Properties

Label 4032.2.k.h.3905.13
Level $4032$
Weight $2$
Character 4032.3905
Analytic conductor $32.196$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4032,2,Mod(3905,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.3905");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.k (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: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.101415451701035401216.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 18x^{12} + 145x^{8} - 72x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 2016)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3905.13
Root \(0.752908 - 0.137538i\) of defining polynomial
Character \(\chi\) \(=\) 4032.3905
Dual form 4032.2.k.h.3905.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.36028 q^{5} +(-2.57794 - 0.595188i) q^{7} +O(q^{10})\) \(q+3.36028 q^{5} +(-2.57794 - 0.595188i) q^{7} -1.64575i q^{11} -3.06871i q^{13} -3.36028 q^{17} -5.53019i q^{19} -1.64575i q^{23} +6.29150 q^{25} +6.06910i q^{29} -4.33981i q^{31} +(-8.66259 - 2.00000i) q^{35} -7.29150 q^{37} +0.979531 q^{41} -5.15587 q^{43} -11.1878 q^{47} +(6.29150 + 3.06871i) q^{49} +4.24264i q^{53} -5.53019i q^{55} +6.13742 q^{59} +11.1878i q^{61} -10.3117i q^{65} -8.48528 q^{67} -5.64575i q^{71} -8.11905i q^{73} +(-0.979531 + 4.24264i) q^{77} -8.48528 q^{79} -11.1878 q^{83} -11.2915 q^{85} +7.70010 q^{89} +(-1.82646 + 7.91094i) q^{91} -18.5830i q^{95} -14.2565i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{25} - 32 q^{37} + 16 q^{49} - 96 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 3.36028 1.50276 0.751382 0.659867i \(-0.229388\pi\)
0.751382 + 0.659867i \(0.229388\pi\)
\(6\) 0 0
\(7\) −2.57794 0.595188i −0.974368 0.224960i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.64575i 0.496213i −0.968733 0.248106i \(-0.920192\pi\)
0.968733 0.248106i \(-0.0798082\pi\)
\(12\) 0 0
\(13\) 3.06871i 0.851108i −0.904933 0.425554i \(-0.860079\pi\)
0.904933 0.425554i \(-0.139921\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.36028 −0.814988 −0.407494 0.913208i \(-0.633597\pi\)
−0.407494 + 0.913208i \(0.633597\pi\)
\(18\) 0 0
\(19\) 5.53019i 1.26871i −0.773041 0.634356i \(-0.781265\pi\)
0.773041 0.634356i \(-0.218735\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.64575i 0.343163i −0.985170 0.171581i \(-0.945112\pi\)
0.985170 0.171581i \(-0.0548877\pi\)
\(24\) 0 0
\(25\) 6.29150 1.25830
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.06910i 1.12700i 0.826115 + 0.563502i \(0.190546\pi\)
−0.826115 + 0.563502i \(0.809454\pi\)
\(30\) 0 0
\(31\) 4.33981i 0.779454i −0.920931 0.389727i \(-0.872569\pi\)
0.920931 0.389727i \(-0.127431\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8.66259 2.00000i −1.46425 0.338062i
\(36\) 0 0
\(37\) −7.29150 −1.19872 −0.599358 0.800481i \(-0.704577\pi\)
−0.599358 + 0.800481i \(0.704577\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.979531 0.152977 0.0764885 0.997070i \(-0.475629\pi\)
0.0764885 + 0.997070i \(0.475629\pi\)
\(42\) 0 0
\(43\) −5.15587 −0.786263 −0.393131 0.919482i \(-0.628608\pi\)
−0.393131 + 0.919482i \(0.628608\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.1878 −1.63190 −0.815951 0.578121i \(-0.803786\pi\)
−0.815951 + 0.578121i \(0.803786\pi\)
\(48\) 0 0
\(49\) 6.29150 + 3.06871i 0.898786 + 0.438387i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.24264i 0.582772i 0.956606 + 0.291386i \(0.0941163\pi\)
−0.956606 + 0.291386i \(0.905884\pi\)
\(54\) 0 0
\(55\) 5.53019i 0.745691i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.13742 0.799025 0.399512 0.916728i \(-0.369179\pi\)
0.399512 + 0.916728i \(0.369179\pi\)
\(60\) 0 0
\(61\) 11.1878i 1.43245i 0.697871 + 0.716223i \(0.254131\pi\)
−0.697871 + 0.716223i \(0.745869\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10.3117i 1.27901i
\(66\) 0 0
\(67\) −8.48528 −1.03664 −0.518321 0.855186i \(-0.673443\pi\)
−0.518321 + 0.855186i \(0.673443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.64575i 0.670027i −0.942213 0.335014i \(-0.891259\pi\)
0.942213 0.335014i \(-0.108741\pi\)
\(72\) 0 0
\(73\) 8.11905i 0.950263i −0.879915 0.475131i \(-0.842401\pi\)
