Properties

Label 2304.2.l.g.575.2
Level $2304$
Weight $2$
Character 2304.575
Analytic conductor $18.398$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,2,Mod(575,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("2304.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.l (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: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 575.2
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 2304.575
Dual form 2304.2.l.g.1727.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.517638 + 0.517638i) q^{5} -1.41421 q^{7} +O(q^{10})\) \(q+(-0.517638 + 0.517638i) q^{5} -1.41421 q^{7} +(2.73205 + 2.73205i) q^{11} +(-1.73205 + 1.73205i) q^{13} -0.378937i q^{17} +(0.378937 + 0.378937i) q^{19} -3.46410i q^{23} +4.46410i q^{25} +(-4.76028 - 4.76028i) q^{29} +0.656339i q^{31} +(0.732051 - 0.732051i) q^{35} +(4.46410 + 4.46410i) q^{37} -10.1769 q^{41} +(6.31319 - 6.31319i) q^{43} -10.3923 q^{47} -5.00000 q^{49} +(-4.00240 + 4.00240i) q^{53} -2.82843 q^{55} +(4.92820 + 4.92820i) q^{59} +(-3.00000 + 3.00000i) q^{61} -1.79315i q^{65} +(1.03528 + 1.03528i) q^{67} -14.0000i q^{71} +8.92820i q^{73} +(-3.86370 - 3.86370i) q^{77} +11.9700i q^{79} +(-7.26795 + 7.26795i) q^{83} +(0.196152 + 0.196152i) q^{85} -13.2827 q^{89} +(2.44949 - 2.44949i) q^{91} -0.392305 q^{95} +2.39230 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{11} - 8 q^{35} + 8 q^{37} - 40 q^{49} - 16 q^{59} - 24 q^{61} - 72 q^{83} - 40 q^{85} + 80 q^{95} - 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.517638 + 0.517638i −0.231495 + 0.231495i −0.813316 0.581822i \(-0.802340\pi\)
0.581822 + 0.813316i \(0.302340\pi\)
\(6\) 0 0
\(7\) −1.41421 −0.534522 −0.267261 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.73205 + 2.73205i 0.823744 + 0.823744i 0.986643 0.162899i \(-0.0520843\pi\)
−0.162899 + 0.986643i \(0.552084\pi\)
\(12\) 0 0
\(13\) −1.73205 + 1.73205i −0.480384 + 0.480384i −0.905254 0.424870i \(-0.860320\pi\)
0.424870 + 0.905254i \(0.360320\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.378937i 0.0919058i −0.998944 0.0459529i \(-0.985368\pi\)
0.998944 0.0459529i \(-0.0146324\pi\)
\(18\) 0 0
\(19\) 0.378937 + 0.378937i 0.0869342 + 0.0869342i 0.749237 0.662302i \(-0.230421\pi\)
−0.662302 + 0.749237i \(0.730421\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.46410i 0.722315i −0.932505 0.361158i \(-0.882382\pi\)
0.932505 0.361158i \(-0.117618\pi\)
\(24\) 0 0
\(25\) 4.46410i 0.892820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.76028 4.76028i −0.883962 0.883962i 0.109973 0.993935i \(-0.464924\pi\)
−0.993935 + 0.109973i \(0.964924\pi\)
\(30\) 0 0
\(31\) 0.656339i 0.117882i 0.998261 + 0.0589410i \(0.0187724\pi\)
−0.998261 + 0.0589410i \(0.981228\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.732051 0.732051i 0.123739 0.123739i
\(36\) 0 0
\(37\) 4.46410 + 4.46410i 0.733894 + 0.733894i 0.971389 0.237495i \(-0.0763263\pi\)
−0.237495 + 0.971389i \(0.576326\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.1769 −1.58936 −0.794682 0.607026i \(-0.792362\pi\)
−0.794682 + 0.607026i \(0.792362\pi\)
\(42\) 0 0
\(43\) 6.31319 6.31319i 0.962753 0.962753i −0.0365779 0.999331i \(-0.511646\pi\)
0.999331 + 0.0365779i \(0.0116457\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.3923 −1.51587 −0.757937 0.652328i \(-0.773792\pi\)
−0.757937 + 0.652328i \(0.773792\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00240 + 4.00240i −0.549772 + 0.549772i −0.926375 0.376602i \(-0.877092\pi\)
0.376602 + 0.926375i \(0.377092\pi\)
\(54\) 0 0
\(55\) −2.82843 −0.381385
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.92820 + 4.92820i 0.641597 + 0.641597i 0.950948 0.309351i \(-0.100112\pi\)
−0.309351 + 0.950948i \(0.600112\pi\)
\(60\) 0 0
\(61\) −3.00000 + 3.00000i −0.384111 + 0.384111i −0.872581 0.488470i \(-0.837555\pi\)
0.488470 + 0.872581i \(0.337555\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.79315i 0.222413i
\(66\) 0 0
\(67\) 1.03528 + 1.03528i 0.126479 + 0.126479i 0.767513 0.641034i \(-0.221494\pi\)
−0.641034 + 0.767513i \(0.721494\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.0000i 1.66149i −0.556650 0.830747i \(-0.687914\pi\)
0.556650 0.830747i \(-0.312086\pi\)
\(72\) 0 0
\(73\) 8.92820i 1.04497i 0.852649 + 0.522484i \(0.174994\pi\)
−0.852649 + 0.522484i \(0.825006\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.86370 3.86370i −0.440310 0.440310i
\(78\) 0 0
\(79\) 11.9700i 1.34674i 0.739308 + 0.673368i \(0.235153\pi\)
