Properties

Label 3328.2.b.u.1665.3
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 1665.3
Root \(1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 3328.1665
Dual form 3328.2.b.u.1665.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.56155i q^{3} -3.56155i q^{5} -3.56155 q^{7} +0.561553 q^{9} +O(q^{10})\) \(q+1.56155i q^{3} -3.56155i q^{5} -3.56155 q^{7} +0.561553 q^{9} +2.00000i q^{11} +1.00000i q^{13} +5.56155 q^{15} +3.56155 q^{17} -6.00000i q^{19} -5.56155i q^{21} -7.68466 q^{25} +5.56155i q^{27} -8.24621i q^{29} -1.12311 q^{31} -3.12311 q^{33} +12.6847i q^{35} +2.68466i q^{37} -1.56155 q^{39} +1.12311 q^{41} +11.8078i q^{43} -2.00000i q^{45} -10.6847 q^{47} +5.68466 q^{49} +5.56155i q^{51} -13.1231i q^{53} +7.12311 q^{55} +9.36932 q^{57} -6.00000i q^{59} +11.3693i q^{61} -2.00000 q^{63} +3.56155 q^{65} -6.00000i q^{67} -10.6847 q^{71} -10.0000 q^{73} -12.0000i q^{75} -7.12311i q^{77} +12.0000 q^{79} -7.00000 q^{81} -7.36932i q^{83} -12.6847i q^{85} +12.8769 q^{87} -8.24621 q^{89} -3.56155i q^{91} -1.75379i q^{93} -21.3693 q^{95} -6.00000 q^{97} +1.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{7} - 6 q^{9}+O(q^{10}) \) 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
\(3\) 1.56155i 0.901563i 0.892634 + 0.450781i \(0.148855\pi\)
−0.892634 + 0.450781i \(0.851145\pi\)
\(4\) 0 0
\(5\) − 3.56155i − 1.59277i −0.604787 0.796387i \(-0.706742\pi\)
0.604787 0.796387i \(-0.293258\pi\)
\(6\) 0 0
\(7\) −3.56155 −1.34614 −0.673070 0.739579i \(-0.735025\pi\)
−0.673070 + 0.739579i \(0.735025\pi\)
\(8\) 0 0
\(9\) 0.561553 0.187184
\(10\) 0 0
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 5.56155 1.43599
\(16\) 0 0
\(17\) 3.56155 0.863803 0.431902 0.901921i \(-0.357843\pi\)
0.431902 + 0.901921i \(0.357843\pi\)
\(18\) 0 0
\(19\) − 6.00000i − 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) 0 0
\(21\) − 5.56155i − 1.21363i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −7.68466 −1.53693
\(26\) 0 0
\(27\) 5.56155i 1.07032i
\(28\) 0 0
\(29\) − 8.24621i − 1.53128i −0.643268 0.765641i \(-0.722422\pi\)
0.643268 0.765641i \(-0.277578\pi\)
\(30\) 0 0
\(31\) −1.12311 −0.201716 −0.100858 0.994901i \(-0.532159\pi\)
−0.100858 + 0.994901i \(0.532159\pi\)
\(32\) 0 0
\(33\) −3.12311 −0.543663
\(34\) 0 0
\(35\) 12.6847i 2.14410i
\(36\) 0 0
\(37\) 2.68466i 0.441355i 0.975347 + 0.220678i \(0.0708268\pi\)
−0.975347 + 0.220678i \(0.929173\pi\)
\(38\) 0 0
\(39\) −1.56155 −0.250049
\(40\) 0 0
\(41\) 1.12311 0.175400 0.0876998 0.996147i \(-0.472048\pi\)
0.0876998 + 0.996147i \(0.472048\pi\)
\(42\) 0 0
\(43\) 11.8078i 1.80067i 0.435200 + 0.900334i \(0.356677\pi\)
−0.435200 + 0.900334i \(0.643323\pi\)
\(44\) 0 0
\(45\) − 2.00000i − 0.298142i
\(46\) 0 0
\(47\) −10.6847 −1.55852 −0.779259 0.626702i \(-0.784404\pi\)
−0.779259 + 0.626702i \(0.784404\pi\)
\(48\) 0 0
\(49\) 5.68466 0.812094
\(50\) 0 0
\(51\) 5.56155i 0.778773i
\(52\) 0 0
\(53\) − 13.1231i − 1.80260i −0.433198 0.901299i \(-0.642615\pi\)
0.433198 0.901299i \(-0.357385\pi\)
\(54\) 0 0
\(55\) 7.12311 0.960479
\(56\) 0 0
\(57\) 9.36932 1.24100
\(58\) 0 0
\(59\) − 6.00000i − 0.781133i −0.920575 0.390567i \(-0.872279\pi\)
0.920575 0.390567i \(-0.127721\pi\)
\(60\) 0 0
\(61\) 11.3693i 1.45569i 0.685741 + 0.727846i \(0.259478\pi\)
−0.685741 + 0.727846i \(0.740522\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) 3.56155 0.441756
\(66\) 0 0
\(67\) − 6.00000i − 0.733017i −0.930415 0.366508i \(-0.880553\pi\)
0.930415 0.366508i \(-0.119447\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.6847 −1.26804 −0.634018 0.773318i \(-0.718595\pi\)
−0.634018 + 0.773318i \(0.718595\pi\)
\(72\) 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
\(77\) − 7.12311i − 0.811753i
\(78\) 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
\(83\) − 7.36932i − 0.808888i −0.914563 0.404444i \(-0.867465\pi\)
0.914563 0.404444i \(-0.132535\pi\)
\(84\) 0 0
\(85\) − 12.6847i − 1.37584i
\(86\) 0 0
\(87\) 12.8769 1.38055
\(88\) 0 0
\(89\) −8.24621 −0.874097 −0.437048 0.899438i \(-0.643976\pi\)
−0.437048 + 0.899438i \(0.643976\pi\)
\(90\) 0 0
\(91\) − 3.56155i − 0.373352i
