Properties

Label 2432.2.c.i.1217.15
Level $2432$
Weight $2$
Character 2432.1217
Analytic conductor $19.420$
Analytic rank $0$
Dimension $16$
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] = [2432,2,Mod(1217,2432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2432.1217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2432 = 2^{7} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2432.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.4196177716\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 8 x^{14} - 8 x^{13} - 3 x^{12} + 20 x^{11} - 24 x^{10} + 28 x^{8} - 96 x^{6} + 160 x^{5} - 48 x^{4} - 256 x^{3} + 512 x^{2} - 512 x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1217.15
Root \(0.569934 - 1.29429i\) of defining polynomial
Character \(\chi\) \(=\) 2432.1217
Dual form 2432.2.c.i.1217.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.63640i q^{3} -2.63403i q^{5} +3.88603 q^{7} -3.95063 q^{9} +O(q^{10})\) \(q+2.63640i q^{3} -2.63403i q^{5} +3.88603 q^{7} -3.95063 q^{9} -5.95063i q^{11} +4.14940i q^{13} +6.94438 q^{15} +1.82843 q^{17} -1.00000i q^{19} +10.2452i q^{21} -4.14940 q^{23} -1.93813 q^{25} -2.50625i q^{27} -8.19638i q^{29} -1.67631 q^{31} +15.6883 q^{33} -10.2359i q^{35} +8.14452i q^{37} -10.9395 q^{39} +0.514201 q^{41} -10.9631i q^{43} +10.4061i q^{45} +13.3035 q^{47} +8.10124 q^{49} +4.82047i q^{51} -1.62446i q^{53} -15.6742 q^{55} +2.63640 q^{57} -12.2234i q^{59} -5.98666i q^{61} -15.3523 q^{63} +10.9297 q^{65} -4.09328i q^{67} -10.9395i q^{69} +11.8696 q^{71} +9.28531 q^{73} -5.10971i q^{75} -23.1243i q^{77} +2.42124 q^{79} -5.24441 q^{81} +0.987504i q^{83} -4.81614i q^{85} +21.6090 q^{87} +2.25389 q^{89} +16.1247i q^{91} -4.41944i q^{93} -2.63403 q^{95} +9.27281 q^{97} +23.5087i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{9} - 16 q^{17} - 40 q^{25} + 48 q^{33} - 16 q^{41} + 16 q^{49} + 8 q^{57} + 16 q^{65} + 16 q^{73} - 64 q^{81} + 16 q^{89} + 80 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times\).

\(n\) \(1407\) \(1921\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 2.63640i 1.52213i 0.648676 + 0.761065i \(0.275323\pi\)
−0.648676 + 0.761065i \(0.724677\pi\)
\(4\) 0 0
\(5\) − 2.63403i − 1.17798i −0.808142 0.588988i \(-0.799527\pi\)
0.808142 0.588988i \(-0.200473\pi\)
\(6\) 0 0
\(7\) 3.88603 1.46878 0.734391 0.678727i \(-0.237468\pi\)
0.734391 + 0.678727i \(0.237468\pi\)
\(8\) 0 0
\(9\) −3.95063 −1.31688
\(10\) 0 0
\(11\) − 5.95063i − 1.79418i −0.441845 0.897091i \(-0.645676\pi\)
0.441845 0.897091i \(-0.354324\pi\)
\(12\) 0 0
\(13\) 4.14940i 1.15084i 0.817859 + 0.575418i \(0.195161\pi\)
−0.817859 + 0.575418i \(0.804839\pi\)
\(14\) 0 0
\(15\) 6.94438 1.79303
\(16\) 0 0
\(17\) 1.82843 0.443459 0.221729 0.975108i \(-0.428830\pi\)
0.221729 + 0.975108i \(0.428830\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i
\(20\) 0 0
\(21\) 10.2452i 2.23568i
\(22\) 0 0
\(23\) −4.14940 −0.865210 −0.432605 0.901584i \(-0.642406\pi\)
−0.432605 + 0.901584i \(0.642406\pi\)
\(24\) 0 0
\(25\) −1.93813 −0.387627
\(26\) 0 0
\(27\) − 2.50625i − 0.482328i
\(28\) 0 0
\(29\) − 8.19638i − 1.52203i −0.648735 0.761015i \(-0.724702\pi\)
0.648735 0.761015i \(-0.275298\pi\)
\(30\) 0 0
\(31\) −1.67631 −0.301075 −0.150537 0.988604i \(-0.548100\pi\)
−0.150537 + 0.988604i \(0.548100\pi\)
\(32\) 0 0
\(33\) 15.6883 2.73098
\(34\) 0 0
\(35\) − 10.2359i − 1.73019i
\(36\) 0 0
\(37\) 8.14452i 1.33895i 0.742834 + 0.669476i \(0.233481\pi\)
−0.742834 + 0.669476i \(0.766519\pi\)
\(38\) 0 0
\(39\) −10.9395 −1.75172
\(40\) 0 0
\(41\) 0.514201 0.0803047 0.0401524 0.999194i \(-0.487216\pi\)
0.0401524 + 0.999194i \(0.487216\pi\)
\(42\) 0 0
\(43\) − 10.9631i − 1.67186i −0.548835 0.835931i \(-0.684929\pi\)
0.548835 0.835931i \(-0.315071\pi\)
\(44\) 0 0
\(45\) 10.4061i 1.55125i
\(46\) 0 0
\(47\) 13.3035 1.94051 0.970257 0.242075i \(-0.0778281\pi\)
0.970257 + 0.242075i \(0.0778281\pi\)
\(48\) 0 0
\(49\) 8.10124 1.15732
\(50\) 0 0
\(51\) 4.82047i 0.675001i
\(52\) 0 0
\(53\) − 1.62446i − 0.223137i −0.993757 0.111568i \(-0.964413\pi\)
0.993757 0.111568i \(-0.0355874\pi\)
\(54\) 0 0
\(55\) −15.6742 −2.11350
\(56\) 0 0
\(57\) 2.63640 0.349200
\(58\) 0 0
\(59\) − 12.2234i − 1.59136i −0.605720 0.795678i \(-0.707115\pi\)
0.605720 0.795678i \(-0.292885\pi\)
\(60\) 0 0
\(61\) − 5.98666i − 0.766513i −0.923642 0.383257i \(-0.874803\pi\)
