) −92.0000 −0.326276 −0.163138 0.986603i \(-0.552162\pi\)
−0.163138 + 0.986603i \(0.552162\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 360.000 1.15389
\(47\) −24.0000 −0.0744843 −0.0372421 0.999306i \(-0.511857\pi\)
−0.0372421 + 0.999306i \(0.511857\pi\)
\(48\) 213.000 0.640498
\(49\) 57.0000 0.166181
\(50\) −75.0000 −0.212132
\(51\) 162.000 0.444795
\(52\) −74.0000 −0.197345
\(53\) 450.000 1.16627 0.583134 0.812376i \(-0.301826\pi\)
0.583134 + 0.812376i \(0.301826\pi\)
\(54\) 81.0000 0.204124
\(55\) 0 0
\(56\) −420.000 −1.00223
\(57\) −372.000 −0.864432
\(58\) −234.000 −0.529754
\(59\) 24.0000 0.0529582 0.0264791 0.999649i \(-0.491570\pi\)
0.0264791 + 0.999649i \(0.491570\pi\)
\(60\) 15.0000 0.0322749
\(61\) 322.000 0.675867 0.337933 0.941170i \(-0.390272\pi\)
0.337933 + 0.941170i \(0.390272\pi\)
\(62\) −600.000 −1.22903
\(63\) −180.000 −0.359966
\(64\) 433.000 0.845703
\(65\) 370.000 0.706044
\(66\) 0 0
\(67\) −196.000 −0.357391 −0.178696 0.983904i \(-0.557188\pi\)
−0.178696 + 0.983904i \(0.557188\pi\)
\(68\) −54.0000 −0.0963009
\(69\) 360.000 0.628100
\(70\) −300.000 −0.512241
\(71\) −288.000 −0.481399 −0.240699 0.970600i \(-0.577377\pi\)
−0.240699 + 0.970600i \(0.577377\pi\)
\(72\) 189.000 0.309359
\(73\) 430.000 0.689420 0.344710 0.938709i \(-0.387977\pi\)
0.344710 + 0.938709i \(0.387977\pi\)
\(74\) 210.000 0.329892
\(75\) −75.0000 −0.115470
\(76\) 124.000 0.187155
\(77\) 0 0
\(78\) −666.000 −0.966790
\(79\) 520.000 0.740564 0.370282 0.928919i \(-0.379261\pi\)
0.370282 + 0.928919i \(0.379261\pi\)
\(80\) 355.000 0.496128
\(81\) 81.0000 0.111111
\(82\) 990.000 1.33326
\(83\) −156.000 −0.206304 −0.103152 0.994666i \(-0.532893\pi\)
−0.103152 + 0.994666i \(0.532893\pi\)
\(84\) 60.0000 0.0779350
\(85\) 270.000 0.344537
\(86\) 276.000 0.346068
\(87\) −234.000 −0.288361
\(88\) 0 0
\(89\) 1026.00 1.22198 0.610988 0.791640i \(-0.290773\pi\)
0.610988 + 0.791640i \(0.290773\pi\)
\(90\) 135.000 0.158114
\(91\) 1480.00 1.70490
\(92\) −120.000 −0.135988
\(93\) −600.000 −0.669001
\(94\) 72.0000 0.0790025
\(95\) −620.000 −0.669586
\(96\) −135.000 −0.143525
\(97\) −286.000 −0.299370 −0.149685 0.988734i \(-0.547826\pi\)
−0.149685 + 0.988734i \(0.547826\pi\)
\(98\) −171.000 −0.176261
\(99\) 0 0
\(100\) 25.0000 0.0250000
\(101\) 1734.00 1.70831 0.854156 0.520017i \(-0.174075\pi\)
0.854156 + 0.520017i \(0.174075\pi\)
\(102\) −486.000 −0.471776
\(103\) 452.000 0.432397 0.216198 0.976349i \(-0.430634\pi\)
0.216198 + 0.976349i \(0.430634\pi\)
\(104\) −1554.00 −1.46521
\(105\) −300.000 −0.278829
\(106\) −1350.00 −1.23702
\(107\) 1404.00 1.26850 0.634251 0.773127i \(-0.281308\pi\)
0.634251 + 0.773127i \(0.281308\pi\)
\(108\) −27.0000 −0.0240563
\(109\) 1474.00 1.29526 0.647631 0.761954i \(-0.275760\pi\)
0.647631 + 0.761954i \(0.275760\pi\)
\(110\) 0 0
\(111\) 210.000 0.179570
\(112\) 1420.00 1.19801
\(113\) 1086.00 0.904091 0.452046 0.891995i \(-0.350694\pi\)
0.452046 + 0.891995i \(0.350694\pi\)
\(114\) 1116.00 0.916868
\(115\) 600.000 0.486524
\(116\) 78.0000 0.0624321
\(117\) −666.000 −0.526254
\(118\) −72.0000 −0.0561707
\(119\) 1080.00 0.831962
\(120\) 315.000 0.239629
\(121\) 0 0
\(122\) −966.000 −0.716865
\(123\) 990.000 0.725734
\(124\) 200.000 0.144843
\(125\) −125.000 −0.0894427
\(126\) 540.000 0.381802
\(127\) −1244.00 −0.869190 −0.434595 0.900626i \(-0.643109\pi\)
−0.434595 + 0.900626i \(0.643109\pi\)
\(128\) −1659.00 −1.14560
\(129\) 276.000 0.188376
\(130\) −1110.00 −0.748873
\(131\) −2328.00 −1.55266 −0.776329 0.630327i \(-0.782921\pi\)
−0.776329 + 0.630327i \(0.782921\pi\)
\(132\) 0 0
\(133\) −2480.00 −1.61687
\(134\) 588.000 0.379071
\(135\) 135.000 0.0860663
\(136\) −1134.00 −0.714998
\(137\) 2118.00 1.32082 0.660412 0.750903i \(-0.270382\pi\)
0.660412 + 0.750903i \(0.270382\pi\)
\(138\) −1080.00 −0.666201
\(139\) −2324.00 −1.41812 −0.709062 0.705147i \(-0.750881\pi\)
−0.709062 + 0.705147i \(0.750881\pi\)
\(140\) 100.000 0.0603682
\(141\) 72.0000 0.0430035
\(142\) 864.000 0.510600
\(143\) 0 0
\(144\) −639.000 −0.369792
\(145\) −390.000 −0.223364
\(146\) −1290.00 −0.731241
\(147\) −171.000 −0.0959445
\(148\) −70.0000 −0.0388781
\(149\) −258.000 −0.141854 −0.0709268 0.997482i \(-0.522596\pi\)
−0.0709268 + 0.997482i \(0.522596\pi\)
\(150\) 225.000 0.122474
\(151\) 808.000 0.435458 0.217729 0.976009i \(-0.430135\pi\)
0.217729 + 0.976009i \(0.430135\pi\)
\(152\) 2604.00 1.38955
\(153\) −486.000 −0.256802
\(154\) 0 0
\(155\) −1000.00 −0.518206
\(156\) 222.000 0.113937
\(157\) 2378.00 1.20882 0.604411 0.796673i \(-0.293408\pi\)
0.604411 + 0.796673i \(0.293408\pi\)
\(158\) −1560.00 −0.785487
