Properties

Label 315.4.a.e.1.1
Level $315$
Weight $4$
Character 315.1
Self dual yes
Analytic conductor $18.586$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,4,Mod(1,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, 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("315.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.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: \(18.5856016518\)
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 \(\chi\) \(=\) 315.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{2} +1.00000 q^{4} +5.00000 q^{5} +7.00000 q^{7} -21.0000 q^{8} +O(q^{10})\) \(q+3.00000 q^{2} +1.00000 q^{4} +5.00000 q^{5} +7.00000 q^{7} -21.0000 q^{8} +15.0000 q^{10} +60.0000 q^{11} +38.0000 q^{13} +21.0000 q^{14} -71.0000 q^{16} -84.0000 q^{17} +110.000 q^{19} +5.00000 q^{20} +180.000 q^{22} +120.000 q^{23} +25.0000 q^{25} +114.000 q^{26} +7.00000 q^{28} +162.000 q^{29} +236.000 q^{31} -45.0000 q^{32} -252.000 q^{34} +35.0000 q^{35} -376.000 q^{37} +330.000 q^{38} -105.000 q^{40} -126.000 q^{41} -34.0000 q^{43} +60.0000 q^{44} +360.000 q^{46} -6.00000 q^{47} +49.0000 q^{49} +75.0000 q^{50} +38.0000 q^{52} +582.000 q^{53} +300.000 q^{55} -147.000 q^{56} +486.000 q^{58} +492.000 q^{59} -880.000 q^{61} +708.000 q^{62} +433.000 q^{64} +190.000 q^{65} -826.000 q^{67} -84.0000 q^{68} +105.000 q^{70} -666.000 q^{71} -826.000 q^{73} -1128.00 q^{74} +110.000 q^{76} +420.000 q^{77} -592.000 q^{79} -355.000 q^{80} -378.000 q^{82} +792.000 q^{83} -420.000 q^{85} -102.000 q^{86} -1260.00 q^{88} +1002.00 q^{89} +266.000 q^{91} +120.000 q^{92} -18.0000 q^{94} +550.000 q^{95} +1442.00 q^{97} +147.000 q^{98} +O(q^{100})\)

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 \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) 0 0
\(4\) 1.00000 0.125000
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) −21.0000 −0.928078
\(9\) 0 0
\(10\) 15.0000 0.474342
\(11\) 60.0000 1.64461 0.822304 0.569049i \(-0.192689\pi\)
0.822304 + 0.569049i \(0.192689\pi\)
\(12\) 0 0
\(13\) 38.0000 0.810716 0.405358 0.914158i \(-0.367147\pi\)
0.405358 + 0.914158i \(0.367147\pi\)
\(14\) 21.0000 0.400892
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) −84.0000 −1.19841 −0.599206 0.800595i \(-0.704517\pi\)
−0.599206 + 0.800595i \(0.704517\pi\)
\(18\) 0 0
\(19\) 110.000 1.32820 0.664098 0.747645i \(-0.268816\pi\)
0.664098 + 0.747645i \(0.268816\pi\)
\(20\) 5.00000 0.0559017
\(21\) 0 0
\(22\) 180.000 1.74437
\(23\) 120.000 1.08790 0.543951 0.839117i \(-0.316928\pi\)
0.543951 + 0.839117i \(0.316928\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 114.000 0.859894
\(27\) 0 0
\(28\) 7.00000 0.0472456
\(29\) 162.000 1.03733 0.518666 0.854977i \(-0.326429\pi\)
0.518666 + 0.854977i \(0.326429\pi\)
\(30\) 0 0
\(31\) 236.000 1.36732 0.683659 0.729802i \(-0.260388\pi\)
0.683659 + 0.729802i \(0.260388\pi\)
\(32\) −45.0000 −0.248592
\(33\) 0 0
\(34\) −252.000 −1.27111
\(35\) 35.0000 0.169031
\(36\) 0 0
\(37\) −376.000 −1.67065 −0.835325 0.549757i \(-0.814720\pi\)
−0.835325 + 0.549757i \(0.814720\pi\)
\(38\) 330.000 1.40876
\(39\) 0 0
\(40\) −105.000 −0.415049
\(41\) −126.000 −0.479949 −0.239974 0.970779i \(-0.577139\pi\)
−0.239974 + 0.970779i \(0.577139\pi\)
\(42\) 0 0
\(43\) −34.0000 −0.120580 −0.0602901 0.998181i \(-0.519203\pi\)
−0.0602901 + 0.998181i \(0.519203\pi\)
\(44\) 60.0000 0.205576
\(45\) 0 0
\(46\) 360.000 1.15389
\(47\) −6.00000 −0.0186211 −0.00931053 0.999957i \(-0.502964\pi\)
−0.00931053 + 0.999957i \(0.502964\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 75.0000 0.212132
\(51\) 0 0
\(52\) 38.0000 0.101339
\(53\) 582.000 1.50837 0.754187 0.656659i \(-0.228031\pi\)
0.754187 + 0.656659i \(0.228031\pi\)
\(54\) 0 0
\(55\) 300.000 0.735491
\(56\) −147.000 −0.350780
\(57\) 0 0
\(58\) 486.000 1.10026
\(59\) 492.000 1.08564 0.542822 0.839848i \(-0.317356\pi\)
0.542822 + 0.839848i \(0.317356\pi\)
\(60\) 0 0
\(61\) −880.000 −1.84709 −0.923545 0.383491i \(-0.874722\pi\)
−0.923545 + 0.383491i \(0.874722\pi\)
\(62\) 708.000 1.45026
\(63\) 0 0
\(64\) 433.000 0.845703
\(65\) 190.000 0.362563
\(66\) 0 0
\(67\) −826.000 −1.50615 −0.753074 0.657935i \(-0.771430\pi\)
−0.753074 + 0.657935i \(0.771430\pi\)
\(68\) −84.0000 −0.149801
\(69\) 0 0
\(70\) 105.000 0.179284
\(71\) −666.000 −1.11323 −0.556617 0.830769i \(-0.687901\pi\)
−0.556617 + 0.830769i \(0.687901\pi\)
\(72\) 0 0
\(73\) −826.000 −1.32433 −0.662164 0.749359i \(-0.730362\pi\)