\(92\) 0 0
\(93\) − 1.75379i − 0.181859i
\(94\) 0 0
\(95\) −21.3693 −2.19245
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 1.12311i 0.112876i
\(100\) 0 0
\(101\) 1.12311i 0.111753i 0.998438 + 0.0558766i \(0.0177953\pi\)
−0.998438 + 0.0558766i \(0.982205\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) −19.8078 −1.93304
\(106\) 0 0
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) − 10.6847i − 1.02340i −0.859163 0.511702i \(-0.829015\pi\)
0.859163 0.511702i \(-0.170985\pi\)
\(110\) 0 0
\(111\) −4.19224 −0.397909
\(112\) 0 0
\(113\) −8.24621 −0.775738 −0.387869 0.921714i \(-0.626789\pi\)
−0.387869 + 0.921714i \(0.626789\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.561553i 0.0519156i
\(118\) 0 0
\(119\) −12.6847 −1.16280
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 1.75379i 0.158134i
\(124\) 0 0
\(125\) 9.56155i 0.855211i
\(126\) 0 0
\(127\) −14.2462 −1.26415 −0.632073 0.774909i \(-0.717796\pi\)
−0.632073 + 0.774909i \(0.717796\pi\)
\(128\) 0 0
\(129\) −18.4384 −1.62341
\(130\) 0 0
\(131\) − 3.31534i − 0.289663i −0.989456 0.144831i \(-0.953736\pi\)
0.989456 0.144831i \(-0.0462640\pi\)
\(132\) 0 0
\(133\) 21.3693i 1.85295i
\(134\) 0 0
\(135\) 19.8078 1.70478
\(136\) 0 0
\(137\) −13.1231 −1.12118 −0.560591 0.828093i \(-0.689426\pi\)
−0.560591 + 0.828093i \(0.689426\pi\)
\(138\) 0 0
\(139\) − 4.68466i − 0.397348i −0.980066 0.198674i \(-0.936337\pi\)
0.980066 0.198674i \(-0.0636634\pi\)
\(140\) 0 0
\(141\) − 16.6847i − 1.40510i
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) −29.3693 −2.43899
\(146\) 0 0
\(147\) 8.87689i 0.732154i
\(148\) 0 0
\(149\) 8.24621i 0.675556i 0.941226 + 0.337778i \(0.109675\pi\)
−0.941226 + 0.337778i \(0.890325\pi\)
\(150\) 0 0
\(151\) −22.6847 −1.84605 −0.923026 0.384738i \(-0.874292\pi\)
−0.923026 + 0.384738i \(0.874292\pi\)
\(152\) 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 \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) 0 0
\(159\) 20.4924 1.62515
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 15.3693i 1.20382i 0.798565 + 0.601909i \(0.205593\pi\)
−0.798565 + 0.601909i \(0.794407\pi\)
\(164\) 0 0
\(165\) 11.1231i 0.865933i
\(166\) 0 0
\(167\) −15.3693 −1.18931 −0.594657 0.803980i \(-0.702712\pi\)
−0.594657 + 0.803980i \(0.702712\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) − 3.36932i − 0.257658i
\(172\) 0 0
\(173\) 3.36932i 0.256164i 0.991764 + 0.128082i \(0.0408821\pi\)
−0.991764 + 0.128082i \(0.959118\pi\)
\(174\) 0 0
\(175\) 27.3693 2.06893
\(176\) 0 0
\(177\) 9.36932 0.704241
\(178\) 0 0
\(179\) − 18.0540i − 1.34942i −0.738084 0.674709i \(-0.764269\pi\)
0.738084 0.674709i \(-0.235731\pi\)
\(180\) 0 0
\(181\) − 7.36932i − 0.547757i −0.961764 0.273879i \(-0.911693\pi\)
0.961764 0.273879i \(-0.0883066\pi\)
\(182\) 0 0
\(183\) −17.7538 −1.31240
\(184\) 0 0
\(185\) 9.56155 0.702979
\(186\) 0 0
\(187\) 7.12311i 0.520893i
\(188\) 0 0
\(189\) − 19.8078i − 1.44080i
\(190\) 0 0
\(191\) 5.36932 0.388510 0.194255 0.980951i \(-0.437771\pi\)
0.194255 + 0.980951i \(0.437771\pi\)
\(192\) 0 0
\(193\) −23.3693 −1.68216 −0.841080 0.540911i \(-0.818080\pi\)
−0.841080 + 0.540911i \(0.818080\pi\)
\(194\) 0 0
\(195\) 5.56155i 0.398271i
\(196\) 0 0
\(197\) 1.31534i 0.0937142i 0.998902 + 0.0468571i \(0.0149205\pi\)
−0.998902 + 0.0468571i \(0.985079\pi\)
\(198\) 0 0
\(199\) 9.36932 0.664173 0.332087 0.943249i \(-0.392247\pi\)
0.332087 + 0.943249i \(0.392247\pi\)
\(200\) 0 0
\(201\) 9.36932 0.660861
\(202\) 0 0
\(203\) 29.3693i 2.06132i
\(204\) 0 0
\(205\) − 4.00000i − 0.279372i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) − 23.8078i − 1.63899i −0.573083 0.819497i \(-0.694253\pi\)
0.573083 0.819497i \(-0.305747\pi\)
\(212\) 0 0
\(213\) − 16.6847i − 1.14321i
\(214\) 0 0
\(215\) 42.0540 2.86806
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) − 15.6155i − 1.05520i
\(220\) 0 0
\(221\) 3.56155i 0.239576i
\(222\) 0 0
\(223\) 6.19224 0.414663 0.207331 0.978271i \(-0.433522\pi\)
0.207331 + 0.978271i \(0.433522\pi\)
\(224\) 0 0
\(225\) −4.31534 −0.287689
\(226\) 0 0
