Properties

Label 3328.2.b.u.1665.4
Level $3328$
Weight $2$
Character 3328.1665
Analytic conductor $26.574$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3328,2,Mod(1665,3328)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3328, 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("3328.1665");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3328.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.5742137927\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 416)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1665.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 3328.1665
Dual form 3328.2.b.u.1665.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.56155i q^{3} -0.561553i q^{5} +0.561553 q^{7} -3.56155 q^{9} +O(q^{10})\) \(q+2.56155i q^{3} -0.561553i q^{5} +0.561553 q^{7} -3.56155 q^{9} -2.00000i q^{11} -1.00000i q^{13} +1.43845 q^{15} -0.561553 q^{17} +6.00000i q^{19} +1.43845i q^{21} +4.68466 q^{25} -1.43845i q^{27} -8.24621i q^{29} +7.12311 q^{31} +5.12311 q^{33} -0.315342i q^{35} +9.68466i q^{37} +2.56155 q^{39} -7.12311 q^{41} +8.80776i q^{43} +2.00000i q^{45} +1.68466 q^{47} -6.68466 q^{49} -1.43845i q^{51} +4.87689i q^{53} -1.12311 q^{55} -15.3693 q^{57} +6.00000i q^{59} +13.3693i q^{61} -2.00000 q^{63} -0.561553 q^{65} +6.00000i q^{67} +1.68466 q^{71} -10.0000 q^{73} +12.0000i q^{75} -1.12311i q^{77} +12.0000 q^{79} -7.00000 q^{81} -17.3693i q^{83} +0.315342i q^{85} +21.1231 q^{87} +8.24621 q^{89} -0.561553i q^{91} +18.2462i q^{93} +3.36932 q^{95} -6.00000 q^{97} +7.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{7} - 6 q^{9} + 14 q^{15} + 6 q^{17} - 6 q^{25} + 12 q^{31} + 4 q^{33} + 2 q^{39} - 12 q^{41} - 18 q^{47} - 2 q^{49} + 12 q^{55} - 12 q^{57} - 8 q^{63} + 6 q^{65} - 18 q^{71} - 40 q^{73} + 48 q^{79} - 28 q^{81} + 68 q^{87} - 36 q^{95} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 2.56155i 1.47891i 0.673204 + 0.739457i \(0.264917\pi\)
−0.673204 + 0.739457i \(0.735083\pi\)
\(4\) 0 0
\(5\) − 0.561553i − 0.251134i −0.992085 0.125567i \(-0.959925\pi\)
0.992085 0.125567i \(-0.0400750\pi\)
\(6\) 0 0
\(7\) 0.561553 0.212247 0.106124 0.994353i \(-0.466156\pi\)
0.106124 + 0.994353i \(0.466156\pi\)
\(8\) 0 0
\(9\) −3.56155 −1.18718
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i
\(14\) 0 0
\(15\) 1.43845 0.371405
\(16\) 0 0
\(17\) −0.561553 −0.136197 −0.0680983 0.997679i \(-0.521693\pi\)
−0.0680983 + 0.997679i \(0.521693\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 0 0
\(21\) 1.43845i 0.313895i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 4.68466 0.936932
\(26\) 0 0
\(27\) − 1.43845i − 0.276829i
\(28\) 0 0
\(29\) − 8.24621i − 1.53128i −0.643268 0.765641i \(-0.722422\pi\)
0.643268 0.765641i \(-0.277578\pi\)
\(30\) 0 0
\(31\) 7.12311 1.27935 0.639674 0.768647i \(-0.279069\pi\)
0.639674 + 0.768647i \(0.279069\pi\)
\(32\) 0 0
\(33\) 5.12311 0.891818
\(34\) 0 0
\(35\) − 0.315342i − 0.0533025i
\(36\) 0 0
\(37\) 9.68466i 1.59215i 0.605199 + 0.796074i \(0.293093\pi\)
−0.605199 + 0.796074i \(0.706907\pi\)
\(38\) 0 0
\(39\) 2.56155 0.410177
\(40\) 0 0
\(41\) −7.12311 −1.11244 −0.556221 0.831034i \(-0.687749\pi\)
−0.556221 + 0.831034i \(0.687749\pi\)
\(42\) 0 0
\(43\) 8.80776i 1.34317i 0.740927 + 0.671586i \(0.234386\pi\)
−0.740927 + 0.671586i \(0.765614\pi\)
\(44\) 0 0
\(45\) 2.00000i 0.298142i
\(46\) 0 0
\(47\) 1.68466 0.245733 0.122866 0.992423i \(-0.460791\pi\)
0.122866 + 0.992423i \(0.460791\pi\)
\(48\) 0 0
\(49\) −6.68466 −0.954951
\(50\) 0 0
\(51\) − 1.43845i − 0.201423i
\(52\) 0 0
\(53\) 4.87689i 0.669893i 0.942237 + 0.334946i \(0.108718\pi\)
−0.942237 + 0.334946i \(0.891282\pi\)
\(54\) 0 0
\(55\) −1.12311 −0.151440
\(56\) 0 0
\(57\) −15.3693 −2.03572
\(58\) 0 0
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 13.3693i 1.71177i 0.517170 + 0.855883i \(0.326986\pi\)
−0.517170 + 0.855883i \(0.673014\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) −0.561553 −0.0696521
\(66\) 0 0
\(67\) 6.00000i 0.733017i 0.930415 + 0.366508i \(0.119447\pi\)
−0.930415 + 0.366508i \(0.880553\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.68466 0.199932 0.0999661 0.994991i \(-0.468127\pi\)
