Properties

Label 912.2.ch.a.47.1
Level $912$
Weight $2$
Character 912.47
Analytic conductor $7.282$
Analytic rank $0$
Dimension $6$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(47,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 0, 9, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.ch (of order \(18\), degree \(6\), 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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label 47.1
Root \(-0.173648 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 912.47
Dual form 912.2.ch.a.815.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.11334 - 1.32683i) q^{3} +(-4.52481 - 2.61240i) q^{7} +(-0.520945 + 2.95442i) q^{9} +O(q^{10})\) \(q+(-1.11334 - 1.32683i) q^{3} +(-4.52481 - 2.61240i) q^{7} +(-0.520945 + 2.95442i) q^{9} +(-0.375515 - 0.315094i) q^{13} +(-0.500000 + 4.33013i) q^{19} +(1.57145 + 8.91215i) q^{21} +(3.83022 + 3.21394i) q^{25} +(4.50000 - 2.59808i) q^{27} +(4.97431 + 2.87192i) q^{31} -8.55943 q^{37} +0.849051i q^{39} +(-2.07738 + 5.70756i) q^{43} +(10.1493 + 17.5791i) q^{49} +(6.30200 - 4.15749i) q^{57} +(10.1356 - 3.68907i) q^{61} +(10.0753 - 12.0073i) q^{63} +(-12.7417 - 2.24670i) q^{67} +(-10.8289 + 9.08651i) q^{73} -8.66025i q^{75} +(10.1814 + 12.1337i) q^{79} +(-8.45723 - 3.07818i) q^{81} +(0.875982 + 2.40674i) q^{91} +(-1.72756 - 9.79747i) q^{93} +(3.29932 + 18.7113i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 15 q^{13} - 3 q^{19} + 9 q^{21} + 27 q^{27} - 39 q^{43} + 21 q^{49} + 42 q^{61} + 36 q^{63} - 33 q^{67} - 51 q^{73} + 12 q^{79} - 48 q^{91} - 54 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{4}{9}\right)\) \(1\) \(-1\) \(-1\)

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
\(3\) −1.11334 1.32683i −0.642788 0.766044i
\(4\) 0 0
\(5\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(6\) 0 0
\(7\) −4.52481 2.61240i −1.71022 0.987396i −0.934246 0.356630i \(-0.883926\pi\)
−0.775974 0.630765i \(-0.782741\pi\)
\(8\) 0 0
\(9\) −0.520945 + 2.95442i −0.173648 + 0.984808i
\(10\) 0 0
\(11\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) −0.375515 0.315094i −0.104149 0.0873915i 0.589226 0.807968i \(-0.299433\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(18\) 0 0
\(19\) −0.500000 + 4.33013i −0.114708 + 0.993399i
\(20\) 0 0
\(21\) 1.57145 + 8.91215i 0.342919 + 1.94479i
\(22\) 0 0
\(23\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(24\) 0 0
\(25\) 3.83022 + 3.21394i 0.766044 + 0.642788i
\(26\) 0 0
\(27\) 4.50000 2.59808i 0.866025 0.500000i
\(28\) 0 0
\(29\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(30\) 0 0
\(31\) 4.97431 + 2.87192i 0.893412 + 0.515812i 0.875057 0.484020i \(-0.160824\pi\)
0.0183550 + 0.999832i \(0.494157\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.55943 −1.40716 −0.703581 0.710615i \(-0.748417\pi\)
−0.703581 + 0.710615i \(0.748417\pi\)
\(38\) 0 0
\(39\) 0.849051i 0.135957i
\(40\) 0 0
\(41\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(42\) 0 0
\(43\) −2.07738 + 5.70756i −0.316798 + 0.870395i 0.674443 + 0.738327i \(0.264384\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(48\) 0 0
\(49\) 10.1493 + 17.5791i 1.44990 + 2.51130i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.30200 4.15749i 0.834721 0.550673i
\(58\) 0 0
\(59\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(60\) 0 0
\(61\) 10.1356 3.68907i 1.29773 0.472337i 0.401476 0.915869i \(-0.368497\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 10.0753 12.0073i 1.26937 1.51278i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.7417 2.24670i −1.55665 0.274479i −0.671932 0.740613i \(-0.734535\pi\)