\(159\) −1350.00 −0.673346
\(160\) −225.000 −0.111174
\(161\) 2400.00 1.17482
\(162\) −243.000 −0.117851
\(163\) −52.0000 −0.0249874 −0.0124937 0.999922i \(-0.503977\pi\)
−0.0124937 + 0.999922i \(0.503977\pi\)
\(164\) −330.000 −0.157126
\(165\) 0 0
\(166\) 468.000 0.218818
\(167\) 3720.00 1.72373 0.861863 0.507141i \(-0.169298\pi\)
0.861863 + 0.507141i \(0.169298\pi\)
\(168\) 1260.00 0.578638
\(169\) 3279.00 1.49249
\(170\) −810.000 −0.365436
\(171\) 1116.00 0.499080
\(172\) −92.0000 −0.0407845
\(173\) −426.000 −0.187215 −0.0936075 0.995609i \(-0.529840\pi\)
−0.0936075 + 0.995609i \(0.529840\pi\)
\(174\) 702.000 0.305853
\(175\) −500.000 −0.215980
\(176\) 0 0
\(177\) −72.0000 −0.0305754
\(178\) −3078.00 −1.29610
\(179\) −1440.00 −0.601289 −0.300644 0.953736i \(-0.597202\pi\)
−0.300644 + 0.953736i \(0.597202\pi\)
\(180\) −45.0000 −0.0186339
\(181\) −3130.00 −1.28537 −0.642683 0.766133i \(-0.722179\pi\)
−0.642683 + 0.766133i \(0.722179\pi\)
\(182\) −4440.00 −1.80832
\(183\) −966.000 −0.390212
\(184\) −2520.00 −1.00966
\(185\) 350.000 0.139095
\(186\) 1800.00 0.709583
\(187\) 0 0
\(188\) −24.0000 −0.00931053
\(189\) 540.000 0.207827
\(190\) 1860.00 0.710203
\(191\) 3576.00 1.35471 0.677357 0.735655i \(-0.263125\pi\)
0.677357 + 0.735655i \(0.263125\pi\)
\(192\) −1299.00 −0.488267
\(193\) −2666.00 −0.994315 −0.497158 0.867660i \(-0.665623\pi\)
−0.497158 + 0.867660i \(0.665623\pi\)
\(194\) 858.000 0.317530
\(195\) −1110.00 −0.407635
\(196\) 57.0000 0.0207726
\(197\) 2718.00 0.982992 0.491496 0.870880i \(-0.336450\pi\)
0.491496 + 0.870880i \(0.336450\pi\)
\(198\) 0 0
\(199\) −3832.00 −1.36504 −0.682521 0.730866i \(-0.739116\pi\)
−0.682521 + 0.730866i \(0.739116\pi\)
\(200\) 525.000 0.185616
\(201\) 588.000 0.206340
\(202\) −5202.00 −1.81194
\(203\) −1560.00 −0.539362
\(204\) 162.000 0.0555994
\(205\) 1650.00 0.562151
\(206\) −1356.00 −0.458626
\(207\) −1080.00 −0.362634
\(208\) 5254.00 1.75144
\(209\) 0 0
\(210\) 900.000 0.295742
\(211\) −1100.00 −0.358896 −0.179448 0.983767i \(-0.557431\pi\)
−0.179448 + 0.983767i \(0.557431\pi\)
\(212\) 450.000 0.145784
\(213\) 864.000 0.277936
\(214\) −4212.00 −1.34545
\(215\) 460.000 0.145915
\(216\) −567.000 −0.178609
\(217\) −4000.00 −1.25133
\(218\) −4422.00 −1.37383
\(219\) −1290.00 −0.398037
\(220\) 0 0
\(221\) 3996.00 1.21629
\(222\) −630.000 −0.190463
\(223\) 1964.00 0.589772 0.294886 0.955532i \(-0.404718\pi\)
0.294886 + 0.955532i \(0.404718\pi\)
\(224\) −900.000 −0.268454
\(225\) 225.000 0.0666667
\(226\) −3258.00 −0.958933
\(227\) −660.000 −0.192977 −0.0964884 0.995334i \(-0.530761\pi\)
−0.0964884 + 0.995334i \(0.530761\pi\)
\(228\) −372.000 −0.108054
\(229\) −1906.00 −0.550009 −0.275004 0.961443i \(-0.588679\pi\)
−0.275004 + 0.961443i \(0.588679\pi\)
\(230\) −1800.00 −0.516037
\(231\) 0 0
\(232\) 1638.00 0.463534
\(233\) 1458.00 0.409943 0.204972 0.978768i \(-0.434290\pi\)
0.204972 + 0.978768i \(0.434290\pi\)
\(234\) 1998.00 0.558177
\(235\) 120.000 0.0333104
\(236\) 24.0000 0.00661978
\(237\) −1560.00 −0.427565
\(238\) −3240.00 −0.882429
\(239\) −1176.00 −0.318281 −0.159140 0.987256i \(-0.550872\pi\)
−0.159140 + 0.987256i \(0.550872\pi\)
\(240\) −1065.00 −0.286439
\(241\) −866.000 −0.231469 −0.115734 0.993280i \(-0.536922\pi\)
−0.115734 + 0.993280i \(0.536922\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 322.000 0.0844834
\(245\) −285.000 −0.0743183
\(246\) −2970.00 −0.769757
\(247\) −9176.00 −2.36379
\(248\) 4200.00 1.07540
\(249\) 468.000 0.119110
\(250\) 375.000 0.0948683
\(251\) 432.000 0.108636 0.0543179 0.998524i \(-0.482702\pi\)
0.0543179 + 0.998524i \(0.482702\pi\)
\(252\) −180.000 −0.0449958
\(253\) 0 0
\(254\) 3732.00 0.921915
\(255\) −810.000 −0.198918
\(256\) 1513.00 0.369385
\(257\) 2526.00 0.613103 0.306552 0.951854i \(-0.400825\pi\)
0.306552 + 0.951854i \(0.400825\pi\)
\(258\) −828.000 −0.199802
\(259\) 1400.00 0.335876
\(260\) 370.000 0.0882555
\(261\) 702.000 0.166485
\(262\) 6984.00 1.64684
\(263\) −5448.00 −1.27733 −0.638666 0.769484i \(-0.720513\pi\)
−0.638666 + 0.769484i \(0.720513\pi\)
\(264\) 0 0
\(265\) −2250.00 −0.521571
\(266\) 7440.00 1.71495
\(267\) −3078.00 −0.705508
\(268\) −196.000 −0.0446739
\(269\) −2574.00 −0.583418 −0.291709 0.956507i \(-0.594224\pi\)
−0.291709 + 0.956507i \(0.594224\pi\)
\(270\) −405.000 −0.0912871
\(271\) 3184.00 0.713706 0.356853 0.934161i \(-0.383850\pi\)
0.356853 + 0.934161i \(0.383850\pi\)
\(272\) 3834.00 0.854671
\(273\) −4440.00 −0.984326
\(274\) −6354.00 −1.40095
\(275\) 0 0
\(276\) 360.000 0.0785125
\(277\) −3962.00 −0.859399 −0.429699 0.902972i \(-0.641380\pi\)