0.0999661 + 0.994991i \(0.468127\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 12.0000i 1.38564i
\(76\) 0 0
\(77\) − 1.12311i − 0.127990i
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) − 17.3693i − 1.90653i −0.302135 0.953265i \(-0.597699\pi\)
0.302135 0.953265i \(-0.402301\pi\)
\(84\) 0 0
\(85\) 0.315342i 0.0342036i
\(86\) 0 0
\(87\) 21.1231 2.26463
\(88\) 0 0
\(89\) 8.24621 0.874097 0.437048 0.899438i \(-0.356024\pi\)
0.437048 + 0.899438i \(0.356024\pi\)
\(90\) 0 0
\(91\) − 0.561553i − 0.0588667i
\(92\) 0 0
\(93\) 18.2462i 1.89204i
\(94\) 0 0
\(95\) 3.36932 0.345685
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 7.12311i 0.715899i
\(100\) 0 0
\(101\) 7.12311i 0.708776i 0.935099 + 0.354388i \(0.115311\pi\)
−0.935099 + 0.354388i \(0.884689\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0.807764 0.0788297
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) − 1.68466i − 0.161361i −0.996740 0.0806805i \(-0.974291\pi\)
0.996740 0.0806805i \(-0.0257093\pi\)
\(110\) 0 0
\(111\) −24.8078 −2.35465
\(112\) 0 0
\(113\) 8.24621 0.775738 0.387869 0.921714i \(-0.373211\pi\)
0.387869 + 0.921714i \(0.373211\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.56155i 0.329266i
\(118\) 0 0
\(119\) −0.315342 −0.0289073
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) − 18.2462i − 1.64521i
\(124\) 0 0
\(125\) − 5.43845i − 0.486430i
\(126\) 0 0
\(127\) 2.24621 0.199319 0.0996595 0.995022i \(-0.468225\pi\)
0.0996595 + 0.995022i \(0.468225\pi\)
\(128\) 0 0
\(129\) −22.5616 −1.98643
\(130\) 0 0
\(131\) 15.6847i 1.37037i 0.728367 + 0.685187i \(0.240280\pi\)
−0.728367 + 0.685187i \(0.759720\pi\)
\(132\) 0 0
\(133\) 3.36932i 0.292157i
\(134\) 0 0
\(135\) −0.807764 −0.0695213
\(136\) 0 0
\(137\) −4.87689 −0.416661 −0.208331 0.978058i \(-0.566803\pi\)
−0.208331 + 0.978058i \(0.566803\pi\)
\(138\) 0 0
\(139\) − 7.68466i − 0.651804i −0.945404 0.325902i \(-0.894332\pi\)
0.945404 0.325902i \(-0.105668\pi\)
\(140\) 0 0
\(141\) 4.31534i 0.363417i
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) −4.63068 −0.384557
\(146\) 0 0
\(147\) − 17.1231i − 1.41229i
\(148\) 0 0
\(149\) 8.24621i 0.675556i 0.941226 + 0.337778i \(0.109675\pi\)
−0.941226 + 0.337778i \(0.890325\pi\)
\(150\) 0 0
\(151\) −10.3153 −0.839451 −0.419725 0.907651i \(-0.637874\pi\)
−0.419725 + 0.907651i \(0.637874\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) − 4.00000i − 0.321288i
\(156\) 0 0
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 0 0
\(159\) −12.4924 −0.990714
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 9.36932i 0.733862i 0.930248 + 0.366931i \(0.119591\pi\)
−0.930248 + 0.366931i \(0.880409\pi\)
\(164\) 0 0
\(165\) − 2.87689i − 0.223966i
\(166\) 0 0
\(167\) 9.36932 0.725020 0.362510 0.931980i \(-0.381920\pi\)
0.362510 + 0.931980i \(0.381920\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) − 21.3693i − 1.63415i
\(172\) 0 0
\(173\) 21.3693i 1.62468i 0.583185 + 0.812340i \(0.301806\pi\)
−0.583185 + 0.812340i \(0.698194\pi\)
\(174\) 0 0
\(175\) 2.63068 0.198861
\(176\) 0 0
\(177\) −15.3693 −1.15523
\(178\) 0 0
\(179\) − 19.0540i − 1.42416i −0.702098 0.712080i \(-0.747753\pi\)
0.702098 0.712080i \(-0.252247\pi\)
\(180\) 0 0
\(181\) − 17.3693i − 1.29105i −0.763739 0.645526i \(-0.776638\pi\)
0.763739 0.645526i \(-0.223362\pi\)
\(182\) 0 0
\(183\) −34.2462 −2.53155
\(184\) 0 0
\(185\) 5.43845 0.399843
\(186\) 0 0
\(187\) 1.12311i 0.0821296i
\(188\) 0 0
\(189\) − 0.807764i − 0.0587562i
\(190\) 0 0
\(191\) −19.3693 −1.40151 −0.700757 0.713400i \(-0.747154\pi\)
−0.700757 + 0.713400i \(0.747154\pi\)
\(192\) 0 0
\(193\) 1.36932 0.0985656 0.0492828 0.998785i \(-0.484306\pi\)
0.0492828 + 0.998785i \(0.484306\pi\)
\(194\) 0 0
\(195\) − 1.43845i − 0.103009i
\(196\) 0 0
\(197\) − 13.6847i − 0.974992i −0.873125 0.487496i \(-0.837910\pi\)
0.873125 0.487496i \(-0.162090\pi\)
\(198\) 0 0
\(199\) −15.3693 −1.08950 −0.544751 0.838598i \(-0.683376\pi\)
−0.544751 + 0.838598i \(0.683376\pi\)
\(200\) 0 0
\(201\) −15.3693 −1.08407
\(202\) 0 0
\(203\) − 4.63068i − 0.325010i
\(204\) 0 0
\(205\) 4.00000i 0.279372i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 3.19224i 0.219763i 0.993945 + 0.109881i \(0.0350471\pi\)
