Properties

Label 3300.2.c.l.1849.2
Level $3300$
Weight $2$
Character 3300.1849
Analytic conductor $26.351$
Analytic rank $0$
Dimension $4$
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] = [3300,2,Mod(1849,3300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3300, 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("3300.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3300.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.3506326670\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 660)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.2
Root \(1.30278i\) of defining polynomial
Character \(\chi\) \(=\) 3300.1849
Dual form 3300.2.c.l.1849.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +2.60555i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +2.60555i q^{7} -1.00000 q^{9} +1.00000 q^{11} +2.60555i q^{13} +0.605551i q^{17} -7.21110 q^{19} +2.60555 q^{21} +1.00000i q^{27} -8.00000 q^{29} -5.21110 q^{31} -1.00000i q^{33} -11.2111i q^{37} +2.60555 q^{39} +8.00000 q^{41} -10.6056i q^{43} -9.21110i q^{47} +0.211103 q^{49} +0.605551 q^{51} +2.00000i q^{53} +7.21110i q^{57} -8.00000 q^{59} -7.21110 q^{61} -2.60555i q^{63} +4.00000i q^{67} +14.4222 q^{71} -6.60555i q^{73} +2.60555i q^{77} -3.21110 q^{79} +1.00000 q^{81} -3.39445i q^{83} +8.00000i q^{87} -6.00000 q^{89} -6.78890 q^{91} +5.21110i q^{93} -16.4222i q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} + 4 q^{11} - 4 q^{21} - 32 q^{29} + 8 q^{31} - 4 q^{39} + 32 q^{41} - 28 q^{49} - 12 q^{51} - 32 q^{59} + 16 q^{79} + 4 q^{81} - 24 q^{89} - 56 q^{91} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3300\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1651\) \(2201\) \(2377\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.60555i 0.984806i 0.870367 + 0.492403i \(0.163881\pi\)
−0.870367 + 0.492403i \(0.836119\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.60555i 0.722650i 0.932440 + 0.361325i \(0.117675\pi\)
−0.932440 + 0.361325i \(0.882325\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.605551i 0.146868i 0.997300 + 0.0734339i \(0.0233958\pi\)
−0.997300 + 0.0734339i \(0.976604\pi\)
\(18\) 0 0
\(19\) −7.21110 −1.65434 −0.827170 0.561951i \(-0.810051\pi\)
−0.827170 + 0.561951i \(0.810051\pi\)
\(20\) 0 0
\(21\) 2.60555 0.568578
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −5.21110 −0.935942 −0.467971 0.883744i \(-0.655015\pi\)
−0.467971 + 0.883744i \(0.655015\pi\)
\(32\) 0 0
\(33\) − 1.00000i − 0.174078i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 11.2111i − 1.84309i −0.388267 0.921547i \(-0.626926\pi\)
0.388267 0.921547i \(-0.373074\pi\)
\(38\) 0 0
\(39\) 2.60555 0.417222
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) − 10.6056i − 1.61733i −0.588268 0.808666i \(-0.700190\pi\)
0.588268 0.808666i \(-0.299810\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 9.21110i − 1.34358i −0.740743 0.671789i \(-0.765526\pi\)
0.740743 0.671789i \(-0.234474\pi\)
\(48\) 0 0
\(49\) 0.211103 0.0301575
\(50\) 0 0
\(51\) 0.605551 0.0847941
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 7.21110i 0.955134i
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −7.21110 −0.923287 −0.461644 0.887066i \(-0.652740\pi\)
−0.461644 + 0.887066i \(0.652740\pi\)
\(62\) 0 0
\(63\) − 2.60555i − 0.328269i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.4222 1.71160 0.855800 0.517306i \(-0.173065\pi\)
0.855800 + 0.517306i \(0.173065\pi\)
\(72\) 0 0
\(73\) − 6.60555i − 0.773121i −0.922264 0.386561i \(-0.873663\pi\)
0.922264 0.386561i \(-0.126337\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.60555i 0.296930i
\(78\) 0 0
\(79\) −3.21110 −0.361277 −0.180639 0.983550i \(-0.557816\pi\)
−0.180639 + 0.983550i \(0.557816\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 3.39445i − 0.372589i −0.982494 0.186295i \(-0.940352\pi\)