−0.739308 + 0.673368i \(0.764847\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.26795 + 7.26795i −0.797761 + 0.797761i −0.982742 0.184981i \(-0.940778\pi\)
0.184981 + 0.982742i \(0.440778\pi\)
\(84\) 0 0
\(85\) 0.196152 + 0.196152i 0.0212757 + 0.0212757i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −13.2827 −1.40797 −0.703983 0.710217i \(-0.748597\pi\)
−0.703983 + 0.710217i \(0.748597\pi\)
\(90\) 0 0
\(91\) 2.44949 2.44949i 0.256776 0.256776i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.392305 −0.0402496
\(96\) 0 0
\(97\) 2.39230 0.242902 0.121451 0.992597i \(-0.461245\pi\)
0.121451 + 0.992597i \(0.461245\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.65926 + 9.65926i −0.961132 + 0.961132i −0.999272 0.0381403i \(-0.987857\pi\)
0.0381403 + 0.999272i \(0.487857\pi\)
\(102\) 0 0
\(103\) −18.9396 −1.86617 −0.933086 0.359653i \(-0.882895\pi\)
−0.933086 + 0.359653i \(0.882895\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.4641 + 13.4641i 1.30162 + 1.30162i 0.927297 + 0.374327i \(0.122126\pi\)
0.374327 + 0.927297i \(0.377874\pi\)
\(108\) 0 0
\(109\) −4.26795 + 4.26795i −0.408795 + 0.408795i −0.881318 0.472523i \(-0.843343\pi\)
0.472523 + 0.881318i \(0.343343\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.24264i 0.399114i −0.979886 0.199557i \(-0.936050\pi\)
0.979886 0.199557i \(-0.0639503\pi\)
\(114\) 0 0
\(115\) 1.79315 + 1.79315i 0.167212 + 0.167212i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.535898i 0.0491257i
\(120\) 0 0
\(121\) 3.92820i 0.357109i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4.89898 4.89898i −0.438178 0.438178i
\(126\) 0 0
\(127\) 5.55532i 0.492955i 0.969149 + 0.246477i \(0.0792731\pi\)
−0.969149 + 0.246477i \(0.920727\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.92820 6.92820i 0.605320 0.605320i −0.336399 0.941719i \(-0.609209\pi\)
0.941719 + 0.336399i \(0.109209\pi\)
\(132\) 0 0
\(133\) −0.535898 0.535898i −0.0464683 0.0464683i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.10634 −0.692572 −0.346286 0.938129i \(-0.612557\pi\)
−0.346286 + 0.938129i \(0.612557\pi\)
\(138\) 0 0
\(139\) 8.76268 8.76268i 0.743241 0.743241i −0.229959 0.973200i \(-0.573859\pi\)
0.973200 + 0.229959i \(0.0738593\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −9.46410 −0.791428
\(144\) 0 0
\(145\) 4.92820 0.409265
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.55291 + 1.55291i −0.127220 + 0.127220i −0.767850 0.640630i \(-0.778673\pi\)
0.640630 + 0.767850i \(0.278673\pi\)
\(150\) 0 0
\(151\) 9.14162 0.743934 0.371967 0.928246i \(-0.378683\pi\)
0.371967 + 0.928246i \(0.378683\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.339746 0.339746i −0.0272891 0.0272891i
\(156\) 0 0
\(157\) −9.92820 + 9.92820i −0.792357 + 0.792357i −0.981877 0.189520i \(-0.939307\pi\)
0.189520 + 0.981877i \(0.439307\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.89898i 0.386094i
\(162\) 0 0
\(163\) −9.14162 9.14162i −0.716027 0.716027i 0.251762 0.967789i \(-0.418990\pi\)
−0.967789 + 0.251762i \(0.918990\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.4641i 1.04188i −0.853592 0.520942i \(-0.825581\pi\)
0.853592 0.520942i \(-0.174419\pi\)
\(168\) 0 0
\(169\) 7.00000i 0.538462i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −12.8666 12.8666i −0.978231 0.978231i 0.0215368 0.999768i \(-0.493144\pi\)
−0.999768 + 0.0215368i \(0.993144\pi\)
\(174\) 0 0
\(175\) 6.31319i 0.477233i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.00000 + 6.00000i −0.448461 + 0.448461i −0.894843 0.446382i \(-0.852712\pi\)
0.446382 + 0.894843i \(0.352712\pi\)
\(180\) 0 0
\(181\) 11.1962 + 11.1962i 0.832203 + 0.832203i 0.987818 0.155614i \(-0.0497357\pi\)
−0.155614 + 0.987818i \(0.549736\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.62158 −0.339785
\(186\) 0 0
\(187\) 1.03528 1.03528i 0.0757069 0.0757069i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.53590 0.472921 0.236461 0.971641i \(-0.424013\pi\)
0.236461 + 0.971641i \(0.424013\pi\)
\(192\) 0 0
\(193\) −18.7846 −1.35215 −0.676073 0.736835i \(-0.736320\pi\)
−0.676073 + 0.736835i \(0.736320\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.48288 4.48288i 0.319392 0.319392i −0.529142 0.848533i \(-0.677486\pi\)
0.848533 + 0.529142i \(0.177486\pi\)
\(198\) 0 0
\(199\) −0.101536 −0.00719769 −0.00359885 0.999994i \(-0.501146\pi\)
−0.00359885 + 0.999994i \(0.501146\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.73205 + 6.73205i 0.472497 + 0.472497i
\(204\) 0 0
\(205\) 5.26795 5.26795i 0.367930 0.367930i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.07055i 0.143223i
