Properties

Label 3380.2.f.i.3041.6
Level $3380$
Weight $2$
Character 3380.3041
Analytic conductor $26.989$
Analytic rank $0$
Dimension $8$
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] = [3380,2,Mod(3041,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3380.3041");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.f (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: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3041.6
Root \(-1.27597 - 0.609843i\) of defining polynomial
Character \(\chi\) \(=\) 3380.3041
Dual form 3380.2.f.i.3041.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.60020 q^{3} +1.00000i q^{5} -4.33225i q^{7} -0.439374 q^{9} +O(q^{10})\) \(q+1.60020 q^{3} +1.00000i q^{5} -4.33225i q^{7} -0.439374 q^{9} +1.73205i q^{11} +1.60020i q^{15} -7.50367 q^{17} -5.37182i q^{19} -6.93244i q^{21} +1.16082 q^{23} -1.00000 q^{25} -5.50367 q^{27} -2.02473 q^{29} +7.86488i q^{31} +2.77162i q^{33} +4.33225 q^{35} -9.53264i q^{37} +7.73205i q^{41} -4.18555 q^{43} -0.439374i q^{45} +3.46410i q^{47} -11.7684 q^{49} -12.0073 q^{51} -12.6471 q^{53} -1.73205 q^{55} -8.59596i q^{57} -6.34709i q^{59} +3.70308 q^{61} +1.90348i q^{63} -5.26045i q^{67} +1.85754 q^{69} +12.5143i q^{71} -5.23898i q^{73} -1.60020 q^{75} +7.50367 q^{77} -8.16719 q^{79} -7.48883 q^{81} -0.456760i q^{83} -7.50367i q^{85} -3.23996 q^{87} -13.2753i q^{89} +12.5854i q^{93} +5.37182 q^{95} +2.81021i q^{97} -0.761018i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 8 q^{9} - 12 q^{17} + 12 q^{23} - 8 q^{25} + 4 q^{27} + 12 q^{35} - 20 q^{43} + 8 q^{49} + 24 q^{53} + 8 q^{61} + 48 q^{69} - 4 q^{75} + 12 q^{77} - 16 q^{79} - 16 q^{81} + 12 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 1.60020 0.923873 0.461937 0.886913i \(-0.347155\pi\)
0.461937 + 0.886913i \(0.347155\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) − 4.33225i − 1.63744i −0.574196 0.818718i \(-0.694685\pi\)
0.574196 0.818718i \(-0.305315\pi\)
\(8\) 0 0
\(9\) −0.439374 −0.146458
\(10\) 0 0
\(11\) 1.73205i 0.522233i 0.965307 + 0.261116i \(0.0840907\pi\)
−0.965307 + 0.261116i \(0.915909\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 1.60020i 0.413169i
\(16\) 0 0
\(17\) −7.50367 −1.81991 −0.909954 0.414710i \(-0.863883\pi\)
−0.909954 + 0.414710i \(0.863883\pi\)
\(18\) 0 0
\(19\) − 5.37182i − 1.23238i −0.787598 0.616190i \(-0.788676\pi\)
0.787598 0.616190i \(-0.211324\pi\)
\(20\) 0 0
\(21\) − 6.93244i − 1.51278i
\(22\) 0 0
\(23\) 1.16082 0.242048 0.121024 0.992650i \(-0.461382\pi\)
0.121024 + 0.992650i \(0.461382\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −5.50367 −1.05918
\(28\) 0 0
\(29\) −2.02473 −0.375983 −0.187991 0.982171i \(-0.560198\pi\)
−0.187991 + 0.982171i \(0.560198\pi\)
\(30\) 0 0
\(31\) 7.86488i 1.41257i 0.707925 + 0.706287i \(0.249631\pi\)
−0.707925 + 0.706287i \(0.750369\pi\)
\(32\) 0 0
\(33\) 2.77162i 0.482477i
\(34\) 0 0
\(35\) 4.33225 0.732283
\(36\) 0 0
\(37\) − 9.53264i − 1.56716i −0.621293 0.783578i \(-0.713392\pi\)
0.621293 0.783578i \(-0.286608\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.73205i 1.20754i 0.797157 + 0.603772i \(0.206336\pi\)
−0.797157 + 0.603772i \(0.793664\pi\)
\(42\) 0 0
\(43\) −4.18555 −0.638290 −0.319145 0.947706i \(-0.603396\pi\)
−0.319145 + 0.947706i \(0.603396\pi\)
\(44\) 0 0
\(45\) − 0.439374i − 0.0654980i
\(46\) 0 0
\(47\) 3.46410i 0.505291i 0.967559 + 0.252646i \(0.0813007\pi\)
−0.967559 + 0.252646i \(0.918699\pi\)
\(48\) 0 0
\(49\) −11.7684 −1.68119
\(50\) 0 0
\(51\) −12.0073 −1.68136
\(52\) 0 0
\(53\) −12.6471 −1.73721 −0.868607 0.495502i \(-0.834984\pi\)
−0.868607 + 0.495502i \(0.834984\pi\)
\(54\) 0 0
\(55\) −1.73205 −0.233550
\(56\) 0 0
\(57\) − 8.59596i − 1.13856i
\(58\) 0 0
\(59\) − 6.34709i − 0.826320i −0.910658 0.413160i \(-0.864425\pi\)
0.910658 0.413160i \(-0.135575\pi\)
\(60\) 0 0
\(61\) 3.70308 0.474131 0.237066 0.971494i \(-0.423814\pi\)
0.237066 + 0.971494i \(0.423814\pi\)
\(62\) 0 0
\(63\) 1.90348i 0.239815i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 5.26045i − 0.642666i −0.946966 0.321333i \(-0.895869\pi\)
0.946966 0.321333i \(-0.104131\pi\)
\(68\) 0 0
\(69\) 1.85754 0.223622
\(70\) 0 0
\(71\) 12.5143i 1.48517i 0.669751 + 0.742586i \(0.266401\pi\)
−0.669751 + 0.742586i \(0.733599\pi\)
\(72\) 0 0
\(73\) − 5.23898i − 0.613177i −0.951842 0.306588i \(-0.900813\pi\)
