Properties

Label 1680.2.t.b.1009.1
Level $1680$
Weight $2$
Character 1680.1009
Analytic conductor $13.415$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1680,2,Mod(1009,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1680.1009");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.t (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: \(13.4148675396\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1680.1009
Dual form 1680.2.t.b.1009.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +2.00000i q^{13} +(-2.00000 + 1.00000i) q^{15} -6.00000 q^{19} -1.00000 q^{21} +(-3.00000 + 4.00000i) q^{25} +1.00000i q^{27} +6.00000 q^{29} -10.0000 q^{31} +2.00000i q^{33} +(-2.00000 + 1.00000i) q^{35} +2.00000 q^{39} +6.00000 q^{41} -8.00000i q^{43} +(1.00000 + 2.00000i) q^{45} +12.0000i q^{47} -1.00000 q^{49} +6.00000i q^{53} +(2.00000 + 4.00000i) q^{55} +6.00000i q^{57} -6.00000 q^{61} +1.00000i q^{63} +(4.00000 - 2.00000i) q^{65} -4.00000i q^{67} -6.00000 q^{71} +14.0000i q^{73} +(4.00000 + 3.00000i) q^{75} +2.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} -6.00000i q^{87} +6.00000 q^{89} +2.00000 q^{91} +10.0000i q^{93} +(6.00000 + 12.0000i) q^{95} -2.00000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{9} - 4 q^{11} - 4 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{25} + 12 q^{29} - 20 q^{31} - 4 q^{35} + 4 q^{39} + 12 q^{41} + 2 q^{45} - 2 q^{49} + 4 q^{55} - 12 q^{61} + 8 q^{65} - 12 q^{71} + 8 q^{75} + 8 q^{79} + 2 q^{81} + 12 q^{89} + 4 q^{91} + 12 q^{95} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) −1.00000 2.00000i −0.447214 0.894427i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) −2.00000 + 1.00000i −0.516398 + 0.258199i
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 0 0
\(35\) −2.00000 + 1.00000i −0.338062 + 0.169031i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 1.00000 + 2.00000i 0.149071 + 0.298142i
\(46\) 0 0
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 2.00000 + 4.00000i 0.269680 + 0.539360i
\(56\) 0 0
\(57\) 6.00000i 0.794719i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 4.00000 2.00000i 0.496139 0.248069i
\(66\) 0 0
\(67\) 4.00000i 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 0 0
\(75\) 4.00000 + 3.00000i 0.461880 + 0.346410i
\(76\) 0 0
\(77\) 2.00000i 0.227921i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 10.0000i 1.03695i
\(94\) 0 0
\(95\) 6.00000 + 12.0000i 0.615587 + 1.23117i
\(96\) 0 0
\(97\) 2.00000i 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) 0 0
\(105\) 1.00000 + 2.00000i 0.0975900 + 0.195180i
\(106\) 0 0
\(107\) 4.00000i 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.0000i 0.940721i −0.882474 0.470360i \(-0.844124\pi\)
0.882474 0.470360i \(-0.155876\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 6.00000i 0.541002i
\(124\) 0 0
\(125\) 11.0000 + 2.00000i 0.983870 + 0.178885i
\(126\) 0 0
\(127\) 4.00000i 0.354943i 0.984126 + 0.177471i \(0.0567917\pi\)
−0.984126 + 0.177471i \(0.943208\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 6.00000i 0.520266i
\(134\) 0 0
\(135\) 2.00000 1.00000i 0.172133 0.0860663i
\(136\) 0 0
\(137\) 2.00000i 0.170872i −0.996344 0.0854358i \(-0.972772\pi\)
0.996344 0.0854358i \(-0.0272282\pi\)
\(138\) 0 0
\(139\) −6.00000 −0.508913 −0.254457 0.967084i \(-0.581897\pi\)
