Properties

Label 1260.2.k.c.1009.2
Level $1260$
Weight $2$
Character 1260.1009
Analytic conductor $10.061$
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] = [1260,2,Mod(1009,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, 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("1260.1009");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.k (of order \(2\), degree \(1\), 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: \(10.0611506547\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1260.1009
Dual form 1260.2.k.c.1009.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.00000 + 1.00000i) q^{5} +1.00000i q^{7} +O(q^{10})\) \(q+(2.00000 + 1.00000i) q^{5} +1.00000i q^{7} -3.00000 q^{11} -1.00000i q^{13} +5.00000i q^{17} +8.00000 q^{19} +2.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -1.00000 q^{29} -2.00000 q^{31} +(-1.00000 + 2.00000i) q^{35} +10.0000i q^{37} +6.00000 q^{41} +4.00000i q^{43} -11.0000i q^{47} -1.00000 q^{49} +6.00000i q^{53} +(-6.00000 - 3.00000i) q^{55} -10.0000 q^{59} +(1.00000 - 2.00000i) q^{65} -10.0000i q^{67} +10.0000i q^{73} -3.00000i q^{77} +7.00000 q^{79} +12.0000i q^{83} +(-5.00000 + 10.0000i) q^{85} +8.00000 q^{89} +1.00000 q^{91} +(16.0000 + 8.00000i) q^{95} +3.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} - 6 q^{11} + 16 q^{19} + 6 q^{25} - 2 q^{29} - 4 q^{31} - 2 q^{35} + 12 q^{41} - 2 q^{49} - 12 q^{55} - 20 q^{59} + 2 q^{65} + 14 q^{79} - 10 q^{85} + 16 q^{89} + 2 q^{91} + 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.00000 + 1.00000i 0.894427 + 0.447214i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.00000i 1.21268i 0.795206 + 0.606339i \(0.207363\pi\)
−0.795206 + 0.606339i \(0.792637\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 + 2.00000i −0.169031 + 0.338062i
\(36\) 0 0
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.0000i 1.60451i −0.596978 0.802257i \(-0.703632\pi\)
0.596978 0.802257i \(-0.296368\pi\)
\(48\) 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
\(55\) −6.00000 3.00000i −0.809040 0.404520i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.00000 2.00000i 0.124035 0.248069i
\(66\) 0 0
\(67\) 10.0000i 1.22169i −0.791748 0.610847i \(-0.790829\pi\)
0.791748 0.610847i \(-0.209171\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000i 0.341882i
\(78\) 0 0
\(79\) 7.00000 0.787562 0.393781 0.919204i \(-0.371167\pi\)
0.393781 + 0.919204i \(0.371167\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) −5.00000 + 10.0000i −0.542326 + 1.08465i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 16.0000 + 8.00000i 1.64157 + 0.820783i
\(96\) 0 0
\(97\) 3.00000i 0.304604i 0.988334 + 0.152302i \(0.0486686\pi\)
−0.988334 + 0.152302i \(0.951331\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) 5.00000i 0.492665i 0.969185 + 0.246332i \(0.0792255\pi\)
−0.969185 + 0.246332i \(0.920775\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000i 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 0 0
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 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
\(115\) −2.00000 + 4.00000i −0.186501 + 0.373002i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.00000 −0.458349
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.00000 + 11.0000i 0.178885 + 0.983870i
\(126\) 0 0
\(127\) 2.00000i 0.177471i −0.996055 0.0887357i \(-0.971717\pi\)
0.996055 0.0887357i \(-0.0282826\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.00000 −0.174741 −0.0873704 0.996176i \(-0.527846\pi\)
−0.0873704 + 0.996176i \(0.527846\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.00000i 0.341743i −0.985293 0.170872i \(-0.945342\pi\)