\(210\) 0 0
\(211\) 15.1774 + 15.1774i 1.04486 + 1.04486i 0.998946 + 0.0459106i \(0.0146189\pi\)
0.0459106 + 0.998946i \(0.485381\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.53590i 0.445745i
\(216\) 0 0
\(217\) 0.928203i 0.0630105i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.656339 + 0.656339i 0.0441501 + 0.0441501i
\(222\) 0 0
\(223\) 12.7279i 0.852325i −0.904647 0.426162i \(-0.859865\pi\)
0.904647 0.426162i \(-0.140135\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.2679 11.2679i 0.747880 0.747880i −0.226201 0.974081i \(-0.572631\pi\)
0.974081 + 0.226201i \(0.0726305\pi\)
\(228\) 0 0
\(229\) −3.19615 3.19615i −0.211208 0.211208i 0.593573 0.804780i \(-0.297717\pi\)
−0.804780 + 0.593573i \(0.797717\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.6574 0.698188 0.349094 0.937088i \(-0.386489\pi\)
0.349094 + 0.937088i \(0.386489\pi\)
\(234\) 0 0
\(235\) 5.37945 5.37945i 0.350917 0.350917i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −14.5359 −0.940249 −0.470125 0.882600i \(-0.655791\pi\)
−0.470125 + 0.882600i \(0.655791\pi\)
\(240\) 0 0
\(241\) 13.4641 0.867299 0.433650 0.901082i \(-0.357226\pi\)
0.433650 + 0.901082i \(0.357226\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.58819 2.58819i 0.165353 0.165353i
\(246\) 0 0
\(247\) −1.31268 −0.0835237
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −9.66025 9.66025i −0.609750 0.609750i 0.333131 0.942881i \(-0.391895\pi\)
−0.942881 + 0.333131i \(0.891895\pi\)
\(252\) 0 0
\(253\) 9.46410 9.46410i 0.595003 0.595003i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.17209i 0.135491i −0.997703 0.0677456i \(-0.978419\pi\)
0.997703 0.0677456i \(-0.0215806\pi\)
\(258\) 0 0
\(259\) −6.31319 6.31319i −0.392283 0.392283i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 10.9282i 0.673862i −0.941529 0.336931i \(-0.890611\pi\)
0.941529 0.336931i \(-0.109389\pi\)
\(264\) 0 0
\(265\) 4.14359i 0.254539i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 14.2808 + 14.2808i 0.870718 + 0.870718i 0.992551 0.121833i \(-0.0388771\pi\)
−0.121833 + 0.992551i \(0.538877\pi\)
\(270\) 0 0
\(271\) 22.5259i 1.36835i 0.729318 + 0.684175i \(0.239838\pi\)
−0.729318 + 0.684175i \(0.760162\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −12.1962 + 12.1962i −0.735456 + 0.735456i
\(276\) 0 0
\(277\) 0.803848 + 0.803848i 0.0482985 + 0.0482985i 0.730844 0.682545i \(-0.239127\pi\)
−0.682545 + 0.730844i \(0.739127\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.656339 0.0391539 0.0195769 0.999808i \(-0.493768\pi\)
0.0195769 + 0.999808i \(0.493768\pi\)
\(282\) 0 0
\(283\) −19.7990 + 19.7990i −1.17693 + 1.17693i −0.196405 + 0.980523i \(0.562927\pi\)
−0.980523 + 0.196405i \(0.937073\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 14.3923 0.849551
\(288\) 0 0
\(289\) 16.8564 0.991553
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.65445 + 1.65445i −0.0966540 + 0.0966540i −0.753780 0.657126i \(-0.771772\pi\)
0.657126 + 0.753780i \(0.271772\pi\)
\(294\) 0 0
\(295\) −5.10205 −0.297053
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.00000 + 6.00000i 0.346989 + 0.346989i
\(300\) 0 0
\(301\) −8.92820 + 8.92820i −0.514613 + 0.514613i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.10583i 0.177839i
\(306\) 0 0
\(307\) −3.86370 3.86370i −0.220513 0.220513i 0.588201 0.808715i \(-0.299836\pi\)
−0.808715 + 0.588201i \(0.799836\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 29.8564i 1.69300i 0.532388 + 0.846501i \(0.321295\pi\)
−0.532388 + 0.846501i \(0.678705\pi\)
\(312\) 0 0
\(313\) 2.00000i 0.113047i 0.998401 + 0.0565233i \(0.0180015\pi\)
−0.998401 + 0.0565233i \(0.981998\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 19.1798 + 19.1798i 1.07725 + 1.07725i 0.996755 + 0.0804904i \(0.0256486\pi\)
0.0804904 + 0.996755i \(0.474351\pi\)
\(318\) 0 0
\(319\) 26.0106i 1.45632i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.143594 0.143594i 0.00798976 0.00798976i
\(324\) 0 0
\(325\) −7.73205 7.73205i −0.428897 0.428897i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 14.6969 0.810268
\(330\) 0 0
\(331\) −8.00481 + 8.00481i −0.439984 + 0.439984i −0.892007 0.452022i \(-0.850703\pi\)
0.452022 + 0.892007i \(0.350703\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.07180 −0.0585585
\(336\) 0 0
\(337\) −18.7846 −1.02326 −0.511631 0.859205i \(-0.670959\pi\)
−0.511631 + 0.859205i \(0.670959\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.79315 + 1.79315i −0.0971046 + 0.0971046i
