Properties

Label 1728.4.c.i.1727.4
Level $1728$
Weight $4$
Character 1728.1727
Analytic conductor $101.955$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,4,Mod(1727,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.1727");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 12x^{10} + 112x^{8} - 368x^{6} + 928x^{4} - 256x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{36}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1727.4
Root \(0.456937 - 0.263813i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.4.c.i.1727.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-13.1987i q^{5} +4.49091i q^{7} +O(q^{10})\) \(q-13.1987i q^{5} +4.49091i q^{7} +22.3519 q^{11} +73.5402 q^{13} +42.6199i q^{17} +122.563i q^{19} -197.965 q^{23} -49.2047 q^{25} +14.6816i q^{29} +147.133i q^{31} +59.2740 q^{35} -234.154 q^{37} +396.340i q^{41} -280.764i q^{43} -534.237 q^{47} +322.832 q^{49} -337.497i q^{53} -295.016i q^{55} -672.928 q^{59} +80.8693 q^{61} -970.632i q^{65} +251.791i q^{67} +95.8124 q^{71} -251.422 q^{73} +100.381i q^{77} +499.839i q^{79} +16.1731 q^{83} +562.526 q^{85} -321.011i q^{89} +330.263i q^{91} +1617.67 q^{95} -210.036 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 36 q^{13} - 132 q^{25} - 516 q^{37} - 720 q^{49} + 972 q^{61} + 660 q^{73} - 1056 q^{85} + 2532 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(5\) − 13.1987i − 1.18052i −0.807212 0.590262i \(-0.799024\pi\)
0.807212 0.590262i \(-0.200976\pi\)
\(6\) 0 0
\(7\) 4.49091i 0.242486i 0.992623 + 0.121243i \(0.0386881\pi\)
−0.992623 + 0.121243i \(0.961312\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 22.3519 0.612669 0.306335 0.951924i \(-0.400897\pi\)
0.306335 + 0.951924i \(0.400897\pi\)
\(12\) 0 0
\(13\) 73.5402 1.56895 0.784476 0.620159i \(-0.212932\pi\)
0.784476 + 0.620159i \(0.212932\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 42.6199i 0.608050i 0.952664 + 0.304025i \(0.0983307\pi\)
−0.952664 + 0.304025i \(0.901669\pi\)
\(18\) 0 0
\(19\) 122.563i 1.47989i 0.672669 + 0.739943i \(0.265148\pi\)
−0.672669 + 0.739943i \(0.734852\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −197.965 −1.79472 −0.897360 0.441299i \(-0.854518\pi\)
−0.897360 + 0.441299i \(0.854518\pi\)
\(24\) 0 0
\(25\) −49.2047 −0.393638
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 14.6816i 0.0940102i 0.998895 + 0.0470051i \(0.0149677\pi\)
−0.998895 + 0.0470051i \(0.985032\pi\)
\(30\) 0 0
\(31\) 147.133i 0.852448i 0.904618 + 0.426224i \(0.140156\pi\)
−0.904618 + 0.426224i \(0.859844\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 59.2740 0.286261
\(36\) 0 0
\(37\) −234.154 −1.04040 −0.520199 0.854045i \(-0.674142\pi\)
−0.520199 + 0.854045i \(0.674142\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 396.340i 1.50970i 0.655895 + 0.754852i \(0.272292\pi\)
−0.655895 + 0.754852i \(0.727708\pi\)
\(42\) 0 0
\(43\) − 280.764i − 0.995725i −0.867256 0.497863i \(-0.834118\pi\)
0.867256 0.497863i \(-0.165882\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −534.237 −1.65801 −0.829006 0.559240i \(-0.811093\pi\)
−0.829006 + 0.559240i \(0.811093\pi\)
\(48\) 0 0
\(49\) 322.832 0.941200
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 337.497i − 0.874695i −0.899293 0.437347i \(-0.855918\pi\)
0.899293 0.437347i \(-0.144082\pi\)
\(54\) 0 0
\(55\) − 295.016i − 0.723271i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −672.928 −1.48488 −0.742439 0.669914i \(-0.766331\pi\)
−0.742439 + 0.669914i \(0.766331\pi\)
\(60\) 0 0
\(61\) 80.8693 0.169742 0.0848709 0.996392i \(-0.472952\pi\)
0.0848709 + 0.996392i \(0.472952\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 970.632i − 1.85219i
\(66\) 0 0
\(67\) 251.791i 0.459121i 0.973294 + 0.229560i \(0.0737289\pi\)
−0.973294 + 0.229560i \(0.926271\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 95.8124 0.160153 0.0800763 0.996789i \(-0.474484\pi\)
0.0800763 + 0.996789i \(0.474484\pi\)
\(72\) 0 0
\(73\) −251.422 −0.403106 −0.201553 0.979478i \(-0.564599\pi\)
−0.201553 + 0.979478i \(0.564599\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 100.381i 0.148564i