−0.993945 + 0.109881i \(0.964953\pi\)
\(212\) 0 0
\(213\) 4.31534i 0.295682i
\(214\) 0 0
\(215\) 4.94602 0.337316
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) − 25.6155i − 1.73094i
\(220\) 0 0
\(221\) 0.561553i 0.0377741i
\(222\) 0 0
\(223\) 26.8078 1.79518 0.897590 0.440831i \(-0.145316\pi\)
0.897590 + 0.440831i \(0.145316\pi\)
\(224\) 0 0
\(225\) −16.6847 −1.11231
\(226\) 0 0
\(227\) 21.3693i 1.41833i 0.705042 + 0.709166i \(0.250928\pi\)
−0.705042 + 0.709166i \(0.749072\pi\)
\(228\) 0 0
\(229\) − 13.6847i − 0.904308i −0.891940 0.452154i \(-0.850656\pi\)
0.891940 0.452154i \(-0.149344\pi\)
\(230\) 0 0
\(231\) 2.87689 0.189286
\(232\) 0 0
\(233\) −13.6847 −0.896512 −0.448256 0.893905i \(-0.647955\pi\)
−0.448256 + 0.893905i \(0.647955\pi\)
\(234\) 0 0
\(235\) − 0.946025i − 0.0617118i
\(236\) 0 0
\(237\) 30.7386i 1.99669i
\(238\) 0 0
\(239\) 21.6847 1.40266 0.701332 0.712835i \(-0.252589\pi\)
0.701332 + 0.712835i \(0.252589\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) − 22.2462i − 1.42710i
\(244\) 0 0
\(245\) 3.75379i 0.239821i
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 0 0
\(249\) 44.4924 2.81959
\(250\) 0 0
\(251\) 16.0000i 1.00991i 0.863145 + 0.504956i \(0.168491\pi\)
−0.863145 + 0.504956i \(0.831509\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −0.807764 −0.0505841
\(256\) 0 0
\(257\) 15.9309 0.993740 0.496870 0.867825i \(-0.334483\pi\)
0.496870 + 0.867825i \(0.334483\pi\)
\(258\) 0 0
\(259\) 5.43845i 0.337929i
\(260\) 0 0
\(261\) 29.3693i 1.81792i
\(262\) 0 0
\(263\) −26.7386 −1.64877 −0.824387 0.566026i \(-0.808480\pi\)
−0.824387 + 0.566026i \(0.808480\pi\)
\(264\) 0 0
\(265\) 2.73863 0.168233
\(266\) 0 0
\(267\) 21.1231i 1.29271i
\(268\) 0 0
\(269\) − 21.3693i − 1.30291i −0.758687 0.651455i \(-0.774159\pi\)
0.758687 0.651455i \(-0.225841\pi\)
\(270\) 0 0
\(271\) 1.68466 0.102336 0.0511679 0.998690i \(-0.483706\pi\)
0.0511679 + 0.998690i \(0.483706\pi\)
\(272\) 0 0
\(273\) 1.43845 0.0870588
\(274\) 0 0
\(275\) − 9.36932i − 0.564991i
\(276\) 0 0
\(277\) 1.36932i 0.0822743i 0.999154 + 0.0411371i \(0.0130981\pi\)
−0.999154 + 0.0411371i \(0.986902\pi\)
\(278\) 0 0
\(279\) −25.3693 −1.51882
\(280\) 0 0
\(281\) −20.2462 −1.20779 −0.603894 0.797065i \(-0.706385\pi\)
−0.603894 + 0.797065i \(0.706385\pi\)
\(282\) 0 0
\(283\) − 2.24621i − 0.133523i −0.997769 0.0667617i \(-0.978733\pi\)
0.997769 0.0667617i \(-0.0212667\pi\)
\(284\) 0 0
\(285\) 8.63068i 0.511238i
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −16.6847 −0.981450
\(290\) 0 0
\(291\) − 15.3693i − 0.900965i
\(292\) 0 0
\(293\) 23.4384i 1.36929i 0.728877 + 0.684644i \(0.240042\pi\)
−0.728877 + 0.684644i \(0.759958\pi\)
\(294\) 0 0
\(295\) 3.36932 0.196169
\(296\) 0 0
\(297\) −2.87689 −0.166934
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 4.94602i 0.285084i
\(302\) 0 0
\(303\) −18.2462 −1.04822
\(304\) 0 0
\(305\) 7.50758 0.429883
\(306\) 0 0
\(307\) − 6.00000i − 0.342438i −0.985233 0.171219i \(-0.945229\pi\)
0.985233 0.171219i \(-0.0547706\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.3693 0.871514 0.435757 0.900064i \(-0.356481\pi\)
0.435757 + 0.900064i \(0.356481\pi\)
\(312\) 0 0
\(313\) −17.0540 −0.963948 −0.481974 0.876186i \(-0.660080\pi\)
−0.481974 + 0.876186i \(0.660080\pi\)
\(314\) 0 0
\(315\) 1.12311i 0.0632798i
\(316\) 0 0
\(317\) − 32.2462i − 1.81113i −0.424210 0.905564i \(-0.639448\pi\)
0.424210 0.905564i \(-0.360552\pi\)
\(318\) 0 0
\(319\) −16.4924 −0.923398
\(320\) 0 0
\(321\) −30.7386 −1.71566
\(322\) 0 0
\(323\) − 3.36932i − 0.187474i
\(324\) 0 0
\(325\) − 4.68466i − 0.259858i
\(326\) 0 0
\(327\) 4.31534 0.238639
\(328\) 0 0
\(329\) 0.946025 0.0521560
\(330\) 0 0
\(331\) − 21.3693i − 1.17456i −0.809382 0.587282i \(-0.800198\pi\)
0.809382 0.587282i \(-0.199802\pi\)
\(332\) 0 0
\(333\) − 34.4924i − 1.89017i
\(334\) 0 0
\(335\) 3.36932 0.184085
\(336\) 0 0
\(337\) 1.05398 0.0574137 0.0287068 0.999588i \(-0.490861\pi\)
0.0287068 + 0.999588i \(0.490861\pi\)
\(338\) 0 0
\(339\) 21.1231i 1.14725i
\(340\) 0 0
\(341\) − 14.2462i − 0.771476i
\(342\) 0 0
\(343\) −7.68466 −0.414933
