Properties

Label 672.2.i.e.209.7
Level $672$
Weight $2$
Character 672.209
Analytic conductor $5.366$
Analytic rank $0$
Dimension $8$
CM discriminant -56
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,2,Mod(209,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.i (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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.10070523904.11
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 10x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 209.7
Root \(1.68014 - 0.420861i\) of defining polynomial
Character \(\chi\) \(=\) 672.209
Dual form 672.2.i.e.209.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.68014 - 0.420861i) q^{3} -3.91044i q^{5} +2.64575 q^{7} +(2.64575 - 1.41421i) q^{9} +O(q^{10})\) \(q+(1.68014 - 0.420861i) q^{3} -3.91044i q^{5} +2.64575 q^{7} +(2.64575 - 1.41421i) q^{9} -4.55066 q^{13} +(-1.64575 - 6.57008i) q^{15} +0.979531 q^{19} +(4.44524 - 1.11349i) q^{21} +7.48331i q^{23} -10.2915 q^{25} +(3.85005 - 3.48957i) q^{27} -10.3460i q^{35} +(-7.64575 + 1.91520i) q^{39} +(-5.53019 - 10.3460i) q^{45} +7.00000 q^{49} +(1.64575 - 0.412247i) q^{57} -5.29570i q^{59} +15.6110 q^{61} +(7.00000 - 3.74166i) q^{63} +17.7951i q^{65} +(3.14944 + 12.5730i) q^{69} +5.65685i q^{71} +(-17.2912 + 4.33130i) q^{75} +5.29150 q^{79} +(5.00000 - 7.48331i) q^{81} +18.1669i q^{83} -12.0399 q^{91} -3.83039i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{15} - 40 q^{25} - 40 q^{39} + 56 q^{49} - 8 q^{57} + 56 q^{63} + 40 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.68014 0.420861i 0.970030 0.242984i
\(4\) 0 0
\(5\) 3.91044i 1.74880i −0.485206 0.874400i \(-0.661255\pi\)
0.485206 0.874400i \(-0.338745\pi\)
\(6\) 0 0
\(7\) 2.64575 1.00000
\(8\) 0 0
\(9\) 2.64575 1.41421i 0.881917 0.471405i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −4.55066 −1.26213 −0.631063 0.775732i \(-0.717381\pi\)
−0.631063 + 0.775732i \(0.717381\pi\)
\(14\) 0 0
\(15\) −1.64575 6.57008i −0.424931 1.69639i
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0.979531 0.224720 0.112360 0.993668i \(-0.464159\pi\)
0.112360 + 0.993668i \(0.464159\pi\)
\(20\) 0 0
\(21\) 4.44524 1.11349i 0.970030 0.242984i
\(22\) 0 0
\(23\) 7.48331i 1.56038i 0.625543 + 0.780189i \(0.284877\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −10.2915 −2.05830
\(26\) 0 0
\(27\) 3.85005 3.48957i 0.740942 0.671569i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 10.3460i 1.74880i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −7.64575 + 1.91520i −1.22430 + 0.306677i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −5.53019 10.3460i −0.824392 1.54230i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.64575 0.412247i 0.217985 0.0546034i
\(58\) 0 0
\(59\) 5.29570i 0.689442i −0.938705 0.344721i \(-0.887974\pi\)
0.938705 0.344721i \(-0.112026\pi\)
\(60\) 0 0
\(61\) 15.6110 1.99879 0.999394 0.0347968i \(-0.0110784\pi\)
0.999394 + 0.0347968i \(0.0110784\pi\)
\(62\) 0 0
\(63\) 7.00000 3.74166i 0.881917 0.471405i
\(64\) 0 0
\(65\) 17.7951i 2.20721i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 3.14944 + 12.5730i 0.379148 + 1.51361i
\(70\) 0 0
\(71\) 5.65685i 0.671345i 0.941979 + 0.335673i \(0.108964\pi\)
−0.941979 + 0.335673i \(0.891036\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −17.2912 + 4.33130i −1.99661 + 0.500135i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.29150 0.595341 0.297670 0.954669i \(-0.403790\pi\)
0.297670 + 0.954669i \(0.403790\pi\)
\(80\) 0 0
\(81\) 5.00000 7.48331i 0.555556 0.831479i
\(82\) 0 0
\(83\) 18.1669i 1.99408i 0.0769020 + 0.997039i \(0.475497\pi\)
−0.0769020 + 0.997039i \(0.524503\pi\)
\(84\) 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
\(91\) −12.0399 −1.26213