0.951842 0.306588i \(-0.0991875\pi\)
\(74\) 0 0
\(75\) −1.60020 −0.184775
\(76\) 0 0
\(77\) 7.50367 0.855123
\(78\) 0 0
\(79\) −8.16719 −0.918880 −0.459440 0.888209i \(-0.651950\pi\)
−0.459440 + 0.888209i \(0.651950\pi\)
\(80\) 0 0
\(81\) −7.48883 −0.832092
\(82\) 0 0
\(83\) − 0.456760i − 0.0501359i −0.999686 0.0250679i \(-0.992020\pi\)
0.999686 0.0250679i \(-0.00798021\pi\)
\(84\) 0 0
\(85\) − 7.50367i − 0.813887i
\(86\) 0 0
\(87\) −3.23996 −0.347360
\(88\) 0 0
\(89\) − 13.2753i − 1.40718i −0.710607 0.703589i \(-0.751580\pi\)
0.710607 0.703589i \(-0.248420\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 12.5854i 1.30504i
\(94\) 0 0
\(95\) 5.37182 0.551137
\(96\) 0 0
\(97\) 2.81021i 0.285334i 0.989771 + 0.142667i \(0.0455678\pi\)
−0.989771 + 0.142667i \(0.954432\pi\)
\(98\) 0 0
\(99\) − 0.761018i − 0.0764852i
\(100\) 0 0
\(101\) 6.63977 0.660681 0.330341 0.943862i \(-0.392836\pi\)
0.330341 + 0.943862i \(0.392836\pi\)
\(102\) 0 0
\(103\) 7.37605 0.726784 0.363392 0.931636i \(-0.381619\pi\)
0.363392 + 0.931636i \(0.381619\pi\)
\(104\) 0 0
\(105\) 6.93244 0.676537
\(106\) 0 0
\(107\) −7.91336 −0.765014 −0.382507 0.923953i \(-0.624939\pi\)
−0.382507 + 0.923953i \(0.624939\pi\)
\(108\) 0 0
\(109\) − 9.18301i − 0.879572i −0.898102 0.439786i \(-0.855054\pi\)
0.898102 0.439786i \(-0.144946\pi\)
\(110\) 0 0
\(111\) − 15.2541i − 1.44785i
\(112\) 0 0
\(113\) −5.77586 −0.543347 −0.271674 0.962389i \(-0.587577\pi\)
−0.271674 + 0.962389i \(0.587577\pi\)
\(114\) 0 0
\(115\) 1.16082i 0.108247i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 32.5078i 2.97998i
\(120\) 0 0
\(121\) 8.00000 0.727273
\(122\) 0 0
\(123\) 12.3728i 1.11562i
\(124\) 0 0
\(125\) − 1.00000i − 0.0894427i
\(126\) 0 0
\(127\) −18.9678 −1.68312 −0.841559 0.540165i \(-0.818362\pi\)
−0.841559 + 0.540165i \(0.818362\pi\)
\(128\) 0 0
\(129\) −6.69770 −0.589699
\(130\) 0 0
\(131\) −1.34637 −0.117633 −0.0588165 0.998269i \(-0.518733\pi\)
−0.0588165 + 0.998269i \(0.518733\pi\)
\(132\) 0 0
\(133\) −23.2720 −2.01794
\(134\) 0 0
\(135\) − 5.50367i − 0.473681i
\(136\) 0 0
\(137\) 12.3728i 1.05708i 0.848909 + 0.528540i \(0.177260\pi\)
−0.848909 + 0.528540i \(0.822740\pi\)
\(138\) 0 0
\(139\) 8.76836 0.743723 0.371861 0.928288i \(-0.378720\pi\)
0.371861 + 0.928288i \(0.378720\pi\)
\(140\) 0 0
\(141\) 5.54324i 0.466825i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) − 2.02473i − 0.168144i
\(146\) 0 0
\(147\) −18.8317 −1.55321
\(148\) 0 0
\(149\) − 4.73939i − 0.388266i −0.980975 0.194133i \(-0.937811\pi\)
0.980975 0.194133i \(-0.0621894\pi\)
\(150\) 0 0
\(151\) − 1.56063i − 0.127002i −0.997982 0.0635010i \(-0.979773\pi\)
0.997982 0.0635010i \(-0.0202266\pi\)
\(152\) 0 0
\(153\) 3.29692 0.266540
\(154\) 0 0
\(155\) −7.86488 −0.631723
\(156\) 0 0
\(157\) −9.15332 −0.730515 −0.365257 0.930907i \(-0.619019\pi\)
−0.365257 + 0.930907i \(0.619019\pi\)
\(158\) 0 0
\(159\) −20.2378 −1.60497
\(160\) 0 0
\(161\) − 5.02897i − 0.396338i
\(162\) 0 0
\(163\) − 2.64303i − 0.207018i −0.994629 0.103509i \(-0.966993\pi\)
0.994629 0.103509i \(-0.0330070\pi\)
\(164\) 0 0
\(165\) −2.77162 −0.215770
\(166\) 0 0
\(167\) 8.90869i 0.689375i 0.938717 + 0.344688i \(0.112015\pi\)
−0.938717 + 0.344688i \(0.887985\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 2.36023i 0.180492i
\(172\) 0 0
\(173\) −13.8079 −1.04980 −0.524899 0.851165i \(-0.675897\pi\)
−0.524899 + 0.851165i \(0.675897\pi\)
\(174\) 0 0
\(175\) 4.33225i 0.327487i
\(176\) 0 0
\(177\) − 10.1566i − 0.763416i
\(178\) 0 0
\(179\) 24.3502 1.82002 0.910009 0.414588i \(-0.136074\pi\)
0.910009 + 0.414588i \(0.136074\pi\)
\(180\) 0 0
\(181\) 24.8336 1.84587 0.922935 0.384956i \(-0.125783\pi\)
0.922935 + 0.384956i \(0.125783\pi\)
\(182\) 0 0
\(183\) 5.92566 0.438037
\(184\) 0 0
\(185\) 9.53264 0.700853
\(186\) 0 0
\(187\) − 12.9967i − 0.950416i
\(188\) 0 0
\(189\) 23.8433i 1.73434i
\(190\) 0 0
\(191\) 5.96875 0.431884 0.215942 0.976406i \(-0.430718\pi\)
0.215942 + 0.976406i \(0.430718\pi\)
\(192\) 0 0
\(193\) − 9.05272i − 0.651629i −0.945434 0.325814i \(-0.894362\pi\)
0.945434 0.325814i \(-0.105638\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 11.3890i − 0.811436i −0.913998 0.405718i \(-0.867022\pi\)
0.913998 0.405718i \(-0.132978\pi\)
\(198\) 0 0
\(199\) −10.7773 −0.763980 −0.381990 0.924166i \(-0.624761\pi\)
