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Properties

Label 180.4.a.b.1.1
Level $180$
Weight $4$
Character 180.1
Self dual yes
Analytic conductor $10.620$
Analytic rank $1$
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] = [180,4,Mod(1,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, 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("180.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 180.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: \(10.6203438010\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 180.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.00000 q^{5} +2.00000 q^{7} +O(q^{10})\) \(q-5.00000 q^{5} +2.00000 q^{7} -30.0000 q^{11} -4.00000 q^{13} -90.0000 q^{17} -28.0000 q^{19} -120.000 q^{23} +25.0000 q^{25} -210.000 q^{29} -4.00000 q^{31} -10.0000 q^{35} +200.000 q^{37} -240.000 q^{41} -136.000 q^{43} +120.000 q^{47} -339.000 q^{49} +30.0000 q^{53} +150.000 q^{55} +450.000 q^{59} -166.000 q^{61} +20.0000 q^{65} +908.000 q^{67} +1020.00 q^{71} -250.000 q^{73} -60.0000 q^{77} -916.000 q^{79} +1140.00 q^{83} +450.000 q^{85} +420.000 q^{89} -8.00000 q^{91} +140.000 q^{95} +1538.00 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 2.00000 0.107990 0.0539949 − 0.998541i \(-0.482805\pi\)
0.0539949 + 0.998541i \(0.482805\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −30.0000 −0.822304 −0.411152 − 0.911567i \(-0.634873\pi\)
−0.411152 + 0.911567i \(0.634873\pi\)
\(12\) 0 0
\(13\) −4.00000 −0.0853385 −0.0426692 − 0.999089i \(-0.513586\pi\)
−0.0426692 + 0.999089i \(0.513586\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −90.0000 −1.28401 −0.642006 − 0.766700i \(-0.721898\pi\)
−0.642006 + 0.766700i \(0.721898\pi\)
\(18\) 0 0
\(19\) −28.0000 −0.338086 −0.169043 − 0.985609i \(-0.554068\pi\)
−0.169043 + 0.985609i \(0.554068\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −120.000 −1.08790 −0.543951 − 0.839117i \(-0.683072\pi\)
−0.543951 + 0.839117i \(0.683072\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −210.000 −1.34469 −0.672345 − 0.740238i \(-0.734713\pi\)
−0.672345 + 0.740238i \(0.734713\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.0231749 −0.0115874 − 0.999933i \(-0.503688\pi\)
−0.0115874 + 0.999933i \(0.503688\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −10.0000 −0.0482945
\(36\) 0 0
\(37\) 200.000 0.888643 0.444322 − 0.895867i \(-0.353445\pi\)
0.444322 + 0.895867i \(0.353445\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −240.000 −0.914188 −0.457094 − 0.889418i \(-0.651110\pi\)
−0.457094 + 0.889418i \(0.651110\pi\)
\(42\) 0 0
\(43\) −136.000 −0.482321 −0.241161 − 0.970485i \(-0.577528\pi\)
−0.241161 + 0.970485i \(0.577528\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 120.000 0.372421 0.186211 − 0.982510i \(-0.440379\pi\)
0.186211 + 0.982510i \(0.440379\pi\)
\(48\) 0 0
\(49\) −339.000 −0.988338
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 30.0000 0.0777513 0.0388756 − 0.999244i \(-0.487622\pi\)
0.0388756 + 0.999244i \(0.487622\pi\)
\(54\) 0 0
\(55\) 150.000 0.367745
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 450.000 0.992966 0.496483 − 0.868046i \(-0.334624\pi\)
0.496483 + 0.868046i \(0.334624\pi\)
\(60\) 0 0
\(61\) −166.000 −0.348428 −0.174214 − 0.984708i \(-0.555738\pi\)
−0.174214 + 0.984708i \(0.555738\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 20.0000 0.0381645
\(66\) 0 0
\(67\) 908.000 1.65567 0.827835 − 0.560972i \(-0.189572\pi\)
0.827835 + 0.560972i \(0.189572\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1020.00 1.70495 0.852477 − 0.522765i \(-0.175099\pi\)
0.852477 + 0.522765i \(0.175099\pi\)
\(72\) 0 0
\(73\) −250.000 −0.400826 −0.200413 − 0.979712i \(-0.564228\pi\)
−0.200413 + 0.979712i \(0.564228\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −60.0000 −0.0888004
\(78\) 0 0
\(79\) −916.000 −1.30453 −0.652266 − 0.757990i \(-0.726182\pi\)
−0.652266 + 0.757990i \(0.726182\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1140.00 1.50761 0.753803 − 0.657101i \(-0.228217\pi\)