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.83039i 0.392990i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.96077i 0.891630i −0.895125 0.445815i \(-0.852914\pi\)
0.895125 0.445815i \(-0.147086\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) −4.35425 17.3828i −0.424931 1.69639i
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.1421i 1.33038i −0.746674 0.665190i \(-0.768350\pi\)
0.746674 0.665190i \(-0.231650\pi\)
\(114\) 0 0
\(115\) 29.2630 2.72879
\(116\) 0 0
\(117\) −12.0399 + 6.43560i −1.11309 + 0.594972i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 20.6921i 1.85076i
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 13.1166i 1.14600i 0.819555 + 0.573000i \(0.194221\pi\)
−0.819555 + 0.573000i \(0.805779\pi\)
\(132\) 0 0
\(133\) 2.59160 0.224720
\(134\) 0 0
\(135\) −13.6458 15.0554i −1.17444 1.29576i
\(136\) 0 0
\(137\) 14.9666i 1.27869i 0.768922 + 0.639343i \(0.220793\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) −19.1822 −1.62701 −0.813505 0.581558i \(-0.802443\pi\)
−0.813505 + 0.581558i \(0.802443\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 11.7610 2.94603i 0.970030 0.242984i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −17.5701 −1.40225 −0.701123 0.713040i \(-0.747318\pi\)
−0.701123 + 0.713040i \(0.747318\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 19.7990i 1.56038i
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 7.70850 0.592961
\(170\) 0 0
\(171\) 2.59160 1.38527i 0.198184 0.105934i
\(172\) 0 0
\(173\) 24.6025i 1.87049i 0.353995 + 0.935247i \(0.384823\pi\)
−0.353995 + 0.935247i \(0.615177\pi\)
\(174\) 0 0
\(175\) −27.2288 −2.05830
\(176\) 0 0
\(177\) −2.22876 8.89753i −0.167524 0.668779i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −24.7124 −1.83686 −0.918428 0.395589i \(-0.870540\pi\)
−0.918428 + 0.395589i \(0.870540\pi\)
\(182\) 0 0
\(183\) 26.2288 6.57008i 1.93889 0.485675i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 10.1863 9.23254i 0.740942 0.671569i
\(190\) 0 0
\(191\) 22.6274i 1.63726i 0.574320 + 0.818631i \(0.305267\pi\)
−0.574320 + 0.818631i \(0.694733\pi\)
\(192\) 0 0
\(193\) 26.4575 1.90445 0.952227 0.305392i \(-0.0987875\pi\)
0.952227 + 0.305392i \(0.0987875\pi\)
\(194\) 0 0
\(195\) 7.48925 + 29.8982i 0.536317 + 2.14106i
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 10.5830 + 19.7990i 0.735570 + 1.37612i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 2.38075 + 9.50432i 0.163126 + 0.651225i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −27.2288 + 14.5544i −1.81525 + 0.970292i
\(226\) 0 0
\(227\) 15.3964i 1.02189i −0.859612 0.510947i \(-0.829295\pi\)
0.859612 0.510947i \(-0.170705\pi\)
\(228\) 0 0
\(229\) 8.46878 0.559633 0.279817 0.960053i \(-0.409726\pi\)
0.279817 + 0.960053i \(0.409726\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 29.9333i 1.96099i −0.196537 0.980497i \(-0.562969\pi\)
0.196537 0.980497i \(-0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.89047 2.22699i 0.577498 0.144659i
\(238\) 0 0
\(239\) 7.48331i 0.484055i 0.970269 + 0.242028i \(0.0778125\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 5.25127 14.6773i 0.336869 0.941551i
\(244\) 0 0
\(245\) 27.3730i 1.74880i
\(246\) 0 0
\(247\) −4.45751 −0.283625
\(248\) 0 0
\(249\) 7.64575 + 30.5230i 0.484530 + 1.93432i
\(250\) 0 0
\(251\) 23.2172i 1.46546i 0.680520 + 0.732730i \(0.261754\pi\)
−0.680520 + 0.732730i \(0.738246\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 28.2843i 1.74408i −0.489432 0.872041i \(-0.662796\pi\)
0.489432 0.872041i \(-0.337204\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 32.4234i 1.97689i −0.151585 0.988444i \(-0.548438\pi\)
0.151585 0.988444i \(-0.451562\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) −20.2288 + 5.06713i −1.22430 + 0.306677i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 14.9666i 0.892834i 0.894825 + 0.446417i \(0.147300\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) −32.2016 −1.91419 −0.957094 0.289779i \(-0.906418\pi\)