−0.429699 + 0.902972i \(0.641380\pi\)
\(278\) 6972.00 1.50415
\(279\) 1800.00 0.386248
\(280\) 2100.00 0.448211
\(281\) 8286.00 1.75908 0.879540 0.475825i \(-0.157851\pi\)
0.879540 + 0.475825i \(0.157851\pi\)
\(282\) −216.000 −0.0456121
\(283\) 2716.00 0.570493 0.285246 0.958454i \(-0.407925\pi\)
0.285246 + 0.958454i \(0.407925\pi\)
\(284\) −288.000 −0.0601748
\(285\) 1860.00 0.386586
\(286\) 0 0
\(287\) 6600.00 1.35744
\(288\) 405.000 0.0828641
\(289\) −1997.00 −0.406473
\(290\) 1170.00 0.236913
\(291\) 858.000 0.172841
\(292\) 430.000 0.0861776
\(293\) −6018.00 −1.19992 −0.599958 0.800032i \(-0.704816\pi\)
−0.599958 + 0.800032i \(0.704816\pi\)
\(294\) 513.000 0.101765
\(295\) −120.000 −0.0236836
\(296\) −1470.00 −0.288655
\(297\) 0 0
\(298\) 774.000 0.150458
\(299\) 8880.00 1.71754
\(300\) −75.0000 −0.0144338
\(301\) 1840.00 0.352345
\(302\) −2424.00 −0.461873
\(303\) −5202.00 −0.986294
\(304\) −8804.00 −1.66100
\(305\) −1610.00 −0.302257
\(306\) 1458.00 0.272380
\(307\) −9236.00 −1.71702 −0.858512 0.512793i \(-0.828611\pi\)
−0.858512 + 0.512793i \(0.828611\pi\)
\(308\) 0 0
\(309\) −1356.00 −0.249644
\(310\) 3000.00 0.549640
\(311\) 1536.00 0.280060 0.140030 0.990147i \(-0.455280\pi\)
0.140030 + 0.990147i \(0.455280\pi\)
\(312\) 4662.00 0.845942
\(313\) −7342.00 −1.32586 −0.662930 0.748681i \(-0.730687\pi\)
−0.662930 + 0.748681i \(0.730687\pi\)
\(314\) −7134.00 −1.28215
\(315\) 900.000 0.160982
\(316\) 520.000 0.0925705
\(317\) −3894.00 −0.689933 −0.344967 0.938615i \(-0.612110\pi\)
−0.344967 + 0.938615i \(0.612110\pi\)
\(318\) 4050.00 0.714191
\(319\) 0 0
\(320\) −2165.00 −0.378210
\(321\) −4212.00 −0.732370
\(322\) −7200.00 −1.24609
\(323\) −6696.00 −1.15348
\(324\) 81.0000 0.0138889
\(325\) −1850.00 −0.315752
\(326\) 156.000 0.0265032
\(327\) −4422.00 −0.747820
\(328\) −6930.00 −1.16660
\(329\) 480.000 0.0804354
\(330\) 0 0
\(331\) 3692.00 0.613084 0.306542 0.951857i \(-0.400828\pi\)
0.306542 + 0.951857i \(0.400828\pi\)
\(332\) −156.000 −0.0257880
\(333\) −630.000 −0.103675
\(334\) −11160.0 −1.82829
\(335\) 980.000 0.159830
\(336\) −4260.00 −0.691673
\(337\) 8998.00 1.45446 0.727229 0.686395i \(-0.240808\pi\)
0.727229 + 0.686395i \(0.240808\pi\)
\(338\) −9837.00 −1.58302
\(339\) −3258.00 −0.521977
\(340\) 270.000 0.0430671
\(341\) 0 0
\(342\) −3348.00 −0.529354
\(343\) 5720.00 0.900440
\(344\) −1932.00 −0.302809
\(345\) −1800.00 −0.280895
\(346\) 1278.00 0.198571
\(347\) −5244.00 −0.811276 −0.405638 0.914034i \(-0.632951\pi\)
−0.405638 + 0.914034i \(0.632951\pi\)
\(348\) −234.000 −0.0360452
\(349\) −6302.00 −0.966585 −0.483293 0.875459i \(-0.660559\pi\)
−0.483293 + 0.875459i \(0.660559\pi\)
\(350\) 1500.00 0.229081
\(351\) 1998.00 0.303833
\(352\) 0 0
\(353\) 3414.00 0.514756 0.257378 0.966311i \(-0.417141\pi\)
0.257378 + 0.966311i \(0.417141\pi\)
\(354\) 216.000 0.0324301
\(355\) 1440.00 0.215288
\(356\) 1026.00 0.152747
\(357\) −3240.00 −0.480333
\(358\) 4320.00 0.637763
\(359\) −4824.00 −0.709195 −0.354597 0.935019i \(-0.615382\pi\)
−0.354597 + 0.935019i \(0.615382\pi\)
\(360\) −945.000 −0.138350
\(361\) 8517.00 1.24173
\(362\) 9390.00 1.36334
\(363\) 0 0
\(364\) 1480.00 0.213113
\(365\) −2150.00 −0.308318
\(366\) 2898.00 0.413882
\(367\) −3508.00 −0.498954 −0.249477 0.968381i \(-0.580259\pi\)
−0.249477 + 0.968381i \(0.580259\pi\)
\(368\) 8520.00 1.20689
\(369\) −2970.00 −0.419003
\(370\) −1050.00 −0.147532
\(371\) −9000.00 −1.25945
\(372\) −600.000 −0.0836251
\(373\) −10802.0 −1.49948 −0.749740 0.661732i \(-0.769822\pi\)
−0.749740 + 0.661732i \(0.769822\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) −504.000 −0.0691272
\(377\) −5772.00 −0.788523
\(378\) −1620.00 −0.220433
\(379\) 1460.00 0.197876 0.0989382 0.995094i \(-0.468455\pi\)
0.0989382 + 0.995094i \(0.468455\pi\)
\(380\) −620.000 −0.0836982
\(381\) 3732.00 0.501827
\(382\) −10728.0 −1.43689
\(383\) −4872.00 −0.649994 −0.324997 0.945715i \(-0.605363\pi\)
−0.324997 + 0.945715i \(0.605363\pi\)
\(384\) 4977.00 0.661410
\(385\) 0 0
\(386\) 7998.00 1.05463
\(387\) −828.000 −0.108759
\(388\) −286.000 −0.0374213
\(389\) −14046.0 −1.83075 −0.915373 0.402606i \(-0.868104\pi\)
−0.915373 + 0.402606i \(0.868104\pi\)
\(390\) 3330.00 0.432362
\(391\) 6480.00 0.838127
\(392\) 1197.00 0.154229
\(393\) 6984.00 0.896428
\(394\) −8154.00 −1.04262
\(395\) −2600.00 −0.331190
\(396\) 0 0
\(397\) −2734.00 −0.345631 −0.172816 0.984954i \(-0.555286\pi\)
−0.172816 + 0.984954i \(0.555286\pi\)
\(398\) 11496.0 1.44785
\(399\) 7440.00 0.933498
\(400\) −1775.00 −0.221875
\(401\) −15942.0 −1.98530 −0.992650 0.121019i \(-0.961384\pi\)