−0.381990 + 0.924166i \(0.624761\pi\)
\(200\) 0 0
\(201\) − 8.41775i − 0.593742i
\(202\) 0 0
\(203\) 8.77162i 0.615647i
\(204\) 0 0
\(205\) −7.73205 −0.540030
\(206\) 0 0
\(207\) −0.510035 −0.0354499
\(208\) 0 0
\(209\) 9.30426 0.643589
\(210\) 0 0
\(211\) −3.22512 −0.222026 −0.111013 0.993819i \(-0.535410\pi\)
−0.111013 + 0.993819i \(0.535410\pi\)
\(212\) 0 0
\(213\) 20.0253i 1.37211i
\(214\) 0 0
\(215\) − 4.18555i − 0.285452i
\(216\) 0 0
\(217\) 34.0726 2.31300
\(218\) 0 0
\(219\) − 8.38340i − 0.566497i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 16.9794i − 1.13702i −0.822676 0.568511i \(-0.807520\pi\)
0.822676 0.568511i \(-0.192480\pi\)
\(224\) 0 0
\(225\) 0.439374 0.0292916
\(226\) 0 0
\(227\) − 2.40052i − 0.159328i −0.996822 0.0796641i \(-0.974615\pi\)
0.996822 0.0796641i \(-0.0253848\pi\)
\(228\) 0 0
\(229\) − 7.35671i − 0.486145i −0.970008 0.243073i \(-0.921845\pi\)
0.970008 0.243073i \(-0.0781553\pi\)
\(230\) 0 0
\(231\) 12.0073 0.790025
\(232\) 0 0
\(233\) −3.37410 −0.221045 −0.110522 0.993874i \(-0.535252\pi\)
−0.110522 + 0.993874i \(0.535252\pi\)
\(234\) 0 0
\(235\) −3.46410 −0.225973
\(236\) 0 0
\(237\) −13.0691 −0.848929
\(238\) 0 0
\(239\) − 23.7807i − 1.53824i −0.639103 0.769121i \(-0.720694\pi\)
0.639103 0.769121i \(-0.279306\pi\)
\(240\) 0 0
\(241\) − 22.3212i − 1.43784i −0.695095 0.718918i \(-0.744638\pi\)
0.695095 0.718918i \(-0.255362\pi\)
\(242\) 0 0
\(243\) 4.52742 0.290434
\(244\) 0 0
\(245\) − 11.7684i − 0.751853i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) − 0.730905i − 0.0463192i
\(250\) 0 0
\(251\) 4.62238 0.291762 0.145881 0.989302i \(-0.453398\pi\)
0.145881 + 0.989302i \(0.453398\pi\)
\(252\) 0 0
\(253\) 2.01060i 0.126406i
\(254\) 0 0
\(255\) − 12.0073i − 0.751929i
\(256\) 0 0
\(257\) −21.8639 −1.36383 −0.681916 0.731430i \(-0.738853\pi\)
−0.681916 + 0.731430i \(0.738853\pi\)
\(258\) 0 0
\(259\) −41.2977 −2.56612
\(260\) 0 0
\(261\) 0.889612 0.0550656
\(262\) 0 0
\(263\) −20.1790 −1.24429 −0.622146 0.782901i \(-0.713739\pi\)
−0.622146 + 0.782901i \(0.713739\pi\)
\(264\) 0 0
\(265\) − 12.6471i − 0.776906i
\(266\) 0 0
\(267\) − 21.2431i − 1.30005i
\(268\) 0 0
\(269\) 7.28687 0.444288 0.222144 0.975014i \(-0.428694\pi\)
0.222144 + 0.975014i \(0.428694\pi\)
\(270\) 0 0
\(271\) − 4.89189i − 0.297161i −0.988900 0.148581i \(-0.952530\pi\)
0.988900 0.148581i \(-0.0474705\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 1.73205i − 0.104447i
\(276\) 0 0
\(277\) −8.80059 −0.528776 −0.264388 0.964416i \(-0.585170\pi\)
−0.264388 + 0.964416i \(0.585170\pi\)
\(278\) 0 0
\(279\) − 3.45562i − 0.206883i
\(280\) 0 0
\(281\) 12.6085i 0.752161i 0.926587 + 0.376081i \(0.122728\pi\)
−0.926587 + 0.376081i \(0.877272\pi\)
\(282\) 0 0
\(283\) 20.9009 1.24243 0.621216 0.783640i \(-0.286639\pi\)
0.621216 + 0.783640i \(0.286639\pi\)
\(284\) 0 0
\(285\) 8.59596 0.509181
\(286\) 0 0
\(287\) 33.4972 1.97727
\(288\) 0 0
\(289\) 39.3051 2.31206
\(290\) 0 0
\(291\) 4.49689i 0.263612i
\(292\) 0 0
\(293\) 0.692481i 0.0404552i 0.999795 + 0.0202276i \(0.00643908\pi\)
−0.999795 + 0.0202276i \(0.993561\pi\)
\(294\) 0 0
\(295\) 6.34709 0.369542
\(296\) 0 0
\(297\) − 9.53264i − 0.553140i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 18.1328i 1.04516i
\(302\) 0 0
\(303\) 10.6249 0.610386
\(304\) 0 0
\(305\) 3.70308i 0.212038i
\(306\) 0 0
\(307\) − 8.83168i − 0.504051i −0.967721 0.252025i \(-0.918903\pi\)
0.967721 0.252025i \(-0.0810966\pi\)
\(308\) 0 0
\(309\) 11.8031 0.671457
\(310\) 0 0
\(311\) −11.3958 −0.646198 −0.323099 0.946365i \(-0.604725\pi\)
−0.323099 + 0.946365i \(0.604725\pi\)
\(312\) 0 0
\(313\) −3.38496 −0.191329 −0.0956647 0.995414i \(-0.530498\pi\)
−0.0956647 + 0.995414i \(0.530498\pi\)
\(314\) 0 0
\(315\) −1.90348 −0.107249
\(316\) 0 0
\(317\) 0.557104i 0.0312901i 0.999878 + 0.0156450i \(0.00498017\pi\)
−0.999878 + 0.0156450i \(0.995020\pi\)
\(318\) 0 0
\(319\) − 3.50693i − 0.196350i
\(320\) 0 0
\(321\) −12.6629 −0.706776
\(322\) 0 0
\(323\) 40.3083i 2.24282i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 14.6946i − 0.812614i
\(328\) 0 0
\(329\) 15.0073 0.827382
\(330\) 0 0
\(331\) − 3.96369i − 0.217864i −0.994049 0.108932i \(-0.965257\pi\)
0.994049 0.108932i \(-0.0347431\pi\)
\(332\) 0 0
\(333\) 4.18839i 0.229522i
\(334\) 0 0
\(335\) 5.26045 0.287409
\(336\) 0 0