−0.957094 + 0.289779i \(0.906418\pi\)
\(284\) 0 0
\(285\) −1.61206 6.43560i −0.0954905 0.381212i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.5018i 0.847206i 0.905848 + 0.423603i \(0.139235\pi\)
−0.905848 + 0.423603i \(0.860765\pi\)
\(294\) 0 0
\(295\) −20.7085 −1.20570
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 34.0540i 1.96939i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −3.77124 15.0554i −0.216652 0.864908i
\(304\) 0 0
\(305\) 61.0460i 3.49548i
\(306\) 0 0
\(307\) 13.9990 0.798964 0.399482 0.916741i \(-0.369190\pi\)
0.399482 + 0.916741i \(0.369190\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −14.6315 27.3730i −0.824392 1.54230i
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 46.8331 2.59783
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 26.4575 1.44123 0.720616 0.693334i \(-0.243859\pi\)
0.720616 + 0.693334i \(0.243859\pi\)
\(338\) 0 0
\(339\) −5.95188 23.7608i −0.323262 1.29051i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 18.5203 1.00000
\(344\) 0 0
\(345\) 49.1660 12.3157i 2.64701 0.663054i
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 28.6305 1.53255 0.766277 0.642510i \(-0.222107\pi\)
0.766277 + 0.642510i \(0.222107\pi\)
\(350\) 0 0
\(351\) −17.5203 + 15.8799i −0.935162 + 0.847604i
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 22.1208 1.17405
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 37.4166i 1.97477i −0.158334 0.987386i \(-0.550612\pi\)
0.158334 0.987386i \(-0.449388\pi\)
\(360\) 0 0
\(361\) −18.0405 −0.949501
\(362\) 0 0
\(363\) −18.4816 + 4.62948i −0.970030 + 0.242984i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 8.70850 + 34.7656i 0.449705 + 1.79529i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −3.36028 + 0.841723i −0.172153 + 0.0431228i
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 5.52026 + 22.0377i 0.278460 + 1.11165i
\(394\) 0 0
\(395\) 20.6921i 1.04113i
\(396\) 0 0
\(397\) −37.7318 −1.89370 −0.946852 0.321668i \(-0.895756\pi\)
−0.946852 + 0.321668i \(0.895756\pi\)
\(398\) 0 0
\(399\) 4.35425 1.09070i 0.217985 0.0546034i
\(400\) 0 0
\(401\) 36.7696i 1.83618i 0.396368 + 0.918092i \(0.370271\pi\)
−0.396368 + 0.918092i \(0.629729\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −29.2630 19.5522i −1.45409 0.971555i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 6.29888 + 25.1461i 0.310701 + 1.24036i
\(412\) 0 0
\(413\) 14.0111i 0.689442i
\(414\) 0 0
\(415\) 71.0405 3.48724
\(416\) 0 0
\(417\) −32.2288 + 8.07303i −1.57825 + 0.395338i
\(418\) 0 0
\(419\) 38.8590i 1.89839i −0.314695 0.949193i \(-0.601902\pi\)
0.314695 0.949193i \(-0.398098\pi\)
\(420\) 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
\(427\) 41.3029 1.99879
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 37.4166i 1.80229i −0.433515 0.901146i \(-0.642727\pi\)
0.433515 0.901146i \(-0.357273\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.33014i 0.350648i
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 18.5203 9.89949i 0.881917 0.471405i
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 29.9333i 1.41264i −0.707894 0.706319i \(-0.750354\pi\)
0.707894 0.706319i \(-0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 16.8014 4.20861i 0.789399 0.197738i
\(454\) 0 0
\(455\) 47.0813i 2.20721i
\(456\) 0 0
\(457\) −5.29150 −0.247526 −0.123763 0.992312i \(-0.539496\pi\)
−0.123763 + 0.992312i \(0.539496\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.19024i 0.288308i 0.989555 + 0.144154i \(0.0460461\pi\)
−0.989555 + 0.144154i \(0.953954\pi\)
\(462\) 0 0
\(463\) −26.4575 −1.22958 −0.614792 0.788689i \(-0.710760\pi\)
−0.614792 + 0.788689i \(0.710760\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 41.6295i 1.92638i 0.268814 + 0.963192i \(0.413368\pi\)
