Properties

Label 4900.2.e.n.2549.2
Level $4900$
Weight $2$
Character 4900.2549
Analytic conductor $39.127$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4900,2,Mod(2549,4900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4900.2549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4900.e (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: \(39.1266969904\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2549.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4900.2549
Dual form 4900.2.e.n.2549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +2.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +2.00000 q^{9} +6.00000 q^{11} +2.00000i q^{13} +6.00000i q^{17} +8.00000 q^{19} -3.00000i q^{23} +5.00000i q^{27} -3.00000 q^{29} -2.00000 q^{31} +6.00000i q^{33} +8.00000i q^{37} -2.00000 q^{39} +3.00000 q^{41} -5.00000i q^{43} -6.00000 q^{51} -12.0000i q^{53} +8.00000i q^{57} +1.00000 q^{61} -7.00000i q^{67} +3.00000 q^{69} -10.0000i q^{73} +4.00000 q^{79} +1.00000 q^{81} +3.00000i q^{83} -3.00000i q^{87} -3.00000 q^{89} -2.00000i q^{93} +10.0000i q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{9} + 12 q^{11} + 16 q^{19} - 6 q^{29} - 4 q^{31} - 4 q^{39} + 6 q^{41} - 12 q^{51} + 2 q^{61} + 6 q^{69} + 8 q^{79} + 2 q^{81} - 6 q^{89} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\) \(2451\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 3.00000i − 0.625543i −0.949828 0.312772i \(-0.898743\pi\)
0.949828 0.312772i \(-0.101257\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 6.00000i 1.04447i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) − 5.00000i − 0.762493i −0.924473 0.381246i \(-0.875495\pi\)
0.924473 0.381246i \(-0.124505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) − 12.0000i − 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000i 1.05963i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 7.00000i − 0.855186i −0.903971 0.427593i \(-0.859362\pi\)
0.903971 0.427593i \(-0.140638\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.00000i 0.329293i 0.986353 + 0.164646i \(0.0526483\pi\)
−0.986353 + 0.164646i \(0.947352\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 3.00000i − 0.321634i
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 2.00000i − 0.207390i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 12.0000 1.20605
\(100\) 0 0
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) − 7.00000i − 0.689730i −0.938652 0.344865i \(-0.887925\pi\)
0.938652 0.344865i \(-0.112075\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 3.00000i − 0.290021i −0.989430 0.145010i \(-0.953678\pi\)
0.989430 0.145010i \(-0.0463216\pi\)
\(108\) 0 0
\(109\) −17.0000 −1.62830 −0.814152 0.580651i \(-0.802798\pi\)
−0.814152 + 0.580651i \(0.802798\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 12.0000i 1.12887i 0.825479 + 0.564433i \(0.190905\pi\)
−0.825479 + 0.564433i \(0.809095\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.00000i 0.369800i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 3.00000i 0.270501i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 0 0
\(129\) 5.00000 0.440225
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.0000i 1.00349i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) 12.0000i 0.970143i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.0000i 1.62503i 0.582941 + 0.812514i \(0.301902\pi\)
−0.582941 + 0.812514i \(0.698098\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 16.0000 1.22355
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) −17.0000 −1.26360 −0.631800 0.775131i \(-0.717684\pi\)
−0.631800 + 0.775131i \(0.717684\pi\)
\(182\) 0 0
\(183\) 1.00000i 0.0739221i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 36.0000i 2.63258i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 7.00000 0.493742
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 6.00000i − 0.417029i
\(208\) 0 0
\(209\) 48.0000 3.32023
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.00000i 0.259828i
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 0 0
\(243\) 16.0000i 1.02640i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 16.0000i 1.01806i
\(248\) 0 0
\(249\) −3.00000 −0.190117
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) − 18.0000i − 1.13165i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.0000i 1.49708i 0.663090 + 0.748539i \(0.269245\pi\)
