Properties

Label 285.4.a.a.1.1
Level $285$
Weight $4$
Character 285.1
Self dual yes
Analytic conductor $16.816$
Analytic rank $0$
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] = [285,4,Mod(1,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, 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("285.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 285.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: \(16.8155443516\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 285.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +5.00000 q^{5} +9.00000 q^{6} +32.0000 q^{7} +21.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +5.00000 q^{5} +9.00000 q^{6} +32.0000 q^{7} +21.0000 q^{8} +9.00000 q^{9} -15.0000 q^{10} -12.0000 q^{11} -3.00000 q^{12} -10.0000 q^{13} -96.0000 q^{14} -15.0000 q^{15} -71.0000 q^{16} -30.0000 q^{17} -27.0000 q^{18} +19.0000 q^{19} +5.00000 q^{20} -96.0000 q^{21} +36.0000 q^{22} -48.0000 q^{23} -63.0000 q^{24} +25.0000 q^{25} +30.0000 q^{26} -27.0000 q^{27} +32.0000 q^{28} +150.000 q^{29} +45.0000 q^{30} +224.000 q^{31} +45.0000 q^{32} +36.0000 q^{33} +90.0000 q^{34} +160.000 q^{35} +9.00000 q^{36} +254.000 q^{37} -57.0000 q^{38} +30.0000 q^{39} +105.000 q^{40} -54.0000 q^{41} +288.000 q^{42} -196.000 q^{43} -12.0000 q^{44} +45.0000 q^{45} +144.000 q^{46} -504.000 q^{47} +213.000 q^{48} +681.000 q^{49} -75.0000 q^{50} +90.0000 q^{51} -10.0000 q^{52} +78.0000 q^{53} +81.0000 q^{54} -60.0000 q^{55} +672.000 q^{56} -57.0000 q^{57} -450.000 q^{58} +132.000 q^{59} -15.0000 q^{60} +230.000 q^{61} -672.000 q^{62} +288.000 q^{63} +433.000 q^{64} -50.0000 q^{65} -108.000 q^{66} +740.000 q^{67} -30.0000 q^{68} +144.000 q^{69} -480.000 q^{70} -120.000 q^{71} +189.000 q^{72} +122.000 q^{73} -762.000 q^{74} -75.0000 q^{75} +19.0000 q^{76} -384.000 q^{77} -90.0000 q^{78} +1184.00 q^{79} -355.000 q^{80} +81.0000 q^{81} +162.000 q^{82} +612.000 q^{83} -96.0000 q^{84} -150.000 q^{85} +588.000 q^{86} -450.000 q^{87} -252.000 q^{88} +1050.00 q^{89} -135.000 q^{90} -320.000 q^{91} -48.0000 q^{92} -672.000 q^{93} +1512.00 q^{94} +95.0000 q^{95} -135.000 q^{96} -1006.00 q^{97} -2043.00 q^{98} -108.000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.00000 −1.06066 −0.530330 0.847791i \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) −3.00000 −0.577350
\(4\) 1.00000 0.125000
\(5\) 5.00000 0.447214
\(6\) 9.00000 0.612372
\(7\) 32.0000 1.72784 0.863919 0.503631i \(-0.168003\pi\)
0.863919 + 0.503631i \(0.168003\pi\)
\(8\) 21.0000 0.928078
\(9\) 9.00000 0.333333
\(10\) −15.0000 −0.474342
\(11\) −12.0000 −0.328921 −0.164461 0.986384i \(-0.552588\pi\)
−0.164461 + 0.986384i \(0.552588\pi\)
\(12\) −3.00000 −0.0721688
\(13\) −10.0000 −0.213346 −0.106673 0.994294i \(-0.534020\pi\)
−0.106673 + 0.994294i \(0.534020\pi\)
\(14\) −96.0000 −1.83265
\(15\) −15.0000 −0.258199
\(16\) −71.0000 −1.10938
\(17\) −30.0000 −0.428004 −0.214002 0.976833i \(-0.568650\pi\)
−0.214002 + 0.976833i \(0.568650\pi\)
\(18\) −27.0000 −0.353553
\(19\) 19.0000 0.229416
\(20\) 5.00000 0.0559017
\(21\) −96.0000 −0.997567
\(22\) 36.0000 0.348874
\(23\) −48.0000 −0.435161 −0.217580 0.976042i \(-0.569816\pi\)
−0.217580 + 0.976042i \(0.569816\pi\)
\(24\) −63.0000 −0.535826
\(25\) 25.0000 0.200000
\(26\) 30.0000 0.226288
\(27\) −27.0000 −0.192450
\(28\) 32.0000 0.215980
\(29\) 150.000 0.960493 0.480247 0.877134i \(-0.340547\pi\)
0.480247 + 0.877134i \(0.340547\pi\)
\(30\) 45.0000 0.273861
\(31\) 224.000 1.29779 0.648897 0.760877i \(-0.275231\pi\)
0.648897 + 0.760877i \(0.275231\pi\)
\(32\) 45.0000 0.248592
\(33\) 36.0000 0.189903
\(34\) 90.0000 0.453967
\(35\) 160.000 0.772712
\(36\) 9.00000 0.0416667
\(37\) 254.000 1.12858 0.564288 0.825578i \(-0.309151\pi\)
0.564288 + 0.825578i \(0.309151\pi\)
\(38\) −57.0000 −0.243332
\(39\) 30.0000 0.123176
\(40\) 105.000 0.415049
\(41\) −54.0000 −0.205692 −0.102846 0.994697i \(-0.532795\pi\)
−0.102846 + 0.994697i \(0.532795\pi\)
\(42\) 288.000 1.05808
\(43\) −196.000 −0.695110 −0.347555 0.937660i \(-0.612988\pi\)
−0.347555 + 0.937660i \(0.612988\pi\)
\(44\) −12.0000 −0.0411152
\(45\) 45.0000 0.149071
\(46\) 144.000 0.461557
\(47\) −504.000 −1.56417 −0.782085 0.623172i \(-0.785844\pi\)
−0.782085 + 0.623172i \(0.785844\pi\)
\(48\) 213.000 0.640498
\(49\) 681.000 1.98542
\(50\) −75.0000 −0.212132
\(51\) 90.0000 0.247108
\(52\) −10.0000 −0.0266683
\(53\) 78.0000 0.202153 0.101077 0.994879i \(-0.467771\pi\)
0.101077 + 0.994879i \(0.467771\pi\)
\(54\) 81.0000 0.204124
\(55\) −60.0000 −0.147098
\(56\) 672.000 1.60357
\(57\) −57.0000 −0.132453
\(58\) −450.000 −1.01876
\(59\) 132.000 0.291270 0.145635 0.989338i \(-0.453477\pi\)
0.145635 + 0.989338i \(0.453477\pi\)
\(60\) −15.0000 −0.0322749
\(61\) 230.000 0.482762 0.241381 0.970430i \(-0.422400\pi\)
0.241381 + 0.970430i \(0.422400\pi\)
\(62\) −672.000 −1.37652
\(63\) 288.000 0.575946
\(64\) 433.000 0.845703
\(65\) −50.0000 −0.0954113
\(66\) −108.000 −0.201422
\(67\) 740.000 1.34933 0.674667 0.738122i \(-0.264287\pi\)
0.674667 + 0.738122i \(0.264287\pi\)
\(68\) −30.0000 −0.0535005
\(69\) 144.000 0.251240
