Properties

Label 1680.2.t.c.1009.1
Level $1680$
Weight $2$
Character 1680.1009
Analytic conductor $13.415$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.c.1009.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +(-1.00000 + 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(-1.00000 + 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +2.00000i q^{13} +(2.00000 + 1.00000i) q^{15} +2.00000 q^{19} -1.00000 q^{21} -8.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +1.00000i q^{27} -2.00000 q^{29} +6.00000 q^{31} +2.00000i q^{33} +(2.00000 + 1.00000i) q^{35} -8.00000i q^{37} +2.00000 q^{39} -10.0000 q^{41} +(1.00000 - 2.00000i) q^{45} -12.0000i q^{47} -1.00000 q^{49} -2.00000i q^{53} +(2.00000 - 4.00000i) q^{55} -2.00000i q^{57} +2.00000 q^{61} +1.00000i q^{63} +(-4.00000 - 2.00000i) q^{65} +4.00000i q^{67} -8.00000 q^{69} -14.0000 q^{71} -2.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} +2.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} -16.0000i q^{83} +2.00000i q^{87} +6.00000 q^{89} +2.00000 q^{91} -6.00000i q^{93} +(-2.00000 + 4.00000i) q^{95} -2.00000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{9} - 4 q^{11} + 4 q^{15} + 4 q^{19} - 2 q^{21} - 6 q^{25} - 4 q^{29} + 12 q^{31} + 4 q^{35} + 4 q^{39} - 20 q^{41} + 2 q^{45} - 2 q^{49} + 4 q^{55} + 4 q^{61} - 8 q^{65} - 16 q^{69} - 28 q^{71} - 8 q^{75} + 8 q^{79} + 2 q^{81} + 12 q^{89} + 4 q^{91} - 4 q^{95} + 4 q^{99}+O(q^{100}) \) 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
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 8.00000i 1.66812i −0.551677 0.834058i \(-0.686012\pi\)
0.551677 0.834058i \(-0.313988\pi\)
\(24\) 0 0
\(25\) −3.00000 4.00000i −0.600000 0.800000i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 0 0
\(35\) 2.00000 + 1.00000i 0.338062 + 0.169031i
\(36\) 0 0
\(37\) 8.00000i 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 1.00000 2.00000i 0.149071 0.298142i
\(46\) 0 0
\(47\) 12.0000i 1.75038i −0.483779 0.875190i \(-0.660736\pi\)
0.483779 0.875190i \(-0.339264\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000i 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 2.00000 4.00000i 0.269680 0.539360i
\(56\) 0 0
\(57\) 2.00000i 0.264906i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 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 \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 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
\(83\) 16.0000i 1.75623i −0.478451 0.878114i \(-0.658802\pi\)
0.478451 0.878114i \(-0.341198\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.00000i 0.214423i
\(88\) 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
\(93\) 6.00000i 0.622171i
\(94\) 0 0
\(95\) −2.00000 + 4.00000i −0.205196 + 0.410391i
\(96\) 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
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 16.0000i 1.57653i −0.615338 0.788263i \(-0.710980\pi\)
0.615338 0.788263i \(-0.289020\pi\)
\(104\) 0 0
\(105\) 1.00000 2.00000i 0.0975900 0.195180i
\(106\) 0 0
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) 16.0000 + 8.00000i 1.49201 + 0.746004i
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 10.0000i 0.901670i
\(124\) 0 0
\(125\) 11.0000 2.00000i 0.983870 0.178885i
\(126\) 0 0
\(127\) 12.0000i 1.06483i −0.846484 0.532414i \(-0.821285\pi\)
0.846484 0.532414i \(-0.178715\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 2.00000i 0.173422i