−0.254457 + 0.967084i \(0.581897\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 4.00000i 0.334497i
\(144\) 0 0
\(145\) −6.00000 12.0000i −0.498273 0.996546i
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 10.0000 + 20.0000i 0.803219 + 1.60644i
\(156\) 0 0
\(157\) 22.0000i 1.75579i −0.478852 0.877896i \(-0.658947\pi\)
0.478852 0.877896i \(-0.341053\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.0000i 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) 0 0
\(165\) 4.00000 2.00000i 0.311400 0.155700i
\(166\) 0 0
\(167\) 24.0000i 1.85718i 0.371113 + 0.928588i \(0.378976\pi\)
−0.371113 + 0.928588i \(0.621024\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 4.00000 + 3.00000i 0.302372 + 0.226779i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) −26.0000 −1.93256 −0.966282 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 0 0
\(183\) 6.00000i 0.443533i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) 16.0000i 1.15171i 0.817554 + 0.575853i \(0.195330\pi\)
−0.817554 + 0.575853i \(0.804670\pi\)
\(194\) 0 0
\(195\) −2.00000 4.00000i −0.143223 0.286446i
\(196\) 0 0
\(197\) 18.0000i 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) −22.0000 −1.55954 −0.779769 0.626067i \(-0.784664\pi\)
−0.779769 + 0.626067i \(0.784664\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 6.00000i 0.421117i
\(204\) 0 0
\(205\) −6.00000 12.0000i −0.419058 0.838116i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 6.00000i 0.411113i
\(214\) 0 0
\(215\) −16.0000 + 8.00000i −1.09119 + 0.545595i
\(216\) 0 0
\(217\) 10.0000i 0.678844i
\(218\) 0 0
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.0000i 1.07144i −0.844396 0.535720i \(-0.820040\pi\)
0.844396 0.535720i \(-0.179960\pi\)
\(224\) 0 0
\(225\) 3.00000 4.00000i 0.200000 0.266667i
\(226\) 0 0
\(227\) 8.00000i 0.530979i −0.964114 0.265489i \(-0.914466\pi\)
0.964114 0.265489i \(-0.0855335\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 0 0
\(235\) 24.0000 12.0000i 1.56559 0.782794i
\(236\) 0 0
\(237\) 4.00000i 0.259828i
\(238\) 0 0
\(239\) 30.0000 1.94054 0.970269 0.242028i \(-0.0778125\pi\)
0.970269 + 0.242028i \(0.0778125\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 1.00000 + 2.00000i 0.0638877 + 0.127775i
\(246\) 0 0
\(247\) 12.0000i 0.763542i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 20.0000i 1.24757i −0.781598 0.623783i \(-0.785595\pi\)
0.781598 0.623783i \(-0.214405\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 8.00000i 0.493301i −0.969104 0.246651i \(-0.920670\pi\)
0.969104 0.246651i \(-0.0793300\pi\)
\(264\) 0 0
\(265\) 12.0000 6.00000i 0.737154 0.368577i
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 0 0
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) −30.0000 −1.82237 −0.911185 0.411997i \(-0.864831\pi\)
−0.911185 + 0.411997i \(0.864831\pi\)
\(272\) 0 0
\(273\) 2.00000i 0.121046i
\(274\) 0 0
\(275\) 6.00000 8.00000i 0.361814 0.482418i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) 12.0000i 0.713326i −0.934233 0.356663i \(-0.883914\pi\)
0.934233 0.356663i \(-0.116086\pi\)
\(284\) 0 0
\(285\) 12.0000 6.00000i 0.710819 0.355409i
\(286\) 0 0
\(287\) 6.00000i 0.354169i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 0 0
\(293\) 16.0000i 0.934730i 0.884064 + 0.467365i \(0.154797\pi\)
−0.884064 + 0.467365i \(0.845203\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.00000i 0.116052i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 10.0000i 0.574485i
