Properties

Label 1840.2.e.c.369.4
Level $1840$
Weight $2$
Character 1840.369
Analytic conductor $14.692$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,2,Mod(369,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, 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("1840.369");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) 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 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 369.4
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 1840.369
Dual form 1840.2.e.c.369.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.61803i q^{3} -2.23607 q^{5} +1.85410i q^{7} +0.381966 q^{9} +O(q^{10})\) \(q+1.61803i q^{3} -2.23607 q^{5} +1.85410i q^{7} +0.381966 q^{9} +5.61803 q^{11} -2.61803i q^{13} -3.61803i q^{15} +0.854102i q^{17} -0.145898 q^{19} -3.00000 q^{21} +1.00000i q^{23} +5.00000 q^{25} +5.47214i q^{27} +9.70820 q^{29} +2.14590 q^{31} +9.09017i q^{33} -4.14590i q^{35} -9.70820i q^{37} +4.23607 q^{39} -5.61803 q^{41} +11.2361i q^{43} -0.854102 q^{45} -1.70820i q^{47} +3.56231 q^{49} -1.38197 q^{51} +2.00000i q^{53} -12.5623 q^{55} -0.236068i q^{57} -6.00000 q^{59} +2.85410 q^{61} +0.708204i q^{63} +5.85410i q^{65} -5.23607i q^{67} -1.61803 q^{69} -0.381966 q^{71} +16.4721i q^{73} +8.09017i q^{75} +10.4164i q^{77} -7.70820 q^{79} -7.70820 q^{81} +7.70820i q^{83} -1.90983i q^{85} +15.7082i q^{87} -3.70820 q^{89} +4.85410 q^{91} +3.47214i q^{93} +0.326238 q^{95} +13.0344i q^{97} +2.14590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{9} + 18 q^{11} - 14 q^{19} - 12 q^{21} + 20 q^{25} + 12 q^{29} + 22 q^{31} + 8 q^{39} - 18 q^{41} + 10 q^{45} - 26 q^{49} - 10 q^{51} - 10 q^{55} - 24 q^{59} - 2 q^{61} - 2 q^{69} - 6 q^{71} - 4 q^{79} - 4 q^{81} + 12 q^{89} + 6 q^{91} - 30 q^{95} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 1.61803i 0.934172i 0.884212 + 0.467086i \(0.154696\pi\)
−0.884212 + 0.467086i \(0.845304\pi\)
\(4\) 0 0
\(5\) −2.23607 −1.00000
\(6\) 0 0
\(7\) 1.85410i 0.700785i 0.936603 + 0.350392i \(0.113952\pi\)
−0.936603 + 0.350392i \(0.886048\pi\)
\(8\) 0 0
\(9\) 0.381966 0.127322
\(10\) 0 0
\(11\) 5.61803 1.69390 0.846950 0.531672i \(-0.178436\pi\)
0.846950 + 0.531672i \(0.178436\pi\)
\(12\) 0 0
\(13\) − 2.61803i − 0.726112i −0.931767 0.363056i \(-0.881733\pi\)
0.931767 0.363056i \(-0.118267\pi\)
\(14\) 0 0
\(15\) − 3.61803i − 0.934172i
\(16\) 0 0
\(17\) 0.854102i 0.207150i 0.994622 + 0.103575i \(0.0330282\pi\)
−0.994622 + 0.103575i \(0.966972\pi\)
\(18\) 0 0
\(19\) −0.145898 −0.0334713 −0.0167357 0.999860i \(-0.505327\pi\)
−0.0167357 + 0.999860i \(0.505327\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 5.47214i 1.05311i
\(28\) 0 0
\(29\) 9.70820 1.80277 0.901384 0.433020i \(-0.142552\pi\)
0.901384 + 0.433020i \(0.142552\pi\)
\(30\) 0 0
\(31\) 2.14590 0.385415 0.192707 0.981256i \(-0.438273\pi\)
0.192707 + 0.981256i \(0.438273\pi\)
\(32\) 0 0
\(33\) 9.09017i 1.58240i
\(34\) 0 0
\(35\) − 4.14590i − 0.700785i
\(36\) 0 0
\(37\) − 9.70820i − 1.59602i −0.602645 0.798009i \(-0.705886\pi\)
0.602645 0.798009i \(-0.294114\pi\)
\(38\) 0 0
\(39\) 4.23607 0.678314
\(40\) 0 0
\(41\) −5.61803 −0.877390 −0.438695 0.898636i \(-0.644559\pi\)
−0.438695 + 0.898636i \(0.644559\pi\)
\(42\) 0 0
\(43\) 11.2361i 1.71348i 0.515745 + 0.856742i \(0.327515\pi\)
−0.515745 + 0.856742i \(0.672485\pi\)
\(44\) 0 0
\(45\) −0.854102 −0.127322
\(46\) 0 0
\(47\) − 1.70820i − 0.249167i −0.992209 0.124584i \(-0.960241\pi\)
0.992209 0.124584i \(-0.0397595\pi\)
\(48\) 0 0
\(49\) 3.56231 0.508901
\(50\) 0 0
\(51\) −1.38197 −0.193514
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) −12.5623 −1.69390
\(56\) 0 0
\(57\) − 0.236068i − 0.0312680i
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 2.85410 0.365430 0.182715 0.983166i \(-0.441511\pi\)
0.182715 + 0.983166i \(0.441511\pi\)
\(62\) 0 0
\(63\) 0.708204i 0.0892253i
\(64\) 0 0
\(65\) 5.85410i 0.726112i
\(66\) 0 0
\(67\) − 5.23607i − 0.639688i −0.947470 0.319844i \(-0.896370\pi\)
0.947470 0.319844i \(-0.103630\pi\)
\(68\) 0 0
\(69\) −1.61803 −0.194788
\(70\) 0 0
\(71\) −0.381966 −0.0453310 −0.0226655 0.999743i \(-0.507215\pi\)
−0.0226655 + 0.999743i \(0.507215\pi\)
\(72\) 0 0
\(73\) 16.4721i 1.92792i 0.266051 + 0.963959i \(0.414281\pi\)