−0.662164 + 0.749359i \(0.730362\pi\)
\(74\) −1128.00 −1.77199
\(75\) 0 0
\(76\) 110.000 0.166025
\(77\) 420.000 0.621603
\(78\) 0 0
\(79\) −592.000 −0.843104 −0.421552 0.906804i \(-0.638514\pi\)
−0.421552 + 0.906804i \(0.638514\pi\)
\(80\) −355.000 −0.496128
\(81\) 0 0
\(82\) −378.000 −0.509062
\(83\) 792.000 1.04739 0.523695 0.851906i \(-0.324553\pi\)
0.523695 + 0.851906i \(0.324553\pi\)
\(84\) 0 0
\(85\) −420.000 −0.535946
\(86\) −102.000 −0.127895
\(87\) 0 0
\(88\) −1260.00 −1.52632
\(89\) 1002.00 1.19339 0.596695 0.802468i \(-0.296480\pi\)
0.596695 + 0.802468i \(0.296480\pi\)
\(90\) 0 0
\(91\) 266.000 0.306422
\(92\) 120.000 0.135988
\(93\) 0 0
\(94\) −18.0000 −0.0197506
\(95\) 550.000 0.593987
\(96\) 0 0
\(97\) 1442.00 1.50941 0.754706 0.656063i \(-0.227780\pi\)
0.754706 + 0.656063i \(0.227780\pi\)
\(98\) 147.000 0.151523
\(99\) 0 0
\(100\) 25.0000 0.0250000
\(101\) −1182.00 −1.16449 −0.582245 0.813014i \(-0.697825\pi\)
−0.582245 + 0.813014i \(0.697825\pi\)
\(102\) 0 0
\(103\) 56.0000 0.0535713 0.0267857 0.999641i \(-0.491473\pi\)
0.0267857 + 0.999641i \(0.491473\pi\)
\(104\) −798.000 −0.752407
\(105\) 0 0
\(106\) 1746.00 1.59987
\(107\) −1572.00 −1.42029 −0.710145 0.704056i \(-0.751371\pi\)
−0.710145 + 0.704056i \(0.751371\pi\)
\(108\) 0 0
\(109\) −1402.00 −1.23199 −0.615997 0.787749i \(-0.711246\pi\)
−0.615997 + 0.787749i \(0.711246\pi\)
\(110\) 900.000 0.780106
\(111\) 0 0
\(112\) −497.000 −0.419304
\(113\) −846.000 −0.704292 −0.352146 0.935945i \(-0.614548\pi\)
−0.352146 + 0.935945i \(0.614548\pi\)
\(114\) 0 0
\(115\) 600.000 0.486524
\(116\) 162.000 0.129667
\(117\) 0 0
\(118\) 1476.00 1.15150
\(119\) −588.000 −0.452957
\(120\) 0 0
\(121\) 2269.00 1.70473
\(122\) −2640.00 −1.95913
\(123\) 0 0
\(124\) 236.000 0.170915
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1096.00 −0.765782 −0.382891 0.923794i \(-0.625071\pi\)
−0.382891 + 0.923794i \(0.625071\pi\)
\(128\) 1659.00 1.14560
\(129\) 0 0
\(130\) 570.000 0.384556
\(131\) −492.000 −0.328139 −0.164070 0.986449i \(-0.552462\pi\)
−0.164070 + 0.986449i \(0.552462\pi\)
\(132\) 0 0
\(133\) 770.000 0.502011
\(134\) −2478.00 −1.59751
\(135\) 0 0
\(136\) 1764.00 1.11222
\(137\) −282.000 −0.175860 −0.0879302 0.996127i \(-0.528025\pi\)
−0.0879302 + 0.996127i \(0.528025\pi\)
\(138\) 0 0
\(139\) 2.00000 0.00122042 0.000610208 1.00000i \(-0.499806\pi\)
0.000610208 1.00000i \(0.499806\pi\)
\(140\) 35.0000 0.0211289
\(141\) 0 0
\(142\) −1998.00 −1.18076
\(143\) 2280.00 1.33331
\(144\) 0 0
\(145\) 810.000 0.463909
\(146\) −2478.00 −1.40466
\(147\) 0 0
\(148\) −376.000 −0.208831
\(149\) 606.000 0.333191 0.166595 0.986025i \(-0.446723\pi\)
0.166595 + 0.986025i \(0.446723\pi\)
\(150\) 0 0
\(151\) −2176.00 −1.17272 −0.586359 0.810051i \(-0.699439\pi\)
−0.586359 + 0.810051i \(0.699439\pi\)
\(152\) −2310.00 −1.23267
\(153\) 0 0
\(154\) 1260.00 0.659310
\(155\) 1180.00 0.611483
\(156\) 0 0
\(157\) 146.000 0.0742170 0.0371085 0.999311i \(-0.488185\pi\)
0.0371085 + 0.999311i \(0.488185\pi\)
\(158\) −1776.00 −0.894247
\(159\) 0 0
\(160\) −225.000 −0.111174
\(161\) 840.000 0.411188
\(162\) 0 0
\(163\) −826.000 −0.396916 −0.198458 0.980109i \(-0.563593\pi\)
−0.198458 + 0.980109i \(0.563593\pi\)
\(164\) −126.000 −0.0599936
\(165\) 0 0
\(166\) 2376.00 1.11092
\(167\) 1206.00 0.558821 0.279410 0.960172i \(-0.409861\pi\)
0.279410 + 0.960172i \(0.409861\pi\)
\(168\) 0 0
\(169\) −753.000 −0.342740
\(170\) −1260.00 −0.568456
\(171\) 0 0
\(172\) −34.0000 −0.0150725
\(173\) 1398.00 0.614381 0.307191 0.951648i \(-0.400611\pi\)
0.307191 + 0.951648i \(0.400611\pi\)
\(174\) 0 0
\(175\) 175.000 0.0755929
\(176\) −4260.00 −1.82449
\(177\) 0 0
\(178\) 3006.00 1.26578
\(179\) −2424.00 −1.01217 −0.506085 0.862484i \(-0.668908\pi\)
−0.506085 + 0.862484i \(0.668908\pi\)
\(180\) 0 0
\(181\) 3728.00 1.53094 0.765470 0.643472i \(-0.222507\pi\)
0.765470 + 0.643472i \(0.222507\pi\)
\(182\) 798.000 0.325009
\(183\) 0 0
\(184\) −2520.00 −1.00966
\(185\) −1880.00 −0.747137
\(186\) 0 0
\(187\) −5040.00 −1.97092
\(188\) −6.00000 −0.00232763
\(189\) 0 0
\(190\) 1650.00 0.630019
\(191\) 2550.00 0.966029 0.483014 0.875612i \(-0.339542\pi\)
0.483014 + 0.875612i \(0.339542\pi\)
\(192\) 0 0
\(193\) −1978.00 −0.737718 −0.368859 0.929485i \(-0.620251\pi\)
−0.368859 + 0.929485i \(0.620251\pi\)
\(194\) 4326.00 1.60097
\(195\) 0 0
\(196\) 49.0000 0.0178571
\(197\) −1170.00 −0.423142 −0.211571 0.977363i \(-0.567858\pi\)
−0.211571 + 0.977363i \(0.567858\pi\)
\(198\) 0 0