\(78\) 0 0
\(79\) 499.839i 0.711852i 0.934514 + 0.355926i \(0.115835\pi\)
−0.934514 + 0.355926i \(0.884165\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.1731 0.0213882 0.0106941 0.999943i \(-0.496596\pi\)
0.0106941 + 0.999943i \(0.496596\pi\)
\(84\) 0 0
\(85\) 562.526 0.717818
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 321.011i − 0.382327i −0.981558 0.191164i \(-0.938774\pi\)
0.981558 0.191164i \(-0.0612261\pi\)
\(90\) 0 0
\(91\) 330.263i 0.380450i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1617.67 1.74704
\(96\) 0 0
\(97\) −210.036 −0.219855 −0.109928 0.993940i \(-0.535062\pi\)
−0.109928 + 0.993940i \(0.535062\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1568.74i 1.54550i 0.634713 + 0.772748i \(0.281118\pi\)
−0.634713 + 0.772748i \(0.718882\pi\)
\(102\) 0 0
\(103\) 544.057i 0.520462i 0.965546 + 0.260231i \(0.0837987\pi\)
−0.965546 + 0.260231i \(0.916201\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1105.78 −0.999064 −0.499532 0.866295i \(-0.666495\pi\)
−0.499532 + 0.866295i \(0.666495\pi\)
\(108\) 0 0
\(109\) −17.6601 −0.0155186 −0.00775932 0.999970i \(-0.502470\pi\)
−0.00775932 + 0.999970i \(0.502470\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1603.29i 1.33473i 0.744729 + 0.667367i \(0.232579\pi\)
−0.744729 + 0.667367i \(0.767421\pi\)
\(114\) 0 0
\(115\) 2612.87i 2.11871i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −191.402 −0.147444
\(120\) 0 0
\(121\) −831.391 −0.624636
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 1000.40i − 0.715825i
\(126\) 0 0
\(127\) 2080.04i 1.45334i 0.686988 + 0.726669i \(0.258932\pi\)
−0.686988 + 0.726669i \(0.741068\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1093.35 0.729213 0.364606 0.931162i \(-0.381204\pi\)
0.364606 + 0.931162i \(0.381204\pi\)
\(132\) 0 0
\(133\) −550.419 −0.358853
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 1344.61i − 0.838522i −0.907866 0.419261i \(-0.862289\pi\)
0.907866 0.419261i \(-0.137711\pi\)
\(138\) 0 0
\(139\) 805.378i 0.491448i 0.969340 + 0.245724i \(0.0790257\pi\)
−0.969340 + 0.245724i \(0.920974\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1643.77 0.961249
\(144\) 0 0
\(145\) 193.777 0.110981
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 879.365i 0.483493i 0.970339 + 0.241746i \(0.0777202\pi\)
−0.970339 + 0.241746i \(0.922280\pi\)
\(150\) 0 0
\(151\) 1826.20i 0.984197i 0.870540 + 0.492098i \(0.163770\pi\)
−0.870540 + 0.492098i \(0.836230\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1941.96 1.00634
\(156\) 0 0
\(157\) −3417.81 −1.73740 −0.868698 0.495342i \(-0.835043\pi\)
−0.868698 + 0.495342i \(0.835043\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 889.044i − 0.435195i
\(162\) 0 0
\(163\) − 1190.78i − 0.572204i −0.958199 0.286102i \(-0.907640\pi\)
0.958199 0.286102i \(-0.0923596\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2123.75 −0.984077 −0.492039 0.870573i \(-0.663748\pi\)
−0.492039 + 0.870573i \(0.663748\pi\)
\(168\) 0 0
\(169\) 3211.16 1.46161
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 1769.12i − 0.777477i −0.921348 0.388738i \(-0.872911\pi\)
0.921348 0.388738i \(-0.127089\pi\)
\(174\) 0 0
\(175\) − 220.974i − 0.0954519i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3685.89 1.53909 0.769543 0.638595i \(-0.220484\pi\)
0.769543 + 0.638595i \(0.220484\pi\)
\(180\) 0 0
\(181\) 2425.51 0.996059 0.498030 0.867160i \(-0.334057\pi\)
0.498030 + 0.867160i \(0.334057\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3090.52i 1.22822i
\(186\) 0 0
\(187\) 952.638i 0.372534i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1367.60 0.518096 0.259048 0.965865i \(-0.416591\pi\)
0.259048 + 0.965865i \(0.416591\pi\)
\(192\) 0 0
\(193\) −1240.32 −0.462591 −0.231296 0.972883i \(-0.574296\pi\)
−0.231296 + 0.972883i \(0.574296\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2889.56i − 1.04504i −0.852628 0.522519i \(-0.824992\pi\)
0.852628 0.522519i \(-0.175008\pi\)
\(198\) 0 0
\(199\) 1851.53i 0.659555i 0.944059 + 0.329777i \(0.106974\pi\)