0.753803 + 0.657101i \(0.228217\pi\)
\(84\) 0 0
\(85\) 450.000 0.574228
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 420.000 0.500224 0.250112 − 0.968217i \(-0.419533\pi\)
0.250112 + 0.968217i \(0.419533\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.00921569
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 140.000 0.151197
\(96\) 0 0
\(97\) 1538.00 1.60990 0.804950 − 0.593343i \(-0.202192\pi\)
0.804950 + 0.593343i \(0.202192\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −450.000 −0.443333 −0.221667 − 0.975122i \(-0.571150\pi\)
−0.221667 + 0.975122i \(0.571150\pi\)
\(102\) 0 0
\(103\) −1150.00 −1.10012 −0.550062 − 0.835124i \(-0.685396\pi\)
−0.550062 + 0.835124i \(0.685396\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1620.00 1.46366 0.731829 − 0.681489i \(-0.238667\pi\)
0.731829 + 0.681489i \(0.238667\pi\)
\(108\) 0 0
\(109\) −1702.00 −1.49561 −0.747807 − 0.663916i \(-0.768893\pi\)
−0.747807 + 0.663916i \(0.768893\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1350.00 −1.12387 −0.561935 − 0.827181i \(-0.689943\pi\)
−0.561935 + 0.827181i \(0.689943\pi\)
\(114\) 0 0
\(115\) 600.000 0.486524
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −180.000 −0.138660
\(120\) 0 0
\(121\) −431.000 −0.323817
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 2450.00 1.71183 0.855915 − 0.517117i \(-0.172995\pi\)
0.855915 + 0.517117i \(0.172995\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −690.000 −0.460195 −0.230098 − 0.973168i \(-0.573905\pi\)
−0.230098 + 0.973168i \(0.573905\pi\)
\(132\) 0 0
\(133\) −56.0000 −0.0365099
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2070.00 −1.29089 −0.645445 − 0.763806i \(-0.723328\pi\)
−0.645445 + 0.763806i \(0.723328\pi\)
\(138\) 0 0
\(139\) −1924.00 −1.17404 −0.587020 − 0.809572i \(-0.699699\pi\)
−0.587020 + 0.809572i \(0.699699\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 120.000 0.0701742
\(144\) 0 0
\(145\) 1050.00 0.601364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2910.00 −1.59998 −0.799988 − 0.600016i \(-0.795161\pi\)
−0.799988 + 0.600016i \(0.795161\pi\)
\(150\) 0 0
\(151\) 176.000 0.0948522 0.0474261 − 0.998875i \(-0.484898\pi\)
0.0474261 + 0.998875i \(0.484898\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 20.0000 0.0103641
\(156\) 0 0
\(157\) 2348.00 1.19357 0.596786 − 0.802400i \(-0.296444\pi\)
0.596786 + 0.802400i \(0.296444\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −240.000 −0.117482
\(162\) 0 0
\(163\) −1996.00 −0.959134 −0.479567 − 0.877505i \(-0.659206\pi\)
−0.479567 + 0.877505i \(0.659206\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3120.00 −1.44571 −0.722853 − 0.691002i \(-0.757170\pi\)
−0.722853 + 0.691002i \(0.757170\pi\)
\(168\) 0 0
\(169\) −2181.00 −0.992717
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1770.00 −0.777865 −0.388932 − 0.921266i \(-0.627156\pi\)
−0.388932 + 0.921266i \(0.627156\pi\)
\(174\) 0 0
\(175\) 50.0000 0.0215980
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2130.00 0.889406 0.444703 − 0.895678i \(-0.353309\pi\)
0.444703 + 0.895678i \(0.353309\pi\)
\(180\) 0 0
\(181\) −1654.00 −0.679231 −0.339616 − 0.940564i \(-0.610297\pi\)
−0.339616 + 0.940564i \(0.610297\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1000.00 −0.397413
\(186\) 0 0
\(187\) 2700.00 1.05585
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1740.00 −0.659173 −0.329586 − 0.944125i \(-0.606909\pi\)
−0.329586 + 0.944125i \(0.606909\pi\)
\(192\) 0 0
\(193\) 86.0000 0.0320747 0.0160373 − 0.999871i \(-0.494895\pi\)
0.0160373 + 0.999871i \(0.494895\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2490.00 0.900534 0.450267 − 0.892894i \(-0.351329\pi\)
0.450267 + 0.892894i \(0.351329\pi\)
\(198\) 0 0
\(199\) −832.000 −0.296376 −0.148188 − 0.988959i \(-0.547344\pi\)
−0.148188 + 0.988959i \(0.547344\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −420.000 −0.145213
\(204\) 0 0
\(205\) 1200.00 0.408837
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 840.000 0.278010
\(210\) 0 0