0.923642 0.383257i \(-0.125197\pi\)
\(62\) 0 0
\(63\) −15.3523 −1.93420
\(64\) 0 0
\(65\) 10.9297 1.35566
\(66\) 0 0
\(67\) − 4.09328i − 0.500074i −0.968236 0.250037i \(-0.919557\pi\)
0.968236 0.250037i \(-0.0804428\pi\)
\(68\) 0 0
\(69\) − 10.9395i − 1.31696i
\(70\) 0 0
\(71\) 11.8696 1.40866 0.704332 0.709871i \(-0.251247\pi\)
0.704332 + 0.709871i \(0.251247\pi\)
\(72\) 0 0
\(73\) 9.28531 1.08676 0.543381 0.839486i \(-0.317144\pi\)
0.543381 + 0.839486i \(0.317144\pi\)
\(74\) 0 0
\(75\) − 5.10971i − 0.590018i
\(76\) 0 0
\(77\) − 23.1243i − 2.63526i
\(78\) 0 0
\(79\) 2.42124 0.272410 0.136205 0.990681i \(-0.456509\pi\)
0.136205 + 0.990681i \(0.456509\pi\)
\(80\) 0 0
\(81\) −5.24441 −0.582712
\(82\) 0 0
\(83\) 0.987504i 0.108393i 0.998530 + 0.0541963i \(0.0172597\pi\)
−0.998530 + 0.0541963i \(0.982740\pi\)
\(84\) 0 0
\(85\) − 4.81614i − 0.522384i
\(86\) 0 0
\(87\) 21.6090 2.31672
\(88\) 0 0
\(89\) 2.25389 0.238912 0.119456 0.992840i \(-0.461885\pi\)
0.119456 + 0.992840i \(0.461885\pi\)
\(90\) 0 0
\(91\) 16.1247i 1.69033i
\(92\) 0 0
\(93\) − 4.41944i − 0.458274i
\(94\) 0 0
\(95\) −2.63403 −0.270246
\(96\) 0 0
\(97\) 9.27281 0.941511 0.470756 0.882264i \(-0.343981\pi\)
0.470756 + 0.882264i \(0.343981\pi\)
\(98\) 0 0
\(99\) 23.5087i 2.36272i
\(100\) 0 0
\(101\) 13.3620i 1.32957i 0.747034 + 0.664785i \(0.231477\pi\)
−0.747034 + 0.664785i \(0.768523\pi\)
\(102\) 0 0
\(103\) −10.0094 −0.986255 −0.493127 0.869957i \(-0.664146\pi\)
−0.493127 + 0.869957i \(0.664146\pi\)
\(104\) 0 0
\(105\) 26.9861 2.63357
\(106\) 0 0
\(107\) 4.85189i 0.469050i 0.972110 + 0.234525i \(0.0753535\pi\)
−0.972110 + 0.234525i \(0.924646\pi\)
\(108\) 0 0
\(109\) 15.1198i 1.44822i 0.689687 + 0.724108i \(0.257748\pi\)
−0.689687 + 0.724108i \(0.742252\pi\)
\(110\) 0 0
\(111\) −21.4723 −2.03806
\(112\) 0 0
\(113\) −3.10175 −0.291789 −0.145894 0.989300i \(-0.546606\pi\)
−0.145894 + 0.989300i \(0.546606\pi\)
\(114\) 0 0
\(115\) 10.9297i 1.01920i
\(116\) 0 0
\(117\) − 16.3928i − 1.51551i
\(118\) 0 0
\(119\) 7.10532 0.651344
\(120\) 0 0
\(121\) −24.4100 −2.21909
\(122\) 0 0
\(123\) 1.35564i 0.122234i
\(124\) 0 0
\(125\) − 8.06506i − 0.721361i
\(126\) 0 0
\(127\) −3.59176 −0.318717 −0.159358 0.987221i \(-0.550943\pi\)
−0.159358 + 0.987221i \(0.550943\pi\)
\(128\) 0 0
\(129\) 28.9032 2.54479
\(130\) 0 0
\(131\) − 9.76906i − 0.853527i −0.904363 0.426763i \(-0.859654\pi\)
0.904363 0.426763i \(-0.140346\pi\)
\(132\) 0 0
\(133\) − 3.88603i − 0.336962i
\(134\) 0 0
\(135\) −6.60154 −0.568170
\(136\) 0 0
\(137\) −4.25690 −0.363692 −0.181846 0.983327i \(-0.558207\pi\)
−0.181846 + 0.983327i \(0.558207\pi\)
\(138\) 0 0
\(139\) 9.50875i 0.806521i 0.915085 + 0.403261i \(0.132123\pi\)
−0.915085 + 0.403261i \(0.867877\pi\)
\(140\) 0 0
\(141\) 35.0734i 2.95371i
\(142\) 0 0
\(143\) 24.6916 2.06481
\(144\) 0 0
\(145\) −21.5895 −1.79291
\(146\) 0 0
\(147\) 21.3581i 1.76159i
\(148\) 0 0
\(149\) − 5.13803i − 0.420924i −0.977602 0.210462i \(-0.932503\pi\)
0.977602 0.210462i \(-0.0674968\pi\)
\(150\) 0 0
\(151\) 5.26807 0.428709 0.214355 0.976756i \(-0.431235\pi\)
0.214355 + 0.976756i \(0.431235\pi\)
\(152\) 0 0
\(153\) −7.22344 −0.583981
\(154\) 0 0
\(155\) 4.41546i 0.354659i
\(156\) 0 0
\(157\) 7.12828i 0.568899i 0.958691 + 0.284449i \(0.0918108\pi\)
−0.958691 + 0.284449i \(0.908189\pi\)
\(158\) 0 0
\(159\) 4.28273 0.339643
\(160\) 0 0
\(161\) −16.1247 −1.27080
\(162\) 0 0
\(163\) − 12.4060i − 0.971712i −0.874039 0.485856i \(-0.838508\pi\)
0.874039 0.485856i \(-0.161492\pi\)
\(164\) 0 0
\(165\) − 41.3234i − 3.21703i
\(166\) 0 0
\(167\) −8.80460 −0.681320 −0.340660 0.940187i \(-0.610651\pi\)
−0.340660 + 0.940187i \(0.610651\pi\)
\(168\) 0 0
\(169\) −4.21753 −0.324426
\(170\) 0 0
\(171\) 3.95063i 0.302112i
\(172\) 0 0
\(173\) 1.52203i 0.115718i 0.998325 + 0.0578590i \(0.0184274\pi\)
−0.998325 + 0.0578590i \(0.981573\pi\)
\(174\) 0 0
\(175\) −7.53165 −0.569339
\(176\) 0 0
\(177\) 32.2259 2.42225
\(178\) 0 0
\(179\) 18.6599i 1.39470i 0.716729 + 0.697352i \(0.245639\pi\)
−0.716729 + 0.697352i \(0.754361\pi\)
\(180\) 0 0
\(181\) − 11.3132i − 0.840907i −0.907314 0.420453i \(-0.861871\pi\)
0.907314 0.420453i \(-0.138129\pi\)
\(182\) 0 0
\(183\) 15.7833 1.16673
\(184\) 0 0
\(185\) 21.4530 1.57725
\(186\) 0 0
\(187\) − 10.8803i − 0.795646i
\(188\) 0 0
\(189\) − 9.73936i − 0.708434i