0.879915 0.475131i \(-0.157599\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.979531 + 4.24264i −0.111628 + 0.483494i
\(78\) 0 0
\(79\) −8.48528 −0.954669 −0.477334 0.878722i \(-0.658397\pi\)
−0.477334 + 0.878722i \(0.658397\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −11.1878 −1.22802 −0.614008 0.789300i \(-0.710444\pi\)
−0.614008 + 0.789300i \(0.710444\pi\)
\(84\) 0 0
\(85\) −11.2915 −1.22474
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.70010 0.816209 0.408104 0.912935i \(-0.366190\pi\)
0.408104 + 0.912935i \(0.366190\pi\)
\(90\) 0 0
\(91\) −1.82646 + 7.91094i −0.191465 + 0.829292i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 18.5830i 1.90658i
\(96\) 0 0
\(97\) 14.2565i 1.44753i −0.690049 0.723763i \(-0.742411\pi\)
0.690049 0.723763i \(-0.257589\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.0399 −1.19802 −0.599008 0.800743i \(-0.704438\pi\)
−0.599008 + 0.800743i \(0.704438\pi\)
\(102\) 0 0
\(103\) 20.1617i 1.98659i 0.115602 + 0.993296i \(0.463120\pi\)
−0.115602 + 0.993296i \(0.536880\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 20.2288i 1.95559i −0.209569 0.977794i \(-0.567206\pi\)
0.209569 0.977794i \(-0.432794\pi\)
\(108\) 0 0
\(109\) −10.5830 −1.01367 −0.506834 0.862044i \(-0.669184\pi\)
−0.506834 + 0.862044i \(0.669184\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.5544i 1.36916i −0.728937 0.684581i \(-0.759985\pi\)
0.728937 0.684581i \(-0.240015\pi\)
\(114\) 0 0
\(115\) 5.53019i 0.515693i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 8.66259 + 2.00000i 0.794099 + 0.183340i
\(120\) 0 0
\(121\) 8.29150 0.753773
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.33981 0.388165
\(126\) 0 0
\(127\) −12.1382 −1.07709 −0.538546 0.842596i \(-0.681026\pi\)
−0.538546 + 0.842596i \(0.681026\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 17.3252 1.51371 0.756854 0.653584i \(-0.226735\pi\)
0.756854 + 0.653584i \(0.226735\pi\)
\(132\) 0 0
\(133\) −3.29150 + 14.2565i −0.285409 + 1.23619i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.06910i 0.518518i −0.965808 0.259259i \(-0.916522\pi\)
0.965808 0.259259i \(-0.0834784\pi\)
\(138\) 0 0
\(139\) 15.8219i 1.34199i 0.741460 + 0.670997i \(0.234134\pi\)
−0.741460 + 0.670997i \(0.765866\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.05034 −0.422330
\(144\) 0 0
\(145\) 20.3939i 1.69362i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.06910i 0.497200i 0.968606 + 0.248600i \(0.0799705\pi\)
−0.968606 + 0.248600i \(0.920030\pi\)
\(150\) 0 0
\(151\) 11.8147 0.961466 0.480733 0.876867i \(-0.340371\pi\)
0.480733 + 0.876867i \(0.340371\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 14.5830i 1.17134i
\(156\) 0 0
\(157\) 17.3252i 1.38270i −0.722520 0.691350i \(-0.757016\pi\)
0.722520 0.691350i \(-0.242984\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.979531 + 4.24264i −0.0771979 + 0.334367i
\(162\) 0 0
\(163\) 8.48528 0.664619 0.332309 0.943170i \(-0.392172\pi\)
0.332309 + 0.943170i \(0.392172\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 23.4626 1.81559 0.907796 0.419413i \(-0.137764\pi\)
0.907796 + 0.419413i \(0.137764\pi\)
\(168\) 0 0
\(169\) 3.58301 0.275616
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 12.4616 0.947438 0.473719 0.880676i \(-0.342911\pi\)
0.473719 + 0.880676i \(0.342911\pi\)
\(174\) 0 0
\(175\) −16.2191 3.74463i −1.22605 0.283067i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.06275i 0.228920i 0.993428 + 0.114460i \(0.0365138\pi\)
−0.993428 + 0.114460i \(0.963486\pi\)
\(180\) 0 0
\(181\) 3.06871i 0.228096i −0.993475 0.114048i \(-0.963618\pi\)
0.993475 0.114048i \(-0.0363817\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −24.5015 −1.80139
\(186\) 0 0
\(187\) 5.53019i 0.404408i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.5203i 0.833577i −0.909004 0.416788i \(-0.863156\pi\)
0.909004 0.416788i \(-0.136844\pi\)
\(192\) 0 0
\(193\) 4.70850 0.338925 0.169463 0.985537i \(-0.445797\pi\)
0.169463 + 0.985537i \(0.445797\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 24.8661i 1.77164i −0.464031 0.885819i \(-0.653597\pi\)
0.464031 0.885819i \(-0.346403\pi\)
\(198\) 0 0
\(199\) 23.3111i 1.65248i 0.563316 + 0.826241i \(0.309525\pi\)