−0.663090 + 0.748539i \(0.730755\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 3.00000i 0.184988i 0.995713 + 0.0924940i \(0.0294839\pi\)
−0.995713 + 0.0924940i \(0.970516\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 3.00000i − 0.183597i
\(268\) 0 0
\(269\) −15.0000 −0.914566 −0.457283 0.889321i \(-0.651177\pi\)
−0.457283 + 0.889321i \(0.651177\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 20.0000i 1.18888i 0.804141 + 0.594438i \(0.202626\pi\)
−0.804141 + 0.594438i \(0.797374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) − 12.0000i − 0.701047i −0.936554 0.350524i \(-0.886004\pi\)
0.936554 0.350524i \(-0.113996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 30.0000i 1.74078i
\(298\) 0 0
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 3.00000i 0.172345i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 19.0000i 1.08439i 0.840254 + 0.542194i \(0.182406\pi\)
−0.840254 + 0.542194i \(0.817594\pi\)
\(308\) 0 0
\(309\) 7.00000 0.398216
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) − 4.00000i − 0.226093i −0.993590 0.113047i \(-0.963939\pi\)
0.993590 0.113047i \(-0.0360610\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) 3.00000 0.167444
\(322\) 0 0
\(323\) 48.0000i 2.67079i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 17.0000i − 0.940102i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −22.0000 −1.20923 −0.604615 0.796518i \(-0.706673\pi\)
−0.604615 + 0.796518i \(0.706673\pi\)
\(332\) 0 0
\(333\) 16.0000i 0.876795i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 20.0000i 1.08947i 0.838608 + 0.544735i \(0.183370\pi\)
−0.838608 + 0.544735i \(0.816630\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 27.0000i − 1.44944i −0.689046 0.724718i \(-0.741970\pi\)
0.689046 0.724718i \(-0.258030\pi\)
\(348\) 0 0
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) −10.0000 −0.533761
\(352\) 0 0
\(353\) − 12.0000i − 0.638696i −0.947638 0.319348i \(-0.896536\pi\)
0.947638 0.319348i \(-0.103464\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 25.0000i 1.31216i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 5.00000i − 0.260998i −0.991448 0.130499i \(-0.958342\pi\)
0.991448 0.130499i \(-0.0416579\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 16.0000i 0.828449i 0.910175 + 0.414224i \(0.135947\pi\)
−0.910175 + 0.414224i \(0.864053\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 6.00000i − 0.309016i
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) 3.00000i 0.153293i 0.997058 + 0.0766464i \(0.0244213\pi\)
−0.997058 + 0.0766464i \(0.975579\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 10.0000i − 0.508329i
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) − 12.0000i − 0.605320i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 26.0000i − 1.30490i −0.757831 0.652451i \(-0.773741\pi\)
0.757831 0.652451i \(-0.226259\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) 0 0
\(403\) − 4.00000i − 0.199254i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 48.0000i 2.37927i
\(408\) 0 0
\(409\) 11.0000 0.543915 0.271957 0.962309i \(-0.412329\pi\)
0.271957 + 0.962309i \(0.412329\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.00000i 0.0979404i
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) 23.0000 1.12095 0.560476 0.828171i \(-0.310618\pi\)
0.560476 + 0.828171i \(0.310618\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 0 0
\(433\) − 28.0000i − 1.34559i −0.739827 0.672797i \(-0.765093\pi\)
0.739827 0.672797i \(-0.234907\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 24.0000i − 1.14808i
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 9.00000i − 0.427603i −0.976877 0.213801i \(-0.931415\pi\)
0.976877 0.213801i \(-0.0685846\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 15.0000i 0.709476i
\(448\) 0 0
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) 0 0
\(453\) − 10.0000i − 0.469841i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 4.00000i − 0.187112i −0.995614 0.0935561i \(-0.970177\pi\)
0.995614 0.0935561i \(-0.0298234\pi\)
\(458\) 0 0
\(459\) −30.0000 −1.40028
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) − 29.0000i − 1.34774i −0.738848 0.673872i \(-0.764630\pi\)