−0.992650 + 0.121019i \(0.961384\pi\)
\(402\) −1764.00 −0.218857
\(403\) −14800.0 −1.82938
\(404\) 1734.00 0.213539
\(405\) −405.000 −0.0496904
\(406\) 4680.00 0.572080
\(407\) 0 0
\(408\) 3402.00 0.412804
\(409\) −8714.00 −1.05350 −0.526748 0.850022i \(-0.676589\pi\)
−0.526748 + 0.850022i \(0.676589\pi\)
\(410\) −4950.00 −0.596251
\(411\) −6354.00 −0.762578
\(412\) 452.000 0.0540496
\(413\) −480.000 −0.0571895
\(414\) 3240.00 0.384631
\(415\) 780.000 0.0922619
\(416\) −3330.00 −0.392468
\(417\) 6972.00 0.818754
\(418\) 0 0
\(419\) 11976.0 1.39634 0.698169 0.715933i \(-0.253998\pi\)
0.698169 + 0.715933i \(0.253998\pi\)
\(420\) −300.000 −0.0348536
\(421\) 11054.0 1.27967 0.639833 0.768514i \(-0.279004\pi\)
0.639833 + 0.768514i \(0.279004\pi\)
\(422\) 3300.00 0.380667
\(423\) −216.000 −0.0248281
\(424\) 9450.00 1.08239
\(425\) −1350.00 −0.154081
\(426\) −2592.00 −0.294795
\(427\) −6440.00 −0.729868
\(428\) 1404.00 0.158563
\(429\) 0 0
\(430\) −1380.00 −0.154766
\(431\) −720.000 −0.0804668 −0.0402334 0.999190i \(-0.512810\pi\)
−0.0402334 + 0.999190i \(0.512810\pi\)
\(432\) 1917.00 0.213499
\(433\) −15622.0 −1.73382 −0.866912 0.498462i \(-0.833898\pi\)
−0.866912 + 0.498462i \(0.833898\pi\)
\(434\) 12000.0 1.32723
\(435\) 1170.00 0.128959
\(436\) 1474.00 0.161908
\(437\) −14880.0 −1.62885
\(438\) 3870.00 0.422182
\(439\) 9880.00 1.07414 0.537069 0.843538i \(-0.319531\pi\)
0.537069 + 0.843538i \(0.319531\pi\)
\(440\) 0 0
\(441\) 513.000 0.0553936
\(442\) −11988.0 −1.29007
\(443\) −16116.0 −1.72843 −0.864215 0.503123i \(-0.832184\pi\)
−0.864215 + 0.503123i \(0.832184\pi\)
\(444\) 210.000 0.0224463
\(445\) −5130.00 −0.546484
\(446\) −5892.00 −0.625548
\(447\) 774.000 0.0818992
\(448\) −8660.00 −0.913274
\(449\) 9018.00 0.947852 0.473926 0.880565i \(-0.342836\pi\)
0.473926 + 0.880565i \(0.342836\pi\)
\(450\) −675.000 −0.0707107
\(451\) 0 0
\(452\) 1086.00 0.113011
\(453\) −2424.00 −0.251412
\(454\) 1980.00 0.204683
\(455\) −7400.00 −0.762456
\(456\) −7812.00 −0.802260
\(457\) 3670.00 0.375657 0.187829 0.982202i \(-0.439855\pi\)
0.187829 + 0.982202i \(0.439855\pi\)
\(458\) 5718.00 0.583372
\(459\) 1458.00 0.148265
\(460\) 600.000 0.0608155
\(461\) −17562.0 −1.77428 −0.887141 0.461499i \(-0.847312\pi\)
−0.887141 + 0.461499i \(0.847312\pi\)
\(462\) 0 0
\(463\) 1172.00 0.117640 0.0588202 0.998269i \(-0.481266\pi\)
0.0588202 + 0.998269i \(0.481266\pi\)
\(464\) −5538.00 −0.554084
\(465\) 3000.00 0.299186
\(466\) −4374.00 −0.434810
\(467\) 6876.00 0.681335 0.340667 0.940184i \(-0.389347\pi\)
0.340667 + 0.940184i \(0.389347\pi\)
\(468\) −666.000 −0.0657818
\(469\) 3920.00 0.385946
\(470\) −360.000 −0.0353310
\(471\) −7134.00 −0.697914
\(472\) 504.000 0.0491493
\(473\) 0 0
\(474\) 4680.00 0.453501
\(475\) 3100.00 0.299448
\(476\) 1080.00 0.103995
\(477\) 4050.00 0.388756
\(478\) 3528.00 0.337588
\(479\) −2280.00 −0.217486 −0.108743 0.994070i \(-0.534683\pi\)
−0.108743 + 0.994070i \(0.534683\pi\)
\(480\) 675.000 0.0641862
\(481\) 5180.00 0.491035
\(482\) 2598.00 0.245510
\(483\) −7200.00 −0.678284
\(484\) 0 0
\(485\) 1430.00 0.133882
\(486\) 729.000 0.0680414
\(487\) −3076.00 −0.286215 −0.143108 0.989707i \(-0.545710\pi\)
−0.143108 + 0.989707i \(0.545710\pi\)
\(488\) 6762.00 0.627257
\(489\) 156.000 0.0144265
\(490\) 855.000 0.0788265
\(491\) 18912.0 1.73826 0.869131 0.494582i \(-0.164679\pi\)
0.869131 + 0.494582i \(0.164679\pi\)
\(492\) 990.000 0.0907168
\(493\) −4212.00 −0.384785
\(494\) 27528.0 2.50717
\(495\) 0 0
\(496\) −14200.0 −1.28548
\(497\) 5760.00 0.519862
\(498\) −1404.00 −0.126335
\(499\) 9956.00 0.893170 0.446585 0.894741i \(-0.352640\pi\)
0.446585 + 0.894741i \(0.352640\pi\)
\(500\) −125.000 −0.0111803
\(501\) −11160.0 −0.995194
\(502\) −1296.00 −0.115226
\(503\) 10656.0 0.944588 0.472294 0.881441i \(-0.343426\pi\)
0.472294 + 0.881441i \(0.343426\pi\)
\(504\) −3780.00 −0.334077
\(505\) −8670.00 −0.763980
\(506\) 0 0
\(507\) −9837.00 −0.861689
\(508\) −1244.00 −0.108649
\(509\) −2766.00 −0.240866 −0.120433 0.992721i \(-0.538428\pi\)
−0.120433 + 0.992721i \(0.538428\pi\)
\(510\) 2430.00 0.210985
\(511\) −8600.00 −0.744504
\(512\) 8733.00 0.753804
\(513\) −3348.00 −0.288144
\(514\) −7578.00 −0.650294
\(515\) −2260.00 −0.193374
\(516\) 276.000 0.0235469
\(517\) 0 0
\(518\) −4200.00 −0.356250
\(519\) 1278.00 0.108089
\(520\) 7770.00 0.655264
\(521\) 10530.0 0.885466 0.442733 0.896654i \(-0.354009\pi\)
0.442733 + 0.896654i \(0.354009\pi\)
\(522\) −2106.00 −0.176585
\(523\) −12692.0 −1.06115 −0.530576 0.847637i \(-0.678024\pi\)
−0.530576 + 0.847637i \(0.678024\pi\)