−0.563316 + 0.826241i \(0.690475\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.61226 15.6458i 0.253531 1.09812i
\(204\) 0 0
\(205\) 3.29150 0.229889
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −9.10132 −0.629551
\(210\) 0 0
\(211\) −15.4676 −1.06483 −0.532417 0.846482i \(-0.678716\pi\)
−0.532417 + 0.846482i \(0.678716\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −17.3252 −1.18157
\(216\) 0 0
\(217\) −2.58301 + 11.1878i −0.175346 + 0.759475i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 10.3117i 0.693643i
\(222\) 0 0
\(223\) 1.19038i 0.0797135i 0.999205 + 0.0398567i \(0.0126902\pi\)
−0.999205 + 0.0398567i \(0.987310\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.13742 −0.407355 −0.203678 0.979038i \(-0.565289\pi\)
−0.203678 + 0.979038i \(0.565289\pi\)
\(228\) 0 0
\(229\) 3.06871i 0.202786i −0.994846 0.101393i \(-0.967670\pi\)
0.994846 0.101393i \(-0.0323300\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.3808i 1.07314i 0.843854 + 0.536572i \(0.180281\pi\)
−0.843854 + 0.536572i \(0.819719\pi\)
\(234\) 0 0
\(235\) −37.5940 −2.45237
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.9373i 1.35432i −0.735837 0.677159i \(-0.763211\pi\)
0.735837 0.677159i \(-0.236789\pi\)
\(240\) 0 0
\(241\) 20.3939i 1.31369i −0.754027 0.656843i \(-0.771891\pi\)
0.754027 0.656843i \(-0.228109\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 21.1412 + 10.3117i 1.35066 + 0.658793i
\(246\) 0 0
\(247\) −16.9706 −1.07981
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5.05034 −0.318774 −0.159387 0.987216i \(-0.550952\pi\)
−0.159387 + 0.987216i \(0.550952\pi\)
\(252\) 0 0
\(253\) −2.70850 −0.170282
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.4616 0.777333 0.388667 0.921378i \(-0.372936\pi\)
0.388667 + 0.921378i \(0.372936\pi\)
\(258\) 0 0
\(259\) 18.7970 + 4.33981i 1.16799 + 0.269663i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 15.0627i 0.928809i −0.885623 0.464404i \(-0.846268\pi\)
0.885623 0.464404i \(-0.153732\pi\)
\(264\) 0 0
\(265\) 14.2565i 0.875768i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.5025 −0.640351 −0.320176 0.947358i \(-0.603742\pi\)
−0.320176 + 0.947358i \(0.603742\pi\)
\(270\) 0 0
\(271\) 13.0194i 0.790875i −0.918493 0.395437i \(-0.870593\pi\)
0.918493 0.395437i \(-0.129407\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.3542i 0.624385i
\(276\) 0 0
\(277\) 27.8745 1.67482 0.837408 0.546578i \(-0.184070\pi\)
0.837408 + 0.546578i \(0.184070\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.9015i 0.650327i −0.945658 0.325163i \(-0.894581\pi\)
0.945658 0.325163i \(-0.105419\pi\)
\(282\) 0 0
\(283\) 12.6724i 0.753299i 0.926356 + 0.376649i \(0.122924\pi\)
−0.926356 + 0.376649i \(0.877076\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.52517 0.583005i −0.149056 0.0344137i
\(288\) 0 0
\(289\) −5.70850 −0.335794
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −7.70010 −0.449845 −0.224922 0.974377i \(-0.572213\pi\)
−0.224922 + 0.974377i \(0.572213\pi\)
\(294\) 0 0
\(295\) 20.6235 1.20075
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5.05034 −0.292069
\(300\) 0 0
\(301\) 13.2915 + 3.06871i 0.766109 + 0.176878i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 37.5940i 2.15263i
\(306\) 0 0
\(307\) 5.53019i 0.315625i −0.987469 0.157812i \(-0.949556\pi\)
0.987469 0.157812i \(-0.0504441\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −6.13742 −0.348021 −0.174011 0.984744i \(-0.555673\pi\)
−0.174011 + 0.984744i \(0.555673\pi\)
\(312\) 0 0
\(313\) 12.2748i 0.693815i −0.937899 0.346908i \(-0.887232\pi\)
0.937899 0.346908i \(-0.112768\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.06910i 0.340875i −0.985369 0.170437i \(-0.945482\pi\)
0.985369 0.170437i \(-0.0545180\pi\)
\(318\) 0 0
\(319\) 9.98823 0.559234
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 18.5830i 1.03399i
\(324\) 0 0
\(325\) 19.3068i 1.07095i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 28.8413 + 6.65882i 1.59007 + 0.367113i
\(330\) 0 0
\(331\) 11.8147 0.649394 0.324697 0.945818i \(-0.394738\pi\)
0.324697 + 0.945818i \(0.394738\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −28.5129 −1.55783
\(336\) 0 0
\(337\) −30.5830 −1.66596 −0.832981 0.553301i \(-0.813368\pi\)