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −25.1244 25.1244i −1.34875 1.34875i −0.887025 0.461721i \(-0.847232\pi\)
−0.461721 0.887025i \(-0.652768\pi\)
\(348\) 0 0
\(349\) 7.39230 7.39230i 0.395701 0.395701i −0.481013 0.876714i \(-0.659731\pi\)
0.876714 + 0.481013i \(0.159731\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 17.4238i 0.927377i −0.885998 0.463688i \(-0.846526\pi\)
0.885998 0.463688i \(-0.153474\pi\)
\(354\) 0 0
\(355\) 7.24693 + 7.24693i 0.384627 + 0.384627i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.85641i 0.203533i 0.994808 + 0.101767i \(0.0324495\pi\)
−0.994808 + 0.101767i \(0.967550\pi\)
\(360\) 0 0
\(361\) 18.7128i 0.984885i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4.62158 4.62158i −0.241904 0.241904i
\(366\) 0 0
\(367\) 18.3848i 0.959678i 0.877357 + 0.479839i \(0.159305\pi\)
−0.877357 + 0.479839i \(0.840695\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 5.66025 5.66025i 0.293866 0.293866i
\(372\) 0 0
\(373\) −22.8564 22.8564i −1.18346 1.18346i −0.978842 0.204618i \(-0.934405\pi\)
−0.204618 0.978842i \(-0.565595\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.4901 0.849283
\(378\) 0 0
\(379\) −8.10634 + 8.10634i −0.416395 + 0.416395i −0.883959 0.467564i \(-0.845132\pi\)
0.467564 + 0.883959i \(0.345132\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 25.4641 1.30115 0.650577 0.759440i \(-0.274527\pi\)
0.650577 + 0.759440i \(0.274527\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 23.6999 23.6999i 1.20163 1.20163i 0.227960 0.973670i \(-0.426794\pi\)
0.973670 0.227960i \(-0.0732055\pi\)
\(390\) 0 0
\(391\) −1.31268 −0.0663850
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −6.19615 6.19615i −0.311762 0.311762i
\(396\) 0 0
\(397\) 16.3205 16.3205i 0.819103 0.819103i −0.166875 0.985978i \(-0.553368\pi\)
0.985978 + 0.166875i \(0.0533677\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 17.3495i 0.866393i 0.901300 + 0.433196i \(0.142614\pi\)
−0.901300 + 0.433196i \(0.857386\pi\)
\(402\) 0 0
\(403\) −1.13681 1.13681i −0.0566286 0.0566286i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.3923i 1.20908i
\(408\) 0 0
\(409\) 14.3923i 0.711654i −0.934552 0.355827i \(-0.884199\pi\)
0.934552 0.355827i \(-0.115801\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.96953 6.96953i −0.342948 0.342948i
\(414\) 0 0
\(415\) 7.52433i 0.369355i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 7.26795 7.26795i 0.355063 0.355063i −0.506927 0.861989i \(-0.669219\pi\)
0.861989 + 0.506927i \(0.169219\pi\)
\(420\) 0 0
\(421\) 17.7321 + 17.7321i 0.864207 + 0.864207i 0.991824 0.127616i \(-0.0407326\pi\)
−0.127616 + 0.991824i \(0.540733\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.69161 0.0820554
\(426\) 0 0
\(427\) 4.24264 4.24264i 0.205316 0.205316i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.67949 −0.129067 −0.0645333 0.997916i \(-0.520556\pi\)
−0.0645333 + 0.997916i \(0.520556\pi\)
\(432\) 0 0
\(433\) −1.07180 −0.0515073 −0.0257536 0.999668i \(-0.508199\pi\)
−0.0257536 + 0.999668i \(0.508199\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.31268 1.31268i 0.0627939 0.0627939i
\(438\) 0 0
\(439\) 29.6985 1.41743 0.708716 0.705494i \(-0.249275\pi\)
0.708716 + 0.705494i \(0.249275\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −26.5885 26.5885i −1.26326 1.26326i −0.949506 0.313750i \(-0.898415\pi\)
−0.313750 0.949506i \(-0.601585\pi\)
\(444\) 0 0
\(445\) 6.87564 6.87564i 0.325937 0.325937i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.6622i 0.880723i 0.897821 + 0.440361i \(0.145150\pi\)
−0.897821 + 0.440361i \(0.854850\pi\)
\(450\) 0 0
\(451\) −27.8038 27.8038i −1.30923 1.30923i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.53590i 0.118885i
\(456\) 0 0
\(457\) 21.3205i 0.997331i −0.866794 0.498666i \(-0.833824\pi\)
0.866794 0.498666i \(-0.166176\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 25.2156 + 25.2156i 1.17441 + 1.17441i 0.981147 + 0.193260i \(0.0619062\pi\)
0.193260 + 0.981147i \(0.438094\pi\)
\(462\) 0 0
\(463\) 35.1523i 1.63366i −0.576875 0.816832i \(-0.695728\pi\)
0.576875 0.816832i \(-0.304272\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.33975 + 4.33975i −0.200819 + 0.200819i −0.800351 0.599532i \(-0.795354\pi\)
0.599532 + 0.800351i \(0.295354\pi\)
\(468\) 0 0
\(469\) −1.46410 1.46410i −0.0676059 0.0676059i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 34.4959 1.58612
\(474\) 0 0
\(475\) −1.69161 + 1.69161i −0.0776166 + 0.0776166i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 30.7846 1.40658 0.703292 0.710901i \(-0.251712\pi\)