\(337\) 16.7243 0.911030 0.455515 0.890228i \(-0.349455\pi\)
0.455515 + 0.890228i \(0.349455\pi\)
\(338\) 0 0
\(339\) −9.24251 −0.501984
\(340\) 0 0
\(341\) −13.6224 −0.737693
\(342\) 0 0
\(343\) 20.6577i 1.11541i
\(344\) 0 0
\(345\) 1.85754i 0.100007i
\(346\) 0 0
\(347\) 7.58085 0.406962 0.203481 0.979079i \(-0.434775\pi\)
0.203481 + 0.979079i \(0.434775\pi\)
\(348\) 0 0
\(349\) 27.3356i 1.46324i 0.681712 + 0.731621i \(0.261236\pi\)
−0.681712 + 0.731621i \(0.738764\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 26.0921i 1.38874i 0.719616 + 0.694372i \(0.244318\pi\)
−0.719616 + 0.694372i \(0.755682\pi\)
\(354\) 0 0
\(355\) −12.5143 −0.664189
\(356\) 0 0
\(357\) 52.0188i 2.75313i
\(358\) 0 0
\(359\) − 0.694176i − 0.0366372i −0.999832 0.0183186i \(-0.994169\pi\)
0.999832 0.0183186i \(-0.00583132\pi\)
\(360\) 0 0
\(361\) −9.85641 −0.518758
\(362\) 0 0
\(363\) 12.8016 0.671908
\(364\) 0 0
\(365\) 5.23898 0.274221
\(366\) 0 0
\(367\) 22.8218 1.19129 0.595644 0.803249i \(-0.296897\pi\)
0.595644 + 0.803249i \(0.296897\pi\)
\(368\) 0 0
\(369\) − 3.39726i − 0.176854i
\(370\) 0 0
\(371\) 54.7904i 2.84458i
\(372\) 0 0
\(373\) −7.88129 −0.408078 −0.204039 0.978963i \(-0.565407\pi\)
−0.204039 + 0.978963i \(0.565407\pi\)
\(374\) 0 0
\(375\) − 1.60020i − 0.0826338i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 31.1426i 1.59969i 0.600209 + 0.799843i \(0.295084\pi\)
−0.600209 + 0.799843i \(0.704916\pi\)
\(380\) 0 0
\(381\) −30.3521 −1.55499
\(382\) 0 0
\(383\) − 34.5123i − 1.76349i −0.471723 0.881747i \(-0.656368\pi\)
0.471723 0.881747i \(-0.343632\pi\)
\(384\) 0 0
\(385\) 7.50367i 0.382422i
\(386\) 0 0
\(387\) 1.83902 0.0934827
\(388\) 0 0
\(389\) −6.18414 −0.313548 −0.156774 0.987634i \(-0.550109\pi\)
−0.156774 + 0.987634i \(0.550109\pi\)
\(390\) 0 0
\(391\) −8.71043 −0.440505
\(392\) 0 0
\(393\) −2.15446 −0.108678
\(394\) 0 0
\(395\) − 8.16719i − 0.410936i
\(396\) 0 0
\(397\) − 22.6563i − 1.13709i −0.822654 0.568543i \(-0.807507\pi\)
0.822654 0.568543i \(-0.192493\pi\)
\(398\) 0 0
\(399\) −37.2398 −1.86432
\(400\) 0 0
\(401\) 3.87803i 0.193660i 0.995301 + 0.0968298i \(0.0308703\pi\)
−0.995301 + 0.0968298i \(0.969130\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 7.48883i − 0.372123i
\(406\) 0 0
\(407\) 16.5110 0.818421
\(408\) 0 0
\(409\) − 13.8606i − 0.685365i −0.939451 0.342682i \(-0.888665\pi\)
0.939451 0.342682i \(-0.111335\pi\)
\(410\) 0 0
\(411\) 19.7989i 0.976607i
\(412\) 0 0
\(413\) −27.4972 −1.35305
\(414\) 0 0
\(415\) 0.456760 0.0224214
\(416\) 0 0
\(417\) 14.0311 0.687106
\(418\) 0 0
\(419\) −30.3502 −1.48270 −0.741352 0.671116i \(-0.765815\pi\)
−0.741352 + 0.671116i \(0.765815\pi\)
\(420\) 0 0
\(421\) − 0.608516i − 0.0296573i −0.999890 0.0148286i \(-0.995280\pi\)
0.999890 0.0148286i \(-0.00472027\pi\)
\(422\) 0 0
\(423\) − 1.52204i − 0.0740039i
\(424\) 0 0
\(425\) 7.50367 0.363982
\(426\) 0 0
\(427\) − 16.0427i − 0.776359i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 39.3544i − 1.89564i −0.318812 0.947818i \(-0.603284\pi\)
0.318812 0.947818i \(-0.396716\pi\)
\(432\) 0 0
\(433\) −6.42949 −0.308981 −0.154491 0.987994i \(-0.549374\pi\)
−0.154491 + 0.987994i \(0.549374\pi\)
\(434\) 0 0
\(435\) − 3.23996i − 0.155344i
\(436\) 0 0
\(437\) − 6.23572i − 0.298295i
\(438\) 0 0
\(439\) −1.47144 −0.0702282 −0.0351141 0.999383i \(-0.511179\pi\)
−0.0351141 + 0.999383i \(0.511179\pi\)
\(440\) 0 0
\(441\) 5.17071 0.246224
\(442\) 0 0
\(443\) −16.9193 −0.803860 −0.401930 0.915670i \(-0.631660\pi\)
−0.401930 + 0.915670i \(0.631660\pi\)
\(444\) 0 0
\(445\) 13.2753 0.629309
\(446\) 0 0
\(447\) − 7.58396i − 0.358709i
\(448\) 0 0
\(449\) 9.11701i 0.430258i 0.976586 + 0.215129i \(0.0690173\pi\)
−0.976586 + 0.215129i \(0.930983\pi\)
\(450\) 0 0
\(451\) −13.3923 −0.630619
\(452\) 0 0
\(453\) − 2.49731i − 0.117334i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 17.5562i − 0.821246i −0.911805 0.410623i \(-0.865311\pi\)
0.911805 0.410623i \(-0.134689\pi\)
\(458\) 0 0
\(459\) 41.2977 1.92761
\(460\) 0 0
\(461\) 15.3544i 0.715127i 0.933889 + 0.357564i \(0.116392\pi\)
−0.933889 + 0.357564i \(0.883608\pi\)
\(462\) 0 0
\(463\) 18.8614i 0.876562i 0.898838 + 0.438281i \(0.144412\pi\)
−0.898838 + 0.438281i \(0.855588\pi\)
\(464\) 0 0
\(465\) −12.5854 −0.583632
\(466\) 0 0
\(467\) 32.9302 1.52383 0.761913 0.647679i \(-0.224260\pi\)