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.3153i 0.875853i 0.899011 + 0.437927i \(0.144287\pi\)
−0.899011 + 0.437927i \(0.855713\pi\)
\(348\) 0 0
\(349\) 29.0540i 1.55522i 0.628745 + 0.777612i \(0.283569\pi\)
−0.628745 + 0.777612i \(0.716431\pi\)
\(350\) 0 0
\(351\) −1.43845 −0.0767786
\(352\) 0 0
\(353\) −25.8617 −1.37648 −0.688241 0.725482i \(-0.741617\pi\)
−0.688241 + 0.725482i \(0.741617\pi\)
\(354\) 0 0
\(355\) − 0.946025i − 0.0502098i
\(356\) 0 0
\(357\) − 0.807764i − 0.0427514i
\(358\) 0 0
\(359\) 17.3693 0.916717 0.458359 0.888767i \(-0.348437\pi\)
0.458359 + 0.888767i \(0.348437\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) 17.9309i 0.941127i
\(364\) 0 0
\(365\) 5.61553i 0.293930i
\(366\) 0 0
\(367\) −27.3693 −1.42867 −0.714333 0.699806i \(-0.753270\pi\)
−0.714333 + 0.699806i \(0.753270\pi\)
\(368\) 0 0
\(369\) 25.3693 1.32067
\(370\) 0 0
\(371\) 2.73863i 0.142183i
\(372\) 0 0
\(373\) 13.3693i 0.692237i 0.938191 + 0.346118i \(0.112500\pi\)
−0.938191 + 0.346118i \(0.887500\pi\)
\(374\) 0 0
\(375\) 13.9309 0.719387
\(376\) 0 0
\(377\) −8.24621 −0.424701
\(378\) 0 0
\(379\) − 16.8769i − 0.866908i −0.901176 0.433454i \(-0.857295\pi\)
0.901176 0.433454i \(-0.142705\pi\)
\(380\) 0 0
\(381\) 5.75379i 0.294776i
\(382\) 0 0
\(383\) −1.68466 −0.0860820 −0.0430410 0.999073i \(-0.513705\pi\)
−0.0430410 + 0.999073i \(0.513705\pi\)
\(384\) 0 0
\(385\) −0.630683 −0.0321426
\(386\) 0 0
\(387\) − 31.3693i − 1.59459i
\(388\) 0 0
\(389\) − 20.2462i − 1.02652i −0.858232 0.513262i \(-0.828437\pi\)
0.858232 0.513262i \(-0.171563\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −40.1771 −2.02667
\(394\) 0 0
\(395\) − 6.73863i − 0.339057i
\(396\) 0 0
\(397\) − 6.00000i − 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) 0 0
\(399\) −8.63068 −0.432075
\(400\) 0 0
\(401\) 31.1231 1.55421 0.777107 0.629369i \(-0.216686\pi\)
0.777107 + 0.629369i \(0.216686\pi\)
\(402\) 0 0
\(403\) − 7.12311i − 0.354827i
\(404\) 0 0
\(405\) 3.93087i 0.195326i
\(406\) 0 0
\(407\) 19.3693 0.960101
\(408\) 0 0
\(409\) 6.63068 0.327866 0.163933 0.986471i \(-0.447582\pi\)
0.163933 + 0.986471i \(0.447582\pi\)
\(410\) 0 0
\(411\) − 12.4924i − 0.616206i
\(412\) 0 0
\(413\) 3.36932i 0.165793i
\(414\) 0 0
\(415\) −9.75379 −0.478795
\(416\) 0 0
\(417\) 19.6847 0.963962
\(418\) 0 0
\(419\) 15.0540i 0.735435i 0.929938 + 0.367717i \(0.119861\pi\)
−0.929938 + 0.367717i \(0.880139\pi\)
\(420\) 0 0
\(421\) 33.6847i 1.64169i 0.571151 + 0.820845i \(0.306497\pi\)
−0.571151 + 0.820845i \(0.693503\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) −2.63068 −0.127607
\(426\) 0 0
\(427\) 7.50758i 0.363317i
\(428\) 0 0
\(429\) − 5.12311i − 0.247346i
\(430\) 0 0
\(431\) 5.68466 0.273820 0.136910 0.990583i \(-0.456283\pi\)
0.136910 + 0.990583i \(0.456283\pi\)
\(432\) 0 0
\(433\) 14.3153 0.687951 0.343976 0.938979i \(-0.388226\pi\)
0.343976 + 0.938979i \(0.388226\pi\)
\(434\) 0 0
\(435\) − 11.8617i − 0.568727i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 41.6155 1.98620 0.993100 0.117267i \(-0.0374134\pi\)
0.993100 + 0.117267i \(0.0374134\pi\)
\(440\) 0 0
\(441\) 23.8078 1.13370
\(442\) 0 0
\(443\) 7.68466i 0.365109i 0.983196 + 0.182555i \(0.0584366\pi\)
−0.983196 + 0.182555i \(0.941563\pi\)
\(444\) 0 0
\(445\) − 4.63068i − 0.219515i
\(446\) 0 0
\(447\) −21.1231 −0.999089
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 14.2462i 0.670828i
\(452\) 0 0
\(453\) − 26.4233i − 1.24147i
\(454\) 0 0
\(455\) −0.315342 −0.0147834
\(456\) 0 0
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) 0 0
\(459\) 0.807764i 0.0377032i
\(460\) 0 0
\(461\) − 21.1922i − 0.987021i −0.869740 0.493510i \(-0.835713\pi\)
0.869740 0.493510i \(-0.164287\pi\)
\(462\) 0 0
\(463\) 23.6155 1.09751 0.548753 0.835984i \(-0.315103\pi\)
0.548753 + 0.835984i \(0.315103\pi\)
\(464\) 0 0
\(465\) 10.2462 0.475157
\(466\) 0 0
\(467\) 14.7386i 0.682023i 0.940059 + 0.341011i \(0.110769\pi\)
−0.940059 + 0.341011i \(0.889231\pi\)
\(468\) 0 0
\(469\) 3.36932i 0.155581i
\(470\) 0 0
\(471\) −5.12311 −0.236060
\(472\) 0 0
\(473\) 17.6155 0.809963
\(474\) 0 0
\(475\) 28.1080i 1.28968i
\(476\) 0 0
\(477\) − 17.3693i − 0.795286i
\(478\) 0 0