−0.266051 + 0.963959i \(0.585719\pi\)
\(74\) 0 0
\(75\) 8.09017i 0.934172i
\(76\) 0 0
\(77\) 10.4164i 1.18706i
\(78\) 0 0
\(79\) −7.70820 −0.867241 −0.433620 0.901096i \(-0.642764\pi\)
−0.433620 + 0.901096i \(0.642764\pi\)
\(80\) 0 0
\(81\) −7.70820 −0.856467
\(82\) 0 0
\(83\) 7.70820i 0.846085i 0.906110 + 0.423043i \(0.139038\pi\)
−0.906110 + 0.423043i \(0.860962\pi\)
\(84\) 0 0
\(85\) − 1.90983i − 0.207150i
\(86\) 0 0
\(87\) 15.7082i 1.68410i
\(88\) 0 0
\(89\) −3.70820 −0.393069 −0.196534 0.980497i \(-0.562969\pi\)
−0.196534 + 0.980497i \(0.562969\pi\)
\(90\) 0 0
\(91\) 4.85410 0.508848
\(92\) 0 0
\(93\) 3.47214i 0.360044i
\(94\) 0 0
\(95\) 0.326238 0.0334713
\(96\) 0 0
\(97\) 13.0344i 1.32345i 0.749748 + 0.661724i \(0.230175\pi\)
−0.749748 + 0.661724i \(0.769825\pi\)
\(98\) 0 0
\(99\) 2.14590 0.215671
\(100\) 0 0
\(101\) −1.52786 −0.152028 −0.0760141 0.997107i \(-0.524219\pi\)
−0.0760141 + 0.997107i \(0.524219\pi\)
\(102\) 0 0
\(103\) 10.8541i 1.06949i 0.845015 + 0.534743i \(0.179592\pi\)
−0.845015 + 0.534743i \(0.820408\pi\)
\(104\) 0 0
\(105\) 6.70820 0.654654
\(106\) 0 0
\(107\) − 11.7082i − 1.13187i −0.824448 0.565937i \(-0.808514\pi\)
0.824448 0.565937i \(-0.191486\pi\)
\(108\) 0 0
\(109\) −7.56231 −0.724338 −0.362169 0.932113i \(-0.617964\pi\)
−0.362169 + 0.932113i \(0.617964\pi\)
\(110\) 0 0
\(111\) 15.7082 1.49096
\(112\) 0 0
\(113\) 13.4164i 1.26211i 0.775738 + 0.631055i \(0.217378\pi\)
−0.775738 + 0.631055i \(0.782622\pi\)
\(114\) 0 0
\(115\) − 2.23607i − 0.208514i
\(116\) 0 0
\(117\) − 1.00000i − 0.0924500i
\(118\) 0 0
\(119\) −1.58359 −0.145168
\(120\) 0 0
\(121\) 20.5623 1.86930
\(122\) 0 0
\(123\) − 9.09017i − 0.819633i
\(124\) 0 0
\(125\) −11.1803 −1.00000
\(126\) 0 0
\(127\) 9.70820i 0.861464i 0.902480 + 0.430732i \(0.141745\pi\)
−0.902480 + 0.430732i \(0.858255\pi\)
\(128\) 0 0
\(129\) −18.1803 −1.60069
\(130\) 0 0
\(131\) 14.1803 1.23894 0.619471 0.785020i \(-0.287347\pi\)
0.619471 + 0.785020i \(0.287347\pi\)
\(132\) 0 0
\(133\) − 0.270510i − 0.0234562i
\(134\) 0 0
\(135\) − 12.2361i − 1.05311i
\(136\) 0 0
\(137\) − 13.8541i − 1.18364i −0.806072 0.591818i \(-0.798410\pi\)
0.806072 0.591818i \(-0.201590\pi\)
\(138\) 0 0
\(139\) 4.29180 0.364025 0.182013 0.983296i \(-0.441739\pi\)
0.182013 + 0.983296i \(0.441739\pi\)
\(140\) 0 0
\(141\) 2.76393 0.232765
\(142\) 0 0
\(143\) − 14.7082i − 1.22996i
\(144\) 0 0
\(145\) −21.7082 −1.80277
\(146\) 0 0
\(147\) 5.76393i 0.475401i
\(148\) 0 0
\(149\) 2.61803 0.214478 0.107239 0.994233i \(-0.465799\pi\)
0.107239 + 0.994233i \(0.465799\pi\)
\(150\) 0 0
\(151\) −14.2705 −1.16132 −0.580659 0.814147i \(-0.697205\pi\)
−0.580659 + 0.814147i \(0.697205\pi\)
\(152\) 0 0
\(153\) 0.326238i 0.0263748i
\(154\) 0 0
\(155\) −4.79837 −0.385415
\(156\) 0 0
\(157\) 2.29180i 0.182905i 0.995809 + 0.0914526i \(0.0291510\pi\)
−0.995809 + 0.0914526i \(0.970849\pi\)
\(158\) 0 0
\(159\) −3.23607 −0.256637
\(160\) 0 0
\(161\) −1.85410 −0.146124
\(162\) 0 0
\(163\) − 22.0344i − 1.72587i −0.505314 0.862935i \(-0.668623\pi\)
0.505314 0.862935i \(-0.331377\pi\)
\(164\) 0 0
\(165\) − 20.3262i − 1.58240i
\(166\) 0 0
\(167\) 9.70820i 0.751243i 0.926773 + 0.375622i \(0.122571\pi\)
−0.926773 + 0.375622i \(0.877429\pi\)
\(168\) 0 0
\(169\) 6.14590 0.472761
\(170\) 0 0
\(171\) −0.0557281 −0.00426163
\(172\) 0 0
\(173\) − 16.5623i − 1.25921i −0.776916 0.629604i \(-0.783217\pi\)
0.776916 0.629604i \(-0.216783\pi\)
\(174\) 0 0
\(175\) 9.27051i 0.700785i
\(176\) 0 0
\(177\) − 9.70820i − 0.729713i
\(178\) 0 0
\(179\) 7.52786 0.562659 0.281329 0.959611i \(-0.409225\pi\)
0.281329 + 0.959611i \(0.409225\pi\)
\(180\) 0 0
\(181\) 15.5623 1.15674 0.578369 0.815776i \(-0.303690\pi\)
0.578369 + 0.815776i \(0.303690\pi\)
\(182\) 0 0
\(183\) 4.61803i 0.341375i
\(184\) 0 0
\(185\) 21.7082i 1.59602i
\(186\) 0 0
\(187\) 4.79837i 0.350892i
\(188\) 0 0
\(189\) −10.1459 −0.738005
\(190\) 0 0
\(191\) −25.4164 −1.83907 −0.919533 0.393012i \(-0.871433\pi\)
−0.919533 + 0.393012i \(0.871433\pi\)
\(192\) 0 0
\(193\) 15.7082i 1.13070i 0.824851 + 0.565351i \(0.191259\pi\)
−0.824851 + 0.565351i \(0.808741\pi\)
\(194\) 0 0
\(195\) −9.47214 −0.678314
\(196\) 0 0
\(197\) − 20.5623i − 1.46500i −0.680765 0.732502i \(-0.738353\pi\)
0.680765 0.732502i \(-0.261647\pi\)