0.703292 + 0.710901i \(0.251712\pi\)
\(480\) 0 0
\(481\) −15.4641 −0.705102
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.23835 + 1.23835i −0.0562305 + 0.0562305i
\(486\) 0 0
\(487\) 25.9091 1.17405 0.587027 0.809567i \(-0.300298\pi\)
0.587027 + 0.809567i \(0.300298\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0.143594 + 0.143594i 0.00648029 + 0.00648029i 0.710340 0.703859i \(-0.248541\pi\)
−0.703859 + 0.710340i \(0.748541\pi\)
\(492\) 0 0
\(493\) −1.80385 + 1.80385i −0.0812412 + 0.0812412i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 19.7990i 0.888106i
\(498\) 0 0
\(499\) 27.3233 + 27.3233i 1.22316 + 1.22316i 0.966502 + 0.256658i \(0.0826213\pi\)
0.256658 + 0.966502i \(0.417379\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 11.0718i 0.493667i 0.969058 + 0.246834i \(0.0793901\pi\)
−0.969058 + 0.246834i \(0.920610\pi\)
\(504\) 0 0
\(505\) 10.0000i 0.444994i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 11.1750 + 11.1750i 0.495324 + 0.495324i 0.909979 0.414655i \(-0.136098\pi\)
−0.414655 + 0.909979i \(0.636098\pi\)
\(510\) 0 0
\(511\) 12.6264i 0.558558i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9.80385 9.80385i 0.432009 0.432009i
\(516\) 0 0
\(517\) −28.3923 28.3923i −1.24869 1.24869i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 21.2875 0.932624 0.466312 0.884620i \(-0.345582\pi\)
0.466312 + 0.884620i \(0.345582\pi\)
\(522\) 0 0
\(523\) 1.13681 1.13681i 0.0497093 0.0497093i −0.681815 0.731524i \(-0.738809\pi\)
0.731524 + 0.681815i \(0.238809\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.248711 0.0108340
\(528\) 0 0
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 17.6269 17.6269i 0.763506 0.763506i
\(534\) 0 0
\(535\) −13.9391 −0.602638
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −13.6603 13.6603i −0.588389 0.588389i
\(540\) 0 0
\(541\) 11.0526 11.0526i 0.475187 0.475187i −0.428402 0.903588i \(-0.640923\pi\)
0.903588 + 0.428402i \(0.140923\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.41851i 0.189268i
\(546\) 0 0
\(547\) −16.1112 16.1112i −0.688863 0.688863i 0.273118 0.961981i \(-0.411945\pi\)
−0.961981 + 0.273118i \(0.911945\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.60770i 0.153693i
\(552\) 0 0
\(553\) 16.9282i 0.719860i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.83032 + 1.83032i 0.0775530 + 0.0775530i 0.744819 0.667266i \(-0.232536\pi\)
−0.667266 + 0.744819i \(0.732536\pi\)
\(558\) 0 0
\(559\) 21.8695i 0.924983i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −18.0526 + 18.0526i −0.760825 + 0.760825i −0.976471 0.215647i \(-0.930814\pi\)
0.215647 + 0.976471i \(0.430814\pi\)
\(564\) 0 0
\(565\) 2.19615 + 2.19615i 0.0923928 + 0.0923928i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.0759 0.632014 0.316007 0.948757i \(-0.397658\pi\)
0.316007 + 0.948757i \(0.397658\pi\)
\(570\) 0 0
\(571\) −19.7990 + 19.7990i −0.828562 + 0.828562i −0.987318 0.158756i \(-0.949252\pi\)
0.158756 + 0.987318i \(0.449252\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 15.4641 0.644898
\(576\) 0 0
\(577\) 38.9282 1.62060 0.810301 0.586014i \(-0.199304\pi\)
0.810301 + 0.586014i \(0.199304\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 10.2784 10.2784i 0.426421 0.426421i
\(582\) 0 0
\(583\) −21.8695 −0.905744
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.07180 + 3.07180i 0.126787 + 0.126787i 0.767653 0.640866i \(-0.221425\pi\)
−0.640866 + 0.767653i \(0.721425\pi\)
\(588\) 0 0
\(589\) −0.248711 + 0.248711i −0.0102480 + 0.0102480i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 32.3238i 1.32738i −0.748007 0.663691i \(-0.768989\pi\)
0.748007 0.663691i \(-0.231011\pi\)
\(594\) 0 0
\(595\) −0.277401 0.277401i −0.0113723 0.0113723i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 33.3205i 1.36144i 0.732544 + 0.680720i \(0.238333\pi\)
−0.732544 + 0.680720i \(0.761667\pi\)
\(600\) 0 0
\(601\) 12.7846i 0.521495i −0.965407 0.260748i \(-0.916031\pi\)
0.965407 0.260748i \(-0.0839690\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.03339 2.03339i −0.0826690 0.0826690i
\(606\) 0 0
\(607\) 17.6269i 0.715454i −0.933826 0.357727i \(-0.883552\pi\)
0.933826 0.357727i \(-0.116448\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 18.0000 18.0000i 0.728202 0.728202i
\(612\) 0 0
\(613\) 12.6077 + 12.6077i 0.509220 + 0.509220i 0.914287 0.405067i \(-0.132752\pi\)
−0.405067 + 0.914287i \(0.632752\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.07107 0.284670 0.142335 0.989819i \(-0.454539\pi\)