0.982494 0.186295i \(-0.0596479\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.00000i 0.857690i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −6.78890 −0.711670
\(92\) 0 0
\(93\) 5.21110i 0.540366i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 16.4222i − 1.66742i −0.552201 0.833711i \(-0.686212\pi\)
0.552201 0.833711i \(-0.313788\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −9.21110 −0.916539 −0.458269 0.888813i \(-0.651531\pi\)
−0.458269 + 0.888813i \(0.651531\pi\)
\(102\) 0 0
\(103\) − 13.2111i − 1.30173i −0.759194 0.650864i \(-0.774407\pi\)
0.759194 0.650864i \(-0.225593\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.81665i 0.562317i 0.959661 + 0.281159i \(0.0907187\pi\)
−0.959661 + 0.281159i \(0.909281\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −11.2111 −1.06411
\(112\) 0 0
\(113\) − 15.2111i − 1.43094i −0.698643 0.715470i \(-0.746213\pi\)
0.698643 0.715470i \(-0.253787\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 2.60555i − 0.240883i
\(118\) 0 0
\(119\) −1.57779 −0.144636
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) − 8.00000i − 0.721336i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.39445i 0.123737i 0.998084 + 0.0618687i \(0.0197060\pi\)
−0.998084 + 0.0618687i \(0.980294\pi\)
\(128\) 0 0
\(129\) −10.6056 −0.933767
\(130\) 0 0
\(131\) −21.2111 −1.85322 −0.926611 0.376021i \(-0.877292\pi\)
−0.926611 + 0.376021i \(0.877292\pi\)
\(132\) 0 0
\(133\) − 18.7889i − 1.62920i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) 12.4222 1.05364 0.526819 0.849978i \(-0.323385\pi\)
0.526819 + 0.849978i \(0.323385\pi\)
\(140\) 0 0
\(141\) −9.21110 −0.775715
\(142\) 0 0
\(143\) 2.60555i 0.217887i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 0.211103i − 0.0174114i
\(148\) 0 0
\(149\) −5.21110 −0.426910 −0.213455 0.976953i \(-0.568472\pi\)
−0.213455 + 0.976953i \(0.568472\pi\)
\(150\) 0 0
\(151\) −4.78890 −0.389715 −0.194857 0.980832i \(-0.562424\pi\)
−0.194857 + 0.980832i \(0.562424\pi\)
\(152\) 0 0
\(153\) − 0.605551i − 0.0489559i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 11.2111i 0.894743i 0.894348 + 0.447372i \(0.147640\pi\)
−0.894348 + 0.447372i \(0.852360\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 9.21110i 0.721469i 0.932669 + 0.360735i \(0.117474\pi\)
−0.932669 + 0.360735i \(0.882526\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 1.81665i − 0.140577i −0.997527 0.0702884i \(-0.977608\pi\)
0.997527 0.0702884i \(-0.0223920\pi\)
\(168\) 0 0
\(169\) 6.21110 0.477777
\(170\) 0 0
\(171\) 7.21110 0.551447
\(172\) 0 0
\(173\) 21.8167i 1.65869i 0.558737 + 0.829345i \(0.311286\pi\)
−0.558737 + 0.829345i \(0.688714\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8.00000i 0.601317i
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 8.78890 0.653274 0.326637 0.945150i \(-0.394085\pi\)
0.326637 + 0.945150i \(0.394085\pi\)
\(182\) 0 0
\(183\) 7.21110i 0.533060i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.605551i 0.0442823i
\(188\) 0 0
\(189\) −2.60555 −0.189526
\(190\) 0 0
\(191\) −10.4222 −0.754124 −0.377062 0.926188i \(-0.623066\pi\)
−0.377062 + 0.926188i \(0.623066\pi\)
\(192\) 0 0
\(193\) − 0.183346i − 0.0131975i −0.999978 0.00659877i \(-0.997900\pi\)
0.999978 0.00659877i \(-0.00210047\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.6056i 1.46808i 0.679104 + 0.734042i \(0.262369\pi\)
−0.679104 + 0.734042i \(0.737631\pi\)
\(198\) 0 0
\(199\) −18.4222 −1.30592 −0.652958 0.757394i \(-0.726472\pi\)
−0.652958 + 0.757394i \(0.726472\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) − 20.8444i − 1.46299i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −7.21110 −0.498802
\(210\) 0 0
\(211\) −8.78890 −0.605053 −0.302526 0.953141i \(-0.597830\pi\)
−0.302526 + 0.953141i \(0.597830\pi\)
\(212\) 0 0
\(213\) − 14.4222i − 0.988193i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 13.5778i − 0.921721i