\(198\) 0 0
\(199\) 11.4164 0.809288 0.404644 0.914474i \(-0.367395\pi\)
0.404644 + 0.914474i \(0.367395\pi\)
\(200\) 0 0
\(201\) 8.47214 0.597578
\(202\) 0 0
\(203\) 18.0000i 1.26335i
\(204\) 0 0
\(205\) 12.5623 0.877390
\(206\) 0 0
\(207\) 0.381966i 0.0265485i
\(208\) 0 0
\(209\) −0.819660 −0.0566971
\(210\) 0 0
\(211\) 7.70820 0.530655 0.265327 0.964158i \(-0.414520\pi\)
0.265327 + 0.964158i \(0.414520\pi\)
\(212\) 0 0
\(213\) − 0.618034i − 0.0423470i
\(214\) 0 0
\(215\) − 25.1246i − 1.71348i
\(216\) 0 0
\(217\) 3.97871i 0.270093i
\(218\) 0 0
\(219\) −26.6525 −1.80101
\(220\) 0 0
\(221\) 2.23607 0.150414
\(222\) 0 0
\(223\) 22.3607i 1.49738i 0.662919 + 0.748691i \(0.269317\pi\)
−0.662919 + 0.748691i \(0.730683\pi\)
\(224\) 0 0
\(225\) 1.90983 0.127322
\(226\) 0 0
\(227\) − 10.2918i − 0.683090i −0.939865 0.341545i \(-0.889050\pi\)
0.939865 0.341545i \(-0.110950\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) −16.8541 −1.10892
\(232\) 0 0
\(233\) − 21.1246i − 1.38392i −0.721936 0.691960i \(-0.756748\pi\)
0.721936 0.691960i \(-0.243252\pi\)
\(234\) 0 0
\(235\) 3.81966i 0.249167i
\(236\) 0 0
\(237\) − 12.4721i − 0.810152i
\(238\) 0 0
\(239\) −10.4721 −0.677386 −0.338693 0.940897i \(-0.609985\pi\)
−0.338693 + 0.940897i \(0.609985\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) 3.94427i 0.253025i
\(244\) 0 0
\(245\) −7.96556 −0.508901
\(246\) 0 0
\(247\) 0.381966i 0.0243039i
\(248\) 0 0
\(249\) −12.4721 −0.790390
\(250\) 0 0
\(251\) 16.7984 1.06030 0.530152 0.847903i \(-0.322135\pi\)
0.530152 + 0.847903i \(0.322135\pi\)
\(252\) 0 0
\(253\) 5.61803i 0.353203i
\(254\) 0 0
\(255\) 3.09017 0.193514
\(256\) 0 0
\(257\) − 11.4164i − 0.712136i −0.934460 0.356068i \(-0.884117\pi\)
0.934460 0.356068i \(-0.115883\pi\)
\(258\) 0 0
\(259\) 18.0000 1.11847
\(260\) 0 0
\(261\) 3.70820 0.229532
\(262\) 0 0
\(263\) − 18.2705i − 1.12661i −0.826250 0.563304i \(-0.809530\pi\)
0.826250 0.563304i \(-0.190470\pi\)
\(264\) 0 0
\(265\) − 4.47214i − 0.274721i
\(266\) 0 0
\(267\) − 6.00000i − 0.367194i
\(268\) 0 0
\(269\) 1.52786 0.0931555 0.0465778 0.998915i \(-0.485168\pi\)
0.0465778 + 0.998915i \(0.485168\pi\)
\(270\) 0 0
\(271\) −25.8541 −1.57052 −0.785262 0.619163i \(-0.787472\pi\)
−0.785262 + 0.619163i \(0.787472\pi\)
\(272\) 0 0
\(273\) 7.85410i 0.475352i
\(274\) 0 0
\(275\) 28.0902 1.69390
\(276\) 0 0
\(277\) 7.52786i 0.452306i 0.974092 + 0.226153i \(0.0726149\pi\)
−0.974092 + 0.226153i \(0.927385\pi\)
\(278\) 0 0
\(279\) 0.819660 0.0490718
\(280\) 0 0
\(281\) 30.6525 1.82857 0.914287 0.405068i \(-0.132752\pi\)
0.914287 + 0.405068i \(0.132752\pi\)
\(282\) 0 0
\(283\) 20.9443i 1.24501i 0.782617 + 0.622504i \(0.213885\pi\)
−0.782617 + 0.622504i \(0.786115\pi\)
\(284\) 0 0
\(285\) 0.527864i 0.0312680i
\(286\) 0 0
\(287\) − 10.4164i − 0.614861i
\(288\) 0 0
\(289\) 16.2705 0.957089
\(290\) 0 0
\(291\) −21.0902 −1.23633
\(292\) 0 0
\(293\) 14.2918i 0.834936i 0.908692 + 0.417468i \(0.137082\pi\)
−0.908692 + 0.417468i \(0.862918\pi\)
\(294\) 0 0
\(295\) 13.4164 0.781133
\(296\) 0 0
\(297\) 30.7426i 1.78387i
\(298\) 0 0
\(299\) 2.61803 0.151405
\(300\) 0 0
\(301\) −20.8328 −1.20078
\(302\) 0 0
\(303\) − 2.47214i − 0.142020i
\(304\) 0 0
\(305\) −6.38197 −0.365430
\(306\) 0 0
\(307\) − 16.8541i − 0.961914i −0.876744 0.480957i \(-0.840289\pi\)
0.876744 0.480957i \(-0.159711\pi\)
\(308\) 0 0
\(309\) −17.5623 −0.999085
\(310\) 0 0
\(311\) 22.4721 1.27428 0.637139 0.770749i \(-0.280118\pi\)
0.637139 + 0.770749i \(0.280118\pi\)
\(312\) 0 0
\(313\) − 25.8541i − 1.46136i −0.682720 0.730680i \(-0.739203\pi\)
0.682720 0.730680i \(-0.260797\pi\)
\(314\) 0 0
\(315\) − 1.58359i − 0.0892253i
\(316\) 0 0
\(317\) − 7.27051i − 0.408353i −0.978934 0.204176i \(-0.934548\pi\)
0.978934 0.204176i \(-0.0654516\pi\)
\(318\) 0 0
\(319\) 54.5410 3.05371
\(320\) 0 0
\(321\) 18.9443 1.05737
\(322\) 0 0
\(323\) − 0.124612i − 0.00693359i
\(324\) 0 0
\(325\) − 13.0902i − 0.726112i
\(326\) 0 0
\(327\) − 12.2361i − 0.676656i
\(328\) 0 0
\(329\) 3.16718 0.174613
\(330\) 0 0
\(331\) 25.1246 1.38097 0.690487 0.723345i \(-0.257396\pi\)
0.690487 + 0.723345i \(0.257396\pi\)
\(332\) 0 0
\(333\) − 3.70820i − 0.203208i
\(334\) 0 0
\(335\) 11.7082i 0.639688i
\(336\) 0 0
\(337\) − 3.38197i − 0.184227i −0.995748 0.0921137i \(-0.970638\pi\)