0.761913 + 0.647679i \(0.224260\pi\)
\(468\) 0 0
\(469\) −22.7896 −1.05232
\(470\) 0 0
\(471\) −14.6471 −0.674903
\(472\) 0 0
\(473\) − 7.24958i − 0.333336i
\(474\) 0 0
\(475\) 5.37182i 0.246476i
\(476\) 0 0
\(477\) 5.55681 0.254429
\(478\) 0 0
\(479\) − 17.2476i − 0.788061i −0.919097 0.394031i \(-0.871080\pi\)
0.919097 0.394031i \(-0.128920\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) − 8.04733i − 0.366166i
\(484\) 0 0
\(485\) −2.81021 −0.127605
\(486\) 0 0
\(487\) 17.9744i 0.814498i 0.913317 + 0.407249i \(0.133512\pi\)
−0.913317 + 0.407249i \(0.866488\pi\)
\(488\) 0 0
\(489\) − 4.22936i − 0.191258i
\(490\) 0 0
\(491\) −26.1953 −1.18218 −0.591089 0.806607i \(-0.701302\pi\)
−0.591089 + 0.806607i \(0.701302\pi\)
\(492\) 0 0
\(493\) 15.1929 0.684253
\(494\) 0 0
\(495\) 0.761018 0.0342052
\(496\) 0 0
\(497\) 54.2149 2.43187
\(498\) 0 0
\(499\) 18.3428i 0.821139i 0.911829 + 0.410569i \(0.134670\pi\)
−0.911829 + 0.410569i \(0.865330\pi\)
\(500\) 0 0
\(501\) 14.2557i 0.636896i
\(502\) 0 0
\(503\) −14.2562 −0.635653 −0.317827 0.948149i \(-0.602953\pi\)
−0.317827 + 0.948149i \(0.602953\pi\)
\(504\) 0 0
\(505\) 6.63977i 0.295466i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 31.7981i 1.40943i 0.709491 + 0.704714i \(0.248925\pi\)
−0.709491 + 0.704714i \(0.751075\pi\)
\(510\) 0 0
\(511\) −22.6966 −1.00404
\(512\) 0 0
\(513\) 29.5647i 1.30531i
\(514\) 0 0
\(515\) 7.37605i 0.325028i
\(516\) 0 0
\(517\) −6.00000 −0.263880
\(518\) 0 0
\(519\) −22.0954 −0.969880
\(520\) 0 0
\(521\) −45.1676 −1.97883 −0.989414 0.145122i \(-0.953642\pi\)
−0.989414 + 0.145122i \(0.953642\pi\)
\(522\) 0 0
\(523\) 26.1146 1.14191 0.570955 0.820981i \(-0.306573\pi\)
0.570955 + 0.820981i \(0.306573\pi\)
\(524\) 0 0
\(525\) 6.93244i 0.302557i
\(526\) 0 0
\(527\) − 59.0155i − 2.57076i
\(528\) 0 0
\(529\) −21.6525 −0.941413
\(530\) 0 0
\(531\) 2.78874i 0.121021i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 7.91336i − 0.342124i
\(536\) 0 0
\(537\) 38.9651 1.68147
\(538\) 0 0
\(539\) − 20.3834i − 0.877975i
\(540\) 0 0
\(541\) − 31.4750i − 1.35321i −0.736344 0.676607i \(-0.763450\pi\)
0.736344 0.676607i \(-0.236550\pi\)
\(542\) 0 0
\(543\) 39.7387 1.70535
\(544\) 0 0
\(545\) 9.18301 0.393357
\(546\) 0 0
\(547\) −5.70391 −0.243881 −0.121941 0.992537i \(-0.538912\pi\)
−0.121941 + 0.992537i \(0.538912\pi\)
\(548\) 0 0
\(549\) −1.62704 −0.0694403
\(550\) 0 0
\(551\) 10.8765i 0.463353i
\(552\) 0 0
\(553\) 35.3823i 1.50461i
\(554\) 0 0
\(555\) 15.2541 0.647500
\(556\) 0 0
\(557\) 20.8720i 0.884374i 0.896923 + 0.442187i \(0.145797\pi\)
−0.896923 + 0.442187i \(0.854203\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) − 20.7973i − 0.878064i
\(562\) 0 0
\(563\) 33.2315 1.40054 0.700270 0.713878i \(-0.253063\pi\)
0.700270 + 0.713878i \(0.253063\pi\)
\(564\) 0 0
\(565\) − 5.77586i − 0.242992i
\(566\) 0 0
\(567\) 32.4435i 1.36250i
\(568\) 0 0
\(569\) 40.0671 1.67970 0.839851 0.542817i \(-0.182642\pi\)
0.839851 + 0.542817i \(0.182642\pi\)
\(570\) 0 0
\(571\) −4.53590 −0.189821 −0.0949107 0.995486i \(-0.530257\pi\)
−0.0949107 + 0.995486i \(0.530257\pi\)
\(572\) 0 0
\(573\) 9.55117 0.399006
\(574\) 0 0
\(575\) −1.16082 −0.0484096
\(576\) 0 0
\(577\) 35.2706i 1.46834i 0.678968 + 0.734168i \(0.262427\pi\)
−0.678968 + 0.734168i \(0.737573\pi\)
\(578\) 0 0
\(579\) − 14.4861i − 0.602023i
\(580\) 0 0
\(581\) −1.97879 −0.0820942
\(582\) 0 0
\(583\) − 21.9054i − 0.907230i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 41.1663i − 1.69912i −0.527496 0.849558i \(-0.676869\pi\)
0.527496 0.849558i \(-0.323131\pi\)
\(588\) 0 0
\(589\) 42.2487 1.74083
\(590\) 0 0
\(591\) − 18.2247i − 0.749664i
\(592\) 0 0
\(593\) − 9.39726i − 0.385899i −0.981209 0.192950i \(-0.938195\pi\)
0.981209 0.192950i \(-0.0618054\pi\)
\(594\) 0 0
\(595\) −32.5078 −1.33269
\(596\) 0 0
\(597\) −17.2457 −0.705821
\(598\) 0 0
\(599\) 21.7881 0.890239 0.445119 0.895471i \(-0.353161\pi\)
0.445119 + 0.895471i \(0.353161\pi\)
\(600\) 0 0
\(601\) 25.0962 1.02370 0.511848 0.859076i \(-0.328961\pi\)
0.511848 + 0.859076i \(0.328961\pi\)
\(602\) 0 0
\(603\) 2.31130i 0.0941236i
\(604\) 0 0
\(605\) 8.00000i 0.325246i
\(606\) 0 0
\(607\) −33.9775 −1.37910 −0.689552 0.724236i \(-0.742193\pi\)
−0.689552 + 0.724236i \(0.742193\pi\)
\(608\) 0 0