\(211\) 2084.00 0.679945 0.339973 − 0.940435i \(-0.389582\pi\)
0.339973 + 0.940435i \(0.389582\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 680.000 0.215701
\(216\) 0 0
\(217\) −8.00000 −0.00250265
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 360.000 0.109576
\(222\) 0 0
\(223\) −1174.00 −0.352542 −0.176271 − 0.984342i \(-0.556403\pi\)
−0.176271 + 0.984342i \(0.556403\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3120.00 0.912254 0.456127 − 0.889915i \(-0.349236\pi\)
0.456127 + 0.889915i \(0.349236\pi\)
\(228\) 0 0
\(229\) −58.0000 −0.0167369 −0.00836845 − 0.999965i \(-0.502664\pi\)
−0.00836845 + 0.999965i \(0.502664\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5910.00 1.66170 0.830852 − 0.556494i \(-0.187854\pi\)
0.830852 + 0.556494i \(0.187854\pi\)
\(234\) 0 0
\(235\) −600.000 −0.166552
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3300.00 −0.893135 −0.446567 − 0.894750i \(-0.647354\pi\)
−0.446567 + 0.894750i \(0.647354\pi\)
\(240\) 0 0
\(241\) −2986.00 −0.798113 −0.399056 − 0.916926i \(-0.630662\pi\)
−0.399056 + 0.916926i \(0.630662\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1695.00 0.441998
\(246\) 0 0
\(247\) 112.000 0.0288518
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6630.00 1.66726 0.833629 − 0.552324i \(-0.186259\pi\)
0.833629 + 0.552324i \(0.186259\pi\)
\(252\) 0 0
\(253\) 3600.00 0.894585
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1530.00 0.371357 0.185679 − 0.982611i \(-0.440552\pi\)
0.185679 + 0.982611i \(0.440552\pi\)
\(258\) 0 0
\(259\) 400.000 0.0959644
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2640.00 −0.618971 −0.309486 − 0.950904i \(-0.600157\pi\)
−0.309486 + 0.950904i \(0.600157\pi\)
\(264\) 0 0
\(265\) −150.000 −0.0347714
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7350.00 1.66594 0.832969 − 0.553319i \(-0.186639\pi\)
0.832969 + 0.553319i \(0.186639\pi\)
\(270\) 0 0
\(271\) 3512.00 0.787228 0.393614 − 0.919276i \(-0.371225\pi\)
0.393614 + 0.919276i \(0.371225\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −750.000 −0.164461
\(276\) 0 0
\(277\) −5368.00 −1.16437 −0.582187 − 0.813055i \(-0.697803\pi\)
−0.582187 + 0.813055i \(0.697803\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3060.00 0.649624 0.324812 − 0.945779i \(-0.394699\pi\)
0.324812 + 0.945779i \(0.394699\pi\)
\(282\) 0 0
\(283\) −5044.00 −1.05949 −0.529743 − 0.848158i \(-0.677712\pi\)
−0.529743 + 0.848158i \(0.677712\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −480.000 −0.0987230
\(288\) 0 0
\(289\) 3187.00 0.648687
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2010.00 0.400769 0.200385 − 0.979717i \(-0.435781\pi\)
0.200385 + 0.979717i \(0.435781\pi\)
\(294\) 0 0
\(295\) −2250.00 −0.444068
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 480.000 0.0928399
\(300\) 0 0
\(301\) −272.000 −0.0520858
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 830.000 0.155822
\(306\) 0 0
\(307\) −2752.00 −0.511612 −0.255806 − 0.966728i \(-0.582341\pi\)
−0.255806 + 0.966728i \(0.582341\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9540.00 −1.73943 −0.869717 − 0.493551i \(-0.835699\pi\)
−0.869717 + 0.493551i \(0.835699\pi\)
\(312\) 0 0
\(313\) 9254.00 1.67114 0.835570 − 0.549384i \(-0.185137\pi\)
0.835570 + 0.549384i \(0.185137\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −150.000 −0.0265768 −0.0132884 − 0.999912i \(-0.504230\pi\)
−0.0132884 + 0.999912i \(0.504230\pi\)
\(318\) 0 0
\(319\) 6300.00 1.10574
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2520.00 0.434107
\(324\) 0 0
\(325\) −100.000 −0.0170677
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 240.000 0.0402177
\(330\) 0 0
\(331\) 1892.00 0.314180 0.157090 − 0.987584i \(-0.449789\pi\)
0.157090 + 0.987584i \(0.449789\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4540.00 −0.740438
\(336\) 0 0
\(337\) −7378.00 −1.19260 −0.596299 − 0.802763i \(-0.703363\pi\)
−0.596299 + 0.802763i \(0.703363\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 120.000 0.0190568
\(342\) 0 0
\(343\) −1364.00 −0.214720