0.985293 0.170872i \(-0.0546583\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.00000i 0.250873i
\(144\) 0 0
\(145\) −2.00000 1.00000i −0.166091 0.0830455i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 9.00000 0.732410 0.366205 0.930534i \(-0.380657\pi\)
0.366205 + 0.930534i \(0.380657\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 2.00000i −0.321288 0.160644i
\(156\) 0 0
\(157\) 18.0000i 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 6.00000i 0.469956i −0.972001 0.234978i \(-0.924498\pi\)
0.972001 0.234978i \(-0.0755019\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.00000i 0.232147i −0.993241 0.116073i \(-0.962969\pi\)
0.993241 0.116073i \(-0.0370308\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.00000i 0.684257i −0.939653 0.342129i \(-0.888852\pi\)
0.939653 0.342129i \(-0.111148\pi\)
\(174\) 0 0
\(175\) −4.00000 + 3.00000i −0.302372 + 0.226779i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −10.0000 + 20.0000i −0.735215 + 1.47043i
\(186\) 0 0
\(187\) 15.0000i 1.09691i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −7.00000 −0.506502 −0.253251 0.967401i \(-0.581500\pi\)
−0.253251 + 0.967401i \(0.581500\pi\)
\(192\) 0 0
\(193\) 8.00000i 0.575853i −0.957653 0.287926i \(-0.907034\pi\)
0.957653 0.287926i \(-0.0929658\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.0000i 0.712470i −0.934396 0.356235i \(-0.884060\pi\)
0.934396 0.356235i \(-0.115940\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.00000i 0.0701862i
\(204\) 0 0
\(205\) 12.0000 + 6.00000i 0.838116 + 0.419058i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.00000 + 8.00000i −0.272798 + 0.545595i
\(216\) 0 0
\(217\) 2.00000i 0.135769i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.00000 0.336336
\(222\) 0 0
\(223\) 19.0000i 1.27233i −0.771551 0.636167i \(-0.780519\pi\)
0.771551 0.636167i \(-0.219481\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 27.0000i 1.79205i −0.444001 0.896026i \(-0.646441\pi\)
0.444001 0.896026i \(-0.353559\pi\)
\(228\) 0 0
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0000i 1.04819i 0.851658 + 0.524097i \(0.175597\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 11.0000 22.0000i 0.717561 1.43512i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.00000 0.323423 0.161712 0.986838i \(-0.448299\pi\)
0.161712 + 0.986838i \(0.448299\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.00000 1.00000i −0.127775 0.0638877i
\(246\) 0 0
\(247\) 8.00000i 0.509028i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) 6.00000i 0.377217i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.00000i 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 24.0000i 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) 0 0
\(265\) −6.00000 + 12.0000i −0.368577 + 0.737154i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −9.00000 12.0000i −0.542720 0.723627i
\(276\) 0 0
\(277\) 14.0000i 0.841178i −0.907251 0.420589i \(-0.861823\pi\)
0.907251 0.420589i \(-0.138177\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 0 0
\(283\) 7.00000i 0.416107i 0.978117 + 0.208053i \(0.0667128\pi\)
−0.978117 + 0.208053i \(0.933287\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000i 0.354169i
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 15.0000i 0.876309i 0.898900 + 0.438155i \(0.144368\pi\)
−0.898900 + 0.438155i \(0.855632\pi\)
\(294\) 0 0
\(295\) −20.0000 10.0000i −1.16445 0.582223i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 19.0000i 1.08439i −0.840254 0.542194i \(-0.817594\pi\)
0.840254 0.542194i \(-0.182406\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 7.00000i 0.395663i 0.980236 + 0.197832i \(0.0633900\pi\)
−0.980236 + 0.197832i \(0.936610\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 28.0000i 1.57264i −0.617822 0.786318i \(-0.711985\pi\)