\(479\) 33.6847 1.53909 0.769546 0.638592i \(-0.220483\pi\)
0.769546 + 0.638592i \(0.220483\pi\)
\(480\) 0 0
\(481\) 9.68466 0.441582
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.36932i 0.152993i
\(486\) 0 0
\(487\) 19.1231 0.866551 0.433275 0.901262i \(-0.357358\pi\)
0.433275 + 0.901262i \(0.357358\pi\)
\(488\) 0 0
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) − 35.6847i − 1.61043i −0.592986 0.805213i \(-0.702051\pi\)
0.592986 0.805213i \(-0.297949\pi\)
\(492\) 0 0
\(493\) 4.63068i 0.208555i
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 0.946025 0.0424350
\(498\) 0 0
\(499\) 16.8769i 0.755514i 0.925905 + 0.377757i \(0.123304\pi\)
−0.925905 + 0.377757i \(0.876696\pi\)
\(500\) 0 0
\(501\) 24.0000i 1.07224i
\(502\) 0 0
\(503\) 4.63068 0.206472 0.103236 0.994657i \(-0.467080\pi\)
0.103236 + 0.994657i \(0.467080\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) 0 0
\(507\) − 2.56155i − 0.113763i
\(508\) 0 0
\(509\) − 10.4924i − 0.465068i −0.972588 0.232534i \(-0.925298\pi\)
0.972588 0.232534i \(-0.0747018\pi\)
\(510\) 0 0
\(511\) −5.61553 −0.248416
\(512\) 0 0
\(513\) 8.63068 0.381054
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 3.36932i − 0.148182i
\(518\) 0 0
\(519\) −54.7386 −2.40276
\(520\) 0 0
\(521\) −12.5616 −0.550332 −0.275166 0.961397i \(-0.588733\pi\)
−0.275166 + 0.961397i \(0.588733\pi\)
\(522\) 0 0
\(523\) 40.4924i 1.77061i 0.465011 + 0.885305i \(0.346050\pi\)
−0.465011 + 0.885305i \(0.653950\pi\)
\(524\) 0 0
\(525\) 6.73863i 0.294098i
\(526\) 0 0
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) − 21.3693i − 0.927349i
\(532\) 0 0
\(533\) 7.12311i 0.308536i
\(534\) 0 0
\(535\) 6.73863 0.291337
\(536\) 0 0
\(537\) 48.8078 2.10621
\(538\) 0 0
\(539\) 13.3693i 0.575857i
\(540\) 0 0
\(541\) − 18.3153i − 0.787438i −0.919231 0.393719i \(-0.871188\pi\)
0.919231 0.393719i \(-0.128812\pi\)
\(542\) 0 0
\(543\) 44.4924 1.90935
\(544\) 0 0
\(545\) −0.946025 −0.0405232
\(546\) 0 0
\(547\) 23.0540i 0.985717i 0.870110 + 0.492858i \(0.164048\pi\)
−0.870110 + 0.492858i \(0.835952\pi\)
\(548\) 0 0
\(549\) − 47.6155i − 2.03218i
\(550\) 0 0
\(551\) 49.4773 2.10780
\(552\) 0 0
\(553\) 6.73863 0.286556
\(554\) 0 0
\(555\) 13.9309i 0.591332i
\(556\) 0 0
\(557\) − 43.3002i − 1.83469i −0.398096 0.917344i \(-0.630329\pi\)
0.398096 0.917344i \(-0.369671\pi\)
\(558\) 0 0
\(559\) 8.80776 0.372529
\(560\) 0 0
\(561\) −2.87689 −0.121463
\(562\) 0 0
\(563\) 43.6847i 1.84109i 0.390638 + 0.920544i \(0.372254\pi\)
−0.390638 + 0.920544i \(0.627746\pi\)
\(564\) 0 0
\(565\) − 4.63068i − 0.194814i
\(566\) 0 0
\(567\) −3.93087 −0.165081
\(568\) 0 0
\(569\) 25.6847 1.07676 0.538378 0.842703i \(-0.319037\pi\)
0.538378 + 0.842703i \(0.319037\pi\)
\(570\) 0 0
\(571\) 17.4384i 0.729776i 0.931051 + 0.364888i \(0.118893\pi\)
−0.931051 + 0.364888i \(0.881107\pi\)
\(572\) 0 0
\(573\) − 49.6155i − 2.07272i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −4.73863 −0.197272 −0.0986360 0.995124i \(-0.531448\pi\)
−0.0986360 + 0.995124i \(0.531448\pi\)
\(578\) 0 0
\(579\) 3.50758i 0.145770i
\(580\) 0 0
\(581\) − 9.75379i − 0.404655i
\(582\) 0 0
\(583\) 9.75379 0.403961
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) 9.36932i 0.386713i 0.981129 + 0.193357i \(0.0619374\pi\)
−0.981129 + 0.193357i \(0.938063\pi\)
\(588\) 0 0
\(589\) 42.7386i 1.76101i
\(590\) 0 0
\(591\) 35.0540 1.44193
\(592\) 0 0
\(593\) 20.2462 0.831412 0.415706 0.909499i \(-0.363534\pi\)
0.415706 + 0.909499i \(0.363534\pi\)
\(594\) 0 0
\(595\) 0.177081i 0.00725961i
\(596\) 0 0
\(597\) − 39.3693i − 1.61128i
\(598\) 0 0
\(599\) −35.3693 −1.44515 −0.722576 0.691292i \(-0.757042\pi\)
−0.722576 + 0.691292i \(0.757042\pi\)
\(600\) 0 0
\(601\) 21.0540 0.858810 0.429405 0.903112i \(-0.358723\pi\)
0.429405 + 0.903112i \(0.358723\pi\)
\(602\) 0 0
\(603\) − 21.3693i − 0.870226i
\(604\) 0 0
\(605\) − 3.93087i − 0.159813i
\(606\) 0 0
\(607\) 31.8617 1.29323 0.646614 0.762817i \(-0.276184\pi\)
0.646614 + 0.762817i \(0.276184\pi\)
\(608\) 0 0
\(609\) 11.8617 0.480662
\(610\) 0 0
\(611\) − 1.68466i − 0.0681540i
\(612\) 0 0
\(613\) − 10.0000i − 0.403896i −0.979396 0.201948i \(-0.935273\pi\)
0.979396 0.201948i \(-0.0647272\pi\)