\(190\) 0 0
\(191\) −11.5963 −0.839078 −0.419539 0.907737i \(-0.637808\pi\)
−0.419539 + 0.907737i \(0.637808\pi\)
\(192\) 0 0
\(193\) −3.78701 −0.272595 −0.136298 0.990668i \(-0.543520\pi\)
−0.136298 + 0.990668i \(0.543520\pi\)
\(194\) 0 0
\(195\) 28.8150i 2.06349i
\(196\) 0 0
\(197\) 26.0803i 1.85814i 0.369902 + 0.929071i \(0.379391\pi\)
−0.369902 + 0.929071i \(0.620609\pi\)
\(198\) 0 0
\(199\) −3.53452 −0.250555 −0.125278 0.992122i \(-0.539982\pi\)
−0.125278 + 0.992122i \(0.539982\pi\)
\(200\) 0 0
\(201\) 10.7916 0.761178
\(202\) 0 0
\(203\) − 31.8514i − 2.23553i
\(204\) 0 0
\(205\) − 1.35442i − 0.0945970i
\(206\) 0 0
\(207\) 16.3928 1.13938
\(208\) 0 0
\(209\) −5.95063 −0.411614
\(210\) 0 0
\(211\) 2.84939i 0.196160i 0.995179 + 0.0980802i \(0.0312702\pi\)
−0.995179 + 0.0980802i \(0.968730\pi\)
\(212\) 0 0
\(213\) 31.2931i 2.14417i
\(214\) 0 0
\(215\) −28.8772 −1.96941
\(216\) 0 0
\(217\) −6.51420 −0.442213
\(218\) 0 0
\(219\) 24.4798i 1.65419i
\(220\) 0 0
\(221\) 7.58688i 0.510349i
\(222\) 0 0
\(223\) 23.4383 1.56954 0.784772 0.619785i \(-0.212780\pi\)
0.784772 + 0.619785i \(0.212780\pi\)
\(224\) 0 0
\(225\) 7.65685 0.510457
\(226\) 0 0
\(227\) 7.43341i 0.493373i 0.969095 + 0.246686i \(0.0793418\pi\)
−0.969095 + 0.246686i \(0.920658\pi\)
\(228\) 0 0
\(229\) 19.2316i 1.27086i 0.772157 + 0.635431i \(0.219178\pi\)
−0.772157 + 0.635431i \(0.780822\pi\)
\(230\) 0 0
\(231\) 60.9651 4.01121
\(232\) 0 0
\(233\) 29.9306 1.96082 0.980411 0.196965i \(-0.0631084\pi\)
0.980411 + 0.196965i \(0.0631084\pi\)
\(234\) 0 0
\(235\) − 35.0419i − 2.28588i
\(236\) 0 0
\(237\) 6.38336i 0.414644i
\(238\) 0 0
\(239\) −16.0479 −1.03805 −0.519026 0.854758i \(-0.673705\pi\)
−0.519026 + 0.854758i \(0.673705\pi\)
\(240\) 0 0
\(241\) 23.9168 1.54061 0.770307 0.637673i \(-0.220103\pi\)
0.770307 + 0.637673i \(0.220103\pi\)
\(242\) 0 0
\(243\) − 21.3451i − 1.36929i
\(244\) 0 0
\(245\) − 21.3389i − 1.36329i
\(246\) 0 0
\(247\) 4.14940 0.264020
\(248\) 0 0
\(249\) −2.60346 −0.164988
\(250\) 0 0
\(251\) − 10.1950i − 0.643505i −0.946824 0.321753i \(-0.895728\pi\)
0.946824 0.321753i \(-0.104272\pi\)
\(252\) 0 0
\(253\) 24.6916i 1.55234i
\(254\) 0 0
\(255\) 12.6973 0.795135
\(256\) 0 0
\(257\) 20.1866 1.25920 0.629602 0.776918i \(-0.283218\pi\)
0.629602 + 0.776918i \(0.283218\pi\)
\(258\) 0 0
\(259\) 31.6499i 1.96663i
\(260\) 0 0
\(261\) 32.3809i 2.00432i
\(262\) 0 0
\(263\) 5.32659 0.328451 0.164226 0.986423i \(-0.447487\pi\)
0.164226 + 0.986423i \(0.447487\pi\)
\(264\) 0 0
\(265\) −4.27888 −0.262850
\(266\) 0 0
\(267\) 5.94216i 0.363654i
\(268\) 0 0
\(269\) − 14.0564i − 0.857032i −0.903534 0.428516i \(-0.859037\pi\)
0.903534 0.428516i \(-0.140963\pi\)
\(270\) 0 0
\(271\) −8.56884 −0.520520 −0.260260 0.965539i \(-0.583808\pi\)
−0.260260 + 0.965539i \(0.583808\pi\)
\(272\) 0 0
\(273\) −42.5112 −2.57290
\(274\) 0 0
\(275\) 11.5331i 0.695474i
\(276\) 0 0
\(277\) 22.7013i 1.36399i 0.731357 + 0.681995i \(0.238887\pi\)
−0.731357 + 0.681995i \(0.761113\pi\)
\(278\) 0 0
\(279\) 6.62249 0.396478
\(280\) 0 0
\(281\) −8.26940 −0.493311 −0.246656 0.969103i \(-0.579332\pi\)
−0.246656 + 0.969103i \(0.579332\pi\)
\(282\) 0 0
\(283\) − 21.7941i − 1.29552i −0.761844 0.647761i \(-0.775706\pi\)
0.761844 0.647761i \(-0.224294\pi\)
\(284\) 0 0
\(285\) − 6.94438i − 0.411350i
\(286\) 0 0
\(287\) 1.99820 0.117950
\(288\) 0 0
\(289\) −13.6569 −0.803344
\(290\) 0 0
\(291\) 24.4469i 1.43310i
\(292\) 0 0
\(293\) − 8.64036i − 0.504775i −0.967626 0.252388i \(-0.918784\pi\)
0.967626 0.252388i \(-0.0812158\pi\)
\(294\) 0 0
\(295\) −32.1970 −1.87458
\(296\) 0 0
\(297\) −14.9138 −0.865384
\(298\) 0 0
\(299\) − 17.2175i − 0.995716i
\(300\) 0 0
\(301\) − 42.6031i − 2.45560i
\(302\) 0 0
\(303\) −35.2277 −2.02378
\(304\) 0 0
\(305\) −15.7691 −0.902934
\(306\) 0 0
\(307\) − 11.3871i − 0.649894i −0.945732 0.324947i \(-0.894654\pi\)
0.945732 0.324947i \(-0.105346\pi\)
\(308\) 0 0
\(309\) − 26.3888i − 1.50121i
\(310\) 0 0
\(311\) −11.3199 −0.641893 −0.320947 0.947097i \(-0.604001\pi\)
−0.320947 + 0.947097i \(0.604001\pi\)
\(312\) 0 0
\(313\) −18.3052 −1.03467 −0.517337 0.855782i \(-0.673076\pi\)
−0.517337 + 0.855782i \(0.673076\pi\)
\(314\) 0 0
\(315\) 40.4384i 2.27845i
\(316\) 0 0
\(317\) 26.0999i 1.46592i 0.680273 + 0.732959i \(0.261861\pi\)
−0.680273 + 0.732959i \(0.738139\pi\)
\(318\) 0 0
\(319\) −48.7736 −2.73080