0.142335 + 0.989819i \(0.454539\pi\)
\(618\) 0 0
\(619\) −20.3538 + 20.3538i −0.818088 + 0.818088i −0.985831 0.167743i \(-0.946352\pi\)
0.167743 + 0.985831i \(0.446352\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 18.7846 0.752589
\(624\) 0 0
\(625\) −17.2487 −0.689948
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.69161 1.69161i 0.0674491 0.0674491i
\(630\) 0 0
\(631\) 29.2923 1.16611 0.583055 0.812433i \(-0.301857\pi\)
0.583055 + 0.812433i \(0.301857\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.87564 2.87564i −0.114116 0.114116i
\(636\) 0 0
\(637\) 8.66025 8.66025i 0.343132 0.343132i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 11.4896i 0.453811i 0.973917 + 0.226905i \(0.0728608\pi\)
−0.973917 + 0.226905i \(0.927139\pi\)
\(642\) 0 0
\(643\) 25.9091 + 25.9091i 1.02176 + 1.02176i 0.999758 + 0.0219976i \(0.00700261\pi\)
0.0219976 + 0.999758i \(0.492997\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.3923i 0.723076i 0.932357 + 0.361538i \(0.117748\pi\)
−0.932357 + 0.361538i \(0.882252\pi\)
\(648\) 0 0
\(649\) 26.9282i 1.05702i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 20.0392 + 20.0392i 0.784196 + 0.784196i 0.980536 0.196340i \(-0.0629057\pi\)
−0.196340 + 0.980536i \(0.562906\pi\)
\(654\) 0 0
\(655\) 7.17260i 0.280257i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −6.39230 + 6.39230i −0.249009 + 0.249009i −0.820564 0.571555i \(-0.806340\pi\)
0.571555 + 0.820564i \(0.306340\pi\)
\(660\) 0 0
\(661\) −21.7846 21.7846i −0.847323 0.847323i 0.142475 0.989798i \(-0.454494\pi\)
−0.989798 + 0.142475i \(0.954494\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.554803 0.0215143
\(666\) 0 0
\(667\) −16.4901 + 16.4901i −0.638499 + 0.638499i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −16.3923 −0.632818
\(672\) 0 0
\(673\) −8.78461 −0.338622 −0.169311 0.985563i \(-0.554154\pi\)
−0.169311 + 0.985563i \(0.554154\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −31.9077 + 31.9077i −1.22631 + 1.22631i −0.260966 + 0.965348i \(0.584041\pi\)
−0.965348 + 0.260966i \(0.915959\pi\)
\(678\) 0 0
\(679\) −3.38323 −0.129836
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −11.2679 11.2679i −0.431156 0.431156i 0.457865 0.889022i \(-0.348614\pi\)
−0.889022 + 0.457865i \(0.848614\pi\)
\(684\) 0 0
\(685\) 4.19615 4.19615i 0.160327 0.160327i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 13.8647i 0.528204i
\(690\) 0 0
\(691\) −23.8386 23.8386i −0.906862 0.906862i 0.0891562 0.996018i \(-0.471583\pi\)
−0.996018 + 0.0891562i \(0.971583\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.07180i 0.344113i
\(696\) 0 0
\(697\) 3.85641i 0.146072i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 16.9062 + 16.9062i 0.638538 + 0.638538i 0.950195 0.311657i \(-0.100884\pi\)
−0.311657 + 0.950195i \(0.600884\pi\)
\(702\) 0 0
\(703\) 3.38323i 0.127601i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13.6603 13.6603i 0.513747 0.513747i
\(708\) 0 0
\(709\) −5.19615 5.19615i −0.195146 0.195146i 0.602770 0.797915i \(-0.294064\pi\)
−0.797915 + 0.602770i \(0.794064\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.27362 0.0851479
\(714\) 0 0
\(715\) 4.89898 4.89898i 0.183211 0.183211i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −25.7128 −0.958926 −0.479463 0.877562i \(-0.659169\pi\)
−0.479463 + 0.877562i \(0.659169\pi\)
\(720\) 0 0
\(721\) 26.7846 0.997511
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 21.2504 21.2504i 0.789219 0.789219i
\(726\) 0 0
\(727\) 18.9396 0.702430 0.351215 0.936295i \(-0.385769\pi\)
0.351215 + 0.936295i \(0.385769\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.39230 2.39230i −0.0884826 0.0884826i
\(732\) 0 0
\(733\) −37.4449 + 37.4449i −1.38306 + 1.38306i −0.543921 + 0.839136i \(0.683061\pi\)
−0.839136 + 0.543921i \(0.816939\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.65685i 0.208373i
\(738\) 0 0
\(739\) −12.6264 12.6264i −0.464469 0.464469i 0.435648 0.900117i \(-0.356519\pi\)
−0.900117 + 0.435648i \(0.856519\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 40.1051i 1.47131i −0.677354 0.735657i \(-0.736874\pi\)
0.677354 0.735657i \(-0.263126\pi\)
\(744\) 0 0
\(745\) 1.60770i 0.0589014i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −19.0411 19.0411i −0.695747 0.695747i
\(750\) 0 0
\(751\) 14.7985i 0.540004i −0.962860 0.270002i \(-0.912976\pi\)
0.962860 0.270002i \(-0.0870244\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −4.73205 + 4.73205i −0.172217 + 0.172217i
\(756\) 0 0