\(199\) 3584.00 1.27670 0.638349 0.769747i \(-0.279618\pi\)
0.638349 + 0.769747i \(0.279618\pi\)
\(200\) −525.000 −0.185616
\(201\) 0 0
\(202\) −3546.00 −1.23513
\(203\) 1134.00 0.392075
\(204\) 0 0
\(205\) −630.000 −0.214640
\(206\) 168.000 0.0568209
\(207\) 0 0
\(208\) −2698.00 −0.899388
\(209\) 6600.00 2.18436
\(210\) 0 0
\(211\) 1640.00 0.535082 0.267541 0.963547i \(-0.413789\pi\)
0.267541 + 0.963547i \(0.413789\pi\)
\(212\) 582.000 0.188547
\(213\) 0 0
\(214\) −4716.00 −1.50644
\(215\) −170.000 −0.0539251
\(216\) 0 0
\(217\) 1652.00 0.516798
\(218\) −4206.00 −1.30673
\(219\) 0 0
\(220\) 300.000 0.0919363
\(221\) −3192.00 −0.971571
\(222\) 0 0
\(223\) −1888.00 −0.566950 −0.283475 0.958980i \(-0.591487\pi\)
−0.283475 + 0.958980i \(0.591487\pi\)
\(224\) −315.000 −0.0939590
\(225\) 0 0
\(226\) −2538.00 −0.747014
\(227\) −2244.00 −0.656121 −0.328061 0.944657i \(-0.606395\pi\)
−0.328061 + 0.944657i \(0.606395\pi\)
\(228\) 0 0
\(229\) −4084.00 −1.17851 −0.589254 0.807948i \(-0.700578\pi\)
−0.589254 + 0.807948i \(0.700578\pi\)
\(230\) 1800.00 0.516037
\(231\) 0 0
\(232\) −3402.00 −0.962725
\(233\) 4026.00 1.13198 0.565991 0.824411i \(-0.308494\pi\)
0.565991 + 0.824411i \(0.308494\pi\)
\(234\) 0 0
\(235\) −30.0000 −0.00832759
\(236\) 492.000 0.135705
\(237\) 0 0
\(238\) −1764.00 −0.480433
\(239\) −4590.00 −1.24227 −0.621135 0.783704i \(-0.713328\pi\)
−0.621135 + 0.783704i \(0.713328\pi\)
\(240\) 0 0
\(241\) 1946.00 0.520136 0.260068 0.965590i \(-0.416255\pi\)
0.260068 + 0.965590i \(0.416255\pi\)
\(242\) 6807.00 1.80814
\(243\) 0 0
\(244\) −880.000 −0.230886
\(245\) 245.000 0.0638877
\(246\) 0 0
\(247\) 4180.00 1.07679
\(248\) −4956.00 −1.26898
\(249\) 0 0
\(250\) 375.000 0.0948683
\(251\) 3636.00 0.914352 0.457176 0.889376i \(-0.348861\pi\)
0.457176 + 0.889376i \(0.348861\pi\)
\(252\) 0 0
\(253\) 7200.00 1.78917
\(254\) −3288.00 −0.812234
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) 4152.00 1.00776 0.503881 0.863773i \(-0.331905\pi\)
0.503881 + 0.863773i \(0.331905\pi\)
\(258\) 0 0
\(259\) −2632.00 −0.631446
\(260\) 190.000 0.0453204
\(261\) 0 0
\(262\) −1476.00 −0.348044
\(263\) 5388.00 1.26326 0.631632 0.775269i \(-0.282385\pi\)
0.631632 + 0.775269i \(0.282385\pi\)
\(264\) 0 0
\(265\) 2910.00 0.674566
\(266\) 2310.00 0.532463
\(267\) 0 0
\(268\) −826.000 −0.188269
\(269\) 1194.00 0.270630 0.135315 0.990803i \(-0.456795\pi\)
0.135315 + 0.990803i \(0.456795\pi\)
\(270\) 0 0
\(271\) 6788.00 1.52156 0.760778 0.649012i \(-0.224818\pi\)
0.760778 + 0.649012i \(0.224818\pi\)
\(272\) 5964.00 1.32949
\(273\) 0 0
\(274\) −846.000 −0.186528
\(275\) 1500.00 0.328921
\(276\) 0 0
\(277\) −8908.00 −1.93224 −0.966119 0.258098i \(-0.916904\pi\)
−0.966119 + 0.258098i \(0.916904\pi\)
\(278\) 6.00000 0.00129445
\(279\) 0 0
\(280\) −735.000 −0.156874
\(281\) 6408.00 1.36039 0.680194 0.733032i \(-0.261895\pi\)
0.680194 + 0.733032i \(0.261895\pi\)
\(282\) 0 0
\(283\) −3652.00 −0.767098 −0.383549 0.923520i \(-0.625298\pi\)
−0.383549 + 0.923520i \(0.625298\pi\)
\(284\) −666.000 −0.139154
\(285\) 0 0
\(286\) 6840.00 1.41419
\(287\) −882.000 −0.181404
\(288\) 0 0
\(289\) 2143.00 0.436190
\(290\) 2430.00 0.492050
\(291\) 0 0
\(292\) −826.000 −0.165541
\(293\) −2214.00 −0.441445 −0.220722 0.975337i \(-0.570841\pi\)
−0.220722 + 0.975337i \(0.570841\pi\)
\(294\) 0 0
\(295\) 2460.00 0.485514
\(296\) 7896.00 1.55049
\(297\) 0 0
\(298\) 1818.00 0.353402
\(299\) 4560.00 0.881979
\(300\) 0 0
\(301\) −238.000 −0.0455751
\(302\) −6528.00 −1.24385
\(303\) 0 0
\(304\) −7810.00 −1.47347
\(305\) −4400.00 −0.826043
\(306\) 0 0
\(307\) −5452.00 −1.01356 −0.506779 0.862076i \(-0.669164\pi\)
−0.506779 + 0.862076i \(0.669164\pi\)
\(308\) 420.000 0.0777004
\(309\) 0 0
\(310\) 3540.00 0.648576
\(311\) −300.000 −0.0546992 −0.0273496 0.999626i \(-0.508707\pi\)
−0.0273496 + 0.999626i \(0.508707\pi\)
\(312\) 0 0
\(313\) −3994.00 −0.721260 −0.360630 0.932709i \(-0.617438\pi\)
−0.360630 + 0.932709i \(0.617438\pi\)
\(314\) 438.000 0.0787190
\(315\) 0 0
\(316\) −592.000 −0.105388
\(317\) 2586.00 0.458184 0.229092 0.973405i \(-0.426424\pi\)
0.229092 + 0.973405i \(0.426424\pi\)
\(318\) 0 0
\(319\) 9720.00 1.70600
\(320\) 2165.00 0.378210
\(321\) 0 0
\(322\) 2520.00 0.436131
\(323\) −9240.00 −1.59173
\(324\) 0 0
\(325\) 950.000 0.162143
\(326\) −2478.00 −0.420993
\(327\) 0 0
\(328\) 2646.00 0.445430
\(329\) −42.0000 −0.00703810
\(330\) 0 0
\(331\) 6248.00 1.03753 0.518763 0.854918i \(-0.326393\pi\)