0.995748 0.0921137i \(-0.0293623\pi\)
\(338\) 0 0
\(339\) −21.7082 −1.17903
\(340\) 0 0
\(341\) 12.0557 0.652854
\(342\) 0 0
\(343\) 19.5836i 1.05741i
\(344\) 0 0
\(345\) 3.61803 0.194788
\(346\) 0 0
\(347\) − 14.5623i − 0.781746i −0.920445 0.390873i \(-0.872173\pi\)
0.920445 0.390873i \(-0.127827\pi\)
\(348\) 0 0
\(349\) −27.7082 −1.48319 −0.741593 0.670850i \(-0.765929\pi\)
−0.741593 + 0.670850i \(0.765929\pi\)
\(350\) 0 0
\(351\) 14.3262 0.764678
\(352\) 0 0
\(353\) − 8.00000i − 0.425797i −0.977074 0.212899i \(-0.931710\pi\)
0.977074 0.212899i \(-0.0682904\pi\)
\(354\) 0 0
\(355\) 0.854102 0.0453310
\(356\) 0 0
\(357\) − 2.56231i − 0.135612i
\(358\) 0 0
\(359\) −13.4164 −0.708091 −0.354045 0.935228i \(-0.615194\pi\)
−0.354045 + 0.935228i \(0.615194\pi\)
\(360\) 0 0
\(361\) −18.9787 −0.998880
\(362\) 0 0
\(363\) 33.2705i 1.74625i
\(364\) 0 0
\(365\) − 36.8328i − 1.92792i
\(366\) 0 0
\(367\) − 12.0000i − 0.626395i −0.949688 0.313197i \(-0.898600\pi\)
0.949688 0.313197i \(-0.101400\pi\)
\(368\) 0 0
\(369\) −2.14590 −0.111711
\(370\) 0 0
\(371\) −3.70820 −0.192520
\(372\) 0 0
\(373\) 20.9443i 1.08445i 0.840232 + 0.542227i \(0.182419\pi\)
−0.840232 + 0.542227i \(0.817581\pi\)
\(374\) 0 0
\(375\) − 18.0902i − 0.934172i
\(376\) 0 0
\(377\) − 25.4164i − 1.30901i
\(378\) 0 0
\(379\) −24.2705 −1.24669 −0.623346 0.781946i \(-0.714227\pi\)
−0.623346 + 0.781946i \(0.714227\pi\)
\(380\) 0 0
\(381\) −15.7082 −0.804756
\(382\) 0 0
\(383\) 0.583592i 0.0298202i 0.999889 + 0.0149101i \(0.00474620\pi\)
−0.999889 + 0.0149101i \(0.995254\pi\)
\(384\) 0 0
\(385\) − 23.2918i − 1.18706i
\(386\) 0 0
\(387\) 4.29180i 0.218164i
\(388\) 0 0
\(389\) 21.3262 1.08128 0.540642 0.841253i \(-0.318182\pi\)
0.540642 + 0.841253i \(0.318182\pi\)
\(390\) 0 0
\(391\) −0.854102 −0.0431938
\(392\) 0 0
\(393\) 22.9443i 1.15739i
\(394\) 0 0
\(395\) 17.2361 0.867241
\(396\) 0 0
\(397\) 15.4377i 0.774796i 0.921913 + 0.387398i \(0.126626\pi\)
−0.921913 + 0.387398i \(0.873374\pi\)
\(398\) 0 0
\(399\) 0.437694 0.0219121
\(400\) 0 0
\(401\) −25.5279 −1.27480 −0.637400 0.770533i \(-0.719990\pi\)
−0.637400 + 0.770533i \(0.719990\pi\)
\(402\) 0 0
\(403\) − 5.61803i − 0.279854i
\(404\) 0 0
\(405\) 17.2361 0.856467
\(406\) 0 0
\(407\) − 54.5410i − 2.70350i
\(408\) 0 0
\(409\) −13.5623 −0.670613 −0.335306 0.942109i \(-0.608840\pi\)
−0.335306 + 0.942109i \(0.608840\pi\)
\(410\) 0 0
\(411\) 22.4164 1.10572
\(412\) 0 0
\(413\) − 11.1246i − 0.547406i
\(414\) 0 0
\(415\) − 17.2361i − 0.846085i
\(416\) 0 0
\(417\) 6.94427i 0.340062i
\(418\) 0 0
\(419\) −29.8885 −1.46015 −0.730075 0.683367i \(-0.760515\pi\)
−0.730075 + 0.683367i \(0.760515\pi\)
\(420\) 0 0
\(421\) 0.145898 0.00711064 0.00355532 0.999994i \(-0.498868\pi\)
0.00355532 + 0.999994i \(0.498868\pi\)
\(422\) 0 0
\(423\) − 0.652476i − 0.0317245i
\(424\) 0 0
\(425\) 4.27051i 0.207150i
\(426\) 0 0
\(427\) 5.29180i 0.256088i
\(428\) 0 0
\(429\) 23.7984 1.14900
\(430\) 0 0
\(431\) 11.2361 0.541222 0.270611 0.962689i \(-0.412774\pi\)
0.270611 + 0.962689i \(0.412774\pi\)
\(432\) 0 0
\(433\) 27.3820i 1.31589i 0.753065 + 0.657947i \(0.228575\pi\)
−0.753065 + 0.657947i \(0.771425\pi\)
\(434\) 0 0
\(435\) − 35.1246i − 1.68410i
\(436\) 0 0
\(437\) − 0.145898i − 0.00697925i
\(438\) 0 0
\(439\) 33.2705 1.58791 0.793957 0.607973i \(-0.208017\pi\)
0.793957 + 0.607973i \(0.208017\pi\)
\(440\) 0 0
\(441\) 1.36068 0.0647943
\(442\) 0 0
\(443\) 0.145898i 0.00693182i 0.999994 + 0.00346591i \(0.00110324\pi\)
−0.999994 + 0.00346591i \(0.998897\pi\)
\(444\) 0 0
\(445\) 8.29180 0.393069
\(446\) 0 0
\(447\) 4.23607i 0.200359i
\(448\) 0 0
\(449\) 17.5623 0.828816 0.414408 0.910091i \(-0.363989\pi\)
0.414408 + 0.910091i \(0.363989\pi\)
\(450\) 0 0
\(451\) −31.5623 −1.48621
\(452\) 0 0
\(453\) − 23.0902i − 1.08487i
\(454\) 0 0
\(455\) −10.8541 −0.508848
\(456\) 0 0
\(457\) 1.41641i 0.0662568i 0.999451 + 0.0331284i \(0.0105470\pi\)
−0.999451 + 0.0331284i \(0.989453\pi\)
\(458\) 0 0
\(459\) −4.67376 −0.218153
\(460\) 0 0
\(461\) 37.3050 1.73746 0.868732 0.495282i \(-0.164935\pi\)
0.868732 + 0.495282i \(0.164935\pi\)
\(462\) 0 0
\(463\) − 15.7082i − 0.730022i −0.931003 0.365011i \(-0.881065\pi\)
0.931003 0.365011i \(-0.118935\pi\)
\(464\) 0 0
\(465\) − 7.76393i − 0.360044i