\(757\) 34.6603 + 34.6603i 1.25975 + 1.25975i 0.951213 + 0.308536i \(0.0998389\pi\)
0.308536 + 0.951213i \(0.400161\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −17.1464 −0.621558 −0.310779 0.950482i \(-0.600590\pi\)
−0.310779 + 0.950482i \(0.600590\pi\)
\(762\) 0 0
\(763\) 6.03579 6.03579i 0.218510 0.218510i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −17.0718 −0.616427
\(768\) 0 0
\(769\) −45.7128 −1.64845 −0.824223 0.566265i \(-0.808388\pi\)
−0.824223 + 0.566265i \(0.808388\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 34.3572 34.3572i 1.23574 1.23574i 0.274019 0.961724i \(-0.411647\pi\)
0.961724 0.274019i \(-0.0883533\pi\)
\(774\) 0 0
\(775\) −2.92996 −0.105247
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.85641 3.85641i −0.138170 0.138170i
\(780\) 0 0
\(781\) 38.2487 38.2487i 1.36865 1.36865i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 10.2784i 0.366853i
\(786\) 0 0
\(787\) 24.3190 + 24.3190i 0.866880 + 0.866880i 0.992126 0.125246i \(-0.0399720\pi\)
−0.125246 + 0.992126i \(0.539972\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.00000i 0.213335i
\(792\) 0 0
\(793\) 10.3923i 0.369042i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −9.76079 9.76079i −0.345745 0.345745i 0.512777 0.858522i \(-0.328617\pi\)
−0.858522 + 0.512777i \(0.828617\pi\)
\(798\) 0 0
\(799\) 3.93803i 0.139318i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −24.3923 + 24.3923i −0.860786 + 0.860786i
\(804\) 0 0
\(805\) −2.53590 2.53590i −0.0893787 0.0893787i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −12.4505 −0.437737 −0.218868 0.975754i \(-0.570237\pi\)
−0.218868 + 0.975754i \(0.570237\pi\)
\(810\) 0 0
\(811\) 21.4906 21.4906i 0.754637 0.754637i −0.220704 0.975341i \(-0.570835\pi\)
0.975341 + 0.220704i \(0.0708354\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9.46410 0.331513
\(816\) 0 0
\(817\) 4.78461 0.167392
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −14.5582 + 14.5582i −0.508086 + 0.508086i −0.913939 0.405853i \(-0.866975\pi\)
0.405853 + 0.913939i \(0.366975\pi\)
\(822\) 0 0
\(823\) −17.4238 −0.607357 −0.303678 0.952775i \(-0.598215\pi\)
−0.303678 + 0.952775i \(0.598215\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 4.39230 + 4.39230i 0.152735 + 0.152735i 0.779339 0.626603i \(-0.215555\pi\)
−0.626603 + 0.779339i \(0.715555\pi\)
\(828\) 0 0
\(829\) 15.1962 15.1962i 0.527784 0.527784i −0.392127 0.919911i \(-0.628261\pi\)
0.919911 + 0.392127i \(0.128261\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.89469i 0.0656470i
\(834\) 0 0
\(835\) 6.96953 + 6.96953i 0.241191 + 0.241191i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9.21539i 0.318151i −0.987266 0.159075i \(-0.949149\pi\)
0.987266 0.159075i \(-0.0508513\pi\)
\(840\) 0 0
\(841\) 16.3205i 0.562776i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3.62347 3.62347i −0.124651 0.124651i
\(846\) 0 0
\(847\) 5.55532i 0.190883i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 15.4641 15.4641i 0.530103 0.530103i
\(852\) 0 0
\(853\) 5.78461 + 5.78461i 0.198061 + 0.198061i 0.799168 0.601107i \(-0.205274\pi\)
−0.601107 + 0.799168i \(0.705274\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9.62209 0.328684 0.164342 0.986403i \(-0.447450\pi\)
0.164342 + 0.986403i \(0.447450\pi\)
\(858\) 0 0
\(859\) 17.4238 17.4238i 0.594493 0.594493i −0.344349 0.938842i \(-0.611900\pi\)
0.938842 + 0.344349i \(0.111900\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 32.3923 1.10265 0.551323 0.834292i \(-0.314123\pi\)
0.551323 + 0.834292i \(0.314123\pi\)
\(864\) 0 0
\(865\) 13.3205 0.452911
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −32.7028 + 32.7028i −1.10937 + 1.10937i
\(870\) 0 0
\(871\) −3.58630 −0.121517
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 6.92820 + 6.92820i 0.234216 + 0.234216i
\(876\) 0 0
\(877\) −18.4641 + 18.4641i −0.623488 + 0.623488i −0.946422 0.322933i \(-0.895331\pi\)
0.322933 + 0.946422i \(0.395331\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 27.4249i 0.923967i 0.886889 + 0.461984i \(0.152862\pi\)
−0.886889 + 0.461984i \(0.847138\pi\)
\(882\) 0 0
\(883\) 8.58682 + 8.58682i 0.288969 + 0.288969i 0.836673 0.547703i \(-0.184498\pi\)
−0.547703 + 0.836673i \(0.684498\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 33.0718i 1.11044i −0.831703 0.555221i \(-0.812634\pi\)
0.831703 0.555221i \(-0.187366\pi\)
\(888\) 0 0
\(889\) 7.85641i 0.263495i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3.93803 3.93803i −0.131781 0.131781i
\(894\) 0 0