\(304\) 0 0
\(305\) 6.00000 + 12.0000i 0.343559 + 0.687118i
\(306\) 0 0
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i −0.918396 0.395663i \(-0.870515\pi\)
0.918396 0.395663i \(-0.129485\pi\)
\(314\) 0 0
\(315\) 2.00000 1.00000i 0.112687 0.0563436i
\(316\) 0 0
\(317\) 26.0000i 1.46031i 0.683284 + 0.730153i \(0.260551\pi\)
−0.683284 + 0.730153i \(0.739449\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −8.00000 6.00000i −0.443760 0.332820i
\(326\) 0 0
\(327\) 10.0000i 0.553001i
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.00000 + 4.00000i −0.437087 + 0.218543i
\(336\) 0 0
\(337\) 16.0000i 0.871576i −0.900049 0.435788i \(-0.856470\pi\)
0.900049 0.435788i \(-0.143530\pi\)
\(338\) 0 0
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) 20.0000 1.08306
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.00000i 0.214731i −0.994220 0.107366i \(-0.965758\pi\)
0.994220 0.107366i \(-0.0342415\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 24.0000i 1.27739i −0.769460 0.638696i \(-0.779474\pi\)
0.769460 0.638696i \(-0.220526\pi\)
\(354\) 0 0
\(355\) 6.00000 + 12.0000i 0.318447 + 0.636894i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −22.0000 −1.16112 −0.580558 0.814219i \(-0.697165\pi\)
−0.580558 + 0.814219i \(0.697165\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 7.00000i 0.367405i
\(364\) 0 0
\(365\) 28.0000 14.0000i 1.46559 0.732793i
\(366\) 0 0
\(367\) 16.0000i 0.835193i −0.908633 0.417597i \(-0.862873\pi\)
0.908633 0.417597i \(-0.137127\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 4.00000i 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) 0 0
\(375\) 2.00000 11.0000i 0.103280 0.568038i
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) 16.0000i 0.817562i −0.912633 0.408781i \(-0.865954\pi\)
0.912633 0.408781i \(-0.134046\pi\)
\(384\) 0 0
\(385\) 4.00000 2.00000i 0.203859 0.101929i
\(386\) 0 0
\(387\) 8.00000i 0.406663i
\(388\) 0 0
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 12.0000i 0.605320i
\(394\) 0 0
\(395\) −4.00000 8.00000i −0.201262 0.402524i
\(396\) 0 0
\(397\) 2.00000i 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 0 0
\(399\) 6.00000 0.300376
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 20.0000i 0.996271i
\(404\) 0 0
\(405\) −1.00000 2.00000i −0.0496904 0.0993808i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.00000i 0.293821i
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 12.0000i 0.583460i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 6.00000i 0.290360i
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −14.0000 −0.674356 −0.337178 0.941441i \(-0.609472\pi\)
−0.337178 + 0.941441i \(0.609472\pi\)
\(432\) 0 0
\(433\) 30.0000i 1.44171i −0.693087 0.720854i \(-0.743750\pi\)
0.693087 0.720854i \(-0.256250\pi\)
\(434\) 0 0
\(435\) −12.0000 + 6.00000i −0.575356 + 0.287678i
\(436\) 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
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 28.0000i 1.33032i −0.746701 0.665160i \(-0.768363\pi\)
0.746701 0.665160i \(-0.231637\pi\)
\(444\) 0 0
\(445\) −6.00000 12.0000i −0.284427 0.568855i
\(446\) 0 0
\(447\) 14.0000i 0.662177i
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.00000 4.00000i −0.0937614 0.187523i
\(456\) 0 0