\(218\) 0 0
\(219\) −6.60555 −0.446362
\(220\) 0 0
\(221\) −1.57779 −0.106134
\(222\) 0 0
\(223\) − 5.21110i − 0.348961i −0.984661 0.174481i \(-0.944175\pi\)
0.984661 0.174481i \(-0.0558246\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 5.81665i − 0.386065i −0.981192 0.193032i \(-0.938168\pi\)
0.981192 0.193032i \(-0.0618323\pi\)
\(228\) 0 0
\(229\) −20.4222 −1.34954 −0.674769 0.738029i \(-0.735757\pi\)
−0.674769 + 0.738029i \(0.735757\pi\)
\(230\) 0 0
\(231\) 2.60555 0.171433
\(232\) 0 0
\(233\) − 3.39445i − 0.222378i −0.993799 0.111189i \(-0.964534\pi\)
0.993799 0.111189i \(-0.0354658\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 3.21110i 0.208584i
\(238\) 0 0
\(239\) −21.2111 −1.37203 −0.686016 0.727586i \(-0.740642\pi\)
−0.686016 + 0.727586i \(0.740642\pi\)
\(240\) 0 0
\(241\) −0.788897 −0.0508174 −0.0254087 0.999677i \(-0.508089\pi\)
−0.0254087 + 0.999677i \(0.508089\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 18.7889i − 1.19551i
\(248\) 0 0
\(249\) −3.39445 −0.215114
\(250\) 0 0
\(251\) 18.4222 1.16280 0.581400 0.813618i \(-0.302505\pi\)
0.581400 + 0.813618i \(0.302505\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.78890i 0.548236i 0.961696 + 0.274118i \(0.0883859\pi\)
−0.961696 + 0.274118i \(0.911614\pi\)
\(258\) 0 0
\(259\) 29.2111 1.81509
\(260\) 0 0
\(261\) 8.00000 0.495188
\(262\) 0 0
\(263\) 19.0278i 1.17330i 0.809840 + 0.586651i \(0.199554\pi\)
−0.809840 + 0.586651i \(0.800446\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 0 0
\(269\) −8.42221 −0.513511 −0.256755 0.966476i \(-0.582653\pi\)
−0.256755 + 0.966476i \(0.582653\pi\)
\(270\) 0 0
\(271\) −19.2111 −1.16699 −0.583496 0.812116i \(-0.698315\pi\)
−0.583496 + 0.812116i \(0.698315\pi\)
\(272\) 0 0
\(273\) 6.78890i 0.410883i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 18.2389i 1.09587i 0.836522 + 0.547933i \(0.184585\pi\)
−0.836522 + 0.547933i \(0.815415\pi\)
\(278\) 0 0
\(279\) 5.21110 0.311981
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) − 2.60555i − 0.154884i −0.996997 0.0774420i \(-0.975325\pi\)
0.996997 0.0774420i \(-0.0246753\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 20.8444i 1.23041i
\(288\) 0 0
\(289\) 16.6333 0.978430
\(290\) 0 0
\(291\) −16.4222 −0.962687
\(292\) 0 0
\(293\) − 1.81665i − 0.106130i −0.998591 0.0530650i \(-0.983101\pi\)
0.998591 0.0530650i \(-0.0168991\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.00000i 0.0580259i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 27.6333 1.59276
\(302\) 0 0
\(303\) 9.21110i 0.529164i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 1.39445i − 0.0795854i −0.999208 0.0397927i \(-0.987330\pi\)
0.999208 0.0397927i \(-0.0126698\pi\)
\(308\) 0 0
\(309\) −13.2111 −0.751553
\(310\) 0 0
\(311\) 1.57779 0.0894685 0.0447343 0.998999i \(-0.485756\pi\)
0.0447343 + 0.998999i \(0.485756\pi\)
\(312\) 0 0
\(313\) 23.2111i 1.31197i 0.754774 + 0.655985i \(0.227746\pi\)
−0.754774 + 0.655985i \(0.772254\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.21110i 0.180353i 0.995926 + 0.0901767i \(0.0287432\pi\)
−0.995926 + 0.0901767i \(0.971257\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 5.81665 0.324654
\(322\) 0 0
\(323\) − 4.36669i − 0.242969i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 10.0000i − 0.553001i
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −21.2111 −1.16587 −0.582934 0.812520i \(-0.698095\pi\)
−0.582934 + 0.812520i \(0.698095\pi\)
\(332\) 0 0
\(333\) 11.2111i 0.614365i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 22.2389i − 1.21143i −0.795683 0.605714i \(-0.792888\pi\)
0.795683 0.605714i \(-0.207112\pi\)
\(338\) 0 0
\(339\) −15.2111 −0.826154
\(340\) 0 0
\(341\) −5.21110 −0.282197
\(342\) 0 0
\(343\) 18.7889i 1.01451i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 7.02776i − 0.377270i −0.982047 0.188635i \(-0.939594\pi\)