0.518763 + 0.854918i \(0.326393\pi\)
\(332\) 792.000 0.130924
\(333\) 0 0
\(334\) 3618.00 0.592719
\(335\) −4130.00 −0.673570
\(336\) 0 0
\(337\) 3062.00 0.494949 0.247474 0.968894i \(-0.420399\pi\)
0.247474 + 0.968894i \(0.420399\pi\)
\(338\) −2259.00 −0.363531
\(339\) 0 0
\(340\) −420.000 −0.0669932
\(341\) 14160.0 2.24870
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 714.000 0.111908
\(345\) 0 0
\(346\) 4194.00 0.651650
\(347\) −5196.00 −0.803850 −0.401925 0.915673i \(-0.631659\pi\)
−0.401925 + 0.915673i \(0.631659\pi\)
\(348\) 0 0
\(349\) 8660.00 1.32825 0.664125 0.747622i \(-0.268804\pi\)
0.664125 + 0.747622i \(0.268804\pi\)
\(350\) 525.000 0.0801784
\(351\) 0 0
\(352\) −2700.00 −0.408837
\(353\) −1128.00 −0.170078 −0.0850388 0.996378i \(-0.527101\pi\)
−0.0850388 + 0.996378i \(0.527101\pi\)
\(354\) 0 0
\(355\) −3330.00 −0.497854
\(356\) 1002.00 0.149174
\(357\) 0 0
\(358\) −7272.00 −1.07357
\(359\) 6618.00 0.972938 0.486469 0.873698i \(-0.338285\pi\)
0.486469 + 0.873698i \(0.338285\pi\)
\(360\) 0 0
\(361\) 5241.00 0.764106
\(362\) 11184.0 1.62381
\(363\) 0 0
\(364\) 266.000 0.0383027
\(365\) −4130.00 −0.592258
\(366\) 0 0
\(367\) 1568.00 0.223022 0.111511 0.993763i \(-0.464431\pi\)
0.111511 + 0.993763i \(0.464431\pi\)
\(368\) −8520.00 −1.20689
\(369\) 0 0
\(370\) −5640.00 −0.792458
\(371\) 4074.00 0.570112
\(372\) 0 0
\(373\) −5200.00 −0.721839 −0.360919 0.932597i \(-0.617537\pi\)
−0.360919 + 0.932597i \(0.617537\pi\)
\(374\) −15120.0 −2.09047
\(375\) 0 0
\(376\) 126.000 0.0172818
\(377\) 6156.00 0.840982
\(378\) 0 0
\(379\) −1096.00 −0.148543 −0.0742714 0.997238i \(-0.523663\pi\)
−0.0742714 + 0.997238i \(0.523663\pi\)
\(380\) 550.000 0.0742484
\(381\) 0 0
\(382\) 7650.00 1.02463
\(383\) −12318.0 −1.64340 −0.821698 0.569924i \(-0.806973\pi\)
−0.821698 + 0.569924i \(0.806973\pi\)
\(384\) 0 0
\(385\) 2100.00 0.277989
\(386\) −5934.00 −0.782468
\(387\) 0 0
\(388\) 1442.00 0.188676
\(389\) 6558.00 0.854766 0.427383 0.904071i \(-0.359436\pi\)
0.427383 + 0.904071i \(0.359436\pi\)
\(390\) 0 0
\(391\) −10080.0 −1.30375
\(392\) −1029.00 −0.132583
\(393\) 0 0
\(394\) −3510.00 −0.448810
\(395\) −2960.00 −0.377048
\(396\) 0 0
\(397\) −1258.00 −0.159036 −0.0795179 0.996833i \(-0.525338\pi\)
−0.0795179 + 0.996833i \(0.525338\pi\)
\(398\) 10752.0 1.35414
\(399\) 0 0
\(400\) −1775.00 −0.221875
\(401\) −2976.00 −0.370609 −0.185305 0.982681i \(-0.559327\pi\)
−0.185305 + 0.982681i \(0.559327\pi\)
\(402\) 0 0
\(403\) 8968.00 1.10851
\(404\) −1182.00 −0.145561
\(405\) 0 0
\(406\) 3402.00 0.415858
\(407\) −22560.0 −2.74756
\(408\) 0 0
\(409\) −9070.00 −1.09653 −0.548267 0.836303i \(-0.684712\pi\)
−0.548267 + 0.836303i \(0.684712\pi\)
\(410\) −1890.00 −0.227660
\(411\) 0 0
\(412\) 56.0000 0.00669641
\(413\) 3444.00 0.410335
\(414\) 0 0
\(415\) 3960.00 0.468407
\(416\) −1710.00 −0.201538
\(417\) 0 0
\(418\) 19800.0 2.31687
\(419\) 6156.00 0.717757 0.358879 0.933384i \(-0.383159\pi\)
0.358879 + 0.933384i \(0.383159\pi\)
\(420\) 0 0
\(421\) −6874.00 −0.795768 −0.397884 0.917436i \(-0.630255\pi\)
−0.397884 + 0.917436i \(0.630255\pi\)
\(422\) 4920.00 0.567540
\(423\) 0 0
\(424\) −12222.0 −1.39989
\(425\) −2100.00 −0.239682
\(426\) 0 0
\(427\) −6160.00 −0.698134
\(428\) −1572.00 −0.177536
\(429\) 0 0
\(430\) −510.000 −0.0571962
\(431\) −10218.0 −1.14196 −0.570979 0.820965i \(-0.693436\pi\)
−0.570979 + 0.820965i \(0.693436\pi\)
\(432\) 0 0
\(433\) −5830.00 −0.647048 −0.323524 0.946220i \(-0.604868\pi\)
−0.323524 + 0.946220i \(0.604868\pi\)
\(434\) 4956.00 0.548147
\(435\) 0 0
\(436\) −1402.00 −0.153999
\(437\) 13200.0 1.44495
\(438\) 0 0
\(439\) 8588.00 0.933674 0.466837 0.884343i \(-0.345393\pi\)
0.466837 + 0.884343i \(0.345393\pi\)
\(440\) −6300.00 −0.682593
\(441\) 0 0
\(442\) −9576.00 −1.03051
\(443\) −12660.0 −1.35778 −0.678888 0.734242i \(-0.737538\pi\)
−0.678888 + 0.734242i \(0.737538\pi\)
\(444\) 0 0
\(445\) 5010.00 0.533701
\(446\) −5664.00 −0.601341
\(447\) 0 0
\(448\) 3031.00 0.319646
\(449\) 3636.00 0.382168 0.191084 0.981574i \(-0.438800\pi\)
0.191084 + 0.981574i \(0.438800\pi\)
\(450\) 0 0
\(451\) −7560.00 −0.789327
\(452\) −846.000 −0.0880365
\(453\) 0 0
\(454\) −6732.00 −0.695922
\(455\) 1330.00 0.137036
\(456\) 0 0
\(457\) −3022.00 −0.309329 −0.154664 0.987967i \(-0.549430\pi\)
−0.154664 + 0.987967i \(0.549430\pi\)
\(458\) −12252.0 −1.25000
\(459\) 0 0
\(460\) 600.000 0.0608155
\(461\) −14742.0 −1.48938 −0.744689 0.667411i \(-0.767402\pi\)
−0.744689 + 0.667411i \(0.767402\pi\)