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6720.00 −1.03962 −0.519811 − 0.854282i \(-0.673997\pi\)
−0.519811 + 0.854282i \(0.673997\pi\)
\(348\) 0 0
\(349\) 5186.00 0.795416 0.397708 − 0.917512i \(-0.369806\pi\)
0.397708 + 0.917512i \(0.369806\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3330.00 −0.502091 −0.251045 − 0.967975i \(-0.580774\pi\)
−0.251045 + 0.967975i \(0.580774\pi\)
\(354\) 0 0
\(355\) −5100.00 −0.762479
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −9000.00 −1.32312 −0.661562 − 0.749890i \(-0.730106\pi\)
−0.661562 + 0.749890i \(0.730106\pi\)
\(360\) 0 0
\(361\) −6075.00 −0.885698
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1250.00 0.179255
\(366\) 0 0
\(367\) −8758.00 −1.24568 −0.622839 − 0.782350i \(-0.714021\pi\)
−0.622839 + 0.782350i \(0.714021\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 60.0000 0.00839635
\(372\) 0 0
\(373\) 4724.00 0.655763 0.327881 − 0.944719i \(-0.393665\pi\)
0.327881 + 0.944719i \(0.393665\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 840.000 0.114754
\(378\) 0 0
\(379\) 7292.00 0.988298 0.494149 − 0.869377i \(-0.335480\pi\)
0.494149 + 0.869377i \(0.335480\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −14520.0 −1.93717 −0.968587 − 0.248676i \(-0.920004\pi\)
−0.968587 + 0.248676i \(0.920004\pi\)
\(384\) 0 0
\(385\) 300.000 0.0397128
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7110.00 −0.926713 −0.463356 − 0.886172i \(-0.653355\pi\)
−0.463356 + 0.886172i \(0.653355\pi\)
\(390\) 0 0
\(391\) 10800.0 1.39688
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4580.00 0.583405
\(396\) 0 0
\(397\) −11488.0 −1.45231 −0.726154 − 0.687532i \(-0.758694\pi\)
−0.726154 + 0.687532i \(0.758694\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −780.000 −0.0971355 −0.0485678 − 0.998820i \(-0.515466\pi\)
−0.0485678 + 0.998820i \(0.515466\pi\)
\(402\) 0 0
\(403\) 16.0000 0.00197771
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6000.00 −0.730735
\(408\) 0 0
\(409\) 5402.00 0.653085 0.326542 − 0.945183i \(-0.394116\pi\)
0.326542 + 0.945183i \(0.394116\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 900.000 0.107230
\(414\) 0 0
\(415\) −5700.00 −0.674222
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2190.00 0.255342 0.127671 − 0.991817i \(-0.459250\pi\)
0.127671 + 0.991817i \(0.459250\pi\)
\(420\) 0 0
\(421\) −7162.00 −0.829108 −0.414554 − 0.910025i \(-0.636062\pi\)
−0.414554 + 0.910025i \(0.636062\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2250.00 −0.256802
\(426\) 0 0
\(427\) −332.000 −0.0376267
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9360.00 −1.04607 −0.523034 − 0.852312i \(-0.675200\pi\)
−0.523034 + 0.852312i \(0.675200\pi\)
\(432\) 0 0
\(433\) 12806.0 1.42129 0.710643 − 0.703552i \(-0.248404\pi\)
0.710643 + 0.703552i \(0.248404\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3360.00 0.367805
\(438\) 0 0
\(439\) 11288.0 1.22721 0.613607 − 0.789612i \(-0.289718\pi\)
0.613607 + 0.789612i \(0.289718\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8520.00 0.913764 0.456882 − 0.889527i \(-0.348966\pi\)
0.456882 + 0.889527i \(0.348966\pi\)
\(444\) 0 0
\(445\) −2100.00 −0.223707
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1260.00 0.132434 0.0662172 − 0.997805i \(-0.478907\pi\)
0.0662172 + 0.997805i \(0.478907\pi\)
\(450\) 0 0
\(451\) 7200.00 0.751740
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 40.0000 0.00412138
\(456\) 0 0
\(457\) −13750.0 −1.40744 −0.703718 − 0.710480i \(-0.748478\pi\)
−0.703718 + 0.710480i \(0.748478\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3210.00 0.324305 0.162152 − 0.986766i \(-0.448156\pi\)
0.162152 + 0.986766i \(0.448156\pi\)
\(462\) 0 0
\(463\) −12850.0 −1.28983 −0.644914 − 0.764255i \(-0.723107\pi\)
−0.644914 + 0.764255i \(0.723107\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8220.00 0.814510 0.407255 − 0.913314i \(-0.366486\pi\)
0.407255 + 0.913314i \(0.366486\pi\)
\(468\) 0 0
\(469\) 1816.00 0.178795
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4080.00 0.396614