\(227\) 3.36932i 0.223629i 0.993729 + 0.111815i \(0.0356663\pi\)
−0.993729 + 0.111815i \(0.964334\pi\)
\(228\) 0 0
\(229\) 1.31534i 0.0869202i 0.999055 + 0.0434601i \(0.0138381\pi\)
−0.999055 + 0.0434601i \(0.986162\pi\)
\(230\) 0 0
\(231\) 11.1231 0.731847
\(232\) 0 0
\(233\) −1.31534 −0.0861709 −0.0430854 0.999071i \(-0.513719\pi\)
−0.0430854 + 0.999071i \(0.513719\pi\)
\(234\) 0 0
\(235\) 38.0540i 2.48237i
\(236\) 0 0
\(237\) 18.7386i 1.21721i
\(238\) 0 0
\(239\) 9.31534 0.602559 0.301280 0.953536i \(-0.402586\pi\)
0.301280 + 0.953536i \(0.402586\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 5.75379i 0.369106i
\(244\) 0 0
\(245\) − 20.2462i − 1.29348i
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 0 0
\(249\) 11.5076 0.729263
\(250\) 0 0
\(251\) − 16.0000i − 1.00991i −0.863145 0.504956i \(-0.831509\pi\)
0.863145 0.504956i \(-0.168491\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 19.8078 1.24041
\(256\) 0 0
\(257\) −12.9309 −0.806605 −0.403303 0.915067i \(-0.632138\pi\)
−0.403303 + 0.915067i \(0.632138\pi\)
\(258\) 0 0
\(259\) − 9.56155i − 0.594126i
\(260\) 0 0
\(261\) − 4.63068i − 0.286632i
\(262\) 0 0
\(263\) 22.7386 1.40212 0.701062 0.713100i \(-0.252710\pi\)
0.701062 + 0.713100i \(0.252710\pi\)
\(264\) 0 0
\(265\) −46.7386 −2.87113
\(266\) 0 0
\(267\) − 12.8769i − 0.788053i
\(268\) 0 0
\(269\) − 3.36932i − 0.205431i −0.994711 0.102715i \(-0.967247\pi\)
0.994711 0.102715i \(-0.0327531\pi\)
\(270\) 0 0
\(271\) −10.6847 −0.649047 −0.324523 0.945878i \(-0.605204\pi\)
−0.324523 + 0.945878i \(0.605204\pi\)
\(272\) 0 0
\(273\) 5.56155 0.336600
\(274\) 0 0
\(275\) − 15.3693i − 0.926805i
\(276\) 0 0
\(277\) 23.3693i 1.40413i 0.712115 + 0.702063i \(0.247738\pi\)
−0.712115 + 0.702063i \(0.752262\pi\)
\(278\) 0 0
\(279\) −0.630683 −0.0377580
\(280\) 0 0
\(281\) −3.75379 −0.223932 −0.111966 0.993712i \(-0.535715\pi\)
−0.111966 + 0.993712i \(0.535715\pi\)
\(282\) 0 0
\(283\) − 14.2462i − 0.846849i −0.905931 0.423425i \(-0.860828\pi\)
0.905931 0.423425i \(-0.139172\pi\)
\(284\) 0 0
\(285\) − 33.3693i − 1.97663i
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −4.31534 −0.253844
\(290\) 0 0
\(291\) − 9.36932i − 0.549239i
\(292\) 0 0
\(293\) − 27.5616i − 1.61016i −0.593164 0.805082i \(-0.702121\pi\)
0.593164 0.805082i \(-0.297879\pi\)
\(294\) 0 0
\(295\) −21.3693 −1.24417
\(296\) 0 0
\(297\) −11.1231 −0.645428
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 42.0540i − 2.42395i
\(302\) 0 0
\(303\) −1.75379 −0.100753
\(304\) 0 0
\(305\) 40.4924 2.31859
\(306\) 0 0
\(307\) 6.00000i 0.342438i 0.985233 + 0.171219i \(0.0547706\pi\)
−0.985233 + 0.171219i \(0.945229\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.36932 −0.531285 −0.265643 0.964072i \(-0.585584\pi\)
−0.265643 + 0.964072i \(0.585584\pi\)
\(312\) 0 0
\(313\) 20.0540 1.13352 0.566759 0.823884i \(-0.308197\pi\)
0.566759 + 0.823884i \(0.308197\pi\)
\(314\) 0 0
\(315\) 7.12311i 0.401342i
\(316\) 0 0
\(317\) 15.7538i 0.884821i 0.896813 + 0.442410i \(0.145877\pi\)
−0.896813 + 0.442410i \(0.854123\pi\)
\(318\) 0 0
\(319\) 16.4924 0.923398
\(320\) 0 0
\(321\) 18.7386 1.04589
\(322\) 0 0
\(323\) − 21.3693i − 1.18902i
\(324\) 0 0
\(325\) − 7.68466i − 0.426268i
\(326\) 0 0
\(327\) 16.6847 0.922664
\(328\) 0 0
\(329\) 38.0540 2.09798
\(330\) 0 0
\(331\) − 3.36932i − 0.185194i −0.995704 0.0925972i \(-0.970483\pi\)
0.995704 0.0925972i \(-0.0295169\pi\)
\(332\) 0 0
\(333\) 1.50758i 0.0826147i
\(334\) 0 0
\(335\) −21.3693 −1.16753
\(336\) 0 0
\(337\) −36.0540 −1.96399 −0.981993 0.188919i \(-0.939501\pi\)
−0.981993 + 0.188919i \(0.939501\pi\)
\(338\) 0 0
\(339\) − 12.8769i − 0.699377i
\(340\) 0 0
\(341\) − 2.24621i − 0.121639i
\(342\) 0 0
\(343\) 4.68466 0.252948
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 28.6847i − 1.53987i −0.638120 0.769937i \(-0.720288\pi\)
0.638120 0.769937i \(-0.279712\pi\)
\(348\) 0 0
\(349\) 8.05398i 0.431119i 0.976491 + 0.215560i \(0.0691576\pi\)
−0.976491 + 0.215560i \(0.930842\pi\)
\(350\) 0 0
\(351\) −5.56155 −0.296854
\(352\) 0 0
\(353\) 31.8617 1.69583 0.847915 0.530133i \(-0.177858\pi\)
0.847915 + 0.530133i \(0.177858\pi\)