\(462\) 0 0
\(463\) −9268.00 −0.930282 −0.465141 0.885237i \(-0.653996\pi\)
−0.465141 + 0.885237i \(0.653996\pi\)
\(464\) −11502.0 −1.15079
\(465\) 0 0
\(466\) 12078.0 1.20065
\(467\) −13920.0 −1.37932 −0.689658 0.724135i \(-0.742239\pi\)
−0.689658 + 0.724135i \(0.742239\pi\)
\(468\) 0 0
\(469\) −5782.00 −0.569271
\(470\) −90.0000 −0.00883275
\(471\) 0 0
\(472\) −10332.0 −1.00756
\(473\) −2040.00 −0.198307
\(474\) 0 0
\(475\) 2750.00 0.265639
\(476\) −588.000 −0.0566196
\(477\) 0 0
\(478\) −13770.0 −1.31763
\(479\) −8220.00 −0.784095 −0.392047 0.919945i \(-0.628233\pi\)
−0.392047 + 0.919945i \(0.628233\pi\)
\(480\) 0 0
\(481\) −14288.0 −1.35442
\(482\) 5838.00 0.551688
\(483\) 0 0
\(484\) 2269.00 0.213092
\(485\) 7210.00 0.675029
\(486\) 0 0
\(487\) 15752.0 1.46569 0.732845 0.680395i \(-0.238192\pi\)
0.732845 + 0.680395i \(0.238192\pi\)
\(488\) 18480.0 1.71424
\(489\) 0 0
\(490\) 735.000 0.0677631
\(491\) 6552.00 0.602215 0.301108 0.953590i \(-0.402644\pi\)
0.301108 + 0.953590i \(0.402644\pi\)
\(492\) 0 0
\(493\) −13608.0 −1.24315
\(494\) 12540.0 1.14211
\(495\) 0 0
\(496\) −16756.0 −1.51687
\(497\) −4662.00 −0.420763
\(498\) 0 0
\(499\) 4160.00 0.373201 0.186600 0.982436i \(-0.440253\pi\)
0.186600 + 0.982436i \(0.440253\pi\)
\(500\) 125.000 0.0111803
\(501\) 0 0
\(502\) 10908.0 0.969816
\(503\) 5166.00 0.457934 0.228967 0.973434i \(-0.426465\pi\)
0.228967 + 0.973434i \(0.426465\pi\)
\(504\) 0 0
\(505\) −5910.00 −0.520775
\(506\) 21600.0 1.89770
\(507\) 0 0
\(508\) −1096.00 −0.0957227
\(509\) −12066.0 −1.05072 −0.525360 0.850880i \(-0.676069\pi\)
−0.525360 + 0.850880i \(0.676069\pi\)
\(510\) 0 0
\(511\) −5782.00 −0.500549
\(512\) −8733.00 −0.753804
\(513\) 0 0
\(514\) 12456.0 1.06889
\(515\) 280.000 0.0239578
\(516\) 0 0
\(517\) −360.000 −0.0306243
\(518\) −7896.00 −0.669750
\(519\) 0 0
\(520\) −3990.00 −0.336487
\(521\) −17886.0 −1.50403 −0.752015 0.659146i \(-0.770918\pi\)
−0.752015 + 0.659146i \(0.770918\pi\)
\(522\) 0 0
\(523\) 5168.00 0.432086 0.216043 0.976384i \(-0.430685\pi\)
0.216043 + 0.976384i \(0.430685\pi\)
\(524\) −492.000 −0.0410174
\(525\) 0 0
\(526\) 16164.0 1.33989
\(527\) −19824.0 −1.63861
\(528\) 0 0
\(529\) 2233.00 0.183529
\(530\) 8730.00 0.715485
\(531\) 0 0
\(532\) 770.000 0.0627514
\(533\) −4788.00 −0.389102
\(534\) 0 0
\(535\) −7860.00 −0.635173
\(536\) 17346.0 1.39782
\(537\) 0 0
\(538\) 3582.00 0.287046
\(539\) 2940.00 0.234944
\(540\) 0 0
\(541\) 7850.00 0.623841 0.311920 0.950108i \(-0.399028\pi\)
0.311920 + 0.950108i \(0.399028\pi\)
\(542\) 20364.0 1.61385
\(543\) 0 0
\(544\) 3780.00 0.297916
\(545\) −7010.00 −0.550964
\(546\) 0 0
\(547\) 2990.00 0.233717 0.116858 0.993149i \(-0.462718\pi\)
0.116858 + 0.993149i \(0.462718\pi\)
\(548\) −282.000 −0.0219826
\(549\) 0 0
\(550\) 4500.00 0.348874
\(551\) 17820.0 1.37778
\(552\) 0 0
\(553\) −4144.00 −0.318663
\(554\) −26724.0 −2.04945
\(555\) 0 0
\(556\) 2.00000 0.000152552 0
\(557\) −12486.0 −0.949818 −0.474909 0.880035i \(-0.657519\pi\)
−0.474909 + 0.880035i \(0.657519\pi\)
\(558\) 0 0
\(559\) −1292.00 −0.0977563
\(560\) −2485.00 −0.187519
\(561\) 0 0
\(562\) 19224.0 1.44291
\(563\) −732.000 −0.0547960 −0.0273980 0.999625i \(-0.508722\pi\)
−0.0273980 + 0.999625i \(0.508722\pi\)
\(564\) 0 0
\(565\) −4230.00 −0.314969
\(566\) −10956.0 −0.813631
\(567\) 0 0
\(568\) 13986.0 1.03317
\(569\) −456.000 −0.0335967 −0.0167983 0.999859i \(-0.505347\pi\)
−0.0167983 + 0.999859i \(0.505347\pi\)
\(570\) 0 0
\(571\) −14092.0 −1.03281 −0.516403 0.856346i \(-0.672729\pi\)
−0.516403 + 0.856346i \(0.672729\pi\)
\(572\) 2280.00 0.166664
\(573\) 0 0
\(574\) −2646.00 −0.192408
\(575\) 3000.00 0.217580
\(576\) 0 0
\(577\) 10190.0 0.735208 0.367604 0.929982i \(-0.380178\pi\)
0.367604 + 0.929982i \(0.380178\pi\)
\(578\) 6429.00 0.462649
\(579\) 0 0
\(580\) 810.000 0.0579887
\(581\) 5544.00 0.395876
\(582\) 0 0
\(583\) 34920.0 2.48068
\(584\) 17346.0 1.22908
\(585\) 0 0
\(586\) −6642.00 −0.468223
\(587\) 23220.0 1.63270 0.816348 0.577561i \(-0.195995\pi\)
0.816348 + 0.577561i \(0.195995\pi\)
\(588\) 0 0
\(589\) 25960.0 1.81607
\(590\) 7380.00 0.514966
\(591\) 0 0
\(592\) 26696.0 1.85338
\(593\) 14916.0 1.03293 0.516464 0.856309i \(-0.327248\pi\)
0.516464 + 0.856309i \(0.327248\pi\)
\(594\) 0 0
\(595\) −2940.00 −0.202568
\(596\) 606.000 0.0416489
\(597\) 0 0
\(598\) 13680.0 0.935480
\(599\) −16914.0 −1.15374 −0.576868 0.816838i \(-0.695725\pi\)
−0.576868 + 0.816838i \(0.695725\pi\)
\(600\) 0 0