\(70\) −480.000 −0.819585
\(71\) −120.000 −0.200583 −0.100291 0.994958i \(-0.531978\pi\)
−0.100291 + 0.994958i \(0.531978\pi\)
\(72\) 189.000 0.309359
\(73\) 122.000 0.195603 0.0978015 0.995206i \(-0.468819\pi\)
0.0978015 + 0.995206i \(0.468819\pi\)
\(74\) −762.000 −1.19704
\(75\) −75.0000 −0.115470
\(76\) 19.0000 0.0286770
\(77\) −384.000 −0.568323
\(78\) −90.0000 −0.130647
\(79\) 1184.00 1.68621 0.843104 0.537751i \(-0.180726\pi\)
0.843104 + 0.537751i \(0.180726\pi\)
\(80\) −355.000 −0.496128
\(81\) 81.0000 0.111111
\(82\) 162.000 0.218170
\(83\) 612.000 0.809346 0.404673 0.914461i \(-0.367385\pi\)
0.404673 + 0.914461i \(0.367385\pi\)
\(84\) −96.0000 −0.124696
\(85\) −150.000 −0.191409
\(86\) 588.000 0.737275
\(87\) −450.000 −0.554541
\(88\) −252.000 −0.305265
\(89\) 1050.00 1.25056 0.625280 0.780401i \(-0.284985\pi\)
0.625280 + 0.780401i \(0.284985\pi\)
\(90\) −135.000 −0.158114
\(91\) −320.000 −0.368628
\(92\) −48.0000 −0.0543951
\(93\) −672.000 −0.749281
\(94\) 1512.00 1.65905
\(95\) 95.0000 0.102598
\(96\) −135.000 −0.143525
\(97\) −1006.00 −1.05303 −0.526515 0.850166i \(-0.676501\pi\)
−0.526515 + 0.850166i \(0.676501\pi\)
\(98\) −2043.00 −2.10586
\(99\) −108.000 −0.109640
\(100\) 25.0000 0.0250000
\(101\) −642.000 −0.632489 −0.316244 0.948678i \(-0.602422\pi\)
−0.316244 + 0.948678i \(0.602422\pi\)
\(102\) −270.000 −0.262098
\(103\) 1472.00 1.40816 0.704080 0.710121i \(-0.251360\pi\)
0.704080 + 0.710121i \(0.251360\pi\)
\(104\) −210.000 −0.198002
\(105\) −480.000 −0.446126
\(106\) −234.000 −0.214416
\(107\) −2052.00 −1.85397 −0.926983 0.375104i \(-0.877607\pi\)
−0.926983 + 0.375104i \(0.877607\pi\)
\(108\) −27.0000 −0.0240563
\(109\) 1334.00 1.17224 0.586119 0.810225i \(-0.300655\pi\)
0.586119 + 0.810225i \(0.300655\pi\)
\(110\) 180.000 0.156021
\(111\) −762.000 −0.651584
\(112\) −2272.00 −1.91682
\(113\) −414.000 −0.344653 −0.172327 0.985040i \(-0.555128\pi\)
−0.172327 + 0.985040i \(0.555128\pi\)
\(114\) 171.000 0.140488
\(115\) −240.000 −0.194610
\(116\) 150.000 0.120062
\(117\) −90.0000 −0.0711154
\(118\) −396.000 −0.308939
\(119\) −960.000 −0.739521
\(120\) −315.000 −0.239629
\(121\) −1187.00 −0.891811
\(122\) −690.000 −0.512046
\(123\) 162.000 0.118756
\(124\) 224.000 0.162224
\(125\) 125.000 0.0894427
\(126\) −864.000 −0.610883
\(127\) 1496.00 1.04526 0.522632 0.852558i \(-0.324950\pi\)
0.522632 + 0.852558i \(0.324950\pi\)
\(128\) −1659.00 −1.14560
\(129\) 588.000 0.401322
\(130\) 150.000 0.101199
\(131\) 1404.00 0.936397 0.468199 0.883623i \(-0.344903\pi\)
0.468199 + 0.883623i \(0.344903\pi\)
\(132\) 36.0000 0.0237379
\(133\) 608.000 0.396393
\(134\) −2220.00 −1.43119
\(135\) −135.000 −0.0860663
\(136\) −630.000 −0.397221
\(137\) 138.000 0.0860594 0.0430297 0.999074i \(-0.486299\pi\)
0.0430297 + 0.999074i \(0.486299\pi\)
\(138\) −432.000 −0.266480
\(139\) −124.000 −0.0756658 −0.0378329 0.999284i \(-0.512045\pi\)
−0.0378329 + 0.999284i \(0.512045\pi\)
\(140\) 160.000 0.0965891
\(141\) 1512.00 0.903074
\(142\) 360.000 0.212750
\(143\) 120.000 0.0701742
\(144\) −639.000 −0.369792
\(145\) 750.000 0.429546
\(146\) −366.000 −0.207468
\(147\) −2043.00 −1.14628
\(148\) 254.000 0.141072
\(149\) 2670.00 1.46802 0.734010 0.679139i \(-0.237647\pi\)
0.734010 + 0.679139i \(0.237647\pi\)
\(150\) 225.000 0.122474
\(151\) −376.000 −0.202639 −0.101319 0.994854i \(-0.532306\pi\)
−0.101319 + 0.994854i \(0.532306\pi\)
\(152\) 399.000 0.212916
\(153\) −270.000 −0.142668
\(154\) 1152.00 0.602797
\(155\) 1120.00 0.580391
\(156\) 30.0000 0.0153969
\(157\) −682.000 −0.346685 −0.173342 0.984862i \(-0.555457\pi\)
−0.173342 + 0.984862i \(0.555457\pi\)
\(158\) −3552.00 −1.78849
\(159\) −234.000 −0.116713
\(160\) 225.000 0.111174
\(161\) −1536.00 −0.751887
\(162\) −243.000 −0.117851
\(163\) 3524.00 1.69338 0.846690 0.532086i \(-0.178592\pi\)
0.846690 + 0.532086i \(0.178592\pi\)
\(164\) −54.0000 −0.0257115
\(165\) 180.000 0.0849272
\(166\) −1836.00 −0.858441
\(167\) −912.000 −0.422591 −0.211295 0.977422i \(-0.567768\pi\)
−0.211295 + 0.977422i \(0.567768\pi\)
\(168\) −2016.00 −0.925820
\(169\) −2097.00 −0.954483
\(170\) 450.000 0.203020
\(171\) 171.000 0.0764719
\(172\) −196.000 −0.0868887
\(173\) 3990.00 1.75349 0.876746 0.480954i \(-0.159710\pi\)
0.876746 + 0.480954i \(0.159710\pi\)
\(174\) 1350.00 0.588180
\(175\) 800.000 0.345568
\(176\) 852.000 0.364897
\(177\) −396.000 −0.168165
\(178\) −3150.00 −1.32642
\(179\) 2316.00 0.967072 0.483536 0.875324i \(-0.339352\pi\)
0.483536 + 0.875324i \(0.339352\pi\)
\(180\) 45.0000 0.0186339
\(181\) −3346.00 −1.37407 −0.687034 0.726625i \(-0.741088\pi\)
−0.687034 + 0.726625i \(0.741088\pi\)
\(182\) 960.000 0.390989
\(183\) −690.000 −0.278723
\(184\) −1008.00 −0.403863
\(185\) 1270.00 0.504715
\(186\) 2016.00 0.794733
\(187\) 360.000 0.140780
\(188\) −504.000 −0.195521
\(189\) −864.000 −0.332522
\(190\) −285.000 −0.108821
\(191\) 960.000 0.363681 0.181841 0.983328i \(-0.441794\pi\)
0.181841 + 0.983328i \(0.441794\pi\)
\(192\) −1299.00 −0.488267
\(193\) −3214.00 −1.19870 −0.599349 0.800488i \(-0.704574\pi\)
−0.599349 + 0.800488i \(0.704574\pi\)
\(194\) 3018.00 1.11691
