Properties

Label 4600.2.e.b.4049.2
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
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] = [4600,2,Mod(4049,4600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4600.4049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.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: \(36.7311849298\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.b.4049.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} -2.00000i q^{7} -6.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} -2.00000i q^{7} -6.00000 q^{9} -1.00000i q^{13} +6.00000 q^{21} -1.00000i q^{23} -9.00000i q^{27} +3.00000 q^{29} +3.00000 q^{31} -8.00000i q^{37} +3.00000 q^{39} +3.00000 q^{41} +2.00000i q^{43} -11.0000i q^{47} +3.00000 q^{49} +14.0000i q^{53} +8.00000 q^{59} -4.00000 q^{61} +12.0000i q^{63} -4.00000i q^{67} +3.00000 q^{69} +7.00000 q^{71} +9.00000i q^{73} +9.00000 q^{81} -4.00000i q^{83} +9.00000i q^{87} +2.00000 q^{89} -2.00000 q^{91} +9.00000i q^{93} +18.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{9} + 12 q^{21} + 6 q^{29} + 6 q^{31} + 6 q^{39} + 6 q^{41} + 6 q^{49} + 16 q^{59} - 8 q^{61} + 6 q^{69} + 14 q^{71} + 18 q^{81} + 4 q^{89} - 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 3.00000i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 9.00000i − 1.73205i
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 11.0000i − 1.60451i −0.596978 0.802257i \(-0.703632\pi\)
0.596978 0.802257i \(-0.296368\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 14.0000i 1.92305i 0.274721 + 0.961524i \(0.411414\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 12.0000i 1.51186i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 7.00000 0.830747 0.415374 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(72\) 0 0
\(73\) 9.00000i 1.05337i 0.850060 + 0.526685i \(0.176565\pi\)
−0.850060 + 0.526685i \(0.823435\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.00000i 0.964901i
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 9.00000i 0.933257i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18.0000i 1.82762i 0.406138 + 0.913812i \(0.366875\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 16.0000i − 1.54678i −0.633932 0.773389i \(-0.718560\pi\)
0.633932 0.773389i \(-0.281440\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) 24.0000 2.27798
\(112\) 0 0
\(113\) − 2.00000i − 0.188144i −0.995565 0.0940721i \(-0.970012\pi\)
0.995565 0.0940721i \(-0.0299884\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 6.00000i 0.554700i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 9.00000i 0.811503i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 11.0000i 0.976092i 0.872818 + 0.488046i \(0.162290\pi\)
−0.872818 + 0.488046i \(0.837710\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −9.00000 −0.786334 −0.393167 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.00000i 0.341743i 0.985293 + 0.170872i \(0.0546583\pi\)
−0.985293 + 0.170872i \(0.945342\pi\)
\(138\) 0 0
\(139\) 11.0000 0.933008 0.466504 0.884519i \(-0.345513\pi\)
0.466504 + 0.884519i \(0.345513\pi\)
\(140\) 0 0
\(141\) 33.0000 2.77910
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 9.00000i 0.742307i
\(148\) 0 0
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) 7.00000 0.569652 0.284826 0.958579i \(-0.408064\pi\)
0.284826 + 0.958579i \(0.408064\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 6.00000i − 0.478852i −0.970915 0.239426i \(-0.923041\pi\)
0.970915 0.239426i \(-0.0769593\pi\)
\(158\) 0 0
\(159\) −42.0000 −3.33082
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) − 7.00000i − 0.548282i −0.961689 0.274141i \(-0.911606\pi\)
0.961689 0.274141i \(-0.0883936\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.0000i 1.23812i 0.785345 + 0.619059i \(0.212486\pi\)
−0.785345 + 0.619059i \(0.787514\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 14.0000i − 1.06440i −0.846619 0.532200i \(-0.821365\pi\)