\(601\) 5762.00 0.391076 0.195538 0.980696i \(-0.437355\pi\)
0.195538 + 0.980696i \(0.437355\pi\)
\(602\) −714.000 −0.0483396
\(603\) 0 0
\(604\) −2176.00 −0.146590
\(605\) 11345.0 0.762380
\(606\) 0 0
\(607\) −28528.0 −1.90760 −0.953802 0.300435i \(-0.902868\pi\)
−0.953802 + 0.300435i \(0.902868\pi\)
\(608\) −4950.00 −0.330179
\(609\) 0 0
\(610\) −13200.0 −0.876151
\(611\) −228.000 −0.0150964
\(612\) 0 0
\(613\) −12976.0 −0.854969 −0.427484 0.904023i \(-0.640600\pi\)
−0.427484 + 0.904023i \(0.640600\pi\)
\(614\) −16356.0 −1.07504
\(615\) 0 0
\(616\) −8820.00 −0.576896
\(617\) 8874.00 0.579017 0.289509 0.957175i \(-0.406508\pi\)
0.289509 + 0.957175i \(0.406508\pi\)
\(618\) 0 0
\(619\) −17170.0 −1.11490 −0.557448 0.830212i \(-0.688219\pi\)
−0.557448 + 0.830212i \(0.688219\pi\)
\(620\) 1180.00 0.0764354
\(621\) 0 0
\(622\) −900.000 −0.0580172
\(623\) 7014.00 0.451059
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −11982.0 −0.765011
\(627\) 0 0
\(628\) 146.000 0.00927712
\(629\) 31584.0 2.00212
\(630\) 0 0
\(631\) −26728.0 −1.68625 −0.843126 0.537716i \(-0.819287\pi\)
−0.843126 + 0.537716i \(0.819287\pi\)
\(632\) 12432.0 0.782466
\(633\) 0 0
\(634\) 7758.00 0.485977
\(635\) −5480.00 −0.342468
\(636\) 0 0
\(637\) 1862.00 0.115817
\(638\) 29160.0 1.80949
\(639\) 0 0
\(640\) 8295.00 0.512326
\(641\) −24492.0 −1.50917 −0.754583 0.656204i \(-0.772161\pi\)
−0.754583 + 0.656204i \(0.772161\pi\)
\(642\) 0 0
\(643\) −1888.00 −0.115794 −0.0578969 0.998323i \(-0.518439\pi\)
−0.0578969 + 0.998323i \(0.518439\pi\)
\(644\) 840.000 0.0513985
\(645\) 0 0
\(646\) −27720.0 −1.68828
\(647\) −13134.0 −0.798069 −0.399035 0.916936i \(-0.630655\pi\)
−0.399035 + 0.916936i \(0.630655\pi\)
\(648\) 0 0
\(649\) 29520.0 1.78546
\(650\) 2850.00 0.171979
\(651\) 0 0
\(652\) −826.000 −0.0496145
\(653\) 12882.0 0.771993 0.385997 0.922500i \(-0.373858\pi\)
0.385997 + 0.922500i \(0.373858\pi\)
\(654\) 0 0
\(655\) −2460.00 −0.146748
\(656\) 8946.00 0.532443
\(657\) 0 0
\(658\) −126.000 −0.00746503
\(659\) 7380.00 0.436243 0.218121 0.975922i \(-0.430007\pi\)
0.218121 + 0.975922i \(0.430007\pi\)
\(660\) 0 0
\(661\) 560.000 0.0329523 0.0164762 0.999864i \(-0.494755\pi\)
0.0164762 + 0.999864i \(0.494755\pi\)
\(662\) 18744.0 1.10046
\(663\) 0 0
\(664\) −16632.0 −0.972058
\(665\) 3850.00 0.224506
\(666\) 0 0
\(667\) 19440.0 1.12852
\(668\) 1206.00 0.0698526
\(669\) 0 0
\(670\) −12390.0 −0.714429
\(671\) −52800.0 −3.03774
\(672\) 0 0
\(673\) −5722.00 −0.327737 −0.163868 0.986482i \(-0.552397\pi\)
−0.163868 + 0.986482i \(0.552397\pi\)
\(674\) 9186.00 0.524973
\(675\) 0 0
\(676\) −753.000 −0.0428425
\(677\) −13650.0 −0.774907 −0.387454 0.921889i \(-0.626645\pi\)
−0.387454 + 0.921889i \(0.626645\pi\)
\(678\) 0 0
\(679\) 10094.0 0.570504
\(680\) 8820.00 0.497399
\(681\) 0 0
\(682\) 42480.0 2.38511
\(683\) 1140.00 0.0638666 0.0319333 0.999490i \(-0.489834\pi\)
0.0319333 + 0.999490i \(0.489834\pi\)
\(684\) 0 0
\(685\) −1410.00 −0.0786472
\(686\) 1029.00 0.0572703
\(687\) 0 0
\(688\) 2414.00 0.133769
\(689\) 22116.0 1.22286
\(690\) 0 0
\(691\) 32510.0 1.78978 0.894891 0.446286i \(-0.147254\pi\)
0.894891 + 0.446286i \(0.147254\pi\)
\(692\) 1398.00 0.0767977
\(693\) 0 0
\(694\) −15588.0 −0.852612
\(695\) 10.0000 0.000545787 0
\(696\) 0 0
\(697\) 10584.0 0.575176
\(698\) 25980.0 1.40882
\(699\) 0 0
\(700\) 175.000 0.00944911
\(701\) 21402.0 1.15313 0.576564 0.817052i \(-0.304393\pi\)
0.576564 + 0.817052i \(0.304393\pi\)
\(702\) 0 0
\(703\) −41360.0 −2.21895
\(704\) 25980.0 1.39085
\(705\) 0 0
\(706\) −3384.00 −0.180395
\(707\) −8274.00 −0.440135
\(708\) 0 0
\(709\) 12170.0 0.644646 0.322323 0.946630i \(-0.395536\pi\)
0.322323 + 0.946630i \(0.395536\pi\)
\(710\) −9990.00 −0.528054
\(711\) 0 0
\(712\) −21042.0 −1.10756
\(713\) 28320.0 1.48751
\(714\) 0 0
\(715\) 11400.0 0.596274
\(716\) −2424.00 −0.126521
\(717\) 0 0
\(718\) 19854.0 1.03196
\(719\) −12012.0 −0.623049 −0.311524 0.950238i \(-0.600840\pi\)
−0.311524 + 0.950238i \(0.600840\pi\)
\(720\) 0 0
\(721\) 392.000 0.0202480
\(722\) 15723.0 0.810456
\(723\) 0 0
\(724\) 3728.00 0.191367
\(725\) 4050.00 0.207467
\(726\) 0 0
\(727\) 8480.00 0.432608 0.216304 0.976326i \(-0.430600\pi\)
0.216304 + 0.976326i \(0.430600\pi\)
\(728\) −5586.00 −0.284383
\(729\) 0 0
\(730\) −12390.0 −0.628184
\(731\) 2856.00 0.144505
\(732\) 0 0
\(733\) −10906.0 −0.549553 −0.274776 0.961508i \(-0.588604\pi\)
−0.274776 + 0.961508i \(0.588604\pi\)
\(734\) 4704.00 0.236550