\(134\) 0 0
\(135\) −2.00000 1.00000i −0.172133 0.0860663i
\(136\) 0 0
\(137\) 10.0000i 0.854358i −0.904167 0.427179i \(-0.859507\pi\)
0.904167 0.427179i \(-0.140493\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 0 0
\(143\) 4.00000i 0.334497i
\(144\) 0 0
\(145\) 2.00000 4.00000i 0.166091 0.332182i
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −6.00000 + 12.0000i −0.481932 + 0.963863i
\(156\) 0 0
\(157\) 22.0000i 1.75579i −0.478852 0.877896i \(-0.658947\pi\)
0.478852 0.877896i \(-0.341053\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) 8.00000i 0.626608i 0.949653 + 0.313304i \(0.101436\pi\)
−0.949653 + 0.313304i \(0.898564\pi\)
\(164\) 0 0
\(165\) −4.00000 2.00000i −0.311400 0.155700i
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 24.0000i 1.82469i 0.409426 + 0.912343i \(0.365729\pi\)
−0.409426 + 0.912343i \(0.634271\pi\)
\(174\) 0 0
\(175\) −4.00000 + 3.00000i −0.302372 + 0.226779i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) 0 0
\(185\) 16.0000 + 8.00000i 1.17634 + 0.588172i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 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
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 2.00000i 0.140372i
\(204\) 0 0
\(205\) 10.0000 20.0000i 0.698430 1.39686i
\(206\) 0 0
\(207\) 8.00000i 0.556038i
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 14.0000i 0.959264i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.00000i 0.407307i
\(218\) 0 0
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 3.00000 + 4.00000i 0.200000 + 0.266667i
\(226\) 0 0
\(227\) 8.00000i 0.530979i 0.964114 + 0.265489i \(0.0855335\pi\)
−0.964114 + 0.265489i \(0.914466\pi\)
\(228\) 0 0
\(229\) 30.0000 1.98246 0.991228 0.132164i \(-0.0421925\pi\)
0.991228 + 0.132164i \(0.0421925\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) 18.0000i 1.17922i −0.807688 0.589610i \(-0.799282\pi\)
0.807688 0.589610i \(-0.200718\pi\)
\(234\) 0 0
\(235\) 24.0000 + 12.0000i 1.56559 + 0.782794i
\(236\) 0 0
\(237\) 4.00000i 0.259828i
\(238\) 0 0
\(239\) −10.0000 −0.646846 −0.323423 0.946254i \(-0.604834\pi\)
−0.323423 + 0.946254i \(0.604834\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 1.00000 2.00000i 0.0638877 0.127775i
\(246\) 0 0
\(247\) 4.00000i 0.254514i
\(248\) 0 0
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) 16.0000i 1.00591i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.0000i 0.748539i 0.927320 + 0.374270i \(0.122107\pi\)
−0.927320 + 0.374270i \(0.877893\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 4.00000 + 2.00000i 0.245718 + 0.122859i
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −14.0000 −0.850439 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 0 0
\(273\) 2.00000i 0.121046i
\(274\) 0 0
\(275\) 6.00000 + 8.00000i 0.361814 + 0.482418i
\(276\) 0 0
\(277\) 24.0000i 1.44202i −0.692925 0.721010i \(-0.743678\pi\)
0.692925 0.721010i \(-0.256322\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 12.0000i 0.713326i −0.934233 0.356663i \(-0.883914\pi\)
0.934233 0.356663i \(-0.116086\pi\)
\(284\) 0 0
\(285\) 4.00000 + 2.00000i 0.236940 + 0.118470i
\(286\) 0 0
\(287\) 10.0000i 0.590281i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 0 0
\(293\) 8.00000i 0.467365i 0.972313 + 0.233682i \(0.0750776\pi\)
−0.972313 + 0.233682i \(0.924922\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.00000i 0.116052i