0.846619 0.532200i \(-0.178635\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 24.0000i 1.80395i
\(178\) 0 0
\(179\) −21.0000 −1.56961 −0.784807 0.619740i \(-0.787238\pi\)
−0.784807 + 0.619740i \(0.787238\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 0 0
\(183\) − 12.0000i − 0.887066i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −18.0000 −1.30931
\(190\) 0 0
\(191\) 2.00000 0.144715 0.0723575 0.997379i \(-0.476948\pi\)
0.0723575 + 0.997379i \(0.476948\pi\)
\(192\) 0 0
\(193\) 1.00000i 0.0719816i 0.999352 + 0.0359908i \(0.0114587\pi\)
−0.999352 + 0.0359908i \(0.988541\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.00000i 0.213741i 0.994273 + 0.106871i \(0.0340831\pi\)
−0.994273 + 0.106871i \(0.965917\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) − 6.00000i − 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.00000i 0.417029i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 21.0000i 1.43890i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 6.00000i − 0.407307i
\(218\) 0 0
\(219\) −27.0000 −1.82449
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.00000i 0.132745i 0.997795 + 0.0663723i \(0.0211425\pi\)
−0.997795 + 0.0663723i \(0.978857\pi\)
\(228\) 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
\(233\) − 21.0000i − 1.37576i −0.725826 0.687878i \(-0.758542\pi\)
0.725826 0.687878i \(-0.241458\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.00000 0.0646846 0.0323423 0.999477i \(-0.489703\pi\)
0.0323423 + 0.999477i \(0.489703\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.00000i 0.311891i 0.987766 + 0.155946i \(0.0498425\pi\)
−0.987766 + 0.155946i \(0.950158\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) 0 0
\(263\) 12.0000i 0.739952i 0.929041 + 0.369976i \(0.120634\pi\)
−0.929041 + 0.369976i \(0.879366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 0 0
\(269\) −17.0000 −1.03651 −0.518254 0.855227i \(-0.673418\pi\)
−0.518254 + 0.855227i \(0.673418\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) − 6.00000i − 0.363137i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 29.0000i − 1.74244i −0.490892 0.871221i \(-0.663329\pi\)
0.490892 0.871221i \(-0.336671\pi\)
\(278\) 0 0
\(279\) −18.0000 −1.07763
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) − 10.0000i − 0.594438i −0.954809 0.297219i \(-0.903941\pi\)
0.954809 0.297219i \(-0.0960592\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 6.00000i − 0.354169i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) −54.0000 −3.16554
\(292\) 0 0
\(293\) 24.0000i 1.40209i 0.713115 + 0.701047i \(0.247284\pi\)
−0.713115 + 0.701047i \(0.752716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.00000 −0.0578315
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) 54.0000i 3.10222i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 20.0000i − 1.14146i −0.821138 0.570730i \(-0.806660\pi\)
0.821138 0.570730i \(-0.193340\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) −29.0000 −1.64444 −0.822220 0.569170i \(-0.807264\pi\)
−0.822220 + 0.569170i \(0.807264\pi\)
\(312\) 0 0
\(313\) 20.0000i 1.13047i 0.824931 + 0.565233i \(0.191214\pi\)
−0.824931 + 0.565233i \(0.808786\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 14.0000i − 0.786318i −0.919470 0.393159i \(-0.871382\pi\)
0.919470 0.393159i \(-0.128618\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 48.0000 2.67910
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 54.0000i 2.98621i
\(328\) 0 0
\(329\) −22.0000 −1.21290
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 0 0
\(333\) 48.0000i 2.63038i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 26.0000i 1.41631i 0.706057 + 0.708155i \(0.250472\pi\)
−0.706057 + 0.708155i \(0.749528\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 20.0000i − 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) 7.00000 0.374701 0.187351 0.982293i \(-0.440010\pi\)