\(524\) −2328.00 −0.194082
\(525\) 1500.00 0.124696
\(526\) 16344.0 1.35481
\(527\) −10800.0 −0.892705
\(528\) 0 0
\(529\) 2233.00 0.183529
\(530\) 6750.00 0.553210
\(531\) 216.000 0.0176527
\(532\) −2480.00 −0.202108
\(533\) 24420.0 1.98452
\(534\) 9234.00 0.748304
\(535\) −7020.00 −0.567292
\(536\) −4116.00 −0.331687
\(537\) 4320.00 0.347154
\(538\) 7722.00 0.618809
\(539\) 0 0
\(540\) 135.000 0.0107583
\(541\) −18110.0 −1.43920 −0.719602 0.694386i \(-0.755676\pi\)
−0.719602 + 0.694386i \(0.755676\pi\)
\(542\) −9552.00 −0.756999
\(543\) 9390.00 0.742106
\(544\) −2430.00 −0.191517
\(545\) −7370.00 −0.579259
\(546\) 13320.0 1.04404
\(547\) −3620.00 −0.282962 −0.141481 0.989941i \(-0.545186\pi\)
−0.141481 + 0.989941i \(0.545186\pi\)
\(548\) 2118.00 0.165103
\(549\) 2898.00 0.225289
\(550\) 0 0
\(551\) 9672.00 0.747806
\(552\) 7560.00 0.582926
\(553\) −10400.0 −0.799734
\(554\) 11886.0 0.911530
\(555\) −1050.00 −0.0803063
\(556\) −2324.00 −0.177265
\(557\) 14166.0 1.07762 0.538809 0.842428i \(-0.318875\pi\)
0.538809 + 0.842428i \(0.318875\pi\)
\(558\) −5400.00 −0.409678
\(559\) 6808.00 0.515112
\(560\) −7100.00 −0.535767
\(561\) 0 0
\(562\) −24858.0 −1.86579
\(563\) 13404.0 1.00339 0.501697 0.865043i \(-0.332709\pi\)
0.501697 + 0.865043i \(0.332709\pi\)
\(564\) 72.0000 0.00537544
\(565\) −5430.00 −0.404322
\(566\) −8148.00 −0.605099
\(567\) −1620.00 −0.119989
\(568\) −6048.00 −0.446775
\(569\) 18654.0 1.37437 0.687185 0.726483i \(-0.258846\pi\)
0.687185 + 0.726483i \(0.258846\pi\)
\(570\) −5580.00 −0.410036
\(571\) 7684.00 0.563162 0.281581 0.959537i \(-0.409141\pi\)
0.281581 + 0.959537i \(0.409141\pi\)
\(572\) 0 0
\(573\) −10728.0 −0.782144
\(574\) −19800.0 −1.43978
\(575\) −3000.00 −0.217580
\(576\) 3897.00 0.281901
\(577\) −1726.00 −0.124531 −0.0622654 0.998060i \(-0.519833\pi\)
−0.0622654 + 0.998060i \(0.519833\pi\)
\(578\) 5991.00 0.431129
\(579\) 7998.00 0.574068
\(580\) −390.000 −0.0279205
\(581\) 3120.00 0.222787
\(582\) −2574.00 −0.183326
\(583\) 0 0
\(584\) 9030.00 0.639836
\(585\) 3330.00 0.235348
\(586\) 18054.0 1.27270
\(587\) 10596.0 0.745049 0.372524 0.928022i \(-0.378492\pi\)
0.372524 + 0.928022i \(0.378492\pi\)
\(588\) −171.000 −0.0119931
\(589\) 24800.0 1.73492
\(590\) 360.000 0.0251203
\(591\) −8154.00 −0.567531
\(592\) 4970.00 0.345043
\(593\) −2862.00 −0.198193 −0.0990963 0.995078i \(-0.531595\pi\)
−0.0990963 + 0.995078i \(0.531595\pi\)
\(594\) 0 0
\(595\) −5400.00 −0.372065
\(596\) −258.000 −0.0177317
\(597\) 11496.0 0.788107
\(598\) −26640.0 −1.82172
\(599\) −23592.0 −1.60925 −0.804627 0.593781i \(-0.797635\pi\)
−0.804627 + 0.593781i \(0.797635\pi\)
\(600\) −1575.00 −0.107165
\(601\) 9574.00 0.649803 0.324902 0.945748i \(-0.394669\pi\)
0.324902 + 0.945748i \(0.394669\pi\)
\(602\) −5520.00 −0.373718
\(603\) −1764.00 −0.119130
\(604\) 808.000 0.0544322
\(605\) 0 0
\(606\) 15606.0 1.04612
\(607\) −17444.0 −1.16644 −0.583221 0.812314i \(-0.698208\pi\)
−0.583221 + 0.812314i \(0.698208\pi\)
\(608\) 5580.00 0.372202
\(609\) 4680.00 0.311401
\(610\) 4830.00 0.320592
\(611\) 1776.00 0.117593
\(612\) −486.000 −0.0321003
\(613\) 2374.00 0.156419 0.0782096 0.996937i \(-0.475080\pi\)
0.0782096 + 0.996937i \(0.475080\pi\)
\(614\) 27708.0 1.82118
\(615\) −4950.00 −0.324558
\(616\) 0 0
\(617\) −12162.0 −0.793555 −0.396778 0.917915i \(-0.629872\pi\)
−0.396778 + 0.917915i \(0.629872\pi\)
\(618\) 4068.00 0.264788
\(619\) 8804.00 0.571668 0.285834 0.958279i \(-0.407729\pi\)
0.285834 + 0.958279i \(0.407729\pi\)
\(620\) −1000.00 −0.0647758
\(621\) 3240.00 0.209367
\(622\) −4608.00 −0.297048
\(623\) −20520.0 −1.31961
\(624\) −15762.0 −1.01119
\(625\) 625.000 0.0400000
\(626\) 22026.0 1.40629
\(627\) 0 0
\(628\) 2378.00 0.151103
\(629\) 3780.00 0.239616
\(630\) −2700.00 −0.170747
\(631\) −12688.0 −0.800478 −0.400239 0.916411i \(-0.631073\pi\)
−0.400239 + 0.916411i \(0.631073\pi\)
\(632\) 10920.0 0.687301
\(633\) 3300.00 0.207209
\(634\) 11682.0 0.731785
\(635\) 6220.00 0.388714
\(636\) −1350.00 −0.0841682
\(637\) −4218.00 −0.262360
\(638\) 0 0
\(639\) −2592.00 −0.160466
\(640\) 8295.00 0.512326
\(641\) −9150.00 −0.563812 −0.281906 0.959442i \(-0.590967\pi\)
−0.281906 + 0.959442i \(0.590967\pi\)
\(642\) 12636.0 0.776796
\(643\) 25292.0 1.55120 0.775598 0.631227i \(-0.217448\pi\)
0.775598 + 0.631227i \(0.217448\pi\)
\(644\) 2400.00 0.146853
\(645\) −1380.00 −0.0842441
\(646\) 20088.0 1.22345
\(647\) −2736.00 −0.166249 −0.0831246 0.996539i \(-0.526490\pi\)
−0.0831246 + 0.996539i \(0.526490\pi\)
\(648\) 1701.00 0.103120
\(649\) 0 0
\(650\) 5550.00 0.334906
\(651\) 12000.0 0.722453