\(320\) 0 0
\(321\) −12.7916 −0.713955
\(322\) 0 0
\(323\) − 1.82843i − 0.101736i
\(324\) 0 0
\(325\) − 8.04210i − 0.446095i
\(326\) 0 0
\(327\) −39.8619 −2.20437
\(328\) 0 0
\(329\) 51.6978 2.85019
\(330\) 0 0
\(331\) 12.2799i 0.674962i 0.941332 + 0.337481i \(0.109575\pi\)
−0.941332 + 0.337481i \(0.890425\pi\)
\(332\) 0 0
\(333\) − 32.1760i − 1.76323i
\(334\) 0 0
\(335\) −10.7818 −0.589075
\(336\) 0 0
\(337\) 15.5263 0.845772 0.422886 0.906183i \(-0.361017\pi\)
0.422886 + 0.906183i \(0.361017\pi\)
\(338\) 0 0
\(339\) − 8.17748i − 0.444140i
\(340\) 0 0
\(341\) 9.97512i 0.540183i
\(342\) 0 0
\(343\) 4.27944 0.231068
\(344\) 0 0
\(345\) −28.8150 −1.55135
\(346\) 0 0
\(347\) 12.5212i 0.672176i 0.941831 + 0.336088i \(0.109104\pi\)
−0.941831 + 0.336088i \(0.890896\pi\)
\(348\) 0 0
\(349\) 1.36237i 0.0729259i 0.999335 + 0.0364630i \(0.0116091\pi\)
−0.999335 + 0.0364630i \(0.988391\pi\)
\(350\) 0 0
\(351\) 10.3994 0.555081
\(352\) 0 0
\(353\) −22.3247 −1.18822 −0.594111 0.804383i \(-0.702496\pi\)
−0.594111 + 0.804383i \(0.702496\pi\)
\(354\) 0 0
\(355\) − 31.2650i − 1.65937i
\(356\) 0 0
\(357\) 18.7325i 0.991430i
\(358\) 0 0
\(359\) 16.2824 0.859351 0.429676 0.902983i \(-0.358628\pi\)
0.429676 + 0.902983i \(0.358628\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) − 64.3547i − 3.37774i
\(364\) 0 0
\(365\) − 24.4578i − 1.28018i
\(366\) 0 0
\(367\) 29.6214 1.54623 0.773113 0.634268i \(-0.218698\pi\)
0.773113 + 0.634268i \(0.218698\pi\)
\(368\) 0 0
\(369\) −2.03142 −0.105751
\(370\) 0 0
\(371\) − 6.31270i − 0.327739i
\(372\) 0 0
\(373\) − 5.18659i − 0.268551i −0.990944 0.134276i \(-0.957129\pi\)
0.990944 0.134276i \(-0.0428708\pi\)
\(374\) 0 0
\(375\) 21.2628 1.09800
\(376\) 0 0
\(377\) 34.0101 1.75161
\(378\) 0 0
\(379\) − 11.6485i − 0.598344i −0.954199 0.299172i \(-0.903290\pi\)
0.954199 0.299172i \(-0.0967104\pi\)
\(380\) 0 0
\(381\) − 9.46932i − 0.485128i
\(382\) 0 0
\(383\) −33.3465 −1.70393 −0.851964 0.523600i \(-0.824589\pi\)
−0.851964 + 0.523600i \(0.824589\pi\)
\(384\) 0 0
\(385\) −60.9103 −3.10428
\(386\) 0 0
\(387\) 43.3113i 2.20164i
\(388\) 0 0
\(389\) 1.51874i 0.0770033i 0.999259 + 0.0385016i \(0.0122585\pi\)
−0.999259 + 0.0385016i \(0.987742\pi\)
\(390\) 0 0
\(391\) −7.58688 −0.383685
\(392\) 0 0
\(393\) 25.7552 1.29918
\(394\) 0 0
\(395\) − 6.37762i − 0.320893i
\(396\) 0 0
\(397\) 29.4592i 1.47852i 0.673422 + 0.739258i \(0.264824\pi\)
−0.673422 + 0.739258i \(0.735176\pi\)
\(398\) 0 0
\(399\) 10.2452 0.512899
\(400\) 0 0
\(401\) −32.0523 −1.60062 −0.800308 0.599589i \(-0.795331\pi\)
−0.800308 + 0.599589i \(0.795331\pi\)
\(402\) 0 0
\(403\) − 6.95569i − 0.346488i
\(404\) 0 0
\(405\) 13.8139i 0.686421i
\(406\) 0 0
\(407\) 48.4651 2.40232
\(408\) 0 0
\(409\) 10.7427 0.531192 0.265596 0.964084i \(-0.414431\pi\)
0.265596 + 0.964084i \(0.414431\pi\)
\(410\) 0 0
\(411\) − 11.2229i − 0.553586i
\(412\) 0 0
\(413\) − 47.5007i − 2.33736i
\(414\) 0 0
\(415\) 2.60112 0.127684
\(416\) 0 0
\(417\) −25.0689 −1.22763
\(418\) 0 0
\(419\) 19.9990i 0.977013i 0.872560 + 0.488507i \(0.162458\pi\)
−0.872560 + 0.488507i \(0.837542\pi\)
\(420\) 0 0
\(421\) 10.5670i 0.515006i 0.966277 + 0.257503i \(0.0828997\pi\)
−0.966277 + 0.257503i \(0.917100\pi\)
\(422\) 0 0
\(423\) −52.5572 −2.55542
\(424\) 0 0
\(425\) −3.54374 −0.171897
\(426\) 0 0
\(427\) − 23.2643i − 1.12584i
\(428\) 0 0
\(429\) 65.0969i 3.14291i
\(430\) 0 0
\(431\) 3.24892 0.156495 0.0782474 0.996934i \(-0.475068\pi\)
0.0782474 + 0.996934i \(0.475068\pi\)
\(432\) 0 0
\(433\) −25.4848 −1.22472 −0.612360 0.790579i \(-0.709780\pi\)
−0.612360 + 0.790579i \(0.709780\pi\)
\(434\) 0 0
\(435\) − 56.9188i − 2.72905i
\(436\) 0 0
\(437\) 4.14940i 0.198493i
\(438\) 0 0
\(439\) −17.9520 −0.856804 −0.428402 0.903588i \(-0.640923\pi\)
−0.428402 + 0.903588i \(0.640923\pi\)
\(440\) 0 0
\(441\) −32.0050 −1.52405
\(442\) 0 0
\(443\) 1.68123i 0.0798777i 0.999202 + 0.0399388i \(0.0127163\pi\)
−0.999202 + 0.0399388i \(0.987284\pi\)
\(444\) 0 0
\(445\) − 5.93682i − 0.281432i
\(446\) 0 0
\(447\) 13.5459 0.640700
\(448\) 0 0
\(449\) −21.8275 −1.03010 −0.515052 0.857159i \(-0.672227\pi\)
−0.515052 + 0.857159i \(0.672227\pi\)
\(450\) 0 0
\(451\) − 3.05982i − 0.144081i
\(452\) 0 0
\(453\) 13.8888i 0.652551i
\(454\) 0 0
\(455\) 42.4730 1.99117
\(456\) 0 0
\(457\) −8.04431 −0.376297 −0.188148 0.982141i \(-0.560249\pi\)