−0.832981 + 0.553301i \(0.813368\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −7.14226 −0.386775
\(342\) 0 0
\(343\) −14.3926 11.6556i −0.777129 0.629342i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.06275i 0.164417i 0.996615 + 0.0822084i \(0.0261973\pi\)
−0.996615 + 0.0822084i \(0.973803\pi\)
\(348\) 0 0
\(349\) 5.05034i 0.270338i −0.990823 0.135169i \(-0.956842\pi\)
0.990823 0.135169i \(-0.0431578\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.3797 0.871805 0.435902 0.899994i \(-0.356429\pi\)
0.435902 + 0.899994i \(0.356429\pi\)
\(354\) 0 0
\(355\) 18.9713i 1.00689i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 28.2288i 1.48986i 0.667144 + 0.744928i \(0.267516\pi\)
−0.667144 + 0.744928i \(0.732484\pi\)
\(360\) 0 0
\(361\) −11.5830 −0.609632
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 27.2823i 1.42802i
\(366\) 0 0
\(367\) 1.19038i 0.0621371i 0.999517 + 0.0310686i \(0.00989102\pi\)
−0.999517 + 0.0310686i \(0.990109\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.52517 10.9373i 0.131100 0.567834i
\(372\) 0 0
\(373\) 12.5830 0.651523 0.325762 0.945452i \(-0.394379\pi\)
0.325762 + 0.945452i \(0.394379\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.6243 0.959202
\(378\) 0 0
\(379\) −11.8147 −0.606880 −0.303440 0.952851i \(-0.598135\pi\)
−0.303440 + 0.952851i \(0.598135\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 29.6000 1.51249 0.756246 0.654288i \(-0.227032\pi\)
0.756246 + 0.654288i \(0.227032\pi\)
\(384\) 0 0
\(385\) −3.29150 + 14.2565i −0.167751 + 0.726577i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 21.2132i 1.07555i 0.843088 + 0.537776i \(0.180735\pi\)
−0.843088 + 0.537776i \(0.819265\pi\)
\(390\) 0 0
\(391\) 5.53019i 0.279674i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −28.5129 −1.43464
\(396\) 0 0
\(397\) 35.7375i 1.79361i 0.442424 + 0.896806i \(0.354119\pi\)
−0.442424 + 0.896806i \(0.645881\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.41618i 0.120658i −0.998179 0.0603291i \(-0.980785\pi\)
0.998179 0.0603291i \(-0.0192150\pi\)
\(402\) 0 0
\(403\) −13.3176 −0.663399
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.0000i 0.594818i
\(408\) 0 0
\(409\) 20.3939i 1.00841i −0.863583 0.504207i \(-0.831785\pi\)
0.863583 0.504207i \(-0.168215\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −15.8219 3.65292i −0.778544 0.179748i
\(414\) 0 0
\(415\) −37.5940 −1.84542
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.1007 0.493450 0.246725 0.969086i \(-0.420646\pi\)
0.246725 + 0.969086i \(0.420646\pi\)
\(420\) 0 0
\(421\) 1.29150 0.0629440 0.0314720 0.999505i \(-0.489980\pi\)
0.0314720 + 0.999505i \(0.489980\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −21.1412 −1.02550
\(426\) 0 0
\(427\) 6.65882 28.8413i 0.322243 1.39573i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 35.5203i 1.71095i −0.517844 0.855475i \(-0.673265\pi\)
0.517844 0.855475i \(-0.326735\pi\)
\(432\) 0 0
\(433\) 6.13742i 0.294946i −0.989066 0.147473i \(-0.952886\pi\)
0.989066 0.147473i \(-0.0471139\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −9.10132 −0.435375
\(438\) 0 0
\(439\) 32.8341i 1.56709i −0.621336 0.783544i \(-0.713410\pi\)
0.621336 0.783544i \(-0.286590\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.9373i 1.18480i −0.805642 0.592402i \(-0.798180\pi\)
0.805642 0.592402i \(-0.201820\pi\)
\(444\) 0 0
\(445\) 25.8745 1.22657
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.72202i 0.458811i 0.973331 + 0.229405i \(0.0736781\pi\)
−0.973331 + 0.229405i \(0.926322\pi\)
\(450\) 0 0
\(451\) 1.61206i 0.0759092i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.13742 + 26.5830i −0.287727 + 1.24623i
\(456\) 0 0
\(457\) −35.7490 −1.67227 −0.836134 0.548525i \(-0.815190\pi\)
−0.836134 + 0.548525i \(0.815190\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −25.4810 −1.18677 −0.593385 0.804919i \(-0.702209\pi\)
−0.593385 + 0.804919i \(0.702209\pi\)
\(462\) 0 0
\(463\) −12.1382 −0.564110 −0.282055 0.959398i \(-0.591016\pi\)
−0.282055 + 0.959398i \(0.591016\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17.3252 −0.801714 −0.400857 0.916141i \(-0.631288\pi\)
−0.400857 + 0.916141i \(0.631288\pi\)
\(468\) 0 0