\(652\) −52.0000 −0.00312343
\(653\) 22218.0 1.33148 0.665741 0.746183i \(-0.268116\pi\)
0.665741 + 0.746183i \(0.268116\pi\)
\(654\) 13266.0 0.793183
\(655\) 11640.0 0.694370
\(656\) 23430.0 1.39449
\(657\) 3870.00 0.229807
\(658\) −1440.00 −0.0853147
\(659\) −14520.0 −0.858299 −0.429149 0.903234i \(-0.641187\pi\)
−0.429149 + 0.903234i \(0.641187\pi\)
\(660\) 0 0
\(661\) −10618.0 −0.624799 −0.312400 0.949951i \(-0.601133\pi\)
−0.312400 + 0.949951i \(0.601133\pi\)
\(662\) −11076.0 −0.650273
\(663\) −11988.0 −0.702225
\(664\) −3276.00 −0.191466
\(665\) 12400.0 0.723085
\(666\) 1890.00 0.109964
\(667\) −9360.00 −0.543359
\(668\) 3720.00 0.215466
\(669\) −5892.00 −0.340505
\(670\) −2940.00 −0.169526
\(671\) 0 0
\(672\) 2700.00 0.154992
\(673\) −1370.00 −0.0784690 −0.0392345 0.999230i \(-0.512492\pi\)
−0.0392345 + 0.999230i \(0.512492\pi\)
\(674\) −26994.0 −1.54269
\(675\) −675.000 −0.0384900
\(676\) 3279.00 0.186561
\(677\) 13758.0 0.781038 0.390519 0.920595i \(-0.372296\pi\)
0.390519 + 0.920595i \(0.372296\pi\)
\(678\) 9774.00 0.553640
\(679\) 5720.00 0.323289
\(680\) 5670.00 0.319757
\(681\) 1980.00 0.111415
\(682\) 0 0
\(683\) 11988.0 0.671608 0.335804 0.941932i \(-0.390992\pi\)
0.335804 + 0.941932i \(0.390992\pi\)
\(684\) 1116.00 0.0623850
\(685\) −10590.0 −0.590691
\(686\) −17160.0 −0.955061
\(687\) 5718.00 0.317548
\(688\) 6532.00 0.361962
\(689\) −33300.0 −1.84126
\(690\) 5400.00 0.297934
\(691\) 32996.0 1.81654 0.908268 0.418388i \(-0.137405\pi\)
0.908268 + 0.418388i \(0.137405\pi\)
\(692\) −426.000 −0.0234019
\(693\) 0 0
\(694\) 15732.0 0.860488
\(695\) 11620.0 0.634204
\(696\) −4914.00 −0.267622
\(697\) 17820.0 0.968408
\(698\) 18906.0 1.02522
\(699\) −4374.00 −0.236681
\(700\) −500.000 −0.0269975
\(701\) 25902.0 1.39558 0.697792 0.716300i \(-0.254166\pi\)
0.697792 + 0.716300i \(0.254166\pi\)
\(702\) −5994.00 −0.322263
\(703\) −8680.00 −0.465679
\(704\) 0 0
\(705\) −360.000 −0.0192318
\(706\) −10242.0 −0.545981
\(707\) −34680.0 −1.84480
\(708\) −72.0000 −0.00382193
\(709\) −27394.0 −1.45106 −0.725531 0.688189i \(-0.758406\pi\)
−0.725531 + 0.688189i \(0.758406\pi\)
\(710\) −4320.00 −0.228347
\(711\) 4680.00 0.246855
\(712\) 21546.0 1.13409
\(713\) −24000.0 −1.26060
\(714\) 9720.00 0.509470
\(715\) 0 0
\(716\) −1440.00 −0.0751611
\(717\) 3528.00 0.183760
\(718\) 14472.0 0.752215
\(719\) 34848.0 1.80753 0.903763 0.428033i \(-0.140793\pi\)
0.903763 + 0.428033i \(0.140793\pi\)
\(720\) 3195.00 0.165376
\(721\) −9040.00 −0.466945
\(722\) −25551.0 −1.31705
\(723\) 2598.00 0.133639
\(724\) −3130.00 −0.160671
\(725\) 1950.00 0.0998913
\(726\) 0 0
\(727\) 28028.0 1.42985 0.714925 0.699201i \(-0.246461\pi\)
0.714925 + 0.699201i \(0.246461\pi\)
\(728\) 31080.0 1.58228
\(729\) 729.000 0.0370370
\(730\) 6450.00 0.327021
\(731\) 4968.00 0.251365
\(732\) −966.000 −0.0487765
\(733\) −18002.0 −0.907120 −0.453560 0.891226i \(-0.649846\pi\)
−0.453560 + 0.891226i \(0.649846\pi\)
\(734\) 10524.0 0.529221
\(735\) 855.000 0.0429077
\(736\) −5400.00 −0.270444
\(737\) 0 0
\(738\) 8910.00 0.444420
\(739\) −15284.0 −0.760800 −0.380400 0.924822i \(-0.624214\pi\)
−0.380400 + 0.924822i \(0.624214\pi\)
\(740\) 350.000 0.0173868
\(741\) 27528.0 1.36473
\(742\) 27000.0 1.33585
\(743\) 18768.0 0.926691 0.463345 0.886178i \(-0.346649\pi\)
0.463345 + 0.886178i \(0.346649\pi\)
\(744\) −12600.0 −0.620885
\(745\) 1290.00 0.0634388
\(746\) 32406.0 1.59044
\(747\) −1404.00 −0.0687680
\(748\) 0 0
\(749\) −28080.0 −1.36985
\(750\) −1125.00 −0.0547723
\(751\) 8696.00 0.422532 0.211266 0.977429i \(-0.432241\pi\)
0.211266 + 0.977429i \(0.432241\pi\)
\(752\) 1704.00 0.0826310
\(753\) −1296.00 −0.0627209
\(754\) 17316.0 0.836355
\(755\) −4040.00 −0.194743
\(756\) 540.000 0.0259783
\(757\) −38662.0 −1.85627 −0.928134 0.372247i \(-0.878587\pi\)
−0.928134 + 0.372247i \(0.878587\pi\)
\(758\) −4380.00 −0.209880
\(759\) 0 0
\(760\) −13020.0 −0.621428
\(761\) −23874.0 −1.13723 −0.568615 0.822604i \(-0.692521\pi\)
−0.568615 + 0.822604i \(0.692521\pi\)
\(762\) −11196.0 −0.532268
\(763\) −29480.0 −1.39875
\(764\) 3576.00 0.169339
\(765\) 2430.00 0.114846
\(766\) 14616.0 0.689422
\(767\) −1776.00 −0.0836084
\(768\) −4539.00 −0.213264
\(769\) −23618.0 −1.10753 −0.553763 0.832675i \(-0.686808\pi\)
−0.553763 + 0.832675i \(0.686808\pi\)
\(770\) 0 0
\(771\) −7578.00 −0.353975
\(772\) −2666.00 −0.124289
\(773\) 11538.0 0.536860 0.268430 0.963299i \(-0.413495\pi\)
0.268430 + 0.963299i \(0.413495\pi\)
\(774\) 2484.00 0.115356
\(775\) 5000.00 0.231749
\(776\) −6006.00 −0.277839
\(777\) −4200.00 −0.193918
\(778\) 42138.0 1.94180
\(779\) −40920.0 −1.88204