−0.188148 + 0.982141i \(0.560249\pi\)
\(458\) 0 0
\(459\) − 4.58249i − 0.213892i
\(460\) 0 0
\(461\) − 6.79339i − 0.316400i −0.987407 0.158200i \(-0.949431\pi\)
0.987407 0.158200i \(-0.0505690\pi\)
\(462\) 0 0
\(463\) 31.2479 1.45221 0.726107 0.687582i \(-0.241328\pi\)
0.726107 + 0.687582i \(0.241328\pi\)
\(464\) 0 0
\(465\) −11.6409 −0.539836
\(466\) 0 0
\(467\) − 36.9113i − 1.70805i −0.520230 0.854026i \(-0.674154\pi\)
0.520230 0.854026i \(-0.325846\pi\)
\(468\) 0 0
\(469\) − 15.9066i − 0.734500i
\(470\) 0 0
\(471\) −18.7930 −0.865938
\(472\) 0 0
\(473\) −65.2375 −2.99962
\(474\) 0 0
\(475\) 1.93813i 0.0889277i
\(476\) 0 0
\(477\) 6.41764i 0.293844i
\(478\) 0 0
\(479\) −7.26161 −0.331792 −0.165896 0.986143i \(-0.553052\pi\)
−0.165896 + 0.986143i \(0.553052\pi\)
\(480\) 0 0
\(481\) −33.7949 −1.54091
\(482\) 0 0
\(483\) − 42.5112i − 1.93433i
\(484\) 0 0
\(485\) − 24.4249i − 1.10908i
\(486\) 0 0
\(487\) −1.00500 −0.0455411 −0.0227705 0.999741i \(-0.507249\pi\)
−0.0227705 + 0.999741i \(0.507249\pi\)
\(488\) 0 0
\(489\) 32.7072 1.47907
\(490\) 0 0
\(491\) − 9.12714i − 0.411902i −0.978562 0.205951i \(-0.933971\pi\)
0.978562 0.205951i \(-0.0660288\pi\)
\(492\) 0 0
\(493\) − 14.9865i − 0.674957i
\(494\) 0 0
\(495\) 61.9228 2.78322
\(496\) 0 0
\(497\) 46.1257 2.06902
\(498\) 0 0
\(499\) − 19.8928i − 0.890524i −0.895400 0.445262i \(-0.853111\pi\)
0.895400 0.445262i \(-0.146889\pi\)
\(500\) 0 0
\(501\) − 23.2125i − 1.03706i
\(502\) 0 0
\(503\) −18.9716 −0.845904 −0.422952 0.906152i \(-0.639006\pi\)
−0.422952 + 0.906152i \(0.639006\pi\)
\(504\) 0 0
\(505\) 35.1960 1.56620
\(506\) 0 0
\(507\) − 11.1191i − 0.493818i
\(508\) 0 0
\(509\) 24.2981i 1.07700i 0.842627 + 0.538498i \(0.181008\pi\)
−0.842627 + 0.538498i \(0.818992\pi\)
\(510\) 0 0
\(511\) 36.0830 1.59622
\(512\) 0 0
\(513\) −2.50625 −0.110654
\(514\) 0 0
\(515\) 26.3651i 1.16178i
\(516\) 0 0
\(517\) − 79.1642i − 3.48164i
\(518\) 0 0
\(519\) −4.01270 −0.176138
\(520\) 0 0
\(521\) 10.1930 0.446563 0.223282 0.974754i \(-0.428323\pi\)
0.223282 + 0.974754i \(0.428323\pi\)
\(522\) 0 0
\(523\) 1.42683i 0.0623907i 0.999513 + 0.0311954i \(0.00993140\pi\)
−0.999513 + 0.0311954i \(0.990069\pi\)
\(524\) 0 0
\(525\) − 19.8565i − 0.866608i
\(526\) 0 0
\(527\) −3.06501 −0.133514
\(528\) 0 0
\(529\) −5.78247 −0.251412
\(530\) 0 0
\(531\) 48.2903i 2.09562i
\(532\) 0 0
\(533\) 2.13363i 0.0924176i
\(534\) 0 0
\(535\) 12.7801 0.552530
\(536\) 0 0
\(537\) −49.1950 −2.12292
\(538\) 0 0
\(539\) − 48.2075i − 2.07644i
\(540\) 0 0
\(541\) − 28.1257i − 1.20922i −0.796522 0.604610i \(-0.793329\pi\)
0.796522 0.604610i \(-0.206671\pi\)
\(542\) 0 0
\(543\) 29.8263 1.27997
\(544\) 0 0
\(545\) 39.8261 1.70596
\(546\) 0 0
\(547\) − 45.2937i − 1.93662i −0.249753 0.968310i \(-0.580349\pi\)
0.249753 0.968310i \(-0.419651\pi\)
\(548\) 0 0
\(549\) 23.6511i 1.00940i
\(550\) 0 0
\(551\) −8.19638 −0.349177
\(552\) 0 0
\(553\) 9.40900 0.400111
\(554\) 0 0
\(555\) 56.5587i 2.40078i
\(556\) 0 0
\(557\) − 13.9151i − 0.589601i −0.955559 0.294801i \(-0.904747\pi\)
0.955559 0.294801i \(-0.0952533\pi\)
\(558\) 0 0
\(559\) 45.4904 1.92404
\(560\) 0 0
\(561\) 28.6849 1.21108
\(562\) 0 0
\(563\) − 10.3367i − 0.435641i −0.975989 0.217820i \(-0.930105\pi\)
0.975989 0.217820i \(-0.0698947\pi\)
\(564\) 0 0
\(565\) 8.17013i 0.343720i
\(566\) 0 0
\(567\) −20.3799 −0.855877
\(568\) 0 0
\(569\) −39.2369 −1.64490 −0.822448 0.568840i \(-0.807392\pi\)
−0.822448 + 0.568840i \(0.807392\pi\)
\(570\) 0 0
\(571\) 33.4374i 1.39931i 0.714479 + 0.699657i \(0.246664\pi\)
−0.714479 + 0.699657i \(0.753336\pi\)
\(572\) 0 0
\(573\) − 30.5725i − 1.27718i
\(574\) 0 0
\(575\) 8.04210 0.335379
\(576\) 0 0
\(577\) −22.3203 −0.929206 −0.464603 0.885519i \(-0.653803\pi\)
−0.464603 + 0.885519i \(0.653803\pi\)
\(578\) 0 0
\(579\) − 9.98409i − 0.414925i
\(580\) 0 0
\(581\) 3.83747i 0.159205i
\(582\) 0 0
\(583\) −9.66656 −0.400348
\(584\) 0 0
\(585\) −43.1791 −1.78523
\(586\) 0 0
\(587\) 43.6753i 1.80267i 0.433119 + 0.901337i \(0.357413\pi\)
−0.433119 + 0.901337i \(0.642587\pi\)
\(588\) 0 0
\(589\) 1.67631i 0.0690712i
\(590\) 0 0
\(591\) −68.7581 −2.82833
\(592\) 0 0
\(593\) −19.6978 −0.808890 −0.404445 0.914562i \(-0.632535\pi\)
−0.404445 + 0.914562i \(0.632535\pi\)
\(594\) 0 0
\(595\) − 18.7157i − 0.767267i
\(596\) 0 0
\(597\) − 9.31842i − 0.381377i