−0.268814 + 0.963192i \(0.586632\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −29.5203 + 7.39458i −1.36022 + 0.340724i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −10.0808 −0.462541
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 8.33263 + 33.2651i 0.379148 + 1.51361i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −38.0000 −1.72194 −0.860972 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14.9666i 0.671345i
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −35.0405 −1.55928
\(506\) 0 0
\(507\) 12.9514 3.24421i 0.575190 0.144080i
\(508\) 0 0
\(509\) 37.4737i 1.66099i −0.557024 0.830497i \(-0.688057\pi\)
0.557024 0.830497i \(-0.311943\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.77124 3.41815i 0.166504 0.150915i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 10.3542 + 41.3357i 0.454501 + 1.81444i
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −25.0594 −1.09577 −0.547885 0.836554i \(-0.684567\pi\)
−0.547885 + 0.836554i \(0.684567\pi\)
\(524\) 0 0
\(525\) −45.7482 + 11.4595i −1.99661 + 0.500135i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −33.0000 −1.43478
\(530\) 0 0
\(531\) −7.48925 14.0111i −0.325006 0.608030i
\(532\) 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
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −41.5203 + 10.4005i −1.78180 + 0.446327i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 41.3029 22.0773i 1.76277 0.942238i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 14.0000 0.595341
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 46.6799i 1.96732i 0.180032 + 0.983661i \(0.442380\pi\)
−0.180032 + 0.983661i \(0.557620\pi\)
\(564\) 0 0
\(565\) −55.3019 −2.32657
\(566\) 0 0
\(567\) 13.2288 19.7990i 0.555556 0.831479i
\(568\) 0 0
\(569\) 2.82843i 0.118574i 0.998241 + 0.0592869i \(0.0188827\pi\)
−0.998241 + 0.0592869i \(0.981117\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 9.52301 + 38.0173i 0.397829 + 1.58819i
\(574\) 0 0
\(575\) 77.0146i 3.21173i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 44.4524 11.1349i 1.84738 0.462753i
\(580\) 0 0
\(581\) 48.0651i 1.99408i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 25.1660 + 47.0813i 1.04049 + 1.94657i
\(586\) 0 0
\(587\) 36.5792i 1.50978i 0.655849 + 0.754892i \(0.272311\pi\)
−0.655849 + 0.754892i \(0.727689\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 11.3137i 0.462266i −0.972922 0.231133i \(-0.925757\pi\)
0.972922 0.231133i \(-0.0742432\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 43.0148i 1.74880i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.1421i 0.569341i −0.958625 0.284670i \(-0.908116\pi\)
0.958625 0.284670i \(-0.0918842\pi\)
\(618\) 0 0
\(619\) −26.3244 −1.05807 −0.529034 0.848601i \(-0.677446\pi\)
−0.529034 + 0.848601i \(0.677446\pi\)
\(620\) 0 0
\(621\) 26.1136 + 28.8111i 1.04790 + 1.15615i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 29.4575 1.17830
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 37.0405 1.47456 0.737280 0.675587i \(-0.236110\pi\)
0.737280 + 0.675587i \(0.236110\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 7.82087i 0.310362i
\(636\) 0 0
\(637\) −31.8546 −1.26213
\(638\) 0 0
\(639\) 8.00000 + 14.9666i 0.316475 + 0.592071i
\(640\) 0 0
\(641\) 48.0833i 1.89917i −0.313503 0.949587i \(-0.601502\pi\)
0.313503 0.949587i \(-0.398498\pi\)
\(642\) 0 0
\(643\) 15.2640 0.601955 0.300978 0.953631i \(-0.402687\pi\)
0.300978 + 0.953631i \(0.402687\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 51.2915 2.00412
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −30.5895 −1.18980 −0.594898 0.803801i \(-0.702807\pi\)
−0.594898 + 0.803801i \(0.702807\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 10.1343i 0.392990i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 0 0