−0.884714 + 0.466134i \(0.845646\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(72\) 0 0
\(73\) −10.8289 + 9.08651i −1.26742 + 1.06350i −0.272575 + 0.962135i \(0.587875\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 8.66025i 1.00000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.1814 + 12.1337i 1.14550 + 1.36515i 0.920478 + 0.390794i \(0.127800\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −8.45723 3.07818i −0.939693 0.342020i
\(82\) 0 0
\(83\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(90\) 0 0
\(91\) 0.875982 + 2.40674i 0.0918278 + 0.252295i
\(92\) 0 0
\(93\) −1.72756 9.79747i −0.179140 1.01595i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.29932 + 18.7113i 0.334995 + 1.89985i 0.427284 + 0.904117i \(0.359470\pi\)
−0.0922897 + 0.995732i \(0.529419\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(102\) 0 0
\(103\) −8.74304 + 5.04780i −0.861477 + 0.497374i −0.864507 0.502621i \(-0.832369\pi\)
0.00302937 + 0.999995i \(0.499036\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(108\) 0 0
\(109\) −17.8542 6.49838i −1.71012 0.622432i −0.713206 0.700954i \(-0.752758\pi\)
−0.996912 + 0.0785223i \(0.974980\pi\)
\(110\) 0 0
\(111\) 9.52956 + 11.3569i 0.904506 + 1.07795i
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.12654 0.945283i 0.104149 0.0873915i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.50000 9.52628i 0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.79339 9.28780i 0.691551 0.824159i −0.299991 0.953942i \(-0.596984\pi\)
0.991542 + 0.129783i \(0.0414282\pi\)
\(128\) 0 0
\(129\) 9.88578 3.59813i 0.870395 0.316798i
\(130\) 0 0
\(131\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(132\) 0 0
\(133\) 13.5744 18.2868i 1.17705 1.58567i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(138\) 0 0
\(139\) −13.9795 + 16.6601i −1.18573 + 1.41310i −0.296866 + 0.954919i \(0.595942\pi\)
−0.888861 + 0.458176i \(0.848503\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 12.0248 33.0379i 0.991790 2.72492i
\(148\) 0 0
\(149\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(150\) 0 0
\(151\) 15.5885i 1.26857i 0.773099 + 0.634285i \(0.218706\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −22.6322 8.23746i −1.80625 0.657421i −0.997609 0.0691164i \(-0.977982\pi\)
−0.808640 0.588304i \(-0.799796\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −17.3640 + 10.0251i −1.36005 + 0.785225i −0.989630 0.143639i \(-0.954120\pi\)
−0.370420 + 0.928864i \(0.620786\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(168\) 0 0
\(169\) −2.21570 12.5659i −0.170438 0.966604i
\(170\) 0 0
\(171\) −12.5326 3.73297i −0.958388 0.285467i
\(172\) 0 0
\(173\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(174\) 0 0
\(175\) −8.93494 24.5486i −0.675418 1.85570i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) 0 0
\(181\) 3.29932 18.7113i 0.245236 1.39080i −0.574707 0.818359i \(-0.694884\pi\)
0.819943 0.572444i \(-0.194005\pi\)
\(182\) 0 0
\(183\) −16.1792 9.34105i −1.19600 0.690510i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −27.1489 −1.97479
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 21.1728 17.7661i 1.52405 1.27883i 0.696261 0.717788i \(-0.254845\pi\)
0.827788 0.561041i \(-0.189599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(198\) 0 0
\(199\) −26.5633 4.68383i −1.88302 0.332028i −0.890589 0.454809i \(-0.849707\pi\)
−0.992434 + 0.122782i \(0.960818\pi\)
\(200\) 0 0
\(201\) 11.2049 + 19.4074i 0.790330 + 1.36889i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 11.3708 2.00497i 0.782796 0.138028i 0.232053 0.972703i \(-0.425456\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −15.0052 25.9898i −1.01862 1.76430i