\(469\) 21.8745 + 5.05034i 1.01007 + 0.233203i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.48528i 0.390154i
\(474\) 0 0
\(475\) 34.7932i 1.59642i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 22.3755i 1.02024i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 47.9058i 2.17529i
\(486\) 0 0
\(487\) 39.0970 1.77165 0.885827 0.464016i \(-0.153592\pi\)
0.885827 + 0.464016i \(0.153592\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10.8118i 0.487928i −0.969784 0.243964i \(-0.921552\pi\)
0.969784 0.243964i \(-0.0784479\pi\)
\(492\) 0 0
\(493\) 20.3939i 0.918495i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.36028 + 14.5544i −0.150729 + 0.652853i
\(498\) 0 0
\(499\) 18.4735 0.826988 0.413494 0.910507i \(-0.364308\pi\)
0.413494 + 0.910507i \(0.364308\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −35.7375 −1.59345 −0.796727 0.604339i \(-0.793437\pi\)
−0.796727 + 0.604339i \(0.793437\pi\)
\(504\) 0 0
\(505\) −40.4575 −1.80034
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8.12179 0.359992 0.179996 0.983667i \(-0.442392\pi\)
0.179996 + 0.983667i \(0.442392\pi\)
\(510\) 0 0
\(511\) −4.83236 + 20.9304i −0.213771 + 0.925906i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 67.7490i 2.98538i
\(516\) 0 0
\(517\) 18.4123i 0.809771i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −13.9990 −0.613306 −0.306653 0.951821i \(-0.599209\pi\)
−0.306653 + 0.951821i \(0.599209\pi\)
\(522\) 0 0
\(523\) 22.9641i 1.00415i 0.864824 + 0.502076i \(0.167430\pi\)
−0.864824 + 0.502076i \(0.832570\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 14.5830i 0.635246i
\(528\) 0 0
\(529\) 20.2915 0.882239
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.00590i 0.130200i
\(534\) 0 0
\(535\) 67.9743i 2.93879i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.05034 10.3542i 0.217533 0.445989i
\(540\) 0 0
\(541\) −1.29150 −0.0555260 −0.0277630 0.999615i \(-0.508838\pi\)
−0.0277630 + 0.999615i \(0.508838\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −35.5619 −1.52330
\(546\) 0 0
\(547\) 32.7617 1.40079 0.700394 0.713756i \(-0.253008\pi\)
0.700394 + 0.713756i \(0.253008\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 33.5633 1.42984
\(552\) 0 0
\(553\) 21.8745 + 5.05034i 0.930199 + 0.214762i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 38.8308i 1.64531i −0.568539 0.822657i \(-0.692491\pi\)
0.568539 0.822657i \(-0.307509\pi\)
\(558\) 0 0
\(559\) 15.8219i 0.669194i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 23.4626 0.988831 0.494416 0.869226i \(-0.335382\pi\)
0.494416 + 0.869226i \(0.335382\pi\)
\(564\) 0 0
\(565\) 48.9068i 2.05753i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 28.5190i 1.19558i −0.801653 0.597790i \(-0.796046\pi\)
0.801653 0.597790i \(-0.203954\pi\)
\(570\) 0 0
\(571\) −46.0793 −1.92836 −0.964180 0.265249i \(-0.914546\pi\)
−0.964180 + 0.265249i \(0.914546\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 10.3542i 0.431802i
\(576\) 0 0
\(577\) 16.2381i 0.676001i 0.941146 + 0.338000i \(0.109751\pi\)
−0.941146 + 0.338000i \(0.890249\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 28.8413 + 6.65882i 1.19654 + 0.276254i
\(582\) 0 0
\(583\) 6.98233 0.289179
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.4626 0.968406 0.484203 0.874956i \(-0.339110\pi\)
0.484203 + 0.874956i \(0.339110\pi\)
\(588\) 0 0
\(589\) −24.0000 −0.988903
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 19.6039 0.805034 0.402517 0.915413i \(-0.368135\pi\)
0.402517 + 0.915413i \(0.368135\pi\)
\(594\) 0 0
\(595\) 29.1088 + 6.72057i 1.19334 + 0.275516i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 18.8118i 0.768628i 0.923202 + 0.384314i \(0.125562\pi\)
−0.923202 + 0.384314i \(0.874438\pi\)
\(600\) 0 0
\(601\) 40.7878i 1.66377i 0.554949 + 0.831884i \(0.312738\pi\)
−0.554949 + 0.831884i \(0.687262\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 27.8618 1.13274
\(606\) 0 0
\(607\) 25.6919i 1.04280i 0.853312 + 0.521401i \(0.174590\pi\)
−0.853312 + 0.521401i \(0.825410\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 34.3320i 1.38892i
\(612\) 0 0
\(613\) 11.4170 0.461128 0.230564 0.973057i \(-0.425943\pi\)