\(195\) 150.000 0.0550858
\(196\) 681.000 0.248178
\(197\) 990.000 0.358044 0.179022 0.983845i \(-0.442707\pi\)
0.179022 + 0.983845i \(0.442707\pi\)
\(198\) 324.000 0.116291
\(199\) 3560.00 1.26815 0.634075 0.773272i \(-0.281381\pi\)
0.634075 + 0.773272i \(0.281381\pi\)
\(200\) 525.000 0.185616
\(201\) −2220.00 −0.779038
\(202\) 1926.00 0.670856
\(203\) 4800.00 1.65958
\(204\) 90.0000 0.0308885
\(205\) −270.000 −0.0919884
\(206\) −4416.00 −1.49358
\(207\) −432.000 −0.145054
\(208\) 710.000 0.236681
\(209\) −228.000 −0.0754598
\(210\) 1440.00 0.473188
\(211\) −2740.00 −0.893978 −0.446989 0.894539i \(-0.647504\pi\)
−0.446989 + 0.894539i \(0.647504\pi\)
\(212\) 78.0000 0.0252692
\(213\) 360.000 0.115807
\(214\) 6156.00 1.96643
\(215\) −980.000 −0.310863
\(216\) −567.000 −0.178609
\(217\) 7168.00 2.24238
\(218\) −4002.00 −1.24335
\(219\) −366.000 −0.112931
\(220\) −60.0000 −0.0183873
\(221\) 300.000 0.0913130
\(222\) 2286.00 0.691109
\(223\) −3448.00 −1.03540 −0.517702 0.855561i \(-0.673212\pi\)
−0.517702 + 0.855561i \(0.673212\pi\)
\(224\) 1440.00 0.429527
\(225\) 225.000 0.0666667
\(226\) 1242.00 0.365560
\(227\) 5124.00 1.49820 0.749101 0.662456i \(-0.230486\pi\)
0.749101 + 0.662456i \(0.230486\pi\)
\(228\) −57.0000 −0.0165567
\(229\) −3010.00 −0.868587 −0.434293 0.900771i \(-0.643002\pi\)
−0.434293 + 0.900771i \(0.643002\pi\)
\(230\) 720.000 0.206415
\(231\) 1152.00 0.328121
\(232\) 3150.00 0.891412
\(233\) −3510.00 −0.986900 −0.493450 0.869774i \(-0.664264\pi\)
−0.493450 + 0.869774i \(0.664264\pi\)
\(234\) 270.000 0.0754293
\(235\) −2520.00 −0.699518
\(236\) 132.000 0.0364088
\(237\) −3552.00 −0.973532
\(238\) 2880.00 0.784381
\(239\) 5280.00 1.42902 0.714508 0.699627i \(-0.246651\pi\)
0.714508 + 0.699627i \(0.246651\pi\)
\(240\) 1065.00 0.286439
\(241\) 1106.00 0.295617 0.147809 0.989016i \(-0.452778\pi\)
0.147809 + 0.989016i \(0.452778\pi\)
\(242\) 3561.00 0.945908
\(243\) −243.000 −0.0641500
\(244\) 230.000 0.0603453
\(245\) 3405.00 0.887908
\(246\) −486.000 −0.125960
\(247\) −190.000 −0.0489450
\(248\) 4704.00 1.20445
\(249\) −1836.00 −0.467276
\(250\) −375.000 −0.0948683
\(251\) −2844.00 −0.715186 −0.357593 0.933878i \(-0.616402\pi\)
−0.357593 + 0.933878i \(0.616402\pi\)
\(252\) 288.000 0.0719932
\(253\) 576.000 0.143134
\(254\) −4488.00 −1.10867
\(255\) 450.000 0.110510
\(256\) 1513.00 0.369385
\(257\) −4014.00 −0.974266 −0.487133 0.873328i \(-0.661957\pi\)
−0.487133 + 0.873328i \(0.661957\pi\)
\(258\) −1764.00 −0.425666
\(259\) 8128.00 1.95000
\(260\) −50.0000 −0.0119264
\(261\) 1350.00 0.320164
\(262\) −4212.00 −0.993199
\(263\) −5472.00 −1.28296 −0.641479 0.767141i \(-0.721679\pi\)
−0.641479 + 0.767141i \(0.721679\pi\)
\(264\) 756.000 0.176245
\(265\) 390.000 0.0904057
\(266\) −1824.00 −0.420438
\(267\) −3150.00 −0.722011
\(268\) 740.000 0.168667
\(269\) 1734.00 0.393025 0.196513 0.980501i \(-0.437038\pi\)
0.196513 + 0.980501i \(0.437038\pi\)
\(270\) 405.000 0.0912871
\(271\) 5456.00 1.22298 0.611492 0.791251i \(-0.290570\pi\)
0.611492 + 0.791251i \(0.290570\pi\)
\(272\) 2130.00 0.474817
\(273\) 960.000 0.212827
\(274\) −414.000 −0.0912798
\(275\) −300.000 −0.0657843
\(276\) 144.000 0.0314050
\(277\) 830.000 0.180036 0.0900178 0.995940i \(-0.471308\pi\)
0.0900178 + 0.995940i \(0.471308\pi\)
\(278\) 372.000 0.0802557
\(279\) 2016.00 0.432598
\(280\) 3360.00 0.717137
\(281\) 3450.00 0.732419 0.366210 0.930532i \(-0.380655\pi\)
0.366210 + 0.930532i \(0.380655\pi\)
\(282\) −4536.00 −0.957854
\(283\) 1964.00 0.412536 0.206268 0.978496i \(-0.433868\pi\)
0.206268 + 0.978496i \(0.433868\pi\)
\(284\) −120.000 −0.0250729
\(285\) −285.000 −0.0592349
\(286\) −360.000 −0.0744309
\(287\) −1728.00 −0.355403
\(288\) 405.000 0.0828641
\(289\) −4013.00 −0.816813
\(290\) −2250.00 −0.455602
\(291\) 3018.00 0.607967
\(292\) 122.000 0.0244504
\(293\) −2466.00 −0.491690 −0.245845 0.969309i \(-0.579065\pi\)
−0.245845 + 0.969309i \(0.579065\pi\)
\(294\) 6129.00 1.21582
\(295\) 660.000 0.130260
\(296\) 5334.00 1.04741
\(297\) 324.000 0.0633010
\(298\) −8010.00 −1.55707
\(299\) 480.000 0.0928399
\(300\) −75.0000 −0.0144338
\(301\) −6272.00 −1.20104
\(302\) 1128.00 0.214931
\(303\) 1926.00 0.365168
\(304\) −1349.00 −0.254508
\(305\) 1150.00 0.215898
\(306\) 810.000 0.151322
\(307\) −7468.00 −1.38834 −0.694171 0.719810i \(-0.744229\pi\)
−0.694171 + 0.719810i \(0.744229\pi\)
\(308\) −384.000 −0.0710404
\(309\) −4416.00 −0.813001
\(310\) −3360.00 −0.615597
\(311\) −3768.00 −0.687021 −0.343511 0.939149i \(-0.611616\pi\)
−0.343511 + 0.939149i \(0.611616\pi\)
\(312\) 630.000 0.114316
\(313\) −6838.00 −1.23485 −0.617423 0.786632i \(-0.711823\pi\)
−0.617423 + 0.786632i \(0.711823\pi\)
\(314\) 2046.00 0.367715
\(315\) 1440.00 0.257571
\(316\) 1184.00 0.210776
\(317\) 7014.00 1.24273 0.621365 0.783521i \(-0.286578\pi\)
0.621365 + 0.783521i \(0.286578\pi\)
\(318\) 702.000 0.123793
\(319\) −1800.00 −0.315927
\(320\) 2165.00 0.378210
\(321\) 6156.00 1.07039
\(322\) 4608.00 0.797496
\(323\) −570.000 −0.0981909
\(324\) 81.0000 0.0138889
\(325\) −250.000 −0.0426692
\(326\) −10572.0 −1.79610
\(327\) −4002.00 −0.676792