\(218\) 0 0
\(219\) 24.1125 + 4.25168i 1.62937 + 0.287302i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 3.43289 9.43178i 0.229883 0.631598i −0.770097 0.637927i \(-0.779792\pi\)
0.999980 + 0.00632846i \(0.00201443\pi\)
\(224\) 0 0
\(225\) −11.4907 + 9.64181i −0.766044 + 0.642788i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −16.6486 −1.10017 −0.550085 0.835109i \(-0.685405\pi\)
−0.550085 + 0.835109i \(0.685405\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.76399 27.0179i 0.309454 1.75500i
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) 5.43036 + 4.55661i 0.349800 + 0.293517i 0.800710 0.599052i \(-0.204456\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 0 0
\(243\) 5.33157 + 14.6484i 0.342020 + 0.939693i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.55216 1.46848i 0.0987613 0.0934371i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(258\) 0 0
\(259\) 38.7298 + 22.3607i 2.40655 + 1.38943i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(270\) 0 0
\(271\) −11.2555 + 30.9243i −0.683725 + 1.87852i −0.313321 + 0.949647i \(0.601442\pi\)
−0.370403 + 0.928871i \(0.620781\pi\)
\(272\) 0 0
\(273\) 2.21806 3.84180i 0.134243 0.232516i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.50000 + 4.33013i 0.150210 + 0.260172i 0.931305 0.364241i \(-0.118672\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 0 0
\(279\) −11.0762 + 13.2001i −0.663115 + 0.790269i
\(280\) 0 0
\(281\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(282\) 0 0
\(283\) 32.4090 5.71458i 1.92652 0.339697i 0.927130 0.374741i \(-0.122268\pi\)
0.999386 + 0.0350443i \(0.0111572\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.9748 + 5.81434i −0.939693 + 0.342020i
\(290\) 0 0
\(291\) 21.1535 25.2097i 1.24004 1.47782i
\(292\) 0 0
\(293\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 24.3102 20.3987i 1.40122 1.17576i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 18.9268 + 22.5561i 1.08021 + 1.28734i 0.955449 + 0.295156i \(0.0953717\pi\)
0.124760 + 0.992187i \(0.460184\pi\)
\(308\) 0 0
\(309\) 16.4315 + 5.98059i 0.934758 + 0.340224i
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) −2.25743 + 12.8025i −0.127597 + 0.723640i 0.852134 + 0.523324i \(0.175308\pi\)
−0.979731 + 0.200316i \(0.935803\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −0.425612 2.41376i −0.0236087 0.133891i
\(326\) 0 0
\(327\) 11.2555 + 30.9243i 0.622432 + 1.71012i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −29.8965 + 17.2608i −1.64326 + 0.948737i −0.663598 + 0.748090i \(0.730971\pi\)
−0.979663 + 0.200648i \(0.935695\pi\)
\(332\) 0 0
\(333\) 4.45899 25.2882i 0.244351 1.38578i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 34.4761 + 12.5483i 1.87803 + 0.683548i 0.951985 + 0.306145i \(0.0990393\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 69.4826i 3.75171i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(348\) 0 0
\(349\) −18.2729 + 31.6496i −0.978126 + 1.69416i −0.308920 + 0.951088i \(0.599967\pi\)
−0.669207 + 0.743076i \(0.733366\pi\)
\(350\) 0 0
\(351\) −2.50846 0.442308i −0.133891 0.0236087i
\(352\) 0 0
\(353\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(360\) 0 0
\(361\) −18.5000 4.33013i −0.973684 0.227901i
\(362\) 0 0
\(363\) −18.7631 + 3.30844i −0.984808 + 0.173648i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.72116 2.05120i 0.0898439 0.107072i −0.719249 0.694752i \(-0.755514\pi\)
0.809093 + 0.587680i \(0.199959\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −19.0000 + 32.9090i −0.983783 + 1.70396i −0.336557 + 0.941663i \(0.609263\pi\)