0.230564 + 0.973057i \(0.425943\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 31.5249i 1.26915i −0.772863 0.634573i \(-0.781176\pi\)
0.772863 0.634573i \(-0.218824\pi\)
\(618\) 0 0
\(619\) 7.14226i 0.287071i −0.989645 0.143536i \(-0.954153\pi\)
0.989645 0.143536i \(-0.0458472\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −19.8504 4.58301i −0.795288 0.183614i
\(624\) 0 0
\(625\) −16.8745 −0.674980
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 24.5015 0.976939
\(630\) 0 0
\(631\) 32.7617 1.30422 0.652111 0.758123i \(-0.273884\pi\)
0.652111 + 0.758123i \(0.273884\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −40.7878 −1.61861
\(636\) 0 0
\(637\) 9.41699 19.3068i 0.373115 0.764964i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.07500i 0.358441i 0.983809 + 0.179220i \(0.0573576\pi\)
−0.983809 + 0.179220i \(0.942642\pi\)
\(642\) 0 0
\(643\) 22.8894i 0.902672i 0.892354 + 0.451336i \(0.149052\pi\)
−0.892354 + 0.451336i \(0.850948\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.2381 0.638386 0.319193 0.947690i \(-0.396588\pi\)
0.319193 + 0.947690i \(0.396588\pi\)
\(648\) 0 0
\(649\) 10.1007i 0.396486i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 16.3808i 0.641032i 0.947243 + 0.320516i \(0.103856\pi\)
−0.947243 + 0.320516i \(0.896144\pi\)
\(654\) 0 0
\(655\) 58.2175 2.27475
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 18.8118i 0.732802i 0.930457 + 0.366401i \(0.119410\pi\)
−0.930457 + 0.366401i \(0.880590\pi\)
\(660\) 0 0
\(661\) 17.3252i 0.673872i 0.941528 + 0.336936i \(0.109391\pi\)
−0.941528 + 0.336936i \(0.890609\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −11.0604 + 47.9058i −0.428903 + 1.85771i
\(666\) 0 0
\(667\) 9.98823 0.386746
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 18.4123 0.710798
\(672\) 0 0
\(673\) −4.70850 −0.181499 −0.0907496 0.995874i \(-0.528926\pi\)
−0.0907496 + 0.995874i \(0.528926\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −21.1412 −0.812523 −0.406262 0.913757i \(-0.633168\pi\)
−0.406262 + 0.913757i \(0.633168\pi\)
\(678\) 0 0
\(679\) −8.48528 + 36.7523i −0.325635 + 1.41042i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 3.77124i 0.144303i 0.997394 + 0.0721513i \(0.0229864\pi\)
−0.997394 + 0.0721513i \(0.977014\pi\)
\(684\) 0 0
\(685\) 20.3939i 0.779211i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 13.0194 0.496001
\(690\) 0 0
\(691\) 41.8608i 1.59246i 0.604995 + 0.796229i \(0.293175\pi\)
−0.604995 + 0.796229i \(0.706825\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 53.1660i 2.01670i
\(696\) 0 0
\(697\) −3.29150 −0.124675
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 48.4955i 1.83165i −0.401578 0.915825i \(-0.631538\pi\)
0.401578 0.915825i \(-0.368462\pi\)
\(702\) 0 0
\(703\) 40.3234i 1.52083i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 31.0381 + 7.16601i 1.16731 + 0.269506i
\(708\) 0 0
\(709\) −28.0000 −1.05156 −0.525781 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −7.14226 −0.267480
\(714\) 0 0
\(715\) −16.9706 −0.634663
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 22.3755 0.834466 0.417233 0.908800i \(-0.363000\pi\)
0.417233 + 0.908800i \(0.363000\pi\)
\(720\) 0 0
\(721\) 12.0000 51.9756i 0.446903 1.93567i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 38.1838i 1.41811i
\(726\) 0 0
\(727\) 2.80244i 0.103937i −0.998649 0.0519684i \(-0.983450\pi\)
0.998649 0.0519684i \(-0.0165495\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 17.3252 0.640795
\(732\) 0 0
\(733\) 19.3068i 0.713113i 0.934274 + 0.356557i \(0.116049\pi\)
−0.934274 + 0.356557i \(0.883951\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13.9647i 0.514395i
\(738\) 0 0
\(739\) 15.7911 0.580886 0.290443 0.956892i \(-0.406197\pi\)
0.290443 + 0.956892i \(0.406197\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8.22876i 0.301884i 0.988543 + 0.150942i \(0.0482306\pi\)
−0.988543 + 0.150942i \(0.951769\pi\)
\(744\) 0 0
\(745\) 20.3939i 0.747175i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −12.0399 + 52.1484i −0.439929 + 1.90546i
\(750\) 0 0
\(751\) −36.0911 −1.31698 −0.658491 0.752588i \(-0.728805\pi\)
−0.658491 + 0.752588i \(0.728805\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 39.7007 1.44486