\(598\) 0 0
\(599\) −34.2570 −1.39970 −0.699851 0.714289i \(-0.746750\pi\)
−0.699851 + 0.714289i \(0.746750\pi\)
\(600\) 0 0
\(601\) 19.1801 0.782373 0.391186 0.920311i \(-0.372065\pi\)
0.391186 + 0.920311i \(0.372065\pi\)
\(602\) 0 0
\(603\) 16.1711i 0.658536i
\(604\) 0 0
\(605\) 64.2968i 2.61404i
\(606\) 0 0
\(607\) 17.1796 0.697297 0.348649 0.937253i \(-0.386641\pi\)
0.348649 + 0.937253i \(0.386641\pi\)
\(608\) 0 0
\(609\) 83.9731 3.40276
\(610\) 0 0
\(611\) 55.2016i 2.23322i
\(612\) 0 0
\(613\) − 13.4502i − 0.543247i −0.962404 0.271623i \(-0.912439\pi\)
0.962404 0.271623i \(-0.0875605\pi\)
\(614\) 0 0
\(615\) 3.57081 0.143989
\(616\) 0 0
\(617\) 15.2065 0.612191 0.306095 0.952001i \(-0.400977\pi\)
0.306095 + 0.952001i \(0.400977\pi\)
\(618\) 0 0
\(619\) 40.1696i 1.61455i 0.590173 + 0.807277i \(0.299060\pi\)
−0.590173 + 0.807277i \(0.700940\pi\)
\(620\) 0 0
\(621\) 10.3994i 0.417315i
\(622\) 0 0
\(623\) 8.75868 0.350909
\(624\) 0 0
\(625\) −30.9343 −1.23737
\(626\) 0 0
\(627\) − 15.6883i − 0.626529i
\(628\) 0 0
\(629\) 14.8917i 0.593770i
\(630\) 0 0
\(631\) 12.0318 0.478980 0.239490 0.970899i \(-0.423020\pi\)
0.239490 + 0.970899i \(0.423020\pi\)
\(632\) 0 0
\(633\) −7.51216 −0.298581
\(634\) 0 0
\(635\) 9.46081i 0.375441i
\(636\) 0 0
\(637\) 33.6153i 1.33189i
\(638\) 0 0
\(639\) −46.8925 −1.85504
\(640\) 0 0
\(641\) 28.1707 1.11267 0.556337 0.830957i \(-0.312206\pi\)
0.556337 + 0.830957i \(0.312206\pi\)
\(642\) 0 0
\(643\) − 38.2190i − 1.50721i −0.657327 0.753605i \(-0.728313\pi\)
0.657327 0.753605i \(-0.271687\pi\)
\(644\) 0 0
\(645\) − 76.1321i − 2.99770i
\(646\) 0 0
\(647\) 7.26170 0.285487 0.142743 0.989760i \(-0.454408\pi\)
0.142743 + 0.989760i \(0.454408\pi\)
\(648\) 0 0
\(649\) −72.7372 −2.85518
\(650\) 0 0
\(651\) − 17.1741i − 0.673105i
\(652\) 0 0
\(653\) 16.0972i 0.629932i 0.949103 + 0.314966i \(0.101993\pi\)
−0.949103 + 0.314966i \(0.898007\pi\)
\(654\) 0 0
\(655\) −25.7320 −1.00543
\(656\) 0 0
\(657\) −36.6828 −1.43113
\(658\) 0 0
\(659\) 17.8679i 0.696035i 0.937488 + 0.348018i \(0.113145\pi\)
−0.937488 + 0.348018i \(0.886855\pi\)
\(660\) 0 0
\(661\) 21.6386i 0.841644i 0.907143 + 0.420822i \(0.138258\pi\)
−0.907143 + 0.420822i \(0.861742\pi\)
\(662\) 0 0
\(663\) −20.0021 −0.776817
\(664\) 0 0
\(665\) −10.2359 −0.396933
\(666\) 0 0
\(667\) 34.0101i 1.31687i
\(668\) 0 0
\(669\) 61.7928i 2.38905i
\(670\) 0 0
\(671\) −35.6244 −1.37526
\(672\) 0 0
\(673\) −43.9881 −1.69561 −0.847807 0.530304i \(-0.822078\pi\)
−0.847807 + 0.530304i \(0.822078\pi\)
\(674\) 0 0
\(675\) 4.85745i 0.186963i
\(676\) 0 0
\(677\) 29.6983i 1.14140i 0.821160 + 0.570699i \(0.193328\pi\)
−0.821160 + 0.570699i \(0.806672\pi\)
\(678\) 0 0
\(679\) 36.0344 1.38287
\(680\) 0 0
\(681\) −19.5975 −0.750977
\(682\) 0 0
\(683\) − 6.20351i − 0.237371i −0.992932 0.118685i \(-0.962132\pi\)
0.992932 0.118685i \(-0.0378680\pi\)
\(684\) 0 0
\(685\) 11.2128i 0.428420i
\(686\) 0 0
\(687\) −50.7024 −1.93442
\(688\) 0 0
\(689\) 6.74053 0.256794
\(690\) 0 0
\(691\) 28.1172i 1.06963i 0.844970 + 0.534814i \(0.179618\pi\)
−0.844970 + 0.534814i \(0.820382\pi\)
\(692\) 0 0
\(693\) 91.3557i 3.47032i
\(694\) 0 0
\(695\) 25.0464 0.950063
\(696\) 0 0
\(697\) 0.940179 0.0356118
\(698\) 0 0
\(699\) 78.9093i 2.98462i
\(700\) 0 0
\(701\) − 8.35403i − 0.315527i −0.987477 0.157764i \(-0.949572\pi\)
0.987477 0.157764i \(-0.0504284\pi\)
\(702\) 0 0
\(703\) 8.14452 0.307177
\(704\) 0 0
\(705\) 92.3846 3.47940
\(706\) 0 0
\(707\) 51.9252i 1.95285i
\(708\) 0 0
\(709\) − 31.7187i − 1.19122i −0.803274 0.595610i \(-0.796910\pi\)
0.803274 0.595610i \(-0.203090\pi\)
\(710\) 0 0
\(711\) −9.56541 −0.358731
\(712\) 0 0
\(713\) 6.95569 0.260493
\(714\) 0 0
\(715\) − 65.0384i − 2.43230i
\(716\) 0 0
\(717\) − 42.3088i − 1.58005i
\(718\) 0 0
\(719\) −33.7972 −1.26042 −0.630211 0.776424i \(-0.717032\pi\)
−0.630211 + 0.776424i \(0.717032\pi\)
\(720\) 0 0
\(721\) −38.8968 −1.44859
\(722\) 0 0
\(723\) 63.0543i 2.34501i
\(724\) 0 0
\(725\) 15.8857i 0.589979i
\(726\) 0 0
\(727\) 32.8637 1.21885 0.609423 0.792845i \(-0.291401\pi\)
0.609423 + 0.792845i \(0.291401\pi\)
\(728\) 0 0
\(729\) 40.5412 1.50153
\(730\) 0 0
\(731\) − 20.0453i − 0.741401i
\(732\) 0 0
\(733\) 14.3118i 0.528618i 0.964438 + 0.264309i \(0.0851439\pi\)
−0.964438 + 0.264309i \(0.914856\pi\)
\(734\) 0 0
\(735\) 56.2581 2.07511
\(736\) 0 0