−0.647225 + 0.762299i \(0.724071\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 33.0709i 1.69874i −0.527798 0.849370i \(-0.676982\pi\)
0.527798 0.849370i \(-0.323018\pi\)
\(380\) 0 0
\(381\) −21.0000 −1.07586
\(382\) 0 0
\(383\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −15.7803 9.11079i −0.802160 0.463127i
\(388\) 0 0
\(389\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 6.78240 + 38.4649i 0.340399 + 1.93050i 0.365493 + 0.930814i \(0.380900\pi\)
−0.0250943 + 0.999685i \(0.507989\pi\)
\(398\) 0 0
\(399\) −39.3764 + 2.34851i −1.97129 + 0.117573i
\(400\) 0 0
\(401\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(402\) 0 0
\(403\) −0.963001 2.64582i −0.0479705 0.131798i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −6.59863 + 37.4227i −0.326281 + 1.85043i 0.174232 + 0.984705i \(0.444256\pi\)
−0.500514 + 0.865729i \(0.666856\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 37.6691 1.84466
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 14.5548 12.2130i 0.709360 0.595223i −0.215060 0.976601i \(-0.568995\pi\)
0.924419 + 0.381377i \(0.124550\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −55.4992 9.78600i −2.68579 0.473578i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(432\) 0 0
\(433\) 23.6691 8.61483i 1.13746 0.414003i 0.296466 0.955043i \(-0.404192\pi\)
0.840996 + 0.541041i \(0.181970\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 38.2827 6.75027i 1.82713 0.322173i 0.848722 0.528839i \(-0.177372\pi\)
0.978412 + 0.206666i \(0.0662612\pi\)
\(440\) 0 0
\(441\) −57.2233 + 20.8276i −2.72492 + 0.991790i
\(442\) 0 0
\(443\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 20.6832 17.3553i 0.971782 0.815422i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.0597 −0.891577 −0.445788 0.895138i \(-0.647077\pi\)
−0.445788 + 0.895138i \(0.647077\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(462\) 0 0
\(463\) −36.9338 21.3238i −1.71646 0.990999i −0.925166 0.379563i \(-0.876074\pi\)
−0.791294 0.611435i \(-0.790592\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) 0 0
\(469\) 51.7845 + 43.4524i 2.39119 + 2.00644i
\(470\) 0 0
\(471\) 14.2677 + 39.2002i 0.657421 + 1.80625i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −15.8319 + 14.9784i −0.726416 + 0.687255i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(480\) 0 0
\(481\) 3.21419 + 2.69703i 0.146555 + 0.122974i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 34.5000 + 19.9186i 1.56334 + 0.902597i 0.996915 + 0.0784867i \(0.0250088\pi\)
0.566429 + 0.824110i \(0.308325\pi\)
\(488\) 0 0
\(489\) 32.6336 + 11.8776i 1.47574 + 0.537126i
\(490\) 0 0
\(491\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 6.27678 17.2453i 0.280987 0.772006i −0.716258 0.697835i \(-0.754147\pi\)
0.997246 0.0741708i \(-0.0236310\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −14.2059 + 16.9299i −0.630906 + 0.751885i
\(508\) 0 0
\(509\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(510\) 0 0
\(511\) 72.7363 12.8254i 3.21766 0.567361i
\(512\) 0 0
\(513\) 9.00000 + 20.7846i 0.397360 + 0.917663i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(522\) 0 0
\(523\) 0.0576190 + 0.0101598i 0.00251950 + 0.000444257i 0.174908 0.984585i \(-0.444037\pi\)
−0.172388 + 0.985029i \(0.555148\pi\)
\(524\) 0 0
\(525\) −22.6241 + 39.1860i −0.987396 + 1.71022i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −17.6190 + 14.7841i −0.766044 + 0.642788i
\(530\) 0 0
\(531\) 0 0
\(532\) 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
\(541\) 5.34570 30.3170i 0.229830 1.30343i −0.623404 0.781900i \(-0.714251\pi\)