\(609\) 14.0363i 0.568780i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 21.4470i 0.866235i 0.901337 + 0.433118i \(0.142587\pi\)
−0.901337 + 0.433118i \(0.857413\pi\)
\(614\) 0 0
\(615\) −12.3728 −0.498919
\(616\) 0 0
\(617\) − 34.0743i − 1.37178i −0.727705 0.685890i \(-0.759413\pi\)
0.727705 0.685890i \(-0.240587\pi\)
\(618\) 0 0
\(619\) 26.9791i 1.08438i 0.840256 + 0.542191i \(0.182405\pi\)
−0.840256 + 0.542191i \(0.817595\pi\)
\(620\) 0 0
\(621\) −6.38878 −0.256373
\(622\) 0 0
\(623\) −57.5118 −2.30416
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 14.8886 0.594595
\(628\) 0 0
\(629\) 71.5298i 2.85208i
\(630\) 0 0
\(631\) 37.9738i 1.51171i 0.654737 + 0.755857i \(0.272779\pi\)
−0.654737 + 0.755857i \(0.727221\pi\)
\(632\) 0 0
\(633\) −5.16082 −0.205124
\(634\) 0 0
\(635\) − 18.9678i − 0.752713i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 5.49844i − 0.217515i
\(640\) 0 0
\(641\) 8.25180 0.325927 0.162963 0.986632i \(-0.447895\pi\)
0.162963 + 0.986632i \(0.447895\pi\)
\(642\) 0 0
\(643\) − 48.1501i − 1.89885i −0.313989 0.949427i \(-0.601666\pi\)
0.313989 0.949427i \(-0.398334\pi\)
\(644\) 0 0
\(645\) − 6.69770i − 0.263722i
\(646\) 0 0
\(647\) 2.17903 0.0856665 0.0428332 0.999082i \(-0.486362\pi\)
0.0428332 + 0.999082i \(0.486362\pi\)
\(648\) 0 0
\(649\) 10.9935 0.431532
\(650\) 0 0
\(651\) 54.5229 2.13692
\(652\) 0 0
\(653\) 33.8682 1.32537 0.662684 0.748900i \(-0.269417\pi\)
0.662684 + 0.748900i \(0.269417\pi\)
\(654\) 0 0
\(655\) − 1.34637i − 0.0526071i
\(656\) 0 0
\(657\) 2.30187i 0.0898046i
\(658\) 0 0
\(659\) 26.4491 1.03031 0.515155 0.857097i \(-0.327734\pi\)
0.515155 + 0.857097i \(0.327734\pi\)
\(660\) 0 0
\(661\) − 30.3985i − 1.18236i −0.806538 0.591182i \(-0.798661\pi\)
0.806538 0.591182i \(-0.201339\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 23.2720i − 0.902451i
\(666\) 0 0
\(667\) −2.35035 −0.0910059
\(668\) 0 0
\(669\) − 27.1703i − 1.05046i
\(670\) 0 0
\(671\) 6.41393i 0.247607i
\(672\) 0 0
\(673\) 36.3952 1.40293 0.701467 0.712702i \(-0.252529\pi\)
0.701467 + 0.712702i \(0.252529\pi\)
\(674\) 0 0
\(675\) 5.50367 0.211836
\(676\) 0 0
\(677\) 36.0043 1.38376 0.691880 0.722013i \(-0.256783\pi\)
0.691880 + 0.722013i \(0.256783\pi\)
\(678\) 0 0
\(679\) 12.1745 0.467215
\(680\) 0 0
\(681\) − 3.84130i − 0.147199i
\(682\) 0 0
\(683\) 12.8296i 0.490909i 0.969408 + 0.245455i \(0.0789372\pi\)
−0.969408 + 0.245455i \(0.921063\pi\)
\(684\) 0 0
\(685\) −12.3728 −0.472740
\(686\) 0 0
\(687\) − 11.7722i − 0.449137i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 32.9792i 1.25459i 0.778783 + 0.627294i \(0.215838\pi\)
−0.778783 + 0.627294i \(0.784162\pi\)
\(692\) 0 0
\(693\) −3.29692 −0.125239
\(694\) 0 0
\(695\) 8.76836i 0.332603i
\(696\) 0 0
\(697\) − 58.0188i − 2.19762i
\(698\) 0 0
\(699\) −5.39922 −0.204217
\(700\) 0 0
\(701\) 40.7352 1.53855 0.769273 0.638921i \(-0.220619\pi\)
0.769273 + 0.638921i \(0.220619\pi\)
\(702\) 0 0
\(703\) −51.2076 −1.93133
\(704\) 0 0
\(705\) −5.54324 −0.208771
\(706\) 0 0
\(707\) − 28.7651i − 1.08182i
\(708\) 0 0
\(709\) 17.6591i 0.663202i 0.943420 + 0.331601i \(0.107589\pi\)
−0.943420 + 0.331601i \(0.892411\pi\)
\(710\) 0 0
\(711\) 3.58845 0.134577
\(712\) 0 0
\(713\) 9.12973i 0.341911i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 38.0537i − 1.42114i
\(718\) 0 0
\(719\) −49.5672 −1.84855 −0.924273 0.381733i \(-0.875327\pi\)
−0.924273 + 0.381733i \(0.875327\pi\)
\(720\) 0 0
\(721\) − 31.9549i − 1.19006i
\(722\) 0 0
\(723\) − 35.7183i − 1.32838i
\(724\) 0 0
\(725\) 2.02473 0.0751965
\(726\) 0 0
\(727\) −36.4738 −1.35274 −0.676370 0.736562i \(-0.736448\pi\)
−0.676370 + 0.736562i \(0.736448\pi\)
\(728\) 0 0
\(729\) 29.7112 1.10042
\(730\) 0 0
\(731\) 31.4070 1.16163
\(732\) 0 0
\(733\) 28.8491i 1.06556i 0.846252 + 0.532782i \(0.178854\pi\)
−0.846252 + 0.532782i \(0.821146\pi\)
\(734\) 0 0
\(735\) − 18.8317i − 0.694617i
\(736\) 0 0
\(737\) 9.11137 0.335621
\(738\) 0 0
\(739\) 19.7835i 0.727746i 0.931448 + 0.363873i \(0.118546\pi\)
−0.931448 + 0.363873i \(0.881454\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 26.4005i − 0.968541i −0.874918 0.484271i \(-0.839085\pi\)
0.874918 0.484271i \(-0.160915\pi\)
\(744\) 0 0
\(745\) 4.73939 0.173638
\(746\) 0 0
\(747\) 0.200688i 0.00734279i
\(748\) 0 0
\(749\) 34.2826i 1.25266i
\(750\) 0 0
\(751\) 13.8034 0.503694 0.251847 0.967767i \(-0.418962\pi\)