\(298\) 0 0
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6.00000i 0.344691i
\(304\) 0 0
\(305\) −2.00000 + 4.00000i −0.114520 + 0.229039i
\(306\) 0 0
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 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
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 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
\(327\) 6.00000i 0.331801i
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) 0 0
\(333\) 8.00000i 0.438397i
\(334\) 0 0
\(335\) −8.00000 4.00000i −0.437087 0.218543i
\(336\) 0 0
\(337\) 32.0000i 1.74315i 0.490261 + 0.871576i \(0.336901\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 8.00000 16.0000i 0.430706 0.861411i
\(346\) 0 0
\(347\) 20.0000i 1.07366i 0.843692 + 0.536828i \(0.180378\pi\)
−0.843692 + 0.536828i \(0.819622\pi\)
\(348\) 0 0
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 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
\(355\) 14.0000 28.0000i 0.743043 1.48609i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 34.0000 1.79445 0.897226 0.441572i \(-0.145579\pi\)
0.897226 + 0.441572i \(0.145579\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 7.00000i 0.367405i
\(364\) 0 0
\(365\) 4.00000 + 2.00000i 0.209370 + 0.104685i
\(366\) 0 0
\(367\) 16.0000i 0.835193i −0.908633 0.417597i \(-0.862873\pi\)
0.908633 0.417597i \(-0.137127\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) 20.0000i 1.03556i 0.855514 + 0.517780i \(0.173242\pi\)
−0.855514 + 0.517780i \(0.826758\pi\)
\(374\) 0 0
\(375\) −2.00000 11.0000i −0.103280 0.568038i
\(376\) 0 0
\(377\) 4.00000i 0.206010i
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) 24.0000i 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) −4.00000 2.00000i −0.203859 0.101929i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 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
\(397\) 30.0000i 1.50566i 0.658217 + 0.752828i \(0.271311\pi\)
−0.658217 + 0.752828i \(0.728689\pi\)
\(398\) 0 0
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 0 0
\(403\) 12.0000i 0.597763i
\(404\) 0 0
\(405\) −1.00000 + 2.00000i −0.0496904 + 0.0993808i
\(406\) 0 0
\(407\) 16.0000i 0.793091i
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 32.0000 + 16.0000i 1.57082 + 0.785409i
\(416\) 0 0
\(417\) 14.0000i 0.685583i
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −38.0000 −1.85201 −0.926003 0.377515i \(-0.876779\pi\)
−0.926003 + 0.377515i \(0.876779\pi\)
\(422\) 0 0
\(423\) 12.0000i 0.583460i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.00000i 0.0967868i
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −22.0000 −1.05970 −0.529851 0.848091i \(-0.677752\pi\)
−0.529851 + 0.848091i \(0.677752\pi\)
\(432\) 0 0
\(433\) 18.0000i 0.865025i 0.901628 + 0.432512i \(0.142373\pi\)
−0.901628 + 0.432512i \(0.857627\pi\)
\(434\) 0 0
\(435\) −4.00000 2.00000i −0.191785 0.0958927i
\(436\) 0 0
\(437\) 16.0000i 0.765384i
\(438\) 0 0
\(439\) −6.00000 −0.286364 −0.143182 0.989696i \(-0.545733\pi\)
−0.143182 + 0.989696i \(0.545733\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) 0 0
\(445\) −6.00000 + 12.0000i −0.284427 + 0.568855i
\(446\) 0 0
\(447\) 6.00000i 0.283790i
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 20.0000 0.941763
\(452\) 0 0
\(453\) 16.0000i 0.751746i
\(454\) 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
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) 0 0