0.853233 0.521529i \(-0.174638\pi\)
\(542\) 0 0
\(543\) −28.5000 + 16.4545i −1.22305 + 0.706129i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.44063 + 6.70557i 0.104354 + 0.286709i 0.980870 0.194662i \(-0.0623610\pi\)
−0.876517 + 0.481371i \(0.840139\pi\)
\(548\) 0 0
\(549\) 5.61897 + 31.8667i 0.239812 + 1.36004i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −14.3708 81.5007i −0.611107 3.46576i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(558\) 0 0
\(559\) 2.57851 1.48870i 0.109059 0.0629654i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 30.2260 + 36.0219i 1.26937 + 1.51278i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 23.5769i 0.986662i −0.869842 0.493331i \(-0.835779\pi\)
0.869842 0.493331i \(-0.164221\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −17.5000 + 30.3109i −0.728535 + 1.26186i 0.228968 + 0.973434i \(0.426465\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) 0 0
\(579\) −47.1450 8.31294i −1.95928 0.345474i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(588\) 0 0
\(589\) −14.9229 + 20.1034i −0.614888 + 0.828347i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 23.3594 + 40.4596i 0.956036 + 1.65590i
\(598\) 0 0
\(599\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(600\) 0 0
\(601\) 19.3248 33.4715i 0.788273 1.36533i −0.138751 0.990327i \(-0.544309\pi\)
0.927024 0.375002i \(-0.122358\pi\)
\(602\) 0 0
\(603\) 13.2754 36.4740i 0.540617 1.48533i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 21.6418i 0.878412i 0.898386 + 0.439206i \(0.144740\pi\)
−0.898386 + 0.439206i \(0.855260\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 34.7686 + 12.6547i 1.40429 + 0.511120i 0.929449 0.368950i \(-0.120283\pi\)
0.474843 + 0.880071i \(0.342505\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(618\) 0 0
\(619\) 7.46601 4.31051i 0.300084 0.173254i −0.342396 0.939556i \(-0.611239\pi\)
0.642481 + 0.766302i \(0.277905\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.34120 + 24.6202i 0.173648 + 0.984808i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −13.3801 36.7615i −0.532653 1.46345i −0.855901 0.517139i \(-0.826997\pi\)
0.323248 0.946314i \(-0.395225\pi\)
\(632\) 0 0
\(633\) −15.3198 12.8548i −0.608907 0.510934i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.72786 9.79920i 0.0684605 0.388259i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(642\) 0 0
\(643\) 1.17540 + 1.40079i 0.0463534 + 0.0552418i 0.788723 0.614749i \(-0.210743\pi\)
−0.742370 + 0.669991i \(0.766298\pi\)
\(644\) 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
\(651\) −17.7781 + 48.8448i −0.696777 + 1.91438i
\(652\) 0 0
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −21.2041 36.7267i −0.827252 1.43284i
\(658\) 0 0
\(659\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(660\) 0 0
\(661\) −35.7083 + 12.9968i −1.38889 + 0.505516i −0.924862 0.380303i \(-0.875820\pi\)
−0.464031 + 0.885819i \(0.653597\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −16.3363 + 5.94593i −0.631598 + 0.229883i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −23.6045 40.8841i −0.909886 1.57597i −0.814221 0.580554i \(-0.802836\pi\)
−0.0956642 0.995414i \(-0.530497\pi\)
\(674\) 0 0
\(675\) 25.5861 + 4.51151i 0.984808 + 0.173648i
\(676\) 0 0
\(677\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) 33.9528 93.2845i 1.30299 3.57993i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 18.5355 + 22.0898i 0.707175 + 0.842779i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 28.5000 + 16.4545i 1.08419 + 0.625958i 0.932024 0.362397i \(-0.118041\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(702\) 0 0