0.982047 0.188635i \(-0.0604063\pi\)
\(348\) 0 0
\(349\) 23.2111 1.24246 0.621231 0.783628i \(-0.286633\pi\)
0.621231 + 0.783628i \(0.286633\pi\)
\(350\) 0 0
\(351\) −2.60555 −0.139074
\(352\) 0 0
\(353\) − 26.0000i − 1.38384i −0.721974 0.691920i \(-0.756765\pi\)
0.721974 0.691920i \(-0.243235\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.57779i 0.0835058i
\(358\) 0 0
\(359\) −33.2111 −1.75281 −0.876407 0.481570i \(-0.840067\pi\)
−0.876407 + 0.481570i \(0.840067\pi\)
\(360\) 0 0
\(361\) 33.0000 1.73684
\(362\) 0 0
\(363\) − 1.00000i − 0.0524864i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 21.2111i − 1.10721i −0.832779 0.553605i \(-0.813252\pi\)
0.832779 0.553605i \(-0.186748\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) −5.21110 −0.270547
\(372\) 0 0
\(373\) − 2.60555i − 0.134910i −0.997722 0.0674552i \(-0.978512\pi\)
0.997722 0.0674552i \(-0.0214880\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 20.8444i − 1.07354i
\(378\) 0 0
\(379\) −10.4222 −0.535353 −0.267676 0.963509i \(-0.586256\pi\)
−0.267676 + 0.963509i \(0.586256\pi\)
\(380\) 0 0
\(381\) 1.39445 0.0714398
\(382\) 0 0
\(383\) 33.2111i 1.69701i 0.529189 + 0.848504i \(0.322496\pi\)
−0.529189 + 0.848504i \(0.677504\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.6056i 0.539110i
\(388\) 0 0
\(389\) 26.8444 1.36107 0.680533 0.732718i \(-0.261748\pi\)
0.680533 + 0.732718i \(0.261748\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 21.2111i 1.06996i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.422205i 0.0211899i 0.999944 + 0.0105949i \(0.00337254\pi\)
−0.999944 + 0.0105949i \(0.996627\pi\)
\(398\) 0 0
\(399\) −18.7889 −0.940621
\(400\) 0 0
\(401\) −8.42221 −0.420585 −0.210292 0.977639i \(-0.567442\pi\)
−0.210292 + 0.977639i \(0.567442\pi\)
\(402\) 0 0
\(403\) − 13.5778i − 0.676358i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 11.2111i − 0.555714i
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) − 20.8444i − 1.02569i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 12.4222i − 0.608318i
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −4.78890 −0.233397 −0.116698 0.993167i \(-0.537231\pi\)
−0.116698 + 0.993167i \(0.537231\pi\)
\(422\) 0 0
\(423\) 9.21110i 0.447859i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 18.7889i − 0.909258i
\(428\) 0 0
\(429\) 2.60555 0.125797
\(430\) 0 0
\(431\) −30.4222 −1.46539 −0.732693 0.680559i \(-0.761737\pi\)
−0.732693 + 0.680559i \(0.761737\pi\)
\(432\) 0 0
\(433\) − 20.4222i − 0.981429i −0.871321 0.490714i \(-0.836736\pi\)
0.871321 0.490714i \(-0.163264\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 0 0
\(441\) −0.211103 −0.0100525
\(442\) 0 0
\(443\) 10.7889i 0.512596i 0.966598 + 0.256298i \(0.0825028\pi\)
−0.966598 + 0.256298i \(0.917497\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 5.21110i 0.246477i
\(448\) 0 0
\(449\) 38.8444 1.83318 0.916591 0.399827i \(-0.130930\pi\)
0.916591 + 0.399827i \(0.130930\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) 4.78890i 0.225002i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 2.60555i − 0.121883i −0.998141 0.0609413i \(-0.980590\pi\)
0.998141 0.0609413i \(-0.0194102\pi\)
\(458\) 0 0
\(459\) −0.605551 −0.0282647
\(460\) 0 0
\(461\) 5.21110 0.242705 0.121353 0.992609i \(-0.461277\pi\)
0.121353 + 0.992609i \(0.461277\pi\)
\(462\) 0 0
\(463\) − 13.2111i − 0.613972i −0.951714 0.306986i \(-0.900680\pi\)
0.951714 0.306986i \(-0.0993205\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.00000i 0.185098i 0.995708 + 0.0925490i \(0.0295015\pi\)
−0.995708 + 0.0925490i \(0.970499\pi\)
\(468\) 0 0
\(469\) −10.4222 −0.481253
\(470\) 0 0
\(471\) 11.2111 0.516580
\(472\) 0 0
\(473\) − 10.6056i − 0.487644i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 2.00000i − 0.0915737i
\(478\) 0 0
\(479\) 29.2111 1.33469 0.667345 0.744749i \(-0.267431\pi\)