\(466\) 0 0
\(467\) − 23.1246i − 1.07008i −0.844827 0.535040i \(-0.820297\pi\)
0.844827 0.535040i \(-0.179703\pi\)
\(468\) 0 0
\(469\) 9.70820 0.448283
\(470\) 0 0
\(471\) −3.70820 −0.170865
\(472\) 0 0
\(473\) 63.1246i 2.90247i
\(474\) 0 0
\(475\) −0.729490 −0.0334713
\(476\) 0 0
\(477\) 0.763932i 0.0349780i
\(478\) 0 0
\(479\) −28.4721 −1.30093 −0.650463 0.759538i \(-0.725425\pi\)
−0.650463 + 0.759538i \(0.725425\pi\)
\(480\) 0 0
\(481\) −25.4164 −1.15889
\(482\) 0 0
\(483\) − 3.00000i − 0.136505i
\(484\) 0 0
\(485\) − 29.1459i − 1.32345i
\(486\) 0 0
\(487\) 8.29180i 0.375737i 0.982194 + 0.187869i \(0.0601579\pi\)
−0.982194 + 0.187869i \(0.939842\pi\)
\(488\) 0 0
\(489\) 35.6525 1.61226
\(490\) 0 0
\(491\) −26.8328 −1.21095 −0.605474 0.795865i \(-0.707016\pi\)
−0.605474 + 0.795865i \(0.707016\pi\)
\(492\) 0 0
\(493\) 8.29180i 0.373444i
\(494\) 0 0
\(495\) −4.79837 −0.215671
\(496\) 0 0
\(497\) − 0.708204i − 0.0317673i
\(498\) 0 0
\(499\) −15.4164 −0.690133 −0.345067 0.938578i \(-0.612144\pi\)
−0.345067 + 0.938578i \(0.612144\pi\)
\(500\) 0 0
\(501\) −15.7082 −0.701791
\(502\) 0 0
\(503\) 10.1459i 0.452383i 0.974083 + 0.226192i \(0.0726276\pi\)
−0.974083 + 0.226192i \(0.927372\pi\)
\(504\) 0 0
\(505\) 3.41641 0.152028
\(506\) 0 0
\(507\) 9.94427i 0.441641i
\(508\) 0 0
\(509\) −6.65248 −0.294866 −0.147433 0.989072i \(-0.547101\pi\)
−0.147433 + 0.989072i \(0.547101\pi\)
\(510\) 0 0
\(511\) −30.5410 −1.35106
\(512\) 0 0
\(513\) − 0.798374i − 0.0352491i
\(514\) 0 0
\(515\) − 24.2705i − 1.06949i
\(516\) 0 0
\(517\) − 9.59675i − 0.422064i
\(518\) 0 0
\(519\) 26.7984 1.17632
\(520\) 0 0
\(521\) −38.0689 −1.66783 −0.833914 0.551894i \(-0.813905\pi\)
−0.833914 + 0.551894i \(0.813905\pi\)
\(522\) 0 0
\(523\) 4.36068i 0.190679i 0.995445 + 0.0953396i \(0.0303937\pi\)
−0.995445 + 0.0953396i \(0.969606\pi\)
\(524\) 0 0
\(525\) −15.0000 −0.654654
\(526\) 0 0
\(527\) 1.83282i 0.0798387i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −2.29180 −0.0994555
\(532\) 0 0
\(533\) 14.7082i 0.637083i
\(534\) 0 0
\(535\) 26.1803i 1.13187i
\(536\) 0 0
\(537\) 12.1803i 0.525620i
\(538\) 0 0
\(539\) 20.0132 0.862028
\(540\) 0 0
\(541\) −0.583592 −0.0250906 −0.0125453 0.999921i \(-0.503993\pi\)
−0.0125453 + 0.999921i \(0.503993\pi\)
\(542\) 0 0
\(543\) 25.1803i 1.08059i
\(544\) 0 0
\(545\) 16.9098 0.724338
\(546\) 0 0
\(547\) 7.85410i 0.335817i 0.985803 + 0.167909i \(0.0537013\pi\)
−0.985803 + 0.167909i \(0.946299\pi\)
\(548\) 0 0
\(549\) 1.09017 0.0465273
\(550\) 0 0
\(551\) −1.41641 −0.0603410
\(552\) 0 0
\(553\) − 14.2918i − 0.607749i
\(554\) 0 0
\(555\) −35.1246 −1.49096
\(556\) 0 0
\(557\) − 30.0000i − 1.27114i −0.772043 0.635570i \(-0.780765\pi\)
0.772043 0.635570i \(-0.219235\pi\)
\(558\) 0 0
\(559\) 29.4164 1.24418
\(560\) 0 0
\(561\) −7.76393 −0.327793
\(562\) 0 0
\(563\) − 15.7082i − 0.662022i −0.943627 0.331011i \(-0.892610\pi\)
0.943627 0.331011i \(-0.107390\pi\)
\(564\) 0 0
\(565\) − 30.0000i − 1.26211i
\(566\) 0 0
\(567\) − 14.2918i − 0.600199i
\(568\) 0 0
\(569\) 16.3607 0.685875 0.342938 0.939358i \(-0.388578\pi\)
0.342938 + 0.939358i \(0.388578\pi\)
\(570\) 0 0
\(571\) −8.27051 −0.346110 −0.173055 0.984912i \(-0.555364\pi\)
−0.173055 + 0.984912i \(0.555364\pi\)
\(572\) 0 0
\(573\) − 41.1246i − 1.71801i
\(574\) 0 0
\(575\) 5.00000i 0.208514i
\(576\) 0 0
\(577\) − 39.7082i − 1.65307i −0.562882 0.826537i \(-0.690308\pi\)
0.562882 0.826537i \(-0.309692\pi\)
\(578\) 0 0
\(579\) −25.4164 −1.05627
\(580\) 0 0
\(581\) −14.2918 −0.592924
\(582\) 0 0
\(583\) 11.2361i 0.465350i
\(584\) 0 0
\(585\) 2.23607i 0.0924500i
\(586\) 0 0
\(587\) 8.85410i 0.365448i 0.983164 + 0.182724i \(0.0584915\pi\)
−0.983164 + 0.182724i \(0.941509\pi\)
\(588\) 0 0
\(589\) −0.313082 −0.0129003
\(590\) 0 0
\(591\) 33.2705 1.36857
\(592\) 0 0
\(593\) − 6.58359i − 0.270356i −0.990821 0.135178i \(-0.956839\pi\)
0.990821 0.135178i \(-0.0431606\pi\)
\(594\) 0 0
\(595\) 3.54102 0.145168
\(596\) 0 0
\(597\) 18.4721i 0.756014i
\(598\) 0 0
\(599\) 23.6180 0.965007 0.482503 0.875894i \(-0.339728\pi\)
0.482503 + 0.875894i \(0.339728\pi\)
\(600\) 0 0
\(601\) −9.85410 −0.401957 −0.200979 0.979596i \(-0.564412\pi\)
−0.200979 + 0.979596i \(0.564412\pi\)
\(602\) 0 0
\(603\) − 2.00000i − 0.0814463i
\(604\) 0 0