\(675\) −39.6228 + 35.9130i −1.52508 + 1.38229i
\(676\) 0 0
\(677\) 9.45150i 0.363251i 0.983368 + 0.181625i \(0.0581358\pi\)
−0.983368 + 0.181625i \(0.941864\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −6.47974 25.8681i −0.248304 0.991267i
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 58.5260 2.23616
\(686\) 0 0
\(687\) 14.2288 3.56418i 0.542861 0.135982i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −52.3633 −1.99199 −0.995997 0.0893857i \(-0.971510\pi\)
−0.995997 + 0.0893857i \(0.971510\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 75.0106i 2.84532i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −12.5978 50.2921i −0.476491 1.90222i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 23.7080i 0.891630i
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 14.0000 7.48331i 0.525041 0.280646i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.14944 + 12.5730i 0.117618 + 0.469548i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 2.64575 26.8701i 0.0979908 0.995187i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 42.9150 1.58510 0.792551 0.609806i \(-0.208753\pi\)
0.792551 + 0.609806i \(0.208753\pi\)
\(734\) 0 0
\(735\) −11.5203 45.9906i −0.424931 1.69639i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) −7.48925 + 1.87600i −0.275125 + 0.0689164i
\(742\) 0 0
\(743\) 7.48331i 0.274536i 0.990534 + 0.137268i \(0.0438322\pi\)
−0.990534 + 0.137268i \(0.956168\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 25.6919 + 48.0651i 0.940017 + 1.75861i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −50.0000 −1.82453 −0.912263 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) 0 0
\(753\) 9.77124 + 39.0083i 0.356084 + 1.42154i
\(754\) 0 0
\(755\) 39.1044i 1.42315i
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0989i 0.870162i
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 53.1155i 1.91043i 0.295912 + 0.955215i \(0.404376\pi\)
−0.295912 + 0.955215i \(0.595624\pi\)
\(774\) 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
\(785\) 68.7067i 2.45225i
\(786\) 0 0
\(787\) −45.2211 −1.61196 −0.805978 0.591945i \(-0.798360\pi\)
−0.805978 + 0.591945i \(0.798360\pi\)
\(788\) 0 0
\(789\) −11.9038 47.5216i −0.423785 1.69181i
\(790\) 0 0
\(791\) 37.4166i 1.33038i
\(792\) 0 0
\(793\) −71.0405 −2.52272
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 55.8860i 1.97958i −0.142521 0.989792i \(-0.545521\pi\)
0.142521 0.989792i \(-0.454479\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 77.4227 2.72879
\(806\) 0 0
\(807\) −13.6458 54.4759i −0.480353 1.91764i
\(808\) 0 0
\(809\) 31.1127i 1.09386i −0.837177 0.546932i \(-0.815796\pi\)
0.837177 0.546932i \(-0.184204\pi\)
\(810\) 0 0
\(811\) 48.4452 1.70114 0.850570 0.525861i \(-0.176257\pi\)
0.850570 + 0.525861i \(0.176257\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −31.8546 + 17.0270i −1.11309 + 0.594972i
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −26.4575 −0.922251 −0.461125 0.887335i \(-0.652554\pi\)
−0.461125 + 0.887335i \(0.652554\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 1.32653 0.0460723 0.0230361 0.999735i \(-0.492667\pi\)
0.0230361 + 0.999735i \(0.492667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 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
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 6.29888 + 25.1461i 0.216945 + 0.866076i
\(844\) 0 0
\(845\) 30.1436i 1.03697i
\(846\) 0 0
\(847\) −29.1033 −1.00000
\(848\) 0 0
\(849\) −54.1033 + 13.5524i −1.85682 + 0.465118i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −57.8935 −1.98224 −0.991118 0.132987i \(-0.957543\pi\)
−0.991118 + 0.132987i \(0.957543\pi\)
\(854\) 0 0
\(855\) −5.41699 10.1343i −0.185257 0.346585i
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 54.3224 1.85346 0.926728 0.375734i \(-0.122609\pi\)