0.667345 + 0.744749i \(0.267431\pi\)
\(480\) 0 0
\(481\) 29.2111 1.33191
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 34.4222i − 1.55982i −0.625892 0.779910i \(-0.715265\pi\)
0.625892 0.779910i \(-0.284735\pi\)
\(488\) 0 0
\(489\) 9.21110 0.416540
\(490\) 0 0
\(491\) −34.4222 −1.55345 −0.776726 0.629838i \(-0.783121\pi\)
−0.776726 + 0.629838i \(0.783121\pi\)
\(492\) 0 0
\(493\) − 4.84441i − 0.218181i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 37.5778i 1.68559i
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −1.81665 −0.0811621
\(502\) 0 0
\(503\) 25.8167i 1.15111i 0.817764 + 0.575554i \(0.195213\pi\)
−0.817764 + 0.575554i \(0.804787\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 6.21110i − 0.275845i
\(508\) 0 0
\(509\) −0.422205 −0.0187139 −0.00935696 0.999956i \(-0.502978\pi\)
−0.00935696 + 0.999956i \(0.502978\pi\)
\(510\) 0 0
\(511\) 17.2111 0.761374
\(512\) 0 0
\(513\) − 7.21110i − 0.318378i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 9.21110i − 0.405104i
\(518\) 0 0
\(519\) 21.8167 0.957645
\(520\) 0 0
\(521\) 20.4222 0.894713 0.447357 0.894356i \(-0.352366\pi\)
0.447357 + 0.894356i \(0.352366\pi\)
\(522\) 0 0
\(523\) − 22.2389i − 0.972437i −0.873837 0.486219i \(-0.838376\pi\)
0.873837 0.486219i \(-0.161624\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 3.15559i − 0.137460i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 20.8444i 0.902872i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 12.0000i 0.517838i
\(538\) 0 0
\(539\) 0.211103 0.00909283
\(540\) 0 0
\(541\) 32.4222 1.39394 0.696970 0.717101i \(-0.254531\pi\)
0.696970 + 0.717101i \(0.254531\pi\)
\(542\) 0 0
\(543\) − 8.78890i − 0.377168i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 35.8167i 1.53141i 0.643192 + 0.765705i \(0.277610\pi\)
−0.643192 + 0.765705i \(0.722390\pi\)
\(548\) 0 0
\(549\) 7.21110 0.307762
\(550\) 0 0
\(551\) 57.6888 2.45763
\(552\) 0 0
\(553\) − 8.36669i − 0.355788i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 43.0278i 1.82314i 0.411140 + 0.911572i \(0.365131\pi\)
−0.411140 + 0.911572i \(0.634869\pi\)
\(558\) 0 0
\(559\) 27.6333 1.16876
\(560\) 0 0
\(561\) 0.605551 0.0255664
\(562\) 0 0
\(563\) 1.81665i 0.0765628i 0.999267 + 0.0382814i \(0.0121883\pi\)
−0.999267 + 0.0382814i \(0.987812\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.60555i 0.109423i
\(568\) 0 0
\(569\) −15.6333 −0.655382 −0.327691 0.944785i \(-0.606271\pi\)
−0.327691 + 0.944785i \(0.606271\pi\)
\(570\) 0 0
\(571\) −16.4222 −0.687248 −0.343624 0.939107i \(-0.611655\pi\)
−0.343624 + 0.939107i \(0.611655\pi\)
\(572\) 0 0
\(573\) 10.4222i 0.435394i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 43.2111i − 1.79890i −0.437022 0.899451i \(-0.643967\pi\)
0.437022 0.899451i \(-0.356033\pi\)
\(578\) 0 0
\(579\) −0.183346 −0.00761961
\(580\) 0 0
\(581\) 8.84441 0.366928
\(582\) 0 0
\(583\) 2.00000i 0.0828315i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.4222i 0.925463i 0.886498 + 0.462732i \(0.153131\pi\)
−0.886498 + 0.462732i \(0.846869\pi\)
\(588\) 0 0
\(589\) 37.5778 1.54837
\(590\) 0 0
\(591\) 20.6056 0.847599
\(592\) 0 0
\(593\) − 23.0278i − 0.945637i −0.881160 0.472818i \(-0.843237\pi\)
0.881160 0.472818i \(-0.156763\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 18.4222i 0.753971i
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −23.2111 −0.946801 −0.473400 0.880847i \(-0.656974\pi\)
−0.473400 + 0.880847i \(0.656974\pi\)
\(602\) 0 0
\(603\) − 4.00000i − 0.162893i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 41.0278i − 1.66527i −0.553826 0.832633i \(-0.686833\pi\)
0.553826 0.832633i \(-0.313167\pi\)
\(608\) 0 0
\(609\) −20.8444 −0.844658
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) − 0.183346i − 0.00740528i −0.999993 0.00370264i \(-0.998821\pi\)