0.187351 + 0.982293i \(0.440010\pi\)
\(350\) 0 0
\(351\) −9.00000 −0.480384
\(352\) 0 0
\(353\) − 19.0000i − 1.01127i −0.862748 0.505634i \(-0.831259\pi\)
0.862748 0.505634i \(-0.168741\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) − 33.0000i − 1.73205i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) 0 0
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 28.0000 1.45369
\(372\) 0 0
\(373\) 14.0000i 0.724893i 0.932005 + 0.362446i \(0.118058\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 3.00000i − 0.154508i
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) −33.0000 −1.69064
\(382\) 0 0
\(383\) − 30.0000i − 1.53293i −0.642287 0.766464i \(-0.722014\pi\)
0.642287 0.766464i \(-0.277986\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 12.0000i − 0.609994i
\(388\) 0 0
\(389\) 16.0000 0.811232 0.405616 0.914044i \(-0.367057\pi\)
0.405616 + 0.914044i \(0.367057\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) − 27.0000i − 1.36197i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 25.0000i 1.25471i 0.778732 + 0.627357i \(0.215863\pi\)
−0.778732 + 0.627357i \(0.784137\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) − 3.00000i − 0.149441i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 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
\(413\) − 16.0000i − 0.787309i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 33.0000i 1.61602i
\(418\) 0 0
\(419\) −22.0000 −1.07477 −0.537385 0.843337i \(-0.680588\pi\)
−0.537385 + 0.843337i \(0.680588\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 66.0000i 3.20903i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.00000i 0.387147i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 34.0000i 1.63394i 0.576683 + 0.816968i \(0.304347\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 7.00000 0.334092 0.167046 0.985949i \(-0.446577\pi\)
0.167046 + 0.985949i \(0.446577\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) 0 0
\(443\) − 33.0000i − 1.56788i −0.620838 0.783939i \(-0.713208\pi\)
0.620838 0.783939i \(-0.286792\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 66.0000i 3.12169i
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 21.0000i 0.986666i
\(454\) 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
\(459\) 0 0
\(460\) 0 0
\(461\) 13.0000 0.605470 0.302735 0.953075i \(-0.402100\pi\)
0.302735 + 0.953075i \(0.402100\pi\)
\(462\) 0 0
\(463\) − 4.00000i − 0.185896i −0.995671 0.0929479i \(-0.970371\pi\)
0.995671 0.0929479i \(-0.0296290\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 42.0000i 1.94353i 0.235954 + 0.971764i \(0.424178\pi\)
−0.235954 + 0.971764i \(0.575822\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 84.0000i − 3.84610i
\(478\) 0 0
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) − 6.00000i − 0.273009i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 25.0000i − 1.13286i −0.824110 0.566429i \(-0.808325\pi\)
0.824110 0.566429i \(-0.191675\pi\)
\(488\) 0 0
\(489\) 21.0000 0.949653
\(490\) 0 0
\(491\) −31.0000 −1.39901 −0.699505 0.714628i \(-0.746596\pi\)
−0.699505 + 0.714628i \(0.746596\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 14.0000i − 0.627986i
\(498\) 0 0
\(499\) 25.0000 1.11915 0.559577 0.828778i \(-0.310964\pi\)
0.559577 + 0.828778i \(0.310964\pi\)
\(500\) 0 0
\(501\) −48.0000 −2.14448
\(502\) 0 0
\(503\) 14.0000i 0.624229i 0.950044 + 0.312115i \(0.101037\pi\)
−0.950044 + 0.312115i \(0.898963\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 36.0000i 1.59882i
\(508\) 0 0
\(509\) −21.0000 −0.930809 −0.465404 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) 0 0
\(511\) 18.0000 0.796273
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 42.0000 1.84360
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 0 0
\(523\) − 42.0000i − 1.83653i −0.395964 0.918266i \(-0.629590\pi\)
0.395964 0.918266i \(-0.370410\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −48.0000 −2.08302