\(737\) −24.3576 −0.897225
\(738\) 0 0
\(739\) 22.6101i 0.831726i 0.909427 + 0.415863i \(0.136520\pi\)
−0.909427 + 0.415863i \(0.863480\pi\)
\(740\) 0 0
\(741\) 10.9395i 0.401873i
\(742\) 0 0
\(743\) 12.6222 0.463062 0.231531 0.972828i \(-0.425627\pi\)
0.231531 + 0.972828i \(0.425627\pi\)
\(744\) 0 0
\(745\) −13.5337 −0.495838
\(746\) 0 0
\(747\) − 3.90126i − 0.142740i
\(748\) 0 0
\(749\) 18.8546i 0.688932i
\(750\) 0 0
\(751\) 4.69943 0.171485 0.0857423 0.996317i \(-0.472674\pi\)
0.0857423 + 0.996317i \(0.472674\pi\)
\(752\) 0 0
\(753\) 26.8783 0.979498
\(754\) 0 0
\(755\) − 13.8763i − 0.505009i
\(756\) 0 0
\(757\) 21.8920i 0.795679i 0.917455 + 0.397839i \(0.130240\pi\)
−0.917455 + 0.397839i \(0.869760\pi\)
\(758\) 0 0
\(759\) −65.0969 −2.36287
\(760\) 0 0
\(761\) 0.164985 0.00598068 0.00299034 0.999996i \(-0.499048\pi\)
0.00299034 + 0.999996i \(0.499048\pi\)
\(762\) 0 0
\(763\) 58.7560i 2.12711i
\(764\) 0 0
\(765\) 19.0268i 0.687915i
\(766\) 0 0
\(767\) 50.7200 1.83139
\(768\) 0 0
\(769\) −8.55812 −0.308614 −0.154307 0.988023i \(-0.549314\pi\)
−0.154307 + 0.988023i \(0.549314\pi\)
\(770\) 0 0
\(771\) 53.2200i 1.91667i
\(772\) 0 0
\(773\) 9.49557i 0.341532i 0.985312 + 0.170766i \(0.0546242\pi\)
−0.985312 + 0.170766i \(0.945376\pi\)
\(774\) 0 0
\(775\) 3.24892 0.116705
\(776\) 0 0
\(777\) −83.4419 −2.99346
\(778\) 0 0
\(779\) − 0.514201i − 0.0184232i
\(780\) 0 0
\(781\) − 70.6317i − 2.52740i
\(782\) 0 0
\(783\) −20.5422 −0.734117
\(784\) 0 0
\(785\) 18.7761 0.670149
\(786\) 0 0
\(787\) − 8.59840i − 0.306500i −0.988187 0.153250i \(-0.951026\pi\)
0.988187 0.153250i \(-0.0489739\pi\)
\(788\) 0 0
\(789\) 14.0430i 0.499945i
\(790\) 0 0
\(791\) −12.0535 −0.428574
\(792\) 0 0
\(793\) 24.8410 0.882131
\(794\) 0 0
\(795\) − 11.2809i − 0.400091i
\(796\) 0 0
\(797\) − 23.3568i − 0.827340i −0.910427 0.413670i \(-0.864247\pi\)
0.910427 0.413670i \(-0.135753\pi\)
\(798\) 0 0
\(799\) 24.3245 0.860538
\(800\) 0 0
\(801\) −8.90428 −0.314617
\(802\) 0 0
\(803\) − 55.2534i − 1.94985i
\(804\) 0 0
\(805\) 42.4730i 1.49698i
\(806\) 0 0
\(807\) 37.0583 1.30451
\(808\) 0 0
\(809\) 11.0634 0.388968 0.194484 0.980906i \(-0.437697\pi\)
0.194484 + 0.980906i \(0.437697\pi\)
\(810\) 0 0
\(811\) − 12.9630i − 0.455193i −0.973756 0.227596i \(-0.926913\pi\)
0.973756 0.227596i \(-0.0730866\pi\)
\(812\) 0 0
\(813\) − 22.5909i − 0.792299i
\(814\) 0 0
\(815\) −32.6778 −1.14465
\(816\) 0 0
\(817\) −10.9631 −0.383551
\(818\) 0 0
\(819\) − 63.7028i − 2.22595i
\(820\) 0 0
\(821\) − 14.8255i − 0.517415i −0.965956 0.258707i \(-0.916703\pi\)
0.965956 0.258707i \(-0.0832965\pi\)
\(822\) 0 0
\(823\) 4.04240 0.140909 0.0704547 0.997515i \(-0.477555\pi\)
0.0704547 + 0.997515i \(0.477555\pi\)
\(824\) 0 0
\(825\) −30.4060 −1.05860
\(826\) 0 0
\(827\) 33.6070i 1.16863i 0.811527 + 0.584314i \(0.198636\pi\)
−0.811527 + 0.584314i \(0.801364\pi\)
\(828\) 0 0
\(829\) 38.3399i 1.33160i 0.746130 + 0.665800i \(0.231910\pi\)
−0.746130 + 0.665800i \(0.768090\pi\)
\(830\) 0 0
\(831\) −59.8498 −2.07617
\(832\) 0 0
\(833\) 14.8125 0.513223
\(834\) 0 0
\(835\) 23.1916i 0.802579i
\(836\) 0 0
\(837\) 4.20125i 0.145217i
\(838\) 0 0
\(839\) −37.6832 −1.30097 −0.650484 0.759520i \(-0.725434\pi\)
−0.650484 + 0.759520i \(0.725434\pi\)
\(840\) 0 0
\(841\) −38.1806 −1.31657
\(842\) 0 0
\(843\) − 21.8015i − 0.750883i
\(844\) 0 0
\(845\) 11.1091i 0.382165i
\(846\) 0 0
\(847\) −94.8581 −3.25936
\(848\) 0 0
\(849\) 57.4580 1.97195
\(850\) 0 0
\(851\) − 33.7949i − 1.15847i
\(852\) 0 0
\(853\) − 7.06647i − 0.241951i −0.992655 0.120976i \(-0.961398\pi\)
0.992655 0.120976i \(-0.0386023\pi\)
\(854\) 0 0
\(855\) 10.4061 0.355881
\(856\) 0 0
\(857\) −7.95910 −0.271878 −0.135939 0.990717i \(-0.543405\pi\)
−0.135939 + 0.990717i \(0.543405\pi\)
\(858\) 0 0
\(859\) − 13.1466i − 0.448558i −0.974525 0.224279i \(-0.927997\pi\)
0.974525 0.224279i \(-0.0720026\pi\)
\(860\) 0 0
\(861\) 5.26807i 0.179535i
\(862\) 0 0
\(863\) −34.9886 −1.19102 −0.595512 0.803346i \(-0.703051\pi\)
−0.595512 + 0.803346i \(0.703051\pi\)
\(864\) 0 0
\(865\) 4.00909 0.136313
\(866\) 0 0
\(867\) − 36.0050i − 1.22279i
\(868\) 0 0
\(869\) − 14.4079i − 0.488754i
\(870\) 0 0
\(871\) 16.9847 0.575504
\(872\) 0 0
\(873\) −36.6335 −1.23985
\(874\) 0 0
\(875\) − 31.3411i − 1.05952i
\(876\) 0 0
\(877\) 49.1816i 1.66075i 0.557208 + 0.830373i \(0.311873\pi\)