0.617822 0.786318i \(-0.288015\pi\)
\(318\) 0 0
\(319\) 3.00000 0.167968
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 40.0000i 2.22566i
\(324\) 0 0
\(325\) 4.00000 3.00000i 0.221880 0.166410i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.0000 0.606450
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 10.0000 20.0000i 0.546358 1.09272i
\(336\) 0 0
\(337\) 22.0000i 1.19842i 0.800593 + 0.599208i \(0.204518\pi\)
−0.800593 + 0.599208i \(0.795482\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.0000i 0.966291i −0.875540 0.483145i \(-0.839494\pi\)
0.875540 0.483145i \(-0.160506\pi\)
\(348\) 0 0
\(349\) −36.0000 −1.92704 −0.963518 0.267644i \(-0.913755\pi\)
−0.963518 + 0.267644i \(0.913755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.00000i 0.479022i −0.970894 0.239511i \(-0.923013\pi\)
0.970894 0.239511i \(-0.0769871\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 28.0000 1.47778 0.738892 0.673824i \(-0.235349\pi\)
0.738892 + 0.673824i \(0.235349\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −10.0000 + 20.0000i −0.523424 + 1.04685i
\(366\) 0 0
\(367\) 19.0000i 0.991792i 0.868382 + 0.495896i \(0.165160\pi\)
−0.868382 + 0.495896i \(0.834840\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 32.0000i 1.65690i −0.560065 0.828449i \(-0.689224\pi\)
0.560065 0.828449i \(-0.310776\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.00000i 0.0515026i
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 3.00000 6.00000i 0.152894 0.305788i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 0 0
\(391\) −10.0000 −0.505722
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 14.0000 + 7.00000i 0.704416 + 0.352208i
\(396\) 0 0
\(397\) 17.0000i 0.853206i −0.904439 0.426603i \(-0.859710\pi\)
0.904439 0.426603i \(-0.140290\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 27.0000 1.34832 0.674158 0.738587i \(-0.264507\pi\)
0.674158 + 0.738587i \(0.264507\pi\)
\(402\) 0 0
\(403\) 2.00000i 0.0996271i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 30.0000i 1.48704i
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.0000i 0.492068i
\(414\) 0 0
\(415\) −12.0000 + 24.0000i −0.589057 + 1.17811i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.00000 0.0977064 0.0488532 0.998806i \(-0.484443\pi\)
0.0488532 + 0.998806i \(0.484443\pi\)
\(420\) 0 0
\(421\) −23.0000 −1.12095 −0.560476 0.828171i \(-0.689382\pi\)
−0.560476 + 0.828171i \(0.689382\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −20.0000 + 15.0000i −0.970143 + 0.727607i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 37.0000 1.78223 0.891114 0.453780i \(-0.149925\pi\)
0.891114 + 0.453780i \(0.149925\pi\)
\(432\) 0 0
\(433\) 38.0000i 1.82616i 0.407777 + 0.913082i \(0.366304\pi\)
−0.407777 + 0.913082i \(0.633696\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 16.0000i 0.765384i
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) 0 0
\(445\) 16.0000 + 8.00000i 0.758473 + 0.379236i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 11.0000 0.519122 0.259561 0.965727i \(-0.416422\pi\)
0.259561 + 0.965727i \(0.416422\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.00000 + 1.00000i 0.0937614 + 0.0468807i
\(456\) 0 0
\(457\) 22.0000i 1.02912i −0.857455 0.514558i \(-0.827956\pi\)
0.857455 0.514558i \(-0.172044\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i −0.995671 0.0929479i \(-0.970371\pi\)
0.995671 0.0929479i \(-0.0296290\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.0000i 1.06431i 0.846646 + 0.532157i \(0.178618\pi\)
−0.846646 + 0.532157i \(0.821382\pi\)
\(468\) 0 0
\(469\) 10.0000 0.461757