\(756\) 0 0
\(757\) 1.16601 0.0423794 0.0211897 0.999775i \(-0.493255\pi\)
0.0211897 + 0.999775i \(0.493255\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.51690 0.0912377 0.0456189 0.998959i \(-0.485474\pi\)
0.0456189 + 0.998959i \(0.485474\pi\)
\(762\) 0 0
\(763\) 27.2823 + 6.29888i 0.987686 + 0.228035i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.8340i 0.680056i
\(768\) 0 0
\(769\) 50.8885i 1.83509i −0.397637 0.917543i \(-0.630170\pi\)
0.397637 0.917543i \(-0.369830\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −16.8014 −0.604305 −0.302152 0.953260i \(-0.597705\pi\)
−0.302152 + 0.953260i \(0.597705\pi\)
\(774\) 0 0
\(775\) 27.3040i 0.980787i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.41699i 0.194084i
\(780\) 0 0
\(781\) −9.29150 −0.332476
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 58.2175i 2.07787i
\(786\) 0 0
\(787\) 31.6438i 1.12798i 0.825782 + 0.563989i \(0.190734\pi\)
−0.825782 + 0.563989i \(0.809266\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −8.66259 + 37.5203i −0.308006 + 1.33407i
\(792\) 0 0
\(793\) 34.3320 1.21917
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12.4616 −0.441413 −0.220706 0.975340i \(-0.570836\pi\)
−0.220706 + 0.975340i \(0.570836\pi\)
\(798\) 0 0
\(799\) 37.5940 1.32998
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −13.3619 −0.471532
\(804\) 0 0
\(805\) −3.29150 + 14.2565i −0.116010 + 0.502475i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 28.5190i 1.00268i 0.865251 + 0.501338i \(0.167159\pi\)
−0.865251 + 0.501338i \(0.832841\pi\)
\(810\) 0 0
\(811\) 17.3593i 0.609566i −0.952422 0.304783i \(-0.901416\pi\)
0.952422 0.304783i \(-0.0985839\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 28.5129 0.998765
\(816\) 0 0
\(817\) 28.5129i 0.997542i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 16.3808i 0.571695i 0.958275 + 0.285848i \(0.0922751\pi\)
−0.958275 + 0.285848i \(0.907725\pi\)
\(822\) 0 0
\(823\) 25.4558 0.887335 0.443667 0.896191i \(-0.353677\pi\)
0.443667 + 0.896191i \(0.353677\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 17.6458i 0.613603i 0.951774 + 0.306801i \(0.0992588\pi\)
−0.951774 + 0.306801i \(0.900741\pi\)
\(828\) 0 0
\(829\) 47.8198i 1.66085i 0.557131 + 0.830424i \(0.311902\pi\)
−0.557131 + 0.830424i \(0.688098\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −21.1412 10.3117i −0.732500 0.357281i
\(834\) 0 0
\(835\) 78.8410 2.72841
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 27.4259 0.946846 0.473423 0.880835i \(-0.343018\pi\)
0.473423 + 0.880835i \(0.343018\pi\)
\(840\) 0 0
\(841\) −7.83399 −0.270138
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.0399 0.414186
\(846\) 0 0
\(847\) −21.3750 4.93500i −0.734452 0.169569i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 12.0000i 0.411355i
\(852\) 0 0
\(853\) 33.5633i 1.14918i 0.818440 + 0.574592i \(0.194839\pi\)
−0.818440 + 0.574592i \(0.805161\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 25.4810 0.870416 0.435208 0.900330i \(-0.356675\pi\)
0.435208 + 0.900330i \(0.356675\pi\)
\(858\) 0 0
\(859\) 12.6724i 0.432378i −0.976352 0.216189i \(-0.930637\pi\)
0.976352 0.216189i \(-0.0693628\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.64575i 0.0560220i −0.999608 0.0280110i \(-0.991083\pi\)
0.999608 0.0280110i \(-0.00891735\pi\)
\(864\) 0 0
\(865\) 41.8745 1.42378
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 13.9647i 0.473719i
\(870\) 0 0
\(871\) 26.0389i 0.882294i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −11.1878 2.58301i −0.378215 0.0873215i
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 52.3633 1.76416 0.882082 0.471095i \(-0.156141\pi\)
0.882082 + 0.471095i \(0.156141\pi\)
\(882\) 0 0
\(883\) −8.80879 −0.296439 −0.148220 0.988954i \(-0.547354\pi\)
−0.148220 + 0.988954i \(0.547354\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −13.3619 −0.448650 −0.224325 0.974514i \(-0.572018\pi\)
−0.224325 + 0.974514i \(0.572018\pi\)
\(888\) 0 0
\(889\) 31.2915 + 7.22451i 1.04948 + 0.242302i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 61.8705i 2.07042i
\(894\) 0 0
\(895\) 10.2917i 0.344013i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 26.3388 0.878447