\(328\) −1134.00 −0.190898
\(329\) −16128.0 −2.70263
\(330\) −540.000 −0.0900789
\(331\) 3092.00 0.513449 0.256725 0.966485i \(-0.417357\pi\)
0.256725 + 0.966485i \(0.417357\pi\)
\(332\) 612.000 0.101168
\(333\) 2286.00 0.376192
\(334\) 2736.00 0.448225
\(335\) 3700.00 0.603441
\(336\) 6816.00 1.10668
\(337\) −10270.0 −1.66007 −0.830033 0.557714i \(-0.811679\pi\)
−0.830033 + 0.557714i \(0.811679\pi\)
\(338\) 6291.00 1.01238
\(339\) 1242.00 0.198986
\(340\) −150.000 −0.0239262
\(341\) −2688.00 −0.426872
\(342\) −513.000 −0.0811107
\(343\) 10816.0 1.70265
\(344\) −4116.00 −0.645116
\(345\) 720.000 0.112358
\(346\) −11970.0 −1.85986
\(347\) 396.000 0.0612634 0.0306317 0.999531i \(-0.490248\pi\)
0.0306317 + 0.999531i \(0.490248\pi\)
\(348\) −450.000 −0.0693176
\(349\) −5146.00 −0.789281 −0.394640 0.918836i \(-0.629131\pi\)
−0.394640 + 0.918836i \(0.629131\pi\)
\(350\) −2400.00 −0.366530
\(351\) 270.000 0.0410585
\(352\) −540.000 −0.0817673
\(353\) −9582.00 −1.44475 −0.722377 0.691499i \(-0.756951\pi\)
−0.722377 + 0.691499i \(0.756951\pi\)
\(354\) 1188.00 0.178366
\(355\) −600.000 −0.0897034
\(356\) 1050.00 0.156320
\(357\) 2880.00 0.426963
\(358\) −6948.00 −1.02574
\(359\) 4968.00 0.730365 0.365182 0.930936i \(-0.381007\pi\)
0.365182 + 0.930936i \(0.381007\pi\)
\(360\) 945.000 0.138350
\(361\) 361.000 0.0526316
\(362\) 10038.0 1.45742
\(363\) 3561.00 0.514887
\(364\) −320.000 −0.0460785
\(365\) 610.000 0.0874763
\(366\) 2070.00 0.295630
\(367\) 4088.00 0.581449 0.290725 0.956807i \(-0.406104\pi\)
0.290725 + 0.956807i \(0.406104\pi\)
\(368\) 3408.00 0.482756
\(369\) −486.000 −0.0685641
\(370\) −3810.00 −0.535331
\(371\) 2496.00 0.349288
\(372\) −672.000 −0.0936602
\(373\) 2990.00 0.415057 0.207529 0.978229i \(-0.433458\pi\)
0.207529 + 0.978229i \(0.433458\pi\)
\(374\) −1080.00 −0.149319
\(375\) −375.000 −0.0516398
\(376\) −10584.0 −1.45167
\(377\) −1500.00 −0.204918
\(378\) 2592.00 0.352693
\(379\) 4436.00 0.601219 0.300610 0.953747i \(-0.402810\pi\)
0.300610 + 0.953747i \(0.402810\pi\)
\(380\) 95.0000 0.0128247
\(381\) −4488.00 −0.603483
\(382\) −2880.00 −0.385742
\(383\) −9048.00 −1.20713 −0.603566 0.797313i \(-0.706254\pi\)
−0.603566 + 0.797313i \(0.706254\pi\)
\(384\) 4977.00 0.661410
\(385\) −1920.00 −0.254162
\(386\) 9642.00 1.27141
\(387\) −1764.00 −0.231703
\(388\) −1006.00 −0.131629
\(389\) −8034.00 −1.04715 −0.523573 0.851981i \(-0.675401\pi\)
−0.523573 + 0.851981i \(0.675401\pi\)
\(390\) −450.000 −0.0584273
\(391\) 1440.00 0.186250
\(392\) 14301.0 1.84263
\(393\) −4212.00 −0.540629
\(394\) −2970.00 −0.379763
\(395\) 5920.00 0.754095
\(396\) −108.000 −0.0137051
\(397\) 11366.0 1.43688 0.718442 0.695587i \(-0.244855\pi\)
0.718442 + 0.695587i \(0.244855\pi\)
\(398\) −10680.0 −1.34508
\(399\) −1824.00 −0.228858
\(400\) −1775.00 −0.221875
\(401\) −8382.00 −1.04383 −0.521917 0.852997i \(-0.674783\pi\)
−0.521917 + 0.852997i \(0.674783\pi\)
\(402\) 6660.00 0.826295
\(403\) −2240.00 −0.276879
\(404\) −642.000 −0.0790611
\(405\) 405.000 0.0496904
\(406\) −14400.0 −1.76025
\(407\) −3048.00 −0.371213
\(408\) 1890.00 0.229336
\(409\) −6550.00 −0.791874 −0.395937 0.918278i \(-0.629580\pi\)
−0.395937 + 0.918278i \(0.629580\pi\)
\(410\) 810.000 0.0975684
\(411\) −414.000 −0.0496864
\(412\) 1472.00 0.176020
\(413\) 4224.00 0.503267
\(414\) 1296.00 0.153852
\(415\) 3060.00 0.361951
\(416\) −450.000 −0.0530362
\(417\) 372.000 0.0436857
\(418\) 684.000 0.0800372
\(419\) −11412.0 −1.33058 −0.665290 0.746585i \(-0.731692\pi\)
−0.665290 + 0.746585i \(0.731692\pi\)
\(420\) −480.000 −0.0557657
\(421\) −8098.00 −0.937464 −0.468732 0.883340i \(-0.655289\pi\)
−0.468732 + 0.883340i \(0.655289\pi\)
\(422\) 8220.00 0.948207
\(423\) −4536.00 −0.521390
\(424\) 1638.00 0.187614
\(425\) −750.000 −0.0856008
\(426\) −1080.00 −0.122831
\(427\) 7360.00 0.834134
\(428\) −2052.00 −0.231746
\(429\) −360.000 −0.0405151
\(430\) 2940.00 0.329720
\(431\) 10368.0 1.15872 0.579361 0.815071i \(-0.303302\pi\)
0.579361 + 0.815071i \(0.303302\pi\)
\(432\) 1917.00 0.213499
\(433\) −4606.00 −0.511201 −0.255601 0.966782i \(-0.582273\pi\)
−0.255601 + 0.966782i \(0.582273\pi\)
\(434\) −21504.0 −2.37840
\(435\) −2250.00 −0.247998
\(436\) 1334.00 0.146530
\(437\) −912.000 −0.0998327
\(438\) 1098.00 0.119782
\(439\) 9944.00 1.08110 0.540548 0.841313i \(-0.318217\pi\)
0.540548 + 0.841313i \(0.318217\pi\)
\(440\) −1260.00 −0.136519
\(441\) 6129.00 0.661808
\(442\) −900.000 −0.0968521
\(443\) −14532.0 −1.55855 −0.779273 0.626684i \(-0.784412\pi\)
−0.779273 + 0.626684i \(0.784412\pi\)
\(444\) −762.000 −0.0814480
\(445\) 5250.00 0.559267
\(446\) 10344.0 1.09821
\(447\) −8010.00 −0.847562
\(448\) 13856.0 1.46124
\(449\) 16050.0 1.68696 0.843481 0.537158i \(-0.180502\pi\)
0.843481 + 0.537158i \(0.180502\pi\)
\(450\) −675.000 −0.0707107
\(451\) 648.000 0.0676566
\(452\) −414.000 −0.0430817
\(453\) 1128.00 0.116994
\(454\) −15372.0 −1.58908
\(455\) −1600.00 −0.164855
\(456\) −1197.00 −0.122927
\(457\) −10918.0 −1.11755 −0.558777 0.829318i \(-0.688729\pi\)
−0.558777 + 0.829318i \(0.688729\pi\)
\(458\) 9030.00 0.921276
\(459\) 810.000 0.0823694