\(474\) 0 0
\(475\) −700.000 −0.0676173
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7020.00 0.669628 0.334814 − 0.942284i \(-0.391326\pi\)
0.334814 + 0.942284i \(0.391326\pi\)
\(480\) 0 0
\(481\) −800.000 −0.0758355
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7690.00 −0.719969
\(486\) 0 0
\(487\) −8122.00 −0.755735 −0.377868 − 0.925860i \(-0.623343\pi\)
−0.377868 + 0.925860i \(0.623343\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 13470.0 1.23807 0.619035 − 0.785363i \(-0.287524\pi\)
0.619035 + 0.785363i \(0.287524\pi\)
\(492\) 0 0
\(493\) 18900.0 1.72660
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2040.00 0.184118
\(498\) 0 0
\(499\) 2468.00 0.221409 0.110704 − 0.993853i \(-0.464689\pi\)
0.110704 + 0.993853i \(0.464689\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4440.00 0.393578 0.196789 − 0.980446i \(-0.436949\pi\)
0.196789 + 0.980446i \(0.436949\pi\)
\(504\) 0 0
\(505\) 2250.00 0.198265
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 11190.0 0.974436 0.487218 − 0.873280i \(-0.338012\pi\)
0.487218 + 0.873280i \(0.338012\pi\)
\(510\) 0 0
\(511\) −500.000 −0.0432851
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5750.00 0.491991
\(516\) 0 0
\(517\) −3600.00 −0.306243
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4020.00 0.338041 0.169021 − 0.985613i \(-0.445940\pi\)
0.169021 + 0.985613i \(0.445940\pi\)
\(522\) 0 0
\(523\) −9076.00 −0.758826 −0.379413 − 0.925228i \(-0.623874\pi\)
−0.379413 + 0.925228i \(0.623874\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 360.000 0.0297568
\(528\) 0 0
\(529\) 2233.00 0.183529
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 960.000 0.0780154
\(534\) 0 0
\(535\) −8100.00 −0.654567
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10170.0 0.812714
\(540\) 0 0
\(541\) −7486.00 −0.594914 −0.297457 − 0.954735i \(-0.596138\pi\)
−0.297457 + 0.954735i \(0.596138\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8510.00 0.668859
\(546\) 0 0
\(547\) 7400.00 0.578430 0.289215 − 0.957264i \(-0.406606\pi\)
0.289215 + 0.957264i \(0.406606\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5880.00 0.454621
\(552\) 0 0
\(553\) −1832.00 −0.140876
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −11490.0 −0.874052 −0.437026 − 0.899449i \(-0.643968\pi\)
−0.437026 + 0.899449i \(0.643968\pi\)
\(558\) 0 0
\(559\) 544.000 0.0411606
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19320.0 1.44625 0.723127 − 0.690715i \(-0.242704\pi\)
0.723127 + 0.690715i \(0.242704\pi\)
\(564\) 0 0
\(565\) 6750.00 0.502610
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8340.00 −0.614466 −0.307233 − 0.951634i \(-0.599403\pi\)
−0.307233 + 0.951634i \(0.599403\pi\)
\(570\) 0 0
\(571\) 21044.0 1.54232 0.771159 − 0.636642i \(-0.219677\pi\)
0.771159 + 0.636642i \(0.219677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3000.00 −0.217580
\(576\) 0 0
\(577\) 1418.00 0.102309 0.0511543 − 0.998691i \(-0.483710\pi\)
0.0511543 + 0.998691i \(0.483710\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2280.00 0.162806
\(582\) 0 0
\(583\) −900.000 −0.0639351
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22020.0 1.54832 0.774159 − 0.632991i \(-0.218173\pi\)
0.774159 + 0.632991i \(0.218173\pi\)
\(588\) 0 0
\(589\) 112.000 0.00783511
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −25230.0 −1.74717 −0.873585 − 0.486671i \(-0.838211\pi\)
−0.873585 + 0.486671i \(0.838211\pi\)
\(594\) 0 0
\(595\) 900.000 0.0620108
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8280.00 −0.564794 −0.282397 − 0.959298i \(-0.591130\pi\)
−0.282397 + 0.959298i \(0.591130\pi\)
\(600\) 0 0
\(601\) −18874.0 −1.28101 −0.640505 − 0.767954i \(-0.721275\pi\)
−0.640505 + 0.767954i \(0.721275\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2155.00 0.144815
\(606\) 0 0
\(607\) 10550.0 0.705455 0.352728 − 0.935726i \(-0.385254\pi\)
0.352728 + 0.935726i \(0.385254\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −480.000 −0.0317819
\(612\) 0 0