\(465\) 12.0000 + 6.00000i 0.556487 + 0.278243i
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −6.00000 8.00000i −0.275299 0.367065i
\(476\) 0 0
\(477\) 2.00000i 0.0915737i
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 0 0
\(483\) 8.00000i 0.364013i
\(484\) 0 0
\(485\) 4.00000 + 2.00000i 0.181631 + 0.0908153i
\(486\) 0 0
\(487\) 20.0000i 0.906287i −0.891438 0.453143i \(-0.850303\pi\)
0.891438 0.453143i \(-0.149697\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) −26.0000 −1.17336 −0.586682 0.809818i \(-0.699566\pi\)
−0.586682 + 0.809818i \(0.699566\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −2.00000 + 4.00000i −0.0898933 + 0.179787i
\(496\) 0 0
\(497\) 14.0000i 0.627986i
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16.0000i 0.713405i −0.934218 0.356702i \(-0.883901\pi\)
0.934218 0.356702i \(-0.116099\pi\)
\(504\) 0 0
\(505\) −6.00000 + 12.0000i −0.266996 + 0.533993i
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 0 0
\(513\) 2.00000i 0.0883022i
\(514\) 0 0
\(515\) 32.0000 + 16.0000i 1.41009 + 0.705044i
\(516\) 0 0
\(517\) 24.0000i 1.05552i
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 0 0
\(525\) 3.00000 + 4.00000i 0.130931 + 0.174574i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 20.0000i 0.866296i
\(534\) 0 0
\(535\) −8.00000 4.00000i −0.345870 0.172935i
\(536\) 0 0
\(537\) 2.00000i 0.0863064i
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 46.0000 1.97769 0.988847 0.148933i \(-0.0475840\pi\)
0.988847 + 0.148933i \(0.0475840\pi\)
\(542\) 0 0
\(543\) 2.00000i 0.0858282i
\(544\) 0 0
\(545\) −6.00000 + 12.0000i −0.257012 + 0.514024i
\(546\) 0 0
\(547\) 4.00000i 0.171028i 0.996337 + 0.0855138i \(0.0272532\pi\)
−0.996337 + 0.0855138i \(0.972747\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) 4.00000i 0.170097i
\(554\) 0 0
\(555\) 8.00000 16.0000i 0.339581 0.679162i
\(556\) 0 0
\(557\) 6.00000i 0.254228i −0.991888 0.127114i \(-0.959429\pi\)
0.991888 0.127114i \(-0.0405714\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) 0 0
\(565\) −28.0000 14.0000i −1.17797 0.588984i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 18.0000i 0.751961i
\(574\) 0 0
\(575\) −32.0000 + 24.0000i −1.33449 + 1.00087i
\(576\) 0 0
\(577\) 10.0000i 0.416305i −0.978096 0.208153i \(-0.933255\pi\)
0.978096 0.208153i \(-0.0667451\pi\)
\(578\) 0 0
\(579\) 16.0000 0.664937
\(580\) 0 0
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) 4.00000i 0.165663i
\(584\) 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
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) 8.00000i 0.328521i −0.986417 0.164260i \(-0.947476\pi\)
0.986417 0.164260i \(-0.0525237\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10.0000i 0.409273i
\(598\) 0 0
\(599\) −14.0000 −0.572024 −0.286012 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) 0 0
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) 7.00000 14.0000i 0.284590 0.569181i
\(606\) 0 0
\(607\) 16.0000i 0.649420i −0.945814 0.324710i \(-0.894733\pi\)
0.945814 0.324710i \(-0.105267\pi\)
\(608\) 0 0
\(609\) 2.00000 0.0810441
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) 16.0000i 0.646234i −0.946359 0.323117i \(-0.895269\pi\)
0.946359 0.323117i \(-0.104731\pi\)
\(614\) 0 0
\(615\) −20.0000 10.0000i −0.806478 0.403239i
\(616\) 0 0
\(617\) 38.0000i 1.52982i −0.644136 0.764911i \(-0.722783\pi\)
0.644136 0.764911i \(-0.277217\pi\)