\(460\) −240.000 −0.0243262
\(461\) 16182.0 1.63486 0.817430 0.576027i \(-0.195398\pi\)
0.817430 + 0.576027i \(0.195398\pi\)
\(462\) −3456.00 −0.348025
\(463\) 3560.00 0.357337 0.178669 0.983909i \(-0.442821\pi\)
0.178669 + 0.983909i \(0.442821\pi\)
\(464\) −10650.0 −1.06555
\(465\) −3360.00 −0.335089
\(466\) 10530.0 1.04677
\(467\) −3372.00 −0.334128 −0.167064 0.985946i \(-0.553429\pi\)
−0.167064 + 0.985946i \(0.553429\pi\)
\(468\) −90.0000 −0.00888943
\(469\) 23680.0 2.33143
\(470\) 7560.00 0.741951
\(471\) 2046.00 0.200159
\(472\) 2772.00 0.270321
\(473\) 2352.00 0.228637
\(474\) 10656.0 1.03259
\(475\) 475.000 0.0458831
\(476\) −960.000 −0.0924402
\(477\) 702.000 0.0673844
\(478\) −15840.0 −1.51570
\(479\) −2448.00 −0.233511 −0.116756 0.993161i \(-0.537249\pi\)
−0.116756 + 0.993161i \(0.537249\pi\)
\(480\) −675.000 −0.0641862
\(481\) −2540.00 −0.240778
\(482\) −3318.00 −0.313549
\(483\) 4608.00 0.434102
\(484\) −1187.00 −0.111476
\(485\) −5030.00 −0.470929
\(486\) 729.000 0.0680414
\(487\) −4048.00 −0.376658 −0.188329 0.982106i \(-0.560307\pi\)
−0.188329 + 0.982106i \(0.560307\pi\)
\(488\) 4830.00 0.448041
\(489\) −10572.0 −0.977674
\(490\) −10215.0 −0.941769
\(491\) −5580.00 −0.512876 −0.256438 0.966561i \(-0.582549\pi\)
−0.256438 + 0.966561i \(0.582549\pi\)
\(492\) 162.000 0.0148446
\(493\) −4500.00 −0.411095
\(494\) 570.000 0.0519140
\(495\) −540.000 −0.0490327
\(496\) −15904.0 −1.43974
\(497\) −3840.00 −0.346575
\(498\) 5508.00 0.495621
\(499\) 476.000 0.0427028 0.0213514 0.999772i \(-0.493203\pi\)
0.0213514 + 0.999772i \(0.493203\pi\)
\(500\) 125.000 0.0111803
\(501\) 2736.00 0.243983
\(502\) 8532.00 0.758569
\(503\) 2832.00 0.251039 0.125520 0.992091i \(-0.459940\pi\)
0.125520 + 0.992091i \(0.459940\pi\)
\(504\) 6048.00 0.534522
\(505\) −3210.00 −0.282858
\(506\) −1728.00 −0.151816
\(507\) 6291.00 0.551071
\(508\) 1496.00 0.130658
\(509\) −2346.00 −0.204292 −0.102146 0.994769i \(-0.532571\pi\)
−0.102146 + 0.994769i \(0.532571\pi\)
\(510\) −1350.00 −0.117214
\(511\) 3904.00 0.337970
\(512\) 8733.00 0.753804
\(513\) −513.000 −0.0441511
\(514\) 12042.0 1.03337
\(515\) 7360.00 0.629748
\(516\) 588.000 0.0501652
\(517\) 6048.00 0.514489
\(518\) −24384.0 −2.06828
\(519\) −11970.0 −1.01238
\(520\) −1050.00 −0.0885491
\(521\) −3990.00 −0.335518 −0.167759 0.985828i \(-0.553653\pi\)
−0.167759 + 0.985828i \(0.553653\pi\)
\(522\) −4050.00 −0.339586
\(523\) 3020.00 0.252496 0.126248 0.991999i \(-0.459706\pi\)
0.126248 + 0.991999i \(0.459706\pi\)
\(524\) 1404.00 0.117050
\(525\) −2400.00 −0.199513
\(526\) 16416.0 1.36078
\(527\) −6720.00 −0.555461
\(528\) −2556.00 −0.210674
\(529\) −9863.00 −0.810635
\(530\) −1170.00 −0.0958897
\(531\) 1188.00 0.0970900
\(532\) 608.000 0.0495491
\(533\) 540.000 0.0438837
\(534\) 9450.00 0.765808
\(535\) −10260.0 −0.829119
\(536\) 15540.0 1.25229
\(537\) −6948.00 −0.558340
\(538\) −5202.00 −0.416866
\(539\) −8172.00 −0.653048
\(540\) −135.000 −0.0107583
\(541\) 5894.00 0.468397 0.234199 0.972189i \(-0.424753\pi\)
0.234199 + 0.972189i \(0.424753\pi\)
\(542\) −16368.0 −1.29717
\(543\) 10038.0 0.793318
\(544\) −1350.00 −0.106398
\(545\) 6670.00 0.524241
\(546\) −2880.00 −0.225737
\(547\) −22780.0 −1.78063 −0.890313 0.455349i \(-0.849514\pi\)
−0.890313 + 0.455349i \(0.849514\pi\)
\(548\) 138.000 0.0107574
\(549\) 2070.00 0.160921
\(550\) 900.000 0.0697748
\(551\) 2850.00 0.220352
\(552\) 3024.00 0.233170
\(553\) 37888.0 2.91349
\(554\) −2490.00 −0.190957
\(555\) −3810.00 −0.291397
\(556\) −124.000 −0.00945822
\(557\) 9366.00 0.712478 0.356239 0.934395i \(-0.384059\pi\)
0.356239 + 0.934395i \(0.384059\pi\)
\(558\) −6048.00 −0.458839
\(559\) 1960.00 0.148299
\(560\) −11360.0 −0.857228
\(561\) −1080.00 −0.0812792
\(562\) −10350.0 −0.776848
\(563\) 19332.0 1.44715 0.723576 0.690245i \(-0.242497\pi\)
0.723576 + 0.690245i \(0.242497\pi\)
\(564\) 1512.00 0.112884
\(565\) −2070.00 −0.154134
\(566\) −5892.00 −0.437560
\(567\) 2592.00 0.191982
\(568\) −2520.00 −0.186156
\(569\) −21222.0 −1.56357 −0.781786 0.623547i \(-0.785691\pi\)
−0.781786 + 0.623547i \(0.785691\pi\)
\(570\) 855.000 0.0628281
\(571\) −26524.0 −1.94395 −0.971974 0.235086i \(-0.924463\pi\)
−0.971974 + 0.235086i \(0.924463\pi\)
\(572\) 120.000 0.00877177
\(573\) −2880.00 −0.209972
\(574\) 5184.00 0.376962
\(575\) −1200.00 −0.0870321
\(576\) 3897.00 0.281901
\(577\) 5186.00 0.374170 0.187085 0.982344i \(-0.440096\pi\)
0.187085 + 0.982344i \(0.440096\pi\)
\(578\) 12039.0 0.866361
\(579\) 9642.00 0.692069
\(580\) 750.000 0.0536932
\(581\) 19584.0 1.39842
\(582\) −9054.00 −0.644846
\(583\) −936.000 −0.0664926
\(584\) 2562.00 0.181535
\(585\) −450.000 −0.0318038
\(586\) 7398.00 0.521516
\(587\) −12516.0 −0.880052 −0.440026 0.897985i \(-0.645031\pi\)
−0.440026 + 0.897985i \(0.645031\pi\)
\(588\) −2043.00 −0.143286
\(589\) 4256.00 0.297734
\(590\) −1980.00 −0.138162
\(591\) −2970.00 −0.206717
\(592\) −18034.0 −1.25201
\(593\) 15042.0 1.04165 0.520827 0.853662i \(-0.325624\pi\)
0.520827 + 0.853662i \(0.325624\pi\)
\(594\) −972.000 −0.0671408
\(595\) −4800.00 −0.330724
\(596\) 2670.00 0.183502
\(597\) −10680.0 −0.732166