0.251847 + 0.967767i \(0.418962\pi\)
\(752\) 0 0
\(753\) 7.39671 0.269551
\(754\) 0 0
\(755\) 1.56063 0.0567970
\(756\) 0 0
\(757\) −5.18160 −0.188328 −0.0941642 0.995557i \(-0.530018\pi\)
−0.0941642 + 0.995557i \(0.530018\pi\)
\(758\) 0 0
\(759\) 3.21736i 0.116783i
\(760\) 0 0
\(761\) − 0.00858275i 0 0.000311124i −1.00000 0.000155562i \(-0.999950\pi\)
1.00000 0.000155562i \(-4.95170e-5\pi\)
\(762\) 0 0
\(763\) −39.7830 −1.44024
\(764\) 0 0
\(765\) 3.29692i 0.119200i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 11.8483i 0.427262i 0.976914 + 0.213631i \(0.0685291\pi\)
−0.976914 + 0.213631i \(0.931471\pi\)
\(770\) 0 0
\(771\) −34.9865 −1.26001
\(772\) 0 0
\(773\) 21.0458i 0.756964i 0.925609 + 0.378482i \(0.123554\pi\)
−0.925609 + 0.378482i \(0.876446\pi\)
\(774\) 0 0
\(775\) − 7.86488i − 0.282515i
\(776\) 0 0
\(777\) −66.0845 −2.37077
\(778\) 0 0
\(779\) 41.5352 1.48815
\(780\) 0 0
\(781\) −21.6754 −0.775605
\(782\) 0 0
\(783\) 11.1434 0.398234
\(784\) 0 0
\(785\) − 9.15332i − 0.326696i
\(786\) 0 0
\(787\) 5.01174i 0.178649i 0.996003 + 0.0893246i \(0.0284708\pi\)
−0.996003 + 0.0893246i \(0.971529\pi\)
\(788\) 0 0
\(789\) −32.2904 −1.14957
\(790\) 0 0
\(791\) 25.0224i 0.889696i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) − 20.2378i − 0.717762i
\(796\) 0 0
\(797\) −40.6566 −1.44013 −0.720065 0.693907i \(-0.755888\pi\)
−0.720065 + 0.693907i \(0.755888\pi\)
\(798\) 0 0
\(799\) − 25.9935i − 0.919583i
\(800\) 0 0
\(801\) 5.83281i 0.206092i
\(802\) 0 0
\(803\) 9.07418 0.320221
\(804\) 0 0
\(805\) 5.02897 0.177248
\(806\) 0 0
\(807\) 11.6604 0.410466
\(808\) 0 0
\(809\) 5.41351 0.190329 0.0951644 0.995462i \(-0.469662\pi\)
0.0951644 + 0.995462i \(0.469662\pi\)
\(810\) 0 0
\(811\) − 0.616994i − 0.0216656i −0.999941 0.0108328i \(-0.996552\pi\)
0.999941 0.0108328i \(-0.00344825\pi\)
\(812\) 0 0
\(813\) − 7.82799i − 0.274540i
\(814\) 0 0
\(815\) 2.64303 0.0925811
\(816\) 0 0
\(817\) 22.4840i 0.786616i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 22.9066i − 0.799445i −0.916636 0.399723i \(-0.869106\pi\)
0.916636 0.399723i \(-0.130894\pi\)
\(822\) 0 0
\(823\) −9.65856 −0.336676 −0.168338 0.985729i \(-0.553840\pi\)
−0.168338 + 0.985729i \(0.553840\pi\)
\(824\) 0 0
\(825\) − 2.77162i − 0.0964954i
\(826\) 0 0
\(827\) 21.2602i 0.739289i 0.929173 + 0.369645i \(0.120521\pi\)
−0.929173 + 0.369645i \(0.879479\pi\)
\(828\) 0 0
\(829\) 23.9895 0.833191 0.416595 0.909092i \(-0.363223\pi\)
0.416595 + 0.909092i \(0.363223\pi\)
\(830\) 0 0
\(831\) −14.0827 −0.488522
\(832\) 0 0
\(833\) 88.3059 3.05962
\(834\) 0 0
\(835\) −8.90869 −0.308298
\(836\) 0 0
\(837\) − 43.2857i − 1.49617i
\(838\) 0 0
\(839\) 21.3055i 0.735548i 0.929915 + 0.367774i \(0.119880\pi\)
−0.929915 + 0.367774i \(0.880120\pi\)
\(840\) 0 0
\(841\) −24.9005 −0.858637
\(842\) 0 0
\(843\) 20.1761i 0.694902i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 34.6580i − 1.19086i
\(848\) 0 0
\(849\) 33.4456 1.14785
\(850\) 0 0
\(851\) − 11.0657i − 0.379327i
\(852\) 0 0
\(853\) 12.1098i 0.414631i 0.978274 + 0.207315i \(0.0664726\pi\)
−0.978274 + 0.207315i \(0.933527\pi\)
\(854\) 0 0
\(855\) −2.36023 −0.0807183
\(856\) 0 0
\(857\) 35.4491 1.21092 0.605459 0.795876i \(-0.292990\pi\)
0.605459 + 0.795876i \(0.292990\pi\)
\(858\) 0 0
\(859\) −47.4600 −1.61931 −0.809657 0.586904i \(-0.800347\pi\)
−0.809657 + 0.586904i \(0.800347\pi\)
\(860\) 0 0
\(861\) 53.6020 1.82675
\(862\) 0 0
\(863\) − 18.7268i − 0.637467i −0.947844 0.318733i \(-0.896743\pi\)
0.947844 0.318733i \(-0.103257\pi\)
\(864\) 0 0
\(865\) − 13.8079i − 0.469484i
\(866\) 0 0
\(867\) 62.8958 2.13605
\(868\) 0 0
\(869\) − 14.1460i − 0.479870i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) − 1.23473i − 0.0417894i
\(874\) 0 0
\(875\) −4.33225 −0.146457
\(876\) 0 0
\(877\) − 1.81517i − 0.0612938i −0.999530 0.0306469i \(-0.990243\pi\)
0.999530 0.0306469i \(-0.00975674\pi\)
\(878\) 0 0
\(879\) 1.10811i 0.0373755i
\(880\) 0 0
\(881\) 50.3437 1.69612 0.848061 0.529899i \(-0.177770\pi\)
0.848061 + 0.529899i \(0.177770\pi\)
\(882\) 0 0
\(883\) 4.42003 0.148746 0.0743729 0.997230i \(-0.476304\pi\)
0.0743729 + 0.997230i \(0.476304\pi\)
\(884\) 0 0
\(885\) 10.1566 0.341410
\(886\) 0 0
\(887\) 2.69134 0.0903662 0.0451831 0.998979i \(-0.485613\pi\)
0.0451831 + 0.998979i \(0.485613\pi\)
\(888\) 0 0
\(889\) 82.1731i 2.75600i