0.738848 0.673872i \(-0.235370\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 33.0000i − 1.52706i −0.645774 0.763529i \(-0.723465\pi\)
0.645774 0.763529i \(-0.276535\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) − 30.0000i − 1.37940i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 24.0000i − 1.09888i
\(478\) 0 0
\(479\) −30.0000 −1.37073 −0.685367 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8.00000i 0.362515i 0.983436 + 0.181257i \(0.0580167\pi\)
−0.983436 + 0.181257i \(0.941983\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 0 0
\(493\) − 18.0000i − 0.810679i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −2.00000 −0.0895323 −0.0447661 0.998997i \(-0.514254\pi\)
−0.0447661 + 0.998997i \(0.514254\pi\)
\(500\) 0 0
\(501\) −21.0000 −0.938211
\(502\) 0 0
\(503\) − 9.00000i − 0.401290i −0.979664 0.200645i \(-0.935696\pi\)
0.979664 0.200645i \(-0.0643038\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) 39.0000 1.72864 0.864322 0.502938i \(-0.167748\pi\)
0.864322 + 0.502938i \(0.167748\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 40.0000i 1.76604i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 44.0000i 1.92399i 0.273075 + 0.961993i \(0.411959\pi\)
−0.273075 + 0.961993i \(0.588041\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 12.0000i − 0.522728i
\(528\) 0 0
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.00000i 0.259889i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 6.00000i 0.258919i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 17.0000 0.730887 0.365444 0.930834i \(-0.380917\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) 0 0
\(543\) − 17.0000i − 0.729540i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 19.0000i − 0.812381i −0.913788 0.406191i \(-0.866857\pi\)
0.913788 0.406191i \(-0.133143\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 0 0
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) −36.0000 −1.51992
\(562\) 0 0
\(563\) − 9.00000i − 0.379305i −0.981851 0.189652i \(-0.939264\pi\)
0.981851 0.189652i \(-0.0607361\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) − 18.0000i − 0.751961i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 10.0000i 0.416305i 0.978096 + 0.208153i \(0.0667451\pi\)
−0.978096 + 0.208153i \(0.933255\pi\)
\(578\) 0 0
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 72.0000i − 2.98194i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 0 0
\(593\) − 12.0000i − 0.492781i −0.969171 0.246390i \(-0.920755\pi\)
0.969171 0.246390i \(-0.0792446\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 20.0000i 0.818546i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) − 14.0000i − 0.570124i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 43.0000i 1.74532i 0.488332 + 0.872658i \(0.337606\pi\)
−0.488332 + 0.872658i \(0.662394\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 12.0000i − 0.483102i −0.970388 0.241551i \(-0.922344\pi\)
0.970388 0.241551i \(-0.0776561\pi\)
\(618\) 0 0
\(619\) 14.0000 0.562708 0.281354 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) 0 0
\(621\) 15.0000 0.601929
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 48.0000i 1.91694i
\(628\) 0 0
\(629\) −48.0000 −1.91389
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) 0 0
\(633\) − 4.00000i − 0.158986i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −39.0000 −1.54041 −0.770204 0.637798i \(-0.779845\pi\)
−0.770204 + 0.637798i \(0.779845\pi\)
\(642\) 0 0
\(643\) 20.0000i 0.788723i 0.918955 + 0.394362i \(0.129034\pi\)
−0.918955 + 0.394362i \(0.870966\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 21.0000i − 0.825595i −0.910823 0.412798i \(-0.864552\pi\)
0.910823 0.412798i \(-0.135448\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.0000i 1.64359i 0.569785 + 0.821794i \(0.307026\pi\)
−0.569785 + 0.821794i \(0.692974\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 20.0000i − 0.780274i
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 25.0000 0.972387 0.486194 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) 0 0
\(663\) − 12.0000i − 0.466041i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.00000i 0.348481i
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) 6.00000 0.231627
\(672\) 0 0