\(457\) 12.0000i 0.561336i 0.959805 + 0.280668i \(0.0905560\pi\)
−0.959805 + 0.280668i \(0.909444\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 36.0000i 1.67306i 0.547920 + 0.836531i \(0.315420\pi\)
−0.547920 + 0.836531i \(0.684580\pi\)
\(464\) 0 0
\(465\) 20.0000 10.0000i 0.927478 0.463739i
\(466\) 0 0
\(467\) 32.0000i 1.48078i −0.672176 0.740392i \(-0.734640\pi\)
0.672176 0.740392i \(-0.265360\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 0 0
\(473\) 16.0000i 0.735681i
\(474\) 0 0
\(475\) 18.0000 24.0000i 0.825897 1.10120i
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.00000 + 2.00000i −0.181631 + 0.0908153i
\(486\) 0 0
\(487\) 12.0000i 0.543772i 0.962329 + 0.271886i \(0.0876473\pi\)
−0.962329 + 0.271886i \(0.912353\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 22.0000 0.992846 0.496423 0.868081i \(-0.334646\pi\)
0.496423 + 0.868081i \(0.334646\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −2.00000 4.00000i −0.0898933 0.179787i
\(496\) 0 0
\(497\) 6.00000i 0.269137i
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) 0 0
\(503\) 24.0000i 1.07011i −0.844818 0.535054i \(-0.820291\pi\)
0.844818 0.535054i \(-0.179709\pi\)
\(504\) 0 0
\(505\) 10.0000 + 20.0000i 0.444994 + 0.889988i
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 0 0
\(513\) 6.00000i 0.264906i
\(514\) 0 0
\(515\) 32.0000 16.0000i 1.41009 0.705044i
\(516\) 0 0
\(517\) 24.0000i 1.05552i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) 0 0
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 0 0
\(525\) 3.00000 4.00000i 0.130931 0.174574i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) −8.00000 + 4.00000i −0.345870 + 0.172935i
\(536\) 0 0
\(537\) 18.0000i 0.776757i
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 26.0000i 1.11577i
\(544\) 0 0
\(545\) 10.0000 + 20.0000i 0.428353 + 0.856706i
\(546\) 0 0
\(547\) 36.0000i 1.53925i −0.638497 0.769624i \(-0.720443\pi\)
0.638497 0.769624i \(-0.279557\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) −36.0000 −1.53365
\(552\) 0 0
\(553\) 4.00000i 0.170097i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.00000i 0.0847427i 0.999102 + 0.0423714i \(0.0134913\pi\)
−0.999102 + 0.0423714i \(0.986509\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 20.0000i 0.842900i −0.906852 0.421450i \(-0.861521\pi\)
0.906852 0.421450i \(-0.138479\pi\)
\(564\) 0 0
\(565\) −20.0000 + 10.0000i −0.841406 + 0.420703i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) −34.0000 −1.42535 −0.712677 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 10.0000i 0.417756i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 22.0000i 0.915872i 0.888985 + 0.457936i \(0.151411\pi\)
−0.888985 + 0.457936i \(0.848589\pi\)
\(578\) 0 0
\(579\) 16.0000 0.664937
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 12.0000i 0.496989i
\(584\) 0 0
\(585\) −4.00000 + 2.00000i −0.165380 + 0.0826898i
\(586\) 0 0
\(587\) 28.0000i 1.15568i 0.816149 + 0.577842i \(0.196105\pi\)
−0.816149 + 0.577842i \(0.803895\pi\)
\(588\) 0 0
\(589\) 60.0000 2.47226
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) 40.0000i 1.64260i −0.570494 0.821302i \(-0.693248\pi\)
0.570494 0.821302i \(-0.306752\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 22.0000i 0.900400i
\(598\) 0 0
\(599\) 42.0000 1.71607 0.858037 0.513588i \(-0.171684\pi\)