\(614\) 0 0
\(615\) −10.2462 −0.413167
\(616\) 0 0
\(617\) 23.6155 0.950725 0.475363 0.879790i \(-0.342317\pi\)
0.475363 + 0.879790i \(0.342317\pi\)
\(618\) 0 0
\(619\) − 9.36932i − 0.376585i −0.982113 0.188292i \(-0.939705\pi\)
0.982113 0.188292i \(-0.0602952\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.63068 0.185524
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) 0 0
\(627\) 30.7386i 1.22758i
\(628\) 0 0
\(629\) − 5.43845i − 0.216845i
\(630\) 0 0
\(631\) −41.0540 −1.63433 −0.817166 0.576402i \(-0.804456\pi\)
−0.817166 + 0.576402i \(0.804456\pi\)
\(632\) 0 0
\(633\) −8.17708 −0.325010
\(634\) 0 0
\(635\) − 1.26137i − 0.0500558i
\(636\) 0 0
\(637\) 6.68466i 0.264856i
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 10.4924 0.414426 0.207213 0.978296i \(-0.433561\pi\)
0.207213 + 0.978296i \(0.433561\pi\)
\(642\) 0 0
\(643\) − 18.0000i − 0.709851i −0.934895 0.354925i \(-0.884506\pi\)
0.934895 0.354925i \(-0.115494\pi\)
\(644\) 0 0
\(645\) 12.6695i 0.498861i
\(646\) 0 0
\(647\) 30.1080 1.18367 0.591833 0.806061i \(-0.298405\pi\)
0.591833 + 0.806061i \(0.298405\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 10.2462i 0.401581i
\(652\) 0 0
\(653\) − 47.6155i − 1.86334i −0.363306 0.931670i \(-0.618352\pi\)
0.363306 0.931670i \(-0.381648\pi\)
\(654\) 0 0
\(655\) 8.80776 0.344148
\(656\) 0 0
\(657\) 35.6155 1.38949
\(658\) 0 0
\(659\) 12.0000i 0.467454i 0.972302 + 0.233727i \(0.0750921\pi\)
−0.972302 + 0.233727i \(0.924908\pi\)
\(660\) 0 0
\(661\) 20.7386i 0.806639i 0.915059 + 0.403320i \(0.132144\pi\)
−0.915059 + 0.403320i \(0.867856\pi\)
\(662\) 0 0
\(663\) −1.43845 −0.0558647
\(664\) 0 0
\(665\) 1.89205 0.0733705
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 68.6695i 2.65492i
\(670\) 0 0
\(671\) 26.7386 1.03223
\(672\) 0 0
\(673\) −21.0540 −0.811571 −0.405786 0.913968i \(-0.633002\pi\)
−0.405786 + 0.913968i \(0.633002\pi\)
\(674\) 0 0
\(675\) − 6.73863i − 0.259370i
\(676\) 0 0
\(677\) − 21.3693i − 0.821290i −0.911795 0.410645i \(-0.865304\pi\)
0.911795 0.410645i \(-0.134696\pi\)
\(678\) 0 0
\(679\) −3.36932 −0.129303
\(680\) 0 0
\(681\) −54.7386 −2.09759
\(682\) 0 0
\(683\) 5.36932i 0.205451i 0.994710 + 0.102726i \(0.0327564\pi\)
−0.994710 + 0.102726i \(0.967244\pi\)
\(684\) 0 0
\(685\) 2.73863i 0.104638i
\(686\) 0 0
\(687\) 35.0540 1.33739
\(688\) 0 0
\(689\) 4.87689 0.185795
\(690\) 0 0
\(691\) − 4.87689i − 0.185526i −0.995688 0.0927629i \(-0.970430\pi\)
0.995688 0.0927629i \(-0.0295699\pi\)
\(692\) 0 0
\(693\) 4.00000i 0.151947i
\(694\) 0 0
\(695\) −4.31534 −0.163690
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) − 35.0540i − 1.32586i
\(700\) 0 0
\(701\) 9.36932i 0.353874i 0.984222 + 0.176937i \(0.0566189\pi\)
−0.984222 + 0.176937i \(0.943381\pi\)
\(702\) 0 0
\(703\) −58.1080 −2.19158
\(704\) 0 0
\(705\) 2.42329 0.0912665
\(706\) 0 0
\(707\) 4.00000i 0.150435i
\(708\) 0 0
\(709\) − 16.7386i − 0.628633i −0.949318 0.314316i \(-0.898225\pi\)
0.949318 0.314316i \(-0.101775\pi\)
\(710\) 0 0
\(711\) −42.7386 −1.60282
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 1.12311i 0.0420018i
\(716\) 0 0
\(717\) 55.5464i 2.07442i
\(718\) 0 0
\(719\) −15.3693 −0.573179 −0.286589 0.958054i \(-0.592522\pi\)
−0.286589 + 0.958054i \(0.592522\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 5.12311i 0.190530i
\(724\) 0 0
\(725\) − 38.6307i − 1.43471i
\(726\) 0 0
\(727\) 41.6155 1.54343 0.771717 0.635966i \(-0.219398\pi\)
0.771717 + 0.635966i \(0.219398\pi\)
\(728\) 0 0
\(729\) 35.9848 1.33277
\(730\) 0 0
\(731\) − 4.94602i − 0.182935i
\(732\) 0 0
\(733\) − 25.0540i − 0.925390i −0.886518 0.462695i \(-0.846883\pi\)
0.886518 0.462695i \(-0.153117\pi\)
\(734\) 0 0
\(735\) −9.61553 −0.354674
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) − 52.1080i − 1.91682i −0.285392 0.958411i \(-0.592124\pi\)
0.285392 0.958411i \(-0.407876\pi\)
\(740\) 0 0
\(741\) 15.3693i 0.564606i
\(742\) 0 0
\(743\) −13.6847 −0.502041 −0.251021 0.967982i \(-0.580766\pi\)
−0.251021 + 0.967982i \(0.580766\pi\)
\(744\) 0 0
\(745\) 4.63068 0.169655
\(746\) 0 0
\(747\) 61.8617i 2.26340i
\(748\) 0 0
\(749\) 6.73863i 0.246224i
\(750\) 0 0