\(354\) 0 0
\(355\) 38.0540i 2.01970i
\(356\) 0 0
\(357\) − 19.8078i − 1.04834i
\(358\) 0 0
\(359\) −7.36932 −0.388938 −0.194469 0.980909i \(-0.562298\pi\)
−0.194469 + 0.980909i \(0.562298\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) 10.9309i 0.573722i
\(364\) 0 0
\(365\) 35.6155i 1.86420i
\(366\) 0 0
\(367\) −2.63068 −0.137321 −0.0686603 0.997640i \(-0.521872\pi\)
−0.0686603 + 0.997640i \(0.521872\pi\)
\(368\) 0 0
\(369\) 0.630683 0.0328321
\(370\) 0 0
\(371\) 46.7386i 2.42655i
\(372\) 0 0
\(373\) 11.3693i 0.588681i 0.955701 + 0.294340i \(0.0951000\pi\)
−0.955701 + 0.294340i \(0.904900\pi\)
\(374\) 0 0
\(375\) −14.9309 −0.771027
\(376\) 0 0
\(377\) 8.24621 0.424701
\(378\) 0 0
\(379\) 25.1231i 1.29049i 0.763977 + 0.645244i \(0.223244\pi\)
−0.763977 + 0.645244i \(0.776756\pi\)
\(380\) 0 0
\(381\) − 22.2462i − 1.13971i
\(382\) 0 0
\(383\) 10.6847 0.545961 0.272980 0.962020i \(-0.411991\pi\)
0.272980 + 0.962020i \(0.411991\pi\)
\(384\) 0 0
\(385\) −25.3693 −1.29294
\(386\) 0 0
\(387\) 6.63068i 0.337057i
\(388\) 0 0
\(389\) 3.75379i 0.190325i 0.995462 + 0.0951623i \(0.0303370\pi\)
−0.995462 + 0.0951623i \(0.969663\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 5.17708 0.261149
\(394\) 0 0
\(395\) − 42.7386i − 2.15041i
\(396\) 0 0
\(397\) 6.00000i 0.301131i 0.988600 + 0.150566i \(0.0481095\pi\)
−0.988600 + 0.150566i \(0.951890\pi\)
\(398\) 0 0
\(399\) −33.3693 −1.67056
\(400\) 0 0
\(401\) 22.8769 1.14242 0.571209 0.820805i \(-0.306475\pi\)
0.571209 + 0.820805i \(0.306475\pi\)
\(402\) 0 0
\(403\) − 1.12311i − 0.0559459i
\(404\) 0 0
\(405\) 24.9309i 1.23882i
\(406\) 0 0
\(407\) −5.36932 −0.266147
\(408\) 0 0
\(409\) 31.3693 1.55111 0.775556 0.631278i \(-0.217469\pi\)
0.775556 + 0.631278i \(0.217469\pi\)
\(410\) 0 0
\(411\) − 20.4924i − 1.01082i
\(412\) 0 0
\(413\) 21.3693i 1.05152i
\(414\) 0 0
\(415\) −26.2462 −1.28838
\(416\) 0 0
\(417\) 7.31534 0.358234
\(418\) 0 0
\(419\) 22.0540i 1.07741i 0.842495 + 0.538704i \(0.181086\pi\)
−0.842495 + 0.538704i \(0.818914\pi\)
\(420\) 0 0
\(421\) − 21.3153i − 1.03885i −0.854517 0.519423i \(-0.826147\pi\)
0.854517 0.519423i \(-0.173853\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) −27.3693 −1.32761
\(426\) 0 0
\(427\) − 40.4924i − 1.95957i
\(428\) 0 0
\(429\) − 3.12311i − 0.150785i
\(430\) 0 0
\(431\) −6.68466 −0.321989 −0.160994 0.986955i \(-0.551470\pi\)
−0.160994 + 0.986955i \(0.551470\pi\)
\(432\) 0 0
\(433\) 26.6847 1.28238 0.641191 0.767381i \(-0.278440\pi\)
0.641191 + 0.767381i \(0.278440\pi\)
\(434\) 0 0
\(435\) − 45.8617i − 2.19890i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0.384472 0.0183498 0.00917492 0.999958i \(-0.497079\pi\)
0.00917492 + 0.999958i \(0.497079\pi\)
\(440\) 0 0
\(441\) 3.19224 0.152011
\(442\) 0 0
\(443\) 4.68466i 0.222575i 0.993788 + 0.111287i \(0.0354974\pi\)
−0.993788 + 0.111287i \(0.964503\pi\)
\(444\) 0 0
\(445\) 29.3693i 1.39224i
\(446\) 0 0
\(447\) −12.8769 −0.609056
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 2.24621i 0.105770i
\(452\) 0 0
\(453\) − 35.4233i − 1.66433i
\(454\) 0 0
\(455\) −12.6847 −0.594666
\(456\) 0 0
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) 0 0
\(459\) 19.8078i 0.924547i
\(460\) 0 0
\(461\) 41.8078i 1.94718i 0.228300 + 0.973591i \(0.426683\pi\)
−0.228300 + 0.973591i \(0.573317\pi\)
\(462\) 0 0
\(463\) −17.6155 −0.818663 −0.409332 0.912386i \(-0.634238\pi\)
−0.409332 + 0.912386i \(0.634238\pi\)
\(464\) 0 0
\(465\) −6.24621 −0.289661
\(466\) 0 0
\(467\) 34.7386i 1.60751i 0.594959 + 0.803756i \(0.297168\pi\)
−0.594959 + 0.803756i \(0.702832\pi\)
\(468\) 0 0
\(469\) 21.3693i 0.986743i
\(470\) 0 0
\(471\) 3.12311 0.143905
\(472\) 0 0
\(473\) −23.6155 −1.08584
\(474\) 0 0
\(475\) 46.1080i 2.11558i
\(476\) 0 0
\(477\) − 7.36932i − 0.337418i
\(478\) 0 0
\(479\) 21.3153 0.973923 0.486961 0.873423i \(-0.338105\pi\)
0.486961 + 0.873423i \(0.338105\pi\)
\(480\) 0 0
\(481\) −2.68466 −0.122410
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 21.3693i 0.970331i
\(486\) 0 0
\(487\) 10.8769 0.492879 0.246440 0.969158i \(-0.420739\pi\)
0.246440 + 0.969158i \(0.420739\pi\)
