Properties

Label 2300.2.i.a.1057.1
Level $2300$
Weight $2$
Character 2300.1057
Analytic conductor $18.366$
Analytic rank $0$
Dimension $8$
CM discriminant -115
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2300,2,Mod(1057,2300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2300, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2300.1057");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2300.i (of order \(4\), degree \(2\), 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: \(18.3655924649\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.928445276160000.122
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 353x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 1057.1
Root \(3.06489 - 3.06489i\) of defining polynomial
Character \(\chi\) \(=\) 2300.1057
Dual form 2300.2.i.a.1793.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.06489 + 3.06489i) q^{7} +3.00000i q^{9} +O(q^{10})\) \(q+(-3.06489 + 3.06489i) q^{7} +3.00000i q^{9} +(2.41234 - 2.41234i) q^{17} +(-3.39116 - 3.39116i) q^{23} +8.78709i q^{29} +10.7871 q^{31} +(-8.54211 + 8.54211i) q^{37} -12.7871 q^{41} +(-6.78233 - 6.78233i) q^{43} -11.7871i q^{49} +(-3.71744 - 3.71744i) q^{53} -14.7871i q^{59} +(-9.19467 - 9.19467i) q^{63} +(7.88956 - 7.88956i) q^{67} -2.78709 q^{71} -9.00000 q^{81} +(1.75978 + 1.75978i) q^{83} +(-13.5647 + 13.5647i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{31} - 28 q^{41} + 52 q^{71} - 72 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(-1\)

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 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.06489 + 3.06489i −1.15842 + 1.15842i −0.173603 + 0.984816i \(0.555541\pi\)
−0.984816 + 0.173603i \(0.944459\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.41234 2.41234i 0.585078 0.585078i −0.351217 0.936294i \(-0.614232\pi\)
0.936294 + 0.351217i \(0.114232\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.39116 3.39116i −0.707107 0.707107i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.78709i 1.63172i 0.578249 + 0.815861i \(0.303736\pi\)
−0.578249 + 0.815861i \(0.696264\pi\)
\(30\) 0 0
\(31\) 10.7871 1.93742 0.968709 0.248199i \(-0.0798387\pi\)
0.968709 + 0.248199i \(0.0798387\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.54211 + 8.54211i −1.40431 + 1.40431i −0.618642 + 0.785673i \(0.712317\pi\)
−0.785673 + 0.618642i \(0.787683\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −12.7871 −1.99701 −0.998504 0.0546823i \(-0.982585\pi\)
−0.998504 + 0.0546823i \(0.982585\pi\)
\(42\) 0 0
\(43\) −6.78233 6.78233i −1.03430 1.03430i −0.999391 0.0349050i \(-0.988887\pi\)
−0.0349050 0.999391i \(-0.511113\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0 0
\(49\) 11.7871i 1.68387i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.71744 3.71744i −0.510630 0.510630i 0.404090 0.914719i \(-0.367588\pi\)
−0.914719 + 0.404090i \(0.867588\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.7871i 1.92511i −0.271078 0.962557i \(-0.587380\pi\)
0.271078 0.962557i \(-0.412620\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −9.19467 9.19467i −1.15842 1.15842i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.88956 7.88956i 0.963863 0.963863i −0.0355060 0.999369i \(-0.511304\pi\)
0.999369 + 0.0355060i \(0.0113043\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.78709 −0.330766 −0.165383 0.986229i \(-0.552886\pi\)
−0.165383 + 0.986229i \(0.552886\pi\)
\(72\) 0 0
\(73\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 1.75978 + 1.75978i 0.193161 + 0.193161i 0.797061 0.603899i \(-0.206387\pi\)
−0.603899 + 0.797061i \(0.706387\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −13.5647 + 13.5647i −1.37728 + 1.37728i −0.528101 + 0.849182i \(0.677096\pi\)