\(532\) 0 0
\(533\) − 3.00000i − 0.129944i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 63.0000i − 2.71865i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7.00000 −0.300954 −0.150477 0.988614i \(-0.548081\pi\)
−0.150477 + 0.988614i \(0.548081\pi\)
\(542\) 0 0
\(543\) 36.0000i 1.54491i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 35.0000i 1.49649i 0.663421 + 0.748246i \(0.269104\pi\)
−0.663421 + 0.748246i \(0.730896\pi\)
\(548\) 0 0
\(549\) 24.0000 1.02430
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.0000i 0.593199i 0.955002 + 0.296600i \(0.0958526\pi\)
−0.955002 + 0.296600i \(0.904147\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 16.0000i 0.674320i 0.941447 + 0.337160i \(0.109466\pi\)
−0.941447 + 0.337160i \(0.890534\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 18.0000i − 0.755929i
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) 6.00000i 0.250654i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 9.00000i − 0.374675i −0.982296 0.187337i \(-0.940014\pi\)
0.982296 0.187337i \(-0.0599858\pi\)
\(578\) 0 0
\(579\) −3.00000 −0.124676
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 33.0000i 1.36206i 0.732257 + 0.681028i \(0.238467\pi\)
−0.732257 + 0.681028i \(0.761533\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −9.00000 −0.370211
\(592\) 0 0
\(593\) − 34.0000i − 1.39621i −0.715994 0.698106i \(-0.754026\pi\)
0.715994 0.698106i \(-0.245974\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 12.0000i 0.491127i
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 37.0000 1.50926 0.754631 0.656150i \(-0.227816\pi\)
0.754631 + 0.656150i \(0.227816\pi\)
\(602\) 0 0
\(603\) 24.0000i 0.977356i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16.0000i 0.649420i 0.945814 + 0.324710i \(0.105267\pi\)
−0.945814 + 0.324710i \(0.894733\pi\)
\(608\) 0 0
\(609\) 18.0000 0.729397
\(610\) 0 0
\(611\) −11.0000 −0.445012
\(612\) 0 0
\(613\) 16.0000i 0.646234i 0.946359 + 0.323117i \(0.104731\pi\)
−0.946359 + 0.323117i \(0.895269\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 48.0000i − 1.93241i −0.257780 0.966204i \(-0.582991\pi\)
0.257780 0.966204i \(-0.417009\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) −9.00000 −0.361158
\(622\) 0 0
\(623\) − 4.00000i − 0.160257i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 14.0000 0.557331 0.278666 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) 0 0
\(633\) − 48.0000i − 1.90783i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.00000i − 0.118864i
\(638\) 0 0
\(639\) −42.0000 −1.66149
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) − 34.0000i − 1.34083i −0.741987 0.670415i \(-0.766116\pi\)
0.741987 0.670415i \(-0.233884\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 39.0000i − 1.53325i −0.642096 0.766624i \(-0.721935\pi\)
0.642096 0.766624i \(-0.278065\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 18.0000 0.705476
\(652\) 0 0
\(653\) 3.00000i 0.117399i 0.998276 + 0.0586995i \(0.0186954\pi\)
−0.998276 + 0.0586995i \(0.981305\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 54.0000i − 2.10674i
\(658\) 0 0
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 3.00000i − 0.116160i
\(668\) 0 0
\(669\) −48.0000 −1.85579
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) − 13.0000i − 0.501113i −0.968102 0.250557i \(-0.919386\pi\)
0.968102 0.250557i \(-0.0806136\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 46.0000i − 1.76792i −0.467559 0.883962i \(-0.654866\pi\)
0.467559 0.883962i \(-0.345134\pi\)
\(678\) 0 0
\(679\) 36.0000 1.38155
\(680\) 0 0
\(681\) −6.00000 −0.229920
\(682\) 0 0
\(683\) − 35.0000i − 1.33924i −0.742705 0.669619i \(-0.766457\pi\)
0.742705 0.669619i \(-0.233543\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 6.00000i 0.228914i
\(688\) 0 0
\(689\) 14.0000 0.533358
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 63.0000 2.38288
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 36.0000i − 1.35392i