\(735\) 0 0
\(736\) −5400.00 −0.270444
\(737\) −49560.0 −2.47702
\(738\) 0 0
\(739\) −11104.0 −0.552730 −0.276365 0.961053i \(-0.589130\pi\)
−0.276365 + 0.961053i \(0.589130\pi\)
\(740\) −1880.00 −0.0933921
\(741\) 0 0
\(742\) 12222.0 0.604695
\(743\) 25416.0 1.25494 0.627471 0.778640i \(-0.284090\pi\)
0.627471 + 0.778640i \(0.284090\pi\)
\(744\) 0 0
\(745\) 3030.00 0.149008
\(746\) −15600.0 −0.765625
\(747\) 0 0
\(748\) −5040.00 −0.246365
\(749\) −11004.0 −0.536819
\(750\) 0 0
\(751\) −30904.0 −1.50160 −0.750801 0.660529i \(-0.770332\pi\)
−0.750801 + 0.660529i \(0.770332\pi\)
\(752\) 426.000 0.0206577
\(753\) 0 0
\(754\) 18468.0 0.891996
\(755\) −10880.0 −0.524455
\(756\) 0 0
\(757\) −16216.0 −0.778574 −0.389287 0.921117i \(-0.627279\pi\)
−0.389287 + 0.921117i \(0.627279\pi\)
\(758\) −3288.00 −0.157553
\(759\) 0 0
\(760\) −11550.0 −0.551266
\(761\) 7230.00 0.344399 0.172199 0.985062i \(-0.444913\pi\)
0.172199 + 0.985062i \(0.444913\pi\)
\(762\) 0 0
\(763\) −9814.00 −0.465650
\(764\) 2550.00 0.120754
\(765\) 0 0
\(766\) −36954.0 −1.74308
\(767\) 18696.0 0.880148
\(768\) 0 0
\(769\) 4790.00 0.224619 0.112309 0.993673i \(-0.464175\pi\)
0.112309 + 0.993673i \(0.464175\pi\)
\(770\) 6300.00 0.294852
\(771\) 0 0
\(772\) −1978.00 −0.0922147
\(773\) 36114.0 1.68038 0.840188 0.542296i \(-0.182445\pi\)
0.840188 + 0.542296i \(0.182445\pi\)
\(774\) 0 0
\(775\) 5900.00 0.273464
\(776\) −30282.0 −1.40085
\(777\) 0 0
\(778\) 19674.0 0.906616
\(779\) −13860.0 −0.637466
\(780\) 0 0
\(781\) −39960.0 −1.83083
\(782\) −30240.0 −1.38284
\(783\) 0 0
\(784\) −3479.00 −0.158482
\(785\) 730.000 0.0331909
\(786\) 0 0
\(787\) 13556.0 0.614002 0.307001 0.951709i \(-0.400675\pi\)
0.307001 + 0.951709i \(0.400675\pi\)
\(788\) −1170.00 −0.0528928
\(789\) 0 0
\(790\) −8880.00 −0.399919
\(791\) −5922.00 −0.266197
\(792\) 0 0
\(793\) −33440.0 −1.49746
\(794\) −3774.00 −0.168683
\(795\) 0 0
\(796\) 3584.00 0.159587
\(797\) 39006.0 1.73358 0.866790 0.498673i \(-0.166179\pi\)
0.866790 + 0.498673i \(0.166179\pi\)
\(798\) 0 0
\(799\) 504.000 0.0223157
\(800\) −1125.00 −0.0497184
\(801\) 0 0
\(802\) −8928.00 −0.393091
\(803\) −49560.0 −2.17800
\(804\) 0 0
\(805\) 4200.00 0.183889
\(806\) 26904.0 1.17575
\(807\) 0 0
\(808\) 24822.0 1.08074
\(809\) 21480.0 0.933494 0.466747 0.884391i \(-0.345426\pi\)
0.466747 + 0.884391i \(0.345426\pi\)
\(810\) 0 0
\(811\) 146.000 0.00632152 0.00316076 0.999995i \(-0.498994\pi\)
0.00316076 + 0.999995i \(0.498994\pi\)
\(812\) 1134.00 0.0490094
\(813\) 0 0
\(814\) −67680.0 −2.91423
\(815\) −4130.00 −0.177506
\(816\) 0 0
\(817\) −3740.00 −0.160154
\(818\) −27210.0 −1.16305
\(819\) 0 0
\(820\) −630.000 −0.0268299
\(821\) 35562.0 1.51172 0.755860 0.654733i \(-0.227219\pi\)
0.755860 + 0.654733i \(0.227219\pi\)
\(822\) 0 0
\(823\) −33964.0 −1.43853 −0.719265 0.694736i \(-0.755521\pi\)
−0.719265 + 0.694736i \(0.755521\pi\)
\(824\) −1176.00 −0.0497183
\(825\) 0 0
\(826\) 10332.0 0.435225
\(827\) −23292.0 −0.979374 −0.489687 0.871898i \(-0.662889\pi\)
−0.489687 + 0.871898i \(0.662889\pi\)
\(828\) 0 0
\(829\) 33716.0 1.41255 0.706276 0.707937i \(-0.250374\pi\)
0.706276 + 0.707937i \(0.250374\pi\)
\(830\) 11880.0 0.496820
\(831\) 0 0
\(832\) 16454.0 0.685625
\(833\) −4116.00 −0.171202
\(834\) 0 0
\(835\) 6030.00 0.249912
\(836\) 6600.00 0.273045
\(837\) 0 0
\(838\) 18468.0 0.761297
\(839\) −35280.0 −1.45173 −0.725865 0.687838i \(-0.758560\pi\)
−0.725865 + 0.687838i \(0.758560\pi\)
\(840\) 0 0
\(841\) 1855.00 0.0760589
\(842\) −20622.0 −0.844039
\(843\) 0 0
\(844\) 1640.00 0.0668852
\(845\) −3765.00 −0.153278
\(846\) 0 0
\(847\) 15883.0 0.644329
\(848\) −41322.0 −1.67335
\(849\) 0 0
\(850\) −6300.00 −0.254221
\(851\) −45120.0 −1.81750
\(852\) 0 0
\(853\) 36902.0 1.48124 0.740622 0.671922i \(-0.234531\pi\)
0.740622 + 0.671922i \(0.234531\pi\)
\(854\) −18480.0 −0.740483
\(855\) 0 0
\(856\) 33012.0 1.31814
\(857\) −30648.0 −1.22161 −0.610803 0.791783i \(-0.709153\pi\)
−0.610803 + 0.791783i \(0.709153\pi\)
\(858\) 0 0
\(859\) 20918.0 0.830865 0.415432 0.909624i \(-0.363630\pi\)
0.415432 + 0.909624i \(0.363630\pi\)
\(860\) −170.000 −0.00674064
\(861\) 0 0
\(862\) −30654.0 −1.21123
\(863\) 19812.0 0.781470 0.390735 0.920503i \(-0.372221\pi\)
0.390735 + 0.920503i \(0.372221\pi\)
\(864\) 0 0
\(865\) 6990.00 0.274760
\(866\) −17490.0 −0.686298
\(867\) 0 0
\(868\) 1652.00 0.0645997
\(869\) −35520.0 −1.38657
\(870\) 0 0
\(871\) −31388.0 −1.22106
\(872\) 29442.0 1.14339
\(873\) 0 0