−0.849182 + 0.528101i \(0.822904\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.787088 0.0783182 0.0391591 0.999233i \(-0.487532\pi\)
0.0391591 + 0.999233i \(0.487532\pi\)
\(102\) 0 0
\(103\) −6.78233 6.78233i −0.668283 0.668283i 0.289036 0.957318i \(-0.406665\pi\)
−0.957318 + 0.289036i \(0.906665\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.0193 + 14.0193i −1.35530 + 1.35530i −0.475685 + 0.879616i \(0.657800\pi\)
−0.879616 + 0.475685i \(0.842200\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −14.6719 14.6719i −1.38022 1.38022i −0.844222 0.535993i \(-0.819937\pi\)
−0.535993 0.844222i \(-0.680063\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 14.7871i 1.35553i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.5647 + 13.5647i −1.15891 + 1.15891i −0.174196 + 0.984711i \(0.555733\pi\)
−0.984711 + 0.174196i \(0.944267\pi\)
\(138\) 0 0
\(139\) 1.21291i 0.102878i −0.998676 0.0514389i \(-0.983619\pi\)
0.998676 0.0514389i \(-0.0163808\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 7.23701 + 7.23701i 0.585078 + 0.585078i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 13.3668 13.3668i 1.06679 1.06679i 0.0691817 0.997604i \(-0.477961\pi\)
0.997604 0.0691817i \(-0.0220388\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 20.7871 1.63825
\(162\) 0 0
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 13.0000i 1.00000i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.0000i 1.19590i 0.801535 + 0.597948i \(0.204017\pi\)
−0.801535 + 0.597948i \(0.795983\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −26.9314 26.9314i −1.89022 1.89022i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 10.1735 10.1735i 0.707107 0.707107i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −22.7871 −1.56873 −0.784364 0.620301i \(-0.787010\pi\)
−0.784364 + 0.620301i \(0.787010\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −33.0612 + 33.0612i −2.24434 + 2.24434i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 20.3470 20.3470i 1.35048 1.35048i 0.465351 0.885126i \(-0.345928\pi\)
0.885126 0.465351i \(-0.154072\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.21291i 0.337195i 0.985685 + 0.168598i \(0.0539238\pi\)
−0.985685 + 0.168598i \(0.946076\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(258\) 0 0
\(259\) 52.3613i 3.25357i
\(260\) 0 0
\(261\) −26.3613 −1.63172
\(262\) 0 0
\(263\) 12.7142 + 12.7142i 0.783993 + 0.783993i 0.980502 0.196509i \(-0.0629604\pi\)
−0.196509 + 0.980502i \(0.562960\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 24.7871i 1.51130i −0.654978 0.755648i \(-0.727322\pi\)
0.654978 0.755648i \(-0.272678\pi\)
\(270\) 0 0
\(271\) 24.3613 1.47984 0.739921 0.672694i \(-0.234863\pi\)
0.739921 + 0.672694i \(0.234863\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(278\) 0 0
\(279\) 32.3613i 1.93742i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 10.4998 + 10.4998i 0.624147 + 0.624147i 0.946589 0.322442i \(-0.104504\pi\)
−0.322442 + 0.946589i \(0.604504\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 39.1910 39.1910i 2.31337 2.31337i
\(288\) 0 0
\(289\) 5.36126i 0.315368i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 15.9770 + 15.9770i 0.933386 + 0.933386i 0.997916 0.0645297i \(-0.0205547\pi\)
−0.0645297 + 0.997916i \(0.520555\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 41.5742 2.39630
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 0 0
\(313\) 5.02255 + 5.02255i 0.283891 + 0.283891i 0.834659 0.550768i \(-0.185665\pi\)
−0.550768 + 0.834659i \(0.685665\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −36.3613 −1.99860 −0.999298 0.0374662i \(-0.988071\pi\)
−0.999298 + 0.0374662i \(0.988071\pi\)
\(332\) 0 0