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12.0000i 0.551761i
\(474\) 0 0
\(475\) 24.0000 + 32.0000i 1.10120 + 1.46826i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.00000 + 6.00000i −0.136223 + 0.272446i
\(486\) 0 0
\(487\) 26.0000i 1.17817i −0.808070 0.589086i \(-0.799488\pi\)
0.808070 0.589086i \(-0.200512\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −33.0000 −1.48927 −0.744635 0.667472i \(-0.767376\pi\)
−0.744635 + 0.667472i \(0.767376\pi\)
\(492\) 0 0
\(493\) 5.00000i 0.225189i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 29.0000 1.29822 0.649109 0.760695i \(-0.275142\pi\)
0.649109 + 0.760695i \(0.275142\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.00000i 0.0445878i 0.999751 + 0.0222939i \(0.00709696\pi\)
−0.999751 + 0.0222939i \(0.992903\pi\)
\(504\) 0 0
\(505\) 24.0000 + 12.0000i 1.06799 + 0.533993i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.00000 + 10.0000i −0.220326 + 0.440653i
\(516\) 0 0
\(517\) 33.0000i 1.45134i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.0000i 0.435607i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.00000i 0.259889i
\(534\) 0 0
\(535\) 8.00000 16.0000i 0.345870 0.691740i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 14.0000 + 7.00000i 0.599694 + 0.299847i
\(546\) 0 0
\(547\) 8.00000i 0.342055i −0.985266 0.171028i \(-0.945291\pi\)
0.985266 0.171028i \(-0.0547087\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 7.00000i 0.297670i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20.0000i 0.847427i −0.905796 0.423714i \(-0.860726\pi\)
0.905796 0.423714i \(-0.139274\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 16.0000i 0.674320i −0.941447 0.337160i \(-0.890534\pi\)
0.941447 0.337160i \(-0.109466\pi\)
\(564\) 0 0
\(565\) 10.0000 20.0000i 0.420703 0.841406i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.00000 + 6.00000i −0.333623 + 0.250217i
\(576\) 0 0
\(577\) 17.0000i 0.707719i 0.935299 + 0.353860i \(0.115131\pi\)
−0.935299 + 0.353860i \(0.884869\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 18.0000i 0.745484i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.0000i 1.15568i −0.816149 0.577842i \(-0.803895\pi\)
0.816149 0.577842i \(-0.196105\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.00000i 0.123195i −0.998101 0.0615976i \(-0.980380\pi\)
0.998101 0.0615976i \(-0.0196196\pi\)
\(594\) 0 0
\(595\) −10.0000 5.00000i −0.409960 0.204980i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −21.0000 −0.858037 −0.429018 0.903296i \(-0.641140\pi\)
−0.429018 + 0.903296i \(0.641140\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.00000 2.00000i −0.162623 0.0813116i
\(606\) 0 0
\(607\) 5.00000i 0.202944i 0.994838 + 0.101472i \(0.0323552\pi\)
−0.994838 + 0.101472i \(0.967645\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −11.0000 −0.445012
\(612\) 0 0
\(613\) 12.0000i 0.484675i 0.970192 + 0.242338i \(0.0779142\pi\)
−0.970192 + 0.242338i \(0.922086\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 34.0000i 1.36879i 0.729112 + 0.684394i \(0.239933\pi\)
−0.729112 + 0.684394i \(0.760067\pi\)
\(618\) 0 0
\(619\) 2.00000 0.0803868 0.0401934 0.999192i \(-0.487203\pi\)
0.0401934 + 0.999192i \(0.487203\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.00000i 0.320513i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −50.0000 −1.99363
\(630\) 0 0
\(631\) 15.0000 0.597141 0.298570 0.954388i \(-0.403490\pi\)
0.298570 + 0.954388i \(0.403490\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.00000 4.00000i 0.0793676 0.158735i
\(636\) 0 0
\(637\) 1.00000i 0.0396214i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) 5.00000i 0.197181i 0.995128 + 0.0985904i \(0.0314334\pi\)