0.858037 + 0.513588i \(0.171684\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) 7.00000 + 14.0000i 0.284590 + 0.569181i
\(606\) 0 0
\(607\) 32.0000i 1.29884i 0.760430 + 0.649420i \(0.224988\pi\)
−0.760430 + 0.649420i \(0.775012\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 24.0000i 0.969351i 0.874694 + 0.484675i \(0.161062\pi\)
−0.874694 + 0.484675i \(0.838938\pi\)
\(614\) 0 0
\(615\) −12.0000 + 6.00000i −0.483887 + 0.241943i
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 0 0
\(619\) −22.0000 −0.884255 −0.442127 0.896952i \(-0.645776\pi\)
−0.442127 + 0.896952i \(0.645776\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.00000i 0.240385i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 12.0000i 0.479234i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) 8.00000i 0.317971i
\(634\) 0 0
\(635\) 8.00000 4.00000i 0.317470 0.158735i
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) 0 0
\(643\) 44.0000i 1.73519i 0.497271 + 0.867595i \(0.334335\pi\)
−0.497271 + 0.867595i \(0.665665\pi\)
\(644\) 0 0
\(645\) 8.00000 + 16.0000i 0.315000 + 0.629999i
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) 0 0
\(653\) 38.0000i 1.48705i −0.668705 0.743527i \(-0.733151\pi\)
0.668705 0.743527i \(-0.266849\pi\)
\(654\) 0 0
\(655\) 12.0000 + 24.0000i 0.468879 + 0.937758i
\(656\) 0 0
\(657\) 14.0000i 0.546192i
\(658\) 0 0
\(659\) 22.0000 0.856998 0.428499 0.903542i \(-0.359042\pi\)
0.428499 + 0.903542i \(0.359042\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 12.0000 6.00000i 0.465340 0.232670i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) 12.0000i 0.462566i 0.972887 + 0.231283i \(0.0742923\pi\)
−0.972887 + 0.231283i \(0.925708\pi\)
\(674\) 0 0
\(675\) −4.00000 3.00000i −0.153960 0.115470i
\(676\) 0 0
\(677\) 48.0000i 1.84479i 0.386248 + 0.922395i \(0.373771\pi\)
−0.386248 + 0.922395i \(0.626229\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) −4.00000 + 2.00000i −0.152832 + 0.0764161i
\(686\) 0 0
\(687\) 6.00000i 0.228914i
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 0 0
\(693\) 2.00000i 0.0759737i
\(694\) 0 0
\(695\) 6.00000 + 12.0000i 0.227593 + 0.455186i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −46.0000 −1.73740 −0.868698 0.495342i \(-0.835043\pi\)
−0.868698 + 0.495342i \(0.835043\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −12.0000 24.0000i −0.451946 0.903892i
\(706\) 0 0
\(707\) 10.0000i 0.376089i
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −8.00000 + 4.00000i −0.299183 + 0.149592i
\(716\) 0 0
\(717\) 30.0000i 1.12037i
\(718\) 0 0
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 26.0000i 0.966950i
\(724\) 0 0
\(725\) −18.0000 + 24.0000i −0.668503 + 0.891338i
\(726\) 0 0
\(727\) 16.0000i 0.593407i 0.954970 + 0.296704i \(0.0958873\pi\)
−0.954970 + 0.296704i \(0.904113\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 14.0000i 0.517102i −0.965998 0.258551i \(-0.916755\pi\)
0.965998 0.258551i \(-0.0832450\pi\)
\(734\) 0 0
\(735\) 2.00000 1.00000i 0.0737711 0.0368856i
\(736\) 0 0
\(737\) 8.00000i 0.294684i
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 0 0
\(743\) 16.0000i 0.586983i 0.955962 + 0.293492i \(0.0948173\pi\)
−0.955962 + 0.293492i \(0.905183\pi\)
\(744\) 0 0
\(745\) 14.0000 + 28.0000i 0.512920 + 1.02584i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 8.00000i 0.291536i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 12.0000i 0.436147i 0.975932 + 0.218074i \(0.0699773\pi\)