\(333\) −25.6263 25.6263i −1.40431 1.40431i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −13.5647 + 13.5647i −0.738914 + 0.738914i −0.972368 0.233454i \(-0.924997\pi\)
0.233454 + 0.972368i \(0.424997\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 14.6719 + 14.6719i 0.792208 + 0.792208i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(348\) 0 0
\(349\) 18.3613i 0.982856i −0.870918 0.491428i \(-0.836475\pi\)
0.870918 0.491428i \(-0.163525\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −24.9738 + 24.9738i −1.30362 + 1.30362i −0.377689 + 0.925933i \(0.623281\pi\)
−0.925933 + 0.377689i \(0.876719\pi\)
\(368\) 0 0
\(369\) 38.3613i 1.99701i
\(370\) 0 0
\(371\) 22.7871 1.18305
\(372\) 0 0
\(373\) 27.1293 + 27.1293i 1.40470 + 1.40470i 0.784220 + 0.620483i \(0.213063\pi\)
0.620483 + 0.784220i \(0.286937\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.454680 0.454680i −0.0232331 0.0232331i 0.695395 0.718628i \(-0.255230\pi\)
−0.718628 + 0.695395i \(0.755230\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 20.3470 20.3470i 1.03430 1.03430i
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −16.3613 −0.827425
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 28.7871i 1.42343i 0.702468 + 0.711715i \(0.252081\pi\)
−0.702468 + 0.711715i \(0.747919\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 45.3208 + 45.3208i 2.23009 + 2.23009i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 26.9314 + 26.9314i 1.29424 + 1.29424i 0.932137 + 0.362106i \(0.117942\pi\)
0.362106 + 0.932137i \(0.382058\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 36.0000i 1.71819i 0.511819 + 0.859093i \(0.328972\pi\)
−0.511819 + 0.859093i \(0.671028\pi\)
\(440\) 0 0
\(441\) 35.3613 1.68387
\(442\) 0 0
\(443\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 11.2129i 0.529170i −0.964362 0.264585i \(-0.914765\pi\)
0.964362 0.264585i \(-0.0852350\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 24.3212 24.3212i 1.13770 1.13770i 0.148838 0.988862i \(-0.452447\pi\)
0.988862 0.148838i \(-0.0475533\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16.6295 16.6295i 0.769524 0.769524i −0.208499 0.978023i \(-0.566858\pi\)
0.978023 + 0.208499i \(0.0668578\pi\)
\(468\) 0 0
\(469\) 48.3613i 2.23312i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 11.1523 11.1523i 0.510630 0.510630i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 30.7871 1.38940 0.694701 0.719299i \(-0.255537\pi\)
0.694701 + 0.719299i \(0.255537\pi\)
\(492\) 0 0
\(493\) 21.1974 + 21.1974i 0.954684 + 0.954684i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.54211 8.54211i 0.383166 0.383166i
\(498\) 0 0
\(499\) 12.3613i 0.553366i 0.960961 + 0.276683i \(0.0892352\pi\)
−0.960961 + 0.276683i \(0.910765\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −31.1036 31.1036i −1.38684 1.38684i −0.831872 0.554968i \(-0.812731\pi\)
−0.554968 0.831872i \(-0.687269\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14.0000i 0.620539i −0.950649 0.310270i \(-0.899581\pi\)
0.950649 0.310270i \(-0.100419\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −6.78233 6.78233i −0.296571 0.296571i 0.543098 0.839669i \(-0.317251\pi\)
−0.839669 + 0.543098i \(0.817251\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 26.0221 26.0221i 1.13354 1.13354i
\(528\) 0 0
\(529\) 23.0000i 1.00000i
\(530\) 0 0
\(531\) 44.3613 1.92511
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −30.4510 + 30.4510i −1.29025 + 1.29025i −0.355621 + 0.934630i \(0.615730\pi\)
−0.934630 + 0.355621i \(0.884270\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 32.4087 + 32.4087i 1.36586 + 1.36586i 0.866247 + 0.499615i \(0.166525\pi\)