\(618\) 0 0
\(619\) −14.0000 −0.562708 −0.281354 0.959604i \(-0.590783\pi\)
−0.281354 + 0.959604i \(0.590783\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 0 0
\(623\) 6.00000i 0.240385i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 4.00000i 0.159745i
\(628\) 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
\(635\) 24.0000 + 12.0000i 0.952411 + 0.476205i
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) 14.0000 0.553831
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 12.0000i 0.473234i 0.971603 + 0.236617i \(0.0760386\pi\)
−0.971603 + 0.236617i \(0.923961\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.00000i 0.314512i −0.987558 0.157256i \(-0.949735\pi\)
0.987558 0.157256i \(-0.0502649\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 0 0
\(653\) 30.0000i 1.17399i −0.809590 0.586995i \(-0.800311\pi\)
0.809590 0.586995i \(-0.199689\pi\)
\(654\) 0 0
\(655\) 12.0000 24.0000i 0.468879 0.937758i
\(656\) 0 0
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) −42.0000 −1.63609 −0.818044 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.00000 + 2.00000i 0.155113 + 0.0775567i
\(666\) 0 0
\(667\) 16.0000i 0.619522i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 20.0000i 0.770943i −0.922720 0.385472i \(-0.874039\pi\)
0.922720 0.385472i \(-0.125961\pi\)
\(674\) 0 0
\(675\) 4.00000 3.00000i 0.153960 0.115470i
\(676\) 0 0
\(677\) 24.0000i 0.922395i 0.887298 + 0.461197i \(0.152580\pi\)
−0.887298 + 0.461197i \(0.847420\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) 4.00000i 0.153056i 0.997067 + 0.0765279i \(0.0243834\pi\)
−0.997067 + 0.0765279i \(0.975617\pi\)
\(684\) 0 0
\(685\) 20.0000 + 10.0000i 0.764161 + 0.382080i
\(686\) 0 0
\(687\) 30.0000i 1.14457i
\(688\) 0 0
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) −30.0000 −1.14125 −0.570627 0.821209i \(-0.693300\pi\)
−0.570627 + 0.821209i \(0.693300\pi\)
\(692\) 0 0
\(693\) 2.00000i 0.0759737i
\(694\) 0 0
\(695\) 14.0000 28.0000i 0.531050 1.06210i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −38.0000 −1.43524 −0.717620 0.696435i \(-0.754769\pi\)
−0.717620 + 0.696435i \(0.754769\pi\)
\(702\) 0 0
\(703\) 16.0000i 0.603451i
\(704\) 0 0
\(705\) 12.0000 24.0000i 0.451946 0.903892i
\(706\) 0 0
\(707\) 6.00000i 0.225653i
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 48.0000i 1.79761i
\(714\) 0 0
\(715\) 8.00000 + 4.00000i 0.299183 + 0.149592i
\(716\) 0 0
\(717\) 10.0000i 0.373457i
\(718\) 0 0
\(719\) 52.0000 1.93927 0.969636 0.244551i \(-0.0786406\pi\)
0.969636 + 0.244551i \(0.0786406\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) 10.0000i 0.371904i
\(724\) 0 0
\(725\) 6.00000 + 8.00000i 0.222834 + 0.297113i
\(726\) 0 0
\(727\) 48.0000i 1.78022i 0.455744 + 0.890111i \(0.349373\pi\)
−0.455744 + 0.890111i \(0.650627\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 30.0000i 1.10808i −0.832492 0.554038i \(-0.813086\pi\)
0.832492 0.554038i \(-0.186914\pi\)
\(734\) 0 0
\(735\) −2.00000 1.00000i −0.0737711 0.0368856i
\(736\) 0 0
\(737\) 8.00000i 0.294684i
\(738\) 0 0
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 0 0
\(745\) 6.00000 12.0000i 0.219823 0.439646i
\(746\) 0 0
\(747\) 16.0000i 0.585409i
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 0 0
\(753\) 8.00000i 0.291536i
\(754\) 0 0
\(755\) 16.0000 32.0000i 0.582300 1.16460i
\(756\) 0 0
\(757\) 44.0000i 1.59921i −0.600528 0.799604i \(-0.705043\pi\)