\(598\) −1440.00 −0.0984715
\(599\) 6600.00 0.450198 0.225099 0.974336i \(-0.427729\pi\)
0.225099 + 0.974336i \(0.427729\pi\)
\(600\) −1575.00 −0.107165
\(601\) −23830.0 −1.61738 −0.808690 0.588234i \(-0.799823\pi\)
−0.808690 + 0.588234i \(0.799823\pi\)
\(602\) 18816.0 1.27389
\(603\) 6660.00 0.449778
\(604\) −376.000 −0.0253298
\(605\) −5935.00 −0.398830
\(606\) −5778.00 −0.387319
\(607\) −4552.00 −0.304382 −0.152191 0.988351i \(-0.548633\pi\)
−0.152191 + 0.988351i \(0.548633\pi\)
\(608\) 855.000 0.0570310
\(609\) −14400.0 −0.958157
\(610\) −3450.00 −0.228994
\(611\) 5040.00 0.333710
\(612\) −270.000 −0.0178335
\(613\) −6802.00 −0.448173 −0.224087 0.974569i \(-0.571940\pi\)
−0.224087 + 0.974569i \(0.571940\pi\)
\(614\) 22404.0 1.47256
\(615\) 810.000 0.0531095
\(616\) −8064.00 −0.527448
\(617\) −13686.0 −0.892995 −0.446497 0.894785i \(-0.647329\pi\)
−0.446497 + 0.894785i \(0.647329\pi\)
\(618\) 13248.0 0.862318
\(619\) 24836.0 1.61267 0.806335 0.591459i \(-0.201448\pi\)
0.806335 + 0.591459i \(0.201448\pi\)
\(620\) 1120.00 0.0725488
\(621\) 1296.00 0.0837467
\(622\) 11304.0 0.728696
\(623\) 33600.0 2.16076
\(624\) −2130.00 −0.136648
\(625\) 625.000 0.0400000
\(626\) 20514.0 1.30975
\(627\) 684.000 0.0435667
\(628\) −682.000 −0.0433356
\(629\) −7620.00 −0.483035
\(630\) −4320.00 −0.273195
\(631\) 30056.0 1.89621 0.948107 0.317953i \(-0.102995\pi\)
0.948107 + 0.317953i \(0.102995\pi\)
\(632\) 24864.0 1.56493
\(633\) 8220.00 0.516138
\(634\) −21042.0 −1.31811
\(635\) 7480.00 0.467456
\(636\) −234.000 −0.0145892
\(637\) −6810.00 −0.423582
\(638\) 5400.00 0.335091
\(639\) −1080.00 −0.0668609
\(640\) −8295.00 −0.512326
\(641\) −3438.00 −0.211845 −0.105923 0.994374i \(-0.533780\pi\)
−0.105923 + 0.994374i \(0.533780\pi\)
\(642\) −18468.0 −1.13532
\(643\) −5596.00 −0.343211 −0.171606 0.985166i \(-0.554895\pi\)
−0.171606 + 0.985166i \(0.554895\pi\)
\(644\) −1536.00 −0.0939858
\(645\) 2940.00 0.179477
\(646\) 1710.00 0.104147
\(647\) 1968.00 0.119583 0.0597914 0.998211i \(-0.480956\pi\)
0.0597914 + 0.998211i \(0.480956\pi\)
\(648\) 1701.00 0.103120
\(649\) −1584.00 −0.0958050
\(650\) 750.000 0.0452576
\(651\) −21504.0 −1.29464
\(652\) 3524.00 0.211673
\(653\) −12138.0 −0.727407 −0.363703 0.931515i \(-0.618488\pi\)
−0.363703 + 0.931515i \(0.618488\pi\)
\(654\) 12006.0 0.717847
\(655\) 7020.00 0.418770
\(656\) 3834.00 0.228190
\(657\) 1098.00 0.0652010
\(658\) 48384.0 2.86657
\(659\) −2004.00 −0.118459 −0.0592297 0.998244i \(-0.518864\pi\)
−0.0592297 + 0.998244i \(0.518864\pi\)
\(660\) 180.000 0.0106159
\(661\) 24590.0 1.44696 0.723480 0.690346i \(-0.242542\pi\)
0.723480 + 0.690346i \(0.242542\pi\)
\(662\) −9276.00 −0.544595
\(663\) −900.000 −0.0527196
\(664\) 12852.0 0.751136
\(665\) 3040.00 0.177272
\(666\) −6858.00 −0.399012
\(667\) −7200.00 −0.417969
\(668\) −912.000 −0.0528239
\(669\) 10344.0 0.597791
\(670\) −11100.0 −0.640045
\(671\) −2760.00 −0.158791
\(672\) −4320.00 −0.247988
\(673\) 11762.0 0.673688 0.336844 0.941561i \(-0.390641\pi\)
0.336844 + 0.941561i \(0.390641\pi\)
\(674\) 30810.0 1.76077
\(675\) −675.000 −0.0384900
\(676\) −2097.00 −0.119310
\(677\) 21438.0 1.21703 0.608515 0.793542i \(-0.291766\pi\)
0.608515 + 0.793542i \(0.291766\pi\)
\(678\) −3726.00 −0.211056
\(679\) −32192.0 −1.81946
\(680\) −3150.00 −0.177643
\(681\) −15372.0 −0.864987
\(682\) 8064.00 0.452766
\(683\) 25644.0 1.43666 0.718331 0.695701i \(-0.244906\pi\)
0.718331 + 0.695701i \(0.244906\pi\)
\(684\) 171.000 0.00955899
\(685\) 690.000 0.0384869
\(686\) −32448.0 −1.80593
\(687\) 9030.00 0.501479
\(688\) 13916.0 0.771137
\(689\) −780.000 −0.0431286
\(690\) −2160.00 −0.119174
\(691\) 12332.0 0.678917 0.339458 0.940621i \(-0.389756\pi\)
0.339458 + 0.940621i \(0.389756\pi\)
\(692\) 3990.00 0.219186
\(693\) −3456.00 −0.189441
\(694\) −1188.00 −0.0649796
\(695\) −620.000 −0.0338388
\(696\) −9450.00 −0.514657
\(697\) 1620.00 0.0880371
\(698\) 15438.0 0.837159
\(699\) 10530.0 0.569787
\(700\) 800.000 0.0431959
\(701\) −1050.00 −0.0565734 −0.0282867 0.999600i \(-0.509005\pi\)
−0.0282867 + 0.999600i \(0.509005\pi\)
\(702\) −810.000 −0.0435491
\(703\) 4826.00 0.258913
\(704\) −5196.00 −0.278170
\(705\) 7560.00 0.403867
\(706\) 28746.0 1.53239
\(707\) −20544.0 −1.09284
\(708\) −396.000 −0.0210206
\(709\) 13790.0 0.730457 0.365229 0.930918i \(-0.380991\pi\)
0.365229 + 0.930918i \(0.380991\pi\)
\(710\) 1800.00 0.0951448
\(711\) 10656.0 0.562069
\(712\) 22050.0 1.16062
\(713\) −10752.0 −0.564748
\(714\) −8640.00 −0.452863
\(715\) 600.000 0.0313828
\(716\) 2316.00 0.120884
\(717\) −15840.0 −0.825043
\(718\) −14904.0 −0.774669
\(719\) −20160.0 −1.04568 −0.522838 0.852432i \(-0.675127\pi\)
−0.522838 + 0.852432i \(0.675127\pi\)
\(720\) −3195.00 −0.165376
\(721\) 47104.0 2.43307
\(722\) −1083.00 −0.0558242
\(723\) −3318.00 −0.170675
\(724\) −3346.00 −0.171758
\(725\) 3750.00 0.192099
\(726\) −10683.0 −0.546120
\(727\) −36976.0 −1.88633 −0.943166 0.332321i \(-0.892168\pi\)
−0.943166 + 0.332321i \(0.892168\pi\)
\(728\) −6720.00 −0.342115
\(729\) 729.000 0.0370370
\(730\) −1830.00 −0.0927827
\(731\) 5880.00 0.297510