−0.995128 + 0.0985904i \(0.968567\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 36.0000i 1.40879i 0.709809 + 0.704394i \(0.248781\pi\)
−0.709809 + 0.704394i \(0.751219\pi\)
\(654\) 0 0
\(655\) −4.00000 2.00000i −0.156293 0.0781465i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 39.0000 1.51922 0.759612 0.650376i \(-0.225389\pi\)
0.759612 + 0.650376i \(0.225389\pi\)
\(660\) 0 0
\(661\) 28.0000 1.08907 0.544537 0.838737i \(-0.316705\pi\)
0.544537 + 0.838737i \(0.316705\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.00000 + 16.0000i −0.310227 + 0.620453i
\(666\) 0 0
\(667\) 2.00000i 0.0774403i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 16.0000i 0.616755i 0.951264 + 0.308377i \(0.0997859\pi\)
−0.951264 + 0.308377i \(0.900214\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11.0000i 0.422764i 0.977403 + 0.211382i \(0.0677965\pi\)
−0.977403 + 0.211382i \(0.932204\pi\)
\(678\) 0 0
\(679\) −3.00000 −0.115129
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 40.0000i 1.53056i 0.643699 + 0.765279i \(0.277399\pi\)
−0.643699 + 0.765279i \(0.722601\pi\)
\(684\) 0 0
\(685\) 4.00000 8.00000i 0.152832 0.305664i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) 40.0000 1.52167 0.760836 0.648944i \(-0.224789\pi\)
0.760836 + 0.648944i \(0.224789\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −20.0000 10.0000i −0.758643 0.379322i
\(696\) 0 0
\(697\) 30.0000i 1.13633i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 25.0000 0.944237 0.472118 0.881535i \(-0.343489\pi\)
0.472118 + 0.881535i \(0.343489\pi\)
\(702\) 0 0
\(703\) 80.0000i 3.01726i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.0000i 0.451306i
\(708\) 0 0
\(709\) −15.0000 −0.563337 −0.281668 0.959512i \(-0.590888\pi\)
−0.281668 + 0.959512i \(0.590888\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.00000i 0.149801i
\(714\) 0 0
\(715\) −3.00000 + 6.00000i −0.112194 + 0.224387i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2.00000 0.0745874 0.0372937 0.999304i \(-0.488126\pi\)
0.0372937 + 0.999304i \(0.488126\pi\)
\(720\) 0 0
\(721\) −5.00000 −0.186210
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.00000 4.00000i −0.111417 0.148556i
\(726\) 0 0
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) 0 0
\(733\) 41.0000i 1.51437i −0.653201 0.757185i \(-0.726574\pi\)
0.653201 0.757185i \(-0.273426\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 30.0000i 1.10506i
\(738\) 0 0
\(739\) 5.00000 0.183928 0.0919640 0.995762i \(-0.470686\pi\)
0.0919640 + 0.995762i \(0.470686\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 30.0000i 1.10059i 0.834969 + 0.550297i \(0.185485\pi\)
−0.834969 + 0.550297i \(0.814515\pi\)
\(744\) 0 0
\(745\) −12.0000 6.00000i −0.439646 0.219823i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) 13.0000 0.474377 0.237188 0.971464i \(-0.423774\pi\)
0.237188 + 0.971464i \(0.423774\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 18.0000 + 9.00000i 0.655087 + 0.327544i
\(756\) 0 0
\(757\) 48.0000i 1.74459i −0.488980 0.872295i \(-0.662631\pi\)
0.488980 0.872295i \(-0.337369\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 38.0000 1.37750 0.688749 0.724999i \(-0.258160\pi\)
0.688749 + 0.724999i \(0.258160\pi\)
\(762\) 0 0
\(763\) 7.00000i 0.253417i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10.0000i 0.361079i
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 27.0000i 0.971123i −0.874203 0.485561i \(-0.838615\pi\)
0.874203 0.485561i \(-0.161385\pi\)
\(774\) 0 0
\(775\) −6.00000 8.00000i −0.215526 0.287368i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 18.0000 36.0000i 0.642448 1.28490i
\(786\) 0 0
\(787\) 3.00000i 0.106938i 0.998569 + 0.0534692i \(0.0170279\pi\)