−0.944059 + 0.329777i \(0.893026\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −65.9336 −0.0227962
\(204\) 0 0
\(205\) 5231.16 1.78224
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2739.52i 0.906681i
\(210\) 0 0
\(211\) 4177.37i 1.36295i 0.731843 + 0.681474i \(0.238661\pi\)
−0.731843 + 0.681474i \(0.761339\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3705.72 −1.17548
\(216\) 0 0
\(217\) −660.762 −0.206707
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3134.28i 0.954002i
\(222\) 0 0
\(223\) − 1155.67i − 0.347039i −0.984830 0.173519i \(-0.944486\pi\)
0.984830 0.173519i \(-0.0555139\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1499.50 0.438437 0.219218 0.975676i \(-0.429649\pi\)
0.219218 + 0.975676i \(0.429649\pi\)
\(228\) 0 0
\(229\) 320.011 0.0923445 0.0461723 0.998933i \(-0.485298\pi\)
0.0461723 + 0.998933i \(0.485298\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 1446.64i − 0.406749i −0.979101 0.203374i \(-0.934809\pi\)
0.979101 0.203374i \(-0.0651909\pi\)
\(234\) 0 0
\(235\) 7051.22i 1.95732i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1432.06 0.387582 0.193791 0.981043i \(-0.437922\pi\)
0.193791 + 0.981043i \(0.437922\pi\)
\(240\) 0 0
\(241\) −1148.97 −0.307102 −0.153551 0.988141i \(-0.549071\pi\)
−0.153551 + 0.988141i \(0.549071\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 4260.95i − 1.11111i
\(246\) 0 0
\(247\) 9013.30i 2.32187i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3652.62 −0.918531 −0.459266 0.888299i \(-0.651887\pi\)
−0.459266 + 0.888299i \(0.651887\pi\)
\(252\) 0 0
\(253\) −4424.90 −1.09957
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 5362.37i − 1.30154i −0.759275 0.650770i \(-0.774446\pi\)
0.759275 0.650770i \(-0.225554\pi\)
\(258\) 0 0
\(259\) − 1051.57i − 0.252283i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1096.98 0.257196 0.128598 0.991697i \(-0.458952\pi\)
0.128598 + 0.991697i \(0.458952\pi\)
\(264\) 0 0
\(265\) −4454.51 −1.03260
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5602.35i 1.26982i 0.772586 + 0.634910i \(0.218963\pi\)
−0.772586 + 0.634910i \(0.781037\pi\)
\(270\) 0 0
\(271\) 1051.62i 0.235725i 0.993030 + 0.117862i \(0.0376041\pi\)
−0.993030 + 0.117862i \(0.962396\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1099.82 −0.241170
\(276\) 0 0
\(277\) −2739.10 −0.594139 −0.297069 0.954856i \(-0.596009\pi\)
−0.297069 + 0.954856i \(0.596009\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 423.244i − 0.0898528i −0.998990 0.0449264i \(-0.985695\pi\)
0.998990 0.0449264i \(-0.0143053\pi\)
\(282\) 0 0
\(283\) 1630.51i 0.342486i 0.985229 + 0.171243i \(0.0547783\pi\)
−0.985229 + 0.171243i \(0.945222\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1779.93 −0.366083
\(288\) 0 0
\(289\) 3096.54 0.630275
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2352.75i 0.469109i 0.972103 + 0.234554i \(0.0753631\pi\)
−0.972103 + 0.234554i \(0.924637\pi\)
\(294\) 0 0
\(295\) 8881.76i 1.75293i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −14558.4 −2.81583
\(300\) 0 0
\(301\) 1260.89 0.241450
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 1067.37i − 0.200384i
\(306\) 0 0
\(307\) 10088.6i 1.87554i 0.347263 + 0.937768i \(0.387111\pi\)
−0.347263 + 0.937768i \(0.612889\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3970.97 −0.724029 −0.362015 0.932172i \(-0.617911\pi\)
−0.362015 + 0.932172i \(0.617911\pi\)
\(312\) 0 0
\(313\) 4539.33 0.819738 0.409869 0.912144i \(-0.365574\pi\)
0.409869 + 0.912144i \(0.365574\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 4211.47i − 0.746181i −0.927795 0.373091i \(-0.878298\pi\)
0.927795 0.373091i \(-0.121702\pi\)
\(318\) 0 0
\(319\) 328.161i 0.0575972i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5223.62 −0.899846
\(324\) 0 0
\(325\) −3618.52 −0.617599
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 2399.21i − 0.402045i
\(330\) 0 0
\(331\) 3143.43i 0.521990i 0.965340 + 0.260995i \(0.0840506\pi\)
−0.965340 + 0.260995i \(0.915949\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3323.30 0.542004