−0.557208 + 0.830373i \(0.688127\pi\)
\(878\) 0 0
\(879\) 22.7795 0.768333
\(880\) 0 0
\(881\) −7.58658 −0.255598 −0.127799 0.991800i \(-0.540791\pi\)
−0.127799 + 0.991800i \(0.540791\pi\)
\(882\) 0 0
\(883\) 15.1875i 0.511101i 0.966796 + 0.255551i \(0.0822568\pi\)
−0.966796 + 0.255551i \(0.917743\pi\)
\(884\) 0 0
\(885\) − 84.8842i − 2.85335i
\(886\) 0 0
\(887\) 32.9000 1.10467 0.552337 0.833621i \(-0.313736\pi\)
0.552337 + 0.833621i \(0.313736\pi\)
\(888\) 0 0
\(889\) −13.9577 −0.468125
\(890\) 0 0
\(891\) 31.2075i 1.04549i
\(892\) 0 0
\(893\) − 13.3035i − 0.445185i
\(894\) 0 0
\(895\) 49.1507 1.64293
\(896\) 0 0
\(897\) 45.3924 1.51561
\(898\) 0 0
\(899\) 13.7397i 0.458244i
\(900\) 0 0
\(901\) − 2.97021i − 0.0989519i
\(902\) 0 0
\(903\) 112.319 3.73774
\(904\) 0 0
\(905\) −29.7995 −0.990568
\(906\) 0 0
\(907\) 55.9392i 1.85743i 0.370793 + 0.928715i \(0.379086\pi\)
−0.370793 + 0.928715i \(0.620914\pi\)
\(908\) 0 0
\(909\) − 52.7884i − 1.75088i
\(910\) 0 0
\(911\) 44.6551 1.47949 0.739745 0.672887i \(-0.234946\pi\)
0.739745 + 0.672887i \(0.234946\pi\)
\(912\) 0 0
\(913\) 5.87627 0.194476
\(914\) 0 0
\(915\) − 41.5736i − 1.37438i
\(916\) 0 0
\(917\) − 37.9629i − 1.25364i
\(918\) 0 0
\(919\) 7.52507 0.248229 0.124115 0.992268i \(-0.460391\pi\)
0.124115 + 0.992268i \(0.460391\pi\)
\(920\) 0 0
\(921\) 30.0209 0.989222
\(922\) 0 0
\(923\) 49.2518i 1.62114i
\(924\) 0 0
\(925\) − 15.7852i − 0.519014i
\(926\) 0 0
\(927\) 39.5434 1.29878
\(928\) 0 0
\(929\) −10.4360 −0.342395 −0.171198 0.985237i \(-0.554764\pi\)
−0.171198 + 0.985237i \(0.554764\pi\)
\(930\) 0 0
\(931\) − 8.10124i − 0.265507i
\(932\) 0 0
\(933\) − 29.8439i − 0.977044i
\(934\) 0 0
\(935\) −28.6591 −0.937252
\(936\) 0 0
\(937\) 19.1073 0.624207 0.312104 0.950048i \(-0.398966\pi\)
0.312104 + 0.950048i \(0.398966\pi\)
\(938\) 0 0
\(939\) − 48.2600i − 1.57491i
\(940\) 0 0
\(941\) 43.2146i 1.40875i 0.709826 + 0.704377i \(0.248774\pi\)
−0.709826 + 0.704377i \(0.751226\pi\)
\(942\) 0 0
\(943\) −2.13363 −0.0694804
\(944\) 0 0
\(945\) −25.6538 −0.834518
\(946\) 0 0
\(947\) 1.25385i 0.0407446i 0.999792 + 0.0203723i \(0.00648516\pi\)
−0.999792 + 0.0203723i \(0.993515\pi\)
\(948\) 0 0
\(949\) 38.5285i 1.25069i
\(950\) 0 0
\(951\) −68.8100 −2.23132
\(952\) 0 0
\(953\) 35.0739 1.13616 0.568078 0.822975i \(-0.307687\pi\)
0.568078 + 0.822975i \(0.307687\pi\)
\(954\) 0 0
\(955\) 30.5450i 0.988413i
\(956\) 0 0
\(957\) − 128.587i − 4.15663i
\(958\) 0 0
\(959\) −16.5425 −0.534184
\(960\) 0 0
\(961\) −28.1900 −0.909354
\(962\) 0 0
\(963\) − 19.1680i − 0.617681i
\(964\) 0 0
\(965\) 9.97512i 0.321110i
\(966\) 0 0
\(967\) 1.66764 0.0536275 0.0268138 0.999640i \(-0.491464\pi\)
0.0268138 + 0.999640i \(0.491464\pi\)
\(968\) 0 0
\(969\) 4.82047 0.154856
\(970\) 0 0
\(971\) − 16.1136i − 0.517110i −0.965996 0.258555i \(-0.916754\pi\)
0.965996 0.258555i \(-0.0832464\pi\)
\(972\) 0 0
\(973\) 36.9513i 1.18460i
\(974\) 0 0
\(975\) 21.2022 0.679015
\(976\) 0 0
\(977\) 33.8857 1.08410 0.542051 0.840346i \(-0.317648\pi\)
0.542051 + 0.840346i \(0.317648\pi\)
\(978\) 0 0
\(979\) − 13.4121i − 0.428651i
\(980\) 0 0
\(981\) − 59.7328i − 1.90712i
\(982\) 0 0
\(983\) 9.89492 0.315599 0.157799 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(984\) 0 0
\(985\) 68.6963 2.18885
\(986\) 0 0
\(987\) 136.296i 4.33836i
\(988\) 0 0
\(989\) 45.4904i 1.44651i
\(990\) 0 0
\(991\) 31.3693 0.996478 0.498239 0.867040i \(-0.333980\pi\)
0.498239 + 0.867040i \(0.333980\pi\)
\(992\) 0 0
\(993\) −32.3747 −1.02738
\(994\) 0 0
\(995\) 9.31004i 0.295148i
\(996\) 0 0
\(997\) − 4.55606i − 0.144292i −0.997394 0.0721459i \(-0.977015\pi\)
0.997394 0.0721459i \(-0.0229847\pi\)
\(998\) 0 0
\(999\) 20.4122 0.645813
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 2432.2.c.i.1217.15 yes 16
4.3 odd 2 inner 2432.2.c.i.1217.1 16
8.3 odd 2 inner 2432.2.c.i.1217.16 yes 16
8.5 even 2 inner 2432.2.c.i.1217.2 yes 16
16.3 odd 4 4864.2.a.br.1.7 8
16.5 even 4 4864.2.a.br.1.8 8
16.11 odd 4 4864.2.a.bm.1.2 8
16.13 even 4 4864.2.a.bm.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.i.1217.1 16 4.3 odd 2 inner
2432.2.c.i.1217.2 yes 16 8.5 even 2 inner
2432.2.c.i.1217.15 yes 16 1.1 even 1 trivial
2432.2.c.i.1217.16 yes 16 8.3 odd 2 inner
4864.2.a.bm.1.1 8 16.13 even 4
4864.2.a.bm.1.2 8 16.11 odd 4
4864.2.a.br.1.7 8 16.3 odd 4
4864.2.a.br.1.8 8 16.5 even 4