\(732\) −690.000 −0.0348403
\(733\) −970.000 −0.0488783 −0.0244391 0.999701i \(-0.507780\pi\)
−0.0244391 + 0.999701i \(0.507780\pi\)
\(734\) −12264.0 −0.616720
\(735\) −10215.0 −0.512634
\(736\) −2160.00 −0.108178
\(737\) −8880.00 −0.443825
\(738\) 1458.00 0.0727232
\(739\) 17900.0 0.891018 0.445509 0.895278i \(-0.353023\pi\)
0.445509 + 0.895278i \(0.353023\pi\)
\(740\) 1270.00 0.0630894
\(741\) 570.000 0.0282584
\(742\) −7488.00 −0.370476
\(743\) 2496.00 0.123243 0.0616214 0.998100i \(-0.480373\pi\)
0.0616214 + 0.998100i \(0.480373\pi\)
\(744\) −14112.0 −0.695391
\(745\) 13350.0 0.656518
\(746\) −8970.00 −0.440235
\(747\) 5508.00 0.269782
\(748\) 360.000 0.0175975
\(749\) −65664.0 −3.20335
\(750\) 1125.00 0.0547723
\(751\) −9520.00 −0.462570 −0.231285 0.972886i \(-0.574293\pi\)
−0.231285 + 0.972886i \(0.574293\pi\)
\(752\) 35784.0 1.73525
\(753\) 8532.00 0.412913
\(754\) 4500.00 0.217348
\(755\) −1880.00 −0.0906228
\(756\) −864.000 −0.0415653
\(757\) −6466.00 −0.310450 −0.155225 0.987879i \(-0.549610\pi\)
−0.155225 + 0.987879i \(0.549610\pi\)
\(758\) −13308.0 −0.637689
\(759\) −1728.00 −0.0826382
\(760\) 1995.00 0.0952188
\(761\) −30198.0 −1.43847 −0.719236 0.694766i \(-0.755508\pi\)
−0.719236 + 0.694766i \(0.755508\pi\)
\(762\) 13464.0 0.640091
\(763\) 42688.0 2.02544
\(764\) 960.000 0.0454602
\(765\) −1350.00 −0.0638031
\(766\) 27144.0 1.28036
\(767\) −1320.00 −0.0621414
\(768\) −4539.00 −0.213264
\(769\) −14590.0 −0.684173 −0.342086 0.939669i \(-0.611134\pi\)
−0.342086 + 0.939669i \(0.611134\pi\)
\(770\) 5760.00 0.269579
\(771\) 12042.0 0.562493
\(772\) −3214.00 −0.149837
\(773\) −34818.0 −1.62007 −0.810036 0.586379i \(-0.800553\pi\)
−0.810036 + 0.586379i \(0.800553\pi\)
\(774\) 5292.00 0.245758
\(775\) 5600.00 0.259559
\(776\) −21126.0 −0.977293
\(777\) −24384.0 −1.12583
\(778\) 24102.0 1.11067
\(779\) −1026.00 −0.0471890
\(780\) 150.000 0.00688572
\(781\) 1440.00 0.0659760
\(782\) −4320.00 −0.197548
\(783\) −4050.00 −0.184847
\(784\) −48351.0 −2.20258
\(785\) −3410.00 −0.155042
\(786\) 12636.0 0.573424
\(787\) −1804.00 −0.0817099 −0.0408549 0.999165i \(-0.513008\pi\)
−0.0408549 + 0.999165i \(0.513008\pi\)
\(788\) 990.000 0.0447554
\(789\) 16416.0 0.740716
\(790\) −17760.0 −0.799839
\(791\) −13248.0 −0.595505
\(792\) −2268.00 −0.101755
\(793\) −2300.00 −0.102995
\(794\) −34098.0 −1.52405
\(795\) −1170.00 −0.0521958
\(796\) 3560.00 0.158519
\(797\) 35718.0 1.58745 0.793724 0.608278i \(-0.208139\pi\)
0.793724 + 0.608278i \(0.208139\pi\)
\(798\) 5472.00 0.242740
\(799\) 15120.0 0.669471
\(800\) 1125.00 0.0497184
\(801\) 9450.00 0.416853
\(802\) 25146.0 1.10715
\(803\) −1464.00 −0.0643380
\(804\) −2220.00 −0.0973798
\(805\) −7680.00 −0.336254
\(806\) 6720.00 0.293675
\(807\) −5202.00 −0.226913
\(808\) −13482.0 −0.586999
\(809\) −44838.0 −1.94860 −0.974302 0.225247i \(-0.927681\pi\)
−0.974302 + 0.225247i \(0.927681\pi\)
\(810\) −1215.00 −0.0527046
\(811\) 17588.0 0.761527 0.380763 0.924673i \(-0.375661\pi\)
0.380763 + 0.924673i \(0.375661\pi\)
\(812\) 4800.00 0.207447
\(813\) −16368.0 −0.706090
\(814\) 9144.00 0.393731
\(815\) 17620.0 0.757303
\(816\) −6390.00 −0.274136
\(817\) −3724.00 −0.159469
\(818\) 19650.0 0.839910
\(819\) −2880.00 −0.122876
\(820\) −270.000 −0.0114985
\(821\) −33522.0 −1.42500 −0.712501 0.701672i \(-0.752437\pi\)
−0.712501 + 0.701672i \(0.752437\pi\)
\(822\) 1242.00 0.0527004
\(823\) −3904.00 −0.165352 −0.0826761 0.996576i \(-0.526347\pi\)
−0.0826761 + 0.996576i \(0.526347\pi\)
\(824\) 30912.0 1.30688
\(825\) 900.000 0.0379806
\(826\) −12672.0 −0.533796
\(827\) 11628.0 0.488930 0.244465 0.969658i \(-0.421388\pi\)
0.244465 + 0.969658i \(0.421388\pi\)
\(828\) −432.000 −0.0181317
\(829\) −19738.0 −0.826935 −0.413467 0.910519i \(-0.635682\pi\)
−0.413467 + 0.910519i \(0.635682\pi\)
\(830\) −9180.00 −0.383907
\(831\) −2490.00 −0.103944
\(832\) −4330.00 −0.180428
\(833\) −20430.0 −0.849769
\(834\) −1116.00 −0.0463356
\(835\) −4560.00 −0.188988
\(836\) −228.000 −0.00943247
\(837\) −6048.00 −0.249760
\(838\) 34236.0 1.41129
\(839\) −31752.0 −1.30656 −0.653278 0.757118i \(-0.726607\pi\)
−0.653278 + 0.757118i \(0.726607\pi\)
\(840\) −10080.0 −0.414039
\(841\) −1889.00 −0.0774530
\(842\) 24294.0 0.994331
\(843\) −10350.0 −0.422862
\(844\) −2740.00 −0.111747
\(845\) −10485.0 −0.426858
\(846\) 13608.0 0.553017
\(847\) −37984.0 −1.54090
\(848\) −5538.00 −0.224264
\(849\) −5892.00 −0.238178
\(850\) 2250.00 0.0907934
\(851\) −12192.0 −0.491112
\(852\) 360.000 0.0144758
\(853\) −10978.0 −0.440656 −0.220328 0.975426i \(-0.570713\pi\)
−0.220328 + 0.975426i \(0.570713\pi\)
\(854\) −22080.0 −0.884733
\(855\) 855.000 0.0341993
\(856\) −43092.0 −1.72062
\(857\) 3450.00 0.137514 0.0687571 0.997633i \(-0.478097\pi\)
0.0687571 + 0.997633i \(0.478097\pi\)
\(858\) 1080.00 0.0429727
\(859\) −46348.0 −1.84095 −0.920473 0.390805i \(-0.872197\pi\)
−0.920473 + 0.390805i \(0.872197\pi\)
\(860\) −980.000 −0.0388578
\(861\) 5184.00 0.205192
\(862\) −31104.0 −1.22901
\(863\) −47736.0 −1.88291 −0.941456 0.337137i \(-0.890541\pi\)
−0.941456 + 0.337137i \(0.890541\pi\)
\(864\) −1215.00 −0.0478416