0.999993 0.00370264i \(-0.00117859\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 39.2111i 1.57858i 0.614021 + 0.789290i \(0.289551\pi\)
−0.614021 + 0.789290i \(0.710449\pi\)
\(618\) 0 0
\(619\) −23.6333 −0.949903 −0.474951 0.880012i \(-0.657534\pi\)
−0.474951 + 0.880012i \(0.657534\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 15.6333i − 0.626335i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 7.21110i 0.287984i
\(628\) 0 0
\(629\) 6.78890 0.270691
\(630\) 0 0
\(631\) −20.8444 −0.829803 −0.414901 0.909866i \(-0.636184\pi\)
−0.414901 + 0.909866i \(0.636184\pi\)
\(632\) 0 0
\(633\) 8.78890i 0.349327i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.550039i 0.0217933i
\(638\) 0 0
\(639\) −14.4222 −0.570534
\(640\) 0 0
\(641\) −38.8444 −1.53426 −0.767131 0.641490i \(-0.778316\pi\)
−0.767131 + 0.641490i \(0.778316\pi\)
\(642\) 0 0
\(643\) 43.6333i 1.72073i 0.509679 + 0.860365i \(0.329764\pi\)
−0.509679 + 0.860365i \(0.670236\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 30.7889i − 1.21044i −0.796060 0.605218i \(-0.793086\pi\)
0.796060 0.605218i \(-0.206914\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) −13.5778 −0.532156
\(652\) 0 0
\(653\) 19.2111i 0.751789i 0.926663 + 0.375894i \(0.122664\pi\)
−0.926663 + 0.375894i \(0.877336\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.60555i 0.257707i
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −41.2666 −1.60509 −0.802543 0.596595i \(-0.796520\pi\)
−0.802543 + 0.596595i \(0.796520\pi\)
\(662\) 0 0
\(663\) 1.57779i 0.0612765i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −5.21110 −0.201473
\(670\) 0 0
\(671\) −7.21110 −0.278382
\(672\) 0 0
\(673\) − 18.6056i − 0.717191i −0.933493 0.358596i \(-0.883256\pi\)
0.933493 0.358596i \(-0.116744\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 12.6056i − 0.484471i −0.970218 0.242235i \(-0.922119\pi\)
0.970218 0.242235i \(-0.0778806\pi\)
\(678\) 0 0
\(679\) 42.7889 1.64209
\(680\) 0 0
\(681\) −5.81665 −0.222895
\(682\) 0 0
\(683\) − 45.2111i − 1.72995i −0.501811 0.864977i \(-0.667333\pi\)
0.501811 0.864977i \(-0.332667\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 20.4222i 0.779156i
\(688\) 0 0
\(689\) −5.21110 −0.198527
\(690\) 0 0
\(691\) −42.4222 −1.61382 −0.806908 0.590677i \(-0.798861\pi\)
−0.806908 + 0.590677i \(0.798861\pi\)
\(692\) 0 0
\(693\) − 2.60555i − 0.0989767i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 4.84441i 0.183495i
\(698\) 0 0
\(699\) −3.39445 −0.128390
\(700\) 0 0
\(701\) 17.5778 0.663904 0.331952 0.943296i \(-0.392293\pi\)
0.331952 + 0.943296i \(0.392293\pi\)
\(702\) 0 0
\(703\) 80.8444i 3.04910i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 24.0000i − 0.902613i
\(708\) 0 0
\(709\) 37.6333 1.41335 0.706674 0.707539i \(-0.250195\pi\)
0.706674 + 0.707539i \(0.250195\pi\)
\(710\) 0 0
\(711\) 3.21110 0.120426
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 21.2111i 0.792143i
\(718\) 0 0
\(719\) −2.42221 −0.0903330 −0.0451665 0.998979i \(-0.514382\pi\)
−0.0451665 + 0.998979i \(0.514382\pi\)
\(720\) 0 0
\(721\) 34.4222 1.28195
\(722\) 0 0
\(723\) 0.788897i 0.0293394i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 28.8444i − 1.06978i −0.844922 0.534890i \(-0.820353\pi\)
0.844922 0.534890i \(-0.179647\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 6.42221 0.237534
\(732\) 0 0
\(733\) − 14.6056i − 0.539468i −0.962935 0.269734i \(-0.913064\pi\)
0.962935 0.269734i \(-0.0869358\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.00000i 0.147342i
\(738\) 0 0
\(739\) 45.2666 1.66516 0.832580 0.553905i \(-0.186863\pi\)
0.832580 + 0.553905i \(0.186863\pi\)
\(740\) 0 0
\(741\) −18.7889 −0.690227
\(742\) 0 0
\(743\) 44.6056i 1.63642i 0.574920 + 0.818209i \(0.305033\pi\)
−0.574920 + 0.818209i \(0.694967\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3.39445i 0.124196i
\(748\) 0 0
\(749\) −15.1556 −0.553773