\(488\) 0 0
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) 23.3153i 1.05221i 0.850421 + 0.526103i \(0.176347\pi\)
−0.850421 + 0.526103i \(0.823653\pi\)
\(492\) 0 0
\(493\) − 29.3693i − 1.32273i
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 38.0540 1.70695
\(498\) 0 0
\(499\) − 25.1231i − 1.12466i −0.826911 0.562332i \(-0.809904\pi\)
0.826911 0.562332i \(-0.190096\pi\)
\(500\) 0 0
\(501\) − 24.0000i − 1.07224i
\(502\) 0 0
\(503\) 29.3693 1.30951 0.654757 0.755840i \(-0.272771\pi\)
0.654757 + 0.755840i \(0.272771\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) 0 0
\(507\) − 1.56155i − 0.0693510i
\(508\) 0 0
\(509\) − 22.4924i − 0.996959i −0.866901 0.498480i \(-0.833892\pi\)
0.866901 0.498480i \(-0.166108\pi\)
\(510\) 0 0
\(511\) 35.6155 1.57554
\(512\) 0 0
\(513\) 33.3693 1.47329
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 21.3693i − 0.939821i
\(518\) 0 0
\(519\) −5.26137 −0.230948
\(520\) 0 0
\(521\) −8.43845 −0.369695 −0.184848 0.982767i \(-0.559179\pi\)
−0.184848 + 0.982767i \(0.559179\pi\)
\(522\) 0 0
\(523\) − 7.50758i − 0.328283i −0.986437 0.164142i \(-0.947515\pi\)
0.986437 0.164142i \(-0.0524854\pi\)
\(524\) 0 0
\(525\) 42.7386i 1.86527i
\(526\) 0 0
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) − 3.36932i − 0.146216i
\(532\) 0 0
\(533\) 1.12311i 0.0486471i
\(534\) 0 0
\(535\) −42.7386 −1.84775
\(536\) 0 0
\(537\) 28.1922 1.21658
\(538\) 0 0
\(539\) 11.3693i 0.489711i
\(540\) 0 0
\(541\) 30.6847i 1.31924i 0.751601 + 0.659618i \(0.229282\pi\)
−0.751601 + 0.659618i \(0.770718\pi\)
\(542\) 0 0
\(543\) 11.5076 0.493837
\(544\) 0 0
\(545\) −38.0540 −1.63005
\(546\) 0 0
\(547\) 14.0540i 0.600905i 0.953797 + 0.300452i \(0.0971376\pi\)
−0.953797 + 0.300452i \(0.902862\pi\)
\(548\) 0 0
\(549\) 6.38447i 0.272483i
\(550\) 0 0
\(551\) −49.4773 −2.10780
\(552\) 0 0
\(553\) −42.7386 −1.81743
\(554\) 0 0
\(555\) 14.9309i 0.633780i
\(556\) 0 0
\(557\) − 10.3002i − 0.436433i −0.975900 0.218216i \(-0.929976\pi\)
0.975900 0.218216i \(-0.0700239\pi\)
\(558\) 0 0
\(559\) −11.8078 −0.499415
\(560\) 0 0
\(561\) −11.1231 −0.469618
\(562\) 0 0
\(563\) − 31.3153i − 1.31978i −0.751360 0.659892i \(-0.770602\pi\)
0.751360 0.659892i \(-0.229398\pi\)
\(564\) 0 0
\(565\) 29.3693i 1.23558i
\(566\) 0 0
\(567\) 24.9309 1.04700
\(568\) 0 0
\(569\) 13.3153 0.558208 0.279104 0.960261i \(-0.409963\pi\)
0.279104 + 0.960261i \(0.409963\pi\)
\(570\) 0 0
\(571\) − 21.5616i − 0.902323i −0.892442 0.451161i \(-0.851010\pi\)
0.892442 0.451161i \(-0.148990\pi\)
\(572\) 0 0
\(573\) 8.38447i 0.350266i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 44.7386 1.86249 0.931247 0.364389i \(-0.118722\pi\)
0.931247 + 0.364389i \(0.118722\pi\)
\(578\) 0 0
\(579\) − 36.4924i − 1.51657i
\(580\) 0 0
\(581\) 26.2462i 1.08888i
\(582\) 0 0
\(583\) 26.2462 1.08701
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) 15.3693i 0.634360i 0.948365 + 0.317180i \(0.102736\pi\)
−0.948365 + 0.317180i \(0.897264\pi\)
\(588\) 0 0
\(589\) 6.73863i 0.277661i
\(590\) 0 0
\(591\) −2.05398 −0.0844893
\(592\) 0 0
\(593\) 3.75379 0.154150 0.0770748 0.997025i \(-0.475442\pi\)
0.0770748 + 0.997025i \(0.475442\pi\)
\(594\) 0 0
\(595\) 45.1771i 1.85208i
\(596\) 0 0
\(597\) 14.6307i 0.598794i
\(598\) 0 0
\(599\) −10.6307 −0.434358 −0.217179 0.976132i \(-0.569686\pi\)
−0.217179 + 0.976132i \(0.569686\pi\)
\(600\) 0 0
\(601\) −16.0540 −0.654855 −0.327428 0.944876i \(-0.606182\pi\)
−0.327428 + 0.944876i \(0.606182\pi\)
\(602\) 0 0
\(603\) − 3.36932i − 0.137209i
\(604\) 0 0
\(605\) − 24.9309i − 1.01358i
\(606\) 0 0
\(607\) −25.8617 −1.04970 −0.524848 0.851196i \(-0.675878\pi\)
−0.524848 + 0.851196i \(0.675878\pi\)
\(608\) 0 0
\(609\) −45.8617 −1.85841
\(610\) 0 0
\(611\) − 10.6847i − 0.432255i
\(612\) 0 0
\(613\) 10.0000i 0.403896i 0.979396 + 0.201948i \(0.0647272\pi\)
−0.979396 + 0.201948i \(0.935273\pi\)
\(614\) 0 0
\(615\) 6.24621 0.251872
\(616\) 0 0
\(617\) −17.6155 −0.709174 −0.354587 0.935023i \(-0.615379\pi\)
−0.354587 + 0.935023i \(0.615379\pi\)
\(618\) 0 0
\(619\) − 15.3693i − 0.617745i −0.951103 0.308873i \(-0.900048\pi\)