\(890\) 0 0
\(891\) − 12.9710i − 0.434546i
\(892\) 0 0
\(893\) 18.6085 0.622710
\(894\) 0 0
\(895\) 24.3502i 0.813937i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 15.9243i − 0.531103i
\(900\) 0 0
\(901\) 94.8997 3.16157
\(902\) 0 0
\(903\) 29.0161i 0.965595i
\(904\) 0 0
\(905\) 24.8336i 0.825498i
\(906\) 0 0
\(907\) −14.4017 −0.478199 −0.239100 0.970995i \(-0.576852\pi\)
−0.239100 + 0.970995i \(0.576852\pi\)
\(908\) 0 0
\(909\) −2.91734 −0.0967620
\(910\) 0 0
\(911\) −8.88723 −0.294447 −0.147223 0.989103i \(-0.547034\pi\)
−0.147223 + 0.989103i \(0.547034\pi\)
\(912\) 0 0
\(913\) 0.791131 0.0261826
\(914\) 0 0
\(915\) 5.92566i 0.195896i
\(916\) 0 0
\(917\) 5.83281i 0.192617i
\(918\) 0 0
\(919\) −42.9747 −1.41760 −0.708802 0.705408i \(-0.750764\pi\)
−0.708802 + 0.705408i \(0.750764\pi\)
\(920\) 0 0
\(921\) − 14.1324i − 0.465679i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 9.53264i 0.313431i
\(926\) 0 0
\(927\) −3.24084 −0.106443
\(928\) 0 0
\(929\) 29.7378i 0.975666i 0.872937 + 0.487833i \(0.162213\pi\)
−0.872937 + 0.487833i \(0.837787\pi\)
\(930\) 0 0
\(931\) 63.2175i 2.07187i
\(932\) 0 0
\(933\) −18.2356 −0.597005
\(934\) 0 0
\(935\) 12.9967 0.425039
\(936\) 0 0
\(937\) 17.1355 0.559793 0.279896 0.960030i \(-0.409700\pi\)
0.279896 + 0.960030i \(0.409700\pi\)
\(938\) 0 0
\(939\) −5.41660 −0.176764
\(940\) 0 0
\(941\) − 23.5767i − 0.768580i −0.923212 0.384290i \(-0.874446\pi\)
0.923212 0.384290i \(-0.125554\pi\)
\(942\) 0 0
\(943\) 8.97553i 0.292284i
\(944\) 0 0
\(945\) −23.8433 −0.775621
\(946\) 0 0
\(947\) 17.6272i 0.572807i 0.958109 + 0.286404i \(0.0924598\pi\)
−0.958109 + 0.286404i \(0.907540\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0.891475i 0.0289080i
\(952\) 0 0
\(953\) 58.7631 1.90352 0.951762 0.306837i \(-0.0992707\pi\)
0.951762 + 0.306837i \(0.0992707\pi\)
\(954\) 0 0
\(955\) 5.96875i 0.193144i
\(956\) 0 0
\(957\) − 5.61178i − 0.181403i
\(958\) 0 0
\(959\) 53.6020 1.73090
\(960\) 0 0
\(961\) −30.8564 −0.995368
\(962\) 0 0
\(963\) 3.47692 0.112042
\(964\) 0 0
\(965\) 9.05272 0.291417
\(966\) 0 0
\(967\) − 2.92168i − 0.0939550i −0.998896 0.0469775i \(-0.985041\pi\)
0.998896 0.0469775i \(-0.0149589\pi\)
\(968\) 0 0
\(969\) 64.5012i 2.07208i
\(970\) 0 0
\(971\) −10.0139 −0.321360 −0.160680 0.987007i \(-0.551369\pi\)
−0.160680 + 0.987007i \(0.551369\pi\)
\(972\) 0 0
\(973\) − 37.9867i − 1.21780i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 26.5301i − 0.848772i −0.905481 0.424386i \(-0.860490\pi\)
0.905481 0.424386i \(-0.139510\pi\)
\(978\) 0 0
\(979\) 22.9935 0.734875
\(980\) 0 0
\(981\) 4.03477i 0.128820i
\(982\) 0 0
\(983\) − 4.69317i − 0.149689i −0.997195 0.0748445i \(-0.976154\pi\)
0.997195 0.0748445i \(-0.0238460\pi\)
\(984\) 0 0
\(985\) 11.3890 0.362885
\(986\) 0 0
\(987\) 24.0147 0.764396
\(988\) 0 0
\(989\) −4.85868 −0.154497
\(990\) 0 0
\(991\) −22.9201 −0.728081 −0.364041 0.931383i \(-0.618603\pi\)
−0.364041 + 0.931383i \(0.618603\pi\)
\(992\) 0 0
\(993\) − 6.34268i − 0.201279i
\(994\) 0 0
\(995\) − 10.7773i − 0.341662i
\(996\) 0 0
\(997\) 49.4426 1.56586 0.782932 0.622108i \(-0.213723\pi\)
0.782932 + 0.622108i \(0.213723\pi\)
\(998\) 0 0
\(999\) 52.4645i 1.65990i
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 3380.2.f.i.3041.6 8
13.5 odd 4 3380.2.a.q.1.3 4
13.8 odd 4 3380.2.a.p.1.3 4
13.9 even 3 260.2.x.a.101.2 8
13.10 even 6 260.2.x.a.121.2 yes 8
13.12 even 2 inner 3380.2.f.i.3041.5 8
39.23 odd 6 2340.2.dj.d.901.4 8
39.35 odd 6 2340.2.dj.d.361.2 8
52.23 odd 6 1040.2.da.c.641.3 8
52.35 odd 6 1040.2.da.c.881.3 8
65.9 even 6 1300.2.y.b.101.3 8
65.22 odd 12 1300.2.ba.c.49.1 8
65.23 odd 12 1300.2.ba.c.849.1 8
65.48 odd 12 1300.2.ba.b.49.4 8
65.49 even 6 1300.2.y.b.901.3 8
65.62 odd 12 1300.2.ba.b.849.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.x.a.101.2 8 13.9 even 3
260.2.x.a.121.2 yes 8 13.10 even 6
1040.2.da.c.641.3 8 52.23 odd 6
1040.2.da.c.881.3 8 52.35 odd 6
1300.2.y.b.101.3 8 65.9 even 6
1300.2.y.b.901.3 8 65.49 even 6
1300.2.ba.b.49.4 8 65.48 odd 12
1300.2.ba.b.849.4 8 65.62 odd 12
1300.2.ba.c.49.1 8 65.22 odd 12
1300.2.ba.c.849.1 8 65.23 odd 12
2340.2.dj.d.361.2 8 39.35 odd 6
2340.2.dj.d.901.4 8 39.23 odd 6
3380.2.a.p.1.3 4 13.8 odd 4
3380.2.a.q.1.3 4 13.5 odd 4
3380.2.f.i.3041.5 8 13.12 even 2 inner
3380.2.f.i.3041.6 8 1.1 even 1 trivial