−0.998569 + 0.0534692i \(0.982972\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 10.0000 0.355559
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 43.0000i 1.52314i −0.648084 0.761569i \(-0.724429\pi\)
0.648084 0.761569i \(-0.275571\pi\)
\(798\) 0 0
\(799\) 55.0000 1.94576
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 30.0000i 1.05868i
\(804\) 0 0
\(805\) −4.00000 2.00000i −0.140981 0.0704907i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −11.0000 −0.386739 −0.193370 0.981126i \(-0.561942\pi\)
−0.193370 + 0.981126i \(0.561942\pi\)
\(810\) 0 0
\(811\) 6.00000 0.210688 0.105344 0.994436i \(-0.466406\pi\)
0.105344 + 0.994436i \(0.466406\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6.00000 12.0000i 0.210171 0.420342i
\(816\) 0 0
\(817\) 32.0000i 1.11954i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −45.0000 −1.57051 −0.785255 0.619172i \(-0.787468\pi\)
−0.785255 + 0.619172i \(0.787468\pi\)
\(822\) 0 0
\(823\) 18.0000i 0.627441i 0.949515 + 0.313720i \(0.101575\pi\)
−0.949515 + 0.313720i \(0.898425\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 22.0000i 0.765015i 0.923952 + 0.382507i \(0.124939\pi\)
−0.923952 + 0.382507i \(0.875061\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.00000i 0.173240i
\(834\) 0 0
\(835\) 3.00000 6.00000i 0.103819 0.207639i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −46.0000 −1.58810 −0.794048 0.607855i \(-0.792030\pi\)
−0.794048 + 0.607855i \(0.792030\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 24.0000 + 12.0000i 0.825625 + 0.412813i
\(846\) 0 0
\(847\) 2.00000i 0.0687208i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −20.0000 −0.685591
\(852\) 0 0
\(853\) 26.0000i 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 22.0000i 0.751506i 0.926720 + 0.375753i \(0.122616\pi\)
−0.926720 + 0.375753i \(0.877384\pi\)
\(858\) 0 0
\(859\) −24.0000 −0.818869 −0.409435 0.912339i \(-0.634274\pi\)
−0.409435 + 0.912339i \(0.634274\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 46.0000i 1.56586i 0.622111 + 0.782929i \(0.286275\pi\)
−0.622111 + 0.782929i \(0.713725\pi\)
\(864\) 0 0
\(865\) 9.00000 18.0000i 0.306009 0.612018i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −21.0000 −0.712376
\(870\) 0 0
\(871\) −10.0000 −0.338837
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −11.0000 + 2.00000i −0.371868 + 0.0676123i
\(876\) 0 0
\(877\) 40.0000i 1.35070i 0.737496 + 0.675352i \(0.236008\pi\)
−0.737496 + 0.675352i \(0.763992\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 48.0000 1.61716 0.808581 0.588386i \(-0.200236\pi\)
0.808581 + 0.588386i \(0.200236\pi\)
\(882\) 0 0
\(883\) 4.00000i 0.134611i −0.997732 0.0673054i \(-0.978560\pi\)
0.997732 0.0673054i \(-0.0214402\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.0000i 0.402921i −0.979497 0.201460i \(-0.935431\pi\)
0.979497 0.201460i \(-0.0645687\pi\)
\(888\) 0 0
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 88.0000i 2.94481i
\(894\) 0 0
\(895\) 8.00000 + 4.00000i 0.267411 + 0.133705i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.00000 0.0667037
\(900\) 0 0
\(901\) −30.0000 −0.999445
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −20.0000 10.0000i −0.664822 0.332411i
\(906\) 0 0
\(907\) 38.0000i 1.26177i 0.775877 + 0.630885i \(0.217308\pi\)
−0.775877 + 0.630885i \(0.782692\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 52.0000 1.72284 0.861418 0.507896i \(-0.169577\pi\)
0.861418 + 0.507896i \(0.169577\pi\)
\(912\) 0 0
\(913\) 36.0000i 1.19143i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.00000i 0.0660458i
\(918\) 0 0
\(919\) 55.0000 1.81428 0.907141 0.420826i \(-0.138260\pi\)