\(605\) −45.9787 −1.86930
\(606\) 0 0
\(607\) − 21.0557i − 0.854626i −0.904104 0.427313i \(-0.859460\pi\)
0.904104 0.427313i \(-0.140540\pi\)
\(608\) 0 0
\(609\) −29.1246 −1.18019
\(610\) 0 0
\(611\) −4.47214 −0.180923
\(612\) 0 0
\(613\) 0.763932i 0.0308549i 0.999881 + 0.0154275i \(0.00491091\pi\)
−0.999881 + 0.0154275i \(0.995089\pi\)
\(614\) 0 0
\(615\) 20.3262i 0.819633i
\(616\) 0 0
\(617\) − 2.56231i − 0.103155i −0.998669 0.0515773i \(-0.983575\pi\)
0.998669 0.0515773i \(-0.0164248\pi\)
\(618\) 0 0
\(619\) −27.8541 −1.11955 −0.559775 0.828644i \(-0.689113\pi\)
−0.559775 + 0.828644i \(0.689113\pi\)
\(620\) 0 0
\(621\) −5.47214 −0.219589
\(622\) 0 0
\(623\) − 6.87539i − 0.275457i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) − 1.32624i − 0.0529648i
\(628\) 0 0
\(629\) 8.29180 0.330616
\(630\) 0 0
\(631\) 23.4164 0.932192 0.466096 0.884734i \(-0.345660\pi\)
0.466096 + 0.884734i \(0.345660\pi\)
\(632\) 0 0
\(633\) 12.4721i 0.495723i
\(634\) 0 0
\(635\) − 21.7082i − 0.861464i
\(636\) 0 0
\(637\) − 9.32624i − 0.369519i
\(638\) 0 0
\(639\) −0.145898 −0.00577164
\(640\) 0 0
\(641\) −0.652476 −0.0257712 −0.0128856 0.999917i \(-0.504102\pi\)
−0.0128856 + 0.999917i \(0.504102\pi\)
\(642\) 0 0
\(643\) − 41.0132i − 1.61740i −0.588221 0.808700i \(-0.700171\pi\)
0.588221 0.808700i \(-0.299829\pi\)
\(644\) 0 0
\(645\) 40.6525 1.60069
\(646\) 0 0
\(647\) 13.7082i 0.538925i 0.963011 + 0.269463i \(0.0868460\pi\)
−0.963011 + 0.269463i \(0.913154\pi\)
\(648\) 0 0
\(649\) −33.7082 −1.32316
\(650\) 0 0
\(651\) −6.43769 −0.252313
\(652\) 0 0
\(653\) − 34.9787i − 1.36882i −0.729096 0.684411i \(-0.760059\pi\)
0.729096 0.684411i \(-0.239941\pi\)
\(654\) 0 0
\(655\) −31.7082 −1.23894
\(656\) 0 0
\(657\) 6.29180i 0.245466i
\(658\) 0 0
\(659\) −17.8885 −0.696839 −0.348419 0.937339i \(-0.613281\pi\)
−0.348419 + 0.937339i \(0.613281\pi\)
\(660\) 0 0
\(661\) −39.3951 −1.53229 −0.766146 0.642666i \(-0.777828\pi\)
−0.766146 + 0.642666i \(0.777828\pi\)
\(662\) 0 0
\(663\) 3.61803i 0.140513i
\(664\) 0 0
\(665\) 0.604878i 0.0234562i
\(666\) 0 0
\(667\) 9.70820i 0.375903i
\(668\) 0 0
\(669\) −36.1803 −1.39881
\(670\) 0 0
\(671\) 16.0344 0.619003
\(672\) 0 0
\(673\) 27.7082i 1.06807i 0.845461 + 0.534036i \(0.179325\pi\)
−0.845461 + 0.534036i \(0.820675\pi\)
\(674\) 0 0
\(675\) 27.3607i 1.05311i
\(676\) 0 0
\(677\) 42.0000i 1.61419i 0.590421 + 0.807096i \(0.298962\pi\)
−0.590421 + 0.807096i \(0.701038\pi\)
\(678\) 0 0
\(679\) −24.1672 −0.927451
\(680\) 0 0
\(681\) 16.6525 0.638124
\(682\) 0 0
\(683\) − 10.9787i − 0.420089i −0.977692 0.210044i \(-0.932639\pi\)
0.977692 0.210044i \(-0.0673609\pi\)
\(684\) 0 0
\(685\) 30.9787i 1.18364i
\(686\) 0 0
\(687\) 16.1803i 0.617318i
\(688\) 0 0
\(689\) 5.23607 0.199478
\(690\) 0 0
\(691\) 21.1246 0.803618 0.401809 0.915723i \(-0.368382\pi\)
0.401809 + 0.915723i \(0.368382\pi\)
\(692\) 0 0
\(693\) 3.97871i 0.151139i
\(694\) 0 0
\(695\) −9.59675 −0.364025
\(696\) 0 0
\(697\) − 4.79837i − 0.181751i
\(698\) 0 0
\(699\) 34.1803 1.29282
\(700\) 0 0
\(701\) −27.9230 −1.05464 −0.527318 0.849668i \(-0.676802\pi\)
−0.527318 + 0.849668i \(0.676802\pi\)
\(702\) 0 0
\(703\) 1.41641i 0.0534208i
\(704\) 0 0
\(705\) −6.18034 −0.232765
\(706\) 0 0
\(707\) − 2.83282i − 0.106539i
\(708\) 0 0
\(709\) −8.56231 −0.321564 −0.160782 0.986990i \(-0.551402\pi\)
−0.160782 + 0.986990i \(0.551402\pi\)
\(710\) 0 0
\(711\) −2.94427 −0.110419
\(712\) 0 0
\(713\) 2.14590i 0.0803645i
\(714\) 0 0
\(715\) 32.8885i 1.22996i
\(716\) 0 0
\(717\) − 16.9443i − 0.632795i
\(718\) 0 0
\(719\) 23.4508 0.874569 0.437285 0.899323i \(-0.355940\pi\)
0.437285 + 0.899323i \(0.355940\pi\)
\(720\) 0 0
\(721\) −20.1246 −0.749480
\(722\) 0 0
\(723\) − 3.23607i − 0.120351i
\(724\) 0 0
\(725\) 48.5410 1.80277
\(726\) 0 0
\(727\) − 40.7426i − 1.51106i −0.655113 0.755531i \(-0.727379\pi\)
0.655113 0.755531i \(-0.272621\pi\)
\(728\) 0 0
\(729\) −29.5066 −1.09284
\(730\) 0 0
\(731\) −9.59675 −0.354949
\(732\) 0 0
\(733\) − 0.111456i − 0.00411673i −0.999998 0.00205836i \(-0.999345\pi\)
0.999998 0.00205836i \(-0.000655198\pi\)
\(734\) 0 0
\(735\) − 12.8885i − 0.475401i
\(736\) 0 0
\(737\) − 29.4164i − 1.08357i
\(738\) 0 0
\(739\) 34.5410 1.27061 0.635306 0.772261i \(-0.280874\pi\)
0.635306 + 0.772261i \(0.280874\pi\)