\(673\) − 32.0000i − 1.23351i −0.787155 0.616755i \(-0.788447\pi\)
0.787155 0.616755i \(-0.211553\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 15.0000i 0.573959i 0.957937 + 0.286980i \(0.0926512\pi\)
−0.957937 + 0.286980i \(0.907349\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.00000i 0.0763048i
\(688\) 0 0
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 18.0000i 0.681799i
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 27.0000 1.01978 0.509888 0.860241i \(-0.329687\pi\)
0.509888 + 0.860241i \(0.329687\pi\)
\(702\) 0 0
\(703\) 64.0000i 2.41381i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.00000 0.0375558 0.0187779 0.999824i \(-0.494022\pi\)
0.0187779 + 0.999824i \(0.494022\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 6.00000i 0.224702i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 6.00000i − 0.224074i
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 26.0000i − 0.966950i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 23.0000i − 0.853023i −0.904482 0.426511i \(-0.859742\pi\)
0.904482 0.426511i \(-0.140258\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 30.0000 1.10959
\(732\) 0 0
\(733\) 2.00000i 0.0738717i 0.999318 + 0.0369358i \(0.0117597\pi\)
−0.999318 + 0.0369358i \(0.988240\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 42.0000i − 1.54709i
\(738\) 0 0
\(739\) 46.0000 1.69214 0.846069 0.533074i \(-0.178963\pi\)
0.846069 + 0.533074i \(0.178963\pi\)
\(740\) 0 0
\(741\) −16.0000 −0.587775
\(742\) 0 0
\(743\) 39.0000i 1.43077i 0.698730 + 0.715386i \(0.253749\pi\)
−0.698730 + 0.715386i \(0.746251\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6.00000i 0.219529i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) 18.0000i 0.655956i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 0 0
\(759\) 18.0000 0.653359
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) −24.0000 −0.864339
\(772\) 0 0
\(773\) − 42.0000i − 1.51064i −0.655359 0.755318i \(-0.727483\pi\)
0.655359 0.755318i \(-0.272517\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 15.0000i − 0.536056i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 31.0000i 1.10503i 0.833503 + 0.552515i \(0.186332\pi\)
−0.833503 + 0.552515i \(0.813668\pi\)
\(788\) 0 0
\(789\) −3.00000 −0.106803
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.00000i 0.0710221i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0000i 0.425062i 0.977154 + 0.212531i \(0.0681706\pi\)
−0.977154 + 0.212531i \(0.931829\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) − 60.0000i − 2.11735i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 15.0000i − 0.528025i
\(808\) 0 0
\(809\) 51.0000 1.79306 0.896532 0.442978i \(-0.146078\pi\)
0.896532 + 0.442978i \(0.146078\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 16.0000i 0.561144i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 40.0000i − 1.39942i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 49.0000i 1.70803i 0.520246 + 0.854016i \(0.325840\pi\)
−0.520246 + 0.854016i \(0.674160\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 45.0000i − 1.56480i −0.622774 0.782402i \(-0.713994\pi\)
0.622774 0.782402i \(-0.286006\pi\)
\(828\) 0 0
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 10.0000i − 0.345651i
\(838\) 0 0
\(839\) −42.0000 −1.45000 −0.725001 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) − 6.00000i − 0.206651i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) 32.0000i 1.09566i 0.836590 + 0.547830i \(0.184546\pi\)
−0.836590 + 0.547830i \(0.815454\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 42.0000i − 1.43469i −0.696717 0.717346i \(-0.745357\pi\)
0.696717 0.717346i \(-0.254643\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 33.0000i − 1.12333i −0.827364 0.561667i \(-0.810160\pi\)
0.827364 0.561667i \(-0.189840\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 19.0000i − 0.645274i
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) 0 0
\(873\) 20.0000i 0.676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 40.0000i − 1.35070i −0.737496 0.675352i \(-0.763992\pi\)
0.737496 0.675352i \(-0.236008\pi\)
\(878\) 0 0
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) 0 0
\(883\) 28.0000i 0.942275i 0.882060 + 0.471138i \(0.156156\pi\)