0.907141 + 0.420826i \(0.138260\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −40.0000 + 30.0000i −1.31519 + 0.986394i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −8.00000 −0.262189
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 15.0000 30.0000i 0.490552 0.981105i
\(936\) 0 0
\(937\) 7.00000i 0.228680i 0.993442 + 0.114340i \(0.0364753\pi\)
−0.993442 + 0.114340i \(0.963525\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 40.0000 1.30396 0.651981 0.758235i \(-0.273938\pi\)
0.651981 + 0.758235i \(0.273938\pi\)
\(942\) 0 0
\(943\) 12.0000i 0.390774i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.00000i 0.194974i −0.995237 0.0974869i \(-0.968920\pi\)
0.995237 0.0974869i \(-0.0310804\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 42.0000i 1.36051i −0.732974 0.680257i \(-0.761868\pi\)
0.732974 0.680257i \(-0.238132\pi\)
\(954\) 0 0
\(955\) −14.0000 7.00000i −0.453029 0.226515i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 8.00000 16.0000i 0.257529 0.515058i
\(966\) 0 0
\(967\) 8.00000i 0.257263i 0.991692 + 0.128631i \(0.0410584\pi\)
−0.991692 + 0.128631i \(0.958942\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −18.0000 −0.577647 −0.288824 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) 0 0
\(973\) 10.0000i 0.320585i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 60.0000i 1.91957i 0.280736 + 0.959785i \(0.409421\pi\)
−0.280736 + 0.959785i \(0.590579\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 13.0000i 0.414636i 0.978274 + 0.207318i \(0.0664734\pi\)
−0.978274 + 0.207318i \(0.933527\pi\)
\(984\) 0 0
\(985\) 10.0000 20.0000i 0.318626 0.637253i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −36.0000 18.0000i −1.14128 0.570638i
\(996\) 0 0
\(997\) 53.0000i 1.67853i −0.543725 0.839263i \(-0.682987\pi\)
0.543725 0.839263i \(-0.317013\pi\)
\(998\) 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 1260.2.k.c.1009.2 2
3.2 odd 2 140.2.e.a.29.1 2
4.3 odd 2 5040.2.t.s.1009.2 2
5.2 odd 4 6300.2.a.c.1.1 1
5.3 odd 4 6300.2.a.t.1.1 1
5.4 even 2 inner 1260.2.k.c.1009.1 2
12.11 even 2 560.2.g.a.449.2 2
15.2 even 4 700.2.a.a.1.1 1
15.8 even 4 700.2.a.j.1.1 1
15.14 odd 2 140.2.e.a.29.2 yes 2
20.19 odd 2 5040.2.t.s.1009.1 2
21.2 odd 6 980.2.q.f.949.2 4
21.5 even 6 980.2.q.c.949.1 4
21.11 odd 6 980.2.q.f.569.1 4
21.17 even 6 980.2.q.c.569.2 4
21.20 even 2 980.2.e.b.589.2 2
24.5 odd 2 2240.2.g.e.449.2 2
24.11 even 2 2240.2.g.f.449.1 2
60.23 odd 4 2800.2.a.a.1.1 1
60.47 odd 4 2800.2.a.bf.1.1 1
60.59 even 2 560.2.g.a.449.1 2
105.44 odd 6 980.2.q.f.949.1 4
105.59 even 6 980.2.q.c.569.1 4
105.62 odd 4 4900.2.a.w.1.1 1
105.74 odd 6 980.2.q.f.569.2 4
105.83 odd 4 4900.2.a.b.1.1 1
105.89 even 6 980.2.q.c.949.2 4
105.104 even 2 980.2.e.b.589.1 2
120.29 odd 2 2240.2.g.e.449.1 2
120.59 even 2 2240.2.g.f.449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.e.a.29.1 2 3.2 odd 2
140.2.e.a.29.2 yes 2 15.14 odd 2
560.2.g.a.449.1 2 60.59 even 2
560.2.g.a.449.2 2 12.11 even 2
700.2.a.a.1.1 1 15.2 even 4
700.2.a.j.1.1 1 15.8 even 4
980.2.e.b.589.1 2 105.104 even 2
980.2.e.b.589.2 2 21.20 even 2
980.2.q.c.569.1 4 105.59 even 6
980.2.q.c.569.2 4 21.17 even 6
980.2.q.c.949.1 4 21.5 even 6
980.2.q.c.949.2 4 105.89 even 6
980.2.q.f.569.1 4 21.11 odd 6
980.2.q.f.569.2 4 105.74 odd 6
980.2.q.f.949.1 4 105.44 odd 6
980.2.q.f.949.2 4 21.2 odd 6
1260.2.k.c.1009.1 2 5.4 even 2 inner
1260.2.k.c.1009.2 2 1.1 even 1 trivial
2240.2.g.e.449.1 2 120.29 odd 2
2240.2.g.e.449.2 2 24.5 odd 2
2240.2.g.f.449.1 2 24.11 even 2
2240.2.g.f.449.2 2 120.59 even 2
2800.2.a.a.1.1 1 60.23 odd 4
2800.2.a.bf.1.1 1 60.47 odd 4
4900.2.a.b.1.1 1 105.83 odd 4
4900.2.a.w.1.1 1 105.62 odd 4
5040.2.t.s.1009.1 2 20.19 odd 2
5040.2.t.s.1009.2 2 4.3 odd 2
6300.2.a.c.1.1 1 5.2 odd 4
6300.2.a.t.1.1 1 5.3 odd 4