\(613\) 11000.0 0.724773 0.362386 − 0.932028i \(-0.381962\pi\)
0.362386 + 0.932028i \(0.381962\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11310.0 0.737963 0.368982 − 0.929437i \(-0.379706\pi\)
0.368982 + 0.929437i \(0.379706\pi\)
\(618\) 0 0
\(619\) −17572.0 −1.14100 −0.570499 − 0.821298i \(-0.693250\pi\)
−0.570499 + 0.821298i \(0.693250\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 840.000 0.0540191
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −18000.0 −1.14103
\(630\) 0 0
\(631\) 1604.00 0.101195 0.0505976 − 0.998719i \(-0.483887\pi\)
0.0505976 + 0.998719i \(0.483887\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −12250.0 −0.765553
\(636\) 0 0
\(637\) 1356.00 0.0843433
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −31320.0 −1.92990 −0.964950 − 0.262435i \(-0.915475\pi\)
−0.964950 + 0.262435i \(0.915475\pi\)
\(642\) 0 0
\(643\) −31300.0 −1.91968 −0.959838 − 0.280555i \(-0.909481\pi\)
−0.959838 + 0.280555i \(0.909481\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −10920.0 −0.663539 −0.331769 − 0.943361i \(-0.607646\pi\)
−0.331769 + 0.943361i \(0.607646\pi\)
\(648\) 0 0
\(649\) −13500.0 −0.816520
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3210.00 0.192369 0.0961845 − 0.995364i \(-0.469336\pi\)
0.0961845 + 0.995364i \(0.469336\pi\)
\(654\) 0 0
\(655\) 3450.00 0.205806
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 11910.0 0.704018 0.352009 − 0.935997i \(-0.385499\pi\)
0.352009 + 0.935997i \(0.385499\pi\)
\(660\) 0 0
\(661\) −3382.00 −0.199008 −0.0995042 − 0.995037i \(-0.531726\pi\)
−0.0995042 + 0.995037i \(0.531726\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 280.000 0.0163277
\(666\) 0 0
\(667\) 25200.0 1.46289
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4980.00 0.286514
\(672\) 0 0
\(673\) 15950.0 0.913562 0.456781 − 0.889579i \(-0.349002\pi\)
0.456781 + 0.889579i \(0.349002\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −32190.0 −1.82742 −0.913709 − 0.406369i \(-0.866795\pi\)
−0.913709 + 0.406369i \(0.866795\pi\)
\(678\) 0 0
\(679\) 3076.00 0.173853
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −22140.0 −1.24036 −0.620178 − 0.784461i \(-0.712940\pi\)
−0.620178 + 0.784461i \(0.712940\pi\)
\(684\) 0 0
\(685\) 10350.0 0.577304
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −120.000 −0.00663518
\(690\) 0 0
\(691\) −6172.00 −0.339789 −0.169894 − 0.985462i \(-0.554343\pi\)
−0.169894 + 0.985462i \(0.554343\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9620.00 0.525047
\(696\) 0 0
\(697\) 21600.0 1.17383
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −19170.0 −1.03287 −0.516434 − 0.856327i \(-0.672741\pi\)
−0.516434 + 0.856327i \(0.672741\pi\)
\(702\) 0 0
\(703\) −5600.00 −0.300438
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −900.000 −0.0478755
\(708\) 0 0
\(709\) −21898.0 −1.15994 −0.579969 − 0.814638i \(-0.696936\pi\)
−0.579969 + 0.814638i \(0.696936\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 480.000 0.0252120
\(714\) 0 0
\(715\) −600.000 −0.0313828
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −16680.0 −0.865173 −0.432586 − 0.901593i \(-0.642399\pi\)
−0.432586 + 0.901593i \(0.642399\pi\)
\(720\) 0 0
\(721\) −2300.00 −0.118802
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5250.00 −0.268938
\(726\) 0 0
\(727\) 6518.00 0.332516 0.166258 − 0.986082i \(-0.446832\pi\)
0.166258 + 0.986082i \(0.446832\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12240.0 0.619306
\(732\) 0 0
\(733\) −23200.0 −1.16905 −0.584524 − 0.811377i \(-0.698719\pi\)
−0.584524 + 0.811377i \(0.698719\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −27240.0 −1.36146
\(738\) 0 0
\(739\) −16324.0 −0.812568 −0.406284 − 0.913747i \(-0.633176\pi\)
−0.406284 + 0.913747i \(0.633176\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 120.000 0.00592513 0.00296257 − 0.999996i \(-0.499057\pi\)
0.00296257 + 0.999996i \(0.499057\pi\)
\(744\) 0 0
\(745\) 14550.0 0.715531
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 3240.00 0.158060
\(750\) 0 0