\(751\) −26.2462 −0.957738 −0.478869 0.877886i \(-0.658953\pi\)
−0.478869 + 0.877886i \(0.658953\pi\)
\(752\) 0 0
\(753\) −40.9848 −1.49357
\(754\) 0 0
\(755\) 5.79261i 0.210815i
\(756\) 0 0
\(757\) 9.36932i 0.340534i 0.985398 + 0.170267i \(0.0544630\pi\)
−0.985398 + 0.170267i \(0.945537\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) − 0.946025i − 0.0342484i
\(764\) 0 0
\(765\) − 1.12311i − 0.0406060i
\(766\) 0 0
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) 9.36932 0.337866 0.168933 0.985628i \(-0.445968\pi\)
0.168933 + 0.985628i \(0.445968\pi\)
\(770\) 0 0
\(771\) 40.8078i 1.46966i
\(772\) 0 0
\(773\) 25.6847i 0.923813i 0.886929 + 0.461906i \(0.152834\pi\)
−0.886929 + 0.461906i \(0.847166\pi\)
\(774\) 0 0
\(775\) 33.3693 1.19866
\(776\) 0 0
\(777\) −13.9309 −0.499767
\(778\) 0 0
\(779\) − 42.7386i − 1.53127i
\(780\) 0 0
\(781\) − 3.36932i − 0.120564i
\(782\) 0 0
\(783\) −11.8617 −0.423904
\(784\) 0 0
\(785\) 1.12311 0.0400854
\(786\) 0 0
\(787\) 35.6155i 1.26956i 0.772694 + 0.634778i \(0.218909\pi\)
−0.772694 + 0.634778i \(0.781091\pi\)
\(788\) 0 0
\(789\) − 68.4924i − 2.43839i
\(790\) 0 0
\(791\) 4.63068 0.164648
\(792\) 0 0
\(793\) 13.3693 0.474758
\(794\) 0 0
\(795\) 7.01515i 0.248802i
\(796\) 0 0
\(797\) − 12.7386i − 0.451226i −0.974217 0.225613i \(-0.927562\pi\)
0.974217 0.225613i \(-0.0724384\pi\)
\(798\) 0 0
\(799\) −0.946025 −0.0334679
\(800\) 0 0
\(801\) −29.3693 −1.03771
\(802\) 0 0
\(803\) 20.0000i 0.705785i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 54.7386 1.92689
\(808\) 0 0
\(809\) 39.9309 1.40389 0.701947 0.712229i \(-0.252314\pi\)
0.701947 + 0.712229i \(0.252314\pi\)
\(810\) 0 0
\(811\) − 25.8617i − 0.908128i −0.890969 0.454064i \(-0.849974\pi\)
0.890969 0.454064i \(-0.150026\pi\)
\(812\) 0 0
\(813\) 4.31534i 0.151346i
\(814\) 0 0
\(815\) 5.26137 0.184298
\(816\) 0 0
\(817\) −52.8466 −1.84887
\(818\) 0 0
\(819\) 2.00000i 0.0698857i
\(820\) 0 0
\(821\) − 5.05398i − 0.176385i −0.996103 0.0881925i \(-0.971891\pi\)
0.996103 0.0881925i \(-0.0281091\pi\)
\(822\) 0 0
\(823\) 39.3693 1.37233 0.686164 0.727447i \(-0.259293\pi\)
0.686164 + 0.727447i \(0.259293\pi\)
\(824\) 0 0
\(825\) 24.0000 0.835573
\(826\) 0 0
\(827\) 36.7386i 1.27753i 0.769403 + 0.638764i \(0.220554\pi\)
−0.769403 + 0.638764i \(0.779446\pi\)
\(828\) 0 0
\(829\) − 32.7386i − 1.13706i −0.822663 0.568530i \(-0.807512\pi\)
0.822663 0.568530i \(-0.192488\pi\)
\(830\) 0 0
\(831\) −3.50758 −0.121677
\(832\) 0 0
\(833\) 3.75379 0.130061
\(834\) 0 0
\(835\) − 5.26137i − 0.182077i
\(836\) 0 0
\(837\) − 10.2462i − 0.354161i
\(838\) 0 0
\(839\) −36.1080 −1.24658 −0.623292 0.781989i \(-0.714205\pi\)
−0.623292 + 0.781989i \(0.714205\pi\)
\(840\) 0 0
\(841\) −39.0000 −1.34483
\(842\) 0 0
\(843\) − 51.8617i − 1.78621i
\(844\) 0 0
\(845\) 0.561553i 0.0193180i
\(846\) 0 0
\(847\) 3.93087 0.135066
\(848\) 0 0
\(849\) 5.75379 0.197470
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 21.6847i − 0.742469i −0.928539 0.371234i \(-0.878935\pi\)
0.928539 0.371234i \(-0.121065\pi\)
\(854\) 0 0
\(855\) −12.0000 −0.410391
\(856\) 0 0
\(857\) 32.2462 1.10151 0.550755 0.834667i \(-0.314340\pi\)
0.550755 + 0.834667i \(0.314340\pi\)
\(858\) 0 0
\(859\) 36.0000i 1.22830i 0.789188 + 0.614152i \(0.210502\pi\)
−0.789188 + 0.614152i \(0.789498\pi\)
\(860\) 0 0
\(861\) − 10.2462i − 0.349190i
\(862\) 0 0
\(863\) −25.0540 −0.852847 −0.426424 0.904524i \(-0.640227\pi\)
−0.426424 + 0.904524i \(0.640227\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) 0 0
\(867\) − 42.7386i − 1.45148i
\(868\) 0 0
\(869\) − 24.0000i − 0.814144i
\(870\) 0 0
\(871\) 6.00000 0.203302
\(872\) 0 0
\(873\) 21.3693 0.723242
\(874\) 0 0
\(875\) − 3.05398i − 0.103243i
\(876\) 0 0
\(877\) − 31.7926i − 1.07356i −0.843722 0.536780i \(-0.819640\pi\)
0.843722 0.536780i \(-0.180360\pi\)
\(878\) 0 0
\(879\) −60.0388 −2.02506
\(880\) 0 0
\(881\) −19.3002 −0.650240 −0.325120 0.945673i \(-0.605405\pi\)
−0.325120 + 0.945673i \(0.605405\pi\)
\(882\) 0 0
\(883\) − 17.4384i − 0.586850i −0.955982 0.293425i \(-0.905205\pi\)
0.955982 0.293425i \(-0.0947952\pi\)
\(884\) 0 0
\(885\) 8.63068i 0.290117i
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 1.26137 0.0423049