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 3.00000i − 0.112351i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.00000i 0.112037i
\(718\) 0 0
\(719\) −28.0000 −1.04422 −0.522112 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 6.00000i 0.223142i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 6.00000i − 0.222528i −0.993791 0.111264i \(-0.964510\pi\)
0.993791 0.111264i \(-0.0354899\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 8.00000i 0.295487i 0.989026 + 0.147743i \(0.0472010\pi\)
−0.989026 + 0.147743i \(0.952799\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −17.0000 −0.625355 −0.312678 0.949859i \(-0.601226\pi\)
−0.312678 + 0.949859i \(0.601226\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.0000i 0.440237i 0.975473 + 0.220119i \(0.0706445\pi\)
−0.975473 + 0.220119i \(0.929356\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 24.0000i 0.878114i
\(748\) 0 0
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) −50.0000 −1.82453 −0.912263 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) 0 0
\(753\) 48.0000i 1.74922i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 29.0000 1.05125 0.525625 0.850717i \(-0.323832\pi\)
0.525625 + 0.850717i \(0.323832\pi\)
\(762\) 0 0
\(763\) − 36.0000i − 1.30329i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 8.00000i − 0.288863i
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) −15.0000 −0.540212
\(772\) 0 0
\(773\) − 14.0000i − 0.503545i −0.967786 0.251773i \(-0.918987\pi\)
0.967786 0.251773i \(-0.0810135\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 48.0000i − 1.72199i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 27.0000i − 0.964901i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 32.0000i − 1.14068i −0.821410 0.570338i \(-0.806812\pi\)
0.821410 0.570338i \(-0.193188\pi\)
\(788\) 0 0
\(789\) −36.0000 −1.28163
\(790\) 0 0
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) 4.00000i 0.142044i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 12.0000i − 0.425062i −0.977154 0.212531i \(-0.931829\pi\)
0.977154 0.212531i \(-0.0681706\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −12.0000 −0.423999
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 51.0000i − 1.79529i
\(808\) 0 0
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) −5.00000 −0.175574 −0.0877869 0.996139i \(-0.527979\pi\)
−0.0877869 + 0.996139i \(0.527979\pi\)
\(812\) 0 0
\(813\) − 60.0000i − 2.10429i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 12.0000 0.419314
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) − 33.0000i − 1.15031i −0.818045 0.575154i \(-0.804942\pi\)
0.818045 0.575154i \(-0.195058\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 36.0000i − 1.25184i −0.779886 0.625921i \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) 0 0
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) 0 0
\(831\) 87.0000 3.01800
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 27.0000i − 0.933257i
\(838\) 0 0
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 18.0000i 0.619953i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 22.0000i 0.755929i
\(848\) 0 0
\(849\) 30.0000 1.02960
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) − 46.0000i − 1.57501i −0.616308 0.787505i \(-0.711372\pi\)
0.616308 0.787505i \(-0.288628\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 41.0000i 1.40053i 0.713881 + 0.700267i \(0.246936\pi\)
−0.713881 + 0.700267i \(0.753064\pi\)
\(858\) 0 0
\(859\) 17.0000 0.580033 0.290016 0.957022i \(-0.406339\pi\)
0.290016 + 0.957022i \(0.406339\pi\)
\(860\) 0 0
\(861\) 18.0000 0.613438
\(862\) 0 0
\(863\) 17.0000i 0.578687i 0.957225 + 0.289343i \(0.0934369\pi\)
−0.957225 + 0.289343i \(0.906563\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 51.0000i 1.73205i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) − 108.000i − 3.65525i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 18.0000i 0.607817i 0.952701 + 0.303908i \(0.0982917\pi\)