\(740\) 0 0
\(741\) −0.618034 −0.0227040
\(742\) 0 0
\(743\) 36.9787i 1.35662i 0.734777 + 0.678309i \(0.237287\pi\)
−0.734777 + 0.678309i \(0.762713\pi\)
\(744\) 0 0
\(745\) −5.85410 −0.214478
\(746\) 0 0
\(747\) 2.94427i 0.107725i
\(748\) 0 0
\(749\) 21.7082 0.793201
\(750\) 0 0
\(751\) 0.875388 0.0319434 0.0159717 0.999872i \(-0.494916\pi\)
0.0159717 + 0.999872i \(0.494916\pi\)
\(752\) 0 0
\(753\) 27.1803i 0.990507i
\(754\) 0 0
\(755\) 31.9098 1.16132
\(756\) 0 0
\(757\) 29.0132i 1.05450i 0.849710 + 0.527251i \(0.176777\pi\)
−0.849710 + 0.527251i \(0.823223\pi\)
\(758\) 0 0
\(759\) −9.09017 −0.329952
\(760\) 0 0
\(761\) 26.5066 0.960863 0.480431 0.877032i \(-0.340480\pi\)
0.480431 + 0.877032i \(0.340480\pi\)
\(762\) 0 0
\(763\) − 14.0213i − 0.507605i
\(764\) 0 0
\(765\) − 0.729490i − 0.0263748i
\(766\) 0 0
\(767\) 15.7082i 0.567190i
\(768\) 0 0
\(769\) 47.7082 1.72040 0.860201 0.509955i \(-0.170338\pi\)
0.860201 + 0.509955i \(0.170338\pi\)
\(770\) 0 0
\(771\) 18.4721 0.665258
\(772\) 0 0
\(773\) − 14.0000i − 0.503545i −0.967786 0.251773i \(-0.918987\pi\)
0.967786 0.251773i \(-0.0810135\pi\)
\(774\) 0 0
\(775\) 10.7295 0.385415
\(776\) 0 0
\(777\) 29.1246i 1.04484i
\(778\) 0 0
\(779\) 0.819660 0.0293674
\(780\) 0 0
\(781\) −2.14590 −0.0767863
\(782\) 0 0
\(783\) 53.1246i 1.89852i
\(784\) 0 0
\(785\) − 5.12461i − 0.182905i
\(786\) 0 0
\(787\) − 4.58359i − 0.163387i −0.996657 0.0816937i \(-0.973967\pi\)
0.996657 0.0816937i \(-0.0260329\pi\)
\(788\) 0 0
\(789\) 29.5623 1.05245
\(790\) 0 0
\(791\) −24.8754 −0.884467
\(792\) 0 0
\(793\) − 7.47214i − 0.265343i
\(794\) 0 0
\(795\) 7.23607 0.256637
\(796\) 0 0
\(797\) 45.4164i 1.60873i 0.594134 + 0.804366i \(0.297495\pi\)
−0.594134 + 0.804366i \(0.702505\pi\)
\(798\) 0 0
\(799\) 1.45898 0.0516150
\(800\) 0 0
\(801\) −1.41641 −0.0500463
\(802\) 0 0
\(803\) 92.5410i 3.26570i
\(804\) 0 0
\(805\) 4.14590 0.146124
\(806\) 0 0
\(807\) 2.47214i 0.0870233i
\(808\) 0 0
\(809\) 42.1591 1.48223 0.741117 0.671376i \(-0.234297\pi\)
0.741117 + 0.671376i \(0.234297\pi\)
\(810\) 0 0
\(811\) −5.70820 −0.200442 −0.100221 0.994965i \(-0.531955\pi\)
−0.100221 + 0.994965i \(0.531955\pi\)
\(812\) 0 0
\(813\) − 41.8328i − 1.46714i
\(814\) 0 0
\(815\) 49.2705i 1.72587i
\(816\) 0 0
\(817\) − 1.63932i − 0.0573526i
\(818\) 0 0
\(819\) 1.85410 0.0647876
\(820\) 0 0
\(821\) 14.9443 0.521559 0.260779 0.965398i \(-0.416020\pi\)
0.260779 + 0.965398i \(0.416020\pi\)
\(822\) 0 0
\(823\) − 32.8328i − 1.14448i −0.820086 0.572240i \(-0.806075\pi\)
0.820086 0.572240i \(-0.193925\pi\)
\(824\) 0 0
\(825\) 45.4508i 1.58240i
\(826\) 0 0
\(827\) 32.2492i 1.12142i 0.828014 + 0.560708i \(0.189471\pi\)
−0.828014 + 0.560708i \(0.810529\pi\)
\(828\) 0 0
\(829\) 34.5410 1.19966 0.599830 0.800128i \(-0.295235\pi\)
0.599830 + 0.800128i \(0.295235\pi\)
\(830\) 0 0
\(831\) −12.1803 −0.422531
\(832\) 0 0
\(833\) 3.04257i 0.105419i
\(834\) 0 0
\(835\) − 21.7082i − 0.751243i
\(836\) 0 0
\(837\) 11.7426i 0.405885i
\(838\) 0 0
\(839\) −11.2361 −0.387912 −0.193956 0.981010i \(-0.562132\pi\)
−0.193956 + 0.981010i \(0.562132\pi\)
\(840\) 0 0
\(841\) 65.2492 2.24997
\(842\) 0 0
\(843\) 49.5967i 1.70820i
\(844\) 0 0
\(845\) −13.7426 −0.472761
\(846\) 0 0
\(847\) 38.1246i 1.30998i
\(848\) 0 0
\(849\) −33.8885 −1.16305
\(850\) 0 0
\(851\) 9.70820 0.332793
\(852\) 0 0
\(853\) − 0.214782i − 0.00735399i −0.999993 0.00367699i \(-0.998830\pi\)
0.999993 0.00367699i \(-0.00117043\pi\)
\(854\) 0 0
\(855\) 0.124612 0.00426163
\(856\) 0 0
\(857\) − 36.0000i − 1.22974i −0.788630 0.614868i \(-0.789209\pi\)
0.788630 0.614868i \(-0.210791\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 16.8541 0.574386
\(862\) 0 0
\(863\) 24.0000i 0.816970i 0.912765 + 0.408485i \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) 0 0
\(865\) 37.0344i 1.25921i
\(866\) 0 0
\(867\) 26.3262i 0.894086i
\(868\) 0 0
\(869\) −43.3050 −1.46902
\(870\) 0 0
\(871\) −13.7082 −0.464485
\(872\) 0 0
\(873\) 4.97871i 0.168504i
\(874\) 0 0
\(875\) − 20.7295i − 0.700785i
\(876\) 0 0
\(877\) 25.6869i 0.867386i 0.901061 + 0.433693i \(0.142790\pi\)
−0.901061 + 0.433693i \(0.857210\pi\)
\(878\) 0 0
\(879\) −23.1246 −0.779974
\(880\) 0 0
\(881\) −51.7082 −1.74209 −0.871047 0.491200i \(-0.836558\pi\)