0.499615 + 0.866247i \(0.333475\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 27.5840 27.5840i 1.15842 1.15842i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −10.7871 −0.447524
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 44.0000i 1.79779i −0.438163 0.898896i \(-0.644371\pi\)
0.438163 0.898896i \(-0.355629\pi\)
\(600\) 0 0
\(601\) −46.3613 −1.89112 −0.945558 0.325455i \(-0.894483\pi\)
−0.945558 + 0.325455i \(0.894483\pi\)
\(602\) 0 0
\(603\) 23.6687 + 23.6687i 0.963863 + 0.963863i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 27.1293 + 27.1293i 1.09574 + 1.09574i 0.994903 + 0.100840i \(0.0321531\pi\)
0.100840 + 0.994903i \(0.467847\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.197872 0.197872i 0.00796603 0.00796603i −0.703113 0.711079i \(-0.748207\pi\)
0.711079 + 0.703113i \(0.248207\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 41.2129i 1.64327i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 8.36126i 0.330766i
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −11.4091 11.4091i −0.449932 0.449932i 0.445400 0.895332i \(-0.353062\pi\)
−0.895332 + 0.445400i \(0.853062\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 29.7985 29.7985i 1.15380 1.15380i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 33.0612 33.0612i 1.27065 1.27065i 0.324897 0.945750i \(-0.394671\pi\)
0.945750 0.324897i \(-0.105329\pi\)
\(678\) 0 0
\(679\) 83.1484i 3.19094i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −48.0000 −1.82601 −0.913003 0.407953i \(-0.866243\pi\)
−0.913003 + 0.407953i \(0.866243\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −30.8468 + 30.8468i −1.16840 + 1.16840i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.41234 + 2.41234i −0.0907253 + 0.0907253i
\(708\) 0 0
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −36.5808 36.5808i −1.36996 1.36996i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 45.9354i 1.71310i 0.516062 + 0.856551i \(0.327398\pi\)
−0.516062 + 0.856551i \(0.672602\pi\)
\(720\) 0 0
\(721\) 41.5742 1.54830
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −11.8049 + 11.8049i −0.437819 + 0.437819i −0.891277 0.453459i \(-0.850190\pi\)
0.453459 + 0.891277i \(0.350190\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) −32.7225 −1.21029
\(732\) 0 0
\(733\) 37.8859 + 37.8859i 1.39935 + 1.39935i 0.801931 + 0.597416i \(0.203806\pi\)
0.597416 + 0.801931i \(0.296194\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 41.9354i 1.54262i −0.636460 0.771310i \(-0.719602\pi\)
0.636460 0.771310i \(-0.280398\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −6.78233 6.78233i −0.248820 0.248820i 0.571667 0.820486i \(-0.306297\pi\)
−0.820486 + 0.571667i \(0.806297\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −5.27935 + 5.27935i −0.193161 + 0.193161i
\(748\) 0 0
\(749\) 85.9354i 3.14001i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −28.2366 + 28.2366i −1.02627 + 1.02627i −0.0266296 + 0.999645i \(0.508477\pi\)
−0.999645 + 0.0266296i \(0.991523\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −39.9354 −1.44766 −0.723829 0.689979i \(-0.757620\pi\)
−0.723829 + 0.689979i \(0.757620\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 27.1293 + 27.1293i 0.975774 + 0.975774i 0.999713 0.0239396i \(-0.00762094\pi\)
−0.0239396 + 0.999713i \(0.507621\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −35.9282 + 35.9282i −1.28070 + 1.28070i −0.340436 + 0.940268i \(0.610575\pi\)
−0.940268 + 0.340436i \(0.889425\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 89.9354 3.19774
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6.32765 + 6.32765i −0.224137 + 0.224137i −0.810238 0.586101i \(-0.800662\pi\)
0.586101 + 0.810238i \(0.300662\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 55.9354i 1.96659i 0.182032 + 0.983293i \(0.441733\pi\)