\(890\) 0 0
\(891\) 14.0000i 0.469018i
\(892\) 0 0
\(893\) 10.1080i 0.338250i
\(894\) 0 0
\(895\) −10.6998 −0.357655
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 58.7386i − 1.95904i
\(900\) 0 0
\(901\) − 2.73863i − 0.0912371i
\(902\) 0 0
\(903\) −12.6695 −0.421615
\(904\) 0 0
\(905\) −9.75379 −0.324227
\(906\) 0 0
\(907\) − 26.4233i − 0.877371i −0.898641 0.438686i \(-0.855444\pi\)
0.898641 0.438686i \(-0.144556\pi\)
\(908\) 0 0
\(909\) − 25.3693i − 0.841447i
\(910\) 0 0
\(911\) −27.3693 −0.906786 −0.453393 0.891311i \(-0.649787\pi\)
−0.453393 + 0.891311i \(0.649787\pi\)
\(912\) 0 0
\(913\) −34.7386 −1.14968
\(914\) 0 0
\(915\) 19.2311i 0.635759i
\(916\) 0 0
\(917\) 8.80776i 0.290858i
\(918\) 0 0
\(919\) −33.7538 −1.11343 −0.556717 0.830702i \(-0.687939\pi\)
−0.556717 + 0.830702i \(0.687939\pi\)
\(920\) 0 0
\(921\) 15.3693 0.506436
\(922\) 0 0
\(923\) − 1.68466i − 0.0554512i
\(924\) 0 0
\(925\) 45.3693i 1.49173i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 35.6155 1.16851 0.584254 0.811571i \(-0.301387\pi\)
0.584254 + 0.811571i \(0.301387\pi\)
\(930\) 0 0
\(931\) − 40.1080i − 1.31448i
\(932\) 0 0
\(933\) 39.3693i 1.28889i
\(934\) 0 0
\(935\) 0.630683 0.0206255
\(936\) 0 0
\(937\) −40.7386 −1.33087 −0.665437 0.746454i \(-0.731755\pi\)
−0.665437 + 0.746454i \(0.731755\pi\)
\(938\) 0 0
\(939\) − 43.6847i − 1.42559i
\(940\) 0 0
\(941\) 37.6847i 1.22848i 0.789117 + 0.614242i \(0.210538\pi\)
−0.789117 + 0.614242i \(0.789462\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −0.453602 −0.0147557
\(946\) 0 0
\(947\) 2.00000i 0.0649913i 0.999472 + 0.0324956i \(0.0103455\pi\)
−0.999472 + 0.0324956i \(0.989654\pi\)
\(948\) 0 0
\(949\) 10.0000i 0.324614i
\(950\) 0 0
\(951\) 82.6004 2.67850
\(952\) 0 0
\(953\) 21.1922 0.686484 0.343242 0.939247i \(-0.388475\pi\)
0.343242 + 0.939247i \(0.388475\pi\)
\(954\) 0 0
\(955\) 10.8769i 0.351968i
\(956\) 0 0
\(957\) − 42.2462i − 1.36563i
\(958\) 0 0
\(959\) −2.73863 −0.0884351
\(960\) 0 0
\(961\) 19.7386 0.636730
\(962\) 0 0
\(963\) − 42.7386i − 1.37723i
\(964\) 0 0
\(965\) − 0.768944i − 0.0247532i
\(966\) 0 0
\(967\) 8.42329 0.270875 0.135437 0.990786i \(-0.456756\pi\)
0.135437 + 0.990786i \(0.456756\pi\)
\(968\) 0 0
\(969\) 8.63068 0.277257
\(970\) 0 0
\(971\) 15.6847i 0.503345i 0.967813 + 0.251672i \(0.0809805\pi\)
−0.967813 + 0.251672i \(0.919019\pi\)
\(972\) 0 0
\(973\) − 4.31534i − 0.138343i
\(974\) 0 0
\(975\) 12.0000 0.384308
\(976\) 0 0
\(977\) 50.9848 1.63115 0.815575 0.578652i \(-0.196421\pi\)
0.815575 + 0.578652i \(0.196421\pi\)
\(978\) 0 0
\(979\) − 16.4924i − 0.527100i
\(980\) 0 0
\(981\) 6.00000i 0.191565i
\(982\) 0 0
\(983\) −40.4233 −1.28930 −0.644651 0.764477i \(-0.722997\pi\)
−0.644651 + 0.764477i \(0.722997\pi\)
\(984\) 0 0
\(985\) −7.68466 −0.244854
\(986\) 0 0
\(987\) 2.42329i 0.0771342i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 3.36932 0.107030 0.0535149 0.998567i \(-0.482958\pi\)
0.0535149 + 0.998567i \(0.482958\pi\)
\(992\) 0 0
\(993\) 54.7386 1.73708
\(994\) 0 0
\(995\) 8.63068i 0.273611i
\(996\) 0 0
\(997\) − 44.7386i − 1.41689i −0.705768 0.708443i \(-0.749398\pi\)
0.705768 0.708443i \(-0.250602\pi\)
\(998\) 0 0
\(999\) 13.9309 0.440753
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 3328.2.b.u.1665.4 4
4.3 odd 2 3328.2.b.ba.1665.1 4
8.3 odd 2 3328.2.b.ba.1665.4 4
8.5 even 2 inner 3328.2.b.u.1665.1 4
16.3 odd 4 832.2.a.o.1.2 2
16.5 even 4 416.2.a.e.1.2 yes 2
16.11 odd 4 416.2.a.c.1.1 2
16.13 even 4 832.2.a.l.1.1 2
48.5 odd 4 3744.2.a.y.1.1 2
48.11 even 4 3744.2.a.x.1.1 2
48.29 odd 4 7488.2.a.cf.1.2 2
48.35 even 4 7488.2.a.ce.1.2 2
208.155 odd 4 5408.2.a.p.1.1 2
208.181 even 4 5408.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
416.2.a.c.1.1 2 16.11 odd 4
416.2.a.e.1.2 yes 2 16.5 even 4
832.2.a.l.1.1 2 16.13 even 4
832.2.a.o.1.2 2 16.3 odd 4
3328.2.b.u.1665.1 4 8.5 even 2 inner
3328.2.b.u.1665.4 4 1.1 even 1 trivial
3328.2.b.ba.1665.1 4 4.3 odd 2
3328.2.b.ba.1665.4 4 8.3 odd 2
3744.2.a.x.1.1 2 48.11 even 4
3744.2.a.y.1.1 2 48.5 odd 4
5408.2.a.p.1.1 2 208.155 odd 4
5408.2.a.bd.1.2 2 208.181 even 4
7488.2.a.ce.1.2 2 48.35 even 4
7488.2.a.cf.1.2 2 48.29 odd 4