0.600528 0.799604i \(-0.294957\pi\)
\(758\) 0 0
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) 6.00000i 0.217215i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 0 0
\(773\) 40.0000i 1.43870i −0.694648 0.719350i \(-0.744440\pi\)
0.694648 0.719350i \(-0.255560\pi\)
\(774\) 0 0
\(775\) −18.0000 24.0000i −0.646579 0.862105i
\(776\) 0 0
\(777\) 8.00000i 0.286998i
\(778\) 0 0
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) 28.0000 1.00192
\(782\) 0 0
\(783\) 2.00000i 0.0714742i
\(784\) 0 0
\(785\) 44.0000 + 22.0000i 1.57043 + 0.785214i
\(786\) 0 0
\(787\) 52.0000i 1.85360i −0.375555 0.926800i \(-0.622548\pi\)
0.375555 0.926800i \(-0.377452\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) 0 0
\(793\) 4.00000i 0.142044i
\(794\) 0 0
\(795\) 2.00000 4.00000i 0.0709327 0.141865i
\(796\) 0 0
\(797\) 24.0000i 0.850124i 0.905164 + 0.425062i \(0.139748\pi\)
−0.905164 + 0.425062i \(0.860252\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 4.00000i 0.141157i
\(804\) 0 0
\(805\) 8.00000 16.0000i 0.281963 0.563926i
\(806\) 0 0
\(807\) 18.0000i 0.633630i
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −18.0000 −0.632065 −0.316033 0.948748i \(-0.602351\pi\)
−0.316033 + 0.948748i \(0.602351\pi\)
\(812\) 0 0
\(813\) 14.0000i 0.491001i
\(814\) 0 0
\(815\) −16.0000 8.00000i −0.560456 0.280228i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 20.0000i 0.697156i −0.937280 0.348578i \(-0.886665\pi\)
0.937280 0.348578i \(-0.113335\pi\)
\(824\) 0 0
\(825\) 8.00000 6.00000i 0.278524 0.208893i
\(826\) 0 0
\(827\) 36.0000i 1.25184i −0.779886 0.625921i \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) 0 0
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) −24.0000 −0.832551
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 6.00000i 0.207390i
\(838\) 0 0
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 6.00000i 0.206651i
\(844\) 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
\(851\) −64.0000 −2.19389
\(852\) 0 0
\(853\) 22.0000i 0.753266i −0.926363 0.376633i \(-0.877082\pi\)
0.926363 0.376633i \(-0.122918\pi\)
\(854\) 0 0
\(855\) 2.00000 4.00000i 0.0683986 0.136797i
\(856\) 0 0
\(857\) 4.00000i 0.136637i 0.997664 + 0.0683187i \(0.0217635\pi\)
−0.997664 + 0.0683187i \(0.978237\pi\)
\(858\) 0 0
\(859\) −26.0000 −0.887109 −0.443554 0.896248i \(-0.646283\pi\)
−0.443554 + 0.896248i \(0.646283\pi\)
\(860\) 0 0
\(861\) 10.0000 0.340799
\(862\) 0 0
\(863\) 24.0000i 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 0 0
\(865\) −48.0000 24.0000i −1.63205 0.816024i
\(866\) 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
\(877\) 32.0000i 1.08056i 0.841484 + 0.540282i \(0.181682\pi\)
−0.841484 + 0.540282i \(0.818318\pi\)
\(878\) 0 0
\(879\) 8.00000 0.269833
\(880\) 0 0
\(881\) 10.0000 0.336909 0.168454 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) 0 0
\(883\) 44.0000i 1.48072i 0.672212 + 0.740359i \(0.265344\pi\)
−0.672212 + 0.740359i \(0.734656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 36.0000i 1.20876i 0.796696 + 0.604381i \(0.206579\pi\)
−0.796696 + 0.604381i \(0.793421\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 0 0
\(893\) 24.0000i 0.803129i
\(894\) 0 0
\(895\) −2.00000 + 4.00000i −0.0668526 + 0.133705i
\(896\) 0 0
\(897\) 16.0000i 0.534224i
\(898\) 0 0
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.00000 4.00000i 0.0664822 0.132964i
\(906\) 0 0