\(780\) −1110.00 −0.0509543
\(781\) 0 0
\(782\) −19440.0 −0.888968
\(783\) −2106.00 −0.0961204
\(784\) −4047.00 −0.184357
\(785\) −11890.0 −0.540602
\(786\) −20952.0 −0.950805
\(787\) 14884.0 0.674152 0.337076 0.941478i \(-0.390562\pi\)
0.337076 + 0.941478i \(0.390562\pi\)
\(788\) 2718.00 0.122874
\(789\) 16344.0 0.737467
\(790\) 7800.00 0.351280
\(791\) −21720.0 −0.976327
\(792\) 0 0
\(793\) −23828.0 −1.06703
\(794\) 8202.00 0.366597
\(795\) 6750.00 0.301129
\(796\) −3832.00 −0.170630
\(797\) −11334.0 −0.503728 −0.251864 0.967763i \(-0.581043\pi\)
−0.251864 + 0.967763i \(0.581043\pi\)
\(798\) −22320.0 −0.990125
\(799\) 1296.00 0.0573832
\(800\) 1125.00 0.0497184
\(801\) 9234.00 0.407325
\(802\) 47826.0 2.10573
\(803\) 0 0
\(804\) 588.000 0.0257925
\(805\) −12000.0 −0.525397
\(806\) 44400.0 1.94035
\(807\) 7722.00 0.336837
\(808\) 36414.0 1.58545
\(809\) −44730.0 −1.94391 −0.971955 0.235167i \(-0.924436\pi\)
−0.971955 + 0.235167i \(0.924436\pi\)
\(810\) 1215.00 0.0527046
\(811\) 42748.0 1.85091 0.925453 0.378862i \(-0.123684\pi\)
0.925453 + 0.378862i \(0.123684\pi\)
\(812\) −1560.00 −0.0674203
\(813\) −9552.00 −0.412058
\(814\) 0 0
\(815\) 260.000 0.0111747
\(816\) −11502.0 −0.493444
\(817\) −11408.0 −0.488513
\(818\) 26142.0 1.11740
\(819\) 13320.0 0.568301
\(820\) 1650.00 0.0702689
\(821\) 31686.0 1.34695 0.673477 0.739208i \(-0.264800\pi\)
0.673477 + 0.739208i \(0.264800\pi\)
\(822\) 19062.0 0.808836
\(823\) 11036.0 0.467425 0.233713 0.972306i \(-0.424913\pi\)
0.233713 + 0.972306i \(0.424913\pi\)
\(824\) 9492.00 0.401298
\(825\) 0 0
\(826\) 1440.00 0.0606586
\(827\) −25884.0 −1.08836 −0.544181 0.838968i \(-0.683159\pi\)
−0.544181 + 0.838968i \(0.683159\pi\)
\(828\) −1080.00 −0.0453292
\(829\) 15950.0 0.668234 0.334117 0.942532i \(-0.391562\pi\)
0.334117 + 0.942532i \(0.391562\pi\)
\(830\) −2340.00 −0.0978585
\(831\) 11886.0 0.496174
\(832\) −32042.0 −1.33516
\(833\) −3078.00 −0.128027
\(834\) −20916.0 −0.868419
\(835\) −18600.0 −0.770874
\(836\) 0 0
\(837\) −5400.00 −0.223000
\(838\) −35928.0 −1.48104
\(839\) 13800.0 0.567853 0.283927 0.958846i \(-0.408363\pi\)
0.283927 + 0.958846i \(0.408363\pi\)
\(840\) −6300.00 −0.258775
\(841\) −18305.0 −0.750543
\(842\) −33162.0 −1.35729
\(843\) −24858.0 −1.01560
\(844\) −1100.00 −0.0448620
\(845\) −16395.0 −0.667462
\(846\) 648.000 0.0263342
\(847\) 0 0
\(848\) −31950.0 −1.29383
\(849\) −8148.00 −0.329374
\(850\) 4050.00 0.163428
\(851\) 8400.00 0.338365
\(852\) 864.000 0.0347420
\(853\) 27862.0 1.11838 0.559189 0.829040i \(-0.311113\pi\)
0.559189 + 0.829040i \(0.311113\pi\)
\(854\) 19320.0 0.774141
\(855\) −5580.00 −0.223195
\(856\) 29484.0 1.17727
\(857\) 7314.00 0.291530 0.145765 0.989319i \(-0.453436\pi\)
0.145765 + 0.989319i \(0.453436\pi\)
\(858\) 0 0
\(859\) −28780.0 −1.14314 −0.571572 0.820552i \(-0.693666\pi\)
−0.571572 + 0.820552i \(0.693666\pi\)
\(860\) 460.000 0.0182394
\(861\) −19800.0 −0.783719
\(862\) 2160.00 0.0853479
\(863\) −32688.0 −1.28935 −0.644677 0.764455i \(-0.723008\pi\)
−0.644677 + 0.764455i \(0.723008\pi\)
\(864\) −1215.00 −0.0478416
\(865\) 2130.00 0.0837251
\(866\) 46866.0 1.83900
\(867\) 5991.00 0.234677
\(868\) −4000.00 −0.156416
\(869\) 0 0
\(870\) −3510.00 −0.136782
\(871\) 14504.0 0.564236
\(872\) 30954.0 1.20210
\(873\) −2574.00 −0.0997900
\(874\) 44640.0 1.72766
\(875\) 2500.00 0.0965891
\(876\) −1290.00 −0.0497546
\(877\) −36650.0 −1.41115 −0.705577 0.708633i \(-0.749312\pi\)
−0.705577 + 0.708633i \(0.749312\pi\)
\(878\) −29640.0 −1.13930
\(879\) 18054.0 0.692772
\(880\) 0 0
\(881\) −2646.00 −0.101187 −0.0505936 0.998719i \(-0.516111\pi\)
−0.0505936 + 0.998719i \(0.516111\pi\)
\(882\) −1539.00 −0.0587538
\(883\) 10892.0 0.415113 0.207557 0.978223i \(-0.433449\pi\)
0.207557 + 0.978223i \(0.433449\pi\)
\(884\) 3996.00 0.152036
\(885\) 360.000 0.0136737
\(886\) 48348.0 1.83328
\(887\) 43464.0 1.64530 0.822648 0.568550i \(-0.192496\pi\)
0.822648 + 0.568550i \(0.192496\pi\)
\(888\) 4410.00 0.166655
\(889\) 24880.0 0.938637
\(890\) 15390.0 0.579634
\(891\) 0 0
\(892\) 1964.00 0.0737215
\(893\) −2976.00 −0.111521
\(894\) −2322.00 −0.0868672
\(895\) 7200.00 0.268904
\(896\) 33180.0 1.23713
\(897\) −26640.0 −0.991621
\(898\) −27054.0 −1.00535
\(899\) 15600.0 0.578742
\(900\) 225.000 0.00833333
\(901\) −24300.0 −0.898502
\(902\) 0 0
\(903\) −5520.00 −0.203426
\(904\) 22806.0 0.839067
\(905\) 15650.0 0.574833
\(906\) 7272.00 0.266662
\(907\) −14884.0 −0.544890 −0.272445 0.962171i \(-0.587832\pi\)
−0.272445 + 0.962171i \(0.587832\pi\)
\(908\) −660.000 −0.0241221
\(909\) 15606.0 0.569437
\(910\) 22200.0 0.808706