\(874\) 39600.0 1.53260
\(875\) 875.000 0.0338062
\(876\) 0 0
\(877\) 38900.0 1.49779 0.748894 0.662690i \(-0.230585\pi\)
0.748894 + 0.662690i \(0.230585\pi\)
\(878\) 25764.0 0.990311
\(879\) 0 0
\(880\) −21300.0 −0.815935
\(881\) 16398.0 0.627086 0.313543 0.949574i \(-0.398484\pi\)
0.313543 + 0.949574i \(0.398484\pi\)
\(882\) 0 0
\(883\) 30422.0 1.15944 0.579718 0.814817i \(-0.303163\pi\)
0.579718 + 0.814817i \(0.303163\pi\)
\(884\) −3192.00 −0.121446
\(885\) 0 0
\(886\) −37980.0 −1.44014
\(887\) 6222.00 0.235529 0.117765 0.993042i \(-0.462427\pi\)
0.117765 + 0.993042i \(0.462427\pi\)
\(888\) 0 0
\(889\) −7672.00 −0.289438
\(890\) 15030.0 0.566075
\(891\) 0 0
\(892\) −1888.00 −0.0708687
\(893\) −660.000 −0.0247324
\(894\) 0 0
\(895\) −12120.0 −0.452656
\(896\) 11613.0 0.432995
\(897\) 0 0
\(898\) 10908.0 0.405350
\(899\) 38232.0 1.41836
\(900\) 0 0
\(901\) −48888.0 −1.80765
\(902\) −22680.0 −0.837208
\(903\) 0 0
\(904\) 17766.0 0.653638
\(905\) 18640.0 0.684657
\(906\) 0 0
\(907\) 21926.0 0.802691 0.401346 0.915927i \(-0.368543\pi\)
0.401346 + 0.915927i \(0.368543\pi\)
\(908\) −2244.00 −0.0820151
\(909\) 0 0
\(910\) 3990.00 0.145349
\(911\) 40662.0 1.47881 0.739403 0.673263i \(-0.235108\pi\)
0.739403 + 0.673263i \(0.235108\pi\)
\(912\) 0 0
\(913\) 47520.0 1.72254
\(914\) −9066.00 −0.328093
\(915\) 0 0
\(916\) −4084.00 −0.147313
\(917\) −3444.00 −0.124025
\(918\) 0 0
\(919\) 25688.0 0.922055 0.461028 0.887386i \(-0.347481\pi\)
0.461028 + 0.887386i \(0.347481\pi\)
\(920\) −12600.0 −0.451532
\(921\) 0 0
\(922\) −44226.0 −1.57972
\(923\) −25308.0 −0.902517
\(924\) 0 0
\(925\) −9400.00 −0.334130
\(926\) −27804.0 −0.986713
\(927\) 0 0
\(928\) −7290.00 −0.257873
\(929\) −16026.0 −0.565981 −0.282990 0.959123i \(-0.591326\pi\)
−0.282990 + 0.959123i \(0.591326\pi\)
\(930\) 0 0
\(931\) 5390.00 0.189742
\(932\) 4026.00 0.141498
\(933\) 0 0
\(934\) −41760.0 −1.46299
\(935\) −25200.0 −0.881420
\(936\) 0 0
\(937\) 18362.0 0.640193 0.320096 0.947385i \(-0.396285\pi\)
0.320096 + 0.947385i \(0.396285\pi\)
\(938\) −17346.0 −0.603803
\(939\) 0 0
\(940\) −30.0000 −0.00104095
\(941\) 8034.00 0.278322 0.139161 0.990270i \(-0.455559\pi\)
0.139161 + 0.990270i \(0.455559\pi\)
\(942\) 0 0
\(943\) −15120.0 −0.522137
\(944\) −34932.0 −1.20439
\(945\) 0 0
\(946\) −6120.00 −0.210337
\(947\) −30732.0 −1.05455 −0.527273 0.849696i \(-0.676786\pi\)
−0.527273 + 0.849696i \(0.676786\pi\)
\(948\) 0 0
\(949\) −31388.0 −1.07365
\(950\) 8250.00 0.281753
\(951\) 0 0
\(952\) 12348.0 0.420379
\(953\) 28242.0 0.959967 0.479983 0.877278i \(-0.340643\pi\)
0.479983 + 0.877278i \(0.340643\pi\)
\(954\) 0 0
\(955\) 12750.0 0.432021
\(956\) −4590.00 −0.155284
\(957\) 0 0
\(958\) −24660.0 −0.831658
\(959\) −1974.00 −0.0664690
\(960\) 0 0
\(961\) 25905.0 0.869558
\(962\) −42864.0 −1.43658
\(963\) 0 0
\(964\) 1946.00 0.0650171
\(965\) −9890.00 −0.329917
\(966\) 0 0
\(967\) 37496.0 1.24694 0.623470 0.781848i \(-0.285723\pi\)
0.623470 + 0.781848i \(0.285723\pi\)
\(968\) −47649.0 −1.58212
\(969\) 0 0
\(970\) 21630.0 0.715977
\(971\) −27204.0 −0.899092 −0.449546 0.893257i \(-0.648414\pi\)
−0.449546 + 0.893257i \(0.648414\pi\)
\(972\) 0 0
\(973\) 14.0000 0.000461274 0
\(974\) 47256.0 1.55460
\(975\) 0 0
\(976\) 62480.0 2.04911
\(977\) 42954.0 1.40657 0.703286 0.710907i \(-0.251715\pi\)
0.703286 + 0.710907i \(0.251715\pi\)
\(978\) 0 0
\(979\) 60120.0 1.96266
\(980\) 245.000 0.00798596
\(981\) 0 0
\(982\) 19656.0 0.638746
\(983\) −10182.0 −0.330372 −0.165186 0.986262i \(-0.552822\pi\)
−0.165186 + 0.986262i \(0.552822\pi\)
\(984\) 0 0
\(985\) −5850.00 −0.189235
\(986\) −40824.0 −1.31856
\(987\) 0 0
\(988\) 4180.00 0.134599
\(989\) −4080.00 −0.131179
\(990\) 0 0
\(991\) 30008.0 0.961893 0.480946 0.876750i \(-0.340293\pi\)
0.480946 + 0.876750i \(0.340293\pi\)
\(992\) −10620.0 −0.339905
\(993\) 0 0
\(994\) −13986.0 −0.446287
\(995\) 17920.0 0.570957
\(996\) 0 0
\(997\) −47554.0 −1.51058 −0.755291 0.655390i \(-0.772504\pi\)
−0.755291 + 0.655390i \(0.772504\pi\)
\(998\) 12480.0 0.395839
\(999\) 0 0
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 315.4.a.e.1.1 yes 1
3.2 odd 2 315.4.a.b.1.1 1
5.4 even 2 1575.4.a.c.1.1 1
7.6 odd 2 2205.4.a.p.1.1 1
15.14 odd 2 1575.4.a.i.1.1 1
21.20 even 2 2205.4.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
315.4.a.b.1.1 1 3.2 odd 2
315.4.a.e.1.1 yes 1 1.1 even 1 trivial
1575.4.a.c.1.1 1 5.4 even 2
1575.4.a.i.1.1 1 15.14 odd 2
2205.4.a.f.1.1 1 21.20 even 2
2205.4.a.p.1.1 1 7.6 odd 2