0.951103 0.308873i \(-0.0999516\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 29.3693 1.17666
\(624\) 0 0
\(625\) −4.36932 −0.174773
\(626\) 0 0
\(627\) 18.7386i 0.748349i
\(628\) 0 0
\(629\) 9.56155i 0.381244i
\(630\) 0 0
\(631\) −3.94602 −0.157089 −0.0785444 0.996911i \(-0.525027\pi\)
−0.0785444 + 0.996911i \(0.525027\pi\)
\(632\) 0 0
\(633\) 37.1771 1.47766
\(634\) 0 0
\(635\) 50.7386i 2.01350i
\(636\) 0 0
\(637\) 5.68466i 0.225234i
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −22.4924 −0.888397 −0.444199 0.895928i \(-0.646512\pi\)
−0.444199 + 0.895928i \(0.646512\pi\)
\(642\) 0 0
\(643\) 18.0000i 0.709851i 0.934895 + 0.354925i \(0.115494\pi\)
−0.934895 + 0.354925i \(0.884506\pi\)
\(644\) 0 0
\(645\) 65.6695i 2.58573i
\(646\) 0 0
\(647\) −44.1080 −1.73406 −0.867031 0.498254i \(-0.833975\pi\)
−0.867031 + 0.498254i \(0.833975\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 6.24621i 0.244808i
\(652\) 0 0
\(653\) 6.38447i 0.249844i 0.992167 + 0.124922i \(0.0398680\pi\)
−0.992167 + 0.124922i \(0.960132\pi\)
\(654\) 0 0
\(655\) −11.8078 −0.461368
\(656\) 0 0
\(657\) −5.61553 −0.219083
\(658\) 0 0
\(659\) − 12.0000i − 0.467454i −0.972302 0.233727i \(-0.924908\pi\)
0.972302 0.233727i \(-0.0750921\pi\)
\(660\) 0 0
\(661\) 28.7386i 1.11780i 0.829234 + 0.558902i \(0.188777\pi\)
−0.829234 + 0.558902i \(0.811223\pi\)
\(662\) 0 0
\(663\) −5.56155 −0.215993
\(664\) 0 0
\(665\) 76.1080 2.95134
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 9.66950i 0.373845i
\(670\) 0 0
\(671\) −22.7386 −0.877815
\(672\) 0 0
\(673\) 16.0540 0.618835 0.309418 0.950926i \(-0.399866\pi\)
0.309418 + 0.950926i \(0.399866\pi\)
\(674\) 0 0
\(675\) − 42.7386i − 1.64501i
\(676\) 0 0
\(677\) − 3.36932i − 0.129493i −0.997902 0.0647467i \(-0.979376\pi\)
0.997902 0.0647467i \(-0.0206239\pi\)
\(678\) 0 0
\(679\) 21.3693 0.820079
\(680\) 0 0
\(681\) −5.26137 −0.201616
\(682\) 0 0
\(683\) 19.3693i 0.741146i 0.928803 + 0.370573i \(0.120839\pi\)
−0.928803 + 0.370573i \(0.879161\pi\)
\(684\) 0 0
\(685\) 46.7386i 1.78579i
\(686\) 0 0
\(687\) −2.05398 −0.0783640
\(688\) 0 0
\(689\) 13.1231 0.499951
\(690\) 0 0
\(691\) 13.1231i 0.499226i 0.968346 + 0.249613i \(0.0803035\pi\)
−0.968346 + 0.249613i \(0.919697\pi\)
\(692\) 0 0
\(693\) − 4.00000i − 0.151947i
\(694\) 0 0
\(695\) −16.6847 −0.632885
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) − 2.05398i − 0.0776885i
\(700\) 0 0
\(701\) 15.3693i 0.580491i 0.956952 + 0.290246i \(0.0937370\pi\)
−0.956952 + 0.290246i \(0.906263\pi\)
\(702\) 0 0
\(703\) 16.1080 0.607523
\(704\) 0 0
\(705\) −59.4233 −2.23801
\(706\) 0 0
\(707\) − 4.00000i − 0.150435i
\(708\) 0 0
\(709\) − 32.7386i − 1.22953i −0.788712 0.614763i \(-0.789252\pi\)
0.788712 0.614763i \(-0.210748\pi\)
\(710\) 0 0
\(711\) 6.73863 0.252719
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 7.12311i 0.266389i
\(716\) 0 0
\(717\) 14.5464i 0.543245i
\(718\) 0 0
\(719\) 9.36932 0.349417 0.174708 0.984620i \(-0.444102\pi\)
0.174708 + 0.984620i \(0.444102\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 3.12311i 0.116150i
\(724\) 0 0
\(725\) 63.3693i 2.35348i
\(726\) 0 0
\(727\) 0.384472 0.0142593 0.00712964 0.999975i \(-0.497731\pi\)
0.00712964 + 0.999975i \(0.497731\pi\)
\(728\) 0 0
\(729\) −29.9848 −1.11055
\(730\) 0 0
\(731\) 42.0540i 1.55542i
\(732\) 0 0
\(733\) − 12.0540i − 0.445224i −0.974907 0.222612i \(-0.928542\pi\)
0.974907 0.222612i \(-0.0714583\pi\)
\(734\) 0 0
\(735\) 31.6155 1.16616
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) − 22.1080i − 0.813254i −0.913594 0.406627i \(-0.866705\pi\)
0.913594 0.406627i \(-0.133295\pi\)
\(740\) 0 0
\(741\) 9.36932i 0.344190i
\(742\) 0 0
\(743\) −1.31534 −0.0482552 −0.0241276 0.999709i \(-0.507681\pi\)
−0.0241276 + 0.999709i \(0.507681\pi\)
\(744\) 0 0
\(745\) 29.3693 1.07601
\(746\) 0 0
\(747\) − 4.13826i − 0.151411i
\(748\) 0 0
\(749\) 42.7386i 1.56164i
\(750\) 0 0
\(751\) −9.75379 −0.355921 −0.177960 0.984038i \(-0.556950\pi\)
−0.177960 + 0.984038i \(0.556950\pi\)
\(752\) 0 0
\(753\) 24.9848 0.910498
\(754\) 0 0
\(755\) 80.7926i 2.94034i