−0.975932 + 0.218074i \(0.930023\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) 10.0000i 0.362024i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 42.0000 1.51456 0.757279 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 0 0
\(771\) −20.0000 −0.720282
\(772\) 0 0
\(773\) 32.0000i 1.15096i −0.817816 0.575480i \(-0.804815\pi\)
0.817816 0.575480i \(-0.195185\pi\)
\(774\) 0 0
\(775\) 30.0000 40.0000i 1.07763 1.43684i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) 0 0
\(785\) −44.0000 + 22.0000i −1.57043 + 0.785214i
\(786\) 0 0
\(787\) 12.0000i 0.427754i 0.976861 + 0.213877i \(0.0686091\pi\)
−0.976861 + 0.213877i \(0.931391\pi\)
\(788\) 0 0
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) 0 0
\(795\) −6.00000 12.0000i −0.212798 0.425596i
\(796\) 0 0
\(797\) 48.0000i 1.70025i 0.526583 + 0.850124i \(0.323473\pi\)
−0.526583 + 0.850124i \(0.676527\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 28.0000i 0.988099i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14.0000i 0.492823i
\(808\) 0 0
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 0 0
\(813\) 30.0000i 1.05215i
\(814\) 0 0
\(815\) −32.0000 + 16.0000i −1.12091 + 0.560456i
\(816\) 0 0
\(817\) 48.0000i 1.67931i
\(818\) 0 0
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 26.0000 0.907406 0.453703 0.891153i \(-0.350103\pi\)
0.453703 + 0.891153i \(0.350103\pi\)
\(822\) 0 0
\(823\) 44.0000i 1.53374i 0.641800 + 0.766872i \(0.278188\pi\)
−0.641800 + 0.766872i \(0.721812\pi\)
\(824\) 0 0
\(825\) −8.00000 6.00000i −0.278524 0.208893i
\(826\) 0 0
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) 0 0
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 48.0000 24.0000i 1.66111 0.830554i
\(836\) 0 0
\(837\) 10.0000i 0.345651i
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 26.0000i 0.895488i
\(844\) 0 0
\(845\) −9.00000 18.0000i −0.309609 0.619219i
\(846\) 0 0
\(847\) 7.00000i 0.240523i
\(848\) 0 0
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 0 0
\(855\) −6.00000 12.0000i −0.205196 0.410391i
\(856\) 0 0
\(857\) 28.0000i 0.956462i −0.878234 0.478231i \(-0.841278\pi\)
0.878234 0.478231i \(-0.158722\pi\)
\(858\) 0 0
\(859\) −2.00000 −0.0682391 −0.0341196 0.999418i \(-0.510863\pi\)
−0.0341196 + 0.999418i \(0.510863\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) 32.0000i 1.08929i 0.838666 + 0.544646i \(0.183336\pi\)
−0.838666 + 0.544646i \(0.816664\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 17.0000i 0.577350i
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) 2.00000i 0.0676897i
\(874\) 0 0
\(875\) 2.00000 11.0000i 0.0676123 0.371868i
\(876\) 0 0
\(877\) 56.0000i 1.89099i −0.325643 0.945493i \(-0.605581\pi\)
0.325643 0.945493i \(-0.394419\pi\)
\(878\) 0 0
\(879\) 16.0000 0.539667
\(880\) 0 0
\(881\) −54.0000 −1.81931 −0.909653 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 0 0
\(883\) 36.0000i 1.21150i 0.795656 + 0.605748i \(0.207126\pi\)
−0.795656 + 0.605748i \(0.792874\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.0000i 0.402921i 0.979497 + 0.201460i \(0.0645687\pi\)
−0.979497 + 0.201460i \(0.935431\pi\)
\(888\) 0 0
\(889\) 4.00000 0.134156
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 0 0
\(893\) 72.0000i 2.40939i
\(894\) 0 0