−0.871047 + 0.491200i \(0.836558\pi\)
\(882\) 0 0
\(883\) − 51.2705i − 1.72539i −0.505725 0.862695i \(-0.668775\pi\)
0.505725 0.862695i \(-0.331225\pi\)
\(884\) 0 0
\(885\) 21.7082i 0.729713i
\(886\) 0 0
\(887\) − 30.5410i − 1.02547i −0.858548 0.512734i \(-0.828633\pi\)
0.858548 0.512734i \(-0.171367\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) −43.3050 −1.45077
\(892\) 0 0
\(893\) 0.249224i 0.00833995i
\(894\) 0 0
\(895\) −16.8328 −0.562659
\(896\) 0 0
\(897\) 4.23607i 0.141438i
\(898\) 0 0
\(899\) 20.8328 0.694813
\(900\) 0 0
\(901\) −1.70820 −0.0569085
\(902\) 0 0
\(903\) − 33.7082i − 1.12174i
\(904\) 0 0
\(905\) −34.7984 −1.15674
\(906\) 0 0
\(907\) 36.5410i 1.21332i 0.794960 + 0.606662i \(0.207492\pi\)
−0.794960 + 0.606662i \(0.792508\pi\)
\(908\) 0 0
\(909\) −0.583592 −0.0193565
\(910\) 0 0
\(911\) 22.3607 0.740842 0.370421 0.928864i \(-0.379213\pi\)
0.370421 + 0.928864i \(0.379213\pi\)
\(912\) 0 0
\(913\) 43.3050i 1.43318i
\(914\) 0 0
\(915\) − 10.3262i − 0.341375i
\(916\) 0 0
\(917\) 26.2918i 0.868232i
\(918\) 0 0
\(919\) −41.4164 −1.36620 −0.683101 0.730324i \(-0.739369\pi\)
−0.683101 + 0.730324i \(0.739369\pi\)
\(920\) 0 0
\(921\) 27.2705 0.898594
\(922\) 0 0
\(923\) 1.00000i 0.0329154i
\(924\) 0 0
\(925\) − 48.5410i − 1.59602i
\(926\) 0 0
\(927\) 4.14590i 0.136169i
\(928\) 0 0
\(929\) 43.3050 1.42079 0.710395 0.703804i \(-0.248516\pi\)
0.710395 + 0.703804i \(0.248516\pi\)
\(930\) 0 0
\(931\) −0.519733 −0.0170336
\(932\) 0 0
\(933\) 36.3607i 1.19040i
\(934\) 0 0
\(935\) − 10.7295i − 0.350892i
\(936\) 0 0
\(937\) − 24.3262i − 0.794704i −0.917666 0.397352i \(-0.869929\pi\)
0.917666 0.397352i \(-0.130071\pi\)
\(938\) 0 0
\(939\) 41.8328 1.36516
\(940\) 0 0
\(941\) 27.2148 0.887177 0.443588 0.896231i \(-0.353705\pi\)
0.443588 + 0.896231i \(0.353705\pi\)
\(942\) 0 0
\(943\) − 5.61803i − 0.182948i
\(944\) 0 0
\(945\) 22.6869 0.738005
\(946\) 0 0
\(947\) − 26.3951i − 0.857726i −0.903369 0.428863i \(-0.858914\pi\)
0.903369 0.428863i \(-0.141086\pi\)
\(948\) 0 0
\(949\) 43.1246 1.39988
\(950\) 0 0
\(951\) 11.7639 0.381472
\(952\) 0 0
\(953\) − 37.3951i − 1.21135i −0.795713 0.605673i \(-0.792904\pi\)
0.795713 0.605673i \(-0.207096\pi\)
\(954\) 0 0
\(955\) 56.8328 1.83907
\(956\) 0 0
\(957\) 88.2492i 2.85269i
\(958\) 0 0
\(959\) 25.6869 0.829474
\(960\) 0 0
\(961\) −26.3951 −0.851456
\(962\) 0 0
\(963\) − 4.47214i − 0.144113i
\(964\) 0 0
\(965\) − 35.1246i − 1.13070i
\(966\) 0 0
\(967\) − 29.2361i − 0.940169i −0.882622 0.470084i \(-0.844224\pi\)
0.882622 0.470084i \(-0.155776\pi\)
\(968\) 0 0
\(969\) 0.201626 0.00647716
\(970\) 0 0
\(971\) −19.1459 −0.614421 −0.307211 0.951642i \(-0.599396\pi\)
−0.307211 + 0.951642i \(0.599396\pi\)
\(972\) 0 0
\(973\) 7.95743i 0.255103i
\(974\) 0 0
\(975\) 21.1803 0.678314
\(976\) 0 0
\(977\) 1.27051i 0.0406472i 0.999793 + 0.0203236i \(0.00646965\pi\)
−0.999793 + 0.0203236i \(0.993530\pi\)
\(978\) 0 0
\(979\) −20.8328 −0.665820
\(980\) 0 0
\(981\) −2.88854 −0.0922241
\(982\) 0 0
\(983\) − 47.3951i − 1.51167i −0.654762 0.755835i \(-0.727231\pi\)
0.654762 0.755835i \(-0.272769\pi\)
\(984\) 0 0
\(985\) 45.9787i 1.46500i
\(986\) 0 0
\(987\) 5.12461i 0.163118i
\(988\) 0 0
\(989\) −11.2361 −0.357286
\(990\) 0 0
\(991\) −17.7295 −0.563196 −0.281598 0.959532i \(-0.590864\pi\)
−0.281598 + 0.959532i \(0.590864\pi\)
\(992\) 0 0
\(993\) 40.6525i 1.29007i
\(994\) 0 0
\(995\) −25.5279 −0.809288
\(996\) 0 0
\(997\) − 62.7214i − 1.98641i −0.116398 0.993203i \(-0.537135\pi\)
0.116398 0.993203i \(-0.462865\pi\)
\(998\) 0 0
\(999\) 53.1246 1.68079
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 1840.2.e.c.369.4 4
4.3 odd 2 230.2.b.a.139.1 4
5.2 odd 4 9200.2.a.by.1.2 2
5.3 odd 4 9200.2.a.bo.1.1 2
5.4 even 2 inner 1840.2.e.c.369.1 4
12.11 even 2 2070.2.d.c.829.4 4
20.3 even 4 1150.2.a.l.1.2 2
20.7 even 4 1150.2.a.n.1.1 2
20.19 odd 2 230.2.b.a.139.4 yes 4
60.59 even 2 2070.2.d.c.829.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.b.a.139.1 4 4.3 odd 2
230.2.b.a.139.4 yes 4 20.19 odd 2
1150.2.a.l.1.2 2 20.3 even 4
1150.2.a.n.1.1 2 20.7 even 4
1840.2.e.c.369.1 4 5.4 even 2 inner
1840.2.e.c.369.4 4 1.1 even 1 trivial
2070.2.d.c.829.2 4 60.59 even 2
2070.2.d.c.829.4 4 12.11 even 2
9200.2.a.bo.1.1 2 5.3 odd 4
9200.2.a.by.1.2 2 5.2 odd 4