\(336\) 0 0
\(337\) 7102.24 1.14802 0.574012 0.818847i \(-0.305386\pi\)
0.574012 + 0.818847i \(0.305386\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3288.71i 0.522269i
\(342\) 0 0
\(343\) 2990.19i 0.470715i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4871.70 0.753678 0.376839 0.926279i \(-0.377011\pi\)
0.376839 + 0.926279i \(0.377011\pi\)
\(348\) 0 0
\(349\) 7245.91 1.11136 0.555680 0.831396i \(-0.312458\pi\)
0.555680 + 0.831396i \(0.312458\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9352.32i 1.41012i 0.709146 + 0.705062i \(0.249081\pi\)
−0.709146 + 0.705062i \(0.750919\pi\)
\(354\) 0 0
\(355\) − 1264.60i − 0.189064i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5162.08 0.758898 0.379449 0.925213i \(-0.376114\pi\)
0.379449 + 0.925213i \(0.376114\pi\)
\(360\) 0 0
\(361\) −8162.66 −1.19006
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3318.44i 0.475877i
\(366\) 0 0
\(367\) 3138.93i 0.446460i 0.974766 + 0.223230i \(0.0716600\pi\)
−0.974766 + 0.223230i \(0.928340\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1515.67 0.212102
\(372\) 0 0
\(373\) −3085.24 −0.428278 −0.214139 0.976803i \(-0.568695\pi\)
−0.214139 + 0.976803i \(0.568695\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1079.68i 0.147498i
\(378\) 0 0
\(379\) − 13468.6i − 1.82542i −0.408611 0.912709i \(-0.633987\pi\)
0.408611 0.912709i \(-0.366013\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1100.69 −0.146847 −0.0734235 0.997301i \(-0.523392\pi\)
−0.0734235 + 0.997301i \(0.523392\pi\)
\(384\) 0 0
\(385\) 1324.89 0.175383
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 10089.1i 1.31501i 0.753452 + 0.657503i \(0.228387\pi\)
−0.753452 + 0.657503i \(0.771613\pi\)
\(390\) 0 0
\(391\) − 8437.26i − 1.09128i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6597.21 0.840359
\(396\) 0 0
\(397\) −1596.47 −0.201825 −0.100912 0.994895i \(-0.532176\pi\)
−0.100912 + 0.994895i \(0.532176\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 4818.54i − 0.600066i −0.953929 0.300033i \(-0.903002\pi\)
0.953929 0.300033i \(-0.0969977\pi\)
\(402\) 0 0
\(403\) 10820.2i 1.33745i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5233.80 −0.637420
\(408\) 0 0
\(409\) −4223.57 −0.510617 −0.255308 0.966860i \(-0.582177\pi\)
−0.255308 + 0.966860i \(0.582177\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 3022.06i − 0.360063i
\(414\) 0 0
\(415\) − 213.463i − 0.0252493i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1165.75 −0.135921 −0.0679604 0.997688i \(-0.521649\pi\)
−0.0679604 + 0.997688i \(0.521649\pi\)
\(420\) 0 0
\(421\) 9114.56 1.05515 0.527573 0.849510i \(-0.323102\pi\)
0.527573 + 0.849510i \(0.323102\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 2097.10i − 0.239352i
\(426\) 0 0
\(427\) 363.177i 0.0411601i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6631.38 −0.741120 −0.370560 0.928809i \(-0.620834\pi\)
−0.370560 + 0.928809i \(0.620834\pi\)
\(432\) 0 0
\(433\) −15681.8 −1.74046 −0.870230 0.492646i \(-0.836030\pi\)
−0.870230 + 0.492646i \(0.836030\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 24263.2i − 2.65598i
\(438\) 0 0
\(439\) − 7923.17i − 0.861395i −0.902496 0.430697i \(-0.858268\pi\)
0.902496 0.430697i \(-0.141732\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 14013.0 1.50289 0.751445 0.659796i \(-0.229357\pi\)
0.751445 + 0.659796i \(0.229357\pi\)
\(444\) 0 0
\(445\) −4236.92 −0.451346
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5967.60i 0.627235i 0.949549 + 0.313617i \(0.101541\pi\)
−0.949549 + 0.313617i \(0.898459\pi\)
\(450\) 0 0
\(451\) 8858.96i 0.924950i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4359.02 0.449130
\(456\) 0 0
\(457\) 12901.4 1.32057 0.660285 0.751015i \(-0.270435\pi\)
0.660285 + 0.751015i \(0.270435\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 9801.20i − 0.990212i −0.868833 0.495106i \(-0.835129\pi\)
0.868833 0.495106i \(-0.164871\pi\)
\(462\) 0 0
\(463\) 3427.68i 0.344056i 0.985092 + 0.172028i \(0.0550320\pi\)