\(756\) 0 0
\(757\) 15.3693i 0.558607i 0.960203 + 0.279304i \(0.0901036\pi\)
−0.960203 + 0.279304i \(0.909896\pi\)
\(758\) 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
\(763\) 38.0540i 1.37765i
\(764\) 0 0
\(765\) − 7.12311i − 0.257536i
\(766\) 0 0
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) −15.3693 −0.554232 −0.277116 0.960837i \(-0.589379\pi\)
−0.277116 + 0.960837i \(0.589379\pi\)
\(770\) 0 0
\(771\) − 20.1922i − 0.727206i
\(772\) 0 0
\(773\) − 13.3153i − 0.478920i −0.970906 0.239460i \(-0.923030\pi\)
0.970906 0.239460i \(-0.0769703\pi\)
\(774\) 0 0
\(775\) 8.63068 0.310023
\(776\) 0 0
\(777\) 14.9309 0.535642
\(778\) 0 0
\(779\) − 6.73863i − 0.241437i
\(780\) 0 0
\(781\) − 21.3693i − 0.764654i
\(782\) 0 0
\(783\) 45.8617 1.63896
\(784\) 0 0
\(785\) −7.12311 −0.254235
\(786\) 0 0
\(787\) 5.61553i 0.200172i 0.994979 + 0.100086i \(0.0319118\pi\)
−0.994979 + 0.100086i \(0.968088\pi\)
\(788\) 0 0
\(789\) 35.5076i 1.26410i
\(790\) 0 0
\(791\) 29.3693 1.04425
\(792\) 0 0
\(793\) −11.3693 −0.403736
\(794\) 0 0
\(795\) − 72.9848i − 2.58851i
\(796\) 0 0
\(797\) − 36.7386i − 1.30135i −0.759357 0.650675i \(-0.774486\pi\)
0.759357 0.650675i \(-0.225514\pi\)
\(798\) 0 0
\(799\) −38.0540 −1.34625
\(800\) 0 0
\(801\) −4.63068 −0.163617
\(802\) 0 0
\(803\) − 20.0000i − 0.705785i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 5.26137 0.185209
\(808\) 0 0
\(809\) 11.0691 0.389170 0.194585 0.980886i \(-0.437664\pi\)
0.194585 + 0.980886i \(0.437664\pi\)
\(810\) 0 0
\(811\) − 31.8617i − 1.11882i −0.828892 0.559408i \(-0.811028\pi\)
0.828892 0.559408i \(-0.188972\pi\)
\(812\) 0 0
\(813\) − 16.6847i − 0.585157i
\(814\) 0 0
\(815\) 54.7386 1.91741
\(816\) 0 0
\(817\) 70.8466 2.47861
\(818\) 0 0
\(819\) − 2.00000i − 0.0698857i
\(820\) 0 0
\(821\) − 32.0540i − 1.11869i −0.828934 0.559346i \(-0.811052\pi\)
0.828934 0.559346i \(-0.188948\pi\)
\(822\) 0 0
\(823\) 14.6307 0.509994 0.254997 0.966942i \(-0.417926\pi\)
0.254997 + 0.966942i \(0.417926\pi\)
\(824\) 0 0
\(825\) 24.0000 0.835573
\(826\) 0 0
\(827\) 12.7386i 0.442966i 0.975164 + 0.221483i \(0.0710897\pi\)
−0.975164 + 0.221483i \(0.928910\pi\)
\(828\) 0 0
\(829\) − 16.7386i − 0.581357i −0.956821 0.290678i \(-0.906119\pi\)
0.956821 0.290678i \(-0.0938810\pi\)
\(830\) 0 0
\(831\) −36.4924 −1.26591
\(832\) 0 0
\(833\) 20.2462 0.701490
\(834\) 0 0
\(835\) 54.7386i 1.89431i
\(836\) 0 0
\(837\) − 6.24621i − 0.215901i
\(838\) 0 0
\(839\) 38.1080 1.31563 0.657816 0.753178i \(-0.271480\pi\)
0.657816 + 0.753178i \(0.271480\pi\)
\(840\) 0 0
\(841\) −39.0000 −1.34483
\(842\) 0 0
\(843\) − 5.86174i − 0.201889i
\(844\) 0 0
\(845\) 3.56155i 0.122521i
\(846\) 0 0
\(847\) −24.9309 −0.856635
\(848\) 0 0
\(849\) 22.2462 0.763488
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 9.31534i 0.318951i 0.987202 + 0.159476i \(0.0509803\pi\)
−0.987202 + 0.159476i \(0.949020\pi\)
\(854\) 0 0
\(855\) −12.0000 −0.410391
\(856\) 0 0
\(857\) 15.7538 0.538139 0.269070 0.963121i \(-0.413284\pi\)
0.269070 + 0.963121i \(0.413284\pi\)
\(858\) 0 0
\(859\) − 36.0000i − 1.22830i −0.789188 0.614152i \(-0.789498\pi\)
0.789188 0.614152i \(-0.210502\pi\)
\(860\) 0 0
\(861\) − 6.24621i − 0.212870i
\(862\) 0 0
\(863\) 12.0540 0.410322 0.205161 0.978728i \(-0.434228\pi\)
0.205161 + 0.978728i \(0.434228\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) 0 0
\(867\) − 6.73863i − 0.228856i
\(868\) 0 0
\(869\) 24.0000i 0.814144i
\(870\) 0 0
\(871\) 6.00000 0.203302
\(872\) 0 0
\(873\) −3.36932 −0.114034
\(874\) 0 0
\(875\) − 34.0540i − 1.15123i
\(876\) 0 0
\(877\) − 54.7926i − 1.85021i −0.379705 0.925107i \(-0.623975\pi\)
0.379705 0.925107i \(-0.376025\pi\)
\(878\) 0 0
\(879\) 43.0388 1.45166
\(880\) 0 0
\(881\) 34.3002 1.15560 0.577801 0.816177i \(-0.303911\pi\)
0.577801 + 0.816177i \(0.303911\pi\)
\(882\) 0 0
\(883\) 21.5616i 0.725604i 0.931866 + 0.362802i \(0.118180\pi\)
−0.931866 + 0.362802i \(0.881820\pi\)
\(884\) 0 0
\(885\) − 33.3693i − 1.12170i
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 50.7386 1.70172
\(890\) 0 0
\(891\) − 14.0000i − 0.469018i
\(892\) 0 0