−0.882060 + 0.471138i \(0.843844\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 57.0000i − 1.91387i −0.290298 0.956936i \(-0.593754\pi\)
0.290298 0.956936i \(-0.406246\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 6.00000 0.201008
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 6.00000i 0.200334i
\(898\) 0 0
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) 72.0000 2.39867
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 23.0000i 0.763702i 0.924224 + 0.381851i \(0.124713\pi\)
−0.924224 + 0.381851i \(0.875287\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 0 0
\(913\) 18.0000i 0.595713i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −19.0000 −0.626071
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 14.0000i − 0.459820i
\(928\) 0 0
\(929\) −45.0000 −1.47640 −0.738201 0.674581i \(-0.764324\pi\)
−0.738201 + 0.674581i \(0.764324\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 44.0000i − 1.43742i −0.695311 0.718709i \(-0.744734\pi\)
0.695311 0.718709i \(-0.255266\pi\)
\(938\) 0 0
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) −54.0000 −1.76035 −0.880175 0.474650i \(-0.842575\pi\)
−0.880175 + 0.474650i \(0.842575\pi\)
\(942\) 0 0
\(943\) − 9.00000i − 0.293080i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9.00000i 0.292461i 0.989251 + 0.146230i \(0.0467141\pi\)
−0.989251 + 0.146230i \(0.953286\pi\)
\(948\) 0 0
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) − 36.0000i − 1.16615i −0.812417 0.583077i \(-0.801849\pi\)
0.812417 0.583077i \(-0.198151\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 18.0000i − 0.581857i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) − 6.00000i − 0.193347i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 7.00000i − 0.225105i −0.993646 0.112552i \(-0.964097\pi\)
0.993646 0.112552i \(-0.0359026\pi\)
\(968\) 0 0
\(969\) −48.0000 −1.54198
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 54.0000i − 1.72761i −0.503824 0.863807i \(-0.668074\pi\)
0.503824 0.863807i \(-0.331926\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) −34.0000 −1.08554
\(982\) 0 0
\(983\) − 39.0000i − 1.24391i −0.783054 0.621953i \(-0.786339\pi\)
0.783054 0.621953i \(-0.213661\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −15.0000 −0.476972
\(990\) 0 0
\(991\) 26.0000 0.825917 0.412959 0.910750i \(-0.364495\pi\)
0.412959 + 0.910750i \(0.364495\pi\)
\(992\) 0 0
\(993\) − 22.0000i − 0.698149i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 32.0000i − 1.01345i −0.862108 0.506725i \(-0.830856\pi\)
0.862108 0.506725i \(-0.169144\pi\)
\(998\) 0 0
\(999\) −40.0000 −1.26554
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 4900.2.e.n.2549.2 2
5.2 odd 4 4900.2.a.q.1.1 1
5.3 odd 4 980.2.a.e.1.1 1
5.4 even 2 inner 4900.2.e.n.2549.1 2
7.3 odd 6 700.2.r.a.149.1 4
7.5 odd 6 700.2.r.a.249.2 4
7.6 odd 2 4900.2.e.m.2549.1 2
15.8 even 4 8820.2.a.a.1.1 1
20.3 even 4 3920.2.a.w.1.1 1
35.3 even 12 140.2.i.a.121.1 yes 2
35.12 even 12 700.2.i.b.501.1 2
35.13 even 4 980.2.a.g.1.1 1
35.17 even 12 700.2.i.b.401.1 2
35.18 odd 12 980.2.i.f.961.1 2
35.19 odd 6 700.2.r.a.249.1 4
35.23 odd 12 980.2.i.f.361.1 2
35.24 odd 6 700.2.r.a.149.2 4
35.27 even 4 4900.2.a.i.1.1 1
35.33 even 12 140.2.i.a.81.1 2
35.34 odd 2 4900.2.e.m.2549.2 2
105.38 odd 12 1260.2.s.c.541.1 2
105.68 odd 12 1260.2.s.c.361.1 2
105.83 odd 4 8820.2.a.p.1.1 1
140.3 odd 12 560.2.q.f.401.1 2
140.83 odd 4 3920.2.a.k.1.1 1
140.103 odd 12 560.2.q.f.81.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.i.a.81.1 2 35.33 even 12
140.2.i.a.121.1 yes 2 35.3 even 12
560.2.q.f.81.1 2 140.103 odd 12
560.2.q.f.401.1 2 140.3 odd 12
700.2.i.b.401.1 2 35.17 even 12
700.2.i.b.501.1 2 35.12 even 12
700.2.r.a.149.1 4 7.3 odd 6
700.2.r.a.149.2 4 35.24 odd 6
700.2.r.a.249.1 4 35.19 odd 6
700.2.r.a.249.2 4 7.5 odd 6
980.2.a.e.1.1 1 5.3 odd 4
980.2.a.g.1.1 1 35.13 even 4
980.2.i.f.361.1 2 35.23 odd 12
980.2.i.f.961.1 2 35.18 odd 12
1260.2.s.c.361.1 2 105.68 odd 12
1260.2.s.c.541.1 2 105.38 odd 12
3920.2.a.k.1.1 1 140.83 odd 4
3920.2.a.w.1.1 1 20.3 even 4
4900.2.a.i.1.1 1 35.27 even 4
4900.2.a.q.1.1 1 5.2 odd 4
4900.2.e.m.2549.1 2 7.6 odd 2
4900.2.e.m.2549.2 2 35.34 odd 2
4900.2.e.n.2549.1 2 5.4 even 2 inner
4900.2.e.n.2549.2 2 1.1 even 1 trivial
8820.2.a.a.1.1 1 15.8 even 4
8820.2.a.p.1.1 1 105.83 odd 4