−0.985092 + 0.172028i \(0.944968\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13777.2 1.36517 0.682583 0.730808i \(-0.260856\pi\)
0.682583 + 0.730808i \(0.260856\pi\)
\(468\) 0 0
\(469\) −1130.77 −0.111331
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 6275.63i − 0.610050i
\(474\) 0 0
\(475\) − 6030.67i − 0.582539i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2793.91 −0.266507 −0.133254 0.991082i \(-0.542542\pi\)
−0.133254 + 0.991082i \(0.542542\pi\)
\(480\) 0 0
\(481\) −17219.8 −1.63234
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2772.20i 0.259545i
\(486\) 0 0
\(487\) 17643.8i 1.64172i 0.571131 + 0.820859i \(0.306505\pi\)
−0.571131 + 0.820859i \(0.693495\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4816.53 −0.442703 −0.221351 0.975194i \(-0.571047\pi\)
−0.221351 + 0.975194i \(0.571047\pi\)
\(492\) 0 0
\(493\) −625.727 −0.0571629
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 430.285i 0.0388349i
\(498\) 0 0
\(499\) 9504.01i 0.852621i 0.904577 + 0.426311i \(0.140187\pi\)
−0.904577 + 0.426311i \(0.859813\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 13840.8 1.22690 0.613452 0.789732i \(-0.289780\pi\)
0.613452 + 0.789732i \(0.289780\pi\)
\(504\) 0 0
\(505\) 20705.2 1.82450
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 8144.19i − 0.709204i −0.935017 0.354602i \(-0.884616\pi\)
0.935017 0.354602i \(-0.115384\pi\)
\(510\) 0 0
\(511\) − 1129.12i − 0.0977478i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7180.83 0.614418
\(516\) 0 0
\(517\) −11941.2 −1.01581
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 21121.1i 1.77607i 0.459779 + 0.888034i \(0.347929\pi\)
−0.459779 + 0.888034i \(0.652071\pi\)
\(522\) 0 0
\(523\) − 15414.6i − 1.28878i −0.764696 0.644391i \(-0.777111\pi\)
0.764696 0.644391i \(-0.222889\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6270.81 −0.518331
\(528\) 0 0
\(529\) 27023.2 2.22102
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 29146.9i 2.36865i
\(534\) 0 0
\(535\) 14594.8i 1.17942i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 7215.91 0.576644
\(540\) 0 0
\(541\) 3269.66 0.259840 0.129920 0.991524i \(-0.458528\pi\)
0.129920 + 0.991524i \(0.458528\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 233.090i 0.0183201i
\(546\) 0 0
\(547\) 2028.46i 0.158557i 0.996853 + 0.0792784i \(0.0252616\pi\)
−0.996853 + 0.0792784i \(0.974738\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1799.41 −0.139124
\(552\) 0 0
\(553\) −2244.74 −0.172615
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5968.99i 0.454065i 0.973887 + 0.227032i \(0.0729023\pi\)
−0.973887 + 0.227032i \(0.927098\pi\)
\(558\) 0 0
\(559\) − 20647.5i − 1.56225i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −13107.0 −0.981161 −0.490581 0.871396i \(-0.663215\pi\)
−0.490581 + 0.871396i \(0.663215\pi\)
\(564\) 0 0
\(565\) 21161.3 1.57569
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 20162.7i − 1.48553i −0.669553 0.742764i \(-0.733514\pi\)
0.669553 0.742764i \(-0.266486\pi\)
\(570\) 0 0
\(571\) − 9396.23i − 0.688652i −0.938850 0.344326i \(-0.888108\pi\)
0.938850 0.344326i \(-0.111892\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 9740.82 0.706470
\(576\) 0 0
\(577\) −3349.25 −0.241648 −0.120824 0.992674i \(-0.538554\pi\)
−0.120824 + 0.992674i \(0.538554\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 72.6318i 0.00518636i
\(582\) 0 0
\(583\) − 7543.72i − 0.535899i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5156.24 0.362557 0.181278 0.983432i \(-0.441977\pi\)
0.181278 + 0.983432i \(0.441977\pi\)
\(588\) 0 0
\(589\) −18033.1 −1.26153
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 19449.0i − 1.34684i −0.739261 0.673419i \(-0.764825\pi\)
0.739261 0.673419i \(-0.235175\pi\)
\(594\) 0 0
\(595\) 2526.26i 0.174061i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 25726.7 1.75486 0.877432 0.479701i \(-0.159255\pi\)
0.877432 + 0.479701i \(0.159255\pi\)
\(600\) 0 0
\(601\) −11668.4 −0.791952 −0.395976 0.918261i \(-0.629594\pi\)
−0.395976 + 0.918261i \(0.629594\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10973.3i 0.737399i