\(893\) 64.1080i 2.14529i
\(894\) 0 0
\(895\) −64.3002 −2.14932
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9.26137i 0.308884i
\(900\) 0 0
\(901\) − 46.7386i − 1.55709i
\(902\) 0 0
\(903\) 65.6695 2.18534
\(904\) 0 0
\(905\) −26.2462 −0.872454
\(906\) 0 0
\(907\) − 35.4233i − 1.17621i −0.808784 0.588106i \(-0.799874\pi\)
0.808784 0.588106i \(-0.200126\pi\)
\(908\) 0 0
\(909\) 0.630683i 0.0209184i
\(910\) 0 0
\(911\) −2.63068 −0.0871584 −0.0435792 0.999050i \(-0.513876\pi\)
−0.0435792 + 0.999050i \(0.513876\pi\)
\(912\) 0 0
\(913\) 14.7386 0.487778
\(914\) 0 0
\(915\) 63.2311i 2.09035i
\(916\) 0 0
\(917\) 11.8078i 0.389927i
\(918\) 0 0
\(919\) −50.2462 −1.65747 −0.828735 0.559642i \(-0.810939\pi\)
−0.828735 + 0.559642i \(0.810939\pi\)
\(920\) 0 0
\(921\) −9.36932 −0.308729
\(922\) 0 0
\(923\) − 10.6847i − 0.351690i
\(924\) 0 0
\(925\) − 20.6307i − 0.678333i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −5.61553 −0.184240 −0.0921198 0.995748i \(-0.529364\pi\)
−0.0921198 + 0.995748i \(0.529364\pi\)
\(930\) 0 0
\(931\) − 34.1080i − 1.11784i
\(932\) 0 0
\(933\) − 14.6307i − 0.478987i
\(934\) 0 0
\(935\) 25.3693 0.829665
\(936\) 0 0
\(937\) 8.73863 0.285479 0.142739 0.989760i \(-0.454409\pi\)
0.142739 + 0.989760i \(0.454409\pi\)
\(938\) 0 0
\(939\) 31.3153i 1.02194i
\(940\) 0 0
\(941\) − 25.3153i − 0.825257i −0.910900 0.412628i \(-0.864611\pi\)
0.910900 0.412628i \(-0.135389\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −70.5464 −2.29487
\(946\) 0 0
\(947\) − 2.00000i − 0.0649913i −0.999472 0.0324956i \(-0.989654\pi\)
0.999472 0.0324956i \(-0.0103455\pi\)
\(948\) 0 0
\(949\) − 10.0000i − 0.324614i
\(950\) 0 0
\(951\) −24.6004 −0.797722
\(952\) 0 0
\(953\) 41.8078 1.35429 0.677143 0.735851i \(-0.263218\pi\)
0.677143 + 0.735851i \(0.263218\pi\)
\(954\) 0 0
\(955\) − 19.1231i − 0.618809i
\(956\) 0 0
\(957\) 25.7538i 0.832502i
\(958\) 0 0
\(959\) 46.7386 1.50927
\(960\) 0 0
\(961\) −29.7386 −0.959311
\(962\) 0 0
\(963\) − 6.73863i − 0.217149i
\(964\) 0 0
\(965\) 83.2311i 2.67930i
\(966\) 0 0
\(967\) −53.4233 −1.71798 −0.858989 0.511995i \(-0.828907\pi\)
−0.858989 + 0.511995i \(0.828907\pi\)
\(968\) 0 0
\(969\) 33.3693 1.07198
\(970\) 0 0
\(971\) − 3.31534i − 0.106394i −0.998584 0.0531972i \(-0.983059\pi\)
0.998584 0.0531972i \(-0.0169412\pi\)
\(972\) 0 0
\(973\) 16.6847i 0.534886i
\(974\) 0 0
\(975\) 12.0000 0.384308
\(976\) 0 0
\(977\) −14.9848 −0.479408 −0.239704 0.970846i \(-0.577050\pi\)
−0.239704 + 0.970846i \(0.577050\pi\)
\(978\) 0 0
\(979\) − 16.4924i − 0.527100i
\(980\) 0 0
\(981\) − 6.00000i − 0.191565i
\(982\) 0 0
\(983\) 21.4233 0.683297 0.341648 0.939828i \(-0.389015\pi\)
0.341648 + 0.939828i \(0.389015\pi\)
\(984\) 0 0
\(985\) 4.68466 0.149266
\(986\) 0 0
\(987\) 59.4233i 1.89146i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −21.3693 −0.678819 −0.339409 0.940639i \(-0.610227\pi\)
−0.339409 + 0.940639i \(0.610227\pi\)
\(992\) 0 0
\(993\) 5.26137 0.166964
\(994\) 0 0
\(995\) − 33.3693i − 1.05788i
\(996\) 0 0
\(997\) − 4.73863i − 0.150074i −0.997181 0.0750370i \(-0.976093\pi\)
0.997181 0.0750370i \(-0.0239075\pi\)
\(998\) 0 0
\(999\) −14.9309 −0.472392
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.1665.3 4
4.3 odd 2 3328.2.b.ba.1665.2 4
8.3 odd 2 3328.2.b.ba.1665.3 4
8.5 even 2 inner 3328.2.b.u.1665.2 4
16.3 odd 4 416.2.a.c.1.2 2
16.5 even 4 832.2.a.l.1.2 2
16.11 odd 4 832.2.a.o.1.1 2
16.13 even 4 416.2.a.e.1.1 yes 2
48.5 odd 4 7488.2.a.cf.1.1 2
48.11 even 4 7488.2.a.ce.1.1 2
48.29 odd 4 3744.2.a.y.1.2 2
48.35 even 4 3744.2.a.x.1.2 2
208.51 odd 4 5408.2.a.p.1.2 2
208.77 even 4 5408.2.a.bd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
416.2.a.c.1.2 2 16.3 odd 4
416.2.a.e.1.1 yes 2 16.13 even 4
832.2.a.l.1.2 2 16.5 even 4
832.2.a.o.1.1 2 16.11 odd 4
3328.2.b.u.1665.2 4 8.5 even 2 inner
3328.2.b.u.1665.3 4 1.1 even 1 trivial
3328.2.b.ba.1665.2 4 4.3 odd 2
3328.2.b.ba.1665.3 4 8.3 odd 2
3744.2.a.x.1.2 2 48.35 even 4
3744.2.a.y.1.2 2 48.29 odd 4
5408.2.a.p.1.2 2 208.51 odd 4
5408.2.a.bd.1.1 2 208.77 even 4
7488.2.a.ce.1.1 2 48.11 even 4
7488.2.a.cf.1.1 2 48.5 odd 4