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) − 18.4222i − 0.671342i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 27.2111i 0.989004i 0.869176 + 0.494502i \(0.164650\pi\)
−0.869176 + 0.494502i \(0.835350\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −15.6333 −0.566707 −0.283353 0.959016i \(-0.591447\pi\)
−0.283353 + 0.959016i \(0.591447\pi\)
\(762\) 0 0
\(763\) 26.0555i 0.943273i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 20.8444i − 0.752648i
\(768\) 0 0
\(769\) 28.7889 1.03815 0.519077 0.854727i \(-0.326276\pi\)
0.519077 + 0.854727i \(0.326276\pi\)
\(770\) 0 0
\(771\) 8.78890 0.316524
\(772\) 0 0
\(773\) 24.0555i 0.865217i 0.901582 + 0.432608i \(0.142407\pi\)
−0.901582 + 0.432608i \(0.857593\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 29.2111i − 1.04794i
\(778\) 0 0
\(779\) −57.6888 −2.06692
\(780\) 0 0
\(781\) 14.4222 0.516067
\(782\) 0 0
\(783\) − 8.00000i − 0.285897i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 2.97224i − 0.105949i −0.998596 0.0529745i \(-0.983130\pi\)
0.998596 0.0529745i \(-0.0168702\pi\)
\(788\) 0 0
\(789\) 19.0278 0.677406
\(790\) 0 0
\(791\) 39.6333 1.40920
\(792\) 0 0
\(793\) − 18.7889i − 0.667213i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 32.0555i 1.13546i 0.823213 + 0.567732i \(0.192179\pi\)
−0.823213 + 0.567732i \(0.807821\pi\)
\(798\) 0 0
\(799\) 5.57779 0.197328
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) − 6.60555i − 0.233105i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 8.42221i 0.296476i
\(808\) 0 0
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) −36.4222 −1.27896 −0.639478 0.768809i \(-0.720850\pi\)
−0.639478 + 0.768809i \(0.720850\pi\)
\(812\) 0 0
\(813\) 19.2111i 0.673763i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 76.4777i 2.67562i
\(818\) 0 0
\(819\) 6.78890 0.237223
\(820\) 0 0
\(821\) 44.8444 1.56508 0.782540 0.622600i \(-0.213924\pi\)
0.782540 + 0.622600i \(0.213924\pi\)
\(822\) 0 0
\(823\) 26.7889i 0.933802i 0.884310 + 0.466901i \(0.154630\pi\)
−0.884310 + 0.466901i \(0.845370\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 4.60555i − 0.160151i −0.996789 0.0800754i \(-0.974484\pi\)
0.996789 0.0800754i \(-0.0255161\pi\)
\(828\) 0 0
\(829\) −49.6333 −1.72384 −0.861918 0.507048i \(-0.830737\pi\)
−0.861918 + 0.507048i \(0.830737\pi\)
\(830\) 0 0
\(831\) 18.2389 0.632699
\(832\) 0 0
\(833\) 0.127833i 0.00442917i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 5.21110i − 0.180122i
\(838\) 0 0
\(839\) −1.57779 −0.0544715 −0.0272358 0.999629i \(-0.508670\pi\)
−0.0272358 + 0.999629i \(0.508670\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.60555i 0.0895278i
\(848\) 0 0
\(849\) −2.60555 −0.0894223
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 10.2389i 0.350572i 0.984518 + 0.175286i \(0.0560850\pi\)
−0.984518 + 0.175286i \(0.943915\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11.0278i 0.376701i 0.982102 + 0.188350i \(0.0603141\pi\)
−0.982102 + 0.188350i \(0.939686\pi\)
\(858\) 0 0
\(859\) −42.7889 −1.45994 −0.729969 0.683480i \(-0.760466\pi\)
−0.729969 + 0.683480i \(0.760466\pi\)
\(860\) 0 0
\(861\) 20.8444 0.710376
\(862\) 0 0
\(863\) − 3.63331i − 0.123679i −0.998086 0.0618396i \(-0.980303\pi\)
0.998086 0.0618396i \(-0.0196967\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 16.6333i − 0.564897i
\(868\) 0 0
\(869\) −3.21110 −0.108929
\(870\) 0 0
\(871\) −10.4222 −0.353143
\(872\) 0 0
\(873\) 16.4222i 0.555807i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 26.2389i − 0.886023i −0.896516 0.443012i \(-0.853910\pi\)
0.896516 0.443012i \(-0.146090\pi\)
\(878\) 0 0
\(879\) −1.81665 −0.0612742
\(880\) 0 0
\(881\) −30.8444 −1.03917 −0.519587 0.854417i \(-0.673914\pi\)
−0.519587 + 0.854417i \(0.673914\pi\)
\(882\) 0 0
\(883\) − 22.4222i − 0.754567i −0.926098 0.377284i \(-0.876858\pi\)