\(606\) 0 0
\(607\) − 26441.9i − 1.76811i −0.467382 0.884055i \(-0.654803\pi\)
0.467382 0.884055i \(-0.345197\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −39287.9 −2.60134
\(612\) 0 0
\(613\) 4001.56 0.263656 0.131828 0.991273i \(-0.457915\pi\)
0.131828 + 0.991273i \(0.457915\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 25511.2i 1.66457i 0.554347 + 0.832286i \(0.312968\pi\)
−0.554347 + 0.832286i \(0.687032\pi\)
\(618\) 0 0
\(619\) − 25105.1i − 1.63014i −0.579361 0.815071i \(-0.696698\pi\)
0.579361 0.815071i \(-0.303302\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1441.63 0.0927091
\(624\) 0 0
\(625\) −19354.5 −1.23869
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 9979.65i − 0.632614i
\(630\) 0 0
\(631\) − 10090.6i − 0.636612i −0.947988 0.318306i \(-0.896886\pi\)
0.947988 0.318306i \(-0.103114\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 27453.8 1.71570
\(636\) 0 0
\(637\) 23741.1 1.47670
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 155.209i − 0.00956376i −0.999989 0.00478188i \(-0.998478\pi\)
0.999989 0.00478188i \(-0.00152213\pi\)
\(642\) 0 0
\(643\) − 8048.26i − 0.493612i −0.969065 0.246806i \(-0.920619\pi\)
0.969065 0.246806i \(-0.0793810\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −17128.5 −1.04079 −0.520394 0.853926i \(-0.674215\pi\)
−0.520394 + 0.853926i \(0.674215\pi\)
\(648\) 0 0
\(649\) −15041.2 −0.909739
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5854.10i 0.350825i 0.984495 + 0.175412i \(0.0561259\pi\)
−0.984495 + 0.175412i \(0.943874\pi\)
\(654\) 0 0
\(655\) − 14430.8i − 0.860853i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −22130.6 −1.30817 −0.654086 0.756420i \(-0.726947\pi\)
−0.654086 + 0.756420i \(0.726947\pi\)
\(660\) 0 0
\(661\) −19160.0 −1.12744 −0.563721 0.825965i \(-0.690631\pi\)
−0.563721 + 0.825965i \(0.690631\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7264.80i 0.423634i
\(666\) 0 0
\(667\) − 2906.44i − 0.168722i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1807.58 0.103996
\(672\) 0 0
\(673\) 24302.6 1.39197 0.695984 0.718057i \(-0.254968\pi\)
0.695984 + 0.718057i \(0.254968\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 16656.4i − 0.945581i −0.881175 0.472791i \(-0.843247\pi\)
0.881175 0.472791i \(-0.156753\pi\)
\(678\) 0 0
\(679\) − 943.255i − 0.0533120i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −24952.6 −1.39793 −0.698964 0.715157i \(-0.746355\pi\)
−0.698964 + 0.715157i \(0.746355\pi\)
\(684\) 0 0
\(685\) −17747.0 −0.989895
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 24819.6i − 1.37235i
\(690\) 0 0
\(691\) − 15536.7i − 0.855346i −0.903934 0.427673i \(-0.859334\pi\)
0.903934 0.427673i \(-0.140666\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10629.9 0.580166
\(696\) 0 0
\(697\) −16892.0 −0.917976
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9683.38i 0.521735i 0.965375 + 0.260868i \(0.0840086\pi\)
−0.965375 + 0.260868i \(0.915991\pi\)
\(702\) 0 0
\(703\) − 28698.6i − 1.53967i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7045.06 −0.374762
\(708\) 0 0
\(709\) 15665.5 0.829801 0.414901 0.909867i \(-0.363816\pi\)
0.414901 + 0.909867i \(0.363816\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 29127.2i − 1.52991i
\(714\) 0 0
\(715\) − 21695.5i − 1.13478i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −15944.3 −0.827015 −0.413507 0.910501i \(-0.635696\pi\)
−0.413507 + 0.910501i \(0.635696\pi\)
\(720\) 0 0
\(721\) −2443.31 −0.126205
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 722.402i − 0.0370060i
\(726\) 0 0
\(727\) 23773.6i 1.21281i 0.795156 + 0.606405i \(0.207389\pi\)
−0.795156 + 0.606405i \(0.792611\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 11966.2 0.605451
\(732\) 0 0
\(733\) 16190.1 0.815818 0.407909 0.913023i \(-0.366258\pi\)
0.407909 + 0.913023i \(0.366258\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5628.00i 0.281289i
\(738\) 0 0
\(739\) 3013.13i 0.149986i 0.997184 + 0.0749930i \(0.0238934\pi\)
−0.997184 + 0.0749930i \(0.976107\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −14363.7 −0.709225 −0.354613 0.935013i \(-0.615387\pi\)