\(895\) −18.0000 36.0000i −0.601674 1.20335i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −60.0000 −2.00111
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 8.00000i 0.266223i
\(904\) 0 0
\(905\) 26.0000 + 52.0000i 0.864269 + 1.72854i
\(906\) 0 0
\(907\) 28.0000i 0.929725i 0.885383 + 0.464862i \(0.153896\pi\)
−0.885383 + 0.464862i \(0.846104\pi\)
\(908\) 0 0
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 12.0000 6.00000i 0.396708 0.198354i
\(916\) 0 0
\(917\) 12.0000i 0.396275i
\(918\) 0 0
\(919\) 44.0000 1.45143 0.725713 0.687998i \(-0.241510\pi\)
0.725713 + 0.687998i \(0.241510\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) 12.0000i 0.394985i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 16.0000i 0.525509i
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) 8.00000i 0.261908i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 18.0000i 0.588034i 0.955800 + 0.294017i \(0.0949923\pi\)
−0.955800 + 0.294017i \(0.905008\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −1.00000 2.00000i −0.0325300 0.0650600i
\(946\) 0 0
\(947\) 4.00000i 0.129983i 0.997886 + 0.0649913i \(0.0207020\pi\)
−0.997886 + 0.0649913i \(0.979298\pi\)
\(948\) 0 0
\(949\) −28.0000 −0.908918
\(950\) 0 0
\(951\) 26.0000 0.843108
\(952\) 0 0
\(953\) 18.0000i 0.583077i −0.956559 0.291539i \(-0.905833\pi\)
0.956559 0.291539i \(-0.0941672\pi\)
\(954\) 0 0
\(955\) 10.0000 + 20.0000i 0.323592 + 0.647185i
\(956\) 0 0
\(957\) 12.0000i 0.387905i
\(958\) 0 0
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 4.00000i 0.128898i
\(964\) 0 0
\(965\) 32.0000 16.0000i 1.03012 0.515058i
\(966\) 0 0
\(967\) 40.0000i 1.28631i −0.765735 0.643157i \(-0.777624\pi\)
0.765735 0.643157i \(-0.222376\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −32.0000 −1.02693 −0.513464 0.858111i \(-0.671638\pi\)
−0.513464 + 0.858111i \(0.671638\pi\)
\(972\) 0 0
\(973\) 6.00000i 0.192351i
\(974\) 0 0
\(975\) −6.00000 + 8.00000i −0.192154 + 0.256205i
\(976\) 0 0
\(977\) 30.0000i 0.959785i −0.877327 0.479893i \(-0.840676\pi\)
0.877327 0.479893i \(-0.159324\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) 28.0000i 0.893061i 0.894768 + 0.446531i \(0.147341\pi\)
−0.894768 + 0.446531i \(0.852659\pi\)
\(984\) 0 0
\(985\) −36.0000 + 18.0000i −1.14706 + 0.573528i
\(986\) 0 0
\(987\) 12.0000i 0.381964i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 22.0000 + 44.0000i 0.697447 + 1.39489i
\(996\) 0 0
\(997\) 22.0000i 0.696747i 0.937356 + 0.348373i \(0.113266\pi\)
−0.937356 + 0.348373i \(0.886734\pi\)
\(998\) 0 0
\(999\) 0 0
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 1680.2.t.b.1009.1 2
3.2 odd 2 5040.2.t.l.1009.2 2
4.3 odd 2 840.2.t.b.169.2 yes 2
5.2 odd 4 8400.2.a.v.1.1 1
5.3 odd 4 8400.2.a.bs.1.1 1
5.4 even 2 inner 1680.2.t.b.1009.2 2
12.11 even 2 2520.2.t.e.1009.2 2
15.14 odd 2 5040.2.t.l.1009.1 2
20.3 even 4 4200.2.a.j.1.1 1
20.7 even 4 4200.2.a.w.1.1 1
20.19 odd 2 840.2.t.b.169.1 2
60.59 even 2 2520.2.t.e.1009.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.b.169.1 2 20.19 odd 2
840.2.t.b.169.2 yes 2 4.3 odd 2
1680.2.t.b.1009.1 2 1.1 even 1 trivial
1680.2.t.b.1009.2 2 5.4 even 2 inner
2520.2.t.e.1009.1 2 60.59 even 2
2520.2.t.e.1009.2 2 12.11 even 2
4200.2.a.j.1.1 1 20.3 even 4
4200.2.a.w.1.1 1 20.7 even 4
5040.2.t.l.1009.1 2 15.14 odd 2
5040.2.t.l.1009.2 2 3.2 odd 2
8400.2.a.v.1.1 1 5.2 odd 4
8400.2.a.bs.1.1 1 5.3 odd 4