\(865\) 19950.0 0.784185
\(866\) 13818.0 0.542211
\(867\) 12039.0 0.471587
\(868\) 7168.00 0.280297
\(869\) −14208.0 −0.554630
\(870\) 6750.00 0.263042
\(871\) −7400.00 −0.287875
\(872\) 28014.0 1.08793
\(873\) −9054.00 −0.351010
\(874\) 2736.00 0.105889
\(875\) 4000.00 0.154542
\(876\) −366.000 −0.0141164
\(877\) −33706.0 −1.29780 −0.648900 0.760874i \(-0.724771\pi\)
−0.648900 + 0.760874i \(0.724771\pi\)
\(878\) −29832.0 −1.14668
\(879\) 7398.00 0.283878
\(880\) 4260.00 0.163187
\(881\) −33774.0 −1.29157 −0.645786 0.763518i \(-0.723470\pi\)
−0.645786 + 0.763518i \(0.723470\pi\)
\(882\) −18387.0 −0.701953
\(883\) −22444.0 −0.855380 −0.427690 0.903925i \(-0.640673\pi\)
−0.427690 + 0.903925i \(0.640673\pi\)
\(884\) 300.000 0.0114141
\(885\) −1980.00 −0.0752056
\(886\) 43596.0 1.65309
\(887\) 30240.0 1.14471 0.572356 0.820005i \(-0.306029\pi\)
0.572356 + 0.820005i \(0.306029\pi\)
\(888\) −16002.0 −0.604721
\(889\) 47872.0 1.80605
\(890\) −15750.0 −0.593192
\(891\) −972.000 −0.0365468
\(892\) −3448.00 −0.129426
\(893\) −9576.00 −0.358845
\(894\) 24030.0 0.898975
\(895\) 11580.0 0.432488
\(896\) −53088.0 −1.97940
\(897\) −1440.00 −0.0536011
\(898\) −48150.0 −1.78929
\(899\) 33600.0 1.24652
\(900\) 225.000 0.00833333
\(901\) −2340.00 −0.0865224
\(902\) −1944.00 −0.0717607
\(903\) 18816.0 0.693419
\(904\) −8694.00 −0.319865
\(905\) −16730.0 −0.614502
\(906\) −3384.00 −0.124090
\(907\) 38972.0 1.42673 0.713365 0.700793i \(-0.247170\pi\)
0.713365 + 0.700793i \(0.247170\pi\)
\(908\) 5124.00 0.187275
\(909\) −5778.00 −0.210830
\(910\) 4800.00 0.174855
\(911\) 16128.0 0.586547 0.293274 0.956029i \(-0.405255\pi\)
0.293274 + 0.956029i \(0.405255\pi\)
\(912\) 4047.00 0.146940
\(913\) −7344.00 −0.266211
\(914\) 32754.0 1.18535
\(915\) −3450.00 −0.124649
\(916\) −3010.00 −0.108573
\(917\) 44928.0 1.61794
\(918\) −2430.00 −0.0873660
\(919\) 23576.0 0.846246 0.423123 0.906072i \(-0.360934\pi\)
0.423123 + 0.906072i \(0.360934\pi\)
\(920\) −5040.00 −0.180613
\(921\) 22404.0 0.801560
\(922\) −48546.0 −1.73403
\(923\) 1200.00 0.0427936
\(924\) 1152.00 0.0410152
\(925\) 6350.00 0.225715
\(926\) −10680.0 −0.379014
\(927\) 13248.0 0.469387
\(928\) 6750.00 0.238771
\(929\) 54882.0 1.93823 0.969117 0.246600i \(-0.0793134\pi\)
0.969117 + 0.246600i \(0.0793134\pi\)
\(930\) 10080.0 0.355415
\(931\) 12939.0 0.455487
\(932\) −3510.00 −0.123363
\(933\) 11304.0 0.396652
\(934\) 10116.0 0.354396
\(935\) 1800.00 0.0629586
\(936\) −1890.00 −0.0660006
\(937\) 33146.0 1.15564 0.577819 0.816165i \(-0.303904\pi\)
0.577819 + 0.816165i \(0.303904\pi\)
\(938\) −71040.0 −2.47286
\(939\) 20514.0 0.712938
\(940\) −2520.00 −0.0874397
\(941\) −33786.0 −1.17045 −0.585224 0.810871i \(-0.698994\pi\)
−0.585224 + 0.810871i \(0.698994\pi\)
\(942\) −6138.00 −0.212300
\(943\) 2592.00 0.0895092
\(944\) −9372.00 −0.323128
\(945\) −4320.00 −0.148709
\(946\) −7056.00 −0.242506
\(947\) −18060.0 −0.619716 −0.309858 0.950783i \(-0.600282\pi\)
−0.309858 + 0.950783i \(0.600282\pi\)
\(948\) −3552.00 −0.121692
\(949\) −1220.00 −0.0417312
\(950\) −1425.00 −0.0486664
\(951\) −21042.0 −0.717491
\(952\) −20160.0 −0.686333
\(953\) 26202.0 0.890625 0.445313 0.895375i \(-0.353093\pi\)
0.445313 + 0.895375i \(0.353093\pi\)
\(954\) −2106.00 −0.0714720
\(955\) 4800.00 0.162643
\(956\) 5280.00 0.178627
\(957\) 5400.00 0.182400
\(958\) 7344.00 0.247676
\(959\) 4416.00 0.148697
\(960\) −6495.00 −0.218360
\(961\) 20385.0 0.684267
\(962\) 7620.00 0.255383
\(963\) −18468.0 −0.617989
\(964\) 1106.00 0.0369521
\(965\) −16070.0 −0.536074
\(966\) −13824.0 −0.460435
\(967\) 13856.0 0.460785 0.230392 0.973098i \(-0.425999\pi\)
0.230392 + 0.973098i \(0.425999\pi\)
\(968\) −24927.0 −0.827670
\(969\) 1710.00 0.0566905
\(970\) 15090.0 0.499496
\(971\) 16644.0 0.550084 0.275042 0.961432i \(-0.411308\pi\)
0.275042 + 0.961432i \(0.411308\pi\)
\(972\) −243.000 −0.00801875
\(973\) −3968.00 −0.130738
\(974\) 12144.0 0.399506
\(975\) 750.000 0.0246351
\(976\) −16330.0 −0.535564
\(977\) −40446.0 −1.32444 −0.662222 0.749308i \(-0.730387\pi\)
−0.662222 + 0.749308i \(0.730387\pi\)
\(978\) 31716.0 1.03698
\(979\) −12600.0 −0.411336
\(980\) 3405.00 0.110989
\(981\) 12006.0 0.390746
\(982\) 16740.0 0.543987
\(983\) 17232.0 0.559120 0.279560 0.960128i \(-0.409811\pi\)
0.279560 + 0.960128i \(0.409811\pi\)
\(984\) 3402.00 0.110215
\(985\) 4950.00 0.160122
\(986\) 13500.0 0.436032
\(987\) 48384.0 1.56036
\(988\) −190.000 −0.00611812
\(989\) 9408.00 0.302484
\(990\) 1620.00 0.0520071
\(991\) 48992.0 1.57042 0.785208 0.619232i \(-0.212556\pi\)
0.785208 + 0.619232i \(0.212556\pi\)
\(992\) 10080.0 0.322621
\(993\) −9276.00 −0.296440
\(994\) 11520.0 0.367598
\(995\) 17800.0 0.567134
\(996\) −1836.00 −0.0584095
\(997\) 7310.00 0.232207 0.116103 0.993237i \(-0.462960\pi\)
0.116103 + 0.993237i \(0.462960\pi\)
\(998\) −1428.00 −0.0452931
\(999\) −6858.00 −0.217195
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 285.4.a.a.1.1 1
3.2 odd 2 855.4.a.f.1.1 1
5.4 even 2 1425.4.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.4.a.a.1.1 1 1.1 even 1 trivial
855.4.a.f.1.1 1 3.2 odd 2
1425.4.a.f.1.1 1 5.4 even 2