−0.182032 + 0.983293i \(0.558267\pi\)
\(810\) 0 0
\(811\) 4.36126 0.153145 0.0765723 0.997064i \(-0.475602\pi\)
0.0765723 + 0.997064i \(0.475602\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) 0 0
\(823\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 38.5385 38.5385i 1.34011 1.34011i 0.444171 0.895942i \(-0.353498\pi\)
0.895942 0.444171i \(-0.146502\pi\)
\(828\) 0 0
\(829\) 35.9354i 1.24809i 0.781389 + 0.624045i \(0.214512\pi\)
−0.781389 + 0.624045i \(0.785488\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −28.4344 28.4344i −0.985194 0.985194i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −48.2129 −1.66251
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −33.7138 + 33.7138i −1.15842 + 1.15842i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 57.9354 1.98600
\(852\) 0 0
\(853\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(858\) 0 0
\(859\) 52.3613i 1.78654i 0.449517 + 0.893272i \(0.351596\pi\)
−0.449517 + 0.893272i \(0.648404\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −40.6940 40.6940i −1.37728 1.37728i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 94.7871i 3.16133i
\(900\) 0 0
\(901\) −17.9354 −0.597516
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −0.850424 + 0.850424i −0.0282379 + 0.0282379i −0.721085 0.692847i \(-0.756356\pi\)
0.692847 + 0.721085i \(0.256356\pi\)
\(908\) 0 0
\(909\) 2.36126i 0.0783182i
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 24.5191 24.5191i 0.809692 0.809692i
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 20.3470 20.3470i 0.668283 0.668283i
\(928\) 0 0
\(929\) 31.9354i 1.04777i −0.851790 0.523884i \(-0.824483\pi\)
0.851790 0.523884i \(-0.175517\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −13.5647 + 13.5647i −0.443138 + 0.443138i −0.893065 0.449927i \(-0.851450\pi\)
0.449927 + 0.893065i \(0.351450\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 43.3631 + 43.3631i 1.41210 + 1.41210i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −40.6940 40.6940i −1.31821 1.31821i −0.915194 0.403013i \(-0.867963\pi\)
−0.403013 0.915194i \(-0.632037\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 83.1484i 2.68500i
\(960\) 0 0
\(961\) 85.3613 2.75359
\(962\) 0 0
\(963\) −42.0580 42.0580i −1.35530 1.35530i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 3.71744 + 3.71744i 0.119176 + 0.119176i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −39.1910 + 39.1910i −1.25383 + 1.25383i −0.299843 + 0.953989i \(0.596934\pi\)
−0.953989 + 0.299843i \(0.903066\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −17.9347 17.9347i −0.572027 0.572027i 0.360668 0.932694i \(-0.382549\pi\)
−0.932694 + 0.360668i \(0.882549\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 46.0000i 1.46271i
\(990\) 0 0
\(991\) −29.9354 −0.950931 −0.475465 0.879734i \(-0.657720\pi\)
−0.475465 + 0.879734i \(0.657720\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(998\) 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 2300.2.i.a.1057.1 8
5.2 odd 4 inner 2300.2.i.a.1793.1 yes 8
5.3 odd 4 inner 2300.2.i.a.1793.4 yes 8
5.4 even 2 inner 2300.2.i.a.1057.4 yes 8
23.22 odd 2 inner 2300.2.i.a.1057.4 yes 8
115.22 even 4 inner 2300.2.i.a.1793.4 yes 8
115.68 even 4 inner 2300.2.i.a.1793.1 yes 8
115.114 odd 2 CM 2300.2.i.a.1057.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2300.2.i.a.1057.1 8 1.1 even 1 trivial
2300.2.i.a.1057.1 8 115.114 odd 2 CM
2300.2.i.a.1057.4 yes 8 5.4 even 2 inner
2300.2.i.a.1057.4 yes 8 23.22 odd 2 inner
2300.2.i.a.1793.1 yes 8 5.2 odd 4 inner
2300.2.i.a.1793.1 yes 8 115.68 even 4 inner
2300.2.i.a.1793.4 yes 8 5.3 odd 4 inner
2300.2.i.a.1793.4 yes 8 115.22 even 4 inner