\(703\) 4.27972 37.0634i 0.161413 1.39787i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.13017 2.62652i −0.117556 0.0986411i 0.582115 0.813107i \(-0.302225\pi\)
−0.699671 + 0.714466i \(0.746670\pi\)
\(710\) 0 0
\(711\) −41.1520 + 23.7591i −1.54332 + 0.891038i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(720\) 0 0
\(721\) 52.7475 1.96442
\(722\) 0 0
\(723\) 12.2782i 0.456632i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −15.1671 + 41.6712i −0.562516 + 1.54550i 0.253419 + 0.967357i \(0.418445\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) 13.5000 23.3827i 0.500000 0.866025i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −21.5000 37.2391i −0.794121 1.37546i −0.923396 0.383849i \(-0.874598\pi\)
0.129275 0.991609i \(-0.458735\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −29.0718 + 5.12614i −1.06942 + 0.188568i −0.680534 0.732717i \(-0.738252\pi\)
−0.388888 + 0.921285i \(0.627141\pi\)
\(740\) 0 0
\(741\) −3.67650 0.424525i −0.135060 0.0155953i
\(742\) 0 0
\(743\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1.48617 + 0.262051i 0.0542310 + 0.00956239i 0.200698 0.979653i \(-0.435679\pi\)
−0.146467 + 0.989216i \(0.546790\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 33.1282 27.7979i 1.20406 1.01033i 0.204560 0.978854i \(-0.434424\pi\)
0.999505 0.0314762i \(-0.0100208\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 63.8104 + 76.0462i 2.31009 + 2.75306i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −3.61814 + 20.5195i −0.130474 + 0.739953i 0.847432 + 0.530904i \(0.178148\pi\)
−0.977905 + 0.209048i \(0.932963\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(774\) 0 0
\(775\) 9.82254 + 26.9872i 0.352836 + 0.969409i
\(776\) 0 0
\(777\) −13.4507 76.2829i −0.482542 2.73663i
\(778\) 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
\(787\) −13.7680 + 7.94897i −0.490777 + 0.283350i −0.724897 0.688858i \(-0.758113\pi\)
0.234120 + 0.972208i \(0.424779\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4.96848 1.80838i −0.176436 0.0642175i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 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
\(809\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(810\) 0 0
\(811\) 34.5136 41.1317i 1.21194 1.44433i 0.350423 0.936592i \(-0.386038\pi\)
0.861512 0.507736i \(-0.169518\pi\)
\(812\) 0 0
\(813\) 53.5625 19.4951i 1.87852 0.683725i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −23.6758 11.8491i −0.828310 0.414548i
\(818\) 0 0
\(819\) −7.56687 + 1.33424i −0.264408 + 0.0466222i
\(820\) 0 0
\(821\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(822\) 0 0
\(823\) 21.1535 25.2097i 0.737364 0.878756i −0.258830 0.965923i \(-0.583337\pi\)
0.996194 + 0.0871670i \(0.0277814\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(828\) 0 0
\(829\) −6.48561 + 11.2334i −0.225255 + 0.390153i −0.956396 0.292074i \(-0.905655\pi\)
0.731141 + 0.682226i \(0.238988\pi\)
\(830\) 0 0
\(831\) 2.96198 8.13798i 0.102750 0.282303i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 29.8458 1.03162
\(838\) 0 0
\(839\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(840\) 0 0
\(841\) −27.2511 9.91858i −0.939693 0.342020i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −49.7730 + 28.7364i −1.71022 + 0.987396i
\(848\) 0 0
\(849\) −43.6645 36.6389i −1.49856 1.25744i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −0.564121 3.19929i −0.0193152 0.109542i 0.973626 0.228150i \(-0.0732677\pi\)
−0.992941 + 0.118609i \(0.962157\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(858\) 0 0