0.926098 0.377284i \(-0.123142\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 23.0278i 0.773196i 0.922248 + 0.386598i \(0.126350\pi\)
−0.922248 + 0.386598i \(0.873650\pi\)
\(888\) 0 0
\(889\) −3.63331 −0.121857
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 66.4222i 2.22273i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 41.6888 1.39040
\(900\) 0 0
\(901\) −1.21110 −0.0403477
\(902\) 0 0
\(903\) − 27.6333i − 0.919579i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 38.0555i − 1.26361i −0.775126 0.631806i \(-0.782314\pi\)
0.775126 0.631806i \(-0.217686\pi\)
\(908\) 0 0
\(909\) 9.21110 0.305513
\(910\) 0 0
\(911\) −10.4222 −0.345303 −0.172652 0.984983i \(-0.555233\pi\)
−0.172652 + 0.984983i \(0.555233\pi\)
\(912\) 0 0
\(913\) − 3.39445i − 0.112340i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 55.2666i − 1.82506i
\(918\) 0 0
\(919\) 26.0000 0.857661 0.428830 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 0 0
\(921\) −1.39445 −0.0459486
\(922\) 0 0
\(923\) 37.5778i 1.23689i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 13.2111i 0.433910i
\(928\) 0 0
\(929\) 30.8444 1.01197 0.505986 0.862542i \(-0.331129\pi\)
0.505986 + 0.862542i \(0.331129\pi\)
\(930\) 0 0
\(931\) −1.52228 −0.0498908
\(932\) 0 0
\(933\) − 1.57779i − 0.0516547i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 57.0278i − 1.86302i −0.363721 0.931508i \(-0.618494\pi\)
0.363721 0.931508i \(-0.381506\pi\)
\(938\) 0 0
\(939\) 23.2111 0.757466
\(940\) 0 0
\(941\) 38.7889 1.26448 0.632241 0.774772i \(-0.282135\pi\)
0.632241 + 0.774772i \(0.282135\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 39.6333i − 1.28791i −0.765064 0.643955i \(-0.777293\pi\)
0.765064 0.643955i \(-0.222707\pi\)
\(948\) 0 0
\(949\) 17.2111 0.558696
\(950\) 0 0
\(951\) 3.21110 0.104127
\(952\) 0 0
\(953\) 29.4500i 0.953978i 0.878909 + 0.476989i \(0.158272\pi\)
−0.878909 + 0.476989i \(0.841728\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 8.00000i 0.258603i
\(958\) 0 0
\(959\) −15.6333 −0.504826
\(960\) 0 0
\(961\) −3.84441 −0.124013
\(962\) 0 0
\(963\) − 5.81665i − 0.187439i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 4.18335i − 0.134527i −0.997735 0.0672637i \(-0.978573\pi\)
0.997735 0.0672637i \(-0.0214269\pi\)
\(968\) 0 0
\(969\) −4.36669 −0.140278
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 32.3667i 1.03763i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30.0000i 0.959785i 0.877327 + 0.479893i \(0.159324\pi\)
−0.877327 + 0.479893i \(0.840676\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 0 0
\(983\) − 48.4777i − 1.54620i −0.634285 0.773100i \(-0.718705\pi\)
0.634285 0.773100i \(-0.281295\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 24.0000i − 0.763928i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 7.63331 0.242480 0.121240 0.992623i \(-0.461313\pi\)
0.121240 + 0.992623i \(0.461313\pi\)
\(992\) 0 0
\(993\) 21.2111i 0.673114i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 21.0278i 0.665956i 0.942935 + 0.332978i \(0.108053\pi\)
−0.942935 + 0.332978i \(0.891947\pi\)
\(998\) 0 0
\(999\) 11.2111 0.354704
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 3300.2.c.l.1849.2 4
3.2 odd 2 9900.2.c.q.5149.3 4
5.2 odd 4 660.2.a.e.1.1 2
5.3 odd 4 3300.2.a.w.1.2 2
5.4 even 2 inner 3300.2.c.l.1849.3 4
15.2 even 4 1980.2.a.h.1.1 2
15.8 even 4 9900.2.a.bl.1.2 2
15.14 odd 2 9900.2.c.q.5149.2 4
20.7 even 4 2640.2.a.bc.1.2 2
55.32 even 4 7260.2.a.w.1.2 2
60.47 odd 4 7920.2.a.bo.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
660.2.a.e.1.1 2 5.2 odd 4
1980.2.a.h.1.1 2 15.2 even 4
2640.2.a.bc.1.2 2 20.7 even 4
3300.2.a.w.1.2 2 5.3 odd 4
3300.2.c.l.1849.2 4 1.1 even 1 trivial
3300.2.c.l.1849.3 4 5.4 even 2 inner
7260.2.a.w.1.2 2 55.32 even 4
7920.2.a.bo.1.2 2 60.47 odd 4
9900.2.a.bl.1.2 2 15.8 even 4
9900.2.c.q.5149.2 4 15.14 odd 2
9900.2.c.q.5149.3 4 3.2 odd 2