−0.354613 + 0.935013i \(0.615387\pi\)
\(744\) 0 0
\(745\) 11606.4 0.570775
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 4965.96i − 0.242260i
\(750\) 0 0
\(751\) − 21754.9i − 1.05705i −0.848917 0.528526i \(-0.822745\pi\)
0.848917 0.528526i \(-0.177255\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 24103.3 1.16187
\(756\) 0 0
\(757\) −25070.4 −1.20370 −0.601848 0.798611i \(-0.705569\pi\)
−0.601848 + 0.798611i \(0.705569\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20326.9i 0.968263i 0.874995 + 0.484132i \(0.160864\pi\)
−0.874995 + 0.484132i \(0.839136\pi\)
\(762\) 0 0
\(763\) − 79.3101i − 0.00376306i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −49487.3 −2.32970
\(768\) 0 0
\(769\) −4680.89 −0.219502 −0.109751 0.993959i \(-0.535005\pi\)
−0.109751 + 0.993959i \(0.535005\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 2385.68i − 0.111005i −0.998459 0.0555025i \(-0.982324\pi\)
0.998459 0.0555025i \(-0.0176761\pi\)
\(774\) 0 0
\(775\) − 7239.65i − 0.335556i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −48576.6 −2.23419
\(780\) 0 0
\(781\) 2141.59 0.0981206
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 45110.6i 2.05104i
\(786\) 0 0
\(787\) − 8714.78i − 0.394725i −0.980331 0.197362i \(-0.936762\pi\)
0.980331 0.197362i \(-0.0632375\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −7200.24 −0.323655
\(792\) 0 0
\(793\) 5947.14 0.266317
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15364.9i 0.682876i 0.939904 + 0.341438i \(0.110914\pi\)
−0.939904 + 0.341438i \(0.889086\pi\)
\(798\) 0 0
\(799\) − 22769.2i − 1.00815i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −5619.77 −0.246971
\(804\) 0 0
\(805\) −11734.2 −0.513759
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 25153.9i − 1.09316i −0.837408 0.546578i \(-0.815930\pi\)
0.837408 0.546578i \(-0.184070\pi\)
\(810\) 0 0
\(811\) − 20366.0i − 0.881807i −0.897554 0.440904i \(-0.854658\pi\)
0.897554 0.440904i \(-0.145342\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −15716.7 −0.675501
\(816\) 0 0
\(817\) 34411.3 1.47356
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 781.431i − 0.0332182i −0.999862 0.0166091i \(-0.994713\pi\)
0.999862 0.0166091i \(-0.00528708\pi\)
\(822\) 0 0
\(823\) 32821.3i 1.39013i 0.718945 + 0.695067i \(0.244625\pi\)
−0.718945 + 0.695067i \(0.755375\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 43717.4 1.83821 0.919106 0.394010i \(-0.128913\pi\)
0.919106 + 0.394010i \(0.128913\pi\)
\(828\) 0 0
\(829\) −13211.9 −0.553521 −0.276761 0.960939i \(-0.589261\pi\)
−0.276761 + 0.960939i \(0.589261\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 13759.1i 0.572297i
\(834\) 0 0
\(835\) 28030.7i 1.16173i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 33876.0 1.39396 0.696978 0.717092i \(-0.254527\pi\)
0.696978 + 0.717092i \(0.254527\pi\)
\(840\) 0 0
\(841\) 24173.5 0.991162
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 42383.0i − 1.72547i
\(846\) 0 0
\(847\) − 3733.70i − 0.151466i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 46354.4 1.86722
\(852\) 0 0
\(853\) 14748.5 0.592005 0.296002 0.955187i \(-0.404346\pi\)
0.296002 + 0.955187i \(0.404346\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24382.3i 0.971859i 0.873998 + 0.485929i \(0.161519\pi\)
−0.873998 + 0.485929i \(0.838481\pi\)
\(858\) 0 0
\(859\) − 21247.2i − 0.843942i −0.906609 0.421971i \(-0.861338\pi\)
0.906609 0.421971i \(-0.138662\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4434.51 0.174916 0.0874581 0.996168i \(-0.472126\pi\)
0.0874581 + 0.996168i \(0.472126\pi\)
\(864\) 0 0
\(865\) −23350.0 −0.917830
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 11172.4i 0.436130i
\(870\) 0 0
\(871\) 18516.7i 0.720339i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4492.69 0.173578
\(876\) 0 0
\(877\) 13985.9 0.538507 0.269253 0.963069i \(-0.413223\pi\)
0.269253 + 0.963069i \(0.413223\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 4428.74i − 0.169362i −0.996408 0.0846812i \(-0.973013\pi\)