−0.952701 + 0.303908i \(0.901708\pi\)
\(878\) 0 0
\(879\) −72.0000 −2.42850
\(880\) 0 0
\(881\) 36.0000 1.21287 0.606435 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(882\) 0 0
\(883\) 36.0000i 1.21150i 0.795656 + 0.605748i \(0.207126\pi\)
−0.795656 + 0.605748i \(0.792874\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 15.0000i − 0.503651i −0.967773 0.251825i \(-0.918969\pi\)
0.967773 0.251825i \(-0.0810309\pi\)
\(888\) 0 0
\(889\) 22.0000 0.737856
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 3.00000i − 0.100167i
\(898\) 0 0
\(899\) 9.00000 0.300167
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 12.0000i 0.399335i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 32.0000i 1.06254i 0.847202 + 0.531271i \(0.178286\pi\)
−0.847202 + 0.531271i \(0.821714\pi\)
\(908\) 0 0
\(909\) −108.000 −3.58213
\(910\) 0 0
\(911\) −44.0000 −1.45779 −0.728893 0.684628i \(-0.759965\pi\)
−0.728893 + 0.684628i \(0.759965\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 18.0000i 0.594412i
\(918\) 0 0
\(919\) 50.0000 1.64935 0.824674 0.565608i \(-0.191359\pi\)
0.824674 + 0.565608i \(0.191359\pi\)
\(920\) 0 0
\(921\) 60.0000 1.97707
\(922\) 0 0
\(923\) − 7.00000i − 0.230408i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 24.0000i − 0.788263i
\(928\) 0 0
\(929\) −19.0000 −0.623370 −0.311685 0.950186i \(-0.600893\pi\)
−0.311685 + 0.950186i \(0.600893\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) − 87.0000i − 2.84825i
\(934\) 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
\(939\) −60.0000 −1.95803
\(940\) 0 0
\(941\) 12.0000 0.391189 0.195594 0.980685i \(-0.437336\pi\)
0.195594 + 0.980685i \(0.437336\pi\)
\(942\) 0 0
\(943\) − 3.00000i − 0.0976934i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 47.0000i − 1.52729i −0.645634 0.763647i \(-0.723407\pi\)
0.645634 0.763647i \(-0.276593\pi\)
\(948\) 0 0
\(949\) 9.00000 0.292152
\(950\) 0 0
\(951\) 42.0000 1.36194
\(952\) 0 0
\(953\) − 18.0000i − 0.583077i −0.956559 0.291539i \(-0.905833\pi\)
0.956559 0.291539i \(-0.0941672\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 8.00000 0.258333
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 96.0000i 3.09356i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 43.0000i 1.38279i 0.722478 + 0.691393i \(0.243003\pi\)
−0.722478 + 0.691393i \(0.756997\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −14.0000 −0.449281 −0.224641 0.974442i \(-0.572121\pi\)
−0.224641 + 0.974442i \(0.572121\pi\)
\(972\) 0 0
\(973\) − 22.0000i − 0.705288i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 48.0000i 1.53566i 0.640656 + 0.767828i \(0.278662\pi\)
−0.640656 + 0.767828i \(0.721338\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −108.000 −3.44817
\(982\) 0 0
\(983\) 14.0000i 0.446531i 0.974758 + 0.223265i \(0.0716716\pi\)
−0.974758 + 0.223265i \(0.928328\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 66.0000i − 2.10080i
\(988\) 0 0
\(989\) 2.00000 0.0635963
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) − 21.0000i − 0.666415i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 58.0000i 1.83688i 0.395562 + 0.918439i \(0.370550\pi\)
−0.395562 + 0.918439i \(0.629450\pi\)
\(998\) 0 0
\(999\) −72.0000 −2.27798
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 4600.2.e.b.4049.2 2
5.2 odd 4 4600.2.a.p.1.1 1
5.3 odd 4 920.2.a.a.1.1 1
5.4 even 2 inner 4600.2.e.b.4049.1 2
15.8 even 4 8280.2.a.d.1.1 1
20.3 even 4 1840.2.a.i.1.1 1
20.7 even 4 9200.2.a.c.1.1 1
40.3 even 4 7360.2.a.a.1.1 1
40.13 odd 4 7360.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.a.1.1 1 5.3 odd 4
1840.2.a.i.1.1 1 20.3 even 4
4600.2.a.p.1.1 1 5.2 odd 4
4600.2.e.b.4049.1 2 5.4 even 2 inner
4600.2.e.b.4049.2 2 1.1 even 1 trivial
7360.2.a.a.1.1 1 40.3 even 4
7360.2.a.ba.1.1 1 40.13 odd 4
8280.2.a.d.1.1 1 15.8 even 4
9200.2.a.c.1.1 1 20.7 even 4