\(751\) 30548.0 1.48430 0.742152 − 0.670232i \(-0.233805\pi\)
0.742152 + 0.670232i \(0.233805\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −880.000 −0.0424192
\(756\) 0 0
\(757\) 16952.0 0.813911 0.406956 − 0.913448i \(-0.366590\pi\)
0.406956 + 0.913448i \(0.366590\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20220.0 0.963173 0.481586 − 0.876399i \(-0.340061\pi\)
0.481586 + 0.876399i \(0.340061\pi\)
\(762\) 0 0
\(763\) −3404.00 −0.161511
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1800.00 −0.0847382
\(768\) 0 0
\(769\) −20722.0 −0.971722 −0.485861 − 0.874036i \(-0.661494\pi\)
−0.485861 + 0.874036i \(0.661494\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −4350.00 −0.202404 −0.101202 − 0.994866i \(-0.532269\pi\)
−0.101202 + 0.994866i \(0.532269\pi\)
\(774\) 0 0
\(775\) −100.000 −0.00463498
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6720.00 0.309074
\(780\) 0 0
\(781\) −30600.0 −1.40199
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −11740.0 −0.533782
\(786\) 0 0
\(787\) 41972.0 1.90107 0.950534 − 0.310621i \(-0.100537\pi\)
0.950534 + 0.310621i \(0.100537\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2700.00 −0.121367
\(792\) 0 0
\(793\) 664.000 0.0297343
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 39510.0 1.75598 0.877990 − 0.478679i \(-0.158884\pi\)
0.877990 + 0.478679i \(0.158884\pi\)
\(798\) 0 0
\(799\) −10800.0 −0.478193
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7500.00 0.329601
\(804\) 0 0
\(805\) 1200.00 0.0525397
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16680.0 −0.724892 −0.362446 − 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(810\) 0 0
\(811\) −15484.0 −0.670428 −0.335214 − 0.942142i \(-0.608809\pi\)
−0.335214 + 0.942142i \(0.608809\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9980.00 0.428938
\(816\) 0 0
\(817\) 3808.00 0.163066
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −4170.00 −0.177264 −0.0886322 − 0.996064i \(-0.528250\pi\)
−0.0886322 + 0.996064i \(0.528250\pi\)
\(822\) 0 0
\(823\) −30226.0 −1.28021 −0.640105 − 0.768288i \(-0.721109\pi\)
−0.640105 + 0.768288i \(0.721109\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 14760.0 0.620623 0.310312 − 0.950635i \(-0.399567\pi\)
0.310312 + 0.950635i \(0.399567\pi\)
\(828\) 0 0
\(829\) −9934.00 −0.416191 −0.208095 − 0.978109i \(-0.566726\pi\)
−0.208095 + 0.978109i \(0.566726\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 30510.0 1.26904
\(834\) 0 0
\(835\) 15600.0 0.646539
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 23520.0 0.967820 0.483910 − 0.875118i \(-0.339216\pi\)
0.483910 + 0.875118i \(0.339216\pi\)
\(840\) 0 0
\(841\) 19711.0 0.808192
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 10905.0 0.443957
\(846\) 0 0
\(847\) −862.000 −0.0349689
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −24000.0 −0.966756
\(852\) 0 0
\(853\) 29816.0 1.19681 0.598406 − 0.801193i \(-0.295801\pi\)
0.598406 + 0.801193i \(0.295801\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 35430.0 1.41221 0.706106 − 0.708106i \(-0.250450\pi\)
0.706106 + 0.708106i \(0.250450\pi\)
\(858\) 0 0
\(859\) −36196.0 −1.43771 −0.718854 − 0.695161i \(-0.755333\pi\)
−0.718854 + 0.695161i \(0.755333\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −480.000 −0.0189332 −0.00946662 − 0.999955i \(-0.503013\pi\)
−0.00946662 + 0.999955i \(0.503013\pi\)
\(864\) 0 0
\(865\) 8850.00 0.347872
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 27480.0 1.07272
\(870\) 0 0
\(871\) −3632.00 −0.141292
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −250.000 −0.00965891
\(876\) 0 0
\(877\) 28532.0 1.09858 0.549291 − 0.835631i \(-0.314898\pi\)
0.549291 + 0.835631i \(0.314898\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 20340.0 0.777834 0.388917 − 0.921273i \(-0.372849\pi\)
0.388917 + 0.921273i \(0.372849\pi\)
\(882\) 0 0
\(883\) −10756.0 −0.409930 −0.204965 − 0.978769i \(-0.565708\pi\)
−0.204965 + 0.978769i \(0.565708\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −600.000 −0.0227125 −0.0113563 − 0.999936i \(-0.503615\pi\)