\(907\) 20.0000i 0.664089i 0.943264 + 0.332045i \(0.107738\pi\)
−0.943264 + 0.332045i \(0.892262\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −42.0000 −1.39152 −0.695761 0.718273i \(-0.744933\pi\)
−0.695761 + 0.718273i \(0.744933\pi\)
\(912\) 0 0
\(913\) 32.0000i 1.05905i
\(914\) 0 0
\(915\) 4.00000 + 2.00000i 0.132236 + 0.0661180i
\(916\) 0 0
\(917\) 12.0000i 0.396275i
\(918\) 0 0
\(919\) −36.0000 −1.18753 −0.593765 0.804638i \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) 28.0000i 0.921631i
\(924\) 0 0
\(925\) −32.0000 + 24.0000i −1.05215 + 0.789115i
\(926\) 0 0
\(927\) 16.0000i 0.525509i
\(928\) 0 0
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 0 0
\(933\) 24.0000i 0.785725i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000i 0.0653372i 0.999466 + 0.0326686i \(0.0104006\pi\)
−0.999466 + 0.0326686i \(0.989599\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) 26.0000 0.847576 0.423788 0.905761i \(-0.360700\pi\)
0.423788 + 0.905761i \(0.360700\pi\)
\(942\) 0 0
\(943\) 80.0000i 2.60516i
\(944\) 0 0
\(945\) −1.00000 + 2.00000i −0.0325300 + 0.0650600i
\(946\) 0 0
\(947\) 36.0000i 1.16984i −0.811090 0.584921i \(-0.801125\pi\)
0.811090 0.584921i \(-0.198875\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 42.0000i 1.36051i −0.732974 0.680257i \(-0.761868\pi\)
0.732974 0.680257i \(-0.238132\pi\)
\(954\) 0 0
\(955\) 18.0000 36.0000i 0.582466 1.16493i
\(956\) 0 0
\(957\) 4.00000i 0.129302i
\(958\) 0 0
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 4.00000i 0.128898i
\(964\) 0 0
\(965\) −32.0000 16.0000i −1.03012 0.515058i
\(966\) 0 0
\(967\) 56.0000i 1.80084i −0.435023 0.900419i \(-0.643260\pi\)
0.435023 0.900419i \(-0.356740\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 48.0000 1.54039 0.770197 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 0 0
\(973\) 14.0000i 0.448819i
\(974\) 0 0
\(975\) −6.00000 8.00000i −0.192154 0.256205i
\(976\) 0 0
\(977\) 26.0000i 0.831814i 0.909407 + 0.415907i \(0.136536\pi\)
−0.909407 + 0.415907i \(0.863464\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 0 0
\(983\) 4.00000i 0.127580i 0.997963 + 0.0637901i \(0.0203188\pi\)
−0.997963 + 0.0637901i \(0.979681\pi\)
\(984\) 0 0
\(985\) −12.0000 6.00000i −0.382352 0.191176i
\(986\) 0 0
\(987\) 12.0000i 0.381964i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) 0 0
\(993\) 32.0000i 1.01549i
\(994\) 0 0
\(995\) −10.0000 + 20.0000i −0.317021 + 0.634043i
\(996\) 0 0
\(997\) 42.0000i 1.33015i −0.746775 0.665077i \(-0.768399\pi\)
0.746775 0.665077i \(-0.231601\pi\)
\(998\) 0 0
\(999\) 8.00000 0.253109
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.c.1009.1 2
3.2 odd 2 5040.2.t.n.1009.1 2
4.3 odd 2 840.2.t.a.169.2 yes 2
5.2 odd 4 8400.2.a.u.1.1 1
5.3 odd 4 8400.2.a.br.1.1 1
5.4 even 2 inner 1680.2.t.c.1009.2 2
12.11 even 2 2520.2.t.c.1009.1 2
15.14 odd 2 5040.2.t.n.1009.2 2
20.3 even 4 4200.2.a.k.1.1 1
20.7 even 4 4200.2.a.x.1.1 1
20.19 odd 2 840.2.t.a.169.1 2
60.59 even 2 2520.2.t.c.1009.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.a.169.1 2 20.19 odd 2
840.2.t.a.169.2 yes 2 4.3 odd 2
1680.2.t.c.1009.1 2 1.1 even 1 trivial
1680.2.t.c.1009.2 2 5.4 even 2 inner
2520.2.t.c.1009.1 2 12.11 even 2
2520.2.t.c.1009.2 2 60.59 even 2
4200.2.a.k.1.1 1 20.3 even 4
4200.2.a.x.1.1 1 20.7 even 4
5040.2.t.n.1009.1 2 3.2 odd 2
5040.2.t.n.1009.2 2 15.14 odd 2
8400.2.a.u.1.1 1 5.2 odd 4
8400.2.a.br.1.1 1 5.3 odd 4