0.926728 + 0.375734i \(0.122609\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 56.5685i 1.92562i 0.270187 + 0.962808i \(0.412914\pi\)
−0.270187 + 0.962808i \(0.587086\pi\)
\(864\) 0 0
\(865\) 96.2065 3.27112
\(866\) 0 0
\(867\) −28.5624 + 7.15464i −0.970030 + 0.242984i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 54.7461i 1.85076i
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 6.10326 + 24.3651i 0.205858 + 0.821816i
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) −34.7932 + 8.71541i −1.16956 + 0.292965i
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −5.29150 −0.177471
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −14.3320 57.2156i −0.478532 1.91037i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 96.6361i 3.21229i
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −12.6724 23.7080i −0.420318 0.786344i
\(910\) 0 0
\(911\) 52.3832i 1.73553i 0.496972 + 0.867766i \(0.334445\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −25.6919 102.566i −0.849348 3.39072i
\(916\) 0 0
\(917\) 34.7032i 1.14600i
\(918\) 0 0
\(919\) −58.2065 −1.92006 −0.960028 0.279904i \(-0.909697\pi\)
−0.960028 + 0.279904i \(0.909697\pi\)
\(920\) 0 0
\(921\) 23.5203 5.89163i 0.775019 0.194136i
\(922\) 0 0
\(923\) 25.7424i 0.847322i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 6.85672 0.224720
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 19.0614i 0.621385i −0.950510 0.310693i \(-0.899439\pi\)
0.950510 0.310693i \(-0.100561\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −36.1033 39.8328i −1.17444 1.29576i
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 29.9333i 0.969633i −0.874616 0.484817i \(-0.838886\pi\)
0.874616 0.484817i \(-0.161114\pi\)
\(954\) 0 0
\(955\) 88.4830 2.86324
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 39.5980i 1.27869i
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 103.460i 3.33051i
\(966\) 0 0
\(967\) 58.0000 1.86515 0.932577 0.360971i \(-0.117555\pi\)
0.932577 + 0.360971i \(0.117555\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 62.3216i 2.00000i −0.00218468 0.999998i \(-0.500695\pi\)
0.00218468 0.999998i \(-0.499305\pi\)
\(972\) 0 0
\(973\) −50.7512 −1.62701
\(974\) 0 0
\(975\) 78.6863 19.7103i 2.51998 0.631233i
\(976\) 0 0
\(977\) 59.8665i 1.91530i 0.287936 + 0.957650i \(0.407031\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 37.0405 1.17663 0.588315 0.808632i \(-0.299791\pi\)
0.588315 + 0.808632i \(0.299791\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 41.6499 1.31907 0.659533 0.751675i \(-0.270754\pi\)
0.659533 + 0.751675i \(0.270754\pi\)
\(998\) 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 672.2.i.e.209.7 8
3.2 odd 2 inner 672.2.i.e.209.8 8
4.3 odd 2 168.2.i.d.125.5 yes 8
7.6 odd 2 inner 672.2.i.e.209.2 8
8.3 odd 2 168.2.i.d.125.8 yes 8
8.5 even 2 inner 672.2.i.e.209.2 8
12.11 even 2 168.2.i.d.125.1 8
21.20 even 2 inner 672.2.i.e.209.1 8
24.5 odd 2 inner 672.2.i.e.209.1 8
24.11 even 2 168.2.i.d.125.4 yes 8
28.27 even 2 168.2.i.d.125.8 yes 8
56.13 odd 2 CM 672.2.i.e.209.7 8
56.27 even 2 168.2.i.d.125.5 yes 8
84.83 odd 2 168.2.i.d.125.4 yes 8
168.83 odd 2 168.2.i.d.125.1 8
168.125 even 2 inner 672.2.i.e.209.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.i.d.125.1 8 12.11 even 2
168.2.i.d.125.1 8 168.83 odd 2
168.2.i.d.125.4 yes 8 24.11 even 2
168.2.i.d.125.4 yes 8 84.83 odd 2
168.2.i.d.125.5 yes 8 4.3 odd 2
168.2.i.d.125.5 yes 8 56.27 even 2
168.2.i.d.125.8 yes 8 8.3 odd 2
168.2.i.d.125.8 yes 8 28.27 even 2
672.2.i.e.209.1 8 21.20 even 2 inner
672.2.i.e.209.1 8 24.5 odd 2 inner
672.2.i.e.209.2 8 7.6 odd 2 inner
672.2.i.e.209.2 8 8.5 even 2 inner
672.2.i.e.209.7 8 1.1 even 1 trivial
672.2.i.e.209.7 8 56.13 odd 2 CM
672.2.i.e.209.8 8 3.2 odd 2 inner
672.2.i.e.209.8 8 168.125 even 2 inner