\(911\) −1248.00 −0.0453876 −0.0226938 0.999742i \(-0.507224\pi\)
−0.0226938 + 0.999742i \(0.507224\pi\)
\(912\) 26412.0 0.958979
\(913\) 0 0
\(914\) −11010.0 −0.398445
\(915\) 4830.00 0.174508
\(916\) −1906.00 −0.0687511
\(917\) 46560.0 1.67671
\(918\) −4374.00 −0.157259
\(919\) 6640.00 0.238339 0.119169 0.992874i \(-0.461977\pi\)
0.119169 + 0.992874i \(0.461977\pi\)
\(920\) 12600.0 0.451532
\(921\) 27708.0 0.991324
\(922\) 52686.0 1.88191
\(923\) 21312.0 0.760014
\(924\) 0 0
\(925\) −1750.00 −0.0622050
\(926\) −3516.00 −0.124776
\(927\) 4068.00 0.144132
\(928\) 3510.00 0.124161
\(929\) 29946.0 1.05758 0.528792 0.848751i \(-0.322645\pi\)
0.528792 + 0.848751i \(0.322645\pi\)
\(930\) −9000.00 −0.317335
\(931\) 7068.00 0.248812
\(932\) 1458.00 0.0512429
\(933\) −4608.00 −0.161693
\(934\) −20628.0 −0.722665
\(935\) 0 0
\(936\) −13986.0 −0.488405
\(937\) −45002.0 −1.56900 −0.784499 0.620130i \(-0.787080\pi\)
−0.784499 + 0.620130i \(0.787080\pi\)
\(938\) −11760.0 −0.409358
\(939\) 22026.0 0.765486
\(940\) 120.000 0.00416380
\(941\) −6090.00 −0.210976 −0.105488 0.994421i \(-0.533640\pi\)
−0.105488 + 0.994421i \(0.533640\pi\)
\(942\) 21402.0 0.740249
\(943\) 39600.0 1.36750
\(944\) −1704.00 −0.0587505
\(945\) −2700.00 −0.0929429
\(946\) 0 0
\(947\) 56388.0 1.93491 0.967457 0.253035i \(-0.0814288\pi\)
0.967457 + 0.253035i \(0.0814288\pi\)
\(948\) −1560.00 −0.0534456
\(949\) −31820.0 −1.08843
\(950\) −9300.00 −0.317612
\(951\) 11682.0 0.398333
\(952\) 22680.0 0.772125
\(953\) −10854.0 −0.368936 −0.184468 0.982839i \(-0.559056\pi\)
−0.184468 + 0.982839i \(0.559056\pi\)
\(954\) −12150.0 −0.412338
\(955\) −17880.0 −0.605846
\(956\) −1176.00 −0.0397851
\(957\) 0 0
\(958\) 6840.00 0.230679
\(959\) −42360.0 −1.42636
\(960\) 6495.00 0.218360
\(961\) 10209.0 0.342687
\(962\) −15540.0 −0.520821
\(963\) 12636.0 0.422834
\(964\) −866.000 −0.0289336
\(965\) 13330.0 0.444671
\(966\) 21600.0 0.719429
\(967\) 42316.0 1.40723 0.703615 0.710582i \(-0.251568\pi\)
0.703615 + 0.710582i \(0.251568\pi\)
\(968\) 0 0
\(969\) 20088.0 0.665964
\(970\) −4290.00 −0.142004
\(971\) 24480.0 0.809063 0.404532 0.914524i \(-0.367435\pi\)
0.404532 + 0.914524i \(0.367435\pi\)
\(972\) −243.000 −0.00801875
\(973\) 46480.0 1.53143
\(974\) 9228.00 0.303577
\(975\) 5550.00 0.182300
\(976\) −22862.0 −0.749790
\(977\) −6906.00 −0.226144 −0.113072 0.993587i \(-0.536069\pi\)
−0.113072 + 0.993587i \(0.536069\pi\)
\(978\) −468.000 −0.0153016
\(979\) 0 0
\(980\) −285.000 −0.00928979
\(981\) 13266.0 0.431754
\(982\) −56736.0 −1.84371
\(983\) 6960.00 0.225829 0.112914 0.993605i \(-0.463981\pi\)
0.112914 + 0.993605i \(0.463981\pi\)
\(984\) 20790.0 0.673538
\(985\) −13590.0 −0.439608
\(986\) 12636.0 0.408126
\(987\) −1440.00 −0.0464394
\(988\) −9176.00 −0.295473
\(989\) 11040.0 0.354956
\(990\) 0 0
\(991\) 47792.0 1.53195 0.765975 0.642870i \(-0.222256\pi\)
0.765975 + 0.642870i \(0.222256\pi\)
\(992\) 9000.00 0.288055
\(993\) −11076.0 −0.353964
\(994\) −17280.0 −0.551397
\(995\) 19160.0 0.610465
\(996\) 468.000 0.0148887
\(997\) −9938.00 −0.315687 −0.157843 0.987464i \(-0.550454\pi\)
−0.157843 + 0.987464i \(0.550454\pi\)
\(998\) −29868.0 −0.947350
\(999\) 1890.00 0.0598568
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 1815.4.a.a.1.1 1
11.10 odd 2 15.4.a.b.1.1 1
33.32 even 2 45.4.a.b.1.1 1
44.43 even 2 240.4.a.f.1.1 1
55.32 even 4 75.4.b.a.49.2 2
55.43 even 4 75.4.b.a.49.1 2
55.54 odd 2 75.4.a.a.1.1 1
77.76 even 2 735.4.a.i.1.1 1
88.21 odd 2 960.4.a.bi.1.1 1
88.43 even 2 960.4.a.l.1.1 1
99.32 even 6 405.4.e.k.136.1 2
99.43 odd 6 405.4.e.d.271.1 2
99.65 even 6 405.4.e.k.271.1 2
99.76 odd 6 405.4.e.d.136.1 2
132.131 odd 2 720.4.a.r.1.1 1
165.32 odd 4 225.4.b.d.199.1 2
165.98 odd 4 225.4.b.d.199.2 2
165.164 even 2 225.4.a.g.1.1 1
220.43 odd 4 1200.4.f.m.49.2 2
220.87 odd 4 1200.4.f.m.49.1 2
220.219 even 2 1200.4.a.o.1.1 1
231.230 odd 2 2205.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 11.10 odd 2
45.4.a.b.1.1 1 33.32 even 2
75.4.a.a.1.1 1 55.54 odd 2
75.4.b.a.49.1 2 55.43 even 4
75.4.b.a.49.2 2 55.32 even 4
225.4.a.g.1.1 1 165.164 even 2
225.4.b.d.199.1 2 165.32 odd 4
225.4.b.d.199.2 2 165.98 odd 4
240.4.a.f.1.1 1 44.43 even 2
405.4.e.d.136.1 2 99.76 odd 6
405.4.e.d.271.1 2 99.43 odd 6
405.4.e.k.136.1 2 99.32 even 6
405.4.e.k.271.1 2 99.65 even 6
720.4.a.r.1.1 1 132.131 odd 2
735.4.a.i.1.1 1 77.76 even 2
960.4.a.l.1.1 1 88.43 even 2
960.4.a.bi.1.1 1 88.21 odd 2
1200.4.a.o.1.1 1 220.219 even 2
1200.4.f.m.49.1 2 220.87 odd 4
1200.4.f.m.49.2 2 220.43 odd 4
1815.4.a.a.1.1 1 1.1 even 1 trivial
2205.4.a.c.1.1 1 231.230 odd 2