−0.0113563 + 0.999936i \(0.503615\pi\)
\(888\) 0 0
\(889\) 4900.00 0.184860
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3360.00 −0.125911
\(894\) 0 0
\(895\) −10650.0 −0.397754
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 840.000 0.0311630
\(900\) 0 0
\(901\) −2700.00 −0.0998336
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 8270.00 0.303761
\(906\) 0 0
\(907\) 25400.0 0.929871 0.464936 − 0.885345i \(-0.346077\pi\)
0.464936 + 0.885345i \(0.346077\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 36240.0 1.31799 0.658993 − 0.752149i \(-0.270983\pi\)
0.658993 + 0.752149i \(0.270983\pi\)
\(912\) 0 0
\(913\) −34200.0 −1.23971
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1380.00 −0.0496964
\(918\) 0 0
\(919\) 6572.00 0.235898 0.117949 − 0.993020i \(-0.462368\pi\)
0.117949 + 0.993020i \(0.462368\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −4080.00 −0.145498
\(924\) 0 0
\(925\) 5000.00 0.177729
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −2340.00 −0.0826404 −0.0413202 − 0.999146i \(-0.513156\pi\)
−0.0413202 + 0.999146i \(0.513156\pi\)
\(930\) 0 0
\(931\) 9492.00 0.334144
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −13500.0 −0.472190
\(936\) 0 0
\(937\) 2522.00 0.0879297 0.0439649 − 0.999033i \(-0.486001\pi\)
0.0439649 + 0.999033i \(0.486001\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 52770.0 1.82811 0.914056 − 0.405589i \(-0.132933\pi\)
0.914056 + 0.405589i \(0.132933\pi\)
\(942\) 0 0
\(943\) 28800.0 0.994546
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −28200.0 −0.967663 −0.483832 − 0.875161i \(-0.660755\pi\)
−0.483832 + 0.875161i \(0.660755\pi\)
\(948\) 0 0
\(949\) 1000.00 0.0342059
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 15570.0 0.529236 0.264618 − 0.964353i \(-0.414754\pi\)
0.264618 + 0.964353i \(0.414754\pi\)
\(954\) 0 0
\(955\) 8700.00 0.294791
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4140.00 −0.139403
\(960\) 0 0
\(961\) −29775.0 −0.999463
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −430.000 −0.0143442
\(966\) 0 0
\(967\) −8350.00 −0.277681 −0.138841 − 0.990315i \(-0.544338\pi\)
−0.138841 + 0.990315i \(0.544338\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −43650.0 −1.44263 −0.721316 − 0.692606i \(-0.756462\pi\)
−0.721316 + 0.692606i \(0.756462\pi\)
\(972\) 0 0
\(973\) −3848.00 −0.126784
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −18810.0 −0.615952 −0.307976 − 0.951394i \(-0.599652\pi\)
−0.307976 + 0.951394i \(0.599652\pi\)
\(978\) 0 0
\(979\) −12600.0 −0.411336
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 25320.0 0.821549 0.410774 − 0.911737i \(-0.365258\pi\)
0.410774 + 0.911737i \(0.365258\pi\)
\(984\) 0 0
\(985\) −12450.0 −0.402731
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16320.0 0.524718
\(990\) 0 0
\(991\) −6736.00 −0.215919 −0.107960 − 0.994155i \(-0.534432\pi\)
−0.107960 + 0.994155i \(0.534432\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4160.00 0.132544
\(996\) 0 0
\(997\) −20500.0 −0.651195 −0.325598 − 0.945508i \(-0.605565\pi\)
−0.325598 + 0.945508i \(0.605565\pi\)
\(998\) 0 0
\(999\) 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 180.4.a.b.1.1 ✓ 1
3.2 odd 2 180.4.a.e.1.1 yes 1
4.3 odd 2 720.4.a.h.1.1 1
5.2 odd 4 900.4.d.d.649.2 2
5.3 odd 4 900.4.d.d.649.1 2
5.4 even 2 900.4.a.i.1.1 1
9.2 odd 6 1620.4.i.c.1081.1 2
9.4 even 3 1620.4.i.i.541.1 2
9.5 odd 6 1620.4.i.c.541.1 2
9.7 even 3 1620.4.i.i.1081.1 2
12.11 even 2 720.4.a.w.1.1 1
15.2 even 4 900.4.d.i.649.2 2
15.8 even 4 900.4.d.i.649.1 2
15.14 odd 2 900.4.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.4.a.b.1.1 ✓ 1 1.1 even 1 trivial
180.4.a.e.1.1 yes 1 3.2 odd 2
720.4.a.h.1.1 1 4.3 odd 2
720.4.a.w.1.1 1 12.11 even 2
900.4.a.i.1.1 1 5.4 even 2
900.4.a.j.1.1 1 15.14 odd 2
900.4.d.d.649.1 2 5.3 odd 4
900.4.d.d.649.2 2 5.2 odd 4
900.4.d.i.649.1 2 15.8 even 4
900.4.d.i.649.2 2 15.2 even 4
1620.4.i.c.541.1 2 9.5 odd 6
1620.4.i.c.1081.1 2 9.2 odd 6
1620.4.i.i.541.1 2 9.4 even 3
1620.4.i.i.1081.1 2 9.7 even 3