\(895\) 6.21166i 0.207633i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.12436 3.12436i 0.104203 0.104203i
\(900\) 0 0
\(901\) 1.51666 + 1.51666i 0.0505273 + 0.0505273i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −11.5911 −0.385302
\(906\) 0 0
\(907\) 8.38375 8.38375i 0.278378 0.278378i −0.554083 0.832461i \(-0.686931\pi\)
0.832461 + 0.554083i \(0.186931\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −49.1769 −1.62930 −0.814652 0.579950i \(-0.803072\pi\)
−0.814652 + 0.579950i \(0.803072\pi\)
\(912\) 0 0
\(913\) −39.7128 −1.31430
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.79796 + 9.79796i −0.323557 + 0.323557i
\(918\) 0 0
\(919\) −2.92996 −0.0966506 −0.0483253 0.998832i \(-0.515388\pi\)
−0.0483253 + 0.998832i \(0.515388\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 24.2487 + 24.2487i 0.798156 + 0.798156i
\(924\) 0 0
\(925\) −19.9282 + 19.9282i −0.655235 + 0.655235i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 17.1464i 0.562556i 0.959626 + 0.281278i \(0.0907583\pi\)
−0.959626 + 0.281278i \(0.909242\pi\)
\(930\) 0 0
\(931\) −1.89469 1.89469i −0.0620959 0.0620959i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.07180i 0.0350515i
\(936\) 0 0
\(937\) 30.0000i 0.980057i 0.871706 + 0.490029i \(0.163014\pi\)
−0.871706 + 0.490029i \(0.836986\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −17.2108 17.2108i −0.561056 0.561056i 0.368551 0.929607i \(-0.379854\pi\)
−0.929607 + 0.368551i \(0.879854\pi\)
\(942\) 0 0
\(943\) 35.2538i 1.14802i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9.60770 + 9.60770i −0.312208 + 0.312208i −0.845765 0.533556i \(-0.820855\pi\)
0.533556 + 0.845765i \(0.320855\pi\)
\(948\) 0 0
\(949\) −15.4641 15.4641i −0.501986 0.501986i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −38.2581 −1.23930 −0.619651 0.784878i \(-0.712726\pi\)
−0.619651 + 0.784878i \(0.712726\pi\)
\(954\) 0 0
\(955\) −3.38323 + 3.38323i −0.109479 + 0.109479i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 11.4641 0.370195
\(960\) 0 0
\(961\) 30.5692 0.986104
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 9.72363 9.72363i 0.313015 0.313015i
\(966\) 0 0
\(967\) −29.2923 −0.941978 −0.470989 0.882139i \(-0.656103\pi\)
−0.470989 + 0.882139i \(0.656103\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 34.9808 + 34.9808i 1.12259 + 1.12259i 0.991351 + 0.131235i \(0.0418942\pi\)
0.131235 + 0.991351i \(0.458106\pi\)
\(972\) 0 0
\(973\) −12.3923 + 12.3923i −0.397279 + 0.397279i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 13.0053i 0.416077i 0.978121 + 0.208039i \(0.0667080\pi\)
−0.978121 + 0.208039i \(0.933292\pi\)
\(978\) 0 0
\(979\) −36.2891 36.2891i −1.15980 1.15980i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 30.6410i 0.977297i −0.872481 0.488648i \(-0.837490\pi\)
0.872481 0.488648i \(-0.162510\pi\)
\(984\) 0 0
\(985\) 4.64102i 0.147875i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −21.8695 21.8695i −0.695411 0.695411i
\(990\) 0 0
\(991\) 45.7081i 1.45197i 0.687713 + 0.725983i \(0.258615\pi\)
−0.687713 + 0.725983i \(0.741385\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0.0525589 0.0525589i 0.00166623 0.00166623i
\(996\) 0 0
\(997\) 21.0000 + 21.0000i 0.665077 + 0.665077i 0.956572 0.291496i \(-0.0941528\pi\)
−0.291496 + 0.956572i \(0.594153\pi\)
\(998\) 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 2304.2.l.g.575.2 yes 8
3.2 odd 2 2304.2.l.c.575.3 yes 8
4.3 odd 2 2304.2.l.c.575.2 yes 8
8.3 odd 2 2304.2.l.f.575.3 yes 8
8.5 even 2 2304.2.l.b.575.3 yes 8
12.11 even 2 inner 2304.2.l.g.575.3 yes 8
16.3 odd 4 2304.2.l.c.1727.3 yes 8
16.5 even 4 2304.2.l.b.1727.2 yes 8
16.11 odd 4 2304.2.l.f.1727.2 yes 8
16.13 even 4 inner 2304.2.l.g.1727.3 yes 8
24.5 odd 2 2304.2.l.f.575.2 yes 8
24.11 even 2 2304.2.l.b.575.2 8
48.5 odd 4 2304.2.l.f.1727.3 yes 8
48.11 even 4 2304.2.l.b.1727.3 yes 8
48.29 odd 4 2304.2.l.c.1727.2 yes 8
48.35 even 4 inner 2304.2.l.g.1727.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2304.2.l.b.575.2 8 24.11 even 2
2304.2.l.b.575.3 yes 8 8.5 even 2
2304.2.l.b.1727.2 yes 8 16.5 even 4
2304.2.l.b.1727.3 yes 8 48.11 even 4
2304.2.l.c.575.2 yes 8 4.3 odd 2
2304.2.l.c.575.3 yes 8 3.2 odd 2
2304.2.l.c.1727.2 yes 8 48.29 odd 4
2304.2.l.c.1727.3 yes 8 16.3 odd 4
2304.2.l.f.575.2 yes 8 24.5 odd 2
2304.2.l.f.575.3 yes 8 8.3 odd 2
2304.2.l.f.1727.2 yes 8 16.11 odd 4
2304.2.l.f.1727.3 yes 8 48.5 odd 4
2304.2.l.g.575.2 yes 8 1.1 even 1 trivial
2304.2.l.g.575.3 yes 8 12.11 even 2 inner
2304.2.l.g.1727.2 yes 8 48.35 even 4 inner
2304.2.l.g.1727.3 yes 8 16.13 even 4 inner