\(900\) 0 0
\(901\) 14.2565i 0.474952i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10.3117i 0.342774i
\(906\) 0 0
\(907\) −1.50295 −0.0499046 −0.0249523 0.999689i \(-0.507943\pi\)
−0.0249523 + 0.999689i \(0.507943\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 18.3542i 0.608103i 0.952656 + 0.304052i \(0.0983396\pi\)
−0.952656 + 0.304052i \(0.901660\pi\)
\(912\) 0 0
\(913\) 18.4123i 0.609357i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −44.6632 10.3117i −1.47491 0.340524i
\(918\) 0 0
\(919\) 12.4617 0.411074 0.205537 0.978649i \(-0.434106\pi\)
0.205537 + 0.978649i \(0.434106\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −17.3252 −0.570265
\(924\) 0 0
\(925\) −45.8745 −1.50834
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.82291 −0.0598077 −0.0299039 0.999553i \(-0.509520\pi\)
−0.0299039 + 0.999553i \(0.509520\pi\)
\(930\) 0 0
\(931\) 16.9706 34.7932i 0.556188 1.14030i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 18.5830i 0.607729i
\(936\) 0 0
\(937\) 24.5497i 0.802004i −0.916077 0.401002i \(-0.868662\pi\)
0.916077 0.401002i \(-0.131338\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 21.9846 0.716678 0.358339 0.933592i \(-0.383343\pi\)
0.358339 + 0.933592i \(0.383343\pi\)
\(942\) 0 0
\(943\) 1.61206i 0.0524961i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 16.2288i 0.527364i −0.964610 0.263682i \(-0.915063\pi\)
0.964610 0.263682i \(-0.0849369\pi\)
\(948\) 0 0
\(949\) −24.9150 −0.808776
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 40.0102i 1.29606i −0.761616 0.648029i \(-0.775593\pi\)
0.761616 0.648029i \(-0.224407\pi\)
\(954\) 0 0
\(955\) 38.7113i 1.25267i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.61226 + 15.6458i −0.116646 + 0.505228i
\(960\) 0 0
\(961\) 12.1660 0.392452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 15.8219 0.509324
\(966\) 0 0
\(967\) 8.48528 0.272868 0.136434 0.990649i \(-0.456436\pi\)
0.136434 + 0.990649i \(0.456436\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 18.4123 0.590878 0.295439 0.955362i \(-0.404534\pi\)
0.295439 + 0.955362i \(0.404534\pi\)
\(972\) 0 0
\(973\) 9.41699 40.7878i 0.301895 1.30760i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.7338i 0.503370i 0.967809 + 0.251685i \(0.0809846\pi\)
−0.967809 + 0.251685i \(0.919015\pi\)
\(978\) 0 0
\(979\) 12.6724i 0.405013i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 23.4626 0.748341 0.374171 0.927360i \(-0.377927\pi\)
0.374171 + 0.927360i \(0.377927\pi\)
\(984\) 0 0
\(985\) 83.5572i 2.66235i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.48528i 0.269816i
\(990\) 0 0
\(991\) −12.4617 −0.395859 −0.197930 0.980216i \(-0.563422\pi\)
−0.197930 + 0.980216i \(0.563422\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 78.3320i 2.48329i
\(996\) 0 0
\(997\) 11.1878i 0.354320i 0.984182 + 0.177160i \(0.0566910\pi\)
−0.984182 + 0.177160i \(0.943309\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4032.2.k.h.3905.13 16
3.2 odd 2 inner 4032.2.k.h.3905.1 16
4.3 odd 2 inner 4032.2.k.h.3905.16 16
7.6 odd 2 inner 4032.2.k.h.3905.2 16
8.3 odd 2 2016.2.k.b.1889.4 yes 16
8.5 even 2 2016.2.k.b.1889.1 16
12.11 even 2 inner 4032.2.k.h.3905.4 16
21.20 even 2 inner 4032.2.k.h.3905.14 16
24.5 odd 2 2016.2.k.b.1889.13 yes 16
24.11 even 2 2016.2.k.b.1889.16 yes 16
28.27 even 2 inner 4032.2.k.h.3905.3 16
56.13 odd 2 2016.2.k.b.1889.14 yes 16
56.27 even 2 2016.2.k.b.1889.15 yes 16
84.83 odd 2 inner 4032.2.k.h.3905.15 16
168.83 odd 2 2016.2.k.b.1889.3 yes 16
168.125 even 2 2016.2.k.b.1889.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2016.2.k.b.1889.1 16 8.5 even 2
2016.2.k.b.1889.2 yes 16 168.125 even 2
2016.2.k.b.1889.3 yes 16 168.83 odd 2
2016.2.k.b.1889.4 yes 16 8.3 odd 2
2016.2.k.b.1889.13 yes 16 24.5 odd 2
2016.2.k.b.1889.14 yes 16 56.13 odd 2
2016.2.k.b.1889.15 yes 16 56.27 even 2
2016.2.k.b.1889.16 yes 16 24.11 even 2
4032.2.k.h.3905.1 16 3.2 odd 2 inner
4032.2.k.h.3905.2 16 7.6 odd 2 inner
4032.2.k.h.3905.3 16 28.27 even 2 inner
4032.2.k.h.3905.4 16 12.11 even 2 inner
4032.2.k.h.3905.13 16 1.1 even 1 trivial
4032.2.k.h.3905.14 16 21.20 even 2 inner
4032.2.k.h.3905.15 16 84.83 odd 2 inner
4032.2.k.h.3905.16 16 4.3 odd 2 inner