0.996408 0.0846812i \(-0.0269872\pi\)
\(882\) 0 0
\(883\) − 40519.1i − 1.54426i −0.635467 0.772128i \(-0.719192\pi\)
0.635467 0.772128i \(-0.280808\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −31161.7 −1.17960 −0.589802 0.807548i \(-0.700794\pi\)
−0.589802 + 0.807548i \(0.700794\pi\)
\(888\) 0 0
\(889\) −9341.29 −0.352415
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 65477.7i − 2.45367i
\(894\) 0 0
\(895\) − 48648.9i − 1.81693i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2160.14 −0.0801388
\(900\) 0 0
\(901\) 14384.1 0.531858
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 32013.5i − 1.17587i
\(906\) 0 0
\(907\) − 24021.4i − 0.879402i −0.898144 0.439701i \(-0.855084\pi\)
0.898144 0.439701i \(-0.144916\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 33057.5 1.20225 0.601123 0.799157i \(-0.294720\pi\)
0.601123 + 0.799157i \(0.294720\pi\)
\(912\) 0 0
\(913\) 361.499 0.0131039
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4910.16i 0.176824i
\(918\) 0 0
\(919\) 10303.3i 0.369831i 0.982754 + 0.184915i \(0.0592011\pi\)
−0.982754 + 0.184915i \(0.940799\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 7046.06 0.251272
\(924\) 0 0
\(925\) 11521.5 0.409540
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 7704.91i − 0.272110i −0.990701 0.136055i \(-0.956558\pi\)
0.990701 0.136055i \(-0.0434424\pi\)
\(930\) 0 0
\(931\) 39567.2i 1.39287i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 12573.5 0.439785
\(936\) 0 0
\(937\) −44189.8 −1.54068 −0.770340 0.637633i \(-0.779914\pi\)
−0.770340 + 0.637633i \(0.779914\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 51448.1i 1.78232i 0.453691 + 0.891159i \(0.350107\pi\)
−0.453691 + 0.891159i \(0.649893\pi\)
\(942\) 0 0
\(943\) − 78461.5i − 2.70950i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −17691.3 −0.607066 −0.303533 0.952821i \(-0.598166\pi\)
−0.303533 + 0.952821i \(0.598166\pi\)
\(948\) 0 0
\(949\) −18489.6 −0.632454
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 27331.2i − 0.929007i −0.885571 0.464504i \(-0.846233\pi\)
0.885571 0.464504i \(-0.153767\pi\)
\(954\) 0 0
\(955\) − 18050.5i − 0.611624i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6038.51 0.203330
\(960\) 0 0
\(961\) 8142.84 0.273332
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 16370.6i 0.546101i
\(966\) 0 0
\(967\) − 48045.3i − 1.59776i −0.601492 0.798879i \(-0.705427\pi\)
0.601492 0.798879i \(-0.294573\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −6999.70 −0.231340 −0.115670 0.993288i \(-0.536901\pi\)
−0.115670 + 0.993288i \(0.536901\pi\)
\(972\) 0 0
\(973\) −3616.88 −0.119169
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 42914.1i 1.40527i 0.711552 + 0.702633i \(0.247992\pi\)
−0.711552 + 0.702633i \(0.752008\pi\)
\(978\) 0 0
\(979\) − 7175.22i − 0.234240i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1615.78 −0.0524266 −0.0262133 0.999656i \(-0.508345\pi\)
−0.0262133 + 0.999656i \(0.508345\pi\)
\(984\) 0 0
\(985\) −38138.3 −1.23369
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 55581.6i 1.78705i
\(990\) 0 0
\(991\) 15725.6i 0.504076i 0.967717 + 0.252038i \(0.0811008\pi\)
−0.967717 + 0.252038i \(0.918899\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 24437.7 0.778620
\(996\) 0 0
\(997\) 22555.5 0.716489 0.358244 0.933628i \(-0.383375\pi\)
0.358244 + 0.933628i \(0.383375\pi\)
\(998\) 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 1728.4.c.i.1727.4 12
3.2 odd 2 inner 1728.4.c.i.1727.10 12
4.3 odd 2 inner 1728.4.c.i.1727.3 12
8.3 odd 2 108.4.b.a.107.10 yes 12
8.5 even 2 108.4.b.a.107.4 yes 12
12.11 even 2 inner 1728.4.c.i.1727.9 12
24.5 odd 2 108.4.b.a.107.9 yes 12
24.11 even 2 108.4.b.a.107.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.4.b.a.107.3 12 24.11 even 2
108.4.b.a.107.4 yes 12 8.5 even 2
108.4.b.a.107.9 yes 12 24.5 odd 2
108.4.b.a.107.10 yes 12 8.3 odd 2
1728.4.c.i.1727.3 12 4.3 odd 2 inner
1728.4.c.i.1727.4 12 1.1 even 1 trivial
1728.4.c.i.1727.9 12 12.11 even 2 inner
1728.4.c.i.1727.10 12 3.2 odd 2 inner