\(859\) −7.77337 21.3572i −0.265224 0.728697i −0.998795 0.0490840i \(-0.984370\pi\)
0.733571 0.679613i \(-0.237852\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 25.5000 + 14.7224i 0.866025 + 0.500000i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 4.07677 + 4.85851i 0.138136 + 0.164624i
\(872\) 0 0
\(873\) −57.0000 −1.92916
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.92830 3.29623i 0.132649 0.111306i −0.574049 0.818821i \(-0.694628\pi\)
0.706698 + 0.707515i \(0.250184\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(882\) 0 0
\(883\) −39.3626 6.94069i −1.32466 0.233573i −0.533820 0.845598i \(-0.679244\pi\)
−0.790838 + 0.612026i \(0.790355\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(888\) 0 0
\(889\) −59.5271 + 21.6661i −1.99647 + 0.726657i
\(890\) 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
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −54.1311 9.54477i −1.80137 0.317630i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 19.5491 53.7106i 0.649116 1.78343i 0.0281394 0.999604i \(-0.491042\pi\)
0.620977 0.783829i \(-0.286736\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −49.0377 28.3119i −1.61760 0.933924i −0.987538 0.157382i \(-0.949695\pi\)
−0.630065 0.776542i \(-0.716972\pi\)
\(920\) 0 0
\(921\) 8.85606 50.2252i 0.291817 1.65498i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −32.7845 27.5095i −1.07795 0.904506i
\(926\) 0 0
\(927\) −10.3587 28.4603i −0.340224 0.934758i
\(928\) 0 0
\(929\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(930\) 0 0
\(931\) −81.1944 + 35.1582i −2.66104 + 1.15226i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −12.0344 10.0980i −0.393146 0.329888i 0.424691 0.905338i \(-0.360383\pi\)
−0.817837 + 0.575450i \(0.804827\pi\)
\(938\) 0 0
\(939\) 19.5000 11.2583i 0.636358 0.367402i
\(940\) 0 0
\(941\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(948\) 0 0
\(949\) 6.92951 0.224941
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.995825 + 1.72482i 0.0321234 + 0.0556393i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 9.32619 1.64446i 0.299910 0.0528822i −0.0216683 0.999765i \(-0.506898\pi\)
0.321578 + 0.946883i \(0.395787\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(972\) 0 0
\(973\) 106.778 38.8639i 3.42314 1.24592i
\(974\) 0 0
\(975\) −2.72880 + 3.25205i −0.0873915 + 0.104149i
\(976\) 0 0
\(977\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 28.5000 49.3634i 0.909935 1.57605i
\(982\) 0 0
\(983\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −17.5279 20.8890i −0.556793 0.663559i 0.412072 0.911151i \(-0.364805\pi\)
−0.968864 + 0.247592i \(0.920361\pi\)
\(992\) 0 0
\(993\) 56.1871 + 20.4504i 1.78304 + 0.648974i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −10.9647 + 62.1837i −0.347255 + 1.96938i −0.158352 + 0.987383i \(0.550618\pi\)
−0.188903 + 0.981996i \(0.560493\pi\)
\(998\) 0 0
\(999\) −38.5174 + 22.2381i −1.21864 + 0.703581i
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 912.2.ch.a.47.1 6
3.2 odd 2 CM 912.2.ch.a.47.1 6
4.3 odd 2 912.2.ch.b.47.1 yes 6
12.11 even 2 912.2.ch.b.47.1 yes 6
19.17 even 9 912.2.ch.b.815.1 yes 6
57.17 odd 18 912.2.ch.b.815.1 yes 6
76.55 odd 18 inner 912.2.ch.a.815.1 yes 6
228.131 even 18 inner 912.2.ch.a.815.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.2.ch.a.47.1 6 1.1 even 1 trivial
912.2.ch.a.47.1 6 3.2 odd 2 CM
912.2.ch.a.815.1 yes 6 76.55 odd 18 inner
912.2.ch.a.815.1 yes 6 228.131 even 18 inner
912.2.ch.b.47.1 yes 6 4.3 odd 2
912.2.ch.b.47.1 yes 6 12.11 even 2
912.2.ch.b.815.1 yes 6 19.17 even 9
912.2.ch.b.815.1 yes 6 57.17 odd 18